[libc][math] Improve tanhf performance.

[lntue]

Summary

Optimize the core part of tanhf implementation that is to compute e^x
similar to https://reviews.llvm.org/D133870. Factor the constants and
polynomial approximation out so that it can be used for exp10f

Performance benchmark using perf tool from the CORE-MATH project on Ryzen 1700:

$ CORE_MATH_PERF_MODE="rdtsc" ./perf.sh tanhf
GNU libc version: 2.35
GNU libc release: stable
CORE-MATH reciprocal throughput   : 13.377
System LIBC reciprocal throughput : 55.046

BEFORE:
LIBC reciprocal throughput        : 75.674
LIBC reciprocal throughput        : 33.242    (with `-msse4.2` flag)
LIBC reciprocal throughput        : 25.927    (with `-mfma` flag)

AFTER:
LIBC reciprocal throughput        : 26.359
LIBC reciprocal throughput        : 18.888    (with `-msse4.2` flag)
LIBC reciprocal throughput        : 14.243    (with `-mfma` flag)

$ CORE_MATH_PERF_MODE="rdtsc" ./perf.sh tanhf --latency
GNU libc version: 2.35
GNU libc release: stable
CORE-MATH latency   : 43.365
System LIBC latency : 123.499

BEFORE
LIBC latency        : 112.968
LIBC latency        : 104.908   (with `-msse4.2` flag)
LIBC latency        : 92.310    (with `-mfma` flag)

AFTER
LIBC latency        : 69.828
LIBC latency        : 63.874    (with `-msse4.2` flag)
LIBC latency        : 57.427    (with `-mfma` flag)

[lntue]

Update docs.

[zimmermann6]

does this patch need to be applied on another one? Or rebased? It does not apply cleanly to main (71e52a1), unless I did something wrong.

[lntue]

Sync to head.

[lntue]

In D134002#3794467, @zimmermann6 wrote:

does this patch need to be applied on another one? Or rebased? It does not apply cleanly to main (71e52a1), unless I did something wrong.

I've just synced the patch to HEAD. Can you try it again? Thanks,

[lntue]

Change to use 32 sub-intervals for middle part to reduce the degree of
polynomial approximation for lower part to 5 from 7. Also update sinhf,
coshf, exp2f to use the same range reduction. This slightly reduce their
latencies and reciprocal throughputs.

[lntue]

Only return lo part in range reduction, separating it with powb_lo
evaluation.

[zimmermann6]

I confirm the function is still correctly rounded, and the timings improved.

libc/docs/math.rst

 | sinf         |        12 |                25 |        51 |                57 | :math:`[-\pi, \pi]`                 | Ryzen 1700 | Ubuntu 20.04 LTS x86_64 | Clang 12.0.0 | FMA           |
 +--------------+-----------+-------------------+-----------+-------------------+-------------------------------------+------------+-------------------------+--------------+---------------+
 | sincosf      |        19 |                30 |        57 |                68 | :math:`[-\pi, \pi]`                 | Ryzen 1700 | Ubuntu 20.04 LTS x86_64 | Clang 12.0.0 | FMA           |
 +--------------+-----------+-------------------+-----------+-------------------+-------------------------------------+------------+-------------------------+--------------+---------------+
 | sinhf        |        14 |                63 |        49 |               137 | :math:`[-10, 10]`                   | Ryzen 1700 | Ubuntu 22.04 LTS x86_64 | Clang 14.0.0 | FMA           |
 +--------------+-----------+-------------------+-----------+-------------------+-------------------------------------+------------+-------------------------+--------------+---------------+
 | tanf         |        19 |                50 |        82 |               107 | :math:`[-\pi, \pi]`                 | Ryzen 1700 | Ubuntu 20.04 LTS x86_64 | Clang 12.0.0 | FMA           |
 +--------------+-----------+-------------------+-----------+-------------------+-------------------------------------+------------+-------------------------+--------------+---------------+
-| tanhf        |        25 |                59 |        95 |               125 | :math:`[-10, 10]`                   | Ryzen 1700 | Ubuntu 20.04 LTS x86_64 | Clang 12.0.0 | FMA           |
+| tanhf        |        14 |                55 |        57 |               123 | :math:`[-10, 10]`                   | Ryzen 1700 | Ubuntu 22.04 LTS x86_64 | Clang 14.0.0 | FMA           |
 +--------------+-----------+-------------------+-----------+-------------------+-------------------------------------+------------+-------------------------+--------------+---------------+
 
 References
 ==========
 
 * `CRLIBM <https://hal-ens-lyon.archives-ouvertes.fr/ensl-01529804/file/crlibm.pdf>`_.
 * `RLIBM <https://people.cs.rutgers.edu/~sn349/rlibm/>`_.
 * `Sollya <https://www.sollya.org/>`_.
 * `The CORE-MATH Project <https://core-math.gitlabpages.inria.fr/>`_.
 * `The GNU C Library (glibc) <https://www.gnu.org/software/libc/>`_.
 * `The GNU MPFR Library <https://www.mpfr.org/>`_.

libc/src/math/generic/explogxf.h

 #include "src/__support/FPUtil/nearest_integer.h"
 #include "src/__support/common.h"
 #include <src/__support/FPUtil/NearestIntegerOperations.h>
 
 #include <errno.h>
 
 namespace __llvm_libc {
 
-static constexpr int EXP_bits_p = 5;
-static constexpr int EXP_num_p = 1 << EXP_bits_p;
-constexpr double mlp = EXP_num_p;
-constexpr double mmld = -1.0 / mlp;
-
-// Wolfram alpha: N[Table[2^x-1,{x,-16/32,15/32,1/32}],27]
-// printf("%.13a,\n", d[i]);
-extern const double EXP_2_POW[EXP_num_p];
+struct ExpBase {
+  // Base = e
+  static constexpr int MID_BITS = 3;
+  static constexpr int MID_MASK = (1 << MID_BITS) - 1;
+  // log2(e) * 2^3
+  static constexpr double LOG2_B = 0x1.71547652b82fep+0 * (1 << MID_BITS);
+  // High and low parts of -log(2) * 2^(-3)
+  static constexpr double M_LOGB_2_HI = -0x1.62e42fefa0000p-1 / (1 << MID_BITS);
+  static constexpr double M_LOGB_2_LO =
+      -0x1.cf79abc9e3b3ap-40 / (1 << MID_BITS);
+  // Look up table for bit fields of 2^(i/8) for i = 0..7, generated by Sollya
+  // with:
+  // > for i from 0 to 7 do printdouble(round(2^(i/8), D, RN));
+  static constexpr int64_t EXP_2_MID[1 << MID_BITS] = {
+      0x3ff0000000000000, 0x3ff172b83c7d517b, 0x3ff306fe0a31b715,
+      0x3ff4bfdad5362a27, 0x3ff6a09e667f3bcd, 0x3ff8ace5422aa0db,
+      0x3ffae89f995ad3ad, 0x3ffd5818dcfba487,
+  };
+
+  // Approximating e^dx with degree-7 minimax polynomial generated by Sollya:
+  // > Q = fpminimax(expm1(x)/x, 6, [|1, D...|], [-log(2)/16, log(2)/16]);
+  // Then:
+  //   e^dx ~ P(dx) = 1 + dx + COEFFS[0] * dx^2 + ... + COEFFS[4] * dx^6.
+  static constexpr double COEFFS[6] = {
+      0x1.00000000000a1p-1, 0x1.5555555555434p-3,  0x1.5555554ad7591p-5,
+      0x1.11111115caf71p-7, 0x1.6c1c38380bd59p-10, 0x1.a01b5f9d8b1d3p-13};
+
+  static constexpr double powb_lo(double dx) {
+    using fputil::multiply_add;
+    double dx2 = dx * dx;
+    double c0 = 1.0 + dx;
+    // c1 = COEFFS[0] + COEFFS[1] * dx
+    double c1 = multiply_add(dx, ExpBase::COEFFS[1], ExpBase::COEFFS[0]);
+    // c2 = COEFFS[2] + COEFFS[3] * dx
+    double c2 = multiply_add(dx, ExpBase::COEFFS[3], ExpBase::COEFFS[2]);
+    // c3 = COEFFS[4] + COEFFS[5] * dx
+    double c3 = multiply_add(dx, ExpBase::COEFFS[5], ExpBase::COEFFS[4]);
+    double dx4 = dx2 * dx2;
+    // c4 = c0 + c1 * dx^2
+    //    = 1 + dx + COEFFS[0] * dx^2 + COEFFS[1] * dx^3
+    double c4 = multiply_add(dx2, c1, c0);
+    // c5 = c2 + c3 * dx^2
+    //    = COEFFS[2] + COEFFS[3] * dx + COEFFS[4] * dx^2 + COEFFS[5] * dx^3
+    double c5 = multiply_add(dx2, c3, c2);
+    // r = c4 + c5 * dx^4
+    //   = 1 + dx + COEFFS[0] * dx^2 + ... + COEFFS[5] * dx^7
+    return multiply_add(dx4, c5, c4);
+  }
+};
 
 // Look up table for bit fields of 2^(i/16) for i = 0..15, generated by Sollya
 // with:
 // > for i from 0 to 15 do printdouble(round(2^(i/16), D, RN));
 inline constexpr int64_t EXP_2_M[16] = {
     0x3ff0000000000000, 0x3ff0b5586cf9890f, 0x3ff172b83c7d517b,
     0x3ff2387a6e756238, 0x3ff306fe0a31b715, 0x3ff3dea64c123422,
     0x3ff4bfdad5362a27, 0x3ff5ab07dd485429, 0x3ff6a09e667f3bcd,

 // N[Table[1/(1 + x), {x, 0/64, 63/64, 1/64}], 40]
 extern const double LOG_P1_1_OVER[LOG_P1_SIZE];
 
 // Taylor series expansion for Log[2, 1 + x] splitted to EVEN AND ODD numbers
 // K_LOG2_ODD starts from x^3
 extern const double K_LOG2_ODD[4];
 extern const double K_LOG2_EVEN[4];
 
-// The algorithm represents exp(x) as
-//   exp(x) = 2^(ln(2) * i) * 2^(ln(2) * j / NUM_P )) * exp(dx)
-// where i integer value, j integer in range [-NUM_P/2, NUM_P/2).
-// 2^(ln(2) * j / NUM_P )) is a table values: 1.0 + EXP_M
-// exp(dx) calculates by taylor expansion.
-
-// Inversion of ln(2). Multiplication by EXP_num_p due to sampling by 1 /
-// EXP_num_p Precise value of the constant is not needed.
-static constexpr double LN2_INV = 0x1.71547652b82fep+0 * EXP_num_p;
-
 // log2(e) * 2^4
 static constexpr double LOG2_E_4 = 0x1.71547652b82fep+4;
 
-// LN2_HIGH + LN2_LOW = ln(2) with precision higher than double(ln(2))
-// Minus sign is to use FMA directly.
-static constexpr double LN2_HIGH = -0x1.62e42fefa0000p-1 / EXP_num_p;
-static constexpr double LN2_LOW = -0x1.cf79abc9e3b3ap-40 / EXP_num_p;
-
 // -log(2) * 2^(-4)
 static constexpr double M_LN2_4_HI = -0x1.62e42fefa0000p-5;
 static constexpr double M_LN2_4_LO = -0x1.cf79abc9e3b3ap-44;
 
-struct exe_eval_result_t {
-  // exp(x) = 2^MULT_POWER2 * mult_exp * (r + 1.0)
-  // where
-  //   MULT_POWER2 template parameter;
+struct exp_b_result_t {
+  // b^x = mult_exp * r
+  // where:
   //   mult_exp = 2^e;
-  //   r in range [~-0.3, ~0.41]
+  //   r
   double mult_exp;
   double r;
 };
 
-// The function correctly calculates exp value with at least float precision
-// in range not narrow than [-log(2^-150), 90]
-template <int MULT_POWER2 = 0>
-inline static exe_eval_result_t exp_eval(double x) {
-  double ps_dbl = fputil::nearest_integer(LN2_INV * x);
-  // Negative sign due to multiply_add optimization
-  double mult_e1, ml;
-  {
-    int ps =
-        static_cast<int>(ps_dbl) + (1 << (EXP_bits_p - 1)) +
-        ((fputil::FPBits<double>::EXPONENT_BIAS + MULT_POWER2) << EXP_bits_p);
-    int table_index = ps & (EXP_num_p - 1);
-    fputil::FPBits<double> bs;
-    bs.set_unbiased_exponent(ps >> EXP_bits_p);
-    ml = EXP_2_POW[table_index];
-    mult_e1 = bs.get_val();
-  }
-  double dx = fputil::multiply_add(ps_dbl, LN2_LOW,
-                                   fputil::multiply_add(ps_dbl, LN2_HIGH, x));
-
-  // Taylor series coefficients
-  double pe = dx * fputil::polyeval(dx, 1.0, 0x1.0p-1, 0x1.5555555555555p-3,
-                                    0x1.5555555555555p-5, 0x1.1111111111111p-7,
-                                    0x1.6c16c16c16c17p-10);
-
-  double r = fputil::multiply_add(ml, pe, pe) + ml;
-  return {mult_e1, r};
+// The function correctly calculates b^x value with at least float precision
+// in a limited range.
+// Range reduction:
+//   b^x = 2^(hi + mid) * b^lo
+// where:
+//   x = (hi + mid) * log_b(2) + lo
+//   hi is an integer,
+//   0 <= mid * 2^MID_BITS < 2^MID_BITS is an integer
+//   -2^(-MID_BITS - 1) <= lo * log2(b) <= 2^(-MID_BITS - 1)
+// Base class needs to provide the following constants and function:
+//   - MID_BITS    : number of bits after decimal points used for mid
+//   - MID_MASK    : 2^MID_BITS - 1, mask to extract mid bits
+//   - LOG2_B      : log2(b) * 2^MID_BITS for scaling
+//   - M_LOGB_2_HI : high part of -log_b(2) * 2^(-MID_BITS)
+//   - M_LOGB_2_LO : low part of -log_b(2) * 2^(-MID_BITS)
+//   - EXP_2_MID   : look up table for bit fields of 2^mid
+//   - powb_lo     : approximating b^lo.
+// Return:
+//   { 2^(hi + mid), b^lo }
+template <class Base> static inline exp_b_result_t exp_b_eval(float x) {
+  double xd = static_cast<double>(x);
+  // kd = round((hi + mid) * log2(b) * 2^MID_BITS)
+  double kd = fputil::nearest_integer(Base::LOG2_B * xd);
+  // k = round((hi + mid) * log2(b) * 2^MID_BITS)
+  int k = static_cast<int>(kd);
+  // hi = floor(kd * 2^(-MID_BITS))
+  // exp_hi = shift hi to the exponent field of double precision.
+  int64_t exp_hi = static_cast<int64_t>((k >> Base::MID_BITS))
+                   << fputil::FloatProperties<double>::MANTISSA_WIDTH;
+  // mh = 2^hi * 2^mid
+  // mh_bits = bit field of mh
+  int64_t mh_bits = Base::EXP_2_MID[k & Base::MID_MASK] + exp_hi;
+  double mh = fputil::FPBits<double>(uint64_t(mh_bits)).get_val();
+  // dx = lo = x - (hi + mid) * log(2)
+  double dx = fputil::multiply_add(
+      kd, Base::M_LOGB_2_LO, fputil::multiply_add(kd, Base::M_LOGB_2_HI, xd));
+  // r = b^lo
+  double r = Base::powb_lo(dx);
+  return {mh, r};
 }
 
 // The function correctly calculates sinh(x) and cosh(x) by calculating exp(x)
 // and exp(-x) simultaneously.
 // To compute e^x, we perform the following range
 // reduction: find hi, mid, lo such that:
 //   x = (hi + mid) * log(2) + lo, in which
 //     hi is an integer,

libc/src/math/generic/explogxf.cpp

 //===-- Single-precision general exp/log functions ------------------------===//
 //
 // Part of the LLVM Project, under the Apache License v2.0 with LLVM Exceptions.
 // See https://llvm.org/LICENSE.txt for license information.
 // SPDX-License-Identifier: Apache-2.0 WITH LLVM-exception
 //
 //===----------------------------------------------------------------------===//
 
 #include "explogxf.h"
 
 namespace __llvm_libc {
 
-// Wolfram alpha: N[Table[2^x-1,{x,-16/32,15/32,1/32}],27]
-// printf("%.13a,\n", d[i]);
-alignas(64) const double EXP_2_POW[EXP_num_p] = {
-    -0x1.2bec333018867p-2, -0x1.1c1142e274118p-2, -0x1.0bdd71829fcf2p-2,
-    -0x1.f69d99accc7b6p-3, -0x1.d4c6af7557c93p-3, -0x1.b23213cc8e86cp-3,
-    -0x1.8edb9f5703dc0p-3, -0x1.6abf137076a8ep-3, -0x1.45d819a94b14bp-3,
-    -0x1.20224341286e4p-3, -0x1.f332113d56b1fp-4, -0x1.a46f918837cb7p-4,
-    -0x1.53f391822dbc7p-4, -0x1.01b466423250ap-4, -0x1.5b505d5b6f268p-5,
-    -0x1.5f134923757f3p-6, 0x0.0000000000000p+0,  0x1.66c34c5615d0fp-6,
-    0x1.6ab0d9f3121ecp-5,  0x1.1301d0125b50ap-4,  0x1.72b83c7d517aep-4,
-    0x1.d4873168b9aa8p-4,  0x1.1c3d373ab11c3p-3,  0x1.4f4efa8fef709p-3,
-    0x1.837f0518db8a9p-3,  0x1.b8d39b9d54e55p-3,  0x1.ef5326091a112p-3,
-    0x1.13821818624b4p-2,  0x1.2ff6b54d8a89cp-2,  0x1.4d0ad5a753e07p-2,
-    0x1.6ac1f752150a5p-2,  0x1.891fac0e95613p-2};
-
 // N[Table[Log[2, 1 + x], {x, 0/64, 63/64, 1/64}], 40]
 alignas(64) const double LOG_P1_LOG2[LOG_P1_SIZE] = {
     0x0.0000000000000p+0, 0x1.6e79685c2d22ap-6, 0x1.6bad3758efd87p-5,
     0x1.0eb389fa29f9bp-4, 0x1.663f6fac91316p-4, 0x1.bc84240adabbap-4,
     0x1.08c588cda79e4p-3, 0x1.32ae9e278ae1ap-3, 0x1.5c01a39fbd688p-3,
     0x1.84c2bd02f03b3p-3, 0x1.acf5e2db4ec94p-3, 0x1.d49ee4c325970p-3,
     0x1.fbc16b902680ap-3, 0x1.11307dad30b76p-2, 0x1.24407ab0e073ap-2,
     0x1.37124cea4cdedp-2, 0x1.49a784bcd1b8bp-2, 0x1.5c01a39fbd688p-2,

libc/src/math/generic/tanhf.cpp

if (unlikely(x_abs <= 0x3da0'0000U)) {
     return fputil::multiply_add(xdbl, pe, xdbl);
   }
 
   if (unlikely(xbits.bits == 0x4058'e0a3U)) {
     if (fputil::get_round() == FE_DOWNWARD)
       return FPBits(0x3f7f'6ad9U).get_val();
   }
 
-  auto ep = exp_eval(2.0f * (sign ? x : -x)); // exp(-2 * x)
-  double result = fputil::multiply_add(ep.mult_exp, ep.r, ep.mult_exp - 1.0) /
-                  (fputil::multiply_add(ep.mult_exp, ep.r, ep.mult_exp + 1.0));
-  return sign ? result : -result;
+  // tanh(x) = (exp(2x) - 1) / (exp(2x) + 1)
+  auto ep = exp_b_eval<ExpBase>(2.0f * x); // exp(2 * x)
+#if defined(LIBC_TARGET_HAS_FMA)
+  return fputil::multiply_add(ep.mult_exp, ep.r, -1.0) /
+         fputil::multiply_add(ep.mult_exp, ep.r, 1.0);
+#else
+  double exp_x = ep.mult_exp * ep.r;
+  return (exp_x - 1.0) / (exp_x + 1.0);
+#endif // LIBC_TARGET_HAS_FMA
 }
 
 } // namespace __llvm_libc

libc/test/src/math/explogxf_test.cpp

 constexpr float def_prec = 0.500001f;
 
 auto f_normal = [](float x) -> bool {
   return !(isnan(x) || isinf(x) || fabs(x) < 2E-38);
 };
 
 TEST(LlvmLibcExpxfTest, InFloatRange) {
   auto fx = [](float x) -> float {
-    auto result = __llvm_libc::exp_eval<-1>(x);
-    return static_cast<float>(2 * result.mult_exp * result.r +
-                              2 * result.mult_exp);
+    auto result = __llvm_libc::exp_b_eval<__llvm_libc::ExpBase>(x);
+    return static_cast<float>(result.mult_exp * result.r);
   };
   auto f_check = [](float x) -> bool {
     return !(
         (isnan(x) || isinf(x) || x < -70 || x > 70 || fabsf(x) < 0x1.0p-10));
   };
   CHECK_DATA(0.0f, neg_inf, mpfr::Operation::Exp, fx, f_check, def_count,
              def_prec);
 }