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

Average Error: 39.76% → 1.67%

Time: 15.4s

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

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

↓

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

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(cos2phi / Float32(alphax * alphax)) + Float32(sin2phi / Float32(alphay * alphay))))
end

Derivation

1. Initial program 39.76

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

2. Simplified 1.67

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

   [Start] 39.76	\[ \frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
   sub-neg [=>] 39.76	\[ \frac{-\log \color{blue}{\left(1 + \left(-u0\right)\right)}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
   log1p-def [=>] 1.67	\[ \frac{-\color{blue}{\mathsf{log1p}\left(-u0\right)}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

3. Final simplification 1.67

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

Alternatives

Alternative 1
Error	7.48%
Cost	3684

\[\begin{array}{l} t_0 := \frac{sin2phi}{alphay \cdot alphay}\\ \mathbf{if}\;t_0 \leq 500000:\\ \;\;\;\;\frac{u0 + \left(u0 \cdot u0\right) \cdot 0.5}{\frac{cos2phi}{alphax \cdot alphax} + t_0}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(-u0\right) \cdot \frac{alphay \cdot \left(-alphay\right)}{sin2phi}\\ \end{array} \]

Alternative 2
Error	12.57%
Cost	608

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

Alternative 3
Error	24.09%
Cost	544

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

Alternative 4
Error	24.17%
Cost	544

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

Alternative 5
Error	24.16%
Cost	544

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

Alternative 6
Error	24.11%
Cost	544

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

Alternative 7
Error	30.85%
Cost	484

\[\begin{array}{l} \mathbf{if}\;sin2phi \leq 3.4999999535528142 \cdot 10^{-19}:\\ \;\;\;\;\left(u0 + \left(u0 \cdot u0\right) \cdot 0.5\right) \cdot \frac{alphax}{\frac{cos2phi}{alphax}}\\ \mathbf{else}:\\ \;\;\;\;alphay \cdot \frac{u0}{\frac{sin2phi}{alphay}}\\ \end{array} \]

Alternative 8
Error	24.23%
Cost	416

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

Alternative 9
Error	32.89%
Cost	292

\[\begin{array}{l} \mathbf{if}\;sin2phi \leq 3.4999999535528142 \cdot 10^{-19}:\\ \;\;\;\;u0 \cdot \left(alphax \cdot \frac{alphax}{cos2phi}\right)\\ \mathbf{else}:\\ \;\;\;\;alphay \cdot \left(u0 \cdot \frac{alphay}{sin2phi}\right)\\ \end{array} \]

Alternative 10
Error	32.97%
Cost	292

\[\begin{array}{l} \mathbf{if}\;sin2phi \leq 3.4999999535528142 \cdot 10^{-19}:\\ \;\;\;\;u0 \cdot \left(alphax \cdot \frac{alphax}{cos2phi}\right)\\ \mathbf{else}:\\ \;\;\;\;alphay \cdot \frac{u0}{\frac{sin2phi}{alphay}}\\ \end{array} \]

Alternative 11
Error	32.96%
Cost	292

\[\begin{array}{l} \mathbf{if}\;sin2phi \leq 3.4999999535528142 \cdot 10^{-19}:\\ \;\;\;\;\frac{u0}{\frac{cos2phi}{alphax \cdot alphax}}\\ \mathbf{else}:\\ \;\;\;\;alphay \cdot \frac{u0}{\frac{sin2phi}{alphay}}\\ \end{array} \]

Alternative 12
Error	41.46%
Cost	224

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

Reproduce

herbie shell --seed 2023115 
(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)))))