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/*
* Double-precision 10^x function.
*
* Copyright (c) 2023-2024, Arm Limited.
* SPDX-License-Identifier: MIT OR Apache-2.0 WITH LLVM-exception
*/
#include "math_config.h"
#include "test_defs.h"
#include "test_sig.h"
#define N (1 << EXP_TABLE_BITS)
#define IndexMask (N - 1)
#define OFlowBound 0x1.34413509f79ffp8 /* log10(DBL_MAX). */
#define UFlowBound -0x1.5ep+8 /* -350. */
#define SmallTop 0x3c6 /* top12(0x1p-57). */
#define BigTop 0x407 /* top12(0x1p8). */
#define Thresh 0x41 /* BigTop - SmallTop. */
#define Shift __exp_data.shift
#define C(i) __exp_data.exp10_poly[i]
static double
special_case (uint64_t sbits, double_t tmp, uint64_t ki)
{
double_t scale, y;
if ((ki & 0x80000000) == 0)
{
/* The exponent of scale might have overflowed by 1. */
sbits -= 1ull << 52;
scale = asdouble (sbits);
y = 2 * (scale + scale * tmp);
return check_oflow (eval_as_double (y));
}
/* n < 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 lo = scale - y + scale * tmp;
double_t 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. */
force_eval_double (opt_barrier_double (0x1p-1022) * 0x1p-1022);
}
y = 0x1p-1022 * y;
return check_uflow (y);
}
/* Double-precision 10^x approximation. Largest observed error is ~0.513 ULP. */
double
exp10 (double x)
{
uint64_t ix = asuint64 (x);
uint32_t abstop = (ix >> 52) & 0x7ff;
if (unlikely (abstop - SmallTop >= Thresh))
{
if (abstop - SmallTop >= 0x80000000)
/* Avoid spurious underflow for tiny x.
Note: 0 is common input. */
return x + 1;
if (abstop == 0x7ff)
return ix == asuint64 (-INFINITY) ? 0.0 : x + 1.0;
if (x >= OFlowBound)
return __math_oflow (0);
if (x < UFlowBound)
return __math_uflow (0);
/* Large x is special-cased below. */
abstop = 0;
}
/* Reduce x: z = x * N / log10(2), k = round(z). */
double_t z = __exp_data.invlog10_2N * x;
double_t kd;
uint64_t ki;
#if TOINT_INTRINSICS
kd = roundtoint (z);
ki = converttoint (z);
#else
kd = eval_as_double (z + Shift);
ki = asuint64 (kd);
kd -= Shift;
#endif
/* r = x - k * log10(2), r in [-0.5, 0.5]. */
double_t r = x;
r = __exp_data.neglog10_2hiN * kd + r;
r = __exp_data.neglog10_2loN * kd + r;
/* exp10(x) = 2^(k/N) * 2^(r/N).
Approximate the two components separately. */
/* s = 2^(k/N), using lookup table. */
uint64_t e = ki << (52 - EXP_TABLE_BITS);
uint64_t i = (ki & IndexMask) * 2;
uint64_t u = __exp_data.tab[i + 1];
uint64_t sbits = u + e;
double_t tail = asdouble (__exp_data.tab[i]);
/* 2^(r/N) ~= 1 + r * Poly(r). */
double_t r2 = r * r;
double_t p = C (0) + r * C (1);
double_t y = C (2) + r * C (3);
y = y + r2 * C (4);
y = p + r2 * y;
y = tail + y * r;
if (unlikely (abstop == 0))
return special_case (sbits, y, ki);
/* Assemble components:
y = 2^(r/N) * 2^(k/N)
~= (y + 1) * s. */
double_t s = asdouble (sbits);
return eval_as_double (s * y + s);
}
#if WANT_EXP10_TESTS
TEST_SIG (S, D, 1, exp10, -9.9, 9.9)
TEST_ULP (exp10, 0.02)
TEST_ULP_NONNEAREST (exp10, 0.5)
TEST_SYM_INTERVAL (exp10, 0, 0x1p-47, 5000)
TEST_SYM_INTERVAL (exp10, 0x1p47, 1, 50000)
TEST_INTERVAL (exp10, 1, OFlowBound, 50000)
TEST_INTERVAL (exp10, -1, UFlowBound, 50000)
TEST_INTERVAL (exp10, OFlowBound, inf, 5000)
TEST_INTERVAL (exp10, UFlowBound, -inf, 5000)
#endif
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