Average Error: 13.5 → 0.5
Time: 17.3s
Precision: binary32
Cost: 19776
\[\left(\left(2.328306437 \cdot 10^{-10} \leq ux \land ux \leq 1\right) \land \left(2.328306437 \cdot 10^{-10} \leq uy \land uy \leq 1\right)\right) \land \left(0 \leq maxCos \land maxCos \leq 1\right)\]
\[\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
↓
\[\sqrt[3]{{\sin \left(uy \cdot \left(2 \cdot \pi\right)\right)}^{3} \cdot {\left(\left(ux - ux \cdot maxCos\right) \cdot \mathsf{fma}\left(ux, maxCos, 2 - ux\right)\right)}^{1.5}} \]
(FPCore (ux uy maxCos)
 :precision binary32
 (*
  (sin (* (* uy 2.0) PI))
  (sqrt
   (- 1.0 (* (+ (- 1.0 ux) (* ux maxCos)) (+ (- 1.0 ux) (* ux maxCos)))))))
↓
(FPCore (ux uy maxCos)
 :precision binary32
 (cbrt
  (*
   (pow (sin (* uy (* 2.0 PI))) 3.0)
   (pow (* (- ux (* ux maxCos)) (fma ux maxCos (- 2.0 ux))) 1.5))))
float code(float ux, float uy, float maxCos) {
	return sinf(((uy * 2.0f) * ((float) M_PI))) * sqrtf((1.0f - (((1.0f - ux) + (ux * maxCos)) * ((1.0f - ux) + (ux * maxCos)))));
}
↓
float code(float ux, float uy, float maxCos) {
	return cbrtf((powf(sinf((uy * (2.0f * ((float) M_PI)))), 3.0f) * powf(((ux - (ux * maxCos)) * fmaf(ux, maxCos, (2.0f - ux))), 1.5f)));
}

function code(ux, uy, maxCos)
	return Float32(sin(Float32(Float32(uy * Float32(2.0)) * Float32(pi))) * sqrt(Float32(Float32(1.0) - Float32(Float32(Float32(Float32(1.0) - ux) + Float32(ux * maxCos)) * Float32(Float32(Float32(1.0) - ux) + Float32(ux * maxCos))))))
end
↓
function code(ux, uy, maxCos)
	return cbrt(Float32((sin(Float32(uy * Float32(Float32(2.0) * Float32(pi)))) ^ Float32(3.0)) * (Float32(Float32(ux - Float32(ux * maxCos)) * fma(ux, maxCos, Float32(Float32(2.0) - ux))) ^ Float32(1.5))))
end

\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)}
↓
\sqrt[3]{{\sin \left(uy \cdot \left(2 \cdot \pi\right)\right)}^{3} \cdot {\left(\left(ux - ux \cdot maxCos\right) \cdot \mathsf{fma}\left(ux, maxCos, 2 - ux\right)\right)}^{1.5}}

Error

Derivation

  1. Initial program 13.5

    \[\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]

  2. Simplified13.6

    \[\leadsto \color{blue}{\sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(ux, maxCos, 1\right) - ux, ux - \mathsf{fma}\left(ux, maxCos, 1\right), 1\right)}} \]

  3. Taylor expanded in ux around 0 0.6

    \[\leadsto \sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\color{blue}{\left(maxCos - 1\right) \cdot \left(\left(1 - maxCos\right) \cdot {ux}^{2}\right) + ux \cdot \left(\left(1 + -1 \cdot \left(maxCos - 1\right)\right) - maxCos\right)}} \]

  4. Simplified0.5

    \[\leadsto \sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\color{blue}{\left(ux - ux \cdot maxCos\right) \cdot \left(2 - \left(ux - ux \cdot maxCos\right)\right)}} \]

  5. Applied egg-rr0.5

    \[\leadsto \color{blue}{\sqrt[3]{{\sin \left(uy \cdot \left(2 \cdot \pi\right)\right)}^{3} \cdot {\left(\left(ux - ux \cdot maxCos\right) \cdot \mathsf{fma}\left(ux, maxCos, 2 - ux\right)\right)}^{1.5}}} \]

  6. Final simplification0.5

    \[\leadsto \sqrt[3]{{\sin \left(uy \cdot \left(2 \cdot \pi\right)\right)}^{3} \cdot {\left(\left(ux - ux \cdot maxCos\right) \cdot \mathsf{fma}\left(ux, maxCos, 2 - ux\right)\right)}^{1.5}} \]

Alternatives

Alternative 1
Error0.6
Cost13376
\[\sqrt{ux - ux \cdot maxCos} \cdot \left(\sqrt{\left(2 + ux \cdot maxCos\right) - ux} \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right)\right) \]
Alternative 2
Error1.4
Cost10308
\[\begin{array}{l} \mathbf{if}\;uy \cdot 2 \leq 0.0008200000156648457:\\ \;\;\;\;\sqrt{ux - ux \cdot maxCos} \cdot \left(2 \cdot \left(\sqrt{\left(2 + ux \cdot maxCos\right) - ux} \cdot \left(uy \cdot \pi\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(2 \cdot \left(uy \cdot \pi\right)\right) \cdot \sqrt{ux \cdot \left(2 - ux\right)}\\ \end{array} \]
Alternative 3
Error1.4
Cost10244
\[\begin{array}{l} \mathbf{if}\;uy \cdot 2 \leq 0.0008200000156648457:\\ \;\;\;\;\sqrt{\left(ux \cdot \left(1 - maxCos\right)\right) \cdot \left(\mathsf{fma}\left(ux, maxCos, 2\right) - ux\right)} \cdot \left(uy \cdot \left(2 \cdot \pi\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(2 \cdot \left(uy \cdot \pi\right)\right) \cdot \sqrt{ux \cdot \left(2 - ux\right)}\\ \end{array} \]
Alternative 4
Error0.5
Cost10176
\[\sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\left(ux \cdot \left(1 - maxCos\right)\right) \cdot \left(2 + ux \cdot \left(maxCos + -1\right)\right)} \]
Alternative 5
Error1.4
Cost10052
\[\begin{array}{l} \mathbf{if}\;uy \cdot 2 \leq 0.0008200000156648457:\\ \;\;\;\;2 \cdot \left(\left(uy \cdot \pi\right) \cdot \sqrt{\left(ux - ux \cdot maxCos\right) \cdot \left(\left(2 + ux \cdot maxCos\right) - ux\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(2 \cdot \left(uy \cdot \pi\right)\right) \cdot \sqrt{ux \cdot \left(2 - ux\right)}\\ \end{array} \]
Alternative 6
Error3.3
Cost9988
\[\begin{array}{l} \mathbf{if}\;uy \cdot 2 \leq 0.0038999998942017555:\\ \;\;\;\;2 \cdot \left(\left(uy \cdot \pi\right) \cdot \sqrt{\left(ux - ux \cdot maxCos\right) \cdot \left(\left(2 + ux \cdot maxCos\right) - ux\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\pi \cdot \left(uy \cdot 2\right)\right) \cdot \sqrt{2 \cdot ux}\\ \end{array} \]
Alternative 7
Error6.0
Cost6976
\[2 \cdot \left(\left(uy \cdot \pi\right) \cdot \sqrt{\left(ux - ux \cdot maxCos\right) \cdot \left(\left(2 + ux \cdot maxCos\right) - ux\right)}\right) \]
Alternative 8
Error7.9
Cost6916
\[\begin{array}{l} \mathbf{if}\;ux \leq 0.0004299999854993075:\\ \;\;\;\;2 \cdot \left(\left(uy \cdot \pi\right) \cdot \sqrt{ux \cdot \left(2 \cdot \left(1 - maxCos\right)\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(uy \cdot \pi\right) \cdot \sqrt{1 + \left(ux - 1\right) \cdot \left(1 - ux\right)}\right)\\ \end{array} \]
Alternative 9
Error11.0
Cost6784
\[2 \cdot \left(\left(uy \cdot \pi\right) \cdot \sqrt{ux \cdot \left(2 \cdot \left(1 - maxCos\right)\right)}\right) \]
Alternative 10
Error11.9
Cost6656
\[2 \cdot \left(\left(uy \cdot \pi\right) \cdot \sqrt{2 \cdot ux}\right) \]
Alternative 11
Error29.7
Cost32
\[0 \]

Error

Reproduce

herbie shell --seed 2022225 
(FPCore (ux uy maxCos)
  :name "UniformSampleCone, y"
  :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)))
  (* (sin (* (* uy 2.0) PI)) (sqrt (- 1.0 (* (+ (- 1.0 ux) (* ux maxCos)) (+ (- 1.0 ux) (* ux maxCos)))))))