Average Error: 12.5 → 0.5
Time: 13.6s
Precision: binary32
\[\left(\left(\left(\left(0.0001 \leq alphax \land alphax \leq 1\right) \land \left(0.0001 \leq alphay \land alphay \leq 1\right)\right) \land \left(2.328306437 \cdot 10^{-10} \leq u0 \land u0 \leq 1\right)\right) \land \left(0 \leq cos2phi \land cos2phi \leq 1\right)\right) \land 0 \leq sin2phi\]
\[\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
↓
\[\frac{-\mathsf{log1p}\left(-u0\right)}{\mathsf{fma}\left(sin2phi, alphax, \left(alphay \cdot alphay\right) \cdot \frac{cos2phi}{alphax}\right)} \cdot \sqrt[3]{{alphax}^{3} \cdot {alphay}^{6}} \]
\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}}
↓
\frac{-\mathsf{log1p}\left(-u0\right)}{\mathsf{fma}\left(sin2phi, alphax, \left(alphay \cdot alphay\right) \cdot \frac{cos2phi}{alphax}\right)} \cdot \sqrt[3]{{alphax}^{3} \cdot {alphay}^{6}}

(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
 (*
  (/
   (- (log1p (- u0)))
   (fma sin2phi alphax (* (* alphay alphay) (/ cos2phi alphax))))
  (cbrt (* (pow alphax 3.0) (pow alphay 6.0)))))
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 (-log1pf(-u0) / fmaf(sin2phi, alphax, ((alphay * alphay) * (cos2phi / alphax)))) * cbrtf(powf(alphax, 3.0f) * powf(alphay, 6.0f));
}

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 12.5

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

  2. Simplified0.5

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

  3. Applied associate-/r*_binary320.5

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

  4. Applied frac-add_binary320.6

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

  5. Applied associate-/r/_binary320.5

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

  6. Simplified0.5

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

  7. Applied add-cbrt-cube_binary320.5

    \[\leadsto \frac{-\mathsf{log1p}\left(-u0\right)}{\mathsf{fma}\left(sin2phi, alphax, \left(alphay \cdot alphay\right) \cdot \frac{cos2phi}{alphax}\right)} \cdot \color{blue}{\sqrt[3]{\left(\left(alphax \cdot \left(alphay \cdot alphay\right)\right) \cdot \left(alphax \cdot \left(alphay \cdot alphay\right)\right)\right) \cdot \left(alphax \cdot \left(alphay \cdot alphay\right)\right)}} \]

  8. Simplified0.5

    \[\leadsto \frac{-\mathsf{log1p}\left(-u0\right)}{\mathsf{fma}\left(sin2phi, alphax, \left(alphay \cdot alphay\right) \cdot \frac{cos2phi}{alphax}\right)} \cdot \sqrt[3]{\color{blue}{{alphax}^{3} \cdot {alphay}^{6}}} \]

  9. Final simplification0.5

    \[\leadsto \frac{-\mathsf{log1p}\left(-u0\right)}{\mathsf{fma}\left(sin2phi, alphax, \left(alphay \cdot alphay\right) \cdot \frac{cos2phi}{alphax}\right)} \cdot \sqrt[3]{{alphax}^{3} \cdot {alphay}^{6}} \]

Reproduce

herbie shell --seed 2021313 
(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)))))