UniformSampleCone, z

Average Accuracy: 99.9% → 99.9%

Time: 6.9s

Precision: binary32

Cost: 3360

\[\left(\left(2.328306437 \cdot 10^{-10} \leq ux \land ux \leq 1\right) \land \left(2.328306437 \cdot 10^{-10} \leq uy \land uy \leq 1\right)\right) \land \left(0 \leq maxCos \land maxCos \leq 1\right)\]

Math

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

↓

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

FPCore

(FPCore (ux uy maxCos) :precision binary32 (+ (- 1.0 ux) (* ux maxCos)))

↓

(FPCore (ux uy maxCos) :precision binary32 (fma ux (+ maxCos -1.0) 1.0))

C

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), 1.0f);
}

Julia

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

↓

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

Derivation

1. Initial program 99.9%

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

2. Simplified 99.9%

   \[\leadsto \color{blue}{\mathsf{fma}\left(ux, maxCos + -1, 1\right)} \]
   Step-by-step derivation

   [Start] 99.9	\[ \left(1 - ux\right) + ux \cdot maxCos \]
   sub-neg [=>] 99.9	\[ \color{blue}{\left(1 + \left(-ux\right)\right)} + ux \cdot maxCos \]
   associate-+l+ [=>] 99.9	\[ \color{blue}{1 + \left(\left(-ux\right) + ux \cdot maxCos\right)} \]
   +-commutative [=>] 99.9	\[ \color{blue}{\left(\left(-ux\right) + ux \cdot maxCos\right) + 1} \]
   neg-mul-1 [=>] 99.9	\[ \left(\color{blue}{-1 \cdot ux} + ux \cdot maxCos\right) + 1 \]
   *-commutative [=>] 99.9	\[ \left(-1 \cdot ux + \color{blue}{maxCos \cdot ux}\right) + 1 \]
   distribute-rgt-out [=>] 99.9	\[ \color{blue}{ux \cdot \left(-1 + maxCos\right)} + 1 \]
   fma-def [=>] 99.9	\[ \color{blue}{\mathsf{fma}\left(ux, -1 + maxCos, 1\right)} \]
   +-commutative [=>] 99.9	\[ \mathsf{fma}\left(ux, \color{blue}{maxCos + -1}, 1\right) \]

3. Final simplification 99.9%

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

Alternatives

Alternative 1
Accuracy	99.9%
Cost	224

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

Alternative 2
Accuracy	99.9%
Cost	224

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

Alternative 3
Accuracy	98.0%
Cost	96

\[1 - ux \]

Alternative 4
Accuracy	71.4%
Cost	32

\[1 \]

Reproduce

herbie shell --seed 2023157 
(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)))