/*
* Copyright (c) 2014,2015 Advanced Micro Devices, Inc.
*
* Permission is hereby granted, free of charge, to any person obtaining a copy
* of this software and associated documentation files (the "Software"), to deal
* in the Software without restriction, including without limitation the rights
* to use, copy, modify, merge, publish, distribute, sublicense, and/or sell
* copies of the Software, and to permit persons to whom the Software is
* furnished to do so, subject to the following conditions:
*
* The above copyright notice and this permission notice shall be included in
* all copies or substantial portions of the Software.
*
* THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR
* IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,
* FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE
* AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER
* LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM,
* OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN
* THE SOFTWARE.
*/
#include "math.h"
/*
Algorithm:
Based on:
Ping-Tak Peter Tang
"Table-driven implementation of the logarithm function in IEEE
floating-point arithmetic"
ACM Transactions on Mathematical Software (TOMS)
Volume 16, Issue 4 (December 1990)
x very close to 1.0 is handled differently, for x everywhere else
a brief explanation is given below
x = (2^m)*A
x = (2^m)*(G+g) with (1 <= G < 2) and (g <= 2^(-8))
x = (2^m)*2*(G/2+g/2)
x = (2^m)*2*(F+f) with (0.5 <= F < 1) and (f <= 2^(-9))
Y = (2^(-1))*(2^(-m))*(2^m)*A
Now, range of Y is: 0.5 <= Y < 1
F = 0x80 + (first 7 mantissa bits) + (8th mantissa bit)
Now, range of F is: 128 <= F <= 256
F = F / 256
Now, range of F is: 0.5 <= F <= 1
f = -(Y-F), with (f <= 2^(-9))
log(x) = m*log(2) + log(2) + log(F-f)
log(x) = m*log(2) + log(2) + log(F) + log(1-(f/F))
log(x) = m*log(2) + log(2*F) + log(1-r)
r = (f/F), with (r <= 2^(-8))
r = f*(1/F) with (1/F) precomputed to avoid division
log(x) = m*log(2) + log(G) - poly
log(G) is precomputed
poly = (r + (r^2)/2 + (r^3)/3 + (r^4)/4) + (r^5)/5))
log(2) and log(G) need to be maintained in extra precision
to avoid losing precision in the calculations
For x close to 1.0, we employ the following technique to
ensure faster convergence.
log(x) = log((1+s)/(1-s)) = 2*s + (2/3)*s^3 + (2/5)*s^5 + (2/7)*s^7
x = ((1+s)/(1-s))
x = 1 + r
s = r/(2+r)
*/
_CLC_OVERLOAD _CLC_DEF float
#if defined(COMPILING_LOG2)
log2(float x)
#elif defined(COMPILING_LOG10)
log10(float x)
#else
log(float x)
#endif
{
#if defined(COMPILING_LOG2)
const float LOG2E = 0x1.715476p+0f; // 1.4426950408889634
const float LOG2E_HEAD = 0x1.700000p+0f; // 1.4375
const float LOG2E_TAIL = 0x1.547652p-8f; // 0.00519504072
#elif defined(COMPILING_LOG10)
const float LOG10E = 0x1.bcb7b2p-2f; // 0.43429448190325182
const float LOG10E_HEAD = 0x1.bc0000p-2f; // 0.43359375
const float LOG10E_TAIL = 0x1.6f62a4p-11f; // 0.0007007319
const float LOG10_2_HEAD = 0x1.340000p-2f; // 0.30078125
const float LOG10_2_TAIL = 0x1.04d426p-12f; // 0.000248745637
#else
const float LOG2_HEAD = 0x1.62e000p-1f; // 0.693115234
const float LOG2_TAIL = 0x1.0bfbe8p-15f; // 0.0000319461833
#endif
uint xi = as_uint(x);
uint ax = xi & EXSIGNBIT_SP32;
// Calculations for |x-1| < 2^-4
float r = x - 1.0f;
int near1 = fabs(r) < 0x1.0p-4f;
float u2 = MATH_DIVIDE(r, 2.0f + r);
float corr = u2 * r;
float u = u2 + u2;
float v = u * u;
float znear1, z1, z2;
// 2/(5 * 2^5), 2/(3 * 2^3)
z2 = mad(u, mad(v, 0x1.99999ap-7f, 0x1.555556p-4f)*v, -corr);
#if defined(COMPILING_LOG2)
z1 = as_float(as_int(r) & 0xffff0000);
z2 = z2 + (r - z1);
znear1 = mad(z1, LOG2E_HEAD, mad(z2, LOG2E_HEAD, mad(z1, LOG2E_TAIL, z2*LOG2E_TAIL)));
#elif defined(COMPILING_LOG10)
z1 = as_float(as_int(r) & 0xffff0000);
z2 = z2 + (r - z1);
znear1 = mad(z1, LOG10E_HEAD, mad(z2, LOG10E_HEAD, mad(z1, LOG10E_TAIL, z2*LOG10E_TAIL)));
#else
znear1 = z2 + r;
#endif
// Calculations for x not near 1
int m = (int)(xi >> EXPSHIFTBITS_SP32) - EXPBIAS_SP32;
// Normalize subnormal
uint xis = as_uint(as_float(xi | 0x3f800000) - 1.0f);
int ms = (int)(xis >> EXPSHIFTBITS_SP32) - 253;
int c = m == -127;
m = c ? ms : m;
uint xin = c ? xis : xi;
float mf = (float)m;
uint indx = (xin & 0x007f0000) + ((xin & 0x00008000) << 1);
// F - Y
float f = as_float(0x3f000000 | indx) - as_float(0x3f000000 | (xin & MANTBITS_SP32));
indx = indx >> 16;
r = f * USE_TABLE(log_inv_tbl, indx);
// 1/3, 1/2
float poly = mad(mad(r, 0x1.555556p-2f, 0.5f), r*r, r);
#if defined(COMPILING_LOG2)
float2 tv = USE_TABLE(log2_tbl, indx);
z1 = tv.s0 + mf;
z2 = mad(poly, -LOG2E, tv.s1);
#elif defined(COMPILING_LOG10)
float2 tv = USE_TABLE(log10_tbl, indx);
z1 = mad(mf, LOG10_2_HEAD, tv.s0);
z2 = mad(poly, -LOG10E, mf*LOG10_2_TAIL) + tv.s1;
#else
float2 tv = USE_TABLE(log_tbl, indx);
z1 = mad(mf, LOG2_HEAD, tv.s0);
z2 = mad(mf, LOG2_TAIL, -poly) + tv.s1;
#endif
float z = z1 + z2;
z = near1 ? znear1 : z;
// Corner cases
z = ax >= PINFBITPATT_SP32 ? x : z;
z = xi != ax ? as_float(QNANBITPATT_SP32) : z;
z = ax == 0 ? as_float(NINFBITPATT_SP32) : z;
return z;
}
#ifdef cl_khr_fp64
_CLC_OVERLOAD _CLC_DEF double
#if defined(COMPILING_LOG2)
log2(double x)
#elif defined(COMPILING_LOG10)
log10(double x)
#else
log(double x)
#endif
{
#ifndef COMPILING_LOG2
// log2_lead and log2_tail sum to an extra-precise version of ln(2)
const double log2_lead = 6.93147122859954833984e-01; /* 0x3fe62e42e0000000 */
const double log2_tail = 5.76999904754328540596e-08; /* 0x3e6efa39ef35793c */
#endif
#if defined(COMPILING_LOG10)
// log10e_lead and log10e_tail sum to an extra-precision version of log10(e) (19 bits in lead)
const double log10e_lead = 4.34293746948242187500e-01; /* 0x3fdbcb7800000000 */
const double log10e_tail = 7.3495500964015109100644e-7; /* 0x3ea8a93728719535 */
#elif defined(COMPILING_LOG2)
// log2e_lead and log2e_tail sum to an extra-precision version of log2(e) (19 bits in lead)
const double log2e_lead = 1.44269180297851562500E+00; /* 0x3FF7154400000000 */
const double log2e_tail = 3.23791044778235969970E-06; /* 0x3ECB295C17F0BBBE */
#endif
// log_thresh1 = 9.39412117004394531250e-1 = 0x3fee0faa00000000
// log_thresh2 = 1.06449508666992187500 = 0x3ff1082c00000000
const double log_thresh1 = 0x1.e0faap-1;
const double log_thresh2 = 0x1.1082cp+0;
bool is_near = x >= log_thresh1 && x <= log_thresh2;
// Near 1 code
double r = x - 1.0;
double u = r / (2.0 + r);
double correction = r * u;
u = u + u;
double v = u * u;
double r1 = r;
const double ca_1 = 8.33333333333317923934e-02; /* 0x3fb55555555554e6 */
const double ca_2 = 1.25000000037717509602e-02; /* 0x3f89999999bac6d4 */
const double ca_3 = 2.23213998791944806202e-03; /* 0x3f62492307f1519f */
const double ca_4 = 4.34887777707614552256e-04; /* 0x3f3c8034c85dfff0 */
double r2 = fma(u*v, fma(v, fma(v, fma(v, ca_4, ca_3), ca_2), ca_1), -correction);
#if defined(COMPILING_LOG10)
r = r1;
r1 = as_double(as_ulong(r1) & 0xffffffff00000000);
r2 = r2 + (r - r1);
double ret_near = fma(log10e_lead, r1, fma(log10e_lead, r2, fma(log10e_tail, r1, log10e_tail * r2)));
#elif defined(COMPILING_LOG2)
r = r1;
r1 = as_double(as_ulong(r1) & 0xffffffff00000000);
r2 = r2 + (r - r1);
double ret_near = fma(log2e_lead, r1, fma(log2e_lead, r2, fma(log2e_tail, r1, log2e_tail*r2)));
#else
double ret_near = r1 + r2;
#endif
// This is the far from 1 code
// Deal with subnormal
ulong ux = as_ulong(x);
ulong uxs = as_ulong(as_double(0x03d0000000000000UL | ux) - 0x1.0p-962);
int c = ux < IMPBIT_DP64;
ux = c ? uxs : ux;
int expadjust = c ? 60 : 0;
int xexp = ((as_int2(ux).hi >> 20) & 0x7ff) - EXPBIAS_DP64 - expadjust;
double f = as_double(HALFEXPBITS_DP64 | (ux & MANTBITS_DP64));
int index = as_int2(ux).hi >> 13;
index = ((0x80 | (index & 0x7e)) >> 1) + (index & 0x1);
double2 tv = USE_TABLE(ln_tbl, index - 64);
double z1 = tv.s0;
double q = tv.s1;
double f1 = index * 0x1.0p-7;
double f2 = f - f1;
u = f2 / fma(f2, 0.5, f1);
v = u * u;
const double cb_1 = 8.33333333333333593622e-02; /* 0x3fb5555555555557 */
const double cb_2 = 1.24999999978138668903e-02; /* 0x3f89999999865ede */
const double cb_3 = 2.23219810758559851206e-03; /* 0x3f6249423bd94741 */
double poly = v * fma(v, fma(v, cb_3, cb_2), cb_1);
double z2 = q + fma(u, poly, u);
double dxexp = (double)xexp;
#if defined (COMPILING_LOG10)
// Add xexp * log(2) to z1,z2 to get log(x)
r1 = fma(dxexp, log2_lead, z1);
r2 = fma(dxexp, log2_tail, z2);
double ret_far = fma(log10e_lead, r1, fma(log10e_lead, r2, fma(log10e_tail, r1, log10e_tail*r2)));
#elif defined(COMPILING_LOG2)
r1 = fma(log2e_lead, z1, dxexp);
r2 = fma(log2e_lead, z2, fma(log2e_tail, z1, log2e_tail*z2));
double ret_far = r1 + r2;
#else
r1 = fma(dxexp, log2_lead, z1);
r2 = fma(dxexp, log2_tail, z2);
double ret_far = r1 + r2;
#endif
double ret = is_near ? ret_near : ret_far;
ret = isinf(x) ? as_double(PINFBITPATT_DP64) : ret;
ret = (isnan(x) | (x < 0.0)) ? as_double(QNANBITPATT_DP64) : ret;
ret = x == 0.0 ? as_double(NINFBITPATT_DP64) : ret;
return ret;
}
#endif // cl_khr_fp64
#ifdef cl_khr_fp16
_CLC_OVERLOAD _CLC_DEF half
#if defined(COMPILING_LOG2)
log2(half x) {
return (half)log2((float)x);
}
#elif defined(COMPILING_LOG10)
log10(half x) {
return (half)log10((float)x);
}
#else
log(half x) {
return (half)log((float)x);
}
#endif
#endif // cl_khr_fp16
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