-/* origin: FreeBSD /usr/src/lib/msun/src/e_exp.c */
/*
- * ====================================================
- * Copyright (C) 2004 by Sun Microsystems, Inc. All rights reserved.
+ * Double-precision e^x function.
*
- * Permission to use, copy, modify, and distribute this
- * software is freely granted, provided that this notice
- * is preserved.
- * ====================================================
- */
-/* exp(x)
- * Returns the exponential of x.
- *
- * Method
- * 1. Argument reduction:
- * Reduce x to an r so that |r| <= 0.5*ln2 ~ 0.34658.
- * Given x, find r and integer k such that
- *
- * x = k*ln2 + r, |r| <= 0.5*ln2.
- *
- * Here r will be represented as r = hi-lo for better
- * accuracy.
- *
- * 2. Approximation of exp(r) by a special rational function on
- * the interval [0,0.34658]:
- * Write
- * R(r**2) = r*(exp(r)+1)/(exp(r)-1) = 2 + r*r/6 - r**4/360 + ...
- * We use a special Remez algorithm on [0,0.34658] to generate
- * a polynomial of degree 5 to approximate R. The maximum error
- * of this polynomial approximation is bounded by 2**-59. In
- * other words,
- * R(z) ~ 2.0 + P1*z + P2*z**2 + P3*z**3 + P4*z**4 + P5*z**5
- * (where z=r*r, and the values of P1 to P5 are listed below)
- * and
- * | 5 | -59
- * | 2.0+P1*z+...+P5*z - R(z) | <= 2
- * | |
- * The computation of exp(r) thus becomes
- * 2*r
- * exp(r) = 1 + ----------
- * R(r) - r
- * r*c(r)
- * = 1 + r + ----------- (for better accuracy)
- * 2 - c(r)
- * where
- * 2 4 10
- * c(r) = r - (P1*r + P2*r + ... + P5*r ).
- *
- * 3. Scale back to obtain exp(x):
- * From step 1, we have
- * exp(x) = 2^k * exp(r)
- *
- * Special cases:
- * exp(INF) is INF, exp(NaN) is NaN;
- * exp(-INF) is 0, and
- * for finite argument, only exp(0)=1 is exact.
- *
- * Accuracy:
- * according to an error analysis, the error is always less than
- * 1 ulp (unit in the last place).
- *
- * Misc. info.
- * For IEEE double
- * if x > 709.782712893383973096 then exp(x) overflows
- * if x < -745.133219101941108420 then exp(x) underflows
+ * Copyright (c) 2018, Arm Limited.
+ * SPDX-License-Identifier: MIT
*/
+#include <math.h>
+#include <stdint.h>
#include "libm.h"
+#include "exp_data.h"
-static const double
-half[2] = {0.5,-0.5},
-ln2hi = 6.93147180369123816490e-01, /* 0x3fe62e42, 0xfee00000 */
-ln2lo = 1.90821492927058770002e-10, /* 0x3dea39ef, 0x35793c76 */
-invln2 = 1.44269504088896338700e+00, /* 0x3ff71547, 0x652b82fe */
-P1 = 1.66666666666666019037e-01, /* 0x3FC55555, 0x5555553E */
-P2 = -2.77777777770155933842e-03, /* 0xBF66C16C, 0x16BEBD93 */
-P3 = 6.61375632143793436117e-05, /* 0x3F11566A, 0xAF25DE2C */
-P4 = -1.65339022054652515390e-06, /* 0xBEBBBD41, 0xC5D26BF1 */
-P5 = 4.13813679705723846039e-08; /* 0x3E663769, 0x72BEA4D0 */
+#define N (1 << EXP_TABLE_BITS)
+#define InvLn2N __exp_data.invln2N
+#define NegLn2hiN __exp_data.negln2hiN
+#define NegLn2loN __exp_data.negln2loN
+#define Shift __exp_data.shift
+#define T __exp_data.tab
+#define C2 __exp_data.poly[5 - EXP_POLY_ORDER]
+#define C3 __exp_data.poly[6 - EXP_POLY_ORDER]
+#define C4 __exp_data.poly[7 - EXP_POLY_ORDER]
+#define C5 __exp_data.poly[8 - EXP_POLY_ORDER]
-double exp(double x)
+/* Handle cases that may overflow or underflow when computing the result that
+ is scale*(1+TMP) without intermediate rounding. The bit representation of
+ scale is in SBITS, however it has a computed exponent that may have
+ overflown into the sign bit so that needs to be adjusted before using it as
+ a double. (int32_t)KI is the k used in the argument reduction and exponent
+ adjustment of scale, positive k here means the result may overflow and
+ negative k means the result may underflow. */
+static inline double specialcase(double_t tmp, uint64_t sbits, uint64_t ki)
{
- double hi, lo, c, xx;
- int k, sign;
- uint32_t hx;
-
- GET_HIGH_WORD(hx, x);
- sign = hx>>31;
- hx &= 0x7fffffff; /* high word of |x| */
+ double_t scale, y;
- /* special cases */
- if (hx >= 0x40862e42) { /* if |x| >= 709.78... */
- if (isnan(x))
- return x;
- if (hx == 0x7ff00000 && sign) /* -inf */
- return 0;
- if (x > 709.782712893383973096) {
- /* overflow if x!=inf */
- STRICT_ASSIGN(double, x, 0x1p1023 * x);
- return x;
- }
- if (x < -745.13321910194110842) {
- /* underflow */
- STRICT_ASSIGN(double, x, 0x1p-1000 * 0x1p-1000);
- return x;
- }
+ if ((ki & 0x80000000) == 0) {
+ /* k > 0, the exponent of scale might have overflowed by <= 460. */
+ sbits -= 1009ull << 52;
+ scale = asdouble(sbits);
+ y = 0x1p1009 * (scale + scale * tmp);
+ return eval_as_double(y);
+ }
+ /* k < 0, need special care in the subnormal range. */
+ sbits += 1022ull << 52;
+ scale = asdouble(sbits);
+ y = scale + scale * tmp;
+ if (y < 1.0) {
+ /* Round y to the right precision before scaling it into the subnormal
+ range to avoid double rounding that can cause 0.5+E/2 ulp error where
+ E is the worst-case ulp error outside the subnormal range. So this
+ is only useful if the goal is better than 1 ulp worst-case error. */
+ double_t hi, lo;
+ lo = scale - y + scale * tmp;
+ hi = 1.0 + y;
+ lo = 1.0 - hi + y + lo;
+ y = eval_as_double(hi + lo) - 1.0;
+ /* Avoid -0.0 with downward rounding. */
+ if (WANT_ROUNDING && y == 0.0)
+ y = 0.0;
+ /* The underflow exception needs to be signaled explicitly. */
+ fp_force_eval(fp_barrier(0x1p-1022) * 0x1p-1022);
}
+ y = 0x1p-1022 * y;
+ return eval_as_double(y);
+}
- /* argument reduction */
- if (hx > 0x3fd62e42) { /* if |x| > 0.5 ln2 */
- if (hx >= 0x3ff0a2b2) /* if |x| >= 1.5 ln2 */
- k = (int)(invln2*x + half[sign]);
- else
- k = 1 - sign - sign;
- hi = x - k*ln2hi; /* k*ln2hi is exact here */
- lo = k*ln2lo;
- STRICT_ASSIGN(double, x, hi - lo);
- } else if (hx > 0x3e300000) { /* if |x| > 2**-28 */
- k = 0;
- hi = x;
- lo = 0;
- } else {
- /* inexact if x!=0 */
- FORCE_EVAL(0x1p1023 + x);
- return 1 + x;
+/* Top 12 bits of a double (sign and exponent bits). */
+static inline uint32_t top12(double x)
+{
+ return asuint64(x) >> 52;
+}
+
+double exp(double x)
+{
+ uint32_t abstop;
+ uint64_t ki, idx, top, sbits;
+ double_t kd, z, r, r2, scale, tail, tmp;
+
+ abstop = top12(x) & 0x7ff;
+ if (predict_false(abstop - top12(0x1p-54) >= top12(512.0) - top12(0x1p-54))) {
+ if (abstop - top12(0x1p-54) >= 0x80000000)
+ /* Avoid spurious underflow for tiny x. */
+ /* Note: 0 is common input. */
+ return WANT_ROUNDING ? 1.0 + x : 1.0;
+ if (abstop >= top12(1024.0)) {
+ if (asuint64(x) == asuint64(-INFINITY))
+ return 0.0;
+ if (abstop >= top12(INFINITY))
+ return 1.0 + x;
+ if (asuint64(x) >> 63)
+ return __math_uflow(0);
+ else
+ return __math_oflow(0);
+ }
+ /* Large x is special cased below. */
+ abstop = 0;
}
- /* x is now in primary range */
- xx = x*x;
- c = x - xx*(P1+xx*(P2+xx*(P3+xx*(P4+xx*P5))));
- x = 1 + (x*c/(2-c) - lo + hi);
- if (k == 0)
- return x;
- return scalbn(x, k);
+ /* exp(x) = 2^(k/N) * exp(r), with exp(r) in [2^(-1/2N),2^(1/2N)]. */
+ /* x = ln2/N*k + r, with int k and r in [-ln2/2N, ln2/2N]. */
+ z = InvLn2N * x;
+#if TOINT_INTRINSICS
+ kd = roundtoint(z);
+ ki = converttoint(z);
+#elif EXP_USE_TOINT_NARROW
+ /* z - kd is in [-0.5-2^-16, 0.5] in all rounding modes. */
+ kd = eval_as_double(z + Shift);
+ ki = asuint64(kd) >> 16;
+ kd = (double_t)(int32_t)ki;
+#else
+ /* z - kd is in [-1, 1] in non-nearest rounding modes. */
+ kd = eval_as_double(z + Shift);
+ ki = asuint64(kd);
+ kd -= Shift;
+#endif
+ r = x + kd * NegLn2hiN + kd * NegLn2loN;
+ /* 2^(k/N) ~= scale * (1 + tail). */
+ idx = 2 * (ki % N);
+ top = ki << (52 - EXP_TABLE_BITS);
+ tail = asdouble(T[idx]);
+ /* This is only a valid scale when -1023*N < k < 1024*N. */
+ sbits = T[idx + 1] + top;
+ /* exp(x) = 2^(k/N) * exp(r) ~= scale + scale * (tail + exp(r) - 1). */
+ /* Evaluation is optimized assuming superscalar pipelined execution. */
+ r2 = r * r;
+ /* Without fma the worst case error is 0.25/N ulp larger. */
+ /* Worst case error is less than 0.5+1.11/N+(abs poly error * 2^53) ulp. */
+ tmp = tail + r + r2 * (C2 + r * C3) + r2 * r2 * (C4 + r * C5);
+ if (predict_false(abstop == 0))
+ return specialcase(tmp, sbits, ki);
+ scale = asdouble(sbits);
+ /* Note: tmp == 0 or |tmp| > 2^-200 and scale > 2^-739, so there
+ is no spurious underflow here even without fma. */
+ return eval_as_double(scale + scale * tmp);
}
projects / musl / blobdiff