Average Error: 0.5 → 0.5
Time: 8.6s
Precision: binary32
\[\left(\left(cosTheta_i > 0.9999 \land cosTheta_i \leq 1\right) \land \left(2.328306437 \cdot 10^{-10} \leq u1 \land u1 \leq 1\right)\right) \land \left(2.328306437 \cdot 10^{-10} \leq u2 \land u2 \leq 1\right)\]
\[\sqrt{\frac{u1}{1 - u1}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]
↓
\[\begin{array}{l} t_0 := \sqrt[3]{6.28318530718} \cdot \sqrt[3]{6.28318530718}\\ \sqrt{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{u1}{1 - u1}\right)\right)} \cdot \sin \left(\sqrt[3]{\left(t_0 \cdot \left(t_0 \cdot t_0\right)\right) \cdot \left(6.28318530718 \cdot {u2}^{3}\right)}\right) \end{array} \]
\sqrt{\frac{u1}{1 - u1}} \cdot \sin \left(6.28318530718 \cdot u2\right)
↓
\begin{array}{l}
t_0 := \sqrt[3]{6.28318530718} \cdot \sqrt[3]{6.28318530718}\\
\sqrt{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{u1}{1 - u1}\right)\right)} \cdot \sin \left(\sqrt[3]{\left(t_0 \cdot \left(t_0 \cdot t_0\right)\right) \cdot \left(6.28318530718 \cdot {u2}^{3}\right)}\right)
\end{array}

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (* (sqrt (/ u1 (- 1.0 u1))) (sin (* 6.28318530718 u2))))
↓
(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (* (cbrt 6.28318530718) (cbrt 6.28318530718))))
   (*
    (sqrt (expm1 (log1p (/ u1 (- 1.0 u1)))))
    (sin (cbrt (* (* t_0 (* t_0 t_0)) (* 6.28318530718 (pow u2 3.0))))))))
float code(float cosTheta_i, float u1, float u2) {
	return sqrtf(u1 / (1.0f - u1)) * sinf(6.28318530718f * u2);
}
↓
float code(float cosTheta_i, float u1, float u2) {
	float t_0 = cbrtf(6.28318530718f) * cbrtf(6.28318530718f);
	return sqrtf(expm1f(log1pf(u1 / (1.0f - u1)))) * sinf(cbrtf((t_0 * (t_0 * t_0)) * (6.28318530718f * powf(u2, 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 0.5

    \[\sqrt{\frac{u1}{1 - u1}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]

  2. Applied add-cube-cbrt_binary320.5

    \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \sin \left(\color{blue}{\left(\left(\sqrt[3]{6.28318530718} \cdot \sqrt[3]{6.28318530718}\right) \cdot \sqrt[3]{6.28318530718}\right)} \cdot u2\right) \]

  3. Applied associate-*l*_binary320.5

    \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \sin \color{blue}{\left(\left(\sqrt[3]{6.28318530718} \cdot \sqrt[3]{6.28318530718}\right) \cdot \left(\sqrt[3]{6.28318530718} \cdot u2\right)\right)} \]

  4. Applied expm1-log1p-u_binary320.5

    \[\leadsto \sqrt{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{u1}{1 - u1}\right)\right)}} \cdot \sin \left(\left(\sqrt[3]{6.28318530718} \cdot \sqrt[3]{6.28318530718}\right) \cdot \left(\sqrt[3]{6.28318530718} \cdot u2\right)\right) \]

  5. Applied add-cbrt-cube_binary320.5

    \[\leadsto \sqrt{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{u1}{1 - u1}\right)\right)} \cdot \sin \left(\left(\sqrt[3]{6.28318530718} \cdot \sqrt[3]{6.28318530718}\right) \cdot \left(\sqrt[3]{6.28318530718} \cdot \color{blue}{\sqrt[3]{\left(u2 \cdot u2\right) \cdot u2}}\right)\right) \]

  6. Applied cbrt-unprod_binary320.6

    \[\leadsto \sqrt{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{u1}{1 - u1}\right)\right)} \cdot \sin \left(\left(\sqrt[3]{6.28318530718} \cdot \sqrt[3]{6.28318530718}\right) \cdot \color{blue}{\sqrt[3]{6.28318530718 \cdot \left(\left(u2 \cdot u2\right) \cdot u2\right)}}\right) \]

  7. Applied add-cbrt-cube_binary320.6

    \[\leadsto \sqrt{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{u1}{1 - u1}\right)\right)} \cdot \sin \left(\color{blue}{\sqrt[3]{\left(\left(\sqrt[3]{6.28318530718} \cdot \sqrt[3]{6.28318530718}\right) \cdot \left(\sqrt[3]{6.28318530718} \cdot \sqrt[3]{6.28318530718}\right)\right) \cdot \left(\sqrt[3]{6.28318530718} \cdot \sqrt[3]{6.28318530718}\right)}} \cdot \sqrt[3]{6.28318530718 \cdot \left(\left(u2 \cdot u2\right) \cdot u2\right)}\right) \]

  8. Applied cbrt-unprod_binary320.5

    \[\leadsto \sqrt{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{u1}{1 - u1}\right)\right)} \cdot \sin \color{blue}{\left(\sqrt[3]{\left(\left(\left(\sqrt[3]{6.28318530718} \cdot \sqrt[3]{6.28318530718}\right) \cdot \left(\sqrt[3]{6.28318530718} \cdot \sqrt[3]{6.28318530718}\right)\right) \cdot \left(\sqrt[3]{6.28318530718} \cdot \sqrt[3]{6.28318530718}\right)\right) \cdot \left(6.28318530718 \cdot \left(\left(u2 \cdot u2\right) \cdot u2\right)\right)}\right)} \]

  9. Taylor expanded in u2 around 0 0.5

    \[\leadsto \sqrt{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{u1}{1 - u1}\right)\right)} \cdot \sin \left(\sqrt[3]{\left(\left(\left(\sqrt[3]{6.28318530718} \cdot \sqrt[3]{6.28318530718}\right) \cdot \left(\sqrt[3]{6.28318530718} \cdot \sqrt[3]{6.28318530718}\right)\right) \cdot \left(\sqrt[3]{6.28318530718} \cdot \sqrt[3]{6.28318530718}\right)\right) \cdot \color{blue}{\left(6.28318530718 \cdot {u2}^{3}\right)}}\right) \]

  10. Final simplification0.5

    \[\leadsto \sqrt{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{u1}{1 - u1}\right)\right)} \cdot \sin \left(\sqrt[3]{\left(\left(\sqrt[3]{6.28318530718} \cdot \sqrt[3]{6.28318530718}\right) \cdot \left(\left(\sqrt[3]{6.28318530718} \cdot \sqrt[3]{6.28318530718}\right) \cdot \left(\sqrt[3]{6.28318530718} \cdot \sqrt[3]{6.28318530718}\right)\right)\right) \cdot \left(6.28318530718 \cdot {u2}^{3}\right)}\right) \]

Reproduce

herbie shell --seed 2022004 
(FPCore (cosTheta_i u1 u2)
  :name "Trowbridge-Reitz 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 (/ u1 (- 1.0 u1))) (sin (* 6.28318530718 u2))))