Average Error: 13.6 → 0.5
Time: 19.6s
Precision: binary32
\[\left(\left(cosTheta_i > 0.9999 \land cosTheta_i \leq 1\right) \land \left(2.328306437 \cdot 10^{-10} \leq u1 \land u1 \leq 1\right)\right) \land \left(2.328306437 \cdot 10^{-10} \leq u2 \land u2 \leq 1\right)\]
\[\sqrt{-\log \left(1 - u1\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
↓
\[\begin{array}{l} t_0 := \sqrt[3]{\pi \cdot \left(\pi \cdot 4\right)}\\ \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \sin \left(\sqrt[3]{\left(\left(t_0 \cdot \left(t_0 \cdot t_0\right)\right) \cdot \left(\pi \cdot 2\right)\right) \cdot \left(u2 \cdot \left(u2 \cdot u2\right)\right)}\right) \end{array} \]
\sqrt{-\log \left(1 - u1\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right)
↓
\begin{array}{l}
t_0 := \sqrt[3]{\pi \cdot \left(\pi \cdot 4\right)}\\
\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \sin \left(\sqrt[3]{\left(\left(t_0 \cdot \left(t_0 \cdot t_0\right)\right) \cdot \left(\pi \cdot 2\right)\right) \cdot \left(u2 \cdot \left(u2 \cdot u2\right)\right)}\right)
\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 (cbrt (* PI (* PI 4.0)))))
   (*
    (sqrt (- (log1p (- u1))))
    (sin (cbrt (* (* (* t_0 (* t_0 t_0)) (* PI 2.0)) (* u2 (* u2 u2))))))))
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 = cbrtf(((float) M_PI) * (((float) M_PI) * 4.0f));
	return sqrtf(-log1pf(-u1)) * sinf(cbrtf(((t_0 * (t_0 * t_0)) * (((float) M_PI) * 2.0f)) * (u2 * (u2 * u2))));
}

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 13.6

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

  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

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

  4. Applied add-cbrt-cube_binary320.5

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

  5. Applied cbrt-unprod_binary320.5

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

  6. Applied add-cube-cbrt_binary320.5

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

  7. Simplified0.5

    \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \sin \left(\sqrt[3]{\left(\left(\color{blue}{\left(\sqrt[3]{\pi \cdot \left(\pi \cdot 4\right)} \cdot \sqrt[3]{\pi \cdot \left(\pi \cdot 4\right)}\right)} \cdot \sqrt[3]{\left(2 \cdot \pi\right) \cdot \left(2 \cdot \pi\right)}\right) \cdot \left(2 \cdot \pi\right)\right) \cdot \left(\left(u2 \cdot u2\right) \cdot u2\right)}\right) \]

  8. Simplified0.5

    \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \sin \left(\sqrt[3]{\left(\left(\left(\sqrt[3]{\pi \cdot \left(\pi \cdot 4\right)} \cdot \sqrt[3]{\pi \cdot \left(\pi \cdot 4\right)}\right) \cdot \color{blue}{\sqrt[3]{\pi \cdot \left(\pi \cdot 4\right)}}\right) \cdot \left(2 \cdot \pi\right)\right) \cdot \left(\left(u2 \cdot u2\right) \cdot u2\right)}\right) \]

  9. Final simplification0.5

    \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \sin \left(\sqrt[3]{\left(\left(\sqrt[3]{\pi \cdot \left(\pi \cdot 4\right)} \cdot \left(\sqrt[3]{\pi \cdot \left(\pi \cdot 4\right)} \cdot \sqrt[3]{\pi \cdot \left(\pi \cdot 4\right)}\right)\right) \cdot \left(\pi \cdot 2\right)\right) \cdot \left(u2 \cdot \left(u2 \cdot u2\right)\right)}\right) \]

Reproduce

herbie shell --seed 2021275 
(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))))