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

Average Error: 12.6 → 0.5

Time: 19.5s

Precision: binary32

Cost: 3904

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

Math

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

↓

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

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
 (*
  (*
   (/ -1.0 (+ (* sin2phi alphax) (* (* alphay alphay) (/ cos2phi alphax))))
   (* alphax (* alphay alphay)))
  (log1p (- u0))))

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 ((-1.0f / ((sin2phi * alphax) + ((alphay * alphay) * (cos2phi / alphax)))) * (alphax * (alphay * alphay))) * log1pf(-u0);
}

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(Float32(Float32(-1.0) / Float32(Float32(sin2phi * alphax) + Float32(Float32(alphay * alphay) * Float32(cos2phi / alphax)))) * Float32(alphax * Float32(alphay * alphay))) * log1p(Float32(-u0)))
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 0.6

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

   [Start] 12.6	\[ \frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
   sub-neg [=>] 12.6	\[ \frac{-\log \color{blue}{\left(1 + \left(-u0\right)\right)}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
   log1p-def [=>] 0.6	\[ \frac{-\color{blue}{\mathsf{log1p}\left(-u0\right)}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

3. Applied egg-rr 0.6

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

4. Applied egg-rr 0.5

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

5. Final simplification 0.5

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

Alternatives

Alternative 1
Error	0.5
Cost	3872

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

Alternative 2
Error	4.1
Cost	3684

\[\begin{array}{l} t_0 := \frac{sin2phi}{alphay \cdot alphay}\\ \mathbf{if}\;t_0 \leq 2.499999936844688 \cdot 10^{-6}:\\ \;\;\;\;\frac{u0}{t_0 + \frac{cos2phi}{alphax \cdot alphax}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\mathsf{log1p}\left(-u0\right)}{\frac{\frac{sin2phi}{alphay}}{alphay}}\\ \end{array} \]

Alternative 3
Error	3.9
Cost	3684

\[\begin{array}{l} t_0 := \frac{sin2phi}{alphay \cdot alphay}\\ \mathbf{if}\;t_0 \leq 4.999999873689376 \cdot 10^{-6}:\\ \;\;\;\;\frac{u0}{t_0 + \frac{cos2phi}{alphax \cdot alphax}}\\ \mathbf{else}:\\ \;\;\;\;\left(-alphay\right) \cdot \left(\mathsf{log1p}\left(-u0\right) \cdot \frac{alphay}{sin2phi}\right)\\ \end{array} \]

Alternative 4
Error	3.9
Cost	3684

\[\begin{array}{l} t_0 := \frac{sin2phi}{alphay \cdot alphay}\\ \mathbf{if}\;t_0 \leq 4.999999873689376 \cdot 10^{-6}:\\ \;\;\;\;\frac{u0}{t_0 + \frac{cos2phi}{alphax \cdot alphax}}\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{log1p}\left(-u0\right) \cdot \left(-alphay\right)\right) \cdot \frac{alphay}{sin2phi}\\ \end{array} \]

Alternative 5
Error	3.9
Cost	3684

\[\begin{array}{l} t_0 := \frac{sin2phi}{alphay \cdot alphay}\\ \mathbf{if}\;t_0 \leq 2.499999936844688 \cdot 10^{-6}:\\ \;\;\;\;\frac{u0}{t_0 + \frac{cos2phi}{alphax \cdot alphax}}\\ \mathbf{else}:\\ \;\;\;\;\frac{alphay \cdot \left(\mathsf{log1p}\left(-u0\right) \cdot \left(-alphay\right)\right)}{sin2phi}\\ \end{array} \]

Alternative 6
Error	0.6
Cost	3680

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

Alternative 7
Error	5.9
Cost	612

\[\begin{array}{l} t_0 := \frac{sin2phi}{alphay \cdot alphay}\\ \mathbf{if}\;t_0 \leq 2.499999936844688 \cdot 10^{-6}:\\ \;\;\;\;\frac{u0}{t_0 + \frac{cos2phi}{alphax \cdot alphax}}\\ \mathbf{else}:\\ \;\;\;\;alphay \cdot \frac{alphay \cdot \left(u0 + 0.5 \cdot \left(u0 \cdot u0\right)\right)}{sin2phi}\\ \end{array} \]

Alternative 8
Error	5.9
Cost	612

\[\begin{array}{l} t_0 := \frac{sin2phi}{alphay \cdot alphay}\\ \mathbf{if}\;t_0 \leq 2.499999936844688 \cdot 10^{-6}:\\ \;\;\;\;\frac{u0}{t_0 + \frac{cos2phi}{alphax \cdot alphax}}\\ \mathbf{else}:\\ \;\;\;\;\left(alphay \cdot \frac{alphay}{sin2phi}\right) \cdot \left(u0 + u0 \cdot \left(u0 \cdot 0.5\right)\right)\\ \end{array} \]

Alternative 9
Error	8.2
Cost	484

\[\begin{array}{l} \mathbf{if}\;sin2phi \leq 9.999999682655225 \cdot 10^{-22}:\\ \;\;\;\;\frac{u0}{\frac{cos2phi}{alphax \cdot alphax}}\\ \mathbf{else}:\\ \;\;\;\;alphay \cdot \frac{alphay \cdot \left(u0 + 0.5 \cdot \left(u0 \cdot u0\right)\right)}{sin2phi}\\ \end{array} \]

Alternative 10
Error	10.8
Cost	420

\[\begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 6.600000101997239 \cdot 10^{-9}:\\ \;\;\;\;\frac{u0}{\frac{cos2phi}{alphax \cdot alphax}}\\ \mathbf{else}:\\ \;\;\;\;u0 \cdot \frac{alphay \cdot alphay}{sin2phi}\\ \end{array} \]

Alternative 11
Error	10.7
Cost	420

\[\begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 9.999999717180685 \cdot 10^{-10}:\\ \;\;\;\;\frac{u0}{\frac{cos2phi}{alphax \cdot alphax}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(alphay \cdot alphay\right) \cdot u0}{sin2phi}\\ \end{array} \]

Alternative 12
Error	10.7
Cost	292

\[\begin{array}{l} \mathbf{if}\;sin2phi \leq 8.00000002901995 \cdot 10^{-15}:\\ \;\;\;\;alphax \cdot \left(alphax \cdot \frac{u0}{cos2phi}\right)\\ \mathbf{else}:\\ \;\;\;\;u0 \cdot \left(alphay \cdot \frac{alphay}{sin2phi}\right)\\ \end{array} \]

Alternative 13
Error	10.7
Cost	292

\[\begin{array}{l} \mathbf{if}\;sin2phi \leq 8.00000002901995 \cdot 10^{-15}:\\ \;\;\;\;alphax \cdot \left(alphax \cdot \frac{u0}{cos2phi}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(alphay \cdot alphay\right) \cdot \frac{u0}{sin2phi}\\ \end{array} \]

Alternative 14
Error	10.7
Cost	292

\[\begin{array}{l} \mathbf{if}\;sin2phi \leq 8.00000002901995 \cdot 10^{-15}:\\ \;\;\;\;alphax \cdot \left(alphax \cdot \frac{u0}{cos2phi}\right)\\ \mathbf{else}:\\ \;\;\;\;u0 \cdot \frac{alphay \cdot alphay}{sin2phi}\\ \end{array} \]

Alternative 15
Error	13.1
Cost	224

\[alphay \cdot \left(alphay \cdot \frac{u0}{sin2phi}\right) \]

Alternative 16
Error	13.1
Cost	224

\[u0 \cdot \left(alphay \cdot \frac{alphay}{sin2phi}\right) \]

Reproduce

herbie shell --seed 2023010 
(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)))))