Average Error: 0.0 → 0.0
Time: 5.7s
Precision: binary32
\[\left(\left(2.328306437 \cdot 10^{-10} \leq ux \land ux \leq 1\right) \land \left(2.328306437 \cdot 10^{-10} \leq uy \land uy \leq 1\right)\right) \land \left(0 \leq maxCos \land maxCos \leq 1\right)\]
\[\left(1 - ux\right) + ux \cdot maxCos \]
↓
\[\mathsf{fma}\left(ux, maxCos, 1 - ux\right) \]
\left(1 - ux\right) + ux \cdot maxCos
↓
\mathsf{fma}\left(ux, maxCos, 1 - ux\right)

(FPCore (ux uy maxCos) :precision binary32 (+ (- 1.0 ux) (* ux maxCos)))
↓
(FPCore (ux uy maxCos) :precision binary32 (fma ux maxCos (- 1.0 ux)))
float code(float ux, float uy, float maxCos) {
	return (1.0f - ux) + (ux * maxCos);
}
↓
float code(float ux, float uy, float maxCos) {
	return fmaf(ux, maxCos, (1.0f - ux));
}

Error

Bits error versus ux

Bits error versus uy

Bits error versus maxCos

Derivation

  1. Initial program 0.0

    \[\left(1 - ux\right) + ux \cdot maxCos \]

  2. Simplified0.0

    \[\leadsto \color{blue}{\mathsf{fma}\left(ux, maxCos, 1 - ux\right)} \]

  3. Final simplification0.0

    \[\leadsto \mathsf{fma}\left(ux, maxCos, 1 - ux\right) \]

Reproduce

herbie shell --seed 2022130 
(FPCore (ux uy maxCos)
  :name "UniformSampleCone, z"
  :precision binary32
  :pre (and (and (and (<= 2.328306437e-10 ux) (<= ux 1.0)) (and (<= 2.328306437e-10 uy) (<= uy 1.0))) (and (<= 0.0 maxCos) (<= maxCos 1.0)))
  (+ (- 1.0 ux) (* ux maxCos)))