Beckmann Distribution sample, tan2theta, alphax != alphay, u1 <= 0.5

Average Error: 12.6 → 0.5

Time: 13.3s

Precision: binary32

Cost: 3680

\[\left(\left(\left(\left(0.0001 \leq alphax \land alphax \leq 1\right) \land \left(0.0001 \leq alphay \land alphay \leq 1\right)\right) \land \left(2.328306437 \cdot 10^{-10} \leq u0 \land u0 \leq 1\right)\right) \land \left(0 \leq cos2phi \land cos2phi \leq 1\right)\right) \land 0 \leq sin2phi\]

Math

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

↓

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

FPCore

(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)))
  (+ (/ (/ cos2phi alphax) alphax) (/ (/ sin2phi alphay) alphay))))

C

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) / (((cos2phi / alphax) / alphax) + ((sin2phi / alphay) / alphay));
}

Julia

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(-log1p(Float32(-u0))) / Float32(Float32(Float32(cos2phi / alphax) / alphax) + Float32(Float32(sin2phi / alphay) / alphay)))
end

Derivation

1. Initial program 12.6

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

2. Simplified 0.5

   \[\leadsto \color{blue}{\frac{-\mathsf{log1p}\left(-u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}}} \]
   Proof
   (/.f32 (neg.f32 (log1p.f32 (neg.f32 u0))) (+.f32 (/.f32 cos2phi (*.f32 alphax alphax)) (/.f32 sin2phi (*.f32 alphay alphay)))): 0 points increase in error, 0 points decrease in error
   (/.f32 (neg.f32 (Rewrite<= log1p-def_binary32 (log.f32 (+.f32 1 (neg.f32 u0))))) (+.f32 (/.f32 cos2phi (*.f32 alphax alphax)) (/.f32 sin2phi (*.f32 alphay alphay)))): 221 points increase in error, 0 points decrease in error
   (/.f32 (neg.f32 (log.f32 (Rewrite<= sub-neg_binary32 (-.f32 1 u0)))) (+.f32 (/.f32 cos2phi (*.f32 alphax alphax)) (/.f32 sin2phi (*.f32 alphay alphay)))): 0 points increase in error, 0 points decrease in error
   (/.f32 (Rewrite<= *-rgt-identity_binary32 (*.f32 (neg.f32 (log.f32 (-.f32 1 u0))) 1)) (+.f32 (/.f32 cos2phi (*.f32 alphax alphax)) (/.f32 sin2phi (*.f32 alphay alphay)))): 0 points increase in error, 0 points decrease in error
   (Rewrite=> associate-/l*_binary32 (/.f32 (neg.f32 (log.f32 (-.f32 1 u0))) (/.f32 (+.f32 (/.f32 cos2phi (*.f32 alphax alphax)) (/.f32 sin2phi (*.f32 alphay alphay))) 1))): 0 points increase in error, 0 points decrease in error
   (/.f32 (neg.f32 (log.f32 (-.f32 1 u0))) (Rewrite=> /-rgt-identity_binary32 (+.f32 (/.f32 cos2phi (*.f32 alphax alphax)) (/.f32 sin2phi (*.f32 alphay alphay))))): 0 points increase in error, 0 points decrease in error

3. Applied egg-rr 0.5

   \[\leadsto \frac{-\mathsf{log1p}\left(-u0\right)}{\color{blue}{\left(-cos2phi\right) \cdot \frac{1}{alphax \cdot \left(-alphax\right)}} + \frac{sin2phi}{alphay \cdot alphay}} \]

4. Simplified 0.5

   \[\leadsto \frac{-\mathsf{log1p}\left(-u0\right)}{\color{blue}{\left(-cos2phi\right) \cdot \frac{\frac{1}{alphax}}{-alphax}} + \frac{sin2phi}{alphay \cdot alphay}} \]
   Proof
   (*.f32 (neg.f32 cos2phi) (/.f32 (/.f32 1 alphax) (neg.f32 alphax))): 0 points increase in error, 0 points decrease in error
   (*.f32 (neg.f32 cos2phi) (Rewrite<= associate-/r*_binary32 (/.f32 1 (*.f32 alphax (neg.f32 alphax))))): 37 points increase in error, 40 points decrease in error

5. Applied egg-rr 0.5

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

6. Applied egg-rr 0.6

   \[\leadsto \frac{-\mathsf{log1p}\left(-u0\right)}{\frac{\frac{cos2phi}{alphax}}{alphax} + \color{blue}{\frac{sin2phi}{alphay} \cdot \frac{1}{alphay}}} \]

7. Applied egg-rr 0.5

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

8. Final simplification 0.5

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

Alternatives

Alternative 1
Error	0.5
Cost	3680

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

Alternative 2
Error	0.5
Cost	3680

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

Alternative 3
Error	4.2
Cost	3588

\[\begin{array}{l} \mathbf{if}\;sin2phi \leq 4.99999991225835 \cdot 10^{-14}:\\ \;\;\;\;\frac{u0}{\frac{sin2phi}{alphay \cdot alphay} + \frac{1}{\frac{alphax}{\frac{cos2phi}{alphax}}}}\\ \mathbf{else}:\\ \;\;\;\;\left(alphay \cdot alphay\right) \cdot \left(\mathsf{log1p}\left(-u0\right) \cdot \frac{-1}{sin2phi}\right)\\ \end{array} \]

Alternative 4
Error	4.2
Cost	3556

\[\begin{array}{l} \mathbf{if}\;sin2phi \leq 4.99999991225835 \cdot 10^{-14}:\\ \;\;\;\;\frac{u0}{\frac{sin2phi}{alphay \cdot alphay} + \frac{1}{\frac{alphax}{\frac{cos2phi}{alphax}}}}\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{log1p}\left(-u0\right) \cdot alphay\right) \cdot \frac{-alphay}{sin2phi}\\ \end{array} \]

Alternative 5
Error	4.2
Cost	3556

\[\begin{array}{l} \mathbf{if}\;sin2phi \leq 4.99999991225835 \cdot 10^{-14}:\\ \;\;\;\;\frac{u0}{\frac{sin2phi}{alphay \cdot alphay} + \frac{1}{\frac{alphax}{\frac{cos2phi}{alphax}}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(-u0\right)}{sin2phi} \cdot \left(alphay \cdot \left(-alphay\right)\right)\\ \end{array} \]

Alternative 6
Error	5.5
Cost	612

\[\begin{array}{l} \mathbf{if}\;sin2phi \leq 3.500000040224799 \cdot 10^{-13}:\\ \;\;\;\;\frac{u0}{\frac{sin2phi}{alphay \cdot alphay} + \frac{1}{\frac{alphax}{\frac{cos2phi}{alphax}}}}\\ \mathbf{else}:\\ \;\;\;\;\left(alphay \cdot alphay\right) \cdot \frac{u0 + \left(u0 \cdot u0\right) \cdot \left(0.5 + u0 \cdot 0.3333333333333333\right)}{sin2phi}\\ \end{array} \]

Alternative 7
Error	6.2
Cost	548

\[\begin{array}{l} \mathbf{if}\;sin2phi \leq 3.500000040224799 \cdot 10^{-13}:\\ \;\;\;\;\frac{u0}{\frac{sin2phi}{alphay \cdot alphay} + \frac{1}{\frac{alphax}{\frac{cos2phi}{alphax}}}}\\ \mathbf{else}:\\ \;\;\;\;\left(alphay \cdot alphay\right) \cdot \frac{u0 + u0 \cdot \left(u0 \cdot 0.5\right)}{sin2phi}\\ \end{array} \]

Alternative 8
Error	6.2
Cost	548

\[\begin{array}{l} \mathbf{if}\;sin2phi \leq 3.500000040224799 \cdot 10^{-13}:\\ \;\;\;\;\frac{u0}{\frac{sin2phi}{alphay \cdot alphay} + \frac{1}{\frac{alphax}{\frac{cos2phi}{alphax}}}}\\ \mathbf{else}:\\ \;\;\;\;\left(alphay \cdot alphay\right) \cdot \left(\frac{u0}{sin2phi} + \frac{u0 \cdot \left(u0 \cdot 0.5\right)}{sin2phi}\right)\\ \end{array} \]

Alternative 9
Error	6.2
Cost	484

\[\begin{array}{l} \mathbf{if}\;sin2phi \leq 3.500000040224799 \cdot 10^{-13}:\\ \;\;\;\;\frac{u0}{\frac{sin2phi}{alphay \cdot alphay} + \frac{cos2phi}{alphax \cdot alphax}}\\ \mathbf{else}:\\ \;\;\;\;\left(alphay \cdot alphay\right) \cdot \frac{u0 \cdot \left(1 + u0 \cdot 0.5\right)}{sin2phi}\\ \end{array} \]

Alternative 10
Error	6.2
Cost	484

\[\begin{array}{l} \mathbf{if}\;sin2phi \leq 3.500000040224799 \cdot 10^{-13}:\\ \;\;\;\;\frac{u0}{\frac{sin2phi}{alphay \cdot alphay} + \frac{cos2phi}{alphax \cdot alphax}}\\ \mathbf{else}:\\ \;\;\;\;\left(alphay \cdot alphay\right) \cdot \frac{u0 + u0 \cdot \left(u0 \cdot 0.5\right)}{sin2phi}\\ \end{array} \]

Alternative 11
Error	7.8
Cost	416

\[\frac{u0}{\frac{sin2phi}{alphay \cdot alphay} + \frac{cos2phi}{alphax \cdot alphax}} \]

Alternative 12
Error	10.9
Cost	292

\[\begin{array}{l} \mathbf{if}\;sin2phi \leq 4.99999991225835 \cdot 10^{-14}:\\ \;\;\;\;\left(alphax \cdot alphax\right) \cdot \frac{u0}{cos2phi}\\ \mathbf{else}:\\ \;\;\;\;\left(alphay \cdot alphay\right) \cdot \frac{u0}{sin2phi}\\ \end{array} \]

Alternative 13
Error	10.9
Cost	292

\[\begin{array}{l} \mathbf{if}\;sin2phi \leq 4.99999991225835 \cdot 10^{-14}:\\ \;\;\;\;\left(alphax \cdot alphax\right) \cdot \frac{u0}{cos2phi}\\ \mathbf{else}:\\ \;\;\;\;u0 \cdot \frac{alphay \cdot alphay}{sin2phi}\\ \end{array} \]

Alternative 14
Error	10.9
Cost	292

\[\begin{array}{l} \mathbf{if}\;sin2phi \leq 4.99999991225835 \cdot 10^{-14}:\\ \;\;\;\;\frac{alphax \cdot alphax}{\frac{cos2phi}{u0}}\\ \mathbf{else}:\\ \;\;\;\;u0 \cdot \frac{alphay \cdot alphay}{sin2phi}\\ \end{array} \]

Alternative 15
Error	24.4
Cost	224

\[\left(alphax \cdot alphax\right) \cdot \frac{u0}{cos2phi} \]

Reproduce

herbie shell --seed 2022329 
(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)))))