Average Error: 13.7 → 0.3
Time: 9.2s
Precision: binary32
\[\left(\left(cosTheta_i > 0.9999 \land cosTheta_i \leq 1\right) \land \left(2.328306437 \cdot 10^{-10} \leq u1 \land u1 \leq 1\right)\right) \land \left(2.328306437 \cdot 10^{-10} \leq u2 \land u2 \leq 1\right)\]
\[\sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
↓
\[\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \mathsf{log1p}\left(\log \left(e^{\mathsf{expm1}\left(\cos \left(2 \cdot \left(u2 \cdot \pi\right)\right)\right)}\right)\right) \]
\sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right)
↓
\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \mathsf{log1p}\left(\log \left(e^{\mathsf{expm1}\left(\cos \left(2 \cdot \left(u2 \cdot \pi\right)\right)\right)}\right)\right)

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (* (sqrt (- (log (- 1.0 u1)))) (cos (* (* 2.0 PI) u2))))
↓
(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (*
  (sqrt (- (log1p (- u1))))
  (log1p (log (exp (expm1 (cos (* 2.0 (* u2 PI)))))))))
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 sqrtf(-log1pf(-u1)) * log1pf(logf(expf(expm1f(cosf(2.0f * (u2 * ((float) M_PI)))))));
}

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 13.7

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

  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 log1p-expm1-u_binary320.3

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

  4. Applied add-exp-log_binary320.4

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

  5. Applied add-exp-log_binary320.4

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

  6. Applied prod-exp_binary320.4

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

  7. Applied add-log-exp_binary320.4

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

  8. Simplified0.3

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

  9. Taylor expanded in u2 around 0 0.3

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

  10. Final simplification0.3

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

Reproduce

herbie shell --seed 2022061 
(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))))