Average Error: 13.4 → 0.3
Time: 11.4s
Precision: binary32
\[\left(\left(cosTheta_i > 0.9999 \land cosTheta_i \leq 1\right) \land \left(2.328306437 \cdot 10^{-10} \leq u1 \land u1 \leq 1\right)\right) \land \left(2.328306437 \cdot 10^{-10} \leq u2 \land u2 \leq 1\right)\]
\[\sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
↓
\[\begin{array}{l} t_0 := \sqrt{\log \left(2 \cdot \pi\right)}\\ \left(\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \sqrt[3]{{\cos \left(2 \cdot \left(u2 \cdot \pi\right)\right)}^{2}}\right) \cdot \sqrt[3]{\cos \left(e^{\mathsf{fma}\left(t_0, t_0, \log u2\right)}\right)} \end{array} \]
\sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right)
↓
\begin{array}{l}
t_0 := \sqrt{\log \left(2 \cdot \pi\right)}\\
\left(\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \sqrt[3]{{\cos \left(2 \cdot \left(u2 \cdot \pi\right)\right)}^{2}}\right) \cdot \sqrt[3]{\cos \left(e^{\mathsf{fma}\left(t_0, t_0, \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 (sqrt (log (* 2.0 PI)))))
   (*
    (* (sqrt (- (log1p (- u1)))) (cbrt (pow (cos (* 2.0 (* u2 PI))) 2.0)))
    (cbrt (cos (exp (fma t_0 t_0 (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 = sqrtf(logf(2.0f * ((float) M_PI)));
	return (sqrtf(-log1pf(-u1)) * cbrtf(powf(cosf(2.0f * (u2 * ((float) M_PI))), 2.0f))) * cbrtf(cosf(expf(fmaf(t_0, t_0, logf(u2)))));
}

Error

Bits error versus cosTheta_i

Bits error versus u1

Bits error versus u2

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-exp-log_binary320.3

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

  4. Applied add-exp-log_binary320.3

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

  5. Applied prod-exp_binary320.3

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

  6. Applied add-cube-cbrt_binary320.4

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

  7. Applied associate-*r*_binary320.5

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

  8. Simplified0.4

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

  9. Applied add-sqr-sqrt_binary320.4

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

  10. Applied fma-def_binary320.4

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

  11. Applied *-un-lft-identity_binary320.4

    \[\leadsto \left(\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(\sqrt[3]{\color{blue}{1 \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right)}} \cdot \sqrt[3]{\cos \left(\left(2 \cdot \pi\right) \cdot u2\right)}\right)\right) \cdot \sqrt[3]{\cos \left(e^{\mathsf{fma}\left(\sqrt{\log \left(2 \cdot \pi\right)}, \sqrt{\log \left(2 \cdot \pi\right)}, \log u2\right)}\right)} \]

  12. Applied cbrt-prod_binary320.4

    \[\leadsto \left(\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(\color{blue}{\left(\sqrt[3]{1} \cdot \sqrt[3]{\cos \left(\left(2 \cdot \pi\right) \cdot u2\right)}\right)} \cdot \sqrt[3]{\cos \left(\left(2 \cdot \pi\right) \cdot u2\right)}\right)\right) \cdot \sqrt[3]{\cos \left(e^{\mathsf{fma}\left(\sqrt{\log \left(2 \cdot \pi\right)}, \sqrt{\log \left(2 \cdot \pi\right)}, \log u2\right)}\right)} \]

  13. Applied associate-*l*_binary320.4

    \[\leadsto \left(\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \color{blue}{\left(\sqrt[3]{1} \cdot \left(\sqrt[3]{\cos \left(\left(2 \cdot \pi\right) \cdot u2\right)} \cdot \sqrt[3]{\cos \left(\left(2 \cdot \pi\right) \cdot u2\right)}\right)\right)}\right) \cdot \sqrt[3]{\cos \left(e^{\mathsf{fma}\left(\sqrt{\log \left(2 \cdot \pi\right)}, \sqrt{\log \left(2 \cdot \pi\right)}, \log u2\right)}\right)} \]

  14. Simplified0.3

    \[\leadsto \left(\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(\sqrt[3]{1} \cdot \color{blue}{\sqrt[3]{{\cos \left(2 \cdot \left(u2 \cdot \pi\right)\right)}^{2}}}\right)\right) \cdot \sqrt[3]{\cos \left(e^{\mathsf{fma}\left(\sqrt{\log \left(2 \cdot \pi\right)}, \sqrt{\log \left(2 \cdot \pi\right)}, \log u2\right)}\right)} \]

  15. Final simplification0.3

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

Reproduce

herbie shell --seed 2022076 
(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))))