Average Error: 12.7 → 0.5
Time: 23.8s
Precision: binary32
\[\left(\left(\left(\left(0.0001 \leq alphax \land alphax \leq 1\right) \land \left(0.0001 \leq alphay \land alphay \leq 1\right)\right) \land \left(2.328306437 \cdot 10^{-10} \leq u0 \land u0 \leq 1\right)\right) \land \left(0 \leq cos2phi \land cos2phi \leq 1\right)\right) \land 0 \leq sin2phi\]
\[\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
↓
\[\frac{\left(\left(alphay \cdot alphay\right) \cdot \left(alphax \cdot alphax\right)\right) \cdot \left(-\mathsf{log1p}\left(-u0\right)\right)}{\mathsf{fma}\left(sin2phi, alphax \cdot alphax, \left(alphay \cdot alphay\right) \cdot cos2phi\right)} \]
\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}}
↓
\frac{\left(\left(alphay \cdot alphay\right) \cdot \left(alphax \cdot alphax\right)\right) \cdot \left(-\mathsf{log1p}\left(-u0\right)\right)}{\mathsf{fma}\left(sin2phi, alphax \cdot alphax, \left(alphay \cdot alphay\right) \cdot cos2phi\right)}

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (/
  (- (log (- 1.0 u0)))
  (+ (/ cos2phi (* alphax alphax)) (/ sin2phi (* alphay alphay)))))
↓
(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (/
  (* (* (* alphay alphay) (* alphax alphax)) (- (log1p (- u0))))
  (fma sin2phi (* alphax alphax) (* (* alphay alphay) cos2phi))))
float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	return -logf(1.0f - u0) / ((cos2phi / (alphax * alphax)) + (sin2phi / (alphay * alphay)));
}
↓
float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	return (((alphay * alphay) * (alphax * alphax)) * -log1pf(-u0)) / fmaf(sin2phi, (alphax * alphax), ((alphay * alphay) * cos2phi));
}

Error

Bits error versus alphax

Bits error versus alphay

Bits error versus u0

Bits error versus cos2phi

Bits error versus sin2phi

Derivation

  1. Initial program 12.7

    \[\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

  2. Simplified0.5

    \[\leadsto \color{blue}{\frac{-\mathsf{log1p}\left(-u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}}} \]

  3. Applied frac-add_binary320.7

    \[\leadsto \frac{-\mathsf{log1p}\left(-u0\right)}{\color{blue}{\frac{cos2phi \cdot \left(alphay \cdot alphay\right) + \left(alphax \cdot alphax\right) \cdot sin2phi}{\left(alphax \cdot alphax\right) \cdot \left(alphay \cdot alphay\right)}}} \]

  4. Applied associate-/r/_binary320.5

    \[\leadsto \color{blue}{\frac{-\mathsf{log1p}\left(-u0\right)}{cos2phi \cdot \left(alphay \cdot alphay\right) + \left(alphax \cdot alphax\right) \cdot sin2phi} \cdot \left(\left(alphax \cdot alphax\right) \cdot \left(alphay \cdot alphay\right)\right)} \]

  5. Simplified0.5

    \[\leadsto \color{blue}{\frac{-\mathsf{log1p}\left(-u0\right)}{\mathsf{fma}\left(sin2phi, alphax \cdot alphax, \left(alphay \cdot alphay\right) \cdot cos2phi\right)}} \cdot \left(\left(alphax \cdot alphax\right) \cdot \left(alphay \cdot alphay\right)\right) \]

  6. Applied associate-*l/_binary320.5

    \[\leadsto \color{blue}{\frac{\left(-\mathsf{log1p}\left(-u0\right)\right) \cdot \left(\left(alphax \cdot alphax\right) \cdot \left(alphay \cdot alphay\right)\right)}{\mathsf{fma}\left(sin2phi, alphax \cdot alphax, \left(alphay \cdot alphay\right) \cdot cos2phi\right)}} \]

  7. Simplified0.5

    \[\leadsto \frac{\color{blue}{\left(\left(alphay \cdot alphay\right) \cdot \left(alphax \cdot alphax\right)\right) \cdot \left(-\mathsf{log1p}\left(-u0\right)\right)}}{\mathsf{fma}\left(sin2phi, alphax \cdot alphax, \left(alphay \cdot alphay\right) \cdot cos2phi\right)} \]

  8. Final simplification0.5

    \[\leadsto \frac{\left(\left(alphay \cdot alphay\right) \cdot \left(alphax \cdot alphax\right)\right) \cdot \left(-\mathsf{log1p}\left(-u0\right)\right)}{\mathsf{fma}\left(sin2phi, alphax \cdot alphax, \left(alphay \cdot alphay\right) \cdot cos2phi\right)} \]

Reproduce

herbie shell --seed 2022024 
(FPCore (alphax alphay u0 cos2phi sin2phi)
  :name "Beckmann Distribution sample, tan2theta, alphax != alphay, u1 <= 0.5"
  :precision binary32
  :pre (and (and (and (and (and (<= 0.0001 alphax) (<= alphax 1.0)) (and (<= 0.0001 alphay) (<= alphay 1.0))) (and (<= 2.328306437e-10 u0) (<= u0 1.0))) (and (<= 0.0 cos2phi) (<= cos2phi 1.0))) (<= 0.0 sin2phi))
  (/ (- (log (- 1.0 u0))) (+ (/ cos2phi (* alphax alphax)) (/ sin2phi (* alphay alphay)))))