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/*
* Copyright (c) 2014,2015 Advanced Micro Devices, Inc.
*
* Copyright (c) 2017 Michal Babej / Tampere University of Technology
*
* 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.
*/
/*
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)
*/
_CL_OVERLOADABLE vtype
#if defined(COMPILING_LOG2)
log2(vtype x)
#elif defined(COMPILING_LOGB)
logb(vtype x)
#elif defined(COMPILING_LOG10)
log10(vtype x)
#else
log(vtype x)
#endif
{
#if defined(COMPILING_LOGB)
#define COMPILING_LOG2
#endif
#if defined(COMPILING_LOG2)
const vtype LOG2E = (vtype)0x1.715476p+0f; // 1.4426950408889634
const vtype LOG2E_HEAD = (vtype)0x1.700000p+0f; // 1.4375
const vtype LOG2E_TAIL = (vtype)0x1.547652p-8f; // 0.00519504072
#elif defined(COMPILING_LOG10)
const vtype LOG10E = (vtype)0x1.bcb7b2p-2f; // 0.43429448190325182
const vtype LOG10E_HEAD = (vtype)0x1.bc0000p-2f; // 0.43359375
const vtype LOG10E_TAIL = (vtype)0x1.6f62a4p-11f; // 0.0007007319
const vtype LOG10_2_HEAD = (vtype)0x1.340000p-2f; // 0.30078125
const vtype LOG10_2_TAIL = (vtype)0x1.04d426p-12f; // 0.000248745637
#else
const vtype LOG2_HEAD = (vtype)0x1.62e000p-1f; // 0.693115234
const vtype LOG2_TAIL = (vtype)0x1.0bfbe8p-15f; // 0.0000319461833
#endif
utype xi = as_utype(x);
utype ax = xi & (utype)EXSIGNBIT_SP32;
// Calculations for |x-1| < 2^-4
vtype r = x - (vtype)1.0f;
itype near1 = (fabs(r) < (vtype)0x1.0p-4f);
vtype u2 = MATH_DIVIDE(r, (vtype)2.0f + r);
vtype corr = u2 * r;
vtype u = u2 + u2;
vtype v = u * u;
vtype znear1, z1, z2;
// 2/(5 * 2^5), 2/(3 * 2^3)
z2 = pocl_fma(u,
pocl_fma(v,
(vtype)0x1.99999ap-7f,
(vtype)0x1.555556p-4f)*v,
-corr);
#if defined(COMPILING_LOG2)
z1 = as_vtype(as_itype(r) & (itype)0xffff0000);
z2 = z2 + (r - z1);
znear1 = pocl_fma(z1, LOG2E_HEAD,
pocl_fma(z2, LOG2E_HEAD,
pocl_fma(z1, LOG2E_TAIL, z2*LOG2E_TAIL)));
#elif defined(COMPILING_LOG10)
z1 = as_vtype(as_itype(r) & (itype)0xffff0000);
z2 = z2 + (r - z1);
znear1 = pocl_fma(z1, LOG10E_HEAD,
pocl_fma(z2, LOG10E_HEAD,
pocl_fma(z1, LOG10E_TAIL, z2*LOG10E_TAIL)));
#else
znear1 = z2 + r;
#endif
// Calculations for x not near 1
itype m = as_itype(xi >> EXPSHIFTBITS_SP32) - (itype)EXPBIAS_SP32;
// Normalize subnormal
utype xis = as_utype(as_vtype(xi | (utype)0x3f800000) - (vtype)1.0f);
itype ms = (as_itype(xis) >> EXPSHIFTBITS_SP32) - (itype)253;
itype c = (m == -127);
m = c ? ms : m;
utype xin = c ? xis : xi;
vtype mf = convert_vtype(m);
utype indx = (xin & (utype)0x007f0000) + ((xin & (utype)0x00008000) << 1);
// F - Y
vtype f = as_vtype((utype)0x3f000000 | indx)
- as_vtype((utype)0x3f000000 | (xin & MANTBITS_SP32));
indx = indx >> 16;
r = f * USE_VTABLE(log_inv_tbl, indx);
// 1/3, 1/2
vtype poly = pocl_fma(
pocl_fma(r, (vtype)0x1.555556p-2f, (vtype)0.5f),
r*r,
r);
#if defined(COMPILING_LOG2)
v2type tv = USE_VTABLE(log2_tbl, indx);
vtype s0 = tv.lo;
vtype s1 = tv.hi;
z1 = s0 + mf;
z2 = pocl_fma(poly, -LOG2E, s1);
#elif defined(COMPILING_LOG10)
v2type tv = USE_VTABLE(log10_tbl, indx);
vtype s0 = tv.lo;
vtype s1 = tv.hi;
z1 = pocl_fma(mf, LOG10_2_HEAD, s0);
z2 = pocl_fma(poly, -LOG10E, mf*LOG10_2_TAIL) + s1;
#else
v2type tv = USE_VTABLE(loge_tbl, indx);
vtype s0 = tv.lo;
vtype s1 = tv.hi;
z1 = pocl_fma(mf, LOG2_HEAD, s0);
z2 = pocl_fma(mf, LOG2_TAIL, -poly) + s1;
#endif
vtype z = z1 + z2;
z = near1 ? znear1 : z;
// Corner cases
z = (ax >= (utype)PINFBITPATT_SP32) ? x : z;
z = (xi != ax) ? as_vtype((utype)QNANBITPATT_SP32) : z;
z = (ax == 0) ? as_vtype((utype)NINFBITPATT_SP32) : z;
return z;
}
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