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

     1  /*
     2   * Double-precision e^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 "exp_data.h"
    12  
    13  #define N (1 << EXP_TABLE_BITS)
    14  #define InvLn2N __exp_data.invln2N
    15  #define NegLn2hiN __exp_data.negln2hiN
    16  #define NegLn2loN __exp_data.negln2loN
    17  #define Shift __exp_data.shift
    18  #define T __exp_data.tab
    19  #define C2 __exp_data.poly[5 - EXP_POLY_ORDER]
    20  #define C3 __exp_data.poly[6 - EXP_POLY_ORDER]
    21  #define C4 __exp_data.poly[7 - EXP_POLY_ORDER]
    22  #define C5 __exp_data.poly[8 - EXP_POLY_ORDER]
    23  
    24  /* Handle cases that may overflow or underflow when computing the result that
    25     is scale*(1+TMP) without intermediate rounding.  The bit representation of
    26     scale is in SBITS, however it has a computed exponent that may have
    27     overflown into the sign bit so that needs to be adjusted before using it as
    28     a double.  (int32_t)KI is the k used in the argument reduction and exponent
    29     adjustment of scale, positive k here means the result may overflow and
    30     negative k means the result may underflow.  */
    31  static inline double specialcase(double_t tmp, uint64_t sbits, uint64_t ki)
    32  {
    33  	double_t scale, y;
    34  
    35  	if ((ki & 0x80000000) == 0) {
    36  		/* k > 0, the exponent of scale might have overflowed by <= 460.  */
    37  		sbits -= 1009ull << 52;
    38  		scale = asdouble(sbits);
    39  		y = 0x1p1009 * (scale + scale * tmp);
    40  		return eval_as_double(y);
    41  	}
    42  	/* k < 0, need special care in the subnormal range.  */
    43  	sbits += 1022ull << 52;
    44  	scale = asdouble(sbits);
    45  	y = scale + scale * tmp;
    46  	if (y < 1.0) {
    47  		/* Round y to the right precision before scaling it into the subnormal
    48  		 range to avoid double rounding that can cause 0.5+E/2 ulp error where
    49  		 E is the worst-case ulp error outside the subnormal range.  So this
    50  		 is only useful if the goal is better than 1 ulp worst-case error.  */
    51  		double_t hi, lo;
    52  		lo = scale - y + scale * tmp;
    53  		hi = 1.0 + y;
    54  		lo = 1.0 - hi + y + lo;
    55  		y = eval_as_double(hi + lo) - 1.0;
    56  		/* Avoid -0.0 with downward rounding.  */
    57  		if (WANT_ROUNDING && y == 0.0)
    58  			y = 0.0;
    59  		/* The underflow exception needs to be signaled explicitly.  */
    60  		fp_force_eval(fp_barrier(0x1p-1022) * 0x1p-1022);
    61  	}
    62  	y = 0x1p-1022 * y;
    63  	return eval_as_double(y);
    64  }
    65  
    66  /* Top 12 bits of a double (sign and exponent bits).  */
    67  static inline uint32_t top12(double x)
    68  {
    69  	return asuint64(x) >> 52;
    70  }
    71  
    72  double exp(double x)
    73  {
    74  	uint32_t abstop;
    75  	uint64_t ki, idx, top, sbits;
    76  	double_t kd, z, r, r2, scale, tail, tmp;
    77  
    78  	abstop = top12(x) & 0x7ff;
    79  	if (predict_false(abstop - top12(0x1p-54) >= top12(512.0) - top12(0x1p-54))) {
    80  		if (abstop - top12(0x1p-54) >= 0x80000000)
    81  			/* Avoid spurious underflow for tiny x.  */
    82  			/* Note: 0 is common input.  */
    83  			return WANT_ROUNDING ? 1.0 + x : 1.0;
    84  		if (abstop >= top12(1024.0)) {
    85  			if (asuint64(x) == asuint64(-INFINITY))
    86  				return 0.0;
    87  			if (abstop >= top12(INFINITY))
    88  				return 1.0 + x;
    89  			if (asuint64(x) >> 63)
    90  				return __math_uflow(0);
    91  			else
    92  				return __math_oflow(0);
    93  		}
    94  		/* Large x is special cased below.  */
    95  		abstop = 0;
    96  	}
    97  
    98  	/* exp(x) = 2^(k/N) * exp(r), with exp(r) in [2^(-1/2N),2^(1/2N)].  */
    99  	/* x = ln2/N*k + r, with int k and r in [-ln2/2N, ln2/2N].  */
   100  	z = InvLn2N * x;
   101  #if TOINT_INTRINSICS
   102  	kd = roundtoint(z);
   103  	ki = converttoint(z);
   104  #elif EXP_USE_TOINT_NARROW
   105  	/* z - kd is in [-0.5-2^-16, 0.5] in all rounding modes.  */
   106  	kd = eval_as_double(z + Shift);
   107  	ki = asuint64(kd) >> 16;
   108  	kd = (double_t)(int32_t)ki;
   109  #else
   110  	/* z - kd is in [-1, 1] in non-nearest rounding modes.  */
   111  	kd = eval_as_double(z + Shift);
   112  	ki = asuint64(kd);
   113  	kd -= Shift;
   114  #endif
   115  	r = x + kd * NegLn2hiN + kd * NegLn2loN;
   116  	/* 2^(k/N) ~= scale * (1 + tail).  */
   117  	idx = 2 * (ki % N);
   118  	top = ki << (52 - EXP_TABLE_BITS);
   119  	tail = asdouble(T[idx]);
   120  	/* This is only a valid scale when -1023*N < k < 1024*N.  */
   121  	sbits = T[idx + 1] + top;
   122  	/* exp(x) = 2^(k/N) * exp(r) ~= scale + scale * (tail + exp(r) - 1).  */
   123  	/* Evaluation is optimized assuming superscalar pipelined execution.  */
   124  	r2 = r * r;
   125  	/* Without fma the worst case error is 0.25/N ulp larger.  */
   126  	/* Worst case error is less than 0.5+1.11/N+(abs poly error * 2^53) ulp.  */
   127  	tmp = tail + r + r2 * (C2 + r * C3) + r2 * r2 * (C4 + r * C5);
   128  	if (predict_false(abstop == 0))
   129  		return specialcase(tmp, sbits, ki);
   130  	scale = asdouble(sbits);
   131  	/* Note: tmp == 0 or |tmp| > 2^-200 and scale > 2^-739, so there
   132  	   is no spurious underflow here even without fma.  */
   133  	return eval_as_double(scale + scale * tmp);
   134  }