?

Average Error: 60.8% → 98.4%
Time: 18.8s
Precision: binary32
Cost: 3680.00

?

\[\left(\left(\left(\left(0.0001 \leq alphax \land alphax \leq 1\right) \land \left(0.0001 \leq alphay \land alphay \leq 1\right)\right) \land \left(2.328306437 \cdot 10^{-10} \leq u0 \land u0 \leq 1\right)\right) \land \left(0 \leq cos2phi \land cos2phi \leq 1\right)\right) \land 0 \leq sin2phi\]
\[\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
↓
\[-\frac{\mathsf{log1p}\left(-u0\right)}{\frac{\frac{cos2phi}{alphax}}{alphax} + \frac{\frac{sin2phi}{alphay}}{alphay}} \]
(FPCore (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)))))
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)));
}

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

\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}}

Error?

Try it out?

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation?

  1. Initial program 60.8

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

  2. Simplified98.4

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

    [Start]60.8

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

    sub-neg [=>]60.8

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

    log1p-def [=>]98.4

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

  3. Applied egg-rr98.4

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

    [Start]98.4

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

    div-inv [=>]98.3

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

    *-commutative [=>]98.3

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

    pow2 [=>]98.3

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

    pow-flip [=>]98.4

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

    metadata-eval [=>]98.4

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

  4. Taylor expanded in alphax around 0 98.4

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

  5. Simplified98.4

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

    [Start]98.4

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

    unpow2 [=>]98.4

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

    unpow2 [=>]98.4

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

    associate-/r* [=>]98.4

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

  6. Applied egg-rr98.3

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

    [Start]98.4

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

    associate-/r* [=>]98.4

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

    div-inv [=>]98.3

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

  7. Applied egg-rr98.4

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

    [Start]98.3

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

    un-div-inv [=>]98.4

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

  8. Final simplification98.4

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

Alternatives

Alternative 1
Error98.4%
Cost3680.00
\[\frac{-\mathsf{log1p}\left(-u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
Alternative 2
Error98.4%
Cost3680.00
\[\frac{-\mathsf{log1p}\left(-u0\right)}{\frac{\frac{sin2phi}{alphay}}{alphay} + \frac{cos2phi}{alphax \cdot alphax}} \]
Alternative 3
Error92.2%
Cost3556.00
\[\begin{array}{l} t_0 := alphax \cdot \frac{alphax}{cos2phi}\\ \mathbf{if}\;sin2phi \leq 0.0005000000237487257:\\ \;\;\;\;\frac{u0 + \left(u0 \cdot u0\right) \cdot 0.5}{\frac{alphay \cdot alphay + sin2phi \cdot t_0}{\left(alphay \cdot alphay\right) \cdot t_0}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\mathsf{log1p}\left(-u0\right)}{\frac{sin2phi}{alphay \cdot alphay}}\\ \end{array} \]
Alternative 4
Error92.2%
Cost3556.00
\[\begin{array}{l} t_0 := alphax \cdot \frac{alphax}{cos2phi}\\ \mathbf{if}\;sin2phi \leq 0.0005000000237487257:\\ \;\;\;\;\frac{u0 + \left(u0 \cdot u0\right) \cdot 0.5}{\frac{alphay \cdot alphay + sin2phi \cdot t_0}{\left(alphay \cdot alphay\right) \cdot t_0}}\\ \mathbf{else}:\\ \;\;\;\;-\frac{\mathsf{log1p}\left(-u0\right)}{\frac{\frac{sin2phi}{alphay}}{alphay}}\\ \end{array} \]
Alternative 5
Error92.7%
Cost3556.00
\[\begin{array}{l} t_0 := alphax \cdot \frac{alphax}{cos2phi}\\ \mathbf{if}\;sin2phi \leq 0.0005000000237487257:\\ \;\;\;\;\frac{u0 + \left(u0 \cdot u0\right) \cdot 0.5}{\frac{alphay \cdot alphay + sin2phi \cdot t_0}{\left(alphay \cdot alphay\right) \cdot t_0}}\\ \mathbf{else}:\\ \;\;\;\;\frac{alphay \cdot \left(\mathsf{log1p}\left(-u0\right) \cdot \left(-alphay\right)\right)}{sin2phi}\\ \end{array} \]
Alternative 6
Error87.4%
Cost800.00
\[\frac{u0 + \left(u0 \cdot u0\right) \cdot 0.5}{\frac{\frac{sin2phi}{alphay} \cdot \left(alphax \cdot alphax\right) + cos2phi \cdot alphay}{alphay \cdot \left(alphax \cdot alphax\right)}} \]
Alternative 7
Error83.5%
Cost644.00
\[\begin{array}{l} \mathbf{if}\;sin2phi \leq 0.0005000000237487257:\\ \;\;\;\;\frac{u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay} \cdot \frac{1}{alphay}}\\ \mathbf{else}:\\ \;\;\;\;\frac{alphay \cdot \left(-alphay\right)}{\left(sin2phi \cdot 0.5 - \frac{sin2phi}{u0}\right) + u0 \cdot \left(sin2phi \cdot 0.08333333333333333\right)}\\ \end{array} \]
Alternative 8
Error87.6%
Cost608.00
\[\frac{u0 + \left(u0 \cdot u0\right) \cdot 0.5}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
Alternative 9
Error81.8%
Cost548.00
\[\begin{array}{l} \mathbf{if}\;sin2phi \leq 0.0005000000237487257:\\ \;\;\;\;\frac{u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay} \cdot \frac{1}{alphay}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-alphay \cdot alphay}{sin2phi \cdot 0.5 - \frac{sin2phi}{u0}}\\ \end{array} \]
Alternative 10
Error77.1%
Cost484.00
\[\begin{array}{l} \mathbf{if}\;sin2phi \leq 4.999999841327613 \cdot 10^{-22}:\\ \;\;\;\;\frac{alphax \cdot alphax}{cos2phi} \cdot \left(u0 + u0 \cdot \left(u0 \cdot 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-alphay \cdot alphay}{sin2phi \cdot 0.5 - \frac{sin2phi}{u0}}\\ \end{array} \]
Alternative 11
Error81.8%
Cost484.00
\[\begin{array}{l} \mathbf{if}\;sin2phi \leq 0.0005000000237487257:\\ \;\;\;\;\frac{u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-alphay \cdot alphay}{sin2phi \cdot 0.5 - \frac{sin2phi}{u0}}\\ \end{array} \]
Alternative 12
Error81.8%
Cost484.00
\[\begin{array}{l} \mathbf{if}\;sin2phi \leq 0.0005000000237487257:\\ \;\;\;\;\frac{u0}{\frac{\frac{sin2phi}{alphay}}{alphay} + \frac{cos2phi}{alphax \cdot alphax}}\\ \mathbf{else}:\\ \;\;\;\;\frac{alphay \cdot \left(-alphay\right)}{sin2phi \cdot 0.5 - \frac{sin2phi}{u0}}\\ \end{array} \]
Alternative 13
Error75.2%
Cost452.00
\[\begin{array}{l} \mathbf{if}\;sin2phi \leq 4.999999841327613 \cdot 10^{-22}:\\ \;\;\;\;\frac{u0}{\frac{cos2phi}{alphax \cdot alphax}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-alphay \cdot alphay}{sin2phi \cdot 0.5 - \frac{sin2phi}{u0}}\\ \end{array} \]
Alternative 14
Error67.1%
Cost292.00
\[\begin{array}{l} \mathbf{if}\;sin2phi \leq 4.999999841327613 \cdot 10^{-22}:\\ \;\;\;\;alphax \cdot \frac{alphax}{\frac{cos2phi}{u0}}\\ \mathbf{else}:\\ \;\;\;\;alphay \cdot \frac{u0 \cdot alphay}{sin2phi}\\ \end{array} \]
Alternative 15
Error67.1%
Cost292.00
\[\begin{array}{l} \mathbf{if}\;sin2phi \leq 4.999999841327613 \cdot 10^{-22}:\\ \;\;\;\;u0 \cdot \left(alphax \cdot \frac{alphax}{cos2phi}\right)\\ \mathbf{else}:\\ \;\;\;\;alphay \cdot \frac{u0 \cdot alphay}{sin2phi}\\ \end{array} \]
Alternative 16
Error67.1%
Cost292.00
\[\begin{array}{l} \mathbf{if}\;sin2phi \leq 4.999999841327613 \cdot 10^{-22}:\\ \;\;\;\;u0 \cdot \left(alphax \cdot \frac{alphax}{cos2phi}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(alphay \cdot alphay\right) \cdot \frac{u0}{sin2phi}\\ \end{array} \]
Alternative 17
Error67.1%
Cost292.00
\[\begin{array}{l} \mathbf{if}\;sin2phi \leq 4.999999841327613 \cdot 10^{-22}:\\ \;\;\;\;\frac{alphax}{\frac{\frac{cos2phi}{alphax}}{u0}}\\ \mathbf{else}:\\ \;\;\;\;\left(alphay \cdot alphay\right) \cdot \frac{u0}{sin2phi}\\ \end{array} \]
Alternative 18
Error67.1%
Cost292.00
\[\begin{array}{l} \mathbf{if}\;sin2phi \leq 4.999999841327613 \cdot 10^{-22}:\\ \;\;\;\;\frac{u0}{\frac{cos2phi}{alphax \cdot alphax}}\\ \mathbf{else}:\\ \;\;\;\;\left(alphay \cdot alphay\right) \cdot \frac{u0}{sin2phi}\\ \end{array} \]
Alternative 19
Error67.1%
Cost292.00
\[\begin{array}{l} \mathbf{if}\;sin2phi \leq 4.999999841327613 \cdot 10^{-22}:\\ \;\;\;\;\frac{u0}{\frac{cos2phi}{alphax \cdot alphax}}\\ \mathbf{else}:\\ \;\;\;\;\frac{u0 \cdot \left(alphay \cdot alphay\right)}{sin2phi}\\ \end{array} \]
Alternative 20
Error23.6%
Cost224.00
\[alphax \cdot \frac{alphax}{\frac{cos2phi}{u0}} \]

Error

Reproduce?

herbie shell --seed 2023121 
(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)))))