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libm_inlines_amd.h
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/* ============================================================
Copyright (c) 2002-2015 Advanced Micro Devices, Inc.
All rights reserved.
Redistribution and use in source and binary forms, with or
without modification, are permitted provided that the
following conditions are met:
+ Redistributions of source code must retain the above
copyright notice, this list of conditions and the
following disclaimer.
+ Redistributions in binary form must reproduce the above
copyright notice, this list of conditions and the
following disclaimer in the documentation and/or other
materials provided with the distribution.
+ Neither the name of Advanced Micro Devices, Inc. nor the
names of its contributors may be used to endorse or
promote products derived from this software without
specific prior written permission.
THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND
CONTRIBUTORS "AS IS" AND ANY EXPRESS OR IMPLIED WARRANTIES,
INCLUDING, BUT NOT LIMITED TO, THE IMPLIED WARRANTIES OF
MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE ARE
DISCLAIMED. IN NO EVENT SHALL ADVANCED MICRO DEVICES,
INC. OR CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT,
INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES
(INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE
GOODS OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR
BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY OF
LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT
(INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT
OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE
POSSIBILITY OF SUCH DAMAGE.
It is licensee's responsibility to comply with any export
regulations applicable in licensee's jurisdiction.
============================================================ */
#ifndef LIBM_INLINES_AMD_H_INCLUDED
#define LIBM_INLINES_AMD_H_INCLUDED 1
#include "libm_util_amd.h"
/* Set defines for inline functions calling other inlines */
#if defined(USE_VAL_WITH_FLAGS) || defined(USE_VALF_WITH_FLAGS) || defined(USE_ZERO_WITH_FLAGS) || defined(USE_ZEROF_WITH_FLAGS) || defined(USE_NAN_WITH_FLAGS) || defined(USE_NANF_WITH_FLAGS) || defined(USE_INDEFINITE_WITH_FLAGS) || defined(USE_INDEFINITEF_WITH_FLAGS) || defined(USE_INFINITY_WITH_FLAGS) || defined(USE_INFINITYF_WITH_FLAGS) || defined(USE_SQRT_AMD_INLINE) || defined(USE_SQRTF_AMD_INLINE)
#undef USE_RAISE_FPSW_FLAGS
#define USE_RAISE_FPSW_FLAGS 1
#endif
#if defined(USE_SPLITDOUBLE)
/* Splits double x into exponent e and mantissa m, where 0.5 <= abs(m) < 1.0.
Assumes that x is not zero, denormal, infinity or NaN, but these conditions
are not checked */
static inline void
splitDouble(double x, int *e, double *m)
{
__UINT8_T ux, uy;
GET_BITS_DP64(x, ux);
uy = ux;
ux &= EXPBITS_DP64;
ux >>= EXPSHIFTBITS_DP64;
*e = (int)ux - EXPBIAS_DP64 + 1;
uy = (uy & (SIGNBIT_DP64 | MANTBITS_DP64)) | HALFEXPBITS_DP64;
PUT_BITS_DP64(uy, x);
*m = x;
}
#endif /* USE_SPLITDOUBLE */
#if defined(USE_SPLITDOUBLE_2)
/* Splits double x into exponent e and mantissa m, where 1.0 <= abs(m) < 4.0.
Assumes that x is not zero, denormal, infinity or NaN, but these conditions
are not checked. Also assumes EXPBIAS_DP is odd. With this
assumption, e will be even on exit. */
static inline void
splitDouble_2(double x, int *e, double *m)
{
__UINT8_T ux, vx;
GET_BITS_DP64(x, ux);
vx = ux;
ux &= EXPBITS_DP64;
ux >>= EXPSHIFTBITS_DP64;
if (ux & 1) {
/* The exponent is odd */
vx = (vx & (SIGNBIT_DP64 | MANTBITS_DP64)) | ONEEXPBITS_DP64;
PUT_BITS_DP64(vx, x);
*m = x;
*e = ux - EXPBIAS_DP64;
} else {
/* The exponent is even */
vx = (vx & (SIGNBIT_DP64 | MANTBITS_DP64)) | TWOEXPBITS_DP64;
PUT_BITS_DP64(vx, x);
*m = x;
*e = ux - EXPBIAS_DP64 - 1;
}
}
#endif /* USE_SPLITDOUBLE_2 */
#if defined(USE_SPLITFLOAT)
/* Splits float x into exponent e and mantissa m, where 0.5 <= abs(m) < 1.0.
Assumes that x is not zero, denormal, infinity or NaN, but these conditions
are not checked */
static inline void
splitFloat(float x, int *e, float *m)
{
unsigned int ux, uy;
GET_BITS_SP32(x, ux);
uy = ux;
ux &= EXPBITS_SP32;
ux >>= EXPSHIFTBITS_SP32;
*e = (int)ux - EXPBIAS_SP32 + 1;
uy = (uy & (SIGNBIT_SP32 | MANTBITS_SP32)) | HALFEXPBITS_SP32;
PUT_BITS_SP32(uy, x);
*m = x;
}
#endif /* USE_SPLITFLOAT */
#if defined(USE_SCALEDOUBLE_1)
/* Scales the double x by 2.0**n.
Assumes EMIN <= n <= EMAX, though this condition is not checked. */
static inline double
scaleDouble_1(double x, int n)
{
double t;
/* Construct the number t = 2.0**n */
PUT_BITS_DP64(((__INT8_T)n + EXPBIAS_DP64) << EXPSHIFTBITS_DP64, t);
return x * t;
}
#endif /* USE_SCALEDOUBLE_1 */
#if defined(USE_SCALEDOUBLE_2)
/* Scales the double x by 2.0**n.
Assumes 2*EMIN <= n <= 2*EMAX, though this condition is not checked. */
static inline double
scaleDouble_2(double x, int n)
{
double t1, t2;
int n1, n2;
n1 = n / 2;
n2 = n - n1;
/* Construct the numbers t1 = 2.0**n1 and t2 = 2.0**n2 */
PUT_BITS_DP64(((__INT8_T)n1 + EXPBIAS_DP64) << EXPSHIFTBITS_DP64, t1);
PUT_BITS_DP64(((__INT8_T)n2 + EXPBIAS_DP64) << EXPSHIFTBITS_DP64, t2);
return (x * t1) * t2;
}
#endif /* USE_SCALEDOUBLE_2 */
#if defined(USE_SCALEDOUBLE_3)
/* Scales the double x by 2.0**n.
Assumes 3*EMIN <= n <= 3*EMAX, though this condition is not checked. */
static inline double
scaleDouble_3(double x, int n)
{
double t1, t2, t3;
int n1, n2, n3;
n1 = n / 3;
n2 = (n - n1) / 2;
n3 = n - n1 - n2;
/* Construct the numbers t1 = 2.0**n1, t2 = 2.0**n2 and t3 = 2.0**n3 */
PUT_BITS_DP64(((__INT8_T)n1 + EXPBIAS_DP64) << EXPSHIFTBITS_DP64, t1);
PUT_BITS_DP64(((__INT8_T)n2 + EXPBIAS_DP64) << EXPSHIFTBITS_DP64, t2);
PUT_BITS_DP64(((__INT8_T)n3 + EXPBIAS_DP64) << EXPSHIFTBITS_DP64, t3);
return ((x * t1) * t2) * t3;
}
#endif /* USE_SCALEDOUBLE_3 */
#if defined(USE_SCALEFLOAT_1)
/* Scales the float x by 2.0**n.
Assumes EMIN <= n <= EMAX, though this condition is not checked. */
static inline float
scaleFloat_1(float x, int n)
{
float t;
/* Construct the number t = 2.0**n */
PUT_BITS_SP32((n + EXPBIAS_SP32) << EXPSHIFTBITS_SP32, t);
return x * t;
}
#endif /* USE_SCALEFLOAT_1 */
#if defined(USE_SCALEFLOAT_2)
/* Scales the float x by 2.0**n.
Assumes 2*EMIN <= n <= 2*EMAX, though this condition is not checked. */
static inline float
scaleFloat_2(float x, int n)
{
float t1, t2;
int n1, n2;
n1 = n / 2;
n2 = n - n1;
/* Construct the numbers t1 = 2.0**n1 and t2 = 2.0**n2 */
PUT_BITS_SP32((n1 + EXPBIAS_SP32) << EXPSHIFTBITS_SP32, t1);
PUT_BITS_SP32((n2 + EXPBIAS_SP32) << EXPSHIFTBITS_SP32, t2);
return (x * t1) * t2;
}
#endif /* USE_SCALEFLOAT_2 */
#if defined(USE_SCALEFLOAT_3)
/* Scales the float x by 2.0**n.
Assumes 3*EMIN <= n <= 3*EMAX, though this condition is not checked. */
static inline float
scaleFloat_3(float x, int n)
{
float t1, t2, t3;
int n1, n2, n3;
n1 = n / 3;
n2 = (n - n1) / 2;
n3 = n - n1 - n2;
/* Construct the numbers t1 = 2.0**n1, t2 = 2.0**n2 and t3 = 2.0**n3 */
PUT_BITS_SP32((n1 + EXPBIAS_SP32) << EXPSHIFTBITS_SP32, t1);
PUT_BITS_SP32((n2 + EXPBIAS_SP32) << EXPSHIFTBITS_SP32, t2);
PUT_BITS_SP32((n3 + EXPBIAS_SP32) << EXPSHIFTBITS_SP32, t3);
return ((x * t1) * t2) * t3;
}
#endif /* USE_SCALEFLOAT_3 */
#if defined(USE_SETPRECISIONDOUBLE)
unsigned int
setPrecisionDouble(void)
{
unsigned int cw, cwold = 0;
/* There is no precision control on Hammer */
return cwold;
}
#endif /* USE_SETPRECISIONDOUBLE */
#if defined(USE_RESTOREPRECISION)
void
restorePrecision(unsigned int cwold)
{
#if defined(linux)
/* There is no precision control on Hammer */
#elif defined(INTERIX86)
#elif defined(TARGET_OSX_X86)
#else
#error Unknown machine
#endif
return;
}
#endif /* USE_RESTOREPRECISION */
#if defined(USE_CLEAR_FPSW_FLAGS)
/* Clears floating-point status flags. The argument should be
the bitwise or of the flags to be cleared, from the
list above, e.g.
clear_fpsw_flags(AMD_F_INEXACT | AMD_F_INVALID);
*/
static inline void
clear_fpsw_flags(int flags)
{
#if defined(DONOTDEFINE_WINDOWS)
unsigned int cw = _mm_getcsr();
cw &= (~flags);
_mm_setcsr(cw);
#elif defined(linux)
unsigned int cw;
/* Get the current floating-point control/status word */
asm volatile("STMXCSR %0" : "=m"(cw));
cw &= (~flags);
asm volatile("LDMXCSR %0" : : "m"(cw));
#elif defined(INTERIX86) || defined(TARGET_WIN)
unsigned int cw;
/* Get the current floating-point control/status word */
asm volatile("STMXCSR %0" : "=m"(cw));
cw &= (~flags);
asm volatile("LDMXCSR %0" : : "m"(cw));
#else
#error Unknown machine
#endif
}
#endif /* USE_CLEAR_FPSW_FLAGS */
#if defined(USE_RAISE_FPSW_FLAGS)
/* Raises floating-point status flags. The argument should be
the bitwise or of the flags to be raised, from the
list above, e.g.
raise_fpsw_flags(AMD_F_INEXACT | AMD_F_INVALID);
*/
static inline void
raise_fpsw_flags(int flags)
{
#if defined(DONOTDEFINE_WINDOWS)
_mm_setcsr(_mm_getcsr() | flags);
#elif defined(linux)
unsigned int cw;
/* Get the current floating-point control/status word */
asm volatile("STMXCSR %0" : "=m"(cw));
cw |= flags;
asm volatile("LDMXCSR %0" : : "m"(cw));
#elif defined(INTERIX86) || defined(__INTERIX) || defined(TARGET_OSX_X86) || defined(TARGET_WIN)
unsigned int cw;
/* Get the current floating-point control/status word */
asm volatile("STMXCSR %0" : "=m"(cw));
cw |= flags;
asm volatile("LDMXCSR %0" : : "m"(cw));
#else
#error Unknown machine
#endif
}
#endif /* USE_RAISE_FPSW_FLAGS */
#if defined(USE_GET_FPSW_INLINE)
/* Return the current floating-point status word */
static inline unsigned int
get_fpsw_inline(void)
{
#if defined(TARGET_WIN)
return _mm_getcsr();
#elif defined(linux)
unsigned int sw;
asm volatile("STMXCSR %0" : "=m"(sw));
return sw;
#elif defined(INTERIX86)
unsigned int sw;
asm volatile("STMXCSR %0" : "=m"(sw));
return sw;
#else
#error Unknown machine
#endif
}
#endif /* USE_GET_FPSW_INLINE */
#if defined(USE_SET_FPSW_INLINE)
/* Set the floating-point status word */
static inline void
set_fpsw_inline(unsigned int sw)
{
#if defined(TARGET_WIN)
_mm_setcsr(sw);
#elif defined(linux)
/* Set the current floating-point control/status word */
asm volatile("LDMXCSR %0" : : "m"(sw));
#elif defined(INTERIX86)
/* Set the current floating-point control/status word */
asm volatile("LDMXCSR %0" : : "m"(sw));
#else
#error Unknown machine
#endif
}
#endif /* USE_SET_FPSW_INLINE */
#if defined(USE_CLEAR_FPSW_INLINE)
/* Clear all exceptions from the floating-point status word */
static inline void
clear_fpsw_inline(void)
{
#if defined(TARGET_WIN)
unsigned int cw;
cw = _mm_getcsr();
cw &= ~(AMD_F_INEXACT | AMD_F_UNDERFLOW | AMD_F_OVERFLOW | AMD_F_DIVBYZERO |
AMD_F_INVALID);
_mm_setcsr(cw);
#elif defined(linux)
unsigned int cw;
/* Get the current floating-point control/status word */
asm volatile("STMXCSR %0" : "=m"(cw));
cw &= ~(AMD_F_INEXACT | AMD_F_UNDERFLOW | AMD_F_OVERFLOW | AMD_F_DIVBYZERO |
AMD_F_INVALID);
asm volatile("LDMXCSR %0" : : "m"(cw));
#elif defined(INTERIX86)
unsigned int cw;
/* Get the current floating-point control/status word */
asm volatile("STMXCSR %0" : "=m"(cw));
cw &= ~(AMD_F_INEXACT | AMD_F_UNDERFLOW | AMD_F_OVERFLOW | AMD_F_DIVBYZERO |
AMD_F_INVALID);
asm volatile("LDMXCSR %0" : : "m"(cw));
#else
#error Unknown machine
#endif
}
#endif /* USE_CLEAR_FPSW_INLINE */
#if defined(USE_VAL_WITH_FLAGS)
/* Returns a double value after raising the given flags,
e.g. val_with_flags(AMD_F_INEXACT);
*/
static inline double
val_with_flags(double val, int flags)
{
raise_fpsw_flags(flags);
return val;
}
#endif /* USE_VAL_WITH_FLAGS */
#if defined(USE_VALF_WITH_FLAGS)
/* Returns a float value after raising the given flags,
e.g. valf_with_flags(AMD_F_INEXACT);
*/
static inline float
valf_with_flags(float val, int flags)
{
raise_fpsw_flags(flags);
return val;
}
#endif /* USE_VALF_WITH_FLAGS */
#if defined(USE_ZERO_WITH_FLAGS)
/* Returns a double +zero after raising the given flags,
e.g. zero_with_flags(AMD_F_INEXACT | AMD_F_INVALID);
*/
static inline double
zero_with_flags(int flags)
{
raise_fpsw_flags(flags);
return 0.0;
}
#endif /* USE_ZERO_WITH_FLAGS */
#if defined(USE_ZEROF_WITH_FLAGS)
/* Returns a float +zero after raising the given flags,
e.g. zerof_with_flags(AMD_F_INEXACT | AMD_F_INVALID);
*/
static inline float
zerof_with_flags(int flags)
{
raise_fpsw_flags(flags);
return 0.0F;
}
#endif /* USE_ZEROF_WITH_FLAGS */
#if defined(USE_NAN_WITH_FLAGS)
/* Returns a double quiet +nan after raising the given flags,
e.g. nan_with_flags(AMD_F_INVALID);
*/
static inline double
nan_with_flags(int flags)
{
double z;
raise_fpsw_flags(flags);
PUT_BITS_DP64(0x7ff8000000000000, z);
return z;
}
#endif /* USE_NAN_WITH_FLAGS */
#if defined(USE_NANF_WITH_FLAGS)
/* Returns a float quiet +nan after raising the given flags,
e.g. nanf_with_flags(AMD_F_INVALID);
*/
static inline float
nanf_with_flags(int flags)
{
float z;
raise_fpsw_flags(flags);
PUT_BITS_SP32(0x7fc00000, z);
return z;
}
#endif /* USE_NANF_WITH_FLAGS */
#if defined(USE_INDEFINITE_WITH_FLAGS)
/* Returns a double indefinite after raising the given flags,
e.g. indefinite_with_flags(AMD_F_INVALID);
*/
static inline double
indefinite_with_flags(int flags)
{
double z;
raise_fpsw_flags(flags);
PUT_BITS_DP64(0xfff8000000000000, z);
return z;
}
#endif /* USE_INDEFINITE_WITH_FLAGS */
#if defined(USE_INDEFINITEF_WITH_FLAGS)
/* Returns a float quiet +indefinite after raising the given flags,
e.g. indefinitef_with_flags(AMD_F_INVALID);
*/
static inline float
indefinitef_with_flags(int flags)
{
float z;
raise_fpsw_flags(flags);
PUT_BITS_SP32(0xffc00000, z);
return z;
}
#endif /* USE_INDEFINITEF_WITH_FLAGS */
#ifdef USE_INFINITY_WITH_FLAGS
/* Returns a positive double infinity after raising the given flags,
e.g. infinity_with_flags(AMD_F_OVERFLOW);
*/
static inline double
infinity_with_flags(int flags)
{
double z;
raise_fpsw_flags(flags);
PUT_BITS_DP64((__UINT8_T)(BIASEDEMAX_DP64 + 1) << EXPSHIFTBITS_DP64, z);
return z;
}
#endif /* USE_INFINITY_WITH_FLAGS */
#ifdef USE_INFINITYF_WITH_FLAGS
/* Returns a positive float infinity after raising the given flags,
e.g. infinityf_with_flags(AMD_F_OVERFLOW);
*/
static inline float
infinityf_with_flags(int flags)
{
float z;
raise_fpsw_flags(flags);
PUT_BITS_SP32((BIASEDEMAX_SP32 + 1) << EXPSHIFTBITS_SP32, z);
return z;
}
#endif /* USE_INFINITYF_WITH_FLAGS */
#if defined(USE_SPLITEXP)
/* Compute the values m, z1, and z2 such that base**x = 2**m * (z1 + z2).
Small arguments abs(x) < 1/(16*ln(base)) and extreme arguments
abs(x) > large/(ln(base)) (where large is the largest representable
floating point number) should be handled separately instead of calling
this function. This function is called by exp_amd, exp2_amd, exp10_amd,
cosh_amd and sinh_amd. */
static inline void
splitexp(double x, double logbase, double thirtytwo_by_logbaseof2,
double logbaseof2_by_32_lead, double logbaseof2_by_32_trail, int *m,
double *z1, double *z2)
{
double q, r, r1, r2, f1, f2;
int n, j;
/* Arrays two_to_jby32_lead_table and two_to_jby32_trail_table contain
leading and trailing parts respectively of precomputed
values of pow(2.0,j/32.0), for j = 0, 1, ..., 31.
two_to_jby32_lead_table contains the first 25 bits of precision,
and two_to_jby32_trail_table contains a further 53 bits precision. */
static const double two_to_jby32_lead_table[32] = {
1.00000000000000000000e+00, /* 0x3ff0000000000000 */
1.02189713716506958008e+00, /* 0x3ff059b0d0000000 */
1.04427373409271240234e+00, /* 0x3ff0b55860000000 */
1.06714040040969848633e+00, /* 0x3ff11301d0000000 */
1.09050768613815307617e+00, /* 0x3ff172b830000000 */
1.11438673734664916992e+00, /* 0x3ff1d48730000000 */
1.13878858089447021484e+00, /* 0x3ff2387a60000000 */
1.16372483968734741211e+00, /* 0x3ff29e9df0000000 */
1.18920707702636718750e+00, /* 0x3ff306fe00000000 */
1.21524733304977416992e+00, /* 0x3ff371a730000000 */
1.24185776710510253906e+00, /* 0x3ff3dea640000000 */
1.26905095577239990234e+00, /* 0x3ff44e0860000000 */
1.29683953523635864258e+00, /* 0x3ff4bfdad0000000 */
1.32523661851882934570e+00, /* 0x3ff5342b50000000 */
1.35425549745559692383e+00, /* 0x3ff5ab07d0000000 */
1.38390988111495971680e+00, /* 0x3ff6247eb0000000 */
1.41421353816986083984e+00, /* 0x3ff6a09e60000000 */
1.44518077373504638672e+00, /* 0x3ff71f75e0000000 */
1.47682613134384155273e+00, /* 0x3ff7a11470000000 */
1.50916439294815063477e+00, /* 0x3ff8258990000000 */
1.54221081733703613281e+00, /* 0x3ff8ace540000000 */
1.57598084211349487305e+00, /* 0x3ff93737b0000000 */
1.61049032211303710938e+00, /* 0x3ff9c49180000000 */
1.64575546979904174805e+00, /* 0x3ffa5503b0000000 */
1.68179279565811157227e+00, /* 0x3ffae89f90000000 */
1.71861928701400756836e+00, /* 0x3ffb7f76f0000000 */
1.75625211000442504883e+00, /* 0x3ffc199bd0000000 */
1.79470902681350708008e+00, /* 0x3ffcb720d0000000 */
1.83400803804397583008e+00, /* 0x3ffd5818d0000000 */
1.87416762113571166992e+00, /* 0x3ffdfc9730000000 */
1.91520655155181884766e+00, /* 0x3ffea4afa0000000 */
1.95714408159255981445e+00}; /* 0x3fff507650000000 */
static const double two_to_jby32_trail_table[32] = {
0.00000000000000000000e+00, /* 0x0000000000000000 */
1.14890470981563546737e-08, /* 0x3e48ac2ba1d73e2a */
4.83347014379782142328e-08, /* 0x3e69f3121ec53172 */
2.67125131841396124714e-10, /* 0x3df25b50a4ebbf1b */
4.65271045830351350190e-08, /* 0x3e68faa2f5b9bef9 */
5.24924336638693782574e-09, /* 0x3e368b9aa7805b80 */
5.38622214388600821910e-08, /* 0x3e6ceac470cd83f6 */
1.90902301017041969782e-08, /* 0x3e547f7b84b09745 */
3.79763538792174980894e-08, /* 0x3e64636e2a5bd1ab */
2.69306947081946450986e-08, /* 0x3e5ceaa72a9c5154 */
4.49683815095311756138e-08, /* 0x3e682468446b6824 */
1.41933332021066904914e-09, /* 0x3e18624b40c4dbd0 */
1.94146510233556266402e-08, /* 0x3e54d8a89c750e5e */
2.46409119489264118569e-08, /* 0x3e5a753e077c2a0f */
4.94812958044698886494e-08, /* 0x3e6a90a852b19260 */
8.48872238075784476136e-10, /* 0x3e0d2ac258f87d03 */
2.42032342089579394887e-08, /* 0x3e59fcef32422cbf */
3.32420002333182569170e-08, /* 0x3e61d8bee7ba46e2 */
1.45956577586525322754e-08, /* 0x3e4f580c36bea881 */
3.46452721050003920866e-08, /* 0x3e62999c25159f11 */
8.07090469079979051284e-09, /* 0x3e415506dadd3e2a */
2.99439161340839520436e-09, /* 0x3e29b8bc9e8a0388 */
9.83621719880452147153e-09, /* 0x3e451f8480e3e236 */
8.35492309647188080486e-09, /* 0x3e41f12ae45a1224 */
3.48493175137966283582e-08, /* 0x3e62b5a75abd0e6a */
1.11084703472699692902e-08, /* 0x3e47daf237553d84 */
5.03688744342840346564e-08, /* 0x3e6b0aa538444196 */
4.81896001063495806249e-08, /* 0x3e69df20d22a0798 */
4.83653666334089557746e-08, /* 0x3e69f7490e4bb40b */
1.29745882314081237628e-08, /* 0x3e4bdcdaf5cb4656 */
9.84532844621636118964e-09, /* 0x3e452486cc2c7b9d */
4.25828404545651943883e-08}; /* 0x3e66dc8a80ce9f09 */
/*
Step 1. Reduce the argument.
To perform argument reduction, we find the integer n such that
x = n * logbaseof2/32 + remainder, |remainder| <= logbaseof2/64.
n is defined by round-to-nearest-integer( x*32/logbaseof2 ) and
remainder by x - n*logbaseof2/32. The calculation of n is
straightforward whereas the computation of x - n*logbaseof2/32
must be carried out carefully.
logbaseof2/32 is so represented in two pieces that
(1) logbaseof2/32 is known to extra precision, (2) the product
of n and the leading piece is a model number and is hence
calculated without error, and (3) the subtraction of the value
obtained in (2) from x is a model number and is hence again
obtained without error.
*/
r = x * thirtytwo_by_logbaseof2;
/* Set n = nearest integer to r */
/* This is faster on Hammer */
if (r > 0)
n = (int)(r + 0.5);
else
n = (int)(r - 0.5);
r1 = x - n * logbaseof2_by_32_lead;
r2 = -n * logbaseof2_by_32_trail;
/* Set j = n mod 32: 5 mod 32 = 5, -5 mod 32 = 27, etc. */
/* j = n % 32;
if (j < 0) j += 32; */
j = n & 0x0000001f;
f1 = two_to_jby32_lead_table[j];
f2 = two_to_jby32_trail_table[j];
*m = (n - j) / 32;
/* Step 2. The following is the core approximation. We approximate
exp(r1+r2)-1 by a polynomial. */
r1 *= logbase;
r2 *= logbase;
r = r1 + r2;
q = r1 + (r2 +
r * r * (5.00000000000000008883e-01 +
r * (1.66666666665260878863e-01 +
r * (4.16666666662260795726e-02 +
r * (8.33336798434219616221e-03 +
r * (1.38889490863777199667e-03))))));
/* Step 3. Function value reconstruction.
We now reconstruct the exponential of the input argument
so that exp(x) = 2**m * (z1 + z2).
The order of the computation below must be strictly observed. */
*z1 = f1;
*z2 = f2 + ((f1 + f2) * q);
}
#endif /* USE_SPLITEXP */
#if defined(USE_SPLITEXPF)
/* Compute the values m, z1, and z2 such that base**x = 2**m * (z1 + z2).
Small arguments abs(x) < 1/(16*ln(base)) and extreme arguments
abs(x) > large/(ln(base)) (where large is the largest representable
floating point number) should be handled separately instead of calling
this function. This function is called by exp_amd, exp2_amd, exp10_amd,
cosh_amd and sinh_amd. */
static inline void
splitexpf(float x, float logbase, float thirtytwo_by_logbaseof2,
float logbaseof2_by_32_lead, float logbaseof2_by_32_trail, int *m,
float *z1, float *z2)
{
float q, r, r1, r2, f1, f2;
int n, j;
/* Arrays two_to_jby32_lead_table and two_to_jby32_trail_table contain
leading and trailing parts respectively of precomputed
values of pow(2.0,j/32.0), for j = 0, 1, ..., 31.
two_to_jby32_lead_table contains the first 10 bits of precision,
and two_to_jby32_trail_table contains a further 24 bits precision. */
static const float two_to_jby32_lead_table[32] = {
1.0000000000E+00F, /* 0x3F800000 */
1.0214843750E+00F, /* 0x3F82C000 */
1.0429687500E+00F, /* 0x3F858000 */
1.0664062500E+00F, /* 0x3F888000 */
1.0898437500E+00F, /* 0x3F8B8000 */
1.1132812500E+00F, /* 0x3F8E8000 */
1.1386718750E+00F, /* 0x3F91C000 */
1.1621093750E+00F, /* 0x3F94C000 */
1.1875000000E+00F, /* 0x3F980000 */
1.2148437500E+00F, /* 0x3F9B8000 */
1.2402343750E+00F, /* 0x3F9EC000 */
1.2675781250E+00F, /* 0x3FA24000 */
1.2949218750E+00F, /* 0x3FA5C000 */
1.3242187500E+00F, /* 0x3FA98000 */
1.3535156250E+00F, /* 0x3FAD4000 */
1.3828125000E+00F, /* 0x3FB10000 */
1.4140625000E+00F, /* 0x3FB50000 */
1.4433593750E+00F, /* 0x3FB8C000 */
1.4765625000E+00F, /* 0x3FBD0000 */
1.5078125000E+00F, /* 0x3FC10000 */
1.5410156250E+00F, /* 0x3FC54000 */
1.5742187500E+00F, /* 0x3FC98000 */
1.6093750000E+00F, /* 0x3FCE0000 */
1.6445312500E+00F, /* 0x3FD28000 */
1.6816406250E+00F, /* 0x3FD74000 */
1.7167968750E+00F, /* 0x3FDBC000 */
1.7558593750E+00F, /* 0x3FE0C000 */
1.7929687500E+00F, /* 0x3FE58000 */
1.8339843750E+00F, /* 0x3FEAC000 */
1.8730468750E+00F, /* 0x3FEFC000 */
1.9140625000E+00F, /* 0x3FF50000 */
1.9570312500E+00F}; /* 0x3FFA8000 */
static const float two_to_jby32_trail_table[32] = {
0.0000000000E+00F, /* 0x00000000 */
4.1277357377E-04F, /* 0x39D86988 */
1.3050324051E-03F, /* 0x3AAB0D9F */
7.3415064253E-04F, /* 0x3A407404 */
6.6398258787E-04F, /* 0x3A2E0F1E */
1.1054925853E-03F, /* 0x3A90E62D */
1.1675967835E-04F, /* 0x38F4DCE0 */
1.6154836630E-03F, /* 0x3AD3BEA3 */
1.7071149778E-03F, /* 0x3ADFC146 */
4.0360994171E-04F, /* 0x39D39B9C */
1.6234370414E-03F, /* 0x3AD4C982 */
1.4728321694E-03F, /* 0x3AC10C0C */
1.9176795613E-03F, /* 0x3AFB5AA6 */
1.0178930825E-03F, /* 0x3A856AD3 */
7.3992193211E-04F, /* 0x3A41F752 */
1.0973819299E-03F, /* 0x3A8FD607 */
1.5106226783E-04F, /* 0x391E6678 */
1.8214319134E-03F, /* 0x3AEEBD1D */
2.6364589576E-04F, /* 0x398A39F4 */
1.3519275235E-03F, /* 0x3AB13329 */
1.1952003697E-03F, /* 0x3A9CA845 */
1.7620950239E-03F, /* 0x3AE6F619 */
1.1153318919E-03F, /* 0x3A923054 */
1.2242280645E-03F, /* 0x3AA07647 */
1.5220546629E-04F, /* 0x391F9958 */
1.8224230735E-03F, /* 0x3AEEDE5F */
3.9278529584E-04F, /* 0x39CDEEC0 */
1.7403248930E-03F, /* 0x3AE41B9D */
2.3711356334E-05F, /* 0x37C6E7C0 */
1.1207590578E-03F, /* 0x3A92E66F */
1.1440613307E-03F, /* 0x3A95F454 */
1.1287408415E-04F}; /* 0x38ECB6D0 */
/*
Step 1. Reduce the argument.
To perform argument reduction, we find the integer n such that
x = n * logbaseof2/32 + remainder, |remainder| <= logbaseof2/64.
n is defined by round-to-nearest-integer( x*32/logbaseof2 ) and
remainder by x - n*logbaseof2/32. The calculation of n is
straightforward whereas the computation of x - n*logbaseof2/32
must be carried out carefully.
logbaseof2/32 is so represented in two pieces that
(1) logbaseof2/32 is known to extra precision, (2) the product
of n and the leading piece is a model number and is hence
calculated without error, and (3) the subtraction of the value
obtained in (2) from x is a model number and is hence again
obtained without error.
*/
r = x * thirtytwo_by_logbaseof2;
/* Set n = nearest integer to r */
/* This is faster on Hammer */
if (r > 0)
n = (int)(r + 0.5F);
else
n = (int)(r - 0.5F);
r1 = x - n * logbaseof2_by_32_lead;
r2 = -n * logbaseof2_by_32_trail;
/* Set j = n mod 32: 5 mod 32 = 5, -5 mod 32 = 27, etc. */
/* j = n % 32;
if (j < 0) j += 32; */
j = n & 0x0000001f;
f1 = two_to_jby32_lead_table[j];
f2 = two_to_jby32_trail_table[j];
*m = (n - j) / 32;
/* Step 2. The following is the core approximation. We approximate
exp(r1+r2)-1 by a polynomial. */
r1 *= logbase;
r2 *= logbase;
r = r1 + r2;
q = r1 + (r2 +
r * r * (5.00000000000000008883e-01F +
r * (1.66666666665260878863e-01F)));
/* Step 3. Function value reconstruction.
We now reconstruct the exponential of the input argument
so that exp(x) = 2**m * (z1 + z2).
The order of the computation below must be strictly observed. */
*z1 = f1;
*z2 = f2 + ((f1 + f2) * q);
}
#endif /* SPLITEXPF */
#if defined(USE_SCALEUPDOUBLE1024)
/* Scales up a double (normal or denormal) whose bit pattern is given
as ux by 2**1024. There are no checks that the input number is
scalable by that amount. */
static inline void
scaleUpDouble1024(__UINT8_T ux, __UINT8_T *ur)
{
__UINT8_T uy;
double y;
if ((ux & EXPBITS_DP64) == 0) {
/* ux is denormalised */
PUT_BITS_DP64(ux | 0x4010000000000000, y);
if (ux & SIGNBIT_DP64)
y += 4.0;
else
y -= 4.0;
GET_BITS_DP64(y, uy);
} else
/* ux is normal */
uy = ux + 0x4000000000000000;
*ur = uy;
return;
}
#endif /* SCALEUPDOUBLE1024 */
#if defined(USE_SCALEDOWNDOUBLE)
/* Scales down a double whose bit pattern is given as ux by 2**k.
There are no checks that the input number is scalable by that amount. */
static inline void
scaleDownDouble(__UINT8_T ux, int k, __UINT8_T *ur)
{
__UINT8_T uy, uk, ax, xsign;
int n, shift;
xsign = ux & SIGNBIT_DP64;
ax = ux & ~SIGNBIT_DP64;
n = (int)((ax & EXPBITS_DP64) >> EXPSHIFTBITS_DP64) - k;
if (n > 0) {
uk = (__UINT8_T)n << EXPSHIFTBITS_DP64;
uy = (ax & ~EXPBITS_DP64) | uk;
} else {
uy = (ax & ~EXPBITS_DP64) | 0x0010000000000000;
shift = (1 - n);
if (shift > MANTLENGTH_DP64 + 1)
/* Sigh. Shifting works mod 64 so be careful not to shift too much */
uy = 0;
else {
/* Make sure we round the result */
uy >>= shift - 1;
uy = (uy >> 1) + (uy & 1);
}
}
*ur = uy | xsign;
}
#endif /* SCALEDOWNDOUBLE */
#if defined(USE_SCALEUPFLOAT128)
/* Scales up a float (normal or denormal) whose bit pattern is given
as ux by 2**128. There are no checks that the input number is
scalable by that amount. */
static inline void
scaleUpFloat128(unsigned int ux, unsigned int *ur)
{
unsigned int uy;
float y;
if ((ux & EXPBITS_SP32) == 0) {
/* ux is denormalised */
PUT_BITS_SP32(ux | 0x40800000, y);
/* Compensate for the implicit bit just added */
if (ux & SIGNBIT_SP32)
y += 4.0F;
else
y -= 4.0F;
GET_BITS_SP32(y, uy);
} else
/* ux is normal */
uy = ux + 0x40000000;
*ur = uy;
}
#endif /* SCALEUPFLOAT128 */
#if defined(USE_SCALEDOWNFLOAT)
/* Scales down a float whose bit pattern is given as ux by 2**k.
There are no checks that the input number is scalable by that amount. */
static inline void
scaleDownFloat(unsigned int ux, int k, unsigned int *ur)
{
unsigned int uy, uk, ax, xsign;
int n, shift;
xsign = ux & SIGNBIT_SP32;
ax = ux & ~SIGNBIT_SP32;
n = ((ax & EXPBITS_SP32) >> EXPSHIFTBITS_SP32) - k;
if (n > 0) {
uk = (unsigned int)n << EXPSHIFTBITS_SP32;
uy = (ax & ~EXPBITS_SP32) | uk;
} else {
uy = (ax & ~EXPBITS_SP32) | 0x00800000;
shift = (1 - n);
if (shift > MANTLENGTH_SP32 + 1)
/* Sigh. Shifting works mod 32 so be careful not to shift too much */
uy = 0;
else {
/* Make sure we round the result */
uy >>= shift - 1;
uy = (uy >> 1) + (uy & 1);
}
}
*ur = uy | xsign;
}
#endif /* SCALEDOWNFLOAT */
#if defined(USE_SQRT_AMD_INLINE)
static inline double
sqrt_amd_inline(double x)
{
/*
Computes the square root of x.
The calculation is carried out in three steps.
Step 1. Reduction.
The input argument is scaled to the interval [1, 4) by
computing
x = 2^e * y, where y in [1,4).
Furthermore y is decomposed as y = c + t where
c = 1 + j/32, j = 0,1,..,96; and |t| <= 1/64.
Step 2. Approximation.
An approximation q = sqrt(1 + (t/c)) - 1 is obtained
from a basic series expansion using precomputed values
stored in rt_jby32_lead_table_dbl and rt_jby32_trail_table_dbl.
Step 3. Reconstruction.
The value of sqrt(x) is reconstructed via
sqrt(x) = 2^(e/2) * sqrt(y)
= 2^(e/2) * sqrt(c) * sqrt(y/c)
= 2^(e/2) * sqrt(c) * sqrt(1 + t/c)
= 2^(e/2) * [ sqrt(c) + sqrt(c)*q ]
*/
__UINT8_T ux, ax, u;
double r1, r2, c, y, p, q, r, twop, z, rtc, rtc_lead, rtc_trail;
int e, denorm = 0, index;
/* Arrays rt_jby32_lead_table_dbl and rt_jby32_trail_table_dbl contain
leading and trailing parts respectively of precomputed
values of sqrt(j/32), for j = 32, 33, ..., 128.
rt_jby32_lead_table_dbl contains the first 21 bits of precision,
and rt_jby32_trail_table_dbl contains a further 53 bits precision. */
static const double rt_jby32_lead_table_dbl[97] = {
1.00000000000000000000e+00, /* 0x3ff0000000000000 */
1.01550388336181640625e+00, /* 0x3ff03f8100000000 */
1.03077602386474609375e+00, /* 0x3ff07e0f00000000 */
1.04582500457763671875e+00, /* 0x3ff0bbb300000000 */
1.06065940856933593750e+00, /* 0x3ff0f87600000000 */
1.07528972625732421875e+00, /* 0x3ff1346300000000 */
1.08972454071044921875e+00, /* 0x3ff16f8300000000 */
1.10396957397460937500e+00, /* 0x3ff1a9dc00000000 */
1.11803340911865234375e+00, /* 0x3ff1e37700000000 */
1.13192272186279296875e+00, /* 0x3ff21c5b00000000 */
1.14564323425292968750e+00, /* 0x3ff2548e00000000 */
1.15920162200927734375e+00, /* 0x3ff28c1700000000 */
1.17260360717773437500e+00, /* 0x3ff2c2fc00000000 */
1.18585395812988281250e+00, /* 0x3ff2f94200000000 */
1.19895744323730468750e+00, /* 0x3ff32eee00000000 */
1.21191978454589843750e+00, /* 0x3ff3640600000000 */
1.22474479675292968750e+00, /* 0x3ff3988e00000000 */
1.23743629455566406250e+00, /* 0x3ff3cc8a00000000 */
1.25000000000000000000e+00, /* 0x3ff4000000000000 */
1.26243782043457031250e+00, /* 0x3ff432f200000000 */
1.27475452423095703125e+00, /* 0x3ff4656500000000 */
1.28695297241210937500e+00, /* 0x3ff4975c00000000 */
1.29903793334960937500e+00, /* 0x3ff4c8dc00000000 */
1.31101036071777343750e+00, /* 0x3ff4f9e600000000 */
1.32287502288818359375e+00, /* 0x3ff52a7f00000000 */
1.33463478088378906250e+00, /* 0x3ff55aaa00000000 */
1.34629058837890625000e+00, /* 0x3ff58a6800000000 */
1.35784721374511718750e+00, /* 0x3ff5b9be00000000 */
1.36930561065673828125e+00, /* 0x3ff5e8ad00000000 */
1.38066959381103515625e+00, /* 0x3ff6173900000000 */
1.39194107055664062500e+00, /* 0x3ff6456400000000 */
1.40312099456787109375e+00, /* 0x3ff6732f00000000 */
1.41421318054199218750e+00, /* 0x3ff6a09e00000000 */
1.42521858215332031250e+00, /* 0x3ff6cdb200000000 */
1.43614006042480468750e+00, /* 0x3ff6fa6e00000000 */
1.44697952270507812500e+00, /* 0x3ff726d400000000 */
1.45773792266845703125e+00, /* 0x3ff752e500000000 */
1.46841716766357421875e+00, /* 0x3ff77ea300000000 */
1.47901916503906250000e+00, /* 0x3ff7aa1000000000 */