Average Error: 13.5 → 0.5
Time: 5.5s
Precision: binary32
\[\left(\left(2.328306437 \cdot 10^{-10} \leq ux \land ux \leq 1\right) \land \left(2.328306437 \cdot 10^{-10} \leq uy \land uy \leq 1\right)\right) \land \left(0 \leq maxCos \land maxCos \leq 1\right)\]
\[\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
↓
\[\sin \left(\sqrt[3]{{\left(uy + uy\right)}^{3} \cdot {\pi}^{3}}\right) \cdot \sqrt{ux \cdot \left(\mathsf{fma}\left(maxCos, -2, 2\right) - ux \cdot {\left(maxCos + -1\right)}^{2}\right)} \]
(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
 (*
  (sin (cbrt (* (pow (+ uy uy) 3.0) (pow PI 3.0))))
  (sqrt (* ux (- (fma maxCos -2.0 2.0) (* ux (pow (+ maxCos -1.0) 2.0)))))))
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 sinf(cbrtf((powf((uy + uy), 3.0f) * powf(((float) M_PI), 3.0f)))) * sqrtf((ux * (fmaf(maxCos, -2.0f, 2.0f) - (ux * powf((maxCos + -1.0f), 2.0f)))));
}

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 Float32(sin(cbrt(Float32((Float32(uy + uy) ^ Float32(3.0)) * (Float32(pi) ^ Float32(3.0))))) * sqrt(Float32(ux * Float32(fma(maxCos, Float32(-2.0), Float32(2.0)) - Float32(ux * (Float32(maxCos + Float32(-1.0)) ^ Float32(2.0)))))))
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)}
↓
\sin \left(\sqrt[3]{{\left(uy + uy\right)}^{3} \cdot {\pi}^{3}}\right) \cdot \sqrt{ux \cdot \left(\mathsf{fma}\left(maxCos, -2, 2\right) - ux \cdot {\left(maxCos + -1\right)}^{2}\right)}

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. Taylor expanded in ux around 0 0.5

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

  3. Simplified0.5

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

  4. Applied egg-rr0.5

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

  5. Final simplification0.5

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

Reproduce

herbie shell --seed 2022210 
(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)))))))