Average Error: 13.6 → 0.5
Time: 13.7s
Precision: binary32
\[\left(\left(2.328306437 \cdot 10^{-10} \leq ux \land ux \leq 1\right) \land \left(2.328306437 \cdot 10^{-10} \leq uy \land uy \leq 1\right)\right) \land \left(0 \leq maxCos \land maxCos \leq 1\right)\]
\[\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
↓
\[\sqrt[3]{{\sin \left(\sqrt[3]{{\pi}^{3} \cdot {\left(uy \cdot 2\right)}^{3}}\right)}^{3} \cdot {\left(2 \cdot \mathsf{fma}\left(maxCos, ux \cdot ux, ux\right) - \mathsf{fma}\left(ux, ux, \mathsf{fma}\left(2, maxCos \cdot ux, {\left(maxCos \cdot ux\right)}^{2}\right)\right)\right)}^{1.5}} \]
\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)}
↓
\sqrt[3]{{\sin \left(\sqrt[3]{{\pi}^{3} \cdot {\left(uy \cdot 2\right)}^{3}}\right)}^{3} \cdot {\left(2 \cdot \mathsf{fma}\left(maxCos, ux \cdot ux, ux\right) - \mathsf{fma}\left(ux, ux, \mathsf{fma}\left(2, maxCos \cdot ux, {\left(maxCos \cdot ux\right)}^{2}\right)\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 (cbrt (* (pow PI 3.0) (pow (* uy 2.0) 3.0)))) 3.0)
   (pow
    (-
     (* 2.0 (fma maxCos (* ux ux) ux))
     (fma ux ux (fma 2.0 (* maxCos ux) (pow (* maxCos ux) 2.0))))
    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(cbrtf((powf(((float) M_PI), 3.0f) * powf((uy * 2.0f), 3.0f)))), 3.0f) * powf(((2.0f * fmaf(maxCos, (ux * ux), ux)) - fmaf(ux, ux, fmaf(2.0f, (maxCos * ux), powf((maxCos * ux), 2.0f)))), 1.5f)));
}

Error

Bits error versus ux

Bits error versus uy

Bits error versus maxCos

Derivation

  1. Initial program 13.6

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

  2. Taylor expanded in ux around 0 0.5

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

  3. Applied egg-rr0.5

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

  4. Applied egg-rr0.5

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

  5. Final simplification0.5

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

Reproduce

herbie shell --seed 2022129 
(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)))))))