// // Copyright (c) 2017 The Khronos Group Inc. // // Licensed under the Apache License, Version 2.0 (the "License"); // you may not use this file except in compliance with the License. // You may obtain a copy of the License at // // http://www.apache.org/licenses/LICENSE-2.0 // // Unless required by applicable law or agreed to in writing, software // distributed under the License is distributed on an "AS IS" BASIS, // WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied. // See the License for the specific language governing permissions and // limitations under the License. // #include "harness/compat.h" #include "reference_math.h" #include #if !defined(_WIN32) #include #endif #include "Utility.h" #if defined( __SSE__ ) || (defined( _MSC_VER ) && (defined(_M_IX86) || defined(_M_X64))) #include #endif #if defined( __SSE2__ ) || (defined( _MSC_VER ) && (defined(_M_IX86) || defined(_M_X64))) #include #endif #ifndef M_PI_4 #define M_PI_4 (M_PI/4) #endif #define EVALUATE( x ) x #define CONCATENATE(x, y) x ## EVALUATE(y) #pragma STDC FP_CONTRACT OFF static void __log2_ep(double *hi, double *lo, double x); typedef union { uint64_t i; double d; }uint64d_t; static const uint64d_t _CL_NAN = { 0x7ff8000000000000ULL }; #define cl_make_nan() _CL_NAN.d static double reduce1( double x ); static double reduce1( double x ) { if( fabs(x) >= HEX_DBL( +, 1, 0, +, 53 ) ) { if( fabs(x) == INFINITY ) return cl_make_nan(); return 0.0; //we patch up the sign for sinPi and cosPi later, since they need different signs } // Find the nearest multiple of 2 const double r = copysign( HEX_DBL( +, 1, 0, +, 53 ), x ); double z = x + r; z -= r; // subtract it from x. Value is now in the range -1 <= x <= 1 return x - z; } /* static double reduceHalf( double x ); static double reduceHalf( double x ) { if( fabs(x) >= HEX_DBL( +, 1, 0, +, 52 ) ) { if( fabs(x) == INFINITY ) return cl_make_nan(); return 0.0; //we patch up the sign for sinPi and cosPi later, since they need different signs } // Find the nearest multiple of 1 const double r = copysign( HEX_DBL( +, 1, 0, +, 52 ), x ); double z = x + r; z -= r; // subtract it from x. Value is now in the range -0.5 <= x <= 0.5 return x - z; } */ double reference_acospi( double x) { return reference_acos( x ) / M_PI; } double reference_asinpi( double x) { return reference_asin( x ) / M_PI; } double reference_atanpi( double x) { return reference_atan( x ) / M_PI; } double reference_atan2pi( double y, double x ) { return reference_atan2( y, x) / M_PI; } double reference_cospi( double x) { if( reference_fabs(x) >= HEX_DBL( +, 1, 0, +, 52 ) ) { if( reference_fabs(x) == INFINITY ) return cl_make_nan(); //Note this probably fails for odd values between 0x1.0p52 and 0x1.0p53. //However, when starting with single precision inputs, there will be no odd values. return 1.0; } x = reduce1(x+0.5); // reduce to [-0.5, 0.5] if( x < -0.5 ) x = -1 - x; else if ( x > 0.5 ) x = 1 - x; // cosPi zeros are all +0 if( x == 0.0 ) return 0.0; return reference_sin( x * M_PI ); } double reference_relaxed_cospi(double x) { return reference_cospi(x); } double reference_relaxed_divide( double x, double y ) { return (float)(((float) x ) / ( (float) y )); } double reference_divide( double x, double y ) { return x / y; } // Add a + b. If the result modulo overflowed, write 1 to *carry, otherwise 0 static inline cl_ulong add_carry( cl_ulong a, cl_ulong b, cl_ulong *carry ) { cl_ulong result = a + b; *carry = result < a; return result; } // Subtract a - b. If the result modulo overflowed, write 1 to *carry, otherwise 0 static inline cl_ulong sub_carry( cl_ulong a, cl_ulong b, cl_ulong *carry ) { cl_ulong result = a - b; *carry = result > a; return result; } static float fallback_frexpf( float x, int *iptr ) { cl_uint u, v; float fu, fv; memcpy( &u, &x, sizeof(u)); cl_uint exponent = u & 0x7f800000U; cl_uint mantissa = u & ~0x7f800000U; // add 1 to the exponent exponent += 0x00800000U; if( (cl_int) exponent < (cl_int) 0x01000000 ) { // subnormal, NaN, Inf mantissa |= 0x3f000000U; v = mantissa & 0xff800000U; u = mantissa; memcpy( &fv, &v, sizeof(v)); memcpy( &fu, &u, sizeof(u)); fu -= fv; memcpy( &v, &fv, sizeof(v)); memcpy( &u, &fu, sizeof(u)); exponent = u & 0x7f800000U; mantissa = u & ~0x7f800000U; *iptr = (exponent >> 23) + (-126 + 1 -126); u = mantissa | 0x3f000000U; memcpy( &fu, &u, sizeof(u)); return fu; } *iptr = (exponent >> 23) - 127; u = mantissa | 0x3f000000U; memcpy( &fu, &u, sizeof(u)); return fu; } static inline int extractf( float, cl_uint * ); static inline int extractf( float x, cl_uint *mant ) { static float (*frexppf)(float, int*) = NULL; int e; // verify that frexp works properly if( NULL == frexppf ) { if( 0.5f == frexpf( HEX_FLT( +, 1, 0, -, 130 ), &e ) && e == -129 ) frexppf = frexpf; else frexppf = fallback_frexpf; } *mant = (cl_uint) (HEX_FLT( +, 1, 0, +, 32 ) * fabsf( frexppf( x, &e ))); return e - 1; } // Shift right by shift bits. Any bits lost on the right side are bitwise OR'd together and ORd into the LSB of the result static inline void shift_right_sticky_64( cl_ulong *p, int shift ); static inline void shift_right_sticky_64( cl_ulong *p, int shift ) { cl_ulong sticky = 0; cl_ulong r = *p; // C doesn't handle shifts greater than the size of the variable dependably if( shift >= 64 ) { sticky |= (0 != r); r = 0; } else { sticky |= (0 != (r << (64-shift))); r >>= shift; } *p = r | sticky; } // Add two 64 bit mantissas. Bits that are below the LSB of the result are OR'd into the LSB of the result static inline void add64( cl_ulong *p, cl_ulong c, int *exponent ); static inline void add64( cl_ulong *p, cl_ulong c, int *exponent ) { cl_ulong carry; c = add_carry(c, *p, &carry); if( carry ) { carry = c & 1; // set aside sticky bit c >>= 1; // right shift to deal with overflow c |= carry | 0x8000000000000000ULL; // or in carry bit, and sticky bit. The latter is to prevent rounding from believing we are exact half way case *exponent = *exponent + 1; // adjust exponent } *p = c; } // IEEE-754 round to nearest, ties to even rounding static float round_to_nearest_even_float( cl_ulong p, int exponent ); static float round_to_nearest_even_float( cl_ulong p, int exponent ) { union{ cl_uint u; cl_float d;} u; // If mantissa is zero, return 0.0f if (p == 0) return 0.0f; // edges if( exponent > 127 ) { volatile float r = exponent * CL_FLT_MAX; // signal overflow // attempt to fool the compiler into not optimizing the above line away if( r > CL_FLT_MAX ) return INFINITY; return r; } if( exponent == -150 && p > 0x8000000000000000ULL) return HEX_FLT( +, 1, 0, -, 149 ); if( exponent <= -150 ) return 0.0f; //Figure out which bits go where int shift = 8 + 32; if( exponent < -126 ) { shift -= 126 + exponent; // subnormal: shift is not 52 exponent = -127; // set exponent to 0 } else p &= 0x7fffffffffffffffULL; // normal: leading bit is implicit. Remove it. // Assemble the double (round toward zero) u.u = (cl_uint)(p >> shift) | ((cl_uint) (exponent + 127) << 23); // put a representation of the residual bits into hi p <<= (64-shift); //round to nearest, ties to even based on the unused portion of p if( p < 0x8000000000000000ULL ) return u.d; if( p == 0x8000000000000000ULL ) u.u += u.u & 1U; else u.u++; return u.d; } static float round_to_nearest_even_float_ftz( cl_ulong p, int exponent ); static float round_to_nearest_even_float_ftz( cl_ulong p, int exponent ) { extern int gCheckTininessBeforeRounding; union{ cl_uint u; cl_float d;} u; int shift = 8 + 32; // If mantissa is zero, return 0.0f if (p == 0) return 0.0f; // edges if( exponent > 127 ) { volatile float r = exponent * CL_FLT_MAX; // signal overflow // attempt to fool the compiler into not optimizing the above line away if( r > CL_FLT_MAX ) return INFINITY; return r; } // Deal with FTZ for gCheckTininessBeforeRounding if( exponent < (gCheckTininessBeforeRounding - 127) ) return 0.0f; if( exponent == -127 ) // only happens for machines that check tininess after rounding p = (p&1) | (p>>1); else p &= 0x7fffffffffffffffULL; // normal: leading bit is implicit. Remove it. cl_ulong q = p; // Assemble the double (round toward zero) u.u = (cl_uint)(q >> shift) | ((cl_uint) (exponent + 127) << 23); // put a representation of the residual bits into hi q <<= (64-shift); //round to nearest, ties to even based on the unused portion of p if( q > 0x8000000000000000ULL ) u.u++; else if( q == 0x8000000000000000ULL ) u.u += u.u & 1U; // Deal with FTZ for ! gCheckTininessBeforeRounding if( 0 == (u.u & 0x7f800000U ) ) return 0.0f; return u.d; } // IEEE-754 round toward zero. static float round_toward_zero_float( cl_ulong p, int exponent ); static float round_toward_zero_float( cl_ulong p, int exponent ) { union{ cl_uint u; cl_float d;} u; // If mantissa is zero, return 0.0f if (p == 0) return 0.0f; // edges if( exponent > 127 ) { volatile float r = exponent * CL_FLT_MAX; // signal overflow // attempt to fool the compiler into not optimizing the above line away if( r > CL_FLT_MAX ) return CL_FLT_MAX; return r; } if( exponent <= -149 ) return 0.0f; //Figure out which bits go where int shift = 8 + 32; if( exponent < -126 ) { shift -= 126 + exponent; // subnormal: shift is not 52 exponent = -127; // set exponent to 0 } else p &= 0x7fffffffffffffffULL; // normal: leading bit is implicit. Remove it. // Assemble the double (round toward zero) u.u = (cl_uint)(p >> shift) | ((cl_uint) (exponent + 127) << 23); return u.d; } static float round_toward_zero_float_ftz( cl_ulong p, int exponent ); static float round_toward_zero_float_ftz( cl_ulong p, int exponent ) { extern int gCheckTininessBeforeRounding; union{ cl_uint u; cl_float d;} u; int shift = 8 + 32; // If mantissa is zero, return 0.0f if (p == 0) return 0.0f; // edges if( exponent > 127 ) { volatile float r = exponent * CL_FLT_MAX; // signal overflow // attempt to fool the compiler into not optimizing the above line away if( r > CL_FLT_MAX ) return CL_FLT_MAX; return r; } // Deal with FTZ for gCheckTininessBeforeRounding if( exponent < -126 ) return 0.0f; cl_ulong q = p &= 0x7fffffffffffffffULL; // normal: leading bit is implicit. Remove it. // Assemble the double (round toward zero) u.u = (cl_uint)(q >> shift) | ((cl_uint) (exponent + 127) << 23); // put a representation of the residual bits into hi q <<= (64-shift); return u.d; } // Subtract two significands. static inline void sub64( cl_ulong *c, cl_ulong p, cl_uint *signC, int *expC ); static inline void sub64( cl_ulong *c, cl_ulong p, cl_uint *signC, int *expC ) { cl_ulong carry; p = sub_carry( *c, p, &carry ); if( carry ) { *signC ^= 0x80000000U; p = -p; } // normalize if( p ) { int shift = 32; cl_ulong test = 1ULL << 32; while( 0 == (p & 0x8000000000000000ULL)) { if( p < test ) { p <<= shift; *expC = *expC - shift; } shift >>= 1; test <<= shift; } } else { // zero result. *expC = -200; *signC = 0; // IEEE rules say a - a = +0 for all rounding modes except -inf } *c = p; } float reference_fma( float a, float b, float c, int shouldFlush ) { static const cl_uint kMSB = 0x80000000U; // Make bits accessible union{ cl_uint u; cl_float d; } ua; ua.d = a; union{ cl_uint u; cl_float d; } ub; ub.d = b; union{ cl_uint u; cl_float d; } uc; uc.d = c; // deal with Nans, infinities and zeros if( isnan( a ) || isnan( b ) || isnan(c) || isinf( a ) || isinf( b ) || isinf(c) || 0 == ( ua.u & ~kMSB) || // a == 0, defeat host FTZ behavior 0 == ( ub.u & ~kMSB) || // b == 0, defeat host FTZ behavior 0 == ( uc.u & ~kMSB) ) // c == 0, defeat host FTZ behavior { FPU_mode_type oldMode; RoundingMode oldRoundMode = kRoundToNearestEven; if( isinf( c ) && !isinf(a) && !isinf(b) ) return (c + a) + b; if (gIsInRTZMode) oldRoundMode = set_round(kRoundTowardZero, kfloat); memset( &oldMode, 0, sizeof( oldMode ) ); if( shouldFlush ) ForceFTZ( &oldMode ); a = (float) reference_multiply( a, b ); // some risk that the compiler will insert a non-compliant fma here on some platforms. a = (float) reference_add( a, c ); // We use STDC FP_CONTRACT OFF above to attempt to defeat that. if( shouldFlush ) RestoreFPState( &oldMode ); if( gIsInRTZMode ) set_round(oldRoundMode, kfloat); return a; } // extract exponent and mantissa // exponent is a standard unbiased signed integer // mantissa is a cl_uint, with leading non-zero bit positioned at the MSB cl_uint mantA, mantB, mantC; int expA = extractf( a, &mantA ); int expB = extractf( b, &mantB ); int expC = extractf( c, &mantC ); cl_uint signC = uc.u & kMSB; // We'll need the sign bit of C later to decide if we are adding or subtracting // exact product of A and B int exponent = expA + expB; cl_uint sign = (ua.u ^ ub.u) & kMSB; cl_ulong product = (cl_ulong) mantA * (cl_ulong) mantB; // renormalize -- 1.m * 1.n yields a number between 1.0 and 3.99999.. // The MSB might not be set. If so, fix that. Otherwise, reflect the fact that we got another power of two from the multiplication if( 0 == (0x8000000000000000ULL & product) ) product <<= 1; else exponent++; // 2**31 * 2**31 gives 2**62. If the MSB was set, then our exponent increased. //infinite precision add cl_ulong addend = (cl_ulong) mantC << 32; if( exponent >= expC ) { // Shift C relative to the product so that their exponents match if( exponent > expC ) shift_right_sticky_64( &addend, exponent - expC ); // Add if( sign ^ signC ) sub64( &product, addend, &sign, &exponent ); else add64( &product, addend, &exponent ); } else { // Shift the product relative to C so that their exponents match shift_right_sticky_64( &product, expC - exponent ); // add if( sign ^ signC ) sub64( &addend, product, &signC, &expC ); else add64( &addend, product, &expC ); product = addend; exponent = expC; sign = signC; } // round to IEEE result -- we do not do flushing to zero here. That part is handled manually in ternary.c. if (gIsInRTZMode) { if( shouldFlush ) ua.d = round_toward_zero_float_ftz( product, exponent); else ua.d = round_toward_zero_float( product, exponent); } else { if( shouldFlush ) ua.d = round_to_nearest_even_float_ftz( product, exponent); else ua.d = round_to_nearest_even_float( product, exponent); } // Set the sign ua.u |= sign; return ua.d; } double reference_relaxed_exp10( double x) { return reference_exp10(x); } double reference_exp10( double x) { return reference_exp2( x * HEX_DBL( +, 1, a934f0979a371, +, 1 ) ); } int reference_ilogb( double x ) { extern int gDeviceILogb0, gDeviceILogbNaN; union { cl_double f; cl_ulong u;} u; u.f = (float) x; cl_int exponent = (cl_int) (u.u >> 52) & 0x7ff; if( exponent == 0x7ff ) { if( u.u & 0x000fffffffffffffULL ) return gDeviceILogbNaN; return CL_INT_MAX; } if( exponent == 0 ) { // deal with denormals u.f = x * HEX_DBL( +, 1, 0, +, 64 ); exponent = (cl_int) (u.u >> 52) & 0x7ff; if( exponent == 0 ) return gDeviceILogb0; return exponent - (1023 + 64); } return exponent - 1023; } double reference_nan( cl_uint x ) { union{ cl_uint u; cl_float f; }u; u.u = x | 0x7fc00000U; return (double) u.f; } double reference_maxmag( double x, double y ) { double fabsx = fabs(x); double fabsy = fabs(y); if( fabsx < fabsy ) return y; if( fabsy < fabsx ) return x; return reference_fmax( x, y ); } double reference_minmag( double x, double y ) { double fabsx = fabs(x); double fabsy = fabs(y); if( fabsx > fabsy ) return y; if( fabsy > fabsx ) return x; return reference_fmin( x, y ); } //double my_nextafter( double x, double y ){ return (double) nextafterf( (float) x, (float) y ); } double reference_relaxed_mad( double a, double b, double c) { return ((float) a )* ((float) b) + (float) c; } double reference_mad( double a, double b, double c ) { return a * b + c; } double reference_recip( double x) { return 1.0 / x; } double reference_rootn( double x, int i ) { //rootn ( x, 0 ) returns a NaN. if( 0 == i ) return cl_make_nan(); //rootn ( x, n ) returns a NaN for x < 0 and n is even. if( x < 0 && 0 == (i&1) ) return cl_make_nan(); if( x == 0.0 ) { switch( i & 0x80000001 ) { //rootn ( +-0, n ) is +0 for even n > 0. case 0: return 0.0f; //rootn ( +-0, n ) is +-0 for odd n > 0. case 1: return x; //rootn ( +-0, n ) is +inf for even n < 0. case 0x80000000: return INFINITY; //rootn ( +-0, n ) is +-inf for odd n < 0. case 0x80000001: return copysign(INFINITY, x); } } double sign = x; x = reference_fabs(x); x = reference_exp2( reference_log2(x) / (double) i ); return reference_copysignd( x, sign ); } double reference_rsqrt( double x) { return 1.0 / reference_sqrt(x); } //double reference_sincos( double x, double *c ){ *c = cos(x); return sin(x); } double reference_sinpi( double x) { double r = reduce1(x); // reduce to [-0.5, 0.5] if( r < -0.5 ) r = -1 - r; else if ( r > 0.5 ) r = 1 - r; // sinPi zeros have the same sign as x if( r == 0.0 ) return reference_copysignd(0.0, x); return reference_sin( r * M_PI ); } double reference_relaxed_sinpi(double x) { return reference_sinpi(x); } double reference_tanpi( double x) { // set aside the sign (allows us to preserve sign of -0) double sign = reference_copysignd( 1.0, x); double z = reference_fabs(x); // if big and even -- caution: only works if x only has single precision if( z >= HEX_DBL( +, 1, 0, +, 24 ) ) { if( z == INFINITY ) return x - x; // nan return reference_copysignd( 0.0, x); // tanpi ( n ) is copysign( 0.0, n) for even integers n. } // reduce to the range [ -0.5, 0.5 ] double nearest = reference_rint( z ); // round to nearest even places n + 0.5 values in the right place for us int i = (int) nearest; // test above against 0x1.0p24 avoids overflow here z -= nearest; //correction for odd integer x for the right sign of zero if( (i&1) && z == 0.0 ) sign = -sign; // track changes to the sign sign *= reference_copysignd(1.0, z); // really should just be an xor z = reference_fabs(z); // remove the sign again // reduce once more // If we don't do this, rounding error in z * M_PI will cause us not to return infinities properly if( z > 0.25 ) { z = 0.5 - z; return sign / reference_tan( z * M_PI ); // use system tan to get the right result } // return sign * reference_tan( z * M_PI ); // use system tan to get the right result } double reference_pown( double x, int i) { return reference_pow( x, (double) i ); } double reference_powr( double x, double y ) { //powr ( x, y ) returns NaN for x < 0. if( x < 0.0 ) return cl_make_nan(); //powr ( x, NaN ) returns the NaN for x >= 0. //powr ( NaN, y ) returns the NaN. if( isnan(x) || isnan(y) ) return x + y; // Note: behavior different here than for pow(1,NaN), pow(NaN, 0) if( x == 1.0 ) { //powr ( +1, +-inf ) returns NaN. if( reference_fabs(y) == INFINITY ) return cl_make_nan(); //powr ( +1, y ) is 1 for finite y. (NaN handled above) return 1.0; } if( y == 0.0 ) { //powr ( +inf, +-0 ) returns NaN. //powr ( +-0, +-0 ) returns NaN. if( x == 0.0 || x == INFINITY ) return cl_make_nan(); //powr ( x, +-0 ) is 1 for finite x > 0. (x <= 0, NaN, INF already handled above) return 1.0; } if( x == 0.0 ) { //powr ( +-0, -inf) is +inf. //powr ( +-0, y ) is +inf for finite y < 0. if( y < 0.0 ) return INFINITY; //powr ( +-0, y ) is +0 for y > 0. (NaN, y==0 handled above) return 0.0; } // x = +inf if( isinf(x) ) { if( y < 0 ) return 0; return INFINITY; } double fabsx = reference_fabs(x); double fabsy = reference_fabs(y); //y = +-inf cases if( isinf(fabsy) ) { if( y < 0 ) { if( fabsx < 1 ) return INFINITY; return 0; } if( fabsx < 1 ) return 0; return INFINITY; } double hi, lo; __log2_ep(&hi, &lo, x); double prod = y * hi; double result = reference_exp2(prod); return result; } double reference_fract( double x, double *ip ) { if(isnan(x)) { *ip = cl_make_nan(); return cl_make_nan(); } float i; float f = modff((float) x, &i ); if( f < 0.0 ) { f = 1.0f + f; i -= 1.0f; if( f == 1.0f ) f = HEX_FLT( +, 1, fffffe, -, 1 ); } *ip = i; return f; } //double my_fdim( double x, double y){ return fdimf( (float) x, (float) y ); } double reference_add( double x, double y ) { volatile float a = (float) x; volatile float b = (float) y; #if defined( __SSE__ ) || (defined( _MSC_VER ) && (defined(_M_IX86) || defined(_M_X64))) // defeat x87 __m128 va = _mm_set_ss( (float) a ); __m128 vb = _mm_set_ss( (float) b ); va = _mm_add_ss( va, vb ); _mm_store_ss( (float*) &a, va ); #elif defined(__PPC__) // Most Power host CPUs do not support the non-IEEE mode (NI) which flushes denorm's to zero. // As such, the reference add with FTZ must be emulated in sw. if (fpu_control & _FPU_MASK_NI) { union{ cl_uint u; cl_float d; } ua; ua.d = a; union{ cl_uint u; cl_float d; } ub; ub.d = b; cl_uint mantA, mantB; cl_ulong addendA, addendB, sum; int expA = extractf( a, &mantA ); int expB = extractf( b, &mantB ); cl_uint signA = ua.u & 0x80000000U; cl_uint signB = ub.u & 0x80000000U; // Force matching exponents if an operand is 0 if (a == 0.0f) { expA = expB; } else if (b == 0.0f) { expB = expA; } addendA = (cl_ulong)mantA << 32; addendB = (cl_ulong)mantB << 32; if (expA >= expB) { // Shift B relative to the A so that their exponents match if( expA > expB ) shift_right_sticky_64( &addendB, expA - expB ); // add if( signA ^ signB ) sub64( &addendA, addendB, &signA, &expA ); else add64( &addendA, addendB, &expA ); } else { // Shift the A relative to B so that their exponents match shift_right_sticky_64( &addendA, expB - expA ); // add if( signA ^ signB ) sub64( &addendB, addendA, &signB, &expB ); else add64( &addendB, addendA, &expB ); addendA = addendB; expA = expB; signA = signB; } // round to IEEE result if (gIsInRTZMode) { ua.d = round_toward_zero_float_ftz( addendA, expA ); } else { ua.d = round_to_nearest_even_float_ftz( addendA, expA ); } // Set the sign ua.u |= signA; a = ua.d; } else { a += b; } #else a += b; #endif return (double) a; } double reference_subtract( double x, double y ) { volatile float a = (float) x; volatile float b = (float) y; #if defined( __SSE__ ) || (defined( _MSC_VER ) && (defined(_M_IX86) || defined(_M_X64))) // defeat x87 __m128 va = _mm_set_ss( (float) a ); __m128 vb = _mm_set_ss( (float) b ); va = _mm_sub_ss( va, vb ); _mm_store_ss( (float*) &a, va ); #else a -= b; #endif return a; } //double reference_divide( double x, double y ){ return (float) x / (float) y; } double reference_multiply( double x, double y) { volatile float a = (float) x; volatile float b = (float) y; #if defined( __SSE__ ) || (defined( _MSC_VER ) && (defined(_M_IX86) || defined(_M_X64))) // defeat x87 __m128 va = _mm_set_ss( (float) a ); __m128 vb = _mm_set_ss( (float) b ); va = _mm_mul_ss( va, vb ); _mm_store_ss( (float*) &a, va ); #elif defined(__PPC__) // Most Power host CPUs do not support the non-IEEE mode (NI) which flushes denorm's to zero. // As such, the reference multiply with FTZ must be emulated in sw. if (fpu_control & _FPU_MASK_NI) { // extract exponent and mantissa // exponent is a standard unbiased signed integer // mantissa is a cl_uint, with leading non-zero bit positioned at the MSB union{ cl_uint u; cl_float d; } ua; ua.d = a; union{ cl_uint u; cl_float d; } ub; ub.d = b; cl_uint mantA, mantB; int expA = extractf( a, &mantA ); int expB = extractf( b, &mantB ); // exact product of A and B int exponent = expA + expB; cl_uint sign = (ua.u ^ ub.u) & 0x80000000U; cl_ulong product = (cl_ulong) mantA * (cl_ulong) mantB; // renormalize -- 1.m * 1.n yields a number between 1.0 and 3.99999.. // The MSB might not be set. If so, fix that. Otherwise, reflect the fact that we got another power of two from the multiplication if( 0 == (0x8000000000000000ULL & product) ) product <<= 1; else exponent++; // 2**31 * 2**31 gives 2**62. If the MSB was set, then our exponent increased. // round to IEEE result -- we do not do flushing to zero here. That part is handled manually in ternary.c. if (gIsInRTZMode) { ua.d = round_toward_zero_float_ftz( product, exponent); } else { ua.d = round_to_nearest_even_float_ftz( product, exponent); } // Set the sign ua.u |= sign; a = ua.d; } else { a *= b; } #else a *= b; #endif return a; } /*double my_remquo( double x, double y, int *iptr ) { if( isnan(x) || isnan(y) || fabs(x) == INFINITY || y == 0.0 ) { *iptr = 0; return NAN; } return (double) remquof( (float) x, (float) y, iptr ); }*/ double reference_lgamma_r( double x, int *signp ) { // This is not currently tested *signp = 0; return x; } int reference_isequal( double x, double y ){ return x == y; } int reference_isfinite( double x ){ return 0 != isfinite(x); } int reference_isgreater( double x, double y ){ return x > y; } int reference_isgreaterequal( double x, double y ){ return x >= y; } int reference_isinf( double x ){ return 0 != isinf(x); } int reference_isless( double x, double y ){ return x < y; } int reference_islessequal( double x, double y ){ return x <= y; } int reference_islessgreater( double x, double y ){ return 0 != islessgreater( x, y ); } int reference_isnan( double x ){ return 0 != isnan( x ); } int reference_isnormal( double x ){ return 0 != isnormal( (float) x ); } int reference_isnotequal( double x, double y ){ return x != y; } int reference_isordered( double x, double y){ return x == x && y == y; } int reference_isunordered( double x, double y ){ return isnan(x) || isnan( y ); } int reference_signbit( float x ){ return 0 != signbit( x ); } #if 1 // defined( _MSC_VER ) //Missing functions for win32 float reference_copysign( float x, float y ) { union { float f; cl_uint u;} ux, uy; ux.f = x; uy.f = y; ux.u &= 0x7fffffffU; ux.u |= uy.u & 0x80000000U; return ux.f; } double reference_copysignd( double x, double y ) { union { double f; cl_ulong u;} ux, uy; ux.f = x; uy.f = y; ux.u &= 0x7fffffffffffffffULL; ux.u |= uy.u & 0x8000000000000000ULL; return ux.f; } double reference_round( double x ) { double absx = reference_fabs(x); if( absx < 0.5 ) return reference_copysignd( 0.0, x ); if( absx < HEX_DBL( +, 1, 0, +, 53 ) ) x = reference_trunc( x + reference_copysignd( 0.5, x ) ); return x; } double reference_trunc( double x ) { if( fabs(x) < HEX_DBL( +, 1, 0, +, 53 ) ) { cl_long l = (cl_long) x; return reference_copysignd( (double) l, x ); } return x; } #ifndef FP_ILOGB0 #define FP_ILOGB0 INT_MIN #endif #ifndef FP_ILOGBNAN #define FP_ILOGBNAN INT_MAX #endif double reference_cbrt(double x){ return reference_copysignd( reference_pow( reference_fabs(x), 1.0/3.0 ), x ); } /* double reference_scalbn(double x, int i) { // suitable for checking single precision scalbnf only if( i > 300 ) return copysign( INFINITY, x); if( i < -300 ) return copysign( 0.0, x); union{ cl_ulong u; double d;} u; u.u = ((cl_ulong) i + 1023) << 52; return x * u.d; } */ double reference_rint( double x ) { if( reference_fabs(x) < HEX_DBL( +, 1, 0, +, 52 ) ) { double magic = reference_copysignd( HEX_DBL( +, 1, 0, +, 52 ), x ); double rounded = (x + magic) - magic; x = reference_copysignd( rounded, x ); } return x; } double reference_acosh( double x ) { // not full precision. Sufficient precision to cover float if( isnan(x) ) return x + x; if( x < 1.0 ) return cl_make_nan(); return reference_log( x + reference_sqrt(x + 1) * reference_sqrt(x-1) ); } double reference_asinh( double x ) { /* * ==================================================== * This function is from fdlibm: http://www.netlib.org * It is Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. * * Developed at SunSoft, a Sun Microsystems, Inc. business. * Permission to use, copy, modify, and distribute this * software is freely granted, provided that this notice * is preserved. * ==================================================== */ if( isnan(x) || isinf(x) ) return x + x; double absx = reference_fabs(x); if( absx < HEX_DBL( +, 1, 0, -, 28 ) ) return x; double sign = reference_copysignd(1.0, x); if( absx > HEX_DBL( +, 1, 0, +, 28 ) ) return sign * (reference_log( absx ) + 0.693147180559945309417232121458176568); // log(2) if( absx > 2.0 ) return sign * reference_log( 2.0 * absx + 1.0 / (reference_sqrt( x * x + 1.0 ) + absx)); return sign * reference_log1p( absx + x*x / (1.0 + reference_sqrt(1.0 + x*x))); } double reference_atanh( double x ) { /* * ==================================================== * This function is from fdlibm: http://www.netlib.org * It is Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. * * Developed at SunSoft, a Sun Microsystems, Inc. business. * Permission to use, copy, modify, and distribute this * software is freely granted, provided that this notice * is preserved. * ==================================================== */ if( isnan(x) ) return x + x; double signed_half = reference_copysignd( 0.5, x ); x = reference_fabs(x); if( x > 1.0 ) return cl_make_nan(); if( x < 0.5 ) return signed_half * reference_log1p( 2.0 * ( x + x*x / (1-x) ) ); return signed_half * reference_log1p(2.0 * x / (1-x)); } double reference_relaxed_atan(double x) { return reference_atan(x); } double reference_relaxed_exp2( double x ) { return reference_exp2(x); } double reference_exp2( double x ) { // Note: only suitable for verifying single precision. Doesn't have range of a full double exp2 implementation. if( x == 0.0 ) return 1.0; // separate x into fractional and integer parts double i = reference_rint( x ); // round to nearest integer if( i < -150 ) return 0.0; if( i > 129 ) return INFINITY; double f = x - i; // -0.5 <= f <= 0.5 // find exp2(f) // calculate as p(f) = (exp2(f)-1)/f // exp2(f) = f * p(f) + 1 // p(f) is a minimax polynomial with error within 0x1.c1fd80f0d1ab7p-50 double p = 0.693147180560184539289 + (0.240226506955902863183 + (0.055504108656833424373 + (0.009618129212846484796 + (0.001333355902958566035 + (0.000154034191902497930 + (0.000015252317761038105 + (0.000001326283129417092 + 0.000000102593187638680 * f)*f)*f)*f)*f)*f)*f)*f; f *= p; f += 1.0; // scale by 2 ** i union{ cl_ulong u; double d; } u; int exponent = (int) i + 1023; u.u = (cl_ulong) exponent << 52; return f * u.d; } double reference_expm1( double x ) { // Note: only suitable for verifying single precision. Doesn't have range of a full double expm1 implementation. It is only accurate to 47 bits or less. // early out for small numbers and NaNs if( ! (reference_fabs(x) > HEX_DBL( +, 1, 0, -, 24 )) ) return x; // early out for large negative numbers if( x < -130.0 ) return -1.0; // early out for large positive numbers if( x > 100.0 ) return INFINITY; // separate x into fractional and integer parts double i = reference_rint( x ); // round to nearest integer double f = x - i; // -0.5 <= f <= 0.5 // reduce f to the range -0.0625 .. f.. 0.0625 int index = (int) (f * 16.0) + 8; // 0...16 static const double reduction[17] = { -0.5, -0.4375, -0.375, -0.3125, -0.25, -0.1875, -0.125, -0.0625, 0.0, +0.0625, +0.125, +0.1875, +0.25, +0.3125, +0.375, +0.4375, +0.5 }; // exponentials[i] = expm1(reduction[i]) static const double exponentials[17] = { HEX_DBL( -, 1, 92e9a0720d3ec, -, 2 ), HEX_DBL( -, 1, 6adb1cd9205ee, -, 2 ), HEX_DBL( -, 1, 40373d42ce2e3, -, 2 ), HEX_DBL( -, 1, 12d35a41ba104, -, 2 ), HEX_DBL( -, 1, c5041854df7d4, -, 3 ), HEX_DBL( -, 1, 5e25fb4fde211, -, 3 ), HEX_DBL( -, 1, e14aed893eef4, -, 4 ), HEX_DBL( -, 1, f0540438fd5c3, -, 5 ), HEX_DBL( +, 0, 0, +, 0 ), HEX_DBL( +, 1, 082b577d34ed8, -, 4 ), HEX_DBL( +, 1, 10b022db7ae68, -, 3 ), HEX_DBL( +, 1, a65c0b85ac1a9, -, 3 ), HEX_DBL( +, 1, 22d78f0fa061a, -, 2 ), HEX_DBL( +, 1, 77a45d8117fd5, -, 2 ), HEX_DBL( +, 1, d1e944f6fbdaa, -, 2 ), HEX_DBL( +, 1, 190048ef6002, -, 1 ), HEX_DBL( +, 1, 4c2531c3c0d38, -, 1 ), }; f -= reduction[index]; // find expm1(f) // calculate as p(f) = (exp(f)-1)/f // expm1(f) = f * p(f) // p(f) is a minimax polynomial with error within 0x1.1d7693618d001p-48 over the range +- 0.0625 double p = 0.999999999999998001599 + (0.499999999999839628284 + (0.166666666672817459505 + (0.041666666612283048687 + (0.008333330214567431435 + (0.001389005319303770070 + 0.000198833381525156667 * f)*f)*f)*f)*f)*f; f *= p; // expm1( reduced f ) // expm1(f) = (exmp1( reduced_f) + 1.0) * ( exponentials[index] + 1 ) - 1 // = exmp1( reduced_f) * exponentials[index] + exmp1( reduced_f) + exponentials[index] + 1 -1 // = exmp1( reduced_f) * exponentials[index] + exmp1( reduced_f) + exponentials[index] f += exponentials[index] + f * exponentials[index]; // scale by e ** i int exponent = (int) i; if( 0 == exponent ) return f; // precise answer for x near 1 // table of e**(i-150) static const double exp_table[128+150+1] = { HEX_DBL( +, 1, 82e16284f5ec5, -, 217 ), HEX_DBL( +, 1, 06e9996332ba1, -, 215 ), HEX_DBL( +, 1, 6555cb289e44b, -, 214 ), HEX_DBL( +, 1, e5ab364643354, -, 213 ), HEX_DBL( +, 1, 4a0bd18e64df7, -, 211 ), HEX_DBL( +, 1, c094499cc578e, -, 210 ), HEX_DBL( +, 1, 30d759323998c, -, 208 ), HEX_DBL( +, 1, 9e5278ab1d4cf, -, 207 ), HEX_DBL( +, 1, 198fa3f30be25, -, 205 ), HEX_DBL( +, 1, 7eae636d6144e, -, 204 ), HEX_DBL( +, 1, 040f1036f4863, -, 202 ), HEX_DBL( +, 1, 6174e477a895f, -, 201 ), HEX_DBL( +, 1, e065b82dd95a, -, 200 ), HEX_DBL( +, 1, 4676be491d129, -, 198 ), HEX_DBL( +, 1, bbb5da5f7c823, -, 197 ), HEX_DBL( +, 1, 2d884eef5fdcb, -, 195 ), HEX_DBL( +, 1, 99d3397ab8371, -, 194 ), HEX_DBL( +, 1, 1681497ed15b3, -, 192 ), HEX_DBL( +, 1, 7a870f597fdbd, -, 191 ), HEX_DBL( +, 1, 013c74edba307, -, 189 ), HEX_DBL( +, 1, 5d9ec4ada7938, -, 188 ), HEX_DBL( +, 1, db2edfd20fa7c, -, 187 ), HEX_DBL( +, 1, 42eb9f39afb0b, -, 185 ), HEX_DBL( +, 1, b6e4f282b43f4, -, 184 ), HEX_DBL( +, 1, 2a42764857b19, -, 182 ), HEX_DBL( +, 1, 9560792d19314, -, 181 ), HEX_DBL( +, 1, 137b6ce8e052c, -, 179 ), HEX_DBL( +, 1, 766b45dd84f18, -, 178 ), HEX_DBL( +, 1, fce362fe6e7d, -, 177 ), HEX_DBL( +, 1, 59d34dd8a5473, -, 175 ), HEX_DBL( +, 1, d606847fc727a, -, 174 ), HEX_DBL( +, 1, 3f6a58b795de3, -, 172 ), HEX_DBL( +, 1, b2216c6efdac1, -, 171 ), HEX_DBL( +, 1, 2705b5b153fb8, -, 169 ), HEX_DBL( +, 1, 90fa1509bd50d, -, 168 ), HEX_DBL( +, 1, 107df698da211, -, 166 ), HEX_DBL( +, 1, 725ae6e7b9d35, -, 165 ), HEX_DBL( +, 1, f75d6040aeff6, -, 164 ), HEX_DBL( +, 1, 56126259e093c, -, 162 ), HEX_DBL( +, 1, d0ec7df4f7bd4, -, 161 ), HEX_DBL( +, 1, 3bf2cf6722e46, -, 159 ), HEX_DBL( +, 1, ad6b22f55db42, -, 158 ), HEX_DBL( +, 1, 23d1f3e5834a, -, 156 ), HEX_DBL( +, 1, 8c9feab89b876, -, 155 ), HEX_DBL( +, 1, 0d88cf37f00dd, -, 153 ), HEX_DBL( +, 1, 6e55d2bf838a7, -, 152 ), HEX_DBL( +, 1, f1e6b68529e33, -, 151 ), HEX_DBL( +, 1, 525be4e4e601d, -, 149 ), HEX_DBL( +, 1, cbe0a45f75eb1, -, 148 ), HEX_DBL( +, 1, 3884e838aea68, -, 146 ), HEX_DBL( +, 1, a8c1f14e2af5d, -, 145 ), HEX_DBL( +, 1, 20a717e64a9bd, -, 143 ), HEX_DBL( +, 1, 8851d84118908, -, 142 ), HEX_DBL( +, 1, 0a9bdfb02d24, -, 140 ), HEX_DBL( +, 1, 6a5bea046b42e, -, 139 ), HEX_DBL( +, 1, ec7f3b269efa8, -, 138 ), HEX_DBL( +, 1, 4eafb87eab0f2, -, 136 ), HEX_DBL( +, 1, c6e2d05bbc, -, 135 ), HEX_DBL( +, 1, 35208867c2683, -, 133 ), HEX_DBL( +, 1, a425b317eeacd, -, 132 ), HEX_DBL( +, 1, 1d8508fa8246a, -, 130 ), HEX_DBL( +, 1, 840fbc08fdc8a, -, 129 ), HEX_DBL( +, 1, 07b7112bc1ffe, -, 127 ), HEX_DBL( +, 1, 666d0dad2961d, -, 126 ), HEX_DBL( +, 1, e726c3f64d0fe, -, 125 ), HEX_DBL( +, 1, 4b0dc07cabf98, -, 123 ), HEX_DBL( +, 1, c1f2daf3b6a46, -, 122 ), HEX_DBL( +, 1, 31c5957a47de2, -, 120 ), HEX_DBL( +, 1, 9f96445648b9f, -, 119 ), HEX_DBL( +, 1, 1a6baeadb4fd1, -, 117 ), HEX_DBL( +, 1, 7fd974d372e45, -, 116 ), HEX_DBL( +, 1, 04da4d1452919, -, 114 ), HEX_DBL( +, 1, 62891f06b345, -, 113 ), HEX_DBL( +, 1, e1dd273aa8a4a, -, 112 ), HEX_DBL( +, 1, 4775e0840bfdd, -, 110 ), HEX_DBL( +, 1, bd109d9d94bda, -, 109 ), HEX_DBL( +, 1, 2e73f53fba844, -, 107 ), HEX_DBL( +, 1, 9b138170d6bfe, -, 106 ), HEX_DBL( +, 1, 175af0cf60ec5, -, 104 ), HEX_DBL( +, 1, 7baee1bffa80b, -, 103 ), HEX_DBL( +, 1, 02057d1245ceb, -, 101 ), HEX_DBL( +, 1, 5eafffb34ba31, -, 100 ), HEX_DBL( +, 1, dca23bae16424, -, 99 ), HEX_DBL( +, 1, 43e7fc88b8056, -, 97 ), HEX_DBL( +, 1, b83bf23a9a9eb, -, 96 ), HEX_DBL( +, 1, 2b2b8dd05b318, -, 94 ), HEX_DBL( +, 1, 969d47321e4cc, -, 93 ), HEX_DBL( +, 1, 1452b7723aed2, -, 91 ), HEX_DBL( +, 1, 778fe2497184c, -, 90 ), HEX_DBL( +, 1, fe7116182e9cc, -, 89 ), HEX_DBL( +, 1, 5ae191a99585a, -, 87 ), HEX_DBL( +, 1, d775d87da854d, -, 86 ), HEX_DBL( +, 1, 4063f8cc8bb98, -, 84 ), HEX_DBL( +, 1, b374b315f87c1, -, 83 ), HEX_DBL( +, 1, 27ec458c65e3c, -, 81 ), HEX_DBL( +, 1, 923372c67a074, -, 80 ), HEX_DBL( +, 1, 1152eaeb73c08, -, 78 ), HEX_DBL( +, 1, 737c5645114b5, -, 77 ), HEX_DBL( +, 1, f8e6c24b5592e, -, 76 ), HEX_DBL( +, 1, 571db733a9d61, -, 74 ), HEX_DBL( +, 1, d257d547e083f, -, 73 ), HEX_DBL( +, 1, 3ce9b9de78f85, -, 71 ), HEX_DBL( +, 1, aebabae3a41b5, -, 70 ), HEX_DBL( +, 1, 24b6031b49bda, -, 68 ), HEX_DBL( +, 1, 8dd5e1bb09d7e, -, 67 ), HEX_DBL( +, 1, 0e5b73d1ff53d, -, 65 ), HEX_DBL( +, 1, 6f741de1748ec, -, 64 ), HEX_DBL( +, 1, f36bd37f42f3e, -, 63 ), HEX_DBL( +, 1, 536452ee2f75c, -, 61 ), HEX_DBL( +, 1, cd480a1b7482, -, 60 ), HEX_DBL( +, 1, 39792499b1a24, -, 58 ), HEX_DBL( +, 1, aa0de4bf35b38, -, 57 ), HEX_DBL( +, 1, 2188ad6ae3303, -, 55 ), HEX_DBL( +, 1, 898471fca6055, -, 54 ), HEX_DBL( +, 1, 0b6c3afdde064, -, 52 ), HEX_DBL( +, 1, 6b7719a59f0e, -, 51 ), HEX_DBL( +, 1, ee001eed62aa, -, 50 ), HEX_DBL( +, 1, 4fb547c775da8, -, 48 ), HEX_DBL( +, 1, c8464f7616468, -, 47 ), HEX_DBL( +, 1, 36121e24d3bba, -, 45 ), HEX_DBL( +, 1, a56e0c2ac7f75, -, 44 ), HEX_DBL( +, 1, 1e642baeb84a, -, 42 ), HEX_DBL( +, 1, 853f01d6d53ba, -, 41 ), HEX_DBL( +, 1, 0885298767e9a, -, 39 ), HEX_DBL( +, 1, 67852a7007e42, -, 38 ), HEX_DBL( +, 1, e8a37a45fc32e, -, 37 ), HEX_DBL( +, 1, 4c1078fe9228a, -, 35 ), HEX_DBL( +, 1, c3527e433fab1, -, 34 ), HEX_DBL( +, 1, 32b48bf117da2, -, 32 ), HEX_DBL( +, 1, a0db0d0ddb3ec, -, 31 ), HEX_DBL( +, 1, 1b48655f37267, -, 29 ), HEX_DBL( +, 1, 81056ff2c5772, -, 28 ), HEX_DBL( +, 1, 05a628c699fa1, -, 26 ), HEX_DBL( +, 1, 639e3175a689d, -, 25 ), HEX_DBL( +, 1, e355bbaee85cb, -, 24 ), HEX_DBL( +, 1, 4875ca227ec38, -, 22 ), HEX_DBL( +, 1, be6c6fdb01612, -, 21 ), HEX_DBL( +, 1, 2f6053b981d98, -, 19 ), HEX_DBL( +, 1, 9c54c3b43bc8b, -, 18 ), HEX_DBL( +, 1, 18354238f6764, -, 16 ), HEX_DBL( +, 1, 7cd79b5647c9b, -, 15 ), HEX_DBL( +, 1, 02cf22526545a, -, 13 ), HEX_DBL( +, 1, 5fc21041027ad, -, 12 ), HEX_DBL( +, 1, de16b9c24a98f, -, 11 ), HEX_DBL( +, 1, 44e51f113d4d6, -, 9 ), HEX_DBL( +, 1, b993fe00d5376, -, 8 ), HEX_DBL( +, 1, 2c155b8213cf4, -, 6 ), HEX_DBL( +, 1, 97db0ccceb0af, -, 5 ), HEX_DBL( +, 1, 152aaa3bf81cc, -, 3 ), HEX_DBL( +, 1, 78b56362cef38, -, 2 ), HEX_DBL( +, 1, 0, +, 0 ), HEX_DBL( +, 1, 5bf0a8b145769, +, 1 ), HEX_DBL( +, 1, d8e64b8d4ddae, +, 2 ), HEX_DBL( +, 1, 415e5bf6fb106, +, 4 ), HEX_DBL( +, 1, b4c902e273a58, +, 5 ), HEX_DBL( +, 1, 28d389970338f, +, 7 ), HEX_DBL( +, 1, 936dc5690c08f, +, 8 ), HEX_DBL( +, 1, 122885aaeddaa, +, 10 ), HEX_DBL( +, 1, 749ea7d470c6e, +, 11 ), HEX_DBL( +, 1, fa7157c470f82, +, 12 ), HEX_DBL( +, 1, 5829dcf95056, +, 14 ), HEX_DBL( +, 1, d3c4488ee4f7f, +, 15 ), HEX_DBL( +, 1, 3de1654d37c9a, +, 17 ), HEX_DBL( +, 1, b00b5916ac955, +, 18 ), HEX_DBL( +, 1, 259ac48bf05d7, +, 20 ), HEX_DBL( +, 1, 8f0ccafad2a87, +, 21 ), HEX_DBL( +, 1, 0f2ebd0a8002, +, 23 ), HEX_DBL( +, 1, 709348c0ea4f9, +, 24 ), HEX_DBL( +, 1, f4f22091940bd, +, 25 ), HEX_DBL( +, 1, 546d8f9ed26e1, +, 27 ), HEX_DBL( +, 1, ceb088b68e804, +, 28 ), HEX_DBL( +, 1, 3a6e1fd9eecfd, +, 30 ), HEX_DBL( +, 1, ab5adb9c436, +, 31 ), HEX_DBL( +, 1, 226af33b1fdc1, +, 33 ), HEX_DBL( +, 1, 8ab7fb5475fb7, +, 34 ), HEX_DBL( +, 1, 0c3d3920962c9, +, 36 ), HEX_DBL( +, 1, 6c932696a6b5d, +, 37 ), HEX_DBL( +, 1, ef822f7f6731d, +, 38 ), HEX_DBL( +, 1, 50bba3796379a, +, 40 ), HEX_DBL( +, 1, c9aae4631c056, +, 41 ), HEX_DBL( +, 1, 370470aec28ed, +, 43 ), HEX_DBL( +, 1, a6b765d8cdf6d, +, 44 ), HEX_DBL( +, 1, 1f43fcc4b662c, +, 46 ), HEX_DBL( +, 1, 866f34a725782, +, 47 ), HEX_DBL( +, 1, 0953e2f3a1ef7, +, 49 ), HEX_DBL( +, 1, 689e221bc8d5b, +, 50 ), HEX_DBL( +, 1, ea215a1d20d76, +, 51 ), HEX_DBL( +, 1, 4d13fbb1a001a, +, 53 ), HEX_DBL( +, 1, c4b334617cc67, +, 54 ), HEX_DBL( +, 1, 33a43d282a519, +, 56 ), HEX_DBL( +, 1, a220d397972eb, +, 57 ), HEX_DBL( +, 1, 1c25c88df6862, +, 59 ), HEX_DBL( +, 1, 8232558201159, +, 60 ), HEX_DBL( +, 1, 0672a3c9eb871, +, 62 ), HEX_DBL( +, 1, 64b41c6d37832, +, 63 ), HEX_DBL( +, 1, e4cf766fe49be, +, 64 ), HEX_DBL( +, 1, 49767bc0483e3, +, 66 ), HEX_DBL( +, 1, bfc951eb8bb76, +, 67 ), HEX_DBL( +, 1, 304d6aeca254b, +, 69 ), HEX_DBL( +, 1, 9d97010884251, +, 70 ), HEX_DBL( +, 1, 19103e4080b45, +, 72 ), HEX_DBL( +, 1, 7e013cd114461, +, 73 ), HEX_DBL( +, 1, 03996528e074c, +, 75 ), HEX_DBL( +, 1, 60d4f6fdac731, +, 76 ), HEX_DBL( +, 1, df8c5af17ba3b, +, 77 ), HEX_DBL( +, 1, 45e3076d61699, +, 79 ), HEX_DBL( +, 1, baed16a6e0da7, +, 80 ), HEX_DBL( +, 1, 2cffdfebde1a1, +, 82 ), HEX_DBL( +, 1, 9919cabefcb69, +, 83 ), HEX_DBL( +, 1, 160345c9953e3, +, 85 ), HEX_DBL( +, 1, 79dbc9dc53c66, +, 86 ), HEX_DBL( +, 1, 00c810d464097, +, 88 ), HEX_DBL( +, 1, 5d009394c5c27, +, 89 ), HEX_DBL( +, 1, da57de8f107a8, +, 90 ), HEX_DBL( +, 1, 425982cf597cd, +, 92 ), HEX_DBL( +, 1, b61e5ca3a5e31, +, 93 ), HEX_DBL( +, 1, 29bb825dfcf87, +, 95 ), HEX_DBL( +, 1, 94a90db0d6fe2, +, 96 ), HEX_DBL( +, 1, 12fec759586fd, +, 98 ), HEX_DBL( +, 1, 75c1dc469e3af, +, 99 ), HEX_DBL( +, 1, fbfd219c43b04, +, 100 ), HEX_DBL( +, 1, 5936d44e1a146, +, 102 ), HEX_DBL( +, 1, d531d8a7ee79c, +, 103 ), HEX_DBL( +, 1, 3ed9d24a2d51b, +, 105 ), HEX_DBL( +, 1, b15cfe5b6e17b, +, 106 ), HEX_DBL( +, 1, 268038c2c0e, +, 108 ), HEX_DBL( +, 1, 9044a73545d48, +, 109 ), HEX_DBL( +, 1, 1002ab6218b38, +, 111 ), HEX_DBL( +, 1, 71b3540cbf921, +, 112 ), HEX_DBL( +, 1, f6799ea9c414a, +, 113 ), HEX_DBL( +, 1, 55779b984f3eb, +, 115 ), HEX_DBL( +, 1, d01a210c44aa4, +, 116 ), HEX_DBL( +, 1, 3b63da8e9121, +, 118 ), HEX_DBL( +, 1, aca8d6b0116b8, +, 119 ), HEX_DBL( +, 1, 234de9e0c74e9, +, 121 ), HEX_DBL( +, 1, 8bec7503ca477, +, 122 ), HEX_DBL( +, 1, 0d0eda9796b9, +, 124 ), HEX_DBL( +, 1, 6db0118477245, +, 125 ), HEX_DBL( +, 1, f1056dc7bf22d, +, 126 ), HEX_DBL( +, 1, 51c2cc3433801, +, 128 ), HEX_DBL( +, 1, cb108ffbec164, +, 129 ), HEX_DBL( +, 1, 37f780991b584, +, 131 ), HEX_DBL( +, 1, a801c0ea8ac4d, +, 132 ), HEX_DBL( +, 1, 20247cc4c46c1, +, 134 ), HEX_DBL( +, 1, 87a0553328015, +, 135 ), HEX_DBL( +, 1, 0a233dee4f9bb, +, 137 ), HEX_DBL( +, 1, 69b7f55b808ba, +, 138 ), HEX_DBL( +, 1, eba064644060a, +, 139 ), HEX_DBL( +, 1, 4e184933d9364, +, 141 ), HEX_DBL( +, 1, c614fe2531841, +, 142 ), HEX_DBL( +, 1, 3494a9b171bf5, +, 144 ), HEX_DBL( +, 1, a36798b9d969b, +, 145 ), HEX_DBL( +, 1, 1d03d8c0c04af, +, 147 ), HEX_DBL( +, 1, 836026385c974, +, 148 ), HEX_DBL( +, 1, 073fbe9ac901d, +, 150 ), HEX_DBL( +, 1, 65cae0969f286, +, 151 ), HEX_DBL( +, 1, e64a58639cae8, +, 152 ), HEX_DBL( +, 1, 4a77f5f9b50f9, +, 154 ), HEX_DBL( +, 1, c12744a3a28e3, +, 155 ), HEX_DBL( +, 1, 313b3b6978e85, +, 157 ), HEX_DBL( +, 1, 9eda3a31e587e, +, 158 ), HEX_DBL( +, 1, 19ebe56b56453, +, 160 ), HEX_DBL( +, 1, 7f2bc6e599b7e, +, 161 ), HEX_DBL( +, 1, 04644610df2ff, +, 163 ), HEX_DBL( +, 1, 61e8b490ac4e6, +, 164 ), HEX_DBL( +, 1, e103201f299b3, +, 165 ), HEX_DBL( +, 1, 46e1b637beaf5, +, 167 ), HEX_DBL( +, 1, bc473cfede104, +, 168 ), HEX_DBL( +, 1, 2deb1b9c85e2d, +, 170 ), HEX_DBL( +, 1, 9a5981ca67d1, +, 171 ), HEX_DBL( +, 1, 16dc8a9ef670b, +, 173 ), HEX_DBL( +, 1, 7b03166942309, +, 174 ), HEX_DBL( +, 1, 0190be03150a7, +, 176 ), HEX_DBL( +, 1, 5e1152f9a8119, +, 177 ), HEX_DBL( +, 1, dbca9263f8487, +, 178 ), HEX_DBL( +, 1, 43556dee93bee, +, 180 ), HEX_DBL( +, 1, b774c12967dfa, +, 181 ), HEX_DBL( +, 1, 2aa4306e922c2, +, 183 ), HEX_DBL( +, 1, 95e54c5dd4217, +, 184 ) }; // scale by e**i -- (expm1(f) + 1)*e**i - 1 = expm1(f) * e**i + e**i - 1 = e**i return exp_table[exponent+150] + (f * exp_table[exponent+150] - 1.0); } double reference_fmax( double x, double y ) { if( isnan(y) ) return x; return x >= y ? x : y; } double reference_fmin( double x, double y ) { if( isnan(y) ) return x; return x <= y ? x : y; } double reference_hypot( double x, double y ) { // Since the inputs are actually floats, we don't have to worry about range here if( isinf(x) || isinf(y) ) return INFINITY; return sqrt( x * x + y * y ); } int reference_ilogbl( long double x) { extern int gDeviceILogb0, gDeviceILogbNaN; // Since we are just using this to verify double precision, we can // use the double precision ilogb here union { double f; cl_ulong u;} u; u.f = (double) x; int exponent = (int)(u.u >> 52) & 0x7ff; if( exponent == 0x7ff ) { if( u.u & 0x000fffffffffffffULL ) return gDeviceILogbNaN; return CL_INT_MAX; } if( exponent == 0 ) { // deal with denormals u.f = x * HEX_DBL( +, 1, 0, +, 64 ); exponent = (cl_uint)(u.u >> 52) & 0x7ff; if( exponent == 0 ) return gDeviceILogb0; exponent -= 1023 + 64; return exponent; } return exponent - 1023; } //double reference_log2( double x ) //{ // return log( x ) * 1.44269504088896340735992468100189214; //} double reference_relaxed_log2( double x ) { return reference_log2(x); } double reference_log2( double x ) { if( isnan(x) || x < 0.0 || x == -INFINITY) return cl_make_nan(); if( x == 0.0f) return -INFINITY; if( x == INFINITY ) return INFINITY; double hi, lo; __log2_ep( &hi, &lo, x ); return hi; } double reference_log1p( double x ) { // This function is suitable only for verifying log1pf(). It produces several double precision ulps of error. // Handle small and NaN if( ! ( reference_fabs(x) > HEX_DBL( +, 1, 0, -, 53 ) ) ) return x; // deal with special values if( x <= -1.0 ) { if( x < -1.0 ) return cl_make_nan(); return -INFINITY; } // infinity if( x == INFINITY ) return INFINITY; // High precision result for when near 0, to avoid problems with the reference result falling in the wrong binade. if( reference_fabs(x) < HEX_DBL( +, 1, 0, -, 28 ) ) return (1.0 - 0.5 * x) * x; // Our polynomial is only good in the region +-2**-4. // If we aren't in that range then we need to reduce to be in that range double correctionLo = -0.0; // correction down stream to compensate for the reduction, if any double correctionHi = -0.0; // correction down stream to compensate for the exponent, if any if( reference_fabs(x) > HEX_DBL( +, 1, 0, -, 4 ) ) { x += 1.0; // double should cover any loss of precision here // separate x into (1+f) * 2**i union{ double d; cl_ulong u;} u; u.d = x; int i = (int) ((u.u >> 52) & 0x7ff) - 1023; u.u &= 0x000fffffffffffffULL; int index = (int) (u.u >> 48 ); u.u |= 0x3ff0000000000000ULL; double f = u.d; // further reduce f to be within 1/16 of 1.0 static const double scale_table[16] = { 1.0, HEX_DBL( +, 1, d2d2d2d6e3f79, -, 1 ), HEX_DBL( +, 1, b8e38e42737a1, -, 1 ), HEX_DBL( +, 1, a1af28711adf3, -, 1 ), HEX_DBL( +, 1, 8cccccd88dd65, -, 1 ), HEX_DBL( +, 1, 79e79e810ec8f, -, 1 ), HEX_DBL( +, 1, 68ba2e94df404, -, 1 ), HEX_DBL( +, 1, 590b216defb29, -, 1 ), HEX_DBL( +, 1, 4aaaaab1500ed, -, 1 ), HEX_DBL( +, 1, 3d70a3e0d6f73, -, 1 ), HEX_DBL( +, 1, 313b13bb39f4f, -, 1 ), HEX_DBL( +, 1, 25ed09823f1cc, -, 1 ), HEX_DBL( +, 1, 1b6db6e77457b, -, 1 ), HEX_DBL( +, 1, 11a7b96a3a34f, -, 1 ), HEX_DBL( +, 1, 0888888e46fea, -, 1 ), HEX_DBL( +, 1, 00000038e9862, -, 1 ) }; // correction_table[i] = -log( scale_table[i] ) // All entries have >= 64 bits of precision (rather than the expected 53) static const double correction_table[16] = { -0.0, HEX_DBL( +, 1, 7a5c722c16058, -, 4 ), HEX_DBL( +, 1, 323db16c89ab1, -, 3 ), HEX_DBL( +, 1, a0f87d180629, -, 3 ), HEX_DBL( +, 1, 050279324e17c, -, 2 ), HEX_DBL( +, 1, 36f885bb270b0, -, 2 ), HEX_DBL( +, 1, 669b771b5cc69, -, 2 ), HEX_DBL( +, 1, 94203a6292a05, -, 2 ), HEX_DBL( +, 1, bfb4f9cb333a4, -, 2 ), HEX_DBL( +, 1, e982376ddb80e, -, 2 ), HEX_DBL( +, 1, 08d5d8769b2b2, -, 1 ), HEX_DBL( +, 1, 1c288bc00e0cf, -, 1 ), HEX_DBL( +, 1, 2ec7535b31ecb, -, 1 ), HEX_DBL( +, 1, 40bed0adc63fb, -, 1 ), HEX_DBL( +, 1, 521a5c0330615, -, 1 ), HEX_DBL( +, 1, 62e42f7dd092c, -, 1 ) }; f *= scale_table[index]; correctionLo = correction_table[index]; // log( 2**(i) ) = i * log(2) correctionHi = (double)i * 0.693147180559945309417232121458176568; x = f - 1.0; } // minmax polynomial for p(x) = (log(x+1) - x)/x valid over the range x = [-1/16, 1/16] // max error HEX_DBL( +, 1, 048f61f9a5eca, -, 52 ) double p = HEX_DBL( -, 1, cc33de97a9d7b, -, 46 ) + (HEX_DBL( -, 1, fffffffff3eb7, -, 2 ) + (HEX_DBL( +, 1, 5555555633ef7, -, 2 ) + (HEX_DBL( -, 1, 00000062c78, -, 2 ) + (HEX_DBL( +, 1, 9999958a3321, -, 3 ) + (HEX_DBL( -, 1, 55534ce65c347, -, 3 ) + (HEX_DBL( +, 1, 24957208391a5, -, 3 ) + (HEX_DBL( -, 1, 02287b9a5b4a1, -, 3 ) + HEX_DBL( +, 1, c757d922180ed, -, 4 ) * x)*x)*x)*x)*x)*x)*x)*x; // log(x+1) = x * p(x) + x x += x * p; return correctionHi + (correctionLo + x); } double reference_logb( double x ) { union { float f; cl_uint u;} u; u.f = (float) x; cl_int exponent = (u.u >> 23) & 0xff; if( exponent == 0xff ) return x * x; if( exponent == 0 ) { // deal with denormals u.u = (u.u & 0x007fffff) | 0x3f800000; u.f -= 1.0f; exponent = (u.u >> 23) & 0xff; if( exponent == 0 ) return -INFINITY; return exponent - (127 + 126); } return exponent - 127; } double reference_relaxed_reciprocal(double x) { return 1.0f / ((float) x); } double reference_reciprocal( double x ) { return 1.0 / x; } double reference_remainder( double x, double y ) { int i; return reference_remquo( x, y, &i ); } double reference_lgamma( double x) { /* * ==================================================== * This function is from fdlibm. http://www.netlib.org * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. * * Developed at SunSoft, a Sun Microsystems, Inc. business. * Permission to use, copy, modify, and distribute this * software is freely granted, provided that this notice * is preserved. * ==================================================== * */ static const double //two52 = 4.50359962737049600000e+15, /* 0x43300000, 0x00000000 */ half= 5.00000000000000000000e-01, /* 0x3FE00000, 0x00000000 */ one = 1.00000000000000000000e+00, /* 0x3FF00000, 0x00000000 */ pi = 3.14159265358979311600e+00, /* 0x400921FB, 0x54442D18 */ a0 = 7.72156649015328655494e-02, /* 0x3FB3C467, 0xE37DB0C8 */ a1 = 3.22467033424113591611e-01, /* 0x3FD4A34C, 0xC4A60FAD */ a2 = 6.73523010531292681824e-02, /* 0x3FB13E00, 0x1A5562A7 */ a3 = 2.05808084325167332806e-02, /* 0x3F951322, 0xAC92547B */ a4 = 7.38555086081402883957e-03, /* 0x3F7E404F, 0xB68FEFE8 */ a5 = 2.89051383673415629091e-03, /* 0x3F67ADD8, 0xCCB7926B */ a6 = 1.19270763183362067845e-03, /* 0x3F538A94, 0x116F3F5D */ a7 = 5.10069792153511336608e-04, /* 0x3F40B6C6, 0x89B99C00 */ a8 = 2.20862790713908385557e-04, /* 0x3F2CF2EC, 0xED10E54D */ a9 = 1.08011567247583939954e-04, /* 0x3F1C5088, 0x987DFB07 */ a10 = 2.52144565451257326939e-05, /* 0x3EFA7074, 0x428CFA52 */ a11 = 4.48640949618915160150e-05, /* 0x3F07858E, 0x90A45837 */ tc = 1.46163214496836224576e+00, /* 0x3FF762D8, 0x6356BE3F */ tf = -1.21486290535849611461e-01, /* 0xBFBF19B9, 0xBCC38A42 */ /* tt = -(tail of tf) */ tt = -3.63867699703950536541e-18, /* 0xBC50C7CA, 0xA48A971F */ t0 = 4.83836122723810047042e-01, /* 0x3FDEF72B, 0xC8EE38A2 */ t1 = -1.47587722994593911752e-01, /* 0xBFC2E427, 0x8DC6C509 */ t2 = 6.46249402391333854778e-02, /* 0x3FB08B42, 0x94D5419B */ t3 = -3.27885410759859649565e-02, /* 0xBFA0C9A8, 0xDF35B713 */ t4 = 1.79706750811820387126e-02, /* 0x3F9266E7, 0x970AF9EC */ t5 = -1.03142241298341437450e-02, /* 0xBF851F9F, 0xBA91EC6A */ t6 = 6.10053870246291332635e-03, /* 0x3F78FCE0, 0xE370E344 */ t7 = -3.68452016781138256760e-03, /* 0xBF6E2EFF, 0xB3E914D7 */ t8 = 2.25964780900612472250e-03, /* 0x3F6282D3, 0x2E15C915 */ t9 = -1.40346469989232843813e-03, /* 0xBF56FE8E, 0xBF2D1AF1 */ t10 = 8.81081882437654011382e-04, /* 0x3F4CDF0C, 0xEF61A8E9 */ t11 = -5.38595305356740546715e-04, /* 0xBF41A610, 0x9C73E0EC */ t12 = 3.15632070903625950361e-04, /* 0x3F34AF6D, 0x6C0EBBF7 */ t13 = -3.12754168375120860518e-04, /* 0xBF347F24, 0xECC38C38 */ t14 = 3.35529192635519073543e-04, /* 0x3F35FD3E, 0xE8C2D3F4 */ u0 = -7.72156649015328655494e-02, /* 0xBFB3C467, 0xE37DB0C8 */ u1 = 6.32827064025093366517e-01, /* 0x3FE4401E, 0x8B005DFF */ u2 = 1.45492250137234768737e+00, /* 0x3FF7475C, 0xD119BD6F */ u3 = 9.77717527963372745603e-01, /* 0x3FEF4976, 0x44EA8450 */ u4 = 2.28963728064692451092e-01, /* 0x3FCD4EAE, 0xF6010924 */ u5 = 1.33810918536787660377e-02, /* 0x3F8B678B, 0xBF2BAB09 */ v1 = 2.45597793713041134822e+00, /* 0x4003A5D7, 0xC2BD619C */ v2 = 2.12848976379893395361e+00, /* 0x40010725, 0xA42B18F5 */ v3 = 7.69285150456672783825e-01, /* 0x3FE89DFB, 0xE45050AF */ v4 = 1.04222645593369134254e-01, /* 0x3FBAAE55, 0xD6537C88 */ v5 = 3.21709242282423911810e-03, /* 0x3F6A5ABB, 0x57D0CF61 */ s0 = -7.72156649015328655494e-02, /* 0xBFB3C467, 0xE37DB0C8 */ s1 = 2.14982415960608852501e-01, /* 0x3FCB848B, 0x36E20878 */ s2 = 3.25778796408930981787e-01, /* 0x3FD4D98F, 0x4F139F59 */ s3 = 1.46350472652464452805e-01, /* 0x3FC2BB9C, 0xBEE5F2F7 */ s4 = 2.66422703033638609560e-02, /* 0x3F9B481C, 0x7E939961 */ s5 = 1.84028451407337715652e-03, /* 0x3F5E26B6, 0x7368F239 */ s6 = 3.19475326584100867617e-05, /* 0x3F00BFEC, 0xDD17E945 */ r1 = 1.39200533467621045958e+00, /* 0x3FF645A7, 0x62C4AB74 */ r2 = 7.21935547567138069525e-01, /* 0x3FE71A18, 0x93D3DCDC */ r3 = 1.71933865632803078993e-01, /* 0x3FC601ED, 0xCCFBDF27 */ r4 = 1.86459191715652901344e-02, /* 0x3F9317EA, 0x742ED475 */ r5 = 7.77942496381893596434e-04, /* 0x3F497DDA, 0xCA41A95B */ r6 = 7.32668430744625636189e-06, /* 0x3EDEBAF7, 0xA5B38140 */ w0 = 4.18938533204672725052e-01, /* 0x3FDACFE3, 0x90C97D69 */ w1 = 8.33333333333329678849e-02, /* 0x3FB55555, 0x5555553B */ w2 = -2.77777777728775536470e-03, /* 0xBF66C16C, 0x16B02E5C */ w3 = 7.93650558643019558500e-04, /* 0x3F4A019F, 0x98CF38B6 */ w4 = -5.95187557450339963135e-04, /* 0xBF4380CB, 0x8C0FE741 */ w5 = 8.36339918996282139126e-04, /* 0x3F4B67BA, 0x4CDAD5D1 */ w6 = -1.63092934096575273989e-03; /* 0xBF5AB89D, 0x0B9E43E4 */ static const double zero= 0.00000000000000000000e+00; double t,y,z,nadj,p,p1,p2,p3,q,r,w; cl_int i,hx,lx,ix; union{ double d; cl_ulong u;}u; u.d = x; hx = (cl_int) (u.u >> 32); lx = (cl_int) (u.u & 0xffffffffULL); /* purge off +-inf, NaN, +-0, and negative arguments */ // *signgamp = 1; ix = hx&0x7fffffff; if(ix>=0x7ff00000) return x*x; if((ix|lx)==0) return INFINITY; if(ix<0x3b900000) { /* |x|<2**-70, return -log(|x|) */ if(hx<0) { // *signgamp = -1; return -reference_log(-x); } else return -reference_log(x); } if(hx<0) { if(ix>=0x43300000) /* |x|>=2**52, must be -integer */ return INFINITY; t = reference_sinpi(x); if(t==zero) return INFINITY; /* -integer */ nadj = reference_log(pi/reference_fabs(t*x)); // if(t=0x3FE76944) {y = 1.0-x; i= 0;} else if(ix>=0x3FCDA661) {y= x-(tc-one); i=1;} else {y = x; i=2;} } else { r = zero; if(ix>=0x3FFBB4C3) {y=2.0-x;i=0;} /* [1.7316,2] */ else if(ix>=0x3FF3B4C4) {y=x-tc;i=1;} /* [1.23,1.73] */ else {y=x-one;i=2;} } switch(i) { case 0: z = y*y; p1 = a0+z*(a2+z*(a4+z*(a6+z*(a8+z*a10)))); p2 = z*(a1+z*(a3+z*(a5+z*(a7+z*(a9+z*a11))))); p = y*p1+p2; r += (p-0.5*y); break; case 1: z = y*y; w = z*y; p1 = t0+w*(t3+w*(t6+w*(t9 +w*t12))); /* parallel comp */ p2 = t1+w*(t4+w*(t7+w*(t10+w*t13))); p3 = t2+w*(t5+w*(t8+w*(t11+w*t14))); p = z*p1-(tt-w*(p2+y*p3)); r += (tf + p); break; case 2: p1 = y*(u0+y*(u1+y*(u2+y*(u3+y*(u4+y*u5))))); p2 = one+y*(v1+y*(v2+y*(v3+y*(v4+y*v5)))); r += (-0.5*y + p1/p2); } } else if(ix<0x40200000) { /* x < 8.0 */ i = (int)x; t = zero; y = x-(double)i; p = y*(s0+y*(s1+y*(s2+y*(s3+y*(s4+y*(s5+y*s6)))))); q = one+y*(r1+y*(r2+y*(r3+y*(r4+y*(r5+y*r6))))); r = half*y+p/q; z = one; /* lgamma(1+s) = log(s) + lgamma(s) */ switch(i) { case 7: z *= (y+6.0); /* FALLTHRU */ case 6: z *= (y+5.0); /* FALLTHRU */ case 5: z *= (y+4.0); /* FALLTHRU */ case 4: z *= (y+3.0); /* FALLTHRU */ case 3: z *= (y+2.0); /* FALLTHRU */ r += reference_log(z); break; } /* 8.0 <= x < 2**58 */ } else if (ix < 0x43900000) { t = reference_log(x); z = one/x; y = z*z; w = w0+z*(w1+y*(w2+y*(w3+y*(w4+y*(w5+y*w6))))); r = (x-half)*(t-one)+w; } else /* 2**58 <= x <= inf */ r = x*(reference_log(x)-one); if(hx<0) r = nadj - r; return r; } #endif // _MSC_VER double reference_assignment( double x ){ return x; } int reference_not( double x ) { int r = !x; return r; } #pragma mark - #pragma mark Double testing #ifndef M_PIL #define M_PIL 3.14159265358979323846264338327950288419716939937510582097494459230781640628620899L #endif static long double reduce1l( long double x ); #ifdef __PPC__ // Since long double on PPC is really extended precision double arithmetic // consisting of two doubles (a high and low). This form of long double has // the potential of representing a number with more than LDBL_MANT_DIG digits // such that reduction algorithm used for other architectures will not work. // Instead and alternate reduction method is used. static long double reduce1l( long double x ) { union { long double ld; double d[2]; } u; // Reduce the high and low halfs separately. u.ld = x; return ((long double)reduce1(u.d[0]) + reduce1(u.d[1])); } #else // !__PPC__ static long double reduce1l( long double x ) { static long double unit_exp = 0; if( 0.0L == unit_exp ) unit_exp = scalbnl( 1.0L, LDBL_MANT_DIG); if( reference_fabsl(x) >= unit_exp ) { if( reference_fabsl(x) == INFINITY ) return cl_make_nan(); return 0.0L; //we patch up the sign for sinPi and cosPi later, since they need different signs } // Find the nearest multiple of 2 const long double r = reference_copysignl( unit_exp, x ); long double z = x + r; z -= r; // subtract it from x. Value is now in the range -1 <= x <= 1 return x - z; } #endif // __PPC__ long double reference_acospil( long double x){ return reference_acosl( x ) / M_PIL; } long double reference_asinpil( long double x){ return reference_asinl( x ) / M_PIL; } long double reference_atanpil( long double x){ return reference_atanl( x ) / M_PIL; } long double reference_atan2pil( long double y, long double x){ return reference_atan2l( y, x) / M_PIL; } long double reference_cospil( long double x) { if( reference_fabsl(x) >= HEX_LDBL( +, 1, 0, +, 54 ) ) { if( reference_fabsl(x) == INFINITY ) return cl_make_nan(); //Note this probably fails for odd values between 0x1.0p52 and 0x1.0p53. //However, when starting with single precision inputs, there will be no odd values. return 1.0L; } x = reduce1l(x); #if DBL_MANT_DIG >= LDBL_MANT_DIG // phase adjust double xhi = 0.0; double xlo = 0.0; xhi = (double) x + 0.5; if(reference_fabsl(x) > 0.5L) { xlo = xhi - x; xlo = 0.5 - xlo; } else { xlo = xhi - 0.5; xlo = x - xlo; } // reduce to [-0.5, 0.5] if( xhi < -0.5 ) { xhi = -1.0 - xhi; xlo = -xlo; } else if ( xhi > 0.5 ) { xhi = 1.0 - xhi; xlo = -xlo; } // cosPi zeros are all +0 if( xhi == 0.0 && xlo == 0.0 ) return 0.0; xhi *= M_PI; xlo *= M_PI; xhi += xlo; return reference_sinl( xhi ); #else // phase adjust x += 0.5L; // reduce to [-0.5, 0.5] if( x < -0.5L ) x = -1.0L - x; else if ( x > 0.5L ) x = 1.0L - x; // cosPi zeros are all +0 if( x == 0.0L ) return 0.0L; return reference_sinl( x * M_PIL ); #endif } long double reference_dividel( long double x, long double y) { double dx = x; double dy = y; return dx/dy; } typedef struct{ double hi, lo; } double_double; // Split doubles_double into a series of consecutive 26-bit precise doubles and a remainder. // Note for later -- for multiplication, it might be better to split each double into a power of two and two 26 bit portions // multiplication of a double double by a known power of two is cheap. The current approach causes some inexact arithmetic in mul_dd. static inline void split_dd( double_double x, double_double *hi, double_double *lo ) { union{ double d; cl_ulong u;}u; u.d = x.hi; u.u &= 0xFFFFFFFFF8000000ULL; hi->hi = u.d; x.hi -= u.d; u.d = x.hi; u.u &= 0xFFFFFFFFF8000000ULL; hi->lo = u.d; x.hi -= u.d; double temp = x.hi; x.hi += x.lo; x.lo -= x.hi - temp; u.d = x.hi; u.u &= 0xFFFFFFFFF8000000ULL; lo->hi = u.d; x.hi -= u.d; lo->lo = x.hi + x.lo; } static inline double_double accum_d( double_double a, double b ) { double temp; if( fabs(b) > fabs(a.hi) ) { temp = a.hi; a.hi += b; a.lo += temp - (a.hi - b); } else { temp = a.hi; a.hi += b; a.lo += b - (a.hi - temp); } if( isnan( a.lo ) ) a.lo = 0.0; return a; } static inline double_double add_dd( double_double a, double_double b ) { double_double r = {-0.0 -0.0 }; if( isinf(a.hi) || isinf( b.hi ) || isnan(a.hi) || isnan( b.hi ) || 0.0 == a.hi || 0.0 == b.hi ) { r.hi = a.hi + b.hi; r.lo = a.lo + b.lo; if( isnan( r.lo ) ) r.lo = 0.0; return r; } //merge sort terms by magnitude -- here we assume that |a.hi| > |a.lo|, |b.hi| > |b.lo|, so we don't have to do the first merge pass double terms[4] = { a.hi, b.hi, a.lo, b.lo }; double temp; //Sort hi terms if( fabs(terms[0]) < fabs(terms[1]) ) { temp = terms[0]; terms[0] = terms[1]; terms[1] = temp; } //sort lo terms if( fabs(terms[2]) < fabs(terms[3]) ) { temp = terms[2]; terms[2] = terms[3]; terms[3] = temp; } // Fix case where small high term is less than large low term if( fabs(terms[1]) < fabs(terms[2]) ) { temp = terms[1]; terms[1] = terms[2]; terms[2] = temp; } // accumulate the results r.hi = terms[2] + terms[3]; r.lo = terms[3] - (r.hi - terms[2]); temp = r.hi; r.hi += terms[1]; r.lo += temp - (r.hi - terms[1]); temp = r.hi; r.hi += terms[0]; r.lo += temp - (r.hi - terms[0]); // canonicalize the result temp = r.hi; r.hi += r.lo; r.lo = r.lo - (r.hi - temp); if( isnan( r.lo ) ) r.lo = 0.0; return r; } static inline double_double mul_dd( double_double a, double_double b ) { double_double result = {-0.0,-0.0}; // Inf, nan and 0 if( isnan( a.hi ) || isnan( b.hi ) || isinf( a.hi ) || isinf( b.hi ) || 0.0 == a.hi || 0.0 == b.hi ) { result.hi = a.hi * b.hi; return result; } double_double ah, al, bh, bl; split_dd( a, &ah, &al ); split_dd( b, &bh, &bl ); double p0 = ah.hi * bh.hi; // exact (52 bits in product) 0 double p1 = ah.hi * bh.lo; // exact (52 bits in product) 26 double p2 = ah.lo * bh.hi; // exact (52 bits in product) 26 double p3 = ah.lo * bh.lo; // exact (52 bits in product) 52 double p4 = al.hi * bh.hi; // exact (52 bits in product) 52 double p5 = al.hi * bh.lo; // exact (52 bits in product) 78 double p6 = al.lo * bh.hi; // inexact (54 bits in product) 78 double p7 = al.lo * bh.lo; // inexact (54 bits in product) 104 double p8 = ah.hi * bl.hi; // exact (52 bits in product) 52 double p9 = ah.hi * bl.lo; // inexact (54 bits in product) 78 double pA = ah.lo * bl.hi; // exact (52 bits in product) 78 double pB = ah.lo * bl.lo; // inexact (54 bits in product) 104 double pC = al.hi * bl.hi; // exact (52 bits in product) 104 // the last 3 terms are two low to appear in the result // accumulate from bottom up #if 0 // works but slow result.hi = pC; result = accum_d( result, pB ); result = accum_d( result, p7 ); result = accum_d( result, pA ); result = accum_d( result, p9 ); result = accum_d( result, p6 ); result = accum_d( result, p5 ); result = accum_d( result, p8 ); result = accum_d( result, p4 ); result = accum_d( result, p3 ); result = accum_d( result, p2 ); result = accum_d( result, p1 ); result = accum_d( result, p0 ); // canonicalize the result double temp = result.hi; result.hi += result.lo; result.lo -= (result.hi - temp); if( isnan( result.lo ) ) result.lo = 0.0; return result; #else // take advantage of the known relative magnitudes of the partial products to avoid some sorting // Combine 2**-78 and 2**-104 terms. Here we are a bit sloppy about canonicalizing the double_doubles double_double t0 = { pA, pC }; double_double t1 = { p9, pB }; double_double t2 = { p6, p7 }; double temp0, temp1, temp2; t0 = accum_d( t0, p5 ); // there is an extra 2**-78 term to deal with // Add in 2**-52 terms. Here we are a bit sloppy about canonicalizing the double_doubles temp0 = t0.hi; temp1 = t1.hi; temp2 = t2.hi; t0.hi += p3; t1.hi += p4; t2.hi += p8; temp0 -= t0.hi-p3; temp1 -= t1.hi-p4; temp2 -= t2.hi - p8; t0.lo += temp0; t1.lo += temp1; t2.lo += temp2; // Add in 2**-26 terms. Here we are a bit sloppy about canonicalizing the double_doubles temp1 = t1.hi; temp2 = t2.hi; t1.hi += p1; t2.hi += p2; temp1 -= t1.hi-p1; temp2 -= t2.hi - p2; t1.lo += temp1; t2.lo += temp2; // Combine accumulators to get the low bits of result t1 = add_dd( t1, add_dd( t2, t0 ) ); // Add in MSB's, and round to precision return accum_d( t1, p0 ); // canonicalizes #endif } long double reference_exp10l( long double z ) { const double_double log2_10 = { HEX_DBL( +, 1, a934f0979a371, +, 1 ), HEX_DBL( +, 1, 7f2495fb7fa6d, -, 53 ) }; double_double x; int j; // Handle NaNs if( isnan(z) ) return z; // init x x.hi = z; x.lo = z - x.hi; // 10**x = exp2( x * log2(10) ) x = mul_dd( x, log2_10); // x * log2(10) //Deal with overflow and underflow for exp2(x) stage next if( x.hi >= 1025 ) return INFINITY; if( x.hi < -1075-24 ) return +0.0; // find nearest integer to x int i = (int) rint(x.hi); // x now holds fractional part. The result would be then 2**i * exp2( x ) x.hi -= i; // We could attempt to find a minimax polynomial for exp2(x) over the range x = [-0.5, 0.5]. // However, this would converge very slowly near the extrema, where 0.5**n is not a lot different // from 0.5**(n+1), thereby requiring something like a 20th order polynomial to get 53 + 24 bits // of precision. Instead we further reduce the range to [-1/32, 1/32] by observing that // // 2**(a+b) = 2**a * 2**b // // We can thus build a table of 2**a values for a = n/16, n = [-8, 8], and reduce the range // of x to [-1/32, 1/32] by subtracting away the nearest value of n/16 from x. const double_double corrections[17] = { { HEX_DBL( +, 1, 6a09e667f3bcd, -, 1 ), HEX_DBL( -, 1, bdd3413b26456, -, 55 ) }, { HEX_DBL( +, 1, 7a11473eb0187, -, 1 ), HEX_DBL( -, 1, 41577ee04992f, -, 56 ) }, { HEX_DBL( +, 1, 8ace5422aa0db, -, 1 ), HEX_DBL( +, 1, 6e9f156864b27, -, 55 ) }, { HEX_DBL( +, 1, 9c49182a3f09, -, 1 ), HEX_DBL( +, 1, c7c46b071f2be, -, 57 ) }, { HEX_DBL( +, 1, ae89f995ad3ad, -, 1 ), HEX_DBL( +, 1, 7a1cd345dcc81, -, 55 ) }, { HEX_DBL( +, 1, c199bdd85529c, -, 1 ), HEX_DBL( +, 1, 11065895048dd, -, 56 ) }, { HEX_DBL( +, 1, d5818dcfba487, -, 1 ), HEX_DBL( +, 1, 2ed02d75b3707, -, 56 ) }, { HEX_DBL( +, 1, ea4afa2a490da, -, 1 ), HEX_DBL( -, 1, e9c23179c2893, -, 55 ) }, { HEX_DBL( +, 1, 0, +, 0 ), HEX_DBL( +, 0, 0, +, 0 ) }, { HEX_DBL( +, 1, 0b5586cf9890f, +, 0 ), HEX_DBL( +, 1, 8a62e4adc610b, -, 54 ) }, { HEX_DBL( +, 1, 172b83c7d517b, +, 0 ), HEX_DBL( -, 1, 19041b9d78a76, -, 55 ) }, { HEX_DBL( +, 1, 2387a6e756238, +, 0 ), HEX_DBL( +, 1, 9b07eb6c70573, -, 54 ) }, { HEX_DBL( +, 1, 306fe0a31b715, +, 0 ), HEX_DBL( +, 1, 6f46ad23182e4, -, 55 ) }, { HEX_DBL( +, 1, 3dea64c123422, +, 0 ), HEX_DBL( +, 1, ada0911f09ebc, -, 55 ) }, { HEX_DBL( +, 1, 4bfdad5362a27, +, 0 ), HEX_DBL( +, 1, d4397afec42e2, -, 56 ) }, { HEX_DBL( +, 1, 5ab07dd485429, +, 0 ), HEX_DBL( +, 1, 6324c054647ad, -, 54 ) }, { HEX_DBL( +, 1, 6a09e667f3bcd, +, 0 ), HEX_DBL( -, 1, bdd3413b26456, -, 54 ) } }; int index = (int) rint( x.hi * 16.0 ); x.hi -= (double) index * 0.0625; // canonicalize x double temp = x.hi; x.hi += x.lo; x.lo -= x.hi - temp; // Minimax polynomial for (exp2(x)-1)/x, over the range [-1/32, 1/32]. Max Error: 2 * 0x1.e112p-87 const double_double c[] = { {HEX_DBL( +, 1, 62e42fefa39ef, -, 1 ), HEX_DBL( +, 1, abc9e3ac1d244, -, 56 )}, {HEX_DBL( +, 1, ebfbdff82c58f, -, 3 ), HEX_DBL( -, 1, 5e4987a631846, -, 57 )}, {HEX_DBL( +, 1, c6b08d704a0c, -, 5 ), HEX_DBL( -, 1, d323200a05713, -, 59 )}, {HEX_DBL( +, 1, 3b2ab6fba4e7a, -, 7 ), HEX_DBL( +, 1, c5ee8f8b9f0c1, -, 63 )}, {HEX_DBL( +, 1, 5d87fe78a672a, -, 10 ), HEX_DBL( +, 1, 884e5e5cc7ecc, -, 64 )}, {HEX_DBL( +, 1, 430912f7e8373, -, 13 ), HEX_DBL( +, 1, 4f1b59514a326, -, 67 )}, {HEX_DBL( +, 1, ffcbfc5985e71, -, 17 ), HEX_DBL( -, 1, db7d6a0953b78, -, 71 )}, {HEX_DBL( +, 1, 62c150eb16465, -, 20 ), HEX_DBL( +, 1, e0767c2d7abf5, -, 80 )}, {HEX_DBL( +, 1, b52502b5e953, -, 24 ), HEX_DBL( +, 1, 6797523f944bc, -, 78 )} }; size_t count = sizeof( c ) / sizeof( c[0] ); // Do polynomial double_double r = c[count-1]; for( j = (int) count-2; j >= 0; j-- ) r = add_dd( c[j], mul_dd( r, x ) ); // unwind approximation r = mul_dd( r, x ); // before: r =(exp2(x)-1)/x; after: r = exp2(x) - 1 // correct for [-0.5, 0.5] -> [-1/32, 1/32] reduction above // exp2(x) = (r + 1) * correction = r * correction + correction r = mul_dd( r, corrections[index+8] ); r = add_dd( r, corrections[index+8] ); // Format result for output: // Get mantissa long double m = ((long double) r.hi + (long double) r.lo ); // Handle a pesky overflow cases when long double = double if( i > 512 ) { m *= HEX_DBL( +, 1, 0, +, 512 ); i -= 512; } else if( i < -512 ) { m *= HEX_DBL( +, 1, 0, -, 512 ); i += 512; } return m * ldexpl( 1.0L, i ); } static double fallback_frexp( double x, int *iptr ) { cl_ulong u, v; double fu, fv; memcpy( &u, &x, sizeof(u)); cl_ulong exponent = u & 0x7ff0000000000000ULL; cl_ulong mantissa = u & ~0x7ff0000000000000ULL; // add 1 to the exponent exponent += 0x0010000000000000ULL; if( (cl_long) exponent < (cl_long) 0x0020000000000000LL ) { // subnormal, NaN, Inf mantissa |= 0x3fe0000000000000ULL; v = mantissa & 0xfff0000000000000ULL; u = mantissa; memcpy( &fv, &v, sizeof(v)); memcpy( &fu, &u, sizeof(u)); fu -= fv; memcpy( &v, &fv, sizeof(v)); memcpy( &u, &fu, sizeof(u)); exponent = u & 0x7ff0000000000000ULL; mantissa = u & ~0x7ff0000000000000ULL; *iptr = (exponent >> 52) + (-1022 + 1 -1022); u = mantissa | 0x3fe0000000000000ULL; memcpy( &fu, &u, sizeof(u)); return fu; } *iptr = (exponent >> 52) - 1023; u = mantissa | 0x3fe0000000000000ULL; memcpy( &fu, &u, sizeof(u)); return fu; } // Assumes zeros, infinities and NaNs handed elsewhere static inline int extract( double x, cl_ulong *mant ); static inline int extract( double x, cl_ulong *mant ) { static double (*frexpp)(double, int*) = NULL; int e; // verify that frexp works properly if( NULL == frexpp ) { if( 0.5 == frexp( HEX_DBL( +, 1, 0, -, 1030 ), &e ) && e == -1029 ) frexpp = frexp; else frexpp = fallback_frexp; } *mant = (cl_ulong) (HEX_DBL( +, 1, 0, +, 64 ) * fabs( frexpp( x, &e ))); return e - 1; } // Return 128-bit product of a*b as (hi << 64) + lo static inline void mul128( cl_ulong a, cl_ulong b, cl_ulong *hi, cl_ulong *lo ); static inline void mul128( cl_ulong a, cl_ulong b, cl_ulong *hi, cl_ulong *lo ) { cl_ulong alo = a & 0xffffffffULL; cl_ulong ahi = a >> 32; cl_ulong blo = b & 0xffffffffULL; cl_ulong bhi = b >> 32; cl_ulong aloblo = alo * blo; cl_ulong alobhi = alo * bhi; cl_ulong ahiblo = ahi * blo; cl_ulong ahibhi = ahi * bhi; alobhi += (aloblo >> 32) + (ahiblo & 0xffffffffULL); // cannot overflow: (2^32-1)^2 + 2 * (2^32-1) = (2^64 - 2^33 + 1) + (2^33 - 2) = 2^64 - 1 *hi = ahibhi + (alobhi >> 32) + (ahiblo >> 32); // cannot overflow: (2^32-1)^2 + 2 * (2^32-1) = (2^64 - 2^33 + 1) + (2^33 - 2) = 2^64 - 1 *lo = (aloblo & 0xffffffffULL) | (alobhi << 32); } // Move the most significant non-zero bit to the MSB // Note: not general. Only works if the most significant non-zero bit is at MSB-1 static inline void renormalize( cl_ulong *hi, cl_ulong *lo, int *exponent ) { if( 0 == (0x8000000000000000ULL & *hi )) { *hi <<= 1; *hi |= *lo >> 63; *lo <<= 1; *exponent -= 1; } } static double round_to_nearest_even_double( cl_ulong hi, cl_ulong lo, int exponent ); static double round_to_nearest_even_double( cl_ulong hi, cl_ulong lo, int exponent ) { union{ cl_ulong u; cl_double d;} u; // edges if( exponent > 1023 ) return INFINITY; if( exponent == -1075 && (hi | (lo!=0)) > 0x8000000000000000ULL ) return HEX_DBL( +, 1, 0, -, 1074 ); if( exponent <= -1075 ) return 0.0; //Figure out which bits go where int shift = 11; if( exponent < -1022 ) { shift -= 1022 + exponent; // subnormal: shift is not 52 exponent = -1023; // set exponent to 0 } else hi &= 0x7fffffffffffffffULL; // normal: leading bit is implicit. Remove it. // Assemble the double (round toward zero) u.u = (hi >> shift) | ((cl_ulong) (exponent + 1023) << 52); // put a representation of the residual bits into hi hi <<= (64-shift); hi |= lo >> shift; lo <<= (64-shift ); hi |= lo != 0; //round to nearest, ties to even if( hi < 0x8000000000000000ULL ) return u.d; if( hi == 0x8000000000000000ULL ) u.u += u.u & 1ULL; else u.u++; return u.d; } // Shift right. Bits lost on the right will be OR'd together and OR'd with the LSB static inline void shift_right_sticky_128( cl_ulong *hi, cl_ulong *lo, int shift ); static inline void shift_right_sticky_128( cl_ulong *hi, cl_ulong *lo, int shift ) { cl_ulong sticky = 0; cl_ulong h = *hi; cl_ulong l = *lo; if( shift >= 64 ) { shift -= 64; sticky = 0 != lo; l = h; h = 0; if( shift >= 64 ) { sticky |= (0 != l); l = 0; } else { sticky |= (0 != (l << (64-shift))); l >>= shift; } } else { sticky |= (0 != (l << (64-shift))); l >>= shift; l |= h << (64-shift); h >>= shift; } *lo = l | sticky; *hi = h; } // 128-bit add of ((*hi << 64) + *lo) + ((chi << 64) + clo) // If the 129 bit result doesn't fit, bits lost off the right end will be OR'd with the LSB static inline void add128( cl_ulong *hi, cl_ulong *lo, cl_ulong chi, cl_ulong clo, int *exp ); static inline void add128( cl_ulong *hi, cl_ulong *lo, cl_ulong chi, cl_ulong clo, int *exponent ) { cl_ulong carry, carry2; // extended precision add clo = add_carry(*lo, clo, &carry); chi = add_carry(*hi, chi, &carry2); chi = add_carry(chi, carry, &carry); //If we overflowed the 128 bit result if( carry || carry2 ) { carry = clo & 1; // set aside low bit clo >>= 1; // right shift low 1 clo |= carry; // or back in the low bit, so we don't come to believe this is an exact half way case for rounding clo |= chi << 63; // move lowest high bit into highest bit of lo chi >>= 1; // right shift hi chi |= 0x8000000000000000ULL; // move the carry bit into hi. *exponent = *exponent + 1; } *hi = chi; *lo = clo; } // 128-bit subtract of ((chi << 64) + clo) - ((*hi << 64) + *lo) static inline void sub128( cl_ulong *chi, cl_ulong *clo, cl_ulong hi, cl_ulong lo, cl_ulong *signC, int *expC ); static inline void sub128( cl_ulong *chi, cl_ulong *clo, cl_ulong hi, cl_ulong lo, cl_ulong *signC, int *expC ) { cl_ulong rHi = *chi; cl_ulong rLo = *clo; cl_ulong carry, carry2; //extended precision subtract rLo = sub_carry(rLo, lo, &carry); rHi = sub_carry(rHi, hi, &carry2); rHi = sub_carry(rHi, carry, &carry); // Check for sign flip if( carry || carry2 ) { *signC ^= 0x8000000000000000ULL; //negate rLo, rHi: -x = (x ^ -1) + 1 rLo ^= -1ULL; rHi ^= -1ULL; rLo++; rHi += 0 == rLo; } // normalize -- move the most significant non-zero bit to the MSB, and adjust exponent accordingly if( rHi == 0 ) { rHi = rLo; *expC = *expC - 64; rLo = 0; } if( rHi ) { int shift = 32; cl_ulong test = 1ULL << 32; while( 0 == (rHi & 0x8000000000000000ULL)) { if( rHi < test ) { rHi <<= shift; rHi |= rLo >> (64-shift); rLo <<= shift; *expC = *expC - shift; } shift >>= 1; test <<= shift; } } else { //zero *expC = INT_MIN; *signC = 0; } *chi = rHi; *clo = rLo; } long double reference_fmal( long double x, long double y, long double z) { static const cl_ulong kMSB = 0x8000000000000000ULL; // cast values back to double. This is an exact function, so double a = x; double b = y; double c = z; // Make bits accessible union{ cl_ulong u; cl_double d; } ua; ua.d = a; union{ cl_ulong u; cl_double d; } ub; ub.d = b; union{ cl_ulong u; cl_double d; } uc; uc.d = c; // deal with Nans, infinities and zeros if( isnan( a ) || isnan( b ) || isnan(c) || isinf( a ) || isinf( b ) || isinf(c) || 0 == ( ua.u & ~kMSB) || // a == 0, defeat host FTZ behavior 0 == ( ub.u & ~kMSB) || // b == 0, defeat host FTZ behavior 0 == ( uc.u & ~kMSB) ) // c == 0, defeat host FTZ behavior { if( isinf( c ) && !isinf(a) && !isinf(b) ) return (c + a) + b; a = (double) reference_multiplyl( a, b ); // some risk that the compiler will insert a non-compliant fma here on some platforms. return reference_addl(a, c); // We use STDC FP_CONTRACT OFF above to attempt to defeat that. } // extract exponent and mantissa // exponent is a standard unbiased signed integer // mantissa is a cl_uint, with leading non-zero bit positioned at the MSB cl_ulong mantA, mantB, mantC; int expA = extract( a, &mantA ); int expB = extract( b, &mantB ); int expC = extract( c, &mantC ); cl_ulong signC = uc.u & kMSB; // We'll need the sign bit of C later to decide if we are adding or subtracting // exact product of A and B int exponent = expA + expB; cl_ulong sign = (ua.u ^ ub.u) & kMSB; cl_ulong hi, lo; mul128( mantA, mantB, &hi, &lo ); // renormalize if( 0 == (kMSB & hi) ) { hi <<= 1; hi |= lo >> 63; lo <<= 1; } else exponent++; // 2**63 * 2**63 gives 2**126. If the MSB was set, then our exponent increased. //infinite precision add cl_ulong chi = mantC; cl_ulong clo = 0; if( exponent >= expC ) { // Normalize C relative to the product if( exponent > expC ) shift_right_sticky_128( &chi, &clo, exponent - expC ); // Add if( sign ^ signC ) sub128( &hi, &lo, chi, clo, &sign, &exponent ); else add128( &hi, &lo, chi, clo, &exponent ); } else { // Shift the product relative to C so that their exponents match shift_right_sticky_128( &hi, &lo, expC - exponent ); // add if( sign ^ signC ) sub128( &chi, &clo, hi, lo, &signC, &expC ); else add128( &chi, &clo, hi, lo, &expC ); hi = chi; lo = clo; exponent = expC; sign = signC; } // round ua.d = round_to_nearest_even_double(hi, lo, exponent); // Set the sign ua.u |= sign; return ua.d; } long double reference_madl( long double a, long double b, long double c) { return a * b + c; } //long double my_nextafterl(long double x, long double y){ return (long double) nextafter( (double) x, (double) y ); } long double reference_recipl( long double x){ return 1.0L / x; } long double reference_rootnl( long double x, int i) { double hi, lo; long double l; //rootn ( x, 0 ) returns a NaN. if( 0 == i ) return cl_make_nan(); //rootn ( x, n ) returns a NaN for x < 0 and n is even. if( x < 0.0L && 0 == (i&1) ) return cl_make_nan(); if( isinf(x) ) { if( i < 0 ) return reference_copysignl(0.0L, x); return x; } if( x == 0.0 ) { switch( i & 0x80000001 ) { //rootn ( +-0, n ) is +0 for even n > 0. case 0: return 0.0L; //rootn ( +-0, n ) is +-0 for odd n > 0. case 1: return x; //rootn ( +-0, n ) is +inf for even n < 0. case 0x80000000: return INFINITY; //rootn ( +-0, n ) is +-inf for odd n < 0. case 0x80000001: return copysign(INFINITY, x); } } if( i == 1 ) return x; if( i == -1 ) return 1.0 / x; long double sign = x; x = reference_fabsl(x); double iHi, iLo; DivideDD(&iHi, &iLo, 1.0, i); x = reference_powl(x, iHi) * reference_powl(x, iLo); return reference_copysignl( x, sign ); } long double reference_rsqrtl( long double x){ return 1.0L / sqrtl(x); } //long double reference_sincosl( long double x, long double *c ){ *c = reference_cosl(x); return reference_sinl(x); } long double reference_sinpil( long double x) { double r = reduce1l(x); // reduce to [-0.5, 0.5] if( r < -0.5L ) r = -1.0L - r; else if ( r > 0.5L ) r = 1.0L - r; // sinPi zeros have the same sign as x if( r == 0.0L ) return reference_copysignl(0.0L, x); return reference_sinl( r * M_PIL ); } long double reference_tanpil( long double x) { // set aside the sign (allows us to preserve sign of -0) long double sign = reference_copysignl( 1.0L, x); long double z = reference_fabsl(x); // if big and even -- caution: only works if x only has single precision if( z >= HEX_LDBL( +, 1, 0, +, 53 ) ) { if( z == INFINITY ) return x - x; // nan return reference_copysignl( 0.0L, x); // tanpi ( n ) is copysign( 0.0, n) for even integers n. } // reduce to the range [ -0.5, 0.5 ] long double nearest = reference_rintl( z ); // round to nearest even places n + 0.5 values in the right place for us int64_t i = (int64_t) nearest; // test above against 0x1.0p53 avoids overflow here z -= nearest; //correction for odd integer x for the right sign of zero if( (i&1) && z == 0.0L ) sign = -sign; // track changes to the sign sign *= reference_copysignl(1.0L, z); // really should just be an xor z = reference_fabsl(z); // remove the sign again // reduce once more // If we don't do this, rounding error in z * M_PI will cause us not to return infinities properly if( z > 0.25L ) { z = 0.5L - z; return sign / reference_tanl( z * M_PIL ); // use system tan to get the right result } // return sign * reference_tanl( z * M_PIL ); // use system tan to get the right result } long double reference_pownl( long double x, int i ){ return reference_powl( x, (long double) i ); } long double reference_powrl( long double x, long double y ) { //powr ( x, y ) returns NaN for x < 0. if( x < 0.0L ) return cl_make_nan(); //powr ( x, NaN ) returns the NaN for x >= 0. //powr ( NaN, y ) returns the NaN. if( isnan(x) || isnan(y) ) return x + y; // Note: behavior different here than for pow(1,NaN), pow(NaN, 0) if( x == 1.0L ) { //powr ( +1, +-inf ) returns NaN. if( reference_fabsl(y) == INFINITY ) return cl_make_nan(); //powr ( +1, y ) is 1 for finite y. (NaN handled above) return 1.0L; } if( y == 0.0L ) { //powr ( +inf, +-0 ) returns NaN. //powr ( +-0, +-0 ) returns NaN. if( x == 0.0L || x == INFINITY ) return cl_make_nan(); //powr ( x, +-0 ) is 1 for finite x > 0. (x <= 0, NaN, INF already handled above) return 1.0L; } if( x == 0.0L ) { //powr ( +-0, -inf) is +inf. //powr ( +-0, y ) is +inf for finite y < 0. if( y < 0.0L ) return INFINITY; //powr ( +-0, y ) is +0 for y > 0. (NaN, y==0 handled above) return 0.0L; } return reference_powl( x, y ); } //long double my_fdiml( long double x, long double y){ return fdim( (double) x, (double) y ); } long double reference_addl( long double x, long double y) { volatile double a = (double) x; volatile double b = (double) y; #if defined( __SSE2__ ) // defeat x87 __m128d va = _mm_set_sd( (double) a ); __m128d vb = _mm_set_sd( (double) b ); va = _mm_add_sd( va, vb ); _mm_store_sd( (double*) &a, va ); #else a += b; #endif return (long double) a; } long double reference_subtractl( long double x, long double y) { volatile double a = (double) x; volatile double b = (double) y; #if defined( __SSE2__ ) // defeat x87 __m128d va = _mm_set_sd( (double) a ); __m128d vb = _mm_set_sd( (double) b ); va = _mm_sub_sd( va, vb ); _mm_store_sd( (double*) &a, va ); #else a -= b; #endif return (long double) a; } long double reference_multiplyl( long double x, long double y) { volatile double a = (double) x; volatile double b = (double) y; #if defined( __SSE2__ ) // defeat x87 __m128d va = _mm_set_sd( (double) a ); __m128d vb = _mm_set_sd( (double) b ); va = _mm_mul_sd( va, vb ); _mm_store_sd( (double*) &a, va ); #else a *= b; #endif return (long double) a; } /*long double my_remquol( long double x, long double y, int *iptr ) { if( isnan(x) || isnan(y) || fabs(x) == INFINITY || y == 0.0 ) { *iptr = 0; return NAN; } return remquo( (double) x, (double) y, iptr ); }*/ long double reference_lgamma_rl( long double x, int *signp ) { // long double lgamma_val = (long double)reference_lgamma( (double)x ); // *signp = signgam; *signp = 0; return x; } int reference_isequall( long double x, long double y){ return x == y; } int reference_isfinitel( long double x){ return 0 != isfinite(x); } int reference_isgreaterl( long double x, long double y){ return x > y; } int reference_isgreaterequall( long double x, long double y){ return x >= y; } int reference_isinfl( long double x){ return 0 != isinf(x); } int reference_islessl( long double x, long double y){ return x < y; } int reference_islessequall( long double x, long double y){ return x <= y; } #if defined(__INTEL_COMPILER) int reference_islessgreaterl(long double x, long double y) { return 0 != islessgreaterl(x, y); } #else int reference_islessgreaterl(long double x, long double y) { return 0 != islessgreater(x, y); } #endif int reference_isnanl( long double x){ return 0 != isnan( x ); } int reference_isnormall( long double x){ return 0 != isnormal( (double) x ); } int reference_isnotequall( long double x, long double y){ return x != y; } int reference_isorderedl( long double x, long double y){ return x == x && y == y; } int reference_isunorderedl( long double x, long double y){ return isnan(x) || isnan( y ); } #if defined( __INTEL_COMPILER ) int reference_signbitl( long double x){ return 0 != signbitl( x ); } #else int reference_signbitl( long double x){ return 0 != signbit( x ); } #endif long double reference_copysignl( long double x, long double y); long double reference_roundl( long double x ); long double reference_cbrtl(long double x); long double reference_copysignl( long double x, long double y ) { // We hope that the long double to double conversion proceeds with sign fidelity, // even for zeros and NaNs union{ double d; cl_ulong u;}u; u.d = (double) y; x = reference_fabsl(x); if( u.u >> 63 ) x = -x; return x; } long double reference_roundl( long double x ) { // Since we are just using this to verify double precision, we can // use the double precision copysign here #if defined(__MINGW32__) && defined(__x86_64__) long double absx = reference_fabsl(x); if (absx < 0.5L) return reference_copysignl(0.0L, x); #endif return round( (double) x ); } long double reference_truncl( long double x ) { // Since we are just using this to verify double precision, we can // use the double precision copysign here return trunc( (double) x ); } static long double reference_scalblnl(long double x, long n); long double reference_cbrtl(long double x) { double yhi = HEX_DBL( +, 1, 5555555555555, -, 2 ); double ylo = HEX_DBL( +, 1, 558, -, 56 ); double fabsx = reference_fabs( x ); if( isnan(x) || fabsx == 1.0 || fabsx == 0.0 || isinf(x) ) return x; double iy = 0.0; double log2x_hi, log2x_lo; // extended precision log .... accurate to at least 64-bits + couple of guard bits __log2_ep(&log2x_hi, &log2x_lo, fabsx); double ylog2x_hi, ylog2x_lo; double y_hi = yhi; double y_lo = ylo; // compute product of y*log2(x) MulDD(&ylog2x_hi, &ylog2x_lo, log2x_hi, log2x_lo, y_hi, y_lo); long double powxy; if(isinf(ylog2x_hi) || (reference_fabs(ylog2x_hi) > 2200)) { powxy = reference_signbit(ylog2x_hi) ? HEX_DBL( +, 0, 0, +, 0 ) : INFINITY; } else { // separate integer + fractional part long int m = lrint(ylog2x_hi); AddDD(&ylog2x_hi, &ylog2x_lo, ylog2x_hi, ylog2x_lo, -m, 0.0); // revert to long double arithemtic long double ylog2x = (long double) ylog2x_hi + (long double) ylog2x_lo; powxy = reference_exp2l( ylog2x ); powxy = reference_scalblnl(powxy, m); } return reference_copysignl( powxy, x ); } /* long double scalbnl( long double x, int i ) { //suitable for checking double precision scalbn only if( i > 3000 ) return copysignl( INFINITY, x); if( i < -3000 ) return copysignl( 0.0L, x); if( i > 0 ) { while( i >= 1000 ) { x *= HEX_LDBL( +, 1, 0, +, 1000 ); i -= 1000; } union{ cl_ulong u; double d;}u; u.u = (cl_ulong)( i + 1023 ) << 52; x *= (long double) u.d; } else if( i < 0 ) { while( i <= -1000 ) { x *= HEX_LDBL( +, 1, 0, -, 1000 ); i += 1000; } union{ cl_ulong u; double d;}u; u.u = (cl_ulong)( i + 1023 ) << 52; x *= (long double) u.d; } return x; } */ long double reference_rintl( long double x ) { #if defined(__PPC__) // On PPC, long doubles are maintained as 2 doubles. Therefore, the combined // mantissa can represent more than LDBL_MANT_DIG binary digits. x = rintl(x); #else static long double magic[2] = { 0.0L, 0.0L}; if( 0.0L == magic[0] ) { magic[0] = scalbnl(0.5L, LDBL_MANT_DIG); magic[1] = scalbnl(-0.5L, LDBL_MANT_DIG); } if( reference_fabsl(x) < magic[0] && x != 0.0L ) { long double m = magic[ x < 0 ]; x += m; x -= m; } #endif // __PPC__ return x; } // extended precision sqrt using newton iteration on 1/sqrt(x). // Final result is computed as x * 1/sqrt(x) static void __sqrt_ep(double *rhi, double *rlo, double xhi, double xlo) { // approximate reciprocal sqrt double thi = 1.0 / sqrt( xhi ); double tlo = 0.0; // One newton iteration in double-double double yhi, ylo; MulDD(&yhi, &ylo, thi, tlo, thi, tlo); MulDD(&yhi, &ylo, yhi, ylo, xhi, xlo); AddDD(&yhi, &ylo, -yhi, -ylo, 3.0, 0.0); MulDD(&yhi, &ylo, yhi, ylo, thi, tlo); MulDD(&yhi, &ylo, yhi, ylo, 0.5, 0.0); MulDD(rhi, rlo, yhi, ylo, xhi, xlo); } long double reference_acoshl( long double x ) { /* * ==================================================== * This function derived from fdlibm http://www.netlib.org * It is Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. * * Developed at SunSoft, a Sun Microsystems, Inc. business. * Permission to use, copy, modify, and distribute this * software is freely granted, provided that this notice * is preserved. * ==================================================== * */ if( isnan(x) || isinf(x)) return x + fabsl(x); if( x < 1.0L ) return cl_make_nan(); if( x == 1.0L ) return 0.0L; if( x > HEX_LDBL( +, 1, 0, +, 60 ) ) return reference_logl(x) + 0.693147180559945309417232121458176568L; if( x > 2.0L ) return reference_logl(2.0L * x - 1.0L / (x + sqrtl(x*x - 1.0L))); double hi, lo; MulD(&hi, &lo, x, x); AddDD(&hi, &lo, hi, lo, -1.0, 0.0); __sqrt_ep(&hi, &lo, hi, lo); AddDD(&hi, &lo, hi, lo, x, 0.0); double correction = lo / hi; __log2_ep(&hi, &lo, hi); double log2Hi = HEX_DBL( +, 1, 62e42fefa39ef, -, 1 ); double log2Lo = HEX_DBL( +, 1, abc9e3b39803f, -, 56 ); MulDD(&hi, &lo, hi, lo, log2Hi, log2Lo); AddDD(&hi, &lo, hi, lo, correction, 0.0); return hi + lo; } long double reference_asinhl( long double x ) { long double cutoff = 0.0L; const long double ln2 = HEX_LDBL( +, b, 17217f7d1cf79ab, -, 4 ); if( cutoff == 0.0L ) cutoff = reference_ldexpl(1.0L, -LDBL_MANT_DIG); if( isnan(x) || isinf(x) ) return x + x; long double absx = reference_fabsl(x); if( absx < cutoff ) return x; long double sign = reference_copysignl(1.0L, x); if( absx <= 4.0/3.0 ) { return sign * reference_log1pl( absx + x*x / (1.0 + sqrtl(1.0 + x*x))); } else if( absx <= HEX_LDBL( +, 1, 0, +, 27 ) ) { return sign * reference_logl( 2.0L * absx + 1.0L / (sqrtl( x * x + 1.0 ) + absx)); } else { return sign * ( reference_logl( absx ) + ln2 ); } } long double reference_atanhl( long double x ) { /* * ==================================================== * This function is from fdlibm: http://www.netlib.org * It is Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. * * Developed at SunSoft, a Sun Microsystems, Inc. business. * Permission to use, copy, modify, and distribute this * software is freely granted, provided that this notice * is preserved. * ==================================================== */ if( isnan(x) ) return x + x; long double signed_half = reference_copysignl( 0.5L, x ); x = reference_fabsl(x); if( x > 1.0L ) return cl_make_nan(); if( x < 0.5L ) return signed_half * reference_log1pl( 2.0L * ( x + x*x / (1-x) ) ); return signed_half * reference_log1pl(2.0L * x / (1-x)); } long double reference_exp2l( long double z) { double_double x; int j; // Handle NaNs if( isnan(z) ) return z; // init x x.hi = z; x.lo = z - x.hi; //Deal with overflow and underflow for exp2(x) stage next if( x.hi >= 1025 ) return INFINITY; if( x.hi < -1075-24 ) return +0.0; // find nearest integer to x int i = (int) rint(x.hi); // x now holds fractional part. The result would be then 2**i * exp2( x ) x.hi -= i; // We could attempt to find a minimax polynomial for exp2(x) over the range x = [-0.5, 0.5]. // However, this would converge very slowly near the extrema, where 0.5**n is not a lot different // from 0.5**(n+1), thereby requiring something like a 20th order polynomial to get 53 + 24 bits // of precision. Instead we further reduce the range to [-1/32, 1/32] by observing that // // 2**(a+b) = 2**a * 2**b // // We can thus build a table of 2**a values for a = n/16, n = [-8, 8], and reduce the range // of x to [-1/32, 1/32] by subtracting away the nearest value of n/16 from x. const double_double corrections[17] = { { HEX_DBL( +, 1, 6a09e667f3bcd, -, 1 ), HEX_DBL( -, 1, bdd3413b26456, -, 55 ) }, { HEX_DBL( +, 1, 7a11473eb0187, -, 1 ), HEX_DBL( -, 1, 41577ee04992f, -, 56 ) }, { HEX_DBL( +, 1, 8ace5422aa0db, -, 1 ), HEX_DBL( +, 1, 6e9f156864b27, -, 55 ) }, { HEX_DBL( +, 1, 9c49182a3f09, -, 1 ), HEX_DBL( +, 1, c7c46b071f2be, -, 57 ) }, { HEX_DBL( +, 1, ae89f995ad3ad, -, 1 ), HEX_DBL( +, 1, 7a1cd345dcc81, -, 55 ) }, { HEX_DBL( +, 1, c199bdd85529c, -, 1 ), HEX_DBL( +, 1, 11065895048dd, -, 56 ) }, { HEX_DBL( +, 1, d5818dcfba487, -, 1 ), HEX_DBL( +, 1, 2ed02d75b3707, -, 56 ) }, { HEX_DBL( +, 1, ea4afa2a490da, -, 1 ), HEX_DBL( -, 1, e9c23179c2893, -, 55 ) }, { HEX_DBL( +, 1, 0, +, 0 ), HEX_DBL( +, 0, 0, +, 0 ) }, { HEX_DBL( +, 1, 0b5586cf9890f, +, 0 ), HEX_DBL( +, 1, 8a62e4adc610b, -, 54 ) }, { HEX_DBL( +, 1, 172b83c7d517b, +, 0 ), HEX_DBL( -, 1, 19041b9d78a76, -, 55 ) }, { HEX_DBL( +, 1, 2387a6e756238, +, 0 ), HEX_DBL( +, 1, 9b07eb6c70573, -, 54 ) }, { HEX_DBL( +, 1, 306fe0a31b715, +, 0 ), HEX_DBL( +, 1, 6f46ad23182e4, -, 55 ) }, { HEX_DBL( +, 1, 3dea64c123422, +, 0 ), HEX_DBL( +, 1, ada0911f09ebc, -, 55 ) }, { HEX_DBL( +, 1, 4bfdad5362a27, +, 0 ), HEX_DBL( +, 1, d4397afec42e2, -, 56 ) }, { HEX_DBL( +, 1, 5ab07dd485429, +, 0 ), HEX_DBL( +, 1, 6324c054647ad, -, 54 ) }, { HEX_DBL( +, 1, 6a09e667f3bcd, +, 0 ), HEX_DBL( -, 1, bdd3413b26456, -, 54 ) } }; int index = (int) rint( x.hi * 16.0 ); x.hi -= (double) index * 0.0625; // canonicalize x double temp = x.hi; x.hi += x.lo; x.lo -= x.hi - temp; // Minimax polynomial for (exp2(x)-1)/x, over the range [-1/32, 1/32]. Max Error: 2 * 0x1.e112p-87 const double_double c[] = { {HEX_DBL( +, 1, 62e42fefa39ef, -, 1 ), HEX_DBL( +, 1, abc9e3ac1d244, -, 56 )}, {HEX_DBL( +, 1, ebfbdff82c58f, -, 3 ), HEX_DBL( -, 1, 5e4987a631846, -, 57 )}, {HEX_DBL( +, 1, c6b08d704a0c, -, 5 ), HEX_DBL( -, 1, d323200a05713, -, 59 )}, {HEX_DBL( +, 1, 3b2ab6fba4e7a, -, 7 ), HEX_DBL( +, 1, c5ee8f8b9f0c1, -, 63 )}, {HEX_DBL( +, 1, 5d87fe78a672a, -, 10 ), HEX_DBL( +, 1, 884e5e5cc7ecc, -, 64 )}, {HEX_DBL( +, 1, 430912f7e8373, -, 13 ), HEX_DBL( +, 1, 4f1b59514a326, -, 67 )}, {HEX_DBL( +, 1, ffcbfc5985e71, -, 17 ), HEX_DBL( -, 1, db7d6a0953b78, -, 71 )}, {HEX_DBL( +, 1, 62c150eb16465, -, 20 ), HEX_DBL( +, 1, e0767c2d7abf5, -, 80 )}, {HEX_DBL( +, 1, b52502b5e953, -, 24 ), HEX_DBL( +, 1, 6797523f944bc, -, 78 )} }; size_t count = sizeof( c ) / sizeof( c[0] ); // Do polynomial double_double r = c[count-1]; for( j = (int) count-2; j >= 0; j-- ) r = add_dd( c[j], mul_dd( r, x ) ); // unwind approximation r = mul_dd( r, x ); // before: r =(exp2(x)-1)/x; after: r = exp2(x) - 1 // correct for [-0.5, 0.5] -> [-1/32, 1/32] reduction above // exp2(x) = (r + 1) * correction = r * correction + correction r = mul_dd( r, corrections[index+8] ); r = add_dd( r, corrections[index+8] ); // Format result for output: // Get mantissa long double m = ((long double) r.hi + (long double) r.lo ); // Handle a pesky overflow cases when long double = double if( i > 512 ) { m *= HEX_DBL( +, 1, 0, +, 512 ); i -= 512; } else if( i < -512 ) { m *= HEX_DBL( +, 1, 0, -, 512 ); i += 512; } return m * ldexpl( 1.0L, i ); } long double reference_expm1l( long double x) { #if defined( _MSC_VER ) && ! defined( __INTEL_COMPILER ) //unimplemented return x; #else union { double f; cl_ulong u;} u; u.f = (double) x; if (reference_isnanl(x)) return x; if ( x > 710 ) return INFINITY; long double y = expm1l(x); // Range of expm1l is -1.0L to +inf. Negative inf // on a few Linux platforms is clearly the wrong sign. if (reference_isinfl(y)) y = INFINITY; return y; #endif } long double reference_fmaxl( long double x, long double y ) { if( isnan(y) ) return x; return x >= y ? x : y; } long double reference_fminl( long double x, long double y ) { if( isnan(y) ) return x; return x <= y ? x : y; } long double reference_hypotl( long double x, long double y ) { static const double tobig = HEX_DBL( +, 1, 0, +, 511 ); static const double big = HEX_DBL( +, 1, 0, +, 513 ); static const double rbig = HEX_DBL( +, 1, 0, -, 513 ); static const double tosmall = HEX_DBL( +, 1, 0, -, 511 ); static const double smalll = HEX_DBL( +, 1, 0, -, 607 ); static const double rsmall = HEX_DBL( +, 1, 0, +, 607 ); long double max, min; if( isinf(x) || isinf(y) ) return INFINITY; if( isnan(x) || isnan(y) ) return x + y; x = reference_fabsl(x); y = reference_fabsl(y); max = reference_fmaxl( x, y ); min = reference_fminl( x, y ); if( max > tobig ) { max *= rbig; min *= rbig; return big * sqrtl( max * max + min * min ); } if( max < tosmall ) { max *= rsmall; min *= rsmall; return smalll * sqrtl( max * max + min * min ); } return sqrtl( x * x + y * y ); } //long double reference_log2l( long double x ) //{ // return log( x ) * 1.44269504088896340735992468100189214L; //} long double reference_log2l( long double x ) { if( isnan(x) || x < 0.0 || x == -INFINITY) return NAN; if( x == 0.0f) return -INFINITY; if( x == INFINITY ) return INFINITY; double hi, lo; __log2_ep( &hi, &lo, x); return (long double) hi + (long double) lo; } long double reference_log1pl( long double x) { #if defined( _MSC_VER ) && ! defined( __INTEL_COMPILER ) //unimplemented return x; #elif defined(__PPC__) // log1pl on PPC inadvertantly returns NaN for very large values. Work // around this limitation by returning logl for large values. return ((x > (long double)(0x1.0p+1022)) ? logl(x) : log1pl(x)); #else return log1pl(x); #endif } long double reference_logbl( long double x ) { // Since we are just using this to verify double precision, we can // use the double precision copysign here union { double f; cl_ulong u;} u; u.f = (double) x; cl_int exponent = (cl_uint)(u.u >> 52) & 0x7ff; if( exponent == 0x7ff ) return x * x; if( exponent == 0 ) { // deal with denormals u.f = x * HEX_DBL( +, 1, 0, +, 64 ); exponent = (cl_int)(u.u >> 52) & 0x7ff; if( exponent == 0 ) return -INFINITY; return exponent - (1023 + 64); } return exponent - 1023; } long double reference_maxmagl( long double x, long double y ) { long double fabsx = fabsl(x); long double fabsy = fabsl(y); if( fabsx < fabsy ) return y; if( fabsy < fabsx ) return x; return reference_fmaxl(x, y); } long double reference_minmagl( long double x, long double y ) { long double fabsx = fabsl(x); long double fabsy = fabsl(y); if( fabsx > fabsy ) return y; if( fabsy > fabsx ) return x; return reference_fminl(x, y); } long double reference_nanl( cl_ulong x ) { union{ cl_ulong u; cl_double f; }u; u.u = x | 0x7ff8000000000000ULL; return (long double) u.f; } long double reference_reciprocall( long double x ) { return 1.0L / x; } long double reference_remainderl( long double x, long double y ); long double reference_remainderl( long double x, long double y ) { int i; return reference_remquol( x, y, &i ); } long double reference_lgammal( long double x); long double reference_lgammal( long double x) { // lgamma is currently not tested return reference_lgamma( x ); } static uint32_t two_over_pi[] = { 0x0, 0x28be60db, 0x24e44152, 0x27f09d5f, 0x11f534dd, 0x3036d8a5, 0x1993c439, 0x107f945, 0x23abdebb, 0x31586dc9, 0x6e3a424, 0x374b8019, 0x92eea09, 0x3464873f, 0x21deb1cb, 0x4a69cfb, 0x288235f5, 0xbaed121, 0xe99c702, 0x1ad17df9, 0x13991d6, 0xe60d4ce, 0x1f49c845, 0x3e2ef7e4, 0x283b1ff8, 0x25fff781, 0x1980fef2, 0x3c462d68, 0xa6d1f6d, 0xd9fb3c9, 0x3cb09b74, 0x3d18fd9a, 0x1e5fea2d, 0x1d49eeb1, 0x3ebe5f17, 0x2cf41ce7, 0x378a5292, 0x3a9afed7, 0x3b11f8d5, 0x3421580c, 0x3046fc7b, 0x1aeafc33, 0x3bc209af, 0x10d876a7, 0x2391615e, 0x3986c219, 0x199855f1, 0x1281a102, 0xdffd880, 0x135cc9cc, 0x10606155 }; static uint32_t pi_over_two[] = { 0x1, 0x2487ed51, 0x42d1846, 0x26263314, 0x1701b839, 0x28948127 }; typedef union { uint64_t u; double d; }d_ui64_t; // radix or base of representation #define RADIX (30) #define DIGITS 6 d_ui64_t two_pow_pradix = { (uint64_t) (1023 + RADIX) << 52 }; d_ui64_t two_pow_mradix = { (uint64_t) (1023 - RADIX) << 52 }; d_ui64_t two_pow_two_mradix = { (uint64_t) (1023-2*RADIX) << 52 }; #define tp_pradix two_pow_pradix.d #define tp_mradix two_pow_mradix.d // extended fixed point representation of double precision // floating point number. // x = sign * [ sum_{i = 0 to 2} ( X[i] * 2^(index - i)*RADIX ) ] typedef struct { uint32_t X[3]; // three 32 bit integers are sufficient to represnt double in base_30 int index; // exponent bias int sign; // sign of double }eprep_t; static eprep_t double_to_eprep(double x); static eprep_t double_to_eprep(double x) { eprep_t result; result.sign = (signbit( x ) == 0) ? 1 : -1; x = fabs( x ); int index = 0; while( x > tp_pradix ) { index++; x *= tp_mradix; } while( x < 1 ) { index--; x *= tp_pradix; } result.index = index; int i = 0; result.X[0] = result.X[1] = result.X[2] = 0; while( x != 0.0 ) { result.X[i] = (uint32_t) x; x = (x - (double) result.X[i]) * tp_pradix; i++; } return result; } /* double eprep_to_double( uint32_t *R, int digits, int index, int sgn ) { d_ui64_t nb, rndcorr; uint64_t lowpart, roundbits, t1; int expo, expofinal, shift; double res; nb.d = (double) R[0]; t1 = R[1]; lowpart = (t1 << RADIX) + R[2]; expo = ((nb.u & 0x7ff0000000000000ULL) >> 52) - 1023; expofinal = expo + RADIX*index; if (expofinal > 1023) { d_ui64_t inf = { 0x7ff0000000000000ULL }; res = inf.d; } else if (expofinal >= -1022){ shift = expo + 2*RADIX - 53; roundbits = lowpart << (64-shift); lowpart = lowpart >> shift; if (lowpart & 0x0000000000000001ULL) { if(roundbits == 0) { int i; for (i=3; i < digits; i++) roundbits = roundbits | R[i]; } if(roundbits == 0) { if (lowpart & 0x0000000000000002ULL) rndcorr.u = (uint64_t) (expo - 52 + 1023) << 52; else rndcorr.d = 0.0; } else rndcorr.u = (uint64_t) (expo - 52 + 1023) << 52; } else{ rndcorr.d = 0.0; } lowpart = lowpart >> 1; nb.u = nb.u | lowpart; res = nb.d + rndcorr.d; if(index*RADIX + 1023 > 0) { nb.u = 0; nb.u = (uint64_t) (index*RADIX + 1023) << 52; res *= nb.d; } else { nb.u = 0; nb.u = (uint64_t) (index*RADIX + 1023 + 2*RADIX) << 52; res *= two_pow_two_mradix.d; res *= nb.d; } } else { if (expofinal < -1022 - 53 ) { res = 0.0; } else { lowpart = lowpart >> (expo + (2*RADIX) - 52); nb.u = nb.u | lowpart; nb.u = (nb.u & 0x000FFFFFFFFFFFFFULL) | 0x0010000000000000ULL; nb.u = nb.u >> (-1023 - expofinal); if(nb.u & 0x0000000000000001ULL) rndcorr.u = 1; else rndcorr.d = 0.0; res = 0.5*(nb.d + rndcorr.d); } } return sgn*res; } */ static double eprep_to_double( eprep_t epx ); static double eprep_to_double( eprep_t epx ) { double res = 0.0; res += ldexp((double) epx.X[0], (epx.index - 0)*RADIX); res += ldexp((double) epx.X[1], (epx.index - 1)*RADIX); res += ldexp((double) epx.X[2], (epx.index - 2)*RADIX); return copysign(res, epx.sign); } static int payne_hanek( double *y, int *exception ); static int payne_hanek( double *y, int *exception ) { double x = *y; // exception cases .. no reduction required if( isnan( x ) || isinf( x ) || (fabs( x ) <= M_PI_4) ) { *exception = 1; return 0; } *exception = 0; // After computation result[0] contains integer part while result[1]....result[DIGITS-1] // contain fractional part. So we are doing computation with (DIGITS-1)*RADIX precision. // Default DIGITS=6 and RADIX=30 so default precision is 150 bits. Kahan-McDonald algorithm // shows that a double precision x, closest to pi/2 is 6381956970095103 x 2^797 which can // cause 61 digits of cancellation in computation of f = x*2/pi - floor(x*2/pi) ... thus we need // at least 114 bits (61 leading zeros + 53 bits of mentissa of f) of precision to accurately compute // f in double precision. Since we are using 150 bits (still an overkill), we should be safe. Extra // bits can act as guard bits for correct rounding. uint64_t result[DIGITS+2]; // compute extended precision representation of x eprep_t epx = double_to_eprep( x ); int index = epx.index; int i, j; // extended precision multiplication of 2/pi*x .... we will loose at max two RADIX=30 bit digits in // the worst case for(i = 0; i < (DIGITS+2); i++) { result[i] = 0; result[i] += ((index + i - 0) >= 0) ? ((uint64_t) two_over_pi[index + i - 0] * (uint64_t) epx.X[0]) : 0; result[i] += ((index + i - 1) >= 0) ? ((uint64_t) two_over_pi[index + i - 1] * (uint64_t) epx.X[1]) : 0; result[i] += ((index + i - 2) >= 0) ? ((uint64_t) two_over_pi[index + i - 2] * (uint64_t) epx.X[2]) : 0; } // Carry propagation. uint64_t tmp; for(i = DIGITS+2-1; i > 0; i--) { tmp = result[i] >> RADIX; result[i - 1] += tmp; result[i] -= (tmp << RADIX); } // we dont ned to normalize the integer part since only last two bits of this will be used // subsequently algorithm which remain unaltered by this normalization. // tmp = result[0] >> RADIX; // result[0] -= (tmp << RADIX); unsigned int N = (unsigned int) result[0]; // if the result is > pi/4, bring it to (-pi/4, pi/4] range. Note that testing if the final // x_star = pi/2*(x*2/pi - k) > pi/4 is equivalent to testing, at this stage, if r[1] (the first fractional // digit) is greater than (2^RADIX)/2 and substracting pi/4 from x_star to bring it to mentioned // range is equivalent to substracting fractional part at this stage from one and changing the sign. int sign = 1; if(result[1] > (uint64_t)(1 << (RADIX - 1))) { for(i = 1; i < (DIGITS + 2); i++) result[i] = (~((unsigned int)result[i]) & 0x3fffffff); N += 1; sign = -1; } // Again as per Kahan-McDonald algorithim there may be 61 leading zeros in the worst case // (when x is multiple of 2/pi very close to an integer) so we need to get rid of these zeros // and adjust the index of final result. So in the worst case, precision of comupted result is // 90 bits (150 bits original bits - 60 lost in cancellation). int ind = 1; for(i = 1; i < (DIGITS+2); i++) { if(result[i] != 0) break; else ind++; } uint64_t r[DIGITS-1]; for(i = 0; i < (DIGITS-1); i++) { r[i] = 0; for(j = 0; j <= i; j++) { r[i] += (result[ind+i-j] * (uint64_t) pi_over_two[j]); } } for(i = (DIGITS-2); i > 0; i--) { tmp = r[i] >> RADIX; r[i - 1] += tmp; r[i] -= (tmp << RADIX); } tmp = r[0] >> RADIX; r[0] -= (tmp << RADIX); eprep_t epr; epr.sign = epx.sign*sign; if(tmp != 0) { epr.index = -ind + 1; epr.X[0] = (uint32_t) tmp; epr.X[1] = (uint32_t) r[0]; epr.X[2] = (uint32_t) r[1]; } else { epr.index = -ind; epr.X[0] = (uint32_t) r[0]; epr.X[1] = (uint32_t) r[1]; epr.X[2] = (uint32_t) r[2]; } *y = eprep_to_double( epr ); return epx.sign*N; } double reference_relaxed_cos(double x) { if(isnan(x)) return NAN; return (float)cos((float)x); } double reference_cos(double x) { int exception; int N = payne_hanek( &x, &exception ); if( exception ) return cos( x ); unsigned int c = N & 3; switch ( c ) { case 0: return cos( x ); case 1: return -sin( x ); case 2: return -cos( x ); case 3: return sin( x ); } return 0.0; } double reference_relaxed_sin(double x){ return (float)sin((float)x); } double reference_sin(double x) { int exception; int N = payne_hanek( &x, &exception ); if( exception ) return sin( x ); int c = N & 3; switch ( c ) { case 0: return sin( x ); case 1: return cos( x ); case 2: return -sin( x ); case 3: return -cos( x ); } return 0.0; } double reference_relaxed_sincos(double x, double * y){ *y = reference_relaxed_cos(x); return reference_relaxed_sin(x); } double reference_sincos(double x, double *y) { int exception; int N = payne_hanek( &x, &exception ); if( exception ) { *y = cos( x ); return sin( x ); } int c = N & 3; switch ( c ) { case 0: *y = cos( x ); return sin( x ); case 1: *y = -sin( x ); return cos( x ); case 2: *y = -cos( x ); return -sin( x ); case 3: *y = sin( x ); return -cos( x ); } return 0.0; } double reference_relaxed_tan(double x){ return ((float) reference_relaxed_sin((float)x))/((float) reference_relaxed_cos((float)x)); } double reference_tan(double x) { int exception; int N = payne_hanek( &x, &exception ); if( exception ) return tan( x ); int c = N & 3; switch ( c ) { case 0: return tan( x ); case 1: return -1.0 / tan( x ); case 2: return tan( x ); case 3: return -1.0 / tan( x ); } return 0.0; } long double reference_cosl(long double xx) { double x = (double) xx; int exception; int N = payne_hanek( &x, &exception ); if( exception ) return cosl( x ); unsigned int c = N & 3; switch ( c ) { case 0: return cosl( x ); case 1: return -sinl( x ); case 2: return -cosl( x ); case 3: return sinl( x ); } return 0.0; } long double reference_sinl(long double xx) { // we use system tanl after reduction which // can flush denorm input to zero so //take care of it here. if(reference_fabsl(xx) < HEX_DBL( +, 1, 0, -, 1022 )) return xx; double x = (double) xx; int exception; int N = payne_hanek( &x, &exception ); if( exception ) return sinl( x ); int c = N & 3; switch ( c ) { case 0: return sinl( x ); case 1: return cosl( x ); case 2: return -sinl( x ); case 3: return -cosl( x ); } return 0.0; } long double reference_sincosl(long double xx, long double *y) { // we use system tanl after reduction which // can flush denorm input to zero so //take care of it here. if(reference_fabsl(xx) < HEX_DBL( +, 1, 0, -, 1022 )) { *y = cosl(xx); return xx; } double x = (double) xx; int exception; int N = payne_hanek( &x, &exception ); if( exception ) { *y = cosl( x ); return sinl( x ); } int c = N & 3; switch ( c ) { case 0: *y = cosl( x ); return sinl( x ); case 1: *y = -sinl( x ); return cosl( x ); case 2: *y = -cosl( x ); return -sinl( x ); case 3: *y = sinl( x ); return -cosl( x ); } return 0.0; } long double reference_tanl(long double xx) { // we use system tanl after reduction which // can flush denorm input to zero so //take care of it here. if(reference_fabsl(xx) < HEX_DBL( +, 1, 0, -, 1022 )) return xx; double x = (double) xx; int exception; int N = payne_hanek( &x, &exception ); if( exception ) return tanl( x ); int c = N & 3; switch ( c ) { case 0: return tanl( x ); case 1: return -1.0 / tanl( x ); case 2: return tanl( x ); case 3: return -1.0 / tanl( x ); } return 0.0; } static double __loglTable1[64][3] = { {HEX_DBL( +, 1, 5390948f40fea, +, 0 ), HEX_DBL( -, 1, a152f142a, -, 2 ), HEX_DBL( +, 1, f93e27b43bd2c, -, 40 )}, {HEX_DBL( +, 1, 5015015015015, +, 0 ), HEX_DBL( -, 1, 921800925, -, 2 ), HEX_DBL( +, 1, 162432a1b8df7, -, 41 )}, {HEX_DBL( +, 1, 4cab88725af6e, +, 0 ), HEX_DBL( -, 1, 8304d90c18, -, 2 ), HEX_DBL( +, 1, 80bb749056fe7, -, 40 )}, {HEX_DBL( +, 1, 49539e3b2d066, +, 0 ), HEX_DBL( -, 1, 7418acebc, -, 2 ), HEX_DBL( +, 1, ceac7f0607711, -, 43 )}, {HEX_DBL( +, 1, 460cbc7f5cf9a, +, 0 ), HEX_DBL( -, 1, 6552b49988, -, 2 ), HEX_DBL( +, 1, d8913d0e89fa, -, 42 )}, {HEX_DBL( +, 1, 42d6625d51f86, +, 0 ), HEX_DBL( -, 1, 56b22e6b58, -, 2 ), HEX_DBL( +, 1, c7eaf515033a1, -, 44 )}, {HEX_DBL( +, 1, 3fb013fb013fb, +, 0 ), HEX_DBL( -, 1, 48365e696, -, 2 ), HEX_DBL( +, 1, 434adcde7edc7, -, 41 )}, {HEX_DBL( +, 1, 3c995a47babe7, +, 0 ), HEX_DBL( -, 1, 39de8e156, -, 2 ), HEX_DBL( +, 1, 8246f8e527754, -, 40 )}, {HEX_DBL( +, 1, 3991c2c187f63, +, 0 ), HEX_DBL( -, 1, 2baa0c34c, -, 2 ), HEX_DBL( +, 1, e1513c28e180d, -, 42 )}, {HEX_DBL( +, 1, 3698df3de0747, +, 0 ), HEX_DBL( -, 1, 1d982c9d58, -, 2 ), HEX_DBL( +, 1, 63ea3fed4b8a2, -, 40 )}, {HEX_DBL( +, 1, 33ae45b57bcb1, +, 0 ), HEX_DBL( -, 1, 0fa848045, -, 2 ), HEX_DBL( +, 1, 32ccbacf1779b, -, 40 )}, {HEX_DBL( +, 1, 30d190130d19, +, 0 ), HEX_DBL( -, 1, 01d9bbcfa8, -, 2 ), HEX_DBL( +, 1, e2bfeb2b884aa, -, 42 )}, {HEX_DBL( +, 1, 2e025c04b8097, +, 0 ), HEX_DBL( -, 1, e857d3d37, -, 3 ), HEX_DBL( +, 1, d9309b4d2ea85, -, 40 )}, {HEX_DBL( +, 1, 2b404ad012b4, +, 0 ), HEX_DBL( -, 1, cd3c712d4, -, 3 ), HEX_DBL( +, 1, ddf360962d7ab, -, 40 )}, {HEX_DBL( +, 1, 288b01288b012, +, 0 ), HEX_DBL( -, 1, b2602497e, -, 3 ), HEX_DBL( +, 1, 597f8a121640f, -, 40 )}, {HEX_DBL( +, 1, 25e22708092f1, +, 0 ), HEX_DBL( -, 1, 97c1cb13d, -, 3 ), HEX_DBL( +, 1, 02807d15580dc, -, 40 )}, {HEX_DBL( +, 1, 23456789abcdf, +, 0 ), HEX_DBL( -, 1, 7d60496d, -, 3 ), HEX_DBL( +, 1, 12ce913d7a827, -, 41 )}, {HEX_DBL( +, 1, 20b470c67c0d8, +, 0 ), HEX_DBL( -, 1, 633a8bf44, -, 3 ), HEX_DBL( +, 1, 0648bca9c96bd, -, 40 )}, {HEX_DBL( +, 1, 1e2ef3b3fb874, +, 0 ), HEX_DBL( -, 1, 494f863b9, -, 3 ), HEX_DBL( +, 1, 066fceb89b0eb, -, 42 )}, {HEX_DBL( +, 1, 1bb4a4046ed29, +, 0 ), HEX_DBL( -, 1, 2f9e32d5c, -, 3 ), HEX_DBL( +, 1, 17b8b6c4f846b, -, 46 )}, {HEX_DBL( +, 1, 19453808ca29c, +, 0 ), HEX_DBL( -, 1, 162593187, -, 3 ), HEX_DBL( +, 1, 2c83506452154, -, 42 )}, {HEX_DBL( +, 1, 16e0689427378, +, 0 ), HEX_DBL( -, 1, f9c95dc1e, -, 4 ), HEX_DBL( +, 1, dd5d2183150f3, -, 41 )}, {HEX_DBL( +, 1, 1485f0e0acd3b, +, 0 ), HEX_DBL( -, 1, c7b528b72, -, 4 ), HEX_DBL( +, 1, 0e43c4f4e619d, -, 40 )}, {HEX_DBL( +, 1, 12358e75d3033, +, 0 ), HEX_DBL( -, 1, 960caf9ac, -, 4 ), HEX_DBL( +, 1, 20fbfd5902a1e, -, 42 )}, {HEX_DBL( +, 1, 0fef010fef01, +, 0 ), HEX_DBL( -, 1, 64ce26c08, -, 4 ), HEX_DBL( +, 1, 8ebeefb4ac467, -, 40 )}, {HEX_DBL( +, 1, 0db20a88f4695, +, 0 ), HEX_DBL( -, 1, 33f7cde16, -, 4 ), HEX_DBL( +, 1, 30b3312da7a7d, -, 40 )}, {HEX_DBL( +, 1, 0b7e6ec259dc7, +, 0 ), HEX_DBL( -, 1, 0387efbcc, -, 4 ), HEX_DBL( +, 1, 796f1632949c3, -, 40 )}, {HEX_DBL( +, 1, 0953f39010953, +, 0 ), HEX_DBL( -, 1, a6f9c378, -, 5 ), HEX_DBL( +, 1, 1687e151172cc, -, 40 )}, {HEX_DBL( +, 1, 073260a47f7c6, +, 0 ), HEX_DBL( -, 1, 47aa07358, -, 5 ), HEX_DBL( +, 1, 1f87e4a9cc778, -, 42 )}, {HEX_DBL( +, 1, 05197f7d73404, +, 0 ), HEX_DBL( -, 1, d23afc498, -, 6 ), HEX_DBL( +, 1, b183a6b628487, -, 40 )}, {HEX_DBL( +, 1, 03091b51f5e1a, +, 0 ), HEX_DBL( -, 1, 16a21e21, -, 6 ), HEX_DBL( +, 1, 7d75c58973ce5, -, 40 )}, {HEX_DBL( +, 1, 0, +, 0 ), HEX_DBL( +, 0, 0, +, 0 ), HEX_DBL( +, 0, 0, +, 0 )}, {HEX_DBL( +, 1, 0, +, 0 ), HEX_DBL( +, 0, 0, +, 0 ), HEX_DBL( +, 0, 0, +, 0 )}, {HEX_DBL( +, 1, f44659e4a4271, -, 1 ), HEX_DBL( +, 1, 11cd1d51, -, 5 ), HEX_DBL( +, 1, 9a0d857e2f4b2, -, 40 )}, {HEX_DBL( +, 1, ecc07b301ecc, -, 1 ), HEX_DBL( +, 1, c4dfab908, -, 5 ), HEX_DBL( +, 1, 55b53fce557fd, -, 40 )}, {HEX_DBL( +, 1, e573ac901e573, -, 1 ), HEX_DBL( +, 1, 3aa2fdd26, -, 4 ), HEX_DBL( +, 1, f1cb0c9532089, -, 40 )}, {HEX_DBL( +, 1, de5d6e3f8868a, -, 1 ), HEX_DBL( +, 1, 918a16e46, -, 4 ), HEX_DBL( +, 1, 9af0dcd65a6e1, -, 43 )}, {HEX_DBL( +, 1, d77b654b82c33, -, 1 ), HEX_DBL( +, 1, e72ec117e, -, 4 ), HEX_DBL( +, 1, a5b93c4ebe124, -, 40 )}, {HEX_DBL( +, 1, d0cb58f6ec074, -, 1 ), HEX_DBL( +, 1, 1dcd19755, -, 3 ), HEX_DBL( +, 1, 5be50e71ddc6c, -, 42 )}, {HEX_DBL( +, 1, ca4b3055ee191, -, 1 ), HEX_DBL( +, 1, 476a9f983, -, 3 ), HEX_DBL( +, 1, ee9a798719e7f, -, 40 )}, {HEX_DBL( +, 1, c3f8f01c3f8f, -, 1 ), HEX_DBL( +, 1, 70742d4ef, -, 3 ), HEX_DBL( +, 1, 3ff1352c1219c, -, 46 )}, {HEX_DBL( +, 1, bdd2b899406f7, -, 1 ), HEX_DBL( +, 1, 98edd077e, -, 3 ), HEX_DBL( +, 1, c383cd11362f4, -, 41 )}, {HEX_DBL( +, 1, b7d6c3dda338b, -, 1 ), HEX_DBL( +, 1, c0db6cdd9, -, 3 ), HEX_DBL( +, 1, 37bd85b1a824e, -, 41 )}, {HEX_DBL( +, 1, b2036406c80d9, -, 1 ), HEX_DBL( +, 1, e840be74e, -, 3 ), HEX_DBL( +, 1, a9334d525e1ec, -, 41 )}, {HEX_DBL( +, 1, ac5701ac5701a, -, 1 ), HEX_DBL( +, 1, 0790adbb, -, 2 ), HEX_DBL( +, 1, 8060bfb6a491, -, 41 )}, {HEX_DBL( +, 1, a6d01a6d01a6d, -, 1 ), HEX_DBL( +, 1, 1ac05b2918, -, 2 ), HEX_DBL( +, 1, c1c161471580a, -, 40 )}, {HEX_DBL( +, 1, a16d3f97a4b01, -, 1 ), HEX_DBL( +, 1, 2db10fc4d8, -, 2 ), HEX_DBL( +, 1, ab1aa62214581, -, 42 )}, {HEX_DBL( +, 1, 9c2d14ee4a101, -, 1 ), HEX_DBL( +, 1, 406463b1b, -, 2 ), HEX_DBL( +, 1, 12e95dbda6611, -, 44 )}, {HEX_DBL( +, 1, 970e4f80cb872, -, 1 ), HEX_DBL( +, 1, 52dbdfc4c8, -, 2 ), HEX_DBL( +, 1, 6b53fee511af, -, 42 )}, {HEX_DBL( +, 1, 920fb49d0e228, -, 1 ), HEX_DBL( +, 1, 6518fe467, -, 2 ), HEX_DBL( +, 1, eea7d7d7d1764, -, 40 )}, {HEX_DBL( +, 1, 8d3018d3018d3, -, 1 ), HEX_DBL( +, 1, 771d2ba7e8, -, 2 ), HEX_DBL( +, 1, ecefa8d4fab97, -, 40 )}, {HEX_DBL( +, 1, 886e5f0abb049, -, 1 ), HEX_DBL( +, 1, 88e9c72e08, -, 2 ), HEX_DBL( +, 1, 913ea3d33fd14, -, 41 )}, {HEX_DBL( +, 1, 83c977ab2bedd, -, 1 ), HEX_DBL( +, 1, 9a802391e, -, 2 ), HEX_DBL( +, 1, 197e845877c94, -, 41 )}, {HEX_DBL( +, 1, 7f405fd017f4, -, 1 ), HEX_DBL( +, 1, abe18797f, -, 2 ), HEX_DBL( +, 1, f4a52f8e8a81, -, 42 )}, {HEX_DBL( +, 1, 7ad2208e0ecc3, -, 1 ), HEX_DBL( +, 1, bd0f2e9e78, -, 2 ), HEX_DBL( +, 1, 031f4336644cc, -, 42 )}, {HEX_DBL( +, 1, 767dce434a9b1, -, 1 ), HEX_DBL( +, 1, ce0a4923a, -, 2 ), HEX_DBL( +, 1, 61f33c897020c, -, 40 )}, {HEX_DBL( +, 1, 724287f46debc, -, 1 ), HEX_DBL( +, 1, ded3fd442, -, 2 ), HEX_DBL( +, 1, b2632e830632, -, 41 )}, {HEX_DBL( +, 1, 6e1f76b4337c6, -, 1 ), HEX_DBL( +, 1, ef6d673288, -, 2 ), HEX_DBL( +, 1, 888ec245a0bf, -, 40 )}, {HEX_DBL( +, 1, 6a13cd153729, -, 1 ), HEX_DBL( +, 1, ffd799a838, -, 2 ), HEX_DBL( +, 1, fe6f3b2f5fc8e, -, 40 )}, {HEX_DBL( +, 1, 661ec6a5122f9, -, 1 ), HEX_DBL( +, 1, 0809cf27f4, -, 1 ), HEX_DBL( +, 1, 81eaa9ef284dd, -, 40 )}, {HEX_DBL( +, 1, 623fa7701623f, -, 1 ), HEX_DBL( +, 1, 10113b153c, -, 1 ), HEX_DBL( +, 1, 1d7b07d6b1143, -, 42 )}, {HEX_DBL( +, 1, 5e75bb8d015e7, -, 1 ), HEX_DBL( +, 1, 18028cf728, -, 1 ), HEX_DBL( +, 1, 76b100b1f6c6, -, 41 )}, {HEX_DBL( +, 1, 5ac056b015ac, -, 1 ), HEX_DBL( +, 1, 1fde3d30e8, -, 1 ), HEX_DBL( +, 1, 26faeb9870945, -, 45 )}, {HEX_DBL( +, 1, 571ed3c506b39, -, 1 ), HEX_DBL( +, 1, 27a4c0585c, -, 1 ), HEX_DBL( +, 1, 7f2c5344d762b, -, 42 )} }; static double __loglTable2[64][3] = { {HEX_DBL( +, 1, 01fbe7f0a1be6, +, 0 ), HEX_DBL( -, 1, 6cf6ddd26112a, -, 7 ), HEX_DBL( +, 1, 0725e5755e314, -, 60 )}, {HEX_DBL( +, 1, 01eba93a97b12, +, 0 ), HEX_DBL( -, 1, 6155b1d99f603, -, 7 ), HEX_DBL( +, 1, 4bcea073117f4, -, 60 )}, {HEX_DBL( +, 1, 01db6c9029cd1, +, 0 ), HEX_DBL( -, 1, 55b54153137ff, -, 7 ), HEX_DBL( +, 1, 21e8faccad0ec, -, 61 )}, {HEX_DBL( +, 1, 01cb31f0f534c, +, 0 ), HEX_DBL( -, 1, 4a158c27245bd, -, 7 ), HEX_DBL( +, 1, 1a5b7bfbf35d3, -, 60 )}, {HEX_DBL( +, 1, 01baf95c9723c, +, 0 ), HEX_DBL( -, 1, 3e76923e3d678, -, 7 ), HEX_DBL( +, 1, eee400eb5fe34, -, 62 )}, {HEX_DBL( +, 1, 01aac2d2acee6, +, 0 ), HEX_DBL( -, 1, 32d85380ce776, -, 7 ), HEX_DBL( +, 1, cbf7a513937bd, -, 61 )}, {HEX_DBL( +, 1, 019a8e52d401e, +, 0 ), HEX_DBL( -, 1, 273acfd74be72, -, 7 ), HEX_DBL( +, 1, 5c64599efa5e6, -, 60 )}, {HEX_DBL( +, 1, 018a5bdca9e42, +, 0 ), HEX_DBL( -, 1, 1b9e072a2e65, -, 7 ), HEX_DBL( +, 1, 364180e0a5d37, -, 60 )}, {HEX_DBL( +, 1, 017a2b6fcc33e, +, 0 ), HEX_DBL( -, 1, 1001f961f3243, -, 7 ), HEX_DBL( +, 1, 63d795746f216, -, 60 )}, {HEX_DBL( +, 1, 0169fd0bd8a8a, +, 0 ), HEX_DBL( -, 1, 0466a6671bca4, -, 7 ), HEX_DBL( +, 1, 4c99ff1907435, -, 60 )}, {HEX_DBL( +, 1, 0159d0b06d129, +, 0 ), HEX_DBL( -, 1, f1981c445cd05, -, 8 ), HEX_DBL( +, 1, 4bfff6366b723, -, 62 )}, {HEX_DBL( +, 1, 0149a65d275a6, +, 0 ), HEX_DBL( -, 1, da6460f76ab8c, -, 8 ), HEX_DBL( +, 1, 9c5404f47589c, -, 61 )}, {HEX_DBL( +, 1, 01397e11a581b, +, 0 ), HEX_DBL( -, 1, c3321ab87f4ef, -, 8 ), HEX_DBL( +, 1, c0da537429cea, -, 61 )}, {HEX_DBL( +, 1, 012957cd85a28, +, 0 ), HEX_DBL( -, 1, ac014958c112c, -, 8 ), HEX_DBL( +, 1, 000c2a1b595e3, -, 64 )}, {HEX_DBL( +, 1, 0119339065ef7, +, 0 ), HEX_DBL( -, 1, 94d1eca95f67a, -, 8 ), HEX_DBL( +, 1, d8d20b0564d5, -, 61 )}, {HEX_DBL( +, 1, 01091159e4b3d, +, 0 ), HEX_DBL( -, 1, 7da4047b92b3e, -, 8 ), HEX_DBL( +, 1, 6194a5d68cf2, -, 66 )}, {HEX_DBL( +, 1, 00f8f129a0535, +, 0 ), HEX_DBL( -, 1, 667790a09bf77, -, 8 ), HEX_DBL( +, 1, ca230e0bea645, -, 61 )}, {HEX_DBL( +, 1, 00e8d2ff374a1, +, 0 ), HEX_DBL( -, 1, 4f4c90e9c4ead, -, 8 ), HEX_DBL( +, 1, 1de3e7f350c1, -, 61 )}, {HEX_DBL( +, 1, 00d8b6da482ce, +, 0 ), HEX_DBL( -, 1, 3823052860649, -, 8 ), HEX_DBL( +, 1, 5789b4c5891b8, -, 64 )}, {HEX_DBL( +, 1, 00c89cba71a8c, +, 0 ), HEX_DBL( -, 1, 20faed2dc9a9e, -, 8 ), HEX_DBL( +, 1, 9e7c40f9839fd, -, 62 )}, {HEX_DBL( +, 1, 00b8849f52834, +, 0 ), HEX_DBL( -, 1, 09d448cb65014, -, 8 ), HEX_DBL( +, 1, 387e3e9b6d02, -, 62 )}, {HEX_DBL( +, 1, 00a86e88899a4, +, 0 ), HEX_DBL( -, 1, e55e2fa53ebf1, -, 9 ), HEX_DBL( +, 1, cdaa71fddfddf, -, 62 )}, {HEX_DBL( +, 1, 00985a75b5e3f, +, 0 ), HEX_DBL( -, 1, b716b429dce0f, -, 9 ), HEX_DBL( +, 1, 2f2af081367bf, -, 63 )}, {HEX_DBL( +, 1, 00884866766ee, +, 0 ), HEX_DBL( -, 1, 88d21ec7a16d7, -, 9 ), HEX_DBL( +, 1, fb95c228d6f16, -, 62 )}, {HEX_DBL( +, 1, 0078385a6a61d, +, 0 ), HEX_DBL( -, 1, 5a906f219a9e8, -, 9 ), HEX_DBL( +, 1, 18aff10a89f29, -, 64 )}, {HEX_DBL( +, 1, 00682a5130fbe, +, 0 ), HEX_DBL( -, 1, 2c51a4dae87f1, -, 9 ), HEX_DBL( +, 1, bcc7e33ddde3, -, 63 )}, {HEX_DBL( +, 1, 00581e4a69944, +, 0 ), HEX_DBL( -, 1, fc2b7f2d782b1, -, 10 ), HEX_DBL( +, 1, fe3ef3300a9fa, -, 64 )}, {HEX_DBL( +, 1, 00481445b39a8, +, 0 ), HEX_DBL( -, 1, 9fb97df0b0b83, -, 10 ), HEX_DBL( +, 1, 0d9a601f2f324, -, 65 )}, {HEX_DBL( +, 1, 00380c42ae963, +, 0 ), HEX_DBL( -, 1, 434d4546227ae, -, 10 ), HEX_DBL( +, 1, 0b9b6a5868f33, -, 63 )}, {HEX_DBL( +, 1, 00280640fa271, +, 0 ), HEX_DBL( -, 1, cdcda8e930c19, -, 11 ), HEX_DBL( +, 1, 3d424ab39f789, -, 64 )}, {HEX_DBL( +, 1, 0018024036051, +, 0 ), HEX_DBL( -, 1, 150c558601261, -, 11 ), HEX_DBL( +, 1, 285bb90327a0f, -, 64 )}, {HEX_DBL( +, 1, 0, +, 0 ), HEX_DBL( +, 0, 0, +, 0 ), HEX_DBL( +, 0, 0, +, 0 )}, {HEX_DBL( +, 1, 0, +, 0 ), HEX_DBL( +, 0, 0, +, 0 ), HEX_DBL( +, 0, 0, +, 0 )}, {HEX_DBL( +, 1, ffa011fca0a1e, -, 1 ), HEX_DBL( +, 1, 14e5640c4197b, -, 10 ), HEX_DBL( +, 1, 95728136ae401, -, 63 )}, {HEX_DBL( +, 1, ff6031f064e07, -, 1 ), HEX_DBL( +, 1, cd61806bf532d, -, 10 ), HEX_DBL( +, 1, 568a4f35d8538, -, 63 )}, {HEX_DBL( +, 1, ff2061d532b9c, -, 1 ), HEX_DBL( +, 1, 42e34af550eda, -, 9 ), HEX_DBL( +, 1, 8f69cee55fec, -, 62 )}, {HEX_DBL( +, 1, fee0a1a513253, -, 1 ), HEX_DBL( +, 1, 9f0a5523902ea, -, 9 ), HEX_DBL( +, 1, daec734b11615, -, 63 )}, {HEX_DBL( +, 1, fea0f15a12139, -, 1 ), HEX_DBL( +, 1, fb25e19f11b26, -, 9 ), HEX_DBL( +, 1, 8bafca62941da, -, 62 )}, {HEX_DBL( +, 1, fe6150ee3e6d4, -, 1 ), HEX_DBL( +, 1, 2b9af9a28e282, -, 8 ), HEX_DBL( +, 1, 0fd3674e1dc5b, -, 61 )}, {HEX_DBL( +, 1, fe21c05baa109, -, 1 ), HEX_DBL( +, 1, 599d4678f24b9, -, 8 ), HEX_DBL( +, 1, dafce1f09937b, -, 61 )}, {HEX_DBL( +, 1, fde23f9c69cf9, -, 1 ), HEX_DBL( +, 1, 8799d8c046eb, -, 8 ), HEX_DBL( +, 1, ffa0ce0bdd217, -, 65 )}, {HEX_DBL( +, 1, fda2ceaa956e8, -, 1 ), HEX_DBL( +, 1, b590b1e5951ee, -, 8 ), HEX_DBL( +, 1, 645a769232446, -, 62 )}, {HEX_DBL( +, 1, fd636d8047a1f, -, 1 ), HEX_DBL( +, 1, e381d3555dbcf, -, 8 ), HEX_DBL( +, 1, 882320d368331, -, 61 )}, {HEX_DBL( +, 1, fd241c179e0cc, -, 1 ), HEX_DBL( +, 1, 08b69f3dccde, -, 7 ), HEX_DBL( +, 1, 01ad5065aba9e, -, 61 )}, {HEX_DBL( +, 1, fce4da6ab93e8, -, 1 ), HEX_DBL( +, 1, 1fa97a61dd298, -, 7 ), HEX_DBL( +, 1, 84cd1f931ae34, -, 60 )}, {HEX_DBL( +, 1, fca5a873bcb19, -, 1 ), HEX_DBL( +, 1, 36997bcc54a3f, -, 7 ), HEX_DBL( +, 1, 1485e97eaee03, -, 60 )}, {HEX_DBL( +, 1, fc66862ccec93, -, 1 ), HEX_DBL( +, 1, 4d86a43264a4f, -, 7 ), HEX_DBL( +, 1, c75e63370988b, -, 61 )}, {HEX_DBL( +, 1, fc27739018cfe, -, 1 ), HEX_DBL( +, 1, 6470f448fb09d, -, 7 ), HEX_DBL( +, 1, d7361eeaed0a1, -, 65 )}, {HEX_DBL( +, 1, fbe87097c6f5a, -, 1 ), HEX_DBL( +, 1, 7b586cc4c2523, -, 7 ), HEX_DBL( +, 1, b3df952cc473c, -, 61 )}, {HEX_DBL( +, 1, fba97d3e084dd, -, 1 ), HEX_DBL( +, 1, 923d0e5a21e06, -, 7 ), HEX_DBL( +, 1, cf56c7b64ae5d, -, 62 )}, {HEX_DBL( +, 1, fb6a997d0ecdc, -, 1 ), HEX_DBL( +, 1, a91ed9bd3df9a, -, 7 ), HEX_DBL( +, 1, b957bdcd89e43, -, 61 )}, {HEX_DBL( +, 1, fb2bc54f0f4ab, -, 1 ), HEX_DBL( +, 1, bffdcfa1f7fbb, -, 7 ), HEX_DBL( +, 1, ea8cad9a21771, -, 62 )}, {HEX_DBL( +, 1, faed00ae41783, -, 1 ), HEX_DBL( +, 1, d6d9f0bbee6f6, -, 7 ), HEX_DBL( +, 1, 5762a9af89c82, -, 60 )}, {HEX_DBL( +, 1, faae4b94dfe64, -, 1 ), HEX_DBL( +, 1, edb33dbe7d335, -, 7 ), HEX_DBL( +, 1, 21e24fc245697, -, 62 )}, {HEX_DBL( +, 1, fa6fa5fd27ff8, -, 1 ), HEX_DBL( +, 1, 0244dbae5ed05, -, 6 ), HEX_DBL( +, 1, 12ef51b967102, -, 60 )}, {HEX_DBL( +, 1, fa310fe15a078, -, 1 ), HEX_DBL( +, 1, 0daeaf24c3529, -, 6 ), HEX_DBL( +, 1, 10d3cfca60b45, -, 59 )}, {HEX_DBL( +, 1, f9f2893bb9192, -, 1 ), HEX_DBL( +, 1, 1917199bb66bc, -, 6 ), HEX_DBL( +, 1, 6cf6034c32e19, -, 60 )}, {HEX_DBL( +, 1, f9b412068b247, -, 1 ), HEX_DBL( +, 1, 247e1b6c615d5, -, 6 ), HEX_DBL( +, 1, 42f0fffa229f7, -, 61 )}, {HEX_DBL( +, 1, f975aa3c18ed6, -, 1 ), HEX_DBL( +, 1, 2fe3b4efcc5ad, -, 6 ), HEX_DBL( +, 1, 70106136a8919, -, 60 )}, {HEX_DBL( +, 1, f93751d6ae09b, -, 1 ), HEX_DBL( +, 1, 3b47e67edea93, -, 6 ), HEX_DBL( +, 1, 38dd5a4f6959a, -, 59 )}, {HEX_DBL( +, 1, f8f908d098df6, -, 1 ), HEX_DBL( +, 1, 46aab0725ea6c, -, 6 ), HEX_DBL( +, 1, 821fc1e799e01, -, 60 )}, {HEX_DBL( +, 1, f8bacf242aa2c, -, 1 ), HEX_DBL( +, 1, 520c1322f1e4e, -, 6 ), HEX_DBL( +, 1, 129dcda3ad563, -, 60 )}, {HEX_DBL( +, 1, f87ca4cbb755, -, 1 ), HEX_DBL( +, 1, 5d6c0ee91d2ab, -, 6 ), HEX_DBL( +, 1, c5b190c04606e, -, 62 )}, {HEX_DBL( +, 1, f83e89c195c25, -, 1 ), HEX_DBL( +, 1, 68caa41d448c3, -, 6 ), HEX_DBL( +, 1, 4723441195ac9, -, 59 )} }; static double __loglTable3[8][3] = { {HEX_DBL( +, 1, 000e00c40ab89, +, 0 ), HEX_DBL( -, 1, 4332be0032168, -, 12 ), HEX_DBL( +, 1, a1003588d217a, -, 65 )}, {HEX_DBL( +, 1, 000a006403e82, +, 0 ), HEX_DBL( -, 1, cdb2987366fcc, -, 13 ), HEX_DBL( +, 1, 5c86001294bbc, -, 67 )}, {HEX_DBL( +, 1, 0006002400d8, +, 0 ), HEX_DBL( -, 1, 150297c90fa6f, -, 13 ), HEX_DBL( +, 1, 01fb4865fae32, -, 66 )}, {HEX_DBL( +, 1, 0, +, 0 ), HEX_DBL( +, 0, 0, +, 0 ), HEX_DBL( +, 0, 0, +, 0 )}, {HEX_DBL( +, 1, 0, +, 0 ), HEX_DBL( +, 0, 0, +, 0 ), HEX_DBL( +, 0, 0, +, 0 )}, {HEX_DBL( +, 1, ffe8011ff280a, -, 1 ), HEX_DBL( +, 1, 14f8daf5e3d3b, -, 12 ), HEX_DBL( +, 1, 3c933b4b6b914, -, 68 )}, {HEX_DBL( +, 1, ffd8031fc184e, -, 1 ), HEX_DBL( +, 1, cd978c38042bb, -, 12 ), HEX_DBL( +, 1, 10f8e642e66fd, -, 65 )}, {HEX_DBL( +, 1, ffc8061f5492b, -, 1 ), HEX_DBL( +, 1, 43183c878274e, -, 11 ), HEX_DBL( +, 1, 5885dd1eb6582, -, 65 )} }; static void __log2_ep(double *hi, double *lo, double x) { union { uint64_t i; double d; } uu; int m; double f = reference_frexp(x, &m); // bring f in [0.75, 1.5) if( f < 0.75 ) { f *= 2.0; m -= 1; } // index first table .... brings down to [1-2^-7, 1+2^6) uu.d = f; int index = (int) (((uu.i + ((uint64_t) 1 << 51)) & 0x000fc00000000000ULL) >> 46); double r1 = __loglTable1[index][0]; double logr1hi = __loglTable1[index][1]; double logr1lo = __loglTable1[index][2]; // since log1rhi has 39 bits of precision, we have 14 bit in hand ... since |m| <= 1023 // which needs 10bits at max, we can directly add m to log1hi without spilling logr1hi += m; // argument reduction needs to be in double-double since reduced argument will form the // leading term of polynomial approximation which sets the precision we eventually achieve double zhi, zlo; MulD(&zhi, &zlo, r1, uu.d); // second index table .... brings down to [1-2^-12, 1+2^-11) uu.d = zhi; index = (int) (((uu.i + ((uint64_t) 1 << 46)) & 0x00007e0000000000ULL) >> 41); double r2 = __loglTable2[index][0]; double logr2hi = __loglTable2[index][1]; double logr2lo = __loglTable2[index][2]; // reduce argument MulDD(&zhi, &zlo, zhi, zlo, r2, 0.0); // third index table .... brings down to [1-2^-14, 1+2^-13) // Actually reduction to 2^-11 would have been sufficient to calculate // second order term in polynomial in double rather than double-double, I // reduced it a bit more to make sure other systematic arithmetic errors // are guarded against .... also this allow lower order product of leading polynomial // term i.e. Ao_hi*z_lo + Ao_lo*z_hi to be done in double rather than double-double ... // hence only term that needs to be done in double-double is Ao_hi*z_hi uu.d = zhi; index = (int) (((uu.i + ((uint64_t) 1 << 41)) & 0x0000038000000000ULL) >> 39); double r3 = __loglTable3[index][0]; double logr3hi = __loglTable3[index][1]; double logr3lo = __loglTable3[index][2]; // log2(x) = m + log2(r1) + log2(r1) + log2(1 + (zh + zlo)) // calculate sum of first three terms ... note that m has already // been added to log2(r1)_hi double log2hi, log2lo; AddDD(&log2hi, &log2lo, logr1hi, logr1lo, logr2hi, logr2lo); AddDD(&log2hi, &log2lo, logr3hi, logr3lo, log2hi, log2lo); // final argument reduction .... zhi will be in [1-2^-14, 1+2^-13) after this MulDD(&zhi, &zlo, zhi, zlo, r3, 0.0); // we dont need to do full double-double substract here. substracting 1.0 for higher // term is exact zhi = zhi - 1.0; // normalize AddD(&zhi, &zlo, zhi, zlo); // polynomail fitting to compute log2(1 + z) ... forth order polynomial fit // to log2(1 + z)/z gives minimax absolute error of O(2^-76) with z in [-2^-14, 2^-13] // log2(1 + z)/z = Ao + A1*z + A2*z^2 + A3*z^3 + A4*z^4 // => log2(1 + z) = Ao*z + A1*z^2 + A2*z^3 + A3*z^4 + A4*z^5 // => log2(1 + z) = (Aohi + Aolo)*(zhi + zlo) + z^2*(A1 + A2*z + A3*z^2 + A4*z^3) // since we are looking for at least 64 digits of precision and z in [-2^-14, 2^-13], final term // can be done in double .... also Aolo*zhi + Aohi*zlo can be done in double .... // Aohi*zhi needs to be done in double-double double Aohi = HEX_DBL( +, 1, 71547652b82fe, +, 0 ); double Aolo = HEX_DBL( +, 1, 777c9cbb675c, -, 56 ); double y; y = HEX_DBL( +, 1, 276d2736fade7, -, 2 ); y = HEX_DBL( -, 1, 7154765782df1, -, 2 ) + y*zhi; y = HEX_DBL( +, 1, ec709dc3a0f67, -, 2 ) + y*zhi; y = HEX_DBL( -, 1, 71547652b82fe, -, 1 ) + y*zhi; double zhisq = zhi*zhi; y = y*zhisq; y = y + zhi*Aolo; y = y + zlo*Aohi; MulD(&zhi, &zlo, Aohi, zhi); AddDD(&zhi, &zlo, zhi, zlo, y, 0.0); AddDD(&zhi, &zlo, zhi, zlo, log2hi, log2lo); *hi = zhi; *lo = zlo; } long double reference_powl( long double x, long double y ) { // this will be used for testing doubles i.e. arguments will // be doubles so cast the input back to double ... returned // result will be long double though .... > 53 bits of precision // if platform allows. // =========== // New finding. // =========== // this function is getting used for computing reference cube root (cbrt) // as follows __powl( x, 1.0L/3.0L ) so if the y are assumed to // be double and is converted from long double to double, truncation // causes errors. So we need to tread y as long double and convert it // to hi, lo doubles when performing y*log2(x). // double x = (double) xx; // double y = (double) yy; static const double neg_epsilon = HEX_DBL( +, 1, 0, +, 53 ); //if x = 1, return x for any y, even NaN if( x == 1.0 ) return x; //if y == 0, return 1 for any x, even NaN if( y == 0.0 ) return 1.0L; //get NaNs out of the way if( x != x || y != y ) return x + y; //do the work required to sort out edge cases double fabsy = reference_fabs( y ); double fabsx = reference_fabs( x ); double iy = reference_rint( fabsy ); //we do round to nearest here so that |fy| <= 0.5 if( iy > fabsy )//convert nearbyint to floor iy -= 1.0; int isOddInt = 0; if( fabsy == iy && !reference_isinf(fabsy) && iy < neg_epsilon ) isOddInt = (int) (iy - 2.0 * rint( 0.5 * iy )); //might be 0, -1, or 1 ///test a few more edge cases //deal with x == 0 cases if( x == 0.0 ) { if( ! isOddInt ) x = 0.0; if( y < 0 ) x = 1.0/ x; return x; } //x == +-Inf cases if( isinf(fabsx) ) { if( x < 0 ) { if( isOddInt ) { if( y < 0 ) return -0.0; else return -INFINITY; } else { if( y < 0 ) return 0.0; else return INFINITY; } } if( y < 0 ) return 0; return INFINITY; } //y = +-inf cases if( isinf(fabsy) ) { if( x == -1 ) return 1; if( y < 0 ) { if( fabsx < 1 ) return INFINITY; return 0; } if( fabsx < 1 ) return 0; return INFINITY; } // x < 0 and y non integer case if( x < 0 && iy != fabsy ) { //return nan; return cl_make_nan(); } //speedy resolution of sqrt and reciprocal sqrt if( fabsy == 0.5 ) { long double xl = sqrtl( x ); if( y < 0 ) xl = 1.0/ xl; return xl; } double log2x_hi, log2x_lo; // extended precision log .... accurate to at least 64-bits + couple of guard bits __log2_ep(&log2x_hi, &log2x_lo, fabsx); double ylog2x_hi, ylog2x_lo; double y_hi = (double) y; double y_lo = (double) ( y - (long double) y_hi); // compute product of y*log2(x) // scale to avoid overflow in double-double multiplication if( reference_fabs( y ) > HEX_DBL( +, 1, 0, +, 970 ) ) { y_hi = reference_ldexp(y_hi, -53); y_lo = reference_ldexp(y_lo, -53); } MulDD(&ylog2x_hi, &ylog2x_lo, log2x_hi, log2x_lo, y_hi, y_lo); if( fabs( y ) > HEX_DBL( +, 1, 0, +, 970 ) ) { ylog2x_hi = reference_ldexp(ylog2x_hi, 53); ylog2x_lo = reference_ldexp(ylog2x_lo, 53); } long double powxy; if(isinf(ylog2x_hi) || (reference_fabs(ylog2x_hi) > 2200)) { powxy = reference_signbit(ylog2x_hi) ? HEX_DBL( +, 0, 0, +, 0 ) : INFINITY; } else { // separate integer + fractional part long int m = lrint(ylog2x_hi); AddDD(&ylog2x_hi, &ylog2x_lo, ylog2x_hi, ylog2x_lo, -m, 0.0); // revert to long double arithemtic long double ylog2x = (long double) ylog2x_hi + (long double) ylog2x_lo; long double tmp = reference_exp2l( ylog2x ); powxy = reference_scalblnl(tmp, m); } // if y is odd integer and x is negative, reverse sign if( isOddInt & reference_signbit(x)) powxy = -powxy; return powxy; } double reference_nextafter(double xx, double yy) { float x = (float) xx; float y = (float) yy; // take care of nans if( x != x ) return x; if( y != y ) return y; if( x == y ) return y; int32f_t a, b; a.f = x; b.f = y; if( a.i & 0x80000000 ) a.i = 0x80000000 - a.i; if(b.i & 0x80000000 ) b.i = 0x80000000 - b.i; a.i += (a.i < b.i) ? 1 : -1; a.i = (a.i < 0) ? (cl_int) 0x80000000 - a.i : a.i; return a.f; } long double reference_nextafterl(long double xx, long double yy) { double x = (double) xx; double y = (double) yy; // take care of nans if( x != x ) return x; if( y != y ) return y; int64d_t a, b; a.d = x; b.d = y; int64_t tmp = 0x8000000000000000LL; if( a.l & tmp ) a.l = tmp - a.l; if(b.l & tmp ) b.l = tmp - b.l; // edge case. if (x == y) or (x = 0.0f and y = -0.0f) or (x = -0.0f and y = 0.0f) // test needs to be done using integer rep because // subnormals may be flushed to zero on some platforms if( a.l == b.l ) return y; a.l += (a.l < b.l) ? 1 : -1; a.l = (a.l < 0) ? tmp - a.l : a.l; return a.d; } double reference_fdim(double xx, double yy) { float x = (float) xx; float y = (float) yy; if( x != x ) return x; if( y != y ) return y; float r = ( x > y ) ? (float) reference_subtract( x, y) : 0.0f; return r; } long double reference_fdiml(long double xx, long double yy) { double x = (double) xx; double y = (double) yy; if( x != x ) return x; if( y != y ) return y; double r = ( x > y ) ? (double) reference_subtractl(x, y) : 0.0; return r; } double reference_remquo(double xd, double yd, int *n) { float xx = (float) xd; float yy = (float) yd; if( isnan(xx) || isnan(yy) || fabsf(xx) == INFINITY || yy == 0.0 ) { *n = 0; return cl_make_nan(); } if( fabsf(yy) == INFINITY || xx == 0.0f ) { *n = 0; return xd; } if( fabsf(xx) == fabsf(yy) ) { *n = (xx == yy) ? 1 : -1; return reference_signbit( xx ) ? -0.0 : 0.0; } int signx = reference_signbit( xx ) ? -1 : 1; int signy = reference_signbit( yy ) ? -1 : 1; int signn = (signx == signy) ? 1 : -1; float x = fabsf(xx); float y = fabsf(yy); int ex, ey; ex = reference_ilogb( x ); ey = reference_ilogb( y ); float xr = x; float yr = y; uint32_t q = 0; if(ex-ey >= -1) { yr = (float) reference_ldexp( y, -ey ); xr = (float) reference_ldexp( x, -ex ); if(ex-ey >= 0) { int i; for(i = ex-ey; i > 0; i--) { q <<= 1; if(xr >= yr) { xr -= yr; q += 1; } xr += xr; } q <<= 1; if( xr > yr ) { xr -= yr; q += 1; } } else //ex-ey = -1 xr = reference_ldexp(xr, ex-ey); } if( (yr < 2.0f*xr) || ( (yr == 2.0f*xr) && (q & 0x00000001) ) ) { xr -= yr; q += 1; } if(ex-ey >= -1) xr = reference_ldexp(xr, ey); int qout = q & 0x0000007f; if( signn < 0) qout = -qout; if( xx < 0.0 ) xr = -xr; *n = qout; return xr; } long double reference_remquol(long double xd, long double yd, int *n) { double xx = (double) xd; double yy = (double) yd; if( isnan(xx) || isnan(yy) || fabs(xx) == INFINITY || yy == 0.0 ) { *n = 0; return cl_make_nan(); } if( reference_fabs(yy) == INFINITY || xx == 0.0 ) { *n = 0; return xd; } if( reference_fabs(xx) == reference_fabs(yy) ) { *n = (xx == yy) ? 1 : -1; return reference_signbit( xx ) ? -0.0 : 0.0; } int signx = reference_signbit( xx ) ? -1 : 1; int signy = reference_signbit( yy ) ? -1 : 1; int signn = (signx == signy) ? 1 : -1; double x = reference_fabs(xx); double y = reference_fabs(yy); int ex, ey; ex = reference_ilogbl( x ); ey = reference_ilogbl( y ); double xr = x; double yr = y; uint32_t q = 0; if(ex-ey >= -1) { yr = reference_ldexp( y, -ey ); xr = reference_ldexp( x, -ex ); int i; if(ex-ey >= 0) { for(i = ex-ey; i > 0; i--) { q <<= 1; if(xr >= yr) { xr -= yr; q += 1; } xr += xr; } q <<= 1; if( xr > yr ) { xr -= yr; q += 1; } } else xr = reference_ldexp(xr, ex-ey); } if( (yr < 2.0*xr) || ( (yr == 2.0*xr) && (q & 0x00000001) ) ) { xr -= yr; q += 1; } if(ex-ey >= -1) xr = reference_ldexp(xr, ey); int qout = q & 0x0000007f; if( signn < 0) qout = -qout; if( xx < 0.0 ) xr = -xr; *n = qout; return xr; } static double reference_scalbn(double x, int n) { if(reference_isinf(x) || reference_isnan(x) || x == 0.0) return x; int bias = 1023; union { double d; cl_long l; } u; u.d = (double) x; int e = (int)((u.l & 0x7ff0000000000000LL) >> 52); if(e == 0) { u.l |= ((cl_long)1023 << 52); u.d -= 1.0; e = (int)((u.l & 0x7ff0000000000000LL) >> 52) - 1022; } e += n; if(e >= 2047 || n >= 2098 ) return reference_copysign(INFINITY, x); if(e < -51 || n <-2097 ) return reference_copysign(0.0, x); if(e <= 0) { bias += (e-1); e = 1; } u.l &= 0x800fffffffffffffLL; u.l |= ((cl_long)e << 52); x = u.d; u.l = ((cl_long)bias << 52); return x * u.d; } static long double reference_scalblnl(long double x, long n) { #if defined(__i386__) || defined(__x86_64__) // INTEL union { long double d; struct{ cl_ulong m; cl_ushort sexp;}u; }u; u.u.m = CL_LONG_MIN; if ( reference_isinf(x) ) return x; if( x == 0.0L || n < -2200) return reference_copysignl( 0.0L, x ); if( n > 2200 ) return reference_copysignl( INFINITY, x ); if( n < 0 ) { u.u.sexp = 0x3fff - 1022; while( n <= -1022 ) { x *= u.d; n += 1022; } u.u.sexp = 0x3fff + n; x *= u.d; return x; } if( n > 0 ) { u.u.sexp = 0x3fff + 1023; while( n >= 1023 ) { x *= u.d; n -= 1023; } u.u.sexp = 0x3fff + n; x *= u.d; return x; } return x; #elif defined(__arm__) // ARM .. sizeof(long double) == sizeof(double) #if __DBL_MAX_EXP__ >= __LDBL_MAX_EXP__ if(reference_isinfl(x) || reference_isnanl(x)) return x; int bias = 1023; union { double d; cl_long l; } u; u.d = (double) x; int e = (int)((u.l & 0x7ff0000000000000LL) >> 52); if(e == 0) { u.l |= ((cl_long)1023 << 52); u.d -= 1.0; e = (int)((u.l & 0x7ff0000000000000LL) >> 52) - 1022; } e += n; if(e >= 2047) return reference_copysignl(INFINITY, x); if(e < -51) return reference_copysignl(0.0, x); if(e <= 0) { bias += (e-1); e = 1; } u.l &= 0x800fffffffffffffLL; u.l |= ((cl_long)e << 52); x = u.d; u.l = ((cl_long)bias << 52); return x * u.d; #endif #else // PPC return scalblnl(x, n); #endif } double reference_relaxed_exp( double x ) { return reference_exp(x); } double reference_exp(double x) { return reference_exp2( x * HEX_DBL( +, 1, 71547652b82fe, +, 0 ) ); } long double reference_expl(long double x) { #if defined(__PPC__) long double scale, bias; // The PPC double long version of expl fails to produce denorm results // and instead generates a 0.0. Compensate for this limitation by // computing expl as: // expl(x + 40) * expl(-40) // Likewise, overflows can prematurely produce an infinity, so we // compute expl as: // expl(x - 40) * expl(40) scale = 1.0L; bias = 0.0L; if (x < -708.0L) { bias = 40.0; scale = expl(-40.0L); } else if (x > 708.0L) { bias = -40.0L; scale = expl(40.0L); } return expl(x + bias) * scale; #else return expl( x ); #endif } double reference_sinh(double x) { return sinh(x); } long double reference_sinhl(long double x) { return sinhl(x); } double reference_fmod(double x, double y) { if( x == 0.0 && fabs(y) > 0.0 ) return x; if( fabs(x) == INFINITY || y == 0 ) return cl_make_nan(); if( fabs(y) == INFINITY ) // we know x is finite from above return x; #if defined(_MSC_VER) && defined(_M_X64) return fmod( x, y ); #else return fmodf( (float) x, (float) y ); #endif } long double reference_fmodl(long double x, long double y) { if( x == 0.0L && fabsl(y) > 0.0L ) return x; if( fabsl(x) == INFINITY || y == 0.0L ) return cl_make_nan(); if( fabsl(y) == INFINITY ) // we know x is finite from above return x; return fmod( (double) x, (double) y ); } double reference_modf(double x, double *n) { if(isnan(x)) { *n = cl_make_nan(); return cl_make_nan(); } float nr; float yr = modff((float) x, &nr); *n = nr; return yr; } long double reference_modfl(long double x, long double *n) { if(isnan(x)) { *n = cl_make_nan(); return cl_make_nan(); } double nr; double yr = modf((double) x, &nr); *n = nr; return yr; } long double reference_fractl(long double x, long double *ip ) { if(isnan(x)) { *ip = cl_make_nan(); return cl_make_nan(); } double i; double f = modf((double) x, &i ); if( f < 0.0 ) { f = 1.0 + f; i -= 1.0; if( f == 1.0 ) f = HEX_DBL( +, 1, fffffffffffff, -, 1 ); } *ip = i; return f; } long double reference_fabsl(long double x) { return fabsl( x ); } double reference_relaxed_log( double x ) { return (float)reference_log((float)x); } double reference_log(double x) { if( x == 0.0 ) return -INFINITY; if( x < 0.0 ) return cl_make_nan(); if( isinf(x) ) return INFINITY; double log2Hi = HEX_DBL( +, 1, 62e42fefa39ef, -, 1 ); double logxHi, logxLo; __log2_ep(&logxHi, &logxLo, x); return logxHi*log2Hi; } long double reference_logl(long double x) { if( x == 0.0 ) return -INFINITY; if( x < 0.0 ) return cl_make_nan(); if( isinf(x) ) return INFINITY; double log2Hi = HEX_DBL( +, 1, 62e42fefa39ef, -, 1 ); double log2Lo = HEX_DBL( +, 1, abc9e3b39803f, -, 56 ); double logxHi, logxLo; __log2_ep(&logxHi, &logxLo, x); //double rhi, rlo; //MulDD(&rhi, &rlo, logxHi, logxLo, log2Hi, log2Lo); //return (long double) rhi + (long double) rlo; long double lg2 = (long double) log2Hi + (long double) log2Lo; long double logx = (long double) logxHi + (long double) logxLo; return logx*lg2; } double reference_relaxed_pow( double x, double y) { return (float)reference_exp2( ((float)y) * (float)reference_log2((float)x)); } double reference_pow( double x, double y ) { static const double neg_epsilon = HEX_DBL( +, 1, 0, +, 53 ); //if x = 1, return x for any y, even NaN if( x == 1.0 ) return x; //if y == 0, return 1 for any x, even NaN if( y == 0.0 ) return 1.0; //get NaNs out of the way if( x != x || y != y ) return x + y; //do the work required to sort out edge cases double fabsy = reference_fabs( y ); double fabsx = reference_fabs( x ); double iy = reference_rint( fabsy ); //we do round to nearest here so that |fy| <= 0.5 if( iy > fabsy )//convert nearbyint to floor iy -= 1.0; int isOddInt = 0; if( fabsy == iy && !reference_isinf(fabsy) && iy < neg_epsilon ) isOddInt = (int) (iy - 2.0 * rint( 0.5 * iy )); //might be 0, -1, or 1 ///test a few more edge cases //deal with x == 0 cases if( x == 0.0 ) { if( ! isOddInt ) x = 0.0; if( y < 0 ) x = 1.0/ x; return x; } //x == +-Inf cases if( isinf(fabsx) ) { if( x < 0 ) { if( isOddInt ) { if( y < 0 ) return -0.0; else return -INFINITY; } else { if( y < 0 ) return 0.0; else return INFINITY; } } if( y < 0 ) return 0; return INFINITY; } //y = +-inf cases if( isinf(fabsy) ) { if( x == -1 ) return 1; if( y < 0 ) { if( fabsx < 1 ) return INFINITY; return 0; } if( fabsx < 1 ) return 0; return INFINITY; } // x < 0 and y non integer case if( x < 0 && iy != fabsy ) { //return nan; return cl_make_nan(); } //speedy resolution of sqrt and reciprocal sqrt if( fabsy == 0.5 ) { long double xl = reference_sqrt( x ); if( y < 0 ) xl = 1.0/ xl; return xl; } double hi, lo; __log2_ep(&hi, &lo, fabsx); double prod = y * hi; double result = reference_exp2(prod); return isOddInt ? reference_copysignd(result, x) : result; } double reference_sqrt(double x) { return sqrt(x); } double reference_floor(double x) { return floorf((float) x); } double reference_ldexp(double value, int exponent) { #ifdef __MINGW32__ /* * ==================================================== * This function is from fdlibm: http://www.netlib.org * It is Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. * * Developed at SunSoft, a Sun Microsystems, Inc. business. * Permission to use, copy, modify, and distribute this * software is freely granted, provided that this notice * is preserved. * ==================================================== */ if(!finite(value)||value==0.0) return value; return scalbn(value,exponent); #else return reference_scalbn(value, exponent); #endif } long double reference_ldexpl(long double x, int n) { return ldexpl( x, n); } long double reference_coshl(long double x) { return coshl(x); } double reference_ceil(double x) { return ceilf((float) x); } long double reference_ceill(long double x) { if( x == 0.0 || reference_isinfl(x) || reference_isnanl(x) ) return x; long double absx = reference_fabsl(x); if( absx >= HEX_LDBL( +, 1, 0, +, 52 ) ) return x; if( absx < 1.0 ) { if( x < 0.0 ) return 0.0; else return 1.0; } long double r = (long double) ((cl_long) x); if( x > 0.0 && r < x ) r += 1.0; return r; } long double reference_acosl(long double x) { long double x2 = x * x; int i; //Prepare a head + tail representation of PI in long double. A good compiler should get rid of all of this work. static const cl_ulong pi_bits[2] = { 0x3243F6A8885A308DULL, 0x313198A2E0370734ULL}; // first 126 bits of pi http://www.super-computing.org/pi-hexa_current.html long double head, tail, temp; #if __LDBL_MANT_DIG__ >= 64 // long double has 64-bits of precision or greater temp = (long double) pi_bits[0] * 0x1.0p64L; head = temp + (long double) pi_bits[1]; temp -= head; // rounding err rounding pi_bits[1] into head tail = (long double) pi_bits[1] + temp; head *= HEX_LDBL( +, 1, 0, -, 125 ); tail *= HEX_LDBL( +, 1, 0, -, 125 ); #else head = (long double) pi_bits[0]; tail = (long double) ((cl_long) pi_bits[0] - (cl_long) head ); // residual part of pi_bits[0] after rounding tail = tail * HEX_LDBL( +, 1, 0, +, 64 ) + (long double) pi_bits[1]; head *= HEX_LDBL( +, 1, 0, -, 61 ); tail *= HEX_LDBL( +, 1, 0, -, 125 ); #endif // oversize values and NaNs go to NaN if( ! (x2 <= 1.0) ) return sqrtl(1.0L - x2 ); // // deal with large |x|: // sqrt( 1 - x**2) // acos(|x| > sqrt(0.5)) = 2 * atan( z ); z = -------------------- ; z in [0, sqrt(0.5)/(1+sqrt(0.5) = .4142135...] // 1 + x if( x2 > 0.5 ) { // we handle the x < 0 case as pi - acos(|x|) long double sign = reference_copysignl( 1.0L, x ); long double fabsx = reference_fabsl( x ); head -= head * sign; // x > 0 ? 0 : pi.hi tail -= tail * sign; // x > 0 ? 0 : pi.low // z = sqrt( 1-x**2 ) / (1+x) = sqrt( (1-x)(1+x) / (1+x)**2 ) = sqrt( (1-x)/(1+x) ) long double z2 = (1.0L - fabsx) / (1.0L + fabsx); // z**2 long double z = sign * sqrtl(z2); // atan(sqrt(q)) // Minimax fit p(x) = ---------------- - 1 // sqrt(q) // // Define q = r*r, and solve for atan(r): // // atan(r) = (p(r) + 1) * r = rp(r) + r static long double atan_coeffs[] = { HEX_LDBL( -, b, 3f52e0c278293b3, -, 67 ), HEX_LDBL( -, a, aaaaaaaaaaa95b8, -, 5 ), HEX_LDBL( +, c, ccccccccc992407, -, 6 ), HEX_LDBL( -, 9, 24924923024398, -, 6 ), HEX_LDBL( +, e, 38e38d6f92c98f3, -, 7 ), HEX_LDBL( -, b, a2e89bfb8393ec6, -, 7 ), HEX_LDBL( +, 9, d89a9f574d412cb, -, 7 ), HEX_LDBL( -, 8, 88580517884c547, -, 7 ), HEX_LDBL( +, f, 0ab6756abdad408, -, 8 ), HEX_LDBL( -, d, 56a5b07a2f15b49, -, 8 ), HEX_LDBL( +, b, 72ab587e46d80b2, -, 8 ), HEX_LDBL( -, 8, 62ea24bb5b2e636, -, 8 ), HEX_LDBL( +, e, d67c16582123937, -, 10 ) }; // minimax fit over [ 0x1.0p-52, 0.18] Max error: 0x1.67ea5c184e5d9p-64 // Calculate y = p(r) const size_t atan_coeff_count = sizeof( atan_coeffs ) / sizeof( atan_coeffs[0] ); long double y = atan_coeffs[ atan_coeff_count - 1]; for( i = (int)atan_coeff_count - 2; i >= 0; i-- ) y = atan_coeffs[i] + y * z2; z *= 2.0L; // fold in 2.0 for 2.0 * atan(z) y *= z; // rp(r) return head + ((y + tail) + z); } // do |x| <= sqrt(0.5) here // acos( sqrt(z) ) - PI/2 // Piecewise minimax polynomial fits for p(z) = 1 + ------------------------; // sqrt(z) // // Define z = x*x, and solve for acos(x) over x in x >= 0: // // acos( sqrt(z) ) = acos(x) = x*(p(z)-1) + PI/2 = xp(x**2) - x + PI/2 // const long double coeffs[4][14] = { { HEX_LDBL( -, a, fa7382e1f347974, -, 10 ), HEX_LDBL( -, b, 4d5a992de1ac4da, -, 6 ), HEX_LDBL( -, a, c526184bd558c17, -, 7 ), HEX_LDBL( -, d, 9ed9b0346ec092a, -, 8 ), HEX_LDBL( -, 9, dca410c1f04b1f, -, 8 ), HEX_LDBL( -, f, 76e411ba9581ee5, -, 9 ), HEX_LDBL( -, c, c71b00479541d8e, -, 9 ), HEX_LDBL( -, a, f527a3f9745c9de, -, 9 ), HEX_LDBL( -, 9, a93060051f48d14, -, 9 ), HEX_LDBL( -, 8, b3d39ad70e06021, -, 9 ), HEX_LDBL( -, f, f2ab95ab84f79c, -, 10 ), HEX_LDBL( -, e, d1af5f5301ccfe4, -, 10 ), HEX_LDBL( -, e, 1b53ba562f0f74a, -, 10 ), HEX_LDBL( -, d, 6a3851330e15526, -, 10 ) }, // x - 0.0625 in [ -0x1.fffffffffp-5, 0x1.0p-4 ] Error: 0x1.97839bf07024p-76 { HEX_LDBL( -, 8, c2f1d638e4c1b48, -, 8 ), HEX_LDBL( -, c, d47ac903c311c2c, -, 6 ), HEX_LDBL( -, d, e020b2dabd5606a, -, 7 ), HEX_LDBL( -, a, 086fafac220f16b, -, 7 ), HEX_LDBL( -, 8, 55b5efaf6b86c3e, -, 7 ), HEX_LDBL( -, f, 05c9774fed2f571, -, 8 ), HEX_LDBL( -, e, 484a93f7f0fc772, -, 8 ), HEX_LDBL( -, e, 1a32baef01626e4, -, 8 ), HEX_LDBL( -, e, 528e525b5c9c73d, -, 8 ), HEX_LDBL( -, e, ddd5d27ad49b2c8, -, 8 ), HEX_LDBL( -, f, b3259e7ae10c6f, -, 8 ), HEX_LDBL( -, 8, 68998170d5b19b7, -, 7 ), HEX_LDBL( -, 9, 4468907f007727, -, 7 ), HEX_LDBL( -, a, 2ad5e4906a8e7b3, -, 7 ) },// x - 0.1875 in [ -0x1.0p-4, 0x1.0p-4 ] Error: 0x1.647af70073457p-73 { HEX_LDBL( -, f, a76585ad399e7ac, -, 8 ), HEX_LDBL( -, e, d665b7dd504ca7c, -, 6 ), HEX_LDBL( -, 9, 4c7c2402bd4bc33, -, 6 ), HEX_LDBL( -, f, ba76b69074ff71c, -, 7 ), HEX_LDBL( -, f, 58117784bdb6d5f, -, 7 ), HEX_LDBL( -, 8, 22ddd8eef53227d, -, 6 ), HEX_LDBL( -, 9, 1d1d3b57a63cdb4, -, 6 ), HEX_LDBL( -, a, 9c4bdc40cca848, -, 6 ), HEX_LDBL( -, c, b673b12794edb24, -, 6 ), HEX_LDBL( -, f, 9290a06e31575bf, -, 6 ), HEX_LDBL( -, 9, b4929c16aeb3d1f, -, 5 ), HEX_LDBL( -, c, 461e725765a7581, -, 5 ), HEX_LDBL( -, 8, 0a59654c98d9207, -, 4 ), HEX_LDBL( -, a, 6de6cbd96c80562, -, 4 ) }, // x - 0.3125 in [ -0x1.0p-4, 0x1.0p-4 ] Error: 0x1.b0246c304ce1ap-70 { HEX_LDBL( -, b, dca8b0359f96342, -, 7 ), HEX_LDBL( -, 8, cd2522fcde9823, -, 5 ), HEX_LDBL( -, d, 2af9397b27ff74d, -, 6 ), HEX_LDBL( -, d, 723f2c2c2409811, -, 6 ), HEX_LDBL( -, f, ea8f8481ecc3cd1, -, 6 ), HEX_LDBL( -, a, 43fd8a7a646b0b2, -, 5 ), HEX_LDBL( -, e, 01b0bf63a4e8d76, -, 5 ), HEX_LDBL( -, 9, f0b7096a2a7b4d, -, 4 ), HEX_LDBL( -, e, 872e7c5a627ab4c, -, 4 ), HEX_LDBL( -, a, dbd760a1882da48, -, 3 ), HEX_LDBL( -, 8, 424e4dea31dd273, -, 2 ), HEX_LDBL( -, c, c05d7730963e793, -, 2 ), HEX_LDBL( -, a, 523d97197cd124a, -, 1 ), HEX_LDBL( -, 8, 307ba943978aaee, +, 0 ) } // x - 0.4375 in [ -0x1.0p-4, 0x1.0p-4 ] Error: 0x1.9ecff73da69c9p-66 }; const long double offsets[4] = { 0.0625, 0.1875, 0.3125, 0.4375 }; const size_t coeff_count = sizeof( coeffs[0] ) / sizeof( coeffs[0][0] ); // reduce the incoming values a bit so that they are in the range [-0x1.0p-4, 0x1.0p-4] const long double *c; i = x2 * 8.0L; c = coeffs[i]; x2 -= offsets[i]; // exact // calcualte p(x2) long double y = c[ coeff_count - 1]; for( i = (int)coeff_count - 2; i >= 0; i-- ) y = c[i] + y * x2; // xp(x2) y *= x; // return xp(x2) - x + PI/2 return head + ((y + tail) - x); } double reference_relaxed_acos(double x) { return reference_acos(x); } double reference_log10(double x) { if( x == 0.0 ) return -INFINITY; if( x < 0.0 ) return cl_make_nan(); if( isinf(x) ) return INFINITY; double log2Hi = HEX_DBL( +, 1, 34413509f79fe, -, 2 ); double logxHi, logxLo; __log2_ep(&logxHi, &logxLo, x); return logxHi*log2Hi; } double reference_relaxed_log10(double x) { return reference_log10(x); } long double reference_log10l(long double x) { if( x == 0.0 ) return -INFINITY; if( x < 0.0 ) return cl_make_nan(); if( isinf(x) ) return INFINITY; double log2Hi = HEX_DBL( +, 1, 34413509f79fe, -, 2 ); double log2Lo = HEX_DBL( +, 1, e623e2566b02d, -, 55 ); double logxHi, logxLo; __log2_ep(&logxHi, &logxLo, x); //double rhi, rlo; //MulDD(&rhi, &rlo, logxHi, logxLo, log2Hi, log2Lo); //return (long double) rhi + (long double) rlo; long double lg2 = (long double) log2Hi + (long double) log2Lo; long double logx = (long double) logxHi + (long double) logxLo; return logx*lg2; } double reference_acos(double x) { return acos( x ); } double reference_atan2(double x, double y) { #if defined(_WIN32) // fix edge cases for Windows if (isinf(x) && isinf(y)) { double retval = (y > 0) ? M_PI_4 : 3.f * M_PI_4; return (x > 0) ? retval : -retval; } #endif // _WIN32 return atan2(x, y); } long double reference_atan2l(long double x, long double y) { #if defined(_WIN32) // fix edge cases for Windows if (isinf(x) && isinf(y)) { long double retval = (y > 0) ? M_PI_4 : 3.f * M_PI_4; return (x > 0) ? retval : -retval; } #endif // _WIN32 return atan2l(x, y); } double reference_frexp(double a, int *exp) { if(isnan(a) || isinf(a) || a == 0.0) { *exp = 0; return a; } union { cl_double d; cl_ulong l; } u; u.d = a; // separate out sign cl_ulong s = u.l & 0x8000000000000000ULL; u.l &= 0x7fffffffffffffffULL; int bias = -1022; if((u.l & 0x7ff0000000000000ULL) == 0) { double d = u.l; u.d = d; bias -= 1074; } int e = (int)((u.l & 0x7ff0000000000000ULL) >> 52); u.l &= 0x000fffffffffffffULL; e += bias; u.l |= ((cl_ulong)1022 << 52); u.l |= s; *exp = e; return u.d; } long double reference_frexpl(long double a, int *exp) { if(isnan(a) || isinf(a) || a == 0.0) { *exp = 0; return a; } if(sizeof(long double) == sizeof(double)) { return reference_frexp(a, exp); } else { return frexpl(a, exp); } } double reference_atan(double x) { return atan( x ); } long double reference_atanl(long double x) { return atanl( x ); } long double reference_asinl(long double x) { return asinl( x ); } double reference_asin(double x) { return asin( x ); } double reference_relaxed_asin(double x) { return reference_asin(x); } double reference_fabs(double x) { return fabs( x); } double reference_cosh(double x) { return cosh( x ); } long double reference_sqrtl(long double x) { #if defined( __SSE2__ ) || (defined( _MSC_VER ) && (defined(_M_IX86) || defined(_M_X64))) __m128d result128 = _mm_set_sd((double) x); result128 = _mm_sqrt_sd(result128, result128); return _mm_cvtsd_f64(result128); #else volatile double dx = x; return sqrt( dx ); #endif } long double reference_tanhl(long double x) { return tanhl( x ); } long double reference_floorl(long double x) { if( x == 0.0 || reference_isinfl(x) || reference_isnanl(x) ) return x; long double absx = reference_fabsl(x); if( absx >= HEX_LDBL( +, 1, 0, +, 52 ) ) return x; if( absx < 1.0 ) { if( x < 0.0 ) return -1.0; else return 0.0; } long double r = (long double) ((cl_long) x); if( x < 0.0 && r > x ) r -= 1.0; return r; } double reference_tanh(double x) { return tanh( x ); } long double reference_assignmentl( long double x ){ return x; } int reference_notl( long double x ) { int r = !x; return r; }