sysdeps/ieee754/flt-32/e_powf.c

   1 /* Single-precision pow function.
   2    Copyright (C) 2017-2018 Free Software Foundation, Inc.
   3    This file is part of the GNU C Library.
   4
   5    The GNU C Library is free software; you can redistribute it and/or
   6    modify it under the terms of the GNU Lesser General Public
   7    License as published by the Free Software Foundation; either
   8    version 2.1 of the License, or (at your option) any later version.
   9
  10    The GNU C Library is distributed in the hope that it will be useful,
  11    but WITHOUT ANY WARRANTY; without even the implied warranty of
  12    MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the GNU
  13    Lesser General Public License for more details.
  14
  15    You should have received a copy of the GNU Lesser General Public
  16    License along with the GNU C Library; if not, see
  17    <http://www.gnu.org/licenses/>.  */
  18
  19 #include <math.h>
  20 #include <math-barriers.h>
  21 #include <math-narrow-eval.h>
  22 #include <stdint.h>
  23 #include <shlib-compat.h>
  24 #include <libm-alias-float.h>
  25 #include "math_config.h"
  26
  27 /*
  28 POWF_LOG2_POLY_ORDER = 5
  29 EXP2F_TABLE_BITS = 5
  30
  31 ULP error: 0.82 (~ 0.5 + relerr*2^24)
  32 relerr: 1.27 * 2^-26 (Relative error ~= 128*Ln2*relerr_log2 + relerr_exp2)
  33 relerr_log2: 1.83 * 2^-33 (Relative error of logx.)
  34 relerr_exp2: 1.69 * 2^-34 (Relative error of exp2(ylogx).)
  35 */
  36
  37 #define N (1 << POWF_LOG2_TABLE_BITS)
  38 #define T __powf_log2_data.tab
  39 #define A __powf_log2_data.poly
  40 #define OFF 0x3f330000
  41
  42 /* Subnormal input is normalized so ix has negative biased exponent.
  43    Output is multiplied by N (POWF_SCALE) if TOINT_INTRINICS is set.  */
  44 static inline double_t
  45 log2_inline (uint32_t ix)
  46 {
  47   /* double_t for better performance on targets with FLT_EVAL_METHOD==2.  */
  48   double_t z, r, r2, r4, p, q, y, y0, invc, logc;
  49   uint32_t iz, top, tmp;
  50   int k, i;
  51
  52   /* x = 2^k z; where z is in range [OFF,2*OFF] and exact.
  53      The range is split into N subintervals.
  54      The ith subinterval contains z and c is near its center.  */
  55   tmp = ix - OFF;
  56   i = (tmp >> (23 - POWF_LOG2_TABLE_BITS)) % N;
  57   top = tmp & 0xff800000;
  58   iz = ix - top;
  59   k = (int32_t) top >> (23 - POWF_SCALE_BITS); /* arithmetic shift */
  60   invc = T[i].invc;
  61   logc = T[i].logc;
  62   z = (double_t) asfloat (iz);
  63
  64   /* log2(x) = log1p(z/c-1)/ln2 + log2(c) + k */
  65   r = z * invc - 1;
  66   y0 = logc + (double_t) k;
  67
  68   /* Pipelined polynomial evaluation to approximate log1p(r)/ln2.  */
  69   r2 = r * r;
  70   y = A[0] * r + A[1];
  71   p = A[2] * r + A[3];
  72   r4 = r2 * r2;
  73   q = A[4] * r + y0;
  74   q = p * r2 + q;
  75   y = y * r4 + q;
  76   return y;
  77 }
  78
  79 #undef N
  80 #undef T
  81 #define N (1 << EXP2F_TABLE_BITS)
  82 #define T __exp2f_data.tab
  83 #define SIGN_BIAS (1 << (EXP2F_TABLE_BITS + 11))
  84
  85 /* The output of log2 and thus the input of exp2 is either scaled by N
  86    (in case of fast toint intrinsics) or not.  The unscaled xd must be
  87    in [-1021,1023], sign_bias sets the sign of the result.  */
  88 static inline double_t
  89 exp2_inline (double_t xd, uint32_t sign_bias)
  90 {
  91   uint64_t ki, ski, t;
  92   /* double_t for better performance on targets with FLT_EVAL_METHOD==2.  */
  93   double_t kd, z, r, r2, y, s;
  94
  95 #if TOINT_INTRINSICS
  96 # define C __exp2f_data.poly_scaled
  97   /* N*x = k + r with r in [-1/2, 1/2] */
  98   kd = roundtoint (xd); /* k */
  99   ki = converttoint (xd);
 100 #else
 101 # define C __exp2f_data.poly
 102 # define SHIFT __exp2f_data.shift_scaled
 103   /* x = k/N + r with r in [-1/(2N), 1/(2N)] */
 104   kd = (double) (xd + SHIFT); /* Rounding to double precision is required.  */
 105   ki = asuint64 (kd);
 106   kd -= SHIFT; /* k/N */
 107 #endif
 108   r = xd - kd;
 109
 110   /* exp2(x) = 2^(k/N) * 2^r ~= s * (C0*r^3 + C1*r^2 + C2*r + 1) */
 111   t = T[ki % N];
 112   ski = ki + sign_bias;
 113   t += ski << (52 - EXP2F_TABLE_BITS);
 114   s = asdouble (t);
 115   z = C[0] * r + C[1];
 116   r2 = r * r;
 117   y = C[2] * r + 1;
 118   y = z * r2 + y;
 119   y = y * s;
 120   return y;
 121 }
 122
 123 /* Returns 0 if not int, 1 if odd int, 2 if even int.  The argument is
 124    the bit representation of a non-zero finite floating-point value.  */
 125 static inline int
 126 checkint (uint32_t iy)
 127 {
 128   int e = iy >> 23 & 0xff;
 129   if (e < 0x7f)
 130     return 0;
 131   if (e > 0x7f + 23)
 132     return 2;
 133   if (iy & ((1 << (0x7f + 23 - e)) - 1))
 134     return 0;
 135   if (iy & (1 << (0x7f + 23 - e)))
 136     return 1;
 137   return 2;
 138 }
 139
 140 static inline int
 141 zeroinfnan (uint32_t ix)
 142 {
 143   return 2 * ix - 1 >= 2u * 0x7f800000 - 1;
 144 }
 145
 146 float
 147 __powf (float x, float y)
 148 {
 149   uint32_t sign_bias = 0;
 150   uint32_t ix, iy;
 151
 152   ix = asuint (x);
 153   iy = asuint (y);
 154   if (__glibc_unlikely (ix - 0x00800000 >= 0x7f800000 - 0x00800000
 155                         || zeroinfnan (iy)))
 156     {
 157       /* Either (x < 0x1p-126 or inf or nan) or (y is 0 or inf or nan).  */
 158       if (__glibc_unlikely (zeroinfnan (iy)))
 159         {
 160           if (2 * iy == 0)
 161             return issignalingf_inline (x) ? x + y : 1.0f;
 162           if (ix == 0x3f800000)
 163             return issignalingf_inline (y) ? x + y : 1.0f;
 164           if (2 * ix > 2u * 0x7f800000 || 2 * iy > 2u * 0x7f800000)
 165             return x + y;
 166           if (2 * ix == 2 * 0x3f800000)
 167             return 1.0f;
 168           if ((2 * ix < 2 * 0x3f800000) == !(iy & 0x80000000))
 169             return 0.0f; /* |x|<1 && y==inf or |x|>1 && y==-inf.  */
 170           return y * y;
 171         }
 172       if (__glibc_unlikely (zeroinfnan (ix)))
 173         {
 174           float_t x2 = x * x;
 175           if (ix & 0x80000000 && checkint (iy) == 1)
 176             {
 177               x2 = -x2;
 178               sign_bias = 1;
 179             }
 180 #if WANT_ERRNO
 181           if (2 * ix == 0 && iy & 0x80000000)
 182             return __math_divzerof (sign_bias);
 183 #endif
 184           return iy & 0x80000000 ? 1 / x2 : x2;
 185         }
 186       /* x and y are non-zero finite.  */
 187       if (ix & 0x80000000)
 188         {
 189           /* Finite x < 0.  */
 190           int yint = checkint (iy);
 191           if (yint == 0)
 192             return __math_invalidf (x);
 193           if (yint == 1)
 194             sign_bias = SIGN_BIAS;
 195           ix &= 0x7fffffff;
 196         }
 197       if (ix < 0x00800000)
 198         {
 199           /* Normalize subnormal x so exponent becomes negative.  */
 200           ix = asuint (x * 0x1p23f);
 201           ix &= 0x7fffffff;
 202           ix -= 23 << 23;
 203         }
 204     }
 205   double_t logx = log2_inline (ix);
 206   double_t ylogx = y * logx; /* Note: cannot overflow, y is single prec.  */
 207   if (__glibc_unlikely ((asuint64 (ylogx) >> 47 & 0xffff)
 208                         >= asuint64 (126.0 * POWF_SCALE) >> 47))
 209     {
 210       /* |y*log(x)| >= 126.  */
 211       if (ylogx > 0x1.fffffffd1d571p+6 * POWF_SCALE)
 212         /* |x^y| > 0x1.ffffffp127.  */
 213         return __math_oflowf (sign_bias);
 214       if (WANT_ROUNDING && WANT_ERRNO
 215           && ylogx > 0x1.fffffffa3aae2p+6 * POWF_SCALE)
 216         /* |x^y| > 0x1.fffffep127, check if we round away from 0.  */
 217         if ((!sign_bias
 218              && math_narrow_eval (1.0f + math_opt_barrier (0x1p-25f)) != 1.0f)
 219             || (sign_bias
 220                 && math_narrow_eval (-1.0f - math_opt_barrier (0x1p-25f))
 221                      != -1.0f))
 222           return __math_oflowf (sign_bias);
 223       if (ylogx <= -150.0 * POWF_SCALE)
 224         return __math_uflowf (sign_bias);
 225 #if WANT_ERRNO_UFLOW
 226       if (ylogx < -149.0 * POWF_SCALE)
 227         return __math_may_uflowf (sign_bias);
 228 #endif
 229     }
 230   return (float) exp2_inline (ylogx, sign_bias);
 231 }
 232 #ifndef __powf
 233 strong_alias (__powf, __ieee754_powf)
 234 strong_alias (__powf, __powf_finite)
 235 versioned_symbol (libm, __powf, powf, GLIBC_2_27);
 236 libm_alias_float_other (__pow, pow)
 237 #endif