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

Average Error: 12.6 → 0.6

Time: 16.0s

Precision: binary32

Cost: 7072

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

Math

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

↓

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

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
 (*
  (* alphax (* alphay (* alphax alphay)))
  (/
   (- (log1p (- u0)))
   (fma cos2phi (* alphay alphay) (* alphax (* alphax sin2phi))))))

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 (alphax * (alphay * (alphax * alphay))) * (-log1pf(-u0) / fmaf(cos2phi, (alphay * alphay), (alphax * (alphax * sin2phi))));
}

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(alphax * Float32(alphay * Float32(alphax * alphay))) * Float32(Float32(-log1p(Float32(-u0))) / fma(cos2phi, Float32(alphay * alphay), Float32(alphax * Float32(alphax * sin2phi)))))
end

Derivation

1. Initial program 12.6

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

2. Applied egg-rr 0.8

   \[\leadsto \color{blue}{\frac{-1}{\mathsf{fma}\left(cos2phi, alphay \cdot alphay, alphax \cdot \left(alphax \cdot sin2phi\right)\right)} \cdot \left(\mathsf{log1p}\left(-u0\right) \cdot {\left(alphax \cdot alphay\right)}^{2}\right)} \]

3. Simplified 0.6

   \[\leadsto \color{blue}{{\left(alphax \cdot alphay\right)}^{2} \cdot \frac{-\mathsf{log1p}\left(-u0\right)}{\mathsf{fma}\left(cos2phi, alphay \cdot alphay, alphax \cdot \left(alphax \cdot sin2phi\right)\right)}} \]
   Proof

   [Start] 0.8	\[ \frac{-1}{\mathsf{fma}\left(cos2phi, alphay \cdot alphay, alphax \cdot \left(alphax \cdot sin2phi\right)\right)} \cdot \left(\mathsf{log1p}\left(-u0\right) \cdot {\left(alphax \cdot alphay\right)}^{2}\right) \]
   associate-*r* [=>] 0.8	\[ \color{blue}{\left(\frac{-1}{\mathsf{fma}\left(cos2phi, alphay \cdot alphay, alphax \cdot \left(alphax \cdot sin2phi\right)\right)} \cdot \mathsf{log1p}\left(-u0\right)\right) \cdot {\left(alphax \cdot alphay\right)}^{2}} \]
   *-commutative [=>] 0.8	\[ \color{blue}{{\left(alphax \cdot alphay\right)}^{2} \cdot \left(\frac{-1}{\mathsf{fma}\left(cos2phi, alphay \cdot alphay, alphax \cdot \left(alphax \cdot sin2phi\right)\right)} \cdot \mathsf{log1p}\left(-u0\right)\right)} \]
   associate-*l/ [=>] 0.6	\[ {\left(alphax \cdot alphay\right)}^{2} \cdot \color{blue}{\frac{-1 \cdot \mathsf{log1p}\left(-u0\right)}{\mathsf{fma}\left(cos2phi, alphay \cdot alphay, alphax \cdot \left(alphax \cdot sin2phi\right)\right)}} \]
   mul-1-neg [=>] 0.6	\[ {\left(alphax \cdot alphay\right)}^{2} \cdot \frac{\color{blue}{-\mathsf{log1p}\left(-u0\right)}}{\mathsf{fma}\left(cos2phi, alphay \cdot alphay, alphax \cdot \left(alphax \cdot sin2phi\right)\right)} \]

4. Applied egg-rr 0.6

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

5. Final simplification 0.6

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

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	2.4
Cost	3556

\[\begin{array}{l} \mathbf{if}\;sin2phi \leq 5.199999941396527 \cdot 10^{-5}:\\ \;\;\;\;\frac{u0 + \left(u0 \cdot u0\right) \cdot 0.5}{\frac{sin2phi}{alphay \cdot alphay} + \frac{cos2phi}{alphax \cdot alphax}}\\ \mathbf{else}:\\ \;\;\;\;alphay \cdot \left(\frac{\mathsf{log1p}\left(-u0\right)}{sin2phi} \cdot \left(-alphay\right)\right)\\ \end{array} \]

Alternative 4
Error	2.4
Cost	3556

\[\begin{array}{l} \mathbf{if}\;sin2phi \leq 5.199999941396527 \cdot 10^{-5}:\\ \;\;\;\;\frac{u0 + \left(u0 \cdot u0\right) \cdot 0.5}{\frac{sin2phi}{alphay \cdot alphay} + \frac{cos2phi}{alphax \cdot alphax}}\\ \mathbf{else}:\\ \;\;\;\;\left(-\mathsf{log1p}\left(-u0\right)\right) \cdot \left(alphay \cdot \frac{alphay}{sin2phi}\right)\\ \end{array} \]

Alternative 5
Error	5.4
Cost	676

\[\begin{array}{l} \mathbf{if}\;sin2phi \leq 5.199999941396527 \cdot 10^{-5}:\\ \;\;\;\;\frac{u0}{alphax \cdot sin2phi + \left(alphay \cdot alphay\right) \cdot \frac{cos2phi}{alphax}} \cdot \left(alphax \cdot \left(alphay \cdot alphay\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{alphay \cdot \left(-alphay\right)}{\left(sin2phi \cdot 0.5 - \frac{sin2phi}{u0}\right) + sin2phi \cdot \left(u0 \cdot 0.08333333333333333\right)}\\ \end{array} \]

Alternative 6
Error	5.4
Cost	644

\[\begin{array}{l} \mathbf{if}\;sin2phi \leq 5.199999941396527 \cdot 10^{-5}:\\ \;\;\;\;\frac{u0 \cdot \left(alphax \cdot alphay\right)}{\frac{alphay}{\frac{alphax}{cos2phi}} + \frac{alphax \cdot sin2phi}{alphay}}\\ \mathbf{else}:\\ \;\;\;\;\frac{alphay \cdot \left(-alphay\right)}{\left(sin2phi \cdot 0.5 - \frac{sin2phi}{u0}\right) + sin2phi \cdot \left(u0 \cdot 0.08333333333333333\right)}\\ \end{array} \]

Alternative 7
Error	6.0
Cost	612

\[\begin{array}{l} \mathbf{if}\;sin2phi \leq 5.199999941396527 \cdot 10^{-5}:\\ \;\;\;\;\left(alphax \cdot alphay\right) \cdot \frac{u0}{alphax \cdot \frac{sin2phi}{alphay} + alphay \cdot \frac{cos2phi}{alphax}}\\ \mathbf{else}:\\ \;\;\;\;\left(alphay \cdot \frac{alphay}{sin2phi}\right) \cdot \left(u0 - u0 \cdot \left(u0 \cdot -0.5\right)\right)\\ \end{array} \]

Alternative 8
Error	6.0
Cost	612

\[\begin{array}{l} \mathbf{if}\;sin2phi \leq 5.199999941396527 \cdot 10^{-5}:\\ \;\;\;\;\left(alphax \cdot alphay\right) \cdot \frac{u0}{\frac{alphax \cdot sin2phi}{alphay} + \frac{alphay \cdot cos2phi}{alphax}}\\ \mathbf{else}:\\ \;\;\;\;\left(alphay \cdot \frac{alphay}{sin2phi}\right) \cdot \left(u0 - u0 \cdot \left(u0 \cdot -0.5\right)\right)\\ \end{array} \]

Alternative 9
Error	6.0
Cost	612

\[\begin{array}{l} \mathbf{if}\;sin2phi \leq 5.199999941396527 \cdot 10^{-5}:\\ \;\;\;\;\frac{u0 \cdot \left(alphax \cdot alphay\right)}{\frac{alphay}{\frac{alphax}{cos2phi}} + \frac{alphax \cdot sin2phi}{alphay}}\\ \mathbf{else}:\\ \;\;\;\;\left(alphay \cdot \frac{alphay}{sin2phi}\right) \cdot \left(u0 - u0 \cdot \left(u0 \cdot -0.5\right)\right)\\ \end{array} \]

Alternative 10
Error	4.1
Cost	608

\[\frac{u0 + \left(u0 \cdot u0\right) \cdot 0.5}{\frac{sin2phi}{alphay \cdot alphay} + \frac{cos2phi}{alphax \cdot alphax}} \]

Alternative 11
Error	8.3
Cost	580

\[\begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 1.9999999920083944 \cdot 10^{-12}:\\ \;\;\;\;u0 \cdot \left(alphax \cdot \frac{alphax}{cos2phi}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{alphay \cdot \left(-alphay\right)}{sin2phi \cdot 0.5 - \frac{sin2phi}{u0}}\\ \end{array} \]

Alternative 12
Error	10.7
Cost	548

\[\begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 1.9999999920083944 \cdot 10^{-12}:\\ \;\;\;\;u0 \cdot \left(alphax \cdot \frac{alphax}{cos2phi}\right)\\ \mathbf{else}:\\ \;\;\;\;alphay \cdot \left(\frac{1}{sin2phi} \cdot \frac{alphay}{\frac{1}{u0}}\right)\\ \end{array} \]

Alternative 13
Error	5.9
Cost	484

\[\begin{array}{l} \mathbf{if}\;sin2phi \leq 5.199999941396527 \cdot 10^{-5}:\\ \;\;\;\;\frac{u0}{\frac{sin2phi}{alphay \cdot alphay} + \frac{cos2phi}{alphax \cdot alphax}}\\ \mathbf{else}:\\ \;\;\;\;\frac{alphay \cdot \left(-alphay\right)}{sin2phi \cdot 0.5 - \frac{sin2phi}{u0}}\\ \end{array} \]

Alternative 14
Error	6.0
Cost	484

\[\begin{array}{l} \mathbf{if}\;sin2phi \leq 5.199999941396527 \cdot 10^{-5}:\\ \;\;\;\;\frac{u0}{\frac{sin2phi}{alphay \cdot alphay} + \frac{cos2phi}{alphax \cdot alphax}}\\ \mathbf{else}:\\ \;\;\;\;\left(alphay \cdot \frac{alphay}{sin2phi}\right) \cdot \left(u0 - u0 \cdot \left(u0 \cdot -0.5\right)\right)\\ \end{array} \]

Alternative 15
Error	10.7
Cost	420

\[\begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 1.9999999920083944 \cdot 10^{-12}:\\ \;\;\;\;u0 \cdot \left(alphax \cdot \frac{alphax}{cos2phi}\right)\\ \mathbf{else}:\\ \;\;\;\;alphay \cdot \left(alphay \cdot \frac{u0}{sin2phi}\right)\\ \end{array} \]

Alternative 16
Error	10.7
Cost	420

\[\begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 1.9999999920083944 \cdot 10^{-12}:\\ \;\;\;\;u0 \cdot \left(alphax \cdot \frac{alphax}{cos2phi}\right)\\ \mathbf{else}:\\ \;\;\;\;u0 \cdot \left(alphay \cdot \frac{alphay}{sin2phi}\right)\\ \end{array} \]

Alternative 17
Error	10.8
Cost	292

\[\begin{array}{l} \mathbf{if}\;sin2phi \leq 9.9999998245167 \cdot 10^{-15}:\\ \;\;\;\;alphax \cdot \frac{alphax}{\frac{cos2phi}{u0}}\\ \mathbf{else}:\\ \;\;\;\;alphay \cdot \left(u0 \cdot \frac{alphay}{sin2phi}\right)\\ \end{array} \]

Alternative 18
Error	10.8
Cost	292

\[\begin{array}{l} \mathbf{if}\;sin2phi \leq 9.9999998245167 \cdot 10^{-15}:\\ \;\;\;\;u0 \cdot \frac{alphax \cdot alphax}{cos2phi}\\ \mathbf{else}:\\ \;\;\;\;alphay \cdot \left(u0 \cdot \frac{alphay}{sin2phi}\right)\\ \end{array} \]

Alternative 19
Error	10.8
Cost	292

\[\begin{array}{l} \mathbf{if}\;sin2phi \leq 9.9999998245167 \cdot 10^{-15}:\\ \;\;\;\;u0 \cdot \left(alphax \cdot \frac{alphax}{cos2phi}\right)\\ \mathbf{else}:\\ \;\;\;\;alphay \cdot \left(u0 \cdot \frac{alphay}{sin2phi}\right)\\ \end{array} \]

Alternative 20
Error	24.4
Cost	224

\[alphax \cdot \frac{alphax}{\frac{cos2phi}{u0}} \]

Reproduce

herbie shell --seed 2023060 
(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)))))