Average Error: 12.5 → 0.5
Time: 17.8s
Precision: binary32
\[\left(\left(\left(\left(0.0001 \leq alphax \land alphax \leq 1\right) \land \left(0.0001 \leq alphay \land alphay \leq 1\right)\right) \land \left(2.328306437 \cdot 10^{-10} \leq u0 \land u0 \leq 1\right)\right) \land \left(0 \leq cos2phi \land cos2phi \leq 1\right)\right) \land 0 \leq sin2phi\]
\[\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
↓
\[\frac{-\mathsf{log1p}\left(-u0\right)}{\mathsf{fma}\left(sin2phi, alphax, \left(alphay \cdot alphay\right) \cdot \frac{cos2phi}{alphax}\right)} \cdot \left(alphax \cdot \left(alphay \cdot alphay\right)\right) \]
(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))))
  (* alphax (* alphay alphay))))
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)))) * (alphax * (alphay * alphay));
}

function code(alphax, alphay, u0, cos2phi, sin2phi)
	return Float32(Float32(-log(Float32(Float32(1.0) - u0))) / Float32(Float32(cos2phi / Float32(alphax * alphax)) + Float32(sin2phi / Float32(alphay * alphay))))
end
↓
function code(alphax, alphay, u0, cos2phi, sin2phi)
	return Float32(Float32(Float32(-log1p(Float32(-u0))) / fma(sin2phi, alphax, Float32(Float32(alphay * alphay) * Float32(cos2phi / alphax)))) * Float32(alphax * Float32(alphay * alphay)))
end

\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 \left(alphax \cdot \left(alphay \cdot alphay\right)\right)

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. Final simplification0.5

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

Reproduce

herbie shell --seed 2022131 
(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)))))