hex_float.h

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// Copyright (c) 2015-2016 The Khronos Group Inc.
//
// Licensed under the Apache License, Version 2.0 (the "License");
// you may not use this file except in compliance with the License.
// You may obtain a copy of the License at
//
// http://www.apache.org/licenses/LICENSE-2.0
//
// Unless required by applicable law or agreed to in writing, software
// distributed under the License is distributed on an "AS IS" BASIS,
// WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
// See the License for the specific language governing permissions and
// limitations under the License.

#ifndef LIBSPIRV_UTIL_HEX_FLOAT_H_
#define LIBSPIRV_UTIL_HEX_FLOAT_H_

#include <cassert>
#include <cctype>
#include <cmath>
#include <cstdint>
#include <iomanip>
#include <limits>
#include <sstream>

#if defined(_MSC_VER) && _MSC_VER < 1800
namespace std {
bool isnan(double f)
{
 return ::_isnan(f) != 0;
}
bool isinf(double f)
{
 return ::_finite(f) == 0;
}
}
#endif

#include "bitutils.h"

namespace spvutils {

class Float16 {
 public:
 Float16(uint16_t v) : val(v) {}
 Float16() {}
 static bool isNan(const Float16& val) {
 return ((val.val & 0x7C00) == 0x7C00) && ((val.val & 0x3FF) != 0);
 }
 // Returns true if the given value is any kind of infinity.
 static bool isInfinity(const Float16& val) {
 return ((val.val & 0x7C00) == 0x7C00) && ((val.val & 0x3FF) == 0);
 }
 Float16(const Float16& other) { val = other.val; }
 uint16_t get_value() const { return val; }

 // Returns the maximum normal value.
 static Float16 max() { return Float16(0x7bff); }
 // Returns the lowest normal value.
 static Float16 lowest() { return Float16(0xfbff); }

 private:
 uint16_t val;
};

// To specialize this type, you must override uint_type to define
// an unsigned integer that can fit your floating point type.
// You must also add a isNan function that returns true if
// a value is Nan.
template <typename T>
struct FloatProxyTraits {
 typedef void uint_type;
};

template <>
struct FloatProxyTraits<float> {
 typedef uint32_t uint_type;
 static bool isNan(float f) { return std::isnan(f); }
 // Returns true if the given value is any kind of infinity.
 static bool isInfinity(float f) { return std::isinf(f); }
 // Returns the maximum normal value.
 static float max() { return std::numeric_limits<float>::max(); }
 // Returns the lowest normal value.
 static float lowest() { return std::numeric_limits<float>::lowest(); }
};

template <>
struct FloatProxyTraits<double> {
 typedef uint64_t uint_type;
 static bool isNan(double f) { return std::isnan(f); }
 // Returns true if the given value is any kind of infinity.
 static bool isInfinity(double f) { return std::isinf(f); }
 // Returns the maximum normal value.
 static double max() { return std::numeric_limits<double>::max(); }
 // Returns the lowest normal value.
 static double lowest() { return std::numeric_limits<double>::lowest(); }
};

template <>
struct FloatProxyTraits<Float16> {
 typedef uint16_t uint_type;
 static bool isNan(Float16 f) { return Float16::isNan(f); }
 // Returns true if the given value is any kind of infinity.
 static bool isInfinity(Float16 f) { return Float16::isInfinity(f); }
 // Returns the maximum normal value.
 static Float16 max() { return Float16::max(); }
 // Returns the lowest normal value.
 static Float16 lowest() { return Float16::lowest(); }
};

// Since copying a floating point number (especially if it is NaN)
// does not guarantee that bits are preserved, this class lets us
// store the type and use it as a float when necessary.
template <typename T>
class FloatProxy {
 public:
 typedef typename FloatProxyTraits<T>::uint_type uint_type;

 // Since this is to act similar to the normal floats,
 // do not initialize the data by default.
 FloatProxy() {}

 // Intentionally non-explicit. This is a proxy type so
 // implicit conversions allow us to use it more transparently.
 FloatProxy(T val) { data_ = BitwiseCast<uint_type>(val); }

 // Intentionally non-explicit. This is a proxy type so
 // implicit conversions allow us to use it more transparently.
 FloatProxy(uint_type val) { data_ = val; }

 // This is helpful to have and is guaranteed not to stomp bits.
 FloatProxy<T> operator-() const {
 return static_cast<uint_type>(data_ ^
 (uint_type(0x1) << (sizeof(T) * 8 - 1)));
 }

 // Returns the data as a floating point value.
 T getAsFloat() const { return BitwiseCast<T>(data_); }

 // Returns the raw data.
 uint_type data() const { return data_; }

 // Returns true if the value represents any type of NaN.
 bool isNan() { return FloatProxyTraits<T>::isNan(getAsFloat()); }
 // Returns true if the value represents any type of infinity.
 bool isInfinity() { return FloatProxyTraits<T>::isInfinity(getAsFloat()); }

 // Returns the maximum normal value.
 static FloatProxy<T> max() {
 return FloatProxy<T>(FloatProxyTraits<T>::max());
 }
 // Returns the lowest normal value.
 static FloatProxy<T> lowest() {
 return FloatProxy<T>(FloatProxyTraits<T>::lowest());
 }

 private:
 uint_type data_;
};

template <typename T>
bool operator==(const FloatProxy<T>& first, const FloatProxy<T>& second) {
 return first.data() == second.data();
}

// Reads a FloatProxy value as a normal float from a stream.
template <typename T>
std::istream& operator>>(std::istream& is, FloatProxy<T>& value) {
 T float_val;
 is >> float_val;
 value = FloatProxy<T>(float_val);
 return is;
}

// This is an example traits. It is not meant to be used in practice, but will
// be the default for any non-specialized type.
template <typename T>
struct HexFloatTraits {
 // Integer type that can store this hex-float.
 typedef void uint_type;
 // Signed integer type that can store this hex-float.
 typedef void int_type;
 // The numerical type that this HexFloat represents.
 typedef void underlying_type;
 // The type needed to construct the underlying type.
 typedef void native_type;
 // The number of bits that are actually relevant in the uint_type.
 // This allows us to deal with, for example, 24-bit values in a 32-bit
 // integer.
 static const uint32_t num_used_bits = 0;
 // Number of bits that represent the exponent.
 static const uint32_t num_exponent_bits = 0;
 // Number of bits that represent the fractional part.
 static const uint32_t num_fraction_bits = 0;
 // The bias of the exponent. (How much we need to subtract from the stored
 // value to get the correct value.)
 static const uint32_t exponent_bias = 0;
};

// Traits for IEEE float.
// 1 sign bit, 8 exponent bits, 23 fractional bits.
template <>
struct HexFloatTraits<FloatProxy<float>> {
 typedef uint32_t uint_type;
 typedef int32_t int_type;
 typedef FloatProxy<float> underlying_type;
 typedef float native_type;
 static const uint_type num_used_bits = 32;
 static const uint_type num_exponent_bits = 8;
 static const uint_type num_fraction_bits = 23;
 static const uint_type exponent_bias = 127;
};

// Traits for IEEE double.
// 1 sign bit, 11 exponent bits, 52 fractional bits.
template <>
struct HexFloatTraits<FloatProxy<double>> {
 typedef uint64_t uint_type;
 typedef int64_t int_type;
 typedef FloatProxy<double> underlying_type;
 typedef double native_type;
 static const uint_type num_used_bits = 64;
 static const uint_type num_exponent_bits = 11;
 static const uint_type num_fraction_bits = 52;
 static const uint_type exponent_bias = 1023;
};

// Traits for IEEE half.
// 1 sign bit, 5 exponent bits, 10 fractional bits.
template <>
struct HexFloatTraits<FloatProxy<Float16>> {
 typedef uint16_t uint_type;
 typedef int16_t int_type;
 typedef uint16_t underlying_type;
 typedef uint16_t native_type;
 static const uint_type num_used_bits = 16;
 static const uint_type num_exponent_bits = 5;
 static const uint_type num_fraction_bits = 10;
 static const uint_type exponent_bias = 15;
};

enum round_direction {
 kRoundToZero,
 kRoundToNearestEven,
 kRoundToPositiveInfinity,
 kRoundToNegativeInfinity
};

// Template class that houses a floating pointer number.
// It exposes a number of constants based on the provided traits to
// assist in interpreting the bits of the value.
template <typename T, typename Traits = HexFloatTraits<T>>
class HexFloat {
 public:
 typedef typename Traits::uint_type uint_type;
 typedef typename Traits::int_type int_type;
 typedef typename Traits::underlying_type underlying_type;
 typedef typename Traits::native_type native_type;

 explicit HexFloat(T f) : value_(f) {}

 T value() const { return value_; }
 void set_value(T f) { value_ = f; }

 // These are all written like this because it is convenient to have
 // compile-time constants for all of these values.

 // Pass-through values to save typing.
 static const uint32_t num_used_bits = Traits::num_used_bits;
 static const uint32_t exponent_bias = Traits::exponent_bias;
 static const uint32_t num_exponent_bits = Traits::num_exponent_bits;
 static const uint32_t num_fraction_bits = Traits::num_fraction_bits;

 // Number of bits to shift left to set the highest relevant bit.
 static const uint32_t top_bit_left_shift = num_used_bits - 1;
 // How many nibbles (hex characters) the fractional part takes up.
 static const uint32_t fraction_nibbles = (num_fraction_bits + 3) / 4;
 // If the fractional part does not fit evenly into a hex character (4-bits)
 // then we have to left-shift to get rid of leading 0s. This is the amount
 // we have to shift (might be 0).
 static const uint32_t num_overflow_bits =
 fraction_nibbles * 4 - num_fraction_bits;

 // The representation of the fraction, not the actual bits. This
 // includes the leading bit that is usually implicit.
 static const uint_type fraction_represent_mask =
 spvutils::SetBits<uint_type, 0,
 num_fraction_bits + num_overflow_bits>::get;

 // The topmost bit in the nibble-aligned fraction.
 static const uint_type fraction_top_bit =
 uint_type(1) << (num_fraction_bits + num_overflow_bits - 1);

 // The least significant bit in the exponent, which is also the bit
 // immediately to the left of the significand.
 static const uint_type first_exponent_bit = uint_type(1)
 << (num_fraction_bits);

 // The mask for the encoded fraction. It does not include the
 // implicit bit.
 static const uint_type fraction_encode_mask =
 spvutils::SetBits<uint_type, 0, num_fraction_bits>::get;

 // The bit that is used as a sign.
 static const uint_type sign_mask = uint_type(1) << top_bit_left_shift;

 // The bits that represent the exponent.
 static const uint_type exponent_mask =
 spvutils::SetBits<uint_type, num_fraction_bits, num_exponent_bits>::get;

 // How far left the exponent is shifted.
 static const uint32_t exponent_left_shift = num_fraction_bits;

 // How far from the right edge the fraction is shifted.
 static const uint32_t fraction_right_shift =
 static_cast<uint32_t>(sizeof(uint_type) * 8) - num_fraction_bits;

 // The maximum representable unbiased exponent.
 static const int_type max_exponent =
 (exponent_mask >> num_fraction_bits) - exponent_bias;
 // The minimum representable exponent for normalized numbers.
 static const int_type min_exponent = -static_cast<int_type>(exponent_bias);

 // Returns the bits associated with the value.
 uint_type getBits() const { return spvutils::BitwiseCast<uint_type>(value_); }

 // Returns the bits associated with the value, without the leading sign bit.
 uint_type getUnsignedBits() const {
 return static_cast<uint_type>(spvutils::BitwiseCast<uint_type>(value_) &
 ~sign_mask);
 }

 // Returns the bits associated with the exponent, shifted to start at the
 // lsb of the type.
 const uint_type getExponentBits() const {
 return static_cast<uint_type>((getBits() & exponent_mask) >>
 num_fraction_bits);
 }

 // Returns the exponent in unbiased form. This is the exponent in the
 // human-friendly form.
 const int_type getUnbiasedExponent() const {
 return static_cast<int_type>(getExponentBits() - exponent_bias);
 }

 // Returns just the significand bits from the value.
 const uint_type getSignificandBits() const {
 return getBits() & fraction_encode_mask;
 }

 // If the number was normalized, returns the unbiased exponent.
 // If the number was denormal, normalize the exponent first.
 const int_type getUnbiasedNormalizedExponent() const {
 if ((getBits() & ~sign_mask) == 0) { // special case if everything is 0
 return 0;
 }
 int_type exp = getUnbiasedExponent();
 if (exp == min_exponent) { // We are in denorm land.
 uint_type significand_bits = getSignificandBits();
 while ((significand_bits & (first_exponent_bit >> 1)) == 0) {
 significand_bits = static_cast<uint_type>(significand_bits << 1);
 exp = static_cast<int_type>(exp - 1);
 }
 significand_bits &= fraction_encode_mask;
 }
 return exp;
 }

 // Returns the signficand after it has been normalized.
 const uint_type getNormalizedSignificand() const {
 int_type unbiased_exponent = getUnbiasedNormalizedExponent();
 uint_type significand = getSignificandBits();
 for (int_type i = unbiased_exponent; i <= min_exponent; ++i) {
 significand = static_cast<uint_type>(significand << 1);
 }
 significand &= fraction_encode_mask;
 return significand;
 }

 // Returns true if this number represents a negative value.
 bool isNegative() const { return (getBits() & sign_mask) != 0; }

 // Sets this HexFloat from the individual components.
 // Note this assumes EVERY significand is normalized, and has an implicit
 // leading one. This means that the only way that this method will set 0,
 // is if you set a number so denormalized that it underflows.
 // Do not use this method with raw bits extracted from a subnormal number,
 // since subnormals do not have an implicit leading 1 in the significand.
 // The significand is also expected to be in the
 // lowest-most num_fraction_bits of the uint_type.
 // The exponent is expected to be unbiased, meaning an exponent of
 // 0 actually means 0.
 // If underflow_round_up is set, then on underflow, if a number is non-0
 // and would underflow, we round up to the smallest denorm.
 void setFromSignUnbiasedExponentAndNormalizedSignificand(
 bool negative, int_type exponent, uint_type significand,
 bool round_denorm_up) {
 bool significand_is_zero = significand == 0;

 if (exponent <= min_exponent) {
 // If this was denormalized, then we have to shift the bit on, meaning
 // the significand is not zero.
 significand_is_zero = false;
 significand |= first_exponent_bit;
 significand = static_cast<uint_type>(significand >> 1);
 }

 while (exponent < min_exponent) {
 significand = static_cast<uint_type>(significand >> 1);
 ++exponent;
 }

 if (exponent == min_exponent) {
 if (significand == 0 && !significand_is_zero && round_denorm_up) {
 significand = static_cast<uint_type>(0x1);
 }
 }

 uint_type new_value = 0;
 if (negative) {
 new_value = static_cast<uint_type>(new_value | sign_mask);
 }
 exponent = static_cast<int_type>(exponent + exponent_bias);
 assert(exponent >= 0);

 // put it all together
 exponent = static_cast<uint_type>((exponent << exponent_left_shift) &
 exponent_mask);
 significand = static_cast<uint_type>(significand & fraction_encode_mask);
 new_value = static_cast<uint_type>(new_value | (exponent | significand));
 value_ = BitwiseCast<T>(new_value);
 }

 // Increments the significand of this number by the given amount.
 // If this would spill the significand into the implicit bit,
 // carry is set to true and the significand is shifted to fit into
 // the correct location, otherwise carry is set to false.
 // All significands and to_increment are assumed to be within the bounds
 // for a valid significand.
 static uint_type incrementSignificand(uint_type significand,
 uint_type to_increment, bool* carry) {
 significand = static_cast<uint_type>(significand + to_increment);
 *carry = false;
 if (significand & first_exponent_bit) {
 *carry = true;
 // The implicit 1-bit will have carried, so we should zero-out the
 // top bit and shift back.
 significand = static_cast<uint_type>(significand & ~first_exponent_bit);
 significand = static_cast<uint_type>(significand >> 1);
 }
 return significand;
 }

 // These exist because MSVC throws warnings on negative right-shifts
 // even if they are not going to be executed. Eg:
 // constant_number < 0? 0: constant_number
 // These convert the negative left-shifts into right shifts.

 template <typename int_type>
 uint_type negatable_left_shift(int_type N, uint_type val)
 {
 if(N >= 0)
 return val << N;

 return val >> -N;
 }

 template <typename int_type>
 uint_type negatable_right_shift(int_type N, uint_type val)
 {
 if(N >= 0)
 return val >> N;

 return val << -N;
 }

 // Returns the significand, rounded to fit in a significand in
 // other_T. This is shifted so that the most significant
 // bit of the rounded number lines up with the most significant bit
 // of the returned significand.
 template <typename other_T>
 typename other_T::uint_type getRoundedNormalizedSignificand(
 round_direction dir, bool* carry_bit) {
 typedef typename other_T::uint_type other_uint_type;
 static const int_type num_throwaway_bits =
 static_cast<int_type>(num_fraction_bits) -
 static_cast<int_type>(other_T::num_fraction_bits);

 static const uint_type last_significant_bit =
 (num_throwaway_bits < 0)
 ? 0
 : negatable_left_shift(num_throwaway_bits, 1u);
 static const uint_type first_rounded_bit =
 (num_throwaway_bits < 1)
 ? 0
 : negatable_left_shift(num_throwaway_bits - 1, 1u);

 static const uint_type throwaway_mask_bits =
 num_throwaway_bits > 0 ? num_throwaway_bits : 0;
 static const uint_type throwaway_mask =
 spvutils::SetBits<uint_type, 0, throwaway_mask_bits>::get;

 *carry_bit = false;
 other_uint_type out_val = 0;
 uint_type significand = getNormalizedSignificand();
 // If we are up-casting, then we just have to shift to the right location.
 if (num_throwaway_bits <= 0) {
 out_val = static_cast<other_uint_type>(significand);
 uint_type shift_amount = static_cast<uint_type>(-num_throwaway_bits);
 out_val = static_cast<other_uint_type>(out_val << shift_amount);
 return out_val;
 }

 // If every non-representable bit is 0, then we don't have any casting to
 // do.
 if ((significand & throwaway_mask) == 0) {
 return static_cast<other_uint_type>(
 negatable_right_shift(num_throwaway_bits, significand));
 }

 bool round_away_from_zero = false;
 // We actually have to narrow the significand here, so we have to follow the
 // rounding rules.
 switch (dir) {
 case kRoundToZero:
 break;
 case kRoundToPositiveInfinity:
 round_away_from_zero = !isNegative();
 break;
 case kRoundToNegativeInfinity:
 round_away_from_zero = isNegative();
 break;
 case kRoundToNearestEven:
 // Have to round down, round bit is 0
 if ((first_rounded_bit & significand) == 0) {
 break;
 }
 if (((significand & throwaway_mask) & ~first_rounded_bit) != 0) {
 // If any subsequent bit of the rounded portion is non-0 then we round
 // up.
 round_away_from_zero = true;
 break;
 }
 // We are exactly half-way between 2 numbers, pick even.
 if ((significand & last_significant_bit) != 0) {
 // 1 for our last bit, round up.
 round_away_from_zero = true;
 break;
 }
 break;
 }

 if (round_away_from_zero) {
 return static_cast<other_uint_type>(
 negatable_right_shift(num_throwaway_bits, incrementSignificand(
 significand, last_significant_bit, carry_bit)));
 } else {
 return static_cast<other_uint_type>(
 negatable_right_shift(num_throwaway_bits, significand));
 }
 }

 // Casts this value to another HexFloat. If the cast is widening,
 // then round_dir is ignored. If the cast is narrowing, then
 // the result is rounded in the direction specified.
 // This number will retain Nan and Inf values.
 // It will also saturate to Inf if the number overflows, and
 // underflow to (0 or min depending on rounding) if the number underflows.
 template <typename other_T>
 void castTo(other_T& other, round_direction round_dir) {
 other = other_T(static_cast<typename other_T::native_type>(0));
 bool negate = isNegative();
 if (getUnsignedBits() == 0) {
 if (negate) {
 other.set_value(-other.value());
 }
 return;
 }
 uint_type significand = getSignificandBits();
 bool carried = false;
 typename other_T::uint_type rounded_significand =
 getRoundedNormalizedSignificand<other_T>(round_dir, &carried);

 int_type exponent = getUnbiasedExponent();
 if (exponent == min_exponent) {
 // If we are denormal, normalize the exponent, so that we can encode
 // easily.
 exponent = static_cast<int_type>(exponent + 1);
 for (uint_type check_bit = first_exponent_bit >> 1; check_bit != 0;
 check_bit = static_cast<uint_type>(check_bit >> 1)) {
 exponent = static_cast<int_type>(exponent - 1);
 if (check_bit & significand) break;
 }
 }

 bool is_nan =
 (getBits() & exponent_mask) == exponent_mask && significand != 0;
 bool is_inf =
 !is_nan &&
 ((exponent + carried) > static_cast<int_type>(other_T::exponent_bias) ||
 (significand == 0 && (getBits() & exponent_mask) == exponent_mask));

 // If we are Nan or Inf we should pass that through.
 if (is_inf) {
 other.set_value(BitwiseCast<typename other_T::underlying_type>(
 static_cast<typename other_T::uint_type>(
 (negate ? other_T::sign_mask : 0) | other_T::exponent_mask)));
 return;
 }
 if (is_nan) {
 typename other_T::uint_type shifted_significand;
 shifted_significand = static_cast<typename other_T::uint_type>(
 negatable_left_shift(
 static_cast<int_type>(other_T::num_fraction_bits) -
 static_cast<int_type>(num_fraction_bits), significand));

 // We are some sort of Nan. We try to keep the bit-pattern of the Nan
 // as close as possible. If we had to shift off bits so we are 0, then we
 // just set the last bit.
 other.set_value(BitwiseCast<typename other_T::underlying_type>(
 static_cast<typename other_T::uint_type>(
 (negate ? other_T::sign_mask : 0) | other_T::exponent_mask |
 (shifted_significand == 0 ? 0x1 : shifted_significand))));
 return;
 }

 bool round_underflow_up =
 isNegative() ? round_dir == kRoundToNegativeInfinity
 : round_dir == kRoundToPositiveInfinity;
 typedef typename other_T::int_type other_int_type;
 // setFromSignUnbiasedExponentAndNormalizedSignificand will
 // zero out any underflowing value (but retain the sign).
 other.setFromSignUnbiasedExponentAndNormalizedSignificand(
 negate, static_cast<other_int_type>(exponent), rounded_significand,
 round_underflow_up);
 return;
 }

 private:
 T value_;

 static_assert(num_used_bits ==
 Traits::num_exponent_bits + Traits::num_fraction_bits + 1,
 "The number of bits do not fit");
 static_assert(sizeof(T) == sizeof(uint_type), "The type sizes do not match");
};

// Returns 4 bits represented by the hex character.
inline uint8_t get_nibble_from_character(int character) {
 const char* dec = "0123456789";
 const char* lower = "abcdef";
 const char* upper = "ABCDEF";
 const char* p = nullptr;
 if ((p = strchr(dec, character))) {
 return static_cast<uint8_t>(p - dec);
 } else if ((p = strchr(lower, character))) {
 return static_cast<uint8_t>(p - lower + 0xa);
 } else if ((p = strchr(upper, character))) {
 return static_cast<uint8_t>(p - upper + 0xa);
 }

 assert(false && "This was called with a non-hex character");
 return 0;
}

// Outputs the given HexFloat to the stream.
template <typename T, typename Traits>
std::ostream& operator<<(std::ostream& os, const HexFloat<T, Traits>& value) {
 typedef HexFloat<T, Traits> HF;
 typedef typename HF::uint_type uint_type;
 typedef typename HF::int_type int_type;

 static_assert(HF::num_used_bits != 0,
 "num_used_bits must be non-zero for a valid float");
 static_assert(HF::num_exponent_bits != 0,
 "num_exponent_bits must be non-zero for a valid float");
 static_assert(HF::num_fraction_bits != 0,
 "num_fractin_bits must be non-zero for a valid float");

 const uint_type bits = spvutils::BitwiseCast<uint_type>(value.value());
 const char* const sign = (bits & HF::sign_mask) ? "-" : "";
 const uint_type exponent = static_cast<uint_type>(
 (bits & HF::exponent_mask) >> HF::num_fraction_bits);

 uint_type fraction = static_cast<uint_type>((bits & HF::fraction_encode_mask)
 << HF::num_overflow_bits);

 const bool is_zero = exponent == 0 && fraction == 0;
 const bool is_denorm = exponent == 0 && !is_zero;

 // exponent contains the biased exponent we have to convert it back into
 // the normal range.
 int_type int_exponent = static_cast<int_type>(exponent - HF::exponent_bias);
 // If the number is all zeros, then we actually have to NOT shift the
 // exponent.
 int_exponent = is_zero ? 0 : int_exponent;

 // If we are denorm, then start shifting, and decreasing the exponent until
 // our leading bit is 1.

 if (is_denorm) {
 while ((fraction & HF::fraction_top_bit) == 0) {
 fraction = static_cast<uint_type>(fraction << 1);
 int_exponent = static_cast<int_type>(int_exponent - 1);
 }
 // Since this is denormalized, we have to consume the leading 1 since it
 // will end up being implicit.
 fraction = static_cast<uint_type>(fraction << 1); // eat the leading 1
 fraction &= HF::fraction_represent_mask;
 }

 uint_type fraction_nibbles = HF::fraction_nibbles;
 // We do not have to display any trailing 0s, since this represents the
 // fractional part.
 while (fraction_nibbles > 0 && (fraction & 0xF) == 0) {
 // Shift off any trailing values;
 fraction = static_cast<uint_type>(fraction >> 4);
 --fraction_nibbles;
 }

 const auto saved_flags = os.flags();
 const auto saved_fill = os.fill();

 os << sign << "0x" << (is_zero ? '0' : '1');
 if (fraction_nibbles) {
 // Make sure to keep the leading 0s in place, since this is the fractional
 // part.
 os << "." << std::setw(static_cast<int>(fraction_nibbles))
 << std::setfill('0') << std::hex << fraction;
 }
 os << "p" << std::dec << (int_exponent >= 0 ? "+" : "") << int_exponent;

 os.flags(saved_flags);
 os.fill(saved_fill);

 return os;
}

// Returns true if negate_value is true and the next character on the
// input stream is a plus or minus sign. In that case we also set the fail bit
// on the stream and set the value to the zero value for its type.
template <typename T, typename Traits>
inline bool RejectParseDueToLeadingSign(std::istream& is, bool negate_value,
 HexFloat<T, Traits>& value) {
 if (negate_value) {
 auto next_char = is.peek();
 if (next_char == '-' || next_char == '+') {
 // Fail the parse. Emulate standard behaviour by setting the value to
 // the zero value, and set the fail bit on the stream.
 value = HexFloat<T, Traits>(typename HexFloat<T, Traits>::uint_type(0));
 is.setstate(std::ios_base::failbit);
 return true;
 }
 }
 return false;
}

// Parses a floating point number from the given stream and stores it into the
// value parameter.
// If negate_value is true then the number may not have a leading minus or
// plus, and if it successfully parses, then the number is negated before
// being stored into the value parameter.
// If the value cannot be correctly parsed or overflows the target floating
// point type, then set the fail bit on the stream.
// TODO(dneto): Promise C++11 standard behavior in how the value is set in
// the error case, but only after all target platforms implement it correctly.
// In particular, the Microsoft C++ runtime appears to be out of spec.
template <typename T, typename Traits>
inline std::istream& ParseNormalFloat(std::istream& is, bool negate_value,
 HexFloat<T, Traits>& value) {
 if (RejectParseDueToLeadingSign(is, negate_value, value)) {
 return is;
 }
 T val;
 is >> val;
 if (negate_value) {
 val = -val;
 }
 value.set_value(val);
 // In the failure case, map -0.0 to 0.0.
 if (is.fail() && value.getUnsignedBits() == 0u) {
 value = HexFloat<T, Traits>(typename HexFloat<T, Traits>::uint_type(0));
 }
 if (val.isInfinity()) {
 // Fail the parse. Emulate standard behaviour by setting the value to
 // the closest normal value, and set the fail bit on the stream.
 value.set_value((value.isNegative() | negate_value) ? T::lowest()
 : T::max());
 is.setstate(std::ios_base::failbit);
 }
 return is;
}

// Specialization of ParseNormalFloat for FloatProxy<Float16> values.
// This will parse the float as it were a 32-bit floating point number,
// and then round it down to fit into a Float16 value.
// The number is rounded towards zero.
// If negate_value is true then the number may not have a leading minus or
// plus, and if it successfully parses, then the number is negated before
// being stored into the value parameter.
// If the value cannot be correctly parsed or overflows the target floating
// point type, then set the fail bit on the stream.
// TODO(dneto): Promise C++11 standard behavior in how the value is set in
// the error case, but only after all target platforms implement it correctly.
// In particular, the Microsoft C++ runtime appears to be out of spec.
template <>
inline std::istream&
ParseNormalFloat<FloatProxy<Float16>, HexFloatTraits<FloatProxy<Float16>>>(
 std::istream& is, bool negate_value,
 HexFloat<FloatProxy<Float16>, HexFloatTraits<FloatProxy<Float16>>>& value) {
 // First parse as a 32-bit float.
 HexFloat<FloatProxy<float>> float_val(0.0f);
 ParseNormalFloat(is, negate_value, float_val);

 // Then convert to 16-bit float, saturating at infinities, and
 // rounding toward zero.
 float_val.castTo(value, kRoundToZero);

 // Overflow on 16-bit behaves the same as for 32- and 64-bit: set the
 // fail bit and set the lowest or highest value.
 if (Float16::isInfinity(value.value().getAsFloat())) {
 value.set_value(value.isNegative() ? Float16::lowest() : Float16::max());
 is.setstate(std::ios_base::failbit);
 }
 return is;
}

// Reads a HexFloat from the given stream.
// If the float is not encoded as a hex-float then it will be parsed
// as a regular float.
// This may fail if your stream does not support at least one unget.
// Nan values can be encoded with "0x1.<not zero>p+exponent_bias".
// This would normally overflow a float and round to
// infinity but this special pattern is the exact representation for a NaN,
// and therefore is actually encoded as the correct NaN. To encode inf,
// either 0x0p+exponent_bias can be specified or any exponent greater than
// exponent_bias.
// Examples using IEEE 32-bit float encoding.
// 0x1.0p+128 (+inf)
// -0x1.0p-128 (-inf)
//
// 0x1.1p+128 (+Nan)
// -0x1.1p+128 (-Nan)
//
// 0x1p+129 (+inf)
// -0x1p+129 (-inf)
template <typename T, typename Traits>
std::istream& operator>>(std::istream& is, HexFloat<T, Traits>& value) {
 using HF = HexFloat<T, Traits>;
 using uint_type = typename HF::uint_type;
 using int_type = typename HF::int_type;

 value.set_value(static_cast<typename HF::native_type>(0.f));

 if (is.flags() & std::ios::skipws) {
 // If the user wants to skip whitespace , then we should obey that.
 while (std::isspace(is.peek())) {
 is.get();
 }
 }

 auto next_char = is.peek();
 bool negate_value = false;

 if (next_char != '-' && next_char != '0') {
 return ParseNormalFloat(is, negate_value, value);
 }

 if (next_char == '-') {
 negate_value = true;
 is.get();
 next_char = is.peek();
 }

 if (next_char == '0') {
 is.get(); // We may have to unget this.
 auto maybe_hex_start = is.peek();
 if (maybe_hex_start != 'x' && maybe_hex_start != 'X') {
 is.unget();
 return ParseNormalFloat(is, negate_value, value);
 } else {
 is.get(); // Throw away the 'x';
 }
 } else {
 return ParseNormalFloat(is, negate_value, value);
 }

 // This "looks" like a hex-float so treat it as one.
 bool seen_p = false;
 bool seen_dot = false;
 uint_type fraction_index = 0;

 uint_type fraction = 0;
 int_type exponent = HF::exponent_bias;

 // Strip off leading zeros so we don't have to special-case them later.
 while ((next_char = is.peek()) == '0') {
 is.get();
 }

 bool is_denorm =
 true; // Assume denorm "representation" until we hear otherwise.
 // NB: This does not mean the value is actually denorm,
 // it just means that it was written 0.
 bool bits_written = false; // Stays false until we write a bit.
 while (!seen_p && !seen_dot) {
 // Handle characters that are left of the fractional part.
 if (next_char == '.') {
 seen_dot = true;
 } else if (next_char == 'p') {
 seen_p = true;
 } else if (::isxdigit(next_char)) {
 // We know this is not denormalized since we have stripped all leading
 // zeroes and we are not a ".".
 is_denorm = false;
 int number = get_nibble_from_character(next_char);
 for (int i = 0; i < 4; ++i, number <<= 1) {
 uint_type write_bit = (number & 0x8) ? 0x1 : 0x0;
 if (bits_written) {
 // If we are here the bits represented belong in the fractional
 // part of the float, and we have to adjust the exponent accordingly.
 fraction = static_cast<uint_type>(
 fraction |
 static_cast<uint_type>(
 write_bit << (HF::top_bit_left_shift - fraction_index++)));
 exponent = static_cast<int_type>(exponent + 1);
 }
 bits_written |= write_bit != 0;
 }
 } else {
 // We have not found our exponent yet, so we have to fail.
 is.setstate(std::ios::failbit);
 return is;
 }
 is.get();
 next_char = is.peek();
 }
 bits_written = false;
 while (seen_dot && !seen_p) {
 // Handle only fractional parts now.
 if (next_char == 'p') {
 seen_p = true;
 } else if (::isxdigit(next_char)) {
 int number = get_nibble_from_character(next_char);
 for (int i = 0; i < 4; ++i, number <<= 1) {
 uint_type write_bit = (number & 0x8) ? 0x01 : 0x00;
 bits_written |= write_bit != 0;
 if (is_denorm && !bits_written) {
 // Handle modifying the exponent here this way we can handle
 // an arbitrary number of hex values without overflowing our
 // integer.
 exponent = static_cast<int_type>(exponent - 1);
 } else {
 fraction = static_cast<uint_type>(
 fraction |
 static_cast<uint_type>(
 write_bit << (HF::top_bit_left_shift - fraction_index++)));
 }
 }
 } else {
 // We still have not found our 'p' exponent yet, so this is not a valid
 // hex-float.
 is.setstate(std::ios::failbit);
 return is;
 }
 is.get();
 next_char = is.peek();
 }

 bool seen_sign = false;
 int8_t exponent_sign = 1;
 int_type written_exponent = 0;
 while (true) {
 if ((next_char == '-' || next_char == '+')) {
 if (seen_sign) {
 is.setstate(std::ios::failbit);
 return is;
 }
 seen_sign = true;
 exponent_sign = (next_char == '-') ? -1 : 1;
 } else if (::isdigit(next_char)) {
 // Hex-floats express their exponent as decimal.
 written_exponent = static_cast<int_type>(written_exponent * 10);
 written_exponent =
 static_cast<int_type>(written_exponent + (next_char - '0'));
 } else {
 break;
 }
 is.get();
 next_char = is.peek();
 }

 written_exponent = static_cast<int_type>(written_exponent * exponent_sign);
 exponent = static_cast<int_type>(exponent + written_exponent);

 bool is_zero = is_denorm && (fraction == 0);
 if (is_denorm && !is_zero) {
 fraction = static_cast<uint_type>(fraction << 1);
 exponent = static_cast<int_type>(exponent - 1);
 } else if (is_zero) {
 exponent = 0;
 }

 if (exponent <= 0 && !is_zero) {
 fraction = static_cast<uint_type>(fraction >> 1);
 fraction |= static_cast<uint_type>(1) << HF::top_bit_left_shift;
 }

 fraction = (fraction >> HF::fraction_right_shift) & HF::fraction_encode_mask;

 const int_type max_exponent =
 SetBits<uint_type, 0, HF::num_exponent_bits>::get;

 // Handle actual denorm numbers
 while (exponent < 0 && !is_zero) {
 fraction = static_cast<uint_type>(fraction >> 1);
 exponent = static_cast<int_type>(exponent + 1);

 fraction &= HF::fraction_encode_mask;
 if (fraction == 0) {
 // We have underflowed our fraction. We should clamp to zero.
 is_zero = true;
 exponent = 0;
 }
 }

 // We have overflowed so we should be inf/-inf.
 if (exponent > max_exponent) {
 exponent = max_exponent;
 fraction = 0;
 }

 uint_type output_bits = static_cast<uint_type>(
 static_cast<uint_type>(negate_value ? 1 : 0) << HF::top_bit_left_shift);
 output_bits |= fraction;

 uint_type shifted_exponent = static_cast<uint_type>(
 static_cast<uint_type>(exponent << HF::exponent_left_shift) &
 HF::exponent_mask);
 output_bits |= shifted_exponent;

 T output_float = spvutils::BitwiseCast<T>(output_bits);
 value.set_value(output_float);

 return is;
}

// Writes a FloatProxy value to a stream.
// Zero and normal numbers are printed in the usual notation, but with
// enough digits to fully reproduce the value. Other values (subnormal,
// NaN, and infinity) are printed as a hex float.
template <typename T>
std::ostream& operator<<(std::ostream& os, const FloatProxy<T>& value) {
 auto float_val = value.getAsFloat();
 switch (std::fpclassify(float_val)) {
 case FP_ZERO:
 case FP_NORMAL: {
 auto saved_precision = os.precision();
 os.precision(std::numeric_limits<T>::digits10);
 os << float_val;
 os.precision(saved_precision);
 } break;
 default:
 os << HexFloat<FloatProxy<T>>(value);
 break;
 }
 return os;
}

template <>
inline std::ostream& operator<<<Float16>(std::ostream& os,
 const FloatProxy<Float16>& value) {
 os << HexFloat<FloatProxy<Float16>>(value);
 return os;
}
}

#endif // LIBSPIRV_UTIL_HEX_FLOAT_H_