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

Average Error: 12.6 → 0.6

Time: 14.3s

Precision: binary32

Cost: 10016

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

Math

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

↓

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

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 (fma sin2phi (pow alphay -2.0) (/ cos2phi (* alphax alphax))))
  (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 / fmaf(sin2phi, powf(alphay, -2.0f), (cos2phi / (alphax * alphax)))) * 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(-1.0) / fma(sin2phi, (alphay ^ Float32(-2.0)), Float32(cos2phi / Float32(alphax * alphax)))) * 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. Applied egg-rr 0.6

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

3. Taylor expanded in cos2phi around 0 0.6

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

4. Simplified 0.6

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

   [Start] 0.6	\[ \frac{-1}{\mathsf{fma}\left(sin2phi, {alphay}^{-2}, \frac{cos2phi}{{alphax}^{2}}\right)} \cdot \mathsf{log1p}\left(-u0\right) \]
   unpow2 [=>] 0.6	\[ \frac{-1}{\mathsf{fma}\left(sin2phi, {alphay}^{-2}, \frac{cos2phi}{\color{blue}{alphax \cdot alphax}}\right)} \cdot \mathsf{log1p}\left(-u0\right) \]

5. Final simplification 0.6

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

Alternatives

Alternative 1
Error	2.4
Cost	3684

\[\begin{array}{l} t_0 := \frac{sin2phi}{alphay \cdot alphay}\\ \mathbf{if}\;t_0 \leq 0.014999999664723873:\\ \;\;\;\;\frac{u0 + \left(u0 \cdot u0\right) \cdot 0.5}{\frac{cos2phi}{alphax \cdot alphax} + t_0}\\ \mathbf{else}:\\ \;\;\;\;alphay \cdot \left(\mathsf{log1p}\left(-u0\right) \cdot \frac{alphay}{-sin2phi}\right)\\ \end{array} \]

Alternative 2
Error	2.4
Cost	3684

\[\begin{array}{l} t_0 := \frac{sin2phi}{alphay \cdot alphay}\\ \mathbf{if}\;t_0 \leq 0.014999999664723873:\\ \;\;\;\;\frac{u0 + \left(u0 \cdot u0\right) \cdot 0.5}{\frac{cos2phi}{alphax \cdot alphax} + t_0}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(-u0\right) \cdot \left(\frac{alphay}{sin2phi} \cdot \left(-alphay\right)\right)\\ \end{array} \]

Alternative 3
Error	0.5
Cost	3680

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

Alternative 4
Error	5.3
Cost	804

\[\begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 0.014999999664723873:\\ \;\;\;\;\frac{u0}{\left(alphax \cdot alphax\right) \cdot \frac{sin2phi}{alphay} + alphay \cdot cos2phi} \cdot \left(alphay \cdot \left(alphax \cdot alphax\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{alphay \cdot \left(-alphay\right)}{\left(sin2phi \cdot 0.5 - \frac{sin2phi}{u0}\right) - sin2phi \cdot \left(u0 \cdot -0.08333333333333333\right)}\\ \end{array} \]

Alternative 5
Error	5.2
Cost	772

\[\begin{array}{l} t_0 := \frac{sin2phi}{alphay \cdot alphay}\\ \mathbf{if}\;t_0 \leq 0.014999999664723873:\\ \;\;\;\;\frac{u0}{t_0 + cos2phi \cdot \frac{-1}{alphax \cdot \left(-alphax\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{alphay \cdot \left(-alphay\right)}{\left(sin2phi \cdot 0.5 - \frac{sin2phi}{u0}\right) - sin2phi \cdot \left(u0 \cdot -0.08333333333333333\right)}\\ \end{array} \]

Alternative 6
Error	5.9
Cost	708

\[\begin{array}{l} t_0 := \frac{sin2phi}{alphay \cdot alphay}\\ \mathbf{if}\;t_0 \leq 0.014999999664723873:\\ \;\;\;\;\frac{u0}{t_0 + cos2phi \cdot \frac{-1}{alphax \cdot \left(-alphax\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{alphay \cdot \left(-alphay\right)}{sin2phi \cdot 0.5 - \frac{sin2phi}{u0}}\\ \end{array} \]

Alternative 7
Error	5.8
Cost	612

\[\begin{array}{l} t_0 := \frac{sin2phi}{alphay \cdot alphay}\\ \mathbf{if}\;t_0 \leq 0.014999999664723873:\\ \;\;\;\;\frac{u0}{\frac{cos2phi}{alphax \cdot alphax} + t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{alphay \cdot \left(-alphay\right)}{sin2phi \cdot 0.5 - \frac{sin2phi}{u0}}\\ \end{array} \]

Alternative 8
Error	4.1
Cost	608

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

Alternative 9
Error	8.1
Cost	452

\[\begin{array}{l} \mathbf{if}\;sin2phi \leq 3.199999999423868 \cdot 10^{-22}:\\ \;\;\;\;alphax \cdot \frac{alphax \cdot u0}{cos2phi}\\ \mathbf{else}:\\ \;\;\;\;\frac{alphay \cdot \left(-alphay\right)}{sin2phi \cdot 0.5 - \frac{sin2phi}{u0}}\\ \end{array} \]

Alternative 10
Error	10.6
Cost	292

\[\begin{array}{l} \mathbf{if}\;sin2phi \leq 3.199999999423868 \cdot 10^{-22}:\\ \;\;\;\;u0 \cdot \left(alphax \cdot \frac{alphax}{cos2phi}\right)\\ \mathbf{else}:\\ \;\;\;\;u0 \cdot \left(alphay \cdot \frac{alphay}{sin2phi}\right)\\ \end{array} \]

Alternative 11
Error	10.6
Cost	292

\[\begin{array}{l} \mathbf{if}\;sin2phi \leq 3.199999999423868 \cdot 10^{-22}:\\ \;\;\;\;u0 \cdot \left(alphax \cdot \frac{alphax}{cos2phi}\right)\\ \mathbf{else}:\\ \;\;\;\;alphay \cdot \left(u0 \cdot \frac{alphay}{sin2phi}\right)\\ \end{array} \]

Alternative 12
Error	10.6
Cost	292

\[\begin{array}{l} \mathbf{if}\;sin2phi \leq 3.199999999423868 \cdot 10^{-22}:\\ \;\;\;\;u0 \cdot \frac{alphax}{\frac{cos2phi}{alphax}}\\ \mathbf{else}:\\ \;\;\;\;alphay \cdot \left(u0 \cdot \frac{alphay}{sin2phi}\right)\\ \end{array} \]

Alternative 13
Error	10.7
Cost	292

\[\begin{array}{l} \mathbf{if}\;sin2phi \leq 3.199999999423868 \cdot 10^{-22}:\\ \;\;\;\;u0 \cdot \frac{alphax \cdot alphax}{cos2phi}\\ \mathbf{else}:\\ \;\;\;\;alphay \cdot \left(u0 \cdot \frac{alphay}{sin2phi}\right)\\ \end{array} \]

Alternative 14
Error	10.7
Cost	292

\[\begin{array}{l} \mathbf{if}\;sin2phi \leq 3.199999999423868 \cdot 10^{-22}:\\ \;\;\;\;alphax \cdot \frac{alphax \cdot u0}{cos2phi}\\ \mathbf{else}:\\ \;\;\;\;alphay \cdot \left(u0 \cdot \frac{alphay}{sin2phi}\right)\\ \end{array} \]

Alternative 15
Error	13.0
Cost	224

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

Reproduce

herbie shell --seed 2023057 
(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)))))