# `mullwx` — Multiply Low Word

> **Category:** [Integer ALU](../categories/alu.md) · **Form:** [XO](../forms/XO.md) · **Opcode:** `0x7c0001d6`

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## Assembler Mnemonics

| Mnemonic | XML entry | Flags | Description |
| --- | --- | --- | --- |
| `mullw` | `mullwx` | — | Multiply Low Word |
| `mullwo` | `mullwx` | OE=1 | Multiply Low Word |
| `mullw.` | `mullwx` | Rc=1 | Multiply Low Word |
| `mullwo.` | `mullwx` | OE=1, Rc=1 | Multiply Low Word |

## Syntax

```asm
mullw[OE][Rc] [RD], [RA], [RB]
```

## Encoding

### `mullwx` — form `XO`

- **Opcode word:** `0x7c0001d6`
- **Primary opcode (bits 0–5):** `31`
- **Extended opcode:** `235`
- **Synchronising:** no

| Bits | Field | Meaning |
| --- | --- | --- |
| 0–5 | `OPCD` | primary opcode (31) |
| 6–10 | `RT` | destination GPR |
| 11–15 | `RA` | source A |
| 16–20 | `RB` | source B |
| 21 | `OE` | overflow-enable flag |
| 22–30 | `XO` | extended opcode (9 bits) |
| 31 | `Rc` | record-form flag |

## Operands

| Field | Role | Description |
| --- | --- | --- |
| `RA` | mullwx: read | Source GPR (`r0`–`r31`). |
| `RB` | mullwx: read | Source GPR. |
| `RD` | mullwx: write | Destination GPR. |
| `CR` | mullwx: write (conditional) | Condition-register update. When `Rc=1`, CR field 0 (or CR6 for vector compares, CR1 for FPU) is updated from the result. |
| `OE` | mullwx: write (conditional) | Overflow-enable bit. When 1, the instruction updates `XER[OV]` and stickies `XER[SO]` on signed overflow. |

## Register Effects

### `mullwx`

- **Reads (always):** `RA`, `RB`
- **Reads (conditional):** _none_
- **Writes (always):** `RD`
- **Writes (conditional):** `CR`, `OE`

## Status-Register Effects

- `mullwx`: **CR0** ← signed-compare(result, 0) with `SO ← XER[SO]`, when `Rc=1`.; **XER[OV]** ← signed-overflow(result); **XER[SO]** stickies, when `OE=1`.

## Operation (pseudocode)

```
RT <- ((RA)[32:63]) * ((RB)[32:63])    ; signed 32×32 → 64
```

## 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

**`mullwx`**
- xenia-canary XML: [`tools/ppc-instructions.xml` — search for `mnem="mullwx"`](../../xenia-canary/tools/ppc-instructions.xml)
- xenia-canary emit: [`src/xenia/cpu/ppc/ppc_emit_alu.cc:390`](../../xenia-canary/src/xenia/cpu/ppc/ppc_emit_alu.cc#L390)
- xenia-rs opcode: [`crates/xenia-cpu/src/opcode.rs:57`](../../xenia-rs/crates/xenia-cpu/src/opcode.rs#L57)
- xenia-rs decoder: [`crates/xenia-cpu/src/decoder.rs:874`](../../xenia-rs/crates/xenia-cpu/src/decoder.rs#L874)
- xenia-rs interpreter: [`crates/xenia-cpu/src/interpreter.rs:357-371`](../../xenia-rs/crates/xenia-cpu/src/interpreter.rs#L357-L371)
<details><summary>xenia-rs interpreter body (frozen snapshot)</summary>

```rust
        PpcOpcode::mullwx => {
            // PPCBUG-009: 32-bit ABI. Truncate product to u32 — overflow detection
            // (mullw_ov) still uses the full i64 product to catch the overflow.
            let ra = ctx.gpr[instr.ra()] as i32 as i64;
            let rb = ctx.gpr[instr.rb()] as i32 as i64;
            let product = ra.wrapping_mul(rb);
            ctx.gpr[instr.rd()] = product as u32 as u64;
            if instr.oe() {
                overflow::apply(ctx, overflow::mullw_ov(product));
            }
            if instr.rc_bit() {
                ctx.update_cr_signed(0, ctx.gpr[instr.rd()] as u32 as i32 as i64);
            }
            ctx.pc += 4;
        }
```
</details>

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## Extended Pseudocode

```
prod64 <- sign_extend_32_to_64((RA)[32:63]) *s sign_extend_32_to_64((RB)[32:63])
RT <- prod64                                         ; 64-bit result
if OE then
    XER[OV] <- (prod64 ≠ sign_extend_32_to_64(prod64[32:63]))  ; set when product doesn't fit in 32 bits
    XER[SO] <- XER[SO] | XER[OV]
if Rc then
    CR0 <- signed_compare(RT, 0) || XER[SO]
```

## Special Cases & Edge Conditions

- **Inputs are the low 32 bits.** `mullw` only looks at `RA[32:63]` and `RB[32:63]`; the high 32 bits of each source are ignored. This is a 32-bit × 32-bit → 64-bit signed multiply. For full 64-bit operands use [`mulldx`](mulldx.md).
- **Result is sign-extended to 64 bits.** The 64-bit product fits into a 64-bit GPR without loss. Subsequent 32-bit consumers see `RT[32:63]` (the low 32 bits of the product); use [`mulhwx`](mulhwx.md) for the signed high 32 bits or [`mulhwux`](mulhwux.md) for the unsigned high 32 bits, computed in parallel without this instruction.
- **`OE` overflow test is 32-bit.** `XER[OV]` is set iff the 64-bit signed product cannot be represented in 32 bits — equivalently, iff `RT[32] ≠ RT[33] = … = RT[63]` (sign bit disagrees with the next 32 bits). Xenia-rs does **not** implement this; `OE` on `mullwo` is a no-op in the interpreter.
- **Xenia-rs CR0 update bug footprint.** The interpreter computes CR0 from `result as i32 as i64` — the low 32 bits sign-extended. For a 32×32→64 multiply the high 32 bits may be non-zero even when the low 32 bits are zero, so xenia's CR0 can differ from the spec's (which compares the full 64-bit product to zero). In practice this matters only for code that relies on `mullw.` to detect overflow via CR0 — extremely rare.
- **Latency.** On the Xenon, `mullw` has higher latency than add/sub; many hot inner loops avoid it by strength-reduction or shift-add chains. This is irrelevant for correctness but sometimes explains surprising instruction sequences in disassembly.

## Related Instructions

- [`mulhwx`](mulhwx.md) — signed high 32 bits of the same 32×32 product.
- [`mulhwux`](mulhwux.md) — unsigned high 32 bits of a 32×32 product.
- [`mulli`](mulli.md) — D-form: `RT ← (RA[32:63]) × SIMM` (low 64 bits, signed).
- [`mulldx`](mulldx.md), [`mulhdx`](mulhdx.md), [`mulhdux`](mulhdux.md) — 64-bit multiplies (low/high, signed/unsigned).
- [`divwx`](divwx.md), [`divwux`](divwux.md) — 32-bit signed / unsigned division.

## IBM Reference

- [AIX 7.3 — `mullw` (Multiply Low Word)](https://www.ibm.com/docs/en/aix/7.3.0?topic=set-mullw-multiply-low-word-instruction)
- [AIX 7.3 — `mulli` (Multiply Low Immediate)](https://www.ibm.com/docs/en/aix/7.3.0?topic=set-mulli-multiply-low-immediate-instruction)