src/__support/FPUtil/dyadic_float.h - llvm-project/libc - Git at Google

 //===-- A class to store high precision floating point numbers --*- 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_DYADIC_FLOAT_H
 #define LLVM_LIBC_SRC___SUPPORT_FPUTIL_DYADIC_FLOAT_H

 #include "FPBits.h"
 #include "multiply_add.h"
 #include "src/__support/CPP/type_traits.h"
 #include "src/__support/big_int.h"
 #include "src/__support/macros/optimization.h" // LIBC_UNLIKELY

 #include <stddef.h>

 namespace LIBC_NAMESPACE::fputil {

 // A generic class to perform computations of high precision floating points.
 // We store the value in dyadic format, including 3 fields:
 //   sign    : boolean value - false means positive, true means negative
 //   exponent: the exponent value of the least significant bit of the mantissa.
 //   mantissa: unsigned integer of length `Bits`.
 // So the real value that is stored is:
 //   real value = (-1)^sign * 2^exponent * (mantissa as unsigned integer)
 // The stored data is normal if for non-zero mantissa, the leading bit is 1.
 // The outputs of the constructors and most functions will be normalized.
 // To simplify and improve the efficiency, many functions will assume that the
 // inputs are normal.
 template <size_t Bits> struct DyadicFloat {
   using MantissaType = LIBC_NAMESPACE::UInt<Bits>;

   Sign sign = Sign::POS;
   int exponent = 0;
   MantissaType mantissa = MantissaType(0);

   LIBC_INLINE constexpr DyadicFloat() = default;

   template <typename T, cpp::enable_if_t<cpp::is_floating_point_v<T>, int> = 0>
   LIBC_INLINE constexpr DyadicFloat(T x) {
     static_assert(FPBits<T>::FRACTION_LEN < Bits);
     FPBits<T> x_bits(x);
     sign = x_bits.sign();
     exponent = x_bits.get_explicit_exponent() - FPBits<T>::FRACTION_LEN;
     mantissa = MantissaType(x_bits.get_explicit_mantissa());
     normalize();
   }

   LIBC_INLINE constexpr DyadicFloat(Sign s, int e, MantissaType m)
       : sign(s), exponent(e), mantissa(m) {
     normalize();
   }

   // Normalizing the mantissa, bringing the leading 1 bit to the most
   // significant bit.
   LIBC_INLINE constexpr DyadicFloat &normalize() {
     if (!mantissa.is_zero()) {
       int shift_length = cpp::countl_zero(mantissa);
       exponent -= shift_length;
       mantissa <<= static_cast<size_t>(shift_length);
     }
     return *this;
   }

   // Used for aligning exponents.  Output might not be normalized.
   LIBC_INLINE constexpr DyadicFloat &shift_left(int shift_length) {
     exponent -= shift_length;
     mantissa <<= static_cast<size_t>(shift_length);
     return *this;
   }

   // Used for aligning exponents.  Output might not be normalized.
   LIBC_INLINE constexpr DyadicFloat &shift_right(int shift_length) {
     exponent += shift_length;
     mantissa >>= static_cast<size_t>(shift_length);
     return *this;
   }

   // Assume that it is already normalized.  Output the unbiased exponent.
   LIBC_INLINE constexpr int get_unbiased_exponent() const {
     return exponent + (Bits - 1);
   }

   // Assume that it is already normalized.
   // Output is rounded correctly with respect to the current rounding mode.
   template <typename T,
             typename = cpp::enable_if_t<cpp::is_floating_point_v<T> &&
                                             (FPBits<T>::FRACTION_LEN < Bits),
                                         void>>
   LIBC_INLINE explicit constexpr operator T() const {
     if (LIBC_UNLIKELY(mantissa.is_zero()))
       return FPBits<T>::zero(sign).get_val();

     // Assume that it is normalized, and output is also normal.
     constexpr uint32_t PRECISION = FPBits<T>::FRACTION_LEN + 1;
     using output_bits_t = typename FPBits<T>::StorageType;
     constexpr output_bits_t IMPLICIT_MASK =
         FPBits<T>::SIG_MASK - FPBits<T>::FRACTION_MASK;

     int exp_hi = exponent + static_cast<int>((Bits - 1) + FPBits<T>::EXP_BIAS);

     if (LIBC_UNLIKELY(exp_hi > 2 * FPBits<T>::EXP_BIAS)) {
       // Results overflow.
       T d_hi =
           FPBits<T>::create_value(sign, 2 * FPBits<T>::EXP_BIAS, IMPLICIT_MASK)
               .get_val();
       return T(2) * d_hi;
     }

     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 = FPBits<T>::EXP_BIAS;
     }

     int exp_lo = exp_hi - static_cast<int>(PRECISION) - 1;

     MantissaType m_hi =
         shift >= MantissaType::BITS ? MantissaType(0) : mantissa >> shift;

     T d_hi = FPBits<T>::create_value(
                  sign, exp_hi,
                  (static_cast<output_bits_t>(m_hi) & FPBits<T>::SIG_MASK) |
                      IMPLICIT_MASK)
                  .get_val();

     MantissaType round_mask =
         shift > MantissaType::BITS ? 0 : MantissaType(1) << (shift - 1);
     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);

     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, FPBits<T>::EXP_BIAS + scale_up_exponent,
                                   IMPLICIT_MASK)
               .get_val();
       T scale_down_factor =
           FPBits<T>::create_value(sign, FPBits<T>::EXP_BIAS - scale_up_exponent,
                                   IMPLICIT_MASK)
               .get_val();

       d_lo = FPBits<T>::create_value(sign, exp_lo + scale_up_exponent,
                                      IMPLICIT_MASK)
                  .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, IMPLICIT_MASK).get_val();

     // Still correct without FMA instructions if `d_lo` is not underflow.
     T r = multiply_add(d_lo, T(round_and_sticky), d_hi);

     if (LIBC_UNLIKELY(denorm)) {
       // Exponent before rounding is in denormal range, simply clear the
       // exponent field.
       output_bits_t clear_exp = (output_bits_t(exp_hi) << FPBits<T>::SIG_LEN);
       output_bits_t r_bits = FPBits<T>(r).uintval() - clear_exp;
       if (!(r_bits & FPBits<T>::EXP_MASK)) {
         // Output is denormal after rounding, clear the implicit bit for 80-bit
         // long double.
         r_bits -= IMPLICIT_MASK;
       }

       return FPBits<T>(r_bits).get_val();
     }

     return r;
   }

   LIBC_INLINE explicit constexpr operator MantissaType() const {
     if (mantissa.is_zero())
       return 0;

     MantissaType new_mant = mantissa;
     if (exponent > 0) {
       new_mant <<= exponent;
     } else {
       new_mant >>= (-exponent);
     }

     if (sign.is_neg()) {
       new_mant = (~new_mant) + 1;
     }

     return new_mant;
   }
 };

 // Quick add - Add 2 dyadic floats with rounding toward 0 and then normalize the
 // output:
 //   - Align the exponents so that:
 //     new a.exponent = new b.exponent = max(a.exponent, b.exponent)
 //   - Add or subtract the mantissas depending on the signs.
 //   - Normalize the result.
 // The absolute errors compared to the mathematical sum is bounded by:
 //   | quick_add(a, b) - (a + b) | < MSB(a + b) * 2^(-Bits + 2),
 // i.e., errors are up to 2 ULPs.
 // Assume inputs are normalized (by constructors or other functions) so that we
 // don't need to normalize the inputs again in this function.  If the inputs are
 // not normalized, the results might lose precision significantly.
 template <size_t Bits>
 LIBC_INLINE constexpr DyadicFloat<Bits> quick_add(DyadicFloat<Bits> a,
                                                   DyadicFloat<Bits> b) {
   if (LIBC_UNLIKELY(a.mantissa.is_zero()))
     return b;
   if (LIBC_UNLIKELY(b.mantissa.is_zero()))
     return a;

   // Align exponents
   if (a.exponent > b.exponent)
     b.shift_right(a.exponent - b.exponent);
   else if (b.exponent > a.exponent)
     a.shift_right(b.exponent - a.exponent);

   DyadicFloat<Bits> result;

   if (a.sign == b.sign) {
     // Addition
     result.sign = a.sign;
     result.exponent = a.exponent;
     result.mantissa = a.mantissa;
     if (result.mantissa.add_overflow(b.mantissa)) {
       // Mantissa addition overflow.
       result.shift_right(1);
       result.mantissa.val[DyadicFloat<Bits>::MantissaType::WORD_COUNT - 1] |=
           (uint64_t(1) << 63);
     }
     // Result is already normalized.
     return result;
   }

   // Subtraction
   if (a.mantissa >= b.mantissa) {
     result.sign = a.sign;
     result.exponent = a.exponent;
     result.mantissa = a.mantissa - b.mantissa;
   } else {
     result.sign = b.sign;
     result.exponent = b.exponent;
     result.mantissa = b.mantissa - a.mantissa;
   }

   return result.normalize();
 }

 // Quick Mul - Slightly less accurate but efficient multiplication of 2 dyadic
 // floats with rounding toward 0 and then normalize the output:
 //   result.exponent = a.exponent + b.exponent + Bits,
 //   result.mantissa = quick_mul_hi(a.mantissa + b.mantissa)
 //                   ~ (full product a.mantissa * b.mantissa) >> Bits.
 // The errors compared to the mathematical product is bounded by:
 //   2 * errors of quick_mul_hi = 2 * (UInt<Bits>::WORD_COUNT - 1) in ULPs.
 // Assume inputs are normalized (by constructors or other functions) so that we
 // don't need to normalize the inputs again in this function.  If the inputs are
 // not normalized, the results might lose precision significantly.
 template <size_t Bits>
 LIBC_INLINE constexpr DyadicFloat<Bits> quick_mul(DyadicFloat<Bits> a,
                                                   DyadicFloat<Bits> b) {
   DyadicFloat<Bits> result;
   result.sign = (a.sign != b.sign) ? Sign::NEG : Sign::POS;
   result.exponent = a.exponent + b.exponent + int(Bits);

   if (!(a.mantissa.is_zero() || b.mantissa.is_zero())) {
     result.mantissa = a.mantissa.quick_mul_hi(b.mantissa);
     // Check the leading bit directly, should be faster than using clz in
     // normalize().
     if (result.mantissa.val[DyadicFloat<Bits>::MantissaType::WORD_COUNT - 1] >>
             63 ==
         0)
       result.shift_left(1);
   } else {
     result.mantissa = (typename DyadicFloat<Bits>::MantissaType)(0);
   }
   return result;
 }

 // Simple polynomial approximation.
 template <size_t Bits>
 LIBC_INLINE 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>
 LIBC_INLINE constexpr DyadicFloat<Bits> pow_n(DyadicFloat<Bits> a,
                                               uint32_t power) {
   DyadicFloat<Bits> result = 1.0;
   DyadicFloat<Bits> cur_power = a;

   while (power > 0) {
     if ((power % 2) > 0) {
       result = quick_mul(result, cur_power);
     }
     power = power >> 1;
     cur_power = quick_mul(cur_power, cur_power);
   }
   return result;
 }

 template <size_t Bits>
 LIBC_INLINE constexpr DyadicFloat<Bits> mul_pow_2(DyadicFloat<Bits> a,
                                                   int32_t pow_2) {
   DyadicFloat<Bits> result = a;
   result.exponent += pow_2;
   return result;
 }

 } // namespace LIBC_NAMESPACE::fputil

 #endif // LLVM_LIBC_SRC___SUPPORT_FPUTIL_DYADIC_FLOAT_H
	//===-- A class to store high precision floating point numbers --- 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_DYADIC_FLOAT_H
	#define LLVM_LIBC_SRC___SUPPORT_FPUTIL_DYADIC_FLOAT_H

	#include "FPBits.h"
	#include "multiply_add.h"
	#include "src/__support/CPP/type_traits.h"
	#include "src/__support/big_int.h"
	#include "src/__support/macros/optimization.h" // LIBC_UNLIKELY

	#include <stddef.h>

	namespace LIBC_NAMESPACE::fputil {

	// A generic class to perform computations of high precision floating points.
	// We store the value in dyadic format, including 3 fields:
	// sign : boolean value - false means positive, true means negative
	// exponent: the exponent value of the least significant bit of the mantissa.
	// mantissa: unsigned integer of length `Bits`.
	// So the real value that is stored is:
	// real value = (-1)^sign * 2^exponent * (mantissa as unsigned integer)
	// The stored data is normal if for non-zero mantissa, the leading bit is 1.
	// The outputs of the constructors and most functions will be normalized.
	// To simplify and improve the efficiency, many functions will assume that the
	// inputs are normal.
	template <size_t Bits> struct DyadicFloat {
	using MantissaType = LIBC_NAMESPACE::UInt<Bits>;

	Sign sign = Sign::POS;
	int exponent = 0;
	MantissaType mantissa = MantissaType(0);

	LIBC_INLINE constexpr DyadicFloat() = default;

	template <typename T, cpp::enable_if_t<cpp::is_floating_point_v<T>, int> = 0>
	LIBC_INLINE constexpr DyadicFloat(T x) {
	static_assert(FPBits<T>::FRACTION_LEN < Bits);
	FPBits<T> x_bits(x);
	sign = x_bits.sign();
	exponent = x_bits.get_explicit_exponent() - FPBits<T>::FRACTION_LEN;
	mantissa = MantissaType(x_bits.get_explicit_mantissa());
	normalize();
	}

	LIBC_INLINE constexpr DyadicFloat(Sign s, int e, MantissaType m)
	: sign(s), exponent(e), mantissa(m) {
	normalize();
	}

	// Normalizing the mantissa, bringing the leading 1 bit to the most
	// significant bit.
	LIBC_INLINE constexpr DyadicFloat &normalize() {
	if (!mantissa.is_zero()) {
	int shift_length = cpp::countl_zero(mantissa);
	exponent -= shift_length;
	mantissa <<= static_cast<size_t>(shift_length);
	}
	return *this;
	}

	// Used for aligning exponents. Output might not be normalized.
	LIBC_INLINE constexpr DyadicFloat &shift_left(int shift_length) {
	exponent -= shift_length;
	mantissa <<= static_cast<size_t>(shift_length);
	return *this;
	}

	// Used for aligning exponents. Output might not be normalized.
	LIBC_INLINE constexpr DyadicFloat &shift_right(int shift_length) {
	exponent += shift_length;
	mantissa >>= static_cast<size_t>(shift_length);
	return *this;
	}

	// Assume that it is already normalized. Output the unbiased exponent.
	LIBC_INLINE constexpr int get_unbiased_exponent() const {
	return exponent + (Bits - 1);
	}

	// Assume that it is already normalized.
	// Output is rounded correctly with respect to the current rounding mode.
	template <typename T,
	typename = cpp::enable_if_t<cpp::is_floating_point_v<T> &&
	(FPBits<T>::FRACTION_LEN < Bits),
	void>>
	LIBC_INLINE explicit constexpr operator T() const {
	if (LIBC_UNLIKELY(mantissa.is_zero()))
	return FPBits<T>::zero(sign).get_val();

	// Assume that it is normalized, and output is also normal.
	constexpr uint32_t PRECISION = FPBits<T>::FRACTION_LEN + 1;
	using output_bits_t = typename FPBits<T>::StorageType;
	constexpr output_bits_t IMPLICIT_MASK =
	FPBits<T>::SIG_MASK - FPBits<T>::FRACTION_MASK;

	int exp_hi = exponent + static_cast<int>((Bits - 1) + FPBits<T>::EXP_BIAS);

	if (LIBC_UNLIKELY(exp_hi > 2 * FPBits<T>::EXP_BIAS)) {
	// Results overflow.
	T d_hi =
	FPBits<T>::create_value(sign, 2 * FPBits<T>::EXP_BIAS, IMPLICIT_MASK)
	.get_val();
	return T(2) * d_hi;
	}

	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 = FPBits<T>::EXP_BIAS;
	}

	int exp_lo = exp_hi - static_cast<int>(PRECISION) - 1;

	MantissaType m_hi =
	shift >= MantissaType::BITS ? MantissaType(0) : mantissa >> shift;

	T d_hi = FPBits<T>::create_value(
	sign, exp_hi,
	(static_cast<output_bits_t>(m_hi) & FPBits<T>::SIG_MASK) \|
	IMPLICIT_MASK)
	.get_val();

	MantissaType round_mask =
	shift > MantissaType::BITS ? 0 : MantissaType(1) << (shift - 1);
	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);

	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, FPBits<T>::EXP_BIAS + scale_up_exponent,
	IMPLICIT_MASK)
	.get_val();
	T scale_down_factor =
	FPBits<T>::create_value(sign, FPBits<T>::EXP_BIAS - scale_up_exponent,
	IMPLICIT_MASK)
	.get_val();

	d_lo = FPBits<T>::create_value(sign, exp_lo + scale_up_exponent,
	IMPLICIT_MASK)
	.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, IMPLICIT_MASK).get_val();

	// Still correct without FMA instructions if `d_lo` is not underflow.
	T r = multiply_add(d_lo, T(round_and_sticky), d_hi);

	if (LIBC_UNLIKELY(denorm)) {
	// Exponent before rounding is in denormal range, simply clear the
	// exponent field.
	output_bits_t clear_exp = (output_bits_t(exp_hi) << FPBits<T>::SIG_LEN);
	output_bits_t r_bits = FPBits<T>(r).uintval() - clear_exp;
	if (!(r_bits & FPBits<T>::EXP_MASK)) {
	// Output is denormal after rounding, clear the implicit bit for 80-bit
	// long double.
	r_bits -= IMPLICIT_MASK;
	}

	return FPBits<T>(r_bits).get_val();
	}

	return r;
	}

	LIBC_INLINE explicit constexpr operator MantissaType() const {
	if (mantissa.is_zero())
	return 0;

	MantissaType new_mant = mantissa;
	if (exponent > 0) {
	new_mant <<= exponent;
	} else {
	new_mant >>= (-exponent);
	}

	if (sign.is_neg()) {
	new_mant = (~new_mant) + 1;
	}

	return new_mant;
	}
	};

	// Quick add - Add 2 dyadic floats with rounding toward 0 and then normalize the
	// output:
	// - Align the exponents so that:
	// new a.exponent = new b.exponent = max(a.exponent, b.exponent)
	// - Add or subtract the mantissas depending on the signs.
	// - Normalize the result.
	// The absolute errors compared to the mathematical sum is bounded by:
	// \| quick_add(a, b) - (a + b) \| < MSB(a + b) * 2^(-Bits + 2),
	// i.e., errors are up to 2 ULPs.
	// Assume inputs are normalized (by constructors or other functions) so that we
	// don't need to normalize the inputs again in this function. If the inputs are
	// not normalized, the results might lose precision significantly.
	template <size_t Bits>
	LIBC_INLINE constexpr DyadicFloat<Bits> quick_add(DyadicFloat<Bits> a,
	DyadicFloat<Bits> b) {
	if (LIBC_UNLIKELY(a.mantissa.is_zero()))
	return b;
	if (LIBC_UNLIKELY(b.mantissa.is_zero()))
	return a;

	// Align exponents
	if (a.exponent > b.exponent)
	b.shift_right(a.exponent - b.exponent);
	else if (b.exponent > a.exponent)
	a.shift_right(b.exponent - a.exponent);

	DyadicFloat<Bits> result;

	if (a.sign == b.sign) {
	// Addition
	result.sign = a.sign;
	result.exponent = a.exponent;
	result.mantissa = a.mantissa;
	if (result.mantissa.add_overflow(b.mantissa)) {
	// Mantissa addition overflow.
	result.shift_right(1);
	result.mantissa.val[DyadicFloat<Bits>::MantissaType::WORD_COUNT - 1] \|=
	(uint64_t(1) << 63);
	}
	// Result is already normalized.
	return result;
	}

	// Subtraction
	if (a.mantissa >= b.mantissa) {
	result.sign = a.sign;
	result.exponent = a.exponent;
	result.mantissa = a.mantissa - b.mantissa;
	} else {
	result.sign = b.sign;
	result.exponent = b.exponent;
	result.mantissa = b.mantissa - a.mantissa;
	}

	return result.normalize();
	}

	// Quick Mul - Slightly less accurate but efficient multiplication of 2 dyadic
	// floats with rounding toward 0 and then normalize the output:
	// result.exponent = a.exponent + b.exponent + Bits,
	// result.mantissa = quick_mul_hi(a.mantissa + b.mantissa)
	// ~ (full product a.mantissa * b.mantissa) >> Bits.
	// The errors compared to the mathematical product is bounded by:
	// 2 * errors of quick_mul_hi = 2 * (UInt<Bits>::WORD_COUNT - 1) in ULPs.
	// Assume inputs are normalized (by constructors or other functions) so that we
	// don't need to normalize the inputs again in this function. If the inputs are
	// not normalized, the results might lose precision significantly.
	template <size_t Bits>
	LIBC_INLINE constexpr DyadicFloat<Bits> quick_mul(DyadicFloat<Bits> a,
	DyadicFloat<Bits> b) {
	DyadicFloat<Bits> result;
	result.sign = (a.sign != b.sign) ? Sign::NEG : Sign::POS;
	result.exponent = a.exponent + b.exponent + int(Bits);

	if (!(a.mantissa.is_zero() \|\| b.mantissa.is_zero())) {
	result.mantissa = a.mantissa.quick_mul_hi(b.mantissa);
	// Check the leading bit directly, should be faster than using clz in
	// normalize().
	if (result.mantissa.val[DyadicFloat<Bits>::MantissaType::WORD_COUNT - 1] >>
	63 ==
	0)
	result.shift_left(1);
	} else {
	result.mantissa = (typename DyadicFloat<Bits>::MantissaType)(0);
	}
	return result;
	}

	// Simple polynomial approximation.
	template <size_t Bits>
	LIBC_INLINE 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>
	LIBC_INLINE constexpr DyadicFloat<Bits> pow_n(DyadicFloat<Bits> a,
	uint32_t power) {
	DyadicFloat<Bits> result = 1.0;
	DyadicFloat<Bits> cur_power = a;

	while (power > 0) {
	if ((power % 2) > 0) {
	result = quick_mul(result, cur_power);
	}
	power = power >> 1;
	cur_power = quick_mul(cur_power, cur_power);
	}
	return result;
	}

	template <size_t Bits>
	LIBC_INLINE constexpr DyadicFloat<Bits> mul_pow_2(DyadicFloat<Bits> a,
	int32_t pow_2) {
	DyadicFloat<Bits> result = a;
	result.exponent += pow_2;
	return result;
	}

	} // namespace LIBC_NAMESPACE::fputil

	#endif // LLVM_LIBC_SRC___SUPPORT_FPUTIL_DYADIC_FLOAT_H