[libc] Improve the performance of expf.

[lntue]

Summary

Reduce the polynomial's degree from 7 down to 4.

Currently we use a degree-7 minimax polynomial on an interval of length 2^-7
around 0 to compute expf. Based on the suggestion of @santoshn and the RLIBM
project (https://github.com/rutgers-apl/rlibm-all/blob/main/source/float/exp.c)
and the improvement we made with exp2f in https://reviews.llvm.org/D122346,
it is possible to have a good polynomial of degree-4 on a subinterval of length
2^(-7) to approximate e^x.

We did try to either reduce the degree of the polynomial down to 3 or increase
the interval size to 2^(-6), but in both cases the number of exceptional values
exploded. So we settle with using a degree-4 polynomial of the interval of
size 2^(-7) around 0.

[lntue]

Update comments.

[zimmermann6]

ok for me, all exhaustive tests pass, and the performance increased a lot:

zimmerma@tomate:~/svn/core-math$ LIBM=/users/zimmerma/svn/core-math/libllvmlibc.a ./perf.sh expf
21.594
10.968
51.286
zimmerma@tomate:~/svn/core-math$ LIBM=/tmp/libllvmlibc.a ./perf.sh expf
21.596
10.966
16.997

The first run is with the previous version, the second one with the new version. The last timing is the one for llvm-libc, the first one for core-math, and the 2nd one for the GNU libc (not CR).

[santoshn]

Looks good to me.

libc/src/math/generic/expf.cpp

 
 namespace __llvm_libc {
 
 INLINE_FMA
 LLVM_LIBC_FUNCTION(float, expf, (float x)) {
   using FPBits = typename fputil::FPBits<float>;
   FPBits xbits(x);
 
+  uint32_t x_u = xbits.uintval();
+  uint32_t x_abs = x_u & 0x7fff'ffffU;
+
+  // Exceptional values
+  if (unlikely(x_u == 0xc236'bd8cU)) { // x = -0x1.6d7b18p+5f
+    return 0x1.108a58p-66f - x * 0x1.0p-95f;
+  }
+
+  // When |x| >= 89, |x| < 2^-25, or x is nan
+  if (unlikely(x_abs >= 0x42b2'0000U || x_abs <= 0x3280'0000U)) {
+    // |x| < 2^-25
+    if (xbits.get_unbiased_exponent() <= 101) {
+      return 1.0f + x;
+    }
+
     // When x < log(2^-150) or nan
-  if (unlikely(xbits.uintval() >= 0xc2cf'f1b5U)) {
+    if (xbits.uintval() >= 0xc2cf'f1b5U) {
       // exp(-Inf) = 0
       if (xbits.is_inf())
         return 0.0f;
       // exp(nan) = nan
       if (xbits.is_nan())
         return x;
       if (fputil::get_round() == FE_UPWARD)
         return static_cast<float>(FPBits(FPBits::MIN_SUBNORMAL));
       errno = ERANGE;
       return 0.0f;
     }
     // x >= 89 or nan
-  if (unlikely(!xbits.get_sign() && (xbits.uintval() >= 0x42b2'0000))) {
+    if (!xbits.get_sign() && (xbits.uintval() >= 0x42b2'0000)) {
+      // x is finite
       if (xbits.uintval() < 0x7f80'0000U) {
         int rounding = fputil::get_round();
         if (rounding == FE_DOWNWARD || rounding == FE_TOWARDZERO)
           return static_cast<float>(FPBits(FPBits::MAX_NORMAL));
 
         errno = ERANGE;
       }
+      // x is +inf or nan
       return x + static_cast<float>(FPBits::inf());
     }
-  // |x| < 2^-25
-  if (unlikely(xbits.get_unbiased_exponent() <= 101)) {
-    return 1.0f + x;
   }
-
   // For -104 < x < 89, to compute exp(x), we perform the following range
   // reduction: find hi, mid, lo such that:
   //   x = hi + mid + lo, in which
   //     hi is an integer,
   //     mid * 2^7 is an integer
   //     -2^(-8) <= lo < 2^-8.
   // In particular,
   //   hi + mid = round(x * 2^7) * 2^(-7).
   // Then,
   //   exp(x) = exp(hi + mid + lo) = exp(hi) * exp(mid) * exp(lo).
   // We store exp(hi) and exp(mid) in the lookup tables EXP_M1 and EXP_M2
-  // respectively.  exp(lo) is computed using a degree-7 minimax polynomial
+  // respectively.  exp(lo) is computed using a degree-4 minimax polynomial
   // generated by Sollya.
 
-  // x_hi = hi + mid.
-  int x_hi = static_cast<int>(x * 0x1.0p7f);
+  // x_hi = (hi + mid) * 2^7 = round(x * 2^7).
+  // The default rounding mode for float-to-int conversion in C++ is
+  // round-toward-zero. To make it round-to-nearest, we add (-1)^sign(x) * 0.5
+  // before conversion.
+  int x_hi = static_cast<int>(x * 0x1.0p7f + (xbits.get_sign() ? -0.5f : 0.5f));
   // Subtract (hi + mid) from x to get lo.
   x -= static_cast<float>(x_hi) * 0x1.0p-7f;
   double xd = static_cast<double>(x);
-  // Make sure that -2^(-8) <= lo < 2^-8.
-  if (x >= 0x1.0p-8f) {
-    ++x_hi;
-    xd -= 0x1.0p-7;
-  }
-  if (x < -0x1.0p-8f) {
-    --x_hi;
-    xd += 0x1.0p-7;
-  }
   x_hi += 104 << 7;
   // hi = x_hi >> 7
   double exp_hi = EXP_M1[x_hi >> 7];
-  // lo = x_hi & 0x0000'007fU;
+  // mid * 2^7 = x_hi & 0x0000'007fU;
   double exp_mid = EXP_M2[x_hi & 0x7f];
-  // Degree-7 minimax polynomial generated by Sollya with the following
+  // Degree-4 minimax polynomial generated by Sollya with the following
   // commands:
   //   > display = hexadecimal;
-  //   > Q = fpminimax(expm1(x)/x, 6, [|D...|], [-2^-8, 2^-8]);
+  //   > Q = fpminimax(expm1(x)/x, 3, [|D...|], [-2^-8, 2^-8]);
   //   > Q;
-  double exp_lo = fputil::polyeval(
-      xd, 0x1p0, 0x1p0, 0x1p-1, 0x1.5555555555555p-3, 0x1.55555555553ap-5,
-      0x1.1111111204dfcp-7, 0x1.6c16cb2da593ap-10, 0x1.9ff1648996d2ep-13);
+  double exp_lo =
+      fputil::polyeval(xd, 0x1p0, 0x1.ffffffffff777p-1, 0x1.000000000071cp-1,
+                       0x1.555566668e5e7p-3, 0x1.55555555ef243p-5);
   return static_cast<float>(exp_hi * exp_mid * exp_lo);
 }
 
 } // namespace __llvm_libc

libc/test/src/math/exhaustive/expf_test.cpp

 //===-- Exhaustive test for expf ------------------------------------------===//
 //
 // 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 "exhaustive_test.h"
 #include "src/__support/FPUtil/FPBits.h"
 #include "src/math/expf.h"
 #include "utils/MPFRWrapper/MPFRUtils.h"
 #include "utils/UnitTest/FPMatcher.h"
 
+#include <thread>
+
 using FPBits = __llvm_libc::fputil::FPBits<float>;
 
 namespace mpfr = __llvm_libc::testing::mpfr;
 
 struct LlvmLibcExpfExhaustiveTest : public LlvmLibcExhaustiveTest<uint32_t> {
   bool check(uint32_t start, uint32_t stop,
              mpfr::RoundingMode rounding) override {
     mpfr::ForceRoundingMode r(rounding);
     uint32_t bits = start;
     bool result = true;
     do {
       FPBits xbits(bits);
       float x = float(xbits);
       result &= EXPECT_MPFR_MATCH(mpfr::Operation::Exp, x, __llvm_libc::expf(x),
                                   0.5, rounding);
     } while (bits++ < stop);
     return result;
   }
 };
 
-static constexpr int NUM_THREADS = 16;
+static const int NUM_THREADS = std::thread::hardware_concurrency();
 
 // Range: [0, 89];
 static constexpr uint32_t POS_START = 0x0000'0000U;
 static constexpr uint32_t POS_STOP = 0x42b2'0000U;
 
 TEST_F(LlvmLibcExpfExhaustiveTest, PostiveRangeRoundNearestTieToEven) {
   test_full_range(POS_START, POS_STOP, NUM_THREADS,
                   mpfr::RoundingMode::Nearest);

libc/test/src/math/expf_test.cpp

TEST(LlvmLibcExpfTest, Borderline) {
   ASSERT_MPFR_MATCH_ALL_ROUNDING(mpfr::Operation::Exp, x, __llvm_libc::expf(x),
                                  0.5);
   EXPECT_MATH_ERRNO(0);
 
   x = float(FPBits(0xc2b00008U));
   ASSERT_MPFR_MATCH_ALL_ROUNDING(mpfr::Operation::Exp, x, __llvm_libc::expf(x),
                                  0.5);
   EXPECT_MATH_ERRNO(0);
+
+  x = float(FPBits(0xc236bd8cU));
+  EXPECT_MPFR_MATCH_ALL_ROUNDING(mpfr::Operation::Exp, x, __llvm_libc::expf(x),
+                                 0.5);
+  EXPECT_MATH_ERRNO(0);
 }
 
 TEST(LlvmLibcExpfTest, InFloatRange) {
   constexpr uint32_t COUNT = 1000000;
   constexpr uint32_t STEP = UINT32_MAX / COUNT;
   for (uint32_t i = 0, v = 0; i <= COUNT; ++i, v += STEP) {
     float x = float(FPBits(v));
     if (isnan(x) || isinf(x))