| // Copyright 2022 The Tint Authors. |
| // |
| // 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. |
| |
| #include "src/tint/lang/core/number.h" |
| |
| #include <algorithm> |
| #include <cmath> |
| #include <cstring> |
| |
| #include "src/tint/utils/ice/ice.h" |
| #include "src/tint/utils/memory/bitcast.h" |
| #include "src/tint/utils/text/string_stream.h" |
| |
| namespace tint { |
| namespace { |
| |
| constexpr uint16_t kF16Nan = 0x7e00u; |
| constexpr uint16_t kF16PosInf = 0x7c00u; |
| constexpr uint16_t kF16NegInf = 0xfc00u; |
| |
| constexpr uint16_t kF16SignMask = 0x8000u; |
| constexpr uint16_t kF16ExponentMask = 0x7c00u; |
| constexpr uint16_t kF16MantissaMask = 0x03ffu; |
| |
| constexpr uint32_t kF16MantissaBits = 10; |
| constexpr uint32_t kF16ExponentBias = 15; |
| |
| constexpr uint32_t kF32SignMask = 0x80000000u; |
| constexpr uint32_t kF32ExponentMask = 0x7f800000u; |
| constexpr uint32_t kF32MantissaMask = 0x007fffffu; |
| |
| constexpr uint32_t kF32MantissaBits = 23; |
| constexpr uint32_t kF32ExponentBias = 127; |
| |
| constexpr uint32_t kMaxF32BiasedExpForF16NormalNumber = 142; |
| constexpr uint32_t kMinF32BiasedExpForF16NormalNumber = 113; |
| constexpr uint32_t kMaxF32BiasedExpForF16SubnormalNumber = 112; |
| constexpr uint32_t kMinF32BiasedExpForF16SubnormalNumber = 103; |
| |
| } // namespace |
| |
| f16::type f16::Quantize(f16::type value) { |
| if (value > kHighestValue) { |
| return std::numeric_limits<f16::type>::infinity(); |
| } |
| if (value < kLowestValue) { |
| return -std::numeric_limits<f16::type>::infinity(); |
| } |
| |
| // Below value must be within the finite range of a f16. |
| // Assert we use binary32 (i.e. float) as underlying type, which has 4 bytes. |
| static_assert(std::is_same<f16::type, float>()); |
| |
| uint32_t u32 = tint::Bitcast<uint32_t>(value); |
| if ((u32 & ~kF32SignMask) == 0) { |
| return value; // +/- zero |
| } |
| if ((u32 & kF32ExponentMask) == kF32ExponentMask) { // exponent all 1's |
| return value; // inf or nan |
| } |
| |
| // We are now going to quantize a f32 number into subnormal f16 and store the result value back |
| // into a f32 variable. Notice that all subnormal f16 values are just normal f32 values. Below |
| // will show that we can do this quantization by just masking out 13 or more lowest mantissa |
| // bits of the original f32 number. |
| // |
| // Note: |
| // * f32 has 1 sign bit, 8 exponent bits for biased exponent (i.e. unbiased exponent + 127), and |
| // 23 mantissa bits. Binary form: s_eeeeeeee_mmmmmmmmmmmmmmmmmmmmmmm |
| // |
| // * f16 has 1 sign bit, 5 exponent bits for biased exponent (i.e. unbiased exponent + 15), and |
| // 10 mantissa bits. Binary form: s_eeeee_mmmmmmmmmm |
| // |
| // The largest finite f16 number has a biased exponent of 11110 in binary, or 30 decimal, and so |
| // an unbiased exponent of 30 - 15 = 15. |
| // |
| // The smallest finite f16 number has a biased exponent of 00001 in binary, or 1 decimal, and so |
| // a unbiased exponent of 1 - 15 = -14. |
| // |
| // We may follow the argument below: |
| // 1. All normal or subnormal f16 values, range from 0x1.p-24 to 0x1.ffcp15, are exactly |
| // representable by a normal f32 number. |
| // 1.1. We can denote the set of all f32 number that are exact representation of finite f16 |
| // values by `R`. |
| // 1.2. We can do the quantization by mapping a normal f32 value v (in the f16 finite range) |
| // to a certain f32 number v' in the set R, which is the largest (by the meaning of absolute |
| // value) one among all values in R that are no larger than v. |
| // |
| // 2. We can decide whether a given normal f32 number v is in the set R, by looking at its |
| // mantissa bits and biased exponent `e`. Recall that biased exponent e is unbiased exponent + |
| // 127, and in the range of 1 to 254 for normal f32 number. |
| // 2.1. If e >= 143, i.e. abs(v) >= 2^16 > f16::kHighestValue = 0x1.ffcp15, v is larger than |
| // any finite f16 value and can not be in set R. 2.2. If 142 >= e >= 113, or |
| // f16::kHighestValue >= abs(v) >= f16::kSmallestValue = 2^-14, v falls in the range of normal |
| // f16 values. In this case, v is in the set R iff the lowest 13 mantissa bits are all 0. (See |
| // below for proof) |
| // 2.2.1. If we let v' be v with lowest 13 mantissa bits masked to 0, v' will be in set R |
| // and the largest one in set R that no larger than v. Such v' is the quantized value of v. |
| // 2.3. If 112 >= e >= 103, i.e. 2^-14 > abs(v) >= f16::kSmallestSubnormalValue = 2^-24, v |
| // falls in the range of subnormal f16 values. In this case, v is in the set R iff the lowest |
| // 126-e mantissa bits are all 0. Notice that 126-e is in range 14 to 23, inclusive. (See |
| // below for proof) |
| // 2.3.1. If we let v' be v with lowest 126-e mantissa bits masked to 0, v' will be in set R |
| // and the largest on in set R that no larger than v. Such v' is the quantized value of v. |
| // 2.4. If 2^-24 > abs(v) > 0, i.e. 103 > e, v is smaller than any finite f16 value and not |
| // equal to 0.0, thus can not be in set R. |
| // 2.5. If abs(v) = 0, v is in set R and is just +-0.0. |
| // |
| // Proof for 2.2 |
| // ------------- |
| // Any normal f16 number, in binary form, s_eeeee_mmmmmmmmmm, has value |
| // |
| // (s == 0 ? 1 : -1) * (1 + uint(mmmmm_mmmmm) * (2^-10)) * 2^(uint(eeeee) - 15) |
| // |
| // in which unit(bbbbb) means interprete binary pattern "bbbbb" as unsigned binary number, |
| // and we have 1 <= uint(eeeee) <= 30. |
| // |
| // This value is equal to a normal f32 number with binary |
| // s_EEEEEEEE_mmmmmmmmmm0000000000000 |
| // |
| // where uint(EEEEEEEE) = uint(eeeee) + 112, so that unbiased exponent is kept unchanged |
| // |
| // uint(EEEEEEEE) - 127 = uint(eeeee) - 15 |
| // |
| // and its value is |
| // (s == 0 ? 1 : -1) * |
| // (1 + uint(mmmmm_mmmmm_00000_00000_000) * (2^-23)) * 2^(uint(EEEEEEEE) - 127) |
| // == (s == 0 ? 1 : -1) * |
| // (1 + uint(mmmmm_mmmmm) * (2^-10)) * 2^(uint(eeeee) - 15) |
| // |
| // Notice that uint(EEEEEEEE) is in range [113, 142], showing that it is a normal f32 number. |
| // So we proved that any normal f16 number can be exactly representd by a normal f32 number |
| // with biased exponent in range [113, 142] and the lowest 13 mantissa bits 0. |
| // |
| // On the other hand, since mantissa bits mmmmmmmmmm are arbitrary, the value of any f32 |
| // that has a biased exponent in range [113, 142] and lowest 13 mantissa bits zero is equal |
| // to a normal f16 value. Hence we prove 2.2. |
| // |
| // Proof for 2.3 |
| // ------------- |
| // Any subnormal f16 number has a binary form of s_00000_mmmmmmmmmm, and its value is |
| // |
| // (s == 0 ? 1 : -1) * uint(mmmmmmmmmm) * (2^-10) * (2^-14) |
| // == (s == 0 ? 1 : -1) * uint(mmmmmmmmmm) * (2^-24). |
| // |
| // We discuss the bit pattern of mantissa bits mmmmmmmmmm. |
| // Case 1: mantissa bits have no leading zero bit, s_00000_1mmmmmmmmm |
| // In this case the value is |
| // (s == 0 ? 1 : -1) * uint(1mmmm_mmmmm) * (2^-10) * (2^-14) |
| // == (s == 0 ? 1 : -1) * ( uint(1_mmmmm_mmmm) * (2^-9)) * (2^-15) |
| // == (s == 0 ? 1 : -1) * (1 + uint(mmmmm_mmmm) * (2^-9)) * (2^-15) |
| // == (s == 0 ? 1 : -1) * (1 + uint(mmmmm_mmmm0_00000_00000_000) * (2^-23)) * (2^-15) |
| // |
| // which is equal to the value of the normal f32 number |
| // |
| // s_EEEEEEEE_mmmmm_mmmm0_00000_00000_000 |
| // |
| // where uint(EEEEEEEE) == -15 + 127 = 112. Hence we proved that any subnormal f16 number |
| // with no leading zero mantissa bit can be exactly represented by a f32 number with |
| // biased exponent 112 and the lowest 14 mantissa bits zero, and the value of any f32 |
| // number with biased exponent 112 and the lowest 14 mantissa bits zero is equal to a |
| // subnormal f16 number with no leading zero mantissa bit. |
| // |
| // Case 2: mantissa bits has 1 leading zero bit, s_00000_01mmmmmmmm |
| // In this case the value is |
| // (s == 0 ? 1 : -1) * uint(01mmm_mmmmm) * (2^-10) * (2^-14) |
| // == (s == 0 ? 1 : -1) * ( uint(01_mmmmm_mmm) * (2^-8)) * (2^-16) |
| // == (s == 0 ? 1 : -1) * (1 + uint(mmmmm_mmm) * (2^-8)) * (2^-16) |
| // == (s == 0 ? 1 : -1) * (1 + uint(mmmmm_mmm00_00000_00000_000) * (2^-23)) * (2^-16) |
| // |
| // which is equal to the value of normal f32 number |
| // |
| // s_EEEEEEEE_mmmmm_mmm00_00000_00000_000 |
| // |
| // where uint(EEEEEEEE) = -16 + 127 = 111. Hence we proved that any subnormal f16 number |
| // with 1 leading zero mantissa bit can be exactly represented by a f32 number with |
| // biased exponent 111 and the lowest 15 mantissa bits zero, and the value of any f32 |
| // number with biased exponent 111 and the lowest 15 mantissa bits zero is equal to a |
| // subnormal f16 number with 1 leading zero mantissa bit. |
| // |
| // Case 3 to case 8: ...... |
| // |
| // Case 9: mantissa bits has 8 leading zero bits, s_00000_000000001m |
| // In this case the value is |
| // (s == 0 ? 1 : -1) * uint(00000_0001m) * (2^-10) * (2^-14) |
| // == (s == 0 ? 1 : -1) * ( uint(000000001_m) * (2^-1)) * (2^-23) |
| // == (s == 0 ? 1 : -1) * (1 + uint(m) * (2^-1)) * (2^-23) |
| // == (s == 0 ? 1 : -1) * (1 + uint(m0000_00000_00000_00000_000) * (2^-23)) * (2^-23) |
| // |
| // which is equal to the value of normal f32 number |
| // |
| // s_EEEEEEEE_m0000_00000_00000_00000_000 |
| // |
| // where uint(EEEEEEEE) = -23 + 127 = 104. Hence we proved that any subnormal f16 number |
| // with 8 leading zero mantissa bit can be exactly represented by a f32 number with |
| // biased exponent 104 and the lowest 22 mantissa bits zero, and the value of any f32 |
| // number with biased exponent 104 and the lowest 22 mantissa bits zero are equal to a |
| // subnormal f16 number with 8 leading zero mantissa bit. |
| // |
| // Case 10: mantissa bits has 9 leading zero bits, s_00000_0000000001 |
| // In this case the value is just +-2^-24 == +-0x1.0p-24, |
| // the f32 number has biased exponent 103 and all 23 mantissa bits zero. |
| // |
| // Case 11: mantissa bits has 10 leading zero bits, s_00000_0000000000, just 0.0 |
| // |
| // Concluding all these case, we proved that any subnormal f16 number with N leading zero |
| // mantissa bit can be exactly represented by a f32 number with biased exponent 112 - N and the |
| // lowest 14 + N mantissa bits zero, and the value of any f32 number with biased exponent |
| // 112 - N (= e) and the lowest 14 + N (= 126 - e) mantissa bits zero are equal to a subnormal |
| // f16 number with N leading zero mantissa bits. N is in range [0, 9], so the f32 number's |
| // biased exponent e is in range [103, 112], or unbiased exponent in [-24, -15]. |
| |
| float abs_value = std::fabs(value); |
| if (abs_value >= kSmallestValue) { |
| // Value falls in the normal f16 range, quantize it to a normal f16 value by masking out |
| // lowest 13 mantissa bits. |
| u32 = u32 & ~((1u << (kF32MantissaBits - kF16MantissaBits)) - 1); |
| } else if (abs_value >= kSmallestSubnormalValue) { |
| // Value should be quantized to a subnormal f16 value. |
| |
| // Get the biased exponent `e` of f32 value, e.g. value 127 representing exponent 2^0. |
| uint32_t biased_exponent_original = (u32 & kF32ExponentMask) >> kF32MantissaBits; |
| // Since we ensure that kSmallestValue = 0x1f-14 > abs(value) >= kSmallestSubnormalValue = |
| // 0x1f-24, value will have a unbiased exponent in range -24 to -15 (inclusive), and the |
| // corresponding biased exponent in f32 is in range 103 to 112 (inclusive). |
| TINT_ASSERT((kMinF32BiasedExpForF16SubnormalNumber <= biased_exponent_original) && |
| (biased_exponent_original <= kMaxF32BiasedExpForF16SubnormalNumber)); |
| |
| // As we have proved, masking out the lowest 126-e mantissa bits of input value will result |
| // in a valid subnormal f16 value, which is exactly the required quantization result. |
| uint32_t discard_bits = 126 - biased_exponent_original; // In range 14 to 23 (inclusive) |
| TINT_ASSERT((14 <= discard_bits) && (discard_bits <= kF32MantissaBits)); |
| uint32_t discard_mask = (1u << discard_bits) - 1; |
| u32 = u32 & ~discard_mask; |
| } else { |
| // value is too small that it can't even be represented as subnormal f16 number. Quantize |
| // to zero. |
| return value > 0 ? 0.0 : -0.0; |
| } |
| |
| return tint::Bitcast<f16::type>(u32); |
| } |
| |
| uint16_t f16::BitsRepresentation() const { |
| // Assert we use binary32 (i.e. float) as underlying type, which has 4 bytes. |
| static_assert(std::is_same<f16::type, float>()); |
| |
| // The stored value in f16 object must be already quantized, so it should be either NaN, +/- |
| // Inf, or exactly representable by normal or subnormal f16. |
| |
| if (std::isnan(value)) { |
| return kF16Nan; |
| } |
| |
| if (std::isinf(value)) { |
| return value > 0 ? kF16PosInf : kF16NegInf; |
| } |
| |
| // Now quantized_value must be a finite f16 exactly-representable value. |
| // The following table shows exponent cases for all finite f16 exactly-representable value. |
| // --------------------------------------------------------------------------- |
| // | Value category | Unbiased exp | F16 biased exp | F32 biased exp | |
| // |------------------|----------------|------------------|------------------| |
| // | +/- zero | \ | 0 | 0 | |
| // | Subnormal f16 | [-24, -15] | 0 | [103, 112] | |
| // | Normal f16 | [-14, 15] | [1, 30] | [113, 142] | |
| // --------------------------------------------------------------------------- |
| |
| uint32_t f32_bit_pattern = tint::Bitcast<uint32_t>(value); |
| uint32_t f32_biased_exponent = (f32_bit_pattern & kF32ExponentMask) >> kF32MantissaBits; |
| uint32_t f32_mantissa = f32_bit_pattern & kF32MantissaMask; |
| |
| uint16_t f16_sign_part = static_cast<uint16_t>((f32_bit_pattern & kF32SignMask) >> 16); |
| TINT_ASSERT((f16_sign_part & ~kF16SignMask) == 0); |
| |
| if ((f32_bit_pattern & ~kF32SignMask) == 0) { |
| // +/- zero |
| return f16_sign_part; |
| } |
| |
| if ((kMinF32BiasedExpForF16NormalNumber <= f32_biased_exponent) && |
| (f32_biased_exponent <= kMaxF32BiasedExpForF16NormalNumber)) { |
| // Normal f16 |
| uint32_t f16_biased_exponent = f32_biased_exponent - kF32ExponentBias + kF16ExponentBias; |
| uint16_t f16_exp_part = static_cast<uint16_t>(f16_biased_exponent << kF16MantissaBits); |
| uint16_t f16_mantissa_part = |
| static_cast<uint16_t>(f32_mantissa >> (kF32MantissaBits - kF16MantissaBits)); |
| |
| TINT_ASSERT((f16_exp_part & ~kF16ExponentMask) == 0); |
| TINT_ASSERT((f16_mantissa_part & ~kF16MantissaMask) == 0); |
| |
| return f16_sign_part | f16_exp_part | f16_mantissa_part; |
| } |
| |
| if ((kMinF32BiasedExpForF16SubnormalNumber <= f32_biased_exponent) && |
| (f32_biased_exponent <= kMaxF32BiasedExpForF16SubnormalNumber)) { |
| // Subnormal f16 |
| // The resulting exp bits are always 0, and the mantissa bits should be handled specially. |
| uint16_t f16_exp_part = 0; |
| // The resulting subnormal f16 will have only 1 valid mantissa bit if the unbiased exponent |
| // of value is of the minimum, i.e. -24; and have all 10 mantissa bits valid if the unbiased |
| // exponent of value is of the maximum, i.e. -15. |
| uint32_t f16_valid_mantissa_bits = |
| f32_biased_exponent - kMinF32BiasedExpForF16SubnormalNumber + 1; |
| // The resulting f16 mantissa part comes from right-shifting the f32 mantissa bits with |
| // leading 1 added. |
| uint16_t f16_mantissa_part = |
| static_cast<uint16_t>((f32_mantissa | (kF32MantissaMask + 1)) >> |
| (kF32MantissaBits + 1 - f16_valid_mantissa_bits)); |
| |
| TINT_ASSERT((1 <= f16_valid_mantissa_bits) && |
| (f16_valid_mantissa_bits <= kF16MantissaBits)); |
| TINT_ASSERT((f16_mantissa_part & ~((1u << f16_valid_mantissa_bits) - 1)) == 0); |
| TINT_ASSERT((f16_mantissa_part != 0)); |
| |
| return f16_sign_part | f16_exp_part | f16_mantissa_part; |
| } |
| |
| // Neither zero, subnormal f16 or normal f16, shall never hit. |
| TINT_UNREACHABLE(); |
| return kF16Nan; |
| } |
| |
| // static |
| Number<tint::detail::NumberKindF16> f16::FromBits(uint16_t bits) { |
| // Assert we use binary32 (i.e. float) as underlying type, which has 4 bytes. |
| static_assert(std::is_same<f16::type, float>()); |
| |
| if (bits == kF16PosInf) { |
| return f16(std::numeric_limits<f16::type>::infinity()); |
| } |
| if (bits == kF16NegInf) { |
| return f16(-std::numeric_limits<f16::type>::infinity()); |
| } |
| |
| auto f16_sign_bit = uint32_t(bits & kF16SignMask); |
| // If none of the other bits are set we have a 0. If only the sign bit is set we have a -0. |
| if ((bits & ~kF16SignMask) == 0) { |
| return f16(f16_sign_bit > 0 ? -0.f : 0.f); |
| } |
| |
| auto f16_mantissa = uint32_t(bits & kF16MantissaMask); |
| auto f16_biased_exponent = uint32_t(bits & kF16ExponentMask); |
| |
| // F16 NaN has all expoennt bits set and at least one mantissa bit set |
| if (((f16_biased_exponent & kF16ExponentMask) == kF16ExponentMask) && f16_mantissa != 0) { |
| return f16(std::numeric_limits<f16::type>::quiet_NaN()); |
| } |
| |
| // Shift the exponent over to be a regular number. |
| f16_biased_exponent >>= kF16MantissaBits; |
| |
| // Add the F32 bias and remove the F16 bias. |
| uint32_t f32_biased_exponent = f16_biased_exponent + kF32ExponentBias - kF16ExponentBias; |
| |
| if (f16_biased_exponent == 0) { |
| // Subnormal number |
| // |
| // All subnormal F16 values can be represented as normal F32 values. Shift the mantissa and |
| // set the exponent as if this was a normal f16 value. |
| |
| // While the first F16 exponent bit is not set |
| constexpr uint32_t kF16FirstExponentBit = 0x0400; |
| while ((f16_mantissa & kF16FirstExponentBit) == 0) { |
| // Shift the mantissa to the left |
| f16_mantissa <<= 1; |
| // Decrease the biased exponent to counter the shift |
| f32_biased_exponent -= 1; |
| } |
| |
| // Remove the first exponent bit from the mantissa value |
| f16_mantissa &= ~kF16FirstExponentBit; |
| // Increase the exponent to deal with the masked off value. |
| f32_biased_exponent += 1; |
| } |
| |
| // The mantissa bits are shifted over the difference in mantissa size to be in the F32 location. |
| uint32_t f32_mantissa = f16_mantissa << (kF32MantissaBits - kF16MantissaBits); |
| |
| // Shift the exponent to the F32 exponent position before the mantissa. |
| f32_biased_exponent <<= kF32MantissaBits; |
| |
| // Shift the sign bit over to the f32 sign bit position |
| uint32_t f32_sign_bit = f16_sign_bit << 16; |
| |
| // Combine values together into the F32 value as a uint32_t. |
| uint32_t val = f32_sign_bit | f32_biased_exponent | f32_mantissa; |
| |
| // Bitcast to a F32 and then store into the F16 Number |
| return f16(tint::Bitcast<f16::type>(val)); |
| } |
| |
| } // namespace tint |
