[PATCH v2] math: Add new exp10 implementation
Adhemerval Zanella Netto
adhemerval.zanella@linaro.org
Mon Dec 4 15:15:43 GMT 2023
On 04/12/23 12:09, Szabolcs Nagy wrote:
> The 12/01/2023 09:49, Joe Ramsay wrote:
>> New implementation is based on the existing exp/exp2, with different
>> reduction constants and polynomial. Worst-case error in round-to-
>> nearest is 0.513 ULP.
>>
>> The exp/exp2 shared table is reused for exp10 - .rodata size of
>> e_exp_data increases by 64 bytes.
>>
>> As for exp/exp2, targets with single-instruction rounding/conversion
>> intrinsics can use them by toggling TOINT_INTRINSICS=1 and adding the
>> necessary code to their math_private.h.
>>
>> Improvements on Neoverse V1 compared to current GLIBC master:
>> exp10 thruput: 3.3x in [-0x1.439b746e36b52p+8 0x1.34413509f79ffp+8]
>> exp10 latency: 1.8x in [-0x1.439b746e36b52p+8 0x1.34413509f79ffp+8]
>>
>> Tested on:
>> aarch64-linux-gnu (TOINT_INTRINSICS, fma contraction) and
>> x86_64-linux-gnu (!TOINT_INTRINSICS, no fma contraction)
>
> thanks, this looks good.
>
> Reviewed-by: Szabolcs Nagy <szabolcs.nagy@arm.com>
>
> i will commit it for you in a bit.
I still this we should provide a benchtest so it would easy to verify
the performance numbers and evaluate whether some architecture might
benefit from providing ifunc variants (as x86 would most likely want
for enable fma).
>
>> ---
>> Differences from v1:
>> * Stop preventing inlining of special-case - no good reason for this,
>> in fact performance is slightly better if it is inlined
>> * Remove configurable wide poly
>> * Update max ULP based on several runs of the subdivide program with
>> threshold=1000000
>> * Define float constant in hex format
>> * No benchtest added, as there is already one for exp10 (I couldn't
>> see a way to have multiple intervals in exp10-inputs?). Anyway the
>> intervals in the previous version's commit message were chosen
>> fairly arbitrarily - added speedup measurement for the interval used
>> in exp10-inputs
>> Thanks,
>> Joe
>> sysdeps/ieee754/dbl-64/e_exp10.c | 144 ++++++++++++++++++++++-----
>> sysdeps/ieee754/dbl-64/e_exp_data.c | 11 ++
>> sysdeps/ieee754/dbl-64/math_config.h | 4 +
>> 3 files changed, 135 insertions(+), 24 deletions(-)
>>
>> diff --git a/sysdeps/ieee754/dbl-64/e_exp10.c b/sysdeps/ieee754/dbl-64/e_exp10.c
>> index fa47f4f922..08069140c0 100644
>> --- a/sysdeps/ieee754/dbl-64/e_exp10.c
>> +++ b/sysdeps/ieee754/dbl-64/e_exp10.c
>> @@ -16,36 +16,132 @@
>> <https://www.gnu.org/licenses/>. */
>>
>> #include <math.h>
>> +#include <math-barriers.h>
>> +#include <math-narrow-eval.h>
>> #include <math_private.h>
>> #include <float.h>
>> #include <libm-alias-finite.h>
>> +#include "math_config.h"
>>
>> -static const double log10_high = 0x2.4d7637p0;
>> -static const double log10_low = 0x7.6aaa2b05ba95cp-28;
>> +#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 - (1ull << 16) < 0x80000000)
>> + {
>> + /* The exponent of scale might have overflowed by 1. */
>> + sbits -= 1ull << 52;
>> + scale = asdouble (sbits);
>> + y = 2 * (scale + scale * tmp);
>> + return check_oflow (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 = math_narrow_eval (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. */
>> + math_force_eval (math_opt_barrier (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
>> -__ieee754_exp10 (double arg)
>> +__ieee754_exp10 (double x)
>> {
>> - int32_t lx;
>> - double arg_high, arg_low;
>> - double exp_high, exp_low;
>> -
>> - if (!isfinite (arg))
>> - return __ieee754_exp (arg);
>> - if (arg < DBL_MIN_10_EXP - DBL_DIG - 10)
>> - return DBL_MIN * DBL_MIN;
>> - else if (arg > DBL_MAX_10_EXP + 1)
>> - return DBL_MAX * DBL_MAX;
>> - else if (fabs (arg) < 0x1p-56)
>> - return 1.0;
>> -
>> - GET_LOW_WORD (lx, arg);
>> - lx &= 0xf8000000;
>> - arg_high = arg;
>> - SET_LOW_WORD (arg_high, lx);
>> - arg_low = arg - arg_high;
>> - exp_high = arg_high * log10_high;
>> - exp_low = arg_high * log10_low + arg_low * M_LN10;
>> - return __ieee754_exp (exp_high) * __ieee754_exp (exp_low);
>> + uint64_t ix = asuint64 (x);
>> + uint32_t abstop = (ix >> 52) & 0x7ff;
>> +
>> + if (__glibc_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;
>> + int64_t ki;
>> +#if TOINT_INTRINSICS
>> + kd = roundtoint (z);
>> + ki = converttoint (z);
>> +#else
>> + kd = math_narrow_eval (z + Shift);
>> + kd -= Shift;
>> + ki = kd;
>> +#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 (__glibc_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 s * y + s;
>> }
>> +
>> libm_alias_finite (__ieee754_exp10, __exp10)
>> diff --git a/sysdeps/ieee754/dbl-64/e_exp_data.c b/sysdeps/ieee754/dbl-64/e_exp_data.c
>> index d498b8bc3b..5c774afbcb 100644
>> --- a/sysdeps/ieee754/dbl-64/e_exp_data.c
>> +++ b/sysdeps/ieee754/dbl-64/e_exp_data.c
>> @@ -58,6 +58,17 @@ const struct exp_data __exp_data = {
>> 0x1.5d7e09b4e3a84p-10,
>> #endif
>> },
>> +.invlog10_2N = 0x1.a934f0979a371p1 * N,
>> +.neglog10_2hiN = -0x1.3441350ap-2 / N,
>> +.neglog10_2loN = 0x1.0c0219dc1da99p-39 / N,
>> +.exp10_poly = {
>> +/* Coeffs generated using Remez in [-log10(2)/256, log10(2)/256 ]. */
>> +0x1.26bb1bbb55516p1,
>> +0x1.53524c73ce9fep1,
>> +0x1.0470591ce4b26p1,
>> +0x1.2bd76577fe684p0,
>> +0x1.1446eeccd0efbp-1
>> +},
>> // 2^(k/N) ~= H[k]*(1 + T[k]) for int k in [0,N)
>> // tab[2*k] = asuint64(T[k])
>> // tab[2*k+1] = asuint64(H[k]) - (k << 52)/N
>> diff --git a/sysdeps/ieee754/dbl-64/math_config.h b/sysdeps/ieee754/dbl-64/math_config.h
>> index 19af33fd86..d617addfbe 100644
>> --- a/sysdeps/ieee754/dbl-64/math_config.h
>> +++ b/sysdeps/ieee754/dbl-64/math_config.h
>> @@ -201,6 +201,10 @@ extern const struct exp_data
>> double poly[4]; /* Last four coefficients. */
>> double exp2_shift;
>> double exp2_poly[EXP2_POLY_ORDER];
>> + double invlog10_2N;
>> + double neglog10_2hiN;
>> + double neglog10_2loN;
>> + double exp10_poly[5];
>> uint64_t tab[2*(1 << EXP_TABLE_BITS)];
>> } __exp_data attribute_hidden;
>>
>> --
>> 2.27.0
>>
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