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

Percentage Accurate: 60.7% → 100.0%

Time: 26.4s

Alternatives: 16

Speedup: 8.9×

Specification

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

Math

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

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (/
  (- (log (- 1.0 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)));
}

Fortran

real(4) function code(alphax, alphay, u0, cos2phi, sin2phi)
    real(4), intent (in) :: alphax
    real(4), intent (in) :: alphay
    real(4), intent (in) :: u0
    real(4), intent (in) :: cos2phi
    real(4), intent (in) :: sin2phi
    code = -log((1.0e0 - u0)) / ((cos2phi / (alphax * alphax)) + (sin2phi / (alphay * alphay)))
end function

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

MATLAB

function tmp = code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = -log((single(1.0) - u0)) / ((cos2phi / (alphax * alphax)) + (sin2phi / (alphay * alphay)));
end

Initial Program: 60.7% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (/
  (- (log (- 1.0 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)));
}

Fortran

real(4) function code(alphax, alphay, u0, cos2phi, sin2phi)
    real(4), intent (in) :: alphax
    real(4), intent (in) :: alphay
    real(4), intent (in) :: u0
    real(4), intent (in) :: cos2phi
    real(4), intent (in) :: sin2phi
    code = -log((1.0e0 - u0)) / ((cos2phi / (alphax * alphax)) + (sin2phi / (alphay * alphay)))
end function

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

MATLAB

function tmp = code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = -log((single(1.0) - u0)) / ((cos2phi / (alphax * alphax)) + (sin2phi / (alphay * alphay)));
end

Alternative 1: 100.0% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \langle \left( \frac{-\mathsf{log1p}\left(-u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \right)_{\text{binary64}} \rangle_{\text{binary32}} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (cast
  (!
   :precision
   binary64
   (/
    (- (log1p (- u0)))
    (+ (/ cos2phi (* alphax alphax)) (/ sin2phi (* alphay alphay)))))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	double tmp = -log1p(-((double) u0)) / ((((double) cos2phi) / (((double) alphax) * ((double) alphax))) + (((double) sin2phi) / (((double) alphay) * ((double) alphay))));
	return (float) tmp;
}

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = Float64(Float64(-log1p(Float64(-Float64(u0)))) / Float64(Float64(Float64(cos2phi) / Float64(Float64(alphax) * Float64(alphax))) + Float64(Float64(sin2phi) / Float64(Float64(alphay) * Float64(alphay)))))
	return Float32(tmp)
end

Derivation

1. Initial program 100.0%

   \[\langle \frac{-\mathsf{log1p}\left(-u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \rangle_{\text{binary64}} \]

2. Final simplification 100.0%

   \[\leadsto \langle \frac{-\mathsf{log1p}\left(-u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \rangle_{\text{binary64}} \]

Alternative 2: 98.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\left(alphax \cdot alphay\right) \cdot \mathsf{log1p}\left(-u0\right)}{\frac{alphax \cdot \left(-sin2phi\right)}{alphay} - \frac{cos2phi \cdot alphay}{alphax}} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (/
  (* (* alphax alphay) (log1p (- u0)))
  (- (/ (* alphax (- sin2phi)) alphay) (/ (* cos2phi alphay) alphax))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	return ((alphax * alphay) * log1pf(-u0)) / (((alphax * -sin2phi) / alphay) - ((cos2phi * alphay) / alphax));
}

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	return Float32(Float32(Float32(alphax * alphay) * log1p(Float32(-u0))) / Float32(Float32(Float32(alphax * Float32(-sin2phi)) / alphay) - Float32(Float32(cos2phi * alphay) / alphax)))
end

Derivation

1. Initial program 57.5%

   \[\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
2. Step-by-step derivation

   1. sub-neg 57.5%

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

   2. log1p-def 98.5%

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

3. Simplified 98.5%

   \[\leadsto \color{blue}{\frac{-\mathsf{log1p}\left(-u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}}} \]
4. Step-by-step derivation

   1. clear-num 98.5%

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

   2. inv-pow 98.5%

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

   3. sqr-pow 98.3%

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

   4. associate-/l* 98.4%

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

   5. frac-2neg 98.4%

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

   6. distribute-frac-neg 98.4%

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

   7. distribute-neg-frac 98.4%

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

   8. div-inv 98.4%

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

   9. clear-num 98.3%

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

   10. distribute-rgt-neg-in 98.3%

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

   11. distribute-neg-frac 98.3%

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

   12. frac-2neg 98.3%

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

   13. metadata-eval 98.3%

       \[\leadsto \frac{-\mathsf{log1p}\left(-u0\right)}{{\left(alphax \cdot \frac{alphax}{cos2phi}\right)}^{\color{blue}{-0.5}} \cdot {\left(\frac{alphax \cdot alphax}{cos2phi}\right)}^{\left(\frac{-1}{2}\right)} + \frac{sin2phi}{alphay \cdot alphay}} \]

   14. associate-/l* 98.4%

       \[\leadsto \frac{-\mathsf{log1p}\left(-u0\right)}{{\left(alphax \cdot \frac{alphax}{cos2phi}\right)}^{-0.5} \cdot {\color{blue}{\left(\frac{alphax}{\frac{cos2phi}{alphax}}\right)}}^{\left(\frac{-1}{2}\right)} + \frac{sin2phi}{alphay \cdot alphay}} \]

   15. frac-2neg 98.4%

       \[\leadsto \frac{-\mathsf{log1p}\left(-u0\right)}{{\left(alphax \cdot \frac{alphax}{cos2phi}\right)}^{-0.5} \cdot {\color{blue}{\left(\frac{-alphax}{-\frac{cos2phi}{alphax}}\right)}}^{\left(\frac{-1}{2}\right)} + \frac{sin2phi}{alphay \cdot alphay}} \]

   16. distribute-frac-neg 98.4%

       \[\leadsto \frac{-\mathsf{log1p}\left(-u0\right)}{{\left(alphax \cdot \frac{alphax}{cos2phi}\right)}^{-0.5} \cdot {\left(\frac{-alphax}{\color{blue}{\frac{-cos2phi}{alphax}}}\right)}^{\left(\frac{-1}{2}\right)} + \frac{sin2phi}{alphay \cdot alphay}} \]

   17. distribute-neg-frac 98.4%

       \[\leadsto \frac{-\mathsf{log1p}\left(-u0\right)}{{\left(alphax \cdot \frac{alphax}{cos2phi}\right)}^{-0.5} \cdot {\color{blue}{\left(-\frac{alphax}{\frac{-cos2phi}{alphax}}\right)}}^{\left(\frac{-1}{2}\right)} + \frac{sin2phi}{alphay \cdot alphay}} \]

   18. div-inv 98.4%

       \[\leadsto \frac{-\mathsf{log1p}\left(-u0\right)}{{\left(alphax \cdot \frac{alphax}{cos2phi}\right)}^{-0.5} \cdot {\left(-\color{blue}{alphax \cdot \frac{1}{\frac{-cos2phi}{alphax}}}\right)}^{\left(\frac{-1}{2}\right)} + \frac{sin2phi}{alphay \cdot alphay}} \]

   19. clear-num 98.4%

       \[\leadsto \frac{-\mathsf{log1p}\left(-u0\right)}{{\left(alphax \cdot \frac{alphax}{cos2phi}\right)}^{-0.5} \cdot {\left(-alphax \cdot \color{blue}{\frac{alphax}{-cos2phi}}\right)}^{\left(\frac{-1}{2}\right)} + \frac{sin2phi}{alphay \cdot alphay}} \]

   20. distribute-rgt-neg-in 98.4%

       \[\leadsto \frac{-\mathsf{log1p}\left(-u0\right)}{{\left(alphax \cdot \frac{alphax}{cos2phi}\right)}^{-0.5} \cdot {\color{blue}{\left(alphax \cdot \left(-\frac{alphax}{-cos2phi}\right)\right)}}^{\left(\frac{-1}{2}\right)} + \frac{sin2phi}{alphay \cdot alphay}} \]

   21. distribute-neg-frac 98.4%

       \[\leadsto \frac{-\mathsf{log1p}\left(-u0\right)}{{\left(alphax \cdot \frac{alphax}{cos2phi}\right)}^{-0.5} \cdot {\left(alphax \cdot \color{blue}{\frac{-alphax}{-cos2phi}}\right)}^{\left(\frac{-1}{2}\right)} + \frac{sin2phi}{alphay \cdot alphay}} \]

   22. frac-2neg 98.4%

       \[\leadsto \frac{-\mathsf{log1p}\left(-u0\right)}{{\left(alphax \cdot \frac{alphax}{cos2phi}\right)}^{-0.5} \cdot {\left(alphax \cdot \color{blue}{\frac{alphax}{cos2phi}}\right)}^{\left(\frac{-1}{2}\right)} + \frac{sin2phi}{alphay \cdot alphay}} \]

5. Applied egg-rr 98.4%

   \[\leadsto \frac{-\mathsf{log1p}\left(-u0\right)}{\color{blue}{{\left(alphax \cdot \frac{alphax}{cos2phi}\right)}^{-0.5} \cdot {\left(alphax \cdot \frac{alphax}{cos2phi}\right)}^{-0.5}} + \frac{sin2phi}{alphay \cdot alphay}} \]
6. Step-by-step derivation

   1. pow-sqr 98.5%

      \[\leadsto \frac{-\mathsf{log1p}\left(-u0\right)}{\color{blue}{{\left(alphax \cdot \frac{alphax}{cos2phi}\right)}^{\left(2 \cdot -0.5\right)}} + \frac{sin2phi}{alphay \cdot alphay}} \]

   2. metadata-eval 98.5%

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

   3. unpow-1 98.5%

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

7. Simplified 98.5%

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

   1. +-commutative 98.5%

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

   2. associate-/r* 98.5%

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

   3. associate-/l/ 98.5%

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

   4. clear-num 98.5%

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

   5. frac-add 98.3%

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

   6. associate-/r/ 98.5%

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

   7. *-commutative 98.5%

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

   8. *-commutative 98.5%

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

   9. frac-2neg 98.5%

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

   10. remove-double-neg 98.5%

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

   11. *-commutative 98.5%

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

   12. fma-def 98.4%

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

   13. associate-*r/ 98.5%

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

   14. *-commutative 98.5%

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

   15. associate-/l* 98.5%

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

9. Applied egg-rr 98.5%

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

    1. associate-*r/ 98.6%

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

    2. associate-/r/ 98.5%

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

    3. *-commutative 98.5%

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

11. Simplified 98.5%

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

12. Taylor expanded in alphax around 0 98.6%

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

13. Final simplification 98.6%

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

Alternative 3: 98.2% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (/
  (log1p (- u0))
  (- (/ -1.0 (* alphax (/ alphax cos2phi))) (/ sin2phi (* alphay alphay)))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	return log1pf(-u0) / ((-1.0f / (alphax * (alphax / cos2phi))) - (sin2phi / (alphay * alphay)));
}

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	return Float32(log1p(Float32(-u0)) / Float32(Float32(Float32(-1.0) / Float32(alphax * Float32(alphax / cos2phi))) - Float32(sin2phi / Float32(alphay * alphay))))
end

Derivation

1. Initial program 57.5%

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

3. Simplified 98.5%

   \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(-u0\right)}{\frac{\frac{-cos2phi}{alphax}}{alphax} - \frac{sin2phi}{alphay \cdot alphay}}} \]
4. Step-by-step derivation

   1. associate-/l/ 98.5%

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

   2. distribute-frac-neg 98.5%

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

   3. frac-2neg 98.5%

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

   4. div-inv 98.4%

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

   5. distribute-lft-neg-in 98.4%

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

   6. remove-double-neg 98.4%

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

   7. distribute-rgt-neg-in 98.4%

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

5. Applied egg-rr 98.4%

   \[\leadsto \frac{\mathsf{log1p}\left(-u0\right)}{\color{blue}{cos2phi \cdot \frac{1}{alphax \cdot \left(-alphax\right)}} - \frac{sin2phi}{alphay \cdot alphay}} \]
6. Step-by-step derivation

   1. un-div-inv 98.5%

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

   2. frac-2neg 98.5%

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

   3. distribute-rgt-neg-out 98.5%

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

   4. remove-double-neg 98.5%

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

   5. distribute-frac-neg 98.5%

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

   6. clear-num 98.5%

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

   7. associate-*r/ 98.5%

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

   8. distribute-neg-frac 98.5%

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

   9. metadata-eval 98.5%

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

7. Applied egg-rr 98.5%

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

8. Final simplification 98.5%

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

Alternative 4: 93.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{sin2phi}{alphay \cdot alphay}\\ t_1 := \frac{\frac{cos2phi}{alphax}}{-alphax} - t_0\\ \mathbf{if}\;t_0 \leq 200:\\ \;\;\;\;-0.5 \cdot \frac{u0}{\frac{t_1}{u0}} - \frac{u0}{t_1}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(-u0\right) \cdot \left(alphay \cdot \frac{alphay}{-sin2phi}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (let* ((t_0 (/ sin2phi (* alphay alphay)))
        (t_1 (- (/ (/ cos2phi alphax) (- alphax)) t_0)))
   (if (<= t_0 200.0)
     (- (* -0.5 (/ u0 (/ t_1 u0))) (/ u0 t_1))
     (* (log1p (- u0)) (* alphay (/ alphay (- sin2phi)))))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float t_0 = sin2phi / (alphay * alphay);
	float t_1 = ((cos2phi / alphax) / -alphax) - t_0;
	float tmp;
	if (t_0 <= 200.0f) {
		tmp = (-0.5f * (u0 / (t_1 / u0))) - (u0 / t_1);
	} else {
		tmp = log1pf(-u0) * (alphay * (alphay / -sin2phi));
	}
	return tmp;
}

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	t_0 = Float32(sin2phi / Float32(alphay * alphay))
	t_1 = Float32(Float32(Float32(cos2phi / alphax) / Float32(-alphax)) - t_0)
	tmp = Float32(0.0)
	if (t_0 <= Float32(200.0))
		tmp = Float32(Float32(Float32(-0.5) * Float32(u0 / Float32(t_1 / u0))) - Float32(u0 / t_1));
	else
		tmp = Float32(log1p(Float32(-u0)) * Float32(alphay * Float32(alphay / Float32(-sin2phi))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 sin2phi (*.f32 alphay alphay)) < 200

   1. Initial program 50.3%

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

   3. Simplified 98.5%

      \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(-u0\right)}{\frac{\frac{-cos2phi}{alphax}}{alphax} - \frac{sin2phi}{alphay \cdot alphay}}} \]
   4. Step-by-step derivation

      1. associate-/l/ 98.5%

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

      2. distribute-frac-neg 98.5%

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

      3. frac-2neg 98.5%

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

      4. div-inv 98.3%

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

      5. distribute-lft-neg-in 98.3%

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

      6. remove-double-neg 98.3%

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

      7. distribute-rgt-neg-in 98.3%

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

   5. Applied egg-rr 98.3%

      \[\leadsto \frac{\mathsf{log1p}\left(-u0\right)}{\color{blue}{cos2phi \cdot \frac{1}{alphax \cdot \left(-alphax\right)}} - \frac{sin2phi}{alphay \cdot alphay}} \]
   6. Step-by-step derivation

      1. un-div-inv 98.5%

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

      2. frac-2neg 98.5%

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

      3. distribute-rgt-neg-out 98.5%

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

      4. remove-double-neg 98.5%

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

      5. distribute-frac-neg 98.5%

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

      6. clear-num 98.5%

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

      7. associate-*r/ 98.6%

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

      8. distribute-neg-frac 98.6%

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

      9. metadata-eval 98.6%

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

   7. Applied egg-rr 98.6%

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

   8. Taylor expanded in u0 around 0 90.0%

      \[\leadsto \color{blue}{-1 \cdot \frac{u0}{-1 \cdot \frac{cos2phi}{{alphax}^{2}} - \frac{sin2phi}{{alphay}^{2}}} + -0.5 \cdot \frac{{u0}^{2}}{-1 \cdot \frac{cos2phi}{{alphax}^{2}} - \frac{sin2phi}{{alphay}^{2}}}} \]
   9. Step-by-step derivation

      1. +-commutative 90.0%

         \[\leadsto \color{blue}{-0.5 \cdot \frac{{u0}^{2}}{-1 \cdot \frac{cos2phi}{{alphax}^{2}} - \frac{sin2phi}{{alphay}^{2}}} + -1 \cdot \frac{u0}{-1 \cdot \frac{cos2phi}{{alphax}^{2}} - \frac{sin2phi}{{alphay}^{2}}}} \]

      2. mul-1-neg 90.0%

         \[\leadsto -0.5 \cdot \frac{{u0}^{2}}{-1 \cdot \frac{cos2phi}{{alphax}^{2}} - \frac{sin2phi}{{alphay}^{2}}} + \color{blue}{\left(-\frac{u0}{-1 \cdot \frac{cos2phi}{{alphax}^{2}} - \frac{sin2phi}{{alphay}^{2}}}\right)} \]

      3. unsub-neg 90.0%

         \[\leadsto \color{blue}{-0.5 \cdot \frac{{u0}^{2}}{-1 \cdot \frac{cos2phi}{{alphax}^{2}} - \frac{sin2phi}{{alphay}^{2}}} - \frac{u0}{-1 \cdot \frac{cos2phi}{{alphax}^{2}} - \frac{sin2phi}{{alphay}^{2}}}} \]

   10. Simplified 90.1%

       \[\leadsto \color{blue}{-0.5 \cdot \frac{u0}{\frac{\frac{\frac{cos2phi}{alphax}}{-alphax} - \frac{sin2phi}{alphay \cdot alphay}}{u0}} - \frac{u0}{\frac{\frac{cos2phi}{alphax}}{-alphax} - \frac{sin2phi}{alphay \cdot alphay}}} \]

   if 200 < (/.f32 sin2phi (*.f32 alphay alphay))

   1. Initial program 64.8%

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

   3. Simplified 98.4%

      \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(-u0\right)}{\frac{\frac{-cos2phi}{alphax}}{alphax} - \frac{sin2phi}{alphay \cdot alphay}}} \]
   4. Step-by-step derivation

      1. associate-/l/ 98.4%

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

      2. distribute-frac-neg 98.4%

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

      3. frac-2neg 98.4%

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

      4. div-inv 98.4%

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

      5. distribute-lft-neg-in 98.4%

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

      6. remove-double-neg 98.4%

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

      7. distribute-rgt-neg-in 98.4%

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

   5. Applied egg-rr 98.4%

      \[\leadsto \frac{\mathsf{log1p}\left(-u0\right)}{\color{blue}{cos2phi \cdot \frac{1}{alphax \cdot \left(-alphax\right)}} - \frac{sin2phi}{alphay \cdot alphay}} \]
   6. Step-by-step derivation

      1. un-div-inv 98.4%

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

      2. frac-2neg 98.4%

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

      3. distribute-rgt-neg-out 98.4%

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

      4. remove-double-neg 98.4%

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

      5. distribute-frac-neg 98.4%

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

      6. clear-num 98.4%

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

      7. associate-*r/ 98.4%

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

      8. distribute-neg-frac 98.4%

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

      9. metadata-eval 98.4%

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

   7. Applied egg-rr 98.4%

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

   8. Taylor expanded in alphax around inf 65.4%

      \[\leadsto \color{blue}{-1 \cdot \frac{{alphay}^{2} \cdot \log \left(1 - u0\right)}{sin2phi}} \]
   9. Step-by-step derivation

      1. mul-1-neg 65.4%

         \[\leadsto \color{blue}{-\frac{{alphay}^{2} \cdot \log \left(1 - u0\right)}{sin2phi}} \]

      2. unpow2 65.4%

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

      3. associate-/l* 64.6%

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

      4. distribute-neg-frac 64.6%

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

      5. distribute-rgt-neg-out 64.6%

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

      6. sub-neg 64.6%

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

      7. log1p-def 97.0%

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

   10. Simplified 97.0%

       \[\leadsto \color{blue}{\frac{alphay \cdot \left(-alphay\right)}{\frac{sin2phi}{\mathsf{log1p}\left(-u0\right)}}} \]
   11. Step-by-step derivation

       1. associate-/r/ 98.5%

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

       2. distribute-rgt-neg-out 98.5%

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

       3. distribute-frac-neg 98.5%

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

       4. associate-*r/ 98.4%

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

       5. distribute-rgt-neg-in 98.4%

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

       6. distribute-frac-neg 98.4%

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

       7. remove-double-neg 98.4%

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

       8. frac-2neg 98.4%

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

   12. Applied egg-rr 98.4%

       \[\leadsto \color{blue}{\left(alphay \cdot \frac{alphay}{-sin2phi}\right) \cdot \mathsf{log1p}\left(-u0\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 94.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 200:\\ \;\;\;\;-0.5 \cdot \frac{u0}{\frac{\frac{\frac{cos2phi}{alphax}}{-alphax} - \frac{sin2phi}{alphay \cdot alphay}}{u0}} - \frac{u0}{\frac{\frac{cos2phi}{alphax}}{-alphax} - \frac{sin2phi}{alphay \cdot alphay}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(-u0\right) \cdot \left(alphay \cdot \frac{alphay}{-sin2phi}\right)\\ \end{array} \]

Alternative 5: 93.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{sin2phi}{alphay \cdot alphay}\\ t_1 := \frac{\frac{cos2phi}{alphax}}{-alphax} - t_0\\ \mathbf{if}\;t_0 \leq 200:\\ \;\;\;\;-0.5 \cdot \frac{u0}{\frac{t_1}{u0}} - \frac{u0}{t_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{alphay}{-sin2phi} \cdot \left(alphay \cdot \mathsf{log1p}\left(-u0\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (let* ((t_0 (/ sin2phi (* alphay alphay)))
        (t_1 (- (/ (/ cos2phi alphax) (- alphax)) t_0)))
   (if (<= t_0 200.0)
     (- (* -0.5 (/ u0 (/ t_1 u0))) (/ u0 t_1))
     (* (/ alphay (- sin2phi)) (* alphay (log1p (- u0)))))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float t_0 = sin2phi / (alphay * alphay);
	float t_1 = ((cos2phi / alphax) / -alphax) - t_0;
	float tmp;
	if (t_0 <= 200.0f) {
		tmp = (-0.5f * (u0 / (t_1 / u0))) - (u0 / t_1);
	} else {
		tmp = (alphay / -sin2phi) * (alphay * log1pf(-u0));
	}
	return tmp;
}

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	t_0 = Float32(sin2phi / Float32(alphay * alphay))
	t_1 = Float32(Float32(Float32(cos2phi / alphax) / Float32(-alphax)) - t_0)
	tmp = Float32(0.0)
	if (t_0 <= Float32(200.0))
		tmp = Float32(Float32(Float32(-0.5) * Float32(u0 / Float32(t_1 / u0))) - Float32(u0 / t_1));
	else
		tmp = Float32(Float32(alphay / Float32(-sin2phi)) * Float32(alphay * log1p(Float32(-u0))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 sin2phi (*.f32 alphay alphay)) < 200

   1. Initial program 50.3%

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

   3. Simplified 98.5%

      \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(-u0\right)}{\frac{\frac{-cos2phi}{alphax}}{alphax} - \frac{sin2phi}{alphay \cdot alphay}}} \]
   4. Step-by-step derivation

      1. associate-/l/ 98.5%

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

      2. distribute-frac-neg 98.5%

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

      3. frac-2neg 98.5%

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

      4. div-inv 98.3%

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

      5. distribute-lft-neg-in 98.3%

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

      6. remove-double-neg 98.3%

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

      7. distribute-rgt-neg-in 98.3%

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

   5. Applied egg-rr 98.3%

      \[\leadsto \frac{\mathsf{log1p}\left(-u0\right)}{\color{blue}{cos2phi \cdot \frac{1}{alphax \cdot \left(-alphax\right)}} - \frac{sin2phi}{alphay \cdot alphay}} \]
   6. Step-by-step derivation

      1. un-div-inv 98.5%

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

      2. frac-2neg 98.5%

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

      3. distribute-rgt-neg-out 98.5%

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

      4. remove-double-neg 98.5%

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

      5. distribute-frac-neg 98.5%

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

      6. clear-num 98.5%

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

      7. associate-*r/ 98.6%

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

      8. distribute-neg-frac 98.6%

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

      9. metadata-eval 98.6%

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

   7. Applied egg-rr 98.6%

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

   8. Taylor expanded in u0 around 0 90.0%

      \[\leadsto \color{blue}{-1 \cdot \frac{u0}{-1 \cdot \frac{cos2phi}{{alphax}^{2}} - \frac{sin2phi}{{alphay}^{2}}} + -0.5 \cdot \frac{{u0}^{2}}{-1 \cdot \frac{cos2phi}{{alphax}^{2}} - \frac{sin2phi}{{alphay}^{2}}}} \]
   9. Step-by-step derivation

      1. +-commutative 90.0%

         \[\leadsto \color{blue}{-0.5 \cdot \frac{{u0}^{2}}{-1 \cdot \frac{cos2phi}{{alphax}^{2}} - \frac{sin2phi}{{alphay}^{2}}} + -1 \cdot \frac{u0}{-1 \cdot \frac{cos2phi}{{alphax}^{2}} - \frac{sin2phi}{{alphay}^{2}}}} \]

      2. mul-1-neg 90.0%

         \[\leadsto -0.5 \cdot \frac{{u0}^{2}}{-1 \cdot \frac{cos2phi}{{alphax}^{2}} - \frac{sin2phi}{{alphay}^{2}}} + \color{blue}{\left(-\frac{u0}{-1 \cdot \frac{cos2phi}{{alphax}^{2}} - \frac{sin2phi}{{alphay}^{2}}}\right)} \]

      3. unsub-neg 90.0%

         \[\leadsto \color{blue}{-0.5 \cdot \frac{{u0}^{2}}{-1 \cdot \frac{cos2phi}{{alphax}^{2}} - \frac{sin2phi}{{alphay}^{2}}} - \frac{u0}{-1 \cdot \frac{cos2phi}{{alphax}^{2}} - \frac{sin2phi}{{alphay}^{2}}}} \]

   10. Simplified 90.1%

       \[\leadsto \color{blue}{-0.5 \cdot \frac{u0}{\frac{\frac{\frac{cos2phi}{alphax}}{-alphax} - \frac{sin2phi}{alphay \cdot alphay}}{u0}} - \frac{u0}{\frac{\frac{cos2phi}{alphax}}{-alphax} - \frac{sin2phi}{alphay \cdot alphay}}} \]

   if 200 < (/.f32 sin2phi (*.f32 alphay alphay))

   1. Initial program 64.8%

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

   3. Simplified 98.4%

      \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(-u0\right)}{\frac{\frac{-cos2phi}{alphax}}{alphax} - \frac{sin2phi}{alphay \cdot alphay}}} \]
   4. Step-by-step derivation

      1. associate-/l/ 98.4%

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

      2. distribute-frac-neg 98.4%

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

      3. frac-2neg 98.4%

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

      4. div-inv 98.4%

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

      5. distribute-lft-neg-in 98.4%

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

      6. remove-double-neg 98.4%

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

      7. distribute-rgt-neg-in 98.4%

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

   5. Applied egg-rr 98.4%

      \[\leadsto \frac{\mathsf{log1p}\left(-u0\right)}{\color{blue}{cos2phi \cdot \frac{1}{alphax \cdot \left(-alphax\right)}} - \frac{sin2phi}{alphay \cdot alphay}} \]
   6. Step-by-step derivation

      1. un-div-inv 98.4%

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

      2. frac-2neg 98.4%

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

      3. distribute-rgt-neg-out 98.4%

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

      4. remove-double-neg 98.4%

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

      5. distribute-frac-neg 98.4%

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

      6. clear-num 98.4%

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

      7. associate-*r/ 98.4%

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

      8. distribute-neg-frac 98.4%

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

      9. metadata-eval 98.4%

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

   7. Applied egg-rr 98.4%

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

   8. Taylor expanded in alphax around inf 65.4%

      \[\leadsto \color{blue}{-1 \cdot \frac{{alphay}^{2} \cdot \log \left(1 - u0\right)}{sin2phi}} \]
   9. Step-by-step derivation

      1. mul-1-neg 65.4%

         \[\leadsto \color{blue}{-\frac{{alphay}^{2} \cdot \log \left(1 - u0\right)}{sin2phi}} \]

      2. unpow2 65.4%

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

      3. associate-/l* 64.6%

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

      4. distribute-neg-frac 64.6%

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

      5. distribute-rgt-neg-out 64.6%

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

      6. sub-neg 64.6%

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

      7. log1p-def 97.0%

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

   10. Simplified 97.0%

       \[\leadsto \color{blue}{\frac{alphay \cdot \left(-alphay\right)}{\frac{sin2phi}{\mathsf{log1p}\left(-u0\right)}}} \]
   11. Step-by-step derivation

       1. *-commutative 97.0%

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

       2. div-inv 96.9%

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

       3. times-frac 98.2%

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

       4. remove-double-neg 98.2%

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

       5. frac-2neg 98.2%

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

       6. div-inv 98.3%

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

       7. clear-num 98.5%

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

       8. /-rgt-identity 98.5%

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

   12. Applied egg-rr 98.5%

       \[\leadsto \color{blue}{\frac{alphay}{-sin2phi} \cdot \left(alphay \cdot \mathsf{log1p}\left(-u0\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 94.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 200:\\ \;\;\;\;-0.5 \cdot \frac{u0}{\frac{\frac{\frac{cos2phi}{alphax}}{-alphax} - \frac{sin2phi}{alphay \cdot alphay}}{u0}} - \frac{u0}{\frac{\frac{cos2phi}{alphax}}{-alphax} - \frac{sin2phi}{alphay \cdot alphay}}\\ \mathbf{else}:\\ \;\;\;\;\frac{alphay}{-sin2phi} \cdot \left(alphay \cdot \mathsf{log1p}\left(-u0\right)\right)\\ \end{array} \]

Alternative 6: 93.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{sin2phi}{alphay \cdot alphay}\\ t_1 := \frac{\frac{cos2phi}{alphax}}{-alphax} - t_0\\ \mathbf{if}\;t_0 \leq 200:\\ \;\;\;\;-0.5 \cdot \frac{u0}{\frac{t_1}{u0}} - \frac{u0}{t_1}\\ \mathbf{else}:\\ \;\;\;\;\left(alphay \cdot alphay\right) \cdot \frac{\mathsf{log1p}\left(-u0\right)}{-sin2phi}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (let* ((t_0 (/ sin2phi (* alphay alphay)))
        (t_1 (- (/ (/ cos2phi alphax) (- alphax)) t_0)))
   (if (<= t_0 200.0)
     (- (* -0.5 (/ u0 (/ t_1 u0))) (/ u0 t_1))
     (* (* alphay alphay) (/ (log1p (- u0)) (- sin2phi))))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float t_0 = sin2phi / (alphay * alphay);
	float t_1 = ((cos2phi / alphax) / -alphax) - t_0;
	float tmp;
	if (t_0 <= 200.0f) {
		tmp = (-0.5f * (u0 / (t_1 / u0))) - (u0 / t_1);
	} else {
		tmp = (alphay * alphay) * (log1pf(-u0) / -sin2phi);
	}
	return tmp;
}

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	t_0 = Float32(sin2phi / Float32(alphay * alphay))
	t_1 = Float32(Float32(Float32(cos2phi / alphax) / Float32(-alphax)) - t_0)
	tmp = Float32(0.0)
	if (t_0 <= Float32(200.0))
		tmp = Float32(Float32(Float32(-0.5) * Float32(u0 / Float32(t_1 / u0))) - Float32(u0 / t_1));
	else
		tmp = Float32(Float32(alphay * alphay) * Float32(log1p(Float32(-u0)) / Float32(-sin2phi)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 sin2phi (*.f32 alphay alphay)) < 200

   1. Initial program 50.3%

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

   3. Simplified 98.5%

      \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(-u0\right)}{\frac{\frac{-cos2phi}{alphax}}{alphax} - \frac{sin2phi}{alphay \cdot alphay}}} \]
   4. Step-by-step derivation

      1. associate-/l/ 98.5%

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

      2. distribute-frac-neg 98.5%

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

      3. frac-2neg 98.5%

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

      4. div-inv 98.3%

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

      5. distribute-lft-neg-in 98.3%

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

      6. remove-double-neg 98.3%

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

      7. distribute-rgt-neg-in 98.3%

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

   5. Applied egg-rr 98.3%

      \[\leadsto \frac{\mathsf{log1p}\left(-u0\right)}{\color{blue}{cos2phi \cdot \frac{1}{alphax \cdot \left(-alphax\right)}} - \frac{sin2phi}{alphay \cdot alphay}} \]
   6. Step-by-step derivation

      1. un-div-inv 98.5%

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

      2. frac-2neg 98.5%

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

      3. distribute-rgt-neg-out 98.5%

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

      4. remove-double-neg 98.5%

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

      5. distribute-frac-neg 98.5%

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

      6. clear-num 98.5%

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

      7. associate-*r/ 98.6%

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

      8. distribute-neg-frac 98.6%

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

      9. metadata-eval 98.6%

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

   7. Applied egg-rr 98.6%

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

   8. Taylor expanded in u0 around 0 90.0%

      \[\leadsto \color{blue}{-1 \cdot \frac{u0}{-1 \cdot \frac{cos2phi}{{alphax}^{2}} - \frac{sin2phi}{{alphay}^{2}}} + -0.5 \cdot \frac{{u0}^{2}}{-1 \cdot \frac{cos2phi}{{alphax}^{2}} - \frac{sin2phi}{{alphay}^{2}}}} \]
   9. Step-by-step derivation

      1. +-commutative 90.0%

         \[\leadsto \color{blue}{-0.5 \cdot \frac{{u0}^{2}}{-1 \cdot \frac{cos2phi}{{alphax}^{2}} - \frac{sin2phi}{{alphay}^{2}}} + -1 \cdot \frac{u0}{-1 \cdot \frac{cos2phi}{{alphax}^{2}} - \frac{sin2phi}{{alphay}^{2}}}} \]

      2. mul-1-neg 90.0%

         \[\leadsto -0.5 \cdot \frac{{u0}^{2}}{-1 \cdot \frac{cos2phi}{{alphax}^{2}} - \frac{sin2phi}{{alphay}^{2}}} + \color{blue}{\left(-\frac{u0}{-1 \cdot \frac{cos2phi}{{alphax}^{2}} - \frac{sin2phi}{{alphay}^{2}}}\right)} \]

      3. unsub-neg 90.0%

         \[\leadsto \color{blue}{-0.5 \cdot \frac{{u0}^{2}}{-1 \cdot \frac{cos2phi}{{alphax}^{2}} - \frac{sin2phi}{{alphay}^{2}}} - \frac{u0}{-1 \cdot \frac{cos2phi}{{alphax}^{2}} - \frac{sin2phi}{{alphay}^{2}}}} \]

   10. Simplified 90.1%

       \[\leadsto \color{blue}{-0.5 \cdot \frac{u0}{\frac{\frac{\frac{cos2phi}{alphax}}{-alphax} - \frac{sin2phi}{alphay \cdot alphay}}{u0}} - \frac{u0}{\frac{\frac{cos2phi}{alphax}}{-alphax} - \frac{sin2phi}{alphay \cdot alphay}}} \]

   if 200 < (/.f32 sin2phi (*.f32 alphay alphay))

   1. Initial program 64.8%

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

   3. Simplified 98.4%

      \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(-u0\right)}{\frac{\frac{-cos2phi}{alphax}}{alphax} - \frac{sin2phi}{alphay \cdot alphay}}} \]
   4. Step-by-step derivation

      1. associate-/l/ 98.4%

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

      2. distribute-frac-neg 98.4%

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

      3. frac-2neg 98.4%

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

      4. div-inv 98.4%

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

      5. distribute-lft-neg-in 98.4%

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

      6. remove-double-neg 98.4%

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

      7. distribute-rgt-neg-in 98.4%

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

   5. Applied egg-rr 98.4%

      \[\leadsto \frac{\mathsf{log1p}\left(-u0\right)}{\color{blue}{cos2phi \cdot \frac{1}{alphax \cdot \left(-alphax\right)}} - \frac{sin2phi}{alphay \cdot alphay}} \]
   6. Step-by-step derivation

      1. un-div-inv 98.4%

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

      2. frac-2neg 98.4%

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

      3. distribute-rgt-neg-out 98.4%

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

      4. remove-double-neg 98.4%

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

      5. distribute-frac-neg 98.4%

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

      6. clear-num 98.4%

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

      7. associate-*r/ 98.4%

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

      8. distribute-neg-frac 98.4%

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

      9. metadata-eval 98.4%

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

   7. Applied egg-rr 98.4%

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

   8. Taylor expanded in alphax around inf 65.4%

      \[\leadsto \color{blue}{-1 \cdot \frac{{alphay}^{2} \cdot \log \left(1 - u0\right)}{sin2phi}} \]
   9. Step-by-step derivation

      1. mul-1-neg 65.4%

         \[\leadsto \color{blue}{-\frac{{alphay}^{2} \cdot \log \left(1 - u0\right)}{sin2phi}} \]

      2. unpow2 65.4%

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

      3. associate-/l* 64.6%

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

      4. distribute-neg-frac 64.6%

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

      5. distribute-rgt-neg-out 64.6%

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

      6. sub-neg 64.6%

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

      7. log1p-def 97.0%

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

   10. Simplified 97.0%

       \[\leadsto \color{blue}{\frac{alphay \cdot \left(-alphay\right)}{\frac{sin2phi}{\mathsf{log1p}\left(-u0\right)}}} \]
   11. Step-by-step derivation

       1. div-inv 97.0%

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

       2. distribute-rgt-neg-out 97.0%

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

       3. distribute-lft-neg-out 97.0%

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

       4. clear-num 98.5%

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

       5. distribute-rgt-neg-in 98.5%

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

       6. *-commutative 98.5%

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

       7. distribute-frac-neg 98.5%

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

       8. frac-2neg 98.5%

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

       9. remove-double-neg 98.5%

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

   12. Applied egg-rr 98.5%

       \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(-u0\right)}{-sin2phi} \cdot \left(alphay \cdot alphay\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 94.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 200:\\ \;\;\;\;-0.5 \cdot \frac{u0}{\frac{\frac{\frac{cos2phi}{alphax}}{-alphax} - \frac{sin2phi}{alphay \cdot alphay}}{u0}} - \frac{u0}{\frac{\frac{cos2phi}{alphax}}{-alphax} - \frac{sin2phi}{alphay \cdot alphay}}\\ \mathbf{else}:\\ \;\;\;\;\left(alphay \cdot alphay\right) \cdot \frac{\mathsf{log1p}\left(-u0\right)}{-sin2phi}\\ \end{array} \]

Alternative 7: 98.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\mathsf{log1p}\left(-u0\right)}{\left(-\frac{\frac{cos2phi}{alphax}}{alphax}\right) - \frac{sin2phi}{alphay \cdot alphay}} \end{array} \]

FPCore

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

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	return Float32(log1p(Float32(-u0)) / Float32(Float32(-Float32(Float32(cos2phi / alphax) / alphax)) - Float32(sin2phi / Float32(alphay * alphay))))
end

Derivation

1. Initial program 57.5%

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

3. Simplified 98.5%

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

4. Final simplification 98.5%

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

Alternative 8: 98.3% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (/
  (- (log1p (- u0)))
  (+ (/ sin2phi (* alphay alphay)) (/ cos2phi (* alphax alphax)))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	return -log1pf(-u0) / ((sin2phi / (alphay * alphay)) + (cos2phi / (alphax * alphax)));
}

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	return Float32(Float32(-log1p(Float32(-u0))) / Float32(Float32(sin2phi / Float32(alphay * alphay)) + Float32(cos2phi / Float32(alphax * alphax))))
end

Derivation

1. Initial program 57.5%

   \[\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
2. Step-by-step derivation

   1. sub-neg 57.5%

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

   2. log1p-def 98.5%

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

3. Simplified 98.5%

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

4. Final simplification 98.5%

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

Alternative 9: 87.5% accurate, 3.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\frac{cos2phi}{alphax}}{-alphax} - \frac{sin2phi}{alphay \cdot alphay}\\ -0.5 \cdot \frac{u0}{\frac{t_0}{u0}} - \frac{u0}{t_0} \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (let* ((t_0
         (- (/ (/ cos2phi alphax) (- alphax)) (/ sin2phi (* alphay alphay)))))
   (- (* -0.5 (/ u0 (/ t_0 u0))) (/ u0 t_0))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float t_0 = ((cos2phi / alphax) / -alphax) - (sin2phi / (alphay * alphay));
	return (-0.5f * (u0 / (t_0 / u0))) - (u0 / t_0);
}

Fortran

real(4) function code(alphax, alphay, u0, cos2phi, sin2phi)
    real(4), intent (in) :: alphax
    real(4), intent (in) :: alphay
    real(4), intent (in) :: u0
    real(4), intent (in) :: cos2phi
    real(4), intent (in) :: sin2phi
    real(4) :: t_0
    t_0 = ((cos2phi / alphax) / -alphax) - (sin2phi / (alphay * alphay))
    code = ((-0.5e0) * (u0 / (t_0 / u0))) - (u0 / t_0)
end function

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	t_0 = Float32(Float32(Float32(cos2phi / alphax) / Float32(-alphax)) - Float32(sin2phi / Float32(alphay * alphay)))
	return Float32(Float32(Float32(-0.5) * Float32(u0 / Float32(t_0 / u0))) - Float32(u0 / t_0))
end

MATLAB

function tmp = code(alphax, alphay, u0, cos2phi, sin2phi)
	t_0 = ((cos2phi / alphax) / -alphax) - (sin2phi / (alphay * alphay));
	tmp = (single(-0.5) * (u0 / (t_0 / u0))) - (u0 / t_0);
end

Derivation

1. Initial program 57.5%

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

3. Simplified 98.5%

   \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(-u0\right)}{\frac{\frac{-cos2phi}{alphax}}{alphax} - \frac{sin2phi}{alphay \cdot alphay}}} \]
4. Step-by-step derivation

   1. associate-/l/ 98.5%

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

   2. distribute-frac-neg 98.5%

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

   3. frac-2neg 98.5%

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

   4. div-inv 98.4%

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

   5. distribute-lft-neg-in 98.4%

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

   6. remove-double-neg 98.4%

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

   7. distribute-rgt-neg-in 98.4%

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

5. Applied egg-rr 98.4%

   \[\leadsto \frac{\mathsf{log1p}\left(-u0\right)}{\color{blue}{cos2phi \cdot \frac{1}{alphax \cdot \left(-alphax\right)}} - \frac{sin2phi}{alphay \cdot alphay}} \]
6. Step-by-step derivation

   1. un-div-inv 98.5%

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

   2. frac-2neg 98.5%

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

   3. distribute-rgt-neg-out 98.5%

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

   4. remove-double-neg 98.5%

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

   5. distribute-frac-neg 98.5%

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

   6. clear-num 98.5%

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

   7. associate-*r/ 98.5%

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

   8. distribute-neg-frac 98.5%

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

   9. metadata-eval 98.5%

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

7. Applied egg-rr 98.5%

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

8. Taylor expanded in u0 around 0 89.2%

   \[\leadsto \color{blue}{-1 \cdot \frac{u0}{-1 \cdot \frac{cos2phi}{{alphax}^{2}} - \frac{sin2phi}{{alphay}^{2}}} + -0.5 \cdot \frac{{u0}^{2}}{-1 \cdot \frac{cos2phi}{{alphax}^{2}} - \frac{sin2phi}{{alphay}^{2}}}} \]
9. Step-by-step derivation

   1. +-commutative 89.2%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{{u0}^{2}}{-1 \cdot \frac{cos2phi}{{alphax}^{2}} - \frac{sin2phi}{{alphay}^{2}}} + -1 \cdot \frac{u0}{-1 \cdot \frac{cos2phi}{{alphax}^{2}} - \frac{sin2phi}{{alphay}^{2}}}} \]

   2. mul-1-neg 89.2%

      \[\leadsto -0.5 \cdot \frac{{u0}^{2}}{-1 \cdot \frac{cos2phi}{{alphax}^{2}} - \frac{sin2phi}{{alphay}^{2}}} + \color{blue}{\left(-\frac{u0}{-1 \cdot \frac{cos2phi}{{alphax}^{2}} - \frac{sin2phi}{{alphay}^{2}}}\right)} \]

   3. unsub-neg 89.2%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{{u0}^{2}}{-1 \cdot \frac{cos2phi}{{alphax}^{2}} - \frac{sin2phi}{{alphay}^{2}}} - \frac{u0}{-1 \cdot \frac{cos2phi}{{alphax}^{2}} - \frac{sin2phi}{{alphay}^{2}}}} \]

10. Simplified 88.9%

    \[\leadsto \color{blue}{-0.5 \cdot \frac{u0}{\frac{\frac{\frac{cos2phi}{alphax}}{-alphax} - \frac{sin2phi}{alphay \cdot alphay}}{u0}} - \frac{u0}{\frac{\frac{cos2phi}{alphax}}{-alphax} - \frac{sin2phi}{alphay \cdot alphay}}} \]

11. Final simplification 88.9%

    \[\leadsto -0.5 \cdot \frac{u0}{\frac{\frac{\frac{cos2phi}{alphax}}{-alphax} - \frac{sin2phi}{alphay \cdot alphay}}{u0}} - \frac{u0}{\frac{\frac{cos2phi}{alphax}}{-alphax} - \frac{sin2phi}{alphay \cdot alphay}} \]

Alternative 10: 83.1% accurate, 4.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 4.999999969612645 \cdot 10^{-9}:\\ \;\;\;\;\frac{u0}{\frac{cos2phi}{alphax \cdot alphax} - \frac{\frac{sin2phi}{alphay}}{-alphay}}\\ \mathbf{else}:\\ \;\;\;\;\frac{alphay \cdot \left(-alphay\right)}{sin2phi \cdot 0.5 - \left(u0 \cdot \left(sin2phi \cdot -0.08333333333333333\right) + \frac{sin2phi}{u0}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (if (<= (/ sin2phi (* alphay alphay)) 4.999999969612645e-9)
   (/ u0 (- (/ cos2phi (* alphax alphax)) (/ (/ sin2phi alphay) (- alphay))))
   (/
    (* alphay (- alphay))
    (-
     (* sin2phi 0.5)
     (+ (* u0 (* sin2phi -0.08333333333333333)) (/ sin2phi u0))))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float tmp;
	if ((sin2phi / (alphay * alphay)) <= 4.999999969612645e-9f) {
		tmp = u0 / ((cos2phi / (alphax * alphax)) - ((sin2phi / alphay) / -alphay));
	} else {
		tmp = (alphay * -alphay) / ((sin2phi * 0.5f) - ((u0 * (sin2phi * -0.08333333333333333f)) + (sin2phi / u0)));
	}
	return tmp;
}

Fortran

real(4) function code(alphax, alphay, u0, cos2phi, sin2phi)
    real(4), intent (in) :: alphax
    real(4), intent (in) :: alphay
    real(4), intent (in) :: u0
    real(4), intent (in) :: cos2phi
    real(4), intent (in) :: sin2phi
    real(4) :: tmp
    if ((sin2phi / (alphay * alphay)) <= 4.999999969612645e-9) then
        tmp = u0 / ((cos2phi / (alphax * alphax)) - ((sin2phi / alphay) / -alphay))
    else
        tmp = (alphay * -alphay) / ((sin2phi * 0.5e0) - ((u0 * (sin2phi * (-0.08333333333333333e0))) + (sin2phi / u0)))
    end if
    code = tmp
end function

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = Float32(0.0)
	if (Float32(sin2phi / Float32(alphay * alphay)) <= Float32(4.999999969612645e-9))
		tmp = Float32(u0 / Float32(Float32(cos2phi / Float32(alphax * alphax)) - Float32(Float32(sin2phi / alphay) / Float32(-alphay))));
	else
		tmp = Float32(Float32(alphay * Float32(-alphay)) / Float32(Float32(sin2phi * Float32(0.5)) - Float32(Float32(u0 * Float32(sin2phi * Float32(-0.08333333333333333))) + Float32(sin2phi / u0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = single(0.0);
	if ((sin2phi / (alphay * alphay)) <= single(4.999999969612645e-9))
		tmp = u0 / ((cos2phi / (alphax * alphax)) - ((sin2phi / alphay) / -alphay));
	else
		tmp = (alphay * -alphay) / ((sin2phi * single(0.5)) - ((u0 * (sin2phi * single(-0.08333333333333333))) + (sin2phi / u0)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 sin2phi (*.f32 alphay alphay)) < 4.99999997e-9

   1. Initial program 46.8%

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

   2. Taylor expanded in u0 around 0 80.0%

      \[\leadsto \color{blue}{\frac{u0}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}}} \]
   3. Step-by-step derivation

      1. +-commutative 80.0%

         \[\leadsto \frac{u0}{\color{blue}{\frac{sin2phi}{{alphay}^{2}} + \frac{cos2phi}{{alphax}^{2}}}} \]

      2. unpow2 80.0%

         \[\leadsto \frac{u0}{\frac{sin2phi}{\color{blue}{alphay \cdot alphay}} + \frac{cos2phi}{{alphax}^{2}}} \]

      3. unpow2 80.0%

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

   4. Simplified 80.0%

      \[\leadsto \color{blue}{\frac{u0}{\frac{sin2phi}{alphay \cdot alphay} + \frac{cos2phi}{alphax \cdot alphax}}} \]
   5. Step-by-step derivation

      1. associate-/r* 80.2%

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

      2. frac-2neg 80.2%

         \[\leadsto \frac{u0}{\color{blue}{\frac{-\frac{sin2phi}{alphay}}{-alphay}} + \frac{cos2phi}{alphax \cdot alphax}} \]

      3. distribute-frac-neg 80.2%

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

   6. Applied egg-rr 80.2%

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

   if 4.99999997e-9 < (/.f32 sin2phi (*.f32 alphay alphay))

   1. Initial program 62.9%

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

   3. Simplified 98.5%

      \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(-u0\right)}{\frac{\frac{-cos2phi}{alphax}}{alphax} - \frac{sin2phi}{alphay \cdot alphay}}} \]
   4. Step-by-step derivation

      1. associate-/l/ 98.5%

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

      2. distribute-frac-neg 98.5%

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

      3. frac-2neg 98.5%

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

      4. div-inv 98.4%

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

      5. distribute-lft-neg-in 98.4%

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

      6. remove-double-neg 98.4%

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

      7. distribute-rgt-neg-in 98.4%

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

   5. Applied egg-rr 98.4%

      \[\leadsto \frac{\mathsf{log1p}\left(-u0\right)}{\color{blue}{cos2phi \cdot \frac{1}{alphax \cdot \left(-alphax\right)}} - \frac{sin2phi}{alphay \cdot alphay}} \]
   6. Step-by-step derivation

      1. un-div-inv 98.5%

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

      2. frac-2neg 98.5%

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

      3. distribute-rgt-neg-out 98.5%

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

      4. remove-double-neg 98.5%

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

      5. distribute-frac-neg 98.5%

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

      6. clear-num 98.5%

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

      7. associate-*r/ 98.5%

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

      8. distribute-neg-frac 98.5%

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

      9. metadata-eval 98.5%

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

   7. Applied egg-rr 98.5%

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

   8. Taylor expanded in alphax around inf 62.2%

      \[\leadsto \color{blue}{-1 \cdot \frac{{alphay}^{2} \cdot \log \left(1 - u0\right)}{sin2phi}} \]
   9. Step-by-step derivation

      1. mul-1-neg 62.2%

         \[\leadsto \color{blue}{-\frac{{alphay}^{2} \cdot \log \left(1 - u0\right)}{sin2phi}} \]

      2. unpow2 62.2%

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

      3. associate-/l* 61.6%

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

      4. distribute-neg-frac 61.6%

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

      5. distribute-rgt-neg-out 61.6%

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

      6. sub-neg 61.6%

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

      7. log1p-def 94.3%

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

   10. Simplified 94.3%

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

   11. Taylor expanded in u0 around 0 89.3%

       \[\leadsto \frac{alphay \cdot \left(-alphay\right)}{\color{blue}{-1 \cdot \left(u0 \cdot \left(-0.3333333333333333 \cdot sin2phi + 0.25 \cdot sin2phi\right)\right) + \left(-1 \cdot \frac{sin2phi}{u0} + 0.5 \cdot sin2phi\right)}} \]
   12. Step-by-step derivation

       1. associate-+r+ 89.3%

          \[\leadsto \frac{alphay \cdot \left(-alphay\right)}{\color{blue}{\left(-1 \cdot \left(u0 \cdot \left(-0.3333333333333333 \cdot sin2phi + 0.25 \cdot sin2phi\right)\right) + -1 \cdot \frac{sin2phi}{u0}\right) + 0.5 \cdot sin2phi}} \]

       2. mul-1-neg 89.3%

          \[\leadsto \frac{alphay \cdot \left(-alphay\right)}{\left(\color{blue}{\left(-u0 \cdot \left(-0.3333333333333333 \cdot sin2phi + 0.25 \cdot sin2phi\right)\right)} + -1 \cdot \frac{sin2phi}{u0}\right) + 0.5 \cdot sin2phi} \]

       3. mul-1-neg 89.3%

          \[\leadsto \frac{alphay \cdot \left(-alphay\right)}{\left(\left(-u0 \cdot \left(-0.3333333333333333 \cdot sin2phi + 0.25 \cdot sin2phi\right)\right) + \color{blue}{\left(-\frac{sin2phi}{u0}\right)}\right) + 0.5 \cdot sin2phi} \]

       4. distribute-neg-out 89.3%

          \[\leadsto \frac{alphay \cdot \left(-alphay\right)}{\color{blue}{\left(-\left(u0 \cdot \left(-0.3333333333333333 \cdot sin2phi + 0.25 \cdot sin2phi\right) + \frac{sin2phi}{u0}\right)\right)} + 0.5 \cdot sin2phi} \]

       5. distribute-rgt-out 89.3%

          \[\leadsto \frac{alphay \cdot \left(-alphay\right)}{\left(-\left(u0 \cdot \color{blue}{\left(sin2phi \cdot \left(-0.3333333333333333 + 0.25\right)\right)} + \frac{sin2phi}{u0}\right)\right) + 0.5 \cdot sin2phi} \]

       6. metadata-eval 89.3%

          \[\leadsto \frac{alphay \cdot \left(-alphay\right)}{\left(-\left(u0 \cdot \left(sin2phi \cdot \color{blue}{-0.08333333333333333}\right) + \frac{sin2phi}{u0}\right)\right) + 0.5 \cdot sin2phi} \]

       7. *-commutative 89.3%

          \[\leadsto \frac{alphay \cdot \left(-alphay\right)}{\left(-\left(u0 \cdot \left(sin2phi \cdot -0.08333333333333333\right) + \frac{sin2phi}{u0}\right)\right) + \color{blue}{sin2phi \cdot 0.5}} \]

   13. Simplified 89.3%

       \[\leadsto \frac{alphay \cdot \left(-alphay\right)}{\color{blue}{\left(-\left(u0 \cdot \left(sin2phi \cdot -0.08333333333333333\right) + \frac{sin2phi}{u0}\right)\right) + sin2phi \cdot 0.5}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 86.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 4.999999969612645 \cdot 10^{-9}:\\ \;\;\;\;\frac{u0}{\frac{cos2phi}{alphax \cdot alphax} - \frac{\frac{sin2phi}{alphay}}{-alphay}}\\ \mathbf{else}:\\ \;\;\;\;\frac{alphay \cdot \left(-alphay\right)}{sin2phi \cdot 0.5 - \left(u0 \cdot \left(sin2phi \cdot -0.08333333333333333\right) + \frac{sin2phi}{u0}\right)}\\ \end{array} \]

Alternative 11: 81.1% accurate, 5.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 4.999999969612645 \cdot 10^{-9}:\\ \;\;\;\;\frac{u0}{\frac{cos2phi}{alphax \cdot alphax} - \frac{\frac{sin2phi}{alphay}}{-alphay}}\\ \mathbf{else}:\\ \;\;\;\;\frac{alphay \cdot \left(-alphay\right)}{sin2phi \cdot 0.5 - \frac{sin2phi}{u0}}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (if (<= (/ sin2phi (* alphay alphay)) 4.999999969612645e-9)
   (/ u0 (- (/ cos2phi (* alphax alphax)) (/ (/ sin2phi alphay) (- alphay))))
   (/ (* alphay (- alphay)) (- (* sin2phi 0.5) (/ sin2phi u0)))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float tmp;
	if ((sin2phi / (alphay * alphay)) <= 4.999999969612645e-9f) {
		tmp = u0 / ((cos2phi / (alphax * alphax)) - ((sin2phi / alphay) / -alphay));
	} else {
		tmp = (alphay * -alphay) / ((sin2phi * 0.5f) - (sin2phi / u0));
	}
	return tmp;
}

Fortran

real(4) function code(alphax, alphay, u0, cos2phi, sin2phi)
    real(4), intent (in) :: alphax
    real(4), intent (in) :: alphay
    real(4), intent (in) :: u0
    real(4), intent (in) :: cos2phi
    real(4), intent (in) :: sin2phi
    real(4) :: tmp
    if ((sin2phi / (alphay * alphay)) <= 4.999999969612645e-9) then
        tmp = u0 / ((cos2phi / (alphax * alphax)) - ((sin2phi / alphay) / -alphay))
    else
        tmp = (alphay * -alphay) / ((sin2phi * 0.5e0) - (sin2phi / u0))
    end if
    code = tmp
end function

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = Float32(0.0)
	if (Float32(sin2phi / Float32(alphay * alphay)) <= Float32(4.999999969612645e-9))
		tmp = Float32(u0 / Float32(Float32(cos2phi / Float32(alphax * alphax)) - Float32(Float32(sin2phi / alphay) / Float32(-alphay))));
	else
		tmp = Float32(Float32(alphay * Float32(-alphay)) / Float32(Float32(sin2phi * Float32(0.5)) - Float32(sin2phi / u0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = single(0.0);
	if ((sin2phi / (alphay * alphay)) <= single(4.999999969612645e-9))
		tmp = u0 / ((cos2phi / (alphax * alphax)) - ((sin2phi / alphay) / -alphay));
	else
		tmp = (alphay * -alphay) / ((sin2phi * single(0.5)) - (sin2phi / u0));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 sin2phi (*.f32 alphay alphay)) < 4.99999997e-9

   1. Initial program 46.8%

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

   2. Taylor expanded in u0 around 0 80.0%

      \[\leadsto \color{blue}{\frac{u0}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}}} \]
   3. Step-by-step derivation

      1. +-commutative 80.0%

         \[\leadsto \frac{u0}{\color{blue}{\frac{sin2phi}{{alphay}^{2}} + \frac{cos2phi}{{alphax}^{2}}}} \]

      2. unpow2 80.0%

         \[\leadsto \frac{u0}{\frac{sin2phi}{\color{blue}{alphay \cdot alphay}} + \frac{cos2phi}{{alphax}^{2}}} \]

      3. unpow2 80.0%

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

   4. Simplified 80.0%

      \[\leadsto \color{blue}{\frac{u0}{\frac{sin2phi}{alphay \cdot alphay} + \frac{cos2phi}{alphax \cdot alphax}}} \]
   5. Step-by-step derivation

      1. associate-/r* 80.2%

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

      2. frac-2neg 80.2%

         \[\leadsto \frac{u0}{\color{blue}{\frac{-\frac{sin2phi}{alphay}}{-alphay}} + \frac{cos2phi}{alphax \cdot alphax}} \]

      3. distribute-frac-neg 80.2%

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

   6. Applied egg-rr 80.2%

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

   if 4.99999997e-9 < (/.f32 sin2phi (*.f32 alphay alphay))

   1. Initial program 62.9%

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

   3. Simplified 98.5%

      \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(-u0\right)}{\frac{\frac{-cos2phi}{alphax}}{alphax} - \frac{sin2phi}{alphay \cdot alphay}}} \]
   4. Step-by-step derivation

      1. associate-/l/ 98.5%

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

      2. distribute-frac-neg 98.5%

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

      3. frac-2neg 98.5%

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

      4. div-inv 98.4%

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

      5. distribute-lft-neg-in 98.4%

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

      6. remove-double-neg 98.4%

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

      7. distribute-rgt-neg-in 98.4%

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

   5. Applied egg-rr 98.4%

      \[\leadsto \frac{\mathsf{log1p}\left(-u0\right)}{\color{blue}{cos2phi \cdot \frac{1}{alphax \cdot \left(-alphax\right)}} - \frac{sin2phi}{alphay \cdot alphay}} \]
   6. Step-by-step derivation

      1. un-div-inv 98.5%

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

      2. frac-2neg 98.5%

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

      3. distribute-rgt-neg-out 98.5%

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

      4. remove-double-neg 98.5%

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

      5. distribute-frac-neg 98.5%

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

      6. clear-num 98.5%

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

      7. associate-*r/ 98.5%

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

      8. distribute-neg-frac 98.5%

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

      9. metadata-eval 98.5%

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

   7. Applied egg-rr 98.5%

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

   8. Taylor expanded in alphax around inf 62.2%

      \[\leadsto \color{blue}{-1 \cdot \frac{{alphay}^{2} \cdot \log \left(1 - u0\right)}{sin2phi}} \]
   9. Step-by-step derivation

      1. mul-1-neg 62.2%

         \[\leadsto \color{blue}{-\frac{{alphay}^{2} \cdot \log \left(1 - u0\right)}{sin2phi}} \]

      2. unpow2 62.2%

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

      3. associate-/l* 61.6%

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

      4. distribute-neg-frac 61.6%

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

      5. distribute-rgt-neg-out 61.6%

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

      6. sub-neg 61.6%

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

      7. log1p-def 94.3%

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

   10. Simplified 94.3%

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

   11. Taylor expanded in u0 around 0 86.4%

       \[\leadsto \frac{alphay \cdot \left(-alphay\right)}{\color{blue}{-1 \cdot \frac{sin2phi}{u0} + 0.5 \cdot sin2phi}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 84.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 4.999999969612645 \cdot 10^{-9}:\\ \;\;\;\;\frac{u0}{\frac{cos2phi}{alphax \cdot alphax} - \frac{\frac{sin2phi}{alphay}}{-alphay}}\\ \mathbf{else}:\\ \;\;\;\;\frac{alphay \cdot \left(-alphay\right)}{sin2phi \cdot 0.5 - \frac{sin2phi}{u0}}\\ \end{array} \]

Alternative 12: 81.1% accurate, 6.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{sin2phi}{alphay \cdot alphay}\\ \mathbf{if}\;t_0 \leq 4.999999969612645 \cdot 10^{-9}:\\ \;\;\;\;\frac{u0}{t_0 + \frac{cos2phi}{alphax \cdot alphax}}\\ \mathbf{else}:\\ \;\;\;\;\frac{alphay \cdot \left(-alphay\right)}{sin2phi \cdot 0.5 - \frac{sin2phi}{u0}}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (let* ((t_0 (/ sin2phi (* alphay alphay))))
   (if (<= t_0 4.999999969612645e-9)
     (/ u0 (+ t_0 (/ cos2phi (* alphax alphax))))
     (/ (* alphay (- alphay)) (- (* sin2phi 0.5) (/ sin2phi u0))))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float t_0 = sin2phi / (alphay * alphay);
	float tmp;
	if (t_0 <= 4.999999969612645e-9f) {
		tmp = u0 / (t_0 + (cos2phi / (alphax * alphax)));
	} else {
		tmp = (alphay * -alphay) / ((sin2phi * 0.5f) - (sin2phi / u0));
	}
	return tmp;
}

Fortran

real(4) function code(alphax, alphay, u0, cos2phi, sin2phi)
    real(4), intent (in) :: alphax
    real(4), intent (in) :: alphay
    real(4), intent (in) :: u0
    real(4), intent (in) :: cos2phi
    real(4), intent (in) :: sin2phi
    real(4) :: t_0
    real(4) :: tmp
    t_0 = sin2phi / (alphay * alphay)
    if (t_0 <= 4.999999969612645e-9) then
        tmp = u0 / (t_0 + (cos2phi / (alphax * alphax)))
    else
        tmp = (alphay * -alphay) / ((sin2phi * 0.5e0) - (sin2phi / u0))
    end if
    code = tmp
end function

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	t_0 = Float32(sin2phi / Float32(alphay * alphay))
	tmp = Float32(0.0)
	if (t_0 <= Float32(4.999999969612645e-9))
		tmp = Float32(u0 / Float32(t_0 + Float32(cos2phi / Float32(alphax * alphax))));
	else
		tmp = Float32(Float32(alphay * Float32(-alphay)) / Float32(Float32(sin2phi * Float32(0.5)) - Float32(sin2phi / u0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alphax, alphay, u0, cos2phi, sin2phi)
	t_0 = sin2phi / (alphay * alphay);
	tmp = single(0.0);
	if (t_0 <= single(4.999999969612645e-9))
		tmp = u0 / (t_0 + (cos2phi / (alphax * alphax)));
	else
		tmp = (alphay * -alphay) / ((sin2phi * single(0.5)) - (sin2phi / u0));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 sin2phi (*.f32 alphay alphay)) < 4.99999997e-9

   1. Initial program 46.8%

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

   2. Taylor expanded in u0 around 0 80.0%

      \[\leadsto \color{blue}{\frac{u0}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}}} \]
   3. Step-by-step derivation

      1. +-commutative 80.0%

         \[\leadsto \frac{u0}{\color{blue}{\frac{sin2phi}{{alphay}^{2}} + \frac{cos2phi}{{alphax}^{2}}}} \]

      2. unpow2 80.0%

         \[\leadsto \frac{u0}{\frac{sin2phi}{\color{blue}{alphay \cdot alphay}} + \frac{cos2phi}{{alphax}^{2}}} \]

      3. unpow2 80.0%

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

   4. Simplified 80.0%

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

   if 4.99999997e-9 < (/.f32 sin2phi (*.f32 alphay alphay))

   1. Initial program 62.9%

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

   3. Simplified 98.5%

      \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(-u0\right)}{\frac{\frac{-cos2phi}{alphax}}{alphax} - \frac{sin2phi}{alphay \cdot alphay}}} \]
   4. Step-by-step derivation

      1. associate-/l/ 98.5%

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

      2. distribute-frac-neg 98.5%

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

      3. frac-2neg 98.5%

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

      4. div-inv 98.4%

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

      5. distribute-lft-neg-in 98.4%

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

      6. remove-double-neg 98.4%

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

      7. distribute-rgt-neg-in 98.4%

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

   5. Applied egg-rr 98.4%

      \[\leadsto \frac{\mathsf{log1p}\left(-u0\right)}{\color{blue}{cos2phi \cdot \frac{1}{alphax \cdot \left(-alphax\right)}} - \frac{sin2phi}{alphay \cdot alphay}} \]
   6. Step-by-step derivation

      1. un-div-inv 98.5%

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

      2. frac-2neg 98.5%

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

      3. distribute-rgt-neg-out 98.5%

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

      4. remove-double-neg 98.5%

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

      5. distribute-frac-neg 98.5%

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

      6. clear-num 98.5%

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

      7. associate-*r/ 98.5%

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

      8. distribute-neg-frac 98.5%

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

      9. metadata-eval 98.5%

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

   7. Applied egg-rr 98.5%

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

   8. Taylor expanded in alphax around inf 62.2%

      \[\leadsto \color{blue}{-1 \cdot \frac{{alphay}^{2} \cdot \log \left(1 - u0\right)}{sin2phi}} \]
   9. Step-by-step derivation

      1. mul-1-neg 62.2%

         \[\leadsto \color{blue}{-\frac{{alphay}^{2} \cdot \log \left(1 - u0\right)}{sin2phi}} \]

      2. unpow2 62.2%

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

      3. associate-/l* 61.6%

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

      4. distribute-neg-frac 61.6%

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

      5. distribute-rgt-neg-out 61.6%

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

      6. sub-neg 61.6%

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

      7. log1p-def 94.3%

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

   10. Simplified 94.3%

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

   11. Taylor expanded in u0 around 0 86.4%

       \[\leadsto \frac{alphay \cdot \left(-alphay\right)}{\color{blue}{-1 \cdot \frac{sin2phi}{u0} + 0.5 \cdot sin2phi}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 84.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 4.999999969612645 \cdot 10^{-9}:\\ \;\;\;\;\frac{u0}{\frac{sin2phi}{alphay \cdot alphay} + \frac{cos2phi}{alphax \cdot alphax}}\\ \mathbf{else}:\\ \;\;\;\;\frac{alphay \cdot \left(-alphay\right)}{sin2phi \cdot 0.5 - \frac{sin2phi}{u0}}\\ \end{array} \]

Alternative 13: 66.9% accurate, 8.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 4.999999918875795 \cdot 10^{-18}:\\ \;\;\;\;\frac{u0}{\frac{cos2phi}{alphax \cdot alphax}}\\ \mathbf{else}:\\ \;\;\;\;alphay \cdot \left(u0 \cdot \frac{alphay}{sin2phi}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (if (<= (/ sin2phi (* alphay alphay)) 4.999999918875795e-18)
   (/ u0 (/ cos2phi (* alphax alphax)))
   (* alphay (* u0 (/ alphay sin2phi)))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float tmp;
	if ((sin2phi / (alphay * alphay)) <= 4.999999918875795e-18f) {
		tmp = u0 / (cos2phi / (alphax * alphax));
	} else {
		tmp = alphay * (u0 * (alphay / sin2phi));
	}
	return tmp;
}

Fortran

real(4) function code(alphax, alphay, u0, cos2phi, sin2phi)
    real(4), intent (in) :: alphax
    real(4), intent (in) :: alphay
    real(4), intent (in) :: u0
    real(4), intent (in) :: cos2phi
    real(4), intent (in) :: sin2phi
    real(4) :: tmp
    if ((sin2phi / (alphay * alphay)) <= 4.999999918875795e-18) then
        tmp = u0 / (cos2phi / (alphax * alphax))
    else
        tmp = alphay * (u0 * (alphay / sin2phi))
    end if
    code = tmp
end function

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = Float32(0.0)
	if (Float32(sin2phi / Float32(alphay * alphay)) <= Float32(4.999999918875795e-18))
		tmp = Float32(u0 / Float32(cos2phi / Float32(alphax * alphax)));
	else
		tmp = Float32(alphay * Float32(u0 * Float32(alphay / sin2phi)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = single(0.0);
	if ((sin2phi / (alphay * alphay)) <= single(4.999999918875795e-18))
		tmp = u0 / (cos2phi / (alphax * alphax));
	else
		tmp = alphay * (u0 * (alphay / sin2phi));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 sin2phi (*.f32 alphay alphay)) < 4.99999992e-18

   1. Initial program 49.0%

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

   2. Taylor expanded in u0 around 0 77.9%

      \[\leadsto \color{blue}{\frac{u0}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}}} \]
   3. Step-by-step derivation

      1. +-commutative 77.9%

         \[\leadsto \frac{u0}{\color{blue}{\frac{sin2phi}{{alphay}^{2}} + \frac{cos2phi}{{alphax}^{2}}}} \]

      2. unpow2 77.9%

         \[\leadsto \frac{u0}{\frac{sin2phi}{\color{blue}{alphay \cdot alphay}} + \frac{cos2phi}{{alphax}^{2}}} \]

      3. unpow2 77.9%

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

   4. Simplified 77.9%

      \[\leadsto \color{blue}{\frac{u0}{\frac{sin2phi}{alphay \cdot alphay} + \frac{cos2phi}{alphax \cdot alphax}}} \]
   5. Step-by-step derivation

      1. +-commutative 77.9%

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

      2. associate-/r* 77.9%

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

      3. frac-add 77.9%

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

   6. Applied egg-rr 77.9%

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

   7. Taylor expanded in cos2phi around inf 57.7%

      \[\leadsto \frac{u0}{\color{blue}{\frac{cos2phi}{{alphax}^{2}}}} \]
   8. Step-by-step derivation

      1. unpow2 57.7%

         \[\leadsto \frac{u0}{\frac{cos2phi}{\color{blue}{alphax \cdot alphax}}} \]

   9. Simplified 57.7%

      \[\leadsto \frac{u0}{\color{blue}{\frac{cos2phi}{alphax \cdot alphax}}} \]

   if 4.99999992e-18 < (/.f32 sin2phi (*.f32 alphay alphay))

   1. Initial program 60.0%

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

   2. Taylor expanded in u0 around 0 77.6%

      \[\leadsto \color{blue}{\frac{u0}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}}} \]
   3. Step-by-step derivation

      1. +-commutative 77.6%

         \[\leadsto \frac{u0}{\color{blue}{\frac{sin2phi}{{alphay}^{2}} + \frac{cos2phi}{{alphax}^{2}}}} \]

      2. unpow2 77.6%

         \[\leadsto \frac{u0}{\frac{sin2phi}{\color{blue}{alphay \cdot alphay}} + \frac{cos2phi}{{alphax}^{2}}} \]

      3. unpow2 77.6%

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

   4. Simplified 77.6%

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

   5. Taylor expanded in sin2phi around inf 71.5%

      \[\leadsto \frac{u0}{\color{blue}{\frac{sin2phi}{{alphay}^{2}}}} \]
   6. Step-by-step derivation

      1. unpow2 71.5%

         \[\leadsto \frac{u0}{\frac{sin2phi}{\color{blue}{alphay \cdot alphay}}} \]

   7. Simplified 71.5%

      \[\leadsto \frac{u0}{\color{blue}{\frac{sin2phi}{alphay \cdot alphay}}} \]
   8. Step-by-step derivation

      1. div-inv 71.5%

         \[\leadsto \color{blue}{u0 \cdot \frac{1}{\frac{sin2phi}{alphay \cdot alphay}}} \]

      2. clear-num 71.8%

         \[\leadsto u0 \cdot \color{blue}{\frac{alphay \cdot alphay}{sin2phi}} \]

      3. associate-*l/ 71.9%

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

      4. *-commutative 71.9%

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

      5. *-commutative 71.9%

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

   9. Applied egg-rr 71.9%

      \[\leadsto \color{blue}{\left(alphay \cdot \frac{alphay}{sin2phi}\right) \cdot u0} \]
   10. Step-by-step derivation

       1. add-exp-log_binary32 70.0%

          \[\leadsto \color{blue}{e^{\log \left(\left(alphay \cdot \frac{alphay}{sin2phi}\right) \cdot u0\right)}} \]

   11. Applied rewrite-once 70.0%

       \[\leadsto \color{blue}{e^{\log \left(\left(alphay \cdot \frac{alphay}{sin2phi}\right) \cdot u0\right)}} \]
   12. Step-by-step derivation

       1. rem-exp-log 71.9%

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

       2. associate-*l* 71.9%

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

   13. Simplified 71.9%

       \[\leadsto \color{blue}{alphay \cdot \left(\frac{alphay}{sin2phi} \cdot u0\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 68.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 4.999999918875795 \cdot 10^{-18}:\\ \;\;\;\;\frac{u0}{\frac{cos2phi}{alphax \cdot alphax}}\\ \mathbf{else}:\\ \;\;\;\;alphay \cdot \left(u0 \cdot \frac{alphay}{sin2phi}\right)\\ \end{array} \]

Alternative 14: 65.6% accurate, 8.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 1.9999999996399175 \cdot 10^{-23}:\\ \;\;\;\;\frac{u0}{\frac{\frac{cos2phi}{alphax}}{alphax}}\\ \mathbf{else}:\\ \;\;\;\;alphay \cdot \left(u0 \cdot \frac{alphay}{sin2phi}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (if (<= (/ sin2phi (* alphay alphay)) 1.9999999996399175e-23)
   (/ u0 (/ (/ cos2phi alphax) alphax))
   (* alphay (* u0 (/ alphay sin2phi)))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float tmp;
	if ((sin2phi / (alphay * alphay)) <= 1.9999999996399175e-23f) {
		tmp = u0 / ((cos2phi / alphax) / alphax);
	} else {
		tmp = alphay * (u0 * (alphay / sin2phi));
	}
	return tmp;
}

Fortran

real(4) function code(alphax, alphay, u0, cos2phi, sin2phi)
    real(4), intent (in) :: alphax
    real(4), intent (in) :: alphay
    real(4), intent (in) :: u0
    real(4), intent (in) :: cos2phi
    real(4), intent (in) :: sin2phi
    real(4) :: tmp
    if ((sin2phi / (alphay * alphay)) <= 1.9999999996399175e-23) then
        tmp = u0 / ((cos2phi / alphax) / alphax)
    else
        tmp = alphay * (u0 * (alphay / sin2phi))
    end if
    code = tmp
end function

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = Float32(0.0)
	if (Float32(sin2phi / Float32(alphay * alphay)) <= Float32(1.9999999996399175e-23))
		tmp = Float32(u0 / Float32(Float32(cos2phi / alphax) / alphax));
	else
		tmp = Float32(alphay * Float32(u0 * Float32(alphay / sin2phi)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = single(0.0);
	if ((sin2phi / (alphay * alphay)) <= single(1.9999999996399175e-23))
		tmp = u0 / ((cos2phi / alphax) / alphax);
	else
		tmp = alphay * (u0 * (alphay / sin2phi));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 sin2phi (*.f32 alphay alphay)) < 2e-23

   1. Initial program 49.3%

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

   2. Taylor expanded in u0 around 0 77.2%

      \[\leadsto \color{blue}{\frac{u0}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}}} \]
   3. Step-by-step derivation

      1. +-commutative 77.2%

         \[\leadsto \frac{u0}{\color{blue}{\frac{sin2phi}{{alphay}^{2}} + \frac{cos2phi}{{alphax}^{2}}}} \]

      2. unpow2 77.2%

         \[\leadsto \frac{u0}{\frac{sin2phi}{\color{blue}{alphay \cdot alphay}} + \frac{cos2phi}{{alphax}^{2}}} \]

      3. unpow2 77.2%

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

   4. Simplified 77.2%

      \[\leadsto \color{blue}{\frac{u0}{\frac{sin2phi}{alphay \cdot alphay} + \frac{cos2phi}{alphax \cdot alphax}}} \]
   5. Step-by-step derivation

      1. +-commutative 77.2%

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

      2. associate-/r* 77.2%

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

      3. frac-add 77.5%

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

   6. Applied egg-rr 77.5%

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

   7. Taylor expanded in cos2phi around inf 64.8%

      \[\leadsto \color{blue}{\frac{{alphax}^{2} \cdot u0}{cos2phi}} \]
   8. Step-by-step derivation

      1. *-commutative 64.8%

         \[\leadsto \frac{\color{blue}{u0 \cdot {alphax}^{2}}}{cos2phi} \]

      2. associate-/l* 64.7%

         \[\leadsto \color{blue}{\frac{u0}{\frac{cos2phi}{{alphax}^{2}}}} \]

      3. unpow2 64.7%

         \[\leadsto \frac{u0}{\frac{cos2phi}{\color{blue}{alphax \cdot alphax}}} \]

      4. associate-/r* 64.8%

         \[\leadsto \frac{u0}{\color{blue}{\frac{\frac{cos2phi}{alphax}}{alphax}}} \]

   9. Simplified 64.8%

      \[\leadsto \color{blue}{\frac{u0}{\frac{\frac{cos2phi}{alphax}}{alphax}}} \]

   if 2e-23 < (/.f32 sin2phi (*.f32 alphay alphay))

   1. Initial program 59.0%

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

   2. Taylor expanded in u0 around 0 77.8%

      \[\leadsto \color{blue}{\frac{u0}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}}} \]
   3. Step-by-step derivation

      1. +-commutative 77.8%

         \[\leadsto \frac{u0}{\color{blue}{\frac{sin2phi}{{alphay}^{2}} + \frac{cos2phi}{{alphax}^{2}}}} \]

      2. unpow2 77.8%

         \[\leadsto \frac{u0}{\frac{sin2phi}{\color{blue}{alphay \cdot alphay}} + \frac{cos2phi}{{alphax}^{2}}} \]

      3. unpow2 77.8%

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

   4. Simplified 77.8%

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

   5. Taylor expanded in sin2phi around inf 69.1%

      \[\leadsto \frac{u0}{\color{blue}{\frac{sin2phi}{{alphay}^{2}}}} \]
   6. Step-by-step derivation

      1. unpow2 69.1%

         \[\leadsto \frac{u0}{\frac{sin2phi}{\color{blue}{alphay \cdot alphay}}} \]

   7. Simplified 69.1%

      \[\leadsto \frac{u0}{\color{blue}{\frac{sin2phi}{alphay \cdot alphay}}} \]
   8. Step-by-step derivation

      1. div-inv 69.1%

         \[\leadsto \color{blue}{u0 \cdot \frac{1}{\frac{sin2phi}{alphay \cdot alphay}}} \]

      2. clear-num 69.4%

         \[\leadsto u0 \cdot \color{blue}{\frac{alphay \cdot alphay}{sin2phi}} \]

      3. associate-*l/ 69.4%

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

      4. *-commutative 69.4%

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

      5. *-commutative 69.4%

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

   9. Applied egg-rr 69.4%

      \[\leadsto \color{blue}{\left(alphay \cdot \frac{alphay}{sin2phi}\right) \cdot u0} \]
   10. Step-by-step derivation

       1. add-exp-log_binary32 67.6%

          \[\leadsto \color{blue}{e^{\log \left(\left(alphay \cdot \frac{alphay}{sin2phi}\right) \cdot u0\right)}} \]

   11. Applied rewrite-once 67.6%

       \[\leadsto \color{blue}{e^{\log \left(\left(alphay \cdot \frac{alphay}{sin2phi}\right) \cdot u0\right)}} \]
   12. Step-by-step derivation

       1. rem-exp-log 69.4%

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

       2. associate-*l* 69.4%

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

   13. Simplified 69.4%

       \[\leadsto \color{blue}{alphay \cdot \left(\frac{alphay}{sin2phi} \cdot u0\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 68.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 1.9999999996399175 \cdot 10^{-23}:\\ \;\;\;\;\frac{u0}{\frac{\frac{cos2phi}{alphax}}{alphax}}\\ \mathbf{else}:\\ \;\;\;\;alphay \cdot \left(u0 \cdot \frac{alphay}{sin2phi}\right)\\ \end{array} \]

Alternative 15: 76.0% accurate, 8.9× speedup

Math

\[\begin{array}{l} \\ \frac{u0}{\frac{sin2phi}{alphay \cdot alphay} + \frac{cos2phi}{alphax \cdot alphax}} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (/ u0 (+ (/ sin2phi (* alphay alphay)) (/ cos2phi (* alphax alphax)))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	return u0 / ((sin2phi / (alphay * alphay)) + (cos2phi / (alphax * alphax)));
}

Fortran

real(4) function code(alphax, alphay, u0, cos2phi, sin2phi)
    real(4), intent (in) :: alphax
    real(4), intent (in) :: alphay
    real(4), intent (in) :: u0
    real(4), intent (in) :: cos2phi
    real(4), intent (in) :: sin2phi
    code = u0 / ((sin2phi / (alphay * alphay)) + (cos2phi / (alphax * alphax)))
end function

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	return Float32(u0 / Float32(Float32(sin2phi / Float32(alphay * alphay)) + Float32(cos2phi / Float32(alphax * alphax))))
end

MATLAB

function tmp = code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = u0 / ((sin2phi / (alphay * alphay)) + (cos2phi / (alphax * alphax)));
end

Derivation

1. Initial program 57.5%

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

2. Taylor expanded in u0 around 0 77.7%

   \[\leadsto \color{blue}{\frac{u0}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}}} \]
3. Step-by-step derivation

   1. +-commutative 77.7%

      \[\leadsto \frac{u0}{\color{blue}{\frac{sin2phi}{{alphay}^{2}} + \frac{cos2phi}{{alphax}^{2}}}} \]

   2. unpow2 77.7%

      \[\leadsto \frac{u0}{\frac{sin2phi}{\color{blue}{alphay \cdot alphay}} + \frac{cos2phi}{{alphax}^{2}}} \]

   3. unpow2 77.7%

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

4. Simplified 77.7%

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

5. Final simplification 77.7%

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

Alternative 16: 59.3% accurate, 16.6× speedup

Math

\[\begin{array}{l} \\ alphay \cdot \left(u0 \cdot \frac{alphay}{sin2phi}\right) \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (* alphay (* u0 (/ alphay sin2phi))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	return alphay * (u0 * (alphay / sin2phi));
}

Fortran

real(4) function code(alphax, alphay, u0, cos2phi, sin2phi)
    real(4), intent (in) :: alphax
    real(4), intent (in) :: alphay
    real(4), intent (in) :: u0
    real(4), intent (in) :: cos2phi
    real(4), intent (in) :: sin2phi
    code = alphay * (u0 * (alphay / sin2phi))
end function

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	return Float32(alphay * Float32(u0 * Float32(alphay / sin2phi)))
end

MATLAB

function tmp = code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = alphay * (u0 * (alphay / sin2phi));
end

Derivation

1. Initial program 57.5%

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

2. Taylor expanded in u0 around 0 77.7%

   \[\leadsto \color{blue}{\frac{u0}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}}} \]
3. Step-by-step derivation

   1. +-commutative 77.7%

      \[\leadsto \frac{u0}{\color{blue}{\frac{sin2phi}{{alphay}^{2}} + \frac{cos2phi}{{alphax}^{2}}}} \]

   2. unpow2 77.7%

      \[\leadsto \frac{u0}{\frac{sin2phi}{\color{blue}{alphay \cdot alphay}} + \frac{cos2phi}{{alphax}^{2}}} \]

   3. unpow2 77.7%

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

4. Simplified 77.7%

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

5. Taylor expanded in sin2phi around inf 61.2%

   \[\leadsto \frac{u0}{\color{blue}{\frac{sin2phi}{{alphay}^{2}}}} \]
6. Step-by-step derivation

   1. unpow2 61.2%

      \[\leadsto \frac{u0}{\frac{sin2phi}{\color{blue}{alphay \cdot alphay}}} \]

7. Simplified 61.2%

   \[\leadsto \frac{u0}{\color{blue}{\frac{sin2phi}{alphay \cdot alphay}}} \]
8. Step-by-step derivation

   1. div-inv 61.2%

      \[\leadsto \color{blue}{u0 \cdot \frac{1}{\frac{sin2phi}{alphay \cdot alphay}}} \]

   2. clear-num 61.4%

      \[\leadsto u0 \cdot \color{blue}{\frac{alphay \cdot alphay}{sin2phi}} \]

   3. associate-*l/ 61.5%

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

   4. *-commutative 61.5%

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

   5. *-commutative 61.5%

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

9. Applied egg-rr 61.5%

   \[\leadsto \color{blue}{\left(alphay \cdot \frac{alphay}{sin2phi}\right) \cdot u0} \]
10. Step-by-step derivation

    1. add-exp-log_binary32 60.0%

       \[\leadsto \color{blue}{e^{\log \left(\left(alphay \cdot \frac{alphay}{sin2phi}\right) \cdot u0\right)}} \]

11. Applied rewrite-once 60.0%

    \[\leadsto \color{blue}{e^{\log \left(\left(alphay \cdot \frac{alphay}{sin2phi}\right) \cdot u0\right)}} \]
12. Step-by-step derivation

    1. rem-exp-log 61.5%

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

    2. associate-*l* 61.5%

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

13. Simplified 61.5%

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

14. Final simplification 61.5%

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

Reproduce

herbie shell --seed 2023297 
(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)))))