-/* origin: FreeBSD /usr/src/lib/msun/src/s_exp2f.c */
-/*-
- * Copyright (c) 2005 David Schultz <das@FreeBSD.ORG>
- * All rights reserved.
- *
- * Redistribution and use in source and binary forms, with or without
- * modification, are permitted provided that the following conditions
- * are met:
- * 1. Redistributions of source code must retain the above copyright
- * notice, this list of conditions and the following disclaimer.
- * 2. Redistributions in binary form must reproduce the above copyright
- * notice, this list of conditions and the following disclaimer in the
- * documentation and/or other materials provided with the distribution.
+/*
+ * Single-precision 2^x function.
*
- * THIS SOFTWARE IS PROVIDED BY THE AUTHOR AND CONTRIBUTORS ``AS IS'' AND
- * ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
- * IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE
- * ARE DISCLAIMED. IN NO EVENT SHALL THE AUTHOR OR CONTRIBUTORS BE LIABLE
- * FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL
- * DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS
- * OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION)
- * HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT
- * LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY
- * OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF
- * SUCH DAMAGE.
+ * Copyright (c) 2017-2018, Arm Limited.
+ * SPDX-License-Identifier: MIT
*/
+#include <math.h>
+#include <stdint.h>
#include "libm.h"
+#include "exp2f_data.h"
-#define TBLSIZE 16
+/*
+EXP2F_TABLE_BITS = 5
+EXP2F_POLY_ORDER = 3
-static const float
-redux = 0x1.8p23f / TBLSIZE,
-P1 = 0x1.62e430p-1f,
-P2 = 0x1.ebfbe0p-3f,
-P3 = 0x1.c6b348p-5f,
-P4 = 0x1.3b2c9cp-7f;
+ULP error: 0.502 (nearest rounding.)
+Relative error: 1.69 * 2^-34 in [-1/64, 1/64] (before rounding.)
+Wrong count: 168353 (all nearest rounding wrong results with fma.)
+Non-nearest ULP error: 1 (rounded ULP error)
+*/
-static const double exp2ft[TBLSIZE] = {
- 0x1.6a09e667f3bcdp-1,
- 0x1.7a11473eb0187p-1,
- 0x1.8ace5422aa0dbp-1,
- 0x1.9c49182a3f090p-1,
- 0x1.ae89f995ad3adp-1,
- 0x1.c199bdd85529cp-1,
- 0x1.d5818dcfba487p-1,
- 0x1.ea4afa2a490dap-1,
- 0x1.0000000000000p+0,
- 0x1.0b5586cf9890fp+0,
- 0x1.172b83c7d517bp+0,
- 0x1.2387a6e756238p+0,
- 0x1.306fe0a31b715p+0,
- 0x1.3dea64c123422p+0,
- 0x1.4bfdad5362a27p+0,
- 0x1.5ab07dd485429p+0,
-};
+#define N (1 << EXP2F_TABLE_BITS)
+#define T __exp2f_data.tab
+#define C __exp2f_data.poly
+#define SHIFT __exp2f_data.shift_scaled
+
+static inline uint32_t top12(float x)
+{
+ return asuint(x) >> 20;
+}
-/*
- * exp2f(x): compute the base 2 exponential of x
- *
- * Accuracy: Peak error < 0.501 ulp; location of peak: -0.030110927.
- *
- * Method: (equally-spaced tables)
- *
- * Reduce x:
- * x = 2**k + y, for integer k and |y| <= 1/2.
- * Thus we have exp2f(x) = 2**k * exp2(y).
- *
- * Reduce y:
- * y = i/TBLSIZE + z for integer i near y * TBLSIZE.
- * Thus we have exp2(y) = exp2(i/TBLSIZE) * exp2(z),
- * with |z| <= 2**-(TBLSIZE+1).
- *
- * We compute exp2(i/TBLSIZE) via table lookup and exp2(z) via a
- * degree-4 minimax polynomial with maximum error under 1.4 * 2**-33.
- * Using double precision for everything except the reduction makes
- * roundoff error insignificant and simplifies the scaling step.
- *
- * This method is due to Tang, but I do not use his suggested parameters:
- *
- * Tang, P. Table-driven Implementation of the Exponential Function
- * in IEEE Floating-Point Arithmetic. TOMS 15(2), 144-157 (1989).
- */
float exp2f(float x)
{
- double tv, twopk, u, z;
- float t;
- uint32_t hx, ix, i0, k;
+ uint32_t abstop;
+ uint64_t ki, t;
+ double_t kd, xd, z, r, r2, y, s;
- /* Filter out exceptional cases. */
- GET_FLOAT_WORD(hx, x);
- ix = hx & 0x7fffffff;
- if (ix >= 0x43000000) { /* |x| >= 128 */
- if (ix >= 0x7f800000) {
- if (hx == 0xff800000) /* -inf */
- return 0;
- return x;
- }
- if (x >= 128) {
- STRICT_ASSIGN(float, x, x * 0x1p127f);
- return x;
- }
- if (x <= -150) {
- STRICT_ASSIGN(float, x, 0x1p-100f*0x1p-100f);
- return x;
- }
- } else if (ix <= 0x33000000) { /* |x| <= 0x1p-25 */
- return 1.0f + x;
+ xd = (double_t)x;
+ abstop = top12(x) & 0x7ff;
+ if (predict_false(abstop >= top12(128.0f))) {
+ /* |x| >= 128 or x is nan. */
+ if (asuint(x) == asuint(-INFINITY))
+ return 0.0f;
+ if (abstop >= top12(INFINITY))
+ return x + x;
+ if (x > 0.0f)
+ return __math_oflowf(0);
+ if (x <= -150.0f)
+ return __math_uflowf(0);
}
- /* Reduce x, computing z, i0, and k. */
- STRICT_ASSIGN(float, t, x + redux);
- GET_FLOAT_WORD(i0, t);
- i0 += TBLSIZE / 2;
- k = (i0 / TBLSIZE) << 20;
- i0 &= TBLSIZE - 1;
- t -= redux;
- z = x - t;
- INSERT_WORDS(twopk, 0x3ff00000 + k, 0);
-
- /* Compute r = exp2(y) = exp2ft[i0] * p(z). */
- tv = exp2ft[i0];
- u = tv * z;
- tv = tv + u * (P1 + z * P2) + u * (z * z) * (P3 + z * P4);
+ /* x = k/N + r with r in [-1/(2N), 1/(2N)] and int k. */
+ kd = eval_as_double(xd + SHIFT);
+ ki = asuint64(kd);
+ kd -= SHIFT; /* k/N for int k. */
+ r = xd - kd;
- /* Scale by 2**(k>>20). */
- return tv * twopk;
+ /* exp2(x) = 2^(k/N) * 2^r ~= s * (C0*r^3 + C1*r^2 + C2*r + 1) */
+ t = T[ki % N];
+ t += ki << (52 - EXP2F_TABLE_BITS);
+ s = asdouble(t);
+ z = C[0] * r + C[1];
+ r2 = r * r;
+ y = C[2] * r + 1;
+ y = z * r2 + y;
+ y = y * s;
+ return eval_as_float(y);
}
projects / musl / blobdiff