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.  */
 31static 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).  */
 67static inline uint32_t top12(double x)
 68{
 69	return asuint64(x) >> 52;
 70}
 71
 72double 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}