src/math/powf.c

   1 /*
   2  * Copyright (c) 2017-2018, Arm Limited.
   3  * SPDX-License-Identifier: MIT
   4  */
   5
   6 #include <math.h>
   7 #include <stdint.h>
   8 #include "libm.h"
   9 #include "exp2f_data.h"
  10 #include "powf_data.h"
  11
  12 /*
  13 POWF_LOG2_POLY_ORDER = 5
  14 EXP2F_TABLE_BITS = 5
  15
  16 ULP error: 0.82 (~ 0.5 + relerr*2^24)
  17 relerr: 1.27 * 2^-26 (Relative error ~= 128*Ln2*relerr_log2 + relerr_exp2)
  18 relerr_log2: 1.83 * 2^-33 (Relative error of logx.)
  19 relerr_exp2: 1.69 * 2^-34 (Relative error of exp2(ylogx).)
  20 */
  21
  22 #define N (1 << POWF_LOG2_TABLE_BITS)
  23 #define T __powf_log2_data.tab
  24 #define A __powf_log2_data.poly
  25 #define OFF 0x3f330000
  26
  27 /* Subnormal input is normalized so ix has negative biased exponent.
  28    Output is multiplied by N (POWF_SCALE) if TOINT_INTRINICS is set.  */
  29 static inline double_t log2_inline(uint32_t ix)
  30 {
  31         double_t z, r, r2, r4, p, q, y, y0, invc, logc;
  32         uint32_t iz, top, tmp;
  33         int k, i;
  34
  35         /* x = 2^k z; where z is in range [OFF,2*OFF] and exact.
  36            The range is split into N subintervals.
  37            The ith subinterval contains z and c is near its center.  */
  38         tmp = ix - OFF;
  39         i = (tmp >> (23 - POWF_LOG2_TABLE_BITS)) % N;
  40         top = tmp & 0xff800000;
  41         iz = ix - top;
  42         k = (int32_t)top >> (23 - POWF_SCALE_BITS); /* arithmetic shift */
  43         invc = T[i].invc;
  44         logc = T[i].logc;
  45         z = (double_t)asfloat(iz);
  46
  47         /* log2(x) = log1p(z/c-1)/ln2 + log2(c) + k */
  48         r = z * invc - 1;
  49         y0 = logc + (double_t)k;
  50
  51         /* Pipelined polynomial evaluation to approximate log1p(r)/ln2.  */
  52         r2 = r * r;
  53         y = A[0] * r + A[1];
  54         p = A[2] * r + A[3];
  55         r4 = r2 * r2;
  56         q = A[4] * r + y0;
  57         q = p * r2 + q;
  58         y = y * r4 + q;
  59         return y;
  60 }
  61
  62 #undef N
  63 #undef T
  64 #define N (1 << EXP2F_TABLE_BITS)
  65 #define T __exp2f_data.tab
  66 #define SIGN_BIAS (1 << (EXP2F_TABLE_BITS + 11))
  67
  68 /* The output of log2 and thus the input of exp2 is either scaled by N
  69    (in case of fast toint intrinsics) or not.  The unscaled xd must be
  70    in [-1021,1023], sign_bias sets the sign of the result.  */
  71 static inline float exp2_inline(double_t xd, uint32_t sign_bias)
  72 {
  73         uint64_t ki, ski, t;
  74         double_t kd, z, r, r2, y, s;
  75
  76 #if TOINT_INTRINSICS
  77 #define C __exp2f_data.poly_scaled
  78         /* N*x = k + r with r in [-1/2, 1/2] */
  79         kd = roundtoint(xd); /* k */
  80         ki = converttoint(xd);
  81 #else
  82 #define C __exp2f_data.poly
  83 #define SHIFT __exp2f_data.shift_scaled
  84         /* x = k/N + r with r in [-1/(2N), 1/(2N)] */
  85         kd = eval_as_double(xd + SHIFT);
  86         ki = asuint64(kd);
  87         kd -= SHIFT; /* k/N */
  88 #endif
  89         r = xd - kd;
  90
  91         /* exp2(x) = 2^(k/N) * 2^r ~= s * (C0*r^3 + C1*r^2 + C2*r + 1) */
  92         t = T[ki % N];
  93         ski = ki + sign_bias;
  94         t += ski << (52 - EXP2F_TABLE_BITS);
  95         s = asdouble(t);
  96         z = C[0] * r + C[1];
  97         r2 = r * r;
  98         y = C[2] * r + 1;
  99         y = z * r2 + y;
 100         y = y * s;
 101         return eval_as_float(y);
 102 }
 103
 104 /* Returns 0 if not int, 1 if odd int, 2 if even int.  The argument is
 105    the bit representation of a non-zero finite floating-point value.  */
 106 static inline int checkint(uint32_t iy)
 107 {
 108         int e = iy >> 23 & 0xff;
 109         if (e < 0x7f)
 110                 return 0;
 111         if (e > 0x7f + 23)
 112                 return 2;
 113         if (iy & ((1 << (0x7f + 23 - e)) - 1))
 114                 return 0;
 115         if (iy & (1 << (0x7f + 23 - e)))
 116                 return 1;
 117         return 2;
 118 }
 119
 120 static inline int zeroinfnan(uint32_t ix)
 121 {
 122         return 2 * ix - 1 >= 2u * 0x7f800000 - 1;
 123 }
 124
 125 float powf(float x, float y)
 126 {
 127         uint32_t sign_bias = 0;
 128         uint32_t ix, iy;
 129
 130         ix = asuint(x);
 131         iy = asuint(y);
 132         if (predict_false(ix - 0x00800000 >= 0x7f800000 - 0x00800000 ||
 133                           zeroinfnan(iy))) {
 134                 /* Either (x < 0x1p-126 or inf or nan) or (y is 0 or inf or nan).  */
 135                 if (predict_false(zeroinfnan(iy))) {
 136                         if (2 * iy == 0)
 137                                 return issignalingf_inline(x) ? x + y : 1.0f;
 138                         if (ix == 0x3f800000)
 139                                 return issignalingf_inline(y) ? x + y : 1.0f;
 140                         if (2 * ix > 2u * 0x7f800000 ||
 141                             2 * iy > 2u * 0x7f800000)
 142                                 return x + y;
 143                         if (2 * ix == 2 * 0x3f800000)
 144                                 return 1.0f;
 145                         if ((2 * ix < 2 * 0x3f800000) == !(iy & 0x80000000))
 146                                 return 0.0f; /* |x|<1 && y==inf or |x|>1 && y==-inf.  */
 147                         return y * y;
 148                 }
 149                 if (predict_false(zeroinfnan(ix))) {
 150                         float_t x2 = x * x;
 151                         if (ix & 0x80000000 && checkint(iy) == 1)
 152                                 x2 = -x2;
 153                         /* Without the barrier some versions of clang hoist the 1/x2 and
 154                            thus division by zero exception can be signaled spuriously.  */
 155                         return iy & 0x80000000 ? fp_barrierf(1 / x2) : x2;
 156                 }
 157                 /* x and y are non-zero finite.  */
 158                 if (ix & 0x80000000) {
 159                         /* Finite x < 0.  */
 160                         int yint = checkint(iy);
 161                         if (yint == 0)
 162                                 return __math_invalidf(x);
 163                         if (yint == 1)
 164                                 sign_bias = SIGN_BIAS;
 165                         ix &= 0x7fffffff;
 166                 }
 167                 if (ix < 0x00800000) {
 168                         /* Normalize subnormal x so exponent becomes negative.  */
 169                         ix = asuint(x * 0x1p23f);
 170                         ix &= 0x7fffffff;
 171                         ix -= 23 << 23;
 172                 }
 173         }
 174         double_t logx = log2_inline(ix);
 175         double_t ylogx = y * logx; /* cannot overflow, y is single prec.  */
 176         if (predict_false((asuint64(ylogx) >> 47 & 0xffff) >=
 177                           asuint64(126.0 * POWF_SCALE) >> 47)) {
 178                 /* |y*log(x)| >= 126.  */
 179                 if (ylogx > 0x1.fffffffd1d571p+6 * POWF_SCALE)
 180                         return __math_oflowf(sign_bias);
 181                 if (ylogx <= -150.0 * POWF_SCALE)
 182                         return __math_uflowf(sign_bias);
 183         }
 184         return exp2_inline(ylogx, sign_bias);
 185 }