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

Percentage Accurate: 60.4% → 98.4%

Time: 19.9s

Alternatives: 27

Speedup: 9.6×

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: 60.4% 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: 98.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 64.8%

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

   1. distribute-frac-neg N/A

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

   2. distribute-neg-frac2 N/A

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

   3. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\log \left(1 - u0\right), \color{blue}{\left(\mathsf{neg}\left(\left(\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\right)\right)\right)}\right) \]

   4. sub-neg N/A

      \[\leadsto \mathsf{/.f32}\left(\log \left(1 + \left(\mathsf{neg}\left(u0\right)\right)\right), \left(\mathsf{neg}\left(\left(\color{blue}{\frac{cos2phi}{alphax \cdot alphax}} + \frac{sin2phi}{alphay \cdot alphay}\right)\right)\right)\right) \]

   5. log1p-define N/A

      \[\leadsto \mathsf{/.f32}\left(\left(\mathsf{log1p}\left(\mathsf{neg}\left(u0\right)\right)\right), \left(\mathsf{neg}\left(\color{blue}{\left(\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\right)}\right)\right)\right) \]

   6. log1p-lowering-log1p.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\left(\mathsf{neg}\left(u0\right)\right)\right), \left(\mathsf{neg}\left(\color{blue}{\left(\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\right)}\right)\right)\right) \]

   7. neg-lowering-neg.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \left(\mathsf{neg}\left(\left(\color{blue}{\frac{cos2phi}{alphax \cdot alphax}} + \frac{sin2phi}{alphay \cdot alphay}\right)\right)\right)\right) \]

   8. distribute-neg-in N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \left(\left(\mathsf{neg}\left(\frac{cos2phi}{alphax \cdot alphax}\right)\right) + \color{blue}{\left(\mathsf{neg}\left(\frac{sin2phi}{alphay \cdot alphay}\right)\right)}\right)\right) \]

   9. unsub-neg N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \left(\left(\mathsf{neg}\left(\frac{cos2phi}{alphax \cdot alphax}\right)\right) - \color{blue}{\frac{sin2phi}{alphay \cdot alphay}}\right)\right) \]

   10. --lowering--.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\left(\mathsf{neg}\left(\frac{cos2phi}{alphax \cdot alphax}\right)\right), \color{blue}{\left(\frac{sin2phi}{alphay \cdot alphay}\right)}\right)\right) \]

   11. neg-sub0 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\left(0 - \frac{cos2phi}{alphax \cdot alphax}\right), \left(\frac{\color{blue}{sin2phi}}{alphay \cdot alphay}\right)\right)\right) \]

   12. --lowering--.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \left(\frac{cos2phi}{alphax \cdot alphax}\right)\right), \left(\frac{\color{blue}{sin2phi}}{alphay \cdot alphay}\right)\right)\right) \]

   13. /-lowering-/.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(cos2phi, \left(alphax \cdot alphax\right)\right)\right), \left(\frac{sin2phi}{alphay \cdot alphay}\right)\right)\right) \]

   14. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right)\right), \left(\frac{sin2phi}{alphay \cdot alphay}\right)\right)\right) \]

   15. /-lowering-/.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right)\right), \mathsf{/.f32}\left(sin2phi, \color{blue}{\left(alphay \cdot alphay\right)}\right)\right)\right) \]

   16. *-lowering-*.f32 98.3%

       \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right)\right), \mathsf{/.f32}\left(sin2phi, \mathsf{*.f32}\left(alphay, \color{blue}{alphay}\right)\right)\right)\right) \]

3. Simplified 98.3%

   \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(-u0\right)}{\left(0 - \frac{cos2phi}{alphax \cdot alphax}\right) - \frac{sin2phi}{alphay \cdot alphay}}} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. associate-/r* N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right)\right), \left(\frac{\frac{sin2phi}{alphay}}{\color{blue}{alphay}}\right)\right)\right) \]

   2. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right)\right), \mathsf{/.f32}\left(\left(\frac{sin2phi}{alphay}\right), \color{blue}{alphay}\right)\right)\right) \]

   3. /-lowering-/.f32 98.3%

      \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right)\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(sin2phi, alphay\right), alphay\right)\right)\right) \]

6. Applied egg-rr 98.3%

   \[\leadsto \frac{\mathsf{log1p}\left(-u0\right)}{\left(0 - \frac{cos2phi}{alphax \cdot alphax}\right) - \color{blue}{\frac{\frac{sin2phi}{alphay}}{alphay}}} \]
7. Step-by-step derivation

   1. clear-num N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \left(\frac{1}{\frac{alphax \cdot alphax}{cos2phi}}\right)\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(sin2phi, \color{blue}{alphay}\right), alphay\right)\right)\right) \]

   2. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(1, \left(\frac{alphax \cdot alphax}{cos2phi}\right)\right)\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(sin2phi, \color{blue}{alphay}\right), alphay\right)\right)\right) \]

   3. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(1, \mathsf{/.f32}\left(\left(alphax \cdot alphax\right), cos2phi\right)\right)\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(sin2phi, alphay\right), alphay\right)\right)\right) \]

   4. *-lowering-*.f32 98.3%

      \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(1, \mathsf{/.f32}\left(\mathsf{*.f32}\left(alphax, alphax\right), cos2phi\right)\right)\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(sin2phi, alphay\right), alphay\right)\right)\right) \]

8. Applied egg-rr 98.3%

   \[\leadsto \frac{\mathsf{log1p}\left(-u0\right)}{\left(0 - \color{blue}{\frac{1}{\frac{alphax \cdot alphax}{cos2phi}}}\right) - \frac{\frac{sin2phi}{alphay}}{alphay}} \]
9. Step-by-step derivation

   1. clear-num N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\left(0 - \frac{cos2phi}{alphax \cdot alphax}\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(sin2phi, \color{blue}{alphay}\right), alphay\right)\right)\right) \]

   2. sub0-neg N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\left(\mathsf{neg}\left(\frac{cos2phi}{alphax \cdot alphax}\right)\right), \mathsf{/.f32}\left(\color{blue}{\mathsf{/.f32}\left(sin2phi, alphay\right)}, alphay\right)\right)\right) \]

   3. associate-/r* N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\left(\mathsf{neg}\left(\frac{\frac{cos2phi}{alphax}}{alphax}\right)\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(\color{blue}{sin2phi}, alphay\right), alphay\right)\right)\right) \]

   4. distribute-neg-frac2 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\left(\frac{\frac{cos2phi}{alphax}}{\mathsf{neg}\left(alphax\right)}\right), \mathsf{/.f32}\left(\color{blue}{\mathsf{/.f32}\left(sin2phi, alphay\right)}, alphay\right)\right)\right) \]

   5. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{/.f32}\left(\left(\frac{cos2phi}{alphax}\right), \left(\mathsf{neg}\left(alphax\right)\right)\right), \mathsf{/.f32}\left(\color{blue}{\mathsf{/.f32}\left(sin2phi, alphay\right)}, alphay\right)\right)\right) \]

   6. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(cos2phi, alphax\right), \left(\mathsf{neg}\left(alphax\right)\right)\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(\color{blue}{sin2phi}, alphay\right), alphay\right)\right)\right) \]

   7. neg-lowering-neg.f32 98.3%

      \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(cos2phi, alphax\right), \mathsf{neg.f32}\left(alphax\right)\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(sin2phi, \color{blue}{alphay}\right), alphay\right)\right)\right) \]

10. Applied egg-rr 98.3%

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

    1. frac-sub N/A

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

    2. associate-/r/ N/A

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

    3. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{*.f32}\left(\left(\frac{\log \left(1 + \left(\mathsf{neg}\left(u0\right)\right)\right)}{\frac{cos2phi}{alphax} \cdot alphay - \left(\mathsf{neg}\left(alphax\right)\right) \cdot \frac{sin2phi}{alphay}}\right), \color{blue}{\left(\left(\mathsf{neg}\left(alphax\right)\right) \cdot alphay\right)}\right) \]

    4. /-lowering-/.f32 N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{/.f32}\left(\log \left(1 + \left(\mathsf{neg}\left(u0\right)\right)\right), \left(\frac{cos2phi}{alphax} \cdot alphay - \left(\mathsf{neg}\left(alphax\right)\right) \cdot \frac{sin2phi}{alphay}\right)\right), \left(\color{blue}{\left(\mathsf{neg}\left(alphax\right)\right)} \cdot alphay\right)\right) \]

    5. log1p-define N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{/.f32}\left(\left(\mathsf{log1p}\left(\mathsf{neg}\left(u0\right)\right)\right), \left(\frac{cos2phi}{alphax} \cdot alphay - \left(\mathsf{neg}\left(alphax\right)\right) \cdot \frac{sin2phi}{alphay}\right)\right), \left(\left(\mathsf{neg}\left(\color{blue}{alphax}\right)\right) \cdot alphay\right)\right) \]

    6. log1p-lowering-log1p.f32 N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\left(\mathsf{neg}\left(u0\right)\right)\right), \left(\frac{cos2phi}{alphax} \cdot alphay - \left(\mathsf{neg}\left(alphax\right)\right) \cdot \frac{sin2phi}{alphay}\right)\right), \left(\left(\mathsf{neg}\left(\color{blue}{alphax}\right)\right) \cdot alphay\right)\right) \]

    7. neg-lowering-neg.f32 N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \left(\frac{cos2phi}{alphax} \cdot alphay - \left(\mathsf{neg}\left(alphax\right)\right) \cdot \frac{sin2phi}{alphay}\right)\right), \left(\left(\mathsf{neg}\left(alphax\right)\right) \cdot alphay\right)\right) \]

    8. cancel-sign-sub N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \left(\frac{cos2phi}{alphax} \cdot alphay + alphax \cdot \frac{sin2phi}{alphay}\right)\right), \left(\left(\mathsf{neg}\left(alphax\right)\right) \cdot alphay\right)\right) \]

    9. +-lowering-+.f32 N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{+.f32}\left(\left(\frac{cos2phi}{alphax} \cdot alphay\right), \left(alphax \cdot \frac{sin2phi}{alphay}\right)\right)\right), \left(\left(\mathsf{neg}\left(alphax\right)\right) \cdot alphay\right)\right) \]

    10. *-lowering-*.f32 N/A

        \[\leadsto \mathsf{*.f32}\left(\mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{+.f32}\left(\mathsf{*.f32}\left(\left(\frac{cos2phi}{alphax}\right), alphay\right), \left(alphax \cdot \frac{sin2phi}{alphay}\right)\right)\right), \left(\left(\mathsf{neg}\left(alphax\right)\right) \cdot alphay\right)\right) \]

    11. /-lowering-/.f32 N/A

        \[\leadsto \mathsf{*.f32}\left(\mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{+.f32}\left(\mathsf{*.f32}\left(\mathsf{/.f32}\left(cos2phi, alphax\right), alphay\right), \left(alphax \cdot \frac{sin2phi}{alphay}\right)\right)\right), \left(\left(\mathsf{neg}\left(alphax\right)\right) \cdot alphay\right)\right) \]

    12. *-lowering-*.f32 N/A

        \[\leadsto \mathsf{*.f32}\left(\mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{+.f32}\left(\mathsf{*.f32}\left(\mathsf{/.f32}\left(cos2phi, alphax\right), alphay\right), \mathsf{*.f32}\left(alphax, \left(\frac{sin2phi}{alphay}\right)\right)\right)\right), \left(\left(\mathsf{neg}\left(alphax\right)\right) \cdot alphay\right)\right) \]

    13. /-lowering-/.f32 N/A

        \[\leadsto \mathsf{*.f32}\left(\mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{+.f32}\left(\mathsf{*.f32}\left(\mathsf{/.f32}\left(cos2phi, alphax\right), alphay\right), \mathsf{*.f32}\left(alphax, \mathsf{/.f32}\left(sin2phi, alphay\right)\right)\right)\right), \left(\left(\mathsf{neg}\left(alphax\right)\right) \cdot alphay\right)\right) \]

    14. distribute-lft-neg-out N/A

        \[\leadsto \mathsf{*.f32}\left(\mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{+.f32}\left(\mathsf{*.f32}\left(\mathsf{/.f32}\left(cos2phi, alphax\right), alphay\right), \mathsf{*.f32}\left(alphax, \mathsf{/.f32}\left(sin2phi, alphay\right)\right)\right)\right), \left(\mathsf{neg}\left(alphax \cdot alphay\right)\right)\right) \]

    15. neg-lowering-neg.f32 N/A

        \[\leadsto \mathsf{*.f32}\left(\mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{+.f32}\left(\mathsf{*.f32}\left(\mathsf{/.f32}\left(cos2phi, alphax\right), alphay\right), \mathsf{*.f32}\left(alphax, \mathsf{/.f32}\left(sin2phi, alphay\right)\right)\right)\right), \mathsf{neg.f32}\left(\left(alphax \cdot alphay\right)\right)\right) \]

    16. *-lowering-*.f32 98.5%

        \[\leadsto \mathsf{*.f32}\left(\mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{+.f32}\left(\mathsf{*.f32}\left(\mathsf{/.f32}\left(cos2phi, alphax\right), alphay\right), \mathsf{*.f32}\left(alphax, \mathsf{/.f32}\left(sin2phi, alphay\right)\right)\right)\right), \mathsf{neg.f32}\left(\mathsf{*.f32}\left(alphax, alphay\right)\right)\right) \]

12. Applied egg-rr 98.5%

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

13. Final simplification 98.5%

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

Alternative 2: 98.2% accurate, 1.0× speedup

Math

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

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

Derivation

1. Initial program 64.8%

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

   1. distribute-frac-neg N/A

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

   2. distribute-neg-frac2 N/A

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

   3. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\log \left(1 - u0\right), \color{blue}{\left(\mathsf{neg}\left(\left(\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\right)\right)\right)}\right) \]

   4. sub-neg N/A

      \[\leadsto \mathsf{/.f32}\left(\log \left(1 + \left(\mathsf{neg}\left(u0\right)\right)\right), \left(\mathsf{neg}\left(\left(\color{blue}{\frac{cos2phi}{alphax \cdot alphax}} + \frac{sin2phi}{alphay \cdot alphay}\right)\right)\right)\right) \]

   5. log1p-define N/A

      \[\leadsto \mathsf{/.f32}\left(\left(\mathsf{log1p}\left(\mathsf{neg}\left(u0\right)\right)\right), \left(\mathsf{neg}\left(\color{blue}{\left(\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\right)}\right)\right)\right) \]

   6. log1p-lowering-log1p.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\left(\mathsf{neg}\left(u0\right)\right)\right), \left(\mathsf{neg}\left(\color{blue}{\left(\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\right)}\right)\right)\right) \]

   7. neg-lowering-neg.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \left(\mathsf{neg}\left(\left(\color{blue}{\frac{cos2phi}{alphax \cdot alphax}} + \frac{sin2phi}{alphay \cdot alphay}\right)\right)\right)\right) \]

   8. distribute-neg-in N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \left(\left(\mathsf{neg}\left(\frac{cos2phi}{alphax \cdot alphax}\right)\right) + \color{blue}{\left(\mathsf{neg}\left(\frac{sin2phi}{alphay \cdot alphay}\right)\right)}\right)\right) \]

   9. unsub-neg N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \left(\left(\mathsf{neg}\left(\frac{cos2phi}{alphax \cdot alphax}\right)\right) - \color{blue}{\frac{sin2phi}{alphay \cdot alphay}}\right)\right) \]

   10. --lowering--.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\left(\mathsf{neg}\left(\frac{cos2phi}{alphax \cdot alphax}\right)\right), \color{blue}{\left(\frac{sin2phi}{alphay \cdot alphay}\right)}\right)\right) \]

   11. neg-sub0 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\left(0 - \frac{cos2phi}{alphax \cdot alphax}\right), \left(\frac{\color{blue}{sin2phi}}{alphay \cdot alphay}\right)\right)\right) \]

   12. --lowering--.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \left(\frac{cos2phi}{alphax \cdot alphax}\right)\right), \left(\frac{\color{blue}{sin2phi}}{alphay \cdot alphay}\right)\right)\right) \]

   13. /-lowering-/.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(cos2phi, \left(alphax \cdot alphax\right)\right)\right), \left(\frac{sin2phi}{alphay \cdot alphay}\right)\right)\right) \]

   14. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right)\right), \left(\frac{sin2phi}{alphay \cdot alphay}\right)\right)\right) \]

   15. /-lowering-/.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right)\right), \mathsf{/.f32}\left(sin2phi, \color{blue}{\left(alphay \cdot alphay\right)}\right)\right)\right) \]

   16. *-lowering-*.f32 98.3%

       \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right)\right), \mathsf{/.f32}\left(sin2phi, \mathsf{*.f32}\left(alphay, \color{blue}{alphay}\right)\right)\right)\right) \]

3. Simplified 98.3%

   \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(-u0\right)}{\left(0 - \frac{cos2phi}{alphax \cdot alphax}\right) - \frac{sin2phi}{alphay \cdot alphay}}} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. associate-/r* N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right)\right), \left(\frac{\frac{sin2phi}{alphay}}{\color{blue}{alphay}}\right)\right)\right) \]

   2. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right)\right), \mathsf{/.f32}\left(\left(\frac{sin2phi}{alphay}\right), \color{blue}{alphay}\right)\right)\right) \]

   3. /-lowering-/.f32 98.3%

      \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right)\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(sin2phi, alphay\right), alphay\right)\right)\right) \]

6. Applied egg-rr 98.3%

   \[\leadsto \frac{\mathsf{log1p}\left(-u0\right)}{\left(0 - \frac{cos2phi}{alphax \cdot alphax}\right) - \color{blue}{\frac{\frac{sin2phi}{alphay}}{alphay}}} \]
7. Step-by-step derivation

   1. clear-num N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \left(\frac{1}{\frac{alphax \cdot alphax}{cos2phi}}\right)\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(sin2phi, \color{blue}{alphay}\right), alphay\right)\right)\right) \]

   2. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(1, \left(\frac{alphax \cdot alphax}{cos2phi}\right)\right)\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(sin2phi, \color{blue}{alphay}\right), alphay\right)\right)\right) \]

   3. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(1, \mathsf{/.f32}\left(\left(alphax \cdot alphax\right), cos2phi\right)\right)\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(sin2phi, alphay\right), alphay\right)\right)\right) \]

   4. *-lowering-*.f32 98.3%

      \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(1, \mathsf{/.f32}\left(\mathsf{*.f32}\left(alphax, alphax\right), cos2phi\right)\right)\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(sin2phi, alphay\right), alphay\right)\right)\right) \]

8. Applied egg-rr 98.3%

   \[\leadsto \frac{\mathsf{log1p}\left(-u0\right)}{\left(0 - \color{blue}{\frac{1}{\frac{alphax \cdot alphax}{cos2phi}}}\right) - \frac{\frac{sin2phi}{alphay}}{alphay}} \]

9. Final simplification 98.3%

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

Alternative 3: 98.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\mathsf{log1p}\left(-u0\right)}{\left(0 - \frac{cos2phi}{alphax \cdot alphax}\right) - \frac{\frac{sin2phi}{alphay}}{alphay}} \end{array} \]

FPCore

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

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	return log1pf(-u0) / ((0.0f - (cos2phi / (alphax * alphax))) - ((sin2phi / alphay) / alphay));
}

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	return Float32(log1p(Float32(-u0)) / Float32(Float32(Float32(0.0) - Float32(cos2phi / Float32(alphax * alphax))) - Float32(Float32(sin2phi / alphay) / alphay)))
end

Derivation

1. Initial program 64.8%

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

   1. distribute-frac-neg N/A

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

   2. distribute-neg-frac2 N/A

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

   3. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\log \left(1 - u0\right), \color{blue}{\left(\mathsf{neg}\left(\left(\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\right)\right)\right)}\right) \]

   4. sub-neg N/A

      \[\leadsto \mathsf{/.f32}\left(\log \left(1 + \left(\mathsf{neg}\left(u0\right)\right)\right), \left(\mathsf{neg}\left(\left(\color{blue}{\frac{cos2phi}{alphax \cdot alphax}} + \frac{sin2phi}{alphay \cdot alphay}\right)\right)\right)\right) \]

   5. log1p-define N/A

      \[\leadsto \mathsf{/.f32}\left(\left(\mathsf{log1p}\left(\mathsf{neg}\left(u0\right)\right)\right), \left(\mathsf{neg}\left(\color{blue}{\left(\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\right)}\right)\right)\right) \]

   6. log1p-lowering-log1p.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\left(\mathsf{neg}\left(u0\right)\right)\right), \left(\mathsf{neg}\left(\color{blue}{\left(\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\right)}\right)\right)\right) \]

   7. neg-lowering-neg.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \left(\mathsf{neg}\left(\left(\color{blue}{\frac{cos2phi}{alphax \cdot alphax}} + \frac{sin2phi}{alphay \cdot alphay}\right)\right)\right)\right) \]

   8. distribute-neg-in N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \left(\left(\mathsf{neg}\left(\frac{cos2phi}{alphax \cdot alphax}\right)\right) + \color{blue}{\left(\mathsf{neg}\left(\frac{sin2phi}{alphay \cdot alphay}\right)\right)}\right)\right) \]

   9. unsub-neg N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \left(\left(\mathsf{neg}\left(\frac{cos2phi}{alphax \cdot alphax}\right)\right) - \color{blue}{\frac{sin2phi}{alphay \cdot alphay}}\right)\right) \]

   10. --lowering--.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\left(\mathsf{neg}\left(\frac{cos2phi}{alphax \cdot alphax}\right)\right), \color{blue}{\left(\frac{sin2phi}{alphay \cdot alphay}\right)}\right)\right) \]

   11. neg-sub0 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\left(0 - \frac{cos2phi}{alphax \cdot alphax}\right), \left(\frac{\color{blue}{sin2phi}}{alphay \cdot alphay}\right)\right)\right) \]

   12. --lowering--.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \left(\frac{cos2phi}{alphax \cdot alphax}\right)\right), \left(\frac{\color{blue}{sin2phi}}{alphay \cdot alphay}\right)\right)\right) \]

   13. /-lowering-/.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(cos2phi, \left(alphax \cdot alphax\right)\right)\right), \left(\frac{sin2phi}{alphay \cdot alphay}\right)\right)\right) \]

   14. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right)\right), \left(\frac{sin2phi}{alphay \cdot alphay}\right)\right)\right) \]

   15. /-lowering-/.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right)\right), \mathsf{/.f32}\left(sin2phi, \color{blue}{\left(alphay \cdot alphay\right)}\right)\right)\right) \]

   16. *-lowering-*.f32 98.3%

       \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right)\right), \mathsf{/.f32}\left(sin2phi, \mathsf{*.f32}\left(alphay, \color{blue}{alphay}\right)\right)\right)\right) \]

3. Simplified 98.3%

   \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(-u0\right)}{\left(0 - \frac{cos2phi}{alphax \cdot alphax}\right) - \frac{sin2phi}{alphay \cdot alphay}}} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. associate-/r* N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right)\right), \left(\frac{\frac{sin2phi}{alphay}}{\color{blue}{alphay}}\right)\right)\right) \]

   2. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right)\right), \mathsf{/.f32}\left(\left(\frac{sin2phi}{alphay}\right), \color{blue}{alphay}\right)\right)\right) \]

   3. /-lowering-/.f32 98.3%

      \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right)\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(sin2phi, alphay\right), alphay\right)\right)\right) \]

6. Applied egg-rr 98.3%

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

Alternative 4: 98.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\mathsf{log1p}\left(-u0\right)}{\frac{0 - \frac{cos2phi}{alphax}}{alphax} - \frac{\frac{sin2phi}{alphay}}{alphay}} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (/
  (log1p (- u0))
  (- (/ (- 0.0 (/ cos2phi alphax)) alphax) (/ (/ sin2phi alphay) alphay))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	return log1pf(-u0) / (((0.0f - (cos2phi / alphax)) / alphax) - ((sin2phi / alphay) / alphay));
}

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	return Float32(log1p(Float32(-u0)) / Float32(Float32(Float32(Float32(0.0) - Float32(cos2phi / alphax)) / alphax) - Float32(Float32(sin2phi / alphay) / alphay)))
end

Derivation

1. Initial program 64.8%

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

   1. distribute-frac-neg N/A

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

   2. distribute-neg-frac2 N/A

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

   3. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\log \left(1 - u0\right), \color{blue}{\left(\mathsf{neg}\left(\left(\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\right)\right)\right)}\right) \]

   4. sub-neg N/A

      \[\leadsto \mathsf{/.f32}\left(\log \left(1 + \left(\mathsf{neg}\left(u0\right)\right)\right), \left(\mathsf{neg}\left(\left(\color{blue}{\frac{cos2phi}{alphax \cdot alphax}} + \frac{sin2phi}{alphay \cdot alphay}\right)\right)\right)\right) \]

   5. log1p-define N/A

      \[\leadsto \mathsf{/.f32}\left(\left(\mathsf{log1p}\left(\mathsf{neg}\left(u0\right)\right)\right), \left(\mathsf{neg}\left(\color{blue}{\left(\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\right)}\right)\right)\right) \]

   6. log1p-lowering-log1p.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\left(\mathsf{neg}\left(u0\right)\right)\right), \left(\mathsf{neg}\left(\color{blue}{\left(\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\right)}\right)\right)\right) \]

   7. neg-lowering-neg.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \left(\mathsf{neg}\left(\left(\color{blue}{\frac{cos2phi}{alphax \cdot alphax}} + \frac{sin2phi}{alphay \cdot alphay}\right)\right)\right)\right) \]

   8. distribute-neg-in N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \left(\left(\mathsf{neg}\left(\frac{cos2phi}{alphax \cdot alphax}\right)\right) + \color{blue}{\left(\mathsf{neg}\left(\frac{sin2phi}{alphay \cdot alphay}\right)\right)}\right)\right) \]

   9. unsub-neg N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \left(\left(\mathsf{neg}\left(\frac{cos2phi}{alphax \cdot alphax}\right)\right) - \color{blue}{\frac{sin2phi}{alphay \cdot alphay}}\right)\right) \]

   10. --lowering--.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\left(\mathsf{neg}\left(\frac{cos2phi}{alphax \cdot alphax}\right)\right), \color{blue}{\left(\frac{sin2phi}{alphay \cdot alphay}\right)}\right)\right) \]

   11. neg-sub0 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\left(0 - \frac{cos2phi}{alphax \cdot alphax}\right), \left(\frac{\color{blue}{sin2phi}}{alphay \cdot alphay}\right)\right)\right) \]

   12. --lowering--.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \left(\frac{cos2phi}{alphax \cdot alphax}\right)\right), \left(\frac{\color{blue}{sin2phi}}{alphay \cdot alphay}\right)\right)\right) \]

   13. /-lowering-/.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(cos2phi, \left(alphax \cdot alphax\right)\right)\right), \left(\frac{sin2phi}{alphay \cdot alphay}\right)\right)\right) \]

   14. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right)\right), \left(\frac{sin2phi}{alphay \cdot alphay}\right)\right)\right) \]

   15. /-lowering-/.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right)\right), \mathsf{/.f32}\left(sin2phi, \color{blue}{\left(alphay \cdot alphay\right)}\right)\right)\right) \]

   16. *-lowering-*.f32 98.3%

       \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right)\right), \mathsf{/.f32}\left(sin2phi, \mathsf{*.f32}\left(alphay, \color{blue}{alphay}\right)\right)\right)\right) \]

3. Simplified 98.3%

   \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(-u0\right)}{\left(0 - \frac{cos2phi}{alphax \cdot alphax}\right) - \frac{sin2phi}{alphay \cdot alphay}}} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. associate-/r* N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right)\right), \left(\frac{\frac{sin2phi}{alphay}}{\color{blue}{alphay}}\right)\right)\right) \]

   2. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right)\right), \mathsf{/.f32}\left(\left(\frac{sin2phi}{alphay}\right), \color{blue}{alphay}\right)\right)\right) \]

   3. /-lowering-/.f32 98.3%

      \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right)\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(sin2phi, alphay\right), alphay\right)\right)\right) \]

6. Applied egg-rr 98.3%

   \[\leadsto \frac{\mathsf{log1p}\left(-u0\right)}{\left(0 - \frac{cos2phi}{alphax \cdot alphax}\right) - \color{blue}{\frac{\frac{sin2phi}{alphay}}{alphay}}} \]
7. Step-by-step derivation

   1. associate-/r* N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \left(\left(0 - \frac{cos2phi}{alphax \cdot alphax}\right) - \frac{sin2phi}{\color{blue}{alphay \cdot alphay}}\right)\right) \]

   2. associate--l- N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \left(0 - \color{blue}{\left(\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\right)}\right)\right) \]

   3. neg-sub0 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \left(\mathsf{neg}\left(\left(\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\right)\right)\right)\right) \]

   4. neg-lowering-neg.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{neg.f32}\left(\left(\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\right)\right)\right) \]

   5. +-lowering-+.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{neg.f32}\left(\mathsf{+.f32}\left(\left(\frac{cos2phi}{alphax \cdot alphax}\right), \left(\frac{sin2phi}{alphay \cdot alphay}\right)\right)\right)\right) \]

   6. associate-/r* N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{neg.f32}\left(\mathsf{+.f32}\left(\left(\frac{\frac{cos2phi}{alphax}}{alphax}\right), \left(\frac{sin2phi}{alphay \cdot alphay}\right)\right)\right)\right) \]

   7. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{neg.f32}\left(\mathsf{+.f32}\left(\mathsf{/.f32}\left(\left(\frac{cos2phi}{alphax}\right), alphax\right), \left(\frac{sin2phi}{alphay \cdot alphay}\right)\right)\right)\right) \]

   8. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{neg.f32}\left(\mathsf{+.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(cos2phi, alphax\right), alphax\right), \left(\frac{sin2phi}{alphay \cdot alphay}\right)\right)\right)\right) \]

   9. associate-/r* N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{neg.f32}\left(\mathsf{+.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(cos2phi, alphax\right), alphax\right), \left(\frac{\frac{sin2phi}{alphay}}{alphay}\right)\right)\right)\right) \]

   10. /-lowering-/.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{neg.f32}\left(\mathsf{+.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(cos2phi, alphax\right), alphax\right), \mathsf{/.f32}\left(\left(\frac{sin2phi}{alphay}\right), alphay\right)\right)\right)\right) \]

   11. /-lowering-/.f32 98.3%

       \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{neg.f32}\left(\mathsf{+.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(cos2phi, alphax\right), alphax\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(sin2phi, alphay\right), alphay\right)\right)\right)\right) \]

8. Applied egg-rr 98.3%

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

9. Final simplification 98.3%

   \[\leadsto \frac{\mathsf{log1p}\left(-u0\right)}{\frac{0 - \frac{cos2phi}{alphax}}{alphax} - \frac{\frac{sin2phi}{alphay}}{alphay}} \]
10. Add Preprocessing

Alternative 5: 98.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\mathsf{log1p}\left(-u0\right)}{\frac{0 - \frac{cos2phi}{alphax}}{alphax} - \frac{sin2phi}{alphay \cdot alphay}} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (/
  (log1p (- u0))
  (- (/ (- 0.0 (/ cos2phi alphax)) alphax) (/ sin2phi (* alphay alphay)))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	return log1pf(-u0) / (((0.0f - (cos2phi / alphax)) / alphax) - (sin2phi / (alphay * alphay)));
}

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	return Float32(log1p(Float32(-u0)) / Float32(Float32(Float32(Float32(0.0) - Float32(cos2phi / alphax)) / alphax) - Float32(sin2phi / Float32(alphay * alphay))))
end

Derivation

1. Initial program 64.8%

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

   1. distribute-frac-neg N/A

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

   2. distribute-neg-frac2 N/A

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

   3. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\log \left(1 - u0\right), \color{blue}{\left(\mathsf{neg}\left(\left(\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\right)\right)\right)}\right) \]

   4. sub-neg N/A

      \[\leadsto \mathsf{/.f32}\left(\log \left(1 + \left(\mathsf{neg}\left(u0\right)\right)\right), \left(\mathsf{neg}\left(\left(\color{blue}{\frac{cos2phi}{alphax \cdot alphax}} + \frac{sin2phi}{alphay \cdot alphay}\right)\right)\right)\right) \]

   5. log1p-define N/A

      \[\leadsto \mathsf{/.f32}\left(\left(\mathsf{log1p}\left(\mathsf{neg}\left(u0\right)\right)\right), \left(\mathsf{neg}\left(\color{blue}{\left(\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\right)}\right)\right)\right) \]

   6. log1p-lowering-log1p.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\left(\mathsf{neg}\left(u0\right)\right)\right), \left(\mathsf{neg}\left(\color{blue}{\left(\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\right)}\right)\right)\right) \]

   7. neg-lowering-neg.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \left(\mathsf{neg}\left(\left(\color{blue}{\frac{cos2phi}{alphax \cdot alphax}} + \frac{sin2phi}{alphay \cdot alphay}\right)\right)\right)\right) \]

   8. distribute-neg-in N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \left(\left(\mathsf{neg}\left(\frac{cos2phi}{alphax \cdot alphax}\right)\right) + \color{blue}{\left(\mathsf{neg}\left(\frac{sin2phi}{alphay \cdot alphay}\right)\right)}\right)\right) \]

   9. unsub-neg N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \left(\left(\mathsf{neg}\left(\frac{cos2phi}{alphax \cdot alphax}\right)\right) - \color{blue}{\frac{sin2phi}{alphay \cdot alphay}}\right)\right) \]

   10. --lowering--.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\left(\mathsf{neg}\left(\frac{cos2phi}{alphax \cdot alphax}\right)\right), \color{blue}{\left(\frac{sin2phi}{alphay \cdot alphay}\right)}\right)\right) \]

   11. neg-sub0 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\left(0 - \frac{cos2phi}{alphax \cdot alphax}\right), \left(\frac{\color{blue}{sin2phi}}{alphay \cdot alphay}\right)\right)\right) \]

   12. --lowering--.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \left(\frac{cos2phi}{alphax \cdot alphax}\right)\right), \left(\frac{\color{blue}{sin2phi}}{alphay \cdot alphay}\right)\right)\right) \]

   13. /-lowering-/.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(cos2phi, \left(alphax \cdot alphax\right)\right)\right), \left(\frac{sin2phi}{alphay \cdot alphay}\right)\right)\right) \]

   14. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right)\right), \left(\frac{sin2phi}{alphay \cdot alphay}\right)\right)\right) \]

   15. /-lowering-/.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right)\right), \mathsf{/.f32}\left(sin2phi, \color{blue}{\left(alphay \cdot alphay\right)}\right)\right)\right) \]

   16. *-lowering-*.f32 98.3%

       \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right)\right), \mathsf{/.f32}\left(sin2phi, \mathsf{*.f32}\left(alphay, \color{blue}{alphay}\right)\right)\right)\right) \]

3. Simplified 98.3%

   \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(-u0\right)}{\left(0 - \frac{cos2phi}{alphax \cdot alphax}\right) - \frac{sin2phi}{alphay \cdot alphay}}} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. associate--l- N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \left(0 - \color{blue}{\left(\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\right)}\right)\right) \]

   2. neg-sub0 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \left(\mathsf{neg}\left(\left(\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\right)\right)\right)\right) \]

   3. neg-lowering-neg.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{neg.f32}\left(\left(\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\right)\right)\right) \]

   4. +-lowering-+.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{neg.f32}\left(\mathsf{+.f32}\left(\left(\frac{cos2phi}{alphax \cdot alphax}\right), \left(\frac{sin2phi}{alphay \cdot alphay}\right)\right)\right)\right) \]

   5. associate-/r* N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{neg.f32}\left(\mathsf{+.f32}\left(\left(\frac{\frac{cos2phi}{alphax}}{alphax}\right), \left(\frac{sin2phi}{alphay \cdot alphay}\right)\right)\right)\right) \]

   6. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{neg.f32}\left(\mathsf{+.f32}\left(\mathsf{/.f32}\left(\left(\frac{cos2phi}{alphax}\right), alphax\right), \left(\frac{sin2phi}{alphay \cdot alphay}\right)\right)\right)\right) \]

   7. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{neg.f32}\left(\mathsf{+.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(cos2phi, alphax\right), alphax\right), \left(\frac{sin2phi}{alphay \cdot alphay}\right)\right)\right)\right) \]

   8. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{neg.f32}\left(\mathsf{+.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(cos2phi, alphax\right), alphax\right), \mathsf{/.f32}\left(sin2phi, \left(alphay \cdot alphay\right)\right)\right)\right)\right) \]

   9. *-lowering-*.f32 98.3%

      \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{neg.f32}\left(\mathsf{+.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(cos2phi, alphax\right), alphax\right), \mathsf{/.f32}\left(sin2phi, \mathsf{*.f32}\left(alphay, alphay\right)\right)\right)\right)\right) \]

6. Applied egg-rr 98.3%

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

7. Final simplification 98.3%

   \[\leadsto \frac{\mathsf{log1p}\left(-u0\right)}{\frac{0 - \frac{cos2phi}{alphax}}{alphax} - \frac{sin2phi}{alphay \cdot alphay}} \]
8. Add Preprocessing

Alternative 6: 94.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u0 \leq 0.0949999988079071:\\ \;\;\;\;\frac{u0 \cdot \left(u0 \cdot \left(u0 \cdot \left(u0 \cdot -0.25 + -0.3333333333333333\right) + -0.5\right) + -1\right)}{\frac{-1}{\frac{alphax \cdot alphax}{cos2phi}} - \frac{\frac{sin2phi}{alphay}}{alphay}}\\ \mathbf{else}:\\ \;\;\;\;alphay \cdot \left(\mathsf{log1p}\left(-u0\right) \cdot \left(0 - \frac{alphay}{sin2phi}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (if (<= u0 0.0949999988079071)
   (/
    (* u0 (+ (* u0 (+ (* u0 (+ (* u0 -0.25) -0.3333333333333333)) -0.5)) -1.0))
    (- (/ -1.0 (/ (* alphax alphax) cos2phi)) (/ (/ sin2phi alphay) alphay)))
   (* alphay (* (log1p (- u0)) (- 0.0 (/ alphay sin2phi))))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float tmp;
	if (u0 <= 0.0949999988079071f) {
		tmp = (u0 * ((u0 * ((u0 * ((u0 * -0.25f) + -0.3333333333333333f)) + -0.5f)) + -1.0f)) / ((-1.0f / ((alphax * alphax) / cos2phi)) - ((sin2phi / alphay) / alphay));
	} else {
		tmp = alphay * (log1pf(-u0) * (0.0f - (alphay / sin2phi)));
	}
	return tmp;
}

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = Float32(0.0)
	if (u0 <= Float32(0.0949999988079071))
		tmp = Float32(Float32(u0 * Float32(Float32(u0 * Float32(Float32(u0 * Float32(Float32(u0 * Float32(-0.25)) + Float32(-0.3333333333333333))) + Float32(-0.5))) + Float32(-1.0))) / Float32(Float32(Float32(-1.0) / Float32(Float32(alphax * alphax) / cos2phi)) - Float32(Float32(sin2phi / alphay) / alphay)));
	else
		tmp = Float32(alphay * Float32(log1p(Float32(-u0)) * Float32(Float32(0.0) - Float32(alphay / sin2phi))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if u0 < 0.0949999988

   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 N/A

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

      2. distribute-neg-frac2 N/A

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

      3. /-lowering-/.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(\log \left(1 - u0\right), \color{blue}{\left(\mathsf{neg}\left(\left(\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\right)\right)\right)}\right) \]

      4. sub-neg N/A

         \[\leadsto \mathsf{/.f32}\left(\log \left(1 + \left(\mathsf{neg}\left(u0\right)\right)\right), \left(\mathsf{neg}\left(\left(\color{blue}{\frac{cos2phi}{alphax \cdot alphax}} + \frac{sin2phi}{alphay \cdot alphay}\right)\right)\right)\right) \]

      5. log1p-define N/A

         \[\leadsto \mathsf{/.f32}\left(\left(\mathsf{log1p}\left(\mathsf{neg}\left(u0\right)\right)\right), \left(\mathsf{neg}\left(\color{blue}{\left(\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\right)}\right)\right)\right) \]

      6. log1p-lowering-log1p.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\left(\mathsf{neg}\left(u0\right)\right)\right), \left(\mathsf{neg}\left(\color{blue}{\left(\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\right)}\right)\right)\right) \]

      7. neg-lowering-neg.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \left(\mathsf{neg}\left(\left(\color{blue}{\frac{cos2phi}{alphax \cdot alphax}} + \frac{sin2phi}{alphay \cdot alphay}\right)\right)\right)\right) \]

      8. distribute-neg-in N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \left(\left(\mathsf{neg}\left(\frac{cos2phi}{alphax \cdot alphax}\right)\right) + \color{blue}{\left(\mathsf{neg}\left(\frac{sin2phi}{alphay \cdot alphay}\right)\right)}\right)\right) \]

      9. unsub-neg N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \left(\left(\mathsf{neg}\left(\frac{cos2phi}{alphax \cdot alphax}\right)\right) - \color{blue}{\frac{sin2phi}{alphay \cdot alphay}}\right)\right) \]

      10. --lowering--.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\left(\mathsf{neg}\left(\frac{cos2phi}{alphax \cdot alphax}\right)\right), \color{blue}{\left(\frac{sin2phi}{alphay \cdot alphay}\right)}\right)\right) \]

      11. neg-sub0 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\left(0 - \frac{cos2phi}{alphax \cdot alphax}\right), \left(\frac{\color{blue}{sin2phi}}{alphay \cdot alphay}\right)\right)\right) \]

      12. --lowering--.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \left(\frac{cos2phi}{alphax \cdot alphax}\right)\right), \left(\frac{\color{blue}{sin2phi}}{alphay \cdot alphay}\right)\right)\right) \]

      13. /-lowering-/.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(cos2phi, \left(alphax \cdot alphax\right)\right)\right), \left(\frac{sin2phi}{alphay \cdot alphay}\right)\right)\right) \]

      14. *-lowering-*.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right)\right), \left(\frac{sin2phi}{alphay \cdot alphay}\right)\right)\right) \]

      15. /-lowering-/.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right)\right), \mathsf{/.f32}\left(sin2phi, \color{blue}{\left(alphay \cdot alphay\right)}\right)\right)\right) \]

      16. *-lowering-*.f32 98.2%

          \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right)\right), \mathsf{/.f32}\left(sin2phi, \mathsf{*.f32}\left(alphay, \color{blue}{alphay}\right)\right)\right)\right) \]

   3. Simplified 98.2%

      \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(-u0\right)}{\left(0 - \frac{cos2phi}{alphax \cdot alphax}\right) - \frac{sin2phi}{alphay \cdot alphay}}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-/r* N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right)\right), \left(\frac{\frac{sin2phi}{alphay}}{\color{blue}{alphay}}\right)\right)\right) \]

      2. /-lowering-/.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right)\right), \mathsf{/.f32}\left(\left(\frac{sin2phi}{alphay}\right), \color{blue}{alphay}\right)\right)\right) \]

      3. /-lowering-/.f32 98.3%

         \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right)\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(sin2phi, alphay\right), alphay\right)\right)\right) \]

   6. Applied egg-rr 98.3%

      \[\leadsto \frac{\mathsf{log1p}\left(-u0\right)}{\left(0 - \frac{cos2phi}{alphax \cdot alphax}\right) - \color{blue}{\frac{\frac{sin2phi}{alphay}}{alphay}}} \]
   7. Step-by-step derivation

      1. clear-num N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \left(\frac{1}{\frac{alphax \cdot alphax}{cos2phi}}\right)\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(sin2phi, \color{blue}{alphay}\right), alphay\right)\right)\right) \]

      2. /-lowering-/.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(1, \left(\frac{alphax \cdot alphax}{cos2phi}\right)\right)\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(sin2phi, \color{blue}{alphay}\right), alphay\right)\right)\right) \]

      3. /-lowering-/.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(1, \mathsf{/.f32}\left(\left(alphax \cdot alphax\right), cos2phi\right)\right)\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(sin2phi, alphay\right), alphay\right)\right)\right) \]

      4. *-lowering-*.f32 98.3%

         \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(1, \mathsf{/.f32}\left(\mathsf{*.f32}\left(alphax, alphax\right), cos2phi\right)\right)\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(sin2phi, alphay\right), alphay\right)\right)\right) \]

   8. Applied egg-rr 98.3%

      \[\leadsto \frac{\mathsf{log1p}\left(-u0\right)}{\left(0 - \color{blue}{\frac{1}{\frac{alphax \cdot alphax}{cos2phi}}}\right) - \frac{\frac{sin2phi}{alphay}}{alphay}} \]

   9. Taylor expanded in u0 around 0

      \[\leadsto \mathsf{/.f32}\left(\color{blue}{\left(u0 \cdot \left(u0 \cdot \left(u0 \cdot \left(\frac{-1}{4} \cdot u0 - \frac{1}{3}\right) - \frac{1}{2}\right) - 1\right)\right)}, \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(1, \mathsf{/.f32}\left(\mathsf{*.f32}\left(alphax, alphax\right), cos2phi\right)\right)\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(sin2phi, alphay\right), alphay\right)\right)\right) \]
   10. Step-by-step derivation

       1. *-lowering-*.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(u0, \left(u0 \cdot \left(u0 \cdot \left(\frac{-1}{4} \cdot u0 - \frac{1}{3}\right) - \frac{1}{2}\right) - 1\right)\right), \mathsf{\_.f32}\left(\color{blue}{\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(1, \mathsf{/.f32}\left(\mathsf{*.f32}\left(alphax, alphax\right), cos2phi\right)\right)\right)}, \mathsf{/.f32}\left(\mathsf{/.f32}\left(sin2phi, alphay\right), alphay\right)\right)\right) \]

       2. sub-neg N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(u0, \left(u0 \cdot \left(u0 \cdot \left(\frac{-1}{4} \cdot u0 - \frac{1}{3}\right) - \frac{1}{2}\right) + \left(\mathsf{neg}\left(1\right)\right)\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \color{blue}{\mathsf{/.f32}\left(1, \mathsf{/.f32}\left(\mathsf{*.f32}\left(alphax, alphax\right), cos2phi\right)\right)}\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(sin2phi, alphay\right), alphay\right)\right)\right) \]

       3. metadata-eval N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(u0, \left(u0 \cdot \left(u0 \cdot \left(\frac{-1}{4} \cdot u0 - \frac{1}{3}\right) - \frac{1}{2}\right) + -1\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(1, \color{blue}{\mathsf{/.f32}\left(\mathsf{*.f32}\left(alphax, alphax\right), cos2phi\right)}\right)\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(sin2phi, alphay\right), alphay\right)\right)\right) \]

       4. +-lowering-+.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\left(u0 \cdot \left(u0 \cdot \left(\frac{-1}{4} \cdot u0 - \frac{1}{3}\right) - \frac{1}{2}\right)\right), -1\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \color{blue}{\mathsf{/.f32}\left(1, \mathsf{/.f32}\left(\mathsf{*.f32}\left(alphax, alphax\right), cos2phi\right)\right)}\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(sin2phi, alphay\right), alphay\right)\right)\right) \]

       5. *-lowering-*.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{*.f32}\left(u0, \left(u0 \cdot \left(\frac{-1}{4} \cdot u0 - \frac{1}{3}\right) - \frac{1}{2}\right)\right), -1\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(\color{blue}{1}, \mathsf{/.f32}\left(\mathsf{*.f32}\left(alphax, alphax\right), cos2phi\right)\right)\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(sin2phi, alphay\right), alphay\right)\right)\right) \]

       6. sub-neg N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{*.f32}\left(u0, \left(u0 \cdot \left(\frac{-1}{4} \cdot u0 - \frac{1}{3}\right) + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right)\right), -1\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(1, \mathsf{/.f32}\left(\mathsf{*.f32}\left(alphax, alphax\right), cos2phi\right)\right)\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(sin2phi, alphay\right), alphay\right)\right)\right) \]

       7. metadata-eval N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{*.f32}\left(u0, \left(u0 \cdot \left(\frac{-1}{4} \cdot u0 - \frac{1}{3}\right) + \frac{-1}{2}\right)\right), -1\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(1, \mathsf{/.f32}\left(\mathsf{*.f32}\left(alphax, alphax\right), cos2phi\right)\right)\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(sin2phi, alphay\right), alphay\right)\right)\right) \]

       8. +-lowering-+.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\left(u0 \cdot \left(\frac{-1}{4} \cdot u0 - \frac{1}{3}\right)\right), \frac{-1}{2}\right)\right), -1\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(1, \mathsf{/.f32}\left(\mathsf{*.f32}\left(alphax, alphax\right), cos2phi\right)\right)\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(sin2phi, alphay\right), alphay\right)\right)\right) \]

       9. *-lowering-*.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{*.f32}\left(u0, \left(\frac{-1}{4} \cdot u0 - \frac{1}{3}\right)\right), \frac{-1}{2}\right)\right), -1\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(1, \mathsf{/.f32}\left(\mathsf{*.f32}\left(alphax, alphax\right), cos2phi\right)\right)\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(sin2phi, alphay\right), alphay\right)\right)\right) \]

       10. sub-neg N/A

           \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{*.f32}\left(u0, \left(\frac{-1}{4} \cdot u0 + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right)\right), \frac{-1}{2}\right)\right), -1\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(1, \mathsf{/.f32}\left(\mathsf{*.f32}\left(alphax, alphax\right), cos2phi\right)\right)\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(sin2phi, alphay\right), alphay\right)\right)\right) \]

       11. metadata-eval N/A

           \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{*.f32}\left(u0, \left(\frac{-1}{4} \cdot u0 + \frac{-1}{3}\right)\right), \frac{-1}{2}\right)\right), -1\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(1, \mathsf{/.f32}\left(\mathsf{*.f32}\left(alphax, alphax\right), cos2phi\right)\right)\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(sin2phi, alphay\right), alphay\right)\right)\right) \]

       12. +-lowering-+.f32 N/A

           \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\left(\frac{-1}{4} \cdot u0\right), \frac{-1}{3}\right)\right), \frac{-1}{2}\right)\right), -1\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(1, \mathsf{/.f32}\left(\mathsf{*.f32}\left(alphax, alphax\right), cos2phi\right)\right)\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(sin2phi, alphay\right), alphay\right)\right)\right) \]

       13. *-commutative N/A

           \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\left(u0 \cdot \frac{-1}{4}\right), \frac{-1}{3}\right)\right), \frac{-1}{2}\right)\right), -1\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(1, \mathsf{/.f32}\left(\mathsf{*.f32}\left(alphax, alphax\right), cos2phi\right)\right)\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(sin2phi, alphay\right), alphay\right)\right)\right) \]

       14. *-lowering-*.f32 97.3%

           \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{*.f32}\left(u0, \frac{-1}{4}\right), \frac{-1}{3}\right)\right), \frac{-1}{2}\right)\right), -1\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(1, \mathsf{/.f32}\left(\mathsf{*.f32}\left(alphax, alphax\right), cos2phi\right)\right)\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(sin2phi, alphay\right), alphay\right)\right)\right) \]

   11. Simplified 97.3%

       \[\leadsto \frac{\color{blue}{u0 \cdot \left(u0 \cdot \left(u0 \cdot \left(u0 \cdot -0.25 + -0.3333333333333333\right) + -0.5\right) + -1\right)}}{\left(0 - \frac{1}{\frac{alphax \cdot alphax}{cos2phi}}\right) - \frac{\frac{sin2phi}{alphay}}{alphay}} \]

   if 0.0949999988 < u0

   1. Initial program 97.8%

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

      1. distribute-frac-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. /-lowering-/.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(\log \left(1 - u0\right), \color{blue}{\left(\mathsf{neg}\left(\left(\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\right)\right)\right)}\right) \]

      4. sub-neg N/A

         \[\leadsto \mathsf{/.f32}\left(\log \left(1 + \left(\mathsf{neg}\left(u0\right)\right)\right), \left(\mathsf{neg}\left(\left(\color{blue}{\frac{cos2phi}{alphax \cdot alphax}} + \frac{sin2phi}{alphay \cdot alphay}\right)\right)\right)\right) \]

      5. log1p-define N/A

         \[\leadsto \mathsf{/.f32}\left(\left(\mathsf{log1p}\left(\mathsf{neg}\left(u0\right)\right)\right), \left(\mathsf{neg}\left(\color{blue}{\left(\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\right)}\right)\right)\right) \]

      6. log1p-lowering-log1p.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\left(\mathsf{neg}\left(u0\right)\right)\right), \left(\mathsf{neg}\left(\color{blue}{\left(\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\right)}\right)\right)\right) \]

      7. neg-lowering-neg.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \left(\mathsf{neg}\left(\left(\color{blue}{\frac{cos2phi}{alphax \cdot alphax}} + \frac{sin2phi}{alphay \cdot alphay}\right)\right)\right)\right) \]

      8. distribute-neg-in N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \left(\left(\mathsf{neg}\left(\frac{cos2phi}{alphax \cdot alphax}\right)\right) + \color{blue}{\left(\mathsf{neg}\left(\frac{sin2phi}{alphay \cdot alphay}\right)\right)}\right)\right) \]

      9. unsub-neg N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \left(\left(\mathsf{neg}\left(\frac{cos2phi}{alphax \cdot alphax}\right)\right) - \color{blue}{\frac{sin2phi}{alphay \cdot alphay}}\right)\right) \]

      10. --lowering--.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\left(\mathsf{neg}\left(\frac{cos2phi}{alphax \cdot alphax}\right)\right), \color{blue}{\left(\frac{sin2phi}{alphay \cdot alphay}\right)}\right)\right) \]

      11. neg-sub0 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\left(0 - \frac{cos2phi}{alphax \cdot alphax}\right), \left(\frac{\color{blue}{sin2phi}}{alphay \cdot alphay}\right)\right)\right) \]

      12. --lowering--.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \left(\frac{cos2phi}{alphax \cdot alphax}\right)\right), \left(\frac{\color{blue}{sin2phi}}{alphay \cdot alphay}\right)\right)\right) \]

      13. /-lowering-/.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(cos2phi, \left(alphax \cdot alphax\right)\right)\right), \left(\frac{sin2phi}{alphay \cdot alphay}\right)\right)\right) \]

      14. *-lowering-*.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right)\right), \left(\frac{sin2phi}{alphay \cdot alphay}\right)\right)\right) \]

      15. /-lowering-/.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right)\right), \mathsf{/.f32}\left(sin2phi, \color{blue}{\left(alphay \cdot alphay\right)}\right)\right)\right) \]

      16. *-lowering-*.f32 99.0%

          \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right)\right), \mathsf{/.f32}\left(sin2phi, \mathsf{*.f32}\left(alphay, \color{blue}{alphay}\right)\right)\right)\right) \]

   3. Simplified 99.0%

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

   5. Taylor expanded in cos2phi around 0

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

      1. mul-1-neg N/A

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

      2. neg-lowering-neg.f32 N/A

         \[\leadsto \mathsf{neg.f32}\left(\left(\frac{{alphay}^{2} \cdot \log \left(1 - u0\right)}{sin2phi}\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{neg.f32}\left(\left(\frac{\log \left(1 - u0\right) \cdot {alphay}^{2}}{sin2phi}\right)\right) \]

      4. associate-/l* N/A

         \[\leadsto \mathsf{neg.f32}\left(\left(\log \left(1 - u0\right) \cdot \frac{{alphay}^{2}}{sin2phi}\right)\right) \]

      5. *-lowering-*.f32 N/A

         \[\leadsto \mathsf{neg.f32}\left(\mathsf{*.f32}\left(\log \left(1 - u0\right), \left(\frac{{alphay}^{2}}{sin2phi}\right)\right)\right) \]

      6. sub-neg N/A

         \[\leadsto \mathsf{neg.f32}\left(\mathsf{*.f32}\left(\log \left(1 + \left(\mathsf{neg}\left(u0\right)\right)\right), \left(\frac{{alphay}^{2}}{sin2phi}\right)\right)\right) \]

      7. mul-1-neg N/A

         \[\leadsto \mathsf{neg.f32}\left(\mathsf{*.f32}\left(\log \left(1 + -1 \cdot u0\right), \left(\frac{{alphay}^{2}}{sin2phi}\right)\right)\right) \]

      8. log1p-define N/A

         \[\leadsto \mathsf{neg.f32}\left(\mathsf{*.f32}\left(\left(\mathsf{log1p}\left(-1 \cdot u0\right)\right), \left(\frac{{alphay}^{2}}{sin2phi}\right)\right)\right) \]

      9. log1p-lowering-log1p.f32 N/A

         \[\leadsto \mathsf{neg.f32}\left(\mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\left(-1 \cdot u0\right)\right), \left(\frac{{alphay}^{2}}{sin2phi}\right)\right)\right) \]

      10. mul-1-neg N/A

          \[\leadsto \mathsf{neg.f32}\left(\mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\left(\mathsf{neg}\left(u0\right)\right)\right), \left(\frac{{alphay}^{2}}{sin2phi}\right)\right)\right) \]

      11. neg-lowering-neg.f32 N/A

          \[\leadsto \mathsf{neg.f32}\left(\mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \left(\frac{{alphay}^{2}}{sin2phi}\right)\right)\right) \]

      12. /-lowering-/.f32 N/A

          \[\leadsto \mathsf{neg.f32}\left(\mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{/.f32}\left(\left({alphay}^{2}\right), sin2phi\right)\right)\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{neg.f32}\left(\mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{/.f32}\left(\left(alphay \cdot alphay\right), sin2phi\right)\right)\right) \]

      14. *-lowering-*.f32 79.9%

          \[\leadsto \mathsf{neg.f32}\left(\mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{/.f32}\left(\mathsf{*.f32}\left(alphay, alphay\right), sin2phi\right)\right)\right) \]

   7. Simplified 79.9%

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

      1. *-commutative N/A

         \[\leadsto \mathsf{neg.f32}\left(\left(\frac{alphay \cdot alphay}{sin2phi} \cdot \log \left(1 + \left(\mathsf{neg}\left(u0\right)\right)\right)\right)\right) \]

      2. associate-/l* N/A

         \[\leadsto \mathsf{neg.f32}\left(\left(\left(alphay \cdot \frac{alphay}{sin2phi}\right) \cdot \log \left(1 + \left(\mathsf{neg}\left(u0\right)\right)\right)\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{neg.f32}\left(\left(alphay \cdot \left(\frac{alphay}{sin2phi} \cdot \log \left(1 + \left(\mathsf{neg}\left(u0\right)\right)\right)\right)\right)\right) \]

      4. *-lowering-*.f32 N/A

         \[\leadsto \mathsf{neg.f32}\left(\mathsf{*.f32}\left(alphay, \left(\frac{alphay}{sin2phi} \cdot \log \left(1 + \left(\mathsf{neg}\left(u0\right)\right)\right)\right)\right)\right) \]

      5. *-lowering-*.f32 N/A

         \[\leadsto \mathsf{neg.f32}\left(\mathsf{*.f32}\left(alphay, \mathsf{*.f32}\left(\left(\frac{alphay}{sin2phi}\right), \log \left(1 + \left(\mathsf{neg}\left(u0\right)\right)\right)\right)\right)\right) \]

      6. /-lowering-/.f32 N/A

         \[\leadsto \mathsf{neg.f32}\left(\mathsf{*.f32}\left(alphay, \mathsf{*.f32}\left(\mathsf{/.f32}\left(alphay, sin2phi\right), \log \left(1 + \left(\mathsf{neg}\left(u0\right)\right)\right)\right)\right)\right) \]

      7. log1p-define N/A

         \[\leadsto \mathsf{neg.f32}\left(\mathsf{*.f32}\left(alphay, \mathsf{*.f32}\left(\mathsf{/.f32}\left(alphay, sin2phi\right), \left(\mathsf{log1p}\left(\mathsf{neg}\left(u0\right)\right)\right)\right)\right)\right) \]

      8. log1p-lowering-log1p.f32 N/A

         \[\leadsto \mathsf{neg.f32}\left(\mathsf{*.f32}\left(alphay, \mathsf{*.f32}\left(\mathsf{/.f32}\left(alphay, sin2phi\right), \mathsf{log1p.f32}\left(\left(\mathsf{neg}\left(u0\right)\right)\right)\right)\right)\right) \]

      9. neg-lowering-neg.f32 80.0%

         \[\leadsto \mathsf{neg.f32}\left(\mathsf{*.f32}\left(alphay, \mathsf{*.f32}\left(\mathsf{/.f32}\left(alphay, sin2phi\right), \mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right)\right)\right)\right) \]

   9. Applied egg-rr 80.0%

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

4. Final simplification 95.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u0 \leq 0.0949999988079071:\\ \;\;\;\;\frac{u0 \cdot \left(u0 \cdot \left(u0 \cdot \left(u0 \cdot -0.25 + -0.3333333333333333\right) + -0.5\right) + -1\right)}{\frac{-1}{\frac{alphax \cdot alphax}{cos2phi}} - \frac{\frac{sin2phi}{alphay}}{alphay}}\\ \mathbf{else}:\\ \;\;\;\;alphay \cdot \left(\mathsf{log1p}\left(-u0\right) \cdot \left(0 - \frac{alphay}{sin2phi}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 95.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;sin2phi \leq 4:\\ \;\;\;\;\frac{u0 \cdot \left(u0 \cdot \left(u0 \cdot \left(u0 \cdot -0.25 + -0.3333333333333333\right) + -0.5\right) + -1\right)}{\frac{-1}{\frac{alphax \cdot alphax}{cos2phi}} - \frac{\frac{sin2phi}{alphay}}{alphay}}\\ \mathbf{else}:\\ \;\;\;\;\frac{alphay \cdot \left(\mathsf{log1p}\left(-u0\right) \cdot \left(-alphay\right)\right)}{sin2phi}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (if (<= sin2phi 4.0)
   (/
    (* u0 (+ (* u0 (+ (* u0 (+ (* u0 -0.25) -0.3333333333333333)) -0.5)) -1.0))
    (- (/ -1.0 (/ (* alphax alphax) cos2phi)) (/ (/ sin2phi alphay) alphay)))
   (/ (* alphay (* (log1p (- u0)) (- alphay))) sin2phi)))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float tmp;
	if (sin2phi <= 4.0f) {
		tmp = (u0 * ((u0 * ((u0 * ((u0 * -0.25f) + -0.3333333333333333f)) + -0.5f)) + -1.0f)) / ((-1.0f / ((alphax * alphax) / cos2phi)) - ((sin2phi / alphay) / alphay));
	} else {
		tmp = (alphay * (log1pf(-u0) * -alphay)) / sin2phi;
	}
	return tmp;
}

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = Float32(0.0)
	if (sin2phi <= Float32(4.0))
		tmp = Float32(Float32(u0 * Float32(Float32(u0 * Float32(Float32(u0 * Float32(Float32(u0 * Float32(-0.25)) + Float32(-0.3333333333333333))) + Float32(-0.5))) + Float32(-1.0))) / Float32(Float32(Float32(-1.0) / Float32(Float32(alphax * alphax) / cos2phi)) - Float32(Float32(sin2phi / alphay) / alphay)));
	else
		tmp = Float32(Float32(alphay * Float32(log1p(Float32(-u0)) * Float32(-alphay))) / sin2phi);
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if sin2phi < 4

   1. Initial program 62.4%

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

      1. distribute-frac-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. /-lowering-/.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(\log \left(1 - u0\right), \color{blue}{\left(\mathsf{neg}\left(\left(\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\right)\right)\right)}\right) \]

      4. sub-neg N/A

         \[\leadsto \mathsf{/.f32}\left(\log \left(1 + \left(\mathsf{neg}\left(u0\right)\right)\right), \left(\mathsf{neg}\left(\left(\color{blue}{\frac{cos2phi}{alphax \cdot alphax}} + \frac{sin2phi}{alphay \cdot alphay}\right)\right)\right)\right) \]

      5. log1p-define N/A

         \[\leadsto \mathsf{/.f32}\left(\left(\mathsf{log1p}\left(\mathsf{neg}\left(u0\right)\right)\right), \left(\mathsf{neg}\left(\color{blue}{\left(\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\right)}\right)\right)\right) \]

      6. log1p-lowering-log1p.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\left(\mathsf{neg}\left(u0\right)\right)\right), \left(\mathsf{neg}\left(\color{blue}{\left(\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\right)}\right)\right)\right) \]

      7. neg-lowering-neg.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \left(\mathsf{neg}\left(\left(\color{blue}{\frac{cos2phi}{alphax \cdot alphax}} + \frac{sin2phi}{alphay \cdot alphay}\right)\right)\right)\right) \]

      8. distribute-neg-in N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \left(\left(\mathsf{neg}\left(\frac{cos2phi}{alphax \cdot alphax}\right)\right) + \color{blue}{\left(\mathsf{neg}\left(\frac{sin2phi}{alphay \cdot alphay}\right)\right)}\right)\right) \]

      9. unsub-neg N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \left(\left(\mathsf{neg}\left(\frac{cos2phi}{alphax \cdot alphax}\right)\right) - \color{blue}{\frac{sin2phi}{alphay \cdot alphay}}\right)\right) \]

      10. --lowering--.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\left(\mathsf{neg}\left(\frac{cos2phi}{alphax \cdot alphax}\right)\right), \color{blue}{\left(\frac{sin2phi}{alphay \cdot alphay}\right)}\right)\right) \]

      11. neg-sub0 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\left(0 - \frac{cos2phi}{alphax \cdot alphax}\right), \left(\frac{\color{blue}{sin2phi}}{alphay \cdot alphay}\right)\right)\right) \]

      12. --lowering--.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \left(\frac{cos2phi}{alphax \cdot alphax}\right)\right), \left(\frac{\color{blue}{sin2phi}}{alphay \cdot alphay}\right)\right)\right) \]

      13. /-lowering-/.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(cos2phi, \left(alphax \cdot alphax\right)\right)\right), \left(\frac{sin2phi}{alphay \cdot alphay}\right)\right)\right) \]

      14. *-lowering-*.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right)\right), \left(\frac{sin2phi}{alphay \cdot alphay}\right)\right)\right) \]

      15. /-lowering-/.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right)\right), \mathsf{/.f32}\left(sin2phi, \color{blue}{\left(alphay \cdot alphay\right)}\right)\right)\right) \]

      16. *-lowering-*.f32 98.7%

          \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right)\right), \mathsf{/.f32}\left(sin2phi, \mathsf{*.f32}\left(alphay, \color{blue}{alphay}\right)\right)\right)\right) \]

   3. Simplified 98.7%

      \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(-u0\right)}{\left(0 - \frac{cos2phi}{alphax \cdot alphax}\right) - \frac{sin2phi}{alphay \cdot alphay}}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-/r* N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right)\right), \left(\frac{\frac{sin2phi}{alphay}}{\color{blue}{alphay}}\right)\right)\right) \]

      2. /-lowering-/.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right)\right), \mathsf{/.f32}\left(\left(\frac{sin2phi}{alphay}\right), \color{blue}{alphay}\right)\right)\right) \]

      3. /-lowering-/.f32 98.7%

         \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right)\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(sin2phi, alphay\right), alphay\right)\right)\right) \]

   6. Applied egg-rr 98.7%

      \[\leadsto \frac{\mathsf{log1p}\left(-u0\right)}{\left(0 - \frac{cos2phi}{alphax \cdot alphax}\right) - \color{blue}{\frac{\frac{sin2phi}{alphay}}{alphay}}} \]
   7. Step-by-step derivation

      1. clear-num N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \left(\frac{1}{\frac{alphax \cdot alphax}{cos2phi}}\right)\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(sin2phi, \color{blue}{alphay}\right), alphay\right)\right)\right) \]

      2. /-lowering-/.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(1, \left(\frac{alphax \cdot alphax}{cos2phi}\right)\right)\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(sin2phi, \color{blue}{alphay}\right), alphay\right)\right)\right) \]

      3. /-lowering-/.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(1, \mathsf{/.f32}\left(\left(alphax \cdot alphax\right), cos2phi\right)\right)\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(sin2phi, alphay\right), alphay\right)\right)\right) \]

      4. *-lowering-*.f32 98.8%

         \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(1, \mathsf{/.f32}\left(\mathsf{*.f32}\left(alphax, alphax\right), cos2phi\right)\right)\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(sin2phi, alphay\right), alphay\right)\right)\right) \]

   8. Applied egg-rr 98.8%

      \[\leadsto \frac{\mathsf{log1p}\left(-u0\right)}{\left(0 - \color{blue}{\frac{1}{\frac{alphax \cdot alphax}{cos2phi}}}\right) - \frac{\frac{sin2phi}{alphay}}{alphay}} \]

   9. Taylor expanded in u0 around 0

      \[\leadsto \mathsf{/.f32}\left(\color{blue}{\left(u0 \cdot \left(u0 \cdot \left(u0 \cdot \left(\frac{-1}{4} \cdot u0 - \frac{1}{3}\right) - \frac{1}{2}\right) - 1\right)\right)}, \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(1, \mathsf{/.f32}\left(\mathsf{*.f32}\left(alphax, alphax\right), cos2phi\right)\right)\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(sin2phi, alphay\right), alphay\right)\right)\right) \]
   10. Step-by-step derivation

       1. *-lowering-*.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(u0, \left(u0 \cdot \left(u0 \cdot \left(\frac{-1}{4} \cdot u0 - \frac{1}{3}\right) - \frac{1}{2}\right) - 1\right)\right), \mathsf{\_.f32}\left(\color{blue}{\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(1, \mathsf{/.f32}\left(\mathsf{*.f32}\left(alphax, alphax\right), cos2phi\right)\right)\right)}, \mathsf{/.f32}\left(\mathsf{/.f32}\left(sin2phi, alphay\right), alphay\right)\right)\right) \]

       2. sub-neg N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(u0, \left(u0 \cdot \left(u0 \cdot \left(\frac{-1}{4} \cdot u0 - \frac{1}{3}\right) - \frac{1}{2}\right) + \left(\mathsf{neg}\left(1\right)\right)\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \color{blue}{\mathsf{/.f32}\left(1, \mathsf{/.f32}\left(\mathsf{*.f32}\left(alphax, alphax\right), cos2phi\right)\right)}\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(sin2phi, alphay\right), alphay\right)\right)\right) \]

       3. metadata-eval N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(u0, \left(u0 \cdot \left(u0 \cdot \left(\frac{-1}{4} \cdot u0 - \frac{1}{3}\right) - \frac{1}{2}\right) + -1\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(1, \color{blue}{\mathsf{/.f32}\left(\mathsf{*.f32}\left(alphax, alphax\right), cos2phi\right)}\right)\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(sin2phi, alphay\right), alphay\right)\right)\right) \]

       4. +-lowering-+.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\left(u0 \cdot \left(u0 \cdot \left(\frac{-1}{4} \cdot u0 - \frac{1}{3}\right) - \frac{1}{2}\right)\right), -1\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \color{blue}{\mathsf{/.f32}\left(1, \mathsf{/.f32}\left(\mathsf{*.f32}\left(alphax, alphax\right), cos2phi\right)\right)}\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(sin2phi, alphay\right), alphay\right)\right)\right) \]

       5. *-lowering-*.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{*.f32}\left(u0, \left(u0 \cdot \left(\frac{-1}{4} \cdot u0 - \frac{1}{3}\right) - \frac{1}{2}\right)\right), -1\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(\color{blue}{1}, \mathsf{/.f32}\left(\mathsf{*.f32}\left(alphax, alphax\right), cos2phi\right)\right)\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(sin2phi, alphay\right), alphay\right)\right)\right) \]

       6. sub-neg N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{*.f32}\left(u0, \left(u0 \cdot \left(\frac{-1}{4} \cdot u0 - \frac{1}{3}\right) + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right)\right), -1\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(1, \mathsf{/.f32}\left(\mathsf{*.f32}\left(alphax, alphax\right), cos2phi\right)\right)\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(sin2phi, alphay\right), alphay\right)\right)\right) \]

       7. metadata-eval N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{*.f32}\left(u0, \left(u0 \cdot \left(\frac{-1}{4} \cdot u0 - \frac{1}{3}\right) + \frac{-1}{2}\right)\right), -1\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(1, \mathsf{/.f32}\left(\mathsf{*.f32}\left(alphax, alphax\right), cos2phi\right)\right)\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(sin2phi, alphay\right), alphay\right)\right)\right) \]

       8. +-lowering-+.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\left(u0 \cdot \left(\frac{-1}{4} \cdot u0 - \frac{1}{3}\right)\right), \frac{-1}{2}\right)\right), -1\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(1, \mathsf{/.f32}\left(\mathsf{*.f32}\left(alphax, alphax\right), cos2phi\right)\right)\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(sin2phi, alphay\right), alphay\right)\right)\right) \]

       9. *-lowering-*.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{*.f32}\left(u0, \left(\frac{-1}{4} \cdot u0 - \frac{1}{3}\right)\right), \frac{-1}{2}\right)\right), -1\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(1, \mathsf{/.f32}\left(\mathsf{*.f32}\left(alphax, alphax\right), cos2phi\right)\right)\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(sin2phi, alphay\right), alphay\right)\right)\right) \]

       10. sub-neg N/A

           \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{*.f32}\left(u0, \left(\frac{-1}{4} \cdot u0 + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right)\right), \frac{-1}{2}\right)\right), -1\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(1, \mathsf{/.f32}\left(\mathsf{*.f32}\left(alphax, alphax\right), cos2phi\right)\right)\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(sin2phi, alphay\right), alphay\right)\right)\right) \]

       11. metadata-eval N/A

           \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{*.f32}\left(u0, \left(\frac{-1}{4} \cdot u0 + \frac{-1}{3}\right)\right), \frac{-1}{2}\right)\right), -1\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(1, \mathsf{/.f32}\left(\mathsf{*.f32}\left(alphax, alphax\right), cos2phi\right)\right)\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(sin2phi, alphay\right), alphay\right)\right)\right) \]

       12. +-lowering-+.f32 N/A

           \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\left(\frac{-1}{4} \cdot u0\right), \frac{-1}{3}\right)\right), \frac{-1}{2}\right)\right), -1\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(1, \mathsf{/.f32}\left(\mathsf{*.f32}\left(alphax, alphax\right), cos2phi\right)\right)\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(sin2phi, alphay\right), alphay\right)\right)\right) \]

       13. *-commutative N/A

           \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\left(u0 \cdot \frac{-1}{4}\right), \frac{-1}{3}\right)\right), \frac{-1}{2}\right)\right), -1\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(1, \mathsf{/.f32}\left(\mathsf{*.f32}\left(alphax, alphax\right), cos2phi\right)\right)\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(sin2phi, alphay\right), alphay\right)\right)\right) \]

       14. *-lowering-*.f32 91.2%

           \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{*.f32}\left(u0, \frac{-1}{4}\right), \frac{-1}{3}\right)\right), \frac{-1}{2}\right)\right), -1\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(1, \mathsf{/.f32}\left(\mathsf{*.f32}\left(alphax, alphax\right), cos2phi\right)\right)\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(sin2phi, alphay\right), alphay\right)\right)\right) \]

   11. Simplified 91.2%

       \[\leadsto \frac{\color{blue}{u0 \cdot \left(u0 \cdot \left(u0 \cdot \left(u0 \cdot -0.25 + -0.3333333333333333\right) + -0.5\right) + -1\right)}}{\left(0 - \frac{1}{\frac{alphax \cdot alphax}{cos2phi}}\right) - \frac{\frac{sin2phi}{alphay}}{alphay}} \]

   if 4 < sin2phi

   1. Initial program 67.1%

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

      1. distribute-frac-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. /-lowering-/.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(\log \left(1 - u0\right), \color{blue}{\left(\mathsf{neg}\left(\left(\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\right)\right)\right)}\right) \]

      4. sub-neg N/A

         \[\leadsto \mathsf{/.f32}\left(\log \left(1 + \left(\mathsf{neg}\left(u0\right)\right)\right), \left(\mathsf{neg}\left(\left(\color{blue}{\frac{cos2phi}{alphax \cdot alphax}} + \frac{sin2phi}{alphay \cdot alphay}\right)\right)\right)\right) \]

      5. log1p-define N/A

         \[\leadsto \mathsf{/.f32}\left(\left(\mathsf{log1p}\left(\mathsf{neg}\left(u0\right)\right)\right), \left(\mathsf{neg}\left(\color{blue}{\left(\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\right)}\right)\right)\right) \]

      6. log1p-lowering-log1p.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\left(\mathsf{neg}\left(u0\right)\right)\right), \left(\mathsf{neg}\left(\color{blue}{\left(\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\right)}\right)\right)\right) \]

      7. neg-lowering-neg.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \left(\mathsf{neg}\left(\left(\color{blue}{\frac{cos2phi}{alphax \cdot alphax}} + \frac{sin2phi}{alphay \cdot alphay}\right)\right)\right)\right) \]

      8. distribute-neg-in N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \left(\left(\mathsf{neg}\left(\frac{cos2phi}{alphax \cdot alphax}\right)\right) + \color{blue}{\left(\mathsf{neg}\left(\frac{sin2phi}{alphay \cdot alphay}\right)\right)}\right)\right) \]

      9. unsub-neg N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \left(\left(\mathsf{neg}\left(\frac{cos2phi}{alphax \cdot alphax}\right)\right) - \color{blue}{\frac{sin2phi}{alphay \cdot alphay}}\right)\right) \]

      10. --lowering--.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\left(\mathsf{neg}\left(\frac{cos2phi}{alphax \cdot alphax}\right)\right), \color{blue}{\left(\frac{sin2phi}{alphay \cdot alphay}\right)}\right)\right) \]

      11. neg-sub0 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\left(0 - \frac{cos2phi}{alphax \cdot alphax}\right), \left(\frac{\color{blue}{sin2phi}}{alphay \cdot alphay}\right)\right)\right) \]

      12. --lowering--.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \left(\frac{cos2phi}{alphax \cdot alphax}\right)\right), \left(\frac{\color{blue}{sin2phi}}{alphay \cdot alphay}\right)\right)\right) \]

      13. /-lowering-/.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(cos2phi, \left(alphax \cdot alphax\right)\right)\right), \left(\frac{sin2phi}{alphay \cdot alphay}\right)\right)\right) \]

      14. *-lowering-*.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right)\right), \left(\frac{sin2phi}{alphay \cdot alphay}\right)\right)\right) \]

      15. /-lowering-/.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right)\right), \mathsf{/.f32}\left(sin2phi, \color{blue}{\left(alphay \cdot alphay\right)}\right)\right)\right) \]

      16. *-lowering-*.f32 97.9%

          \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right)\right), \mathsf{/.f32}\left(sin2phi, \mathsf{*.f32}\left(alphay, \color{blue}{alphay}\right)\right)\right)\right) \]

   3. Simplified 97.9%

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

   5. Taylor expanded in cos2phi around 0

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

      1. mul-1-neg N/A

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

      2. neg-lowering-neg.f32 N/A

         \[\leadsto \mathsf{neg.f32}\left(\left(\frac{{alphay}^{2} \cdot \log \left(1 - u0\right)}{sin2phi}\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{neg.f32}\left(\left(\frac{\log \left(1 - u0\right) \cdot {alphay}^{2}}{sin2phi}\right)\right) \]

      4. associate-/l* N/A

         \[\leadsto \mathsf{neg.f32}\left(\left(\log \left(1 - u0\right) \cdot \frac{{alphay}^{2}}{sin2phi}\right)\right) \]

      5. *-lowering-*.f32 N/A

         \[\leadsto \mathsf{neg.f32}\left(\mathsf{*.f32}\left(\log \left(1 - u0\right), \left(\frac{{alphay}^{2}}{sin2phi}\right)\right)\right) \]

      6. sub-neg N/A

         \[\leadsto \mathsf{neg.f32}\left(\mathsf{*.f32}\left(\log \left(1 + \left(\mathsf{neg}\left(u0\right)\right)\right), \left(\frac{{alphay}^{2}}{sin2phi}\right)\right)\right) \]

      7. mul-1-neg N/A

         \[\leadsto \mathsf{neg.f32}\left(\mathsf{*.f32}\left(\log \left(1 + -1 \cdot u0\right), \left(\frac{{alphay}^{2}}{sin2phi}\right)\right)\right) \]

      8. log1p-define N/A

         \[\leadsto \mathsf{neg.f32}\left(\mathsf{*.f32}\left(\left(\mathsf{log1p}\left(-1 \cdot u0\right)\right), \left(\frac{{alphay}^{2}}{sin2phi}\right)\right)\right) \]

      9. log1p-lowering-log1p.f32 N/A

         \[\leadsto \mathsf{neg.f32}\left(\mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\left(-1 \cdot u0\right)\right), \left(\frac{{alphay}^{2}}{sin2phi}\right)\right)\right) \]

      10. mul-1-neg N/A

          \[\leadsto \mathsf{neg.f32}\left(\mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\left(\mathsf{neg}\left(u0\right)\right)\right), \left(\frac{{alphay}^{2}}{sin2phi}\right)\right)\right) \]

      11. neg-lowering-neg.f32 N/A

          \[\leadsto \mathsf{neg.f32}\left(\mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \left(\frac{{alphay}^{2}}{sin2phi}\right)\right)\right) \]

      12. /-lowering-/.f32 N/A

          \[\leadsto \mathsf{neg.f32}\left(\mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{/.f32}\left(\left({alphay}^{2}\right), sin2phi\right)\right)\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{neg.f32}\left(\mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{/.f32}\left(\left(alphay \cdot alphay\right), sin2phi\right)\right)\right) \]

      14. *-lowering-*.f32 98.8%

          \[\leadsto \mathsf{neg.f32}\left(\mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{/.f32}\left(\mathsf{*.f32}\left(alphay, alphay\right), sin2phi\right)\right)\right) \]

   7. Simplified 98.8%

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

      1. *-commutative N/A

         \[\leadsto \mathsf{neg.f32}\left(\left(\frac{alphay \cdot alphay}{sin2phi} \cdot \log \left(1 + \left(\mathsf{neg}\left(u0\right)\right)\right)\right)\right) \]

      2. associate-/l* N/A

         \[\leadsto \mathsf{neg.f32}\left(\left(\left(alphay \cdot \frac{alphay}{sin2phi}\right) \cdot \log \left(1 + \left(\mathsf{neg}\left(u0\right)\right)\right)\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{neg.f32}\left(\left(alphay \cdot \left(\frac{alphay}{sin2phi} \cdot \log \left(1 + \left(\mathsf{neg}\left(u0\right)\right)\right)\right)\right)\right) \]

      4. *-lowering-*.f32 N/A

         \[\leadsto \mathsf{neg.f32}\left(\mathsf{*.f32}\left(alphay, \left(\frac{alphay}{sin2phi} \cdot \log \left(1 + \left(\mathsf{neg}\left(u0\right)\right)\right)\right)\right)\right) \]

      5. *-lowering-*.f32 N/A

         \[\leadsto \mathsf{neg.f32}\left(\mathsf{*.f32}\left(alphay, \mathsf{*.f32}\left(\left(\frac{alphay}{sin2phi}\right), \log \left(1 + \left(\mathsf{neg}\left(u0\right)\right)\right)\right)\right)\right) \]

      6. /-lowering-/.f32 N/A

         \[\leadsto \mathsf{neg.f32}\left(\mathsf{*.f32}\left(alphay, \mathsf{*.f32}\left(\mathsf{/.f32}\left(alphay, sin2phi\right), \log \left(1 + \left(\mathsf{neg}\left(u0\right)\right)\right)\right)\right)\right) \]

      7. log1p-define N/A

         \[\leadsto \mathsf{neg.f32}\left(\mathsf{*.f32}\left(alphay, \mathsf{*.f32}\left(\mathsf{/.f32}\left(alphay, sin2phi\right), \left(\mathsf{log1p}\left(\mathsf{neg}\left(u0\right)\right)\right)\right)\right)\right) \]

      8. log1p-lowering-log1p.f32 N/A

         \[\leadsto \mathsf{neg.f32}\left(\mathsf{*.f32}\left(alphay, \mathsf{*.f32}\left(\mathsf{/.f32}\left(alphay, sin2phi\right), \mathsf{log1p.f32}\left(\left(\mathsf{neg}\left(u0\right)\right)\right)\right)\right)\right) \]

      9. neg-lowering-neg.f32 98.7%

         \[\leadsto \mathsf{neg.f32}\left(\mathsf{*.f32}\left(alphay, \mathsf{*.f32}\left(\mathsf{/.f32}\left(alphay, sin2phi\right), \mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right)\right)\right)\right) \]

   9. Applied egg-rr 98.7%

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

       1. distribute-lft-neg-in N/A

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

       2. associate-*l/ N/A

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

       3. associate-*r/ N/A

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

       4. /-lowering-/.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(\left(\left(\mathsf{neg}\left(alphay\right)\right) \cdot \left(alphay \cdot \log \left(1 + \left(\mathsf{neg}\left(u0\right)\right)\right)\right)\right), \color{blue}{sin2phi}\right) \]

       5. *-lowering-*.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(\left(\mathsf{neg}\left(alphay\right)\right), \left(alphay \cdot \log \left(1 + \left(\mathsf{neg}\left(u0\right)\right)\right)\right)\right), sin2phi\right) \]

       6. neg-lowering-neg.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(\mathsf{neg.f32}\left(alphay\right), \left(alphay \cdot \log \left(1 + \left(\mathsf{neg}\left(u0\right)\right)\right)\right)\right), sin2phi\right) \]

       7. *-commutative N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(\mathsf{neg.f32}\left(alphay\right), \left(\log \left(1 + \left(\mathsf{neg}\left(u0\right)\right)\right) \cdot alphay\right)\right), sin2phi\right) \]

       8. *-lowering-*.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(\mathsf{neg.f32}\left(alphay\right), \mathsf{*.f32}\left(\log \left(1 + \left(\mathsf{neg}\left(u0\right)\right)\right), alphay\right)\right), sin2phi\right) \]

       9. log1p-define N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(\mathsf{neg.f32}\left(alphay\right), \mathsf{*.f32}\left(\left(\mathsf{log1p}\left(\mathsf{neg}\left(u0\right)\right)\right), alphay\right)\right), sin2phi\right) \]

       10. log1p-lowering-log1p.f32 N/A

           \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(\mathsf{neg.f32}\left(alphay\right), \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\left(\mathsf{neg}\left(u0\right)\right)\right), alphay\right)\right), sin2phi\right) \]

       11. neg-lowering-neg.f32 99.1%

           \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(\mathsf{neg.f32}\left(alphay\right), \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), alphay\right)\right), sin2phi\right) \]

   11. Applied egg-rr 99.1%

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

4. Final simplification 95.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;sin2phi \leq 4:\\ \;\;\;\;\frac{u0 \cdot \left(u0 \cdot \left(u0 \cdot \left(u0 \cdot -0.25 + -0.3333333333333333\right) + -0.5\right) + -1\right)}{\frac{-1}{\frac{alphax \cdot alphax}{cos2phi}} - \frac{\frac{sin2phi}{alphay}}{alphay}}\\ \mathbf{else}:\\ \;\;\;\;\frac{alphay \cdot \left(\mathsf{log1p}\left(-u0\right) \cdot \left(-alphay\right)\right)}{sin2phi}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 95.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;sin2phi \leq 4:\\ \;\;\;\;\frac{u0 \cdot \left(u0 \cdot \left(u0 \cdot \left(u0 \cdot -0.25 + -0.3333333333333333\right) + -0.5\right) + -1\right)}{\frac{-1}{\frac{alphax \cdot alphax}{cos2phi}} - \frac{\frac{sin2phi}{alphay}}{alphay}}\\ \mathbf{else}:\\ \;\;\;\;\frac{alphay}{sin2phi} \cdot \left(\mathsf{log1p}\left(-u0\right) \cdot \left(-alphay\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (if (<= sin2phi 4.0)
   (/
    (* u0 (+ (* u0 (+ (* u0 (+ (* u0 -0.25) -0.3333333333333333)) -0.5)) -1.0))
    (- (/ -1.0 (/ (* alphax alphax) cos2phi)) (/ (/ sin2phi alphay) alphay)))
   (* (/ alphay sin2phi) (* (log1p (- u0)) (- alphay)))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float tmp;
	if (sin2phi <= 4.0f) {
		tmp = (u0 * ((u0 * ((u0 * ((u0 * -0.25f) + -0.3333333333333333f)) + -0.5f)) + -1.0f)) / ((-1.0f / ((alphax * alphax) / cos2phi)) - ((sin2phi / alphay) / alphay));
	} else {
		tmp = (alphay / sin2phi) * (log1pf(-u0) * -alphay);
	}
	return tmp;
}

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = Float32(0.0)
	if (sin2phi <= Float32(4.0))
		tmp = Float32(Float32(u0 * Float32(Float32(u0 * Float32(Float32(u0 * Float32(Float32(u0 * Float32(-0.25)) + Float32(-0.3333333333333333))) + Float32(-0.5))) + Float32(-1.0))) / Float32(Float32(Float32(-1.0) / Float32(Float32(alphax * alphax) / cos2phi)) - Float32(Float32(sin2phi / alphay) / alphay)));
	else
		tmp = Float32(Float32(alphay / sin2phi) * Float32(log1p(Float32(-u0)) * Float32(-alphay)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if sin2phi < 4

   1. Initial program 62.4%

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

      1. distribute-frac-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. /-lowering-/.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(\log \left(1 - u0\right), \color{blue}{\left(\mathsf{neg}\left(\left(\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\right)\right)\right)}\right) \]

      4. sub-neg N/A

         \[\leadsto \mathsf{/.f32}\left(\log \left(1 + \left(\mathsf{neg}\left(u0\right)\right)\right), \left(\mathsf{neg}\left(\left(\color{blue}{\frac{cos2phi}{alphax \cdot alphax}} + \frac{sin2phi}{alphay \cdot alphay}\right)\right)\right)\right) \]

      5. log1p-define N/A

         \[\leadsto \mathsf{/.f32}\left(\left(\mathsf{log1p}\left(\mathsf{neg}\left(u0\right)\right)\right), \left(\mathsf{neg}\left(\color{blue}{\left(\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\right)}\right)\right)\right) \]

      6. log1p-lowering-log1p.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\left(\mathsf{neg}\left(u0\right)\right)\right), \left(\mathsf{neg}\left(\color{blue}{\left(\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\right)}\right)\right)\right) \]

      7. neg-lowering-neg.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \left(\mathsf{neg}\left(\left(\color{blue}{\frac{cos2phi}{alphax \cdot alphax}} + \frac{sin2phi}{alphay \cdot alphay}\right)\right)\right)\right) \]

      8. distribute-neg-in N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \left(\left(\mathsf{neg}\left(\frac{cos2phi}{alphax \cdot alphax}\right)\right) + \color{blue}{\left(\mathsf{neg}\left(\frac{sin2phi}{alphay \cdot alphay}\right)\right)}\right)\right) \]

      9. unsub-neg N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \left(\left(\mathsf{neg}\left(\frac{cos2phi}{alphax \cdot alphax}\right)\right) - \color{blue}{\frac{sin2phi}{alphay \cdot alphay}}\right)\right) \]

      10. --lowering--.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\left(\mathsf{neg}\left(\frac{cos2phi}{alphax \cdot alphax}\right)\right), \color{blue}{\left(\frac{sin2phi}{alphay \cdot alphay}\right)}\right)\right) \]

      11. neg-sub0 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\left(0 - \frac{cos2phi}{alphax \cdot alphax}\right), \left(\frac{\color{blue}{sin2phi}}{alphay \cdot alphay}\right)\right)\right) \]

      12. --lowering--.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \left(\frac{cos2phi}{alphax \cdot alphax}\right)\right), \left(\frac{\color{blue}{sin2phi}}{alphay \cdot alphay}\right)\right)\right) \]

      13. /-lowering-/.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(cos2phi, \left(alphax \cdot alphax\right)\right)\right), \left(\frac{sin2phi}{alphay \cdot alphay}\right)\right)\right) \]

      14. *-lowering-*.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right)\right), \left(\frac{sin2phi}{alphay \cdot alphay}\right)\right)\right) \]

      15. /-lowering-/.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right)\right), \mathsf{/.f32}\left(sin2phi, \color{blue}{\left(alphay \cdot alphay\right)}\right)\right)\right) \]

      16. *-lowering-*.f32 98.7%

          \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right)\right), \mathsf{/.f32}\left(sin2phi, \mathsf{*.f32}\left(alphay, \color{blue}{alphay}\right)\right)\right)\right) \]

   3. Simplified 98.7%

      \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(-u0\right)}{\left(0 - \frac{cos2phi}{alphax \cdot alphax}\right) - \frac{sin2phi}{alphay \cdot alphay}}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-/r* N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right)\right), \left(\frac{\frac{sin2phi}{alphay}}{\color{blue}{alphay}}\right)\right)\right) \]

      2. /-lowering-/.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right)\right), \mathsf{/.f32}\left(\left(\frac{sin2phi}{alphay}\right), \color{blue}{alphay}\right)\right)\right) \]

      3. /-lowering-/.f32 98.7%

         \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right)\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(sin2phi, alphay\right), alphay\right)\right)\right) \]

   6. Applied egg-rr 98.7%

      \[\leadsto \frac{\mathsf{log1p}\left(-u0\right)}{\left(0 - \frac{cos2phi}{alphax \cdot alphax}\right) - \color{blue}{\frac{\frac{sin2phi}{alphay}}{alphay}}} \]
   7. Step-by-step derivation

      1. clear-num N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \left(\frac{1}{\frac{alphax \cdot alphax}{cos2phi}}\right)\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(sin2phi, \color{blue}{alphay}\right), alphay\right)\right)\right) \]

      2. /-lowering-/.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(1, \left(\frac{alphax \cdot alphax}{cos2phi}\right)\right)\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(sin2phi, \color{blue}{alphay}\right), alphay\right)\right)\right) \]

      3. /-lowering-/.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(1, \mathsf{/.f32}\left(\left(alphax \cdot alphax\right), cos2phi\right)\right)\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(sin2phi, alphay\right), alphay\right)\right)\right) \]

      4. *-lowering-*.f32 98.8%

         \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(1, \mathsf{/.f32}\left(\mathsf{*.f32}\left(alphax, alphax\right), cos2phi\right)\right)\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(sin2phi, alphay\right), alphay\right)\right)\right) \]

   8. Applied egg-rr 98.8%

      \[\leadsto \frac{\mathsf{log1p}\left(-u0\right)}{\left(0 - \color{blue}{\frac{1}{\frac{alphax \cdot alphax}{cos2phi}}}\right) - \frac{\frac{sin2phi}{alphay}}{alphay}} \]

   9. Taylor expanded in u0 around 0

      \[\leadsto \mathsf{/.f32}\left(\color{blue}{\left(u0 \cdot \left(u0 \cdot \left(u0 \cdot \left(\frac{-1}{4} \cdot u0 - \frac{1}{3}\right) - \frac{1}{2}\right) - 1\right)\right)}, \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(1, \mathsf{/.f32}\left(\mathsf{*.f32}\left(alphax, alphax\right), cos2phi\right)\right)\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(sin2phi, alphay\right), alphay\right)\right)\right) \]
   10. Step-by-step derivation

       1. *-lowering-*.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(u0, \left(u0 \cdot \left(u0 \cdot \left(\frac{-1}{4} \cdot u0 - \frac{1}{3}\right) - \frac{1}{2}\right) - 1\right)\right), \mathsf{\_.f32}\left(\color{blue}{\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(1, \mathsf{/.f32}\left(\mathsf{*.f32}\left(alphax, alphax\right), cos2phi\right)\right)\right)}, \mathsf{/.f32}\left(\mathsf{/.f32}\left(sin2phi, alphay\right), alphay\right)\right)\right) \]

       2. sub-neg N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(u0, \left(u0 \cdot \left(u0 \cdot \left(\frac{-1}{4} \cdot u0 - \frac{1}{3}\right) - \frac{1}{2}\right) + \left(\mathsf{neg}\left(1\right)\right)\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \color{blue}{\mathsf{/.f32}\left(1, \mathsf{/.f32}\left(\mathsf{*.f32}\left(alphax, alphax\right), cos2phi\right)\right)}\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(sin2phi, alphay\right), alphay\right)\right)\right) \]

       3. metadata-eval N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(u0, \left(u0 \cdot \left(u0 \cdot \left(\frac{-1}{4} \cdot u0 - \frac{1}{3}\right) - \frac{1}{2}\right) + -1\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(1, \color{blue}{\mathsf{/.f32}\left(\mathsf{*.f32}\left(alphax, alphax\right), cos2phi\right)}\right)\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(sin2phi, alphay\right), alphay\right)\right)\right) \]

       4. +-lowering-+.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\left(u0 \cdot \left(u0 \cdot \left(\frac{-1}{4} \cdot u0 - \frac{1}{3}\right) - \frac{1}{2}\right)\right), -1\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \color{blue}{\mathsf{/.f32}\left(1, \mathsf{/.f32}\left(\mathsf{*.f32}\left(alphax, alphax\right), cos2phi\right)\right)}\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(sin2phi, alphay\right), alphay\right)\right)\right) \]

       5. *-lowering-*.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{*.f32}\left(u0, \left(u0 \cdot \left(\frac{-1}{4} \cdot u0 - \frac{1}{3}\right) - \frac{1}{2}\right)\right), -1\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(\color{blue}{1}, \mathsf{/.f32}\left(\mathsf{*.f32}\left(alphax, alphax\right), cos2phi\right)\right)\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(sin2phi, alphay\right), alphay\right)\right)\right) \]

       6. sub-neg N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{*.f32}\left(u0, \left(u0 \cdot \left(\frac{-1}{4} \cdot u0 - \frac{1}{3}\right) + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right)\right), -1\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(1, \mathsf{/.f32}\left(\mathsf{*.f32}\left(alphax, alphax\right), cos2phi\right)\right)\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(sin2phi, alphay\right), alphay\right)\right)\right) \]

       7. metadata-eval N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{*.f32}\left(u0, \left(u0 \cdot \left(\frac{-1}{4} \cdot u0 - \frac{1}{3}\right) + \frac{-1}{2}\right)\right), -1\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(1, \mathsf{/.f32}\left(\mathsf{*.f32}\left(alphax, alphax\right), cos2phi\right)\right)\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(sin2phi, alphay\right), alphay\right)\right)\right) \]

       8. +-lowering-+.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\left(u0 \cdot \left(\frac{-1}{4} \cdot u0 - \frac{1}{3}\right)\right), \frac{-1}{2}\right)\right), -1\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(1, \mathsf{/.f32}\left(\mathsf{*.f32}\left(alphax, alphax\right), cos2phi\right)\right)\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(sin2phi, alphay\right), alphay\right)\right)\right) \]

       9. *-lowering-*.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{*.f32}\left(u0, \left(\frac{-1}{4} \cdot u0 - \frac{1}{3}\right)\right), \frac{-1}{2}\right)\right), -1\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(1, \mathsf{/.f32}\left(\mathsf{*.f32}\left(alphax, alphax\right), cos2phi\right)\right)\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(sin2phi, alphay\right), alphay\right)\right)\right) \]

       10. sub-neg N/A

           \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{*.f32}\left(u0, \left(\frac{-1}{4} \cdot u0 + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right)\right), \frac{-1}{2}\right)\right), -1\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(1, \mathsf{/.f32}\left(\mathsf{*.f32}\left(alphax, alphax\right), cos2phi\right)\right)\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(sin2phi, alphay\right), alphay\right)\right)\right) \]

       11. metadata-eval N/A

           \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{*.f32}\left(u0, \left(\frac{-1}{4} \cdot u0 + \frac{-1}{3}\right)\right), \frac{-1}{2}\right)\right), -1\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(1, \mathsf{/.f32}\left(\mathsf{*.f32}\left(alphax, alphax\right), cos2phi\right)\right)\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(sin2phi, alphay\right), alphay\right)\right)\right) \]

       12. +-lowering-+.f32 N/A

           \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\left(\frac{-1}{4} \cdot u0\right), \frac{-1}{3}\right)\right), \frac{-1}{2}\right)\right), -1\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(1, \mathsf{/.f32}\left(\mathsf{*.f32}\left(alphax, alphax\right), cos2phi\right)\right)\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(sin2phi, alphay\right), alphay\right)\right)\right) \]

       13. *-commutative N/A

           \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\left(u0 \cdot \frac{-1}{4}\right), \frac{-1}{3}\right)\right), \frac{-1}{2}\right)\right), -1\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(1, \mathsf{/.f32}\left(\mathsf{*.f32}\left(alphax, alphax\right), cos2phi\right)\right)\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(sin2phi, alphay\right), alphay\right)\right)\right) \]

       14. *-lowering-*.f32 91.2%

           \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{*.f32}\left(u0, \frac{-1}{4}\right), \frac{-1}{3}\right)\right), \frac{-1}{2}\right)\right), -1\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(1, \mathsf{/.f32}\left(\mathsf{*.f32}\left(alphax, alphax\right), cos2phi\right)\right)\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(sin2phi, alphay\right), alphay\right)\right)\right) \]

   11. Simplified 91.2%

       \[\leadsto \frac{\color{blue}{u0 \cdot \left(u0 \cdot \left(u0 \cdot \left(u0 \cdot -0.25 + -0.3333333333333333\right) + -0.5\right) + -1\right)}}{\left(0 - \frac{1}{\frac{alphax \cdot alphax}{cos2phi}}\right) - \frac{\frac{sin2phi}{alphay}}{alphay}} \]

   if 4 < sin2phi

   1. Initial program 67.1%

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

      1. distribute-frac-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. /-lowering-/.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(\log \left(1 - u0\right), \color{blue}{\left(\mathsf{neg}\left(\left(\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\right)\right)\right)}\right) \]

      4. sub-neg N/A

         \[\leadsto \mathsf{/.f32}\left(\log \left(1 + \left(\mathsf{neg}\left(u0\right)\right)\right), \left(\mathsf{neg}\left(\left(\color{blue}{\frac{cos2phi}{alphax \cdot alphax}} + \frac{sin2phi}{alphay \cdot alphay}\right)\right)\right)\right) \]

      5. log1p-define N/A

         \[\leadsto \mathsf{/.f32}\left(\left(\mathsf{log1p}\left(\mathsf{neg}\left(u0\right)\right)\right), \left(\mathsf{neg}\left(\color{blue}{\left(\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\right)}\right)\right)\right) \]

      6. log1p-lowering-log1p.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\left(\mathsf{neg}\left(u0\right)\right)\right), \left(\mathsf{neg}\left(\color{blue}{\left(\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\right)}\right)\right)\right) \]

      7. neg-lowering-neg.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \left(\mathsf{neg}\left(\left(\color{blue}{\frac{cos2phi}{alphax \cdot alphax}} + \frac{sin2phi}{alphay \cdot alphay}\right)\right)\right)\right) \]

      8. distribute-neg-in N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \left(\left(\mathsf{neg}\left(\frac{cos2phi}{alphax \cdot alphax}\right)\right) + \color{blue}{\left(\mathsf{neg}\left(\frac{sin2phi}{alphay \cdot alphay}\right)\right)}\right)\right) \]

      9. unsub-neg N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \left(\left(\mathsf{neg}\left(\frac{cos2phi}{alphax \cdot alphax}\right)\right) - \color{blue}{\frac{sin2phi}{alphay \cdot alphay}}\right)\right) \]

      10. --lowering--.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\left(\mathsf{neg}\left(\frac{cos2phi}{alphax \cdot alphax}\right)\right), \color{blue}{\left(\frac{sin2phi}{alphay \cdot alphay}\right)}\right)\right) \]

      11. neg-sub0 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\left(0 - \frac{cos2phi}{alphax \cdot alphax}\right), \left(\frac{\color{blue}{sin2phi}}{alphay \cdot alphay}\right)\right)\right) \]

      12. --lowering--.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \left(\frac{cos2phi}{alphax \cdot alphax}\right)\right), \left(\frac{\color{blue}{sin2phi}}{alphay \cdot alphay}\right)\right)\right) \]

      13. /-lowering-/.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(cos2phi, \left(alphax \cdot alphax\right)\right)\right), \left(\frac{sin2phi}{alphay \cdot alphay}\right)\right)\right) \]

      14. *-lowering-*.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right)\right), \left(\frac{sin2phi}{alphay \cdot alphay}\right)\right)\right) \]

      15. /-lowering-/.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right)\right), \mathsf{/.f32}\left(sin2phi, \color{blue}{\left(alphay \cdot alphay\right)}\right)\right)\right) \]

      16. *-lowering-*.f32 97.9%

          \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right)\right), \mathsf{/.f32}\left(sin2phi, \mathsf{*.f32}\left(alphay, \color{blue}{alphay}\right)\right)\right)\right) \]

   3. Simplified 97.9%

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

   5. Taylor expanded in cos2phi around 0

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

      1. mul-1-neg N/A

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

      2. neg-lowering-neg.f32 N/A

         \[\leadsto \mathsf{neg.f32}\left(\left(\frac{{alphay}^{2} \cdot \log \left(1 - u0\right)}{sin2phi}\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{neg.f32}\left(\left(\frac{\log \left(1 - u0\right) \cdot {alphay}^{2}}{sin2phi}\right)\right) \]

      4. associate-/l* N/A

         \[\leadsto \mathsf{neg.f32}\left(\left(\log \left(1 - u0\right) \cdot \frac{{alphay}^{2}}{sin2phi}\right)\right) \]

      5. *-lowering-*.f32 N/A

         \[\leadsto \mathsf{neg.f32}\left(\mathsf{*.f32}\left(\log \left(1 - u0\right), \left(\frac{{alphay}^{2}}{sin2phi}\right)\right)\right) \]

      6. sub-neg N/A

         \[\leadsto \mathsf{neg.f32}\left(\mathsf{*.f32}\left(\log \left(1 + \left(\mathsf{neg}\left(u0\right)\right)\right), \left(\frac{{alphay}^{2}}{sin2phi}\right)\right)\right) \]

      7. mul-1-neg N/A

         \[\leadsto \mathsf{neg.f32}\left(\mathsf{*.f32}\left(\log \left(1 + -1 \cdot u0\right), \left(\frac{{alphay}^{2}}{sin2phi}\right)\right)\right) \]

      8. log1p-define N/A

         \[\leadsto \mathsf{neg.f32}\left(\mathsf{*.f32}\left(\left(\mathsf{log1p}\left(-1 \cdot u0\right)\right), \left(\frac{{alphay}^{2}}{sin2phi}\right)\right)\right) \]

      9. log1p-lowering-log1p.f32 N/A

         \[\leadsto \mathsf{neg.f32}\left(\mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\left(-1 \cdot u0\right)\right), \left(\frac{{alphay}^{2}}{sin2phi}\right)\right)\right) \]

      10. mul-1-neg N/A

          \[\leadsto \mathsf{neg.f32}\left(\mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\left(\mathsf{neg}\left(u0\right)\right)\right), \left(\frac{{alphay}^{2}}{sin2phi}\right)\right)\right) \]

      11. neg-lowering-neg.f32 N/A

          \[\leadsto \mathsf{neg.f32}\left(\mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \left(\frac{{alphay}^{2}}{sin2phi}\right)\right)\right) \]

      12. /-lowering-/.f32 N/A

          \[\leadsto \mathsf{neg.f32}\left(\mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{/.f32}\left(\left({alphay}^{2}\right), sin2phi\right)\right)\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{neg.f32}\left(\mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{/.f32}\left(\left(alphay \cdot alphay\right), sin2phi\right)\right)\right) \]

      14. *-lowering-*.f32 98.8%

          \[\leadsto \mathsf{neg.f32}\left(\mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{/.f32}\left(\mathsf{*.f32}\left(alphay, alphay\right), sin2phi\right)\right)\right) \]

   7. Simplified 98.8%

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

      1. associate-/l* N/A

         \[\leadsto \mathsf{neg.f32}\left(\left(\log \left(1 + \left(\mathsf{neg}\left(u0\right)\right)\right) \cdot \left(alphay \cdot \frac{alphay}{sin2phi}\right)\right)\right) \]

      2. associate-*r* N/A

         \[\leadsto \mathsf{neg.f32}\left(\left(\left(\log \left(1 + \left(\mathsf{neg}\left(u0\right)\right)\right) \cdot alphay\right) \cdot \frac{alphay}{sin2phi}\right)\right) \]

      3. *-lowering-*.f32 N/A

         \[\leadsto \mathsf{neg.f32}\left(\mathsf{*.f32}\left(\left(\log \left(1 + \left(\mathsf{neg}\left(u0\right)\right)\right) \cdot alphay\right), \left(\frac{alphay}{sin2phi}\right)\right)\right) \]

      4. *-lowering-*.f32 N/A

         \[\leadsto \mathsf{neg.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(\log \left(1 + \left(\mathsf{neg}\left(u0\right)\right)\right), alphay\right), \left(\frac{alphay}{sin2phi}\right)\right)\right) \]

      5. log1p-define N/A

         \[\leadsto \mathsf{neg.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(\left(\mathsf{log1p}\left(\mathsf{neg}\left(u0\right)\right)\right), alphay\right), \left(\frac{alphay}{sin2phi}\right)\right)\right) \]

      6. log1p-lowering-log1p.f32 N/A

         \[\leadsto \mathsf{neg.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\left(\mathsf{neg}\left(u0\right)\right)\right), alphay\right), \left(\frac{alphay}{sin2phi}\right)\right)\right) \]

      7. neg-lowering-neg.f32 N/A

         \[\leadsto \mathsf{neg.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), alphay\right), \left(\frac{alphay}{sin2phi}\right)\right)\right) \]

      8. /-lowering-/.f32 99.0%

         \[\leadsto \mathsf{neg.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), alphay\right), \mathsf{/.f32}\left(alphay, sin2phi\right)\right)\right) \]

   9. Applied egg-rr 99.0%

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

4. Final simplification 95.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;sin2phi \leq 4:\\ \;\;\;\;\frac{u0 \cdot \left(u0 \cdot \left(u0 \cdot \left(u0 \cdot -0.25 + -0.3333333333333333\right) + -0.5\right) + -1\right)}{\frac{-1}{\frac{alphax \cdot alphax}{cos2phi}} - \frac{\frac{sin2phi}{alphay}}{alphay}}\\ \mathbf{else}:\\ \;\;\;\;\frac{alphay}{sin2phi} \cdot \left(\mathsf{log1p}\left(-u0\right) \cdot \left(-alphay\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 92.6% accurate, 4.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 64.8%

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

   1. distribute-frac-neg N/A

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

   2. distribute-neg-frac2 N/A

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

   3. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\log \left(1 - u0\right), \color{blue}{\left(\mathsf{neg}\left(\left(\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\right)\right)\right)}\right) \]

   4. sub-neg N/A

      \[\leadsto \mathsf{/.f32}\left(\log \left(1 + \left(\mathsf{neg}\left(u0\right)\right)\right), \left(\mathsf{neg}\left(\left(\color{blue}{\frac{cos2phi}{alphax \cdot alphax}} + \frac{sin2phi}{alphay \cdot alphay}\right)\right)\right)\right) \]

   5. log1p-define N/A

      \[\leadsto \mathsf{/.f32}\left(\left(\mathsf{log1p}\left(\mathsf{neg}\left(u0\right)\right)\right), \left(\mathsf{neg}\left(\color{blue}{\left(\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\right)}\right)\right)\right) \]

   6. log1p-lowering-log1p.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\left(\mathsf{neg}\left(u0\right)\right)\right), \left(\mathsf{neg}\left(\color{blue}{\left(\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\right)}\right)\right)\right) \]

   7. neg-lowering-neg.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \left(\mathsf{neg}\left(\left(\color{blue}{\frac{cos2phi}{alphax \cdot alphax}} + \frac{sin2phi}{alphay \cdot alphay}\right)\right)\right)\right) \]

   8. distribute-neg-in N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \left(\left(\mathsf{neg}\left(\frac{cos2phi}{alphax \cdot alphax}\right)\right) + \color{blue}{\left(\mathsf{neg}\left(\frac{sin2phi}{alphay \cdot alphay}\right)\right)}\right)\right) \]

   9. unsub-neg N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \left(\left(\mathsf{neg}\left(\frac{cos2phi}{alphax \cdot alphax}\right)\right) - \color{blue}{\frac{sin2phi}{alphay \cdot alphay}}\right)\right) \]

   10. --lowering--.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\left(\mathsf{neg}\left(\frac{cos2phi}{alphax \cdot alphax}\right)\right), \color{blue}{\left(\frac{sin2phi}{alphay \cdot alphay}\right)}\right)\right) \]

   11. neg-sub0 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\left(0 - \frac{cos2phi}{alphax \cdot alphax}\right), \left(\frac{\color{blue}{sin2phi}}{alphay \cdot alphay}\right)\right)\right) \]

   12. --lowering--.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \left(\frac{cos2phi}{alphax \cdot alphax}\right)\right), \left(\frac{\color{blue}{sin2phi}}{alphay \cdot alphay}\right)\right)\right) \]

   13. /-lowering-/.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(cos2phi, \left(alphax \cdot alphax\right)\right)\right), \left(\frac{sin2phi}{alphay \cdot alphay}\right)\right)\right) \]

   14. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right)\right), \left(\frac{sin2phi}{alphay \cdot alphay}\right)\right)\right) \]

   15. /-lowering-/.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right)\right), \mathsf{/.f32}\left(sin2phi, \color{blue}{\left(alphay \cdot alphay\right)}\right)\right)\right) \]

   16. *-lowering-*.f32 98.3%

       \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right)\right), \mathsf{/.f32}\left(sin2phi, \mathsf{*.f32}\left(alphay, \color{blue}{alphay}\right)\right)\right)\right) \]

3. Simplified 98.3%

   \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(-u0\right)}{\left(0 - \frac{cos2phi}{alphax \cdot alphax}\right) - \frac{sin2phi}{alphay \cdot alphay}}} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. associate-/r* N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right)\right), \left(\frac{\frac{sin2phi}{alphay}}{\color{blue}{alphay}}\right)\right)\right) \]

   2. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right)\right), \mathsf{/.f32}\left(\left(\frac{sin2phi}{alphay}\right), \color{blue}{alphay}\right)\right)\right) \]

   3. /-lowering-/.f32 98.3%

      \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right)\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(sin2phi, alphay\right), alphay\right)\right)\right) \]

6. Applied egg-rr 98.3%

   \[\leadsto \frac{\mathsf{log1p}\left(-u0\right)}{\left(0 - \frac{cos2phi}{alphax \cdot alphax}\right) - \color{blue}{\frac{\frac{sin2phi}{alphay}}{alphay}}} \]
7. Step-by-step derivation

   1. clear-num N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \left(\frac{1}{\frac{alphax \cdot alphax}{cos2phi}}\right)\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(sin2phi, \color{blue}{alphay}\right), alphay\right)\right)\right) \]

   2. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(1, \left(\frac{alphax \cdot alphax}{cos2phi}\right)\right)\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(sin2phi, \color{blue}{alphay}\right), alphay\right)\right)\right) \]

   3. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(1, \mathsf{/.f32}\left(\left(alphax \cdot alphax\right), cos2phi\right)\right)\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(sin2phi, alphay\right), alphay\right)\right)\right) \]

   4. *-lowering-*.f32 98.3%

      \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(1, \mathsf{/.f32}\left(\mathsf{*.f32}\left(alphax, alphax\right), cos2phi\right)\right)\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(sin2phi, alphay\right), alphay\right)\right)\right) \]

8. Applied egg-rr 98.3%

   \[\leadsto \frac{\mathsf{log1p}\left(-u0\right)}{\left(0 - \color{blue}{\frac{1}{\frac{alphax \cdot alphax}{cos2phi}}}\right) - \frac{\frac{sin2phi}{alphay}}{alphay}} \]

9. Taylor expanded in u0 around 0

   \[\leadsto \mathsf{/.f32}\left(\color{blue}{\left(u0 \cdot \left(u0 \cdot \left(u0 \cdot \left(\frac{-1}{4} \cdot u0 - \frac{1}{3}\right) - \frac{1}{2}\right) - 1\right)\right)}, \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(1, \mathsf{/.f32}\left(\mathsf{*.f32}\left(alphax, alphax\right), cos2phi\right)\right)\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(sin2phi, alphay\right), alphay\right)\right)\right) \]
10. Step-by-step derivation

    1. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(u0, \left(u0 \cdot \left(u0 \cdot \left(\frac{-1}{4} \cdot u0 - \frac{1}{3}\right) - \frac{1}{2}\right) - 1\right)\right), \mathsf{\_.f32}\left(\color{blue}{\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(1, \mathsf{/.f32}\left(\mathsf{*.f32}\left(alphax, alphax\right), cos2phi\right)\right)\right)}, \mathsf{/.f32}\left(\mathsf{/.f32}\left(sin2phi, alphay\right), alphay\right)\right)\right) \]

    2. sub-neg N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(u0, \left(u0 \cdot \left(u0 \cdot \left(\frac{-1}{4} \cdot u0 - \frac{1}{3}\right) - \frac{1}{2}\right) + \left(\mathsf{neg}\left(1\right)\right)\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \color{blue}{\mathsf{/.f32}\left(1, \mathsf{/.f32}\left(\mathsf{*.f32}\left(alphax, alphax\right), cos2phi\right)\right)}\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(sin2phi, alphay\right), alphay\right)\right)\right) \]

    3. metadata-eval N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(u0, \left(u0 \cdot \left(u0 \cdot \left(\frac{-1}{4} \cdot u0 - \frac{1}{3}\right) - \frac{1}{2}\right) + -1\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(1, \color{blue}{\mathsf{/.f32}\left(\mathsf{*.f32}\left(alphax, alphax\right), cos2phi\right)}\right)\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(sin2phi, alphay\right), alphay\right)\right)\right) \]

    4. +-lowering-+.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\left(u0 \cdot \left(u0 \cdot \left(\frac{-1}{4} \cdot u0 - \frac{1}{3}\right) - \frac{1}{2}\right)\right), -1\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \color{blue}{\mathsf{/.f32}\left(1, \mathsf{/.f32}\left(\mathsf{*.f32}\left(alphax, alphax\right), cos2phi\right)\right)}\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(sin2phi, alphay\right), alphay\right)\right)\right) \]

    5. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{*.f32}\left(u0, \left(u0 \cdot \left(\frac{-1}{4} \cdot u0 - \frac{1}{3}\right) - \frac{1}{2}\right)\right), -1\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(\color{blue}{1}, \mathsf{/.f32}\left(\mathsf{*.f32}\left(alphax, alphax\right), cos2phi\right)\right)\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(sin2phi, alphay\right), alphay\right)\right)\right) \]

    6. sub-neg N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{*.f32}\left(u0, \left(u0 \cdot \left(\frac{-1}{4} \cdot u0 - \frac{1}{3}\right) + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right)\right), -1\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(1, \mathsf{/.f32}\left(\mathsf{*.f32}\left(alphax, alphax\right), cos2phi\right)\right)\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(sin2phi, alphay\right), alphay\right)\right)\right) \]

    7. metadata-eval N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{*.f32}\left(u0, \left(u0 \cdot \left(\frac{-1}{4} \cdot u0 - \frac{1}{3}\right) + \frac{-1}{2}\right)\right), -1\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(1, \mathsf{/.f32}\left(\mathsf{*.f32}\left(alphax, alphax\right), cos2phi\right)\right)\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(sin2phi, alphay\right), alphay\right)\right)\right) \]

    8. +-lowering-+.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\left(u0 \cdot \left(\frac{-1}{4} \cdot u0 - \frac{1}{3}\right)\right), \frac{-1}{2}\right)\right), -1\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(1, \mathsf{/.f32}\left(\mathsf{*.f32}\left(alphax, alphax\right), cos2phi\right)\right)\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(sin2phi, alphay\right), alphay\right)\right)\right) \]

    9. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{*.f32}\left(u0, \left(\frac{-1}{4} \cdot u0 - \frac{1}{3}\right)\right), \frac{-1}{2}\right)\right), -1\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(1, \mathsf{/.f32}\left(\mathsf{*.f32}\left(alphax, alphax\right), cos2phi\right)\right)\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(sin2phi, alphay\right), alphay\right)\right)\right) \]

    10. sub-neg N/A

        \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{*.f32}\left(u0, \left(\frac{-1}{4} \cdot u0 + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right)\right), \frac{-1}{2}\right)\right), -1\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(1, \mathsf{/.f32}\left(\mathsf{*.f32}\left(alphax, alphax\right), cos2phi\right)\right)\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(sin2phi, alphay\right), alphay\right)\right)\right) \]

    11. metadata-eval N/A

        \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{*.f32}\left(u0, \left(\frac{-1}{4} \cdot u0 + \frac{-1}{3}\right)\right), \frac{-1}{2}\right)\right), -1\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(1, \mathsf{/.f32}\left(\mathsf{*.f32}\left(alphax, alphax\right), cos2phi\right)\right)\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(sin2phi, alphay\right), alphay\right)\right)\right) \]

    12. +-lowering-+.f32 N/A

        \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\left(\frac{-1}{4} \cdot u0\right), \frac{-1}{3}\right)\right), \frac{-1}{2}\right)\right), -1\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(1, \mathsf{/.f32}\left(\mathsf{*.f32}\left(alphax, alphax\right), cos2phi\right)\right)\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(sin2phi, alphay\right), alphay\right)\right)\right) \]

    13. *-commutative N/A

        \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\left(u0 \cdot \frac{-1}{4}\right), \frac{-1}{3}\right)\right), \frac{-1}{2}\right)\right), -1\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(1, \mathsf{/.f32}\left(\mathsf{*.f32}\left(alphax, alphax\right), cos2phi\right)\right)\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(sin2phi, alphay\right), alphay\right)\right)\right) \]

    14. *-lowering-*.f32 91.7%

        \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{*.f32}\left(u0, \frac{-1}{4}\right), \frac{-1}{3}\right)\right), \frac{-1}{2}\right)\right), -1\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(1, \mathsf{/.f32}\left(\mathsf{*.f32}\left(alphax, alphax\right), cos2phi\right)\right)\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(sin2phi, alphay\right), alphay\right)\right)\right) \]

11. Simplified 91.7%

    \[\leadsto \frac{\color{blue}{u0 \cdot \left(u0 \cdot \left(u0 \cdot \left(u0 \cdot -0.25 + -0.3333333333333333\right) + -0.5\right) + -1\right)}}{\left(0 - \frac{1}{\frac{alphax \cdot alphax}{cos2phi}}\right) - \frac{\frac{sin2phi}{alphay}}{alphay}} \]

12. Final simplification 91.7%

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

Alternative 10: 92.7% accurate, 4.0× speedup

Math

\[\begin{array}{l} \\ \frac{u0 \cdot \left(u0 \cdot \left(u0 \cdot \left(u0 \cdot -0.25 + -0.3333333333333333\right) + -0.5\right) + -1\right)}{\frac{0 - \frac{cos2phi}{alphax}}{alphax} - \frac{\frac{sin2phi}{alphay}}{alphay}} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (/
  (* u0 (+ (* u0 (+ (* u0 (+ (* u0 -0.25) -0.3333333333333333)) -0.5)) -1.0))
  (- (/ (- 0.0 (/ cos2phi alphax)) alphax) (/ (/ sin2phi alphay) alphay))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	return (u0 * ((u0 * ((u0 * ((u0 * -0.25f) + -0.3333333333333333f)) + -0.5f)) + -1.0f)) / (((0.0f - (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 = (u0 * ((u0 * ((u0 * ((u0 * (-0.25e0)) + (-0.3333333333333333e0))) + (-0.5e0))) + (-1.0e0))) / (((0.0e0 - (cos2phi / alphax)) / alphax) - ((sin2phi / alphay) / alphay))
end function

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	return Float32(Float32(u0 * Float32(Float32(u0 * Float32(Float32(u0 * Float32(Float32(u0 * Float32(-0.25)) + Float32(-0.3333333333333333))) + Float32(-0.5))) + Float32(-1.0))) / Float32(Float32(Float32(Float32(0.0) - Float32(cos2phi / alphax)) / alphax) - Float32(Float32(sin2phi / alphay) / alphay)))
end

MATLAB

function tmp = code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = (u0 * ((u0 * ((u0 * ((u0 * single(-0.25)) + single(-0.3333333333333333))) + single(-0.5))) + single(-1.0))) / (((single(0.0) - (cos2phi / alphax)) / alphax) - ((sin2phi / alphay) / alphay));
end

Derivation

1. Initial program 64.8%

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

   1. distribute-frac-neg N/A

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

   2. distribute-neg-frac2 N/A

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

   3. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\log \left(1 - u0\right), \color{blue}{\left(\mathsf{neg}\left(\left(\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\right)\right)\right)}\right) \]

   4. sub-neg N/A

      \[\leadsto \mathsf{/.f32}\left(\log \left(1 + \left(\mathsf{neg}\left(u0\right)\right)\right), \left(\mathsf{neg}\left(\left(\color{blue}{\frac{cos2phi}{alphax \cdot alphax}} + \frac{sin2phi}{alphay \cdot alphay}\right)\right)\right)\right) \]

   5. log1p-define N/A

      \[\leadsto \mathsf{/.f32}\left(\left(\mathsf{log1p}\left(\mathsf{neg}\left(u0\right)\right)\right), \left(\mathsf{neg}\left(\color{blue}{\left(\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\right)}\right)\right)\right) \]

   6. log1p-lowering-log1p.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\left(\mathsf{neg}\left(u0\right)\right)\right), \left(\mathsf{neg}\left(\color{blue}{\left(\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\right)}\right)\right)\right) \]

   7. neg-lowering-neg.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \left(\mathsf{neg}\left(\left(\color{blue}{\frac{cos2phi}{alphax \cdot alphax}} + \frac{sin2phi}{alphay \cdot alphay}\right)\right)\right)\right) \]

   8. distribute-neg-in N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \left(\left(\mathsf{neg}\left(\frac{cos2phi}{alphax \cdot alphax}\right)\right) + \color{blue}{\left(\mathsf{neg}\left(\frac{sin2phi}{alphay \cdot alphay}\right)\right)}\right)\right) \]

   9. unsub-neg N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \left(\left(\mathsf{neg}\left(\frac{cos2phi}{alphax \cdot alphax}\right)\right) - \color{blue}{\frac{sin2phi}{alphay \cdot alphay}}\right)\right) \]

   10. --lowering--.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\left(\mathsf{neg}\left(\frac{cos2phi}{alphax \cdot alphax}\right)\right), \color{blue}{\left(\frac{sin2phi}{alphay \cdot alphay}\right)}\right)\right) \]

   11. neg-sub0 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\left(0 - \frac{cos2phi}{alphax \cdot alphax}\right), \left(\frac{\color{blue}{sin2phi}}{alphay \cdot alphay}\right)\right)\right) \]

   12. --lowering--.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \left(\frac{cos2phi}{alphax \cdot alphax}\right)\right), \left(\frac{\color{blue}{sin2phi}}{alphay \cdot alphay}\right)\right)\right) \]

   13. /-lowering-/.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(cos2phi, \left(alphax \cdot alphax\right)\right)\right), \left(\frac{sin2phi}{alphay \cdot alphay}\right)\right)\right) \]

   14. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right)\right), \left(\frac{sin2phi}{alphay \cdot alphay}\right)\right)\right) \]

   15. /-lowering-/.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right)\right), \mathsf{/.f32}\left(sin2phi, \color{blue}{\left(alphay \cdot alphay\right)}\right)\right)\right) \]

   16. *-lowering-*.f32 98.3%

       \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right)\right), \mathsf{/.f32}\left(sin2phi, \mathsf{*.f32}\left(alphay, \color{blue}{alphay}\right)\right)\right)\right) \]

3. Simplified 98.3%

   \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(-u0\right)}{\left(0 - \frac{cos2phi}{alphax \cdot alphax}\right) - \frac{sin2phi}{alphay \cdot alphay}}} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. associate-/r* N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right)\right), \left(\frac{\frac{sin2phi}{alphay}}{\color{blue}{alphay}}\right)\right)\right) \]

   2. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right)\right), \mathsf{/.f32}\left(\left(\frac{sin2phi}{alphay}\right), \color{blue}{alphay}\right)\right)\right) \]

   3. /-lowering-/.f32 98.3%

      \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right)\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(sin2phi, alphay\right), alphay\right)\right)\right) \]

6. Applied egg-rr 98.3%

   \[\leadsto \frac{\mathsf{log1p}\left(-u0\right)}{\left(0 - \frac{cos2phi}{alphax \cdot alphax}\right) - \color{blue}{\frac{\frac{sin2phi}{alphay}}{alphay}}} \]
7. Step-by-step derivation

   1. clear-num N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \left(\frac{1}{\frac{alphax \cdot alphax}{cos2phi}}\right)\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(sin2phi, \color{blue}{alphay}\right), alphay\right)\right)\right) \]

   2. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(1, \left(\frac{alphax \cdot alphax}{cos2phi}\right)\right)\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(sin2phi, \color{blue}{alphay}\right), alphay\right)\right)\right) \]

   3. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(1, \mathsf{/.f32}\left(\left(alphax \cdot alphax\right), cos2phi\right)\right)\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(sin2phi, alphay\right), alphay\right)\right)\right) \]

   4. *-lowering-*.f32 98.3%

      \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(1, \mathsf{/.f32}\left(\mathsf{*.f32}\left(alphax, alphax\right), cos2phi\right)\right)\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(sin2phi, alphay\right), alphay\right)\right)\right) \]

8. Applied egg-rr 98.3%

   \[\leadsto \frac{\mathsf{log1p}\left(-u0\right)}{\left(0 - \color{blue}{\frac{1}{\frac{alphax \cdot alphax}{cos2phi}}}\right) - \frac{\frac{sin2phi}{alphay}}{alphay}} \]
9. Step-by-step derivation

   1. clear-num N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\left(0 - \frac{cos2phi}{alphax \cdot alphax}\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(sin2phi, \color{blue}{alphay}\right), alphay\right)\right)\right) \]

   2. sub0-neg N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\left(\mathsf{neg}\left(\frac{cos2phi}{alphax \cdot alphax}\right)\right), \mathsf{/.f32}\left(\color{blue}{\mathsf{/.f32}\left(sin2phi, alphay\right)}, alphay\right)\right)\right) \]

   3. associate-/r* N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\left(\mathsf{neg}\left(\frac{\frac{cos2phi}{alphax}}{alphax}\right)\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(\color{blue}{sin2phi}, alphay\right), alphay\right)\right)\right) \]

   4. distribute-neg-frac2 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\left(\frac{\frac{cos2phi}{alphax}}{\mathsf{neg}\left(alphax\right)}\right), \mathsf{/.f32}\left(\color{blue}{\mathsf{/.f32}\left(sin2phi, alphay\right)}, alphay\right)\right)\right) \]

   5. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{/.f32}\left(\left(\frac{cos2phi}{alphax}\right), \left(\mathsf{neg}\left(alphax\right)\right)\right), \mathsf{/.f32}\left(\color{blue}{\mathsf{/.f32}\left(sin2phi, alphay\right)}, alphay\right)\right)\right) \]

   6. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(cos2phi, alphax\right), \left(\mathsf{neg}\left(alphax\right)\right)\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(\color{blue}{sin2phi}, alphay\right), alphay\right)\right)\right) \]

   7. neg-lowering-neg.f32 98.3%

      \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(cos2phi, alphax\right), \mathsf{neg.f32}\left(alphax\right)\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(sin2phi, \color{blue}{alphay}\right), alphay\right)\right)\right) \]

10. Applied egg-rr 98.3%

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

11. Taylor expanded in u0 around 0

    \[\leadsto \mathsf{/.f32}\left(\color{blue}{\left(u0 \cdot \left(u0 \cdot \left(u0 \cdot \left(\frac{-1}{4} \cdot u0 - \frac{1}{3}\right) - \frac{1}{2}\right) - 1\right)\right)}, \mathsf{\_.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(cos2phi, alphax\right), \mathsf{neg.f32}\left(alphax\right)\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(sin2phi, alphay\right), alphay\right)\right)\right) \]
12. Step-by-step derivation

    1. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(u0, \left(u0 \cdot \left(u0 \cdot \left(\frac{-1}{4} \cdot u0 - \frac{1}{3}\right) - \frac{1}{2}\right) - 1\right)\right), \mathsf{\_.f32}\left(\color{blue}{\mathsf{/.f32}\left(\mathsf{/.f32}\left(cos2phi, alphax\right), \mathsf{neg.f32}\left(alphax\right)\right)}, \mathsf{/.f32}\left(\mathsf{/.f32}\left(sin2phi, alphay\right), alphay\right)\right)\right) \]

    2. sub-neg N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(u0, \left(u0 \cdot \left(u0 \cdot \left(\frac{-1}{4} \cdot u0 - \frac{1}{3}\right) - \frac{1}{2}\right) + \left(\mathsf{neg}\left(1\right)\right)\right)\right), \mathsf{\_.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(cos2phi, alphax\right), \color{blue}{\mathsf{neg.f32}\left(alphax\right)}\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(sin2phi, alphay\right), alphay\right)\right)\right) \]

    3. metadata-eval N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(u0, \left(u0 \cdot \left(u0 \cdot \left(\frac{-1}{4} \cdot u0 - \frac{1}{3}\right) - \frac{1}{2}\right) + -1\right)\right), \mathsf{\_.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(cos2phi, alphax\right), \mathsf{neg.f32}\left(alphax\right)\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(sin2phi, alphay\right), alphay\right)\right)\right) \]

    4. +-lowering-+.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\left(u0 \cdot \left(u0 \cdot \left(\frac{-1}{4} \cdot u0 - \frac{1}{3}\right) - \frac{1}{2}\right)\right), -1\right)\right), \mathsf{\_.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(cos2phi, alphax\right), \color{blue}{\mathsf{neg.f32}\left(alphax\right)}\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(sin2phi, alphay\right), alphay\right)\right)\right) \]

    5. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{*.f32}\left(u0, \left(u0 \cdot \left(\frac{-1}{4} \cdot u0 - \frac{1}{3}\right) - \frac{1}{2}\right)\right), -1\right)\right), \mathsf{\_.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(cos2phi, alphax\right), \mathsf{neg.f32}\left(\color{blue}{alphax}\right)\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(sin2phi, alphay\right), alphay\right)\right)\right) \]

    6. sub-neg N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{*.f32}\left(u0, \left(u0 \cdot \left(\frac{-1}{4} \cdot u0 - \frac{1}{3}\right) + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right)\right), -1\right)\right), \mathsf{\_.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(cos2phi, alphax\right), \mathsf{neg.f32}\left(alphax\right)\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(sin2phi, alphay\right), alphay\right)\right)\right) \]

    7. metadata-eval N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{*.f32}\left(u0, \left(u0 \cdot \left(\frac{-1}{4} \cdot u0 - \frac{1}{3}\right) + \frac{-1}{2}\right)\right), -1\right)\right), \mathsf{\_.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(cos2phi, alphax\right), \mathsf{neg.f32}\left(alphax\right)\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(sin2phi, alphay\right), alphay\right)\right)\right) \]

    8. +-lowering-+.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\left(u0 \cdot \left(\frac{-1}{4} \cdot u0 - \frac{1}{3}\right)\right), \frac{-1}{2}\right)\right), -1\right)\right), \mathsf{\_.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(cos2phi, alphax\right), \mathsf{neg.f32}\left(alphax\right)\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(sin2phi, alphay\right), alphay\right)\right)\right) \]

    9. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{*.f32}\left(u0, \left(\frac{-1}{4} \cdot u0 - \frac{1}{3}\right)\right), \frac{-1}{2}\right)\right), -1\right)\right), \mathsf{\_.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(cos2phi, alphax\right), \mathsf{neg.f32}\left(alphax\right)\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(sin2phi, alphay\right), alphay\right)\right)\right) \]

    10. sub-neg N/A

        \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{*.f32}\left(u0, \left(\frac{-1}{4} \cdot u0 + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right)\right), \frac{-1}{2}\right)\right), -1\right)\right), \mathsf{\_.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(cos2phi, alphax\right), \mathsf{neg.f32}\left(alphax\right)\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(sin2phi, alphay\right), alphay\right)\right)\right) \]

    11. metadata-eval N/A

        \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{*.f32}\left(u0, \left(\frac{-1}{4} \cdot u0 + \frac{-1}{3}\right)\right), \frac{-1}{2}\right)\right), -1\right)\right), \mathsf{\_.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(cos2phi, alphax\right), \mathsf{neg.f32}\left(alphax\right)\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(sin2phi, alphay\right), alphay\right)\right)\right) \]

    12. +-lowering-+.f32 N/A

        \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\left(\frac{-1}{4} \cdot u0\right), \frac{-1}{3}\right)\right), \frac{-1}{2}\right)\right), -1\right)\right), \mathsf{\_.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(cos2phi, alphax\right), \mathsf{neg.f32}\left(alphax\right)\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(sin2phi, alphay\right), alphay\right)\right)\right) \]

    13. *-commutative N/A

        \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\left(u0 \cdot \frac{-1}{4}\right), \frac{-1}{3}\right)\right), \frac{-1}{2}\right)\right), -1\right)\right), \mathsf{\_.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(cos2phi, alphax\right), \mathsf{neg.f32}\left(alphax\right)\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(sin2phi, alphay\right), alphay\right)\right)\right) \]

    14. *-lowering-*.f32 91.7%

        \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{*.f32}\left(u0, \frac{-1}{4}\right), \frac{-1}{3}\right)\right), \frac{-1}{2}\right)\right), -1\right)\right), \mathsf{\_.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(cos2phi, alphax\right), \mathsf{neg.f32}\left(alphax\right)\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(sin2phi, alphay\right), alphay\right)\right)\right) \]

13. Simplified 91.7%

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

14. Final simplification 91.7%

    \[\leadsto \frac{u0 \cdot \left(u0 \cdot \left(u0 \cdot \left(u0 \cdot -0.25 + -0.3333333333333333\right) + -0.5\right) + -1\right)}{\frac{0 - \frac{cos2phi}{alphax}}{alphax} - \frac{\frac{sin2phi}{alphay}}{alphay}} \]
15. Add Preprocessing

Alternative 11: 92.7% accurate, 4.3× speedup

Math

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

FPCore

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

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	return (u0 * (1.0f + (u0 * (0.5f + (u0 * (0.3333333333333333f + (u0 * 0.25f))))))) / ((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 = (u0 * (1.0e0 + (u0 * (0.5e0 + (u0 * (0.3333333333333333e0 + (u0 * 0.25e0))))))) / ((cos2phi / (alphax * alphax)) + (sin2phi / (alphay * 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(0.3333333333333333) + Float32(u0 * Float32(0.25)))))))) / Float32(Float32(cos2phi / Float32(alphax * alphax)) + Float32(sin2phi / Float32(alphay * alphay))))
end

MATLAB

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

Derivation

1. Initial program 64.8%

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

3. Taylor expanded in u0 around 0

   \[\leadsto \mathsf{/.f32}\left(\color{blue}{\left(u0 \cdot \left(1 + u0 \cdot \left(\frac{1}{2} + u0 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right)\right)\right)\right)}, \mathsf{+.f32}\left(\mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right), \mathsf{/.f32}\left(sin2phi, \mathsf{*.f32}\left(alphay, alphay\right)\right)\right)\right) \]
4. Step-by-step derivation

   1. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(u0, \left(1 + u0 \cdot \left(\frac{1}{2} + u0 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right)\right)\right)\right), \mathsf{+.f32}\left(\color{blue}{\mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right)}, \mathsf{/.f32}\left(sin2phi, \mathsf{*.f32}\left(alphay, alphay\right)\right)\right)\right) \]

   2. +-lowering-+.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(1, \left(u0 \cdot \left(\frac{1}{2} + u0 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right)\right)\right)\right)\right), \mathsf{+.f32}\left(\mathsf{/.f32}\left(cos2phi, \color{blue}{\mathsf{*.f32}\left(alphax, alphax\right)}\right), \mathsf{/.f32}\left(sin2phi, \mathsf{*.f32}\left(alphay, alphay\right)\right)\right)\right) \]

   3. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(u0, \left(\frac{1}{2} + u0 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right)\right)\right)\right)\right), \mathsf{+.f32}\left(\mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, \color{blue}{alphax}\right)\right), \mathsf{/.f32}\left(sin2phi, \mathsf{*.f32}\left(alphay, alphay\right)\right)\right)\right) \]

   4. +-lowering-+.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\frac{1}{2}, \left(u0 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right)\right)\right)\right)\right)\right), \mathsf{+.f32}\left(\mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right), \mathsf{/.f32}\left(sin2phi, \mathsf{*.f32}\left(alphay, alphay\right)\right)\right)\right) \]

   5. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\frac{1}{2}, \mathsf{*.f32}\left(u0, \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right)\right)\right)\right)\right)\right), \mathsf{+.f32}\left(\mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right), \mathsf{/.f32}\left(sin2phi, \mathsf{*.f32}\left(alphay, alphay\right)\right)\right)\right) \]

   6. +-lowering-+.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\frac{1}{2}, \mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\frac{1}{3}, \left(\frac{1}{4} \cdot u0\right)\right)\right)\right)\right)\right)\right), \mathsf{+.f32}\left(\mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right), \mathsf{/.f32}\left(sin2phi, \mathsf{*.f32}\left(alphay, alphay\right)\right)\right)\right) \]

   7. *-commutative N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\frac{1}{2}, \mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\frac{1}{3}, \left(u0 \cdot \frac{1}{4}\right)\right)\right)\right)\right)\right)\right), \mathsf{+.f32}\left(\mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right), \mathsf{/.f32}\left(sin2phi, \mathsf{*.f32}\left(alphay, alphay\right)\right)\right)\right) \]

   8. *-lowering-*.f32 91.6%

      \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\frac{1}{2}, \mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\frac{1}{3}, \mathsf{*.f32}\left(u0, \frac{1}{4}\right)\right)\right)\right)\right)\right)\right), \mathsf{+.f32}\left(\mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right), \mathsf{/.f32}\left(sin2phi, \mathsf{*.f32}\left(alphay, alphay\right)\right)\right)\right) \]

5. Simplified 91.6%

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

Alternative 12: 90.8% accurate, 4.6× speedup

Math

\[\begin{array}{l} \\ \frac{u0 \cdot \left(-1 + u0 \cdot \left(-0.5 + u0 \cdot -0.3333333333333333\right)\right)}{\frac{0 - \frac{cos2phi}{alphax}}{alphax} - \frac{\frac{sin2phi}{alphay}}{alphay}} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (/
  (* u0 (+ -1.0 (* u0 (+ -0.5 (* u0 -0.3333333333333333)))))
  (- (/ (- 0.0 (/ cos2phi alphax)) alphax) (/ (/ sin2phi alphay) alphay))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	return (u0 * (-1.0f + (u0 * (-0.5f + (u0 * -0.3333333333333333f))))) / (((0.0f - (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 = (u0 * ((-1.0e0) + (u0 * ((-0.5e0) + (u0 * (-0.3333333333333333e0)))))) / (((0.0e0 - (cos2phi / alphax)) / alphax) - ((sin2phi / alphay) / 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(Float32(Float32(0.0) - Float32(cos2phi / alphax)) / alphax) - Float32(Float32(sin2phi / alphay) / alphay)))
end

MATLAB

function tmp = code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = (u0 * (single(-1.0) + (u0 * (single(-0.5) + (u0 * single(-0.3333333333333333)))))) / (((single(0.0) - (cos2phi / alphax)) / alphax) - ((sin2phi / alphay) / alphay));
end

Derivation

1. Initial program 64.8%

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

   1. distribute-frac-neg N/A

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

   2. distribute-neg-frac2 N/A

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

   3. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\log \left(1 - u0\right), \color{blue}{\left(\mathsf{neg}\left(\left(\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\right)\right)\right)}\right) \]

   4. sub-neg N/A

      \[\leadsto \mathsf{/.f32}\left(\log \left(1 + \left(\mathsf{neg}\left(u0\right)\right)\right), \left(\mathsf{neg}\left(\left(\color{blue}{\frac{cos2phi}{alphax \cdot alphax}} + \frac{sin2phi}{alphay \cdot alphay}\right)\right)\right)\right) \]

   5. log1p-define N/A

      \[\leadsto \mathsf{/.f32}\left(\left(\mathsf{log1p}\left(\mathsf{neg}\left(u0\right)\right)\right), \left(\mathsf{neg}\left(\color{blue}{\left(\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\right)}\right)\right)\right) \]

   6. log1p-lowering-log1p.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\left(\mathsf{neg}\left(u0\right)\right)\right), \left(\mathsf{neg}\left(\color{blue}{\left(\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\right)}\right)\right)\right) \]

   7. neg-lowering-neg.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \left(\mathsf{neg}\left(\left(\color{blue}{\frac{cos2phi}{alphax \cdot alphax}} + \frac{sin2phi}{alphay \cdot alphay}\right)\right)\right)\right) \]

   8. distribute-neg-in N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \left(\left(\mathsf{neg}\left(\frac{cos2phi}{alphax \cdot alphax}\right)\right) + \color{blue}{\left(\mathsf{neg}\left(\frac{sin2phi}{alphay \cdot alphay}\right)\right)}\right)\right) \]

   9. unsub-neg N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \left(\left(\mathsf{neg}\left(\frac{cos2phi}{alphax \cdot alphax}\right)\right) - \color{blue}{\frac{sin2phi}{alphay \cdot alphay}}\right)\right) \]

   10. --lowering--.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\left(\mathsf{neg}\left(\frac{cos2phi}{alphax \cdot alphax}\right)\right), \color{blue}{\left(\frac{sin2phi}{alphay \cdot alphay}\right)}\right)\right) \]

   11. neg-sub0 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\left(0 - \frac{cos2phi}{alphax \cdot alphax}\right), \left(\frac{\color{blue}{sin2phi}}{alphay \cdot alphay}\right)\right)\right) \]

   12. --lowering--.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \left(\frac{cos2phi}{alphax \cdot alphax}\right)\right), \left(\frac{\color{blue}{sin2phi}}{alphay \cdot alphay}\right)\right)\right) \]

   13. /-lowering-/.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(cos2phi, \left(alphax \cdot alphax\right)\right)\right), \left(\frac{sin2phi}{alphay \cdot alphay}\right)\right)\right) \]

   14. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right)\right), \left(\frac{sin2phi}{alphay \cdot alphay}\right)\right)\right) \]

   15. /-lowering-/.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right)\right), \mathsf{/.f32}\left(sin2phi, \color{blue}{\left(alphay \cdot alphay\right)}\right)\right)\right) \]

   16. *-lowering-*.f32 98.3%

       \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right)\right), \mathsf{/.f32}\left(sin2phi, \mathsf{*.f32}\left(alphay, \color{blue}{alphay}\right)\right)\right)\right) \]

3. Simplified 98.3%

   \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(-u0\right)}{\left(0 - \frac{cos2phi}{alphax \cdot alphax}\right) - \frac{sin2phi}{alphay \cdot alphay}}} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. associate-/r* N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right)\right), \left(\frac{\frac{sin2phi}{alphay}}{\color{blue}{alphay}}\right)\right)\right) \]

   2. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right)\right), \mathsf{/.f32}\left(\left(\frac{sin2phi}{alphay}\right), \color{blue}{alphay}\right)\right)\right) \]

   3. /-lowering-/.f32 98.3%

      \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right)\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(sin2phi, alphay\right), alphay\right)\right)\right) \]

6. Applied egg-rr 98.3%

   \[\leadsto \frac{\mathsf{log1p}\left(-u0\right)}{\left(0 - \frac{cos2phi}{alphax \cdot alphax}\right) - \color{blue}{\frac{\frac{sin2phi}{alphay}}{alphay}}} \]
7. Step-by-step derivation

   1. clear-num N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \left(\frac{1}{\frac{alphax \cdot alphax}{cos2phi}}\right)\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(sin2phi, \color{blue}{alphay}\right), alphay\right)\right)\right) \]

   2. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(1, \left(\frac{alphax \cdot alphax}{cos2phi}\right)\right)\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(sin2phi, \color{blue}{alphay}\right), alphay\right)\right)\right) \]

   3. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(1, \mathsf{/.f32}\left(\left(alphax \cdot alphax\right), cos2phi\right)\right)\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(sin2phi, alphay\right), alphay\right)\right)\right) \]

   4. *-lowering-*.f32 98.3%

      \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(1, \mathsf{/.f32}\left(\mathsf{*.f32}\left(alphax, alphax\right), cos2phi\right)\right)\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(sin2phi, alphay\right), alphay\right)\right)\right) \]

8. Applied egg-rr 98.3%

   \[\leadsto \frac{\mathsf{log1p}\left(-u0\right)}{\left(0 - \color{blue}{\frac{1}{\frac{alphax \cdot alphax}{cos2phi}}}\right) - \frac{\frac{sin2phi}{alphay}}{alphay}} \]
9. Step-by-step derivation

   1. clear-num N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\left(0 - \frac{cos2phi}{alphax \cdot alphax}\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(sin2phi, \color{blue}{alphay}\right), alphay\right)\right)\right) \]

   2. sub0-neg N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\left(\mathsf{neg}\left(\frac{cos2phi}{alphax \cdot alphax}\right)\right), \mathsf{/.f32}\left(\color{blue}{\mathsf{/.f32}\left(sin2phi, alphay\right)}, alphay\right)\right)\right) \]

   3. associate-/r* N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\left(\mathsf{neg}\left(\frac{\frac{cos2phi}{alphax}}{alphax}\right)\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(\color{blue}{sin2phi}, alphay\right), alphay\right)\right)\right) \]

   4. distribute-neg-frac2 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\left(\frac{\frac{cos2phi}{alphax}}{\mathsf{neg}\left(alphax\right)}\right), \mathsf{/.f32}\left(\color{blue}{\mathsf{/.f32}\left(sin2phi, alphay\right)}, alphay\right)\right)\right) \]

   5. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{/.f32}\left(\left(\frac{cos2phi}{alphax}\right), \left(\mathsf{neg}\left(alphax\right)\right)\right), \mathsf{/.f32}\left(\color{blue}{\mathsf{/.f32}\left(sin2phi, alphay\right)}, alphay\right)\right)\right) \]

   6. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(cos2phi, alphax\right), \left(\mathsf{neg}\left(alphax\right)\right)\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(\color{blue}{sin2phi}, alphay\right), alphay\right)\right)\right) \]

   7. neg-lowering-neg.f32 98.3%

      \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(cos2phi, alphax\right), \mathsf{neg.f32}\left(alphax\right)\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(sin2phi, \color{blue}{alphay}\right), alphay\right)\right)\right) \]

10. Applied egg-rr 98.3%

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

11. Taylor expanded in u0 around 0

    \[\leadsto \mathsf{/.f32}\left(\color{blue}{\left(u0 \cdot \left(u0 \cdot \left(\frac{-1}{3} \cdot u0 - \frac{1}{2}\right) - 1\right)\right)}, \mathsf{\_.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(cos2phi, alphax\right), \mathsf{neg.f32}\left(alphax\right)\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(sin2phi, alphay\right), alphay\right)\right)\right) \]
12. Step-by-step derivation

    1. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(u0, \left(u0 \cdot \left(\frac{-1}{3} \cdot u0 - \frac{1}{2}\right) - 1\right)\right), \mathsf{\_.f32}\left(\color{blue}{\mathsf{/.f32}\left(\mathsf{/.f32}\left(cos2phi, alphax\right), \mathsf{neg.f32}\left(alphax\right)\right)}, \mathsf{/.f32}\left(\mathsf{/.f32}\left(sin2phi, alphay\right), alphay\right)\right)\right) \]

    2. sub-neg N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(u0, \left(u0 \cdot \left(\frac{-1}{3} \cdot u0 - \frac{1}{2}\right) + \left(\mathsf{neg}\left(1\right)\right)\right)\right), \mathsf{\_.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(cos2phi, alphax\right), \color{blue}{\mathsf{neg.f32}\left(alphax\right)}\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(sin2phi, alphay\right), alphay\right)\right)\right) \]

    3. metadata-eval N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(u0, \left(u0 \cdot \left(\frac{-1}{3} \cdot u0 - \frac{1}{2}\right) + -1\right)\right), \mathsf{\_.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(cos2phi, alphax\right), \mathsf{neg.f32}\left(alphax\right)\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(sin2phi, alphay\right), alphay\right)\right)\right) \]

    4. +-lowering-+.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\left(u0 \cdot \left(\frac{-1}{3} \cdot u0 - \frac{1}{2}\right)\right), -1\right)\right), \mathsf{\_.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(cos2phi, alphax\right), \color{blue}{\mathsf{neg.f32}\left(alphax\right)}\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(sin2phi, alphay\right), alphay\right)\right)\right) \]

    5. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{*.f32}\left(u0, \left(\frac{-1}{3} \cdot u0 - \frac{1}{2}\right)\right), -1\right)\right), \mathsf{\_.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(cos2phi, alphax\right), \mathsf{neg.f32}\left(\color{blue}{alphax}\right)\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(sin2phi, alphay\right), alphay\right)\right)\right) \]

    6. sub-neg N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{*.f32}\left(u0, \left(\frac{-1}{3} \cdot u0 + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right)\right), -1\right)\right), \mathsf{\_.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(cos2phi, alphax\right), \mathsf{neg.f32}\left(alphax\right)\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(sin2phi, alphay\right), alphay\right)\right)\right) \]

    7. metadata-eval N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{*.f32}\left(u0, \left(\frac{-1}{3} \cdot u0 + \frac{-1}{2}\right)\right), -1\right)\right), \mathsf{\_.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(cos2phi, alphax\right), \mathsf{neg.f32}\left(alphax\right)\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(sin2phi, alphay\right), alphay\right)\right)\right) \]

    8. +-lowering-+.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\left(\frac{-1}{3} \cdot u0\right), \frac{-1}{2}\right)\right), -1\right)\right), \mathsf{\_.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(cos2phi, alphax\right), \mathsf{neg.f32}\left(alphax\right)\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(sin2phi, alphay\right), alphay\right)\right)\right) \]

    9. *-commutative N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\left(u0 \cdot \frac{-1}{3}\right), \frac{-1}{2}\right)\right), -1\right)\right), \mathsf{\_.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(cos2phi, alphax\right), \mathsf{neg.f32}\left(alphax\right)\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(sin2phi, alphay\right), alphay\right)\right)\right) \]

    10. *-lowering-*.f32 89.4%

        \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{*.f32}\left(u0, \frac{-1}{3}\right), \frac{-1}{2}\right)\right), -1\right)\right), \mathsf{\_.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(cos2phi, alphax\right), \mathsf{neg.f32}\left(alphax\right)\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(sin2phi, alphay\right), alphay\right)\right)\right) \]

13. Simplified 89.4%

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

14. Final simplification 89.4%

    \[\leadsto \frac{u0 \cdot \left(-1 + u0 \cdot \left(-0.5 + u0 \cdot -0.3333333333333333\right)\right)}{\frac{0 - \frac{cos2phi}{alphax}}{alphax} - \frac{\frac{sin2phi}{alphay}}{alphay}} \]
15. Add Preprocessing

Alternative 13: 82.8% accurate, 5.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;sin2phi \leq 4.999999969612645 \cdot 10^{-9}:\\ \;\;\;\;\frac{u0 \cdot u0}{u0 \cdot \left(\frac{\frac{sin2phi}{alphay}}{alphay} + \frac{\frac{cos2phi}{alphax}}{alphax}\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(u0 \cdot \left(-1 + u0 \cdot \left(-0.5 + u0 \cdot -0.3333333333333333\right)\right)\right) \cdot \left(-\frac{alphay \cdot alphay}{sin2phi}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (if (<= sin2phi 4.999999969612645e-9)
   (/
    (* u0 u0)
    (* u0 (+ (/ (/ sin2phi alphay) alphay) (/ (/ cos2phi alphax) alphax))))
   (*
    (* u0 (+ -1.0 (* u0 (+ -0.5 (* u0 -0.3333333333333333)))))
    (- (/ (* alphay alphay) sin2phi)))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float tmp;
	if (sin2phi <= 4.999999969612645e-9f) {
		tmp = (u0 * u0) / (u0 * (((sin2phi / alphay) / alphay) + ((cos2phi / alphax) / alphax)));
	} else {
		tmp = (u0 * (-1.0f + (u0 * (-0.5f + (u0 * -0.3333333333333333f))))) * -((alphay * 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 <= 4.999999969612645e-9) then
        tmp = (u0 * u0) / (u0 * (((sin2phi / alphay) / alphay) + ((cos2phi / alphax) / alphax)))
    else
        tmp = (u0 * ((-1.0e0) + (u0 * ((-0.5e0) + (u0 * (-0.3333333333333333e0)))))) * -((alphay * alphay) / sin2phi)
    end if
    code = tmp
end function

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = Float32(0.0)
	if (sin2phi <= Float32(4.999999969612645e-9))
		tmp = Float32(Float32(u0 * u0) / Float32(u0 * Float32(Float32(Float32(sin2phi / alphay) / alphay) + Float32(Float32(cos2phi / alphax) / alphax))));
	else
		tmp = Float32(Float32(u0 * Float32(Float32(-1.0) + Float32(u0 * Float32(Float32(-0.5) + Float32(u0 * Float32(-0.3333333333333333)))))) * Float32(-Float32(Float32(alphay * alphay) / sin2phi)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = single(0.0);
	if (sin2phi <= single(4.999999969612645e-9))
		tmp = (u0 * u0) / (u0 * (((sin2phi / alphay) / alphay) + ((cos2phi / alphax) / alphax)));
	else
		tmp = (u0 * (single(-1.0) + (u0 * (single(-0.5) + (u0 * single(-0.3333333333333333)))))) * -((alphay * alphay) / sin2phi);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if sin2phi < 4.99999997e-9

   1. Initial program 64.5%

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

      1. distribute-frac-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. /-lowering-/.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(\log \left(1 - u0\right), \color{blue}{\left(\mathsf{neg}\left(\left(\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\right)\right)\right)}\right) \]

      4. sub-neg N/A

         \[\leadsto \mathsf{/.f32}\left(\log \left(1 + \left(\mathsf{neg}\left(u0\right)\right)\right), \left(\mathsf{neg}\left(\left(\color{blue}{\frac{cos2phi}{alphax \cdot alphax}} + \frac{sin2phi}{alphay \cdot alphay}\right)\right)\right)\right) \]

      5. log1p-define N/A

         \[\leadsto \mathsf{/.f32}\left(\left(\mathsf{log1p}\left(\mathsf{neg}\left(u0\right)\right)\right), \left(\mathsf{neg}\left(\color{blue}{\left(\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\right)}\right)\right)\right) \]

      6. log1p-lowering-log1p.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\left(\mathsf{neg}\left(u0\right)\right)\right), \left(\mathsf{neg}\left(\color{blue}{\left(\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\right)}\right)\right)\right) \]

      7. neg-lowering-neg.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \left(\mathsf{neg}\left(\left(\color{blue}{\frac{cos2phi}{alphax \cdot alphax}} + \frac{sin2phi}{alphay \cdot alphay}\right)\right)\right)\right) \]

      8. distribute-neg-in N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \left(\left(\mathsf{neg}\left(\frac{cos2phi}{alphax \cdot alphax}\right)\right) + \color{blue}{\left(\mathsf{neg}\left(\frac{sin2phi}{alphay \cdot alphay}\right)\right)}\right)\right) \]

      9. unsub-neg N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \left(\left(\mathsf{neg}\left(\frac{cos2phi}{alphax \cdot alphax}\right)\right) - \color{blue}{\frac{sin2phi}{alphay \cdot alphay}}\right)\right) \]

      10. --lowering--.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\left(\mathsf{neg}\left(\frac{cos2phi}{alphax \cdot alphax}\right)\right), \color{blue}{\left(\frac{sin2phi}{alphay \cdot alphay}\right)}\right)\right) \]

      11. neg-sub0 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\left(0 - \frac{cos2phi}{alphax \cdot alphax}\right), \left(\frac{\color{blue}{sin2phi}}{alphay \cdot alphay}\right)\right)\right) \]

      12. --lowering--.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \left(\frac{cos2phi}{alphax \cdot alphax}\right)\right), \left(\frac{\color{blue}{sin2phi}}{alphay \cdot alphay}\right)\right)\right) \]

      13. /-lowering-/.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(cos2phi, \left(alphax \cdot alphax\right)\right)\right), \left(\frac{sin2phi}{alphay \cdot alphay}\right)\right)\right) \]

      14. *-lowering-*.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right)\right), \left(\frac{sin2phi}{alphay \cdot alphay}\right)\right)\right) \]

      15. /-lowering-/.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right)\right), \mathsf{/.f32}\left(sin2phi, \color{blue}{\left(alphay \cdot alphay\right)}\right)\right)\right) \]

      16. *-lowering-*.f32 98.6%

          \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right)\right), \mathsf{/.f32}\left(sin2phi, \mathsf{*.f32}\left(alphay, \color{blue}{alphay}\right)\right)\right)\right) \]

   3. Simplified 98.6%

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

   5. Taylor expanded in u0 around 0

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

      1. /-lowering-/.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(u0, \color{blue}{\left(\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}\right)}\right) \]

      2. +-lowering-+.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{+.f32}\left(\left(\frac{cos2phi}{{alphax}^{2}}\right), \color{blue}{\left(\frac{sin2phi}{{alphay}^{2}}\right)}\right)\right) \]

      3. /-lowering-/.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{/.f32}\left(cos2phi, \left({alphax}^{2}\right)\right), \left(\frac{\color{blue}{sin2phi}}{{alphay}^{2}}\right)\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{/.f32}\left(cos2phi, \left(alphax \cdot alphax\right)\right), \left(\frac{sin2phi}{{alphay}^{2}}\right)\right)\right) \]

      5. *-lowering-*.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right), \left(\frac{sin2phi}{{alphay}^{2}}\right)\right)\right) \]

      6. /-lowering-/.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right), \mathsf{/.f32}\left(sin2phi, \color{blue}{\left({alphay}^{2}\right)}\right)\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right), \mathsf{/.f32}\left(sin2phi, \left(alphay \cdot \color{blue}{alphay}\right)\right)\right)\right) \]

      8. *-lowering-*.f32 68.0%

         \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right), \mathsf{/.f32}\left(sin2phi, \mathsf{*.f32}\left(alphay, \color{blue}{alphay}\right)\right)\right)\right) \]

   7. Simplified 68.0%

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

   9. Applied egg-rr 68.3%

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

   if 4.99999997e-9 < sin2phi

   1. Initial program 65.0%

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

      1. distribute-frac-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. /-lowering-/.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(\log \left(1 - u0\right), \color{blue}{\left(\mathsf{neg}\left(\left(\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\right)\right)\right)}\right) \]

      4. sub-neg N/A

         \[\leadsto \mathsf{/.f32}\left(\log \left(1 + \left(\mathsf{neg}\left(u0\right)\right)\right), \left(\mathsf{neg}\left(\left(\color{blue}{\frac{cos2phi}{alphax \cdot alphax}} + \frac{sin2phi}{alphay \cdot alphay}\right)\right)\right)\right) \]

      5. log1p-define N/A

         \[\leadsto \mathsf{/.f32}\left(\left(\mathsf{log1p}\left(\mathsf{neg}\left(u0\right)\right)\right), \left(\mathsf{neg}\left(\color{blue}{\left(\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\right)}\right)\right)\right) \]

      6. log1p-lowering-log1p.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\left(\mathsf{neg}\left(u0\right)\right)\right), \left(\mathsf{neg}\left(\color{blue}{\left(\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\right)}\right)\right)\right) \]

      7. neg-lowering-neg.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \left(\mathsf{neg}\left(\left(\color{blue}{\frac{cos2phi}{alphax \cdot alphax}} + \frac{sin2phi}{alphay \cdot alphay}\right)\right)\right)\right) \]

      8. distribute-neg-in N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \left(\left(\mathsf{neg}\left(\frac{cos2phi}{alphax \cdot alphax}\right)\right) + \color{blue}{\left(\mathsf{neg}\left(\frac{sin2phi}{alphay \cdot alphay}\right)\right)}\right)\right) \]

      9. unsub-neg N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \left(\left(\mathsf{neg}\left(\frac{cos2phi}{alphax \cdot alphax}\right)\right) - \color{blue}{\frac{sin2phi}{alphay \cdot alphay}}\right)\right) \]

      10. --lowering--.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\left(\mathsf{neg}\left(\frac{cos2phi}{alphax \cdot alphax}\right)\right), \color{blue}{\left(\frac{sin2phi}{alphay \cdot alphay}\right)}\right)\right) \]

      11. neg-sub0 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\left(0 - \frac{cos2phi}{alphax \cdot alphax}\right), \left(\frac{\color{blue}{sin2phi}}{alphay \cdot alphay}\right)\right)\right) \]

      12. --lowering--.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \left(\frac{cos2phi}{alphax \cdot alphax}\right)\right), \left(\frac{\color{blue}{sin2phi}}{alphay \cdot alphay}\right)\right)\right) \]

      13. /-lowering-/.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(cos2phi, \left(alphax \cdot alphax\right)\right)\right), \left(\frac{sin2phi}{alphay \cdot alphay}\right)\right)\right) \]

      14. *-lowering-*.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right)\right), \left(\frac{sin2phi}{alphay \cdot alphay}\right)\right)\right) \]

      15. /-lowering-/.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right)\right), \mathsf{/.f32}\left(sin2phi, \color{blue}{\left(alphay \cdot alphay\right)}\right)\right)\right) \]

      16. *-lowering-*.f32 98.0%

          \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right)\right), \mathsf{/.f32}\left(sin2phi, \mathsf{*.f32}\left(alphay, \color{blue}{alphay}\right)\right)\right)\right) \]

   3. Simplified 98.0%

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

   5. Taylor expanded in cos2phi around 0

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

      1. mul-1-neg N/A

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

      2. neg-lowering-neg.f32 N/A

         \[\leadsto \mathsf{neg.f32}\left(\left(\frac{{alphay}^{2} \cdot \log \left(1 - u0\right)}{sin2phi}\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{neg.f32}\left(\left(\frac{\log \left(1 - u0\right) \cdot {alphay}^{2}}{sin2phi}\right)\right) \]

      4. associate-/l* N/A

         \[\leadsto \mathsf{neg.f32}\left(\left(\log \left(1 - u0\right) \cdot \frac{{alphay}^{2}}{sin2phi}\right)\right) \]

      5. *-lowering-*.f32 N/A

         \[\leadsto \mathsf{neg.f32}\left(\mathsf{*.f32}\left(\log \left(1 - u0\right), \left(\frac{{alphay}^{2}}{sin2phi}\right)\right)\right) \]

      6. sub-neg N/A

         \[\leadsto \mathsf{neg.f32}\left(\mathsf{*.f32}\left(\log \left(1 + \left(\mathsf{neg}\left(u0\right)\right)\right), \left(\frac{{alphay}^{2}}{sin2phi}\right)\right)\right) \]

      7. mul-1-neg N/A

         \[\leadsto \mathsf{neg.f32}\left(\mathsf{*.f32}\left(\log \left(1 + -1 \cdot u0\right), \left(\frac{{alphay}^{2}}{sin2phi}\right)\right)\right) \]

      8. log1p-define N/A

         \[\leadsto \mathsf{neg.f32}\left(\mathsf{*.f32}\left(\left(\mathsf{log1p}\left(-1 \cdot u0\right)\right), \left(\frac{{alphay}^{2}}{sin2phi}\right)\right)\right) \]

      9. log1p-lowering-log1p.f32 N/A

         \[\leadsto \mathsf{neg.f32}\left(\mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\left(-1 \cdot u0\right)\right), \left(\frac{{alphay}^{2}}{sin2phi}\right)\right)\right) \]

      10. mul-1-neg N/A

          \[\leadsto \mathsf{neg.f32}\left(\mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\left(\mathsf{neg}\left(u0\right)\right)\right), \left(\frac{{alphay}^{2}}{sin2phi}\right)\right)\right) \]

      11. neg-lowering-neg.f32 N/A

          \[\leadsto \mathsf{neg.f32}\left(\mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \left(\frac{{alphay}^{2}}{sin2phi}\right)\right)\right) \]

      12. /-lowering-/.f32 N/A

          \[\leadsto \mathsf{neg.f32}\left(\mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{/.f32}\left(\left({alphay}^{2}\right), sin2phi\right)\right)\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{neg.f32}\left(\mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{/.f32}\left(\left(alphay \cdot alphay\right), sin2phi\right)\right)\right) \]

      14. *-lowering-*.f32 97.0%

          \[\leadsto \mathsf{neg.f32}\left(\mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{/.f32}\left(\mathsf{*.f32}\left(alphay, alphay\right), sin2phi\right)\right)\right) \]

   7. Simplified 97.0%

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

   8. Taylor expanded in u0 around 0

      \[\leadsto \mathsf{neg.f32}\left(\mathsf{*.f32}\left(\color{blue}{\left(u0 \cdot \left(u0 \cdot \left(\frac{-1}{3} \cdot u0 - \frac{1}{2}\right) - 1\right)\right)}, \mathsf{/.f32}\left(\mathsf{*.f32}\left(alphay, alphay\right), sin2phi\right)\right)\right) \]
   9. Step-by-step derivation

      1. *-lowering-*.f32 N/A

         \[\leadsto \mathsf{neg.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(u0, \left(u0 \cdot \left(\frac{-1}{3} \cdot u0 - \frac{1}{2}\right) - 1\right)\right), \mathsf{/.f32}\left(\mathsf{*.f32}\left(alphay, alphay\right), sin2phi\right)\right)\right) \]

      2. sub-neg N/A

         \[\leadsto \mathsf{neg.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(u0, \left(u0 \cdot \left(\frac{-1}{3} \cdot u0 - \frac{1}{2}\right) + \left(\mathsf{neg}\left(1\right)\right)\right)\right), \mathsf{/.f32}\left(\mathsf{*.f32}\left(alphay, alphay\right), sin2phi\right)\right)\right) \]

      3. metadata-eval N/A

         \[\leadsto \mathsf{neg.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(u0, \left(u0 \cdot \left(\frac{-1}{3} \cdot u0 - \frac{1}{2}\right) + -1\right)\right), \mathsf{/.f32}\left(\mathsf{*.f32}\left(alphay, alphay\right), sin2phi\right)\right)\right) \]

      4. +-lowering-+.f32 N/A

         \[\leadsto \mathsf{neg.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\left(u0 \cdot \left(\frac{-1}{3} \cdot u0 - \frac{1}{2}\right)\right), -1\right)\right), \mathsf{/.f32}\left(\mathsf{*.f32}\left(alphay, alphay\right), sin2phi\right)\right)\right) \]

      5. *-lowering-*.f32 N/A

         \[\leadsto \mathsf{neg.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{*.f32}\left(u0, \left(\frac{-1}{3} \cdot u0 - \frac{1}{2}\right)\right), -1\right)\right), \mathsf{/.f32}\left(\mathsf{*.f32}\left(alphay, alphay\right), sin2phi\right)\right)\right) \]

      6. sub-neg N/A

         \[\leadsto \mathsf{neg.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{*.f32}\left(u0, \left(\frac{-1}{3} \cdot u0 + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right)\right), -1\right)\right), \mathsf{/.f32}\left(\mathsf{*.f32}\left(alphay, alphay\right), sin2phi\right)\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{neg.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{*.f32}\left(u0, \left(\frac{-1}{3} \cdot u0 + \frac{-1}{2}\right)\right), -1\right)\right), \mathsf{/.f32}\left(\mathsf{*.f32}\left(alphay, alphay\right), sin2phi\right)\right)\right) \]

      8. +-lowering-+.f32 N/A

         \[\leadsto \mathsf{neg.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\left(\frac{-1}{3} \cdot u0\right), \frac{-1}{2}\right)\right), -1\right)\right), \mathsf{/.f32}\left(\mathsf{*.f32}\left(alphay, alphay\right), sin2phi\right)\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{neg.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\left(u0 \cdot \frac{-1}{3}\right), \frac{-1}{2}\right)\right), -1\right)\right), \mathsf{/.f32}\left(\mathsf{*.f32}\left(alphay, alphay\right), sin2phi\right)\right)\right) \]

      10. *-lowering-*.f32 90.0%

          \[\leadsto \mathsf{neg.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{*.f32}\left(u0, \frac{-1}{3}\right), \frac{-1}{2}\right)\right), -1\right)\right), \mathsf{/.f32}\left(\mathsf{*.f32}\left(alphay, alphay\right), sin2phi\right)\right)\right) \]

   10. Simplified 90.0%

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

4. Final simplification 81.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;sin2phi \leq 4.999999969612645 \cdot 10^{-9}:\\ \;\;\;\;\frac{u0 \cdot u0}{u0 \cdot \left(\frac{\frac{sin2phi}{alphay}}{alphay} + \frac{\frac{cos2phi}{alphax}}{alphax}\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(u0 \cdot \left(-1 + u0 \cdot \left(-0.5 + u0 \cdot -0.3333333333333333\right)\right)\right) \cdot \left(-\frac{alphay \cdot alphay}{sin2phi}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 82.8% accurate, 5.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;sin2phi \leq 4.999999969612645 \cdot 10^{-9}:\\ \;\;\;\;\frac{u0 \cdot u0}{u0 \cdot \left(\frac{\frac{sin2phi}{alphay}}{alphay} + \frac{\frac{cos2phi}{alphax}}{alphax}\right)}\\ \mathbf{else}:\\ \;\;\;\;alphay \cdot \left(\frac{alphay}{sin2phi} \cdot \left(u0 \cdot \left(\left(--1\right) - u0 \cdot \left(-0.5 + u0 \cdot -0.3333333333333333\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (if (<= sin2phi 4.999999969612645e-9)
   (/
    (* u0 u0)
    (* u0 (+ (/ (/ sin2phi alphay) alphay) (/ (/ cos2phi alphax) alphax))))
   (*
    alphay
    (*
     (/ alphay sin2phi)
     (* u0 (- (- -1.0) (* u0 (+ -0.5 (* u0 -0.3333333333333333)))))))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float tmp;
	if (sin2phi <= 4.999999969612645e-9f) {
		tmp = (u0 * u0) / (u0 * (((sin2phi / alphay) / alphay) + ((cos2phi / alphax) / alphax)));
	} else {
		tmp = alphay * ((alphay / sin2phi) * (u0 * (-(-1.0f) - (u0 * (-0.5f + (u0 * -0.3333333333333333f))))));
	}
	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 <= 4.999999969612645e-9) then
        tmp = (u0 * u0) / (u0 * (((sin2phi / alphay) / alphay) + ((cos2phi / alphax) / alphax)))
    else
        tmp = alphay * ((alphay / sin2phi) * (u0 * (-(-1.0e0) - (u0 * ((-0.5e0) + (u0 * (-0.3333333333333333e0)))))))
    end if
    code = tmp
end function

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = Float32(0.0)
	if (sin2phi <= Float32(4.999999969612645e-9))
		tmp = Float32(Float32(u0 * u0) / Float32(u0 * Float32(Float32(Float32(sin2phi / alphay) / alphay) + Float32(Float32(cos2phi / alphax) / alphax))));
	else
		tmp = Float32(alphay * Float32(Float32(alphay / sin2phi) * Float32(u0 * Float32(Float32(-Float32(-1.0)) - Float32(u0 * Float32(Float32(-0.5) + Float32(u0 * Float32(-0.3333333333333333))))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = single(0.0);
	if (sin2phi <= single(4.999999969612645e-9))
		tmp = (u0 * u0) / (u0 * (((sin2phi / alphay) / alphay) + ((cos2phi / alphax) / alphax)));
	else
		tmp = alphay * ((alphay / sin2phi) * (u0 * (-single(-1.0) - (u0 * (single(-0.5) + (u0 * single(-0.3333333333333333)))))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if sin2phi < 4.99999997e-9

   1. Initial program 64.5%

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

      1. distribute-frac-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. /-lowering-/.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(\log \left(1 - u0\right), \color{blue}{\left(\mathsf{neg}\left(\left(\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\right)\right)\right)}\right) \]

      4. sub-neg N/A

         \[\leadsto \mathsf{/.f32}\left(\log \left(1 + \left(\mathsf{neg}\left(u0\right)\right)\right), \left(\mathsf{neg}\left(\left(\color{blue}{\frac{cos2phi}{alphax \cdot alphax}} + \frac{sin2phi}{alphay \cdot alphay}\right)\right)\right)\right) \]

      5. log1p-define N/A

         \[\leadsto \mathsf{/.f32}\left(\left(\mathsf{log1p}\left(\mathsf{neg}\left(u0\right)\right)\right), \left(\mathsf{neg}\left(\color{blue}{\left(\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\right)}\right)\right)\right) \]

      6. log1p-lowering-log1p.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\left(\mathsf{neg}\left(u0\right)\right)\right), \left(\mathsf{neg}\left(\color{blue}{\left(\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\right)}\right)\right)\right) \]

      7. neg-lowering-neg.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \left(\mathsf{neg}\left(\left(\color{blue}{\frac{cos2phi}{alphax \cdot alphax}} + \frac{sin2phi}{alphay \cdot alphay}\right)\right)\right)\right) \]

      8. distribute-neg-in N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \left(\left(\mathsf{neg}\left(\frac{cos2phi}{alphax \cdot alphax}\right)\right) + \color{blue}{\left(\mathsf{neg}\left(\frac{sin2phi}{alphay \cdot alphay}\right)\right)}\right)\right) \]

      9. unsub-neg N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \left(\left(\mathsf{neg}\left(\frac{cos2phi}{alphax \cdot alphax}\right)\right) - \color{blue}{\frac{sin2phi}{alphay \cdot alphay}}\right)\right) \]

      10. --lowering--.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\left(\mathsf{neg}\left(\frac{cos2phi}{alphax \cdot alphax}\right)\right), \color{blue}{\left(\frac{sin2phi}{alphay \cdot alphay}\right)}\right)\right) \]

      11. neg-sub0 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\left(0 - \frac{cos2phi}{alphax \cdot alphax}\right), \left(\frac{\color{blue}{sin2phi}}{alphay \cdot alphay}\right)\right)\right) \]

      12. --lowering--.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \left(\frac{cos2phi}{alphax \cdot alphax}\right)\right), \left(\frac{\color{blue}{sin2phi}}{alphay \cdot alphay}\right)\right)\right) \]

      13. /-lowering-/.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(cos2phi, \left(alphax \cdot alphax\right)\right)\right), \left(\frac{sin2phi}{alphay \cdot alphay}\right)\right)\right) \]

      14. *-lowering-*.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right)\right), \left(\frac{sin2phi}{alphay \cdot alphay}\right)\right)\right) \]

      15. /-lowering-/.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right)\right), \mathsf{/.f32}\left(sin2phi, \color{blue}{\left(alphay \cdot alphay\right)}\right)\right)\right) \]

      16. *-lowering-*.f32 98.6%

          \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right)\right), \mathsf{/.f32}\left(sin2phi, \mathsf{*.f32}\left(alphay, \color{blue}{alphay}\right)\right)\right)\right) \]

   3. Simplified 98.6%

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

   5. Taylor expanded in u0 around 0

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

      1. /-lowering-/.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(u0, \color{blue}{\left(\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}\right)}\right) \]

      2. +-lowering-+.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{+.f32}\left(\left(\frac{cos2phi}{{alphax}^{2}}\right), \color{blue}{\left(\frac{sin2phi}{{alphay}^{2}}\right)}\right)\right) \]

      3. /-lowering-/.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{/.f32}\left(cos2phi, \left({alphax}^{2}\right)\right), \left(\frac{\color{blue}{sin2phi}}{{alphay}^{2}}\right)\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{/.f32}\left(cos2phi, \left(alphax \cdot alphax\right)\right), \left(\frac{sin2phi}{{alphay}^{2}}\right)\right)\right) \]

      5. *-lowering-*.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right), \left(\frac{sin2phi}{{alphay}^{2}}\right)\right)\right) \]

      6. /-lowering-/.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right), \mathsf{/.f32}\left(sin2phi, \color{blue}{\left({alphay}^{2}\right)}\right)\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right), \mathsf{/.f32}\left(sin2phi, \left(alphay \cdot \color{blue}{alphay}\right)\right)\right)\right) \]

      8. *-lowering-*.f32 68.0%

         \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right), \mathsf{/.f32}\left(sin2phi, \mathsf{*.f32}\left(alphay, \color{blue}{alphay}\right)\right)\right)\right) \]

   7. Simplified 68.0%

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

   9. Applied egg-rr 68.3%

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

   if 4.99999997e-9 < sin2phi

   1. Initial program 65.0%

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

      1. distribute-frac-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. /-lowering-/.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(\log \left(1 - u0\right), \color{blue}{\left(\mathsf{neg}\left(\left(\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\right)\right)\right)}\right) \]

      4. sub-neg N/A

         \[\leadsto \mathsf{/.f32}\left(\log \left(1 + \left(\mathsf{neg}\left(u0\right)\right)\right), \left(\mathsf{neg}\left(\left(\color{blue}{\frac{cos2phi}{alphax \cdot alphax}} + \frac{sin2phi}{alphay \cdot alphay}\right)\right)\right)\right) \]

      5. log1p-define N/A

         \[\leadsto \mathsf{/.f32}\left(\left(\mathsf{log1p}\left(\mathsf{neg}\left(u0\right)\right)\right), \left(\mathsf{neg}\left(\color{blue}{\left(\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\right)}\right)\right)\right) \]

      6. log1p-lowering-log1p.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\left(\mathsf{neg}\left(u0\right)\right)\right), \left(\mathsf{neg}\left(\color{blue}{\left(\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\right)}\right)\right)\right) \]

      7. neg-lowering-neg.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \left(\mathsf{neg}\left(\left(\color{blue}{\frac{cos2phi}{alphax \cdot alphax}} + \frac{sin2phi}{alphay \cdot alphay}\right)\right)\right)\right) \]

      8. distribute-neg-in N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \left(\left(\mathsf{neg}\left(\frac{cos2phi}{alphax \cdot alphax}\right)\right) + \color{blue}{\left(\mathsf{neg}\left(\frac{sin2phi}{alphay \cdot alphay}\right)\right)}\right)\right) \]

      9. unsub-neg N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \left(\left(\mathsf{neg}\left(\frac{cos2phi}{alphax \cdot alphax}\right)\right) - \color{blue}{\frac{sin2phi}{alphay \cdot alphay}}\right)\right) \]

      10. --lowering--.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\left(\mathsf{neg}\left(\frac{cos2phi}{alphax \cdot alphax}\right)\right), \color{blue}{\left(\frac{sin2phi}{alphay \cdot alphay}\right)}\right)\right) \]

      11. neg-sub0 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\left(0 - \frac{cos2phi}{alphax \cdot alphax}\right), \left(\frac{\color{blue}{sin2phi}}{alphay \cdot alphay}\right)\right)\right) \]

      12. --lowering--.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \left(\frac{cos2phi}{alphax \cdot alphax}\right)\right), \left(\frac{\color{blue}{sin2phi}}{alphay \cdot alphay}\right)\right)\right) \]

      13. /-lowering-/.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(cos2phi, \left(alphax \cdot alphax\right)\right)\right), \left(\frac{sin2phi}{alphay \cdot alphay}\right)\right)\right) \]

      14. *-lowering-*.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right)\right), \left(\frac{sin2phi}{alphay \cdot alphay}\right)\right)\right) \]

      15. /-lowering-/.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right)\right), \mathsf{/.f32}\left(sin2phi, \color{blue}{\left(alphay \cdot alphay\right)}\right)\right)\right) \]

      16. *-lowering-*.f32 98.0%

          \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right)\right), \mathsf{/.f32}\left(sin2phi, \mathsf{*.f32}\left(alphay, \color{blue}{alphay}\right)\right)\right)\right) \]

   3. Simplified 98.0%

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

   5. Taylor expanded in cos2phi around 0

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

      1. mul-1-neg N/A

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

      2. neg-lowering-neg.f32 N/A

         \[\leadsto \mathsf{neg.f32}\left(\left(\frac{{alphay}^{2} \cdot \log \left(1 - u0\right)}{sin2phi}\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{neg.f32}\left(\left(\frac{\log \left(1 - u0\right) \cdot {alphay}^{2}}{sin2phi}\right)\right) \]

      4. associate-/l* N/A

         \[\leadsto \mathsf{neg.f32}\left(\left(\log \left(1 - u0\right) \cdot \frac{{alphay}^{2}}{sin2phi}\right)\right) \]

      5. *-lowering-*.f32 N/A

         \[\leadsto \mathsf{neg.f32}\left(\mathsf{*.f32}\left(\log \left(1 - u0\right), \left(\frac{{alphay}^{2}}{sin2phi}\right)\right)\right) \]

      6. sub-neg N/A

         \[\leadsto \mathsf{neg.f32}\left(\mathsf{*.f32}\left(\log \left(1 + \left(\mathsf{neg}\left(u0\right)\right)\right), \left(\frac{{alphay}^{2}}{sin2phi}\right)\right)\right) \]

      7. mul-1-neg N/A

         \[\leadsto \mathsf{neg.f32}\left(\mathsf{*.f32}\left(\log \left(1 + -1 \cdot u0\right), \left(\frac{{alphay}^{2}}{sin2phi}\right)\right)\right) \]

      8. log1p-define N/A

         \[\leadsto \mathsf{neg.f32}\left(\mathsf{*.f32}\left(\left(\mathsf{log1p}\left(-1 \cdot u0\right)\right), \left(\frac{{alphay}^{2}}{sin2phi}\right)\right)\right) \]

      9. log1p-lowering-log1p.f32 N/A

         \[\leadsto \mathsf{neg.f32}\left(\mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\left(-1 \cdot u0\right)\right), \left(\frac{{alphay}^{2}}{sin2phi}\right)\right)\right) \]

      10. mul-1-neg N/A

          \[\leadsto \mathsf{neg.f32}\left(\mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\left(\mathsf{neg}\left(u0\right)\right)\right), \left(\frac{{alphay}^{2}}{sin2phi}\right)\right)\right) \]

      11. neg-lowering-neg.f32 N/A

          \[\leadsto \mathsf{neg.f32}\left(\mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \left(\frac{{alphay}^{2}}{sin2phi}\right)\right)\right) \]

      12. /-lowering-/.f32 N/A

          \[\leadsto \mathsf{neg.f32}\left(\mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{/.f32}\left(\left({alphay}^{2}\right), sin2phi\right)\right)\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{neg.f32}\left(\mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{/.f32}\left(\left(alphay \cdot alphay\right), sin2phi\right)\right)\right) \]

      14. *-lowering-*.f32 97.0%

          \[\leadsto \mathsf{neg.f32}\left(\mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{/.f32}\left(\mathsf{*.f32}\left(alphay, alphay\right), sin2phi\right)\right)\right) \]

   7. Simplified 97.0%

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

      1. *-commutative N/A

         \[\leadsto \mathsf{neg.f32}\left(\left(\frac{alphay \cdot alphay}{sin2phi} \cdot \log \left(1 + \left(\mathsf{neg}\left(u0\right)\right)\right)\right)\right) \]

      2. associate-/l* N/A

         \[\leadsto \mathsf{neg.f32}\left(\left(\left(alphay \cdot \frac{alphay}{sin2phi}\right) \cdot \log \left(1 + \left(\mathsf{neg}\left(u0\right)\right)\right)\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{neg.f32}\left(\left(alphay \cdot \left(\frac{alphay}{sin2phi} \cdot \log \left(1 + \left(\mathsf{neg}\left(u0\right)\right)\right)\right)\right)\right) \]

      4. *-lowering-*.f32 N/A

         \[\leadsto \mathsf{neg.f32}\left(\mathsf{*.f32}\left(alphay, \left(\frac{alphay}{sin2phi} \cdot \log \left(1 + \left(\mathsf{neg}\left(u0\right)\right)\right)\right)\right)\right) \]

      5. *-lowering-*.f32 N/A

         \[\leadsto \mathsf{neg.f32}\left(\mathsf{*.f32}\left(alphay, \mathsf{*.f32}\left(\left(\frac{alphay}{sin2phi}\right), \log \left(1 + \left(\mathsf{neg}\left(u0\right)\right)\right)\right)\right)\right) \]

      6. /-lowering-/.f32 N/A

         \[\leadsto \mathsf{neg.f32}\left(\mathsf{*.f32}\left(alphay, \mathsf{*.f32}\left(\mathsf{/.f32}\left(alphay, sin2phi\right), \log \left(1 + \left(\mathsf{neg}\left(u0\right)\right)\right)\right)\right)\right) \]

      7. log1p-define N/A

         \[\leadsto \mathsf{neg.f32}\left(\mathsf{*.f32}\left(alphay, \mathsf{*.f32}\left(\mathsf{/.f32}\left(alphay, sin2phi\right), \left(\mathsf{log1p}\left(\mathsf{neg}\left(u0\right)\right)\right)\right)\right)\right) \]

      8. log1p-lowering-log1p.f32 N/A

         \[\leadsto \mathsf{neg.f32}\left(\mathsf{*.f32}\left(alphay, \mathsf{*.f32}\left(\mathsf{/.f32}\left(alphay, sin2phi\right), \mathsf{log1p.f32}\left(\left(\mathsf{neg}\left(u0\right)\right)\right)\right)\right)\right) \]

      9. neg-lowering-neg.f32 96.9%

         \[\leadsto \mathsf{neg.f32}\left(\mathsf{*.f32}\left(alphay, \mathsf{*.f32}\left(\mathsf{/.f32}\left(alphay, sin2phi\right), \mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right)\right)\right)\right) \]

   9. Applied egg-rr 96.9%

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

   10. Taylor expanded in u0 around 0

       \[\leadsto \mathsf{neg.f32}\left(\mathsf{*.f32}\left(alphay, \mathsf{*.f32}\left(\mathsf{/.f32}\left(alphay, sin2phi\right), \color{blue}{\left(u0 \cdot \left(u0 \cdot \left(\frac{-1}{3} \cdot u0 - \frac{1}{2}\right) - 1\right)\right)}\right)\right)\right) \]
   11. Step-by-step derivation

       1. *-lowering-*.f32 N/A

          \[\leadsto \mathsf{neg.f32}\left(\mathsf{*.f32}\left(alphay, \mathsf{*.f32}\left(\mathsf{/.f32}\left(alphay, sin2phi\right), \mathsf{*.f32}\left(u0, \left(u0 \cdot \left(\frac{-1}{3} \cdot u0 - \frac{1}{2}\right) - 1\right)\right)\right)\right)\right) \]

       2. sub-neg N/A

          \[\leadsto \mathsf{neg.f32}\left(\mathsf{*.f32}\left(alphay, \mathsf{*.f32}\left(\mathsf{/.f32}\left(alphay, sin2phi\right), \mathsf{*.f32}\left(u0, \left(u0 \cdot \left(\frac{-1}{3} \cdot u0 - \frac{1}{2}\right) + \left(\mathsf{neg}\left(1\right)\right)\right)\right)\right)\right)\right) \]

       3. metadata-eval N/A

          \[\leadsto \mathsf{neg.f32}\left(\mathsf{*.f32}\left(alphay, \mathsf{*.f32}\left(\mathsf{/.f32}\left(alphay, sin2phi\right), \mathsf{*.f32}\left(u0, \left(u0 \cdot \left(\frac{-1}{3} \cdot u0 - \frac{1}{2}\right) + -1\right)\right)\right)\right)\right) \]

       4. +-lowering-+.f32 N/A

          \[\leadsto \mathsf{neg.f32}\left(\mathsf{*.f32}\left(alphay, \mathsf{*.f32}\left(\mathsf{/.f32}\left(alphay, sin2phi\right), \mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\left(u0 \cdot \left(\frac{-1}{3} \cdot u0 - \frac{1}{2}\right)\right), -1\right)\right)\right)\right)\right) \]

       5. *-lowering-*.f32 N/A

          \[\leadsto \mathsf{neg.f32}\left(\mathsf{*.f32}\left(alphay, \mathsf{*.f32}\left(\mathsf{/.f32}\left(alphay, sin2phi\right), \mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{*.f32}\left(u0, \left(\frac{-1}{3} \cdot u0 - \frac{1}{2}\right)\right), -1\right)\right)\right)\right)\right) \]

       6. sub-neg N/A

          \[\leadsto \mathsf{neg.f32}\left(\mathsf{*.f32}\left(alphay, \mathsf{*.f32}\left(\mathsf{/.f32}\left(alphay, sin2phi\right), \mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{*.f32}\left(u0, \left(\frac{-1}{3} \cdot u0 + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right)\right), -1\right)\right)\right)\right)\right) \]

       7. metadata-eval N/A

          \[\leadsto \mathsf{neg.f32}\left(\mathsf{*.f32}\left(alphay, \mathsf{*.f32}\left(\mathsf{/.f32}\left(alphay, sin2phi\right), \mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{*.f32}\left(u0, \left(\frac{-1}{3} \cdot u0 + \frac{-1}{2}\right)\right), -1\right)\right)\right)\right)\right) \]

       8. +-lowering-+.f32 N/A

          \[\leadsto \mathsf{neg.f32}\left(\mathsf{*.f32}\left(alphay, \mathsf{*.f32}\left(\mathsf{/.f32}\left(alphay, sin2phi\right), \mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\left(\frac{-1}{3} \cdot u0\right), \frac{-1}{2}\right)\right), -1\right)\right)\right)\right)\right) \]

       9. *-commutative N/A

          \[\leadsto \mathsf{neg.f32}\left(\mathsf{*.f32}\left(alphay, \mathsf{*.f32}\left(\mathsf{/.f32}\left(alphay, sin2phi\right), \mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\left(u0 \cdot \frac{-1}{3}\right), \frac{-1}{2}\right)\right), -1\right)\right)\right)\right)\right) \]

       10. *-lowering-*.f32 89.9%

           \[\leadsto \mathsf{neg.f32}\left(\mathsf{*.f32}\left(alphay, \mathsf{*.f32}\left(\mathsf{/.f32}\left(alphay, sin2phi\right), \mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{*.f32}\left(u0, \frac{-1}{3}\right), \frac{-1}{2}\right)\right), -1\right)\right)\right)\right)\right) \]

   12. Simplified 89.9%

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

4. Final simplification 81.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;sin2phi \leq 4.999999969612645 \cdot 10^{-9}:\\ \;\;\;\;\frac{u0 \cdot u0}{u0 \cdot \left(\frac{\frac{sin2phi}{alphay}}{alphay} + \frac{\frac{cos2phi}{alphax}}{alphax}\right)}\\ \mathbf{else}:\\ \;\;\;\;alphay \cdot \left(\frac{alphay}{sin2phi} \cdot \left(u0 \cdot \left(\left(--1\right) - u0 \cdot \left(-0.5 + u0 \cdot -0.3333333333333333\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 90.8% accurate, 5.0× speedup

Math

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

FPCore

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

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	return (u0 * (1.0f + (u0 * (0.5f + (u0 * 0.3333333333333333f))))) / ((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 = (u0 * (1.0e0 + (u0 * (0.5e0 + (u0 * 0.3333333333333333e0))))) / ((cos2phi / (alphax * alphax)) + (sin2phi / (alphay * 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(cos2phi / Float32(alphax * alphax)) + Float32(sin2phi / Float32(alphay * alphay))))
end

MATLAB

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

Derivation

1. Initial program 64.8%

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

3. Taylor expanded in u0 around 0

   \[\leadsto \mathsf{/.f32}\left(\color{blue}{\left(u0 \cdot \left(1 + u0 \cdot \left(\frac{1}{2} + \frac{1}{3} \cdot u0\right)\right)\right)}, \mathsf{+.f32}\left(\mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right), \mathsf{/.f32}\left(sin2phi, \mathsf{*.f32}\left(alphay, alphay\right)\right)\right)\right) \]
4. Step-by-step derivation

   1. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(u0, \left(1 + u0 \cdot \left(\frac{1}{2} + \frac{1}{3} \cdot u0\right)\right)\right), \mathsf{+.f32}\left(\color{blue}{\mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right)}, \mathsf{/.f32}\left(sin2phi, \mathsf{*.f32}\left(alphay, alphay\right)\right)\right)\right) \]

   2. +-lowering-+.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(1, \left(u0 \cdot \left(\frac{1}{2} + \frac{1}{3} \cdot u0\right)\right)\right)\right), \mathsf{+.f32}\left(\mathsf{/.f32}\left(cos2phi, \color{blue}{\mathsf{*.f32}\left(alphax, alphax\right)}\right), \mathsf{/.f32}\left(sin2phi, \mathsf{*.f32}\left(alphay, alphay\right)\right)\right)\right) \]

   3. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(u0, \left(\frac{1}{2} + \frac{1}{3} \cdot u0\right)\right)\right)\right), \mathsf{+.f32}\left(\mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, \color{blue}{alphax}\right)\right), \mathsf{/.f32}\left(sin2phi, \mathsf{*.f32}\left(alphay, alphay\right)\right)\right)\right) \]

   4. +-lowering-+.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\frac{1}{2}, \left(\frac{1}{3} \cdot u0\right)\right)\right)\right)\right), \mathsf{+.f32}\left(\mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right), \mathsf{/.f32}\left(sin2phi, \mathsf{*.f32}\left(alphay, alphay\right)\right)\right)\right) \]

   5. *-commutative N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\frac{1}{2}, \left(u0 \cdot \frac{1}{3}\right)\right)\right)\right)\right), \mathsf{+.f32}\left(\mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right), \mathsf{/.f32}\left(sin2phi, \mathsf{*.f32}\left(alphay, alphay\right)\right)\right)\right) \]

   6. *-lowering-*.f32 89.3%

      \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\frac{1}{2}, \mathsf{*.f32}\left(u0, \frac{1}{3}\right)\right)\right)\right)\right), \mathsf{+.f32}\left(\mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right), \mathsf{/.f32}\left(sin2phi, \mathsf{*.f32}\left(alphay, alphay\right)\right)\right)\right) \]

5. Simplified 89.3%

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

Alternative 16: 80.7% accurate, 5.3× speedup

Math

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

FPCore

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

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float tmp;
	if (sin2phi <= 4.999999969612645e-9f) {
		tmp = (u0 * u0) / (u0 * (((sin2phi / alphay) / alphay) + ((cos2phi / alphax) / alphax)));
	} else {
		tmp = u0 * (((alphay * alphay) / sin2phi) + ((0.5f * (u0 * (alphay * 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 <= 4.999999969612645e-9) then
        tmp = (u0 * u0) / (u0 * (((sin2phi / alphay) / alphay) + ((cos2phi / alphax) / alphax)))
    else
        tmp = u0 * (((alphay * alphay) / sin2phi) + ((0.5e0 * (u0 * (alphay * alphay))) / sin2phi))
    end if
    code = tmp
end function

Julia

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

MATLAB

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

Derivation

1. Split input into 2 regimes

2. if sin2phi < 4.99999997e-9

   1. Initial program 64.5%

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

      1. distribute-frac-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. /-lowering-/.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(\log \left(1 - u0\right), \color{blue}{\left(\mathsf{neg}\left(\left(\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\right)\right)\right)}\right) \]

      4. sub-neg N/A

         \[\leadsto \mathsf{/.f32}\left(\log \left(1 + \left(\mathsf{neg}\left(u0\right)\right)\right), \left(\mathsf{neg}\left(\left(\color{blue}{\frac{cos2phi}{alphax \cdot alphax}} + \frac{sin2phi}{alphay \cdot alphay}\right)\right)\right)\right) \]

      5. log1p-define N/A

         \[\leadsto \mathsf{/.f32}\left(\left(\mathsf{log1p}\left(\mathsf{neg}\left(u0\right)\right)\right), \left(\mathsf{neg}\left(\color{blue}{\left(\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\right)}\right)\right)\right) \]

      6. log1p-lowering-log1p.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\left(\mathsf{neg}\left(u0\right)\right)\right), \left(\mathsf{neg}\left(\color{blue}{\left(\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\right)}\right)\right)\right) \]

      7. neg-lowering-neg.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \left(\mathsf{neg}\left(\left(\color{blue}{\frac{cos2phi}{alphax \cdot alphax}} + \frac{sin2phi}{alphay \cdot alphay}\right)\right)\right)\right) \]

      8. distribute-neg-in N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \left(\left(\mathsf{neg}\left(\frac{cos2phi}{alphax \cdot alphax}\right)\right) + \color{blue}{\left(\mathsf{neg}\left(\frac{sin2phi}{alphay \cdot alphay}\right)\right)}\right)\right) \]

      9. unsub-neg N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \left(\left(\mathsf{neg}\left(\frac{cos2phi}{alphax \cdot alphax}\right)\right) - \color{blue}{\frac{sin2phi}{alphay \cdot alphay}}\right)\right) \]

      10. --lowering--.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\left(\mathsf{neg}\left(\frac{cos2phi}{alphax \cdot alphax}\right)\right), \color{blue}{\left(\frac{sin2phi}{alphay \cdot alphay}\right)}\right)\right) \]

      11. neg-sub0 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\left(0 - \frac{cos2phi}{alphax \cdot alphax}\right), \left(\frac{\color{blue}{sin2phi}}{alphay \cdot alphay}\right)\right)\right) \]

      12. --lowering--.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \left(\frac{cos2phi}{alphax \cdot alphax}\right)\right), \left(\frac{\color{blue}{sin2phi}}{alphay \cdot alphay}\right)\right)\right) \]

      13. /-lowering-/.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(cos2phi, \left(alphax \cdot alphax\right)\right)\right), \left(\frac{sin2phi}{alphay \cdot alphay}\right)\right)\right) \]

      14. *-lowering-*.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right)\right), \left(\frac{sin2phi}{alphay \cdot alphay}\right)\right)\right) \]

      15. /-lowering-/.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right)\right), \mathsf{/.f32}\left(sin2phi, \color{blue}{\left(alphay \cdot alphay\right)}\right)\right)\right) \]

      16. *-lowering-*.f32 98.6%

          \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right)\right), \mathsf{/.f32}\left(sin2phi, \mathsf{*.f32}\left(alphay, \color{blue}{alphay}\right)\right)\right)\right) \]

   3. Simplified 98.6%

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

   5. Taylor expanded in u0 around 0

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

      1. /-lowering-/.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(u0, \color{blue}{\left(\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}\right)}\right) \]

      2. +-lowering-+.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{+.f32}\left(\left(\frac{cos2phi}{{alphax}^{2}}\right), \color{blue}{\left(\frac{sin2phi}{{alphay}^{2}}\right)}\right)\right) \]

      3. /-lowering-/.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{/.f32}\left(cos2phi, \left({alphax}^{2}\right)\right), \left(\frac{\color{blue}{sin2phi}}{{alphay}^{2}}\right)\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{/.f32}\left(cos2phi, \left(alphax \cdot alphax\right)\right), \left(\frac{sin2phi}{{alphay}^{2}}\right)\right)\right) \]

      5. *-lowering-*.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right), \left(\frac{sin2phi}{{alphay}^{2}}\right)\right)\right) \]

      6. /-lowering-/.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right), \mathsf{/.f32}\left(sin2phi, \color{blue}{\left({alphay}^{2}\right)}\right)\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right), \mathsf{/.f32}\left(sin2phi, \left(alphay \cdot \color{blue}{alphay}\right)\right)\right)\right) \]

      8. *-lowering-*.f32 68.0%

         \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right), \mathsf{/.f32}\left(sin2phi, \mathsf{*.f32}\left(alphay, \color{blue}{alphay}\right)\right)\right)\right) \]

   7. Simplified 68.0%

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

   9. Applied egg-rr 68.3%

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

   if 4.99999997e-9 < sin2phi

   1. Initial program 65.0%

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

      1. distribute-frac-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. /-lowering-/.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(\log \left(1 - u0\right), \color{blue}{\left(\mathsf{neg}\left(\left(\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\right)\right)\right)}\right) \]

      4. sub-neg N/A

         \[\leadsto \mathsf{/.f32}\left(\log \left(1 + \left(\mathsf{neg}\left(u0\right)\right)\right), \left(\mathsf{neg}\left(\left(\color{blue}{\frac{cos2phi}{alphax \cdot alphax}} + \frac{sin2phi}{alphay \cdot alphay}\right)\right)\right)\right) \]

      5. log1p-define N/A

         \[\leadsto \mathsf{/.f32}\left(\left(\mathsf{log1p}\left(\mathsf{neg}\left(u0\right)\right)\right), \left(\mathsf{neg}\left(\color{blue}{\left(\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\right)}\right)\right)\right) \]

      6. log1p-lowering-log1p.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\left(\mathsf{neg}\left(u0\right)\right)\right), \left(\mathsf{neg}\left(\color{blue}{\left(\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\right)}\right)\right)\right) \]

      7. neg-lowering-neg.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \left(\mathsf{neg}\left(\left(\color{blue}{\frac{cos2phi}{alphax \cdot alphax}} + \frac{sin2phi}{alphay \cdot alphay}\right)\right)\right)\right) \]

      8. distribute-neg-in N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \left(\left(\mathsf{neg}\left(\frac{cos2phi}{alphax \cdot alphax}\right)\right) + \color{blue}{\left(\mathsf{neg}\left(\frac{sin2phi}{alphay \cdot alphay}\right)\right)}\right)\right) \]

      9. unsub-neg N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \left(\left(\mathsf{neg}\left(\frac{cos2phi}{alphax \cdot alphax}\right)\right) - \color{blue}{\frac{sin2phi}{alphay \cdot alphay}}\right)\right) \]

      10. --lowering--.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\left(\mathsf{neg}\left(\frac{cos2phi}{alphax \cdot alphax}\right)\right), \color{blue}{\left(\frac{sin2phi}{alphay \cdot alphay}\right)}\right)\right) \]

      11. neg-sub0 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\left(0 - \frac{cos2phi}{alphax \cdot alphax}\right), \left(\frac{\color{blue}{sin2phi}}{alphay \cdot alphay}\right)\right)\right) \]

      12. --lowering--.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \left(\frac{cos2phi}{alphax \cdot alphax}\right)\right), \left(\frac{\color{blue}{sin2phi}}{alphay \cdot alphay}\right)\right)\right) \]

      13. /-lowering-/.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(cos2phi, \left(alphax \cdot alphax\right)\right)\right), \left(\frac{sin2phi}{alphay \cdot alphay}\right)\right)\right) \]

      14. *-lowering-*.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right)\right), \left(\frac{sin2phi}{alphay \cdot alphay}\right)\right)\right) \]

      15. /-lowering-/.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right)\right), \mathsf{/.f32}\left(sin2phi, \color{blue}{\left(alphay \cdot alphay\right)}\right)\right)\right) \]

      16. *-lowering-*.f32 98.0%

          \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right)\right), \mathsf{/.f32}\left(sin2phi, \mathsf{*.f32}\left(alphay, \color{blue}{alphay}\right)\right)\right)\right) \]

   3. Simplified 98.0%

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

   5. Taylor expanded in cos2phi around 0

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

      1. mul-1-neg N/A

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

      2. neg-lowering-neg.f32 N/A

         \[\leadsto \mathsf{neg.f32}\left(\left(\frac{{alphay}^{2} \cdot \log \left(1 - u0\right)}{sin2phi}\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{neg.f32}\left(\left(\frac{\log \left(1 - u0\right) \cdot {alphay}^{2}}{sin2phi}\right)\right) \]

      4. associate-/l* N/A

         \[\leadsto \mathsf{neg.f32}\left(\left(\log \left(1 - u0\right) \cdot \frac{{alphay}^{2}}{sin2phi}\right)\right) \]

      5. *-lowering-*.f32 N/A

         \[\leadsto \mathsf{neg.f32}\left(\mathsf{*.f32}\left(\log \left(1 - u0\right), \left(\frac{{alphay}^{2}}{sin2phi}\right)\right)\right) \]

      6. sub-neg N/A

         \[\leadsto \mathsf{neg.f32}\left(\mathsf{*.f32}\left(\log \left(1 + \left(\mathsf{neg}\left(u0\right)\right)\right), \left(\frac{{alphay}^{2}}{sin2phi}\right)\right)\right) \]

      7. mul-1-neg N/A

         \[\leadsto \mathsf{neg.f32}\left(\mathsf{*.f32}\left(\log \left(1 + -1 \cdot u0\right), \left(\frac{{alphay}^{2}}{sin2phi}\right)\right)\right) \]

      8. log1p-define N/A

         \[\leadsto \mathsf{neg.f32}\left(\mathsf{*.f32}\left(\left(\mathsf{log1p}\left(-1 \cdot u0\right)\right), \left(\frac{{alphay}^{2}}{sin2phi}\right)\right)\right) \]

      9. log1p-lowering-log1p.f32 N/A

         \[\leadsto \mathsf{neg.f32}\left(\mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\left(-1 \cdot u0\right)\right), \left(\frac{{alphay}^{2}}{sin2phi}\right)\right)\right) \]

      10. mul-1-neg N/A

          \[\leadsto \mathsf{neg.f32}\left(\mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\left(\mathsf{neg}\left(u0\right)\right)\right), \left(\frac{{alphay}^{2}}{sin2phi}\right)\right)\right) \]

      11. neg-lowering-neg.f32 N/A

          \[\leadsto \mathsf{neg.f32}\left(\mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \left(\frac{{alphay}^{2}}{sin2phi}\right)\right)\right) \]

      12. /-lowering-/.f32 N/A

          \[\leadsto \mathsf{neg.f32}\left(\mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{/.f32}\left(\left({alphay}^{2}\right), sin2phi\right)\right)\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{neg.f32}\left(\mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{/.f32}\left(\left(alphay \cdot alphay\right), sin2phi\right)\right)\right) \]

      14. *-lowering-*.f32 97.0%

          \[\leadsto \mathsf{neg.f32}\left(\mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{/.f32}\left(\mathsf{*.f32}\left(alphay, alphay\right), sin2phi\right)\right)\right) \]

   7. Simplified 97.0%

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

      1. *-commutative N/A

         \[\leadsto \mathsf{neg.f32}\left(\left(\frac{alphay \cdot alphay}{sin2phi} \cdot \log \left(1 + \left(\mathsf{neg}\left(u0\right)\right)\right)\right)\right) \]

      2. associate-/l* N/A

         \[\leadsto \mathsf{neg.f32}\left(\left(\left(alphay \cdot \frac{alphay}{sin2phi}\right) \cdot \log \left(1 + \left(\mathsf{neg}\left(u0\right)\right)\right)\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{neg.f32}\left(\left(alphay \cdot \left(\frac{alphay}{sin2phi} \cdot \log \left(1 + \left(\mathsf{neg}\left(u0\right)\right)\right)\right)\right)\right) \]

      4. *-lowering-*.f32 N/A

         \[\leadsto \mathsf{neg.f32}\left(\mathsf{*.f32}\left(alphay, \left(\frac{alphay}{sin2phi} \cdot \log \left(1 + \left(\mathsf{neg}\left(u0\right)\right)\right)\right)\right)\right) \]

      5. *-lowering-*.f32 N/A

         \[\leadsto \mathsf{neg.f32}\left(\mathsf{*.f32}\left(alphay, \mathsf{*.f32}\left(\left(\frac{alphay}{sin2phi}\right), \log \left(1 + \left(\mathsf{neg}\left(u0\right)\right)\right)\right)\right)\right) \]

      6. /-lowering-/.f32 N/A

         \[\leadsto \mathsf{neg.f32}\left(\mathsf{*.f32}\left(alphay, \mathsf{*.f32}\left(\mathsf{/.f32}\left(alphay, sin2phi\right), \log \left(1 + \left(\mathsf{neg}\left(u0\right)\right)\right)\right)\right)\right) \]

      7. log1p-define N/A

         \[\leadsto \mathsf{neg.f32}\left(\mathsf{*.f32}\left(alphay, \mathsf{*.f32}\left(\mathsf{/.f32}\left(alphay, sin2phi\right), \left(\mathsf{log1p}\left(\mathsf{neg}\left(u0\right)\right)\right)\right)\right)\right) \]

      8. log1p-lowering-log1p.f32 N/A

         \[\leadsto \mathsf{neg.f32}\left(\mathsf{*.f32}\left(alphay, \mathsf{*.f32}\left(\mathsf{/.f32}\left(alphay, sin2phi\right), \mathsf{log1p.f32}\left(\left(\mathsf{neg}\left(u0\right)\right)\right)\right)\right)\right) \]

      9. neg-lowering-neg.f32 96.9%

         \[\leadsto \mathsf{neg.f32}\left(\mathsf{*.f32}\left(alphay, \mathsf{*.f32}\left(\mathsf{/.f32}\left(alphay, sin2phi\right), \mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right)\right)\right)\right) \]

   9. Applied egg-rr 96.9%

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

   10. Taylor expanded in u0 around 0

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

       1. *-lowering-*.f32 N/A

          \[\leadsto \mathsf{*.f32}\left(u0, \color{blue}{\left(\frac{1}{2} \cdot \frac{{alphay}^{2} \cdot u0}{sin2phi} - -1 \cdot \frac{{alphay}^{2}}{sin2phi}\right)}\right) \]

       2. cancel-sign-sub-inv N/A

          \[\leadsto \mathsf{*.f32}\left(u0, \left(\frac{1}{2} \cdot \frac{{alphay}^{2} \cdot u0}{sin2phi} + \color{blue}{\left(\mathsf{neg}\left(-1\right)\right) \cdot \frac{{alphay}^{2}}{sin2phi}}\right)\right) \]

       3. metadata-eval N/A

          \[\leadsto \mathsf{*.f32}\left(u0, \left(\frac{1}{2} \cdot \frac{{alphay}^{2} \cdot u0}{sin2phi} + 1 \cdot \frac{\color{blue}{{alphay}^{2}}}{sin2phi}\right)\right) \]

       4. *-lft-identity N/A

          \[\leadsto \mathsf{*.f32}\left(u0, \left(\frac{1}{2} \cdot \frac{{alphay}^{2} \cdot u0}{sin2phi} + \frac{{alphay}^{2}}{\color{blue}{sin2phi}}\right)\right) \]

       5. +-lowering-+.f32 N/A

          \[\leadsto \mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\left(\frac{1}{2} \cdot \frac{{alphay}^{2} \cdot u0}{sin2phi}\right), \color{blue}{\left(\frac{{alphay}^{2}}{sin2phi}\right)}\right)\right) \]

       6. associate-*r/ N/A

          \[\leadsto \mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\left(\frac{\frac{1}{2} \cdot \left({alphay}^{2} \cdot u0\right)}{sin2phi}\right), \left(\frac{\color{blue}{{alphay}^{2}}}{sin2phi}\right)\right)\right) \]

       7. /-lowering-/.f32 N/A

          \[\leadsto \mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{/.f32}\left(\left(\frac{1}{2} \cdot \left({alphay}^{2} \cdot u0\right)\right), sin2phi\right), \left(\frac{\color{blue}{{alphay}^{2}}}{sin2phi}\right)\right)\right) \]

       8. *-lowering-*.f32 N/A

          \[\leadsto \mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(\frac{1}{2}, \left({alphay}^{2} \cdot u0\right)\right), sin2phi\right), \left(\frac{{\color{blue}{alphay}}^{2}}{sin2phi}\right)\right)\right) \]

       9. *-commutative N/A

          \[\leadsto \mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(\frac{1}{2}, \left(u0 \cdot {alphay}^{2}\right)\right), sin2phi\right), \left(\frac{{alphay}^{2}}{sin2phi}\right)\right)\right) \]

       10. *-lowering-*.f32 N/A

           \[\leadsto \mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(\frac{1}{2}, \mathsf{*.f32}\left(u0, \left({alphay}^{2}\right)\right)\right), sin2phi\right), \left(\frac{{alphay}^{2}}{sin2phi}\right)\right)\right) \]

       11. unpow2 N/A

           \[\leadsto \mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(\frac{1}{2}, \mathsf{*.f32}\left(u0, \left(alphay \cdot alphay\right)\right)\right), sin2phi\right), \left(\frac{{alphay}^{2}}{sin2phi}\right)\right)\right) \]

       12. *-lowering-*.f32 N/A

           \[\leadsto \mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(\frac{1}{2}, \mathsf{*.f32}\left(u0, \mathsf{*.f32}\left(alphay, alphay\right)\right)\right), sin2phi\right), \left(\frac{{alphay}^{2}}{sin2phi}\right)\right)\right) \]

       13. /-lowering-/.f32 N/A

           \[\leadsto \mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(\frac{1}{2}, \mathsf{*.f32}\left(u0, \mathsf{*.f32}\left(alphay, alphay\right)\right)\right), sin2phi\right), \mathsf{/.f32}\left(\left({alphay}^{2}\right), \color{blue}{sin2phi}\right)\right)\right) \]

       14. unpow2 N/A

           \[\leadsto \mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(\frac{1}{2}, \mathsf{*.f32}\left(u0, \mathsf{*.f32}\left(alphay, alphay\right)\right)\right), sin2phi\right), \mathsf{/.f32}\left(\left(alphay \cdot alphay\right), sin2phi\right)\right)\right) \]

       15. *-lowering-*.f32 86.9%

           \[\leadsto \mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(\frac{1}{2}, \mathsf{*.f32}\left(u0, \mathsf{*.f32}\left(alphay, alphay\right)\right)\right), sin2phi\right), \mathsf{/.f32}\left(\mathsf{*.f32}\left(alphay, alphay\right), sin2phi\right)\right)\right) \]

   12. Simplified 86.9%

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

4. Final simplification 79.7%

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

Alternative 17: 80.8% accurate, 5.3× speedup

Math

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

FPCore

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

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float tmp;
	if (sin2phi <= 4.999999969612645e-9f) {
		tmp = u0 / (((sin2phi / alphay) / alphay) + ((cos2phi / alphax) / alphax));
	} else {
		tmp = u0 * (((alphay * alphay) / sin2phi) + ((0.5f * (u0 * (alphay * 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 <= 4.999999969612645e-9) then
        tmp = u0 / (((sin2phi / alphay) / alphay) + ((cos2phi / alphax) / alphax))
    else
        tmp = u0 * (((alphay * alphay) / sin2phi) + ((0.5e0 * (u0 * (alphay * alphay))) / sin2phi))
    end if
    code = tmp
end function

Julia

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

MATLAB

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

Derivation

1. Split input into 2 regimes

2. if sin2phi < 4.99999997e-9

   1. Initial program 64.5%

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

      1. distribute-frac-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. /-lowering-/.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(\log \left(1 - u0\right), \color{blue}{\left(\mathsf{neg}\left(\left(\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\right)\right)\right)}\right) \]

      4. sub-neg N/A

         \[\leadsto \mathsf{/.f32}\left(\log \left(1 + \left(\mathsf{neg}\left(u0\right)\right)\right), \left(\mathsf{neg}\left(\left(\color{blue}{\frac{cos2phi}{alphax \cdot alphax}} + \frac{sin2phi}{alphay \cdot alphay}\right)\right)\right)\right) \]

      5. log1p-define N/A

         \[\leadsto \mathsf{/.f32}\left(\left(\mathsf{log1p}\left(\mathsf{neg}\left(u0\right)\right)\right), \left(\mathsf{neg}\left(\color{blue}{\left(\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\right)}\right)\right)\right) \]

      6. log1p-lowering-log1p.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\left(\mathsf{neg}\left(u0\right)\right)\right), \left(\mathsf{neg}\left(\color{blue}{\left(\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\right)}\right)\right)\right) \]

      7. neg-lowering-neg.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \left(\mathsf{neg}\left(\left(\color{blue}{\frac{cos2phi}{alphax \cdot alphax}} + \frac{sin2phi}{alphay \cdot alphay}\right)\right)\right)\right) \]

      8. distribute-neg-in N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \left(\left(\mathsf{neg}\left(\frac{cos2phi}{alphax \cdot alphax}\right)\right) + \color{blue}{\left(\mathsf{neg}\left(\frac{sin2phi}{alphay \cdot alphay}\right)\right)}\right)\right) \]

      9. unsub-neg N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \left(\left(\mathsf{neg}\left(\frac{cos2phi}{alphax \cdot alphax}\right)\right) - \color{blue}{\frac{sin2phi}{alphay \cdot alphay}}\right)\right) \]

      10. --lowering--.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\left(\mathsf{neg}\left(\frac{cos2phi}{alphax \cdot alphax}\right)\right), \color{blue}{\left(\frac{sin2phi}{alphay \cdot alphay}\right)}\right)\right) \]

      11. neg-sub0 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\left(0 - \frac{cos2phi}{alphax \cdot alphax}\right), \left(\frac{\color{blue}{sin2phi}}{alphay \cdot alphay}\right)\right)\right) \]

      12. --lowering--.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \left(\frac{cos2phi}{alphax \cdot alphax}\right)\right), \left(\frac{\color{blue}{sin2phi}}{alphay \cdot alphay}\right)\right)\right) \]

      13. /-lowering-/.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(cos2phi, \left(alphax \cdot alphax\right)\right)\right), \left(\frac{sin2phi}{alphay \cdot alphay}\right)\right)\right) \]

      14. *-lowering-*.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right)\right), \left(\frac{sin2phi}{alphay \cdot alphay}\right)\right)\right) \]

      15. /-lowering-/.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right)\right), \mathsf{/.f32}\left(sin2phi, \color{blue}{\left(alphay \cdot alphay\right)}\right)\right)\right) \]

      16. *-lowering-*.f32 98.6%

          \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right)\right), \mathsf{/.f32}\left(sin2phi, \mathsf{*.f32}\left(alphay, \color{blue}{alphay}\right)\right)\right)\right) \]

   3. Simplified 98.6%

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

   5. Taylor expanded in u0 around 0

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

      1. /-lowering-/.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(u0, \color{blue}{\left(\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}\right)}\right) \]

      2. +-lowering-+.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{+.f32}\left(\left(\frac{cos2phi}{{alphax}^{2}}\right), \color{blue}{\left(\frac{sin2phi}{{alphay}^{2}}\right)}\right)\right) \]

      3. /-lowering-/.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{/.f32}\left(cos2phi, \left({alphax}^{2}\right)\right), \left(\frac{\color{blue}{sin2phi}}{{alphay}^{2}}\right)\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{/.f32}\left(cos2phi, \left(alphax \cdot alphax\right)\right), \left(\frac{sin2phi}{{alphay}^{2}}\right)\right)\right) \]

      5. *-lowering-*.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right), \left(\frac{sin2phi}{{alphay}^{2}}\right)\right)\right) \]

      6. /-lowering-/.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right), \mathsf{/.f32}\left(sin2phi, \color{blue}{\left({alphay}^{2}\right)}\right)\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right), \mathsf{/.f32}\left(sin2phi, \left(alphay \cdot \color{blue}{alphay}\right)\right)\right)\right) \]

      8. *-lowering-*.f32 68.0%

         \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right), \mathsf{/.f32}\left(sin2phi, \mathsf{*.f32}\left(alphay, \color{blue}{alphay}\right)\right)\right)\right) \]

   7. Simplified 68.0%

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

      1. +-commutative N/A

         \[\leadsto \mathsf{/.f32}\left(u0, \left(\frac{sin2phi}{alphay \cdot alphay} + \color{blue}{\frac{cos2phi}{alphax \cdot alphax}}\right)\right) \]

      2. +-lowering-+.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{+.f32}\left(\left(\frac{sin2phi}{alphay \cdot alphay}\right), \color{blue}{\left(\frac{cos2phi}{alphax \cdot alphax}\right)}\right)\right) \]

      3. associate-/r* N/A

         \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{+.f32}\left(\left(\frac{\frac{sin2phi}{alphay}}{alphay}\right), \left(\frac{\color{blue}{cos2phi}}{alphax \cdot alphax}\right)\right)\right) \]

      4. /-lowering-/.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{/.f32}\left(\left(\frac{sin2phi}{alphay}\right), alphay\right), \left(\frac{\color{blue}{cos2phi}}{alphax \cdot alphax}\right)\right)\right) \]

      5. /-lowering-/.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(sin2phi, alphay\right), alphay\right), \left(\frac{cos2phi}{alphax \cdot alphax}\right)\right)\right) \]

      6. associate-/r* N/A

         \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(sin2phi, alphay\right), alphay\right), \left(\frac{\frac{cos2phi}{alphax}}{\color{blue}{alphax}}\right)\right)\right) \]

      7. /-lowering-/.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(sin2phi, alphay\right), alphay\right), \mathsf{/.f32}\left(\left(\frac{cos2phi}{alphax}\right), \color{blue}{alphax}\right)\right)\right) \]

      8. /-lowering-/.f32 68.2%

         \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(sin2phi, alphay\right), alphay\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(cos2phi, alphax\right), alphax\right)\right)\right) \]

   9. Applied egg-rr 68.2%

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

   if 4.99999997e-9 < sin2phi

   1. Initial program 65.0%

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

      1. distribute-frac-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. /-lowering-/.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(\log \left(1 - u0\right), \color{blue}{\left(\mathsf{neg}\left(\left(\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\right)\right)\right)}\right) \]

      4. sub-neg N/A

         \[\leadsto \mathsf{/.f32}\left(\log \left(1 + \left(\mathsf{neg}\left(u0\right)\right)\right), \left(\mathsf{neg}\left(\left(\color{blue}{\frac{cos2phi}{alphax \cdot alphax}} + \frac{sin2phi}{alphay \cdot alphay}\right)\right)\right)\right) \]

      5. log1p-define N/A

         \[\leadsto \mathsf{/.f32}\left(\left(\mathsf{log1p}\left(\mathsf{neg}\left(u0\right)\right)\right), \left(\mathsf{neg}\left(\color{blue}{\left(\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\right)}\right)\right)\right) \]

      6. log1p-lowering-log1p.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\left(\mathsf{neg}\left(u0\right)\right)\right), \left(\mathsf{neg}\left(\color{blue}{\left(\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\right)}\right)\right)\right) \]

      7. neg-lowering-neg.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \left(\mathsf{neg}\left(\left(\color{blue}{\frac{cos2phi}{alphax \cdot alphax}} + \frac{sin2phi}{alphay \cdot alphay}\right)\right)\right)\right) \]

      8. distribute-neg-in N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \left(\left(\mathsf{neg}\left(\frac{cos2phi}{alphax \cdot alphax}\right)\right) + \color{blue}{\left(\mathsf{neg}\left(\frac{sin2phi}{alphay \cdot alphay}\right)\right)}\right)\right) \]

      9. unsub-neg N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \left(\left(\mathsf{neg}\left(\frac{cos2phi}{alphax \cdot alphax}\right)\right) - \color{blue}{\frac{sin2phi}{alphay \cdot alphay}}\right)\right) \]

      10. --lowering--.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\left(\mathsf{neg}\left(\frac{cos2phi}{alphax \cdot alphax}\right)\right), \color{blue}{\left(\frac{sin2phi}{alphay \cdot alphay}\right)}\right)\right) \]

      11. neg-sub0 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\left(0 - \frac{cos2phi}{alphax \cdot alphax}\right), \left(\frac{\color{blue}{sin2phi}}{alphay \cdot alphay}\right)\right)\right) \]

      12. --lowering--.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \left(\frac{cos2phi}{alphax \cdot alphax}\right)\right), \left(\frac{\color{blue}{sin2phi}}{alphay \cdot alphay}\right)\right)\right) \]

      13. /-lowering-/.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(cos2phi, \left(alphax \cdot alphax\right)\right)\right), \left(\frac{sin2phi}{alphay \cdot alphay}\right)\right)\right) \]

      14. *-lowering-*.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right)\right), \left(\frac{sin2phi}{alphay \cdot alphay}\right)\right)\right) \]

      15. /-lowering-/.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right)\right), \mathsf{/.f32}\left(sin2phi, \color{blue}{\left(alphay \cdot alphay\right)}\right)\right)\right) \]

      16. *-lowering-*.f32 98.0%

          \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right)\right), \mathsf{/.f32}\left(sin2phi, \mathsf{*.f32}\left(alphay, \color{blue}{alphay}\right)\right)\right)\right) \]

   3. Simplified 98.0%

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

   5. Taylor expanded in cos2phi around 0

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

      1. mul-1-neg N/A

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

      2. neg-lowering-neg.f32 N/A

         \[\leadsto \mathsf{neg.f32}\left(\left(\frac{{alphay}^{2} \cdot \log \left(1 - u0\right)}{sin2phi}\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{neg.f32}\left(\left(\frac{\log \left(1 - u0\right) \cdot {alphay}^{2}}{sin2phi}\right)\right) \]

      4. associate-/l* N/A

         \[\leadsto \mathsf{neg.f32}\left(\left(\log \left(1 - u0\right) \cdot \frac{{alphay}^{2}}{sin2phi}\right)\right) \]

      5. *-lowering-*.f32 N/A

         \[\leadsto \mathsf{neg.f32}\left(\mathsf{*.f32}\left(\log \left(1 - u0\right), \left(\frac{{alphay}^{2}}{sin2phi}\right)\right)\right) \]

      6. sub-neg N/A

         \[\leadsto \mathsf{neg.f32}\left(\mathsf{*.f32}\left(\log \left(1 + \left(\mathsf{neg}\left(u0\right)\right)\right), \left(\frac{{alphay}^{2}}{sin2phi}\right)\right)\right) \]

      7. mul-1-neg N/A

         \[\leadsto \mathsf{neg.f32}\left(\mathsf{*.f32}\left(\log \left(1 + -1 \cdot u0\right), \left(\frac{{alphay}^{2}}{sin2phi}\right)\right)\right) \]

      8. log1p-define N/A

         \[\leadsto \mathsf{neg.f32}\left(\mathsf{*.f32}\left(\left(\mathsf{log1p}\left(-1 \cdot u0\right)\right), \left(\frac{{alphay}^{2}}{sin2phi}\right)\right)\right) \]

      9. log1p-lowering-log1p.f32 N/A

         \[\leadsto \mathsf{neg.f32}\left(\mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\left(-1 \cdot u0\right)\right), \left(\frac{{alphay}^{2}}{sin2phi}\right)\right)\right) \]

      10. mul-1-neg N/A

          \[\leadsto \mathsf{neg.f32}\left(\mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\left(\mathsf{neg}\left(u0\right)\right)\right), \left(\frac{{alphay}^{2}}{sin2phi}\right)\right)\right) \]

      11. neg-lowering-neg.f32 N/A

          \[\leadsto \mathsf{neg.f32}\left(\mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \left(\frac{{alphay}^{2}}{sin2phi}\right)\right)\right) \]

      12. /-lowering-/.f32 N/A

          \[\leadsto \mathsf{neg.f32}\left(\mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{/.f32}\left(\left({alphay}^{2}\right), sin2phi\right)\right)\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{neg.f32}\left(\mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{/.f32}\left(\left(alphay \cdot alphay\right), sin2phi\right)\right)\right) \]

      14. *-lowering-*.f32 97.0%

          \[\leadsto \mathsf{neg.f32}\left(\mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{/.f32}\left(\mathsf{*.f32}\left(alphay, alphay\right), sin2phi\right)\right)\right) \]

   7. Simplified 97.0%

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

      1. *-commutative N/A

         \[\leadsto \mathsf{neg.f32}\left(\left(\frac{alphay \cdot alphay}{sin2phi} \cdot \log \left(1 + \left(\mathsf{neg}\left(u0\right)\right)\right)\right)\right) \]

      2. associate-/l* N/A

         \[\leadsto \mathsf{neg.f32}\left(\left(\left(alphay \cdot \frac{alphay}{sin2phi}\right) \cdot \log \left(1 + \left(\mathsf{neg}\left(u0\right)\right)\right)\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{neg.f32}\left(\left(alphay \cdot \left(\frac{alphay}{sin2phi} \cdot \log \left(1 + \left(\mathsf{neg}\left(u0\right)\right)\right)\right)\right)\right) \]

      4. *-lowering-*.f32 N/A

         \[\leadsto \mathsf{neg.f32}\left(\mathsf{*.f32}\left(alphay, \left(\frac{alphay}{sin2phi} \cdot \log \left(1 + \left(\mathsf{neg}\left(u0\right)\right)\right)\right)\right)\right) \]

      5. *-lowering-*.f32 N/A

         \[\leadsto \mathsf{neg.f32}\left(\mathsf{*.f32}\left(alphay, \mathsf{*.f32}\left(\left(\frac{alphay}{sin2phi}\right), \log \left(1 + \left(\mathsf{neg}\left(u0\right)\right)\right)\right)\right)\right) \]

      6. /-lowering-/.f32 N/A

         \[\leadsto \mathsf{neg.f32}\left(\mathsf{*.f32}\left(alphay, \mathsf{*.f32}\left(\mathsf{/.f32}\left(alphay, sin2phi\right), \log \left(1 + \left(\mathsf{neg}\left(u0\right)\right)\right)\right)\right)\right) \]

      7. log1p-define N/A

         \[\leadsto \mathsf{neg.f32}\left(\mathsf{*.f32}\left(alphay, \mathsf{*.f32}\left(\mathsf{/.f32}\left(alphay, sin2phi\right), \left(\mathsf{log1p}\left(\mathsf{neg}\left(u0\right)\right)\right)\right)\right)\right) \]

      8. log1p-lowering-log1p.f32 N/A

         \[\leadsto \mathsf{neg.f32}\left(\mathsf{*.f32}\left(alphay, \mathsf{*.f32}\left(\mathsf{/.f32}\left(alphay, sin2phi\right), \mathsf{log1p.f32}\left(\left(\mathsf{neg}\left(u0\right)\right)\right)\right)\right)\right) \]

      9. neg-lowering-neg.f32 96.9%

         \[\leadsto \mathsf{neg.f32}\left(\mathsf{*.f32}\left(alphay, \mathsf{*.f32}\left(\mathsf{/.f32}\left(alphay, sin2phi\right), \mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right)\right)\right)\right) \]

   9. Applied egg-rr 96.9%

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

   10. Taylor expanded in u0 around 0

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

       1. *-lowering-*.f32 N/A

          \[\leadsto \mathsf{*.f32}\left(u0, \color{blue}{\left(\frac{1}{2} \cdot \frac{{alphay}^{2} \cdot u0}{sin2phi} - -1 \cdot \frac{{alphay}^{2}}{sin2phi}\right)}\right) \]

       2. cancel-sign-sub-inv N/A

          \[\leadsto \mathsf{*.f32}\left(u0, \left(\frac{1}{2} \cdot \frac{{alphay}^{2} \cdot u0}{sin2phi} + \color{blue}{\left(\mathsf{neg}\left(-1\right)\right) \cdot \frac{{alphay}^{2}}{sin2phi}}\right)\right) \]

       3. metadata-eval N/A

          \[\leadsto \mathsf{*.f32}\left(u0, \left(\frac{1}{2} \cdot \frac{{alphay}^{2} \cdot u0}{sin2phi} + 1 \cdot \frac{\color{blue}{{alphay}^{2}}}{sin2phi}\right)\right) \]

       4. *-lft-identity N/A

          \[\leadsto \mathsf{*.f32}\left(u0, \left(\frac{1}{2} \cdot \frac{{alphay}^{2} \cdot u0}{sin2phi} + \frac{{alphay}^{2}}{\color{blue}{sin2phi}}\right)\right) \]

       5. +-lowering-+.f32 N/A

          \[\leadsto \mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\left(\frac{1}{2} \cdot \frac{{alphay}^{2} \cdot u0}{sin2phi}\right), \color{blue}{\left(\frac{{alphay}^{2}}{sin2phi}\right)}\right)\right) \]

       6. associate-*r/ N/A

          \[\leadsto \mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\left(\frac{\frac{1}{2} \cdot \left({alphay}^{2} \cdot u0\right)}{sin2phi}\right), \left(\frac{\color{blue}{{alphay}^{2}}}{sin2phi}\right)\right)\right) \]

       7. /-lowering-/.f32 N/A

          \[\leadsto \mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{/.f32}\left(\left(\frac{1}{2} \cdot \left({alphay}^{2} \cdot u0\right)\right), sin2phi\right), \left(\frac{\color{blue}{{alphay}^{2}}}{sin2phi}\right)\right)\right) \]

       8. *-lowering-*.f32 N/A

          \[\leadsto \mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(\frac{1}{2}, \left({alphay}^{2} \cdot u0\right)\right), sin2phi\right), \left(\frac{{\color{blue}{alphay}}^{2}}{sin2phi}\right)\right)\right) \]

       9. *-commutative N/A

          \[\leadsto \mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(\frac{1}{2}, \left(u0 \cdot {alphay}^{2}\right)\right), sin2phi\right), \left(\frac{{alphay}^{2}}{sin2phi}\right)\right)\right) \]

       10. *-lowering-*.f32 N/A

           \[\leadsto \mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(\frac{1}{2}, \mathsf{*.f32}\left(u0, \left({alphay}^{2}\right)\right)\right), sin2phi\right), \left(\frac{{alphay}^{2}}{sin2phi}\right)\right)\right) \]

       11. unpow2 N/A

           \[\leadsto \mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(\frac{1}{2}, \mathsf{*.f32}\left(u0, \left(alphay \cdot alphay\right)\right)\right), sin2phi\right), \left(\frac{{alphay}^{2}}{sin2phi}\right)\right)\right) \]

       12. *-lowering-*.f32 N/A

           \[\leadsto \mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(\frac{1}{2}, \mathsf{*.f32}\left(u0, \mathsf{*.f32}\left(alphay, alphay\right)\right)\right), sin2phi\right), \left(\frac{{alphay}^{2}}{sin2phi}\right)\right)\right) \]

       13. /-lowering-/.f32 N/A

           \[\leadsto \mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(\frac{1}{2}, \mathsf{*.f32}\left(u0, \mathsf{*.f32}\left(alphay, alphay\right)\right)\right), sin2phi\right), \mathsf{/.f32}\left(\left({alphay}^{2}\right), \color{blue}{sin2phi}\right)\right)\right) \]

       14. unpow2 N/A

           \[\leadsto \mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(\frac{1}{2}, \mathsf{*.f32}\left(u0, \mathsf{*.f32}\left(alphay, alphay\right)\right)\right), sin2phi\right), \mathsf{/.f32}\left(\left(alphay \cdot alphay\right), sin2phi\right)\right)\right) \]

       15. *-lowering-*.f32 86.9%

           \[\leadsto \mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(\frac{1}{2}, \mathsf{*.f32}\left(u0, \mathsf{*.f32}\left(alphay, alphay\right)\right)\right), sin2phi\right), \mathsf{/.f32}\left(\mathsf{*.f32}\left(alphay, alphay\right), sin2phi\right)\right)\right) \]

   12. Simplified 86.9%

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

4. Final simplification 79.7%

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

Alternative 18: 87.1% accurate, 5.5× speedup

Math

\[\begin{array}{l} \\ \frac{u0 \cdot \left(-1 + u0 \cdot -0.5\right)}{\frac{0 - \frac{cos2phi}{alphax}}{alphax} - \frac{\frac{sin2phi}{alphay}}{alphay}} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (/
  (* u0 (+ -1.0 (* u0 -0.5)))
  (- (/ (- 0.0 (/ cos2phi alphax)) alphax) (/ (/ sin2phi alphay) alphay))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	return (u0 * (-1.0f + (u0 * -0.5f))) / (((0.0f - (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 = (u0 * ((-1.0e0) + (u0 * (-0.5e0)))) / (((0.0e0 - (cos2phi / alphax)) / alphax) - ((sin2phi / alphay) / 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(Float32(Float32(0.0) - Float32(cos2phi / alphax)) / alphax) - Float32(Float32(sin2phi / alphay) / alphay)))
end

MATLAB

function tmp = code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = (u0 * (single(-1.0) + (u0 * single(-0.5)))) / (((single(0.0) - (cos2phi / alphax)) / alphax) - ((sin2phi / alphay) / alphay));
end

Derivation

1. Initial program 64.8%

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

   1. distribute-frac-neg N/A

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

   2. distribute-neg-frac2 N/A

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

   3. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\log \left(1 - u0\right), \color{blue}{\left(\mathsf{neg}\left(\left(\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\right)\right)\right)}\right) \]

   4. sub-neg N/A

      \[\leadsto \mathsf{/.f32}\left(\log \left(1 + \left(\mathsf{neg}\left(u0\right)\right)\right), \left(\mathsf{neg}\left(\left(\color{blue}{\frac{cos2phi}{alphax \cdot alphax}} + \frac{sin2phi}{alphay \cdot alphay}\right)\right)\right)\right) \]

   5. log1p-define N/A

      \[\leadsto \mathsf{/.f32}\left(\left(\mathsf{log1p}\left(\mathsf{neg}\left(u0\right)\right)\right), \left(\mathsf{neg}\left(\color{blue}{\left(\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\right)}\right)\right)\right) \]

   6. log1p-lowering-log1p.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\left(\mathsf{neg}\left(u0\right)\right)\right), \left(\mathsf{neg}\left(\color{blue}{\left(\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\right)}\right)\right)\right) \]

   7. neg-lowering-neg.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \left(\mathsf{neg}\left(\left(\color{blue}{\frac{cos2phi}{alphax \cdot alphax}} + \frac{sin2phi}{alphay \cdot alphay}\right)\right)\right)\right) \]

   8. distribute-neg-in N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \left(\left(\mathsf{neg}\left(\frac{cos2phi}{alphax \cdot alphax}\right)\right) + \color{blue}{\left(\mathsf{neg}\left(\frac{sin2phi}{alphay \cdot alphay}\right)\right)}\right)\right) \]

   9. unsub-neg N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \left(\left(\mathsf{neg}\left(\frac{cos2phi}{alphax \cdot alphax}\right)\right) - \color{blue}{\frac{sin2phi}{alphay \cdot alphay}}\right)\right) \]

   10. --lowering--.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\left(\mathsf{neg}\left(\frac{cos2phi}{alphax \cdot alphax}\right)\right), \color{blue}{\left(\frac{sin2phi}{alphay \cdot alphay}\right)}\right)\right) \]

   11. neg-sub0 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\left(0 - \frac{cos2phi}{alphax \cdot alphax}\right), \left(\frac{\color{blue}{sin2phi}}{alphay \cdot alphay}\right)\right)\right) \]

   12. --lowering--.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \left(\frac{cos2phi}{alphax \cdot alphax}\right)\right), \left(\frac{\color{blue}{sin2phi}}{alphay \cdot alphay}\right)\right)\right) \]

   13. /-lowering-/.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(cos2phi, \left(alphax \cdot alphax\right)\right)\right), \left(\frac{sin2phi}{alphay \cdot alphay}\right)\right)\right) \]

   14. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right)\right), \left(\frac{sin2phi}{alphay \cdot alphay}\right)\right)\right) \]

   15. /-lowering-/.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right)\right), \mathsf{/.f32}\left(sin2phi, \color{blue}{\left(alphay \cdot alphay\right)}\right)\right)\right) \]

   16. *-lowering-*.f32 98.3%

       \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right)\right), \mathsf{/.f32}\left(sin2phi, \mathsf{*.f32}\left(alphay, \color{blue}{alphay}\right)\right)\right)\right) \]

3. Simplified 98.3%

   \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(-u0\right)}{\left(0 - \frac{cos2phi}{alphax \cdot alphax}\right) - \frac{sin2phi}{alphay \cdot alphay}}} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. associate-/r* N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right)\right), \left(\frac{\frac{sin2phi}{alphay}}{\color{blue}{alphay}}\right)\right)\right) \]

   2. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right)\right), \mathsf{/.f32}\left(\left(\frac{sin2phi}{alphay}\right), \color{blue}{alphay}\right)\right)\right) \]

   3. /-lowering-/.f32 98.3%

      \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right)\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(sin2phi, alphay\right), alphay\right)\right)\right) \]

6. Applied egg-rr 98.3%

   \[\leadsto \frac{\mathsf{log1p}\left(-u0\right)}{\left(0 - \frac{cos2phi}{alphax \cdot alphax}\right) - \color{blue}{\frac{\frac{sin2phi}{alphay}}{alphay}}} \]
7. Step-by-step derivation

   1. clear-num N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \left(\frac{1}{\frac{alphax \cdot alphax}{cos2phi}}\right)\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(sin2phi, \color{blue}{alphay}\right), alphay\right)\right)\right) \]

   2. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(1, \left(\frac{alphax \cdot alphax}{cos2phi}\right)\right)\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(sin2phi, \color{blue}{alphay}\right), alphay\right)\right)\right) \]

   3. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(1, \mathsf{/.f32}\left(\left(alphax \cdot alphax\right), cos2phi\right)\right)\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(sin2phi, alphay\right), alphay\right)\right)\right) \]

   4. *-lowering-*.f32 98.3%

      \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(1, \mathsf{/.f32}\left(\mathsf{*.f32}\left(alphax, alphax\right), cos2phi\right)\right)\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(sin2phi, alphay\right), alphay\right)\right)\right) \]

8. Applied egg-rr 98.3%

   \[\leadsto \frac{\mathsf{log1p}\left(-u0\right)}{\left(0 - \color{blue}{\frac{1}{\frac{alphax \cdot alphax}{cos2phi}}}\right) - \frac{\frac{sin2phi}{alphay}}{alphay}} \]
9. Step-by-step derivation

   1. clear-num N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\left(0 - \frac{cos2phi}{alphax \cdot alphax}\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(sin2phi, \color{blue}{alphay}\right), alphay\right)\right)\right) \]

   2. sub0-neg N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\left(\mathsf{neg}\left(\frac{cos2phi}{alphax \cdot alphax}\right)\right), \mathsf{/.f32}\left(\color{blue}{\mathsf{/.f32}\left(sin2phi, alphay\right)}, alphay\right)\right)\right) \]

   3. associate-/r* N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\left(\mathsf{neg}\left(\frac{\frac{cos2phi}{alphax}}{alphax}\right)\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(\color{blue}{sin2phi}, alphay\right), alphay\right)\right)\right) \]

   4. distribute-neg-frac2 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\left(\frac{\frac{cos2phi}{alphax}}{\mathsf{neg}\left(alphax\right)}\right), \mathsf{/.f32}\left(\color{blue}{\mathsf{/.f32}\left(sin2phi, alphay\right)}, alphay\right)\right)\right) \]

   5. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{/.f32}\left(\left(\frac{cos2phi}{alphax}\right), \left(\mathsf{neg}\left(alphax\right)\right)\right), \mathsf{/.f32}\left(\color{blue}{\mathsf{/.f32}\left(sin2phi, alphay\right)}, alphay\right)\right)\right) \]

   6. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(cos2phi, alphax\right), \left(\mathsf{neg}\left(alphax\right)\right)\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(\color{blue}{sin2phi}, alphay\right), alphay\right)\right)\right) \]

   7. neg-lowering-neg.f32 98.3%

      \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(cos2phi, alphax\right), \mathsf{neg.f32}\left(alphax\right)\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(sin2phi, \color{blue}{alphay}\right), alphay\right)\right)\right) \]

10. Applied egg-rr 98.3%

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

11. Taylor expanded in u0 around 0

    \[\leadsto \mathsf{/.f32}\left(\color{blue}{\left(u0 \cdot \left(\frac{-1}{2} \cdot u0 - 1\right)\right)}, \mathsf{\_.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(cos2phi, alphax\right), \mathsf{neg.f32}\left(alphax\right)\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(sin2phi, alphay\right), alphay\right)\right)\right) \]
12. Step-by-step derivation

    1. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(u0, \left(\frac{-1}{2} \cdot u0 - 1\right)\right), \mathsf{\_.f32}\left(\color{blue}{\mathsf{/.f32}\left(\mathsf{/.f32}\left(cos2phi, alphax\right), \mathsf{neg.f32}\left(alphax\right)\right)}, \mathsf{/.f32}\left(\mathsf{/.f32}\left(sin2phi, alphay\right), alphay\right)\right)\right) \]

    2. sub-neg N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(u0, \left(\frac{-1}{2} \cdot u0 + \left(\mathsf{neg}\left(1\right)\right)\right)\right), \mathsf{\_.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(cos2phi, alphax\right), \color{blue}{\mathsf{neg.f32}\left(alphax\right)}\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(sin2phi, alphay\right), alphay\right)\right)\right) \]

    3. metadata-eval N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(u0, \left(\frac{-1}{2} \cdot u0 + -1\right)\right), \mathsf{\_.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(cos2phi, alphax\right), \mathsf{neg.f32}\left(alphax\right)\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(sin2phi, alphay\right), alphay\right)\right)\right) \]

    4. +-lowering-+.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\left(\frac{-1}{2} \cdot u0\right), -1\right)\right), \mathsf{\_.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(cos2phi, alphax\right), \color{blue}{\mathsf{neg.f32}\left(alphax\right)}\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(sin2phi, alphay\right), alphay\right)\right)\right) \]

    5. *-commutative N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\left(u0 \cdot \frac{-1}{2}\right), -1\right)\right), \mathsf{\_.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(cos2phi, alphax\right), \mathsf{neg.f32}\left(\color{blue}{alphax}\right)\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(sin2phi, alphay\right), alphay\right)\right)\right) \]

    6. *-lowering-*.f32 85.2%

       \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{*.f32}\left(u0, \frac{-1}{2}\right), -1\right)\right), \mathsf{\_.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(cos2phi, alphax\right), \mathsf{neg.f32}\left(\color{blue}{alphax}\right)\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(sin2phi, alphay\right), alphay\right)\right)\right) \]

13. Simplified 85.2%

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

14. Final simplification 85.2%

    \[\leadsto \frac{u0 \cdot \left(-1 + u0 \cdot -0.5\right)}{\frac{0 - \frac{cos2phi}{alphax}}{alphax} - \frac{\frac{sin2phi}{alphay}}{alphay}} \]
15. Add Preprocessing

Alternative 19: 80.8% accurate, 5.8× speedup

Math

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

FPCore

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

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float tmp;
	if (sin2phi <= 4.999999969612645e-9f) {
		tmp = u0 / (((sin2phi / alphay) / alphay) + ((cos2phi / alphax) / alphax));
	} else {
		tmp = alphay * (u0 * ((alphay / sin2phi) - ((-0.5f * (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 <= 4.999999969612645e-9) then
        tmp = u0 / (((sin2phi / alphay) / alphay) + ((cos2phi / alphax) / alphax))
    else
        tmp = alphay * (u0 * ((alphay / sin2phi) - (((-0.5e0) * (u0 * alphay)) / sin2phi)))
    end if
    code = tmp
end function

Julia

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

MATLAB

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

Derivation

1. Split input into 2 regimes

2. if sin2phi < 4.99999997e-9

   1. Initial program 64.5%

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

      1. distribute-frac-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. /-lowering-/.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(\log \left(1 - u0\right), \color{blue}{\left(\mathsf{neg}\left(\left(\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\right)\right)\right)}\right) \]

      4. sub-neg N/A

         \[\leadsto \mathsf{/.f32}\left(\log \left(1 + \left(\mathsf{neg}\left(u0\right)\right)\right), \left(\mathsf{neg}\left(\left(\color{blue}{\frac{cos2phi}{alphax \cdot alphax}} + \frac{sin2phi}{alphay \cdot alphay}\right)\right)\right)\right) \]

      5. log1p-define N/A

         \[\leadsto \mathsf{/.f32}\left(\left(\mathsf{log1p}\left(\mathsf{neg}\left(u0\right)\right)\right), \left(\mathsf{neg}\left(\color{blue}{\left(\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\right)}\right)\right)\right) \]

      6. log1p-lowering-log1p.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\left(\mathsf{neg}\left(u0\right)\right)\right), \left(\mathsf{neg}\left(\color{blue}{\left(\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\right)}\right)\right)\right) \]

      7. neg-lowering-neg.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \left(\mathsf{neg}\left(\left(\color{blue}{\frac{cos2phi}{alphax \cdot alphax}} + \frac{sin2phi}{alphay \cdot alphay}\right)\right)\right)\right) \]

      8. distribute-neg-in N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \left(\left(\mathsf{neg}\left(\frac{cos2phi}{alphax \cdot alphax}\right)\right) + \color{blue}{\left(\mathsf{neg}\left(\frac{sin2phi}{alphay \cdot alphay}\right)\right)}\right)\right) \]

      9. unsub-neg N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \left(\left(\mathsf{neg}\left(\frac{cos2phi}{alphax \cdot alphax}\right)\right) - \color{blue}{\frac{sin2phi}{alphay \cdot alphay}}\right)\right) \]

      10. --lowering--.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\left(\mathsf{neg}\left(\frac{cos2phi}{alphax \cdot alphax}\right)\right), \color{blue}{\left(\frac{sin2phi}{alphay \cdot alphay}\right)}\right)\right) \]

      11. neg-sub0 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\left(0 - \frac{cos2phi}{alphax \cdot alphax}\right), \left(\frac{\color{blue}{sin2phi}}{alphay \cdot alphay}\right)\right)\right) \]

      12. --lowering--.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \left(\frac{cos2phi}{alphax \cdot alphax}\right)\right), \left(\frac{\color{blue}{sin2phi}}{alphay \cdot alphay}\right)\right)\right) \]

      13. /-lowering-/.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(cos2phi, \left(alphax \cdot alphax\right)\right)\right), \left(\frac{sin2phi}{alphay \cdot alphay}\right)\right)\right) \]

      14. *-lowering-*.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right)\right), \left(\frac{sin2phi}{alphay \cdot alphay}\right)\right)\right) \]

      15. /-lowering-/.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right)\right), \mathsf{/.f32}\left(sin2phi, \color{blue}{\left(alphay \cdot alphay\right)}\right)\right)\right) \]

      16. *-lowering-*.f32 98.6%

          \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right)\right), \mathsf{/.f32}\left(sin2phi, \mathsf{*.f32}\left(alphay, \color{blue}{alphay}\right)\right)\right)\right) \]

   3. Simplified 98.6%

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

   5. Taylor expanded in u0 around 0

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

      1. /-lowering-/.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(u0, \color{blue}{\left(\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}\right)}\right) \]

      2. +-lowering-+.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{+.f32}\left(\left(\frac{cos2phi}{{alphax}^{2}}\right), \color{blue}{\left(\frac{sin2phi}{{alphay}^{2}}\right)}\right)\right) \]

      3. /-lowering-/.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{/.f32}\left(cos2phi, \left({alphax}^{2}\right)\right), \left(\frac{\color{blue}{sin2phi}}{{alphay}^{2}}\right)\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{/.f32}\left(cos2phi, \left(alphax \cdot alphax\right)\right), \left(\frac{sin2phi}{{alphay}^{2}}\right)\right)\right) \]

      5. *-lowering-*.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right), \left(\frac{sin2phi}{{alphay}^{2}}\right)\right)\right) \]

      6. /-lowering-/.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right), \mathsf{/.f32}\left(sin2phi, \color{blue}{\left({alphay}^{2}\right)}\right)\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right), \mathsf{/.f32}\left(sin2phi, \left(alphay \cdot \color{blue}{alphay}\right)\right)\right)\right) \]

      8. *-lowering-*.f32 68.0%

         \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right), \mathsf{/.f32}\left(sin2phi, \mathsf{*.f32}\left(alphay, \color{blue}{alphay}\right)\right)\right)\right) \]

   7. Simplified 68.0%

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

      1. +-commutative N/A

         \[\leadsto \mathsf{/.f32}\left(u0, \left(\frac{sin2phi}{alphay \cdot alphay} + \color{blue}{\frac{cos2phi}{alphax \cdot alphax}}\right)\right) \]

      2. +-lowering-+.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{+.f32}\left(\left(\frac{sin2phi}{alphay \cdot alphay}\right), \color{blue}{\left(\frac{cos2phi}{alphax \cdot alphax}\right)}\right)\right) \]

      3. associate-/r* N/A

         \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{+.f32}\left(\left(\frac{\frac{sin2phi}{alphay}}{alphay}\right), \left(\frac{\color{blue}{cos2phi}}{alphax \cdot alphax}\right)\right)\right) \]

      4. /-lowering-/.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{/.f32}\left(\left(\frac{sin2phi}{alphay}\right), alphay\right), \left(\frac{\color{blue}{cos2phi}}{alphax \cdot alphax}\right)\right)\right) \]

      5. /-lowering-/.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(sin2phi, alphay\right), alphay\right), \left(\frac{cos2phi}{alphax \cdot alphax}\right)\right)\right) \]

      6. associate-/r* N/A

         \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(sin2phi, alphay\right), alphay\right), \left(\frac{\frac{cos2phi}{alphax}}{\color{blue}{alphax}}\right)\right)\right) \]

      7. /-lowering-/.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(sin2phi, alphay\right), alphay\right), \mathsf{/.f32}\left(\left(\frac{cos2phi}{alphax}\right), \color{blue}{alphax}\right)\right)\right) \]

      8. /-lowering-/.f32 68.2%

         \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(sin2phi, alphay\right), alphay\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(cos2phi, alphax\right), alphax\right)\right)\right) \]

   9. Applied egg-rr 68.2%

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

   if 4.99999997e-9 < sin2phi

   1. Initial program 65.0%

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

      1. distribute-frac-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. /-lowering-/.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(\log \left(1 - u0\right), \color{blue}{\left(\mathsf{neg}\left(\left(\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\right)\right)\right)}\right) \]

      4. sub-neg N/A

         \[\leadsto \mathsf{/.f32}\left(\log \left(1 + \left(\mathsf{neg}\left(u0\right)\right)\right), \left(\mathsf{neg}\left(\left(\color{blue}{\frac{cos2phi}{alphax \cdot alphax}} + \frac{sin2phi}{alphay \cdot alphay}\right)\right)\right)\right) \]

      5. log1p-define N/A

         \[\leadsto \mathsf{/.f32}\left(\left(\mathsf{log1p}\left(\mathsf{neg}\left(u0\right)\right)\right), \left(\mathsf{neg}\left(\color{blue}{\left(\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\right)}\right)\right)\right) \]

      6. log1p-lowering-log1p.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\left(\mathsf{neg}\left(u0\right)\right)\right), \left(\mathsf{neg}\left(\color{blue}{\left(\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\right)}\right)\right)\right) \]

      7. neg-lowering-neg.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \left(\mathsf{neg}\left(\left(\color{blue}{\frac{cos2phi}{alphax \cdot alphax}} + \frac{sin2phi}{alphay \cdot alphay}\right)\right)\right)\right) \]

      8. distribute-neg-in N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \left(\left(\mathsf{neg}\left(\frac{cos2phi}{alphax \cdot alphax}\right)\right) + \color{blue}{\left(\mathsf{neg}\left(\frac{sin2phi}{alphay \cdot alphay}\right)\right)}\right)\right) \]

      9. unsub-neg N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \left(\left(\mathsf{neg}\left(\frac{cos2phi}{alphax \cdot alphax}\right)\right) - \color{blue}{\frac{sin2phi}{alphay \cdot alphay}}\right)\right) \]

      10. --lowering--.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\left(\mathsf{neg}\left(\frac{cos2phi}{alphax \cdot alphax}\right)\right), \color{blue}{\left(\frac{sin2phi}{alphay \cdot alphay}\right)}\right)\right) \]

      11. neg-sub0 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\left(0 - \frac{cos2phi}{alphax \cdot alphax}\right), \left(\frac{\color{blue}{sin2phi}}{alphay \cdot alphay}\right)\right)\right) \]

      12. --lowering--.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \left(\frac{cos2phi}{alphax \cdot alphax}\right)\right), \left(\frac{\color{blue}{sin2phi}}{alphay \cdot alphay}\right)\right)\right) \]

      13. /-lowering-/.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(cos2phi, \left(alphax \cdot alphax\right)\right)\right), \left(\frac{sin2phi}{alphay \cdot alphay}\right)\right)\right) \]

      14. *-lowering-*.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right)\right), \left(\frac{sin2phi}{alphay \cdot alphay}\right)\right)\right) \]

      15. /-lowering-/.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right)\right), \mathsf{/.f32}\left(sin2phi, \color{blue}{\left(alphay \cdot alphay\right)}\right)\right)\right) \]

      16. *-lowering-*.f32 98.0%

          \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right)\right), \mathsf{/.f32}\left(sin2phi, \mathsf{*.f32}\left(alphay, \color{blue}{alphay}\right)\right)\right)\right) \]

   3. Simplified 98.0%

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

   5. Taylor expanded in cos2phi around 0

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

      1. mul-1-neg N/A

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

      2. neg-lowering-neg.f32 N/A

         \[\leadsto \mathsf{neg.f32}\left(\left(\frac{{alphay}^{2} \cdot \log \left(1 - u0\right)}{sin2phi}\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{neg.f32}\left(\left(\frac{\log \left(1 - u0\right) \cdot {alphay}^{2}}{sin2phi}\right)\right) \]

      4. associate-/l* N/A

         \[\leadsto \mathsf{neg.f32}\left(\left(\log \left(1 - u0\right) \cdot \frac{{alphay}^{2}}{sin2phi}\right)\right) \]

      5. *-lowering-*.f32 N/A

         \[\leadsto \mathsf{neg.f32}\left(\mathsf{*.f32}\left(\log \left(1 - u0\right), \left(\frac{{alphay}^{2}}{sin2phi}\right)\right)\right) \]

      6. sub-neg N/A

         \[\leadsto \mathsf{neg.f32}\left(\mathsf{*.f32}\left(\log \left(1 + \left(\mathsf{neg}\left(u0\right)\right)\right), \left(\frac{{alphay}^{2}}{sin2phi}\right)\right)\right) \]

      7. mul-1-neg N/A

         \[\leadsto \mathsf{neg.f32}\left(\mathsf{*.f32}\left(\log \left(1 + -1 \cdot u0\right), \left(\frac{{alphay}^{2}}{sin2phi}\right)\right)\right) \]

      8. log1p-define N/A

         \[\leadsto \mathsf{neg.f32}\left(\mathsf{*.f32}\left(\left(\mathsf{log1p}\left(-1 \cdot u0\right)\right), \left(\frac{{alphay}^{2}}{sin2phi}\right)\right)\right) \]

      9. log1p-lowering-log1p.f32 N/A

         \[\leadsto \mathsf{neg.f32}\left(\mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\left(-1 \cdot u0\right)\right), \left(\frac{{alphay}^{2}}{sin2phi}\right)\right)\right) \]

      10. mul-1-neg N/A

          \[\leadsto \mathsf{neg.f32}\left(\mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\left(\mathsf{neg}\left(u0\right)\right)\right), \left(\frac{{alphay}^{2}}{sin2phi}\right)\right)\right) \]

      11. neg-lowering-neg.f32 N/A

          \[\leadsto \mathsf{neg.f32}\left(\mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \left(\frac{{alphay}^{2}}{sin2phi}\right)\right)\right) \]

      12. /-lowering-/.f32 N/A

          \[\leadsto \mathsf{neg.f32}\left(\mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{/.f32}\left(\left({alphay}^{2}\right), sin2phi\right)\right)\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{neg.f32}\left(\mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{/.f32}\left(\left(alphay \cdot alphay\right), sin2phi\right)\right)\right) \]

      14. *-lowering-*.f32 97.0%

          \[\leadsto \mathsf{neg.f32}\left(\mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{/.f32}\left(\mathsf{*.f32}\left(alphay, alphay\right), sin2phi\right)\right)\right) \]

   7. Simplified 97.0%

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

      1. *-commutative N/A

         \[\leadsto \mathsf{neg.f32}\left(\left(\frac{alphay \cdot alphay}{sin2phi} \cdot \log \left(1 + \left(\mathsf{neg}\left(u0\right)\right)\right)\right)\right) \]

      2. associate-/l* N/A

         \[\leadsto \mathsf{neg.f32}\left(\left(\left(alphay \cdot \frac{alphay}{sin2phi}\right) \cdot \log \left(1 + \left(\mathsf{neg}\left(u0\right)\right)\right)\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{neg.f32}\left(\left(alphay \cdot \left(\frac{alphay}{sin2phi} \cdot \log \left(1 + \left(\mathsf{neg}\left(u0\right)\right)\right)\right)\right)\right) \]

      4. *-lowering-*.f32 N/A

         \[\leadsto \mathsf{neg.f32}\left(\mathsf{*.f32}\left(alphay, \left(\frac{alphay}{sin2phi} \cdot \log \left(1 + \left(\mathsf{neg}\left(u0\right)\right)\right)\right)\right)\right) \]

      5. *-lowering-*.f32 N/A

         \[\leadsto \mathsf{neg.f32}\left(\mathsf{*.f32}\left(alphay, \mathsf{*.f32}\left(\left(\frac{alphay}{sin2phi}\right), \log \left(1 + \left(\mathsf{neg}\left(u0\right)\right)\right)\right)\right)\right) \]

      6. /-lowering-/.f32 N/A

         \[\leadsto \mathsf{neg.f32}\left(\mathsf{*.f32}\left(alphay, \mathsf{*.f32}\left(\mathsf{/.f32}\left(alphay, sin2phi\right), \log \left(1 + \left(\mathsf{neg}\left(u0\right)\right)\right)\right)\right)\right) \]

      7. log1p-define N/A

         \[\leadsto \mathsf{neg.f32}\left(\mathsf{*.f32}\left(alphay, \mathsf{*.f32}\left(\mathsf{/.f32}\left(alphay, sin2phi\right), \left(\mathsf{log1p}\left(\mathsf{neg}\left(u0\right)\right)\right)\right)\right)\right) \]

      8. log1p-lowering-log1p.f32 N/A

         \[\leadsto \mathsf{neg.f32}\left(\mathsf{*.f32}\left(alphay, \mathsf{*.f32}\left(\mathsf{/.f32}\left(alphay, sin2phi\right), \mathsf{log1p.f32}\left(\left(\mathsf{neg}\left(u0\right)\right)\right)\right)\right)\right) \]

      9. neg-lowering-neg.f32 96.9%

         \[\leadsto \mathsf{neg.f32}\left(\mathsf{*.f32}\left(alphay, \mathsf{*.f32}\left(\mathsf{/.f32}\left(alphay, sin2phi\right), \mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right)\right)\right)\right) \]

   9. Applied egg-rr 96.9%

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

   10. Taylor expanded in u0 around 0

       \[\leadsto \mathsf{neg.f32}\left(\mathsf{*.f32}\left(alphay, \color{blue}{\left(u0 \cdot \left(-1 \cdot \frac{alphay}{sin2phi} + \frac{-1}{2} \cdot \frac{alphay \cdot u0}{sin2phi}\right)\right)}\right)\right) \]
   11. Step-by-step derivation

       1. *-lowering-*.f32 N/A

          \[\leadsto \mathsf{neg.f32}\left(\mathsf{*.f32}\left(alphay, \mathsf{*.f32}\left(u0, \left(-1 \cdot \frac{alphay}{sin2phi} + \frac{-1}{2} \cdot \frac{alphay \cdot u0}{sin2phi}\right)\right)\right)\right) \]

       2. +-commutative N/A

          \[\leadsto \mathsf{neg.f32}\left(\mathsf{*.f32}\left(alphay, \mathsf{*.f32}\left(u0, \left(\frac{-1}{2} \cdot \frac{alphay \cdot u0}{sin2phi} + -1 \cdot \frac{alphay}{sin2phi}\right)\right)\right)\right) \]

       3. mul-1-neg N/A

          \[\leadsto \mathsf{neg.f32}\left(\mathsf{*.f32}\left(alphay, \mathsf{*.f32}\left(u0, \left(\frac{-1}{2} \cdot \frac{alphay \cdot u0}{sin2phi} + \left(\mathsf{neg}\left(\frac{alphay}{sin2phi}\right)\right)\right)\right)\right)\right) \]

       4. unsub-neg N/A

          \[\leadsto \mathsf{neg.f32}\left(\mathsf{*.f32}\left(alphay, \mathsf{*.f32}\left(u0, \left(\frac{-1}{2} \cdot \frac{alphay \cdot u0}{sin2phi} - \frac{alphay}{sin2phi}\right)\right)\right)\right) \]

       5. --lowering--.f32 N/A

          \[\leadsto \mathsf{neg.f32}\left(\mathsf{*.f32}\left(alphay, \mathsf{*.f32}\left(u0, \mathsf{\_.f32}\left(\left(\frac{-1}{2} \cdot \frac{alphay \cdot u0}{sin2phi}\right), \left(\frac{alphay}{sin2phi}\right)\right)\right)\right)\right) \]

       6. associate-*r/ N/A

          \[\leadsto \mathsf{neg.f32}\left(\mathsf{*.f32}\left(alphay, \mathsf{*.f32}\left(u0, \mathsf{\_.f32}\left(\left(\frac{\frac{-1}{2} \cdot \left(alphay \cdot u0\right)}{sin2phi}\right), \left(\frac{alphay}{sin2phi}\right)\right)\right)\right)\right) \]

       7. /-lowering-/.f32 N/A

          \[\leadsto \mathsf{neg.f32}\left(\mathsf{*.f32}\left(alphay, \mathsf{*.f32}\left(u0, \mathsf{\_.f32}\left(\mathsf{/.f32}\left(\left(\frac{-1}{2} \cdot \left(alphay \cdot u0\right)\right), sin2phi\right), \left(\frac{alphay}{sin2phi}\right)\right)\right)\right)\right) \]

       8. *-lowering-*.f32 N/A

          \[\leadsto \mathsf{neg.f32}\left(\mathsf{*.f32}\left(alphay, \mathsf{*.f32}\left(u0, \mathsf{\_.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(\frac{-1}{2}, \left(alphay \cdot u0\right)\right), sin2phi\right), \left(\frac{alphay}{sin2phi}\right)\right)\right)\right)\right) \]

       9. *-commutative N/A

          \[\leadsto \mathsf{neg.f32}\left(\mathsf{*.f32}\left(alphay, \mathsf{*.f32}\left(u0, \mathsf{\_.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(\frac{-1}{2}, \left(u0 \cdot alphay\right)\right), sin2phi\right), \left(\frac{alphay}{sin2phi}\right)\right)\right)\right)\right) \]

       10. *-lowering-*.f32 N/A

           \[\leadsto \mathsf{neg.f32}\left(\mathsf{*.f32}\left(alphay, \mathsf{*.f32}\left(u0, \mathsf{\_.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(\frac{-1}{2}, \mathsf{*.f32}\left(u0, alphay\right)\right), sin2phi\right), \left(\frac{alphay}{sin2phi}\right)\right)\right)\right)\right) \]

       11. /-lowering-/.f32 86.8%

           \[\leadsto \mathsf{neg.f32}\left(\mathsf{*.f32}\left(alphay, \mathsf{*.f32}\left(u0, \mathsf{\_.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(\frac{-1}{2}, \mathsf{*.f32}\left(u0, alphay\right)\right), sin2phi\right), \mathsf{/.f32}\left(alphay, sin2phi\right)\right)\right)\right)\right) \]

   12. Simplified 86.8%

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

4. Final simplification 79.6%

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

Alternative 20: 80.7% accurate, 6.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;sin2phi \leq 4.999999969612645 \cdot 10^{-9}:\\ \;\;\;\;\frac{u0}{\frac{\frac{sin2phi}{alphay}}{alphay} + \frac{\frac{cos2phi}{alphax}}{alphax}}\\ \mathbf{else}:\\ \;\;\;\;\left(u0 \cdot \left(-1 + u0 \cdot -0.5\right)\right) \cdot \left(-\frac{alphay \cdot alphay}{sin2phi}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (if (<= sin2phi 4.999999969612645e-9)
   (/ u0 (+ (/ (/ sin2phi alphay) alphay) (/ (/ cos2phi alphax) alphax)))
   (* (* u0 (+ -1.0 (* u0 -0.5))) (- (/ (* alphay alphay) sin2phi)))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float tmp;
	if (sin2phi <= 4.999999969612645e-9f) {
		tmp = u0 / (((sin2phi / alphay) / alphay) + ((cos2phi / alphax) / alphax));
	} else {
		tmp = (u0 * (-1.0f + (u0 * -0.5f))) * -((alphay * 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 <= 4.999999969612645e-9) then
        tmp = u0 / (((sin2phi / alphay) / alphay) + ((cos2phi / alphax) / alphax))
    else
        tmp = (u0 * ((-1.0e0) + (u0 * (-0.5e0)))) * -((alphay * alphay) / sin2phi)
    end if
    code = tmp
end function

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = Float32(0.0)
	if (sin2phi <= Float32(4.999999969612645e-9))
		tmp = Float32(u0 / Float32(Float32(Float32(sin2phi / alphay) / alphay) + Float32(Float32(cos2phi / alphax) / alphax)));
	else
		tmp = Float32(Float32(u0 * Float32(Float32(-1.0) + Float32(u0 * Float32(-0.5)))) * Float32(-Float32(Float32(alphay * alphay) / sin2phi)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = single(0.0);
	if (sin2phi <= single(4.999999969612645e-9))
		tmp = u0 / (((sin2phi / alphay) / alphay) + ((cos2phi / alphax) / alphax));
	else
		tmp = (u0 * (single(-1.0) + (u0 * single(-0.5)))) * -((alphay * alphay) / sin2phi);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if sin2phi < 4.99999997e-9

   1. Initial program 64.5%

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

      1. distribute-frac-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. /-lowering-/.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(\log \left(1 - u0\right), \color{blue}{\left(\mathsf{neg}\left(\left(\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\right)\right)\right)}\right) \]

      4. sub-neg N/A

         \[\leadsto \mathsf{/.f32}\left(\log \left(1 + \left(\mathsf{neg}\left(u0\right)\right)\right), \left(\mathsf{neg}\left(\left(\color{blue}{\frac{cos2phi}{alphax \cdot alphax}} + \frac{sin2phi}{alphay \cdot alphay}\right)\right)\right)\right) \]

      5. log1p-define N/A

         \[\leadsto \mathsf{/.f32}\left(\left(\mathsf{log1p}\left(\mathsf{neg}\left(u0\right)\right)\right), \left(\mathsf{neg}\left(\color{blue}{\left(\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\right)}\right)\right)\right) \]

      6. log1p-lowering-log1p.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\left(\mathsf{neg}\left(u0\right)\right)\right), \left(\mathsf{neg}\left(\color{blue}{\left(\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\right)}\right)\right)\right) \]

      7. neg-lowering-neg.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \left(\mathsf{neg}\left(\left(\color{blue}{\frac{cos2phi}{alphax \cdot alphax}} + \frac{sin2phi}{alphay \cdot alphay}\right)\right)\right)\right) \]

      8. distribute-neg-in N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \left(\left(\mathsf{neg}\left(\frac{cos2phi}{alphax \cdot alphax}\right)\right) + \color{blue}{\left(\mathsf{neg}\left(\frac{sin2phi}{alphay \cdot alphay}\right)\right)}\right)\right) \]

      9. unsub-neg N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \left(\left(\mathsf{neg}\left(\frac{cos2phi}{alphax \cdot alphax}\right)\right) - \color{blue}{\frac{sin2phi}{alphay \cdot alphay}}\right)\right) \]

      10. --lowering--.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\left(\mathsf{neg}\left(\frac{cos2phi}{alphax \cdot alphax}\right)\right), \color{blue}{\left(\frac{sin2phi}{alphay \cdot alphay}\right)}\right)\right) \]

      11. neg-sub0 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\left(0 - \frac{cos2phi}{alphax \cdot alphax}\right), \left(\frac{\color{blue}{sin2phi}}{alphay \cdot alphay}\right)\right)\right) \]

      12. --lowering--.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \left(\frac{cos2phi}{alphax \cdot alphax}\right)\right), \left(\frac{\color{blue}{sin2phi}}{alphay \cdot alphay}\right)\right)\right) \]

      13. /-lowering-/.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(cos2phi, \left(alphax \cdot alphax\right)\right)\right), \left(\frac{sin2phi}{alphay \cdot alphay}\right)\right)\right) \]

      14. *-lowering-*.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right)\right), \left(\frac{sin2phi}{alphay \cdot alphay}\right)\right)\right) \]

      15. /-lowering-/.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right)\right), \mathsf{/.f32}\left(sin2phi, \color{blue}{\left(alphay \cdot alphay\right)}\right)\right)\right) \]

      16. *-lowering-*.f32 98.6%

          \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right)\right), \mathsf{/.f32}\left(sin2phi, \mathsf{*.f32}\left(alphay, \color{blue}{alphay}\right)\right)\right)\right) \]

   3. Simplified 98.6%

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

   5. Taylor expanded in u0 around 0

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

      1. /-lowering-/.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(u0, \color{blue}{\left(\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}\right)}\right) \]

      2. +-lowering-+.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{+.f32}\left(\left(\frac{cos2phi}{{alphax}^{2}}\right), \color{blue}{\left(\frac{sin2phi}{{alphay}^{2}}\right)}\right)\right) \]

      3. /-lowering-/.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{/.f32}\left(cos2phi, \left({alphax}^{2}\right)\right), \left(\frac{\color{blue}{sin2phi}}{{alphay}^{2}}\right)\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{/.f32}\left(cos2phi, \left(alphax \cdot alphax\right)\right), \left(\frac{sin2phi}{{alphay}^{2}}\right)\right)\right) \]

      5. *-lowering-*.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right), \left(\frac{sin2phi}{{alphay}^{2}}\right)\right)\right) \]

      6. /-lowering-/.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right), \mathsf{/.f32}\left(sin2phi, \color{blue}{\left({alphay}^{2}\right)}\right)\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right), \mathsf{/.f32}\left(sin2phi, \left(alphay \cdot \color{blue}{alphay}\right)\right)\right)\right) \]

      8. *-lowering-*.f32 68.0%

         \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right), \mathsf{/.f32}\left(sin2phi, \mathsf{*.f32}\left(alphay, \color{blue}{alphay}\right)\right)\right)\right) \]

   7. Simplified 68.0%

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

      1. +-commutative N/A

         \[\leadsto \mathsf{/.f32}\left(u0, \left(\frac{sin2phi}{alphay \cdot alphay} + \color{blue}{\frac{cos2phi}{alphax \cdot alphax}}\right)\right) \]

      2. +-lowering-+.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{+.f32}\left(\left(\frac{sin2phi}{alphay \cdot alphay}\right), \color{blue}{\left(\frac{cos2phi}{alphax \cdot alphax}\right)}\right)\right) \]

      3. associate-/r* N/A

         \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{+.f32}\left(\left(\frac{\frac{sin2phi}{alphay}}{alphay}\right), \left(\frac{\color{blue}{cos2phi}}{alphax \cdot alphax}\right)\right)\right) \]

      4. /-lowering-/.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{/.f32}\left(\left(\frac{sin2phi}{alphay}\right), alphay\right), \left(\frac{\color{blue}{cos2phi}}{alphax \cdot alphax}\right)\right)\right) \]

      5. /-lowering-/.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(sin2phi, alphay\right), alphay\right), \left(\frac{cos2phi}{alphax \cdot alphax}\right)\right)\right) \]

      6. associate-/r* N/A

         \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(sin2phi, alphay\right), alphay\right), \left(\frac{\frac{cos2phi}{alphax}}{\color{blue}{alphax}}\right)\right)\right) \]

      7. /-lowering-/.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(sin2phi, alphay\right), alphay\right), \mathsf{/.f32}\left(\left(\frac{cos2phi}{alphax}\right), \color{blue}{alphax}\right)\right)\right) \]

      8. /-lowering-/.f32 68.2%

         \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(sin2phi, alphay\right), alphay\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(cos2phi, alphax\right), alphax\right)\right)\right) \]

   9. Applied egg-rr 68.2%

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

   if 4.99999997e-9 < sin2phi

   1. Initial program 65.0%

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

      1. distribute-frac-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. /-lowering-/.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(\log \left(1 - u0\right), \color{blue}{\left(\mathsf{neg}\left(\left(\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\right)\right)\right)}\right) \]

      4. sub-neg N/A

         \[\leadsto \mathsf{/.f32}\left(\log \left(1 + \left(\mathsf{neg}\left(u0\right)\right)\right), \left(\mathsf{neg}\left(\left(\color{blue}{\frac{cos2phi}{alphax \cdot alphax}} + \frac{sin2phi}{alphay \cdot alphay}\right)\right)\right)\right) \]

      5. log1p-define N/A

         \[\leadsto \mathsf{/.f32}\left(\left(\mathsf{log1p}\left(\mathsf{neg}\left(u0\right)\right)\right), \left(\mathsf{neg}\left(\color{blue}{\left(\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\right)}\right)\right)\right) \]

      6. log1p-lowering-log1p.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\left(\mathsf{neg}\left(u0\right)\right)\right), \left(\mathsf{neg}\left(\color{blue}{\left(\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\right)}\right)\right)\right) \]

      7. neg-lowering-neg.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \left(\mathsf{neg}\left(\left(\color{blue}{\frac{cos2phi}{alphax \cdot alphax}} + \frac{sin2phi}{alphay \cdot alphay}\right)\right)\right)\right) \]

      8. distribute-neg-in N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \left(\left(\mathsf{neg}\left(\frac{cos2phi}{alphax \cdot alphax}\right)\right) + \color{blue}{\left(\mathsf{neg}\left(\frac{sin2phi}{alphay \cdot alphay}\right)\right)}\right)\right) \]

      9. unsub-neg N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \left(\left(\mathsf{neg}\left(\frac{cos2phi}{alphax \cdot alphax}\right)\right) - \color{blue}{\frac{sin2phi}{alphay \cdot alphay}}\right)\right) \]

      10. --lowering--.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\left(\mathsf{neg}\left(\frac{cos2phi}{alphax \cdot alphax}\right)\right), \color{blue}{\left(\frac{sin2phi}{alphay \cdot alphay}\right)}\right)\right) \]

      11. neg-sub0 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\left(0 - \frac{cos2phi}{alphax \cdot alphax}\right), \left(\frac{\color{blue}{sin2phi}}{alphay \cdot alphay}\right)\right)\right) \]

      12. --lowering--.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \left(\frac{cos2phi}{alphax \cdot alphax}\right)\right), \left(\frac{\color{blue}{sin2phi}}{alphay \cdot alphay}\right)\right)\right) \]

      13. /-lowering-/.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(cos2phi, \left(alphax \cdot alphax\right)\right)\right), \left(\frac{sin2phi}{alphay \cdot alphay}\right)\right)\right) \]

      14. *-lowering-*.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right)\right), \left(\frac{sin2phi}{alphay \cdot alphay}\right)\right)\right) \]

      15. /-lowering-/.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right)\right), \mathsf{/.f32}\left(sin2phi, \color{blue}{\left(alphay \cdot alphay\right)}\right)\right)\right) \]

      16. *-lowering-*.f32 98.0%

          \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right)\right), \mathsf{/.f32}\left(sin2phi, \mathsf{*.f32}\left(alphay, \color{blue}{alphay}\right)\right)\right)\right) \]

   3. Simplified 98.0%

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

   5. Taylor expanded in cos2phi around 0

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

      1. mul-1-neg N/A

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

      2. neg-lowering-neg.f32 N/A

         \[\leadsto \mathsf{neg.f32}\left(\left(\frac{{alphay}^{2} \cdot \log \left(1 - u0\right)}{sin2phi}\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{neg.f32}\left(\left(\frac{\log \left(1 - u0\right) \cdot {alphay}^{2}}{sin2phi}\right)\right) \]

      4. associate-/l* N/A

         \[\leadsto \mathsf{neg.f32}\left(\left(\log \left(1 - u0\right) \cdot \frac{{alphay}^{2}}{sin2phi}\right)\right) \]

      5. *-lowering-*.f32 N/A

         \[\leadsto \mathsf{neg.f32}\left(\mathsf{*.f32}\left(\log \left(1 - u0\right), \left(\frac{{alphay}^{2}}{sin2phi}\right)\right)\right) \]

      6. sub-neg N/A

         \[\leadsto \mathsf{neg.f32}\left(\mathsf{*.f32}\left(\log \left(1 + \left(\mathsf{neg}\left(u0\right)\right)\right), \left(\frac{{alphay}^{2}}{sin2phi}\right)\right)\right) \]

      7. mul-1-neg N/A

         \[\leadsto \mathsf{neg.f32}\left(\mathsf{*.f32}\left(\log \left(1 + -1 \cdot u0\right), \left(\frac{{alphay}^{2}}{sin2phi}\right)\right)\right) \]

      8. log1p-define N/A

         \[\leadsto \mathsf{neg.f32}\left(\mathsf{*.f32}\left(\left(\mathsf{log1p}\left(-1 \cdot u0\right)\right), \left(\frac{{alphay}^{2}}{sin2phi}\right)\right)\right) \]

      9. log1p-lowering-log1p.f32 N/A

         \[\leadsto \mathsf{neg.f32}\left(\mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\left(-1 \cdot u0\right)\right), \left(\frac{{alphay}^{2}}{sin2phi}\right)\right)\right) \]

      10. mul-1-neg N/A

          \[\leadsto \mathsf{neg.f32}\left(\mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\left(\mathsf{neg}\left(u0\right)\right)\right), \left(\frac{{alphay}^{2}}{sin2phi}\right)\right)\right) \]

      11. neg-lowering-neg.f32 N/A

          \[\leadsto \mathsf{neg.f32}\left(\mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \left(\frac{{alphay}^{2}}{sin2phi}\right)\right)\right) \]

      12. /-lowering-/.f32 N/A

          \[\leadsto \mathsf{neg.f32}\left(\mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{/.f32}\left(\left({alphay}^{2}\right), sin2phi\right)\right)\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{neg.f32}\left(\mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{/.f32}\left(\left(alphay \cdot alphay\right), sin2phi\right)\right)\right) \]

      14. *-lowering-*.f32 97.0%

          \[\leadsto \mathsf{neg.f32}\left(\mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{/.f32}\left(\mathsf{*.f32}\left(alphay, alphay\right), sin2phi\right)\right)\right) \]

   7. Simplified 97.0%

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

   8. Taylor expanded in u0 around 0

      \[\leadsto \mathsf{neg.f32}\left(\mathsf{*.f32}\left(\color{blue}{\left(u0 \cdot \left(\frac{-1}{2} \cdot u0 - 1\right)\right)}, \mathsf{/.f32}\left(\mathsf{*.f32}\left(alphay, alphay\right), sin2phi\right)\right)\right) \]
   9. Step-by-step derivation

      1. *-lowering-*.f32 N/A

         \[\leadsto \mathsf{neg.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(u0, \left(\frac{-1}{2} \cdot u0 - 1\right)\right), \mathsf{/.f32}\left(\mathsf{*.f32}\left(alphay, alphay\right), sin2phi\right)\right)\right) \]

      2. sub-neg N/A

         \[\leadsto \mathsf{neg.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(u0, \left(\frac{-1}{2} \cdot u0 + \left(\mathsf{neg}\left(1\right)\right)\right)\right), \mathsf{/.f32}\left(\mathsf{*.f32}\left(alphay, alphay\right), sin2phi\right)\right)\right) \]

      3. metadata-eval N/A

         \[\leadsto \mathsf{neg.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(u0, \left(\frac{-1}{2} \cdot u0 + -1\right)\right), \mathsf{/.f32}\left(\mathsf{*.f32}\left(alphay, alphay\right), sin2phi\right)\right)\right) \]

      4. +-lowering-+.f32 N/A

         \[\leadsto \mathsf{neg.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\left(\frac{-1}{2} \cdot u0\right), -1\right)\right), \mathsf{/.f32}\left(\mathsf{*.f32}\left(alphay, alphay\right), sin2phi\right)\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{neg.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\left(u0 \cdot \frac{-1}{2}\right), -1\right)\right), \mathsf{/.f32}\left(\mathsf{*.f32}\left(alphay, alphay\right), sin2phi\right)\right)\right) \]

      6. *-lowering-*.f32 86.6%

         \[\leadsto \mathsf{neg.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{*.f32}\left(u0, \frac{-1}{2}\right), -1\right)\right), \mathsf{/.f32}\left(\mathsf{*.f32}\left(alphay, alphay\right), sin2phi\right)\right)\right) \]

   10. Simplified 86.6%

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

4. Final simplification 79.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;sin2phi \leq 4.999999969612645 \cdot 10^{-9}:\\ \;\;\;\;\frac{u0}{\frac{\frac{sin2phi}{alphay}}{alphay} + \frac{\frac{cos2phi}{alphax}}{alphax}}\\ \mathbf{else}:\\ \;\;\;\;\left(u0 \cdot \left(-1 + u0 \cdot -0.5\right)\right) \cdot \left(-\frac{alphay \cdot alphay}{sin2phi}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 21: 80.7% accurate, 6.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;sin2phi \leq 4.999999969612645 \cdot 10^{-9}:\\ \;\;\;\;\frac{u0}{\frac{\frac{sin2phi}{alphay}}{alphay} + \frac{\frac{cos2phi}{alphax}}{alphax}}\\ \mathbf{else}:\\ \;\;\;\;alphay \cdot \left(\frac{alphay}{sin2phi} \cdot \left(u0 \cdot \left(\left(--1\right) - u0 \cdot -0.5\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (if (<= sin2phi 4.999999969612645e-9)
   (/ u0 (+ (/ (/ sin2phi alphay) alphay) (/ (/ cos2phi alphax) alphax)))
   (* alphay (* (/ alphay sin2phi) (* u0 (- (- -1.0) (* u0 -0.5)))))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float tmp;
	if (sin2phi <= 4.999999969612645e-9f) {
		tmp = u0 / (((sin2phi / alphay) / alphay) + ((cos2phi / alphax) / alphax));
	} else {
		tmp = alphay * ((alphay / sin2phi) * (u0 * (-(-1.0f) - (u0 * -0.5f))));
	}
	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 <= 4.999999969612645e-9) then
        tmp = u0 / (((sin2phi / alphay) / alphay) + ((cos2phi / alphax) / alphax))
    else
        tmp = alphay * ((alphay / sin2phi) * (u0 * (-(-1.0e0) - (u0 * (-0.5e0)))))
    end if
    code = tmp
end function

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = Float32(0.0)
	if (sin2phi <= Float32(4.999999969612645e-9))
		tmp = Float32(u0 / Float32(Float32(Float32(sin2phi / alphay) / alphay) + Float32(Float32(cos2phi / alphax) / alphax)));
	else
		tmp = Float32(alphay * Float32(Float32(alphay / sin2phi) * Float32(u0 * Float32(Float32(-Float32(-1.0)) - Float32(u0 * Float32(-0.5))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = single(0.0);
	if (sin2phi <= single(4.999999969612645e-9))
		tmp = u0 / (((sin2phi / alphay) / alphay) + ((cos2phi / alphax) / alphax));
	else
		tmp = alphay * ((alphay / sin2phi) * (u0 * (-single(-1.0) - (u0 * single(-0.5)))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if sin2phi < 4.99999997e-9

   1. Initial program 64.5%

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

      1. distribute-frac-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. /-lowering-/.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(\log \left(1 - u0\right), \color{blue}{\left(\mathsf{neg}\left(\left(\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\right)\right)\right)}\right) \]

      4. sub-neg N/A

         \[\leadsto \mathsf{/.f32}\left(\log \left(1 + \left(\mathsf{neg}\left(u0\right)\right)\right), \left(\mathsf{neg}\left(\left(\color{blue}{\frac{cos2phi}{alphax \cdot alphax}} + \frac{sin2phi}{alphay \cdot alphay}\right)\right)\right)\right) \]

      5. log1p-define N/A

         \[\leadsto \mathsf{/.f32}\left(\left(\mathsf{log1p}\left(\mathsf{neg}\left(u0\right)\right)\right), \left(\mathsf{neg}\left(\color{blue}{\left(\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\right)}\right)\right)\right) \]

      6. log1p-lowering-log1p.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\left(\mathsf{neg}\left(u0\right)\right)\right), \left(\mathsf{neg}\left(\color{blue}{\left(\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\right)}\right)\right)\right) \]

      7. neg-lowering-neg.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \left(\mathsf{neg}\left(\left(\color{blue}{\frac{cos2phi}{alphax \cdot alphax}} + \frac{sin2phi}{alphay \cdot alphay}\right)\right)\right)\right) \]

      8. distribute-neg-in N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \left(\left(\mathsf{neg}\left(\frac{cos2phi}{alphax \cdot alphax}\right)\right) + \color{blue}{\left(\mathsf{neg}\left(\frac{sin2phi}{alphay \cdot alphay}\right)\right)}\right)\right) \]

      9. unsub-neg N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \left(\left(\mathsf{neg}\left(\frac{cos2phi}{alphax \cdot alphax}\right)\right) - \color{blue}{\frac{sin2phi}{alphay \cdot alphay}}\right)\right) \]

      10. --lowering--.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\left(\mathsf{neg}\left(\frac{cos2phi}{alphax \cdot alphax}\right)\right), \color{blue}{\left(\frac{sin2phi}{alphay \cdot alphay}\right)}\right)\right) \]

      11. neg-sub0 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\left(0 - \frac{cos2phi}{alphax \cdot alphax}\right), \left(\frac{\color{blue}{sin2phi}}{alphay \cdot alphay}\right)\right)\right) \]

      12. --lowering--.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \left(\frac{cos2phi}{alphax \cdot alphax}\right)\right), \left(\frac{\color{blue}{sin2phi}}{alphay \cdot alphay}\right)\right)\right) \]

      13. /-lowering-/.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(cos2phi, \left(alphax \cdot alphax\right)\right)\right), \left(\frac{sin2phi}{alphay \cdot alphay}\right)\right)\right) \]

      14. *-lowering-*.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right)\right), \left(\frac{sin2phi}{alphay \cdot alphay}\right)\right)\right) \]

      15. /-lowering-/.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right)\right), \mathsf{/.f32}\left(sin2phi, \color{blue}{\left(alphay \cdot alphay\right)}\right)\right)\right) \]

      16. *-lowering-*.f32 98.6%

          \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right)\right), \mathsf{/.f32}\left(sin2phi, \mathsf{*.f32}\left(alphay, \color{blue}{alphay}\right)\right)\right)\right) \]

   3. Simplified 98.6%

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

   5. Taylor expanded in u0 around 0

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

      1. /-lowering-/.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(u0, \color{blue}{\left(\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}\right)}\right) \]

      2. +-lowering-+.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{+.f32}\left(\left(\frac{cos2phi}{{alphax}^{2}}\right), \color{blue}{\left(\frac{sin2phi}{{alphay}^{2}}\right)}\right)\right) \]

      3. /-lowering-/.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{/.f32}\left(cos2phi, \left({alphax}^{2}\right)\right), \left(\frac{\color{blue}{sin2phi}}{{alphay}^{2}}\right)\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{/.f32}\left(cos2phi, \left(alphax \cdot alphax\right)\right), \left(\frac{sin2phi}{{alphay}^{2}}\right)\right)\right) \]

      5. *-lowering-*.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right), \left(\frac{sin2phi}{{alphay}^{2}}\right)\right)\right) \]

      6. /-lowering-/.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right), \mathsf{/.f32}\left(sin2phi, \color{blue}{\left({alphay}^{2}\right)}\right)\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right), \mathsf{/.f32}\left(sin2phi, \left(alphay \cdot \color{blue}{alphay}\right)\right)\right)\right) \]

      8. *-lowering-*.f32 68.0%

         \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right), \mathsf{/.f32}\left(sin2phi, \mathsf{*.f32}\left(alphay, \color{blue}{alphay}\right)\right)\right)\right) \]

   7. Simplified 68.0%

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

      1. +-commutative N/A

         \[\leadsto \mathsf{/.f32}\left(u0, \left(\frac{sin2phi}{alphay \cdot alphay} + \color{blue}{\frac{cos2phi}{alphax \cdot alphax}}\right)\right) \]

      2. +-lowering-+.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{+.f32}\left(\left(\frac{sin2phi}{alphay \cdot alphay}\right), \color{blue}{\left(\frac{cos2phi}{alphax \cdot alphax}\right)}\right)\right) \]

      3. associate-/r* N/A

         \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{+.f32}\left(\left(\frac{\frac{sin2phi}{alphay}}{alphay}\right), \left(\frac{\color{blue}{cos2phi}}{alphax \cdot alphax}\right)\right)\right) \]

      4. /-lowering-/.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{/.f32}\left(\left(\frac{sin2phi}{alphay}\right), alphay\right), \left(\frac{\color{blue}{cos2phi}}{alphax \cdot alphax}\right)\right)\right) \]

      5. /-lowering-/.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(sin2phi, alphay\right), alphay\right), \left(\frac{cos2phi}{alphax \cdot alphax}\right)\right)\right) \]

      6. associate-/r* N/A

         \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(sin2phi, alphay\right), alphay\right), \left(\frac{\frac{cos2phi}{alphax}}{\color{blue}{alphax}}\right)\right)\right) \]

      7. /-lowering-/.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(sin2phi, alphay\right), alphay\right), \mathsf{/.f32}\left(\left(\frac{cos2phi}{alphax}\right), \color{blue}{alphax}\right)\right)\right) \]

      8. /-lowering-/.f32 68.2%

         \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(sin2phi, alphay\right), alphay\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(cos2phi, alphax\right), alphax\right)\right)\right) \]

   9. Applied egg-rr 68.2%

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

   if 4.99999997e-9 < sin2phi

   1. Initial program 65.0%

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

      1. distribute-frac-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. /-lowering-/.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(\log \left(1 - u0\right), \color{blue}{\left(\mathsf{neg}\left(\left(\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\right)\right)\right)}\right) \]

      4. sub-neg N/A

         \[\leadsto \mathsf{/.f32}\left(\log \left(1 + \left(\mathsf{neg}\left(u0\right)\right)\right), \left(\mathsf{neg}\left(\left(\color{blue}{\frac{cos2phi}{alphax \cdot alphax}} + \frac{sin2phi}{alphay \cdot alphay}\right)\right)\right)\right) \]

      5. log1p-define N/A

         \[\leadsto \mathsf{/.f32}\left(\left(\mathsf{log1p}\left(\mathsf{neg}\left(u0\right)\right)\right), \left(\mathsf{neg}\left(\color{blue}{\left(\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\right)}\right)\right)\right) \]

      6. log1p-lowering-log1p.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\left(\mathsf{neg}\left(u0\right)\right)\right), \left(\mathsf{neg}\left(\color{blue}{\left(\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\right)}\right)\right)\right) \]

      7. neg-lowering-neg.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \left(\mathsf{neg}\left(\left(\color{blue}{\frac{cos2phi}{alphax \cdot alphax}} + \frac{sin2phi}{alphay \cdot alphay}\right)\right)\right)\right) \]

      8. distribute-neg-in N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \left(\left(\mathsf{neg}\left(\frac{cos2phi}{alphax \cdot alphax}\right)\right) + \color{blue}{\left(\mathsf{neg}\left(\frac{sin2phi}{alphay \cdot alphay}\right)\right)}\right)\right) \]

      9. unsub-neg N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \left(\left(\mathsf{neg}\left(\frac{cos2phi}{alphax \cdot alphax}\right)\right) - \color{blue}{\frac{sin2phi}{alphay \cdot alphay}}\right)\right) \]

      10. --lowering--.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\left(\mathsf{neg}\left(\frac{cos2phi}{alphax \cdot alphax}\right)\right), \color{blue}{\left(\frac{sin2phi}{alphay \cdot alphay}\right)}\right)\right) \]

      11. neg-sub0 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\left(0 - \frac{cos2phi}{alphax \cdot alphax}\right), \left(\frac{\color{blue}{sin2phi}}{alphay \cdot alphay}\right)\right)\right) \]

      12. --lowering--.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \left(\frac{cos2phi}{alphax \cdot alphax}\right)\right), \left(\frac{\color{blue}{sin2phi}}{alphay \cdot alphay}\right)\right)\right) \]

      13. /-lowering-/.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(cos2phi, \left(alphax \cdot alphax\right)\right)\right), \left(\frac{sin2phi}{alphay \cdot alphay}\right)\right)\right) \]

      14. *-lowering-*.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right)\right), \left(\frac{sin2phi}{alphay \cdot alphay}\right)\right)\right) \]

      15. /-lowering-/.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right)\right), \mathsf{/.f32}\left(sin2phi, \color{blue}{\left(alphay \cdot alphay\right)}\right)\right)\right) \]

      16. *-lowering-*.f32 98.0%

          \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right)\right), \mathsf{/.f32}\left(sin2phi, \mathsf{*.f32}\left(alphay, \color{blue}{alphay}\right)\right)\right)\right) \]

   3. Simplified 98.0%

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

   5. Taylor expanded in cos2phi around 0

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

      1. mul-1-neg N/A

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

      2. neg-lowering-neg.f32 N/A

         \[\leadsto \mathsf{neg.f32}\left(\left(\frac{{alphay}^{2} \cdot \log \left(1 - u0\right)}{sin2phi}\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{neg.f32}\left(\left(\frac{\log \left(1 - u0\right) \cdot {alphay}^{2}}{sin2phi}\right)\right) \]

      4. associate-/l* N/A

         \[\leadsto \mathsf{neg.f32}\left(\left(\log \left(1 - u0\right) \cdot \frac{{alphay}^{2}}{sin2phi}\right)\right) \]

      5. *-lowering-*.f32 N/A

         \[\leadsto \mathsf{neg.f32}\left(\mathsf{*.f32}\left(\log \left(1 - u0\right), \left(\frac{{alphay}^{2}}{sin2phi}\right)\right)\right) \]

      6. sub-neg N/A

         \[\leadsto \mathsf{neg.f32}\left(\mathsf{*.f32}\left(\log \left(1 + \left(\mathsf{neg}\left(u0\right)\right)\right), \left(\frac{{alphay}^{2}}{sin2phi}\right)\right)\right) \]

      7. mul-1-neg N/A

         \[\leadsto \mathsf{neg.f32}\left(\mathsf{*.f32}\left(\log \left(1 + -1 \cdot u0\right), \left(\frac{{alphay}^{2}}{sin2phi}\right)\right)\right) \]

      8. log1p-define N/A

         \[\leadsto \mathsf{neg.f32}\left(\mathsf{*.f32}\left(\left(\mathsf{log1p}\left(-1 \cdot u0\right)\right), \left(\frac{{alphay}^{2}}{sin2phi}\right)\right)\right) \]

      9. log1p-lowering-log1p.f32 N/A

         \[\leadsto \mathsf{neg.f32}\left(\mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\left(-1 \cdot u0\right)\right), \left(\frac{{alphay}^{2}}{sin2phi}\right)\right)\right) \]

      10. mul-1-neg N/A

          \[\leadsto \mathsf{neg.f32}\left(\mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\left(\mathsf{neg}\left(u0\right)\right)\right), \left(\frac{{alphay}^{2}}{sin2phi}\right)\right)\right) \]

      11. neg-lowering-neg.f32 N/A

          \[\leadsto \mathsf{neg.f32}\left(\mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \left(\frac{{alphay}^{2}}{sin2phi}\right)\right)\right) \]

      12. /-lowering-/.f32 N/A

          \[\leadsto \mathsf{neg.f32}\left(\mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{/.f32}\left(\left({alphay}^{2}\right), sin2phi\right)\right)\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{neg.f32}\left(\mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{/.f32}\left(\left(alphay \cdot alphay\right), sin2phi\right)\right)\right) \]

      14. *-lowering-*.f32 97.0%

          \[\leadsto \mathsf{neg.f32}\left(\mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{/.f32}\left(\mathsf{*.f32}\left(alphay, alphay\right), sin2phi\right)\right)\right) \]

   7. Simplified 97.0%

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

      1. *-commutative N/A

         \[\leadsto \mathsf{neg.f32}\left(\left(\frac{alphay \cdot alphay}{sin2phi} \cdot \log \left(1 + \left(\mathsf{neg}\left(u0\right)\right)\right)\right)\right) \]

      2. associate-/l* N/A

         \[\leadsto \mathsf{neg.f32}\left(\left(\left(alphay \cdot \frac{alphay}{sin2phi}\right) \cdot \log \left(1 + \left(\mathsf{neg}\left(u0\right)\right)\right)\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{neg.f32}\left(\left(alphay \cdot \left(\frac{alphay}{sin2phi} \cdot \log \left(1 + \left(\mathsf{neg}\left(u0\right)\right)\right)\right)\right)\right) \]

      4. *-lowering-*.f32 N/A

         \[\leadsto \mathsf{neg.f32}\left(\mathsf{*.f32}\left(alphay, \left(\frac{alphay}{sin2phi} \cdot \log \left(1 + \left(\mathsf{neg}\left(u0\right)\right)\right)\right)\right)\right) \]

      5. *-lowering-*.f32 N/A

         \[\leadsto \mathsf{neg.f32}\left(\mathsf{*.f32}\left(alphay, \mathsf{*.f32}\left(\left(\frac{alphay}{sin2phi}\right), \log \left(1 + \left(\mathsf{neg}\left(u0\right)\right)\right)\right)\right)\right) \]

      6. /-lowering-/.f32 N/A

         \[\leadsto \mathsf{neg.f32}\left(\mathsf{*.f32}\left(alphay, \mathsf{*.f32}\left(\mathsf{/.f32}\left(alphay, sin2phi\right), \log \left(1 + \left(\mathsf{neg}\left(u0\right)\right)\right)\right)\right)\right) \]

      7. log1p-define N/A

         \[\leadsto \mathsf{neg.f32}\left(\mathsf{*.f32}\left(alphay, \mathsf{*.f32}\left(\mathsf{/.f32}\left(alphay, sin2phi\right), \left(\mathsf{log1p}\left(\mathsf{neg}\left(u0\right)\right)\right)\right)\right)\right) \]

      8. log1p-lowering-log1p.f32 N/A

         \[\leadsto \mathsf{neg.f32}\left(\mathsf{*.f32}\left(alphay, \mathsf{*.f32}\left(\mathsf{/.f32}\left(alphay, sin2phi\right), \mathsf{log1p.f32}\left(\left(\mathsf{neg}\left(u0\right)\right)\right)\right)\right)\right) \]

      9. neg-lowering-neg.f32 96.9%

         \[\leadsto \mathsf{neg.f32}\left(\mathsf{*.f32}\left(alphay, \mathsf{*.f32}\left(\mathsf{/.f32}\left(alphay, sin2phi\right), \mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right)\right)\right)\right) \]

   9. Applied egg-rr 96.9%

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

   10. Taylor expanded in u0 around 0

       \[\leadsto \mathsf{neg.f32}\left(\mathsf{*.f32}\left(alphay, \mathsf{*.f32}\left(\mathsf{/.f32}\left(alphay, sin2phi\right), \color{blue}{\left(u0 \cdot \left(\frac{-1}{2} \cdot u0 - 1\right)\right)}\right)\right)\right) \]
   11. Step-by-step derivation

       1. *-lowering-*.f32 N/A

          \[\leadsto \mathsf{neg.f32}\left(\mathsf{*.f32}\left(alphay, \mathsf{*.f32}\left(\mathsf{/.f32}\left(alphay, sin2phi\right), \mathsf{*.f32}\left(u0, \left(\frac{-1}{2} \cdot u0 - 1\right)\right)\right)\right)\right) \]

       2. sub-neg N/A

          \[\leadsto \mathsf{neg.f32}\left(\mathsf{*.f32}\left(alphay, \mathsf{*.f32}\left(\mathsf{/.f32}\left(alphay, sin2phi\right), \mathsf{*.f32}\left(u0, \left(\frac{-1}{2} \cdot u0 + \left(\mathsf{neg}\left(1\right)\right)\right)\right)\right)\right)\right) \]

       3. metadata-eval N/A

          \[\leadsto \mathsf{neg.f32}\left(\mathsf{*.f32}\left(alphay, \mathsf{*.f32}\left(\mathsf{/.f32}\left(alphay, sin2phi\right), \mathsf{*.f32}\left(u0, \left(\frac{-1}{2} \cdot u0 + -1\right)\right)\right)\right)\right) \]

       4. +-lowering-+.f32 N/A

          \[\leadsto \mathsf{neg.f32}\left(\mathsf{*.f32}\left(alphay, \mathsf{*.f32}\left(\mathsf{/.f32}\left(alphay, sin2phi\right), \mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\left(\frac{-1}{2} \cdot u0\right), -1\right)\right)\right)\right)\right) \]

       5. *-commutative N/A

          \[\leadsto \mathsf{neg.f32}\left(\mathsf{*.f32}\left(alphay, \mathsf{*.f32}\left(\mathsf{/.f32}\left(alphay, sin2phi\right), \mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\left(u0 \cdot \frac{-1}{2}\right), -1\right)\right)\right)\right)\right) \]

       6. *-lowering-*.f32 86.6%

          \[\leadsto \mathsf{neg.f32}\left(\mathsf{*.f32}\left(alphay, \mathsf{*.f32}\left(\mathsf{/.f32}\left(alphay, sin2phi\right), \mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{*.f32}\left(u0, \frac{-1}{2}\right), -1\right)\right)\right)\right)\right) \]

   12. Simplified 86.6%

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

4. Final simplification 79.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;sin2phi \leq 4.999999969612645 \cdot 10^{-9}:\\ \;\;\;\;\frac{u0}{\frac{\frac{sin2phi}{alphay}}{alphay} + \frac{\frac{cos2phi}{alphax}}{alphax}}\\ \mathbf{else}:\\ \;\;\;\;alphay \cdot \left(\frac{alphay}{sin2phi} \cdot \left(u0 \cdot \left(\left(--1\right) - u0 \cdot -0.5\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 22: 87.1% accurate, 6.1× speedup

Math

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

FPCore

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

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	return (u0 * (1.0f + (u0 * 0.5f))) / ((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 = (u0 * (1.0e0 + (u0 * 0.5e0))) / ((cos2phi / (alphax * alphax)) + (sin2phi / (alphay * 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(cos2phi / Float32(alphax * alphax)) + Float32(sin2phi / Float32(alphay * alphay))))
end

MATLAB

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

Derivation

1. Initial program 64.8%

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

3. Taylor expanded in u0 around 0

   \[\leadsto \mathsf{/.f32}\left(\color{blue}{\left(u0 \cdot \left(1 + \frac{1}{2} \cdot u0\right)\right)}, \mathsf{+.f32}\left(\mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right), \mathsf{/.f32}\left(sin2phi, \mathsf{*.f32}\left(alphay, alphay\right)\right)\right)\right) \]
4. Step-by-step derivation

   1. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(u0, \left(1 + \frac{1}{2} \cdot u0\right)\right), \mathsf{+.f32}\left(\color{blue}{\mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right)}, \mathsf{/.f32}\left(sin2phi, \mathsf{*.f32}\left(alphay, alphay\right)\right)\right)\right) \]

   2. +-lowering-+.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(1, \left(\frac{1}{2} \cdot u0\right)\right)\right), \mathsf{+.f32}\left(\mathsf{/.f32}\left(cos2phi, \color{blue}{\mathsf{*.f32}\left(alphax, alphax\right)}\right), \mathsf{/.f32}\left(sin2phi, \mathsf{*.f32}\left(alphay, alphay\right)\right)\right)\right) \]

   3. *-commutative N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(1, \left(u0 \cdot \frac{1}{2}\right)\right)\right), \mathsf{+.f32}\left(\mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, \color{blue}{alphax}\right)\right), \mathsf{/.f32}\left(sin2phi, \mathsf{*.f32}\left(alphay, alphay\right)\right)\right)\right) \]

   4. *-lowering-*.f32 85.0%

      \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(u0, \frac{1}{2}\right)\right)\right), \mathsf{+.f32}\left(\mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, \color{blue}{alphax}\right)\right), \mathsf{/.f32}\left(sin2phi, \mathsf{*.f32}\left(alphay, alphay\right)\right)\right)\right) \]

5. Simplified 85.0%

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

Alternative 23: 75.5% accurate, 8.9× speedup

Math

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

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(Float32(sin2phi / alphay) / alphay) + Float32(Float32(cos2phi / 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 64.8%

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

   1. distribute-frac-neg N/A

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

   2. distribute-neg-frac2 N/A

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

   3. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\log \left(1 - u0\right), \color{blue}{\left(\mathsf{neg}\left(\left(\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\right)\right)\right)}\right) \]

   4. sub-neg N/A

      \[\leadsto \mathsf{/.f32}\left(\log \left(1 + \left(\mathsf{neg}\left(u0\right)\right)\right), \left(\mathsf{neg}\left(\left(\color{blue}{\frac{cos2phi}{alphax \cdot alphax}} + \frac{sin2phi}{alphay \cdot alphay}\right)\right)\right)\right) \]

   5. log1p-define N/A

      \[\leadsto \mathsf{/.f32}\left(\left(\mathsf{log1p}\left(\mathsf{neg}\left(u0\right)\right)\right), \left(\mathsf{neg}\left(\color{blue}{\left(\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\right)}\right)\right)\right) \]

   6. log1p-lowering-log1p.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\left(\mathsf{neg}\left(u0\right)\right)\right), \left(\mathsf{neg}\left(\color{blue}{\left(\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\right)}\right)\right)\right) \]

   7. neg-lowering-neg.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \left(\mathsf{neg}\left(\left(\color{blue}{\frac{cos2phi}{alphax \cdot alphax}} + \frac{sin2phi}{alphay \cdot alphay}\right)\right)\right)\right) \]

   8. distribute-neg-in N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \left(\left(\mathsf{neg}\left(\frac{cos2phi}{alphax \cdot alphax}\right)\right) + \color{blue}{\left(\mathsf{neg}\left(\frac{sin2phi}{alphay \cdot alphay}\right)\right)}\right)\right) \]

   9. unsub-neg N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \left(\left(\mathsf{neg}\left(\frac{cos2phi}{alphax \cdot alphax}\right)\right) - \color{blue}{\frac{sin2phi}{alphay \cdot alphay}}\right)\right) \]

   10. --lowering--.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\left(\mathsf{neg}\left(\frac{cos2phi}{alphax \cdot alphax}\right)\right), \color{blue}{\left(\frac{sin2phi}{alphay \cdot alphay}\right)}\right)\right) \]

   11. neg-sub0 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\left(0 - \frac{cos2phi}{alphax \cdot alphax}\right), \left(\frac{\color{blue}{sin2phi}}{alphay \cdot alphay}\right)\right)\right) \]

   12. --lowering--.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \left(\frac{cos2phi}{alphax \cdot alphax}\right)\right), \left(\frac{\color{blue}{sin2phi}}{alphay \cdot alphay}\right)\right)\right) \]

   13. /-lowering-/.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(cos2phi, \left(alphax \cdot alphax\right)\right)\right), \left(\frac{sin2phi}{alphay \cdot alphay}\right)\right)\right) \]

   14. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right)\right), \left(\frac{sin2phi}{alphay \cdot alphay}\right)\right)\right) \]

   15. /-lowering-/.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right)\right), \mathsf{/.f32}\left(sin2phi, \color{blue}{\left(alphay \cdot alphay\right)}\right)\right)\right) \]

   16. *-lowering-*.f32 98.3%

       \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right)\right), \mathsf{/.f32}\left(sin2phi, \mathsf{*.f32}\left(alphay, \color{blue}{alphay}\right)\right)\right)\right) \]

3. Simplified 98.3%

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

5. Taylor expanded in u0 around 0

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

   1. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(u0, \color{blue}{\left(\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}\right)}\right) \]

   2. +-lowering-+.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{+.f32}\left(\left(\frac{cos2phi}{{alphax}^{2}}\right), \color{blue}{\left(\frac{sin2phi}{{alphay}^{2}}\right)}\right)\right) \]

   3. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{/.f32}\left(cos2phi, \left({alphax}^{2}\right)\right), \left(\frac{\color{blue}{sin2phi}}{{alphay}^{2}}\right)\right)\right) \]

   4. unpow2 N/A

      \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{/.f32}\left(cos2phi, \left(alphax \cdot alphax\right)\right), \left(\frac{sin2phi}{{alphay}^{2}}\right)\right)\right) \]

   5. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right), \left(\frac{sin2phi}{{alphay}^{2}}\right)\right)\right) \]

   6. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right), \mathsf{/.f32}\left(sin2phi, \color{blue}{\left({alphay}^{2}\right)}\right)\right)\right) \]

   7. unpow2 N/A

      \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right), \mathsf{/.f32}\left(sin2phi, \left(alphay \cdot \color{blue}{alphay}\right)\right)\right)\right) \]

   8. *-lowering-*.f32 74.0%

      \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right), \mathsf{/.f32}\left(sin2phi, \mathsf{*.f32}\left(alphay, \color{blue}{alphay}\right)\right)\right)\right) \]

7. Simplified 74.0%

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

   1. +-commutative N/A

      \[\leadsto \mathsf{/.f32}\left(u0, \left(\frac{sin2phi}{alphay \cdot alphay} + \color{blue}{\frac{cos2phi}{alphax \cdot alphax}}\right)\right) \]

   2. +-lowering-+.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{+.f32}\left(\left(\frac{sin2phi}{alphay \cdot alphay}\right), \color{blue}{\left(\frac{cos2phi}{alphax \cdot alphax}\right)}\right)\right) \]

   3. associate-/r* N/A

      \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{+.f32}\left(\left(\frac{\frac{sin2phi}{alphay}}{alphay}\right), \left(\frac{\color{blue}{cos2phi}}{alphax \cdot alphax}\right)\right)\right) \]

   4. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{/.f32}\left(\left(\frac{sin2phi}{alphay}\right), alphay\right), \left(\frac{\color{blue}{cos2phi}}{alphax \cdot alphax}\right)\right)\right) \]

   5. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(sin2phi, alphay\right), alphay\right), \left(\frac{cos2phi}{alphax \cdot alphax}\right)\right)\right) \]

   6. associate-/r* N/A

      \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(sin2phi, alphay\right), alphay\right), \left(\frac{\frac{cos2phi}{alphax}}{\color{blue}{alphax}}\right)\right)\right) \]

   7. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(sin2phi, alphay\right), alphay\right), \mathsf{/.f32}\left(\left(\frac{cos2phi}{alphax}\right), \color{blue}{alphax}\right)\right)\right) \]

   8. /-lowering-/.f32 74.2%

      \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(sin2phi, alphay\right), alphay\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(cos2phi, alphax\right), alphax\right)\right)\right) \]

9. Applied egg-rr 74.2%

   \[\leadsto \frac{u0}{\color{blue}{\frac{\frac{sin2phi}{alphay}}{alphay} + \frac{\frac{cos2phi}{alphax}}{alphax}}} \]
10. Add Preprocessing

Alternative 24: 75.5% accurate, 8.9× speedup

Math

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

FPCore

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

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	return 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 = u0 / ((cos2phi / (alphax * alphax)) + (sin2phi / (alphay * alphay)))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 64.8%

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

   1. distribute-frac-neg N/A

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

   2. distribute-neg-frac2 N/A

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

   3. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\log \left(1 - u0\right), \color{blue}{\left(\mathsf{neg}\left(\left(\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\right)\right)\right)}\right) \]

   4. sub-neg N/A

      \[\leadsto \mathsf{/.f32}\left(\log \left(1 + \left(\mathsf{neg}\left(u0\right)\right)\right), \left(\mathsf{neg}\left(\left(\color{blue}{\frac{cos2phi}{alphax \cdot alphax}} + \frac{sin2phi}{alphay \cdot alphay}\right)\right)\right)\right) \]

   5. log1p-define N/A

      \[\leadsto \mathsf{/.f32}\left(\left(\mathsf{log1p}\left(\mathsf{neg}\left(u0\right)\right)\right), \left(\mathsf{neg}\left(\color{blue}{\left(\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\right)}\right)\right)\right) \]

   6. log1p-lowering-log1p.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\left(\mathsf{neg}\left(u0\right)\right)\right), \left(\mathsf{neg}\left(\color{blue}{\left(\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\right)}\right)\right)\right) \]

   7. neg-lowering-neg.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \left(\mathsf{neg}\left(\left(\color{blue}{\frac{cos2phi}{alphax \cdot alphax}} + \frac{sin2phi}{alphay \cdot alphay}\right)\right)\right)\right) \]

   8. distribute-neg-in N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \left(\left(\mathsf{neg}\left(\frac{cos2phi}{alphax \cdot alphax}\right)\right) + \color{blue}{\left(\mathsf{neg}\left(\frac{sin2phi}{alphay \cdot alphay}\right)\right)}\right)\right) \]

   9. unsub-neg N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \left(\left(\mathsf{neg}\left(\frac{cos2phi}{alphax \cdot alphax}\right)\right) - \color{blue}{\frac{sin2phi}{alphay \cdot alphay}}\right)\right) \]

   10. --lowering--.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\left(\mathsf{neg}\left(\frac{cos2phi}{alphax \cdot alphax}\right)\right), \color{blue}{\left(\frac{sin2phi}{alphay \cdot alphay}\right)}\right)\right) \]

   11. neg-sub0 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\left(0 - \frac{cos2phi}{alphax \cdot alphax}\right), \left(\frac{\color{blue}{sin2phi}}{alphay \cdot alphay}\right)\right)\right) \]

   12. --lowering--.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \left(\frac{cos2phi}{alphax \cdot alphax}\right)\right), \left(\frac{\color{blue}{sin2phi}}{alphay \cdot alphay}\right)\right)\right) \]

   13. /-lowering-/.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(cos2phi, \left(alphax \cdot alphax\right)\right)\right), \left(\frac{sin2phi}{alphay \cdot alphay}\right)\right)\right) \]

   14. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right)\right), \left(\frac{sin2phi}{alphay \cdot alphay}\right)\right)\right) \]

   15. /-lowering-/.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right)\right), \mathsf{/.f32}\left(sin2phi, \color{blue}{\left(alphay \cdot alphay\right)}\right)\right)\right) \]

   16. *-lowering-*.f32 98.3%

       \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right)\right), \mathsf{/.f32}\left(sin2phi, \mathsf{*.f32}\left(alphay, \color{blue}{alphay}\right)\right)\right)\right) \]

3. Simplified 98.3%

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

5. Taylor expanded in u0 around 0

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

   1. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(u0, \color{blue}{\left(\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}\right)}\right) \]

   2. +-lowering-+.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{+.f32}\left(\left(\frac{cos2phi}{{alphax}^{2}}\right), \color{blue}{\left(\frac{sin2phi}{{alphay}^{2}}\right)}\right)\right) \]

   3. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{/.f32}\left(cos2phi, \left({alphax}^{2}\right)\right), \left(\frac{\color{blue}{sin2phi}}{{alphay}^{2}}\right)\right)\right) \]

   4. unpow2 N/A

      \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{/.f32}\left(cos2phi, \left(alphax \cdot alphax\right)\right), \left(\frac{sin2phi}{{alphay}^{2}}\right)\right)\right) \]

   5. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right), \left(\frac{sin2phi}{{alphay}^{2}}\right)\right)\right) \]

   6. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right), \mathsf{/.f32}\left(sin2phi, \color{blue}{\left({alphay}^{2}\right)}\right)\right)\right) \]

   7. unpow2 N/A

      \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right), \mathsf{/.f32}\left(sin2phi, \left(alphay \cdot \color{blue}{alphay}\right)\right)\right)\right) \]

   8. *-lowering-*.f32 74.0%

      \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right), \mathsf{/.f32}\left(sin2phi, \mathsf{*.f32}\left(alphay, \color{blue}{alphay}\right)\right)\right)\right) \]

7. Simplified 74.0%

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

Alternative 25: 66.4% accurate, 9.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;sin2phi \leq 1.000000031374395 \cdot 10^{-22}:\\ \;\;\;\;u0 \cdot \frac{alphax}{\frac{cos2phi}{alphax}}\\ \mathbf{else}:\\ \;\;\;\;\frac{u0 \cdot \left(alphay \cdot alphay\right)}{sin2phi}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (if (<= sin2phi 1.000000031374395e-22)
   (* u0 (/ alphax (/ cos2phi alphax)))
   (/ (* u0 (* alphay alphay)) sin2phi)))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float tmp;
	if (sin2phi <= 1.000000031374395e-22f) {
		tmp = u0 * (alphax / (cos2phi / alphax));
	} else {
		tmp = (u0 * (alphay * 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 <= 1.000000031374395e-22) then
        tmp = u0 * (alphax / (cos2phi / alphax))
    else
        tmp = (u0 * (alphay * alphay)) / sin2phi
    end if
    code = tmp
end function

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = Float32(0.0)
	if (sin2phi <= Float32(1.000000031374395e-22))
		tmp = Float32(u0 * Float32(alphax / Float32(cos2phi / alphax)));
	else
		tmp = Float32(Float32(u0 * Float32(alphay * alphay)) / sin2phi);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = single(0.0);
	if (sin2phi <= single(1.000000031374395e-22))
		tmp = u0 * (alphax / (cos2phi / alphax));
	else
		tmp = (u0 * (alphay * alphay)) / sin2phi;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if sin2phi < 1.00000003e-22

   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 N/A

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

      2. distribute-neg-frac2 N/A

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

      3. /-lowering-/.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(\log \left(1 - u0\right), \color{blue}{\left(\mathsf{neg}\left(\left(\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\right)\right)\right)}\right) \]

      4. sub-neg N/A

         \[\leadsto \mathsf{/.f32}\left(\log \left(1 + \left(\mathsf{neg}\left(u0\right)\right)\right), \left(\mathsf{neg}\left(\left(\color{blue}{\frac{cos2phi}{alphax \cdot alphax}} + \frac{sin2phi}{alphay \cdot alphay}\right)\right)\right)\right) \]

      5. log1p-define N/A

         \[\leadsto \mathsf{/.f32}\left(\left(\mathsf{log1p}\left(\mathsf{neg}\left(u0\right)\right)\right), \left(\mathsf{neg}\left(\color{blue}{\left(\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\right)}\right)\right)\right) \]

      6. log1p-lowering-log1p.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\left(\mathsf{neg}\left(u0\right)\right)\right), \left(\mathsf{neg}\left(\color{blue}{\left(\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\right)}\right)\right)\right) \]

      7. neg-lowering-neg.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \left(\mathsf{neg}\left(\left(\color{blue}{\frac{cos2phi}{alphax \cdot alphax}} + \frac{sin2phi}{alphay \cdot alphay}\right)\right)\right)\right) \]

      8. distribute-neg-in N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \left(\left(\mathsf{neg}\left(\frac{cos2phi}{alphax \cdot alphax}\right)\right) + \color{blue}{\left(\mathsf{neg}\left(\frac{sin2phi}{alphay \cdot alphay}\right)\right)}\right)\right) \]

      9. unsub-neg N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \left(\left(\mathsf{neg}\left(\frac{cos2phi}{alphax \cdot alphax}\right)\right) - \color{blue}{\frac{sin2phi}{alphay \cdot alphay}}\right)\right) \]

      10. --lowering--.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\left(\mathsf{neg}\left(\frac{cos2phi}{alphax \cdot alphax}\right)\right), \color{blue}{\left(\frac{sin2phi}{alphay \cdot alphay}\right)}\right)\right) \]

      11. neg-sub0 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\left(0 - \frac{cos2phi}{alphax \cdot alphax}\right), \left(\frac{\color{blue}{sin2phi}}{alphay \cdot alphay}\right)\right)\right) \]

      12. --lowering--.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \left(\frac{cos2phi}{alphax \cdot alphax}\right)\right), \left(\frac{\color{blue}{sin2phi}}{alphay \cdot alphay}\right)\right)\right) \]

      13. /-lowering-/.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(cos2phi, \left(alphax \cdot alphax\right)\right)\right), \left(\frac{sin2phi}{alphay \cdot alphay}\right)\right)\right) \]

      14. *-lowering-*.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right)\right), \left(\frac{sin2phi}{alphay \cdot alphay}\right)\right)\right) \]

      15. /-lowering-/.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right)\right), \mathsf{/.f32}\left(sin2phi, \color{blue}{\left(alphay \cdot alphay\right)}\right)\right)\right) \]

      16. *-lowering-*.f32 98.5%

          \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right)\right), \mathsf{/.f32}\left(sin2phi, \mathsf{*.f32}\left(alphay, \color{blue}{alphay}\right)\right)\right)\right) \]

   3. Simplified 98.5%

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

   5. Taylor expanded in cos2phi around inf

      \[\leadsto \color{blue}{-1 \cdot \frac{{alphax}^{2} \cdot \log \left(1 - u0\right)}{cos2phi}} \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(\frac{{alphax}^{2} \cdot \log \left(1 - u0\right)}{cos2phi}\right) \]

      2. associate-/l* N/A

         \[\leadsto \mathsf{neg}\left({alphax}^{2} \cdot \frac{\log \left(1 - u0\right)}{cos2phi}\right) \]

      3. distribute-rgt-neg-in N/A

         \[\leadsto {alphax}^{2} \cdot \color{blue}{\left(\mathsf{neg}\left(\frac{\log \left(1 - u0\right)}{cos2phi}\right)\right)} \]

      4. mul-1-neg N/A

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

      5. *-lowering-*.f32 N/A

         \[\leadsto \mathsf{*.f32}\left(\left({alphax}^{2}\right), \color{blue}{\left(-1 \cdot \frac{\log \left(1 - u0\right)}{cos2phi}\right)}\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{*.f32}\left(\left(alphax \cdot alphax\right), \left(\color{blue}{-1} \cdot \frac{\log \left(1 - u0\right)}{cos2phi}\right)\right) \]

      7. *-lowering-*.f32 N/A

         \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(alphax, alphax\right), \left(\color{blue}{-1} \cdot \frac{\log \left(1 - u0\right)}{cos2phi}\right)\right) \]

      8. mul-1-neg N/A

         \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(alphax, alphax\right), \left(\mathsf{neg}\left(\frac{\log \left(1 - u0\right)}{cos2phi}\right)\right)\right) \]

      9. distribute-neg-frac2 N/A

         \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(alphax, alphax\right), \left(\frac{\log \left(1 - u0\right)}{\color{blue}{\mathsf{neg}\left(cos2phi\right)}}\right)\right) \]

      10. /-lowering-/.f32 N/A

          \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(alphax, alphax\right), \mathsf{/.f32}\left(\log \left(1 - u0\right), \color{blue}{\left(\mathsf{neg}\left(cos2phi\right)\right)}\right)\right) \]

      11. sub-neg N/A

          \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(alphax, alphax\right), \mathsf{/.f32}\left(\log \left(1 + \left(\mathsf{neg}\left(u0\right)\right)\right), \left(\mathsf{neg}\left(cos2phi\right)\right)\right)\right) \]

      12. mul-1-neg N/A

          \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(alphax, alphax\right), \mathsf{/.f32}\left(\log \left(1 + -1 \cdot u0\right), \left(\mathsf{neg}\left(cos2phi\right)\right)\right)\right) \]

      13. log1p-define N/A

          \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(alphax, alphax\right), \mathsf{/.f32}\left(\left(\mathsf{log1p}\left(-1 \cdot u0\right)\right), \left(\mathsf{neg}\left(\color{blue}{cos2phi}\right)\right)\right)\right) \]

      14. log1p-lowering-log1p.f32 N/A

          \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(alphax, alphax\right), \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\left(-1 \cdot u0\right)\right), \left(\mathsf{neg}\left(\color{blue}{cos2phi}\right)\right)\right)\right) \]

      15. mul-1-neg N/A

          \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(alphax, alphax\right), \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\left(\mathsf{neg}\left(u0\right)\right)\right), \left(\mathsf{neg}\left(cos2phi\right)\right)\right)\right) \]

      16. neg-lowering-neg.f32 N/A

          \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(alphax, alphax\right), \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \left(\mathsf{neg}\left(cos2phi\right)\right)\right)\right) \]

      17. neg-lowering-neg.f32 74.1%

          \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(alphax, alphax\right), \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{neg.f32}\left(cos2phi\right)\right)\right) \]

   7. Simplified 74.1%

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

   8. Taylor expanded in u0 around 0

      \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(alphax, alphax\right), \color{blue}{\left(\frac{u0}{cos2phi}\right)}\right) \]
   9. Step-by-step derivation

      1. /-lowering-/.f32 54.1%

         \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(alphax, alphax\right), \mathsf{/.f32}\left(u0, \color{blue}{cos2phi}\right)\right) \]

   10. Simplified 54.1%

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

       1. *-commutative N/A

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

       2. div-inv N/A

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

       3. associate-*l* N/A

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

       4. associate-/r/ N/A

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

       5. clear-num N/A

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

       6. *-lowering-*.f32 N/A

          \[\leadsto \mathsf{*.f32}\left(u0, \color{blue}{\left(\frac{alphax \cdot alphax}{cos2phi}\right)}\right) \]

       7. clear-num N/A

          \[\leadsto \mathsf{*.f32}\left(u0, \left(\frac{1}{\color{blue}{\frac{cos2phi}{alphax \cdot alphax}}}\right)\right) \]

       8. associate-/r* N/A

          \[\leadsto \mathsf{*.f32}\left(u0, \left(\frac{1}{\frac{\frac{cos2phi}{alphax}}{\color{blue}{alphax}}}\right)\right) \]

       9. clear-num N/A

          \[\leadsto \mathsf{*.f32}\left(u0, \left(\frac{alphax}{\color{blue}{\frac{cos2phi}{alphax}}}\right)\right) \]

       10. /-lowering-/.f32 N/A

           \[\leadsto \mathsf{*.f32}\left(u0, \mathsf{/.f32}\left(alphax, \color{blue}{\left(\frac{cos2phi}{alphax}\right)}\right)\right) \]

       11. /-lowering-/.f32 54.2%

           \[\leadsto \mathsf{*.f32}\left(u0, \mathsf{/.f32}\left(alphax, \mathsf{/.f32}\left(cos2phi, \color{blue}{alphax}\right)\right)\right) \]

   12. Applied egg-rr 54.2%

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

   if 1.00000003e-22 < sin2phi

   1. Initial program 64.7%

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

      1. distribute-frac-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. /-lowering-/.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(\log \left(1 - u0\right), \color{blue}{\left(\mathsf{neg}\left(\left(\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\right)\right)\right)}\right) \]

      4. sub-neg N/A

         \[\leadsto \mathsf{/.f32}\left(\log \left(1 + \left(\mathsf{neg}\left(u0\right)\right)\right), \left(\mathsf{neg}\left(\left(\color{blue}{\frac{cos2phi}{alphax \cdot alphax}} + \frac{sin2phi}{alphay \cdot alphay}\right)\right)\right)\right) \]

      5. log1p-define N/A

         \[\leadsto \mathsf{/.f32}\left(\left(\mathsf{log1p}\left(\mathsf{neg}\left(u0\right)\right)\right), \left(\mathsf{neg}\left(\color{blue}{\left(\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\right)}\right)\right)\right) \]

      6. log1p-lowering-log1p.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\left(\mathsf{neg}\left(u0\right)\right)\right), \left(\mathsf{neg}\left(\color{blue}{\left(\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\right)}\right)\right)\right) \]

      7. neg-lowering-neg.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \left(\mathsf{neg}\left(\left(\color{blue}{\frac{cos2phi}{alphax \cdot alphax}} + \frac{sin2phi}{alphay \cdot alphay}\right)\right)\right)\right) \]

      8. distribute-neg-in N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \left(\left(\mathsf{neg}\left(\frac{cos2phi}{alphax \cdot alphax}\right)\right) + \color{blue}{\left(\mathsf{neg}\left(\frac{sin2phi}{alphay \cdot alphay}\right)\right)}\right)\right) \]

      9. unsub-neg N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \left(\left(\mathsf{neg}\left(\frac{cos2phi}{alphax \cdot alphax}\right)\right) - \color{blue}{\frac{sin2phi}{alphay \cdot alphay}}\right)\right) \]

      10. --lowering--.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\left(\mathsf{neg}\left(\frac{cos2phi}{alphax \cdot alphax}\right)\right), \color{blue}{\left(\frac{sin2phi}{alphay \cdot alphay}\right)}\right)\right) \]

      11. neg-sub0 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\left(0 - \frac{cos2phi}{alphax \cdot alphax}\right), \left(\frac{\color{blue}{sin2phi}}{alphay \cdot alphay}\right)\right)\right) \]

      12. --lowering--.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \left(\frac{cos2phi}{alphax \cdot alphax}\right)\right), \left(\frac{\color{blue}{sin2phi}}{alphay \cdot alphay}\right)\right)\right) \]

      13. /-lowering-/.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(cos2phi, \left(alphax \cdot alphax\right)\right)\right), \left(\frac{sin2phi}{alphay \cdot alphay}\right)\right)\right) \]

      14. *-lowering-*.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right)\right), \left(\frac{sin2phi}{alphay \cdot alphay}\right)\right)\right) \]

      15. /-lowering-/.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right)\right), \mathsf{/.f32}\left(sin2phi, \color{blue}{\left(alphay \cdot alphay\right)}\right)\right)\right) \]

      16. *-lowering-*.f32 98.2%

          \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right)\right), \mathsf{/.f32}\left(sin2phi, \mathsf{*.f32}\left(alphay, \color{blue}{alphay}\right)\right)\right)\right) \]

   3. Simplified 98.2%

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

   5. Taylor expanded in u0 around 0

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

      1. /-lowering-/.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(u0, \color{blue}{\left(\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}\right)}\right) \]

      2. +-lowering-+.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{+.f32}\left(\left(\frac{cos2phi}{{alphax}^{2}}\right), \color{blue}{\left(\frac{sin2phi}{{alphay}^{2}}\right)}\right)\right) \]

      3. /-lowering-/.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{/.f32}\left(cos2phi, \left({alphax}^{2}\right)\right), \left(\frac{\color{blue}{sin2phi}}{{alphay}^{2}}\right)\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{/.f32}\left(cos2phi, \left(alphax \cdot alphax\right)\right), \left(\frac{sin2phi}{{alphay}^{2}}\right)\right)\right) \]

      5. *-lowering-*.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right), \left(\frac{sin2phi}{{alphay}^{2}}\right)\right)\right) \]

      6. /-lowering-/.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right), \mathsf{/.f32}\left(sin2phi, \color{blue}{\left({alphay}^{2}\right)}\right)\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right), \mathsf{/.f32}\left(sin2phi, \left(alphay \cdot \color{blue}{alphay}\right)\right)\right)\right) \]

      8. *-lowering-*.f32 75.7%

         \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right), \mathsf{/.f32}\left(sin2phi, \mathsf{*.f32}\left(alphay, \color{blue}{alphay}\right)\right)\right)\right) \]

   7. Simplified 75.7%

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

   8. Taylor expanded in cos2phi around 0

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

      1. /-lowering-/.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(\left({alphay}^{2} \cdot u0\right), \color{blue}{sin2phi}\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{/.f32}\left(\left(u0 \cdot {alphay}^{2}\right), sin2phi\right) \]

      3. *-lowering-*.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(u0, \left({alphay}^{2}\right)\right), sin2phi\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(u0, \left(alphay \cdot alphay\right)\right), sin2phi\right) \]

      5. *-lowering-*.f32 72.0%

         \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(u0, \mathsf{*.f32}\left(alphay, alphay\right)\right), sin2phi\right) \]

   10. Simplified 72.0%

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

Alternative 26: 66.4% accurate, 9.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;sin2phi \leq 1.000000031374395 \cdot 10^{-22}:\\ \;\;\;\;u0 \cdot \frac{alphax}{\frac{cos2phi}{alphax}}\\ \mathbf{else}:\\ \;\;\;\;u0 \cdot \frac{alphay \cdot alphay}{sin2phi}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (if (<= sin2phi 1.000000031374395e-22)
   (* u0 (/ alphax (/ cos2phi alphax)))
   (* u0 (/ (* alphay alphay) sin2phi))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float tmp;
	if (sin2phi <= 1.000000031374395e-22f) {
		tmp = u0 * (alphax / (cos2phi / alphax));
	} else {
		tmp = u0 * ((alphay * 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 <= 1.000000031374395e-22) then
        tmp = u0 * (alphax / (cos2phi / alphax))
    else
        tmp = u0 * ((alphay * alphay) / sin2phi)
    end if
    code = tmp
end function

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = Float32(0.0)
	if (sin2phi <= Float32(1.000000031374395e-22))
		tmp = Float32(u0 * Float32(alphax / Float32(cos2phi / alphax)));
	else
		tmp = Float32(u0 * Float32(Float32(alphay * alphay) / sin2phi));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = single(0.0);
	if (sin2phi <= single(1.000000031374395e-22))
		tmp = u0 * (alphax / (cos2phi / alphax));
	else
		tmp = u0 * ((alphay * alphay) / sin2phi);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if sin2phi < 1.00000003e-22

   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 N/A

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

      2. distribute-neg-frac2 N/A

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

      3. /-lowering-/.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(\log \left(1 - u0\right), \color{blue}{\left(\mathsf{neg}\left(\left(\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\right)\right)\right)}\right) \]

      4. sub-neg N/A

         \[\leadsto \mathsf{/.f32}\left(\log \left(1 + \left(\mathsf{neg}\left(u0\right)\right)\right), \left(\mathsf{neg}\left(\left(\color{blue}{\frac{cos2phi}{alphax \cdot alphax}} + \frac{sin2phi}{alphay \cdot alphay}\right)\right)\right)\right) \]

      5. log1p-define N/A

         \[\leadsto \mathsf{/.f32}\left(\left(\mathsf{log1p}\left(\mathsf{neg}\left(u0\right)\right)\right), \left(\mathsf{neg}\left(\color{blue}{\left(\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\right)}\right)\right)\right) \]

      6. log1p-lowering-log1p.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\left(\mathsf{neg}\left(u0\right)\right)\right), \left(\mathsf{neg}\left(\color{blue}{\left(\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\right)}\right)\right)\right) \]

      7. neg-lowering-neg.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \left(\mathsf{neg}\left(\left(\color{blue}{\frac{cos2phi}{alphax \cdot alphax}} + \frac{sin2phi}{alphay \cdot alphay}\right)\right)\right)\right) \]

      8. distribute-neg-in N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \left(\left(\mathsf{neg}\left(\frac{cos2phi}{alphax \cdot alphax}\right)\right) + \color{blue}{\left(\mathsf{neg}\left(\frac{sin2phi}{alphay \cdot alphay}\right)\right)}\right)\right) \]

      9. unsub-neg N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \left(\left(\mathsf{neg}\left(\frac{cos2phi}{alphax \cdot alphax}\right)\right) - \color{blue}{\frac{sin2phi}{alphay \cdot alphay}}\right)\right) \]

      10. --lowering--.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\left(\mathsf{neg}\left(\frac{cos2phi}{alphax \cdot alphax}\right)\right), \color{blue}{\left(\frac{sin2phi}{alphay \cdot alphay}\right)}\right)\right) \]

      11. neg-sub0 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\left(0 - \frac{cos2phi}{alphax \cdot alphax}\right), \left(\frac{\color{blue}{sin2phi}}{alphay \cdot alphay}\right)\right)\right) \]

      12. --lowering--.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \left(\frac{cos2phi}{alphax \cdot alphax}\right)\right), \left(\frac{\color{blue}{sin2phi}}{alphay \cdot alphay}\right)\right)\right) \]

      13. /-lowering-/.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(cos2phi, \left(alphax \cdot alphax\right)\right)\right), \left(\frac{sin2phi}{alphay \cdot alphay}\right)\right)\right) \]

      14. *-lowering-*.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right)\right), \left(\frac{sin2phi}{alphay \cdot alphay}\right)\right)\right) \]

      15. /-lowering-/.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right)\right), \mathsf{/.f32}\left(sin2phi, \color{blue}{\left(alphay \cdot alphay\right)}\right)\right)\right) \]

      16. *-lowering-*.f32 98.5%

          \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right)\right), \mathsf{/.f32}\left(sin2phi, \mathsf{*.f32}\left(alphay, \color{blue}{alphay}\right)\right)\right)\right) \]

   3. Simplified 98.5%

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

   5. Taylor expanded in cos2phi around inf

      \[\leadsto \color{blue}{-1 \cdot \frac{{alphax}^{2} \cdot \log \left(1 - u0\right)}{cos2phi}} \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(\frac{{alphax}^{2} \cdot \log \left(1 - u0\right)}{cos2phi}\right) \]

      2. associate-/l* N/A

         \[\leadsto \mathsf{neg}\left({alphax}^{2} \cdot \frac{\log \left(1 - u0\right)}{cos2phi}\right) \]

      3. distribute-rgt-neg-in N/A

         \[\leadsto {alphax}^{2} \cdot \color{blue}{\left(\mathsf{neg}\left(\frac{\log \left(1 - u0\right)}{cos2phi}\right)\right)} \]

      4. mul-1-neg N/A

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

      5. *-lowering-*.f32 N/A

         \[\leadsto \mathsf{*.f32}\left(\left({alphax}^{2}\right), \color{blue}{\left(-1 \cdot \frac{\log \left(1 - u0\right)}{cos2phi}\right)}\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{*.f32}\left(\left(alphax \cdot alphax\right), \left(\color{blue}{-1} \cdot \frac{\log \left(1 - u0\right)}{cos2phi}\right)\right) \]

      7. *-lowering-*.f32 N/A

         \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(alphax, alphax\right), \left(\color{blue}{-1} \cdot \frac{\log \left(1 - u0\right)}{cos2phi}\right)\right) \]

      8. mul-1-neg N/A

         \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(alphax, alphax\right), \left(\mathsf{neg}\left(\frac{\log \left(1 - u0\right)}{cos2phi}\right)\right)\right) \]

      9. distribute-neg-frac2 N/A

         \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(alphax, alphax\right), \left(\frac{\log \left(1 - u0\right)}{\color{blue}{\mathsf{neg}\left(cos2phi\right)}}\right)\right) \]

      10. /-lowering-/.f32 N/A

          \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(alphax, alphax\right), \mathsf{/.f32}\left(\log \left(1 - u0\right), \color{blue}{\left(\mathsf{neg}\left(cos2phi\right)\right)}\right)\right) \]

      11. sub-neg N/A

          \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(alphax, alphax\right), \mathsf{/.f32}\left(\log \left(1 + \left(\mathsf{neg}\left(u0\right)\right)\right), \left(\mathsf{neg}\left(cos2phi\right)\right)\right)\right) \]

      12. mul-1-neg N/A

          \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(alphax, alphax\right), \mathsf{/.f32}\left(\log \left(1 + -1 \cdot u0\right), \left(\mathsf{neg}\left(cos2phi\right)\right)\right)\right) \]

      13. log1p-define N/A

          \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(alphax, alphax\right), \mathsf{/.f32}\left(\left(\mathsf{log1p}\left(-1 \cdot u0\right)\right), \left(\mathsf{neg}\left(\color{blue}{cos2phi}\right)\right)\right)\right) \]

      14. log1p-lowering-log1p.f32 N/A

          \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(alphax, alphax\right), \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\left(-1 \cdot u0\right)\right), \left(\mathsf{neg}\left(\color{blue}{cos2phi}\right)\right)\right)\right) \]

      15. mul-1-neg N/A

          \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(alphax, alphax\right), \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\left(\mathsf{neg}\left(u0\right)\right)\right), \left(\mathsf{neg}\left(cos2phi\right)\right)\right)\right) \]

      16. neg-lowering-neg.f32 N/A

          \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(alphax, alphax\right), \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \left(\mathsf{neg}\left(cos2phi\right)\right)\right)\right) \]

      17. neg-lowering-neg.f32 74.1%

          \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(alphax, alphax\right), \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{neg.f32}\left(cos2phi\right)\right)\right) \]

   7. Simplified 74.1%

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

   8. Taylor expanded in u0 around 0

      \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(alphax, alphax\right), \color{blue}{\left(\frac{u0}{cos2phi}\right)}\right) \]
   9. Step-by-step derivation

      1. /-lowering-/.f32 54.1%

         \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(alphax, alphax\right), \mathsf{/.f32}\left(u0, \color{blue}{cos2phi}\right)\right) \]

   10. Simplified 54.1%

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

       1. *-commutative N/A

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

       2. div-inv N/A

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

       3. associate-*l* N/A

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

       4. associate-/r/ N/A

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

       5. clear-num N/A

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

       6. *-lowering-*.f32 N/A

          \[\leadsto \mathsf{*.f32}\left(u0, \color{blue}{\left(\frac{alphax \cdot alphax}{cos2phi}\right)}\right) \]

       7. clear-num N/A

          \[\leadsto \mathsf{*.f32}\left(u0, \left(\frac{1}{\color{blue}{\frac{cos2phi}{alphax \cdot alphax}}}\right)\right) \]

       8. associate-/r* N/A

          \[\leadsto \mathsf{*.f32}\left(u0, \left(\frac{1}{\frac{\frac{cos2phi}{alphax}}{\color{blue}{alphax}}}\right)\right) \]

       9. clear-num N/A

          \[\leadsto \mathsf{*.f32}\left(u0, \left(\frac{alphax}{\color{blue}{\frac{cos2phi}{alphax}}}\right)\right) \]

       10. /-lowering-/.f32 N/A

           \[\leadsto \mathsf{*.f32}\left(u0, \mathsf{/.f32}\left(alphax, \color{blue}{\left(\frac{cos2phi}{alphax}\right)}\right)\right) \]

       11. /-lowering-/.f32 54.2%

           \[\leadsto \mathsf{*.f32}\left(u0, \mathsf{/.f32}\left(alphax, \mathsf{/.f32}\left(cos2phi, \color{blue}{alphax}\right)\right)\right) \]

   12. Applied egg-rr 54.2%

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

   if 1.00000003e-22 < sin2phi

   1. Initial program 64.7%

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

      1. distribute-frac-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. /-lowering-/.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(\log \left(1 - u0\right), \color{blue}{\left(\mathsf{neg}\left(\left(\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\right)\right)\right)}\right) \]

      4. sub-neg N/A

         \[\leadsto \mathsf{/.f32}\left(\log \left(1 + \left(\mathsf{neg}\left(u0\right)\right)\right), \left(\mathsf{neg}\left(\left(\color{blue}{\frac{cos2phi}{alphax \cdot alphax}} + \frac{sin2phi}{alphay \cdot alphay}\right)\right)\right)\right) \]

      5. log1p-define N/A

         \[\leadsto \mathsf{/.f32}\left(\left(\mathsf{log1p}\left(\mathsf{neg}\left(u0\right)\right)\right), \left(\mathsf{neg}\left(\color{blue}{\left(\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\right)}\right)\right)\right) \]

      6. log1p-lowering-log1p.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\left(\mathsf{neg}\left(u0\right)\right)\right), \left(\mathsf{neg}\left(\color{blue}{\left(\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\right)}\right)\right)\right) \]

      7. neg-lowering-neg.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \left(\mathsf{neg}\left(\left(\color{blue}{\frac{cos2phi}{alphax \cdot alphax}} + \frac{sin2phi}{alphay \cdot alphay}\right)\right)\right)\right) \]

      8. distribute-neg-in N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \left(\left(\mathsf{neg}\left(\frac{cos2phi}{alphax \cdot alphax}\right)\right) + \color{blue}{\left(\mathsf{neg}\left(\frac{sin2phi}{alphay \cdot alphay}\right)\right)}\right)\right) \]

      9. unsub-neg N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \left(\left(\mathsf{neg}\left(\frac{cos2phi}{alphax \cdot alphax}\right)\right) - \color{blue}{\frac{sin2phi}{alphay \cdot alphay}}\right)\right) \]

      10. --lowering--.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\left(\mathsf{neg}\left(\frac{cos2phi}{alphax \cdot alphax}\right)\right), \color{blue}{\left(\frac{sin2phi}{alphay \cdot alphay}\right)}\right)\right) \]

      11. neg-sub0 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\left(0 - \frac{cos2phi}{alphax \cdot alphax}\right), \left(\frac{\color{blue}{sin2phi}}{alphay \cdot alphay}\right)\right)\right) \]

      12. --lowering--.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \left(\frac{cos2phi}{alphax \cdot alphax}\right)\right), \left(\frac{\color{blue}{sin2phi}}{alphay \cdot alphay}\right)\right)\right) \]

      13. /-lowering-/.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(cos2phi, \left(alphax \cdot alphax\right)\right)\right), \left(\frac{sin2phi}{alphay \cdot alphay}\right)\right)\right) \]

      14. *-lowering-*.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right)\right), \left(\frac{sin2phi}{alphay \cdot alphay}\right)\right)\right) \]

      15. /-lowering-/.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right)\right), \mathsf{/.f32}\left(sin2phi, \color{blue}{\left(alphay \cdot alphay\right)}\right)\right)\right) \]

      16. *-lowering-*.f32 98.2%

          \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right)\right), \mathsf{/.f32}\left(sin2phi, \mathsf{*.f32}\left(alphay, \color{blue}{alphay}\right)\right)\right)\right) \]

   3. Simplified 98.2%

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

   5. Taylor expanded in u0 around 0

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

      1. /-lowering-/.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(u0, \color{blue}{\left(\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}\right)}\right) \]

      2. +-lowering-+.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{+.f32}\left(\left(\frac{cos2phi}{{alphax}^{2}}\right), \color{blue}{\left(\frac{sin2phi}{{alphay}^{2}}\right)}\right)\right) \]

      3. /-lowering-/.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{/.f32}\left(cos2phi, \left({alphax}^{2}\right)\right), \left(\frac{\color{blue}{sin2phi}}{{alphay}^{2}}\right)\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{/.f32}\left(cos2phi, \left(alphax \cdot alphax\right)\right), \left(\frac{sin2phi}{{alphay}^{2}}\right)\right)\right) \]

      5. *-lowering-*.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right), \left(\frac{sin2phi}{{alphay}^{2}}\right)\right)\right) \]

      6. /-lowering-/.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right), \mathsf{/.f32}\left(sin2phi, \color{blue}{\left({alphay}^{2}\right)}\right)\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right), \mathsf{/.f32}\left(sin2phi, \left(alphay \cdot \color{blue}{alphay}\right)\right)\right)\right) \]

      8. *-lowering-*.f32 75.7%

         \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right), \mathsf{/.f32}\left(sin2phi, \mathsf{*.f32}\left(alphay, \color{blue}{alphay}\right)\right)\right)\right) \]

   7. Simplified 75.7%

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

      1. div-inv N/A

         \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right), \left(sin2phi \cdot \color{blue}{\frac{1}{alphay \cdot alphay}}\right)\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right), \left(\frac{1}{alphay \cdot alphay} \cdot \color{blue}{sin2phi}\right)\right)\right) \]

      3. *-lowering-*.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right), \mathsf{*.f32}\left(\left(\frac{1}{alphay \cdot alphay}\right), \color{blue}{sin2phi}\right)\right)\right) \]

      4. /-lowering-/.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right), \mathsf{*.f32}\left(\mathsf{/.f32}\left(1, \left(alphay \cdot alphay\right)\right), sin2phi\right)\right)\right) \]

      5. *-lowering-*.f32 75.7%

         \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right), \mathsf{*.f32}\left(\mathsf{/.f32}\left(1, \mathsf{*.f32}\left(alphay, alphay\right)\right), sin2phi\right)\right)\right) \]

   9. Applied egg-rr 75.7%

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

       1. flip3-+ N/A

          \[\leadsto \frac{u0}{\frac{{\left(\frac{cos2phi}{alphax \cdot alphax}\right)}^{3} + {\left(\frac{1}{alphay \cdot alphay} \cdot sin2phi\right)}^{3}}{\color{blue}{\frac{cos2phi}{alphax \cdot alphax} \cdot \frac{cos2phi}{alphax \cdot alphax} + \left(\left(\frac{1}{alphay \cdot alphay} \cdot sin2phi\right) \cdot \left(\frac{1}{alphay \cdot alphay} \cdot sin2phi\right) - \frac{cos2phi}{alphax \cdot alphax} \cdot \left(\frac{1}{alphay \cdot alphay} \cdot sin2phi\right)\right)}}} \]

       2. associate-/r/ N/A

          \[\leadsto \frac{u0}{{\left(\frac{cos2phi}{alphax \cdot alphax}\right)}^{3} + {\left(\frac{1}{alphay \cdot alphay} \cdot sin2phi\right)}^{3}} \cdot \color{blue}{\left(\frac{cos2phi}{alphax \cdot alphax} \cdot \frac{cos2phi}{alphax \cdot alphax} + \left(\left(\frac{1}{alphay \cdot alphay} \cdot sin2phi\right) \cdot \left(\frac{1}{alphay \cdot alphay} \cdot sin2phi\right) - \frac{cos2phi}{alphax \cdot alphax} \cdot \left(\frac{1}{alphay \cdot alphay} \cdot sin2phi\right)\right)\right)} \]

       3. associate-/r* N/A

          \[\leadsto \frac{u0}{{\left(\frac{\frac{cos2phi}{alphax}}{alphax}\right)}^{3} + {\left(\frac{1}{alphay \cdot alphay} \cdot sin2phi\right)}^{3}} \cdot \left(\frac{cos2phi}{alphax \cdot alphax} \cdot \frac{cos2phi}{alphax \cdot alphax} + \left(\left(\frac{1}{alphay \cdot alphay} \cdot sin2phi\right) \cdot \left(\frac{1}{alphay \cdot alphay} \cdot sin2phi\right) - \frac{cos2phi}{alphax \cdot alphax} \cdot \left(\frac{1}{alphay \cdot alphay} \cdot sin2phi\right)\right)\right) \]

       4. *-commutative N/A

          \[\leadsto \frac{u0}{{\left(\frac{\frac{cos2phi}{alphax}}{alphax}\right)}^{3} + {\left(sin2phi \cdot \frac{1}{alphay \cdot alphay}\right)}^{3}} \cdot \left(\frac{cos2phi}{alphax \cdot alphax} \cdot \frac{cos2phi}{\color{blue}{alphax} \cdot alphax} + \left(\left(\frac{1}{alphay \cdot alphay} \cdot sin2phi\right) \cdot \left(\frac{1}{alphay \cdot alphay} \cdot sin2phi\right) - \frac{cos2phi}{alphax \cdot alphax} \cdot \left(\frac{1}{alphay \cdot alphay} \cdot sin2phi\right)\right)\right) \]

       5. div-inv N/A

          \[\leadsto \frac{u0}{{\left(\frac{\frac{cos2phi}{alphax}}{alphax}\right)}^{3} + {\left(\frac{sin2phi}{alphay \cdot alphay}\right)}^{3}} \cdot \left(\frac{cos2phi}{alphax \cdot alphax} \cdot \frac{cos2phi}{\color{blue}{alphax} \cdot alphax} + \left(\left(\frac{1}{alphay \cdot alphay} \cdot sin2phi\right) \cdot \left(\frac{1}{alphay \cdot alphay} \cdot sin2phi\right) - \frac{cos2phi}{alphax \cdot alphax} \cdot \left(\frac{1}{alphay \cdot alphay} \cdot sin2phi\right)\right)\right) \]

       6. associate-/l/ N/A

          \[\leadsto \frac{u0}{{\left(\frac{\frac{cos2phi}{alphax}}{alphax}\right)}^{3} + {\left(\frac{\frac{sin2phi}{alphay}}{alphay}\right)}^{3}} \cdot \left(\frac{cos2phi}{alphax \cdot alphax} \cdot \frac{cos2phi}{\color{blue}{alphax} \cdot alphax} + \left(\left(\frac{1}{alphay \cdot alphay} \cdot sin2phi\right) \cdot \left(\frac{1}{alphay \cdot alphay} \cdot sin2phi\right) - \frac{cos2phi}{alphax \cdot alphax} \cdot \left(\frac{1}{alphay \cdot alphay} \cdot sin2phi\right)\right)\right) \]

       7. associate-+r- N/A

          \[\leadsto \frac{u0}{{\left(\frac{\frac{cos2phi}{alphax}}{alphax}\right)}^{3} + {\left(\frac{\frac{sin2phi}{alphay}}{alphay}\right)}^{3}} \cdot \left(\left(\frac{cos2phi}{alphax \cdot alphax} \cdot \frac{cos2phi}{alphax \cdot alphax} + \left(\frac{1}{alphay \cdot alphay} \cdot sin2phi\right) \cdot \left(\frac{1}{alphay \cdot alphay} \cdot sin2phi\right)\right) - \color{blue}{\frac{cos2phi}{alphax \cdot alphax} \cdot \left(\frac{1}{alphay \cdot alphay} \cdot sin2phi\right)}\right) \]

   11. Applied egg-rr 75.7%

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

   12. Taylor expanded in sin2phi around inf

       \[\leadsto \mathsf{*.f32}\left(\color{blue}{\left(\frac{{alphay}^{2}}{sin2phi}\right)}, u0\right) \]
   13. Step-by-step derivation

       1. /-lowering-/.f32 N/A

          \[\leadsto \mathsf{*.f32}\left(\mathsf{/.f32}\left(\left({alphay}^{2}\right), sin2phi\right), u0\right) \]

       2. unpow2 N/A

          \[\leadsto \mathsf{*.f32}\left(\mathsf{/.f32}\left(\left(alphay \cdot alphay\right), sin2phi\right), u0\right) \]

       3. *-lowering-*.f32 71.9%

          \[\leadsto \mathsf{*.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(alphay, alphay\right), sin2phi\right), u0\right) \]

   14. Simplified 71.9%

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

4. Final simplification 67.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;sin2phi \leq 1.000000031374395 \cdot 10^{-22}:\\ \;\;\;\;u0 \cdot \frac{alphax}{\frac{cos2phi}{alphax}}\\ \mathbf{else}:\\ \;\;\;\;u0 \cdot \frac{alphay \cdot alphay}{sin2phi}\\ \end{array} \]
5. Add Preprocessing

Alternative 27: 23.9% accurate, 16.6× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 64.8%

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

   1. distribute-frac-neg N/A

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

   2. distribute-neg-frac2 N/A

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

   3. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\log \left(1 - u0\right), \color{blue}{\left(\mathsf{neg}\left(\left(\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\right)\right)\right)}\right) \]

   4. sub-neg N/A

      \[\leadsto \mathsf{/.f32}\left(\log \left(1 + \left(\mathsf{neg}\left(u0\right)\right)\right), \left(\mathsf{neg}\left(\left(\color{blue}{\frac{cos2phi}{alphax \cdot alphax}} + \frac{sin2phi}{alphay \cdot alphay}\right)\right)\right)\right) \]

   5. log1p-define N/A

      \[\leadsto \mathsf{/.f32}\left(\left(\mathsf{log1p}\left(\mathsf{neg}\left(u0\right)\right)\right), \left(\mathsf{neg}\left(\color{blue}{\left(\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\right)}\right)\right)\right) \]

   6. log1p-lowering-log1p.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\left(\mathsf{neg}\left(u0\right)\right)\right), \left(\mathsf{neg}\left(\color{blue}{\left(\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\right)}\right)\right)\right) \]

   7. neg-lowering-neg.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \left(\mathsf{neg}\left(\left(\color{blue}{\frac{cos2phi}{alphax \cdot alphax}} + \frac{sin2phi}{alphay \cdot alphay}\right)\right)\right)\right) \]

   8. distribute-neg-in N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \left(\left(\mathsf{neg}\left(\frac{cos2phi}{alphax \cdot alphax}\right)\right) + \color{blue}{\left(\mathsf{neg}\left(\frac{sin2phi}{alphay \cdot alphay}\right)\right)}\right)\right) \]

   9. unsub-neg N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \left(\left(\mathsf{neg}\left(\frac{cos2phi}{alphax \cdot alphax}\right)\right) - \color{blue}{\frac{sin2phi}{alphay \cdot alphay}}\right)\right) \]

   10. --lowering--.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\left(\mathsf{neg}\left(\frac{cos2phi}{alphax \cdot alphax}\right)\right), \color{blue}{\left(\frac{sin2phi}{alphay \cdot alphay}\right)}\right)\right) \]

   11. neg-sub0 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\left(0 - \frac{cos2phi}{alphax \cdot alphax}\right), \left(\frac{\color{blue}{sin2phi}}{alphay \cdot alphay}\right)\right)\right) \]

   12. --lowering--.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \left(\frac{cos2phi}{alphax \cdot alphax}\right)\right), \left(\frac{\color{blue}{sin2phi}}{alphay \cdot alphay}\right)\right)\right) \]

   13. /-lowering-/.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(cos2phi, \left(alphax \cdot alphax\right)\right)\right), \left(\frac{sin2phi}{alphay \cdot alphay}\right)\right)\right) \]

   14. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right)\right), \left(\frac{sin2phi}{alphay \cdot alphay}\right)\right)\right) \]

   15. /-lowering-/.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right)\right), \mathsf{/.f32}\left(sin2phi, \color{blue}{\left(alphay \cdot alphay\right)}\right)\right)\right) \]

   16. *-lowering-*.f32 98.3%

       \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right)\right), \mathsf{/.f32}\left(sin2phi, \mathsf{*.f32}\left(alphay, \color{blue}{alphay}\right)\right)\right)\right) \]

3. Simplified 98.3%

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

5. Taylor expanded in cos2phi around inf

   \[\leadsto \color{blue}{-1 \cdot \frac{{alphax}^{2} \cdot \log \left(1 - u0\right)}{cos2phi}} \]
6. Step-by-step derivation

   1. mul-1-neg N/A

      \[\leadsto \mathsf{neg}\left(\frac{{alphax}^{2} \cdot \log \left(1 - u0\right)}{cos2phi}\right) \]

   2. associate-/l* N/A

      \[\leadsto \mathsf{neg}\left({alphax}^{2} \cdot \frac{\log \left(1 - u0\right)}{cos2phi}\right) \]

   3. distribute-rgt-neg-in N/A

      \[\leadsto {alphax}^{2} \cdot \color{blue}{\left(\mathsf{neg}\left(\frac{\log \left(1 - u0\right)}{cos2phi}\right)\right)} \]

   4. mul-1-neg N/A

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

   5. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{*.f32}\left(\left({alphax}^{2}\right), \color{blue}{\left(-1 \cdot \frac{\log \left(1 - u0\right)}{cos2phi}\right)}\right) \]

   6. unpow2 N/A

      \[\leadsto \mathsf{*.f32}\left(\left(alphax \cdot alphax\right), \left(\color{blue}{-1} \cdot \frac{\log \left(1 - u0\right)}{cos2phi}\right)\right) \]

   7. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(alphax, alphax\right), \left(\color{blue}{-1} \cdot \frac{\log \left(1 - u0\right)}{cos2phi}\right)\right) \]

   8. mul-1-neg N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(alphax, alphax\right), \left(\mathsf{neg}\left(\frac{\log \left(1 - u0\right)}{cos2phi}\right)\right)\right) \]

   9. distribute-neg-frac2 N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(alphax, alphax\right), \left(\frac{\log \left(1 - u0\right)}{\color{blue}{\mathsf{neg}\left(cos2phi\right)}}\right)\right) \]

   10. /-lowering-/.f32 N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(alphax, alphax\right), \mathsf{/.f32}\left(\log \left(1 - u0\right), \color{blue}{\left(\mathsf{neg}\left(cos2phi\right)\right)}\right)\right) \]

   11. sub-neg N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(alphax, alphax\right), \mathsf{/.f32}\left(\log \left(1 + \left(\mathsf{neg}\left(u0\right)\right)\right), \left(\mathsf{neg}\left(cos2phi\right)\right)\right)\right) \]

   12. mul-1-neg N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(alphax, alphax\right), \mathsf{/.f32}\left(\log \left(1 + -1 \cdot u0\right), \left(\mathsf{neg}\left(cos2phi\right)\right)\right)\right) \]

   13. log1p-define N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(alphax, alphax\right), \mathsf{/.f32}\left(\left(\mathsf{log1p}\left(-1 \cdot u0\right)\right), \left(\mathsf{neg}\left(\color{blue}{cos2phi}\right)\right)\right)\right) \]

   14. log1p-lowering-log1p.f32 N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(alphax, alphax\right), \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\left(-1 \cdot u0\right)\right), \left(\mathsf{neg}\left(\color{blue}{cos2phi}\right)\right)\right)\right) \]

   15. mul-1-neg N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(alphax, alphax\right), \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\left(\mathsf{neg}\left(u0\right)\right)\right), \left(\mathsf{neg}\left(cos2phi\right)\right)\right)\right) \]

   16. neg-lowering-neg.f32 N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(alphax, alphax\right), \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \left(\mathsf{neg}\left(cos2phi\right)\right)\right)\right) \]

   17. neg-lowering-neg.f32 27.1%

       \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(alphax, alphax\right), \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{neg.f32}\left(cos2phi\right)\right)\right) \]

7. Simplified 27.1%

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

8. Taylor expanded in u0 around 0

   \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(alphax, alphax\right), \color{blue}{\left(\frac{u0}{cos2phi}\right)}\right) \]
9. Step-by-step derivation

   1. /-lowering-/.f32 21.5%

      \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(alphax, alphax\right), \mathsf{/.f32}\left(u0, \color{blue}{cos2phi}\right)\right) \]

10. Simplified 21.5%

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

    1. *-commutative N/A

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

    2. div-inv N/A

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

    3. associate-*l* N/A

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

    4. associate-/r/ N/A

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

    5. clear-num N/A

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

    6. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{*.f32}\left(u0, \color{blue}{\left(\frac{alphax \cdot alphax}{cos2phi}\right)}\right) \]

    7. clear-num N/A

       \[\leadsto \mathsf{*.f32}\left(u0, \left(\frac{1}{\color{blue}{\frac{cos2phi}{alphax \cdot alphax}}}\right)\right) \]

    8. associate-/r* N/A

       \[\leadsto \mathsf{*.f32}\left(u0, \left(\frac{1}{\frac{\frac{cos2phi}{alphax}}{\color{blue}{alphax}}}\right)\right) \]

    9. clear-num N/A

       \[\leadsto \mathsf{*.f32}\left(u0, \left(\frac{alphax}{\color{blue}{\frac{cos2phi}{alphax}}}\right)\right) \]

    10. /-lowering-/.f32 N/A

        \[\leadsto \mathsf{*.f32}\left(u0, \mathsf{/.f32}\left(alphax, \color{blue}{\left(\frac{cos2phi}{alphax}\right)}\right)\right) \]

    11. /-lowering-/.f32 21.5%

        \[\leadsto \mathsf{*.f32}\left(u0, \mathsf{/.f32}\left(alphax, \mathsf{/.f32}\left(cos2phi, \color{blue}{alphax}\right)\right)\right) \]

12. Applied egg-rr 21.5%

    \[\leadsto \color{blue}{u0 \cdot \frac{alphax}{\frac{cos2phi}{alphax}}} \]
13. Add Preprocessing

Reproduce

herbie shell --seed 2024145 
(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)))))