## Floating-point to Integer Conversion Overview

<div id="fpr-to-gpr-conversion-mode"></div>

IEEE 754 does not specify what results are obtained when converting a NaN
or out-of-range floating-point value to integer, so different programming
languages and ISAs have made different choices.  The different conversion
modes supported by the `cffpr` instruction are as follows:

* P-Type: <div id="fpr-to-gpr-conversion-p-type"></div>
    Used by most other PowerISA instructions,
    as well as commonly used floating-point to integer conversions on x86.

* S-Type: <div id="fpr-to-gpr-conversion-s-type"></div>
    Used for several notable programming languages:

    * Java's conversion from `float`/`double` to `long`/`int`[^java_fp_to_int]
    * Rust's `as` operator[^rust_as_operator]
    * LLVM's `llvm.fptosi.sat`[^llvm_fptosi_sat] and
        `llvm.fptoui.sat`[^llvm_fptoui_sat] intrinsics
    * SPIR-V's OpenCL dialect's `OpConvertFToU`[^SPIRV_OpConvertFToU] and
        `OpConvertFToS`[^SPIRV_OpConvertFToS] instructions when decorated with
        the `SaturatedConversion`[^SPIRV_SaturatedConversion] decorator.
    * Also WebAssembly's `trunc_sat_u`[^trunc_sat_u] and
        `trunc_sat_s`[^trunc_sat_s] instructions,


* E-Type: <div id="fpr-to-gpr-conversion-e-type"></div>
    Used for ECMAScript's `ToInt32` abstract operation[^ECMAScript_ToInt32].
    Also implemented in ARMv8.3A as the `FJCVTZS` instruction[^ARM_FJCVTZS].

[^trunc_sat_u]: WASM's `trunc_sat_u`:
    <https://webassembly.github.io/spec/core/exec/numerics.html#op-trunc-sat-u>
[^trunc_sat_s]: WASM's `trunc_sat_s`:
    <https://webassembly.github.io/spec/core/exec/numerics.html#op-trunc-sat-s>
[^java_fp_to_int]: Java `float`/`double` to `long`/`int` conversion:
    <https://docs.oracle.com/javase/specs/jls/se16/html/jls-5.html#jls-5.1.3>
[^rust_as_operator]: Rust's `as` operator:
    <https://doc.rust-lang.org/1.70.0/reference/expressions/operator-expr.html#numeric-cast>
[^llvm_fptosi_sat]: LLVM's `llvm.fptosi.sat` intrinsic:
    <https://llvm.org/docs/LangRef.html#llvm-fptosi-sat-intrinsic>
[^llvm_fptoui_sat]: LLVM's `llvm.fptoui.sat` intrinsic:
    <https://llvm.org/docs/LangRef.html#llvm-fptoui-sat-intrinsic>
[^SPIRV_OpConvertFToU]: SPIR-V's `OpConvertFToU` instruction:
    <https://www.khronos.org/registry/spir-v/specs/unified1/SPIRV.html#OpConvertFToU>
[^SPIRV_OpConvertFToS]: SPIR-V's `OpConvertFToS` instruction:
    <https://www.khronos.org/registry/spir-v/specs/unified1/SPIRV.html#OpConvertFToS>
[^SPIRV_SaturatedConversion]: SPIR-V's `SaturatedConversion` decorator:<br/>
    <https://www.khronos.org/registry/spir-v/specs/unified1/SPIRV.html#_a_id_decoration_a_decoration>
[^ECMAScript_ToInt32]: ECMAScript's `ToInt32` abstract operation:
    <https://262.ecma-international.org/14.0/#sec-toint32>
[^ARM_FJCVTZS]: ARM's `FJCVTZS` instruction:
    <https://developer.arm.com/documentation/dui0801/g/hko1477562192868>

### Floating-point to Integer Conversion Semantics Summary

Let `round` be the result of `bfp_ROUND_TO_INTEGER(rmode, input)`.
Let `w` be the number of bits in the result's type.
The result of Floating-point to Integer conversion is as follows:

```
+------+------+---------------------------------------------------------------+
|Type| Result | Category of rounding                                          |
|    | Sign   +----------+-----------+----------+-----------+---------+-------+
|    |        | NaN      | +Inf      | -Inf     | > Max     | < Min   | Else  |
|    |        |          |           |          | Possible  | Possible|       |
|    |        |          |           |          | Result    | Result  |       |
+----+--------+----------+-----------+----------+-----------+---------+-------+
|  P |Unsigned| 0        | 2^w - 1   | 0        | 2^w - 1   | 0       | round |
|    +--------+----------+-----------+----------+-----------+---------+-------+
|    | Signed | -2^(w-1) | 2^(w-1)-1 | -2^(w-1) | 2^(w-1)-1 | -2^(w-1)| round |
+----+--------+----------+-----------+----------+-----------+---------+-------+
|  S |Unsigned| 0        | 2^w - 1   | 0        | 2^w - 1   | 0       | round |
|    +--------+----------+-----------+----------+-----------+---------+-------+
|    | Signed | 0        | 2^(w-1)-1 | -2^(w-1) | 2^(w-1)-1 | -2^(w-1)| round |
+----+--------+----------+-----------+----------+-----------+---------+-------+
|  E | Either | 0                               | round & (2^w - 1)           |
+----+--------+---------------------------------+-----------------------------+
```