#include "sv_math.h"
-/* For x < -Thres, the result is subnormal and not handled correctly by
- FEXPA. */
-#define Thres 37.9
+/* For x < -Thres (-log10(2^126)), the result is subnormal and not handled
+ correctly by FEXPA. */
+#define Thres 0x1.2f702p+5
static const struct data
{
- float log2_10_lo, c0, c2, c4;
- float c1, c3, log10_2;
- float shift, log2_10_hi, thres;
+ float log10_2, log2_10_hi, log2_10_lo, c1;
+ float c0, shift, thres;
} data = {
/* Coefficients generated using Remez algorithm with minimisation of relative
- error.
- rel error: 0x1.89dafa3p-24
- abs error: 0x1.167d55p-23 in [-log10(2)/2, log10(2)/2]
- maxerr: 0.52 +0.5 ulp. */
- .c0 = 0x1.26bb16p+1f,
- .c1 = 0x1.5350d2p+1f,
- .c2 = 0x1.04744ap+1f,
- .c3 = 0x1.2d8176p+0f,
- .c4 = 0x1.12b41ap-1f,
+ error. */
+ .c0 = 0x1.26bb62p1,
+ .c1 = 0x1.53524cp1,
/* 1.5*2^17 + 127, a shift value suitable for FEXPA. */
.shift = 0x1.803f8p17f,
.log10_2 = 0x1.a934fp+1,
/* exp10(x) = 2^(n/N) * 10^r = 2^n * (1 + poly (r)),
with poly(r) in [1/sqrt(2), sqrt(2)] and
x = r + n * log10(2) / N, with r in [-log10(2)/2N, log10(2)/2N]. */
-
- svfloat32_t lane_consts = svld1rq (svptrue_b32 (), &d->log2_10_lo);
+ svfloat32_t lane_consts = svld1rq (svptrue_b32 (), &d->log10_2);
/* n = round(x/(log10(2)/N)). */
svfloat32_t shift = sv_f32 (d->shift);
- svfloat32_t z = svmad_x (pg, sv_f32 (d->log10_2), x, shift);
- svfloat32_t n = svsub_x (svptrue_b32 (), z, shift);
+ svfloat32_t z = svmla_lane (shift, x, lane_consts, 0);
+ svfloat32_t n = svsub_x (pg, z, shift);
/* r = x - n*log10(2)/N. */
- svfloat32_t r = svmsb_x (pg, sv_f32 (d->log2_10_hi), n, x);
- r = svmls_lane (r, n, lane_consts, 0);
+ svfloat32_t r = x;
+ r = svmls_lane (r, n, lane_consts, 1);
+ r = svmls_lane (r, n, lane_consts, 2);
svfloat32_t scale = svexpa (svreinterpret_u32 (z));
/* Polynomial evaluation: poly(r) ~ exp10(r)-1. */
- svfloat32_t p12 = svmla_lane (sv_f32 (d->c1), r, lane_consts, 2);
- svfloat32_t p34 = svmla_lane (sv_f32 (d->c3), r, lane_consts, 3);
- svfloat32_t r2 = svmul_x (svptrue_b32 (), r, r);
- svfloat32_t p14 = svmla_x (pg, p12, p34, r2);
- svfloat32_t p0 = svmul_lane (r, lane_consts, 1);
- svfloat32_t poly = svmla_x (pg, p0, r2, p14);
-
+ svfloat32_t poly = svmla_lane (sv_f32 (d->c0), r, lane_consts, 3);
+ poly = svmul_x (pg, poly, r);
return svmla_x (pg, scale, scale, poly);
}
special);
}
-/* Single-precision SVE exp10f routine. Implements the same algorithm
- as AdvSIMD exp10f.
- Worst case error is 1.02 ULPs.
- _ZGVsMxv_exp10f(-0x1.040488p-4) got 0x1.ba5f9ep-1
- want 0x1.ba5f9cp-1. */
+/* Single-precision SVE exp10f routine. Based on the FEXPA instruction.
+ Worst case error is 1.10 ULP.
+ _ZGVsMxv_exp10f (0x1.cc76dep+3) got 0x1.be0172p+47
+ want 0x1.be017p+47. */
svfloat32_t SV_NAME_F1 (exp10) (svfloat32_t x, const svbool_t pg)
{
const struct data *d = ptr_barrier (&data);
#include "sv_math.h"
-#define N (1 << V_EXP_TABLE_BITS)
-
#define BigBound 1022
#define UOFlowBound 1280
static const struct data
{
- double c0, c2;
- double c1, c3;
+ double c2, c4;
+ double c0, c1, c3;
double shift, big_bound, uoflow_bound;
} data = {
/* Coefficients are computed using Remez algorithm with
minimisation of the absolute error. */
- .c0 = 0x1.62e42fefa3686p-1, .c1 = 0x1.ebfbdff82c241p-3,
- .c2 = 0x1.c6b09b16de99ap-5, .c3 = 0x1.3b2abf5571ad8p-7,
- .shift = 0x1.8p52 / N, .uoflow_bound = UOFlowBound,
- .big_bound = BigBound,
+ .c0 = 0x1.62e42fefa39efp-1, .c1 = 0x1.ebfbdff82a31bp-3,
+ .c2 = 0x1.c6b08d706c8a5p-5, .c3 = 0x1.3b2ad2ff7d2f3p-7,
+ .c4 = 0x1.5d8761184beb3p-10, .shift = 0x1.800000000ffc0p+46,
+ .uoflow_bound = UOFlowBound, .big_bound = BigBound,
};
#define SpecialOffset 0x6000000000000000 /* 0x1p513. */
svadd_x (pg, svsub_x (pg, svreinterpret_u64 (s), SpecialBias2), b));
/* |n| > 1280 => 2^(n) overflows. */
- svbool_t p_cmp = svacgt (pg, n, d->uoflow_bound);
+ svbool_t p_cmp = svacle (pg, n, d->uoflow_bound);
svfloat64_t r1 = svmul_x (svptrue_b64 (), s1, s1);
svfloat64_t r2 = svmla_x (pg, s2, s2, y);
svfloat64_t r0 = svmul_x (svptrue_b64 (), r2, s1);
- return svsel (p_cmp, r1, r0);
+ return svsel (p_cmp, r0, r1);
}
/* Fast vector implementation of exp2.
- Maximum measured error is 1.65 ulp.
- _ZGVsMxv_exp2(-0x1.4c264ab5b559bp-6) got 0x1.f8db0d4df721fp-1
- want 0x1.f8db0d4df721dp-1. */
+ Maximum measured error is 0.52 + 0.5 ulp.
+ _ZGVsMxv_exp2 (0x1.3b72ad5b701bfp-1) got 0x1.8861641b49e08p+0
+ want 0x1.8861641b49e07p+0. */
svfloat64_t SV_NAME_D1 (exp2) (svfloat64_t x, svbool_t pg)
{
const struct data *d = ptr_barrier (&data);
- svbool_t no_big_scale = svacle (pg, x, d->big_bound);
- svbool_t special = svnot_z (pg, no_big_scale);
-
- /* Reduce x to k/N + r, where k is integer and r in [-1/2N, 1/2N]. */
- svfloat64_t shift = sv_f64 (d->shift);
- svfloat64_t kd = svadd_x (pg, x, shift);
- svuint64_t ki = svreinterpret_u64 (kd);
- /* kd = k/N. */
- kd = svsub_x (pg, kd, shift);
- svfloat64_t r = svsub_x (pg, x, kd);
-
- /* scale ~= 2^(k/N). */
- svuint64_t idx = svand_x (pg, ki, N - 1);
- svuint64_t sbits = svld1_gather_index (pg, __v_exp_data, idx);
- /* This is only a valid scale when -1023*N < k < 1024*N. */
- svuint64_t top = svlsl_x (pg, ki, 52 - V_EXP_TABLE_BITS);
- svfloat64_t scale = svreinterpret_f64 (svadd_x (pg, sbits, top));
-
- svfloat64_t c13 = svld1rq (svptrue_b64 (), &d->c1);
- /* Approximate exp2(r) using polynomial. */
- /* y = exp2(r) - 1 ~= C0 r + C1 r^2 + C2 r^3 + C3 r^4. */
+ svbool_t special = svacge (pg, x, d->big_bound);
+
+ svfloat64_t z = svadd_x (svptrue_b64 (), x, d->shift);
+ svfloat64_t n = svsub_x (svptrue_b64 (), z, d->shift);
+ svfloat64_t r = svsub_x (svptrue_b64 (), x, n);
+
+ svfloat64_t scale = svexpa (svreinterpret_u64 (z));
+
svfloat64_t r2 = svmul_x (svptrue_b64 (), r, r);
- svfloat64_t p01 = svmla_lane (sv_f64 (d->c0), r, c13, 0);
- svfloat64_t p23 = svmla_lane (sv_f64 (d->c2), r, c13, 1);
- svfloat64_t p = svmla_x (pg, p01, p23, r2);
+ svfloat64_t c24 = svld1rq (svptrue_b64 (), &d->c2);
+
+ /* Approximate exp2(r) using polynomial. */
+ /* y = exp2(r) - 1 ~= r * (C0 + C1 r + C2 r^2 + C3 r^3 + C4 r^4). */
+ svfloat64_t p12 = svmla_lane (sv_f64 (d->c1), r, c24, 0);
+ svfloat64_t p34 = svmla_lane (sv_f64 (d->c3), r, c24, 1);
+ svfloat64_t p = svmla_x (pg, p12, p34, r2);
+ p = svmad_x (pg, p, r, d->c0);
svfloat64_t y = svmul_x (svptrue_b64 (), r, p);
+
/* Assemble exp2(x) = exp2(r) * scale. */
if (__glibc_unlikely (svptest_any (pg, special)))
- return special_case (pg, scale, y, kd, d);
+ {
+ /* FEXPA zeroes the sign bit, however the sign is meaningful to the
+ special case function so needs to be copied.
+ e = sign bit of u << 46. */
+ svuint64_t e = svand_x (pg, svlsl_x (pg, svreinterpret_u64 (z), 46),
+ 0x8000000000000000);
+ scale = svreinterpret_f64 (svadd_x (pg, e, svreinterpret_u64 (scale)));
+ return special_case (pg, scale, y, n, d);
+ }
+
return svmla_x (pg, scale, scale, y);
}
<https://www.gnu.org/licenses/>. */
#include "sv_math.h"
-#include "poly_sve_f64.h"
-#define SpecialBound 0x1.62b7d369a5aa9p+9
-#define ExponentBias 0x3ff0000000000000
+#define FexpaBound 0x1.4cb5ecef28adap-3 /* 15*ln2/64. */
+#define SpecialBound 0x1.628c2855bfaddp+9 /* ln(2^(1023 + 1/128)). */
static const struct data
{
- double poly[11];
- double shift, inv_ln2, special_bound;
- /* To be loaded in one quad-word. */
+ double c2, c4;
+ double inv_ln2;
double ln2_hi, ln2_lo;
+ double c0, c1, c3;
+ double shift, thres;
+ uint64_t expm1_data[32];
} data = {
- /* Generated using fpminimax. */
- .poly = { 0x1p-1, 0x1.5555555555559p-3, 0x1.555555555554bp-5,
- 0x1.111111110f663p-7, 0x1.6c16c16c1b5f3p-10, 0x1.a01a01affa35dp-13,
- 0x1.a01a018b4ecbbp-16, 0x1.71ddf82db5bb4p-19, 0x1.27e517fc0d54bp-22,
- 0x1.af5eedae67435p-26, 0x1.1f143d060a28ap-29, },
-
- .special_bound = SpecialBound,
- .inv_ln2 = 0x1.71547652b82fep0,
- .ln2_hi = 0x1.62e42fefa39efp-1,
- .ln2_lo = 0x1.abc9e3b39803fp-56,
- .shift = 0x1.8p52,
+ /* Table emulating FEXPA - 1, for values of FEXPA close to 1.
+ The table holds values of 2^(i/64) - 1, computed in arbitrary precision.
+ The first half of the table stores values associated to i from 0 to 15.
+ The second half of the table stores values associated to i from 0 to -15. */
+ .expm1_data = {
+ 0x0000000000000000, 0x3f864d1f3bc03077, 0x3f966c34c5615d0f, 0x3fa0e8a30eb37901,
+ 0x3fa6ab0d9f3121ec, 0x3fac7d865a7a3440, 0x3fb1301d0125b50a, 0x3fb429aaea92ddfb,
+ 0x3fb72b83c7d517ae, 0x3fba35beb6fcb754, 0x3fbd4873168b9aa8, 0x3fc031dc431466b2,
+ 0x3fc1c3d373ab11c3, 0x3fc35a2b2f13e6e9, 0x3fc4f4efa8fef709, 0x3fc6942d3720185a,
+ 0x0000000000000000, 0xbfc331751ec3a814, 0xbfc20224341286e4, 0xbfc0cf85bed0f8b7,
+ 0xbfbf332113d56b1f, 0xbfbcc0768d4175a6, 0xbfba46f918837cb7, 0xbfb7c695afc3b424,
+ 0xbfb53f391822dbc7, 0xbfb2b0cfe1266bd4, 0xbfb01b466423250a, 0xbfaafd11874c009e,
+ 0xbfa5b505d5b6f268, 0xbfa05e4119ea5d89, 0xbf95f134923757f3, 0xbf860f9f985bc9f4,
+ },
+
+ /* Generated using Remez, in [-log(2)/128, log(2)/128]. */
+ .c0 = 0x1p-1,
+ .c1 = 0x1.55555555548f9p-3,
+ .c2 = 0x1.5555555554c22p-5,
+ .c3 = 0x1.111123aaa2fb2p-7,
+ .c4 = 0x1.6c16d77d98e5bp-10,
+ .ln2_hi = 0x1.62e42fefa3800p-1,
+ .ln2_lo = 0x1.ef35793c76730p-45,
+ .inv_ln2 = 0x1.71547652b82fep+0,
+ .shift = 0x1.800000000ffc0p+46, /* 1.5*2^46+1023. */
+ .thres = SpecialBound,
};
-static svfloat64_t NOINLINE
-special_case (svfloat64_t x, svfloat64_t y, svbool_t pg)
+#define SpecialOffset 0x6000000000000000 /* 0x1p513. */
+/* SpecialBias1 + SpecialBias1 = asuint(1.0). */
+#define SpecialBias1 0x7000000000000000 /* 0x1p769. */
+#define SpecialBias2 0x3010000000000000 /* 0x1p-254. */
+
+static NOINLINE svfloat64_t
+special_case (svbool_t pg, svfloat64_t y, svfloat64_t s, svfloat64_t p,
+ svfloat64_t n)
{
- return sv_call_f64 (expm1, x, y, pg);
+ /* s=2^n may overflow, break it up into s=s1*s2,
+ such that exp = s + s*y can be computed as s1*(s2+s2*y)
+ and s1*s1 overflows only if n>0. */
+
+ /* If n<=0 then set b to 0x6, 0 otherwise. */
+ svbool_t p_sign = svcmple (pg, n, 0.0); /* n <= 0. */
+ svuint64_t b
+ = svdup_u64_z (p_sign, SpecialOffset); /* Inactive lanes set to 0. */
+
+ /* Set s1 to generate overflow depending on sign of exponent n,
+ ie. s1 = 0x70...0 - b. */
+ svfloat64_t s1 = svreinterpret_f64 (svsubr_x (pg, b, SpecialBias1));
+ /* Offset s to avoid overflow in final result if n is below threshold.
+ ie. s2 = as_u64 (s) - 0x3010...0 + b. */
+ svfloat64_t s2 = svreinterpret_f64 (
+ svadd_x (pg, svsub_x (pg, svreinterpret_u64 (s), SpecialBias2), b));
+
+ /* |n| > 1280 => 2^(n) overflows. */
+ svbool_t p_cmp = svacgt (pg, n, 1280.0);
+
+ svfloat64_t r1 = svmul_x (svptrue_b64 (), s1, s1);
+ svfloat64_t r2 = svmla_x (pg, s2, s2, p);
+ svfloat64_t r0 = svmul_x (svptrue_b64 (), r2, s1);
+
+ svbool_t is_safe = svacle (pg, n, 1023); /* Only correct special lanes. */
+ return svsel (is_safe, y, svsub_x (pg, svsel (p_cmp, r1, r0), 1.0));
}
-/* Double-precision vector exp(x) - 1 function.
- The maximum error observed error is 2.18 ULP:
- _ZGVsMxv_expm1(0x1.634ba0c237d7bp-2) got 0x1.a8b9ea8d66e22p-2
- want 0x1.a8b9ea8d66e2p-2. */
+/* FEXPA based SVE expm1 algorithm.
+ Maximum measured error is 2.81 + 0.5 ULP:
+ _ZGVsMxv_expm1 (0x1.974060e619bfp-3) got 0x1.c290e5858bb53p-3
+ want 0x1.c290e5858bb5p-3. */
svfloat64_t SV_NAME_D1 (expm1) (svfloat64_t x, svbool_t pg)
{
const struct data *d = ptr_barrier (&data);
- /* Large, Nan/Inf. */
- svbool_t special = svnot_z (pg, svaclt (pg, x, d->special_bound));
-
- /* Reduce argument to smaller range:
- Let i = round(x / ln2)
- and f = x - i * ln2, then f is in [-ln2/2, ln2/2].
- exp(x) - 1 = 2^i * (expm1(f) + 1) - 1
- where 2^i is exact because i is an integer. */
- svfloat64_t shift = sv_f64 (d->shift);
- svfloat64_t n = svsub_x (pg, svmla_x (pg, shift, x, d->inv_ln2), shift);
- svint64_t i = svcvt_s64_x (pg, n);
- svfloat64_t ln2 = svld1rq (svptrue_b64 (), &d->ln2_hi);
- svfloat64_t f = svmls_lane (x, n, ln2, 0);
- f = svmls_lane (f, n, ln2, 1);
-
- /* Approximate expm1(f) using polynomial.
- Taylor expansion for expm1(x) has the form:
- x + ax^2 + bx^3 + cx^4 ....
- So we calculate the polynomial P(f) = a + bf + cf^2 + ...
- and assemble the approximation expm1(f) ~= f + f^2 * P(f). */
- svfloat64_t f2 = svmul_x (pg, f, f);
- svfloat64_t f4 = svmul_x (pg, f2, f2);
- svfloat64_t f8 = svmul_x (pg, f4, f4);
- svfloat64_t p
- = svmla_x (pg, f, f2, sv_estrin_10_f64_x (pg, f, f2, f4, f8, d->poly));
-
- /* Assemble the result.
- expm1(x) ~= 2^i * (p + 1) - 1
- Let t = 2^i. */
- svint64_t u = svadd_x (pg, svlsl_x (pg, i, 52), ExponentBias);
- svfloat64_t t = svreinterpret_f64 (u);
-
- /* expm1(x) ~= p * t + (t - 1). */
- svfloat64_t y = svmla_x (pg, svsub_x (pg, t, 1), p, t);
+ svbool_t special = svacgt (pg, x, d->thres);
- if (__glibc_unlikely (svptest_any (pg, special)))
- return special_case (x, y, special);
+ svfloat64_t z = svmla_x (pg, sv_f64 (d->shift), x, d->inv_ln2);
+ svuint64_t u = svreinterpret_u64 (z);
+ svfloat64_t n = svsub_x (pg, z, d->shift);
+ /* r = x - n * ln2, r is in [-ln2/128, ln2/128]. */
+ svfloat64_t ln2 = svld1rq (svptrue_b64 (), &d->ln2_hi);
+ svfloat64_t r = x;
+ r = svmls_lane (r, n, ln2, 0);
+ r = svmls_lane (r, n, ln2, 1);
+
+ /* y = exp(r) - 1 ~= r + C0 r^2 + C1 r^3 + C2 r^4 + C3 r^5 + C4 r^6. */
+ svfloat64_t r2 = svmul_x (svptrue_b64 (), r, r);
+ svfloat64_t c24 = svld1rq (svptrue_b64 (), &d->c2);
+
+ svfloat64_t p;
+ svfloat64_t c12 = svmla_lane (sv_f64 (d->c1), r, c24, 0);
+ svfloat64_t c34 = svmla_lane (sv_f64 (d->c3), r, c24, 1);
+ p = svmad_x (pg, c34, r2, c12);
+ p = svmad_x (pg, p, r, sv_f64 (d->c0));
+ p = svmad_x (pg, p, r2, r);
+
+ svfloat64_t scale = svexpa (u);
+ svfloat64_t scalem1 = svsub_x (pg, scale, sv_f64 (1.0));
+
+ /* We want to construct expm1(x) = (scale - 1) + scale * poly.
+ However, for values of scale close to 1, scale-1 causes large ULP errors
+ due to cancellation.
+
+ This can be circumvented by using a small lookup for scale-1
+ when our input is below a certain bound, otherwise we can use FEXPA.
+
+ This bound is based upon the table size:
+ Bound = (TableSize-1/64) * ln2.
+ The current bound is based upon a table size of 16. */
+ svbool_t is_small = svaclt (pg, x, FexpaBound);
+
+ if (svptest_any (pg, is_small))
+ {
+ /* Index via the input of FEXPA, but we only care about the lower 4 bits.
+ */
+ svuint64_t base_idx = svand_x (pg, u, 0xf);
+
+ /* We can use the sign of x as a fifth bit to account for the asymmetry
+ of e^x around 0. */
+ svuint64_t signBit
+ = svlsl_x (pg, svlsr_x (pg, svreinterpret_u64 (x), 63), 4);
+ svuint64_t idx = svorr_x (pg, base_idx, signBit);
+
+ /* Lookup values for scale - 1 for small x. */
+ svfloat64_t lookup = svreinterpret_f64 (
+ svld1_gather_index (is_small, d->expm1_data, idx));
+
+ /* Select the appropriate scale - 1 value based on x. */
+ scalem1 = svsel (is_small, lookup, scalem1);
+ }
+
+ svfloat64_t y = svmla_x (pg, scalem1, scale, p);
+
+ /* FEXPA returns nan for large inputs so we special case those. */
+ if (__glibc_unlikely (svptest_any (pg, special)))
+ {
+ /* FEXPA zeroes the sign bit, however the sign is meaningful to the
+ special case function so needs to be copied.
+ e = sign bit of u << 46. */
+ svuint64_t e = svand_x (pg, svlsl_x (pg, u, 46), 0x8000000000000000);
+ /* Copy sign to s. */
+ scale = svreinterpret_f64 (svadd_x (pg, e, svreinterpret_u64 (scale)));
+ return special_case (pg, y, scale, p, n);
+ }
+
+ /* return expm1 = (scale - 1) + (scale * poly). */
return y;
}
projects / thirdparty / glibc.git / commitdiff
