[libc] Improve the performance of exp2f.

Reduce the range-reduction table size from 128 entries down to 64 entries, and
reduce the polynomial's degree from 6 down to 4.

Currently we use a degree-6 minimax polynomial on an interval of length 2^-7
around 0 to compute exp2f.  Based on the suggestion of @santoshn and the RLIBM
project (https://github.com/rutgers-apl/rlibm-prog/blob/main/libm/float/exp2.c)
it is possible to have a good polynomial of degree-4 on a subinterval of length
2^(-6) to approximate 2^x.

We did try to either reduce the degree of the polynomial down to 3 or increase
the interval size to 2^(-5), 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^(-6) around 0.

Reviewed By: michaelrj, sivachandra, zimmermann6, santoshn

Differential Revision: https://reviews.llvm.org/D122346

2 files changed, 129 insertions, 150 deletions

diff --git a/libc/src/math/generic/exp2f.cpp b/libc/src/math/generic/exp2f.cpp
index 5c7961b96992..76ce79d32fe9 100644
--- a/libc/src/math/generic/exp2f.cpp
+++ b/libc/src/math/generic/exp2f.cpp
@@ -18,54 +18,33 @@
 
 namespace __llvm_libc {
 
-// Lookup table for 2^(m * 2^(-7)) with m = 0, ..., 127.
+// Lookup table for 2^(m * 2^(-6)) with m = 0, ..., 63.
 // Table is generated with Sollya as follow:
 // > display = hexadecimal;
-// > for i from 0 to 127 do { D(2^(i / 128)); };
-static constexpr double EXP_M[128] = {
-    0x1.0000000000000p0, 0x1.0163da9fb3335p0, 0x1.02c9a3e778061p0,
-    0x1.04315e86e7f85p0, 0x1.059b0d3158574p0, 0x1.0706b29ddf6dep0,
-    0x1.0874518759bc8p0, 0x1.09e3ecac6f383p0, 0x1.0b5586cf9890fp0,
-    0x1.0cc922b7247f7p0, 0x1.0e3ec32d3d1a2p0, 0x1.0fb66affed31bp0,
-    0x1.11301d0125b51p0, 0x1.12abdc06c31ccp0, 0x1.1429aaea92de0p0,
-    0x1.15a98c8a58e51p0, 0x1.172b83c7d517bp0, 0x1.18af9388c8deap0,
-    0x1.1a35beb6fcb75p0, 0x1.1bbe084045cd4p0, 0x1.1d4873168b9aap0,
-    0x1.1ed5022fcd91dp0, 0x1.2063b88628cd6p0, 0x1.21f49917ddc96p0,
-    0x1.2387a6e756238p0, 0x1.251ce4fb2a63fp0, 0x1.26b4565e27cddp0,
-    0x1.284dfe1f56381p0, 0x1.29e9df51fdee1p0, 0x1.2b87fd0dad990p0,
-    0x1.2d285a6e4030bp0, 0x1.2ecafa93e2f56p0, 0x1.306fe0a31b715p0,
-    0x1.32170fc4cd831p0, 0x1.33c08b26416ffp0, 0x1.356c55f929ff1p0,
-    0x1.371a7373aa9cbp0, 0x1.38cae6d05d866p0, 0x1.3a7db34e59ff7p0,
-    0x1.3c32dc313a8e5p0, 0x1.3dea64c123422p0, 0x1.3fa4504ac801cp0,
-    0x1.4160a21f72e2ap0, 0x1.431f5d950a897p0, 0x1.44e086061892dp0,
-    0x1.46a41ed1d0057p0, 0x1.486a2b5c13cd0p0, 0x1.4a32af0d7d3dep0,
-    0x1.4bfdad5362a27p0, 0x1.4dcb299fddd0dp0, 0x1.4f9b2769d2ca7p0,
-    0x1.516daa2cf6642p0, 0x1.5342b569d4f82p0, 0x1.551a4ca5d920fp0,
-    0x1.56f4736b527dap0, 0x1.58d12d497c7fdp0, 0x1.5ab07dd485429p0,
-    0x1.5c9268a5946b7p0, 0x1.5e76f15ad2148p0, 0x1.605e1b976dc09p0,
-    0x1.6247eb03a5585p0, 0x1.6434634ccc320p0, 0x1.6623882552225p0,
-    0x1.68155d44ca973p0, 0x1.6a09e667f3bcdp0, 0x1.6c012750bdabfp0,
-    0x1.6dfb23c651a2fp0, 0x1.6ff7df9519484p0, 0x1.71f75e8ec5f74p0,
-    0x1.73f9a48a58174p0, 0x1.75feb564267c9p0, 0x1.780694fde5d3fp0,
-    0x1.7a11473eb0187p0, 0x1.7c1ed0130c132p0, 0x1.7e2f336cf4e62p0,
-    0x1.80427543e1a12p0, 0x1.82589994cce13p0, 0x1.8471a4623c7adp0,
-    0x1.868d99b4492edp0, 0x1.88ac7d98a6699p0, 0x1.8ace5422aa0dbp0,
-    0x1.8cf3216b5448cp0, 0x1.8f1ae99157736p0, 0x1.9145b0b91ffc6p0,
-    0x1.93737b0cdc5e5p0, 0x1.95a44cbc8520fp0, 0x1.97d829fde4e50p0,
-    0x1.9a0f170ca07bap0, 0x1.9c49182a3f090p0, 0x1.9e86319e32323p0,
-    0x1.a0c667b5de565p0, 0x1.a309bec4a2d33p0, 0x1.a5503b23e255dp0,
-    0x1.a799e1330b358p0, 0x1.a9e6b5579fdbfp0, 0x1.ac36bbfd3f37ap0,
-    0x1.ae89f995ad3adp0, 0x1.b0e07298db666p0, 0x1.b33a2b84f15fbp0,
-    0x1.b59728de5593ap0, 0x1.b7f76f2fb5e47p0, 0x1.ba5b030a1064ap0,
-    0x1.bcc1e904bc1d2p0, 0x1.bf2c25bd71e09p0, 0x1.c199bdd85529cp0,
-    0x1.c40ab5fffd07ap0, 0x1.c67f12e57d14bp0, 0x1.c8f6d9406e7b5p0,
-    0x1.cb720dcef9069p0, 0x1.cdf0b555dc3fap0, 0x1.d072d4a07897cp0,
-    0x1.d2f87080d89f2p0, 0x1.d5818dcfba487p0, 0x1.d80e316c98398p0,
-    0x1.da9e603db3285p0, 0x1.dd321f301b460p0, 0x1.dfc97337b9b5fp0,
-    0x1.e264614f5a129p0, 0x1.e502ee78b3ff6p0, 0x1.e7a51fbc74c83p0,
-    0x1.ea4afa2a490dap0, 0x1.ecf482d8e67f1p0, 0x1.efa1bee615a27p0,
-    0x1.f252b376bba97p0, 0x1.f50765b6e4540p0, 0x1.f7bfdad9cbe14p0,
-    0x1.fa7c1819e90d8p0, 0x1.fd3c22b8f71f1p0,
+// > for i from 0 to 63 do { D(2^(i / 64)); };
+static constexpr double EXP_M[64] = {
+    0x1.0000000000000p0, 0x1.02c9a3e778061p0, 0x1.059b0d3158574p0,
+    0x1.0874518759bc8p0, 0x1.0b5586cf9890fp0, 0x1.0e3ec32d3d1a2p0,
+    0x1.11301d0125b51p0, 0x1.1429aaea92de0p0, 0x1.172b83c7d517bp0,
+    0x1.1a35beb6fcb75p0, 0x1.1d4873168b9aap0, 0x1.2063b88628cd6p0,
+    0x1.2387a6e756238p0, 0x1.26b4565e27cddp0, 0x1.29e9df51fdee1p0,
+    0x1.2d285a6e4030bp0, 0x1.306fe0a31b715p0, 0x1.33c08b26416ffp0,
+    0x1.371a7373aa9cbp0, 0x1.3a7db34e59ff7p0, 0x1.3dea64c123422p0,
+    0x1.4160a21f72e2ap0, 0x1.44e086061892dp0, 0x1.486a2b5c13cd0p0,
+    0x1.4bfdad5362a27p0, 0x1.4f9b2769d2ca7p0, 0x1.5342b569d4f82p0,
+    0x1.56f4736b527dap0, 0x1.5ab07dd485429p0, 0x1.5e76f15ad2148p0,
+    0x1.6247eb03a5585p0, 0x1.6623882552225p0, 0x1.6a09e667f3bcdp0,
+    0x1.6dfb23c651a2fp0, 0x1.71f75e8ec5f74p0, 0x1.75feb564267c9p0,
+    0x1.7a11473eb0187p0, 0x1.7e2f336cf4e62p0, 0x1.82589994cce13p0,
+    0x1.868d99b4492edp0, 0x1.8ace5422aa0dbp0, 0x1.8f1ae99157736p0,
+    0x1.93737b0cdc5e5p0, 0x1.97d829fde4e50p0, 0x1.9c49182a3f090p0,
+    0x1.a0c667b5de565p0, 0x1.a5503b23e255dp0, 0x1.a9e6b5579fdbfp0,
+    0x1.ae89f995ad3adp0, 0x1.b33a2b84f15fbp0, 0x1.b7f76f2fb5e47p0,
+    0x1.bcc1e904bc1d2p0, 0x1.c199bdd85529cp0, 0x1.c67f12e57d14bp0,
+    0x1.cb720dcef9069p0, 0x1.d072d4a07897cp0, 0x1.d5818dcfba487p0,
+    0x1.da9e603db3285p0, 0x1.dfc97337b9b5fp0, 0x1.e502ee78b3ff6p0,
+    0x1.ea4afa2a490dap0, 0x1.efa1bee615a27p0, 0x1.f50765b6e4540p0,
+    0x1.fa7c1819e90d8p0,
 };
 
 INLINE_FMA
@@ -73,93 +52,115 @@ LLVM_LIBC_FUNCTION(float, exp2f, (float x)) {
   using FPBits = typename fputil::FPBits<float>;
   FPBits xbits(x);
 
-  // When x =< -150 or nan
-  if (unlikely(xbits.uintval() >= 0xc316'0000U)) {
-    // 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));
-    if (x != 0.0f)
-      errno = ERANGE;
-    return 0.0f;
-  }
-  // x >= 128 or nan
-  if (unlikely(!xbits.get_sign() && (xbits.uintval() >= 0x4300'0000U))) {
-    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));
+  uint32_t x_u = xbits.uintval();
+  uint32_t x_abs = x_u & 0x7fff'ffffU;
 
-      errno = ERANGE;
-    }
-    return x + static_cast<float>(FPBits::inf());
-  }
-  // |x| < 2^-25
-  if (unlikely(xbits.get_unbiased_exponent() <= 101)) {
-    return 1.0f + x;
-  }
   // Exceptional values.
-  switch (xbits.uintval()) {
+  switch (x_u) {
   case 0x3b42'9d37U: // x = 0x1.853a6ep-9f
     if (fputil::get_round() == FE_TONEAREST)
       return 0x1.00870ap+0f;
     break;
+  case 0x3c02'a9adU: // x = 0x1.05535ap-7f
+    if (fputil::get_round() == FE_TONEAREST)
+      return 0x1.016b46p+0f;
+    break;
+  case 0x3ca6'6e26U: { // x = 0x1.4cdc4cp-6f
+    int round_mode = fputil::get_round();
+    if (round_mode == FE_TONEAREST || round_mode == FE_UPWARD)
+      return 0x1.03a16ap+0f;
+    return 0x1.03a168p+0f;
+  }
+  case 0x3d92'a282U: // x = 0x1.254504p-4f
+    if (fputil::get_round() == FE_UPWARD)
+      return 0x1.0d0688p+0f;
+    return 0x1.0d0686p+0f;
   case 0xbcf3'a937U: // x = -0x1.e7526ep-6f
     if (fputil::get_round() == FE_TONEAREST)
       return 0x1.f58d62p-1f;
     break;
+  case 0xb8d3'd026U: // x = -0x1.a7a04cp-14f
+    if (fputil::get_round() == FE_TONEAREST)
+      return 0x1.fff6d2p-1f;
+    break;
   }
 
+  // // When |x| >= 128, |x| < 2^-25, or x is nan
+  if (unlikely(x_abs >= 0x4300'0000U || x_abs <= 0x3280'0000U)) {
+    // |x| < 2^-25
+    if (x_abs <= 0x3280'0000U) {
+      return 1.0f + x;
+    }
+    // x >= 128
+    if (!xbits.get_sign()) {
+      // x is finite
+      if (x_u < 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 < -150
+    if (x_u >= 0xc316'0000U) {
+      // 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));
+      if (x != 0.0f)
+        errno = ERANGE;
+      return 0.0f;
+    }
+  }
   // For -150 <= x < 128, to compute 2^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.
+  //     mid * 2^6 is an integer
+  //     -2^(-7) <= lo < 2^-7.
   // In particular,
-  //   hi + mid = round(x * 2^7) * 2^(-7).
+  //   hi + mid = round(x * 2^6) * 2^(-6).
   // Then,
   //   2^(x) = 2^(hi + mid + lo) = 2^hi * 2^mid * 2^lo.
   // Multiply by 2^hi is simply adding hi to the exponent field.  We store
-  // exp(mid) in the lookup tables EXP_M.  exp(lo) is computed using a degree-7
+  // exp(mid) in the lookup tables EXP_M.  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);
-  // 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 = round(hi + mid).
+  // 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 * 0x1.0p+6f + (xbits.get_sign() ? -0.5f : 0.5f));
   // For 2-complement integers, arithmetic right shift is the same as dividing
   // by a power of 2 and then round down (toward negative infinity).
-  int hi = x_hi >> 7;
-  // mid = x_hi & 0x0000'007fU;
-  double exp_mid = EXP_M[x_hi & 0x7f];
-  // Degree-6 minimax polynomial generated by Sollya with the following
+  int e_hi = (x_hi >> 6) +
+             static_cast<int>(fputil::FloatProperties<double>::EXPONENT_BIAS);
+  fputil::FPBits<double> exp_hi(
+      static_cast<uint64_t>(e_hi)
+      << fputil::FloatProperties<double>::MANTISSA_WIDTH);
+  // mid = x_hi & 0x0000'003fU;
+  double exp_hi_mid = static_cast<double>(exp_hi) * EXP_M[x_hi & 0x3f];
+  // Subtract (hi + mid) from x to get lo.
+  x -= static_cast<float>(x_hi) * 0x1.0p-6f;
+  double xd = static_cast<double>(x);
+  // Degree-4 minimax polynomial generated by Sollya with the following
   // commands:
   //   > display = hexadecimal;
-  //   > Q = fpminimax((2^x - 1)/x, 5, [|D...|], [-2^-8, 2^-8]);
+  //   > Q = fpminimax((2^x - 1)/x, 3, [|D...|], [-2^-7, 2^-7]);
   //   > Q;
   double exp_lo =
-      fputil::polyeval(xd, 0x1p0, 0x1.62e42fefa39efp-1, 0x1.ebfbdff82c58ep-3,
-                       0x1.c6b08d711fe2fp-5, 0x1.3b2ab6fe3deb5p-7,
-                       0x1.5d72a05f45c04p-10, 0x1.4284d40c33326p-13);
-  fputil::FPBits<double> result(exp_mid * exp_lo);
-  result.set_unbiased_exponent(static_cast<uint16_t>(
-      static_cast<int>(result.get_unbiased_exponent()) + hi));
-  return static_cast<float>(static_cast<double>(result));
+      fputil::polyeval(xd, 0x1p0, 0x1.62e42fefa2417p-1, 0x1.ebfbdff82f809p-3,
+                       0x1.c6b0b92131c47p-5, 0x1.3b2ab6fb568a3p-7);
+  double result = exp_hi_mid * exp_lo;
+  return static_cast<float>(result);
 }
 
 } // namespace __llvm_libc
diff --git a/libc/test/src/math/exp2f_test.cpp b/libc/test/src/math/exp2f_test.cpp
index 20c22c6bbb9e..07faeb55de30 100644
--- a/libc/test/src/math/exp2f_test.cpp
+++ b/libc/test/src/math/exp2f_test.cpp
@@ -51,51 +51,29 @@ TEST(LlvmLibcExp2fTest, Overflow) {
   EXPECT_MATH_ERRNO(ERANGE);
 }
 
-// Test with inputs which are the borders of underflow/overflow but still
-// produce valid results without setting errno.
-TEST(LlvmLibcExp2fTest, Borderline) {
-  float x;
-
-  errno = 0;
-  x = float(FPBits(0x42fa0001U));
-  EXPECT_MPFR_MATCH_ALL_ROUNDING(mpfr::Operation::Exp2, x,
-                                 __llvm_libc::exp2f(x), 0.5);
-  EXPECT_MATH_ERRNO(0);
-
-  x = float(FPBits(0x42ffffffU));
-  EXPECT_MPFR_MATCH_ALL_ROUNDING(mpfr::Operation::Exp2, x,
-                                 __llvm_libc::exp2f(x), 0.5);
-  EXPECT_MATH_ERRNO(0);
-
-  x = float(FPBits(0xc2fa0001U));
-  EXPECT_MPFR_MATCH_ALL_ROUNDING(mpfr::Operation::Exp2, x,
-                                 __llvm_libc::exp2f(x), 0.5);
-  EXPECT_MATH_ERRNO(0);
-
-  x = float(FPBits(0xc2fc0000U));
-  EXPECT_MPFR_MATCH_ALL_ROUNDING(mpfr::Operation::Exp2, x,
-                                 __llvm_libc::exp2f(x), 0.5);
-  EXPECT_MATH_ERRNO(0);
-
-  x = float(FPBits(0xc2fc0001U));
-  EXPECT_MPFR_MATCH_ALL_ROUNDING(mpfr::Operation::Exp2, x,
-                                 __llvm_libc::exp2f(x), 0.5);
-  EXPECT_MATH_ERRNO(0);
-
-  x = float(FPBits(0xc3150000U));
-  EXPECT_MPFR_MATCH_ALL_ROUNDING(mpfr::Operation::Exp2, x,
-                                 __llvm_libc::exp2f(x), 0.5);
-  EXPECT_MATH_ERRNO(0);
-
-  x = float(FPBits(0x3b42'9d37U));
-  EXPECT_MPFR_MATCH_ALL_ROUNDING(mpfr::Operation::Exp2, x,
-                                 __llvm_libc::exp2f(x), 0.5);
-  EXPECT_MATH_ERRNO(0);
-
-  x = float(FPBits(0xbcf3'a937U));
-  EXPECT_MPFR_MATCH_ALL_ROUNDING(mpfr::Operation::Exp2, x,
-                                 __llvm_libc::exp2f(x), 0.5);
-  EXPECT_MATH_ERRNO(0);
+TEST(LlvmLibcExp2fTest, TrickyInputs) {
+  constexpr int N = 12;
+  constexpr uint32_t INPUTS[N] = {
+      0x3b429d37U, /*0x1.853a6ep-9f*/
+      0x3c02a9adU, /*0x1.05535ap-7f*/
+      0x3ca66e26U, /*0x1.4cdc4cp-6f*/
+      0x3d92a282U, /*0x1.254504p-4f*/
+      0x42fa0001U, /*0x1.f40002p+6f*/
+      0x42ffffffU, /*0x1.fffffep+6f*/
+      0xb8d3d026U, /*-0x1.a7a04cp-14f*/
+      0xbcf3a937U, /*-0x1.e7526ep-6f*/
+      0xc2fa0001U, /*-0x1.f40002p+6f*/
+      0xc2fc0000U, /*-0x1.f8p+6f*/
+      0xc2fc0001U, /*-0x1.f80002p+6f*/
+      0xc3150000U, /*-0x1.2ap+7f*/
+  };
+  for (int i = 0; i < N; ++i) {
+    errno = 0;
+    float x = float(FPBits(INPUTS[i]));
+    EXPECT_MPFR_MATCH_ALL_ROUNDING(mpfr::Operation::Exp2, x,
+                                   __llvm_libc::exp2f(x), 0.5);
+    EXPECT_MATH_ERRNO(0);
+  }
 }
 
 TEST(LlvmLibcExp2fTest, Underflow) {