libclc/clc/lib/generic/math/clc_log_base.h

//===----------------------------------------------------------------------===//
//
// Part of the LLVM Project, under the Apache License v2.0 with LLVM Exceptions.
// See https://llvm.org/LICENSE.txt for license information.
// SPDX-License-Identifier: Apache-2.0 WITH LLVM-exception
//
//===----------------------------------------------------------------------===//

#include <clc/math/clc_fabs.h>
#include <clc/math/clc_fma.h>
#include <clc/math/clc_mad.h>
#include <clc/math/math.h>
#include <clc/relational/clc_isinf.h>
#include <clc/relational/clc_isnan.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)
__clc_log2(float x)
#elif defined(COMPILING_LOG10)
__clc_log10(float x)
#else
__clc_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 = __clc_as_uint(x);
  uint ax = xi & EXSIGNBIT_SP32;

  // Calculations for |x-1| < 2^-4
  float r = x - 1.0f;
  int near1 = __clc_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 = __clc_mad(u, __clc_mad(v, 0x1.99999ap-7f, 0x1.555556p-4f) * v, -corr);

#if defined(COMPILING_LOG2)
  z1 = __clc_as_float(__clc_as_int(r) & 0xffff0000);
  z2 = z2 + (r - z1);
  znear1 = __clc_mad(
      z1, LOG2E_HEAD,
      __clc_mad(z2, LOG2E_HEAD, __clc_mad(z1, LOG2E_TAIL, z2 * LOG2E_TAIL)));
#elif defined(COMPILING_LOG10)
  z1 = __clc_as_float(__clc_as_int(r) & 0xffff0000);
  z2 = z2 + (r - z1);
  znear1 = __clc_mad(
      z1, LOG10E_HEAD,
      __clc_mad(z2, LOG10E_HEAD, __clc_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 = __clc_as_uint(__clc_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 = __clc_as_float(0x3f000000 | indx) -
            __clc_as_float(0x3f000000 | (xin & MANTBITS_SP32));

  indx = indx >> 16;
  r = f * __CLC_USE_TABLE(log_inv_tbl, indx);

  // 1/3,  1/2
  float poly = __clc_mad(__clc_mad(r, 0x1.555556p-2f, 0.5f), r * r, r);

#if defined(COMPILING_LOG2)
  float2 tv = __CLC_USE_TABLE(log2_tbl, indx);
  z1 = tv.s0 + mf;
  z2 = __clc_mad(poly, -LOG2E, tv.s1);
#elif defined(COMPILING_LOG10)
  float2 tv = __CLC_USE_TABLE(log10_tbl, indx);
  z1 = __clc_mad(mf, LOG10_2_HEAD, tv.s0);
  z2 = __clc_mad(poly, -LOG10E, mf * LOG10_2_TAIL) + tv.s1;
#else
  float2 tv = __CLC_USE_TABLE(log_tbl, indx);
  z1 = __clc_mad(mf, LOG2_HEAD, tv.s0);
  z2 = __clc_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 ? __clc_as_float(QNANBITPATT_SP32) : z;
  z = ax == 0 ? __clc_as_float(NINFBITPATT_SP32) : z;

  return z;
}

#ifdef cl_khr_fp64

_CLC_OVERLOAD _CLC_DEF double
#if defined(COMPILING_LOG2)
__clc_log2(double x)
#elif defined(COMPILING_LOG10)
__clc_log10(double x)
#else
__clc_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 = __clc_fma(
      u * v, __clc_fma(v, __clc_fma(v, __clc_fma(v, ca_4, ca_3), ca_2), ca_1),
      -correction);

#if defined(COMPILING_LOG10)
  r = r1;
  r1 = __clc_as_double(__clc_as_ulong(r1) & 0xffffffff00000000);
  r2 = r2 + (r - r1);
  double ret_near = __clc_fma(
      log10e_lead, r1,
      __clc_fma(log10e_lead, r2, __clc_fma(log10e_tail, r1, log10e_tail * r2)));
#elif defined(COMPILING_LOG2)
  r = r1;
  r1 = __clc_as_double(__clc_as_ulong(r1) & 0xffffffff00000000);
  r2 = r2 + (r - r1);
  double ret_near = __clc_fma(
      log2e_lead, r1,
      __clc_fma(log2e_lead, r2, __clc_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 = __clc_as_ulong(x);
  ulong uxs =
      __clc_as_ulong(__clc_as_double(0x03d0000000000000UL | ux) - 0x1.0p-962);
  int c = ux < IMPBIT_DP64;
  ux = c ? uxs : ux;
  int expadjust = c ? 60 : 0;

  int xexp = ((__clc_as_int2(ux).hi >> 20) & 0x7ff) - EXPBIAS_DP64 - expadjust;
  double f = __clc_as_double(HALFEXPBITS_DP64 | (ux & MANTBITS_DP64));
  int index = __clc_as_int2(ux).hi >> 13;
  index = ((0x80 | (index & 0x7e)) >> 1) + (index & 0x1);

  double z1 = __CLC_USE_TABLE(ln_tbl_lo, index - 64);
  double q = __CLC_USE_TABLE(ln_tbl_hi, index - 64);

  double f1 = index * 0x1.0p-7;
  double f2 = f - f1;
  u = f2 / __clc_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 * __clc_fma(v, __clc_fma(v, cb_3, cb_2), cb_1);
  double z2 = q + __clc_fma(u, poly, u);

  double dxexp = (double)xexp;
#if defined(COMPILING_LOG10)
  // Add xexp * log(2) to z1,z2 to get log(x)
  r1 = __clc_fma(dxexp, log2_lead, z1);
  r2 = __clc_fma(dxexp, log2_tail, z2);
  double ret_far = __clc_fma(
      log10e_lead, r1,
      __clc_fma(log10e_lead, r2, __clc_fma(log10e_tail, r1, log10e_tail * r2)));
#elif defined(COMPILING_LOG2)
  r1 = __clc_fma(log2e_lead, z1, dxexp);
  r2 = __clc_fma(log2e_lead, z2, __clc_fma(log2e_tail, z1, log2e_tail * z2));
  double ret_far = r1 + r2;
#else
  r1 = __clc_fma(dxexp, log2_lead, z1);
  r2 = __clc_fma(dxexp, log2_tail, z2);
  double ret_far = r1 + r2;
#endif

  double ret = is_near ? ret_near : ret_far;

  ret = __clc_isinf(x) ? __clc_as_double(PINFBITPATT_DP64) : ret;
  ret = (__clc_isnan(x) | (x < 0.0)) ? __clc_as_double(QNANBITPATT_DP64) : ret;
  ret = x == 0.0 ? __clc_as_double(NINFBITPATT_DP64) : ret;
  return ret;
}

#endif // cl_khr_fp64

#ifdef cl_khr_fp16

_CLC_OVERLOAD _CLC_DEF half
#if defined(COMPILING_LOG2)
__clc_log2(half x) {
  return (half)__clc_log2((float)x);
}
#elif defined(COMPILING_LOG10)
__clc_log10(half x) {
  return (half)__clc_log10((float)x);
}
#else
__clc_log(half x) {
  return (half)__clc_log((float)x);
}
#endif

#endif // cl_khr_fp16