## Floating-point to Integer Conversion Overview

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

IEEE 754 doesn't 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>
    This type of conversion is that 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>
    This type of conversion is used for WebAssembly's
    [`trunc_sat_u`](https://webassembly.github.io/spec/core/exec/numerics.html#op-trunc-sat-u)
    and
    [trunc_sat_s](https://webassembly.github.io/spec/core/exec/numerics.html#op-trunc-sat-s)
    instructions, as well as several notable programming languages:

    * Java's conversion from
        [`float`/`double` to `long`/`int`](https://docs.oracle.com/javase/specs/jls/se16/html/jls-5.html#jls-5.1.3)
    * Rust's [`as` operator](https://doc.rust-lang.org/reference/expressions/operator-expr.html#semantics)
    * LLVM's [`llvm.fptosi.sat`](https://llvm.org/docs/LangRef.html#llvm-fptosi-sat-intrinsic) and
        [`llvm.fptoui.sat`](https://llvm.org/docs/LangRef.html#llvm-fptoui-sat-intrinsic) intrinsics
    * SPIR-V's OpenCL dialect's
        [`OpConvertFToU`](https://www.khronos.org/registry/spir-v/specs/unified1/SPIRV.html#OpConvertFToU) and
        [`OpConvertFToS`](https://www.khronos.org/registry/spir-v/specs/unified1/SPIRV.html#OpConvertFToS)
        instructions when decorated with
        [the `SaturatedConversion` decorator](https://www.khronos.org/registry/spir-v/specs/unified1/SPIRV.html#_a_id_decoration_a_decoration).

* E-Type: <div id="fpr-to-gpr-conversion-e-type"></div>
    This type of conversion is used for
    [ECMAScript's `ToInt32` abstract operation](https://262.ecma-international.org/14.0/#sec-toint32).

    This type of conversion is also implemented in ARMv8.3A as the `FJCVTZS`
    instruction.
    <https://developer.arm.com/documentation/dui0801/g/hko1477562192868>

### Floating-point to Integer Conversion Semantics Summary

Let `rounded` 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:

```
+--------+------------+-----------------------------------------------------------------------+
| Kind   | Result's   | Category of rounded                                                   |
|        | Signedness +-----------+-----------+-----------+-----------+-----------+-----------+
|        |            | NaN       | +Infinity | -Infinity | > Maximum | < Minimum | Otherwise |
|        |            |           |           |           | Possible  | Possible  |           |
|        |            |           |           |           | Result    | Result    |           |
+--------+------------+-----------+-----------+-----------+-----------+-----------+-----------+
| P-Type | Unsigned   | 0         | 2^w - 1   | 0         | 2^w - 1   | 0         | rounded   |
|        +------------+-----------+-----------+-----------+-----------+-----------+-----------+
|        | Signed     | -2^(w-1)  | 2^(w-1)-1 | -2^(w-1)  | 2^(w-1)-1 | -2^(w-1)  | rounded   |
+--------+------------+-----------+-----------+-----------+-----------+-----------+-----------+
| S-Type | Unsigned   | 0         | 2^w - 1   | 0         | 2^w - 1   | 0         | rounded   |
|        +------------+-----------+-----------+-----------+-----------+-----------+-----------+
|        | Signed     | 0         | 2^(w-1)-1 | -2^(w-1)  | 2^(w-1)-1 | -2^(w-1)  | rounded   |
+--------+------------+-----------+-----------+-----------+-----------+-----------+-----------+
| E-Type | Either     | 0                                 | rounded & (2^w - 1)               |
+--------+------------+-----------------------------------+-----------------------------------+
```