# `fresx` — Floating Reciprocal Estimate Single

> **Category:** [Floating-Point](../categories/fpu.md) · **Form:** [A](../forms/A.md) · **Opcode:** `0xec000030`

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## Assembler Mnemonics

| Mnemonic | XML entry | Flags | Description |
| --- | --- | --- | --- |
| `fres` | `fresx` | — | Floating Reciprocal Estimate Single |
| `fres.` | `fresx` | Rc=1 | Floating Reciprocal Estimate Single |

## Syntax

```asm
fres[Rc] [FD], [FB]
```

## Encoding

### `fresx` — form `A`

- **Opcode word:** `0xec000030`
- **Primary opcode (bits 0–5):** `59`
- **Extended opcode:** `24`
- **Synchronising:** no

| Bits | Field | Meaning |
| --- | --- | --- |
| 0–5 | `OPCD` | primary opcode (59 or 63) |
| 6–10 | `FRT` | destination FPR |
| 11–15 | `FRA` | source A FPR |
| 16–20 | `FRB` | source B FPR |
| 21–25 | `FRC` | source C FPR (multiplier for madd-style ops) |
| 26–30 | `XO` | extended opcode (5 bits) |
| 31 | `Rc` | record-form flag (updates CR1) |

## Operands

| Field | Role | Description |
| --- | --- | --- |
| `FB` | fresx: read | Source B floating-point register. |
| `FD` | fresx: write | Destination floating-point register. |
| `CR` | fresx: write (conditional) | Condition-register update. When `Rc=1`, CR field 0 (or CR6 for vector compares, CR1 for FPU) is updated from the result. |
| `FPSCR` | fresx: write | Floating-Point Status and Control Register. |

## Register Effects

### `fresx`

- **Reads (always):** `FB`
- **Reads (conditional):** _none_
- **Writes (always):** `FD`, `FPSCR`
- **Writes (conditional):** `CR`

## Status-Register Effects

- `fresx`: **CR1** ← FPSCR[FX, FEX, VX, OX] when `Rc=1`.; **FPSCR** updated per IEEE-754 flags (FX, FEX, FPRF, FR, FI, exceptions).

## Operation (pseudocode)

```
; Pseudocode derives directly from the xenia-rs interpreter
; arm (see Implementation References). Operation semantics:
;   - Read source operands from the fields listed under Operands.
;   - Apply the arithmetic / logical / memory action described
;     in the Description field above.
;   - Write results to the destination register(s); update any
;     status bits enumerated under Status-Register Effects.
; Consult the IBM AIX reference link under IBM Reference for
; canonical PPC-style pseudocode where xenia's expression is
; terse.
```

## C Translation Example

```c
/* C translation: the xenia-rs interpreter arm below in           */
/* Implementation References is the authoritative semantic        */
/* snapshot. Translate it line-by-line:                            */
/*   - ctx.gpr[N]  -> r[N]       (or f[]/v[] for FPRs/VRs)        */
/*   - mem.read_u*/write_u* -> mem_read_u*_be / mem_write_u*_be   */
/*   - ctx.update_cr_signed(fld, v) -> update_cr_signed(fld, v)   */
/*   - ctx.xer_ca / xer_ov / xer_so -> xer.CA / xer.OV / xer.SO   */
/* The Register Effects and Status-Register Effects tables above  */
/* enumerate every side effect a faithful translation must emit.  */
```

## Implementation References

**`fresx`**
- xenia-canary XML: [`tools/ppc-instructions.xml` — search for `mnem="fresx"`](../../xenia-canary/tools/ppc-instructions.xml)
- xenia-canary emit: [`src/xenia/cpu/ppc/ppc_emit_fpu.cc:106`](../../xenia-canary/src/xenia/cpu/ppc/ppc_emit_fpu.cc#L106)
- xenia-rs opcode: [`crates/xenia-cpu/src/opcode.rs:29`](../../xenia-rs/crates/xenia-cpu/src/opcode.rs#L29)
- xenia-rs decoder: [`crates/xenia-cpu/src/decoder.rs:390`](../../xenia-rs/crates/xenia-cpu/src/decoder.rs#L390)
- xenia-rs interpreter: [`crates/xenia-cpu/src/interpreter.rs:2815-2835`](../../xenia-rs/crates/xenia-cpu/src/interpreter.rs#L2815-L2835)
<details><summary>xenia-rs interpreter body (frozen snapshot)</summary>

```rust
        PpcOpcode::fresx => {
            // Single-precision reciprocal estimate: frD = 1.0 / frB.
            // PPCBUG-184: pre-quantize input to f32 to match canary's
            // `f.Recip(f.Convert(frB, FLOAT32_TYPE))` behavior. Hardware
            // produces a ~12-bit LUT estimate; both emulators produce a
            // fully-IEEE single reciprocal, but the f32 quantization at
            // least makes the input precision match.
            let b_full = ctx.fpr[instr.rb()];
            let b = b_full as f32 as f64;
            if b == 0.0 {
                fpscr::set_exception(ctx, fpscr::ZX);
            }
            if fpscr::is_snan(b_full) {
                fpscr::set_exception(ctx, fpscr::VXSNAN);
            }
            let result = to_single(ctx, 1.0 / b);
            ctx.fpr[instr.rd()] = result;
            fpscr::update_after_op(ctx, result, b.is_finite() && b != 0.0);
            if instr.rc_bit() { update_cr1_from_fpscr(ctx); }
            ctx.pc += 4;
        }
```
</details>

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## Special Cases & Edge Conditions

- **Single-precision reciprocal estimate.** PowerISA specifies a *low-precision* approximation of `1/FRB` accurate to roughly 12–14 bits of significand, intended as the seed for a Newton-Raphson refinement step. **xenia quirk:** xenia-rs computes the *full-precision* `1.0 / b` then rounds to single, so it produces a far more accurate result than hardware. Title code that depends on the limited precision of `fres` to trigger refinement loops will still work (the loops just refine an already-correct value), but bit-exact correlation with hardware is impossible.
- **Single precision result.** Final value is rounded to binary32 then re-encoded into the FPR.
- **Divide by zero.** `1/±0` → ±∞ and sets `FPSCR[ZX, FX]`. xenia returns the host ±∞ but does not update FPSCR.
- **`fres(±∞) = ±0`** (correctly signed).
- **`fres(NaN) = NaN`**; signalling NaNs are quietened.
- **Overflow / underflow.** May set `OX`/`UX`/`XX`/`FX`. xenia does not update FPSCR.
- **`Rc=1` (`fres.`)** copies `FPSCR[FX, FEX, VX, OX]` into CR1.
- **Encoding.** A-form, primary 59, XO 24. Reads `FRB` only; `FRA`/`FRC` are don't-care.
- **Use case.** Software reciprocal: `1/d ≈ x = fres(d); x = x*(2 - d*x);` (one Newton-Raphson step recovers full single precision). Two iterations recover full double precision. The `(2 - d*x)` step compiles to `fnmsub`.
- **Performance.** Cheap on Xenon (single-cycle issue) — divides by `fres` + 1–2 NR steps + `fmul` are far faster than `fdiv`/`fdivs`.

## Related Instructions

- [`frsqrtex`](frsqrtex.md) — reciprocal-square-root estimate; same NR refinement approach.
- [`fdivx`](fdivx.md), [`fdivsx`](fdivsx.md) — true divide; alternative when refinement isn't needed.
- [`fnmsubx`](fnmsubx.md), [`fnmsubsx`](fnmsubsx.md) — the workhorse for the `(2 - d*x)` step.
- [`fmulx`](fmulx.md), [`fmulsx`](fmulsx.md) — final multiply to apply the reciprocal.
- [`fmaddsx`](fmaddsx.md) — alternate refinement formulation.

## IBM Reference

- [AIX 7.3 — `fres` (Floating Reciprocal Estimate Single)](https://www.ibm.com/docs/en/aix/7.3.0?topic=set-fres-floating-reciprocal-estimate-single-instruction)
- [PowerISA v2.07B, Book I, Chapter 4 — Floating-Point Processor](https://openpowerfoundation.org/specifications/isa/) (relative-error bound for `fres`; intended Newton-Raphson refinement pattern).