Average Error: 0.0 → 0.0
Time: 3.8s
Precision: binary32
\[\left(\left(2.328306437 \cdot 10^{-10} \leq ux \land ux \leq 1\right) \land \left(2.328306437 \cdot 10^{-10} \leq uy \land uy \leq 1\right)\right) \land \left(0 \leq maxCos \land maxCos \leq 1\right)\]
\[\left(1 - ux\right) + ux \cdot maxCos \]
↓
\[\mathsf{fma}\left(maxCos \cdot \sqrt{ux}, \sqrt{ux}, 1 - ux\right) \]
(FPCore (ux uy maxCos) :precision binary32 (+ (- 1.0 ux) (* ux maxCos)))
↓
(FPCore (ux uy maxCos)
 :precision binary32
 (fma (* maxCos (sqrt ux)) (sqrt ux) (- 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((maxCos * sqrtf(ux)), sqrtf(ux), (1.0f - ux));
}

function code(ux, uy, maxCos)
	return Float32(Float32(Float32(1.0) - ux) + Float32(ux * maxCos))
end
↓
function code(ux, uy, maxCos)
	return fma(Float32(maxCos * sqrt(ux)), sqrt(ux), Float32(Float32(1.0) - ux))
end

\left(1 - ux\right) + ux \cdot maxCos
↓
\mathsf{fma}\left(maxCos \cdot \sqrt{ux}, \sqrt{ux}, 1 - ux\right)

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. Taylor expanded in ux around 0 0.1

    \[\leadsto \color{blue}{\left(1 + maxCos \cdot ux\right) - ux} \]

  4. Simplified0.1

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

  5. Applied egg-rr0.0

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

  6. Final simplification0.0

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

Reproduce

herbie shell --seed 2022133 
(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)))