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+# RFC: Float8E4M3 and Float8E3M4
+
+Status: In Review<br/>
+Initial version: 8/8/2024<br/>
+Last updated: 8/9/2024<br/>
+Discussion thread: [PR-2486](https://github.com/openxla/stablehlo/pull/2486)
+[RFC] Add f8E4M3 and f8E3M4 types support
+
+## Summary
+
+Amazon has proposed two new FP8 types, Float8E4M3 and Float8E3M4. These
+types are implemented in commercially available hardware[^1], and added to MLIR
+builtin types[^2]˒[^3] and LLVM APFloat[^4]˒[^5].
+
+Both Float8E4M3 and Float8E3M4 follows IEEE 754 convention similar to existing
+type Float8E5M2.
+
+### Float8E4M3
+
+8-bit floating point type with 1 sign bit, 4 bits exponent and 3 bits mantissa
+following IEEE-754 conventions with bit layout S1E4M3.
+
+```c
+f8E4M3 (IEEE 754)
+- Exponent bias: 7
+- Minimum stored exponent value: 1 (binary 0001)
+- Maximum stored exponent value: 14 (binary 1110)
+- Minimum unbiased exponent value: 1 − 7 = −6
+- Maximum unbiased exponent value: 14 - 7 = 7
+- Precision specifies the total number of bits used for the significand
+    (mantisa), including implicit leading integer bit = 3 + 1 = 4
+- Follows IEEE 754 conventions for representation of special values
+- Has Positive and Negative zero
+- Has Positive and Negative infinity
+- Has NaNs
+
+Additional details:
+- Min exp (unbiased): -6
+- Max exp (unbiased): 7
+- Infinities (+/-): S.1111.000
+- Zeros (+/-): S.0000.000
+- NaNs: S.1111.{001, 010, 011, 100, 101, 110, 111}
+- Min normal number: S.0001.000 = +/-2^(1 - 7) x (1 + 0) = +/-2^(-6)
+- Max normal number: S.1110.111 = +/-2^(14 - 7) x (1 + 7/8) = +/-240
+- Min subnormal number: S.0000.001 = +/-2^(-6) x 1/8 = +/-2^(-9)
+- Max subnormal number: S.0000.111 = +/-2^(-6) x 7/8 = +/-2^(-9) x 7
+```
+
+#### Comparison of Float8E4M3FN and Float8E4M3
+
+|                   |Float8E4M3FN                                                            |Float8E4M3                                                               |
+|-------------------|------------------------------------------------------------------------|-------------------------------------------------------------------------|
+|Bias               |7                                                                       |7                                                                        |
+|Min Normal Value   |`0bS0001000` = -1<sup>S</sup> $\times$ 1.0 $\times$ 2<sup>-6</sup>      |`0bS0001000` = -1<sup>S</sup> $\times$ 1.0 $\times$ 2<sup>-6</sup>       |
+|Max Normal Value   |`0bS1111110` = -1<sup>S</sup> $\times$ 1.75 $\times$ 2<sup>8</sup> = 448|`0bS1110111` = -1<sup>S</sup> $\times$ 1.875 $\times$ 2<sup>7</sup> = 240|
+|Min Subnormal Value|`0bS0000001` = -1<sup>S</sup> $\times$ 0.125 $\times$ 2<sup>-6</sup>    |`0bS0000001` = -1<sup>S</sup> $\times$ 0.125 $\times$ 2<sup>-6</sup>     |
+|Max Subnormal Value|`0bS0000111` = -1<sup>S</sup> $\times$ 0.875 $\times$ 2<sup>-6</sup>    |`0bS0000111` = -1<sup>S</sup> $\times$ 0.875 $\times$ 2<sup>-6</sup>     |
+|NaN                |`0bS1111111`                                                            |`0bS1111MMM`, where `MMM` is non-zero.                                   |
+|Infinity           |N/A                                                                     |`0bS1111000`                                                             |
+|-0.0               |`0b10000000`                                                            |`0b10000000`                                                             |
+
+### Float8E3M4
+
+8-bit floating point type with 1 sign bit, 3 bits exponent and 4 bits mantissa
+following IEEE-754 conventions with bit layout S1E3M4.
+
+```c
+f8E3M4 (IEEE 754)
+- Exponent bias: 3
+- Minimum stored exponent value: 1 (binary 001)
+- Maximum stored exponent value: 6 (binary 110)
+- Minimum unbiased exponent value: 1 − 3 = −2
+- Maximum unbiased exponent value: 6 - 3 = 3
+- Precision specifies the total number of bits used for the significand
+    (mantissa), including implicit leading integer bit = 4 + 1 = 5
+- Follows IEEE 754 conventions for representation of special values
+- Has Positive and Negative zero
+- Has Positive and Negative infinity
+- Has NaNs
+
+Additional details:
+- Min exp (unbiased): -2
+- Max exp (unbiased): 3
+- Infinities (+/-): S.111.0000
+- Zeros (+/-): S.000.0000
+- NaNs: S.111.{0,1}⁴ except S.111.0000
+- Min normal number: S.001.0000 = +/-2^(1 - 3) x (1 + 0) = +/-0.25
+- Max normal number: S.110.1111 = +/-2^(6 - 3) x (1 + 15/16) = +/-15.5
+- Min subnormal number: S.000.0001 = +/-2^(-2) x 1/16 = +/-2^(-6)
+- Max subnormal number: S.000.1111 = +/-2^(-2) x 15/16 = +/-2^(-6) x 15
+```
+
+### Comparison of Float8E5M2, Float8E4M3 and Float8E3M4
+
+|                   |Float8E5M2                                                                  |Float8E4M3                                                               |Float8E3M4                                                                 |
+|-------------------|----------------------------------------------------------------------------|-------------------------------------------------------------------------|---------------------------------------------------------------------------|
+|Bias               |15                                                                          |7                                                                        |3                                                                          |
+|Min Normal Value   |`0bS0000100` = -1<sup>S</sup> $\times$ 1.0 $\times$ 2<sup>-14</sup>         |`0bS0001000` = -1<sup>S</sup> $\times$ 1.0 $\times$ 2<sup>-6</sup>       |`0bS0010000` = -1<sup>S</sup> $\times$ 1.0 $\times$ 2<sup>-2</sup>         |
+|Max Normal Value   |`0bS1111011` = -1<sup>S</sup> $\times$ 1.75 $\times$ 2<sup>15</sup> = 57344 |`0bS1110111` = -1<sup>S</sup> $\times$ 1.875 $\times$ 2<sup>7</sup> = 240|`0bS1101111` = -1<sup>S</sup> $\times$ 1.9375 $\times$ 2<sup>3</sup> = 15.5|
+|Min Subnormal Value|`0bS0000001` = -1<sup>S</sup> $\times$ 0.25 $\times$ 2<sup>-14</sup>        |`0bS0000001` = -1<sup>S</sup> $\times$ 0.125 $\times$ 2<sup>-6</sup>     |`0bS0000001` = -1<sup>S</sup> $\times$ 0.0625 $\times$ 2<sup>-2</sup>      |
+|Max Subnormal Value|`0bS0000011` = -1<sup>S</sup> $\times$ 0.75 $\times$ 2<sup>-14</sup>        |`0bS0000111` = -1<sup>S</sup> $\times$ 0.875 $\times$ 2<sup>-6</sup>     |`0bS0001111` = -1<sup>S</sup> $\times$ 0.9375 $\times$ 2<sup>-2</sup>      |
+|NaN                |`0bS11111MM`, where `MM` is non-zero.                                       |`0bS1111MMM`, where `MMM` is non-zero.                                   |`0bS111MMMM`, where `MMMM` is non-zero.                                    |
+|Infinity           |`0bS1111100`                                                                |`0bS1111000`                                                             |`0bS1110000`                                                               |
+|-0.0               |`0b10000000`                                                                |`0b10000000`                                                             |`0b10000000`                                                               |
+
+## Changes in StableHLO
+
+I propose adding Float8E4M3 and Float8E3M4 types to StableHLO similar to the
+previously introduces FP8 types (below) with some differences:
+
+- [FP8 RFC](https://github.com/openxla/xla/discussions/22)
+- [[RFC] Add Float8E4M3FNUZ and Float8E5M2FNUZ to StableHLO](https://github.com/openxla/stablehlo/pull/1342)
+
+### StableHLO Interpreter
+
+To provide a reference implementation, I intend to add support for
+Float8E4M3 and Float8E3M4 in the StableHLO interpreter. This will be
+useful for testing other backends and validating new implementations. This will
+be achieved in two ways:
+
+1. Map directly to the appropriate APFloat operation.
+2. Cast up to the appropriate type, use that implementation, cast back down.
+
+### Float8E4M3 and Float8E3M4 Arithmetic
+
+I intend for Float8E4M3 and Float8E3M4 to be types that support the
+appropriate arithmetic operations, like any other floating point type. For
+platforms that don't have hardware support for these types, they may either
+throw an error and reject the program or cast up to an appropriate higher
+precision type that is supported, compute the answer, and cast back down.
+
+This is a simple approach that aligns with user expectations of a floating
+point data type, and is the approach taken by BFloat16. This also gives
+backends freedom to exploit any hardware support.
+
+Here's an example of a real JAX program (logging the MLIR) computing a simple
+dot product in Float8E4M3. Note the answer is slightly "wrong", as expected
+due to the lower precision (round-to-nearest).
+
+```python
+>>> import jax
+>>> import jax.numpy as jnp
+>>> x = jnp.arange(8, dtype=jnp.float8_e4m3)
+module @jit_iota {
+  func.func public @main() -> tensor<8xf8E4M3> {
+    %0 = stablehlo.iota dim = 0 : tensor<8xf8E4M3>
+    return %0 : tensor<8xf8E4M3>
+  }
+}
+>>> x
+Array([0, 1, 2, 3, 4, 5, 6, 7], dtype=float8_e4m3)
+>>> x @ x
+module @jit_matmul {
+  func.func public @main(%arg0: tensor<8xf8E4M3> {mhlo.sharding = ""}, %arg1: tensor<8xf8E4M3> {mhlo.sharding = ""}) -> tensor<f8E4M3> {
+    %0 = "stablehlo.dot_general"(%arg0, %arg1) {dot_dimension_numbers = #stablehlo.dot<lhs_contracting_dimensions = [0], rhs_contracting_dimensions = [0]>, precision_config = [#stablehlo<precision DEFAULT>, #stablehlo<precision DEFAULT>]} : (tensor<8xf8E4M3>, tensor<8xf8E4M3>) -> tensor<f8E4M3>
+    return %0 : tensor<f8E4M3>
+  }
+}
+Array(144, dtype=float8_e4m3)
+```
+
+### Testing
+
+Built on the StableHLO interpreter, I intend to introduce tests for all
+possible operations with Float8E4M3 and Float8E3M4 inputs. This will at
+a minimum mean adding additional cases to the `interpret_X.mlir` family of
+tests.
+
+### References and Links
+
+- [RFC: FP8 in StableHLO](https://github.com/openxla/stablehlo/blob/main/rfcs/20221031-fp8.md)
+- [RFC: Float8E4M3FNUZ and Float8E5M2FNUZ](https://github.com/openxla/stablehlo/blob/main/rfcs/20230321-fp8_fnuz.md)
+
+[^1]: [Amazon EC2 Trn1 Instances](https://aws.amazon.com/ec2/instance-types/trn1/)
+[^2]: LLVM [PR-97118](https://github.com/llvm/llvm-project/pull/97118) [MLIR] Add f8E4M3 IEEE 754 type (Merged)
+[^3]: LLVM [PR-101230](https://github.com/llvm/llvm-project/pull/101230) [MLIR] Add f8E3M4 IEEE 754 type (Merged)
+[^4]: LLVM [PR-97179](https://github.com/llvm/llvm-project/pull/97179) [APFloat] Add support for f8E4M3 IEEE 754 type (Merged)
+[^5]: LLVM [PR-99698](https://github.com/llvm/llvm-project/pull/99698) [APFloat] Add support for f8E3M4 IEEE 754 type (Merged)