Average Error: 13.4 → 0.3
Time: 16.4s
Precision: binary32
\[\left(\left(cosTheta_i > 0.9999 \land cosTheta_i \leq 1\right) \land \left(2.328306437 \cdot 10^{-10} \leq u1 \land u1 \leq 1\right)\right) \land \left(2.328306437 \cdot 10^{-10} \leq u2 \land u2 \leq 1\right)\]
\[\sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
↓
\[\sqrt[3]{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(-\mathsf{log1p}\left(-u1\right)\right)}^{1.5}\right)\right) \cdot {\cos \left(2 \cdot \left(\pi \cdot u2\right)\right)}^{3}} \]
\sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right)
↓
\sqrt[3]{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(-\mathsf{log1p}\left(-u1\right)\right)}^{1.5}\right)\right) \cdot {\cos \left(2 \cdot \left(\pi \cdot u2\right)\right)}^{3}}

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (* (sqrt (- (log (- 1.0 u1)))) (cos (* (* 2.0 PI) u2))))
↓
(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (cbrt
  (*
   (expm1 (log1p (pow (- (log1p (- u1))) 1.5)))
   (pow (cos (* 2.0 (* PI u2))) 3.0))))
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) {
	return cbrtf(expm1f(log1pf(powf(-log1pf(-u1), 1.5f))) * powf(cosf(2.0f * (((float) M_PI) * 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 13.4

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

  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

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

  4. Simplified0.3

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

  5. Applied pow1/2_binary320.3

    \[\leadsto \sqrt[3]{{\color{blue}{\left({\left(-\mathsf{log1p}\left(-u1\right)\right)}^{0.5}\right)}}^{3}} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

  6. Applied pow-pow_binary320.3

    \[\leadsto \sqrt[3]{\color{blue}{{\left(-\mathsf{log1p}\left(-u1\right)\right)}^{\left(0.5 \cdot 3\right)}}} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

  7. Simplified0.3

    \[\leadsto \sqrt[3]{{\left(-\mathsf{log1p}\left(-u1\right)\right)}^{\color{blue}{1.5}}} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

  8. Applied add-cbrt-cube_binary320.4

    \[\leadsto \color{blue}{\sqrt[3]{\left(\left(\sqrt[3]{{\left(-\mathsf{log1p}\left(-u1\right)\right)}^{1.5}} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right)\right) \cdot \left(\sqrt[3]{{\left(-\mathsf{log1p}\left(-u1\right)\right)}^{1.5}} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right)\right)\right) \cdot \left(\sqrt[3]{{\left(-\mathsf{log1p}\left(-u1\right)\right)}^{1.5}} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right)\right)}} \]

  9. Simplified0.3

    \[\leadsto \sqrt[3]{\color{blue}{{\left(-\mathsf{log1p}\left(-u1\right)\right)}^{1.5} \cdot {\cos \left(2 \cdot \left(\pi \cdot u2\right)\right)}^{3}}} \]

  10. Applied expm1-log1p-u_binary320.3

    \[\leadsto \sqrt[3]{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(-\mathsf{log1p}\left(-u1\right)\right)}^{1.5}\right)\right)} \cdot {\cos \left(2 \cdot \left(\pi \cdot u2\right)\right)}^{3}} \]

  11. Final simplification0.3

    \[\leadsto \sqrt[3]{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(-\mathsf{log1p}\left(-u1\right)\right)}^{1.5}\right)\right) \cdot {\cos \left(2 \cdot \left(\pi \cdot u2\right)\right)}^{3}} \]

Reproduce

herbie shell --seed 2021357 
(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))))