Average Error: 13.5 → 0.5
Time: 6.8s
Precision: binary32
\[\left(\left(2.328306437 \cdot 10^{-10} \leq ux \land ux \leq 1\right) \land \left(2.328306437 \cdot 10^{-10} \leq uy \land uy \leq 1\right)\right) \land \left(0 \leq maxCos \land maxCos \leq 1\right)\]
\[\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
↓
\[\sqrt[3]{{\sin \left(2 \cdot \left(uy \cdot \pi\right)\right)}^{3} \cdot \sqrt{{\left(\mathsf{fma}\left(-2, maxCos, 2\right) - {\left(maxCos + -1\right)}^{2} \cdot ux\right)}^{3} \cdot {ux}^{3}}} \]
(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 (* 2.0 (* uy PI))) 3.0)
   (sqrt
    (*
     (pow (- (fma -2.0 maxCos 2.0) (* (pow (+ maxCos -1.0) 2.0) ux)) 3.0)
     (pow ux 3.0))))))
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 cbrtf((powf(sinf((2.0f * (uy * ((float) M_PI)))), 3.0f) * sqrtf((powf((fmaf(-2.0f, maxCos, 2.0f) - (powf((maxCos + -1.0f), 2.0f) * ux)), 3.0f) * powf(ux, 3.0f)))));
}

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

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

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

  4. Applied egg-rr0.5

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

  5. Taylor expanded in uy around inf 0.5

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

  6. Simplified0.5

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

  7. Final simplification0.5

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

Reproduce

herbie shell --seed 2022192 
(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)))))))