Average Error: 13.5 → 0.3
Time: 19.0s
Precision: binary32
\[2.328306437 \cdot 10^{-10} \leq ux \land ux \leq 1 \land 2.328306437 \cdot 10^{-10} \leq uy \land uy \leq 1 \land 0 \leq maxCos \land maxCos \leq 1\]
\[\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)}\]
↓
\[\cos \left(\sqrt[3]{{\pi}^{2} \cdot {\left(2 \cdot uy\right)}^{3}} \cdot \sqrt[3]{\pi}\right) \cdot \sqrt{\left(2 \cdot \left(maxCos \cdot {ux}^{2}\right) + 2 \cdot ux\right) - \left({ux}^{2} \cdot {maxCos}^{2} + \left({ux}^{2} + 2 \cdot \left(maxCos \cdot ux\right)\right)\right)}\]
\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]{{\pi}^{2} \cdot {\left(2 \cdot uy\right)}^{3}} \cdot \sqrt[3]{\pi}\right) \cdot \sqrt{\left(2 \cdot \left(maxCos \cdot {ux}^{2}\right) + 2 \cdot ux\right) - \left({ux}^{2} \cdot {maxCos}^{2} + \left({ux}^{2} + 2 \cdot \left(maxCos \cdot ux\right)\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 PI 2.0) (pow (* 2.0 uy) 3.0))) (cbrt PI)))
  (sqrt
   (-
    (+ (* 2.0 (* maxCos (pow ux 2.0))) (* 2.0 ux))
    (+
     (* (pow ux 2.0) (pow maxCos 2.0))
     (+ (pow ux 2.0) (* 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(((float) M_PI), 2.0f) * powf((2.0f * uy), 3.0f)) * cbrtf((float) M_PI)) * sqrtf(((2.0f * (maxCos * powf(ux, 2.0f))) + (2.0f * ux)) - ((powf(ux, 2.0f) * powf(maxCos, 2.0f)) + (powf(ux, 2.0f) + (2.0f * (maxCos * ux)))));
}

Error

Bits error versus ux

Bits error versus uy

Bits error versus maxCos

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 13.5

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

  2. Simplified0.3

    \[\leadsto \color{blue}{\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{\left(ux - ux \cdot maxCos\right) \cdot \left(\left(\left(1 - ux\right) + ux \cdot maxCos\right) + 1\right)}}\]
  3. Using strategy rm

  4. Applied add-cube-cbrt_binary320.4

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

  5. Applied associate-*r*_binary320.4

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

  6. Simplified0.4

    \[\leadsto \cos \left(\color{blue}{\left(\left(2 \cdot uy\right) \cdot \left(\sqrt[3]{\pi} \cdot \sqrt[3]{\pi}\right)\right)} \cdot \sqrt[3]{\pi}\right) \cdot \sqrt{\left(ux - ux \cdot maxCos\right) \cdot \left(\left(\left(1 - ux\right) + ux \cdot maxCos\right) + 1\right)}\]
  7. Using strategy rm

  8. Applied add-cbrt-cube_binary320.4

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

  9. Simplified0.3

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

  10. Taylor expanded around 0 0.3

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

  11. Final simplification0.3

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

Reproduce

herbie shell --seed 2021173 
(FPCore (ux uy maxCos)
  :name "UniformSampleCone, x"
  :precision binary32
  :pre (and (<= 2.328306437e-10 ux 1.0) (<= 2.328306437e-10 uy 1.0) (<= 0.0 maxCos 1.0))
  (* (cos (* (* uy 2.0) PI)) (sqrt (- 1.0 (* (+ (- 1.0 ux) (* ux maxCos)) (+ (- 1.0 ux) (* ux maxCos)))))))