github.com/afumu/libc@v0.0.6/musl/src/math/log2.c (about)

     1  /*
     2   * Double-precision log2(x) function.
     3   *
     4   * Copyright (c) 2018, Arm Limited.
     5   * SPDX-License-Identifier: MIT
     6   */
     7  
     8  #include <math.h>
     9  #include <stdint.h>
    10  #include "libm.h"
    11  #include "log2_data.h"
    12  
    13  #define T __log2_data.tab
    14  #define T2 __log2_data.tab2
    15  #define B __log2_data.poly1
    16  #define A __log2_data.poly
    17  #define InvLn2hi __log2_data.invln2hi
    18  #define InvLn2lo __log2_data.invln2lo
    19  #define N (1 << LOG2_TABLE_BITS)
    20  #define OFF 0x3fe6000000000000
    21  
    22  /* Top 16 bits of a double.  */
    23  static inline uint32_t top16(double x)
    24  {
    25  	return asuint64(x) >> 48;
    26  }
    27  
    28  double log2(double x)
    29  {
    30  	double_t z, r, r2, r4, y, invc, logc, kd, hi, lo, t1, t2, t3, p;
    31  	uint64_t ix, iz, tmp;
    32  	uint32_t top;
    33  	int k, i;
    34  
    35  	ix = asuint64(x);
    36  	top = top16(x);
    37  #define LO asuint64(1.0 - 0x1.5b51p-5)
    38  #define HI asuint64(1.0 + 0x1.6ab2p-5)
    39  	if (predict_false(ix - LO < HI - LO)) {
    40  		/* Handle close to 1.0 inputs separately.  */
    41  		/* Fix sign of zero with downward rounding when x==1.  */
    42  		if (WANT_ROUNDING && predict_false(ix == asuint64(1.0)))
    43  			return 0;
    44  		r = x - 1.0;
    45  #if __FP_FAST_FMA
    46  		hi = r * InvLn2hi;
    47  		lo = r * InvLn2lo + __builtin_fma(r, InvLn2hi, -hi);
    48  #else
    49  		double_t rhi, rlo;
    50  		rhi = asdouble(asuint64(r) & -1ULL << 32);
    51  		rlo = r - rhi;
    52  		hi = rhi * InvLn2hi;
    53  		lo = rlo * InvLn2hi + r * InvLn2lo;
    54  #endif
    55  		r2 = r * r; /* rounding error: 0x1p-62.  */
    56  		r4 = r2 * r2;
    57  		/* Worst-case error is less than 0.54 ULP (0.55 ULP without fma).  */
    58  		p = r2 * (B[0] + r * B[1]);
    59  		y = hi + p;
    60  		lo += hi - y + p;
    61  		lo += r4 * (B[2] + r * B[3] + r2 * (B[4] + r * B[5]) +
    62  			    r4 * (B[6] + r * B[7] + r2 * (B[8] + r * B[9])));
    63  		y += lo;
    64  		return eval_as_double(y);
    65  	}
    66  	if (predict_false(top - 0x0010 >= 0x7ff0 - 0x0010)) {
    67  		/* x < 0x1p-1022 or inf or nan.  */
    68  		if (ix * 2 == 0)
    69  			return __math_divzero(1);
    70  		if (ix == asuint64(INFINITY)) /* log(inf) == inf.  */
    71  			return x;
    72  		if ((top & 0x8000) || (top & 0x7ff0) == 0x7ff0)
    73  			return __math_invalid(x);
    74  		/* x is subnormal, normalize it.  */
    75  		ix = asuint64(x * 0x1p52);
    76  		ix -= 52ULL << 52;
    77  	}
    78  
    79  	/* x = 2^k z; where z is in range [OFF,2*OFF) and exact.
    80  	   The range is split into N subintervals.
    81  	   The ith subinterval contains z and c is near its center.  */
    82  	tmp = ix - OFF;
    83  	i = (tmp >> (52 - LOG2_TABLE_BITS)) % N;
    84  	k = (int64_t)tmp >> 52; /* arithmetic shift */
    85  	iz = ix - (tmp & 0xfffULL << 52);
    86  	invc = T[i].invc;
    87  	logc = T[i].logc;
    88  	z = asdouble(iz);
    89  	kd = (double_t)k;
    90  
    91  	/* log2(x) = log2(z/c) + log2(c) + k.  */
    92  	/* r ~= z/c - 1, |r| < 1/(2*N).  */
    93  #if __FP_FAST_FMA
    94  	/* rounding error: 0x1p-55/N.  */
    95  	r = __builtin_fma(z, invc, -1.0);
    96  	t1 = r * InvLn2hi;
    97  	t2 = r * InvLn2lo + __builtin_fma(r, InvLn2hi, -t1);
    98  #else
    99  	double_t rhi, rlo;
   100  	/* rounding error: 0x1p-55/N + 0x1p-65.  */
   101  	r = (z - T2[i].chi - T2[i].clo) * invc;
   102  	rhi = asdouble(asuint64(r) & -1ULL << 32);
   103  	rlo = r - rhi;
   104  	t1 = rhi * InvLn2hi;
   105  	t2 = rlo * InvLn2hi + r * InvLn2lo;
   106  #endif
   107  
   108  	/* hi + lo = r/ln2 + log2(c) + k.  */
   109  	t3 = kd + logc;
   110  	hi = t3 + t1;
   111  	lo = t3 - hi + t1 + t2;
   112  
   113  	/* log2(r+1) = r/ln2 + r^2*poly(r).  */
   114  	/* Evaluation is optimized assuming superscalar pipelined execution.  */
   115  	r2 = r * r; /* rounding error: 0x1p-54/N^2.  */
   116  	r4 = r2 * r2;
   117  	/* Worst-case error if |y| > 0x1p-4: 0.547 ULP (0.550 ULP without fma).
   118  	   ~ 0.5 + 2/N/ln2 + abs-poly-error*0x1p56 ULP (+ 0.003 ULP without fma).  */
   119  	p = A[0] + r * A[1] + r2 * (A[2] + r * A[3]) + r4 * (A[4] + r * A[5]);
   120  	y = lo + r2 * p + hi;
   121  	return eval_as_double(y);
   122  }