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

Percentage Accurate: 60.5% → 98.3%

Time: 19.6s

Alternatives: 11

Speedup: 5.5×

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: 60.5% 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: 98.3% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 59.9%

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

   1. distribute-frac-neg 59.9%

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

   2. distribute-neg-frac2 59.9%

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

   3. neg-mul-1 59.9%

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

   4. associate-/r* 59.9%

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

   5. metadata-eval 59.9%

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

   6. distribute-neg-frac2 59.9%

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

   7. /-rgt-identity 59.9%

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

   8. sub-neg 59.9%

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

   9. log1p-define 98.0%

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

3. Simplified 98.0%

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

   1. associate-/r* 98.1%

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

   2. div-inv 97.9%

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

6. Applied egg-rr 97.9%

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

   1. associate-*r/ 98.1%

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

   2. *-rgt-identity 98.1%

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

8. Simplified 98.1%

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

   1. associate-/r* 98.0%

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

   2. +-commutative 98.0%

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

   3. associate-/r* 98.1%

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

   4. clear-num 97.9%

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

   5. associate-/r* 97.8%

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

   6. frac-add 97.6%

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

   7. *-un-lft-identity 97.6%

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

10. Applied egg-rr 97.6%

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

    1. associate-/r/ 98.2%

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

    2. +-commutative 98.2%

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

    3. fma-define 98.2%

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

    4. associate-/r/ 98.1%

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

    5. associate-*l/ 98.0%

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

12. Applied egg-rr 98.0%

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

    1. div-inv 97.9%

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

    2. clear-num 98.2%

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

14. Applied egg-rr 98.2%

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

15. Final simplification 98.2%

    \[\leadsto \frac{\mathsf{log1p}\left(-u0\right)}{-\mathsf{fma}\left(alphay \cdot \frac{alphay}{sin2phi}, \frac{cos2phi}{alphax}, alphax\right)} \cdot \left(\frac{alphay}{sin2phi} \cdot \left(alphay \cdot alphax\right)\right) \]
16. Add Preprocessing

Alternative 2: 98.4% accurate, 1.0× speedup

Math

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

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(Float32(cos2phi / alphax) / alphax)))
end

Derivation

1. Initial program 59.9%

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

   1. distribute-frac-neg 59.9%

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

   2. distribute-neg-frac2 59.9%

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

   3. sub-neg 59.9%

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

   4. log1p-define 98.0%

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

   5. neg-sub0 98.0%

      \[\leadsto \frac{\mathsf{log1p}\left(-u0\right)}{\color{blue}{0 - \left(\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\right)}} \]

   6. associate--r+ 98.0%

      \[\leadsto \frac{\mathsf{log1p}\left(-u0\right)}{\color{blue}{\left(0 - \frac{cos2phi}{alphax \cdot alphax}\right) - \frac{sin2phi}{alphay \cdot alphay}}} \]

   7. neg-sub0 98.0%

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

   8. associate-/r* 97.9%

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

   9. distribute-neg-frac2 97.9%

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

3. Simplified 97.9%

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

5. Final simplification 97.9%

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

Alternative 3: 98.4% accurate, 1.0× speedup

Math

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

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

Derivation

1. Initial program 59.9%

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

   1. distribute-frac-neg 59.9%

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

   2. distribute-neg-frac2 59.9%

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

   3. neg-mul-1 59.9%

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

   4. associate-/r* 59.9%

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

   5. metadata-eval 59.9%

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

   6. distribute-neg-frac2 59.9%

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

   7. /-rgt-identity 59.9%

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

   8. sub-neg 59.9%

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

   9. log1p-define 98.0%

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

3. Simplified 98.0%

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

5. Final simplification 98.0%

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

Alternative 4: 98.4% accurate, 1.0× speedup

Math

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

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

Derivation

1. Initial program 59.9%

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

   1. distribute-frac-neg 59.9%

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

   2. distribute-neg-frac2 59.9%

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

   3. neg-mul-1 59.9%

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

   4. associate-/r* 59.9%

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

   5. metadata-eval 59.9%

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

   6. distribute-neg-frac2 59.9%

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

   7. /-rgt-identity 59.9%

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

   8. sub-neg 59.9%

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

   9. log1p-define 98.0%

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

3. Simplified 98.0%

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

   1. associate-/r* 98.1%

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

   2. div-inv 97.9%

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

6. Applied egg-rr 97.9%

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

   1. associate-*r/ 98.1%

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

   2. *-rgt-identity 98.1%

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

8. Simplified 98.1%

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

9. Final simplification 98.1%

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

Alternative 5: 92.9% accurate, 3.3× speedup

Math

\[\begin{array}{l} \\ \frac{u0 \cdot \left(1 + u0 \cdot \left(0.5 - u0 \cdot \left(u0 \cdot -0.25 - 0.3333333333333333\right)\right)\right)}{\frac{alphax + \frac{alphay \cdot cos2phi}{alphax \cdot \frac{sin2phi}{alphay}}}{alphax \cdot \frac{alphay}{\frac{sin2phi}{alphay}}}} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (/
  (* u0 (+ 1.0 (* u0 (- 0.5 (* u0 (- (* u0 -0.25) 0.3333333333333333))))))
  (/
   (+ alphax (/ (* alphay cos2phi) (* alphax (/ sin2phi alphay))))
   (* alphax (/ alphay (/ sin2phi alphay))))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	return (u0 * (1.0f + (u0 * (0.5f - (u0 * ((u0 * -0.25f) - 0.3333333333333333f)))))) / ((alphax + ((alphay * cos2phi) / (alphax * (sin2phi / alphay)))) / (alphax * (alphay / (sin2phi / 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 = (u0 * (1.0e0 + (u0 * (0.5e0 - (u0 * ((u0 * (-0.25e0)) - 0.3333333333333333e0)))))) / ((alphax + ((alphay * cos2phi) / (alphax * (sin2phi / alphay)))) / (alphax * (alphay / (sin2phi / alphay))))
end function

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	return Float32(Float32(u0 * Float32(Float32(1.0) + Float32(u0 * Float32(Float32(0.5) - Float32(u0 * Float32(Float32(u0 * Float32(-0.25)) - Float32(0.3333333333333333))))))) / Float32(Float32(alphax + Float32(Float32(alphay * cos2phi) / Float32(alphax * Float32(sin2phi / alphay)))) / Float32(alphax * Float32(alphay / Float32(sin2phi / alphay)))))
end

MATLAB

function tmp = code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = (u0 * (single(1.0) + (u0 * (single(0.5) - (u0 * ((u0 * single(-0.25)) - single(0.3333333333333333))))))) / ((alphax + ((alphay * cos2phi) / (alphax * (sin2phi / alphay)))) / (alphax * (alphay / (sin2phi / alphay))));
end

Derivation

1. Initial program 59.9%

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

   1. distribute-frac-neg 59.9%

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

   2. distribute-neg-frac2 59.9%

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

   3. neg-mul-1 59.9%

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

   4. associate-/r* 59.9%

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

   5. metadata-eval 59.9%

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

   6. distribute-neg-frac2 59.9%

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

   7. /-rgt-identity 59.9%

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

   8. sub-neg 59.9%

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

   9. log1p-define 98.0%

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

3. Simplified 98.0%

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

   1. associate-/r* 98.1%

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

   2. div-inv 97.9%

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

6. Applied egg-rr 97.9%

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

   1. associate-*r/ 98.1%

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

   2. *-rgt-identity 98.1%

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

8. Simplified 98.1%

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

   1. associate-/r* 98.0%

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

   2. +-commutative 98.0%

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

   3. associate-/r* 98.1%

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

   4. clear-num 97.9%

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

   5. associate-/r* 97.8%

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

   6. frac-add 97.6%

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

   7. *-un-lft-identity 97.6%

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

10. Applied egg-rr 97.6%

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

    1. frac-times 97.6%

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

12. Applied egg-rr 97.6%

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

13. Taylor expanded in u0 around 0 92.4%

    \[\leadsto \frac{-\color{blue}{u0 \cdot \left(u0 \cdot \left(u0 \cdot \left(-0.25 \cdot u0 - 0.3333333333333333\right) - 0.5\right) - 1\right)}}{\frac{alphax + \frac{alphay \cdot cos2phi}{\frac{sin2phi}{alphay} \cdot alphax}}{\frac{alphay}{\frac{sin2phi}{alphay}} \cdot alphax}} \]

14. Final simplification 92.4%

    \[\leadsto \frac{u0 \cdot \left(1 + u0 \cdot \left(0.5 - u0 \cdot \left(u0 \cdot -0.25 - 0.3333333333333333\right)\right)\right)}{\frac{alphax + \frac{alphay \cdot cos2phi}{alphax \cdot \frac{sin2phi}{alphay}}}{alphax \cdot \frac{alphay}{\frac{sin2phi}{alphay}}}} \]
15. Add Preprocessing

Alternative 6: 91.0% accurate, 3.7× speedup

Math

\[\begin{array}{l} \\ \frac{u0 \cdot \left(1 + u0 \cdot \left(0.5 - u0 \cdot -0.3333333333333333\right)\right)}{\frac{alphax + \frac{alphay \cdot cos2phi}{alphax \cdot \frac{sin2phi}{alphay}}}{alphax \cdot \frac{alphay}{\frac{sin2phi}{alphay}}}} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (/
  (* u0 (+ 1.0 (* u0 (- 0.5 (* u0 -0.3333333333333333)))))
  (/
   (+ alphax (/ (* alphay cos2phi) (* alphax (/ sin2phi alphay))))
   (* alphax (/ alphay (/ sin2phi alphay))))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	return (u0 * (1.0f + (u0 * (0.5f - (u0 * -0.3333333333333333f))))) / ((alphax + ((alphay * cos2phi) / (alphax * (sin2phi / alphay)))) / (alphax * (alphay / (sin2phi / 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 = (u0 * (1.0e0 + (u0 * (0.5e0 - (u0 * (-0.3333333333333333e0)))))) / ((alphax + ((alphay * cos2phi) / (alphax * (sin2phi / alphay)))) / (alphax * (alphay / (sin2phi / alphay))))
end function

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	return Float32(Float32(u0 * Float32(Float32(1.0) + Float32(u0 * Float32(Float32(0.5) - Float32(u0 * Float32(-0.3333333333333333)))))) / Float32(Float32(alphax + Float32(Float32(alphay * cos2phi) / Float32(alphax * Float32(sin2phi / alphay)))) / Float32(alphax * Float32(alphay / Float32(sin2phi / alphay)))))
end

MATLAB

function tmp = code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = (u0 * (single(1.0) + (u0 * (single(0.5) - (u0 * single(-0.3333333333333333)))))) / ((alphax + ((alphay * cos2phi) / (alphax * (sin2phi / alphay)))) / (alphax * (alphay / (sin2phi / alphay))));
end

Derivation

1. Initial program 59.9%

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

   1. distribute-frac-neg 59.9%

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

   2. distribute-neg-frac2 59.9%

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

   3. neg-mul-1 59.9%

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

   4. associate-/r* 59.9%

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

   5. metadata-eval 59.9%

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

   6. distribute-neg-frac2 59.9%

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

   7. /-rgt-identity 59.9%

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

   8. sub-neg 59.9%

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

   9. log1p-define 98.0%

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

3. Simplified 98.0%

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

   1. associate-/r* 98.1%

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

   2. div-inv 97.9%

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

6. Applied egg-rr 97.9%

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

   1. associate-*r/ 98.1%

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

   2. *-rgt-identity 98.1%

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

8. Simplified 98.1%

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

   1. associate-/r* 98.0%

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

   2. +-commutative 98.0%

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

   3. associate-/r* 98.1%

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

   4. clear-num 97.9%

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

   5. associate-/r* 97.8%

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

   6. frac-add 97.6%

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

   7. *-un-lft-identity 97.6%

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

10. Applied egg-rr 97.6%

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

    1. frac-times 97.6%

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

12. Applied egg-rr 97.6%

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

13. Taylor expanded in u0 around 0 90.5%

    \[\leadsto \frac{-\color{blue}{u0 \cdot \left(u0 \cdot \left(-0.3333333333333333 \cdot u0 - 0.5\right) - 1\right)}}{\frac{alphax + \frac{alphay \cdot cos2phi}{\frac{sin2phi}{alphay} \cdot alphax}}{\frac{alphay}{\frac{sin2phi}{alphay}} \cdot alphax}} \]

14. Final simplification 90.5%

    \[\leadsto \frac{u0 \cdot \left(1 + u0 \cdot \left(0.5 - u0 \cdot -0.3333333333333333\right)\right)}{\frac{alphax + \frac{alphay \cdot cos2phi}{alphax \cdot \frac{sin2phi}{alphay}}}{alphax \cdot \frac{alphay}{\frac{sin2phi}{alphay}}}} \]
15. Add Preprocessing

Alternative 7: 83.1% accurate, 3.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;sin2phi \leq 1.9999999949504854 \cdot 10^{-6}:\\ \;\;\;\;\frac{u0}{\frac{alphax + \frac{alphay \cdot cos2phi}{alphax \cdot \frac{sin2phi}{alphay}}}{alphax \cdot \frac{alphay}{\frac{sin2phi}{alphay}}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{u0 \cdot \left(-1 - u0 \cdot \left(0.5 - u0 \cdot -0.3333333333333333\right)\right)}{\frac{\frac{cos2phi}{alphax}}{alphax} + \frac{sin2phi}{alphay} \cdot \frac{-1}{alphay}}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (if (<= sin2phi 1.9999999949504854e-6)
   (/
    u0
    (/
     (+ alphax (/ (* alphay cos2phi) (* alphax (/ sin2phi alphay))))
     (* alphax (/ alphay (/ sin2phi alphay)))))
   (/
    (* u0 (- -1.0 (* u0 (- 0.5 (* u0 -0.3333333333333333)))))
    (+ (/ (/ cos2phi alphax) alphax) (* (/ sin2phi alphay) (/ -1.0 alphay))))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float tmp;
	if (sin2phi <= 1.9999999949504854e-6f) {
		tmp = u0 / ((alphax + ((alphay * cos2phi) / (alphax * (sin2phi / alphay)))) / (alphax * (alphay / (sin2phi / alphay))));
	} else {
		tmp = (u0 * (-1.0f - (u0 * (0.5f - (u0 * -0.3333333333333333f))))) / (((cos2phi / alphax) / alphax) + ((sin2phi / alphay) * (-1.0f / alphay)));
	}
	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 <= 1.9999999949504854e-6) then
        tmp = u0 / ((alphax + ((alphay * cos2phi) / (alphax * (sin2phi / alphay)))) / (alphax * (alphay / (sin2phi / alphay))))
    else
        tmp = (u0 * ((-1.0e0) - (u0 * (0.5e0 - (u0 * (-0.3333333333333333e0)))))) / (((cos2phi / alphax) / alphax) + ((sin2phi / alphay) * ((-1.0e0) / alphay)))
    end if
    code = tmp
end function

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = Float32(0.0)
	if (sin2phi <= Float32(1.9999999949504854e-6))
		tmp = Float32(u0 / Float32(Float32(alphax + Float32(Float32(alphay * cos2phi) / Float32(alphax * Float32(sin2phi / alphay)))) / Float32(alphax * Float32(alphay / Float32(sin2phi / alphay)))));
	else
		tmp = Float32(Float32(u0 * Float32(Float32(-1.0) - Float32(u0 * Float32(Float32(0.5) - Float32(u0 * Float32(-0.3333333333333333)))))) / Float32(Float32(Float32(cos2phi / alphax) / alphax) + Float32(Float32(sin2phi / alphay) * Float32(Float32(-1.0) / alphay))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = single(0.0);
	if (sin2phi <= single(1.9999999949504854e-6))
		tmp = u0 / ((alphax + ((alphay * cos2phi) / (alphax * (sin2phi / alphay)))) / (alphax * (alphay / (sin2phi / alphay))));
	else
		tmp = (u0 * (single(-1.0) - (u0 * (single(0.5) - (u0 * single(-0.3333333333333333)))))) / (((cos2phi / alphax) / alphax) + ((sin2phi / alphay) * (single(-1.0) / alphay)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if sin2phi < 1.99999999e-6

   1. Initial program 52.0%

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

      1. distribute-frac-neg 52.0%

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

      2. distribute-neg-frac2 52.0%

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

      3. neg-mul-1 52.0%

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

      4. associate-/r* 52.0%

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

      5. metadata-eval 52.0%

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

      6. distribute-neg-frac2 52.0%

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

      7. /-rgt-identity 52.0%

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

      8. sub-neg 52.0%

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

      9. log1p-define 98.7%

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

   3. Simplified 98.7%

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

      1. associate-/r* 98.8%

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

      2. div-inv 98.5%

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

   6. Applied egg-rr 98.5%

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

      1. associate-*r/ 98.8%

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

      2. *-rgt-identity 98.8%

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

   8. Simplified 98.8%

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

      1. associate-/r* 98.7%

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

      2. +-commutative 98.7%

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

      3. associate-/r* 98.8%

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

      4. clear-num 98.7%

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

      5. associate-/r* 98.5%

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

      6. frac-add 98.0%

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

      7. *-un-lft-identity 98.0%

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

   10. Applied egg-rr 98.0%

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

       1. frac-times 97.9%

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

   12. Applied egg-rr 97.9%

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

   13. Taylor expanded in u0 around 0 76.4%

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

       1. neg-mul-1 30.4%

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

   15. Simplified 76.4%

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

   if 1.99999999e-6 < sin2phi

   1. Initial program 65.3%

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

      1. distribute-frac-neg 65.3%

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

      2. distribute-neg-frac2 65.3%

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

      3. sub-neg 65.3%

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

      4. log1p-define 97.5%

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

      5. neg-sub0 97.5%

         \[\leadsto \frac{\mathsf{log1p}\left(-u0\right)}{\color{blue}{0 - \left(\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\right)}} \]

      6. associate--r+ 97.5%

         \[\leadsto \frac{\mathsf{log1p}\left(-u0\right)}{\color{blue}{\left(0 - \frac{cos2phi}{alphax \cdot alphax}\right) - \frac{sin2phi}{alphay \cdot alphay}}} \]

      7. neg-sub0 97.5%

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

      8. associate-/r* 97.5%

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

      9. distribute-neg-frac2 97.5%

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

   3. Simplified 97.5%

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

      1. add-sqr-sqrt -0.0%

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

      2. sqrt-unprod 96.2%

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

      3. sqr-neg 96.2%

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

      4. sqrt-prod 96.2%

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

      5. add-sqr-sqrt 96.2%

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

      6. div-inv 96.2%

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

   6. Applied egg-rr 96.2%

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

      1. associate-*r/ 96.2%

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

      2. *-rgt-identity 96.2%

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

   8. Simplified 96.2%

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

      1. associate-/r* 97.6%

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

      2. div-inv 97.4%

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

   10. Applied egg-rr 96.2%

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

   11. Taylor expanded in u0 around 0 88.7%

       \[\leadsto \frac{\color{blue}{u0 \cdot \left(u0 \cdot \left(-0.3333333333333333 \cdot u0 - 0.5\right) - 1\right)}}{\frac{\frac{cos2phi}{alphax}}{alphax} - \frac{sin2phi}{alphay} \cdot \frac{1}{alphay}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 83.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;sin2phi \leq 1.9999999949504854 \cdot 10^{-6}:\\ \;\;\;\;\frac{u0}{\frac{alphax + \frac{alphay \cdot cos2phi}{alphax \cdot \frac{sin2phi}{alphay}}}{alphax \cdot \frac{alphay}{\frac{sin2phi}{alphay}}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{u0 \cdot \left(-1 - u0 \cdot \left(0.5 - u0 \cdot -0.3333333333333333\right)\right)}{\frac{\frac{cos2phi}{alphax}}{alphax} + \frac{sin2phi}{alphay} \cdot \frac{-1}{alphay}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 87.4% accurate, 4.3× speedup

Math

\[\begin{array}{l} \\ \frac{u0 \cdot \left(1 - u0 \cdot -0.5\right)}{\frac{alphax + \frac{alphay \cdot cos2phi}{alphax \cdot \frac{sin2phi}{alphay}}}{alphax \cdot \frac{alphay}{\frac{sin2phi}{alphay}}}} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (/
  (* u0 (- 1.0 (* u0 -0.5)))
  (/
   (+ alphax (/ (* alphay cos2phi) (* alphax (/ sin2phi alphay))))
   (* alphax (/ alphay (/ sin2phi alphay))))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	return (u0 * (1.0f - (u0 * -0.5f))) / ((alphax + ((alphay * cos2phi) / (alphax * (sin2phi / alphay)))) / (alphax * (alphay / (sin2phi / 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 = (u0 * (1.0e0 - (u0 * (-0.5e0)))) / ((alphax + ((alphay * cos2phi) / (alphax * (sin2phi / alphay)))) / (alphax * (alphay / (sin2phi / alphay))))
end function

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	return Float32(Float32(u0 * Float32(Float32(1.0) - Float32(u0 * Float32(-0.5)))) / Float32(Float32(alphax + Float32(Float32(alphay * cos2phi) / Float32(alphax * Float32(sin2phi / alphay)))) / Float32(alphax * Float32(alphay / Float32(sin2phi / alphay)))))
end

MATLAB

function tmp = code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = (u0 * (single(1.0) - (u0 * single(-0.5)))) / ((alphax + ((alphay * cos2phi) / (alphax * (sin2phi / alphay)))) / (alphax * (alphay / (sin2phi / alphay))));
end

Derivation

1. Initial program 59.9%

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

   1. distribute-frac-neg 59.9%

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

   2. distribute-neg-frac2 59.9%

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

   3. neg-mul-1 59.9%

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

   4. associate-/r* 59.9%

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

   5. metadata-eval 59.9%

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

   6. distribute-neg-frac2 59.9%

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

   7. /-rgt-identity 59.9%

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

   8. sub-neg 59.9%

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

   9. log1p-define 98.0%

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

3. Simplified 98.0%

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

   1. associate-/r* 98.1%

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

   2. div-inv 97.9%

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

6. Applied egg-rr 97.9%

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

   1. associate-*r/ 98.1%

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

   2. *-rgt-identity 98.1%

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

8. Simplified 98.1%

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

   1. associate-/r* 98.0%

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

   2. +-commutative 98.0%

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

   3. associate-/r* 98.1%

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

   4. clear-num 97.9%

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

   5. associate-/r* 97.8%

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

   6. frac-add 97.6%

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

   7. *-un-lft-identity 97.6%

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

10. Applied egg-rr 97.6%

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

    1. frac-times 97.6%

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

12. Applied egg-rr 97.6%

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

13. Taylor expanded in u0 around 0 86.8%

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

14. Final simplification 86.8%

    \[\leadsto \frac{u0 \cdot \left(1 - u0 \cdot -0.5\right)}{\frac{alphax + \frac{alphay \cdot cos2phi}{alphax \cdot \frac{sin2phi}{alphay}}}{alphax \cdot \frac{alphay}{\frac{sin2phi}{alphay}}}} \]
15. Add Preprocessing

Alternative 9: 81.1% accurate, 4.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;sin2phi \leq 1.9999999949504854 \cdot 10^{-6}:\\ \;\;\;\;\frac{u0}{\frac{alphax + \frac{alphay \cdot cos2phi}{alphax \cdot \frac{sin2phi}{alphay}}}{alphax \cdot \frac{alphay}{\frac{sin2phi}{alphay}}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{u0 \cdot \left(u0 \cdot -0.5 + -1\right)}{\frac{\frac{cos2phi}{alphax}}{alphax} + \frac{sin2phi}{alphay} \cdot \frac{-1}{alphay}}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (if (<= sin2phi 1.9999999949504854e-6)
   (/
    u0
    (/
     (+ alphax (/ (* alphay cos2phi) (* alphax (/ sin2phi alphay))))
     (* alphax (/ alphay (/ sin2phi alphay)))))
   (/
    (* u0 (+ (* u0 -0.5) -1.0))
    (+ (/ (/ cos2phi alphax) alphax) (* (/ sin2phi alphay) (/ -1.0 alphay))))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float tmp;
	if (sin2phi <= 1.9999999949504854e-6f) {
		tmp = u0 / ((alphax + ((alphay * cos2phi) / (alphax * (sin2phi / alphay)))) / (alphax * (alphay / (sin2phi / alphay))));
	} else {
		tmp = (u0 * ((u0 * -0.5f) + -1.0f)) / (((cos2phi / alphax) / alphax) + ((sin2phi / alphay) * (-1.0f / alphay)));
	}
	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 <= 1.9999999949504854e-6) then
        tmp = u0 / ((alphax + ((alphay * cos2phi) / (alphax * (sin2phi / alphay)))) / (alphax * (alphay / (sin2phi / alphay))))
    else
        tmp = (u0 * ((u0 * (-0.5e0)) + (-1.0e0))) / (((cos2phi / alphax) / alphax) + ((sin2phi / alphay) * ((-1.0e0) / alphay)))
    end if
    code = tmp
end function

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = Float32(0.0)
	if (sin2phi <= Float32(1.9999999949504854e-6))
		tmp = Float32(u0 / Float32(Float32(alphax + Float32(Float32(alphay * cos2phi) / Float32(alphax * Float32(sin2phi / alphay)))) / Float32(alphax * Float32(alphay / Float32(sin2phi / alphay)))));
	else
		tmp = Float32(Float32(u0 * Float32(Float32(u0 * Float32(-0.5)) + Float32(-1.0))) / Float32(Float32(Float32(cos2phi / alphax) / alphax) + Float32(Float32(sin2phi / alphay) * Float32(Float32(-1.0) / alphay))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = single(0.0);
	if (sin2phi <= single(1.9999999949504854e-6))
		tmp = u0 / ((alphax + ((alphay * cos2phi) / (alphax * (sin2phi / alphay)))) / (alphax * (alphay / (sin2phi / alphay))));
	else
		tmp = (u0 * ((u0 * single(-0.5)) + single(-1.0))) / (((cos2phi / alphax) / alphax) + ((sin2phi / alphay) * (single(-1.0) / alphay)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if sin2phi < 1.99999999e-6

   1. Initial program 52.0%

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

      1. distribute-frac-neg 52.0%

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

      2. distribute-neg-frac2 52.0%

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

      3. neg-mul-1 52.0%

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

      4. associate-/r* 52.0%

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

      5. metadata-eval 52.0%

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

      6. distribute-neg-frac2 52.0%

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

      7. /-rgt-identity 52.0%

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

      8. sub-neg 52.0%

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

      9. log1p-define 98.7%

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

   3. Simplified 98.7%

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

      1. associate-/r* 98.8%

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

      2. div-inv 98.5%

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

   6. Applied egg-rr 98.5%

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

      1. associate-*r/ 98.8%

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

      2. *-rgt-identity 98.8%

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

   8. Simplified 98.8%

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

      1. associate-/r* 98.7%

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

      2. +-commutative 98.7%

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

      3. associate-/r* 98.8%

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

      4. clear-num 98.7%

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

      5. associate-/r* 98.5%

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

      6. frac-add 98.0%

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

      7. *-un-lft-identity 98.0%

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

   10. Applied egg-rr 98.0%

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

       1. frac-times 97.9%

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

   12. Applied egg-rr 97.9%

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

   13. Taylor expanded in u0 around 0 76.4%

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

       1. neg-mul-1 30.4%

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

   15. Simplified 76.4%

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

   if 1.99999999e-6 < sin2phi

   1. Initial program 65.3%

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

      1. distribute-frac-neg 65.3%

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

      2. distribute-neg-frac2 65.3%

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

      3. sub-neg 65.3%

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

      4. log1p-define 97.5%

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

      5. neg-sub0 97.5%

         \[\leadsto \frac{\mathsf{log1p}\left(-u0\right)}{\color{blue}{0 - \left(\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\right)}} \]

      6. associate--r+ 97.5%

         \[\leadsto \frac{\mathsf{log1p}\left(-u0\right)}{\color{blue}{\left(0 - \frac{cos2phi}{alphax \cdot alphax}\right) - \frac{sin2phi}{alphay \cdot alphay}}} \]

      7. neg-sub0 97.5%

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

      8. associate-/r* 97.5%

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

      9. distribute-neg-frac2 97.5%

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

   3. Simplified 97.5%

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

      1. add-sqr-sqrt -0.0%

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

      2. sqrt-unprod 96.2%

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

      3. sqr-neg 96.2%

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

      4. sqrt-prod 96.2%

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

      5. add-sqr-sqrt 96.2%

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

      6. div-inv 96.2%

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

   6. Applied egg-rr 96.2%

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

      1. associate-*r/ 96.2%

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

      2. *-rgt-identity 96.2%

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

   8. Simplified 96.2%

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

      1. associate-/r* 97.6%

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

      2. div-inv 97.4%

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

   10. Applied egg-rr 96.2%

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

   11. Taylor expanded in u0 around 0 85.3%

       \[\leadsto \frac{\color{blue}{u0 \cdot \left(-0.5 \cdot u0 - 1\right)}}{\frac{\frac{cos2phi}{alphax}}{alphax} - \frac{sin2phi}{alphay} \cdot \frac{1}{alphay}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 81.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;sin2phi \leq 1.9999999949504854 \cdot 10^{-6}:\\ \;\;\;\;\frac{u0}{\frac{alphax + \frac{alphay \cdot cos2phi}{alphax \cdot \frac{sin2phi}{alphay}}}{alphax \cdot \frac{alphay}{\frac{sin2phi}{alphay}}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{u0 \cdot \left(u0 \cdot -0.5 + -1\right)}{\frac{\frac{cos2phi}{alphax}}{alphax} + \frac{sin2phi}{alphay} \cdot \frac{-1}{alphay}}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 63.9% accurate, 5.5× speedup

Math

\[\begin{array}{l} \\ \frac{u0 \cdot \left(u0 \cdot -0.5 + -1\right)}{\frac{\frac{cos2phi}{alphax}}{alphax} + \frac{sin2phi}{alphay} \cdot \frac{-1}{alphay}} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (/
  (* u0 (+ (* u0 -0.5) -1.0))
  (+ (/ (/ cos2phi alphax) alphax) (* (/ sin2phi alphay) (/ -1.0 alphay)))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	return (u0 * ((u0 * -0.5f) + -1.0f)) / (((cos2phi / alphax) / alphax) + ((sin2phi / alphay) * (-1.0f / 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 = (u0 * ((u0 * (-0.5e0)) + (-1.0e0))) / (((cos2phi / alphax) / alphax) + ((sin2phi / alphay) * ((-1.0e0) / alphay)))
end function

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	return Float32(Float32(u0 * Float32(Float32(u0 * Float32(-0.5)) + Float32(-1.0))) / Float32(Float32(Float32(cos2phi / alphax) / alphax) + Float32(Float32(sin2phi / alphay) * Float32(Float32(-1.0) / alphay))))
end

MATLAB

function tmp = code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = (u0 * ((u0 * single(-0.5)) + single(-1.0))) / (((cos2phi / alphax) / alphax) + ((sin2phi / alphay) * (single(-1.0) / alphay)));
end

Derivation

1. Initial program 59.9%

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

   1. distribute-frac-neg 59.9%

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

   2. distribute-neg-frac2 59.9%

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

   3. sub-neg 59.9%

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

   4. log1p-define 98.0%

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

   5. neg-sub0 98.0%

      \[\leadsto \frac{\mathsf{log1p}\left(-u0\right)}{\color{blue}{0 - \left(\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\right)}} \]

   6. associate--r+ 98.0%

      \[\leadsto \frac{\mathsf{log1p}\left(-u0\right)}{\color{blue}{\left(0 - \frac{cos2phi}{alphax \cdot alphax}\right) - \frac{sin2phi}{alphay \cdot alphay}}} \]

   7. neg-sub0 98.0%

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

   8. associate-/r* 97.9%

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

   9. distribute-neg-frac2 97.9%

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

3. Simplified 97.9%

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

   1. add-sqr-sqrt -0.0%

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

   2. sqrt-unprod 72.7%

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

   3. sqr-neg 72.7%

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

   4. sqrt-prod 72.7%

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

   5. add-sqr-sqrt 72.7%

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

   6. div-inv 72.7%

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

6. Applied egg-rr 72.7%

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

   1. associate-*r/ 72.7%

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

   2. *-rgt-identity 72.7%

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

8. Simplified 72.7%

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

   1. associate-/r* 98.1%

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

   2. div-inv 97.9%

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

10. Applied egg-rr 72.6%

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

11. Taylor expanded in u0 around 0 64.7%

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

12. Final simplification 64.7%

    \[\leadsto \frac{u0 \cdot \left(u0 \cdot -0.5 + -1\right)}{\frac{\frac{cos2phi}{alphax}}{alphax} + \frac{sin2phi}{alphay} \cdot \frac{-1}{alphay}} \]
13. Add Preprocessing

Alternative 11: 56.4% accurate, 7.7× speedup

Math

\[\begin{array}{l} \\ \frac{u0}{\frac{sin2phi}{alphay} \cdot \frac{1}{alphay} - \frac{\frac{cos2phi}{alphax}}{alphax}} \end{array} \]

FPCore

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

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	return u0 / (((sin2phi / alphay) * (1.0f / 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) * (1.0e0 / alphay)) - ((cos2phi / alphax) / alphax))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 59.9%

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

   1. distribute-frac-neg 59.9%

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

   2. distribute-neg-frac2 59.9%

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

   3. sub-neg 59.9%

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

   4. log1p-define 98.0%

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

   5. neg-sub0 98.0%

      \[\leadsto \frac{\mathsf{log1p}\left(-u0\right)}{\color{blue}{0 - \left(\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\right)}} \]

   6. associate--r+ 98.0%

      \[\leadsto \frac{\mathsf{log1p}\left(-u0\right)}{\color{blue}{\left(0 - \frac{cos2phi}{alphax \cdot alphax}\right) - \frac{sin2phi}{alphay \cdot alphay}}} \]

   7. neg-sub0 98.0%

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

   8. associate-/r* 97.9%

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

   9. distribute-neg-frac2 97.9%

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

3. Simplified 97.9%

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

   1. add-sqr-sqrt -0.0%

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

   2. sqrt-unprod 72.7%

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

   3. sqr-neg 72.7%

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

   4. sqrt-prod 72.7%

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

   5. add-sqr-sqrt 72.7%

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

   6. div-inv 72.7%

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

6. Applied egg-rr 72.7%

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

   1. associate-*r/ 72.7%

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

   2. *-rgt-identity 72.7%

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

8. Simplified 72.7%

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

   1. associate-/r* 98.1%

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

   2. div-inv 97.9%

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

10. Applied egg-rr 72.6%

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

11. Taylor expanded in u0 around 0 57.1%

    \[\leadsto \frac{\color{blue}{-1 \cdot u0}}{\frac{\frac{cos2phi}{alphax}}{alphax} - \frac{sin2phi}{alphay} \cdot \frac{1}{alphay}} \]
12. Step-by-step derivation

    1. neg-mul-1 57.1%

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

13. Simplified 57.1%

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

14. Final simplification 57.1%

    \[\leadsto \frac{u0}{\frac{sin2phi}{alphay} \cdot \frac{1}{alphay} - \frac{\frac{cos2phi}{alphax}}{alphax}} \]
15. Add Preprocessing

Reproduce

herbie shell --seed 2024089 
(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)))))