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/*
* Single-precision log(1+x) function.
*
* Copyright (c) 2022-2023, Arm Limited.
* SPDX-License-Identifier: MIT OR Apache-2.0 WITH LLVM-exception
*/
#include "hornerf.h"
#include "math_config.h"
#include "pl_sig.h"
#include "pl_test.h"
#define Ln2 (0x1.62e43p-1f)
#define SignMask (0x80000000)
/* Biased exponent of the largest float m for which m^8 underflows. */
#define M8UFLOW_BOUND_BEXP 112
/* Biased exponent of the largest float for which we just return x. */
#define TINY_BOUND_BEXP 103
#define C(i) __log1pf_data.coeffs[i]
static inline float
eval_poly (float m, uint32_t e)
{
#ifdef LOG1PF_2U5
/* 2.5 ulp variant. Approximate log(1+m) on [-0.25, 0.5] using
slightly modified Estrin scheme (no x^0 term, and x term is just x). */
float p_12 = fmaf (m, C (1), C (0));
float p_34 = fmaf (m, C (3), C (2));
float p_56 = fmaf (m, C (5), C (4));
float p_78 = fmaf (m, C (7), C (6));
float m2 = m * m;
float p_02 = fmaf (m2, p_12, m);
float p_36 = fmaf (m2, p_56, p_34);
float p_79 = fmaf (m2, C (8), p_78);
float m4 = m2 * m2;
float p_06 = fmaf (m4, p_36, p_02);
if (unlikely (e < M8UFLOW_BOUND_BEXP))
return p_06;
float m8 = m4 * m4;
return fmaf (m8, p_79, p_06);
#elif defined(LOG1PF_1U3)
/* 1.3 ulp variant. Approximate log(1+m) on [-0.25, 0.5] using Horner
scheme. Our polynomial approximation for log1p has the form
x + C1 * x^2 + C2 * x^3 + C3 * x^4 + ...
Hence approximation has the form m + m^2 * P(m)
where P(x) = C1 + C2 * x + C3 * x^2 + ... . */
return fmaf (m, m * HORNER_8 (m, C), m);
#else
#error No log1pf approximation exists with the requested precision. Options are 13 or 25.
#endif
}
static inline uint32_t
biased_exponent (uint32_t ix)
{
return (ix & 0x7f800000) >> 23;
}
/* log1pf approximation using polynomial on reduced interval. Worst-case error
when using Estrin is roughly 2.02 ULP:
log1pf(0x1.21e13ap-2) got 0x1.fe8028p-3 want 0x1.fe802cp-3. */
float
log1pf (float x)
{
uint32_t ix = asuint (x);
uint32_t ia = ix & ~SignMask;
uint32_t ia12 = ia >> 20;
uint32_t e = biased_exponent (ix);
/* Handle special cases first. */
if (unlikely (ia12 >= 0x7f8 || ix >= 0xbf800000 || ix == 0x80000000
|| e <= TINY_BOUND_BEXP))
{
if (ix == 0xff800000)
{
/* x == -Inf => log1pf(x) = NaN. */
return NAN;
}
if ((ix == 0x7f800000 || e <= TINY_BOUND_BEXP) && ia12 <= 0x7f8)
{
/* |x| < TinyBound => log1p(x) = x.
x == Inf => log1pf(x) = Inf. */
return x;
}
if (ix == 0xbf800000)
{
/* x == -1.0 => log1pf(x) = -Inf. */
return __math_divzerof (-1);
}
if (ia12 >= 0x7f8)
{
/* x == +/-NaN => log1pf(x) = NaN. */
return __math_invalidf (asfloat (ia));
}
/* x < -1.0 => log1pf(x) = NaN. */
return __math_invalidf (x);
}
/* With x + 1 = t * 2^k (where t = m + 1 and k is chosen such that m
is in [-0.25, 0.5]):
log1p(x) = log(t) + log(2^k) = log1p(m) + k*log(2).
We approximate log1p(m) with a polynomial, then scale by
k*log(2). Instead of doing this directly, we use an intermediate
scale factor s = 4*k*log(2) to ensure the scale is representable
as a normalised fp32 number. */
if (ix <= 0x3f000000 || ia <= 0x3e800000)
{
/* If x is in [-0.25, 0.5] then we can shortcut all the logic
below, as k = 0 and m = x. All we need is to return the
polynomial. */
return eval_poly (x, e);
}
float m = x + 1.0f;
/* k is used scale the input. 0x3f400000 is chosen as we are trying to
reduce x to the range [-0.25, 0.5]. Inside this range, k is 0.
Outside this range, if k is reinterpreted as (NOT CONVERTED TO) float:
let k = sign * 2^p where sign = -1 if x < 0
1 otherwise
and p is a negative integer whose magnitude increases with the
magnitude of x. */
int k = (asuint (m) - 0x3f400000) & 0xff800000;
/* By using integer arithmetic, we obtain the necessary scaling by
subtracting the unbiased exponent of k from the exponent of x. */
float m_scale = asfloat (asuint (x) - k);
/* Scale up to ensure that the scale factor is representable as normalised
fp32 number (s in [2**-126,2**26]), and scale m down accordingly. */
float s = asfloat (asuint (4.0f) - k);
m_scale = m_scale + fmaf (0.25f, s, -1.0f);
float p = eval_poly (m_scale, biased_exponent (asuint (m_scale)));
/* The scale factor to be applied back at the end - by multiplying float(k)
by 2^-23 we get the unbiased exponent of k. */
float scale_back = (float) k * 0x1.0p-23f;
/* Apply the scaling back. */
return fmaf (scale_back, Ln2, p);
}
PL_SIG (S, F, 1, log1p, -0.9, 10.0)
PL_TEST_ULP (log1pf, 1.52)
PL_TEST_INTERVAL (log1pf, -10.0, 10.0, 10000)
PL_TEST_INTERVAL (log1pf, 0.0, 0x1p-23, 50000)
PL_TEST_INTERVAL (log1pf, 0x1p-23, 0.001, 50000)
PL_TEST_INTERVAL (log1pf, 0.001, 1.0, 50000)
PL_TEST_INTERVAL (log1pf, 0.0, -0x1p-23, 50000)
PL_TEST_INTERVAL (log1pf, -0x1p-23, -0.001, 50000)
PL_TEST_INTERVAL (log1pf, -0.001, -1.0, 50000)
PL_TEST_INTERVAL (log1pf, -1.0, inf, 5000)
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