Average Error: 13.6 → 0.3
Time: 9.8s
Precision: binary32
\[\left(\left(cosTheta_i > 0.9999 \land cosTheta_i \leq 1\right) \land \left(2.328306437 \cdot 10^{-10} \leq u1 \land u1 \leq 1\right)\right) \land \left(2.328306437 \cdot 10^{-10} \leq u2 \land u2 \leq 1\right)\]
\[\sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
↓
\[\begin{array}{l} t_0 := \mathsf{log1p}\left(-u1\right)\\ t_1 := \cos \left(\left(2 \cdot \pi\right) \cdot u2\right)\\ \sqrt[3]{\left(\sqrt{t_0 \cdot t_0} \cdot \sqrt{-t_0}\right) \cdot \left(\left(t_1 \cdot t_1\right) \cdot \cos \left(e^{\log \left(2 \cdot \pi\right) + \log u2}\right)\right)} \end{array} \]
\sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right)
↓
\begin{array}{l}
t_0 := \mathsf{log1p}\left(-u1\right)\\
t_1 := \cos \left(\left(2 \cdot \pi\right) \cdot u2\right)\\
\sqrt[3]{\left(\sqrt{t_0 \cdot t_0} \cdot \sqrt{-t_0}\right) \cdot \left(\left(t_1 \cdot t_1\right) \cdot \cos \left(e^{\log \left(2 \cdot \pi\right) + \log u2}\right)\right)}
\end{array}

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (* (sqrt (- (log (- 1.0 u1)))) (cos (* (* 2.0 PI) u2))))
↓
(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (log1p (- u1))) (t_1 (cos (* (* 2.0 PI) u2))))
   (cbrt
    (*
     (* (sqrt (* t_0 t_0)) (sqrt (- t_0)))
     (* (* t_1 t_1) (cos (exp (+ (log (* 2.0 PI)) (log u2)))))))))
float code(float cosTheta_i, float u1, float u2) {
	return sqrtf(-logf(1.0f - u1)) * cosf((2.0f * ((float) M_PI)) * u2);
}
↓
float code(float cosTheta_i, float u1, float u2) {
	float t_0 = log1pf(-u1);
	float t_1 = cosf((2.0f * ((float) M_PI)) * u2);
	return cbrtf((sqrtf(t_0 * t_0) * sqrtf(-t_0)) * ((t_1 * t_1) * cosf(expf(logf(2.0f * ((float) M_PI)) + logf(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

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

  2. Simplified0.3

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

  3. Applied add-cbrt-cube_binary320.3

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

  4. Applied add-cbrt-cube_binary320.3

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

  5. Applied cbrt-unprod_binary320.3

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

  6. Applied add-exp-log_binary320.3

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

  7. Applied add-exp-log_binary320.3

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

  8. Applied prod-exp_binary320.3

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

  9. Applied sqrt-unprod_binary320.3

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

  10. Final simplification0.3

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

Reproduce

herbie shell --seed 2022081 
(FPCore (cosTheta_i u1 u2)
  :name "Beckmann Sample, near normal, slope_x"
  :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)))) (cos (* (* 2.0 PI) u2))))