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

Percentage Accurate: 60.5% → 98.3%

Time: 21.4s

Alternatives: 21

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: 60.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Alternative 1: 98.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 57.1%

   \[\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
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 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. 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 98.5%

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

6. Applied egg-rr 98.5%

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

7. Final simplification 98.5%

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

Alternative 2: 93.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{sin2phi}{alphay \cdot alphay} + \frac{cos2phi}{alphax \cdot alphax}\\ \frac{u0}{sin2phi \cdot \left(alphax \cdot alphax\right) + cos2phi \cdot \left(alphay \cdot alphay\right)} \cdot \left(alphay \cdot \left(alphay \cdot \left(alphax \cdot alphax\right)\right)\right) + \left(\frac{0.5}{t\_0} + u0 \cdot \left(\frac{1}{t\_0} \cdot \left(u0 \cdot 0.25 + 0.3333333333333333\right)\right)\right) \cdot \left(u0 \cdot u0\right) \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (let* ((t_0 (+ (/ sin2phi (* alphay alphay)) (/ cos2phi (* alphax alphax)))))
   (+
    (*
     (/ u0 (+ (* sin2phi (* alphax alphax)) (* cos2phi (* alphay alphay))))
     (* alphay (* alphay (* alphax alphax))))
    (*
     (+ (/ 0.5 t_0) (* u0 (* (/ 1.0 t_0) (+ (* u0 0.25) 0.3333333333333333))))
     (* u0 u0)))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float t_0 = (sin2phi / (alphay * alphay)) + (cos2phi / (alphax * alphax));
	return ((u0 / ((sin2phi * (alphax * alphax)) + (cos2phi * (alphay * alphay)))) * (alphay * (alphay * (alphax * alphax)))) + (((0.5f / t_0) + (u0 * ((1.0f / t_0) * ((u0 * 0.25f) + 0.3333333333333333f)))) * (u0 * u0));
}

Fortran

real(4) function code(alphax, alphay, u0, cos2phi, sin2phi)
    real(4), intent (in) :: alphax
    real(4), intent (in) :: alphay
    real(4), intent (in) :: u0
    real(4), intent (in) :: cos2phi
    real(4), intent (in) :: sin2phi
    real(4) :: t_0
    t_0 = (sin2phi / (alphay * alphay)) + (cos2phi / (alphax * alphax))
    code = ((u0 / ((sin2phi * (alphax * alphax)) + (cos2phi * (alphay * alphay)))) * (alphay * (alphay * (alphax * alphax)))) + (((0.5e0 / t_0) + (u0 * ((1.0e0 / t_0) * ((u0 * 0.25e0) + 0.3333333333333333e0)))) * (u0 * u0))
end function

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	t_0 = Float32(Float32(sin2phi / Float32(alphay * alphay)) + Float32(cos2phi / Float32(alphax * alphax)))
	return Float32(Float32(Float32(u0 / Float32(Float32(sin2phi * Float32(alphax * alphax)) + Float32(cos2phi * Float32(alphay * alphay)))) * Float32(alphay * Float32(alphay * Float32(alphax * alphax)))) + Float32(Float32(Float32(Float32(0.5) / t_0) + Float32(u0 * Float32(Float32(Float32(1.0) / t_0) * Float32(Float32(u0 * Float32(0.25)) + Float32(0.3333333333333333))))) * Float32(u0 * u0)))
end

MATLAB

function tmp = code(alphax, alphay, u0, cos2phi, sin2phi)
	t_0 = (sin2phi / (alphay * alphay)) + (cos2phi / (alphax * alphax));
	tmp = ((u0 / ((sin2phi * (alphax * alphax)) + (cos2phi * (alphay * alphay)))) * (alphay * (alphay * (alphax * alphax)))) + (((single(0.5) / t_0) + (u0 * ((single(1.0) / t_0) * ((u0 * single(0.25)) + single(0.3333333333333333))))) * (u0 * u0));
end

Derivation

1. Initial program 57.1%

   \[\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
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 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 u0 around 0

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

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

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

   2. +-commutative N/A

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

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

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

7. Simplified 94.1%

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

   1. distribute-rgt-in N/A

      \[\leadsto \frac{1}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \cdot u0 + \color{blue}{\left(u0 \cdot \left(\frac{\frac{1}{2}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} + u0 \cdot \left(\frac{u0 \cdot \frac{1}{4}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} + \frac{\frac{1}{3}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}}\right)\right)\right) \cdot u0} \]

   2. *-commutative N/A

      \[\leadsto \frac{1}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \cdot u0 + u0 \cdot \color{blue}{\left(u0 \cdot \left(\frac{\frac{1}{2}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} + u0 \cdot \left(\frac{u0 \cdot \frac{1}{4}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} + \frac{\frac{1}{3}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}}\right)\right)\right)} \]

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

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

9. Applied egg-rr 94.1%

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

    1. frac-add N/A

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

    2. associate-/r/ N/A

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

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

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

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

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

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

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

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

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

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

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

    8. *-commutative N/A

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

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

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

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

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

    11. associate-*l* N/A

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

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

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

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

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

    14. *-lowering-*.f32 94.2%

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

11. Applied egg-rr 94.2%

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

Alternative 3: 93.2% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{cos2phi}{alphax \cdot alphax}\\ t_1 := t\_0 + \frac{\frac{sin2phi}{alphay}}{alphay}\\ \frac{u0}{\frac{sin2phi}{alphay \cdot alphay} + t\_0} + \left(u0 \cdot u0\right) \cdot \left(\frac{u0 \cdot \left(u0 \cdot 0.25 + 0.3333333333333333\right)}{t\_1} - \frac{-0.5}{t\_1}\right) \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (let* ((t_0 (/ cos2phi (* alphax alphax)))
        (t_1 (+ t_0 (/ (/ sin2phi alphay) alphay))))
   (+
    (/ u0 (+ (/ sin2phi (* alphay alphay)) t_0))
    (*
     (* u0 u0)
     (- (/ (* u0 (+ (* u0 0.25) 0.3333333333333333)) t_1) (/ -0.5 t_1))))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float t_0 = cos2phi / (alphax * alphax);
	float t_1 = t_0 + ((sin2phi / alphay) / alphay);
	return (u0 / ((sin2phi / (alphay * alphay)) + t_0)) + ((u0 * u0) * (((u0 * ((u0 * 0.25f) + 0.3333333333333333f)) / t_1) - (-0.5f / t_1)));
}

Fortran

real(4) function code(alphax, alphay, u0, cos2phi, sin2phi)
    real(4), intent (in) :: alphax
    real(4), intent (in) :: alphay
    real(4), intent (in) :: u0
    real(4), intent (in) :: cos2phi
    real(4), intent (in) :: sin2phi
    real(4) :: t_0
    real(4) :: t_1
    t_0 = cos2phi / (alphax * alphax)
    t_1 = t_0 + ((sin2phi / alphay) / alphay)
    code = (u0 / ((sin2phi / (alphay * alphay)) + t_0)) + ((u0 * u0) * (((u0 * ((u0 * 0.25e0) + 0.3333333333333333e0)) / t_1) - ((-0.5e0) / t_1)))
end function

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	t_0 = Float32(cos2phi / Float32(alphax * alphax))
	t_1 = Float32(t_0 + Float32(Float32(sin2phi / alphay) / alphay))
	return Float32(Float32(u0 / Float32(Float32(sin2phi / Float32(alphay * alphay)) + t_0)) + Float32(Float32(u0 * u0) * Float32(Float32(Float32(u0 * Float32(Float32(u0 * Float32(0.25)) + Float32(0.3333333333333333))) / t_1) - Float32(Float32(-0.5) / t_1))))
end

MATLAB

function tmp = code(alphax, alphay, u0, cos2phi, sin2phi)
	t_0 = cos2phi / (alphax * alphax);
	t_1 = t_0 + ((sin2phi / alphay) / alphay);
	tmp = (u0 / ((sin2phi / (alphay * alphay)) + t_0)) + ((u0 * u0) * (((u0 * ((u0 * single(0.25)) + single(0.3333333333333333))) / t_1) - (single(-0.5) / t_1)));
end

Derivation

1. Initial program 57.1%

   \[\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
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 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 u0 around 0

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

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

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

   2. +-commutative N/A

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

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

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

7. Simplified 94.1%

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

   1. distribute-rgt-in N/A

      \[\leadsto \frac{1}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \cdot u0 + \color{blue}{\left(u0 \cdot \left(\frac{\frac{1}{2}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} + u0 \cdot \left(\frac{u0 \cdot \frac{1}{4}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} + \frac{\frac{1}{3}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}}\right)\right)\right) \cdot u0} \]

   2. *-commutative N/A

      \[\leadsto \frac{1}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \cdot u0 + u0 \cdot \color{blue}{\left(u0 \cdot \left(\frac{\frac{1}{2}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} + u0 \cdot \left(\frac{u0 \cdot \frac{1}{4}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} + \frac{\frac{1}{3}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}}\right)\right)\right)} \]

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

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

9. Applied egg-rr 94.1%

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

    1. +-commutative N/A

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

    2. fma-define N/A

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

    3. frac-2neg N/A

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

    4. distribute-frac-neg2 N/A

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

    5. fmm-undef N/A

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

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

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

11. Applied egg-rr 94.1%

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

12. Final simplification 94.1%

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

Alternative 4: 93.0% accurate, 4.1× speedup

Math

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

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (/
  (*
   u0
   (- (* u0 (- (* (- u0) (+ (* u0 -0.25) -0.3333333333333333)) -0.5)) -1.0))
  (+ (/ (/ 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)) / (((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))) / (((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(Float32(-u0) * Float32(Float32(u0 * Float32(-0.25)) + Float32(-0.3333333333333333))) - Float32(-0.5))) - Float32(-1.0))) / Float32(Float32(Float32(cos2phi / alphax) / alphax) + Float32(sin2phi / Float32(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))) / (((cos2phi / alphax) / alphax) + (sin2phi / (alphay * alphay)));
end

Derivation

1. Initial program 57.1%

   \[\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
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 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. 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 98.5%

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

6. Applied egg-rr 98.5%

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

7. Taylor expanded in u0 around 0

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

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

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

   2. sub-neg N/A

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

   3. metadata-eval N/A

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

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

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

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

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

   6. sub-neg N/A

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

   7. metadata-eval N/A

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

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

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

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

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

   10. sub-neg N/A

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

   11. metadata-eval N/A

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

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

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

   13. *-lowering-*.f32 94.0%

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

9. Simplified 94.0%

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

10. Final simplification 94.0%

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

Alternative 5: 93.0% accurate, 4.3× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 57.1%

   \[\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
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 93.9%

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

5. Simplified 93.9%

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

6. Final simplification 93.9%

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

Alternative 6: 91.2% accurate, 4.8× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 57.1%

   \[\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
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 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. 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 98.5%

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

6. Applied egg-rr 98.5%

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

   1. associate-/r* N/A

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

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

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

   3. /-lowering-/.f32 98.5%

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

8. Applied egg-rr 98.5%

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

9. Taylor expanded in u0 around 0

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

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

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

    2. sub-neg N/A

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

    3. metadata-eval N/A

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

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

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

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

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

    6. sub-neg N/A

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

    7. metadata-eval N/A

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

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

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

    9. *-lowering-*.f32 92.2%

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

11. Simplified 92.2%

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

12. Final simplification 92.2%

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

Alternative 7: 91.2% accurate, 4.8× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 57.1%

   \[\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
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 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. 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 98.5%

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

6. Applied egg-rr 98.5%

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

7. Taylor expanded in u0 around 0

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

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

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

   2. sub-neg N/A

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

   3. metadata-eval N/A

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

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

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

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

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

   6. sub-neg N/A

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

   7. metadata-eval N/A

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

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

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

   9. *-lowering-*.f32 92.2%

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

9. Simplified 92.2%

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

10. Final simplification 92.2%

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

Alternative 8: 91.2% accurate, 5.0× speedup

Math

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

FPCore

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

C

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

MATLAB

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

Derivation

1. Initial program 57.1%

   \[\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
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 92.1%

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

5. Simplified 92.1%

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

6. Final simplification 92.1%

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

Alternative 9: 81.6% accurate, 5.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;sin2phi \leq 0.00024300000222865492:\\ \;\;\;\;\frac{u0}{\frac{\frac{alphax}{\frac{alphay}{sin2phi}} + \frac{cos2phi}{alphax} \cdot alphay}{alphax \cdot alphay}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-1 + u0 \cdot -0.5\right) \cdot \left(u0 \cdot \left(0 - alphay \cdot alphay\right)\right)}{sin2phi}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (if (<= sin2phi 0.00024300000222865492)
   (/
    u0
    (/
     (+ (/ alphax (/ alphay sin2phi)) (* (/ cos2phi alphax) alphay))
     (* alphax alphay)))
   (/ (* (+ -1.0 (* u0 -0.5)) (* u0 (- 0.0 (* alphay alphay)))) sin2phi)))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float tmp;
	if (sin2phi <= 0.00024300000222865492f) {
		tmp = u0 / (((alphax / (alphay / sin2phi)) + ((cos2phi / alphax) * alphay)) / (alphax * alphay));
	} else {
		tmp = ((-1.0f + (u0 * -0.5f)) * (u0 * (0.0f - (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 <= 0.00024300000222865492e0) then
        tmp = u0 / (((alphax / (alphay / sin2phi)) + ((cos2phi / alphax) * alphay)) / (alphax * alphay))
    else
        tmp = (((-1.0e0) + (u0 * (-0.5e0))) * (u0 * (0.0e0 - (alphay * alphay)))) / sin2phi
    end if
    code = tmp
end function

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = Float32(0.0)
	if (sin2phi <= Float32(0.00024300000222865492))
		tmp = Float32(u0 / Float32(Float32(Float32(alphax / Float32(alphay / sin2phi)) + Float32(Float32(cos2phi / alphax) * alphay)) / Float32(alphax * alphay)));
	else
		tmp = Float32(Float32(Float32(Float32(-1.0) + Float32(u0 * Float32(-0.5))) * Float32(u0 * Float32(Float32(0.0) - 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(0.00024300000222865492))
		tmp = u0 / (((alphax / (alphay / sin2phi)) + ((cos2phi / alphax) * alphay)) / (alphax * alphay));
	else
		tmp = ((single(-1.0) + (u0 * single(-0.5))) * (u0 * (single(0.0) - (alphay * alphay)))) / sin2phi;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if sin2phi < 2.43000002e-4

   1. Initial program 52.9%

      \[\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
   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 98.7%

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

   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. 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 75.9%

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

   7. Simplified 75.9%

      \[\leadsto \color{blue}{\frac{u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}}} \]
   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. associate-/r* N/A

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

      3. associate-/r* N/A

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

      4. frac-add N/A

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

      5. *-commutative N/A

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

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

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

      7. *-commutative N/A

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

      8. *-commutative N/A

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

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

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

      10. clear-num N/A

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

      11. un-div-inv N/A

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

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

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

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

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

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

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

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

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

      16. *-lowering-*.f32 76.1%

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

   9. Applied egg-rr 76.1%

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

   if 2.43000002e-4 < sin2phi

   1. Initial program 61.2%

      \[\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
   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. flip3-- N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \left(\frac{{0}^{3} - {\left(\frac{cos2phi}{alphax \cdot alphax}\right)}^{3}}{0 \cdot 0 + \left(\frac{cos2phi}{alphax \cdot alphax} \cdot \frac{cos2phi}{alphax \cdot alphax} + 0 \cdot \frac{cos2phi}{alphax \cdot alphax}\right)} - \frac{\color{blue}{sin2phi}}{alphay \cdot alphay}\right)\right) \]

   6. Applied egg-rr 68.9%

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

   7. Taylor expanded in u0 around 0

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

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

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

      2. sub-neg N/A

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

      3. metadata-eval N/A

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

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

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

      5. *-lowering-*.f32 62.6%

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

   9. Simplified 62.6%

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

   10. Taylor expanded in alphax around inf

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

       1. mul-1-neg N/A

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

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

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

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

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

       4. associate-*r* N/A

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

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

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

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

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

       7. unpow2 N/A

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

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

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

       9. sub-neg N/A

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

       10. metadata-eval N/A

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

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

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

       12. *-lowering-*.f32 87.4%

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

   12. Simplified 87.4%

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

4. Final simplification 81.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;sin2phi \leq 0.00024300000222865492:\\ \;\;\;\;\frac{u0}{\frac{\frac{alphax}{\frac{alphay}{sin2phi}} + \frac{cos2phi}{alphax} \cdot alphay}{alphax \cdot alphay}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-1 + u0 \cdot -0.5\right) \cdot \left(u0 \cdot \left(0 - alphay \cdot alphay\right)\right)}{sin2phi}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 87.5% accurate, 5.5× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 57.1%

   \[\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
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 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. 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 98.5%

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

6. Applied egg-rr 98.5%

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

   1. associate-/r* N/A

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

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

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

   3. /-lowering-/.f32 98.5%

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

8. Applied egg-rr 98.5%

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

9. Taylor expanded in u0 around 0

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

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

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

    2. sub-neg N/A

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

    3. metadata-eval N/A

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

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

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

    5. *-lowering-*.f32 88.5%

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

11. Simplified 88.5%

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

12. Final simplification 88.5%

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

Alternative 11: 81.6% accurate, 5.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;sin2phi \leq 0.00024300000222865492:\\ \;\;\;\;u0 \cdot \frac{1}{\frac{sin2phi}{alphay \cdot alphay} + \frac{cos2phi}{alphax \cdot alphax}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-1 + u0 \cdot -0.5\right) \cdot \left(u0 \cdot \left(0 - alphay \cdot alphay\right)\right)}{sin2phi}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (if (<= sin2phi 0.00024300000222865492)
   (*
    u0
    (/ 1.0 (+ (/ sin2phi (* alphay alphay)) (/ cos2phi (* alphax alphax)))))
   (/ (* (+ -1.0 (* u0 -0.5)) (* u0 (- 0.0 (* alphay alphay)))) sin2phi)))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float tmp;
	if (sin2phi <= 0.00024300000222865492f) {
		tmp = u0 * (1.0f / ((sin2phi / (alphay * alphay)) + (cos2phi / (alphax * alphax))));
	} else {
		tmp = ((-1.0f + (u0 * -0.5f)) * (u0 * (0.0f - (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 <= 0.00024300000222865492e0) then
        tmp = u0 * (1.0e0 / ((sin2phi / (alphay * alphay)) + (cos2phi / (alphax * alphax))))
    else
        tmp = (((-1.0e0) + (u0 * (-0.5e0))) * (u0 * (0.0e0 - (alphay * alphay)))) / sin2phi
    end if
    code = tmp
end function

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = Float32(0.0)
	if (sin2phi <= Float32(0.00024300000222865492))
		tmp = Float32(u0 * Float32(Float32(1.0) / Float32(Float32(sin2phi / Float32(alphay * alphay)) + Float32(cos2phi / Float32(alphax * alphax)))));
	else
		tmp = Float32(Float32(Float32(Float32(-1.0) + Float32(u0 * Float32(-0.5))) * Float32(u0 * Float32(Float32(0.0) - 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(0.00024300000222865492))
		tmp = u0 * (single(1.0) / ((sin2phi / (alphay * alphay)) + (cos2phi / (alphax * alphax))));
	else
		tmp = ((single(-1.0) + (u0 * single(-0.5))) * (u0 * (single(0.0) - (alphay * alphay)))) / sin2phi;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if sin2phi < 2.43000002e-4

   1. Initial program 52.9%

      \[\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
   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 98.7%

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

   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. 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 75.9%

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

   7. Simplified 75.9%

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

      1. div-inv N/A

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

      2. *-commutative N/A

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

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

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

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

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

      5. +-commutative N/A

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

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

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

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

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

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

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

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

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

      10. *-lowering-*.f32 76.0%

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

   9. Applied egg-rr 76.0%

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

   if 2.43000002e-4 < sin2phi

   1. Initial program 61.2%

      \[\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
   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. flip3-- N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \left(\frac{{0}^{3} - {\left(\frac{cos2phi}{alphax \cdot alphax}\right)}^{3}}{0 \cdot 0 + \left(\frac{cos2phi}{alphax \cdot alphax} \cdot \frac{cos2phi}{alphax \cdot alphax} + 0 \cdot \frac{cos2phi}{alphax \cdot alphax}\right)} - \frac{\color{blue}{sin2phi}}{alphay \cdot alphay}\right)\right) \]

   6. Applied egg-rr 68.9%

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

   7. Taylor expanded in u0 around 0

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

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

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

      2. sub-neg N/A

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

      3. metadata-eval N/A

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

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

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

      5. *-lowering-*.f32 62.6%

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

   9. Simplified 62.6%

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

   10. Taylor expanded in alphax around inf

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

       1. mul-1-neg N/A

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

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

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

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

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

       4. associate-*r* N/A

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

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

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

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

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

       7. unpow2 N/A

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

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

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

       9. sub-neg N/A

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

       10. metadata-eval N/A

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

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

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

       12. *-lowering-*.f32 87.4%

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

   12. Simplified 87.4%

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

4. Final simplification 81.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;sin2phi \leq 0.00024300000222865492:\\ \;\;\;\;u0 \cdot \frac{1}{\frac{sin2phi}{alphay \cdot alphay} + \frac{cos2phi}{alphax \cdot alphax}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-1 + u0 \cdot -0.5\right) \cdot \left(u0 \cdot \left(0 - alphay \cdot alphay\right)\right)}{sin2phi}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 81.6% accurate, 5.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;sin2phi \leq 0.00024300000222865492:\\ \;\;\;\;\frac{u0}{\frac{\frac{cos2phi}{alphax}}{alphax} + \frac{sin2phi}{alphay \cdot alphay}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-1 + u0 \cdot -0.5\right) \cdot \left(u0 \cdot \left(0 - alphay \cdot alphay\right)\right)}{sin2phi}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (if (<= sin2phi 0.00024300000222865492)
   (/ u0 (+ (/ (/ cos2phi alphax) alphax) (/ sin2phi (* alphay alphay))))
   (/ (* (+ -1.0 (* u0 -0.5)) (* u0 (- 0.0 (* alphay alphay)))) sin2phi)))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float tmp;
	if (sin2phi <= 0.00024300000222865492f) {
		tmp = u0 / (((cos2phi / alphax) / alphax) + (sin2phi / (alphay * alphay)));
	} else {
		tmp = ((-1.0f + (u0 * -0.5f)) * (u0 * (0.0f - (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 <= 0.00024300000222865492e0) then
        tmp = u0 / (((cos2phi / alphax) / alphax) + (sin2phi / (alphay * alphay)))
    else
        tmp = (((-1.0e0) + (u0 * (-0.5e0))) * (u0 * (0.0e0 - (alphay * alphay)))) / sin2phi
    end if
    code = tmp
end function

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = Float32(0.0)
	if (sin2phi <= Float32(0.00024300000222865492))
		tmp = Float32(u0 / Float32(Float32(Float32(cos2phi / alphax) / alphax) + Float32(sin2phi / Float32(alphay * alphay))));
	else
		tmp = Float32(Float32(Float32(Float32(-1.0) + Float32(u0 * Float32(-0.5))) * Float32(u0 * Float32(Float32(0.0) - 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(0.00024300000222865492))
		tmp = u0 / (((cos2phi / alphax) / alphax) + (sin2phi / (alphay * alphay)));
	else
		tmp = ((single(-1.0) + (u0 * single(-0.5))) * (u0 * (single(0.0) - (alphay * alphay)))) / sin2phi;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if sin2phi < 2.43000002e-4

   1. Initial program 52.9%

      \[\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
   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 98.7%

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

   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. 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 75.9%

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

   7. Simplified 75.9%

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

      1. associate-/r* N/A

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

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

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

      3. /-lowering-/.f32 76.0%

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

   9. Applied egg-rr 76.0%

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

   if 2.43000002e-4 < sin2phi

   1. Initial program 61.2%

      \[\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
   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. flip3-- N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \left(\frac{{0}^{3} - {\left(\frac{cos2phi}{alphax \cdot alphax}\right)}^{3}}{0 \cdot 0 + \left(\frac{cos2phi}{alphax \cdot alphax} \cdot \frac{cos2phi}{alphax \cdot alphax} + 0 \cdot \frac{cos2phi}{alphax \cdot alphax}\right)} - \frac{\color{blue}{sin2phi}}{alphay \cdot alphay}\right)\right) \]

   6. Applied egg-rr 68.9%

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

   7. Taylor expanded in u0 around 0

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

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

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

      2. sub-neg N/A

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

      3. metadata-eval N/A

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

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

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

      5. *-lowering-*.f32 62.6%

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

   9. Simplified 62.6%

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

   10. Taylor expanded in alphax around inf

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

       1. mul-1-neg N/A

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

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

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

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

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

       4. associate-*r* N/A

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

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

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

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

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

       7. unpow2 N/A

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

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

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

       9. sub-neg N/A

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

       10. metadata-eval N/A

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

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

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

       12. *-lowering-*.f32 87.4%

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

   12. Simplified 87.4%

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

4. Final simplification 81.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;sin2phi \leq 0.00024300000222865492:\\ \;\;\;\;\frac{u0}{\frac{\frac{cos2phi}{alphax}}{alphax} + \frac{sin2phi}{alphay \cdot alphay}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-1 + u0 \cdot -0.5\right) \cdot \left(u0 \cdot \left(0 - alphay \cdot alphay\right)\right)}{sin2phi}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 87.5% accurate, 5.8× speedup

Math

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

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(Float32(cos2phi / alphax) / Float32(-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 57.1%

   \[\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
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 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. 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 98.5%

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

6. Applied egg-rr 98.5%

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

7. Taylor expanded in u0 around 0

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

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

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

   2. sub-neg N/A

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

   3. metadata-eval N/A

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

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

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

   5. *-lowering-*.f32 88.5%

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

9. Simplified 88.5%

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

10. Final simplification 88.5%

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

Alternative 14: 87.5% accurate, 6.1× speedup

Math

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

FPCore

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

C

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

MATLAB

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

Derivation

1. Initial program 57.1%

   \[\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
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 88.4%

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

5. Simplified 88.4%

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

6. Final simplification 88.4%

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

Alternative 15: 76.1% accurate, 8.9× speedup

Math

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

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(Float32(sin2phi / 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 57.1%

   \[\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
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 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 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 76.2%

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

7. Simplified 76.2%

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

   1. associate-/r* N/A

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

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

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

   3. /-lowering-/.f32 76.3%

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

9. Applied egg-rr 76.3%

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

Alternative 16: 76.1% accurate, 8.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 57.1%

   \[\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
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 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 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 76.2%

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

7. Simplified 76.2%

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

8. Final simplification 76.2%

   \[\leadsto \frac{u0}{\frac{sin2phi}{alphay \cdot alphay} + \frac{cos2phi}{alphax \cdot alphax}} \]
9. Add Preprocessing

Alternative 17: 65.8% accurate, 9.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;sin2phi \leq 9.200000173618653 \cdot 10^{-27}:\\ \;\;\;\;u0 \cdot \frac{alphax}{\frac{cos2phi}{alphax}}\\ \mathbf{else}:\\ \;\;\;\;alphay \cdot \left(alphay \cdot \frac{u0}{sin2phi}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (if (<= sin2phi 9.200000173618653e-27)
   (* u0 (/ alphax (/ cos2phi alphax)))
   (* alphay (* alphay (/ u0 sin2phi)))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float tmp;
	if (sin2phi <= 9.200000173618653e-27f) {
		tmp = u0 * (alphax / (cos2phi / alphax));
	} else {
		tmp = alphay * (alphay * (u0 / 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 <= 9.200000173618653e-27) then
        tmp = u0 * (alphax / (cos2phi / alphax))
    else
        tmp = alphay * (alphay * (u0 / sin2phi))
    end if
    code = tmp
end function

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = Float32(0.0)
	if (sin2phi <= Float32(9.200000173618653e-27))
		tmp = Float32(u0 * Float32(alphax / Float32(cos2phi / alphax)));
	else
		tmp = Float32(alphay * Float32(alphay * Float32(u0 / sin2phi)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = single(0.0);
	if (sin2phi <= single(9.200000173618653e-27))
		tmp = u0 * (alphax / (cos2phi / alphax));
	else
		tmp = alphay * (alphay * (u0 / sin2phi));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if sin2phi < 9.20000017e-27

   1. Initial program 59.2%

      \[\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
   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. 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 71.1%

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

   7. Simplified 71.1%

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

   8. Taylor expanded in cos2phi around inf

      \[\leadsto \mathsf{/.f32}\left(u0, \color{blue}{\left(\frac{cos2phi}{{alphax}^{2}}\right)}\right) \]
   9. Step-by-step derivation

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

         \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{/.f32}\left(cos2phi, \color{blue}{\left({alphax}^{2}\right)}\right)\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{/.f32}\left(cos2phi, \left(alphax \cdot \color{blue}{alphax}\right)\right)\right) \]

      3. *-lowering-*.f32 61.4%

         \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, \color{blue}{alphax}\right)\right)\right) \]

   10. Simplified 61.4%

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

       1. clear-num N/A

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

       2. associate-/r* N/A

          \[\leadsto \frac{1}{\frac{\frac{\frac{cos2phi}{alphax}}{alphax}}{u0}} \]

       3. associate-/r/ N/A

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

       4. clear-num N/A

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

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

          \[\leadsto \mathsf{*.f32}\left(\left(\frac{alphax}{\frac{cos2phi}{alphax}}\right), \color{blue}{u0}\right) \]

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

          \[\leadsto \mathsf{*.f32}\left(\mathsf{/.f32}\left(alphax, \left(\frac{cos2phi}{alphax}\right)\right), u0\right) \]

       7. /-lowering-/.f32 62.0%

          \[\leadsto \mathsf{*.f32}\left(\mathsf{/.f32}\left(alphax, \mathsf{/.f32}\left(cos2phi, alphax\right)\right), u0\right) \]

   12. Applied egg-rr 62.0%

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

   if 9.20000017e-27 < sin2phi

   1. Initial program 56.8%

      \[\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
   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 98.4%

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

   3. Simplified 98.4%

      \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(-u0\right)}{\left(0 - \frac{cos2phi}{alphax \cdot alphax}\right) - \frac{sin2phi}{alphay \cdot alphay}}} \]
   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 77.1%

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

   7. Simplified 77.1%

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

   8. Taylor expanded in alphay around 0

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

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

         \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{/.f32}\left(\left(sin2phi + \frac{{alphay}^{2} \cdot cos2phi}{{alphax}^{2}}\right), \color{blue}{\left({alphay}^{2}\right)}\right)\right) \]

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

         \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{/.f32}\left(\mathsf{+.f32}\left(sin2phi, \left(\frac{{alphay}^{2} \cdot cos2phi}{{alphax}^{2}}\right)\right), \left({\color{blue}{alphay}}^{2}\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{/.f32}\left(\mathsf{+.f32}\left(sin2phi, \mathsf{/.f32}\left(\left({alphay}^{2} \cdot cos2phi\right), \left({alphax}^{2}\right)\right)\right), \left({alphay}^{2}\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{/.f32}\left(\mathsf{+.f32}\left(sin2phi, \mathsf{/.f32}\left(\mathsf{*.f32}\left(\left({alphay}^{2}\right), cos2phi\right), \left({alphax}^{2}\right)\right)\right), \left({alphay}^{2}\right)\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{/.f32}\left(\mathsf{+.f32}\left(sin2phi, \mathsf{/.f32}\left(\mathsf{*.f32}\left(\left(alphay \cdot alphay\right), cos2phi\right), \left({alphax}^{2}\right)\right)\right), \left({alphay}^{2}\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{/.f32}\left(\mathsf{+.f32}\left(sin2phi, \mathsf{/.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(alphay, alphay\right), cos2phi\right), \left({alphax}^{2}\right)\right)\right), \left({alphay}^{2}\right)\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{/.f32}\left(\mathsf{+.f32}\left(sin2phi, \mathsf{/.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(alphay, alphay\right), cos2phi\right), \left(alphax \cdot alphax\right)\right)\right), \left({alphay}^{2}\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{/.f32}\left(\mathsf{+.f32}\left(sin2phi, \mathsf{/.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(alphay, alphay\right), cos2phi\right), \mathsf{*.f32}\left(alphax, alphax\right)\right)\right), \left({alphay}^{2}\right)\right)\right) \]

      9. unpow2 N/A

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

      10. *-lowering-*.f32 77.1%

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

   10. Simplified 77.1%

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

       1. associate-/r* N/A

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

       2. associate-/r/ N/A

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

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

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

   12. Applied egg-rr 77.3%

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

   13. Taylor expanded in sin2phi around inf

       \[\leadsto \mathsf{*.f32}\left(\color{blue}{\left(\frac{alphay \cdot u0}{sin2phi}\right)}, alphay\right) \]
   14. Step-by-step derivation

       1. associate-/l* N/A

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

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

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

       3. /-lowering-/.f32 66.9%

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

   15. Simplified 66.9%

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

4. Final simplification 66.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;sin2phi \leq 9.200000173618653 \cdot 10^{-27}:\\ \;\;\;\;u0 \cdot \frac{alphax}{\frac{cos2phi}{alphax}}\\ \mathbf{else}:\\ \;\;\;\;alphay \cdot \left(alphay \cdot \frac{u0}{sin2phi}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 65.8% accurate, 9.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;sin2phi \leq 9.200000173618653 \cdot 10^{-27}:\\ \;\;\;\;alphax \cdot \left(alphax \cdot \frac{u0}{cos2phi}\right)\\ \mathbf{else}:\\ \;\;\;\;alphay \cdot \left(alphay \cdot \frac{u0}{sin2phi}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (if (<= sin2phi 9.200000173618653e-27)
   (* alphax (* alphax (/ u0 cos2phi)))
   (* alphay (* alphay (/ u0 sin2phi)))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float tmp;
	if (sin2phi <= 9.200000173618653e-27f) {
		tmp = alphax * (alphax * (u0 / cos2phi));
	} else {
		tmp = alphay * (alphay * (u0 / 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 <= 9.200000173618653e-27) then
        tmp = alphax * (alphax * (u0 / cos2phi))
    else
        tmp = alphay * (alphay * (u0 / sin2phi))
    end if
    code = tmp
end function

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = Float32(0.0)
	if (sin2phi <= Float32(9.200000173618653e-27))
		tmp = Float32(alphax * Float32(alphax * Float32(u0 / cos2phi)));
	else
		tmp = Float32(alphay * Float32(alphay * Float32(u0 / sin2phi)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = single(0.0);
	if (sin2phi <= single(9.200000173618653e-27))
		tmp = alphax * (alphax * (u0 / cos2phi));
	else
		tmp = alphay * (alphay * (u0 / sin2phi));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if sin2phi < 9.20000017e-27

   1. Initial program 59.2%

      \[\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
   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. 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 71.1%

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

   7. Simplified 71.1%

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

   8. Taylor expanded in cos2phi around inf

      \[\leadsto \mathsf{/.f32}\left(u0, \color{blue}{\left(\frac{cos2phi}{{alphax}^{2}}\right)}\right) \]
   9. Step-by-step derivation

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

         \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{/.f32}\left(cos2phi, \color{blue}{\left({alphax}^{2}\right)}\right)\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{/.f32}\left(cos2phi, \left(alphax \cdot \color{blue}{alphax}\right)\right)\right) \]

      3. *-lowering-*.f32 61.4%

         \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, \color{blue}{alphax}\right)\right)\right) \]

   10. Simplified 61.4%

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

       1. associate-/r/ N/A

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

       2. associate-*r* N/A

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

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

          \[\leadsto \mathsf{*.f32}\left(\left(\frac{u0}{cos2phi} \cdot alphax\right), \color{blue}{alphax}\right) \]

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

          \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(\left(\frac{u0}{cos2phi}\right), alphax\right), alphax\right) \]

       5. /-lowering-/.f32 61.8%

          \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{/.f32}\left(u0, cos2phi\right), alphax\right), alphax\right) \]

   12. Applied egg-rr 61.8%

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

   if 9.20000017e-27 < sin2phi

   1. Initial program 56.8%

      \[\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
   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 98.4%

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

   3. Simplified 98.4%

      \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(-u0\right)}{\left(0 - \frac{cos2phi}{alphax \cdot alphax}\right) - \frac{sin2phi}{alphay \cdot alphay}}} \]
   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 77.1%

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

   7. Simplified 77.1%

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

   8. Taylor expanded in alphay around 0

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

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

         \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{/.f32}\left(\left(sin2phi + \frac{{alphay}^{2} \cdot cos2phi}{{alphax}^{2}}\right), \color{blue}{\left({alphay}^{2}\right)}\right)\right) \]

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

         \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{/.f32}\left(\mathsf{+.f32}\left(sin2phi, \left(\frac{{alphay}^{2} \cdot cos2phi}{{alphax}^{2}}\right)\right), \left({\color{blue}{alphay}}^{2}\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{/.f32}\left(\mathsf{+.f32}\left(sin2phi, \mathsf{/.f32}\left(\left({alphay}^{2} \cdot cos2phi\right), \left({alphax}^{2}\right)\right)\right), \left({alphay}^{2}\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{/.f32}\left(\mathsf{+.f32}\left(sin2phi, \mathsf{/.f32}\left(\mathsf{*.f32}\left(\left({alphay}^{2}\right), cos2phi\right), \left({alphax}^{2}\right)\right)\right), \left({alphay}^{2}\right)\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{/.f32}\left(\mathsf{+.f32}\left(sin2phi, \mathsf{/.f32}\left(\mathsf{*.f32}\left(\left(alphay \cdot alphay\right), cos2phi\right), \left({alphax}^{2}\right)\right)\right), \left({alphay}^{2}\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{/.f32}\left(\mathsf{+.f32}\left(sin2phi, \mathsf{/.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(alphay, alphay\right), cos2phi\right), \left({alphax}^{2}\right)\right)\right), \left({alphay}^{2}\right)\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{/.f32}\left(\mathsf{+.f32}\left(sin2phi, \mathsf{/.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(alphay, alphay\right), cos2phi\right), \left(alphax \cdot alphax\right)\right)\right), \left({alphay}^{2}\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{/.f32}\left(\mathsf{+.f32}\left(sin2phi, \mathsf{/.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(alphay, alphay\right), cos2phi\right), \mathsf{*.f32}\left(alphax, alphax\right)\right)\right), \left({alphay}^{2}\right)\right)\right) \]

      9. unpow2 N/A

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

      10. *-lowering-*.f32 77.1%

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

   10. Simplified 77.1%

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

       1. associate-/r* N/A

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

       2. associate-/r/ N/A

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

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

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

   12. Applied egg-rr 77.3%

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

   13. Taylor expanded in sin2phi around inf

       \[\leadsto \mathsf{*.f32}\left(\color{blue}{\left(\frac{alphay \cdot u0}{sin2phi}\right)}, alphay\right) \]
   14. Step-by-step derivation

       1. associate-/l* N/A

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

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

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

       3. /-lowering-/.f32 66.9%

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

   15. Simplified 66.9%

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

4. Final simplification 66.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;sin2phi \leq 9.200000173618653 \cdot 10^{-27}:\\ \;\;\;\;alphax \cdot \left(alphax \cdot \frac{u0}{cos2phi}\right)\\ \mathbf{else}:\\ \;\;\;\;alphay \cdot \left(alphay \cdot \frac{u0}{sin2phi}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 65.8% accurate, 9.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;sin2phi \leq 9.200000173618653 \cdot 10^{-27}:\\ \;\;\;\;u0 \cdot \frac{alphax \cdot alphax}{cos2phi}\\ \mathbf{else}:\\ \;\;\;\;alphay \cdot \left(alphay \cdot \frac{u0}{sin2phi}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (if (<= sin2phi 9.200000173618653e-27)
   (* u0 (/ (* alphax alphax) cos2phi))
   (* alphay (* alphay (/ u0 sin2phi)))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float tmp;
	if (sin2phi <= 9.200000173618653e-27f) {
		tmp = u0 * ((alphax * alphax) / cos2phi);
	} else {
		tmp = alphay * (alphay * (u0 / 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 <= 9.200000173618653e-27) then
        tmp = u0 * ((alphax * alphax) / cos2phi)
    else
        tmp = alphay * (alphay * (u0 / sin2phi))
    end if
    code = tmp
end function

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = Float32(0.0)
	if (sin2phi <= Float32(9.200000173618653e-27))
		tmp = Float32(u0 * Float32(Float32(alphax * alphax) / cos2phi));
	else
		tmp = Float32(alphay * Float32(alphay * Float32(u0 / sin2phi)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = single(0.0);
	if (sin2phi <= single(9.200000173618653e-27))
		tmp = u0 * ((alphax * alphax) / cos2phi);
	else
		tmp = alphay * (alphay * (u0 / sin2phi));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if sin2phi < 9.20000017e-27

   1. Initial program 59.2%

      \[\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
   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. Taylor expanded in u0 around 0

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

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

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

      2. +-commutative N/A

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

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

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

   7. Simplified 94.0%

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

   8. Taylor expanded in alphax around 0

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

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

         \[\leadsto \mathsf{*.f32}\left(u0, \mathsf{*.f32}\left(\left({alphax}^{2}\right), \color{blue}{\left(u0 \cdot \left(u0 \cdot \left(\frac{1}{4} \cdot \frac{u0}{cos2phi} + \frac{1}{3} \cdot \frac{1}{cos2phi}\right) + \frac{1}{2} \cdot \frac{1}{cos2phi}\right) + \frac{1}{cos2phi}\right)}\right)\right) \]

      2. unpow2 N/A

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

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

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

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

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

   10. Simplified 78.8%

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

   11. Taylor expanded in u0 around 0

       \[\leadsto \mathsf{*.f32}\left(u0, \color{blue}{\left(\frac{{alphax}^{2}}{cos2phi}\right)}\right) \]
   12. Step-by-step derivation

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

          \[\leadsto \mathsf{*.f32}\left(u0, \mathsf{/.f32}\left(\left({alphax}^{2}\right), \color{blue}{cos2phi}\right)\right) \]

       2. unpow2 N/A

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

       3. *-lowering-*.f32 61.7%

          \[\leadsto \mathsf{*.f32}\left(u0, \mathsf{/.f32}\left(\mathsf{*.f32}\left(alphax, alphax\right), cos2phi\right)\right) \]

   13. Simplified 61.7%

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

   if 9.20000017e-27 < sin2phi

   1. Initial program 56.8%

      \[\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
   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 98.4%

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

   3. Simplified 98.4%

      \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(-u0\right)}{\left(0 - \frac{cos2phi}{alphax \cdot alphax}\right) - \frac{sin2phi}{alphay \cdot alphay}}} \]
   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 77.1%

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

   7. Simplified 77.1%

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

   8. Taylor expanded in alphay around 0

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

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

         \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{/.f32}\left(\left(sin2phi + \frac{{alphay}^{2} \cdot cos2phi}{{alphax}^{2}}\right), \color{blue}{\left({alphay}^{2}\right)}\right)\right) \]

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

         \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{/.f32}\left(\mathsf{+.f32}\left(sin2phi, \left(\frac{{alphay}^{2} \cdot cos2phi}{{alphax}^{2}}\right)\right), \left({\color{blue}{alphay}}^{2}\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{/.f32}\left(\mathsf{+.f32}\left(sin2phi, \mathsf{/.f32}\left(\left({alphay}^{2} \cdot cos2phi\right), \left({alphax}^{2}\right)\right)\right), \left({alphay}^{2}\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{/.f32}\left(\mathsf{+.f32}\left(sin2phi, \mathsf{/.f32}\left(\mathsf{*.f32}\left(\left({alphay}^{2}\right), cos2phi\right), \left({alphax}^{2}\right)\right)\right), \left({alphay}^{2}\right)\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{/.f32}\left(\mathsf{+.f32}\left(sin2phi, \mathsf{/.f32}\left(\mathsf{*.f32}\left(\left(alphay \cdot alphay\right), cos2phi\right), \left({alphax}^{2}\right)\right)\right), \left({alphay}^{2}\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{/.f32}\left(\mathsf{+.f32}\left(sin2phi, \mathsf{/.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(alphay, alphay\right), cos2phi\right), \left({alphax}^{2}\right)\right)\right), \left({alphay}^{2}\right)\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{/.f32}\left(\mathsf{+.f32}\left(sin2phi, \mathsf{/.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(alphay, alphay\right), cos2phi\right), \left(alphax \cdot alphax\right)\right)\right), \left({alphay}^{2}\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{/.f32}\left(\mathsf{+.f32}\left(sin2phi, \mathsf{/.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(alphay, alphay\right), cos2phi\right), \mathsf{*.f32}\left(alphax, alphax\right)\right)\right), \left({alphay}^{2}\right)\right)\right) \]

      9. unpow2 N/A

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

      10. *-lowering-*.f32 77.1%

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

   10. Simplified 77.1%

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

       1. associate-/r* N/A

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

       2. associate-/r/ N/A

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

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

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

   12. Applied egg-rr 77.3%

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

   13. Taylor expanded in sin2phi around inf

       \[\leadsto \mathsf{*.f32}\left(\color{blue}{\left(\frac{alphay \cdot u0}{sin2phi}\right)}, alphay\right) \]
   14. Step-by-step derivation

       1. associate-/l* N/A

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

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

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

       3. /-lowering-/.f32 66.9%

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

   15. Simplified 66.9%

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

4. Final simplification 66.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;sin2phi \leq 9.200000173618653 \cdot 10^{-27}:\\ \;\;\;\;u0 \cdot \frac{alphax \cdot alphax}{cos2phi}\\ \mathbf{else}:\\ \;\;\;\;alphay \cdot \left(alphay \cdot \frac{u0}{sin2phi}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 65.8% accurate, 9.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;sin2phi \leq 9.200000173618653 \cdot 10^{-27}:\\ \;\;\;\;u0 \cdot \frac{alphax \cdot alphax}{cos2phi}\\ \mathbf{else}:\\ \;\;\;\;\left(alphay \cdot alphay\right) \cdot \frac{u0}{sin2phi}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (if (<= sin2phi 9.200000173618653e-27)
   (* u0 (/ (* alphax alphax) cos2phi))
   (* (* alphay alphay) (/ u0 sin2phi))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float tmp;
	if (sin2phi <= 9.200000173618653e-27f) {
		tmp = u0 * ((alphax * alphax) / cos2phi);
	} else {
		tmp = (alphay * alphay) * (u0 / 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 <= 9.200000173618653e-27) then
        tmp = u0 * ((alphax * alphax) / cos2phi)
    else
        tmp = (alphay * alphay) * (u0 / sin2phi)
    end if
    code = tmp
end function

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = Float32(0.0)
	if (sin2phi <= Float32(9.200000173618653e-27))
		tmp = Float32(u0 * Float32(Float32(alphax * alphax) / cos2phi));
	else
		tmp = Float32(Float32(alphay * alphay) * Float32(u0 / sin2phi));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = single(0.0);
	if (sin2phi <= single(9.200000173618653e-27))
		tmp = u0 * ((alphax * alphax) / cos2phi);
	else
		tmp = (alphay * alphay) * (u0 / sin2phi);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if sin2phi < 9.20000017e-27

   1. Initial program 59.2%

      \[\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
   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. Taylor expanded in u0 around 0

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

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

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

      2. +-commutative N/A

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

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

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

   7. Simplified 94.0%

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

   8. Taylor expanded in alphax around 0

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

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

         \[\leadsto \mathsf{*.f32}\left(u0, \mathsf{*.f32}\left(\left({alphax}^{2}\right), \color{blue}{\left(u0 \cdot \left(u0 \cdot \left(\frac{1}{4} \cdot \frac{u0}{cos2phi} + \frac{1}{3} \cdot \frac{1}{cos2phi}\right) + \frac{1}{2} \cdot \frac{1}{cos2phi}\right) + \frac{1}{cos2phi}\right)}\right)\right) \]

      2. unpow2 N/A

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

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

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

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

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

   10. Simplified 78.8%

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

   11. Taylor expanded in u0 around 0

       \[\leadsto \mathsf{*.f32}\left(u0, \color{blue}{\left(\frac{{alphax}^{2}}{cos2phi}\right)}\right) \]
   12. Step-by-step derivation

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

          \[\leadsto \mathsf{*.f32}\left(u0, \mathsf{/.f32}\left(\left({alphax}^{2}\right), \color{blue}{cos2phi}\right)\right) \]

       2. unpow2 N/A

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

       3. *-lowering-*.f32 61.7%

          \[\leadsto \mathsf{*.f32}\left(u0, \mathsf{/.f32}\left(\mathsf{*.f32}\left(alphax, alphax\right), cos2phi\right)\right) \]

   13. Simplified 61.7%

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

   if 9.20000017e-27 < sin2phi

   1. Initial program 56.8%

      \[\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
   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 98.4%

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

   3. Simplified 98.4%

      \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(-u0\right)}{\left(0 - \frac{cos2phi}{alphax \cdot alphax}\right) - \frac{sin2phi}{alphay \cdot alphay}}} \]
   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 77.1%

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

   7. Simplified 77.1%

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

   8. Taylor expanded in alphay around 0

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

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

         \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{/.f32}\left(\left(sin2phi + \frac{{alphay}^{2} \cdot cos2phi}{{alphax}^{2}}\right), \color{blue}{\left({alphay}^{2}\right)}\right)\right) \]

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

         \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{/.f32}\left(\mathsf{+.f32}\left(sin2phi, \left(\frac{{alphay}^{2} \cdot cos2phi}{{alphax}^{2}}\right)\right), \left({\color{blue}{alphay}}^{2}\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{/.f32}\left(\mathsf{+.f32}\left(sin2phi, \mathsf{/.f32}\left(\left({alphay}^{2} \cdot cos2phi\right), \left({alphax}^{2}\right)\right)\right), \left({alphay}^{2}\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{/.f32}\left(\mathsf{+.f32}\left(sin2phi, \mathsf{/.f32}\left(\mathsf{*.f32}\left(\left({alphay}^{2}\right), cos2phi\right), \left({alphax}^{2}\right)\right)\right), \left({alphay}^{2}\right)\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{/.f32}\left(\mathsf{+.f32}\left(sin2phi, \mathsf{/.f32}\left(\mathsf{*.f32}\left(\left(alphay \cdot alphay\right), cos2phi\right), \left({alphax}^{2}\right)\right)\right), \left({alphay}^{2}\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{/.f32}\left(\mathsf{+.f32}\left(sin2phi, \mathsf{/.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(alphay, alphay\right), cos2phi\right), \left({alphax}^{2}\right)\right)\right), \left({alphay}^{2}\right)\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{/.f32}\left(\mathsf{+.f32}\left(sin2phi, \mathsf{/.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(alphay, alphay\right), cos2phi\right), \left(alphax \cdot alphax\right)\right)\right), \left({alphay}^{2}\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{/.f32}\left(\mathsf{+.f32}\left(sin2phi, \mathsf{/.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(alphay, alphay\right), cos2phi\right), \mathsf{*.f32}\left(alphax, alphax\right)\right)\right), \left({alphay}^{2}\right)\right)\right) \]

      9. unpow2 N/A

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

      10. *-lowering-*.f32 77.1%

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

   10. Simplified 77.1%

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

       1. associate-/r* N/A

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

       2. associate-/r/ N/A

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

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

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

   12. Applied egg-rr 77.3%

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

   13. Taylor expanded in sin2phi around inf

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

       1. associate-/l* N/A

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

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

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

       3. unpow2 N/A

          \[\leadsto \mathsf{*.f32}\left(\left(alphay \cdot alphay\right), \left(\frac{\color{blue}{u0}}{sin2phi}\right)\right) \]

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

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

       5. /-lowering-/.f32 66.9%

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

   15. Simplified 66.9%

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

Alternative 21: 23.7% accurate, 16.6× speedup

Math

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

FPCore

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

C

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

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 * alphax) / cos2phi)
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 57.1%

   \[\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
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 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 u0 around 0

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

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

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

   2. +-commutative N/A

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

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

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

7. Simplified 94.1%

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

8. Taylor expanded in alphax around 0

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

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

      \[\leadsto \mathsf{*.f32}\left(u0, \mathsf{*.f32}\left(\left({alphax}^{2}\right), \color{blue}{\left(u0 \cdot \left(u0 \cdot \left(\frac{1}{4} \cdot \frac{u0}{cos2phi} + \frac{1}{3} \cdot \frac{1}{cos2phi}\right) + \frac{1}{2} \cdot \frac{1}{cos2phi}\right) + \frac{1}{cos2phi}\right)}\right)\right) \]

   2. unpow2 N/A

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

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

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

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

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

10. Simplified 26.3%

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

11. Taylor expanded in u0 around 0

    \[\leadsto \mathsf{*.f32}\left(u0, \color{blue}{\left(\frac{{alphax}^{2}}{cos2phi}\right)}\right) \]
12. Step-by-step derivation

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

       \[\leadsto \mathsf{*.f32}\left(u0, \mathsf{/.f32}\left(\left({alphax}^{2}\right), \color{blue}{cos2phi}\right)\right) \]

    2. unpow2 N/A

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

    3. *-lowering-*.f32 22.8%

       \[\leadsto \mathsf{*.f32}\left(u0, \mathsf{/.f32}\left(\mathsf{*.f32}\left(alphax, alphax\right), cos2phi\right)\right) \]

13. Simplified 22.8%

    \[\leadsto u0 \cdot \color{blue}{\frac{alphax \cdot alphax}{cos2phi}} \]
14. Add Preprocessing

Reproduce

herbie shell --seed 2024150 
(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)))))