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AArch64: Improve codegen in users of ADVSIMD expm1 helper
Add inline helper for expm1 and rearrange operations so MOV is not necessary in reduction or around the special-case handler. Reduce memory access by using more indexed MLAs in polynomial. Speedup on Neoverse V1 for expm1 (19%), sinh (8.5%), and tanh (7.5%).
This commit is contained in:
Pierre Blanchard
Wilco Dijkstra
parent
ca0c0d0f26
commit
13a7ef5999
5 changed files with 135 additions and 162 deletions
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@ -18,31 +18,18 @@
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<https://www.gnu.org/licenses/>. */
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#include "v_math.h"
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#include "poly_advsimd_f64.h"
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#include "v_expm1_inline.h"
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static const struct data
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{
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float64x2_t poly[11];
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float64x2_t invln2;
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double ln2[2];
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float64x2_t shift;
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int64x2_t exponent_bias;
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struct v_expm1_data d;
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#if WANT_SIMD_EXCEPT
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uint64x2_t thresh, tiny_bound;
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#else
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float64x2_t oflow_bound;
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#endif
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} data = {
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/* Generated using fpminimax, with degree=12 in [log(2)/2, log(2)/2]. */
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.poly = { V2 (0x1p-1), V2 (0x1.5555555555559p-3), V2 (0x1.555555555554bp-5),
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V2 (0x1.111111110f663p-7), V2 (0x1.6c16c16c1b5f3p-10),
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V2 (0x1.a01a01affa35dp-13), V2 (0x1.a01a018b4ecbbp-16),
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V2 (0x1.71ddf82db5bb4p-19), V2 (0x1.27e517fc0d54bp-22),
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V2 (0x1.af5eedae67435p-26), V2 (0x1.1f143d060a28ap-29) },
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.invln2 = V2 (0x1.71547652b82fep0),
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.ln2 = { 0x1.62e42fefa39efp-1, 0x1.abc9e3b39803fp-56 },
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.shift = V2 (0x1.8p52),
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.exponent_bias = V2 (0x3ff0000000000000),
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.d = V_EXPM1_DATA,
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#if WANT_SIMD_EXCEPT
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/* asuint64(oflow_bound) - asuint64(0x1p-51), shifted left by 1 for abs
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compare. */
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@ -58,67 +45,36 @@ static const struct data
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};
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static float64x2_t VPCS_ATTR NOINLINE
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special_case (float64x2_t x, float64x2_t y, uint64x2_t special)
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special_case (float64x2_t x, uint64x2_t special, const struct data *d)
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{
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return v_call_f64 (expm1, x, y, special);
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return v_call_f64 (expm1, x, expm1_inline (v_zerofy_f64 (x, special), &d->d),
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special);
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}
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/* Double-precision vector exp(x) - 1 function.
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The maximum error observed error is 2.18 ULP:
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_ZGVnN2v_expm1 (0x1.634ba0c237d7bp-2) got 0x1.a8b9ea8d66e22p-2
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want 0x1.a8b9ea8d66e2p-2. */
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The maximum error observed error is 2.05 ULP:
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_ZGVnN2v_expm1(0x1.634902eaff3adp-2) got 0x1.a8b636e2a9388p-2
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want 0x1.a8b636e2a9386p-2. */
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float64x2_t VPCS_ATTR V_NAME_D1 (expm1) (float64x2_t x)
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{
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const struct data *d = ptr_barrier (&data);
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uint64x2_t ix = vreinterpretq_u64_f64 (x);
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#if WANT_SIMD_EXCEPT
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uint64x2_t ix = vreinterpretq_u64_f64 (x);
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/* If fp exceptions are to be triggered correctly, fall back to scalar for
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|x| < 2^-51, |x| > oflow_bound, Inf & NaN. Add ix to itself for
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shift-left by 1, and compare with thresh which was left-shifted offline -
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this is effectively an absolute compare. */
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uint64x2_t special
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= vcgeq_u64 (vsubq_u64 (vaddq_u64 (ix, ix), d->tiny_bound), d->thresh);
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if (__glibc_unlikely (v_any_u64 (special)))
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x = v_zerofy_f64 (x, special);
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#else
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/* Large input, NaNs and Infs. */
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uint64x2_t special = vcageq_f64 (x, d->oflow_bound);
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#endif
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/* Reduce argument to smaller range:
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Let i = round(x / ln2)
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and f = x - i * ln2, then f is in [-ln2/2, ln2/2].
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exp(x) - 1 = 2^i * (expm1(f) + 1) - 1
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where 2^i is exact because i is an integer. */
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float64x2_t n = vsubq_f64 (vfmaq_f64 (d->shift, d->invln2, x), d->shift);
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int64x2_t i = vcvtq_s64_f64 (n);
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float64x2_t ln2 = vld1q_f64 (&d->ln2[0]);
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float64x2_t f = vfmsq_laneq_f64 (x, n, ln2, 0);
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f = vfmsq_laneq_f64 (f, n, ln2, 1);
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/* Approximate expm1(f) using polynomial.
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Taylor expansion for expm1(x) has the form:
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x + ax^2 + bx^3 + cx^4 ....
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So we calculate the polynomial P(f) = a + bf + cf^2 + ...
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and assemble the approximation expm1(f) ~= f + f^2 * P(f). */
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float64x2_t f2 = vmulq_f64 (f, f);
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float64x2_t f4 = vmulq_f64 (f2, f2);
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float64x2_t f8 = vmulq_f64 (f4, f4);
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float64x2_t p = vfmaq_f64 (f, f2, v_estrin_10_f64 (f, f2, f4, f8, d->poly));
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/* Assemble the result.
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expm1(x) ~= 2^i * (p + 1) - 1
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Let t = 2^i. */
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int64x2_t u = vaddq_s64 (vshlq_n_s64 (i, 52), d->exponent_bias);
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float64x2_t t = vreinterpretq_f64_s64 (u);
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if (__glibc_unlikely (v_any_u64 (special)))
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return special_case (vreinterpretq_f64_u64 (ix),
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vfmaq_f64 (vsubq_f64 (t, v_f64 (1.0)), p, t),
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special);
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return special_case (x, special, d);
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/* expm1(x) ~= p * t + (t - 1). */
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return vfmaq_f64 (vsubq_f64 (t, v_f64 (1.0)), p, t);
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return expm1_inline (x, &d->d);
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}
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@ -18,72 +18,31 @@
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<https://www.gnu.org/licenses/>. */
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#include "v_math.h"
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#include "poly_advsimd_f64.h"
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#include "v_expm1_inline.h"
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static const struct data
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{
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float64x2_t poly[11], inv_ln2;
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double m_ln2[2];
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float64x2_t shift;
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struct v_expm1_data d;
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uint64x2_t halff;
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int64x2_t onef;
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#if WANT_SIMD_EXCEPT
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uint64x2_t tiny_bound, thresh;
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#else
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uint64x2_t large_bound;
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float64x2_t large_bound;
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#endif
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} data = {
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/* Generated using Remez, deg=12 in [-log(2)/2, log(2)/2]. */
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.poly = { V2 (0x1p-1), V2 (0x1.5555555555559p-3), V2 (0x1.555555555554bp-5),
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V2 (0x1.111111110f663p-7), V2 (0x1.6c16c16c1b5f3p-10),
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V2 (0x1.a01a01affa35dp-13), V2 (0x1.a01a018b4ecbbp-16),
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V2 (0x1.71ddf82db5bb4p-19), V2 (0x1.27e517fc0d54bp-22),
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V2 (0x1.af5eedae67435p-26), V2 (0x1.1f143d060a28ap-29), },
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.inv_ln2 = V2 (0x1.71547652b82fep0),
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.m_ln2 = {-0x1.62e42fefa39efp-1, -0x1.abc9e3b39803fp-56},
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.shift = V2 (0x1.8p52),
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.d = V_EXPM1_DATA,
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.halff = V2 (0x3fe0000000000000),
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.onef = V2 (0x3ff0000000000000),
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#if WANT_SIMD_EXCEPT
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/* 2^-26, below which sinh(x) rounds to x. */
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.tiny_bound = V2 (0x3e50000000000000),
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/* asuint(large_bound) - asuint(tiny_bound). */
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.thresh = V2 (0x0230000000000000),
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#else
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/* 2^9. expm1 helper overflows for large input. */
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.large_bound = V2 (0x4080000000000000),
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/* 2^9. expm1 helper overflows for large input. */
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.large_bound = V2 (0x1p+9),
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#endif
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};
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static inline float64x2_t
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expm1_inline (float64x2_t x)
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{
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const struct data *d = ptr_barrier (&data);
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/* Reduce argument:
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exp(x) - 1 = 2^i * (expm1(f) + 1) - 1
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where i = round(x / ln2)
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and f = x - i * ln2 (f in [-ln2/2, ln2/2]). */
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float64x2_t j = vsubq_f64 (vfmaq_f64 (d->shift, d->inv_ln2, x), d->shift);
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int64x2_t i = vcvtq_s64_f64 (j);
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float64x2_t m_ln2 = vld1q_f64 (d->m_ln2);
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float64x2_t f = vfmaq_laneq_f64 (x, j, m_ln2, 0);
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f = vfmaq_laneq_f64 (f, j, m_ln2, 1);
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/* Approximate expm1(f) using polynomial. */
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float64x2_t f2 = vmulq_f64 (f, f);
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float64x2_t f4 = vmulq_f64 (f2, f2);
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float64x2_t f8 = vmulq_f64 (f4, f4);
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float64x2_t p = vfmaq_f64 (f, f2, v_estrin_10_f64 (f, f2, f4, f8, d->poly));
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/* t = 2^i. */
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float64x2_t t = vreinterpretq_f64_u64 (
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vreinterpretq_u64_s64 (vaddq_s64 (vshlq_n_s64 (i, 52), d->onef)));
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/* expm1(x) ~= p * t + (t - 1). */
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return vfmaq_f64 (vsubq_f64 (t, v_f64 (1.0)), p, t);
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}
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static float64x2_t NOINLINE VPCS_ATTR
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special_case (float64x2_t x)
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{
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@ -92,23 +51,23 @@ special_case (float64x2_t x)
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/* Approximation for vector double-precision sinh(x) using expm1.
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sinh(x) = (exp(x) - exp(-x)) / 2.
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The greatest observed error is 2.57 ULP:
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_ZGVnN2v_sinh (0x1.9fb1d49d1d58bp-2) got 0x1.ab34e59d678dcp-2
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want 0x1.ab34e59d678d9p-2. */
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The greatest observed error is 2.52 ULP:
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_ZGVnN2v_sinh(-0x1.a098a2177a2b9p-2) got -0x1.ac2f05bb66fccp-2
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want -0x1.ac2f05bb66fc9p-2. */
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float64x2_t VPCS_ATTR V_NAME_D1 (sinh) (float64x2_t x)
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{
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const struct data *d = ptr_barrier (&data);
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float64x2_t ax = vabsq_f64 (x);
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uint64x2_t sign
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= veorq_u64 (vreinterpretq_u64_f64 (x), vreinterpretq_u64_f64 (ax));
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float64x2_t halfsign = vreinterpretq_f64_u64 (vorrq_u64 (sign, d->halff));
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uint64x2_t ix = vreinterpretq_u64_f64 (x);
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float64x2_t halfsign = vreinterpretq_f64_u64 (
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vbslq_u64 (v_u64 (0x8000000000000000), ix, d->halff));
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#if WANT_SIMD_EXCEPT
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uint64x2_t special = vcgeq_u64 (
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vsubq_u64 (vreinterpretq_u64_f64 (ax), d->tiny_bound), d->thresh);
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#else
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uint64x2_t special = vcgeq_u64 (vreinterpretq_u64_f64 (ax), d->large_bound);
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uint64x2_t special = vcageq_f64 (x, d->large_bound);
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#endif
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/* Fall back to scalar variant for all lanes if any of them are special. */
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@ -118,7 +77,7 @@ float64x2_t VPCS_ATTR V_NAME_D1 (sinh) (float64x2_t x)
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/* Up to the point that expm1 overflows, we can use it to calculate sinh
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using a slight rearrangement of the definition of sinh. This allows us to
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retain acceptable accuracy for very small inputs. */
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float64x2_t t = expm1_inline (ax);
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float64x2_t t = expm1_inline (ax, &d->d);
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t = vaddq_f64 (t, vdivq_f64 (t, vaddq_f64 (t, v_f64 (1.0))));
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return vmulq_f64 (t, halfsign);
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}
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@ -18,68 +18,30 @@
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<https://www.gnu.org/licenses/>. */
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#include "v_math.h"
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#include "poly_advsimd_f64.h"
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#include "v_expm1_inline.h"
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static const struct data
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{
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float64x2_t poly[11];
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float64x2_t inv_ln2, ln2_hi, ln2_lo, shift;
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uint64x2_t onef;
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struct v_expm1_data d;
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uint64x2_t thresh, tiny_bound;
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} data = {
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/* Generated using Remez, deg=12 in [-log(2)/2, log(2)/2]. */
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.poly = { V2 (0x1p-1), V2 (0x1.5555555555559p-3), V2 (0x1.555555555554bp-5),
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V2 (0x1.111111110f663p-7), V2 (0x1.6c16c16c1b5f3p-10),
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V2 (0x1.a01a01affa35dp-13), V2 (0x1.a01a018b4ecbbp-16),
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V2 (0x1.71ddf82db5bb4p-19), V2 (0x1.27e517fc0d54bp-22),
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V2 (0x1.af5eedae67435p-26), V2 (0x1.1f143d060a28ap-29), },
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.inv_ln2 = V2 (0x1.71547652b82fep0),
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.ln2_hi = V2 (-0x1.62e42fefa39efp-1),
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.ln2_lo = V2 (-0x1.abc9e3b39803fp-56),
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.shift = V2 (0x1.8p52),
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.onef = V2 (0x3ff0000000000000),
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.d = V_EXPM1_DATA,
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.tiny_bound = V2 (0x3e40000000000000), /* asuint64 (0x1p-27). */
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/* asuint64(0x1.241bf835f9d5fp+4) - asuint64(tiny_bound). */
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.thresh = V2 (0x01f241bf835f9d5f),
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};
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static inline float64x2_t
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expm1_inline (float64x2_t x, const struct data *d)
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{
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/* Helper routine for calculating exp(x) - 1. Vector port of the helper from
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the scalar variant of tanh. */
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/* Reduce argument: f in [-ln2/2, ln2/2], i is exact. */
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float64x2_t j = vsubq_f64 (vfmaq_f64 (d->shift, d->inv_ln2, x), d->shift);
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int64x2_t i = vcvtq_s64_f64 (j);
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float64x2_t f = vfmaq_f64 (x, j, d->ln2_hi);
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f = vfmaq_f64 (f, j, d->ln2_lo);
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/* Approximate expm1(f) using polynomial. */
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float64x2_t f2 = vmulq_f64 (f, f);
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float64x2_t f4 = vmulq_f64 (f2, f2);
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float64x2_t p = vfmaq_f64 (
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f, f2, v_estrin_10_f64 (f, f2, f4, vmulq_f64 (f4, f4), d->poly));
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/* t = 2 ^ i. */
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float64x2_t t = vreinterpretq_f64_u64 (
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vaddq_u64 (vreinterpretq_u64_s64 (i << 52), d->onef));
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/* expm1(x) = p * t + (t - 1). */
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return vfmaq_f64 (vsubq_f64 (t, v_f64 (1)), p, t);
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}
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static float64x2_t NOINLINE VPCS_ATTR
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special_case (float64x2_t x, float64x2_t y, uint64x2_t special)
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special_case (float64x2_t x, float64x2_t q, float64x2_t qp2,
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uint64x2_t special)
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{
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return v_call_f64 (tanh, x, y, special);
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return v_call_f64 (tanh, x, vdivq_f64 (q, qp2), special);
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}
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/* Vector approximation for double-precision tanh(x), using a simplified
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version of expm1. The greatest observed error is 2.77 ULP:
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_ZGVnN2v_tanh(-0x1.c4a4ca0f9f3b7p-3) got -0x1.bd6a21a163627p-3
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want -0x1.bd6a21a163624p-3. */
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version of expm1. The greatest observed error is 2.70 ULP:
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_ZGVnN2v_tanh(-0x1.c59aa220cb177p-3) got -0x1.be5452a6459fep-3
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want -0x1.be5452a6459fbp-3. */
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float64x2_t VPCS_ATTR V_NAME_D1 (tanh) (float64x2_t x)
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{
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const struct data *d = ptr_barrier (&data);
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@ -100,10 +62,10 @@ float64x2_t VPCS_ATTR V_NAME_D1 (tanh) (float64x2_t x)
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u = vaddq_f64 (u, u);
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/* tanh(x) = (e^2x - 1) / (e^2x + 1). */
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float64x2_t q = expm1_inline (u, d);
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float64x2_t qp2 = vaddq_f64 (q, v_f64 (2));
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float64x2_t q = expm1_inline (u, &d->d);
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float64x2_t qp2 = vaddq_f64 (q, v_f64 (2.0));
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if (__glibc_unlikely (v_any_u64 (special)))
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return special_case (x, vdivq_f64 (q, qp2), special);
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return special_case (x, q, qp2, special);
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return vdivq_f64 (q, qp2);
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||||
}
|
||||
|
|
|
|||
97
sysdeps/aarch64/fpu/v_expm1_inline.h
Normal file
97
sysdeps/aarch64/fpu/v_expm1_inline.h
Normal file
|
|
@ -0,0 +1,97 @@
|
|||
/* Double-precision inline helper for vector (Advanced SIMD) expm1 function
|
||||
|
||||
Copyright (C) 2024 Free Software Foundation, Inc.
|
||||
This file is part of the GNU C Library.
|
||||
|
||||
The GNU C Library is free software; you can redistribute it and/or
|
||||
modify it under the terms of the GNU Lesser General Public
|
||||
License as published by the Free Software Foundation; either
|
||||
version 2.1 of the License, or (at your option) any later version.
|
||||
|
||||
The GNU C Library is distributed in the hope that it will be useful,
|
||||
but WITHOUT ANY WARRANTY; without even the implied warranty of
|
||||
MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU
|
||||
Lesser General Public License for more details.
|
||||
|
||||
You should have received a copy of the GNU Lesser General Public
|
||||
License along with the GNU C Library; if not, see
|
||||
<https://www.gnu.org/licenses/>. */
|
||||
|
||||
#ifndef AARCH64_FPU_V_EXPM1_INLINE_H
|
||||
#define AARCH64_FPU_V_EXPM1_INLINE_H
|
||||
|
||||
#include "v_math.h"
|
||||
|
||||
struct v_expm1_data
|
||||
{
|
||||
float64x2_t c2, c4, c6, c8;
|
||||
float64x2_t invln2;
|
||||
int64x2_t exponent_bias;
|
||||
double c1, c3, c5, c7, c9, c10;
|
||||
double ln2[2];
|
||||
};
|
||||
|
||||
/* Generated using fpminimax, with degree=12 in [log(2)/2, log(2)/2]. */
|
||||
#define V_EXPM1_DATA \
|
||||
{ \
|
||||
.c1 = 0x1.5555555555559p-3, .c2 = V2 (0x1.555555555554bp-5), \
|
||||
.c3 = 0x1.111111110f663p-7, .c4 = V2 (0x1.6c16c16c1b5f3p-10), \
|
||||
.c5 = 0x1.a01a01affa35dp-13, .c6 = V2 (0x1.a01a018b4ecbbp-16), \
|
||||
.c7 = 0x1.71ddf82db5bb4p-19, .c8 = V2 (0x1.27e517fc0d54bp-22), \
|
||||
.c9 = 0x1.af5eedae67435p-26, .c10 = 0x1.1f143d060a28ap-29, \
|
||||
.ln2 = { 0x1.62e42fefa39efp-1, 0x1.abc9e3b39803fp-56 }, \
|
||||
.invln2 = V2 (0x1.71547652b82fep0), \
|
||||
.exponent_bias = V2 (0x3ff0000000000000), \
|
||||
}
|
||||
|
||||
static inline float64x2_t
|
||||
expm1_inline (float64x2_t x, const struct v_expm1_data *d)
|
||||
{
|
||||
/* Helper routine for calculating exp(x) - 1. */
|
||||
|
||||
float64x2_t ln2 = vld1q_f64 (&d->ln2[0]);
|
||||
|
||||
/* 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. */
|
||||
float64x2_t n = vrndaq_f64 (vmulq_f64 (x, d->invln2));
|
||||
int64x2_t i = vcvtq_s64_f64 (n);
|
||||
float64x2_t f = vfmsq_laneq_f64 (x, n, ln2, 0);
|
||||
f = vfmsq_laneq_f64 (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). */
|
||||
float64x2_t f2 = vmulq_f64 (f, f);
|
||||
float64x2_t f4 = vmulq_f64 (f2, f2);
|
||||
float64x2_t lane_consts_13 = vld1q_f64 (&d->c1);
|
||||
float64x2_t lane_consts_57 = vld1q_f64 (&d->c5);
|
||||
float64x2_t lane_consts_910 = vld1q_f64 (&d->c9);
|
||||
float64x2_t p01 = vfmaq_laneq_f64 (v_f64 (0.5), f, lane_consts_13, 0);
|
||||
float64x2_t p23 = vfmaq_laneq_f64 (d->c2, f, lane_consts_13, 1);
|
||||
float64x2_t p45 = vfmaq_laneq_f64 (d->c4, f, lane_consts_57, 0);
|
||||
float64x2_t p67 = vfmaq_laneq_f64 (d->c6, f, lane_consts_57, 1);
|
||||
float64x2_t p03 = vfmaq_f64 (p01, f2, p23);
|
||||
float64x2_t p47 = vfmaq_f64 (p45, f2, p67);
|
||||
float64x2_t p89 = vfmaq_laneq_f64 (d->c8, f, lane_consts_910, 0);
|
||||
float64x2_t p = vfmaq_laneq_f64 (p89, f2, lane_consts_910, 1);
|
||||
p = vfmaq_f64 (p47, f4, p);
|
||||
p = vfmaq_f64 (p03, f4, p);
|
||||
|
||||
p = vfmaq_f64 (f, f2, p);
|
||||
|
||||
/* Assemble the result.
|
||||
expm1(x) ~= 2^i * (p + 1) - 1
|
||||
Let t = 2^i. */
|
||||
int64x2_t u = vaddq_s64 (vshlq_n_s64 (i, 52), d->exponent_bias);
|
||||
float64x2_t t = vreinterpretq_f64_s64 (u);
|
||||
|
||||
/* expm1(x) ~= p * t + (t - 1). */
|
||||
return vfmaq_f64 (vsubq_f64 (t, v_f64 (1.0)), p, t);
|
||||
}
|
||||
|
||||
#endif
|
||||
|
|
@ -21,7 +21,6 @@
|
|||
#define AARCH64_FPU_V_EXPM1F_INLINE_H
|
||||
|
||||
#include "v_math.h"
|
||||
#include "math_config.h"
|
||||
|
||||
struct v_expm1f_data
|
||||
{
|
||||
|
|
|
|||
Loading…
Add table
Reference in a new issue