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

Percentage Accurate: 60.8% → 98.4%

Time: 18.4s

Alternatives: 14

Speedup: 8.9×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Initial Program: 60.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Alternative 1: 98.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 58.1%

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

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

   12. distribute-neg-frac2 N/A

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

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

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

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

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

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

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

   16. /-lowering-/.f32 N/A

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

   17. *-lowering-*.f32 97.8%

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

3. Simplified 97.8%

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

   1. distribute-frac-neg2 N/A

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

   2. remove-double-neg N/A

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

   3. remove-double-neg N/A

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

   4. associate-/l/ N/A

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

   5. distribute-frac-neg N/A

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

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

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

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

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

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

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

   9. remove-double-neg N/A

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

   10. remove-double-neg 97.9%

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

6. Applied egg-rr 97.9%

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

7. Final simplification 97.9%

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

Alternative 2: 93.1% accurate, 3.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 58.1%

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

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

   12. distribute-neg-frac2 N/A

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

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

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

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

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

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

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

   16. /-lowering-/.f32 N/A

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

   17. *-lowering-*.f32 97.8%

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

3. Simplified 97.8%

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

   1. associate-/r* N/A

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

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

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

   3. /-lowering-/.f32 97.8%

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

6. Applied egg-rr 97.8%

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

7. Taylor expanded in u0 around 0

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

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

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

   2. sub-neg N/A

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

   3. metadata-eval N/A

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

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

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

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

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

   6. sub-neg N/A

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

   7. metadata-eval N/A

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

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

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

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

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

   10. sub-neg N/A

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

   11. metadata-eval N/A

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

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

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

   13. *-commutative N/A

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

   14. *-lowering-*.f32 92.9%

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

9. Simplified 92.9%

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

    1. frac-sub N/A

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

    2. associate-/r/ N/A

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

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

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

11. Applied egg-rr 93.9%

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

12. Final simplification 93.9%

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

Alternative 3: 90.2% accurate, 4.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;sin2phi \leq 120:\\ \;\;\;\;\frac{u0 \cdot \left(1 + u0 \cdot 0.5\right)}{\frac{sin2phi}{alphay \cdot alphay} + \frac{cos2phi}{alphax \cdot alphax}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(u0 \cdot \left(alphay \cdot alphay\right)\right) \cdot \left(u0 \cdot \left(\left(-u0\right) \cdot \left(u0 \cdot -0.25 + -0.3333333333333333\right) - -0.5\right) - -1\right)}{sin2phi}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (if (<= sin2phi 120.0)
   (/
    (* u0 (+ 1.0 (* u0 0.5)))
    (+ (/ sin2phi (* alphay alphay)) (/ cos2phi (* alphax alphax))))
   (/
    (*
     (* u0 (* alphay alphay))
     (- (* u0 (- (* (- u0) (+ (* u0 -0.25) -0.3333333333333333)) -0.5)) -1.0))
    sin2phi)))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float tmp;
	if (sin2phi <= 120.0f) {
		tmp = (u0 * (1.0f + (u0 * 0.5f))) / ((sin2phi / (alphay * alphay)) + (cos2phi / (alphax * alphax)));
	} else {
		tmp = ((u0 * (alphay * alphay)) * ((u0 * ((-u0 * ((u0 * -0.25f) + -0.3333333333333333f)) - -0.5f)) - -1.0f)) / 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 <= 120.0e0) then
        tmp = (u0 * (1.0e0 + (u0 * 0.5e0))) / ((sin2phi / (alphay * alphay)) + (cos2phi / (alphax * alphax)))
    else
        tmp = ((u0 * (alphay * alphay)) * ((u0 * ((-u0 * ((u0 * (-0.25e0)) + (-0.3333333333333333e0))) - (-0.5e0))) - (-1.0e0))) / sin2phi
    end if
    code = tmp
end function

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = Float32(0.0)
	if (sin2phi <= Float32(120.0))
		tmp = Float32(Float32(u0 * Float32(Float32(1.0) + Float32(u0 * Float32(0.5)))) / Float32(Float32(sin2phi / Float32(alphay * alphay)) + Float32(cos2phi / Float32(alphax * alphax))));
	else
		tmp = Float32(Float32(Float32(u0 * Float32(alphay * alphay)) * Float32(Float32(u0 * Float32(Float32(Float32(-u0) * Float32(Float32(u0 * Float32(-0.25)) + Float32(-0.3333333333333333))) - Float32(-0.5))) - Float32(-1.0))) / sin2phi);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = single(0.0);
	if (sin2phi <= single(120.0))
		tmp = (u0 * (single(1.0) + (u0 * single(0.5)))) / ((sin2phi / (alphay * alphay)) + (cos2phi / (alphax * alphax)));
	else
		tmp = ((u0 * (alphay * alphay)) * ((u0 * ((-u0 * ((u0 * single(-0.25)) + single(-0.3333333333333333))) - single(-0.5))) - single(-1.0))) / sin2phi;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if sin2phi < 120

   1. Initial program 51.7%

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

      1. distribute-frac-neg N/A

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

      2. distribute-neg-frac2 N/A

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

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

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

      4. sub-neg N/A

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

      5. log1p-define N/A

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

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

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

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

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

      8. distribute-neg-in N/A

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

      9. unsub-neg N/A

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

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

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

      11. associate-/r* N/A

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

      12. distribute-neg-frac2 N/A

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

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

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

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

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

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

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

      16. /-lowering-/.f32 N/A

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

      17. *-lowering-*.f32 98.7%

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

   3. Simplified 98.7%

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

      1. frac-2neg N/A

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

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

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

      3. neg-log N/A

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

      4. flip3-+ N/A

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

      5. clear-num N/A

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

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

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

      7. clear-num N/A

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

      8. flip3-+ N/A

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

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

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

      10. unsub-neg N/A

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

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

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

      12. sub-neg N/A

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

   6. Applied egg-rr 49.6%

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

   7. Taylor expanded in u0 around 0

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

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

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

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

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

      3. *-commutative N/A

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

      4. *-lowering-*.f32 89.2%

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

   9. Simplified 89.2%

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

   if 120 < sin2phi

   1. Initial program 66.5%

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

      1. distribute-frac-neg N/A

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

      2. distribute-neg-frac2 N/A

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

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

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

      4. sub-neg N/A

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

      5. log1p-define N/A

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

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

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

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

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

      8. distribute-neg-in N/A

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

      9. unsub-neg N/A

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

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

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

      11. associate-/r* N/A

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

      12. distribute-neg-frac2 N/A

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

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

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

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

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

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

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

      16. /-lowering-/.f32 N/A

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

      17. *-lowering-*.f32 96.6%

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

   3. Simplified 96.6%

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

      1. associate-/r* N/A

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

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

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

      3. /-lowering-/.f32 96.6%

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

   6. Applied egg-rr 96.6%

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

   7. Taylor expanded in u0 around 0

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

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

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

      2. sub-neg N/A

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

      3. metadata-eval N/A

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

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

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

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

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

      6. sub-neg N/A

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

      7. metadata-eval N/A

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

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

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

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

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

      10. sub-neg N/A

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

      11. metadata-eval N/A

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

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

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

      13. *-commutative N/A

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

      14. *-lowering-*.f32 92.0%

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

   9. Simplified 92.0%

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

   10. Taylor expanded in cos2phi around 0

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

       1. mul-1-neg N/A

          \[\leadsto \mathsf{neg}\left(\frac{{alphay}^{2} \cdot \left(u0 \cdot \left(u0 \cdot \left(u0 \cdot \left(\frac{-1}{4} \cdot u0 - \frac{1}{3}\right) - \frac{1}{2}\right) - 1\right)\right)}{sin2phi}\right) \]

       2. distribute-neg-frac2 N/A

          \[\leadsto \frac{{alphay}^{2} \cdot \left(u0 \cdot \left(u0 \cdot \left(u0 \cdot \left(\frac{-1}{4} \cdot u0 - \frac{1}{3}\right) - \frac{1}{2}\right) - 1\right)\right)}{\color{blue}{\mathsf{neg}\left(sin2phi\right)}} \]

       3. neg-mul-1 N/A

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

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

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

   12. Simplified 94.6%

       \[\leadsto \color{blue}{\frac{\left(u0 \cdot \left(alphay \cdot alphay\right)\right) \cdot \left(u0 \cdot \left(u0 \cdot \left(u0 \cdot -0.25 + -0.3333333333333333\right) + -0.5\right) + -1\right)}{-sin2phi}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 91.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;sin2phi \leq 120:\\ \;\;\;\;\frac{u0 \cdot \left(1 + u0 \cdot 0.5\right)}{\frac{sin2phi}{alphay \cdot alphay} + \frac{cos2phi}{alphax \cdot alphax}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(u0 \cdot \left(alphay \cdot alphay\right)\right) \cdot \left(u0 \cdot \left(\left(-u0\right) \cdot \left(u0 \cdot -0.25 + -0.3333333333333333\right) - -0.5\right) - -1\right)}{sin2phi}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 93.0% accurate, 4.3× speedup

Math

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

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (/
  (* u0 (+ 1.0 (* u0 (+ 0.5 (* u0 (+ 0.3333333333333333 (* u0 0.25)))))))
  (+ (/ 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 + (u0 * 0.25f))))))) / ((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 + (u0 * 0.25e0))))))) / ((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(0.3333333333333333) + Float32(u0 * Float32(0.25)))))))) / 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) + (u0 * single(0.25)))))))) / ((sin2phi / (alphay * alphay)) + (cos2phi / (alphax * alphax)));
end

Derivation

1. Initial program 58.1%

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

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

   12. distribute-neg-frac2 N/A

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

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

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

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

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

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

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

   16. /-lowering-/.f32 N/A

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

   17. *-lowering-*.f32 97.8%

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

3. Simplified 97.8%

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

   1. frac-2neg N/A

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

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

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

   3. neg-log N/A

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

   4. flip3-+ N/A

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

   5. clear-num N/A

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

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

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

   7. clear-num N/A

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

   8. flip3-+ N/A

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

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

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

   10. unsub-neg N/A

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

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

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

   12. sub-neg N/A

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

6. Applied egg-rr 56.2%

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

7. Taylor expanded in u0 around 0

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

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

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

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

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

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

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

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

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

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

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

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

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

   7. *-commutative N/A

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

   8. *-lowering-*.f32 93.0%

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

9. Simplified 93.0%

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

Alternative 5: 83.2% accurate, 4.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{sin2phi}{alphay \cdot alphay}\\ \mathbf{if}\;t\_0 \leq 1.199999957179898 \cdot 10^{-7}:\\ \;\;\;\;\frac{u0}{t\_0 + \frac{cos2phi}{alphax \cdot alphax}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + u0 \cdot \left(0.5 + u0 \cdot 0.3333333333333333\right)\right) \cdot \left(u0 \cdot \left(alphay \cdot alphay\right)\right)}{sin2phi}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (let* ((t_0 (/ sin2phi (* alphay alphay))))
   (if (<= t_0 1.199999957179898e-7)
     (/ u0 (+ t_0 (/ cos2phi (* alphax alphax))))
     (/
      (*
       (+ 1.0 (* u0 (+ 0.5 (* u0 0.3333333333333333))))
       (* u0 (* alphay alphay)))
      sin2phi))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float t_0 = sin2phi / (alphay * alphay);
	float tmp;
	if (t_0 <= 1.199999957179898e-7f) {
		tmp = u0 / (t_0 + (cos2phi / (alphax * alphax)));
	} else {
		tmp = ((1.0f + (u0 * (0.5f + (u0 * 0.3333333333333333f)))) * (u0 * (alphay * alphay))) / sin2phi;
	}
	return tmp;
}

Fortran

real(4) function code(alphax, alphay, u0, cos2phi, sin2phi)
    real(4), intent (in) :: alphax
    real(4), intent (in) :: alphay
    real(4), intent (in) :: u0
    real(4), intent (in) :: cos2phi
    real(4), intent (in) :: sin2phi
    real(4) :: t_0
    real(4) :: tmp
    t_0 = sin2phi / (alphay * alphay)
    if (t_0 <= 1.199999957179898e-7) then
        tmp = u0 / (t_0 + (cos2phi / (alphax * alphax)))
    else
        tmp = ((1.0e0 + (u0 * (0.5e0 + (u0 * 0.3333333333333333e0)))) * (u0 * (alphay * alphay))) / sin2phi
    end if
    code = tmp
end function

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	t_0 = Float32(sin2phi / Float32(alphay * alphay))
	tmp = Float32(0.0)
	if (t_0 <= Float32(1.199999957179898e-7))
		tmp = Float32(u0 / Float32(t_0 + Float32(cos2phi / Float32(alphax * alphax))));
	else
		tmp = Float32(Float32(Float32(Float32(1.0) + Float32(u0 * Float32(Float32(0.5) + Float32(u0 * Float32(0.3333333333333333))))) * Float32(u0 * Float32(alphay * alphay))) / sin2phi);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alphax, alphay, u0, cos2phi, sin2phi)
	t_0 = sin2phi / (alphay * alphay);
	tmp = single(0.0);
	if (t_0 <= single(1.199999957179898e-7))
		tmp = u0 / (t_0 + (cos2phi / (alphax * alphax)));
	else
		tmp = ((single(1.0) + (u0 * (single(0.5) + (u0 * single(0.3333333333333333))))) * (u0 * (alphay * alphay))) / sin2phi;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 sin2phi (*.f32 alphay alphay)) < 1.19999996e-7

   1. Initial program 54.0%

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

      1. distribute-frac-neg N/A

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

      2. distribute-neg-frac2 N/A

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

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

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

      4. sub-neg N/A

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

      5. log1p-define N/A

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

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

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

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

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

      8. distribute-neg-in N/A

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

      9. unsub-neg N/A

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

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

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

      11. associate-/r* N/A

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

      12. distribute-neg-frac2 N/A

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

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

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

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

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

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

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

      16. /-lowering-/.f32 N/A

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

      17. *-lowering-*.f32 98.8%

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

   3. Simplified 98.8%

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

      1. frac-2neg N/A

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

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

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

      3. neg-log N/A

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

      4. flip3-+ N/A

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

      5. clear-num N/A

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

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

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

      7. clear-num N/A

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

      8. flip3-+ N/A

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

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

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

      10. unsub-neg N/A

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

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

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

      12. sub-neg N/A

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

   6. Applied egg-rr 52.0%

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

   7. Taylor expanded in u0 around 0

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

   9. Simplified 76.7%

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

   if 1.19999996e-7 < (/.f32 sin2phi (*.f32 alphay alphay))

   1. Initial program 60.5%

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

      1. distribute-frac-neg N/A

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

      2. distribute-neg-frac2 N/A

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

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

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

      4. sub-neg N/A

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

      5. log1p-define N/A

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

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

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

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

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

      8. distribute-neg-in N/A

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

      9. unsub-neg N/A

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

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

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

      11. associate-/r* N/A

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

      12. distribute-neg-frac2 N/A

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

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

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

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

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

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

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

      16. /-lowering-/.f32 N/A

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

      17. *-lowering-*.f32 97.3%

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

   3. Simplified 97.3%

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

      1. frac-2neg N/A

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

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

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

      3. neg-log N/A

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

      4. flip3-+ N/A

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

      5. clear-num N/A

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

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

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

      7. clear-num N/A

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

      8. flip3-+ N/A

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

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

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

      10. unsub-neg N/A

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

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

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

      12. sub-neg N/A

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

   6. Applied egg-rr 58.6%

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

   7. Taylor expanded in u0 around 0

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

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

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

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

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

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

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

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

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

      5. *-commutative N/A

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

      6. *-lowering-*.f32 90.7%

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

   9. Simplified 90.7%

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

   10. Taylor expanded in sin2phi around inf

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

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

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

       2. associate-*r* N/A

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

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

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

       4. *-commutative N/A

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

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

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

       6. unpow2 N/A

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

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

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

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

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

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

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

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

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

       11. *-commutative N/A

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

       12. *-lowering-*.f32 89.4%

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

   12. Simplified 89.4%

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

4. Final simplification 84.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 1.199999957179898 \cdot 10^{-7}:\\ \;\;\;\;\frac{u0}{\frac{sin2phi}{alphay \cdot alphay} + \frac{cos2phi}{alphax \cdot alphax}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + u0 \cdot \left(0.5 + u0 \cdot 0.3333333333333333\right)\right) \cdot \left(u0 \cdot \left(alphay \cdot alphay\right)\right)}{sin2phi}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 91.4% accurate, 5.0× speedup

Math

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

FPCore

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

Derivation

1. Initial program 58.1%

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

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

   12. distribute-neg-frac2 N/A

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

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

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

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

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

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

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

   16. /-lowering-/.f32 N/A

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

   17. *-lowering-*.f32 97.8%

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

3. Simplified 97.8%

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

   1. frac-2neg N/A

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

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

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

   3. neg-log N/A

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

   4. flip3-+ N/A

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

   5. clear-num N/A

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

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

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

   7. clear-num N/A

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

   8. flip3-+ N/A

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

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

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

   10. unsub-neg N/A

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

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

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

   12. sub-neg N/A

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

6. Applied egg-rr 56.2%

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

7. Taylor expanded in u0 around 0

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

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

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

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

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

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

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

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

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

   5. *-commutative N/A

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

   6. *-lowering-*.f32 91.4%

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

9. Simplified 91.4%

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

    1. +-commutative N/A

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

    2. distribute-lft-in N/A

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

    3. *-rgt-identity N/A

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

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

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

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

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

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

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

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

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

    8. *-lowering-*.f32 91.6%

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

11. Applied egg-rr 91.6%

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

12. Final simplification 91.6%

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

Alternative 7: 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 58.1%

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

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

   12. distribute-neg-frac2 N/A

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

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

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

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

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

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

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

   16. /-lowering-/.f32 N/A

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

   17. *-lowering-*.f32 97.8%

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

3. Simplified 97.8%

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

   1. frac-2neg N/A

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

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

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

   3. neg-log N/A

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

   4. flip3-+ N/A

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

   5. clear-num N/A

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

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

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

   7. clear-num N/A

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

   8. flip3-+ N/A

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

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

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

   10. unsub-neg N/A

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

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

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

   12. sub-neg N/A

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

6. Applied egg-rr 56.2%

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

7. Taylor expanded in u0 around 0

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

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

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

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

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

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

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

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

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

   5. *-commutative N/A

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

   6. *-lowering-*.f32 91.4%

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

9. Simplified 91.4%

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

Alternative 8: 91.2% accurate, 5.0× speedup

Math

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

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(u0 * Float32(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 58.1%

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

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

   12. distribute-neg-frac2 N/A

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

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

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

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

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

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

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

   16. /-lowering-/.f32 N/A

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

   17. *-lowering-*.f32 97.8%

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

3. Simplified 97.8%

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

   1. frac-2neg N/A

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

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

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

   3. neg-log N/A

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

   4. flip3-+ N/A

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

   5. clear-num N/A

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

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

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

   7. clear-num N/A

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

   8. flip3-+ N/A

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

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

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

   10. unsub-neg N/A

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

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

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

   12. sub-neg N/A

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

6. Applied egg-rr 56.2%

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

7. Taylor expanded in u0 around 0

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

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

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

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

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

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

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

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

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

   5. *-commutative N/A

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

   6. *-lowering-*.f32 91.4%

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

9. Simplified 91.4%

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

    1. associate-/l* N/A

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

    2. *-commutative N/A

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

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

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

11. Applied egg-rr 91.3%

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

12. Final simplification 91.3%

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

Alternative 9: 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 58.1%

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

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

   12. distribute-neg-frac2 N/A

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

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

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

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

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

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

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

   16. /-lowering-/.f32 N/A

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

   17. *-lowering-*.f32 97.8%

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

3. Simplified 97.8%

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

   1. frac-2neg N/A

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

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

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

   3. neg-log N/A

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

   4. flip3-+ N/A

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

   5. clear-num N/A

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

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

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

   7. clear-num N/A

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

   8. flip3-+ N/A

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

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

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

   10. unsub-neg N/A

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

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

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

   12. sub-neg N/A

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

6. Applied egg-rr 56.2%

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

7. Taylor expanded in u0 around 0

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

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

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

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

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

   3. *-commutative N/A

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

   4. *-lowering-*.f32 88.2%

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

9. Simplified 88.2%

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

Alternative 10: 67.2% accurate, 7.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 1.99999996490334 \cdot 10^{-14}:\\ \;\;\;\;\frac{alphax \cdot alphax}{\frac{cos2phi}{u0}}\\ \mathbf{else}:\\ \;\;\;\;\frac{u0 \cdot \left(alphay \cdot alphay\right)}{sin2phi}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (if (<= (/ sin2phi (* alphay alphay)) 1.99999996490334e-14)
   (/ (* alphax alphax) (/ cos2phi u0))
   (/ (* u0 (* alphay alphay)) sin2phi)))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float tmp;
	if ((sin2phi / (alphay * alphay)) <= 1.99999996490334e-14f) {
		tmp = (alphax * alphax) / (cos2phi / u0);
	} else {
		tmp = (u0 * (alphay * alphay)) / sin2phi;
	}
	return tmp;
}

Fortran

real(4) function code(alphax, alphay, u0, cos2phi, sin2phi)
    real(4), intent (in) :: alphax
    real(4), intent (in) :: alphay
    real(4), intent (in) :: u0
    real(4), intent (in) :: cos2phi
    real(4), intent (in) :: sin2phi
    real(4) :: tmp
    if ((sin2phi / (alphay * alphay)) <= 1.99999996490334e-14) then
        tmp = (alphax * alphax) / (cos2phi / u0)
    else
        tmp = (u0 * (alphay * alphay)) / sin2phi
    end if
    code = tmp
end function

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = Float32(0.0)
	if (Float32(sin2phi / Float32(alphay * alphay)) <= Float32(1.99999996490334e-14))
		tmp = Float32(Float32(alphax * alphax) / Float32(cos2phi / u0));
	else
		tmp = Float32(Float32(u0 * Float32(alphay * alphay)) / sin2phi);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = single(0.0);
	if ((sin2phi / (alphay * alphay)) <= single(1.99999996490334e-14))
		tmp = (alphax * alphax) / (cos2phi / u0);
	else
		tmp = (u0 * (alphay * alphay)) / sin2phi;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 sin2phi (*.f32 alphay alphay)) < 1.99999996e-14

   1. Initial program 52.1%

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

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

      12. distribute-neg-frac2 N/A

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

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

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

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

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

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

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

      16. /-lowering-/.f32 N/A

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

      17. *-lowering-*.f32 98.7%

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

   3. Simplified 98.7%

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

   5. Taylor expanded in u0 around 0

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

      1. associate-*r/ N/A

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

      2. sub-neg N/A

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

      3. mul-1-neg N/A

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

      4. distribute-lft-out N/A

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

      5. times-frac N/A

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

      6. metadata-eval N/A

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

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

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

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

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

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

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

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

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

      11. unpow2 N/A

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

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

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

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

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

      14. unpow2 N/A

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

      15. *-lowering-*.f32 77.2%

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

   7. Simplified 77.2%

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

   8. Taylor expanded in cos2phi around inf

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

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

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

      2. *-commutative N/A

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

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

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

      4. unpow2 N/A

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

      5. *-lowering-*.f32 56.3%

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

   10. Simplified 56.3%

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

       1. *-commutative N/A

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

       2. associate-/l* N/A

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

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

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

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

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

       5. /-lowering-/.f32 56.3%

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

   12. Applied egg-rr 56.3%

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

       1. clear-num N/A

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

       2. un-div-inv N/A

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

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

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

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

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

       5. /-lowering-/.f32 56.5%

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

   14. Applied egg-rr 56.5%

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

   if 1.99999996e-14 < (/.f32 sin2phi (*.f32 alphay alphay))

   1. Initial program 60.5%

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

      1. distribute-frac-neg N/A

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

      2. distribute-neg-frac2 N/A

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

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

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

      4. sub-neg N/A

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

      5. log1p-define N/A

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

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

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

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

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

      8. distribute-neg-in N/A

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

      9. unsub-neg N/A

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

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

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

      11. associate-/r* N/A

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

      12. distribute-neg-frac2 N/A

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

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

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

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

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

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

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

      16. /-lowering-/.f32 N/A

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

      17. *-lowering-*.f32 97.5%

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

   3. Simplified 97.5%

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

   5. Taylor expanded in u0 around 0

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

      1. associate-*r/ N/A

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

      2. sub-neg N/A

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

      3. mul-1-neg N/A

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

      4. distribute-lft-out N/A

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

      5. times-frac N/A

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

      6. metadata-eval N/A

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

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

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

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

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

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

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

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

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

      11. unpow2 N/A

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

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

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

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

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

      14. unpow2 N/A

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

      15. *-lowering-*.f32 77.0%

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

   7. Simplified 77.0%

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

   8. Taylor expanded in cos2phi around 0

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

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

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

      2. *-commutative N/A

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

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

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

      4. unpow2 N/A

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

      5. *-lowering-*.f32 73.0%

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

   10. Simplified 73.0%

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

Alternative 11: 75.8% 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 58.1%

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

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

   12. distribute-neg-frac2 N/A

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

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

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

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

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

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

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

   16. /-lowering-/.f32 N/A

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

   17. *-lowering-*.f32 97.8%

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

3. Simplified 97.8%

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

   1. frac-2neg N/A

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

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

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

   3. neg-log N/A

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

   4. flip3-+ N/A

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

   5. clear-num N/A

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

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

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

   7. clear-num N/A

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

   8. flip3-+ N/A

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

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

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

   10. unsub-neg N/A

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

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

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

   12. sub-neg N/A

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

6. Applied egg-rr 56.2%

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

7. Taylor expanded in u0 around 0

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

9. Simplified 77.0%

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

Alternative 12: 23.5% accurate, 16.6× speedup

Math

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

FPCore

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

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	return (alphax * alphax) / (cos2phi / 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
    code = (alphax * alphax) / (cos2phi / u0)
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 58.1%

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

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

   12. distribute-neg-frac2 N/A

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

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

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

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

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

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

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

   16. /-lowering-/.f32 N/A

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

   17. *-lowering-*.f32 97.8%

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

3. Simplified 97.8%

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

5. Taylor expanded in u0 around 0

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

   1. associate-*r/ N/A

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

   2. sub-neg N/A

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

   3. mul-1-neg N/A

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

   4. distribute-lft-out N/A

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

   5. times-frac N/A

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

   6. metadata-eval N/A

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

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

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

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

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

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

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

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

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

   11. unpow2 N/A

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

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

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

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

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

   14. unpow2 N/A

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

   15. *-lowering-*.f32 77.0%

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

7. Simplified 77.0%

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

8. Taylor expanded in cos2phi around inf

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

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

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

   2. *-commutative N/A

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

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

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

   4. unpow2 N/A

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

   5. *-lowering-*.f32 24.2%

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

10. Simplified 24.2%

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

    1. *-commutative N/A

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

    2. associate-/l* N/A

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

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

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

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

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

    5. /-lowering-/.f32 24.2%

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

12. Applied egg-rr 24.2%

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

    1. clear-num N/A

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

    2. un-div-inv N/A

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

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

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

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

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

    5. /-lowering-/.f32 24.3%

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

14. Applied egg-rr 24.3%

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

Alternative 13: 23.5% accurate, 16.6× speedup

Math

\[\begin{array}{l} \\ \left(u0 \cdot alphax\right) \cdot \frac{alphax}{cos2phi} \end{array} \]

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(Float32(u0 * alphax) * Float32(alphax / cos2phi))
end

MATLAB

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

Derivation

1. Initial program 58.1%

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

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

   12. distribute-neg-frac2 N/A

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

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

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

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

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

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

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

   16. /-lowering-/.f32 N/A

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

   17. *-lowering-*.f32 97.8%

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

3. Simplified 97.8%

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

5. Taylor expanded in u0 around 0

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

   1. associate-*r/ N/A

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

   2. sub-neg N/A

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

   3. mul-1-neg N/A

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

   4. distribute-lft-out N/A

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

   5. times-frac N/A

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

   6. metadata-eval N/A

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

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

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

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

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

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

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

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

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

   11. unpow2 N/A

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

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

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

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

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

   14. unpow2 N/A

       \[\leadsto \mathsf{*.f32}\left(1, \mathsf{/.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right), \mathsf{/.f32}\left(sin2phi, \left(alphay \cdot \color{blue}{alphay}\right)\right)\right)\right)\right) \]

   15. *-lowering-*.f32 77.0%

       \[\leadsto \mathsf{*.f32}\left(1, \mathsf{/.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right), \mathsf{/.f32}\left(sin2phi, \mathsf{*.f32}\left(alphay, \color{blue}{alphay}\right)\right)\right)\right)\right) \]

7. Simplified 77.0%

   \[\leadsto \color{blue}{1 \cdot \frac{u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}}} \]

8. Taylor expanded in cos2phi around inf

   \[\leadsto \color{blue}{\frac{{alphax}^{2} \cdot u0}{cos2phi}} \]
9. Step-by-step derivation

   1. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\left({alphax}^{2} \cdot u0\right), \color{blue}{cos2phi}\right) \]

   2. *-commutative N/A

      \[\leadsto \mathsf{/.f32}\left(\left(u0 \cdot {alphax}^{2}\right), cos2phi\right) \]

   3. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(u0, \left({alphax}^{2}\right)\right), cos2phi\right) \]

   4. unpow2 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(u0, \left(alphax \cdot alphax\right)\right), cos2phi\right) \]

   5. *-lowering-*.f32 24.2%

      \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(u0, \mathsf{*.f32}\left(alphax, alphax\right)\right), cos2phi\right) \]

10. Simplified 24.2%

    \[\leadsto \color{blue}{\frac{u0 \cdot \left(alphax \cdot alphax\right)}{cos2phi}} \]
11. Step-by-step derivation

    1. associate-*r* N/A

       \[\leadsto \frac{\left(u0 \cdot alphax\right) \cdot alphax}{cos2phi} \]

    2. associate-/l* N/A

       \[\leadsto \left(u0 \cdot alphax\right) \cdot \color{blue}{\frac{alphax}{cos2phi}} \]

    3. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{*.f32}\left(\left(u0 \cdot alphax\right), \color{blue}{\left(\frac{alphax}{cos2phi}\right)}\right) \]

    4. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(u0, alphax\right), \left(\frac{\color{blue}{alphax}}{cos2phi}\right)\right) \]

    5. /-lowering-/.f32 24.2%

       \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(u0, alphax\right), \mathsf{/.f32}\left(alphax, \color{blue}{cos2phi}\right)\right) \]

12. Applied egg-rr 24.2%

    \[\leadsto \color{blue}{\left(u0 \cdot alphax\right) \cdot \frac{alphax}{cos2phi}} \]
13. Add Preprocessing

Alternative 14: 23.5% accurate, 16.6× speedup

Math

\[\begin{array}{l} \\ \left(alphax \cdot alphax\right) \cdot \frac{u0}{cos2phi} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (* (* alphax alphax) (/ u0 cos2phi)))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	return (alphax * alphax) * (u0 / 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 = (alphax * alphax) * (u0 / cos2phi)
end function

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	return Float32(Float32(alphax * alphax) * Float32(u0 / cos2phi))
end

MATLAB

function tmp = code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = (alphax * alphax) * (u0 / cos2phi);
end

Derivation

1. Initial program 58.1%

   \[\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
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. associate-/r* N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\left(\mathsf{neg}\left(\frac{\frac{cos2phi}{alphax}}{alphax}\right)\right), \left(\frac{sin2phi}{alphay \cdot alphay}\right)\right)\right) \]

   12. distribute-neg-frac2 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\left(\frac{\frac{cos2phi}{alphax}}{\mathsf{neg}\left(alphax\right)}\right), \left(\frac{\color{blue}{sin2phi}}{alphay \cdot alphay}\right)\right)\right) \]

   13. /-lowering-/.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{/.f32}\left(\left(\frac{cos2phi}{alphax}\right), \left(\mathsf{neg}\left(alphax\right)\right)\right), \left(\frac{\color{blue}{sin2phi}}{alphay \cdot alphay}\right)\right)\right) \]

   14. /-lowering-/.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(cos2phi, alphax\right), \left(\mathsf{neg}\left(alphax\right)\right)\right), \left(\frac{sin2phi}{alphay \cdot alphay}\right)\right)\right) \]

   15. neg-lowering-neg.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(cos2phi, alphax\right), \mathsf{neg.f32}\left(alphax\right)\right), \left(\frac{sin2phi}{alphay \cdot alphay}\right)\right)\right) \]

   16. /-lowering-/.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(cos2phi, alphax\right), \mathsf{neg.f32}\left(alphax\right)\right), \mathsf{/.f32}\left(sin2phi, \color{blue}{\left(alphay \cdot alphay\right)}\right)\right)\right) \]

   17. *-lowering-*.f32 97.8%

       \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(cos2phi, alphax\right), \mathsf{neg.f32}\left(alphax\right)\right), \mathsf{/.f32}\left(sin2phi, \mathsf{*.f32}\left(alphay, \color{blue}{alphay}\right)\right)\right)\right) \]

3. Simplified 97.8%

   \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(-u0\right)}{\frac{\frac{cos2phi}{alphax}}{-alphax} - \frac{sin2phi}{alphay \cdot alphay}}} \]
4. Add Preprocessing

5. Taylor expanded in u0 around 0

   \[\leadsto \color{blue}{-1 \cdot \frac{u0}{-1 \cdot \frac{cos2phi}{{alphax}^{2}} - \frac{sin2phi}{{alphay}^{2}}}} \]
6. Step-by-step derivation

   1. associate-*r/ N/A

      \[\leadsto \frac{-1 \cdot u0}{\color{blue}{-1 \cdot \frac{cos2phi}{{alphax}^{2}} - \frac{sin2phi}{{alphay}^{2}}}} \]

   2. sub-neg N/A

      \[\leadsto \frac{-1 \cdot u0}{-1 \cdot \frac{cos2phi}{{alphax}^{2}} + \color{blue}{\left(\mathsf{neg}\left(\frac{sin2phi}{{alphay}^{2}}\right)\right)}} \]

   3. mul-1-neg N/A

      \[\leadsto \frac{-1 \cdot u0}{-1 \cdot \frac{cos2phi}{{alphax}^{2}} + -1 \cdot \color{blue}{\frac{sin2phi}{{alphay}^{2}}}} \]

   4. distribute-lft-out N/A

      \[\leadsto \frac{-1 \cdot u0}{-1 \cdot \color{blue}{\left(\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}\right)}} \]

   5. times-frac N/A

      \[\leadsto \frac{-1}{-1} \cdot \color{blue}{\frac{u0}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}}} \]

   6. metadata-eval N/A

      \[\leadsto 1 \cdot \frac{\color{blue}{u0}}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}} \]

   7. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{*.f32}\left(1, \color{blue}{\left(\frac{u0}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}}\right)}\right) \]

   8. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{*.f32}\left(1, \mathsf{/.f32}\left(u0, \color{blue}{\left(\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}\right)}\right)\right) \]

   9. +-lowering-+.f32 N/A

      \[\leadsto \mathsf{*.f32}\left(1, \mathsf{/.f32}\left(u0, \mathsf{+.f32}\left(\left(\frac{cos2phi}{{alphax}^{2}}\right), \color{blue}{\left(\frac{sin2phi}{{alphay}^{2}}\right)}\right)\right)\right) \]

   10. /-lowering-/.f32 N/A

       \[\leadsto \mathsf{*.f32}\left(1, \mathsf{/.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{/.f32}\left(cos2phi, \left({alphax}^{2}\right)\right), \left(\frac{\color{blue}{sin2phi}}{{alphay}^{2}}\right)\right)\right)\right) \]

   11. unpow2 N/A

       \[\leadsto \mathsf{*.f32}\left(1, \mathsf{/.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{/.f32}\left(cos2phi, \left(alphax \cdot alphax\right)\right), \left(\frac{sin2phi}{{alphay}^{2}}\right)\right)\right)\right) \]

   12. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{*.f32}\left(1, \mathsf{/.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right), \left(\frac{sin2phi}{{alphay}^{2}}\right)\right)\right)\right) \]

   13. /-lowering-/.f32 N/A

       \[\leadsto \mathsf{*.f32}\left(1, \mathsf{/.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right), \mathsf{/.f32}\left(sin2phi, \color{blue}{\left({alphay}^{2}\right)}\right)\right)\right)\right) \]

   14. unpow2 N/A

       \[\leadsto \mathsf{*.f32}\left(1, \mathsf{/.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right), \mathsf{/.f32}\left(sin2phi, \left(alphay \cdot \color{blue}{alphay}\right)\right)\right)\right)\right) \]

   15. *-lowering-*.f32 77.0%

       \[\leadsto \mathsf{*.f32}\left(1, \mathsf{/.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right), \mathsf{/.f32}\left(sin2phi, \mathsf{*.f32}\left(alphay, \color{blue}{alphay}\right)\right)\right)\right)\right) \]

7. Simplified 77.0%

   \[\leadsto \color{blue}{1 \cdot \frac{u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}}} \]

8. Taylor expanded in cos2phi around inf

   \[\leadsto \color{blue}{\frac{{alphax}^{2} \cdot u0}{cos2phi}} \]
9. Step-by-step derivation

   1. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\left({alphax}^{2} \cdot u0\right), \color{blue}{cos2phi}\right) \]

   2. *-commutative N/A

      \[\leadsto \mathsf{/.f32}\left(\left(u0 \cdot {alphax}^{2}\right), cos2phi\right) \]

   3. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(u0, \left({alphax}^{2}\right)\right), cos2phi\right) \]

   4. unpow2 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(u0, \left(alphax \cdot alphax\right)\right), cos2phi\right) \]

   5. *-lowering-*.f32 24.2%

      \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(u0, \mathsf{*.f32}\left(alphax, alphax\right)\right), cos2phi\right) \]

10. Simplified 24.2%

    \[\leadsto \color{blue}{\frac{u0 \cdot \left(alphax \cdot alphax\right)}{cos2phi}} \]
11. Step-by-step derivation

    1. *-commutative N/A

       \[\leadsto \frac{\left(alphax \cdot alphax\right) \cdot u0}{cos2phi} \]

    2. associate-/l* N/A

       \[\leadsto \left(alphax \cdot alphax\right) \cdot \color{blue}{\frac{u0}{cos2phi}} \]

    3. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{*.f32}\left(\left(alphax \cdot alphax\right), \color{blue}{\left(\frac{u0}{cos2phi}\right)}\right) \]

    4. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(alphax, alphax\right), \left(\frac{\color{blue}{u0}}{cos2phi}\right)\right) \]

    5. /-lowering-/.f32 24.2%

       \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(alphax, alphax\right), \mathsf{/.f32}\left(u0, \color{blue}{cos2phi}\right)\right) \]

12. Applied egg-rr 24.2%

    \[\leadsto \color{blue}{\left(alphax \cdot alphax\right) \cdot \frac{u0}{cos2phi}} \]
13. Add Preprocessing

Reproduce

herbie shell --seed 2024288 
(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)))))