Average Error: 13.5 → 0.6
Time: 10.7s
Precision: binary32
\[\left(\left(cosTheta_i > 0.9999 \land cosTheta_i \leq 1\right) \land \left(2.328306437 \cdot 10^{-10} \leq u1 \land u1 \leq 1\right)\right) \land \left(2.328306437 \cdot 10^{-10} \leq u2 \land u2 \leq 1\right)\]
\[\sqrt{-\log \left(1 - u1\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
↓
\[\begin{array}{l} t_0 := \cos \left(\pi \cdot u2\right) \cdot \sin \left(\pi \cdot u2\right)\\ \sqrt[3]{\left({\left(\sqrt{-\mathsf{log1p}\left(-u1\right)}\right)}^{3} \cdot \left(8 \cdot {t_0}^{2}\right)\right) \cdot \sqrt[3]{{t_0}^{3}}} \end{array} \]
\sqrt{-\log \left(1 - u1\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right)
↓
\begin{array}{l}
t_0 := \cos \left(\pi \cdot u2\right) \cdot \sin \left(\pi \cdot u2\right)\\
\sqrt[3]{\left({\left(\sqrt{-\mathsf{log1p}\left(-u1\right)}\right)}^{3} \cdot \left(8 \cdot {t_0}^{2}\right)\right) \cdot \sqrt[3]{{t_0}^{3}}}
\end{array}

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (* (sqrt (- (log (- 1.0 u1)))) (sin (* (* 2.0 PI) u2))))
↓
(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (* (cos (* PI u2)) (sin (* PI u2)))))
   (cbrt
    (*
     (* (pow (sqrt (- (log1p (- u1)))) 3.0) (* 8.0 (pow t_0 2.0)))
     (cbrt (pow t_0 3.0))))))
float code(float cosTheta_i, float u1, float u2) {
	return sqrtf(-logf(1.0f - u1)) * sinf((2.0f * ((float) M_PI)) * u2);
}
↓
float code(float cosTheta_i, float u1, float u2) {
	float t_0 = cosf(((float) M_PI) * u2) * sinf(((float) M_PI) * u2);
	return cbrtf((powf(sqrtf(-log1pf(-u1)), 3.0f) * (8.0f * powf(t_0, 2.0f))) * cbrtf(powf(t_0, 3.0f)));
}

Error

Bits error versus cosTheta_i

Bits error versus u1

Bits error versus u2

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 13.5

    \[\sqrt{-\log \left(1 - u1\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

  2. Simplified0.5

    \[\leadsto \color{blue}{\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right)} \]

  3. Applied add-cbrt-cube_binary320.5

    \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \color{blue}{\sqrt[3]{\left(\sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right)\right) \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right)}} \]

  4. Applied add-cbrt-cube_binary320.5

    \[\leadsto \color{blue}{\sqrt[3]{\left(\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \sqrt{-\mathsf{log1p}\left(-u1\right)}\right) \cdot \sqrt{-\mathsf{log1p}\left(-u1\right)}}} \cdot \sqrt[3]{\left(\sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right)\right) \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right)} \]

  5. Applied cbrt-unprod_binary320.6

    \[\leadsto \color{blue}{\sqrt[3]{\left(\left(\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \sqrt{-\mathsf{log1p}\left(-u1\right)}\right) \cdot \sqrt{-\mathsf{log1p}\left(-u1\right)}\right) \cdot \left(\left(\sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right)\right) \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right)\right)}} \]

  6. Taylor expanded in u2 around inf 0.6

    \[\leadsto \sqrt[3]{\left(\left(\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \sqrt{-\mathsf{log1p}\left(-u1\right)}\right) \cdot \sqrt{-\mathsf{log1p}\left(-u1\right)}\right) \cdot \color{blue}{{\sin \left(2 \cdot \left(\pi \cdot u2\right)\right)}^{3}}} \]

  7. Applied sin-2_binary320.6

    \[\leadsto \sqrt[3]{\left(\left(\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \sqrt{-\mathsf{log1p}\left(-u1\right)}\right) \cdot \sqrt{-\mathsf{log1p}\left(-u1\right)}\right) \cdot {\color{blue}{\left(2 \cdot \left(\sin \left(\pi \cdot u2\right) \cdot \cos \left(\pi \cdot u2\right)\right)\right)}}^{3}} \]

  8. Applied unpow-prod-down_binary320.6

    \[\leadsto \sqrt[3]{\left(\left(\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \sqrt{-\mathsf{log1p}\left(-u1\right)}\right) \cdot \sqrt{-\mathsf{log1p}\left(-u1\right)}\right) \cdot \color{blue}{\left({2}^{3} \cdot {\left(\sin \left(\pi \cdot u2\right) \cdot \cos \left(\pi \cdot u2\right)\right)}^{3}\right)}} \]

  9. Applied associate-*r*_binary320.6

    \[\leadsto \sqrt[3]{\color{blue}{\left(\left(\left(\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \sqrt{-\mathsf{log1p}\left(-u1\right)}\right) \cdot \sqrt{-\mathsf{log1p}\left(-u1\right)}\right) \cdot {2}^{3}\right) \cdot {\left(\sin \left(\pi \cdot u2\right) \cdot \cos \left(\pi \cdot u2\right)\right)}^{3}}} \]

  10. Simplified0.6

    \[\leadsto \sqrt[3]{\color{blue}{\left(8 \cdot {\left(\sqrt{-\mathsf{log1p}\left(-u1\right)}\right)}^{3}\right)} \cdot {\left(\sin \left(\pi \cdot u2\right) \cdot \cos \left(\pi \cdot u2\right)\right)}^{3}} \]

  11. Applied add-cube-cbrt_binary320.6

    \[\leadsto \sqrt[3]{\left(8 \cdot {\left(\sqrt{-\mathsf{log1p}\left(-u1\right)}\right)}^{3}\right) \cdot \color{blue}{\left(\left(\sqrt[3]{{\left(\sin \left(\pi \cdot u2\right) \cdot \cos \left(\pi \cdot u2\right)\right)}^{3}} \cdot \sqrt[3]{{\left(\sin \left(\pi \cdot u2\right) \cdot \cos \left(\pi \cdot u2\right)\right)}^{3}}\right) \cdot \sqrt[3]{{\left(\sin \left(\pi \cdot u2\right) \cdot \cos \left(\pi \cdot u2\right)\right)}^{3}}\right)}} \]

  12. Applied associate-*r*_binary320.6

    \[\leadsto \sqrt[3]{\color{blue}{\left(\left(8 \cdot {\left(\sqrt{-\mathsf{log1p}\left(-u1\right)}\right)}^{3}\right) \cdot \left(\sqrt[3]{{\left(\sin \left(\pi \cdot u2\right) \cdot \cos \left(\pi \cdot u2\right)\right)}^{3}} \cdot \sqrt[3]{{\left(\sin \left(\pi \cdot u2\right) \cdot \cos \left(\pi \cdot u2\right)\right)}^{3}}\right)\right) \cdot \sqrt[3]{{\left(\sin \left(\pi \cdot u2\right) \cdot \cos \left(\pi \cdot u2\right)\right)}^{3}}}} \]

  13. Simplified0.6

    \[\leadsto \sqrt[3]{\color{blue}{\left({\left(\sqrt{-\mathsf{log1p}\left(-u1\right)}\right)}^{3} \cdot \left(8 \cdot {\left(\cos \left(\pi \cdot u2\right) \cdot \sin \left(\pi \cdot u2\right)\right)}^{2}\right)\right)} \cdot \sqrt[3]{{\left(\sin \left(\pi \cdot u2\right) \cdot \cos \left(\pi \cdot u2\right)\right)}^{3}}} \]

  14. Final simplification0.6

    \[\leadsto \sqrt[3]{\left({\left(\sqrt{-\mathsf{log1p}\left(-u1\right)}\right)}^{3} \cdot \left(8 \cdot {\left(\cos \left(\pi \cdot u2\right) \cdot \sin \left(\pi \cdot u2\right)\right)}^{2}\right)\right) \cdot \sqrt[3]{{\left(\cos \left(\pi \cdot u2\right) \cdot \sin \left(\pi \cdot u2\right)\right)}^{3}}} \]

Reproduce

herbie shell --seed 2022081 
(FPCore (cosTheta_i u1 u2)
  :name "Beckmann Sample, near normal, slope_y"
  :precision binary32
  :pre (and (and (and (> cosTheta_i 0.9999) (<= cosTheta_i 1.0)) (and (<= 2.328306437e-10 u1) (<= u1 1.0))) (and (<= 2.328306437e-10 u2) (<= u2 1.0)))
  (* (sqrt (- (log (- 1.0 u1)))) (sin (* (* 2.0 PI) u2))))