Average Error: 12.8 → 0.5
Time: 15.5s
Precision: binary32
Cost: 3744
\[\left(\left(\left(\left(0.0001 \leq alphax \land alphax \leq 1\right) \land \left(0.0001 \leq alphay \land alphay \leq 1\right)\right) \land \left(2.328306437 \cdot 10^{-10} \leq u0 \land u0 \leq 1\right)\right) \land \left(0 \leq cos2phi \land cos2phi \leq 1\right)\right) \land 0 \leq sin2phi\]
\[\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
↓
\[\frac{-\mathsf{log1p}\left(-u0\right)}{\frac{1}{alphax \cdot \frac{alphax}{cos2phi}} + \frac{sin2phi}{alphay \cdot alphay}} \]
(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)))
  (+ (/ 1.0 (* alphax (/ alphax cos2phi))) (/ 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) / ((1.0f / (alphax * (alphax / cos2phi))) + (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(1.0) / Float32(alphax * Float32(alphax / cos2phi))) + Float32(sin2phi / Float32(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{1}{alphax \cdot \frac{alphax}{cos2phi}} + \frac{sin2phi}{alphay \cdot alphay}}

Error

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 12.8

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

  2. Simplified0.5

    \[\leadsto \color{blue}{\frac{-\mathsf{log1p}\left(-u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}}} \]
    Proof
    (/.f32 (neg.f32 (log1p.f32 (neg.f32 u0))) (+.f32 (/.f32 cos2phi (*.f32 alphax alphax)) (/.f32 sin2phi (*.f32 alphay alphay)))): 0 points increase in error, 0 points decrease in error
    (/.f32 (neg.f32 (Rewrite<= log1p-def_binary32 (log.f32 (+.f32 1 (neg.f32 u0))))) (+.f32 (/.f32 cos2phi (*.f32 alphax alphax)) (/.f32 sin2phi (*.f32 alphay alphay)))): 212 points increase in error, 2 points decrease in error
    (/.f32 (neg.f32 (log.f32 (Rewrite<= sub-neg_binary32 (-.f32 1 u0)))) (+.f32 (/.f32 cos2phi (*.f32 alphax alphax)) (/.f32 sin2phi (*.f32 alphay alphay)))): 0 points increase in error, 0 points decrease in error
    (/.f32 (Rewrite<= *-rgt-identity_binary32 (*.f32 (neg.f32 (log.f32 (-.f32 1 u0))) 1)) (+.f32 (/.f32 cos2phi (*.f32 alphax alphax)) (/.f32 sin2phi (*.f32 alphay alphay)))): 0 points increase in error, 0 points decrease in error
    (Rewrite=> associate-/l*_binary32 (/.f32 (neg.f32 (log.f32 (-.f32 1 u0))) (/.f32 (+.f32 (/.f32 cos2phi (*.f32 alphax alphax)) (/.f32 sin2phi (*.f32 alphay alphay))) 1))): 0 points increase in error, 0 points decrease in error
    (/.f32 (neg.f32 (log.f32 (-.f32 1 u0))) (Rewrite=> /-rgt-identity_binary32 (+.f32 (/.f32 cos2phi (*.f32 alphax alphax)) (/.f32 sin2phi (*.f32 alphay alphay))))): 0 points increase in error, 0 points decrease in error

  3. Applied egg-rr0.5

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

  4. Applied egg-rr0.5

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

  5. Final simplification0.5

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

Alternatives

Alternative 1
Error0.5
Cost3680
\[\frac{-\mathsf{log1p}\left(-u0\right)}{\frac{sin2phi}{alphay \cdot alphay} + \frac{cos2phi}{alphax \cdot alphax}} \]
Alternative 2
Error0.5
Cost3680
\[\frac{-\mathsf{log1p}\left(-u0\right)}{\frac{sin2phi}{alphay \cdot alphay} + \frac{\frac{cos2phi}{alphax}}{alphax}} \]
Alternative 3
Error2.4
Cost3556
\[\begin{array}{l} \mathbf{if}\;u0 \leq 0.009999999776482582:\\ \;\;\;\;\frac{u0 - u0 \cdot \left(u0 \cdot -0.5\right)}{\frac{sin2phi}{alphay \cdot alphay} + \frac{cos2phi}{alphax \cdot alphax}}\\ \mathbf{else}:\\ \;\;\;\;alphay \cdot \frac{-alphay}{\frac{sin2phi}{\mathsf{log1p}\left(-u0\right)}}\\ \end{array} \]
Alternative 4
Error2.2
Cost3556
\[\begin{array}{l} \mathbf{if}\;sin2phi \leq 0.019999999552965164:\\ \;\;\;\;\frac{u0 - u0 \cdot \left(u0 \cdot -0.5\right)}{\frac{sin2phi}{alphay \cdot alphay} + \frac{cos2phi}{alphax \cdot alphax}}\\ \mathbf{else}:\\ \;\;\;\;\left(alphay \cdot alphay\right) \cdot \frac{-\mathsf{log1p}\left(-u0\right)}{sin2phi}\\ \end{array} \]
Alternative 5
Error2.2
Cost3556
\[\begin{array}{l} \mathbf{if}\;sin2phi \leq 0.05000000074505806:\\ \;\;\;\;\frac{u0 - u0 \cdot \left(u0 \cdot -0.5\right)}{\frac{sin2phi}{alphay \cdot alphay} + \frac{cos2phi}{alphax \cdot alphax}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\mathsf{log1p}\left(-u0\right) \cdot alphay\right) \cdot \left(-alphay\right)}{sin2phi}\\ \end{array} \]
Alternative 6
Error5.0
Cost740
\[\begin{array}{l} t_0 := \frac{sin2phi}{alphay \cdot alphay}\\ \mathbf{if}\;t_0 \leq 50:\\ \;\;\;\;\frac{3 \cdot \left(u0 \cdot 0.3333333333333333\right)}{t_0 + \frac{cos2phi}{alphax \cdot alphax}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{alphay \cdot \left(-alphay\right)}{sin2phi}}{0.5 + \left(u0 \cdot 0.08333333333333333 - \frac{1}{u0}\right)}\\ \end{array} \]
Alternative 7
Error5.3
Cost708
\[\begin{array}{l} t_0 := \frac{sin2phi}{alphay \cdot alphay}\\ \mathbf{if}\;t_0 \leq 50:\\ \;\;\;\;\frac{u0}{t_0 + \frac{\frac{1}{alphax}}{\frac{alphax}{cos2phi}}}\\ \mathbf{else}:\\ \;\;\;\;alphay \cdot \frac{-alphay}{sin2phi \cdot \left(0.5 + \left(u0 \cdot 0.08333333333333333 - \frac{1}{u0}\right)\right)}\\ \end{array} \]
Alternative 8
Error5.0
Cost708
\[\begin{array}{l} t_0 := \frac{sin2phi}{alphay \cdot alphay}\\ \mathbf{if}\;t_0 \leq 50:\\ \;\;\;\;\frac{u0}{t_0 + \frac{\frac{1}{alphax}}{\frac{alphax}{cos2phi}}}\\ \mathbf{else}:\\ \;\;\;\;alphay \cdot \frac{\frac{-alphay}{sin2phi}}{0.5 + \left(u0 \cdot 0.08333333333333333 - \frac{1}{u0}\right)}\\ \end{array} \]
Alternative 9
Error5.0
Cost708
\[\begin{array}{l} t_0 := \frac{sin2phi}{alphay \cdot alphay}\\ \mathbf{if}\;t_0 \leq 50:\\ \;\;\;\;\frac{u0}{t_0 + \frac{\frac{1}{alphax}}{\frac{alphax}{cos2phi}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{alphay \cdot \left(-alphay\right)}{sin2phi}}{0.5 + \left(u0 \cdot 0.08333333333333333 - \frac{1}{u0}\right)}\\ \end{array} \]
Alternative 10
Error5.0
Cost708
\[\begin{array}{l} t_0 := \frac{sin2phi}{alphay \cdot alphay}\\ \mathbf{if}\;t_0 \leq 50:\\ \;\;\;\;\frac{u0}{t_0 - cos2phi \cdot \frac{1}{alphax \cdot \left(-alphax\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{alphay \cdot \left(-alphay\right)}{sin2phi}}{0.5 + \left(u0 \cdot 0.08333333333333333 - \frac{1}{u0}\right)}\\ \end{array} \]
Alternative 11
Error5.8
Cost676
\[\begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 50:\\ \;\;\;\;\frac{u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay} \cdot \frac{1}{alphay}}\\ \mathbf{else}:\\ \;\;\;\;\frac{alphay \cdot \left(-alphay\right)}{sin2phi \cdot 0.5 - \frac{sin2phi}{u0}}\\ \end{array} \]
Alternative 12
Error5.8
Cost676
\[\begin{array}{l} t_0 := \frac{sin2phi}{alphay \cdot alphay}\\ \mathbf{if}\;t_0 \leq 50:\\ \;\;\;\;\frac{u0}{t_0 + \frac{\frac{1}{alphax}}{\frac{alphax}{cos2phi}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{alphay \cdot \left(-alphay\right)}{sin2phi \cdot 0.5 - \frac{sin2phi}{u0}}\\ \end{array} \]
Alternative 13
Error5.8
Cost612
\[\begin{array}{l} t_0 := \frac{sin2phi}{alphay \cdot alphay}\\ \mathbf{if}\;t_0 \leq 50:\\ \;\;\;\;\frac{u0}{t_0 + \frac{cos2phi}{alphax \cdot alphax}}\\ \mathbf{else}:\\ \;\;\;\;\frac{alphay \cdot \left(-alphay\right)}{sin2phi \cdot 0.5 - \frac{sin2phi}{u0}}\\ \end{array} \]
Alternative 14
Error3.9
Cost608
\[\frac{u0 - u0 \cdot \left(u0 \cdot -0.5\right)}{\frac{sin2phi}{alphay \cdot alphay} + \frac{cos2phi}{alphax \cdot alphax}} \]
Alternative 15
Error8.2
Cost452
\[\begin{array}{l} \mathbf{if}\;sin2phi \leq 1.999999967550318 \cdot 10^{-17}:\\ \;\;\;\;\left(u0 \cdot alphax\right) \cdot \left(alphax \cdot \frac{1}{cos2phi}\right)\\ \mathbf{else}:\\ \;\;\;\;alphay \cdot \frac{-alphay}{sin2phi \cdot 0.5 - \frac{sin2phi}{u0}}\\ \end{array} \]
Alternative 16
Error8.2
Cost452
\[\begin{array}{l} \mathbf{if}\;sin2phi \leq 1.999999967550318 \cdot 10^{-17}:\\ \;\;\;\;\left(u0 \cdot alphax\right) \cdot \left(alphax \cdot \frac{1}{cos2phi}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{alphay \cdot \left(-alphay\right)}{sin2phi \cdot 0.5 - \frac{sin2phi}{u0}}\\ \end{array} \]
Alternative 17
Error10.6
Cost356
\[\begin{array}{l} \mathbf{if}\;sin2phi \leq 1.999999967550318 \cdot 10^{-17}:\\ \;\;\;\;\left(u0 \cdot alphax\right) \cdot \left(alphax \cdot \frac{1}{cos2phi}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{u0 \cdot \left(alphay \cdot alphay\right)}{sin2phi}\\ \end{array} \]
Alternative 18
Error21.1
Cost292
\[\begin{array}{l} \mathbf{if}\;sin2phi \leq 400000000:\\ \;\;\;\;u0 \cdot \left(alphax \cdot \frac{alphax}{cos2phi}\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
Alternative 19
Error10.6
Cost292
\[\begin{array}{l} \mathbf{if}\;sin2phi \leq 1.999999967550318 \cdot 10^{-17}:\\ \;\;\;\;\left(alphax \cdot alphax\right) \cdot \frac{u0}{cos2phi}\\ \mathbf{else}:\\ \;\;\;\;\left(alphay \cdot alphay\right) \cdot \frac{u0}{sin2phi}\\ \end{array} \]
Alternative 20
Error10.6
Cost292
\[\begin{array}{l} \mathbf{if}\;sin2phi \leq 1.999999967550318 \cdot 10^{-17}:\\ \;\;\;\;\frac{u0}{\frac{cos2phi}{alphax \cdot alphax}}\\ \mathbf{else}:\\ \;\;\;\;\left(alphay \cdot alphay\right) \cdot \frac{u0}{sin2phi}\\ \end{array} \]
Alternative 21
Error10.6
Cost292
\[\begin{array}{l} \mathbf{if}\;sin2phi \leq 1.999999967550318 \cdot 10^{-17}:\\ \;\;\;\;\frac{u0}{\frac{cos2phi}{alphax \cdot alphax}}\\ \mathbf{else}:\\ \;\;\;\;\frac{u0 \cdot \left(alphay \cdot alphay\right)}{sin2phi}\\ \end{array} \]
Alternative 22
Error26.7
Cost32
\[0 \]

Error

Reproduce

herbie shell --seed 2022331 
(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)))))