[libc][math] Implement double precision exp function correctly rounded for all rounding modes.

[lntue]

Summary

Implement double precision exp function correctly rounded for all
rounding modes. Using 4 stages:

• Range reduction: reduce to exp(x) = 2^hi * 2^mid1 * 2^mid2 * exp(lo).
• Use 64 + 64 LUT for 2^mid1 and 2^mid2, and use cubic Taylor polynomial to

approximate (exp(lo) - 1) / lo in double precision. Relative error in this
step is bounded by 1.5 * 2^-63.

• If the rounding test fails, use degree-6 Taylor polynomial to approximate

exp(lo) in double-double precision. Relative error in this step is bounded by
2^-99.

• If the rounding test still fails, use degree-7 Taylor polynomial to compute

exp(lo) in ~128-bit precision.

[lntue]

Remove unused template.

[zimmermann6]

when I apply this patch on main (commit 96b5ea6), and I build libc.a, I don't find the symbol exp into libc.a:

zimmerma@biscotte:/localdisk/zimmerma/llvm-project$ nm /localdisk/zimmerma/llvm-project/build/projects/libc/lib/libc.a | grep -w exp
zimmerma@biscotte:/localdisk/zimmerma/llvm-project$

Maybe I did something wrong?

[lntue]

Hi Paul, do you mind sending me the build commands that you used and their
logs, and/or attaching the generated libc.a.

Thanks,

Tue

[zimmermann6]

In D158551#4609925, @lntue wrote:

Hi Paul, do you mind sending me the build commands that you used and their
logs, and/or attaching the generated libc.a.

Thanks,

Tue

sure, I'll do that in a private mail.

[zimmermann6]

after help from Tue I was able to use this patch with the core-math tools. All tests pass, and on my machine (AMD EPYC 7282) I get the following timings:

zimmerma@biscotte:/tmp/core-math$ LIBM=/localdisk/zimmerma/llvm-project/build/projects/libc/lib/libm.a CORE_MATH_PERF_MODE=rdtsc ./perf.sh exp
GNU libc version: 2.37
GNU libc release: stable
[####################] 100 %
Ntrial = 20 ; Min = 18.639 + 0.385 clc/call; Median-Min = 0.327 clc/call; Max = 21.116 clc/call;
[####################] 100 %
Ntrial = 20 ; Min = 13.164 + 0.253 clc/call; Median-Min = 0.280 clc/call; Max = 13.590 clc/call;
[####################] 100 %
Ntrial = 20 ; Min = 19.464 + 0.307 clc/call; Median-Min = 0.308 clc/call; Max = 19.828 clc/call;

and for latency:

zimmerma@biscotte:/tmp/core-math$ LIBM=/localdisk/zimmerma/llvm-project/build/projects/libc/lib/libm.a CORE_MATH_PERF_MODE=rdtsc PERF_ARGS=--latency ./perf.sh exp
GNU libc version: 2.37
GNU libc release: stable
[####################] 100 %
Ntrial = 20 ; Min = 49.458 + 0.339 clc/call; Median-Min = 0.269 clc/call; Max = 50.109 clc/call;
[####################] 100 %
Ntrial = 20 ; Min = 39.380 + 0.326 clc/call; Median-Min = 0.299 clc/call; Max = 39.948 clc/call;
[####################] 100 %
Ntrial = 20 ; Min = 54.919 + 0.347 clc/call; Median-Min = 0.310 clc/call; Max = 55.561 clc/call;

libc/config/darwin/arm/entrypoints.txt

set(TARGET_LIBM_ENTRYPOINTS
     libc.src.math.copysignf
     libc.src.math.copysignl
     libc.src.math.ceil
     libc.src.math.ceilf
     libc.src.math.ceill
     libc.src.math.coshf
     libc.src.math.cosf
     libc.src.math.erff
+    libc.src.math.exp
     libc.src.math.expf
     libc.src.math.exp10f
     libc.src.math.exp2f
     libc.src.math.expm1f
     libc.src.math.fabs
     libc.src.math.fabsf
     libc.src.math.fabsl
     libc.src.math.fdim

libc/config/linux/aarch64/entrypoints.txt

set(TARGET_LIBM_ENTRYPOINTS
     libc.src.math.copysignf
     libc.src.math.copysignl
     libc.src.math.ceil
     libc.src.math.ceilf
     libc.src.math.ceill
     libc.src.math.coshf
     libc.src.math.cosf
     libc.src.math.erff
+    libc.src.math.exp
     libc.src.math.expf
     libc.src.math.exp10f
     libc.src.math.exp2f
     libc.src.math.expm1f
     libc.src.math.fabs
     libc.src.math.fabsf
     libc.src.math.fabsl
     libc.src.math.fdim

libc/config/linux/riscv64/entrypoints.txt

set(TARGET_LIBM_ENTRYPOINTS
     libc.src.math.copysignf
     libc.src.math.copysignl
     libc.src.math.ceil
     libc.src.math.ceilf
     libc.src.math.ceill
     libc.src.math.coshf
     libc.src.math.cosf
     libc.src.math.erff
+    libc.src.math.exp
     libc.src.math.expf
     libc.src.math.exp10f
     libc.src.math.exp2f
     libc.src.math.expm1f
     libc.src.math.fabs
     libc.src.math.fabsf
     libc.src.math.fabsl
     libc.src.math.fdim

libc/config/linux/x86_64/entrypoints.txt

set(TARGET_LIBM_ENTRYPOINTS
     libc.src.math.copysignl
     libc.src.math.ceil
     libc.src.math.ceilf
     libc.src.math.ceill
     libc.src.math.cos
     libc.src.math.coshf
     libc.src.math.cosf
     libc.src.math.erff
+    libc.src.math.exp
     libc.src.math.expf
     libc.src.math.exp10f
     libc.src.math.exp2f
     libc.src.math.expm1f
     libc.src.math.fabs
     libc.src.math.fabsf
     libc.src.math.fabsl
     libc.src.math.fdim

libc/config/windows/entrypoints.txt

set(TARGET_LIBM_ENTRYPOINTS
     libc.src.math.copysignl
     libc.src.math.ceil
     libc.src.math.ceilf
     libc.src.math.ceill
     libc.src.math.cos
     libc.src.math.cosf
     libc.src.math.coshf
     libc.src.math.erff
+    libc.src.math.exp
     libc.src.math.expf
     libc.src.math.exp10f
     libc.src.math.exp2f
     libc.src.math.expm1f
     libc.src.math.fabs
     libc.src.math.fabsf
     libc.src.math.fabsl
     libc.src.math.fdim

libc/docs/math/index.rst

 | erfl       |         |         |         |         |         |         |         |         |         |         |         |         |
 +------------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+
 | erfc       |         |         |         |         |         |         |         |         |         |         |         |         |
 +------------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+
 | erfcf      |         |         |         |         |         |         |         |         |         |         |         |         |
 +------------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+
 | erfcl      |         |         |         |         |         |         |         |         |         |         |         |         |
 +------------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+
-| exp        |         |         |         |         |         |         |         |         |         |         |         |         |
+| exp        | |check| | |check| |         | |check| | |check| |         |         | |check| |         |         |         |         |
 +------------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+
 | expf       | |check| | |check| |         | |check| | |check| |         |         | |check| |         |         |         |         |
 +------------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+
 | expl       |         |         |         |         |         |         |         |         |         |         |         |         |
 +------------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+
 | exp10      |         |         |         |         |         |         |         |         |         |         |         |         |
 +------------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+
 | exp10f     | |check| | |check| |         | |check| | |check| |         |         | |check| |         |         |         |         |

 acosh          |check|
 asin           |check|
 asinh          |check|
 atan           |check|
 atanh          |check|
 cos            |check|          large
 cosh           |check|
 erf            |check|
-exp            |check|
+exp            |check|          |check|
 exp10          |check|
 exp2           |check|
 expm1          |check|
 fma            |check|          |check|
 hypot          |check|          |check|
 log            |check|          |check|
 log10          |check|          |check|
 log1p          |check|          |check|

libc/spec/stdc.td

HeaderSpec Math = HeaderSpec<
           FunctionSpec<"cosf", RetValSpec<FloatType>, [ArgSpec<FloatType>]>,
           FunctionSpec<"sin", RetValSpec<DoubleType>, [ArgSpec<DoubleType>]>,
           FunctionSpec<"sinf", RetValSpec<FloatType>, [ArgSpec<FloatType>]>,
           FunctionSpec<"tan", RetValSpec<DoubleType>, [ArgSpec<DoubleType>]>,
           FunctionSpec<"tanf", RetValSpec<FloatType>, [ArgSpec<FloatType>]>,
 
           FunctionSpec<"erff", RetValSpec<FloatType>, [ArgSpec<FloatType>]>,
 
+          FunctionSpec<"exp", RetValSpec<DoubleType>, [ArgSpec<DoubleType>]>,
           FunctionSpec<"expf", RetValSpec<FloatType>, [ArgSpec<FloatType>]>,
+
           FunctionSpec<"exp2f", RetValSpec<FloatType>, [ArgSpec<FloatType>]>,
           FunctionSpec<"expm1f", RetValSpec<FloatType>, [ArgSpec<FloatType>]>,
 
           FunctionSpec<"remainderf", RetValSpec<FloatType>, [ArgSpec<FloatType>, ArgSpec<FloatType>]>,
           FunctionSpec<"remainder", RetValSpec<DoubleType>, [ArgSpec<DoubleType>, ArgSpec<DoubleType>]>,
           FunctionSpec<"remainderl", RetValSpec<LongDoubleType>, [ArgSpec<LongDoubleType>, ArgSpec<LongDoubleType>]>,
 
           FunctionSpec<"remquof", RetValSpec<FloatType>, [ArgSpec<FloatType>, ArgSpec<FloatType>, ArgSpec<IntPtr>]>,

libc/src/__support/FPUtil/PolyEval.h

 //===-- Common header for PolyEval implementations --------------*- C++ -*-===//
 //
 // 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
 //
 //===----------------------------------------------------------------------===//
 
 #ifndef LLVM_LIBC_SRC_SUPPORT_FPUTIL_POLYEVAL_H
 #define LLVM_LIBC_SRC_SUPPORT_FPUTIL_POLYEVAL_H
 
 #include "multiply_add.h"
+#include "src/__support/CPP/type_traits.h"
 #include "src/__support/common.h"
 
 // Evaluate polynomial using Horner's Scheme:
 // With polyeval(x, a_0, a_1, ..., a_n) = a_n * x^n + ... + a_1 * x + a_0, we
 // evaluated it as:  a_0 + x * (a_1 + x * ( ... (a_(n-1) + x * a_n) ... ) ) ).
 // We will use FMA instructions if available.
 // Example: to evaluate x^3 + 2*x^2 + 3*x + 4, call
 //   polyeval( x, 4.0, 3.0, 2.0, 1.0 )
 
 namespace __llvm_libc {
 namespace fputil {
 
-template <typename T> LIBC_INLINE T polyeval(T, T a0) { return a0; }
+template <typename T> LIBC_INLINE T polyeval(const T &, const T &a0) {
+  return a0;
+}
 
 template <typename T, typename... Ts>
-LIBC_INLINE T polyeval(T x, T a0, Ts... a) {
+LIBC_INLINE T polyeval(const T &x, const T &a0, const Ts &...a) {
   return multiply_add(x, polyeval(x, a...), a0);
 }
 
 } // namespace fputil
 } // namespace __llvm_libc
 
 #endif // LLVM_LIBC_SRC_SUPPORT_FPUTIL_POLYEVAL_H

libc/src/__support/FPUtil/double_double.h

LIBC_INLINE constexpr DoubleDouble exact_add(double a, double b) {
   DoubleDouble r{0.0, 0.0};
   r.hi = a + b;
   double t = r.hi - a;
   r.lo = b - t;
   return r;
 }
 
 // Assumption: |a.hi| >= |b.hi|
-LIBC_INLINE constexpr DoubleDouble add(DoubleDouble a, DoubleDouble b) {
+LIBC_INLINE constexpr DoubleDouble add(const DoubleDouble &a,
+                                       const DoubleDouble &b) {
   DoubleDouble r = exact_add(a.hi, b.hi);
   double lo = a.lo + b.lo;
   return exact_add(r.hi, r.lo + lo);
 }
 
 // Assumption: |a.hi| >= |b|
-LIBC_INLINE constexpr DoubleDouble add(DoubleDouble a, double b) {
+LIBC_INLINE constexpr DoubleDouble add(const DoubleDouble &a, double b) {
   DoubleDouble r = exact_add(a.hi, b);
   return exact_add(r.hi, r.lo + a.lo);
 }
 
 // Velkamp's Splitting for double precision.
 LIBC_INLINE constexpr DoubleDouble split(double a) {
   DoubleDouble r{0.0, 0.0};
   // Splitting constant = 2^ceil(prec(double)/2) + 1 = 2^27 + 1.
#else
   double t2 = as.hi * bs.lo + t1;
   double t3 = as.lo * bs.hi + t2;
   r.lo = as.lo * bs.lo + t3;
 #endif // LIBC_TARGET_CPU_HAS_FMA
 
   return r;
 }
 
-LIBC_INLINE DoubleDouble quick_mult(DoubleDouble a, DoubleDouble b) {
+LIBC_INLINE DoubleDouble quick_mult(double a, const DoubleDouble &b) {
+  DoubleDouble r = exact_mult(a, b.hi);
+  r.lo = multiply_add(a, b.lo, r.lo);
+  return r;
+}
+
+LIBC_INLINE DoubleDouble quick_mult(const DoubleDouble &a,
+                                    const DoubleDouble &b) {
   DoubleDouble r = exact_mult(a.hi, b.hi);
-  double t1 = fputil::multiply_add(a.hi, b.lo, r.lo);
-  double t2 = fputil::multiply_add(a.lo, b.hi, t1);
+  double t1 = multiply_add(a.hi, b.lo, r.lo);
+  double t2 = multiply_add(a.lo, b.hi, t1);
   r.lo = t2;
   return r;
 }
 
+// Assuming |c| >= |a * b|.
+template <>
+LIBC_INLINE DoubleDouble multiply_add<DoubleDouble>(const DoubleDouble &a,
+                                                    const DoubleDouble &b,
+                                                    const DoubleDouble &c) {
+  return add(c, quick_mult(a, b));
+}
+
 } // namespace __llvm_libc::fputil
 
 #endif // LLVM_LIBC_SRC_SUPPORT_FPUTIL_DOUBLEDOUBLE_H

libc/src/__support/FPUtil/dyadic_float.h

template <size_t Bits> struct DyadicFloat {
 
   // Used for aligning exponents.  Output might not be normalized.
   DyadicFloat &shift_right(int shift_length) {
     exponent += shift_length;
     mantissa >>= static_cast<size_t>(shift_length);
     return *this;
   }
 
-  // Assume that it is already normalized and output is also normal.
+  // Assume that it is already normalized and output is not underflow.
   // Output is rounded correctly with respect to the current rounding mode.
-  // TODO(lntue): Test or add support for denormal output.
+  // TODO(lntue): Add support for underflow.
   // TODO(lntue): Test or add specialization for x86 long double.
   template <typename T, typename = cpp::enable_if_t<
                             cpp::is_floating_point_v<T> &&
                                 (FloatProperties<T>::MANTISSA_WIDTH < Bits),
                             void>>
   explicit operator T() const {
     // TODO(lntue): Do we need to treat signed zeros properly?
     if (mantissa.is_zero())
       return 0.0;
 
     // Assume that it is normalized, and output is also normal.
     constexpr size_t PRECISION = FloatProperties<T>::MANTISSA_WIDTH + 1;
     using output_bits_t = typename FPBits<T>::UIntType;
 
-    MantissaType m_hi(mantissa >> (Bits - PRECISION));
-    auto d_hi = FPBits<T>::create_value(
-        sign, exponent + (Bits - 1) + FloatProperties<T>::EXPONENT_BIAS,
-        output_bits_t(m_hi) & FloatProperties<T>::MANTISSA_MASK);
+    int exp_hi = exponent + static_cast<int>((Bits - 1) +
+                                             FloatProperties<T>::EXPONENT_BIAS);
 
-    const MantissaType round_mask = MantissaType(1) << (Bits - PRECISION - 1);
+    bool denorm = false;
+    uint32_t shift = Bits - PRECISION;
+    if (LIBC_UNLIKELY(exp_hi <= 0)) {
+      // Output is denormal.
+      denorm = true;
+      shift = (Bits - PRECISION) + static_cast<uint32_t>(1 - exp_hi);
+
+      exp_hi = FloatProperties<T>::EXPONENT_BIAS;
+    }
+
+    int exp_lo = exp_hi - PRECISION - 1;
+
+    MantissaType m_hi(mantissa >> shift);
+
+    T d_hi = FPBits<T>::create_value(sign, exp_hi,
+                                     output_bits_t(m_hi) &
+                                         FloatProperties<T>::MANTISSA_MASK)
+                 .get_val();
+
+    const MantissaType round_mask = MantissaType(1) << (shift - 1);
     const MantissaType sticky_mask = round_mask - MantissaType(1);
 
     bool round_bit = !(mantissa & round_mask).is_zero();
     bool sticky_bit = !(mantissa & sticky_mask).is_zero();
     int round_and_sticky = int(round_bit) * 2 + int(sticky_bit);
-    auto d_lo = FPBits<T>::create_value(sign,
-                                        exponent + (Bits - PRECISION - 2) +
-                                            FloatProperties<T>::EXPONENT_BIAS,
-                                        output_bits_t(0));
+
+    T d_lo;
+    if (LIBC_UNLIKELY(exp_lo <= 0)) {
+      // d_lo is denormal, but the output is normal.
+      int scale_up_exponent = 2 * PRECISION;
+      T scale_up_factor =
+          FPBits<T>::create_value(
+              sign, FloatProperties<T>::EXPONENT_BIAS + scale_up_exponent,
+              output_bits_t(0))
+              .get_val();
+      T scale_down_factor =
+          FPBits<T>::create_value(
+              sign, FloatProperties<T>::EXPONENT_BIAS - scale_up_exponent,
+              output_bits_t(0))
+              .get_val();
+
+      d_lo = FPBits<T>::create_value(sign, exp_lo + scale_up_exponent,
+                                     output_bits_t(0))
+                 .get_val();
+
+      return multiply_add(d_lo, T(round_and_sticky), d_hi * scale_up_factor) *
+             scale_down_factor;
+    }
+
+    d_lo = FPBits<T>::create_value(sign, exp_lo, output_bits_t(0)).get_val();
 
     // Still correct without FMA instructions if `d_lo` is not underflow.
-    return multiply_add(d_lo.get_val(), T(round_and_sticky), d_hi.get_val());
+    T r = multiply_add(d_lo, T(round_and_sticky), d_hi);
+
+    if (LIBC_UNLIKELY(denorm)) {
+      // Output is denormal, simply clear the exponent field.
+      output_bits_t clear_exp = output_bits_t(exp_hi)
+                                << FloatProperties<T>::MANTISSA_WIDTH;
+      output_bits_t r_bits = FPBits<T>(r).uintval() - clear_exp;
+      return FPBits<T>(r_bits).get_val();
+    }
+
+    return r;
   }
 
   explicit operator MantissaType() const {
     if (mantissa.is_zero())
       return 0;
 
     MantissaType new_mant = mantissa;
     if (exponent > 0) {
if (result.mantissa.val[DyadicFloat<Bits>::MantissaType::WORDCOUNT - 1] >>
         0)
       result.shift_left(1);
   } else {
     result.mantissa = (typename DyadicFloat<Bits>::MantissaType)(0);
   }
   return result;
 }
 
+// Simple polynomial approximation.
+template <size_t Bits>
+constexpr DyadicFloat<Bits> multiply_add(const DyadicFloat<Bits> &a,
+                                         const DyadicFloat<Bits> &b,
+                                         const DyadicFloat<Bits> &c) {
+  return quick_add(c, quick_mul(a, b));
+}
+
 // Simple exponentiation implementation for printf. Only handles positive
 // exponents, since division isn't implemented.
 template <size_t Bits>
 constexpr DyadicFloat<Bits> pow_n(DyadicFloat<Bits> a, uint32_t power) {
   DyadicFloat<Bits> result = 1.0;
   DyadicFloat<Bits> cur_power = a;
 
   while (power > 0) {

libc/src/__support/FPUtil/multiply_add.h

 
 namespace __llvm_libc {
 namespace fputil {
 
 // Implement a simple wrapper for multiply-add operation:
 //   multiply_add(x, y, z) = x*y + z
 // which uses FMA instructions to speed up if available.
 
-template <typename T> LIBC_INLINE T multiply_add(T x, T y, T z) {
+template <typename T>
+LIBC_INLINE T multiply_add(const T &x, const T &y, const T &z) {
   return x * y + z;
 }
 
 } // namespace fputil
 } // namespace __llvm_libc
 
 #if defined(LIBC_TARGET_CPU_HAS_FMA)
 
 // FMA instructions are available.
 #include "FMA.h"
 
 namespace __llvm_libc {
 namespace fputil {
 
-template <> LIBC_INLINE float multiply_add<float>(float x, float y, float z) {
+LIBC_INLINE float multiply_add(float x, float y, float z) {
   return fma(x, y, z);
 }
 
-template <>
-LIBC_INLINE double multiply_add<double>(double x, double y, double z) {
+LIBC_INLINE double multiply_add(double x, double y, double z) {
   return fma(x, y, z);
 }
 
 } // namespace fputil
 } // namespace __llvm_libc
 
 #endif // LIBC_TARGET_CPU_HAS_FMA
 
 #endif // LLVM_LIBC_SRC_SUPPORT_FPUTIL_MULTIPLY_ADD_H

libc/src/math/CMakeLists.txt

 
 add_math_entrypoint_object(cos)
 add_math_entrypoint_object(cosf)
 add_math_entrypoint_object(cosh)
 add_math_entrypoint_object(coshf)
 
 add_math_entrypoint_object(erff)
 
+add_math_entrypoint_object(exp)
 add_math_entrypoint_object(expf)
 
 add_math_entrypoint_object(exp2f)
 
 add_math_entrypoint_object(exp10f)
 
 add_math_entrypoint_object(expm1f)
 

libc/src/math/exp.h

+//===-- Implementation header for exp ---------------------------*- C++ -*-===//
+//
+// 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
+//
+//===----------------------------------------------------------------------===//
+
+#ifndef LLVM_LIBC_SRC_MATH_EXP_H
+#define LLVM_LIBC_SRC_MATH_EXP_H
+
+namespace __llvm_libc {
+
+double exp(double x);
+
+} // namespace __llvm_libc
+
+#endif // LLVM_LIBC_SRC_MATH_EXP_H

libc/src/math/generic/CMakeLists.txt

DEPENDS
     libc.src.__support.FPUtil.polyeval
     libc.src.__support.macros.optimization
     libc.include.math
   COMPILE_OPTIONS
     -O3
 )
 
 add_entrypoint_object(
+  exp
+  SRCS
+    exp.cpp
+  HDRS
+    ../exp.h
+  DEPENDS
+    .common_constants
+    libc.src.__support.FPUtil.basic_operations
+    libc.src.__support.FPUtil.fenv_impl
+    libc.src.__support.FPUtil.fp_bits
+    libc.src.__support.FPUtil.multiply_add
+    libc.src.__support.FPUtil.nearest_integer
+    libc.src.__support.FPUtil.polyeval
+    libc.src.__support.FPUtil.rounding_mode
+    libc.src.__support.macros.optimization
+    libc.include.errno
+    libc.src.errno.errno
+    libc.include.math
+  COMPILE_OPTIONS
+    -O3
+)
+
+add_entrypoint_object(
   expf
   SRCS
     expf.cpp
   HDRS
     ../expf.h
   DEPENDS
     .common_constants
     libc.src.__support.FPUtil.basic_operations

libc/src/math/generic/exp.cpp

+//===-- Double-precision e^x function -------------------------------------===//
+//
+// 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 "src/math/exp.h"
+#include "common_constants.h" // Lookup tables EXP_M1 and EXP_M2.
+#include "src/__support/CPP/bit.h"
+#include "src/__support/CPP/optional.h"
+#include "src/__support/FPUtil/FEnvImpl.h"
+#include "src/__support/FPUtil/FPBits.h"
+#include "src/__support/FPUtil/PolyEval.h"
+#include "src/__support/FPUtil/double_double.h"
+#include "src/__support/FPUtil/dyadic_float.h"
+#include "src/__support/FPUtil/multiply_add.h"
+#include "src/__support/FPUtil/nearest_integer.h"
+#include "src/__support/FPUtil/rounding_mode.h"
+#include "src/__support/common.h"
+#include "src/__support/macros/optimization.h" // LIBC_UNLIKELY
+
+#include <errno.h>
+
+namespace __llvm_libc {
+
+using fputil::DoubleDouble;
+using Float128 = typename fputil::DyadicFloat<128>;
+
+// 2^12 * log2(e)
+constexpr double LOG2_E = 0x1.71547652b82fep+0;
+
+// Error bounds:
+// Errors when using double precision.
+constexpr double ERR_D = 0x1.8p-63;
+// Errors when using double-double precision.
+constexpr double ERR_DD = 0x1.0p-99;
+
+struct TripleDouble {
+  double hi = 0.0;
+  double mid = 0.0;
+  double lo = 0.0;
+};
+
+// -2^-12 * log(2)
+// > a = -2^-12 * log(2);
+// > b = round(a, 30, RN);
+// > c = round(a - b, 30, RN);
+// > d = round(a - b - c, D, RN);
+// Errors < 1.5 * 2^-133
+constexpr double MLOG_2_EXP2_M12_HI = -0x1.62e42ffp-13;
+constexpr double MLOG_2_EXP2_M12_MID = 0x1.718432a1b0e26p-47;
+constexpr double MLOG_2_EXP2_M12_MID_30 = 0x1.718432ap-47;
+constexpr double MLOG_2_EXP2_M12_LO = 0x1.b0e2633fe0685p-79;
+
+// 2^(k * 2^-6), for k = 0..63.
+constexpr TripleDouble EXP_MID1[64] = {
+    {0x1p0, 0, 0},
+    {0x1.02c9a3e778061p0, -0x1.19083535b085dp-56, -0x1.9085b0a3d74d5p-110},
+    {0x1.059b0d3158574p0, 0x1.d73e2a475b465p-55, 0x1.05ff94f8d257ep-110},
+    {0x1.0874518759bc8p0, 0x1.186be4bb284ffp-57, 0x1.15820d96b414fp-111},
+    {0x1.0b5586cf9890fp0, 0x1.8a62e4adc610bp-54, -0x1.67c9bd6ebf74cp-108},
+    {0x1.0e3ec32d3d1a2p0, 0x1.03a1727c57b53p-59, -0x1.5aa76994e9ddbp-113},
+    {0x1.11301d0125b51p0, -0x1.6c51039449b3ap-54, 0x1.9d58b988f562dp-109},
+    {0x1.1429aaea92dep0, -0x1.32fbf9af1369ep-54, -0x1.2fe7bb4c76416p-108},
+    {0x1.172b83c7d517bp0, -0x1.19041b9d78a76p-55, 0x1.4f2406aa13ffp-109},
+    {0x1.1a35beb6fcb75p0, 0x1.e5b4c7b4968e4p-55, 0x1.ad36183926ae8p-111},
+    {0x1.1d4873168b9aap0, 0x1.e016e00a2643cp-54, 0x1.ea62d0881b918p-110},
+    {0x1.2063b88628cd6p0, 0x1.dc775814a8495p-55, -0x1.781dbc16f1ea4p-111},
+    {0x1.2387a6e756238p0, 0x1.9b07eb6c70573p-54, -0x1.4d89f9af532ep-109},
+    {0x1.26b4565e27cddp0, 0x1.2bd339940e9d9p-55, 0x1.277393a461b77p-110},
+    {0x1.29e9df51fdee1p0, 0x1.612e8afad1255p-55, 0x1.de5448560469p-111},
+    {0x1.2d285a6e4030bp0, 0x1.0024754db41d5p-54, -0x1.ee9d8f8cb9307p-110},
+    {0x1.306fe0a31b715p0, 0x1.6f46ad23182e4p-55, 0x1.7b7b2f09cd0d9p-110},
+    {0x1.33c08b26416ffp0, 0x1.32721843659a6p-54, -0x1.406a2ea6cfc6bp-108},
+    {0x1.371a7373aa9cbp0, -0x1.63aeabf42eae2p-54, 0x1.87e3e12516bfap-108},
+    {0x1.3a7db34e59ff7p0, -0x1.5e436d661f5e3p-56, 0x1.9b0b1ff17c296p-111},
+    {0x1.3dea64c123422p0, 0x1.ada0911f09ebcp-55, -0x1.808ba68fa8fb7p-109},
+    {0x1.4160a21f72e2ap0, -0x1.ef3691c309278p-58, -0x1.32b43eafc6518p-114},
+    {0x1.44e086061892dp0, 0x1.89b7a04ef80dp-59, -0x1.0ac312de3d922p-114},
+    {0x1.486a2b5c13cdp0, 0x1.3c1a3b69062fp-56, 0x1.e1eebae743acp-111},
+    {0x1.4bfdad5362a27p0, 0x1.d4397afec42e2p-56, 0x1.c06c7745c2b39p-113},
+    {0x1.4f9b2769d2ca7p0, -0x1.4b309d25957e3p-54, -0x1.1aa1fd7b685cdp-112},
+    {0x1.5342b569d4f82p0, -0x1.07abe1db13cadp-55, 0x1.fa733951f214cp-111},
+    {0x1.56f4736b527dap0, 0x1.9bb2c011d93adp-54, -0x1.ff86852a613ffp-111},
+    {0x1.5ab07dd485429p0, 0x1.6324c054647adp-54, -0x1.744ee506fdafep-109},
+    {0x1.5e76f15ad2148p0, 0x1.ba6f93080e65ep-54, -0x1.95f9ab75fa7d6p-108},
+    {0x1.6247eb03a5585p0, -0x1.383c17e40b497p-54, 0x1.5d8e757cfb991p-111},
+    {0x1.6623882552225p0, -0x1.bb60987591c34p-54, 0x1.4a337f4dc0a3bp-108},
+    {0x1.6a09e667f3bcdp0, -0x1.bdd3413b26456p-54, 0x1.57d3e3adec175p-108},
+    {0x1.6dfb23c651a2fp0, -0x1.bbe3a683c88abp-57, 0x1.a59f88abbe778p-115},
+    {0x1.71f75e8ec5f74p0, -0x1.16e4786887a99p-55, -0x1.269796953a4c3p-109},
+    {0x1.75feb564267c9p0, -0x1.0245957316dd3p-54, -0x1.8f8e7fa19e5e8p-108},
+    {0x1.7a11473eb0187p0, -0x1.41577ee04992fp-55, -0x1.4217a932d10d4p-113},
+    {0x1.7e2f336cf4e62p0, 0x1.05d02ba15797ep-56, 0x1.70a1427f8fcdfp-112},
+    {0x1.82589994cce13p0, -0x1.d4c1dd41532d8p-54, 0x1.0f6ad65cbbac1p-112},
+    {0x1.868d99b4492edp0, -0x1.fc6f89bd4f6bap-54, -0x1.f16f65181d921p-109},
+    {0x1.8ace5422aa0dbp0, 0x1.6e9f156864b27p-54, -0x1.30644a7836333p-110},
+    {0x1.8f1ae99157736p0, 0x1.5cc13a2e3976cp-55, 0x1.3bf26d2b85163p-114},
+    {0x1.93737b0cdc5e5p0, -0x1.75fc781b57ebcp-57, 0x1.697e257ac0db2p-111},
+    {0x1.97d829fde4e5p0, -0x1.d185b7c1b85d1p-54, 0x1.7edb9d7144b6fp-108},
+    {0x1.9c49182a3f09p0, 0x1.c7c46b071f2bep-56, 0x1.6376b7943085cp-110},
+    {0x1.a0c667b5de565p0, -0x1.359495d1cd533p-54, 0x1.354084551b4fbp-109},
+    {0x1.a5503b23e255dp0, -0x1.d2f6edb8d41e1p-54, -0x1.bfd7adfd63f48p-111},
+    {0x1.a9e6b5579fdbfp0, 0x1.0fac90ef7fd31p-54, 0x1.8b16ae39e8cb9p-109},
+    {0x1.ae89f995ad3adp0, 0x1.7a1cd345dcc81p-54, 0x1.a7fbc3ae675eap-108},
+    {0x1.b33a2b84f15fbp0, -0x1.2805e3084d708p-57, 0x1.2babc0edda4d9p-111},
+    {0x1.b7f76f2fb5e47p0, -0x1.5584f7e54ac3bp-56, 0x1.aa64481e1ab72p-111},
+    {0x1.bcc1e904bc1d2p0, 0x1.23dd07a2d9e84p-55, 0x1.9a164050e1258p-109},
+    {0x1.c199bdd85529cp0, 0x1.11065895048ddp-55, 0x1.99e51125928dap-110},
+    {0x1.c67f12e57d14bp0, 0x1.2884dff483cadp-54, -0x1.fc44c329d5cb2p-109},
+    {0x1.cb720dcef9069p0, 0x1.503cbd1e949dbp-56, 0x1.d8765566b032ep-110},
+    {0x1.d072d4a07897cp0, -0x1.cbc3743797a9cp-54, -0x1.e7044039da0f6p-108},
+    {0x1.d5818dcfba487p0, 0x1.2ed02d75b3707p-55, -0x1.ab053b05531fcp-111},
+    {0x1.da9e603db3285p0, 0x1.c2300696db532p-54, 0x1.7f6246f0ec615p-108},
+    {0x1.dfc97337b9b5fp0, -0x1.1a5cd4f184b5cp-54, 0x1.b7225a944efd6p-108},
+    {0x1.e502ee78b3ff6p0, 0x1.39e8980a9cc8fp-55, 0x1.1e92cb3c2d278p-109},
+    {0x1.ea4afa2a490dap0, -0x1.e9c23179c2893p-54, -0x1.fc0f242bbf3dep-109},
+    {0x1.efa1bee615a27p0, 0x1.dc7f486a4b6bp-54, 0x1.f6dd5d229ff69p-108},
+    {0x1.f50765b6e454p0, 0x1.9d3e12dd8a18bp-54, -0x1.4019bffc80ef3p-110},
+    {0x1.fa7c1819e90d8p0, 0x1.74853f3a5931ep-55, 0x1.dc060c36f7651p-112},
+};
+
+// 2^(k * 2^-12), for k = 0..63.
+constexpr TripleDouble EXP_MID2[64] = {
+    {0x1p0, 0, 0},
+    {0x1.000b175effdc7p0, 0x1.ae8e38c59c72ap-54, 0x1.39726694630e3p-108},
+    {0x1.00162f3904052p0, -0x1.7b5d0d58ea8f4p-58, 0x1.e5e06ddd31156p-112},
+    {0x1.0021478e11ce6p0, 0x1.4115cb6b16a8ep-54, 0x1.5a0768b51f609p-111},
+    {0x1.002c605e2e8cfp0, -0x1.d7c96f201bb2fp-55, 0x1.d008403605217p-111},
+    {0x1.003779a95f959p0, 0x1.84711d4c35e9fp-54, 0x1.89bc16f765708p-109},
+    {0x1.0042936faa3d8p0, -0x1.0484245243777p-55, -0x1.4535b7f8c1e2dp-109},
+    {0x1.004dadb113dap0, -0x1.4b237da2025f9p-54, -0x1.8ba92f6b25456p-108},
+    {0x1.0058c86da1c0ap0, -0x1.5e00e62d6b30dp-56, -0x1.30c72e81f4294p-113},
+    {0x1.0063e3a559473p0, 0x1.a1d6cedbb9481p-54, -0x1.34a5384e6f0b9p-110},
+    {0x1.006eff583fc3dp0, -0x1.4acf197a00142p-54, 0x1.f8d0580865d2ep-108},
+    {0x1.007a1b865a8cap0, -0x1.eaf2ea42391a5p-57, -0x1.002bcb3ae9a99p-111},
+    {0x1.0085382faef83p0, 0x1.da93f90835f75p-56, 0x1.c3c5aedee9851p-111},
+    {0x1.00905554425d4p0, -0x1.6a79084ab093cp-55, 0x1.7217851d1ec6ep-109},
+    {0x1.009b72f41a12bp0, 0x1.86364f8fbe8f8p-54, -0x1.80cbca335a7c3p-110},
+    {0x1.00a6910f3b6fdp0, -0x1.82e8e14e3110ep-55, -0x1.706bd4eb22595p-110},
+    {0x1.00b1afa5abcbfp0, -0x1.4f6b2a7609f71p-55, -0x1.b55dd523f3c08p-111},
+    {0x1.00bcceb7707ecp0, -0x1.e1a258ea8f71bp-56, 0x1.90a1e207cced1p-110},
+    {0x1.00c7ee448ee02p0, 0x1.4362ca5bc26f1p-56, 0x1.78d0472db37c5p-110},
+    {0x1.00d30e4d0c483p0, 0x1.095a56c919d02p-54, -0x1.bcd4db3cb52fep-109},
+    {0x1.00de2ed0ee0f5p0, -0x1.406ac4e81a645p-57, -0x1.cf1b131575ec2p-112},
+    {0x1.00e94fd0398ep0, 0x1.b5a6902767e09p-54, -0x1.6aaa1fa7ff913p-112},
+    {0x1.00f4714af41d3p0, -0x1.91b2060859321p-54, 0x1.68f236dff3218p-110},
+    {0x1.00ff93412315cp0, 0x1.427068ab22306p-55, -0x1.e8bb58067e60ap-109},
+    {0x1.010ab5b2cbd11p0, 0x1.c1d0660524e08p-54, 0x1.d4cd5e1d71fdfp-108},
+    {0x1.0115d89ff3a8bp0, -0x1.e7bdfb3204be8p-54, 0x1.e4ecf350ebe88p-108},
+    {0x1.0120fc089ff63p0, 0x1.843aa8b9cbbc6p-55, 0x1.6a2aa2c89c4f8p-109},
+    {0x1.012c1fecd613bp0, -0x1.34104ee7edae9p-56, 0x1.1ca368a20ed05p-110},
+    {0x1.0137444c9b5b5p0, -0x1.2b6aeb6176892p-56, 0x1.edb1095d925cfp-114},
+    {0x1.01426927f5278p0, 0x1.a8cd33b8a1bb3p-56, -0x1.488c78eded75fp-111},
+    {0x1.014d8e7ee8d2fp0, 0x1.2edc08e5da99ap-56, -0x1.7480f5ea1b3c9p-113},
+    {0x1.0158b4517bb88p0, 0x1.57ba2dc7e0c73p-55, -0x1.ae45989a04dd5p-111},
+    {0x1.0163da9fb3335p0, 0x1.b61299ab8cdb7p-54, 0x1.bf48007d80987p-109},
+    {0x1.016f0169949edp0, -0x1.90565902c5f44p-54, 0x1.1aa91a059292cp-109},
+    {0x1.017a28af25567p0, 0x1.70fc41c5c2d53p-55, 0x1.b6663292855f5p-110},
+    {0x1.018550706ab62p0, 0x1.4b9a6e145d76cp-54, 0x1.e7fbca6793d94p-108},
+    {0x1.019078ad6a19fp0, -0x1.008eff5142bf9p-56, -0x1.5b9f5c7de3b93p-110},
+    {0x1.019ba16628de2p0, -0x1.77669f033c7dep-54, 0x1.4638bf2f6acabp-110},
+    {0x1.01a6ca9aac5f3p0, -0x1.09bb78eeead0ap-54, -0x1.ab237b9a069c5p-109},
+    {0x1.01b1f44af9f9ep0, 0x1.371231477ece5p-54, 0x1.3ab358be97cefp-108},
+    {0x1.01bd1e77170b4p0, 0x1.5e7626621eb5bp-56, -0x1.4027b2294bb64p-110},
+    {0x1.01c8491f08f08p0, -0x1.bc72b100828a5p-54, 0x1.656394426c99p-111},
+    {0x1.01d37442d507p0, -0x1.ce39cbbab8bbep-57, 0x1.bf9785189bdd8p-111},
+    {0x1.01de9fe280ac8p0, 0x1.16996709da2e2p-55, 0x1.7c12f86114fe3p-109},
+    {0x1.01e9cbfe113efp0, -0x1.c11f5239bf535p-55, -0x1.653d5d24b5d28p-109},
+    {0x1.01f4f8958c1c6p0, 0x1.e1d4eb5edc6b3p-55, 0x1.04a0cdc1d86d7p-109},
+    {0x1.020025a8f6a35p0, -0x1.afb99946ee3fp-54, 0x1.c678c46149782p-109},
+    {0x1.020b533856324p0, -0x1.8f06d8a148a32p-54, 0x1.48524e1e9df7p-108},
+    {0x1.02168143b0281p0, -0x1.2bf310fc54eb6p-55, 0x1.9953ea727ff0bp-109},
+    {0x1.0221afcb09e3ep0, -0x1.c95a035eb4175p-54, -0x1.ccfbbec22d28ep-108},
+    {0x1.022cdece68c4fp0, -0x1.491793e46834dp-54, 0x1.9e2bb6e181de1p-108},
+    {0x1.02380e4dd22adp0, -0x1.3e8d0d9c49091p-56, 0x1.f17609ae29308p-110},
+    {0x1.02433e494b755p0, -0x1.314aa16278aa3p-54, -0x1.c7dc2c476bfb8p-110},
+    {0x1.024e6ec0da046p0, 0x1.48daf888e9651p-55, -0x1.fab994971d4a3p-109},
+    {0x1.02599fb483385p0, 0x1.56dc8046821f4p-55, 0x1.848b62cbdd0afp-109},
+    {0x1.0264d1244c719p0, 0x1.45b42356b9d47p-54, -0x1.bf603ba715d0cp-109},
+    {0x1.027003103b10ep0, -0x1.082ef51b61d7ep-56, 0x1.89434e751e1aap-110},
+    {0x1.027b357854772p0, 0x1.2106ed0920a34p-56, -0x1.03b54fd64e8acp-110},
+    {0x1.0286685c9e059p0, -0x1.fd4cf26ea5d0fp-54, 0x1.7785ea0acc486p-109},
+    {0x1.02919bbd1d1d8p0, -0x1.09f8775e78084p-54, -0x1.ce447fdb35ff9p-109},
+    {0x1.029ccf99d720ap0, 0x1.64cbba902ca27p-58, 0x1.5b884aab5642ap-112},
+    {0x1.02a803f2d170dp0, 0x1.4383ef231d207p-54, -0x1.cfb3e46d7c1cp-108},
+    {0x1.02b338c811703p0, 0x1.4a47a505b3a47p-54, -0x1.0d40cee4b81afp-112},
+    {0x1.02be6e199c811p0, 0x1.e47120223467fp-54, 0x1.6ae7d36d7c1f7p-109},
+};
+
+// Polynomial approximations with double precision:
+// Return expm1(dx) / x ~ 1 + dx / 2 + dx^2 / 6 + dx^3 / 24.
+// For |dx| < 2^-13 + 2^-30:
+//   | output - expm1(dx) / dx | < 2^-51.
+LIBC_INLINE double poly_approx_d(double dx) {
+  // dx^2
+  double dx2 = dx * dx;
+  // c0 = 1 + dx / 2
+  double c0 = fputil::multiply_add(dx, 0.5, 1.0);
+  // c1 = 1/6 + dx / 24
+  double c1 =
+      fputil::multiply_add(dx, 0x1.5555555555555p-5, 0x1.5555555555555p-3);
+  // p = dx^2 * c1 + c0 = 1 + dx / 2 + dx^2 / 6 + dx^3 / 24
+  double p = fputil::multiply_add(dx2, c1, c0);
+  return p;
+}
+
+// Polynomial approximation with double-double precision:
+// Return exp(dx) ~ 1 + dx + dx^2 / 2 + ... + dx^6 / 720
+// For |dx| < 2^-13 + 2^-30:
+//   | output - exp(dx) | < 2^-101
+DoubleDouble poly_approx_dd(const DoubleDouble &dx) {
+  // Taylor polynomial.
+  constexpr DoubleDouble COEFFS[] = {
+      {0, 0x1p0},                                      // 1
+      {0, 0x1p0},                                      // 1
+      {0, 0x1p-1},                                     // 1/2
+      {0x1.5555555555555p-57, 0x1.5555555555555p-3},   // 1/6
+      {0x1.5555555555555p-59, 0x1.5555555555555p-5},   // 1/24
+      {0x1.1111111111111p-63, 0x1.1111111111111p-7},   // 1/120
+      {-0x1.f49f49f49f49fp-65, 0x1.6c16c16c16c17p-10}, // 1/720
+  };
+
+  DoubleDouble p = fputil::polyeval(dx, COEFFS[0], COEFFS[1], COEFFS[2],
+                                    COEFFS[3], COEFFS[4], COEFFS[5], COEFFS[6]);
+  return p;
+}
+
+// Polynomial approximation with 128-bit precision:
+// Return exp(dx) ~ 1 + dx + dx^2 / 2 + ... + dx^7 / 5040
+// For |dx| < 2^-13 + 2^-30:
+//   | output - exp(dx) | < 2^-126.
+Float128 poly_approx_f128(const Float128 &dx) {
+  using MType = typename Float128::MantissaType;
+
+  constexpr Float128 COEFFS_128[]{
+      {false, -127, MType({0, 0x8000000000000000})},                  // 1.0
+      {false, -127, MType({0, 0x8000000000000000})},                  // 1.0
+      {false, -128, MType({0, 0x8000000000000000})},                  // 0.5
+      {false, -130, MType({0xaaaaaaaaaaaaaaab, 0xaaaaaaaaaaaaaaaa})}, // 1/6
+      {false, -132, MType({0xaaaaaaaaaaaaaaab, 0xaaaaaaaaaaaaaaaa})}, // 1/24
+      {false, -134, MType({0x8888888888888889, 0x8888888888888888})}, // 1/120
+      {false, -137, MType({0x60b60b60b60b60b6, 0xb60b60b60b60b60b})}, // 1/720
+      {false, -140, MType({0x00b00b00b00b00b0, 0xb00b00b00b00b00b})}, // 1/5040
+  };
+
+  Float128 p = fputil::polyeval(dx, COEFFS_128[0], COEFFS_128[1], COEFFS_128[2],
+                                COEFFS_128[3], COEFFS_128[4], COEFFS_128[5],
+                                COEFFS_128[6], COEFFS_128[7]);
+  return p;
+}
+
+// Compute exp(x) using 128-bit precision.
+// TODO(lntue): investigate triple-double precision implementation for this
+// step.
+Float128 exp_f128(double x, double kd, int idx1, int idx2) {
+  // Recalculate dx:
+
+  double t1 = fputil::multiply_add(kd, MLOG_2_EXP2_M12_HI, x); // exact
+  double t2 = kd * MLOG_2_EXP2_M12_MID_30;                     // exact
+  double t3 = kd * MLOG_2_EXP2_M12_LO;                         // Error < 2^-133
+
+  Float128 dx = fputil::quick_add(
+      Float128(t1), fputil::quick_add(Float128(t2), Float128(t3)));
+
+  // TODO: Skip recalculating exp_mid1 and exp_mid2.
+  Float128 exp_mid1 =
+      fputil::quick_add(Float128(EXP_MID1[idx1].hi),
+                        fputil::quick_add(Float128(EXP_MID1[idx1].mid),
+                                          Float128(EXP_MID1[idx1].lo)));
+
+  Float128 exp_mid2 =
+      fputil::quick_add(Float128(EXP_MID2[idx2].hi),
+                        fputil::quick_add(Float128(EXP_MID2[idx2].mid),
+                                          Float128(EXP_MID2[idx2].lo)));
+
+  Float128 exp_mid = fputil::quick_mul(exp_mid1, exp_mid2);
+
+  Float128 p = poly_approx_f128(dx);
+
+  Float128 r = fputil::quick_mul(exp_mid, p);
+
+  r.exponent += static_cast<int>(kd) >> 12;
+
+  return r;
+}
+
+// Compute exp(x) with double-double precision.
+DoubleDouble exp_double_double(double x, double kd,
+                               const DoubleDouble &exp_mid) {
+  // Recalculate dx:
+  //   dx = x - k * 2^-12 * log(2)
+  double t1 = fputil::multiply_add(kd, MLOG_2_EXP2_M12_HI, x); // exact
+  double t2 = kd * MLOG_2_EXP2_M12_MID_30;                     // exact
+  double t3 = kd * MLOG_2_EXP2_M12_LO;                         // Error < 2^-130
+
+  DoubleDouble dx = fputil::exact_add(t1, t2);
+  dx.lo += t3;
+
+  // Degree-6 Taylor polynomial approximation in double-double precision.
+  // | p - exp(x) | < 2^-100.
+  DoubleDouble p = poly_approx_dd(dx);
+
+  // Error bounds: 2^-99.
+  DoubleDouble r = fputil::quick_mult(exp_mid, p);
+
+  return r;
+}
+
+// Rounding tests when the output might be denormal.
+cpp::optional<double> ziv_test_denorm(int hi, double mid, double lo,
+                                      double err) {
+  using FloatProp = typename fputil::FloatProperties<double>;
+
+  // Scaling factor = 1/(min normal number) = 2^1022
+  int64_t exp_hi = static_cast<int64_t>(hi + 1022) << FloatProp::MANTISSA_WIDTH;
+  double mid_hi = cpp::bit_cast<double>(exp_hi + cpp::bit_cast<int64_t>(mid));
+
+  // Extra errors from another rounding step.
+  err += 0x1.0p-52;
+
+  double lo_u = lo + err;
+  double lo_l = lo - err;
+  double mid_lo_u =
+      cpp::bit_cast<double>(exp_hi + cpp::bit_cast<int64_t>(lo_u));
+  double mid_lo_l =
+      cpp::bit_cast<double>(exp_hi + cpp::bit_cast<int64_t>(lo_l));
+
+  // By adding 2^-511, the results will have similar rounding points as denormal
+  // outputs.
+  double upper = (mid_hi + mid_lo_u);
+  double lower = (mid_hi + mid_lo_l);
+
+  uint64_t scale_down = 0;
+
+  if (upper < 1.0) {
+    // Upper bound is in denormal range, need extra rounding.
+    upper += 1.0;
+    lower += 1.0;
+    scale_down = 0x3FF0'0000'0000'0000; // 1.0
+  }
+
+  if (LIBC_LIKELY(upper == lower)) {
+    return cpp::bit_cast<double>(cpp::bit_cast<uint64_t>(upper) - scale_down);
+  }
+
+  return cpp::nullopt;
+}
+
+// Check for exceptional cases when
+// |x| < 2^-53
+double set_exceptional(double x) {
+  using FPBits = typename fputil::FPBits<double>;
+  using FloatProp = typename fputil::FloatProperties<double>;
+  FPBits xbits(x);
+
+  uint64_t x_u = xbits.uintval();
+  uint64_t x_abs = x_u & FloatProp::EXP_MANT_MASK;
+
+  // |x| < 2^-53
+  if (x_abs <= 0x3ca0'0000'0000'0000ULL) {
+    // exp(x) ~ 1 + x
+    return 1 + x;
+  }
+
+  // x <= log(2^-1075) || x >= 0x1.6232bdd7abcd3p+9 or inf/nan.
+
+  // x <= log(2^-1075) or -inf/nan
+  if (x_u >= 0xc087'4910'd52d'3052ULL) {
+    // exp(-Inf) = 0
+    if (xbits.is_inf())
+      return 0.0;
+
+    // exp(nan) = nan
+    if (xbits.is_nan())
+      return x;
+
+    if (fputil::quick_get_round() == FE_UPWARD)
+      return static_cast<double>(FPBits(FPBits::MIN_SUBNORMAL));
+    fputil::set_errno_if_required(ERANGE);
+    fputil::raise_except_if_required(FE_UNDERFLOW);
+    return 0.0;
+  }
+
+  // x >= round(log(MAX_NORMAL), D, RU) = 0x1.62e42fefa39fp+9 or +inf/nan
+  // x is finite
+  if (x_u < 0x7ff0'0000'0000'0000ULL) {
+    int rounding = fputil::quick_get_round();
+    if (rounding == FE_DOWNWARD || rounding == FE_TOWARDZERO)
+      return static_cast<double>(FPBits(FPBits::MAX_NORMAL));
+
+    fputil::set_errno_if_required(ERANGE);
+    fputil::raise_except_if_required(FE_OVERFLOW);
+  }
+  // x is +inf or nan
+  return x + static_cast<double>(FPBits::inf());
+}
+
+LLVM_LIBC_FUNCTION(double, exp, (double x)) {
+  using FPBits = typename fputil::FPBits<double>;
+  using FloatProp = typename fputil::FloatProperties<double>;
+  FPBits xbits(x);
+
+  uint64_t x_u = xbits.uintval();
+
+  // Upper bound: max normal number = 2^1023 * (2 - 2^-52)
+  // > round(log (2^1023 ( 2 - 2^-52 )), D, RU) = 0x1.62e42fefa39fp+9
+  // > round(log (2^1023 ( 2 - 2^-52 )), D, RD) = 0x1.62e42fefa39efp+9
+  // > round(log (2^1023 ( 2 - 2^-52 )), D, RN) = 0x1.62e42fefa39efp+9
+  // > round(exp(0x1.62e42fefa39fp+9), D, RN) = infty
+
+  // Lower bound: min denormal number / 2 = 2^-1075
+  // > round(log(2^-1075), D, RN) = -0x1.74910d52d3052p9
+
+  // Another lower bound: min normal number = 2^-1022
+  // > round(log(2^-1022), D, RN) = -0x1.6232bdd7abcd2p9
+
+  // x < log(2^-1075) or x >= 0x1.6232bdd7abcd3p+9 or |x| < 2^-53.
+  if (LIBC_UNLIKELY(x_u >= 0xc0874910d52d3052 ||
+                    (x_u < 0xbca0000000000000 && x_u >= 0x40862e42fefa39f0) ||
+                    x_u < 0x3ca0000000000000)) {
+    return set_exceptional(x);
+  }
+
+  // Now log(2^-1022) <= x <= -2^-53 or 2^-53 <= x < log(2^1023 * (2 - 2^-52))
+
+  // Range reduction:
+  // Let x = log(2) * (hi + mid1 + mid2) + lo
+  // in which:
+  //   hi is an integer
+  //   mid1 * 2^6 is an integer
+  //   mid2 * 2^12 is an integer
+  // then:
+  //   exp(x) = 2^hi * 2^(mid1) * 2^(mid2) * exp(lo).
+  // With this formula:
+  //   - multiplying by 2^hi is exact and cheap, simply by adding the exponent
+  //     field.
+  //   - 2^(mid1) and 2^(mid2) are stored in 2 x 64-element tables.
+  //   - exp(lo) ~ 1 + lo + a0 * lo^2 + ...
+  //
+  // They can be defined by:
+  //   hi + mid1 + mid2 = 2^(-12) * round(2^12 * log_2(e) * x)
+  // If we store L2E = round(log2(e), D, RN), then:
+  //   log2(e) - L2E ~ 1.5 * 2^(-56)
+  // So the errors when computing in double precision is:
+  //   | x * 2^12 * log_2(e) - D(x * 2^12 * L2E) | <=
+  //  <= | x * 2^12 * log_2(e) - x * 2^12 * L2E | +
+  //     + | x * 2^12 * L2E - D(x * 2^12 * L2E) |
+  //  <= 2^12 * ( |x| * 1.5 * 2^-56 + eps(x))  for RN
+  //     2^12 * ( |x| * 1.5 * 2^-56 + 2*eps(x)) for other rounding modes.
+  // So if:
+  //   hi + mid1 + mid2 = 2^(-12) * round(x * 2^12 * L2E) is computed entirely
+  // in double precision, the reduced argument:
+  //   lo = x - log(2) * (hi + mid1 + mid2) is bounded by:
+  //   |lo| <= 2^-13 + (|x| * 1.5 * 2^-56 + 2*eps(x))
+  //         < 2^-13 + (1.5 * 2^9 * 1.5 * 2^-56 + 2*2^(9 - 52))
+  //         < 2^-13 + 2^-41
+  //
+
+  // The following trick computes the round(x * L2E) more efficiently
+  // than using the rounding instructions, with the tradeoff for less accuracy,
+  // and hence a slightly larger range for the reduced argument `lo`.
+  //
+  // To be precise, since |x| < |log(2^-1075)| < 1.5 * 2^9,
+  //   |x * 2^12 * L2E| < 1.5 * 2^9 * 1.5 < 2^23,
+  // So we can fit the rounded result round(x * 2^12 * L2E) in int32_t.
+  // Thus, the goal is to be able to use an additional addition and fixed width
+  // shift to get an int32_t representing round(x * 2^12 * L2E).
+  //
+  // Assuming int32_t using 2-complement representation, since the mantissa part
+  // of a double precision is unsigned with the leading bit hidden, if we add an
+  // extra constant C = 2^e1 + 2^e2 with e1 > e2 >= 2^25 to the product, the
+  // part that are < 2^e2 in resulted mantissa of (x*2^12*L2E + C) can be
+  // considered as a proper 2-complement representations of x*2^12*L2E.
+  //
+  // One small problem with this approach is that the sum (x*2^12*L2E + C) in
+  // double precision is rounded to the least significant bit of the dorminant
+  // factor C.  In order to minimize the rounding errors from this addition, we
+  // want to minimize e1.  Another constraint that we want is that after
+  // shifting the mantissa so that the least significant bit of int32_t
+  // corresponds to the unit bit of (x*2^12*L2E), the sign is correct without
+  // any adjustment.  So combining these 2 requirements, we can choose
+  //   C = 2^33 + 2^32, so that the sign bit corresponds to 2^31 bit, and hence
+  // after right shifting the mantissa, the resulting int32_t has correct sign.
+  // With this choice of C, the number of mantissa bits we need to shift to the
+  // right is: 52 - 33 = 19.
+  //
+  // Moreover, since the integer right shifts are equivalent to rounding down,
+  // we can add an extra 0.5 so that it will become round-to-nearest, tie-to-
+  // +infinity.  So in particular, we can compute:
+  //   hmm = x * 2^12 * L2E + C,
+  // where C = 2^33 + 2^32 + 2^-1, then if
+  //   k = int32_t(lower 51 bits of double(x * 2^12 * L2E + C) >> 19),
+  // the reduced argument:
+  //   lo = x - log(2) * 2^-12 * k is bounded by:
+  //   |lo| <= 2^-13 + 2^-41 + 2^-12*2^-19
+  //         = 2^-13 + 2^-31 + 2^-41.
+  //
+  // Finally, notice that k only uses the mantissa of x * 2^12 * L2E, so the
+  // exponent 2^12 is not needed.  So we can simply define
+  //   C = 2^(33 - 12) + 2^(32 - 12) + 2^(-13 - 12), and
+  //   k = int32_t(lower 51 bits of double(x * L2E + C) >> 19).
+
+  // Rounding errors <= 2^-31 + 2^-41.
+  double tmp = fputil::multiply_add(x, LOG2_E, 0x1.8000'0000'4p21);
+  int k = static_cast<int>(cpp::bit_cast<uint64_t>(tmp) >> 19);
+  double kd = static_cast<double>(k);
+
+  uint32_t idx1 = (k >> 6) & 0x3f;
+  uint32_t idx2 = k & 0x3f;
+  int hi = k >> 12;
+
+  bool denorm = (hi <= -1022);
+
+  DoubleDouble exp_mid1{EXP_MID1[idx1].mid, EXP_MID1[idx1].hi};
+  DoubleDouble exp_mid2{EXP_MID2[idx2].mid, EXP_MID2[idx2].hi};
+
+  DoubleDouble exp_mid = fputil::quick_mult(exp_mid1, exp_mid2);
+
+  // |x - (hi + mid1 + mid2) * log(2) - dx| < 2^11 * eps(M_LOG_2_EXP2_M12.lo)
+  //                                        = 2^11 * 2^-13 * 2^-52
+  //                                        = 2^-54.
+  // |dx| < 2^-13 + 2^-30.
+  double lo_h = fputil::multiply_add(kd, MLOG_2_EXP2_M12_HI, x); // exact
+  double dx = fputil::multiply_add(kd, MLOG_2_EXP2_M12_MID, lo_h);
+
+  // We use the degree-4 Taylor polynomial to approximate exp(lo):
+  //   exp(lo) ~ 1 + lo + lo^2 / 2 + lo^3 / 6 + lo^4 / 24 = 1 + lo * P(lo)
+  // So that the errors are bounded by:
+  //   |P(lo) - expm1(lo)/lo| < |lo|^4 / 64 < 2^(-13 * 4) / 64 = 2^-58
+  // Let P_ be an evaluation of P where all intermediate computations are in
+  // double precision.  Using either Horner's or Estrin's schemes, the evaluated
+  // errors can be bounded by:
+  //      |P_(dx) - P(dx)| < 2^-51
+  //   => |dx * P_(dx) - expm1(lo) | < 1.5 * 2^-64
+  //   => 2^(mid1 + mid2) * |dx * P_(dx) - expm1(lo)| < 1.5 * 2^-63.
+  // Since we approximate
+  //   2^(mid1 + mid2) ~ exp_mid.hi + exp_mid.lo,
+  // We use the expression:
+  //    (exp_mid.hi + exp_mid.lo) * (1 + dx * P_(dx)) ~
+  //  ~ exp_mid.hi + (exp_mid.hi * dx * P_(dx) + exp_mid.lo)
+  // with errors bounded by 1.5 * 2^-63.
+
+  double mid_lo = dx * exp_mid.hi;
+
+  // Approximate expm1(dx)/dx ~ 1 + dx / 2 + dx^2 / 6 + dx^3 / 24.
+  double p = poly_approx_d(dx);
+
+  double lo = fputil::multiply_add(p, mid_lo, exp_mid.lo);
+
+  if (LIBC_UNLIKELY(denorm)) {
+    if (auto r = ziv_test_denorm(hi, exp_mid.hi, lo, ERR_D);
+        LIBC_LIKELY(r.has_value()))
+      return r.value();
+  } else {
+    double upper = exp_mid.hi + (lo + ERR_D);
+    double lower = exp_mid.hi + (lo - ERR_D);
+
+    if (LIBC_LIKELY(upper == lower)) {
+      // to multiply by 2^hi, a fast way is to simply add hi to the exponent
+      // field.
+      int64_t exp_hi = static_cast<int64_t>(hi) << FloatProp::MANTISSA_WIDTH;
+      double r = cpp::bit_cast<double>(exp_hi + cpp::bit_cast<int64_t>(upper));
+      return r;
+    }
+  }
+
+  // Use double-double
+  DoubleDouble r_dd = exp_double_double(x, kd, exp_mid);
+
+  if (LIBC_UNLIKELY(denorm)) {
+    if (auto r = ziv_test_denorm(hi, r_dd.hi, r_dd.lo, ERR_DD);
+        LIBC_LIKELY(r.has_value()))
+      return r.value();
+  } else {
+    double upper_dd = r_dd.hi + (r_dd.lo + ERR_DD);
+    double lower_dd = r_dd.hi + (r_dd.lo - ERR_DD);
+
+    if (LIBC_LIKELY(upper_dd == lower_dd)) {
+      int64_t exp_hi = static_cast<int64_t>(hi) << FloatProp::MANTISSA_WIDTH;
+      double r =
+          cpp::bit_cast<double>(exp_hi + cpp::bit_cast<int64_t>(upper_dd));
+      return r;
+    }
+  }
+
+  // Use 128-bit precision
+  Float128 r_f128 = exp_f128(x, kd, idx1, idx2);
+
+  return static_cast<double>(r_f128);
+}
+
+} // namespace __llvm_libc

libc/test/src/math/CMakeLists.txt

add_fp_unittest(
   DEPENDS
     libc.src.errno.errno
     libc.include.math
     libc.src.math.expf
     libc.src.__support.FPUtil.fp_bits
 )
 
 add_fp_unittest(
+ exp_test
+ NEED_MPFR
+ SUITE
+   libc_math_unittests
+ SRCS
+   exp_test.cpp
+ DEPENDS
+   libc.src.errno.errno
+   libc.include.math
+   libc.src.math.exp
+   libc.src.__support.FPUtil.fp_bits
+)
+
+add_fp_unittest(
   exp2f_test
   NEED_MPFR
   SUITE
     libc_math_unittests
   SRCS
     exp2f_test.cpp
   DEPENDS
     libc.src.errno.errno

libc/test/src/math/exp_test.cpp

+//===-- Unittests for exp -------------------------------------------------===//
+//
+// 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 "src/__support/FPUtil/FPBits.h"
+#include "src/errno/libc_errno.h"
+#include "src/math/exp.h"
+#include "test/UnitTest/FPMatcher.h"
+#include "test/UnitTest/Test.h"
+#include "utils/MPFRWrapper/MPFRUtils.h"
+#include <math.h>
+
+#include <errno.h>
+#include <stdint.h>
+
+namespace mpfr = __llvm_libc::testing::mpfr;
+using __llvm_libc::testing::tlog;
+
+DECLARE_SPECIAL_CONSTANTS(double)
+
+TEST(LlvmLibcExpTest, SpecialNumbers) {
+  EXPECT_FP_EQ(aNaN, __llvm_libc::exp(aNaN));
+  EXPECT_FP_EQ(inf, __llvm_libc::exp(inf));
+  EXPECT_FP_EQ_ALL_ROUNDING(zero, __llvm_libc::exp(neg_inf));
+  EXPECT_FP_EQ_WITH_EXCEPTION(zero, __llvm_libc::exp(-0x1.0p20), FE_UNDERFLOW);
+  EXPECT_FP_EQ_WITH_EXCEPTION(inf, __llvm_libc::exp(0x1.0p20), FE_OVERFLOW);
+  EXPECT_FP_EQ_ALL_ROUNDING(1.0, __llvm_libc::exp(0.0));
+  EXPECT_FP_EQ_ALL_ROUNDING(1.0, __llvm_libc::exp(-0.0));
+}
+
+TEST(LlvmLibcExpTest, TrickyInputs) {
+  constexpr int N = 14;
+  constexpr uint64_t INPUTS[N] = {
+      0x3FD79289C6E6A5C0,
+      0x3FD05DE80A173EA0, // 0x1.05de80a173eap-2
+      0xbf1eb7a4cb841fcc, // -0x1.eb7a4cb841fccp-14
+      0xbf19a61fb925970d,
+      0x3fda7b764e2cf47a, // 0x1.a7b764e2cf47ap-2
+      0xc04757852a4b93aa, // -0x1.757852a4b93aap+5
+      0x4044c19e5712e377, // x=0x1.4c19e5712e377p+5
+      0xbf19a61fb925970d, // x=-0x1.9a61fb925970dp-14
+      0xc039a74cdab36c28, // x=-0x1.9a74cdab36c28p+4
+      0xc085b3e4e2e3bba9, // x=-0x1.5b3e4e2e3bba9p+9
+      0xc086960d591aec34, // x=-0x1.6960d591aec34p+9
+      0xc086232c09d58d91, // x=-0x1.6232c09d58d91p+9
+      0xc0874910d52d3051, // x=-0x1.74910d52d3051p9
+      0xc0867a172ceb0990, // x=-0x1.67a172ceb099p+9
+  };
+  for (int i = 0; i < N; ++i) {
+    double x = double(FPBits(INPUTS[i]));
+    EXPECT_MPFR_MATCH_ALL_ROUNDING(mpfr::Operation::Exp, x, __llvm_libc::exp(x),
+                                   0.5);
+  }
+}
+
+TEST(LlvmLibcExpTest, InDoubleRange) {
+  constexpr uint64_t COUNT = 1'231;
+  uint64_t START = __llvm_libc::fputil::FPBits<double>(0.25).uintval();
+  uint64_t STOP = __llvm_libc::fputil::FPBits<double>(4.0).uintval();
+  uint64_t STEP = (STOP - START) / COUNT;
+
+  auto test = [&](mpfr::RoundingMode rounding_mode) {
+    mpfr::ForceRoundingMode __r(rounding_mode);
+    if (!__r.success)
+      return;
+
+    uint64_t fails = 0;
+    uint64_t count = 0;
+    uint64_t cc = 0;
+    double mx, mr = 0.0;
+    double tol = 0.5;
+
+    for (uint64_t i = 0, v = START; i <= COUNT; ++i, v += STEP) {
+      double x = FPBits(v).get_val();
+      if (isnan(x) || isinf(x) || x < 0.0)
+        continue;
+      libc_errno = 0;
+      double result = __llvm_libc::exp(x);
+      ++cc;
+      if (isnan(result) || isinf(result))
+        continue;
+
+      ++count;
+      // ASSERT_MPFR_MATCH(mpfr::Operation::Log, x, result, 0.5);
+      if (!TEST_MPFR_MATCH_ROUNDING_SILENTLY(mpfr::Operation::Exp, x, result,
+                                             0.5, rounding_mode)) {
+        ++fails;
+        while (!TEST_MPFR_MATCH_ROUNDING_SILENTLY(mpfr::Operation::Exp, x,
+                                                  result, tol, rounding_mode)) {
+          mx = x;
+          mr = result;
+
+          if (tol > 1000.0)
+            break;
+
+          tol *= 2.0;
+        }
+      }
+    }
+    tlog << " Exp failed: " << fails << "/" << count << "/" << cc
+         << " tests.\n";
+    tlog << "   Max ULPs is at most: " << static_cast<uint64_t>(tol) << ".\n";
+    if (fails) {
+      EXPECT_MPFR_MATCH(mpfr::Operation::Exp, mx, mr, 0.5, rounding_mode);
+    }
+  };
+
+  tlog << " Test Rounding To Nearest...\n";
+  test(mpfr::RoundingMode::Nearest);
+
+  tlog << " Test Rounding Downward...\n";
+  test(mpfr::RoundingMode::Downward);
+
+  tlog << " Test Rounding Upward...\n";
+  test(mpfr::RoundingMode::Upward);
+
+  tlog << " Test Rounding Toward Zero...\n";
+  test(mpfr::RoundingMode::TowardZero);
+}

libc/test/src/math/log10_test.cpp

TEST(LlvmLibcLog10Test, SpecialNumbers) {
   EXPECT_FP_IS_NAN_WITH_EXCEPTION(__llvm_libc::log10(neg_inf), FE_INVALID);
   EXPECT_FP_EQ_WITH_EXCEPTION(neg_inf, __llvm_libc::log10(0.0), FE_DIVBYZERO);
   EXPECT_FP_EQ_WITH_EXCEPTION(neg_inf, __llvm_libc::log10(-0.0), FE_DIVBYZERO);
   EXPECT_FP_IS_NAN_WITH_EXCEPTION(__llvm_libc::log10(-1.0), FE_INVALID);
   EXPECT_FP_EQ_ALL_ROUNDING(zero, __llvm_libc::log10(1.0));
 }
 
 TEST(LlvmLibcLog10Test, TrickyInputs) {
-  constexpr int N = 35;
+  constexpr int N = 36;
   constexpr uint64_t INPUTS[N] = {
       0x3ff0000000000000, // x = 1.0
       0x4024000000000000, // x = 10.0
       0x4059000000000000, // x = 10^2
       0x408f400000000000, // x = 10^3
       0x40c3880000000000, // x = 10^4
       0x40f86a0000000000, // x = 10^5
       0x412e848000000000, // x = 10^6
constexpr uint64_t INPUTS[N] = {
       0x43abc16d674ec800, // x = 10^18
       0x43e158e460913d00, // x = 10^19
       0x4415af1d78b58c40, // x = 10^20
       0x444b1ae4d6e2ef50, // x = 10^21
       0x4480f0cf064dd592, // x = 10^22
       0x3fefffffffef06ad, 0x3fefde0f22c7d0eb, 0x225e7812faadb32f,
       0x3fee1076964c2903, 0x3fdfe93fff7fceb0, 0x3ff012631ad8df10,
       0x3fefbfdaa448ed98, 0x44b0c9705a25ce02, 0x2c88d301065c7f9b,
-      0x30160580e7268a99, 0x5ca04103b7eaa345, 0x19ad77dc4a40093f};
+      0x30160580e7268a99, 0x5ca04103b7eaa345, 0x19ad77dc4a40093f,
+      0x0000449fb5c8a96e};
   for (int i = 0; i < N; ++i) {
     double x = double(FPBits(INPUTS[i]));
     EXPECT_MPFR_MATCH_ALL_ROUNDING(mpfr::Operation::Log10, x,
                                    __llvm_libc::log10(x), 0.5);
   }
 }
 
 TEST(LlvmLibcLog10Test, AllExponents) {