Average Error: 13.4 → 0.3
Time: 14.4s
Precision: binary32
\[\left(\left(2.328306437 \cdot 10^{-10} \leq ux \land ux \leq 1\right) \land \left(2.328306437 \cdot 10^{-10} \leq uy \land uy \leq 1\right)\right) \land \left(0 \leq maxCos \land maxCos \leq 1\right)\]
\[\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
↓
\[\cos \left(\sqrt[3]{{\left({\pi}^{\left(\sqrt{3}\right)}\right)}^{\left(\sqrt{3}\right)} \cdot \left({uy}^{3} \cdot 8\right)}\right) \cdot \sqrt{2 \cdot \mathsf{fma}\left(maxCos, ux \cdot ux, ux\right) - \mathsf{fma}\left(ux, ux, \left(maxCos \cdot ux\right) \cdot \left(2 + maxCos \cdot ux\right)\right)} \]
\cos \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)}
↓
\cos \left(\sqrt[3]{{\left({\pi}^{\left(\sqrt{3}\right)}\right)}^{\left(\sqrt{3}\right)} \cdot \left({uy}^{3} \cdot 8\right)}\right) \cdot \sqrt{2 \cdot \mathsf{fma}\left(maxCos, ux \cdot ux, ux\right) - \mathsf{fma}\left(ux, ux, \left(maxCos \cdot ux\right) \cdot \left(2 + maxCos \cdot ux\right)\right)}

(FPCore (ux uy maxCos)
 :precision binary32
 (*
  (cos (* (* uy 2.0) PI))
  (sqrt
   (- 1.0 (* (+ (- 1.0 ux) (* ux maxCos)) (+ (- 1.0 ux) (* ux maxCos)))))))
↓
(FPCore (ux uy maxCos)
 :precision binary32
 (*
  (cos (cbrt (* (pow (pow PI (sqrt 3.0)) (sqrt 3.0)) (* (pow uy 3.0) 8.0))))
  (sqrt
   (-
    (* 2.0 (fma maxCos (* ux ux) ux))
    (fma ux ux (* (* maxCos ux) (+ 2.0 (* maxCos ux))))))))
float code(float ux, float uy, float maxCos) {
	return cosf((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 cosf(cbrtf(powf(powf(((float) M_PI), sqrtf(3.0f)), sqrtf(3.0f)) * (powf(uy, 3.0f) * 8.0f))) * sqrtf((2.0f * fmaf(maxCos, (ux * ux), ux)) - fmaf(ux, ux, ((maxCos * ux) * (2.0f + (maxCos * ux)))));
}

Error

Bits error versus ux

Bits error versus uy

Bits error versus maxCos

Derivation

  1. Initial program 13.4

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

  2. Taylor expanded in ux around 0 0.3

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

  3. Simplified0.3

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

  4. Applied add-cbrt-cube_binary320.3

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

  5. Applied add-cbrt-cube_binary320.3

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

  6. Applied add-cbrt-cube_binary320.3

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

  7. Applied cbrt-unprod_binary320.3

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

  8. Applied cbrt-unprod_binary320.3

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

  9. Simplified0.3

    \[\leadsto \cos \left(\sqrt[3]{\color{blue}{{\pi}^{3} \cdot \left({uy}^{3} \cdot 8\right)}}\right) \cdot \sqrt{2 \cdot \mathsf{fma}\left(maxCos, ux \cdot ux, ux\right) - \mathsf{fma}\left(ux, ux, \left(maxCos \cdot ux\right) \cdot \left(maxCos \cdot ux + 2\right)\right)} \]

  10. Applied add-sqr-sqrt_binary320.3

    \[\leadsto \cos \left(\sqrt[3]{{\pi}^{\color{blue}{\left(\sqrt{3} \cdot \sqrt{3}\right)}} \cdot \left({uy}^{3} \cdot 8\right)}\right) \cdot \sqrt{2 \cdot \mathsf{fma}\left(maxCos, ux \cdot ux, ux\right) - \mathsf{fma}\left(ux, ux, \left(maxCos \cdot ux\right) \cdot \left(maxCos \cdot ux + 2\right)\right)} \]

  11. Applied pow-unpow_binary320.3

    \[\leadsto \cos \left(\sqrt[3]{\color{blue}{{\left({\pi}^{\left(\sqrt{3}\right)}\right)}^{\left(\sqrt{3}\right)}} \cdot \left({uy}^{3} \cdot 8\right)}\right) \cdot \sqrt{2 \cdot \mathsf{fma}\left(maxCos, ux \cdot ux, ux\right) - \mathsf{fma}\left(ux, ux, \left(maxCos \cdot ux\right) \cdot \left(maxCos \cdot ux + 2\right)\right)} \]

  12. Final simplification0.3

    \[\leadsto \cos \left(\sqrt[3]{{\left({\pi}^{\left(\sqrt{3}\right)}\right)}^{\left(\sqrt{3}\right)} \cdot \left({uy}^{3} \cdot 8\right)}\right) \cdot \sqrt{2 \cdot \mathsf{fma}\left(maxCos, ux \cdot ux, ux\right) - \mathsf{fma}\left(ux, ux, \left(maxCos \cdot ux\right) \cdot \left(2 + maxCos \cdot ux\right)\right)} \]

Reproduce

herbie shell --seed 2022004 
(FPCore (ux uy maxCos)
  :name "UniformSampleCone, x"
  :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)))
  (* (cos (* (* uy 2.0) PI)) (sqrt (- 1.0 (* (+ (- 1.0 ux) (* ux maxCos)) (+ (- 1.0 ux) (* ux maxCos)))))))