D993 - A fused multiply-add rounds once

Status: decided Date: 2026-10-25 known_by: published

Every multiply-add the instruction set defines as fused is translated as one rounding of a * b + c. Every one it defines as unfused rounds the product first. The host evaluator computes both the same way, so a trace stays an exact oracle of either (D991).

Which is which (AMD's published RDNA 2 instruction set, the VOP2, VOP3 and VOP3P opcodes):

• Fused, in single precision: V_FMA_F32, V_FMAC_F32, V_FMAAK_F32, V_FMAMK_F32, V_DIV_FMAS_F32 and V_FMA_MIX_F32. They are emitted as GLSL.std.450 Fma (Model::f32_fma). Before this, they were a multiply and an add the driver could fuse or not.
• Fused, in half precision: V_FMA_F16, V_FMAC_F16, V_PK_FMA_F16, V_PK_FMAC_F16, and V_FMA_MIXLO_F16 and V_FMA_MIXHI_F16 where every source is a half. These round once to the half they write (fused_to_half). The product of two halves is exact in single precision, so the sum is the only rounding. Its error is found exactly (TwoSum), and an inexact sum whose last bit is even is moved one step toward the exact value: it is rounded to odd. Rounding to odd at 24 bits and then to nearest at 11 is the one rounding to nearest at 11 (Boldo and Melquiond, IEEE Trans. Computers 57(4), 2008). Before this, the sum was rounded to nearest twice, which can round a tie the wrong way.
• A mix narrowing to a half with a single-precision source rounds once to single precision, then to the half.
• Unfused: V_MAD_F32, V_MAC_F32, V_MADAK_F32 and V_MADMK_F32. The product and the sum each round, denormals flushed (legacy_multiply_add, unchanged). ACO selects these for an fmul and an fadd kept apart, and the fused opcodes for ffma.
• V_FMA_F64 is not translated.

The host evaluator. It computes Fma with the host's own fused multiply-add in every run, not only in a trace: the operation is exact on every device, so a run that forms addresses may rely on it.

The test. fused_multiply_add.rs runs each form on the device and on the host, on operands where fused and unfused differ, against values worked out by hand:

• In single precision, a = b = 1 + 2^-12 and c = -(1 + 2^-11). The product 1 + 2^-11 + 2^-24 rounds to 1 + 2^-11, a tie to even. Fused, the result is 2^-24; unfused, zero.
• In half precision, a = 2^-11 (1 + 2^-10), b = 1 - 2^-10 and c = 1 + 2^-10. The exact result 1 + 3 * 2^-11 - 2^-31 lies just below the tie between 0x3c01 and 0x3c02. Rounded once it is 0x3c01; rounded first to single precision it lands on the tie, which rounds to 0x3c02.

Rejected:

• Decorating the multiply and the add NoContraction for the fused forms: that rounds twice, as the unfused forms do.
• Computing the half forms in single precision and narrowing: the double rounding above.