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

Percentage Accurate: 60.6% → 98.3%

Time: 24.0s

Alternatives: 29

Speedup: 9.6×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Initial Program: 60.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Alternative 1: 98.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 60.9%

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

   1. distribute-frac-neg N/A

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

   2. distribute-neg-frac2 N/A

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

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

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

   4. sub-neg N/A

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

   5. log1p-define N/A

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

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

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

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

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

   8. distribute-neg-in N/A

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

   9. unsub-neg N/A

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

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

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

   11. neg-sub0 N/A

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

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

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

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

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

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

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

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

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

   16. *-lowering-*.f32 98.4%

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

3. Simplified 98.4%

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

   1. associate-/r* N/A

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

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

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

   3. /-lowering-/.f32 98.4%

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

6. Applied egg-rr 98.4%

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

   1. sub0-neg N/A

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

   2. distribute-neg-frac2 N/A

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

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

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

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

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

   5. neg-lowering-neg.f32 98.4%

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

8. Applied egg-rr 98.4%

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

9. Final simplification 98.4%

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

Alternative 2: 96.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{sin2phi}{alphay \cdot alphay}\\ t_1 := t\_0 + \frac{cos2phi}{alphax \cdot alphax}\\ \mathbf{if}\;t\_0 \leq 0.30000001192092896:\\ \;\;\;\;\frac{-1}{\frac{u0 \cdot \left(u0 \cdot \left(t\_1 \cdot 0.08333333333333333 - u0 \cdot \left(0.5 \cdot \left(t\_1 \cdot -0.08333333333333333\right)\right)\right) + 0.5 \cdot t\_1\right) - t\_1}{u0}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(-u0\right) \cdot \frac{0 - alphay \cdot alphay}{sin2phi}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (let* ((t_0 (/ sin2phi (* alphay alphay)))
        (t_1 (+ t_0 (/ cos2phi (* alphax alphax)))))
   (if (<= t_0 0.30000001192092896)
     (/
      -1.0
      (/
       (-
        (*
         u0
         (+
          (*
           u0
           (-
            (* t_1 0.08333333333333333)
            (* u0 (* 0.5 (* t_1 -0.08333333333333333)))))
          (* 0.5 t_1)))
        t_1)
       u0))
     (* (log1p (- u0)) (/ (- 0.0 (* alphay alphay)) sin2phi)))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float t_0 = sin2phi / (alphay * alphay);
	float t_1 = t_0 + (cos2phi / (alphax * alphax));
	float tmp;
	if (t_0 <= 0.30000001192092896f) {
		tmp = -1.0f / (((u0 * ((u0 * ((t_1 * 0.08333333333333333f) - (u0 * (0.5f * (t_1 * -0.08333333333333333f))))) + (0.5f * t_1))) - t_1) / u0);
	} else {
		tmp = log1pf(-u0) * ((0.0f - (alphay * alphay)) / sin2phi);
	}
	return tmp;
}

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	t_0 = Float32(sin2phi / Float32(alphay * alphay))
	t_1 = Float32(t_0 + Float32(cos2phi / Float32(alphax * alphax)))
	tmp = Float32(0.0)
	if (t_0 <= Float32(0.30000001192092896))
		tmp = Float32(Float32(-1.0) / Float32(Float32(Float32(u0 * Float32(Float32(u0 * Float32(Float32(t_1 * Float32(0.08333333333333333)) - Float32(u0 * Float32(Float32(0.5) * Float32(t_1 * Float32(-0.08333333333333333)))))) + Float32(Float32(0.5) * t_1))) - t_1) / u0));
	else
		tmp = Float32(log1p(Float32(-u0)) * Float32(Float32(Float32(0.0) - Float32(alphay * alphay)) / sin2phi));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 sin2phi (*.f32 alphay alphay)) < 0.300000012

   1. Initial program 55.9%

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

      1. distribute-frac-neg N/A

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

      2. distribute-neg-frac2 N/A

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

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

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

      4. sub-neg N/A

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

      5. log1p-define N/A

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

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

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

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

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

      8. distribute-neg-in N/A

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

      9. unsub-neg N/A

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

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

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

      11. neg-sub0 N/A

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

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

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

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

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

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

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

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

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

      16. *-lowering-*.f32 98.6%

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

   3. Simplified 98.6%

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

      1. sub-neg N/A

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

      2. associate--l- N/A

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

      3. neg-sub0 N/A

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

      4. distribute-frac-neg2 N/A

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

      5. neg-mul-1 N/A

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

      6. clear-num N/A

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

      7. un-div-inv N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      15. sub-neg N/A

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

      16. log1p-define N/A

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

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

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

      18. neg-lowering-neg.f32 98.4%

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

   6. Applied egg-rr 98.4%

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

   7. Taylor expanded in u0 around 0

      \[\leadsto \mathsf{/.f32}\left(-1, \color{blue}{\left(\frac{-1 \cdot \left(\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}\right) + u0 \cdot \left(u0 \cdot \left(-1 \cdot \left(u0 \cdot \left(\frac{-1}{2} \cdot \left(\frac{-1}{3} \cdot \left(\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}\right) + \frac{1}{4} \cdot \left(\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}\right)\right) + \left(\frac{-1}{4} \cdot \left(\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}\right) + \frac{1}{6} \cdot \left(\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}\right)\right)\right)\right) - \left(\frac{-1}{3} \cdot \left(\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}\right) + \frac{1}{4} \cdot \left(\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}\right)\right)\right) - \frac{-1}{2} \cdot \left(\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}\right)\right)}{u0}\right)}\right) \]

   9. Simplified 95.3%

      \[\leadsto \frac{-1}{\color{blue}{\frac{u0 \cdot \left(u0 \cdot \left(\left(0.5 \cdot \left(\left(\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\right) \cdot -0.08333333333333333\right)\right) \cdot \left(-u0\right) + \left(\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\right) \cdot 0.08333333333333333\right) + \left(\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\right) \cdot 0.5\right) - \left(\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\right)}{u0}}} \]

   if 0.300000012 < (/.f32 sin2phi (*.f32 alphay alphay))

   1. Initial program 65.1%

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

      1. distribute-frac-neg N/A

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

      2. distribute-neg-frac2 N/A

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

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

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

      4. sub-neg N/A

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

      5. log1p-define N/A

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

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

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

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

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

      8. distribute-neg-in N/A

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

      9. unsub-neg N/A

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

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

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

      11. neg-sub0 N/A

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

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

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

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

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

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

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

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

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

      16. *-lowering-*.f32 98.2%

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

   3. Simplified 98.2%

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

   5. Taylor expanded in cos2phi around 0

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

      1. mul-1-neg N/A

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

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

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

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

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

      6. sub-neg N/A

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

      7. mul-1-neg N/A

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

      8. log1p-define N/A

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

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

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

      10. mul-1-neg N/A

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

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

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

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

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

      13. unpow2 N/A

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

      14. *-lowering-*.f32 98.4%

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

   7. Simplified 98.4%

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

4. Final simplification 97.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 0.30000001192092896:\\ \;\;\;\;\frac{-1}{\frac{u0 \cdot \left(u0 \cdot \left(\left(\frac{sin2phi}{alphay \cdot alphay} + \frac{cos2phi}{alphax \cdot alphax}\right) \cdot 0.08333333333333333 - u0 \cdot \left(0.5 \cdot \left(\left(\frac{sin2phi}{alphay \cdot alphay} + \frac{cos2phi}{alphax \cdot alphax}\right) \cdot -0.08333333333333333\right)\right)\right) + 0.5 \cdot \left(\frac{sin2phi}{alphay \cdot alphay} + \frac{cos2phi}{alphax \cdot alphax}\right)\right) - \left(\frac{sin2phi}{alphay \cdot alphay} + \frac{cos2phi}{alphax \cdot alphax}\right)}{u0}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(-u0\right) \cdot \frac{0 - alphay \cdot alphay}{sin2phi}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 96.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{sin2phi}{alphay \cdot alphay}\\ t_1 := t\_0 + \frac{cos2phi}{alphax \cdot alphax}\\ \mathbf{if}\;t\_0 \leq 10:\\ \;\;\;\;\frac{-1}{\frac{u0 \cdot \left(u0 \cdot \left(t\_1 \cdot 0.08333333333333333 - u0 \cdot \left(0.5 \cdot \left(t\_1 \cdot -0.08333333333333333\right)\right)\right) + 0.5 \cdot t\_1\right) - t\_1}{u0}}\\ \mathbf{else}:\\ \;\;\;\;\left(alphay \cdot alphay\right) \cdot \frac{\mathsf{log1p}\left(-u0\right)}{-sin2phi}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (let* ((t_0 (/ sin2phi (* alphay alphay)))
        (t_1 (+ t_0 (/ cos2phi (* alphax alphax)))))
   (if (<= t_0 10.0)
     (/
      -1.0
      (/
       (-
        (*
         u0
         (+
          (*
           u0
           (-
            (* t_1 0.08333333333333333)
            (* u0 (* 0.5 (* t_1 -0.08333333333333333)))))
          (* 0.5 t_1)))
        t_1)
       u0))
     (* (* alphay alphay) (/ (log1p (- u0)) (- sin2phi))))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float t_0 = sin2phi / (alphay * alphay);
	float t_1 = t_0 + (cos2phi / (alphax * alphax));
	float tmp;
	if (t_0 <= 10.0f) {
		tmp = -1.0f / (((u0 * ((u0 * ((t_1 * 0.08333333333333333f) - (u0 * (0.5f * (t_1 * -0.08333333333333333f))))) + (0.5f * t_1))) - t_1) / u0);
	} else {
		tmp = (alphay * alphay) * (log1pf(-u0) / -sin2phi);
	}
	return tmp;
}

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	t_0 = Float32(sin2phi / Float32(alphay * alphay))
	t_1 = Float32(t_0 + Float32(cos2phi / Float32(alphax * alphax)))
	tmp = Float32(0.0)
	if (t_0 <= Float32(10.0))
		tmp = Float32(Float32(-1.0) / Float32(Float32(Float32(u0 * Float32(Float32(u0 * Float32(Float32(t_1 * Float32(0.08333333333333333)) - Float32(u0 * Float32(Float32(0.5) * Float32(t_1 * Float32(-0.08333333333333333)))))) + Float32(Float32(0.5) * t_1))) - t_1) / u0));
	else
		tmp = Float32(Float32(alphay * alphay) * Float32(log1p(Float32(-u0)) / Float32(-sin2phi)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 sin2phi (*.f32 alphay alphay)) < 10

   1. Initial program 55.6%

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

      1. distribute-frac-neg N/A

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

      2. distribute-neg-frac2 N/A

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

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

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

      4. sub-neg N/A

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

      5. log1p-define N/A

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

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

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

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

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

      8. distribute-neg-in N/A

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

      9. unsub-neg N/A

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

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

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

      11. neg-sub0 N/A

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

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

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

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

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

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

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

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

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

      16. *-lowering-*.f32 98.6%

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

   3. Simplified 98.6%

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

      1. sub-neg N/A

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

      2. associate--l- N/A

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

      3. neg-sub0 N/A

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

      4. distribute-frac-neg2 N/A

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

      5. neg-mul-1 N/A

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

      6. clear-num N/A

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

      7. un-div-inv N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      15. sub-neg N/A

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

      16. log1p-define N/A

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

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

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

      18. neg-lowering-neg.f32 98.3%

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

   6. Applied egg-rr 98.3%

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

   7. Taylor expanded in u0 around 0

      \[\leadsto \mathsf{/.f32}\left(-1, \color{blue}{\left(\frac{-1 \cdot \left(\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}\right) + u0 \cdot \left(u0 \cdot \left(-1 \cdot \left(u0 \cdot \left(\frac{-1}{2} \cdot \left(\frac{-1}{3} \cdot \left(\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}\right) + \frac{1}{4} \cdot \left(\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}\right)\right) + \left(\frac{-1}{4} \cdot \left(\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}\right) + \frac{1}{6} \cdot \left(\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}\right)\right)\right)\right) - \left(\frac{-1}{3} \cdot \left(\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}\right) + \frac{1}{4} \cdot \left(\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}\right)\right)\right) - \frac{-1}{2} \cdot \left(\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}\right)\right)}{u0}\right)}\right) \]

   9. Simplified 94.9%

      \[\leadsto \frac{-1}{\color{blue}{\frac{u0 \cdot \left(u0 \cdot \left(\left(0.5 \cdot \left(\left(\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\right) \cdot -0.08333333333333333\right)\right) \cdot \left(-u0\right) + \left(\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\right) \cdot 0.08333333333333333\right) + \left(\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\right) \cdot 0.5\right) - \left(\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\right)}{u0}}} \]

   if 10 < (/.f32 sin2phi (*.f32 alphay alphay))

   1. Initial program 65.9%

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

      1. distribute-frac-neg N/A

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

      2. distribute-neg-frac2 N/A

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

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

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

      4. sub-neg N/A

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

      5. log1p-define N/A

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

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

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

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

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

      8. distribute-neg-in N/A

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

      9. unsub-neg N/A

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

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

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

      11. neg-sub0 N/A

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

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

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

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

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

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

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

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

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

      16. *-lowering-*.f32 98.3%

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

   3. Simplified 98.3%

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

   5. Taylor expanded in cos2phi around 0

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

      1. mul-1-neg N/A

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

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

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

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

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

      6. sub-neg N/A

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

      7. mul-1-neg N/A

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

      8. log1p-define N/A

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

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

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

      10. mul-1-neg N/A

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

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

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

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

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

      13. unpow2 N/A

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

      14. *-lowering-*.f32 98.9%

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

   7. Simplified 98.9%

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

      1. clear-num N/A

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

      2. associate-*r/ N/A

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

      3. div-inv N/A

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

      4. times-frac N/A

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

      5. *-lft-identity N/A

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

      6. associate-*l/ N/A

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

      7. remove-double-div N/A

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

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

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

      9. associate-*l/ N/A

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

      10. *-lft-identity N/A

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

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

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

      12. log1p-define N/A

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

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

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

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

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

      15. *-lowering-*.f32 99.1%

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

   9. Applied egg-rr 99.1%

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

4. Final simplification 97.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 10:\\ \;\;\;\;\frac{-1}{\frac{u0 \cdot \left(u0 \cdot \left(\left(\frac{sin2phi}{alphay \cdot alphay} + \frac{cos2phi}{alphax \cdot alphax}\right) \cdot 0.08333333333333333 - u0 \cdot \left(0.5 \cdot \left(\left(\frac{sin2phi}{alphay \cdot alphay} + \frac{cos2phi}{alphax \cdot alphax}\right) \cdot -0.08333333333333333\right)\right)\right) + 0.5 \cdot \left(\frac{sin2phi}{alphay \cdot alphay} + \frac{cos2phi}{alphax \cdot alphax}\right)\right) - \left(\frac{sin2phi}{alphay \cdot alphay} + \frac{cos2phi}{alphax \cdot alphax}\right)}{u0}}\\ \mathbf{else}:\\ \;\;\;\;\left(alphay \cdot alphay\right) \cdot \frac{\mathsf{log1p}\left(-u0\right)}{-sin2phi}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 93.1% accurate, 2.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 60.9%

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

   1. distribute-frac-neg N/A

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

   2. distribute-neg-frac2 N/A

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

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

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

   4. sub-neg N/A

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

   5. log1p-define N/A

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

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

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

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

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

   8. distribute-neg-in N/A

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

   9. unsub-neg N/A

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

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

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

   11. neg-sub0 N/A

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

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

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

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

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

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

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

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

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

   16. *-lowering-*.f32 98.4%

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

3. Simplified 98.4%

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

5. Taylor expanded in u0 around 0

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

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

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

   2. +-commutative N/A

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

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

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

7. Simplified 93.3%

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

   1. distribute-lft-in N/A

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

   2. *-commutative N/A

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

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

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

9. Applied egg-rr 93.5%

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

10. Final simplification 93.5%

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

Alternative 5: 91.0% accurate, 3.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{sin2phi}{alphay \cdot alphay}\\ \mathbf{if}\;t\_0 \leq 1.2000000476837158:\\ \;\;\;\;\frac{-1}{\frac{\left(t\_0 + \frac{cos2phi}{alphax \cdot alphax}\right) \cdot \left(-1 + u0 \cdot 0.5\right)}{u0}}\\ \mathbf{else}:\\ \;\;\;\;u0 \cdot \frac{alphay \cdot alphay + \left(0.5 + u0 \cdot \left(u0 \cdot 0.25 + 0.3333333333333333\right)\right) \cdot \left(u0 \cdot \left(alphay \cdot alphay\right)\right)}{sin2phi}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (let* ((t_0 (/ sin2phi (* alphay alphay))))
   (if (<= t_0 1.2000000476837158)
     (/
      -1.0
      (/ (* (+ t_0 (/ cos2phi (* alphax alphax))) (+ -1.0 (* u0 0.5))) u0))
     (*
      u0
      (/
       (+
        (* alphay alphay)
        (*
         (+ 0.5 (* u0 (+ (* u0 0.25) 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 <= 1.2000000476837158f) {
		tmp = -1.0f / (((t_0 + (cos2phi / (alphax * alphax))) * (-1.0f + (u0 * 0.5f))) / u0);
	} else {
		tmp = u0 * (((alphay * alphay) + ((0.5f + (u0 * ((u0 * 0.25f) + 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 <= 1.2000000476837158e0) then
        tmp = (-1.0e0) / (((t_0 + (cos2phi / (alphax * alphax))) * ((-1.0e0) + (u0 * 0.5e0))) / u0)
    else
        tmp = u0 * (((alphay * alphay) + ((0.5e0 + (u0 * ((u0 * 0.25e0) + 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(1.2000000476837158))
		tmp = Float32(Float32(-1.0) / Float32(Float32(Float32(t_0 + Float32(cos2phi / Float32(alphax * alphax))) * Float32(Float32(-1.0) + Float32(u0 * Float32(0.5)))) / u0));
	else
		tmp = Float32(u0 * Float32(Float32(Float32(alphay * alphay) + Float32(Float32(Float32(0.5) + Float32(u0 * Float32(Float32(u0 * Float32(0.25)) + 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(1.2000000476837158))
		tmp = single(-1.0) / (((t_0 + (cos2phi / (alphax * alphax))) * (single(-1.0) + (u0 * single(0.5)))) / u0);
	else
		tmp = u0 * (((alphay * alphay) + ((single(0.5) + (u0 * ((u0 * single(0.25)) + 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)) < 1.20000005

   1. Initial program 55.7%

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

      1. distribute-frac-neg N/A

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

      2. distribute-neg-frac2 N/A

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

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

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

      4. sub-neg N/A

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

      5. log1p-define N/A

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

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

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

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

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

      8. distribute-neg-in N/A

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

      9. unsub-neg N/A

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

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

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

      11. neg-sub0 N/A

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

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

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

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

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

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

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

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

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

      16. *-lowering-*.f32 98.6%

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

   3. Simplified 98.6%

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

      1. sub-neg N/A

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

      2. associate--l- N/A

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

      3. neg-sub0 N/A

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

      4. distribute-frac-neg2 N/A

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

      5. neg-mul-1 N/A

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

      6. clear-num N/A

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

      7. un-div-inv N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      15. sub-neg N/A

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

      16. log1p-define N/A

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

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

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

      18. neg-lowering-neg.f32 98.3%

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

   6. Applied egg-rr 98.3%

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

   7. Taylor expanded in u0 around 0

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

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

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

      2. associate-*r* N/A

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

      3. distribute-rgt-out N/A

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

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

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

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

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

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

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

      7. unpow2 N/A

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

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

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

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

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

      10. unpow2 N/A

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

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

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

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

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

      13. *-lowering-*.f32 88.3%

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

   9. Simplified 88.3%

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

   if 1.20000005 < (/.f32 sin2phi (*.f32 alphay alphay))

   1. Initial program 65.5%

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

      1. distribute-frac-neg N/A

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

      2. distribute-neg-frac2 N/A

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

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

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

      4. sub-neg N/A

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

      5. log1p-define N/A

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

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

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

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

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

      8. distribute-neg-in N/A

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

      9. unsub-neg N/A

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

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

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

      11. neg-sub0 N/A

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

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

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

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

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

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

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

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

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

      16. *-lowering-*.f32 98.3%

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

   3. Simplified 98.3%

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

   5. Taylor expanded in u0 around 0

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

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

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

      2. +-commutative N/A

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

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

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

   7. Simplified 92.9%

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

   8. Taylor expanded in sin2phi around inf

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

      1. associate-/l* N/A

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

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

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

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

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

   10. Simplified 93.7%

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

   11. Taylor expanded in u0 around 0

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

       1. *-commutative N/A

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

       2. +-commutative N/A

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

       3. distribute-lft-in N/A

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

       4. *-commutative N/A

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

       5. associate-*r* N/A

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

       6. *-commutative N/A

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

       7. *-commutative N/A

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

       8. associate-*r* N/A

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

       9. *-commutative N/A

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

       10. distribute-rgt-in N/A

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

       11. associate-*r* N/A

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

       12. *-commutative N/A

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

       13. distribute-rgt-in N/A

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

       14. associate-*r* N/A

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

   13. Simplified 93.7%

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

4. Final simplification 91.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 1.2000000476837158:\\ \;\;\;\;\frac{-1}{\frac{\left(\frac{sin2phi}{alphay \cdot alphay} + \frac{cos2phi}{alphax \cdot alphax}\right) \cdot \left(-1 + u0 \cdot 0.5\right)}{u0}}\\ \mathbf{else}:\\ \;\;\;\;u0 \cdot \frac{alphay \cdot alphay + \left(0.5 + u0 \cdot \left(u0 \cdot 0.25 + 0.3333333333333333\right)\right) \cdot \left(u0 \cdot \left(alphay \cdot alphay\right)\right)}{sin2phi}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 91.0% accurate, 3.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{sin2phi}{alphay \cdot alphay}\\ \mathbf{if}\;t\_0 \leq 1.2000000476837158:\\ \;\;\;\;\frac{-1}{\frac{\left(t\_0 + \frac{cos2phi}{alphax \cdot alphax}\right) \cdot \left(-1 + u0 \cdot 0.5\right)}{u0}}\\ \mathbf{else}:\\ \;\;\;\;\frac{alphay \cdot alphay}{sin2phi} \cdot \left(u0 - \left(u0 \cdot u0\right) \cdot \left(u0 \cdot \left(u0 \cdot -0.25 + -0.3333333333333333\right) + -0.5\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (let* ((t_0 (/ sin2phi (* alphay alphay))))
   (if (<= t_0 1.2000000476837158)
     (/
      -1.0
      (/ (* (+ t_0 (/ cos2phi (* alphax alphax))) (+ -1.0 (* u0 0.5))) u0))
     (*
      (/ (* alphay alphay) sin2phi)
      (-
       u0
       (* (* u0 u0) (+ (* u0 (+ (* u0 -0.25) -0.3333333333333333)) -0.5)))))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float t_0 = sin2phi / (alphay * alphay);
	float tmp;
	if (t_0 <= 1.2000000476837158f) {
		tmp = -1.0f / (((t_0 + (cos2phi / (alphax * alphax))) * (-1.0f + (u0 * 0.5f))) / u0);
	} else {
		tmp = ((alphay * alphay) / sin2phi) * (u0 - ((u0 * u0) * ((u0 * ((u0 * -0.25f) + -0.3333333333333333f)) + -0.5f)));
	}
	return tmp;
}

Fortran

real(4) function code(alphax, alphay, u0, cos2phi, sin2phi)
    real(4), intent (in) :: alphax
    real(4), intent (in) :: alphay
    real(4), intent (in) :: u0
    real(4), intent (in) :: cos2phi
    real(4), intent (in) :: sin2phi
    real(4) :: t_0
    real(4) :: tmp
    t_0 = sin2phi / (alphay * alphay)
    if (t_0 <= 1.2000000476837158e0) then
        tmp = (-1.0e0) / (((t_0 + (cos2phi / (alphax * alphax))) * ((-1.0e0) + (u0 * 0.5e0))) / u0)
    else
        tmp = ((alphay * alphay) / sin2phi) * (u0 - ((u0 * u0) * ((u0 * ((u0 * (-0.25e0)) + (-0.3333333333333333e0))) + (-0.5e0))))
    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(1.2000000476837158))
		tmp = Float32(Float32(-1.0) / Float32(Float32(Float32(t_0 + Float32(cos2phi / Float32(alphax * alphax))) * Float32(Float32(-1.0) + Float32(u0 * Float32(0.5)))) / u0));
	else
		tmp = Float32(Float32(Float32(alphay * alphay) / sin2phi) * Float32(u0 - Float32(Float32(u0 * u0) * Float32(Float32(u0 * Float32(Float32(u0 * Float32(-0.25)) + Float32(-0.3333333333333333))) + Float32(-0.5)))));
	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(1.2000000476837158))
		tmp = single(-1.0) / (((t_0 + (cos2phi / (alphax * alphax))) * (single(-1.0) + (u0 * single(0.5)))) / u0);
	else
		tmp = ((alphay * alphay) / sin2phi) * (u0 - ((u0 * u0) * ((u0 * ((u0 * single(-0.25)) + single(-0.3333333333333333))) + single(-0.5))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 sin2phi (*.f32 alphay alphay)) < 1.20000005

   1. Initial program 55.7%

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

      1. distribute-frac-neg N/A

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

      2. distribute-neg-frac2 N/A

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

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

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

      4. sub-neg N/A

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

      5. log1p-define N/A

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

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

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

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

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

      8. distribute-neg-in N/A

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

      9. unsub-neg N/A

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

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

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

      11. neg-sub0 N/A

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

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

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

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

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

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

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

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

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

      16. *-lowering-*.f32 98.6%

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

   3. Simplified 98.6%

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

      1. sub-neg N/A

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

      2. associate--l- N/A

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

      3. neg-sub0 N/A

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

      4. distribute-frac-neg2 N/A

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

      5. neg-mul-1 N/A

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

      6. clear-num N/A

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

      7. un-div-inv N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      15. sub-neg N/A

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

      16. log1p-define N/A

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

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

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

      18. neg-lowering-neg.f32 98.3%

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

   6. Applied egg-rr 98.3%

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

   7. Taylor expanded in u0 around 0

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

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

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

      2. associate-*r* N/A

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

      3. distribute-rgt-out N/A

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

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

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

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

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

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

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

      7. unpow2 N/A

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

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

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

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

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

      10. unpow2 N/A

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

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

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

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

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

      13. *-lowering-*.f32 88.3%

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

   9. Simplified 88.3%

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

   if 1.20000005 < (/.f32 sin2phi (*.f32 alphay alphay))

   1. Initial program 65.5%

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

      1. distribute-frac-neg N/A

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

      2. distribute-neg-frac2 N/A

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

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

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

      4. sub-neg N/A

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

      5. log1p-define N/A

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

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

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

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

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

      8. distribute-neg-in N/A

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

      9. unsub-neg N/A

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

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

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

      11. neg-sub0 N/A

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

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

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

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

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

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

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

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

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

      16. *-lowering-*.f32 98.3%

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

   3. Simplified 98.3%

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

   5. Taylor expanded in cos2phi around 0

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

      1. mul-1-neg N/A

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

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

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

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

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

      6. sub-neg N/A

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

      7. mul-1-neg N/A

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

      8. log1p-define N/A

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

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

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

      10. mul-1-neg N/A

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

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

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

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

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

      13. unpow2 N/A

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

      14. *-lowering-*.f32 98.8%

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

   7. Simplified 98.8%

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

   8. Taylor expanded in u0 around 0

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

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

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

      2. sub-neg N/A

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

      3. metadata-eval N/A

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

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

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

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

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

      6. sub-neg N/A

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

      7. metadata-eval N/A

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

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

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

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

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

      10. sub-neg N/A

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

      11. metadata-eval N/A

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

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

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

      13. *-commutative N/A

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

      14. *-lowering-*.f32 93.3%

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

   10. Simplified 93.3%

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

       1. distribute-rgt-in N/A

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

       2. neg-mul-1 N/A

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

       3. unsub-neg N/A

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

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

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

       5. *-commutative N/A

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

       6. associate-*l* N/A

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

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

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

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

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

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

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

       10. +-commutative N/A

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

       11. metadata-eval N/A

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

       12. distribute-rgt-neg-in N/A

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

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

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

       14. distribute-rgt-neg-in N/A

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

       15. metadata-eval N/A

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

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

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

       17. *-lowering-*.f32 93.6%

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

   12. Applied egg-rr 93.6%

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

4. Final simplification 91.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 1.2000000476837158:\\ \;\;\;\;\frac{-1}{\frac{\left(\frac{sin2phi}{alphay \cdot alphay} + \frac{cos2phi}{alphax \cdot alphax}\right) \cdot \left(-1 + u0 \cdot 0.5\right)}{u0}}\\ \mathbf{else}:\\ \;\;\;\;\frac{alphay \cdot alphay}{sin2phi} \cdot \left(u0 - \left(u0 \cdot u0\right) \cdot \left(u0 \cdot \left(u0 \cdot -0.25 + -0.3333333333333333\right) + -0.5\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 91.1% accurate, 3.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{sin2phi}{alphay \cdot alphay}\\ \mathbf{if}\;t\_0 \leq 40:\\ \;\;\;\;\frac{-1}{\frac{\left(t\_0 + \frac{cos2phi}{alphax \cdot alphax}\right) \cdot \left(-1 + u0 \cdot 0.5\right)}{u0}}\\ \mathbf{else}:\\ \;\;\;\;u0 \cdot \frac{\left(alphay \cdot alphay\right) \cdot \left(u0 \cdot \left(0.5 + u0 \cdot \left(u0 \cdot 0.25 + 0.3333333333333333\right)\right) + 1\right)}{sin2phi}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (let* ((t_0 (/ sin2phi (* alphay alphay))))
   (if (<= t_0 40.0)
     (/
      -1.0
      (/ (* (+ t_0 (/ cos2phi (* alphax alphax))) (+ -1.0 (* u0 0.5))) u0))
     (*
      u0
      (/
       (*
        (* alphay alphay)
        (+ (* u0 (+ 0.5 (* u0 (+ (* u0 0.25) 0.3333333333333333)))) 1.0))
       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 <= 40.0f) {
		tmp = -1.0f / (((t_0 + (cos2phi / (alphax * alphax))) * (-1.0f + (u0 * 0.5f))) / u0);
	} else {
		tmp = u0 * (((alphay * alphay) * ((u0 * (0.5f + (u0 * ((u0 * 0.25f) + 0.3333333333333333f)))) + 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) :: t_0
    real(4) :: tmp
    t_0 = sin2phi / (alphay * alphay)
    if (t_0 <= 40.0e0) then
        tmp = (-1.0e0) / (((t_0 + (cos2phi / (alphax * alphax))) * ((-1.0e0) + (u0 * 0.5e0))) / u0)
    else
        tmp = u0 * (((alphay * alphay) * ((u0 * (0.5e0 + (u0 * ((u0 * 0.25e0) + 0.3333333333333333e0)))) + 1.0e0)) / 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(40.0))
		tmp = Float32(Float32(-1.0) / Float32(Float32(Float32(t_0 + Float32(cos2phi / Float32(alphax * alphax))) * Float32(Float32(-1.0) + Float32(u0 * Float32(0.5)))) / u0));
	else
		tmp = Float32(u0 * Float32(Float32(Float32(alphay * alphay) * Float32(Float32(u0 * Float32(Float32(0.5) + Float32(u0 * Float32(Float32(u0 * Float32(0.25)) + Float32(0.3333333333333333))))) + Float32(1.0))) / 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(40.0))
		tmp = single(-1.0) / (((t_0 + (cos2phi / (alphax * alphax))) * (single(-1.0) + (u0 * single(0.5)))) / u0);
	else
		tmp = u0 * (((alphay * alphay) * ((u0 * (single(0.5) + (u0 * ((u0 * single(0.25)) + single(0.3333333333333333))))) + single(1.0))) / sin2phi);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 sin2phi (*.f32 alphay alphay)) < 40

   1. Initial program 55.7%

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

      1. distribute-frac-neg N/A

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

      2. distribute-neg-frac2 N/A

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

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

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

      4. sub-neg N/A

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

      5. log1p-define N/A

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

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

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

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

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

      8. distribute-neg-in N/A

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

      9. unsub-neg N/A

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

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

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

      11. neg-sub0 N/A

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

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

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

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

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

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

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

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

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

      16. *-lowering-*.f32 98.6%

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

   3. Simplified 98.6%

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

      1. sub-neg N/A

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

      2. associate--l- N/A

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

      3. neg-sub0 N/A

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

      4. distribute-frac-neg2 N/A

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

      5. neg-mul-1 N/A

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

      6. clear-num N/A

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

      7. un-div-inv N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      15. sub-neg N/A

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

      16. log1p-define N/A

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

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

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

      18. neg-lowering-neg.f32 98.4%

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

   6. Applied egg-rr 98.4%

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

   7. Taylor expanded in u0 around 0

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

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

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

      2. associate-*r* N/A

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

      3. distribute-rgt-out N/A

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

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

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

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

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

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

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

      7. unpow2 N/A

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

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

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

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

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

      10. unpow2 N/A

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

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

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

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

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

      13. *-lowering-*.f32 88.4%

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

   9. Simplified 88.4%

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

   if 40 < (/.f32 sin2phi (*.f32 alphay alphay))

   1. Initial program 65.9%

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

      1. distribute-frac-neg N/A

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

      2. distribute-neg-frac2 N/A

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

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

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

      4. sub-neg N/A

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

      5. log1p-define N/A

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

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

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

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

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

      8. distribute-neg-in N/A

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

      9. unsub-neg N/A

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

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

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

      11. neg-sub0 N/A

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

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

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

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

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

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

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

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

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

      16. *-lowering-*.f32 98.2%

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

   3. Simplified 98.2%

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

   5. Taylor expanded in u0 around 0

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

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

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

      2. +-commutative N/A

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

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

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

   7. Simplified 92.9%

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

   8. Taylor expanded in sin2phi around inf

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

      1. associate-/l* N/A

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

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

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

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

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

   10. Simplified 93.8%

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

   11. Taylor expanded in alphay around 0

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

   13. Simplified 93.5%

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

4. Final simplification 91.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 40:\\ \;\;\;\;\frac{-1}{\frac{\left(\frac{sin2phi}{alphay \cdot alphay} + \frac{cos2phi}{alphax \cdot alphax}\right) \cdot \left(-1 + u0 \cdot 0.5\right)}{u0}}\\ \mathbf{else}:\\ \;\;\;\;u0 \cdot \frac{\left(alphay \cdot alphay\right) \cdot \left(u0 \cdot \left(0.5 + u0 \cdot \left(u0 \cdot 0.25 + 0.3333333333333333\right)\right) + 1\right)}{sin2phi}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 90.1% accurate, 3.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{sin2phi}{alphay \cdot alphay}\\ \mathbf{if}\;t\_0 \leq 1.2000000476837158:\\ \;\;\;\;\frac{u0 \cdot \left(u0 \cdot 0.5 + 1\right)}{t\_0 + \frac{cos2phi}{alphax \cdot alphax}}\\ \mathbf{else}:\\ \;\;\;\;u0 \cdot \frac{\left(alphay \cdot alphay\right) \cdot \left(u0 \cdot \left(0.5 + u0 \cdot \left(u0 \cdot 0.25 + 0.3333333333333333\right)\right) + 1\right)}{sin2phi}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (let* ((t_0 (/ sin2phi (* alphay alphay))))
   (if (<= t_0 1.2000000476837158)
     (/ (* u0 (+ (* u0 0.5) 1.0)) (+ t_0 (/ cos2phi (* alphax alphax))))
     (*
      u0
      (/
       (*
        (* alphay alphay)
        (+ (* u0 (+ 0.5 (* u0 (+ (* u0 0.25) 0.3333333333333333)))) 1.0))
       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 <= 1.2000000476837158f) {
		tmp = (u0 * ((u0 * 0.5f) + 1.0f)) / (t_0 + (cos2phi / (alphax * alphax)));
	} else {
		tmp = u0 * (((alphay * alphay) * ((u0 * (0.5f + (u0 * ((u0 * 0.25f) + 0.3333333333333333f)))) + 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) :: t_0
    real(4) :: tmp
    t_0 = sin2phi / (alphay * alphay)
    if (t_0 <= 1.2000000476837158e0) then
        tmp = (u0 * ((u0 * 0.5e0) + 1.0e0)) / (t_0 + (cos2phi / (alphax * alphax)))
    else
        tmp = u0 * (((alphay * alphay) * ((u0 * (0.5e0 + (u0 * ((u0 * 0.25e0) + 0.3333333333333333e0)))) + 1.0e0)) / 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(1.2000000476837158))
		tmp = Float32(Float32(u0 * Float32(Float32(u0 * Float32(0.5)) + Float32(1.0))) / Float32(t_0 + Float32(cos2phi / Float32(alphax * alphax))));
	else
		tmp = Float32(u0 * Float32(Float32(Float32(alphay * alphay) * Float32(Float32(u0 * Float32(Float32(0.5) + Float32(u0 * Float32(Float32(u0 * Float32(0.25)) + Float32(0.3333333333333333))))) + Float32(1.0))) / 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(1.2000000476837158))
		tmp = (u0 * ((u0 * single(0.5)) + single(1.0))) / (t_0 + (cos2phi / (alphax * alphax)));
	else
		tmp = u0 * (((alphay * alphay) * ((u0 * (single(0.5) + (u0 * ((u0 * single(0.25)) + single(0.3333333333333333))))) + single(1.0))) / sin2phi);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 sin2phi (*.f32 alphay alphay)) < 1.20000005

   1. Initial program 55.7%

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

   3. Taylor expanded in u0 around 0

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

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

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

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

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

      3. *-commutative N/A

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

      4. *-lowering-*.f32 86.5%

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

   5. Simplified 86.5%

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

   if 1.20000005 < (/.f32 sin2phi (*.f32 alphay alphay))

   1. Initial program 65.5%

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

      1. distribute-frac-neg N/A

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

      2. distribute-neg-frac2 N/A

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

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

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

      4. sub-neg N/A

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

      5. log1p-define N/A

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

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

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

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

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

      8. distribute-neg-in N/A

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

      9. unsub-neg N/A

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

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

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

      11. neg-sub0 N/A

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

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

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

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

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

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

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

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

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

      16. *-lowering-*.f32 98.3%

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

   3. Simplified 98.3%

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

   5. Taylor expanded in u0 around 0

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

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

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

      2. +-commutative N/A

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

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

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

   7. Simplified 92.9%

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

   8. Taylor expanded in sin2phi around inf

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

      1. associate-/l* N/A

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

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

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

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

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

   10. Simplified 93.7%

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

   11. Taylor expanded in alphay around 0

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

   13. Simplified 93.4%

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

4. Final simplification 90.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 1.2000000476837158:\\ \;\;\;\;\frac{u0 \cdot \left(u0 \cdot 0.5 + 1\right)}{\frac{sin2phi}{alphay \cdot alphay} + \frac{cos2phi}{alphax \cdot alphax}}\\ \mathbf{else}:\\ \;\;\;\;u0 \cdot \frac{\left(alphay \cdot alphay\right) \cdot \left(u0 \cdot \left(0.5 + u0 \cdot \left(u0 \cdot 0.25 + 0.3333333333333333\right)\right) + 1\right)}{sin2phi}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 93.0% accurate, 4.3× speedup

Math

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

FPCore

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

MATLAB

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

Derivation

1. Initial program 60.9%

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

   1. distribute-frac-neg N/A

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

   2. distribute-neg-frac2 N/A

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

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

      \[\leadsto \mathsf{/.f32}\left(\log \left(1 - u0\right), \color{blue}{\left(\mathsf{neg}\left(\left(\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\right)\right)\right)}\right) \]

   4. sub-neg N/A

      \[\leadsto \mathsf{/.f32}\left(\log \left(1 + \left(\mathsf{neg}\left(u0\right)\right)\right), \left(\mathsf{neg}\left(\left(\color{blue}{\frac{cos2phi}{alphax \cdot alphax}} + \frac{sin2phi}{alphay \cdot alphay}\right)\right)\right)\right) \]

   5. log1p-define N/A

      \[\leadsto \mathsf{/.f32}\left(\left(\mathsf{log1p}\left(\mathsf{neg}\left(u0\right)\right)\right), \left(\mathsf{neg}\left(\color{blue}{\left(\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\right)}\right)\right)\right) \]

   6. log1p-lowering-log1p.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\left(\mathsf{neg}\left(u0\right)\right)\right), \left(\mathsf{neg}\left(\color{blue}{\left(\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\right)}\right)\right)\right) \]

   7. neg-lowering-neg.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \left(\mathsf{neg}\left(\left(\color{blue}{\frac{cos2phi}{alphax \cdot alphax}} + \frac{sin2phi}{alphay \cdot alphay}\right)\right)\right)\right) \]

   8. distribute-neg-in N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \left(\left(\mathsf{neg}\left(\frac{cos2phi}{alphax \cdot alphax}\right)\right) + \color{blue}{\left(\mathsf{neg}\left(\frac{sin2phi}{alphay \cdot alphay}\right)\right)}\right)\right) \]

   9. unsub-neg N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \left(\left(\mathsf{neg}\left(\frac{cos2phi}{alphax \cdot alphax}\right)\right) - \color{blue}{\frac{sin2phi}{alphay \cdot alphay}}\right)\right) \]

   10. --lowering--.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\left(\mathsf{neg}\left(\frac{cos2phi}{alphax \cdot alphax}\right)\right), \color{blue}{\left(\frac{sin2phi}{alphay \cdot alphay}\right)}\right)\right) \]

   11. neg-sub0 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\left(0 - \frac{cos2phi}{alphax \cdot alphax}\right), \left(\frac{\color{blue}{sin2phi}}{alphay \cdot alphay}\right)\right)\right) \]

   12. --lowering--.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \left(\frac{cos2phi}{alphax \cdot alphax}\right)\right), \left(\frac{\color{blue}{sin2phi}}{alphay \cdot alphay}\right)\right)\right) \]

   13. /-lowering-/.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(cos2phi, \left(alphax \cdot alphax\right)\right)\right), \left(\frac{sin2phi}{alphay \cdot alphay}\right)\right)\right) \]

   14. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right)\right), \left(\frac{sin2phi}{alphay \cdot alphay}\right)\right)\right) \]

   15. /-lowering-/.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right)\right), \mathsf{/.f32}\left(sin2phi, \color{blue}{\left(alphay \cdot alphay\right)}\right)\right)\right) \]

   16. *-lowering-*.f32 98.4%

       \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right)\right), \mathsf{/.f32}\left(sin2phi, \mathsf{*.f32}\left(alphay, \color{blue}{alphay}\right)\right)\right)\right) \]

3. Simplified 98.4%

   \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(-u0\right)}{\left(0 - \frac{cos2phi}{alphax \cdot alphax}\right) - \frac{sin2phi}{alphay \cdot alphay}}} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. associate-/r* N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \left(\frac{\frac{cos2phi}{alphax}}{alphax}\right)\right), \mathsf{/.f32}\left(sin2phi, \mathsf{*.f32}\left(alphay, alphay\right)\right)\right)\right) \]

   2. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(\left(\frac{cos2phi}{alphax}\right), alphax\right)\right), \mathsf{/.f32}\left(sin2phi, \mathsf{*.f32}\left(alphay, alphay\right)\right)\right)\right) \]

   3. /-lowering-/.f32 98.4%

      \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(\mathsf{/.f32}\left(cos2phi, alphax\right), alphax\right)\right), \mathsf{/.f32}\left(sin2phi, \mathsf{*.f32}\left(alphay, alphay\right)\right)\right)\right) \]

6. Applied egg-rr 98.4%

   \[\leadsto \frac{\mathsf{log1p}\left(-u0\right)}{\left(0 - \color{blue}{\frac{\frac{cos2phi}{alphax}}{alphax}}\right) - \frac{sin2phi}{alphay \cdot alphay}} \]
7. Step-by-step derivation

   1. sub0-neg N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\left(\mathsf{neg}\left(\frac{\frac{cos2phi}{alphax}}{alphax}\right)\right), \mathsf{/.f32}\left(\color{blue}{sin2phi}, \mathsf{*.f32}\left(alphay, alphay\right)\right)\right)\right) \]

   2. distribute-neg-frac2 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\left(\frac{\frac{cos2phi}{alphax}}{\mathsf{neg}\left(alphax\right)}\right), \mathsf{/.f32}\left(\color{blue}{sin2phi}, \mathsf{*.f32}\left(alphay, alphay\right)\right)\right)\right) \]

   3. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{/.f32}\left(\left(\frac{cos2phi}{alphax}\right), \left(\mathsf{neg}\left(alphax\right)\right)\right), \mathsf{/.f32}\left(\color{blue}{sin2phi}, \mathsf{*.f32}\left(alphay, alphay\right)\right)\right)\right) \]

   4. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(cos2phi, alphax\right), \left(\mathsf{neg}\left(alphax\right)\right)\right), \mathsf{/.f32}\left(sin2phi, \mathsf{*.f32}\left(alphay, alphay\right)\right)\right)\right) \]

   5. neg-lowering-neg.f32 98.4%

      \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(cos2phi, alphax\right), \mathsf{neg.f32}\left(alphax\right)\right), \mathsf{/.f32}\left(sin2phi, \mathsf{*.f32}\left(alphay, alphay\right)\right)\right)\right) \]

8. Applied egg-rr 98.4%

   \[\leadsto \frac{\mathsf{log1p}\left(-u0\right)}{\color{blue}{\frac{\frac{cos2phi}{alphax}}{-alphax}} - \frac{sin2phi}{alphay \cdot alphay}} \]
9. Step-by-step derivation

   1. associate-/r* 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 98.4%

      \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(cos2phi, alphax\right), \mathsf{neg.f32}\left(alphax\right)\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(sin2phi, alphay\right), alphay\right)\right)\right) \]

10. Applied egg-rr 98.4%

    \[\leadsto \frac{\mathsf{log1p}\left(-u0\right)}{\frac{\frac{cos2phi}{alphax}}{-alphax} - \color{blue}{\frac{\frac{sin2phi}{alphay}}{alphay}}} \]

11. Taylor expanded in u0 around 0

    \[\leadsto \mathsf{/.f32}\left(\color{blue}{\left(u0 \cdot \left(u0 \cdot \left(u0 \cdot \left(\frac{-1}{4} \cdot u0 - \frac{1}{3}\right) - \frac{1}{2}\right) - 1\right)\right)}, \mathsf{\_.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(cos2phi, alphax\right), \mathsf{neg.f32}\left(alphax\right)\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(sin2phi, alphay\right), alphay\right)\right)\right) \]
12. Step-by-step derivation

    1. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(u0, \left(u0 \cdot \left(u0 \cdot \left(\frac{-1}{4} \cdot u0 - \frac{1}{3}\right) - \frac{1}{2}\right) - 1\right)\right), \mathsf{\_.f32}\left(\color{blue}{\mathsf{/.f32}\left(\mathsf{/.f32}\left(cos2phi, alphax\right), \mathsf{neg.f32}\left(alphax\right)\right)}, \mathsf{/.f32}\left(\mathsf{/.f32}\left(sin2phi, alphay\right), alphay\right)\right)\right) \]

    2. sub-neg N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(u0, \left(u0 \cdot \left(u0 \cdot \left(\frac{-1}{4} \cdot u0 - \frac{1}{3}\right) - \frac{1}{2}\right) + \left(\mathsf{neg}\left(1\right)\right)\right)\right), \mathsf{\_.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(cos2phi, alphax\right), \color{blue}{\mathsf{neg.f32}\left(alphax\right)}\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(sin2phi, alphay\right), alphay\right)\right)\right) \]

    3. metadata-eval N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(u0, \left(u0 \cdot \left(u0 \cdot \left(\frac{-1}{4} \cdot u0 - \frac{1}{3}\right) - \frac{1}{2}\right) + -1\right)\right), \mathsf{\_.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(cos2phi, alphax\right), \mathsf{neg.f32}\left(alphax\right)\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(sin2phi, alphay\right), alphay\right)\right)\right) \]

    4. +-lowering-+.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\left(u0 \cdot \left(u0 \cdot \left(\frac{-1}{4} \cdot u0 - \frac{1}{3}\right) - \frac{1}{2}\right)\right), -1\right)\right), \mathsf{\_.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(cos2phi, alphax\right), \color{blue}{\mathsf{neg.f32}\left(alphax\right)}\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(sin2phi, alphay\right), alphay\right)\right)\right) \]

    5. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{*.f32}\left(u0, \left(u0 \cdot \left(\frac{-1}{4} \cdot u0 - \frac{1}{3}\right) - \frac{1}{2}\right)\right), -1\right)\right), \mathsf{\_.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(cos2phi, alphax\right), \mathsf{neg.f32}\left(\color{blue}{alphax}\right)\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(sin2phi, alphay\right), alphay\right)\right)\right) \]

    6. sub-neg N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{*.f32}\left(u0, \left(u0 \cdot \left(\frac{-1}{4} \cdot u0 - \frac{1}{3}\right) + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right)\right), -1\right)\right), \mathsf{\_.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(cos2phi, alphax\right), \mathsf{neg.f32}\left(alphax\right)\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(sin2phi, alphay\right), alphay\right)\right)\right) \]

    7. metadata-eval N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{*.f32}\left(u0, \left(u0 \cdot \left(\frac{-1}{4} \cdot u0 - \frac{1}{3}\right) + \frac{-1}{2}\right)\right), -1\right)\right), \mathsf{\_.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(cos2phi, alphax\right), \mathsf{neg.f32}\left(alphax\right)\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(sin2phi, alphay\right), alphay\right)\right)\right) \]

    8. +-lowering-+.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\left(u0 \cdot \left(\frac{-1}{4} \cdot u0 - \frac{1}{3}\right)\right), \frac{-1}{2}\right)\right), -1\right)\right), \mathsf{\_.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(cos2phi, alphax\right), \mathsf{neg.f32}\left(alphax\right)\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(sin2phi, alphay\right), alphay\right)\right)\right) \]

    9. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{*.f32}\left(u0, \left(\frac{-1}{4} \cdot u0 - \frac{1}{3}\right)\right), \frac{-1}{2}\right)\right), -1\right)\right), \mathsf{\_.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(cos2phi, alphax\right), \mathsf{neg.f32}\left(alphax\right)\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(sin2phi, alphay\right), alphay\right)\right)\right) \]

    10. sub-neg N/A

        \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{*.f32}\left(u0, \left(\frac{-1}{4} \cdot u0 + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right)\right), \frac{-1}{2}\right)\right), -1\right)\right), \mathsf{\_.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(cos2phi, alphax\right), \mathsf{neg.f32}\left(alphax\right)\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(sin2phi, alphay\right), alphay\right)\right)\right) \]

    11. metadata-eval N/A

        \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{*.f32}\left(u0, \left(\frac{-1}{4} \cdot u0 + \frac{-1}{3}\right)\right), \frac{-1}{2}\right)\right), -1\right)\right), \mathsf{\_.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(cos2phi, alphax\right), \mathsf{neg.f32}\left(alphax\right)\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(sin2phi, alphay\right), alphay\right)\right)\right) \]

    12. +-lowering-+.f32 N/A

        \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\left(\frac{-1}{4} \cdot u0\right), \frac{-1}{3}\right)\right), \frac{-1}{2}\right)\right), -1\right)\right), \mathsf{\_.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(cos2phi, alphax\right), \mathsf{neg.f32}\left(alphax\right)\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(sin2phi, alphay\right), alphay\right)\right)\right) \]

    13. *-commutative N/A

        \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\left(u0 \cdot \frac{-1}{4}\right), \frac{-1}{3}\right)\right), \frac{-1}{2}\right)\right), -1\right)\right), \mathsf{\_.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(cos2phi, alphax\right), \mathsf{neg.f32}\left(alphax\right)\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(sin2phi, alphay\right), alphay\right)\right)\right) \]

    14. *-lowering-*.f32 93.2%

        \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{*.f32}\left(u0, \frac{-1}{4}\right), \frac{-1}{3}\right)\right), \frac{-1}{2}\right)\right), -1\right)\right), \mathsf{\_.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(cos2phi, alphax\right), \mathsf{neg.f32}\left(alphax\right)\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(sin2phi, alphay\right), alphay\right)\right)\right) \]

13. Simplified 93.2%

    \[\leadsto \frac{\color{blue}{u0 \cdot \left(u0 \cdot \left(u0 \cdot \left(u0 \cdot -0.25 + -0.3333333333333333\right) + -0.5\right) + -1\right)}}{\frac{\frac{cos2phi}{alphax}}{-alphax} - \frac{\frac{sin2phi}{alphay}}{alphay}} \]

14. Final simplification 93.2%

    \[\leadsto \frac{u0 \cdot \left(1 - u0 \cdot \left(u0 \cdot \left(u0 \cdot -0.25 + -0.3333333333333333\right) + -0.5\right)\right)}{\frac{\frac{sin2phi}{alphay}}{alphay} + \frac{\frac{cos2phi}{alphax}}{alphax}} \]
15. Add Preprocessing

Alternative 10: 93.0% accurate, 4.3× speedup

Math

\[\begin{array}{l} \\ \frac{u0 \cdot \left(1 - u0 \cdot \left(u0 \cdot \left(u0 \cdot -0.25 + -0.3333333333333333\right) + -0.5\right)\right)}{\frac{sin2phi}{alphay \cdot alphay} + \frac{\frac{cos2phi}{alphax}}{alphax}} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (/
  (* u0 (- 1.0 (* u0 (+ (* u0 (+ (* u0 -0.25) -0.3333333333333333)) -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 * ((u0 * ((u0 * -0.25f) + -0.3333333333333333f)) + -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 * ((u0 * ((u0 * (-0.25e0)) + (-0.3333333333333333e0))) + (-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(Float32(u0 * Float32(Float32(u0 * Float32(-0.25)) + Float32(-0.3333333333333333))) + Float32(-0.5))))) / Float32(Float32(sin2phi / Float32(alphay * alphay)) + Float32(Float32(cos2phi / alphax) / alphax)))
end

MATLAB

function tmp = code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = (u0 * (single(1.0) - (u0 * ((u0 * ((u0 * single(-0.25)) + single(-0.3333333333333333))) + single(-0.5))))) / ((sin2phi / (alphay * alphay)) + ((cos2phi / alphax) / alphax));
end

Derivation

1. Initial program 60.9%

   \[\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
2. Step-by-step derivation

   1. distribute-frac-neg N/A

      \[\leadsto \mathsf{neg}\left(\frac{\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}}\right) \]

   2. distribute-neg-frac2 N/A

      \[\leadsto \frac{\log \left(1 - u0\right)}{\color{blue}{\mathsf{neg}\left(\left(\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\right)\right)}} \]

   3. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\log \left(1 - u0\right), \color{blue}{\left(\mathsf{neg}\left(\left(\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\right)\right)\right)}\right) \]

   4. sub-neg N/A

      \[\leadsto \mathsf{/.f32}\left(\log \left(1 + \left(\mathsf{neg}\left(u0\right)\right)\right), \left(\mathsf{neg}\left(\left(\color{blue}{\frac{cos2phi}{alphax \cdot alphax}} + \frac{sin2phi}{alphay \cdot alphay}\right)\right)\right)\right) \]

   5. log1p-define N/A

      \[\leadsto \mathsf{/.f32}\left(\left(\mathsf{log1p}\left(\mathsf{neg}\left(u0\right)\right)\right), \left(\mathsf{neg}\left(\color{blue}{\left(\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\right)}\right)\right)\right) \]

   6. log1p-lowering-log1p.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\left(\mathsf{neg}\left(u0\right)\right)\right), \left(\mathsf{neg}\left(\color{blue}{\left(\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\right)}\right)\right)\right) \]

   7. neg-lowering-neg.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \left(\mathsf{neg}\left(\left(\color{blue}{\frac{cos2phi}{alphax \cdot alphax}} + \frac{sin2phi}{alphay \cdot alphay}\right)\right)\right)\right) \]

   8. distribute-neg-in N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \left(\left(\mathsf{neg}\left(\frac{cos2phi}{alphax \cdot alphax}\right)\right) + \color{blue}{\left(\mathsf{neg}\left(\frac{sin2phi}{alphay \cdot alphay}\right)\right)}\right)\right) \]

   9. unsub-neg N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \left(\left(\mathsf{neg}\left(\frac{cos2phi}{alphax \cdot alphax}\right)\right) - \color{blue}{\frac{sin2phi}{alphay \cdot alphay}}\right)\right) \]

   10. --lowering--.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\left(\mathsf{neg}\left(\frac{cos2phi}{alphax \cdot alphax}\right)\right), \color{blue}{\left(\frac{sin2phi}{alphay \cdot alphay}\right)}\right)\right) \]

   11. neg-sub0 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\left(0 - \frac{cos2phi}{alphax \cdot alphax}\right), \left(\frac{\color{blue}{sin2phi}}{alphay \cdot alphay}\right)\right)\right) \]

   12. --lowering--.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \left(\frac{cos2phi}{alphax \cdot alphax}\right)\right), \left(\frac{\color{blue}{sin2phi}}{alphay \cdot alphay}\right)\right)\right) \]

   13. /-lowering-/.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(cos2phi, \left(alphax \cdot alphax\right)\right)\right), \left(\frac{sin2phi}{alphay \cdot alphay}\right)\right)\right) \]

   14. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right)\right), \left(\frac{sin2phi}{alphay \cdot alphay}\right)\right)\right) \]

   15. /-lowering-/.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right)\right), \mathsf{/.f32}\left(sin2phi, \color{blue}{\left(alphay \cdot alphay\right)}\right)\right)\right) \]

   16. *-lowering-*.f32 98.4%

       \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right)\right), \mathsf{/.f32}\left(sin2phi, \mathsf{*.f32}\left(alphay, \color{blue}{alphay}\right)\right)\right)\right) \]

3. Simplified 98.4%

   \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(-u0\right)}{\left(0 - \frac{cos2phi}{alphax \cdot alphax}\right) - \frac{sin2phi}{alphay \cdot alphay}}} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. associate-/r* N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \left(\frac{\frac{cos2phi}{alphax}}{alphax}\right)\right), \mathsf{/.f32}\left(sin2phi, \mathsf{*.f32}\left(alphay, alphay\right)\right)\right)\right) \]

   2. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(\left(\frac{cos2phi}{alphax}\right), alphax\right)\right), \mathsf{/.f32}\left(sin2phi, \mathsf{*.f32}\left(alphay, alphay\right)\right)\right)\right) \]

   3. /-lowering-/.f32 98.4%

      \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(\mathsf{/.f32}\left(cos2phi, alphax\right), alphax\right)\right), \mathsf{/.f32}\left(sin2phi, \mathsf{*.f32}\left(alphay, alphay\right)\right)\right)\right) \]

6. Applied egg-rr 98.4%

   \[\leadsto \frac{\mathsf{log1p}\left(-u0\right)}{\left(0 - \color{blue}{\frac{\frac{cos2phi}{alphax}}{alphax}}\right) - \frac{sin2phi}{alphay \cdot alphay}} \]
7. Step-by-step derivation

   1. sub0-neg N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\left(\mathsf{neg}\left(\frac{\frac{cos2phi}{alphax}}{alphax}\right)\right), \mathsf{/.f32}\left(\color{blue}{sin2phi}, \mathsf{*.f32}\left(alphay, alphay\right)\right)\right)\right) \]

   2. distribute-neg-frac2 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\left(\frac{\frac{cos2phi}{alphax}}{\mathsf{neg}\left(alphax\right)}\right), \mathsf{/.f32}\left(\color{blue}{sin2phi}, \mathsf{*.f32}\left(alphay, alphay\right)\right)\right)\right) \]

   3. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{/.f32}\left(\left(\frac{cos2phi}{alphax}\right), \left(\mathsf{neg}\left(alphax\right)\right)\right), \mathsf{/.f32}\left(\color{blue}{sin2phi}, \mathsf{*.f32}\left(alphay, alphay\right)\right)\right)\right) \]

   4. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(cos2phi, alphax\right), \left(\mathsf{neg}\left(alphax\right)\right)\right), \mathsf{/.f32}\left(sin2phi, \mathsf{*.f32}\left(alphay, alphay\right)\right)\right)\right) \]

   5. neg-lowering-neg.f32 98.4%

      \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(cos2phi, alphax\right), \mathsf{neg.f32}\left(alphax\right)\right), \mathsf{/.f32}\left(sin2phi, \mathsf{*.f32}\left(alphay, alphay\right)\right)\right)\right) \]

8. Applied egg-rr 98.4%

   \[\leadsto \frac{\mathsf{log1p}\left(-u0\right)}{\color{blue}{\frac{\frac{cos2phi}{alphax}}{-alphax}} - \frac{sin2phi}{alphay \cdot alphay}} \]

9. Taylor expanded in u0 around 0

   \[\leadsto \mathsf{/.f32}\left(\color{blue}{\left(u0 \cdot \left(u0 \cdot \left(u0 \cdot \left(\frac{-1}{4} \cdot u0 - \frac{1}{3}\right) - \frac{1}{2}\right) - 1\right)\right)}, \mathsf{\_.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(cos2phi, alphax\right), \mathsf{neg.f32}\left(alphax\right)\right), \mathsf{/.f32}\left(sin2phi, \mathsf{*.f32}\left(alphay, alphay\right)\right)\right)\right) \]
10. Step-by-step derivation

    1. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(u0, \left(u0 \cdot \left(u0 \cdot \left(\frac{-1}{4} \cdot u0 - \frac{1}{3}\right) - \frac{1}{2}\right) - 1\right)\right), \mathsf{\_.f32}\left(\color{blue}{\mathsf{/.f32}\left(\mathsf{/.f32}\left(cos2phi, alphax\right), \mathsf{neg.f32}\left(alphax\right)\right)}, \mathsf{/.f32}\left(sin2phi, \mathsf{*.f32}\left(alphay, alphay\right)\right)\right)\right) \]

    2. sub-neg N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(u0, \left(u0 \cdot \left(u0 \cdot \left(\frac{-1}{4} \cdot u0 - \frac{1}{3}\right) - \frac{1}{2}\right) + \left(\mathsf{neg}\left(1\right)\right)\right)\right), \mathsf{\_.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(cos2phi, alphax\right), \color{blue}{\mathsf{neg.f32}\left(alphax\right)}\right), \mathsf{/.f32}\left(sin2phi, \mathsf{*.f32}\left(alphay, alphay\right)\right)\right)\right) \]

    3. metadata-eval N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(u0, \left(u0 \cdot \left(u0 \cdot \left(\frac{-1}{4} \cdot u0 - \frac{1}{3}\right) - \frac{1}{2}\right) + -1\right)\right), \mathsf{\_.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(cos2phi, alphax\right), \mathsf{neg.f32}\left(alphax\right)\right), \mathsf{/.f32}\left(sin2phi, \mathsf{*.f32}\left(alphay, alphay\right)\right)\right)\right) \]

    4. +-lowering-+.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\left(u0 \cdot \left(u0 \cdot \left(\frac{-1}{4} \cdot u0 - \frac{1}{3}\right) - \frac{1}{2}\right)\right), -1\right)\right), \mathsf{\_.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(cos2phi, alphax\right), \color{blue}{\mathsf{neg.f32}\left(alphax\right)}\right), \mathsf{/.f32}\left(sin2phi, \mathsf{*.f32}\left(alphay, alphay\right)\right)\right)\right) \]

    5. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{*.f32}\left(u0, \left(u0 \cdot \left(\frac{-1}{4} \cdot u0 - \frac{1}{3}\right) - \frac{1}{2}\right)\right), -1\right)\right), \mathsf{\_.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(cos2phi, alphax\right), \mathsf{neg.f32}\left(\color{blue}{alphax}\right)\right), \mathsf{/.f32}\left(sin2phi, \mathsf{*.f32}\left(alphay, alphay\right)\right)\right)\right) \]

    6. sub-neg N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{*.f32}\left(u0, \left(u0 \cdot \left(\frac{-1}{4} \cdot u0 - \frac{1}{3}\right) + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right)\right), -1\right)\right), \mathsf{\_.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(cos2phi, alphax\right), \mathsf{neg.f32}\left(alphax\right)\right), \mathsf{/.f32}\left(sin2phi, \mathsf{*.f32}\left(alphay, alphay\right)\right)\right)\right) \]

    7. metadata-eval N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{*.f32}\left(u0, \left(u0 \cdot \left(\frac{-1}{4} \cdot u0 - \frac{1}{3}\right) + \frac{-1}{2}\right)\right), -1\right)\right), \mathsf{\_.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(cos2phi, alphax\right), \mathsf{neg.f32}\left(alphax\right)\right), \mathsf{/.f32}\left(sin2phi, \mathsf{*.f32}\left(alphay, alphay\right)\right)\right)\right) \]

    8. +-lowering-+.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\left(u0 \cdot \left(\frac{-1}{4} \cdot u0 - \frac{1}{3}\right)\right), \frac{-1}{2}\right)\right), -1\right)\right), \mathsf{\_.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(cos2phi, alphax\right), \mathsf{neg.f32}\left(alphax\right)\right), \mathsf{/.f32}\left(sin2phi, \mathsf{*.f32}\left(alphay, alphay\right)\right)\right)\right) \]

    9. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{*.f32}\left(u0, \left(\frac{-1}{4} \cdot u0 - \frac{1}{3}\right)\right), \frac{-1}{2}\right)\right), -1\right)\right), \mathsf{\_.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(cos2phi, alphax\right), \mathsf{neg.f32}\left(alphax\right)\right), \mathsf{/.f32}\left(sin2phi, \mathsf{*.f32}\left(alphay, alphay\right)\right)\right)\right) \]

    10. sub-neg N/A

        \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{*.f32}\left(u0, \left(\frac{-1}{4} \cdot u0 + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right)\right), \frac{-1}{2}\right)\right), -1\right)\right), \mathsf{\_.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(cos2phi, alphax\right), \mathsf{neg.f32}\left(alphax\right)\right), \mathsf{/.f32}\left(sin2phi, \mathsf{*.f32}\left(alphay, alphay\right)\right)\right)\right) \]

    11. metadata-eval N/A

        \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{*.f32}\left(u0, \left(\frac{-1}{4} \cdot u0 + \frac{-1}{3}\right)\right), \frac{-1}{2}\right)\right), -1\right)\right), \mathsf{\_.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(cos2phi, alphax\right), \mathsf{neg.f32}\left(alphax\right)\right), \mathsf{/.f32}\left(sin2phi, \mathsf{*.f32}\left(alphay, alphay\right)\right)\right)\right) \]

    12. +-lowering-+.f32 N/A

        \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\left(\frac{-1}{4} \cdot u0\right), \frac{-1}{3}\right)\right), \frac{-1}{2}\right)\right), -1\right)\right), \mathsf{\_.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(cos2phi, alphax\right), \mathsf{neg.f32}\left(alphax\right)\right), \mathsf{/.f32}\left(sin2phi, \mathsf{*.f32}\left(alphay, alphay\right)\right)\right)\right) \]

    13. *-commutative N/A

        \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\left(u0 \cdot \frac{-1}{4}\right), \frac{-1}{3}\right)\right), \frac{-1}{2}\right)\right), -1\right)\right), \mathsf{\_.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(cos2phi, alphax\right), \mathsf{neg.f32}\left(alphax\right)\right), \mathsf{/.f32}\left(sin2phi, \mathsf{*.f32}\left(alphay, alphay\right)\right)\right)\right) \]

    14. *-lowering-*.f32 93.1%

        \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{*.f32}\left(u0, \frac{-1}{4}\right), \frac{-1}{3}\right)\right), \frac{-1}{2}\right)\right), -1\right)\right), \mathsf{\_.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(cos2phi, alphax\right), \mathsf{neg.f32}\left(alphax\right)\right), \mathsf{/.f32}\left(sin2phi, \mathsf{*.f32}\left(alphay, alphay\right)\right)\right)\right) \]

11. Simplified 93.1%

    \[\leadsto \frac{\color{blue}{u0 \cdot \left(u0 \cdot \left(u0 \cdot \left(u0 \cdot -0.25 + -0.3333333333333333\right) + -0.5\right) + -1\right)}}{\frac{\frac{cos2phi}{alphax}}{-alphax} - \frac{sin2phi}{alphay \cdot alphay}} \]

12. Final simplification 93.1%

    \[\leadsto \frac{u0 \cdot \left(1 - u0 \cdot \left(u0 \cdot \left(u0 \cdot -0.25 + -0.3333333333333333\right) + -0.5\right)\right)}{\frac{sin2phi}{alphay \cdot alphay} + \frac{\frac{cos2phi}{alphax}}{alphax}} \]
13. Add Preprocessing

Alternative 11: 93.0% accurate, 4.3× speedup

Math

\[\begin{array}{l} \\ \frac{u0 \cdot \left(u0 \cdot \left(0.5 + u0 \cdot \left(u0 \cdot 0.25 + 0.3333333333333333\right)\right) + 1\right)}{\frac{sin2phi}{alphay \cdot alphay} + \frac{cos2phi}{alphax \cdot alphax}} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (/
  (* u0 (+ (* u0 (+ 0.5 (* u0 (+ (* u0 0.25) 0.3333333333333333)))) 1.0))
  (+ (/ sin2phi (* alphay alphay)) (/ cos2phi (* alphax alphax)))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	return (u0 * ((u0 * (0.5f + (u0 * ((u0 * 0.25f) + 0.3333333333333333f)))) + 1.0f)) / ((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 * (0.5e0 + (u0 * ((u0 * 0.25e0) + 0.3333333333333333e0)))) + 1.0e0)) / ((sin2phi / (alphay * alphay)) + (cos2phi / (alphax * alphax)))
end function

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	return Float32(Float32(u0 * Float32(Float32(u0 * Float32(Float32(0.5) + Float32(u0 * Float32(Float32(u0 * Float32(0.25)) + Float32(0.3333333333333333))))) + Float32(1.0))) / Float32(Float32(sin2phi / Float32(alphay * alphay)) + Float32(cos2phi / Float32(alphax * alphax))))
end

MATLAB

function tmp = code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = (u0 * ((u0 * (single(0.5) + (u0 * ((u0 * single(0.25)) + single(0.3333333333333333))))) + single(1.0))) / ((sin2phi / (alphay * alphay)) + (cos2phi / (alphax * alphax)));
end

Derivation

1. Initial program 60.9%

   \[\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
2. Add Preprocessing

3. Taylor expanded in u0 around 0

   \[\leadsto \mathsf{/.f32}\left(\color{blue}{\left(u0 \cdot \left(1 + u0 \cdot \left(\frac{1}{2} + u0 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right)\right)\right)\right)}, \mathsf{+.f32}\left(\mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right), \mathsf{/.f32}\left(sin2phi, \mathsf{*.f32}\left(alphay, alphay\right)\right)\right)\right) \]
4. Step-by-step derivation

   1. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(u0, \left(1 + u0 \cdot \left(\frac{1}{2} + u0 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right)\right)\right)\right), \mathsf{+.f32}\left(\color{blue}{\mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right)}, \mathsf{/.f32}\left(sin2phi, \mathsf{*.f32}\left(alphay, alphay\right)\right)\right)\right) \]

   2. +-lowering-+.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(1, \left(u0 \cdot \left(\frac{1}{2} + u0 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right)\right)\right)\right)\right), \mathsf{+.f32}\left(\mathsf{/.f32}\left(cos2phi, \color{blue}{\mathsf{*.f32}\left(alphax, alphax\right)}\right), \mathsf{/.f32}\left(sin2phi, \mathsf{*.f32}\left(alphay, alphay\right)\right)\right)\right) \]

   3. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(u0, \left(\frac{1}{2} + u0 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right)\right)\right)\right)\right), \mathsf{+.f32}\left(\mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, \color{blue}{alphax}\right)\right), \mathsf{/.f32}\left(sin2phi, \mathsf{*.f32}\left(alphay, alphay\right)\right)\right)\right) \]

   4. +-lowering-+.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\frac{1}{2}, \left(u0 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right)\right)\right)\right)\right)\right), \mathsf{+.f32}\left(\mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right), \mathsf{/.f32}\left(sin2phi, \mathsf{*.f32}\left(alphay, alphay\right)\right)\right)\right) \]

   5. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\frac{1}{2}, \mathsf{*.f32}\left(u0, \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right)\right)\right)\right)\right)\right), \mathsf{+.f32}\left(\mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right), \mathsf{/.f32}\left(sin2phi, \mathsf{*.f32}\left(alphay, alphay\right)\right)\right)\right) \]

   6. +-lowering-+.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\frac{1}{2}, \mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\frac{1}{3}, \left(\frac{1}{4} \cdot u0\right)\right)\right)\right)\right)\right)\right), \mathsf{+.f32}\left(\mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right), \mathsf{/.f32}\left(sin2phi, \mathsf{*.f32}\left(alphay, alphay\right)\right)\right)\right) \]

   7. *-commutative N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\frac{1}{2}, \mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\frac{1}{3}, \left(u0 \cdot \frac{1}{4}\right)\right)\right)\right)\right)\right)\right), \mathsf{+.f32}\left(\mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right), \mathsf{/.f32}\left(sin2phi, \mathsf{*.f32}\left(alphay, alphay\right)\right)\right)\right) \]

   8. *-lowering-*.f32 93.1%

      \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\frac{1}{2}, \mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\frac{1}{3}, \mathsf{*.f32}\left(u0, \frac{1}{4}\right)\right)\right)\right)\right)\right)\right), \mathsf{+.f32}\left(\mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right), \mathsf{/.f32}\left(sin2phi, \mathsf{*.f32}\left(alphay, alphay\right)\right)\right)\right) \]

5. Simplified 93.1%

   \[\leadsto \frac{\color{blue}{u0 \cdot \left(1 + u0 \cdot \left(0.5 + u0 \cdot \left(0.3333333333333333 + u0 \cdot 0.25\right)\right)\right)}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

6. Final simplification 93.1%

   \[\leadsto \frac{u0 \cdot \left(u0 \cdot \left(0.5 + u0 \cdot \left(u0 \cdot 0.25 + 0.3333333333333333\right)\right) + 1\right)}{\frac{sin2phi}{alphay \cdot alphay} + \frac{cos2phi}{alphax \cdot alphax}} \]
7. Add Preprocessing

Alternative 12: 83.5% accurate, 4.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{sin2phi}{alphay \cdot alphay}\\ \mathbf{if}\;t\_0 \leq 0.30000001192092896:\\ \;\;\;\;\frac{u0}{t\_0 + \frac{\frac{1}{alphax}}{\frac{alphax}{cos2phi}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{alphay \cdot alphay}{sin2phi} \cdot \left(u0 \cdot \left(1 - u0 \cdot \left(-0.5 + u0 \cdot -0.3333333333333333\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (let* ((t_0 (/ sin2phi (* alphay alphay))))
   (if (<= t_0 0.30000001192092896)
     (/ u0 (+ t_0 (/ (/ 1.0 alphax) (/ alphax cos2phi))))
     (*
      (/ (* alphay alphay) sin2phi)
      (* u0 (- 1.0 (* u0 (+ -0.5 (* u0 -0.3333333333333333)))))))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float t_0 = sin2phi / (alphay * alphay);
	float tmp;
	if (t_0 <= 0.30000001192092896f) {
		tmp = u0 / (t_0 + ((1.0f / alphax) / (alphax / cos2phi)));
	} else {
		tmp = ((alphay * alphay) / sin2phi) * (u0 * (1.0f - (u0 * (-0.5f + (u0 * -0.3333333333333333f)))));
	}
	return tmp;
}

Fortran

real(4) function code(alphax, alphay, u0, cos2phi, sin2phi)
    real(4), intent (in) :: alphax
    real(4), intent (in) :: alphay
    real(4), intent (in) :: u0
    real(4), intent (in) :: cos2phi
    real(4), intent (in) :: sin2phi
    real(4) :: t_0
    real(4) :: tmp
    t_0 = sin2phi / (alphay * alphay)
    if (t_0 <= 0.30000001192092896e0) then
        tmp = u0 / (t_0 + ((1.0e0 / alphax) / (alphax / cos2phi)))
    else
        tmp = ((alphay * alphay) / sin2phi) * (u0 * (1.0e0 - (u0 * ((-0.5e0) + (u0 * (-0.3333333333333333e0))))))
    end if
    code = tmp
end function

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	t_0 = Float32(sin2phi / Float32(alphay * alphay))
	tmp = Float32(0.0)
	if (t_0 <= Float32(0.30000001192092896))
		tmp = Float32(u0 / Float32(t_0 + Float32(Float32(Float32(1.0) / alphax) / Float32(alphax / cos2phi))));
	else
		tmp = Float32(Float32(Float32(alphay * alphay) / sin2phi) * Float32(u0 * Float32(Float32(1.0) - Float32(u0 * Float32(Float32(-0.5) + Float32(u0 * Float32(-0.3333333333333333)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alphax, alphay, u0, cos2phi, sin2phi)
	t_0 = sin2phi / (alphay * alphay);
	tmp = single(0.0);
	if (t_0 <= single(0.30000001192092896))
		tmp = u0 / (t_0 + ((single(1.0) / alphax) / (alphax / cos2phi)));
	else
		tmp = ((alphay * alphay) / sin2phi) * (u0 * (single(1.0) - (u0 * (single(-0.5) + (u0 * single(-0.3333333333333333))))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 sin2phi (*.f32 alphay alphay)) < 0.300000012

   1. Initial program 55.9%

      \[\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
   2. Step-by-step derivation

      1. distribute-frac-neg N/A

         \[\leadsto \mathsf{neg}\left(\frac{\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}}\right) \]

      2. distribute-neg-frac2 N/A

         \[\leadsto \frac{\log \left(1 - u0\right)}{\color{blue}{\mathsf{neg}\left(\left(\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\right)\right)}} \]

      3. /-lowering-/.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(\log \left(1 - u0\right), \color{blue}{\left(\mathsf{neg}\left(\left(\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\right)\right)\right)}\right) \]

      4. sub-neg N/A

         \[\leadsto \mathsf{/.f32}\left(\log \left(1 + \left(\mathsf{neg}\left(u0\right)\right)\right), \left(\mathsf{neg}\left(\left(\color{blue}{\frac{cos2phi}{alphax \cdot alphax}} + \frac{sin2phi}{alphay \cdot alphay}\right)\right)\right)\right) \]

      5. log1p-define N/A

         \[\leadsto \mathsf{/.f32}\left(\left(\mathsf{log1p}\left(\mathsf{neg}\left(u0\right)\right)\right), \left(\mathsf{neg}\left(\color{blue}{\left(\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\right)}\right)\right)\right) \]

      6. log1p-lowering-log1p.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\left(\mathsf{neg}\left(u0\right)\right)\right), \left(\mathsf{neg}\left(\color{blue}{\left(\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\right)}\right)\right)\right) \]

      7. neg-lowering-neg.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \left(\mathsf{neg}\left(\left(\color{blue}{\frac{cos2phi}{alphax \cdot alphax}} + \frac{sin2phi}{alphay \cdot alphay}\right)\right)\right)\right) \]

      8. distribute-neg-in N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \left(\left(\mathsf{neg}\left(\frac{cos2phi}{alphax \cdot alphax}\right)\right) + \color{blue}{\left(\mathsf{neg}\left(\frac{sin2phi}{alphay \cdot alphay}\right)\right)}\right)\right) \]

      9. unsub-neg N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \left(\left(\mathsf{neg}\left(\frac{cos2phi}{alphax \cdot alphax}\right)\right) - \color{blue}{\frac{sin2phi}{alphay \cdot alphay}}\right)\right) \]

      10. --lowering--.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\left(\mathsf{neg}\left(\frac{cos2phi}{alphax \cdot alphax}\right)\right), \color{blue}{\left(\frac{sin2phi}{alphay \cdot alphay}\right)}\right)\right) \]

      11. neg-sub0 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\left(0 - \frac{cos2phi}{alphax \cdot alphax}\right), \left(\frac{\color{blue}{sin2phi}}{alphay \cdot alphay}\right)\right)\right) \]

      12. --lowering--.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \left(\frac{cos2phi}{alphax \cdot alphax}\right)\right), \left(\frac{\color{blue}{sin2phi}}{alphay \cdot alphay}\right)\right)\right) \]

      13. /-lowering-/.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(cos2phi, \left(alphax \cdot alphax\right)\right)\right), \left(\frac{sin2phi}{alphay \cdot alphay}\right)\right)\right) \]

      14. *-lowering-*.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right)\right), \left(\frac{sin2phi}{alphay \cdot alphay}\right)\right)\right) \]

      15. /-lowering-/.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right)\right), \mathsf{/.f32}\left(sin2phi, \color{blue}{\left(alphay \cdot alphay\right)}\right)\right)\right) \]

      16. *-lowering-*.f32 98.6%

          \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right)\right), \mathsf{/.f32}\left(sin2phi, \mathsf{*.f32}\left(alphay, \color{blue}{alphay}\right)\right)\right)\right) \]

   3. Simplified 98.6%

      \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(-u0\right)}{\left(0 - \frac{cos2phi}{alphax \cdot alphax}\right) - \frac{sin2phi}{alphay \cdot alphay}}} \]
   4. Add Preprocessing

   5. Taylor expanded in u0 around 0

      \[\leadsto \color{blue}{\frac{u0}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}}} \]
   6. Step-by-step derivation

      1. /-lowering-/.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(u0, \color{blue}{\left(\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}\right)}\right) \]

      2. +-lowering-+.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{+.f32}\left(\left(\frac{cos2phi}{{alphax}^{2}}\right), \color{blue}{\left(\frac{sin2phi}{{alphay}^{2}}\right)}\right)\right) \]

      3. /-lowering-/.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{/.f32}\left(cos2phi, \left({alphax}^{2}\right)\right), \left(\frac{\color{blue}{sin2phi}}{{alphay}^{2}}\right)\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{/.f32}\left(cos2phi, \left(alphax \cdot alphax\right)\right), \left(\frac{sin2phi}{{alphay}^{2}}\right)\right)\right) \]

      5. *-lowering-*.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right), \left(\frac{sin2phi}{{alphay}^{2}}\right)\right)\right) \]

      6. /-lowering-/.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right), \mathsf{/.f32}\left(sin2phi, \color{blue}{\left({alphay}^{2}\right)}\right)\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right), \mathsf{/.f32}\left(sin2phi, \left(alphay \cdot \color{blue}{alphay}\right)\right)\right)\right) \]

      8. *-lowering-*.f32 73.7%

         \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right), \mathsf{/.f32}\left(sin2phi, \mathsf{*.f32}\left(alphay, \color{blue}{alphay}\right)\right)\right)\right) \]

   7. Simplified 73.7%

      \[\leadsto \color{blue}{\frac{u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}}} \]
   8. Step-by-step derivation

      1. associate-/r* N/A

         \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{+.f32}\left(\left(\frac{\frac{cos2phi}{alphax}}{alphax}\right), \mathsf{/.f32}\left(\color{blue}{sin2phi}, \mathsf{*.f32}\left(alphay, alphay\right)\right)\right)\right) \]

      2. div-inv N/A

         \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{+.f32}\left(\left(\frac{cos2phi}{alphax} \cdot \frac{1}{alphax}\right), \mathsf{/.f32}\left(\color{blue}{sin2phi}, \mathsf{*.f32}\left(alphay, alphay\right)\right)\right)\right) \]

      3. clear-num N/A

         \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{+.f32}\left(\left(\frac{1}{\frac{alphax}{cos2phi}} \cdot \frac{1}{alphax}\right), \mathsf{/.f32}\left(sin2phi, \mathsf{*.f32}\left(alphay, alphay\right)\right)\right)\right) \]

      4. associate-*l/ N/A

         \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{+.f32}\left(\left(\frac{1 \cdot \frac{1}{alphax}}{\frac{alphax}{cos2phi}}\right), \mathsf{/.f32}\left(\color{blue}{sin2phi}, \mathsf{*.f32}\left(alphay, alphay\right)\right)\right)\right) \]

      5. div-inv N/A

         \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{+.f32}\left(\left(\frac{\frac{1}{alphax}}{\frac{alphax}{cos2phi}}\right), \mathsf{/.f32}\left(sin2phi, \mathsf{*.f32}\left(alphay, alphay\right)\right)\right)\right) \]

      6. /-lowering-/.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{/.f32}\left(\left(\frac{1}{alphax}\right), \left(\frac{alphax}{cos2phi}\right)\right), \mathsf{/.f32}\left(\color{blue}{sin2phi}, \mathsf{*.f32}\left(alphay, alphay\right)\right)\right)\right) \]

      7. /-lowering-/.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(1, alphax\right), \left(\frac{alphax}{cos2phi}\right)\right), \mathsf{/.f32}\left(sin2phi, \mathsf{*.f32}\left(alphay, alphay\right)\right)\right)\right) \]

      8. /-lowering-/.f32 73.7%

         \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(1, alphax\right), \mathsf{/.f32}\left(alphax, cos2phi\right)\right), \mathsf{/.f32}\left(sin2phi, \mathsf{*.f32}\left(alphay, alphay\right)\right)\right)\right) \]

   9. Applied egg-rr 73.7%

      \[\leadsto \frac{u0}{\color{blue}{\frac{\frac{1}{alphax}}{\frac{alphax}{cos2phi}}} + \frac{sin2phi}{alphay \cdot alphay}} \]

   if 0.300000012 < (/.f32 sin2phi (*.f32 alphay alphay))

   1. Initial program 65.1%

      \[\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
   2. Step-by-step derivation

      1. distribute-frac-neg N/A

         \[\leadsto \mathsf{neg}\left(\frac{\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}}\right) \]

      2. distribute-neg-frac2 N/A

         \[\leadsto \frac{\log \left(1 - u0\right)}{\color{blue}{\mathsf{neg}\left(\left(\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\right)\right)}} \]

      3. /-lowering-/.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(\log \left(1 - u0\right), \color{blue}{\left(\mathsf{neg}\left(\left(\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\right)\right)\right)}\right) \]

      4. sub-neg N/A

         \[\leadsto \mathsf{/.f32}\left(\log \left(1 + \left(\mathsf{neg}\left(u0\right)\right)\right), \left(\mathsf{neg}\left(\left(\color{blue}{\frac{cos2phi}{alphax \cdot alphax}} + \frac{sin2phi}{alphay \cdot alphay}\right)\right)\right)\right) \]

      5. log1p-define N/A

         \[\leadsto \mathsf{/.f32}\left(\left(\mathsf{log1p}\left(\mathsf{neg}\left(u0\right)\right)\right), \left(\mathsf{neg}\left(\color{blue}{\left(\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\right)}\right)\right)\right) \]

      6. log1p-lowering-log1p.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\left(\mathsf{neg}\left(u0\right)\right)\right), \left(\mathsf{neg}\left(\color{blue}{\left(\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\right)}\right)\right)\right) \]

      7. neg-lowering-neg.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \left(\mathsf{neg}\left(\left(\color{blue}{\frac{cos2phi}{alphax \cdot alphax}} + \frac{sin2phi}{alphay \cdot alphay}\right)\right)\right)\right) \]

      8. distribute-neg-in N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \left(\left(\mathsf{neg}\left(\frac{cos2phi}{alphax \cdot alphax}\right)\right) + \color{blue}{\left(\mathsf{neg}\left(\frac{sin2phi}{alphay \cdot alphay}\right)\right)}\right)\right) \]

      9. unsub-neg N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \left(\left(\mathsf{neg}\left(\frac{cos2phi}{alphax \cdot alphax}\right)\right) - \color{blue}{\frac{sin2phi}{alphay \cdot alphay}}\right)\right) \]

      10. --lowering--.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\left(\mathsf{neg}\left(\frac{cos2phi}{alphax \cdot alphax}\right)\right), \color{blue}{\left(\frac{sin2phi}{alphay \cdot alphay}\right)}\right)\right) \]

      11. neg-sub0 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\left(0 - \frac{cos2phi}{alphax \cdot alphax}\right), \left(\frac{\color{blue}{sin2phi}}{alphay \cdot alphay}\right)\right)\right) \]

      12. --lowering--.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \left(\frac{cos2phi}{alphax \cdot alphax}\right)\right), \left(\frac{\color{blue}{sin2phi}}{alphay \cdot alphay}\right)\right)\right) \]

      13. /-lowering-/.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(cos2phi, \left(alphax \cdot alphax\right)\right)\right), \left(\frac{sin2phi}{alphay \cdot alphay}\right)\right)\right) \]

      14. *-lowering-*.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right)\right), \left(\frac{sin2phi}{alphay \cdot alphay}\right)\right)\right) \]

      15. /-lowering-/.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right)\right), \mathsf{/.f32}\left(sin2phi, \color{blue}{\left(alphay \cdot alphay\right)}\right)\right)\right) \]

      16. *-lowering-*.f32 98.2%

          \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right)\right), \mathsf{/.f32}\left(sin2phi, \mathsf{*.f32}\left(alphay, \color{blue}{alphay}\right)\right)\right)\right) \]

   3. Simplified 98.2%

      \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(-u0\right)}{\left(0 - \frac{cos2phi}{alphax \cdot alphax}\right) - \frac{sin2phi}{alphay \cdot alphay}}} \]
   4. Add Preprocessing

   5. Taylor expanded in cos2phi around 0

      \[\leadsto \color{blue}{-1 \cdot \frac{{alphay}^{2} \cdot \log \left(1 - u0\right)}{sin2phi}} \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(\frac{{alphay}^{2} \cdot \log \left(1 - u0\right)}{sin2phi}\right) \]

      2. neg-lowering-neg.f32 N/A

         \[\leadsto \mathsf{neg.f32}\left(\left(\frac{{alphay}^{2} \cdot \log \left(1 - u0\right)}{sin2phi}\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{neg.f32}\left(\left(\frac{\log \left(1 - u0\right) \cdot {alphay}^{2}}{sin2phi}\right)\right) \]

      4. associate-/l* N/A

         \[\leadsto \mathsf{neg.f32}\left(\left(\log \left(1 - u0\right) \cdot \frac{{alphay}^{2}}{sin2phi}\right)\right) \]

      5. *-lowering-*.f32 N/A

         \[\leadsto \mathsf{neg.f32}\left(\mathsf{*.f32}\left(\log \left(1 - u0\right), \left(\frac{{alphay}^{2}}{sin2phi}\right)\right)\right) \]

      6. sub-neg N/A

         \[\leadsto \mathsf{neg.f32}\left(\mathsf{*.f32}\left(\log \left(1 + \left(\mathsf{neg}\left(u0\right)\right)\right), \left(\frac{{alphay}^{2}}{sin2phi}\right)\right)\right) \]

      7. mul-1-neg N/A

         \[\leadsto \mathsf{neg.f32}\left(\mathsf{*.f32}\left(\log \left(1 + -1 \cdot u0\right), \left(\frac{{alphay}^{2}}{sin2phi}\right)\right)\right) \]

      8. log1p-define N/A

         \[\leadsto \mathsf{neg.f32}\left(\mathsf{*.f32}\left(\left(\mathsf{log1p}\left(-1 \cdot u0\right)\right), \left(\frac{{alphay}^{2}}{sin2phi}\right)\right)\right) \]

      9. log1p-lowering-log1p.f32 N/A

         \[\leadsto \mathsf{neg.f32}\left(\mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\left(-1 \cdot u0\right)\right), \left(\frac{{alphay}^{2}}{sin2phi}\right)\right)\right) \]

      10. mul-1-neg N/A

          \[\leadsto \mathsf{neg.f32}\left(\mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\left(\mathsf{neg}\left(u0\right)\right)\right), \left(\frac{{alphay}^{2}}{sin2phi}\right)\right)\right) \]

      11. neg-lowering-neg.f32 N/A

          \[\leadsto \mathsf{neg.f32}\left(\mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \left(\frac{{alphay}^{2}}{sin2phi}\right)\right)\right) \]

      12. /-lowering-/.f32 N/A

          \[\leadsto \mathsf{neg.f32}\left(\mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{/.f32}\left(\left({alphay}^{2}\right), sin2phi\right)\right)\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{neg.f32}\left(\mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{/.f32}\left(\left(alphay \cdot alphay\right), sin2phi\right)\right)\right) \]

      14. *-lowering-*.f32 98.4%

          \[\leadsto \mathsf{neg.f32}\left(\mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{/.f32}\left(\mathsf{*.f32}\left(alphay, alphay\right), sin2phi\right)\right)\right) \]

   7. Simplified 98.4%

      \[\leadsto \color{blue}{-\mathsf{log1p}\left(-u0\right) \cdot \frac{alphay \cdot alphay}{sin2phi}} \]

   8. Taylor expanded in u0 around 0

      \[\leadsto \mathsf{neg.f32}\left(\mathsf{*.f32}\left(\color{blue}{\left(u0 \cdot \left(u0 \cdot \left(\frac{-1}{3} \cdot u0 - \frac{1}{2}\right) - 1\right)\right)}, \mathsf{/.f32}\left(\mathsf{*.f32}\left(alphay, alphay\right), sin2phi\right)\right)\right) \]
   9. Step-by-step derivation

      1. *-lowering-*.f32 N/A

         \[\leadsto \mathsf{neg.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(u0, \left(u0 \cdot \left(\frac{-1}{3} \cdot u0 - \frac{1}{2}\right) - 1\right)\right), \mathsf{/.f32}\left(\mathsf{*.f32}\left(alphay, alphay\right), sin2phi\right)\right)\right) \]

      2. sub-neg N/A

         \[\leadsto \mathsf{neg.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(u0, \left(u0 \cdot \left(\frac{-1}{3} \cdot u0 - \frac{1}{2}\right) + \left(\mathsf{neg}\left(1\right)\right)\right)\right), \mathsf{/.f32}\left(\mathsf{*.f32}\left(alphay, alphay\right), sin2phi\right)\right)\right) \]

      3. metadata-eval N/A

         \[\leadsto \mathsf{neg.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(u0, \left(u0 \cdot \left(\frac{-1}{3} \cdot u0 - \frac{1}{2}\right) + -1\right)\right), \mathsf{/.f32}\left(\mathsf{*.f32}\left(alphay, alphay\right), sin2phi\right)\right)\right) \]

      4. +-lowering-+.f32 N/A

         \[\leadsto \mathsf{neg.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\left(u0 \cdot \left(\frac{-1}{3} \cdot u0 - \frac{1}{2}\right)\right), -1\right)\right), \mathsf{/.f32}\left(\mathsf{*.f32}\left(alphay, alphay\right), sin2phi\right)\right)\right) \]

      5. *-lowering-*.f32 N/A

         \[\leadsto \mathsf{neg.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{*.f32}\left(u0, \left(\frac{-1}{3} \cdot u0 - \frac{1}{2}\right)\right), -1\right)\right), \mathsf{/.f32}\left(\mathsf{*.f32}\left(alphay, alphay\right), sin2phi\right)\right)\right) \]

      6. sub-neg N/A

         \[\leadsto \mathsf{neg.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{*.f32}\left(u0, \left(\frac{-1}{3} \cdot u0 + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right)\right), -1\right)\right), \mathsf{/.f32}\left(\mathsf{*.f32}\left(alphay, alphay\right), sin2phi\right)\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{neg.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{*.f32}\left(u0, \left(\frac{-1}{3} \cdot u0 + \frac{-1}{2}\right)\right), -1\right)\right), \mathsf{/.f32}\left(\mathsf{*.f32}\left(alphay, alphay\right), sin2phi\right)\right)\right) \]

      8. +-lowering-+.f32 N/A

         \[\leadsto \mathsf{neg.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\left(\frac{-1}{3} \cdot u0\right), \frac{-1}{2}\right)\right), -1\right)\right), \mathsf{/.f32}\left(\mathsf{*.f32}\left(alphay, alphay\right), sin2phi\right)\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{neg.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\left(u0 \cdot \frac{-1}{3}\right), \frac{-1}{2}\right)\right), -1\right)\right), \mathsf{/.f32}\left(\mathsf{*.f32}\left(alphay, alphay\right), sin2phi\right)\right)\right) \]

      10. *-lowering-*.f32 90.8%

          \[\leadsto \mathsf{neg.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{*.f32}\left(u0, \frac{-1}{3}\right), \frac{-1}{2}\right)\right), -1\right)\right), \mathsf{/.f32}\left(\mathsf{*.f32}\left(alphay, alphay\right), sin2phi\right)\right)\right) \]

   10. Simplified 90.8%

       \[\leadsto -\color{blue}{\left(u0 \cdot \left(u0 \cdot \left(u0 \cdot -0.3333333333333333 + -0.5\right) + -1\right)\right)} \cdot \frac{alphay \cdot alphay}{sin2phi} \]
3. Recombined 2 regimes into one program.

4. Final simplification 83.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 0.30000001192092896:\\ \;\;\;\;\frac{u0}{\frac{sin2phi}{alphay \cdot alphay} + \frac{\frac{1}{alphax}}{\frac{alphax}{cos2phi}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{alphay \cdot alphay}{sin2phi} \cdot \left(u0 \cdot \left(1 - u0 \cdot \left(-0.5 + u0 \cdot -0.3333333333333333\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 81.7% accurate, 4.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{sin2phi}{alphay \cdot alphay}\\ \mathbf{if}\;t\_0 \leq 0.30000001192092896:\\ \;\;\;\;\frac{u0}{t\_0 + \frac{\frac{1}{alphax}}{\frac{alphax}{cos2phi}}}\\ \mathbf{else}:\\ \;\;\;\;u0 \cdot \frac{alphay \cdot alphay + u0 \cdot \left(\left(alphay \cdot alphay\right) \cdot 0.5\right)}{sin2phi}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (let* ((t_0 (/ sin2phi (* alphay alphay))))
   (if (<= t_0 0.30000001192092896)
     (/ u0 (+ t_0 (/ (/ 1.0 alphax) (/ alphax cos2phi))))
     (*
      u0
      (/ (+ (* alphay alphay) (* u0 (* (* alphay alphay) 0.5))) 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 <= 0.30000001192092896f) {
		tmp = u0 / (t_0 + ((1.0f / alphax) / (alphax / cos2phi)));
	} else {
		tmp = u0 * (((alphay * alphay) + (u0 * ((alphay * alphay) * 0.5f))) / 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 <= 0.30000001192092896e0) then
        tmp = u0 / (t_0 + ((1.0e0 / alphax) / (alphax / cos2phi)))
    else
        tmp = u0 * (((alphay * alphay) + (u0 * ((alphay * alphay) * 0.5e0))) / 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(0.30000001192092896))
		tmp = Float32(u0 / Float32(t_0 + Float32(Float32(Float32(1.0) / alphax) / Float32(alphax / cos2phi))));
	else
		tmp = Float32(u0 * Float32(Float32(Float32(alphay * alphay) + Float32(u0 * Float32(Float32(alphay * alphay) * Float32(0.5)))) / 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(0.30000001192092896))
		tmp = u0 / (t_0 + ((single(1.0) / alphax) / (alphax / cos2phi)));
	else
		tmp = u0 * (((alphay * alphay) + (u0 * ((alphay * alphay) * single(0.5)))) / sin2phi);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 sin2phi (*.f32 alphay alphay)) < 0.300000012

   1. Initial program 55.9%

      \[\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
   2. Step-by-step derivation

      1. distribute-frac-neg N/A

         \[\leadsto \mathsf{neg}\left(\frac{\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}}\right) \]

      2. distribute-neg-frac2 N/A

         \[\leadsto \frac{\log \left(1 - u0\right)}{\color{blue}{\mathsf{neg}\left(\left(\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\right)\right)}} \]

      3. /-lowering-/.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(\log \left(1 - u0\right), \color{blue}{\left(\mathsf{neg}\left(\left(\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\right)\right)\right)}\right) \]

      4. sub-neg N/A

         \[\leadsto \mathsf{/.f32}\left(\log \left(1 + \left(\mathsf{neg}\left(u0\right)\right)\right), \left(\mathsf{neg}\left(\left(\color{blue}{\frac{cos2phi}{alphax \cdot alphax}} + \frac{sin2phi}{alphay \cdot alphay}\right)\right)\right)\right) \]

      5. log1p-define N/A

         \[\leadsto \mathsf{/.f32}\left(\left(\mathsf{log1p}\left(\mathsf{neg}\left(u0\right)\right)\right), \left(\mathsf{neg}\left(\color{blue}{\left(\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\right)}\right)\right)\right) \]

      6. log1p-lowering-log1p.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\left(\mathsf{neg}\left(u0\right)\right)\right), \left(\mathsf{neg}\left(\color{blue}{\left(\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\right)}\right)\right)\right) \]

      7. neg-lowering-neg.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \left(\mathsf{neg}\left(\left(\color{blue}{\frac{cos2phi}{alphax \cdot alphax}} + \frac{sin2phi}{alphay \cdot alphay}\right)\right)\right)\right) \]

      8. distribute-neg-in N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \left(\left(\mathsf{neg}\left(\frac{cos2phi}{alphax \cdot alphax}\right)\right) + \color{blue}{\left(\mathsf{neg}\left(\frac{sin2phi}{alphay \cdot alphay}\right)\right)}\right)\right) \]

      9. unsub-neg N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \left(\left(\mathsf{neg}\left(\frac{cos2phi}{alphax \cdot alphax}\right)\right) - \color{blue}{\frac{sin2phi}{alphay \cdot alphay}}\right)\right) \]

      10. --lowering--.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\left(\mathsf{neg}\left(\frac{cos2phi}{alphax \cdot alphax}\right)\right), \color{blue}{\left(\frac{sin2phi}{alphay \cdot alphay}\right)}\right)\right) \]

      11. neg-sub0 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\left(0 - \frac{cos2phi}{alphax \cdot alphax}\right), \left(\frac{\color{blue}{sin2phi}}{alphay \cdot alphay}\right)\right)\right) \]

      12. --lowering--.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \left(\frac{cos2phi}{alphax \cdot alphax}\right)\right), \left(\frac{\color{blue}{sin2phi}}{alphay \cdot alphay}\right)\right)\right) \]

      13. /-lowering-/.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(cos2phi, \left(alphax \cdot alphax\right)\right)\right), \left(\frac{sin2phi}{alphay \cdot alphay}\right)\right)\right) \]

      14. *-lowering-*.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right)\right), \left(\frac{sin2phi}{alphay \cdot alphay}\right)\right)\right) \]

      15. /-lowering-/.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right)\right), \mathsf{/.f32}\left(sin2phi, \color{blue}{\left(alphay \cdot alphay\right)}\right)\right)\right) \]

      16. *-lowering-*.f32 98.6%

          \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right)\right), \mathsf{/.f32}\left(sin2phi, \mathsf{*.f32}\left(alphay, \color{blue}{alphay}\right)\right)\right)\right) \]

   3. Simplified 98.6%

      \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(-u0\right)}{\left(0 - \frac{cos2phi}{alphax \cdot alphax}\right) - \frac{sin2phi}{alphay \cdot alphay}}} \]
   4. Add Preprocessing

   5. Taylor expanded in u0 around 0

      \[\leadsto \color{blue}{\frac{u0}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}}} \]
   6. Step-by-step derivation

      1. /-lowering-/.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(u0, \color{blue}{\left(\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}\right)}\right) \]

      2. +-lowering-+.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{+.f32}\left(\left(\frac{cos2phi}{{alphax}^{2}}\right), \color{blue}{\left(\frac{sin2phi}{{alphay}^{2}}\right)}\right)\right) \]

      3. /-lowering-/.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{/.f32}\left(cos2phi, \left({alphax}^{2}\right)\right), \left(\frac{\color{blue}{sin2phi}}{{alphay}^{2}}\right)\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{/.f32}\left(cos2phi, \left(alphax \cdot alphax\right)\right), \left(\frac{sin2phi}{{alphay}^{2}}\right)\right)\right) \]

      5. *-lowering-*.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right), \left(\frac{sin2phi}{{alphay}^{2}}\right)\right)\right) \]

      6. /-lowering-/.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right), \mathsf{/.f32}\left(sin2phi, \color{blue}{\left({alphay}^{2}\right)}\right)\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right), \mathsf{/.f32}\left(sin2phi, \left(alphay \cdot \color{blue}{alphay}\right)\right)\right)\right) \]

      8. *-lowering-*.f32 73.7%

         \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right), \mathsf{/.f32}\left(sin2phi, \mathsf{*.f32}\left(alphay, \color{blue}{alphay}\right)\right)\right)\right) \]

   7. Simplified 73.7%

      \[\leadsto \color{blue}{\frac{u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}}} \]
   8. Step-by-step derivation

      1. associate-/r* N/A

         \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{+.f32}\left(\left(\frac{\frac{cos2phi}{alphax}}{alphax}\right), \mathsf{/.f32}\left(\color{blue}{sin2phi}, \mathsf{*.f32}\left(alphay, alphay\right)\right)\right)\right) \]

      2. div-inv N/A

         \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{+.f32}\left(\left(\frac{cos2phi}{alphax} \cdot \frac{1}{alphax}\right), \mathsf{/.f32}\left(\color{blue}{sin2phi}, \mathsf{*.f32}\left(alphay, alphay\right)\right)\right)\right) \]

      3. clear-num N/A

         \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{+.f32}\left(\left(\frac{1}{\frac{alphax}{cos2phi}} \cdot \frac{1}{alphax}\right), \mathsf{/.f32}\left(sin2phi, \mathsf{*.f32}\left(alphay, alphay\right)\right)\right)\right) \]

      4. associate-*l/ N/A

         \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{+.f32}\left(\left(\frac{1 \cdot \frac{1}{alphax}}{\frac{alphax}{cos2phi}}\right), \mathsf{/.f32}\left(\color{blue}{sin2phi}, \mathsf{*.f32}\left(alphay, alphay\right)\right)\right)\right) \]

      5. div-inv N/A

         \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{+.f32}\left(\left(\frac{\frac{1}{alphax}}{\frac{alphax}{cos2phi}}\right), \mathsf{/.f32}\left(sin2phi, \mathsf{*.f32}\left(alphay, alphay\right)\right)\right)\right) \]

      6. /-lowering-/.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{/.f32}\left(\left(\frac{1}{alphax}\right), \left(\frac{alphax}{cos2phi}\right)\right), \mathsf{/.f32}\left(\color{blue}{sin2phi}, \mathsf{*.f32}\left(alphay, alphay\right)\right)\right)\right) \]

      7. /-lowering-/.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(1, alphax\right), \left(\frac{alphax}{cos2phi}\right)\right), \mathsf{/.f32}\left(sin2phi, \mathsf{*.f32}\left(alphay, alphay\right)\right)\right)\right) \]

      8. /-lowering-/.f32 73.7%

         \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(1, alphax\right), \mathsf{/.f32}\left(alphax, cos2phi\right)\right), \mathsf{/.f32}\left(sin2phi, \mathsf{*.f32}\left(alphay, alphay\right)\right)\right)\right) \]

   9. Applied egg-rr 73.7%

      \[\leadsto \frac{u0}{\color{blue}{\frac{\frac{1}{alphax}}{\frac{alphax}{cos2phi}}} + \frac{sin2phi}{alphay \cdot alphay}} \]

   if 0.300000012 < (/.f32 sin2phi (*.f32 alphay alphay))

   1. Initial program 65.1%

      \[\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
   2. Step-by-step derivation

      1. distribute-frac-neg N/A

         \[\leadsto \mathsf{neg}\left(\frac{\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}}\right) \]

      2. distribute-neg-frac2 N/A

         \[\leadsto \frac{\log \left(1 - u0\right)}{\color{blue}{\mathsf{neg}\left(\left(\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\right)\right)}} \]

      3. /-lowering-/.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(\log \left(1 - u0\right), \color{blue}{\left(\mathsf{neg}\left(\left(\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\right)\right)\right)}\right) \]

      4. sub-neg N/A

         \[\leadsto \mathsf{/.f32}\left(\log \left(1 + \left(\mathsf{neg}\left(u0\right)\right)\right), \left(\mathsf{neg}\left(\left(\color{blue}{\frac{cos2phi}{alphax \cdot alphax}} + \frac{sin2phi}{alphay \cdot alphay}\right)\right)\right)\right) \]

      5. log1p-define N/A

         \[\leadsto \mathsf{/.f32}\left(\left(\mathsf{log1p}\left(\mathsf{neg}\left(u0\right)\right)\right), \left(\mathsf{neg}\left(\color{blue}{\left(\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\right)}\right)\right)\right) \]

      6. log1p-lowering-log1p.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\left(\mathsf{neg}\left(u0\right)\right)\right), \left(\mathsf{neg}\left(\color{blue}{\left(\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\right)}\right)\right)\right) \]

      7. neg-lowering-neg.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \left(\mathsf{neg}\left(\left(\color{blue}{\frac{cos2phi}{alphax \cdot alphax}} + \frac{sin2phi}{alphay \cdot alphay}\right)\right)\right)\right) \]

      8. distribute-neg-in N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \left(\left(\mathsf{neg}\left(\frac{cos2phi}{alphax \cdot alphax}\right)\right) + \color{blue}{\left(\mathsf{neg}\left(\frac{sin2phi}{alphay \cdot alphay}\right)\right)}\right)\right) \]

      9. unsub-neg N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \left(\left(\mathsf{neg}\left(\frac{cos2phi}{alphax \cdot alphax}\right)\right) - \color{blue}{\frac{sin2phi}{alphay \cdot alphay}}\right)\right) \]

      10. --lowering--.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\left(\mathsf{neg}\left(\frac{cos2phi}{alphax \cdot alphax}\right)\right), \color{blue}{\left(\frac{sin2phi}{alphay \cdot alphay}\right)}\right)\right) \]

      11. neg-sub0 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\left(0 - \frac{cos2phi}{alphax \cdot alphax}\right), \left(\frac{\color{blue}{sin2phi}}{alphay \cdot alphay}\right)\right)\right) \]

      12. --lowering--.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \left(\frac{cos2phi}{alphax \cdot alphax}\right)\right), \left(\frac{\color{blue}{sin2phi}}{alphay \cdot alphay}\right)\right)\right) \]

      13. /-lowering-/.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(cos2phi, \left(alphax \cdot alphax\right)\right)\right), \left(\frac{sin2phi}{alphay \cdot alphay}\right)\right)\right) \]

      14. *-lowering-*.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right)\right), \left(\frac{sin2phi}{alphay \cdot alphay}\right)\right)\right) \]

      15. /-lowering-/.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right)\right), \mathsf{/.f32}\left(sin2phi, \color{blue}{\left(alphay \cdot alphay\right)}\right)\right)\right) \]

      16. *-lowering-*.f32 98.2%

          \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right)\right), \mathsf{/.f32}\left(sin2phi, \mathsf{*.f32}\left(alphay, \color{blue}{alphay}\right)\right)\right)\right) \]

   3. Simplified 98.2%

      \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(-u0\right)}{\left(0 - \frac{cos2phi}{alphax \cdot alphax}\right) - \frac{sin2phi}{alphay \cdot alphay}}} \]
   4. Add Preprocessing

   5. Taylor expanded in u0 around 0

      \[\leadsto \color{blue}{u0 \cdot \left(u0 \cdot \left(u0 \cdot \left(\frac{1}{4} \cdot \frac{u0}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}} + \frac{1}{3} \cdot \frac{1}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}}\right) + \frac{1}{2} \cdot \frac{1}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}}\right) + \frac{1}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}}\right)} \]
   6. Step-by-step derivation

      1. *-lowering-*.f32 N/A

         \[\leadsto \mathsf{*.f32}\left(u0, \color{blue}{\left(u0 \cdot \left(u0 \cdot \left(\frac{1}{4} \cdot \frac{u0}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}} + \frac{1}{3} \cdot \frac{1}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}}\right) + \frac{1}{2} \cdot \frac{1}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}}\right) + \frac{1}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}}\right)}\right) \]

      2. +-commutative N/A

         \[\leadsto \mathsf{*.f32}\left(u0, \left(\frac{1}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}} + \color{blue}{u0 \cdot \left(u0 \cdot \left(\frac{1}{4} \cdot \frac{u0}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}} + \frac{1}{3} \cdot \frac{1}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}}\right) + \frac{1}{2} \cdot \frac{1}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}}\right)}\right)\right) \]

      3. +-lowering-+.f32 N/A

         \[\leadsto \mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\left(\frac{1}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}}\right), \color{blue}{\left(u0 \cdot \left(u0 \cdot \left(\frac{1}{4} \cdot \frac{u0}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}} + \frac{1}{3} \cdot \frac{1}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}}\right) + \frac{1}{2} \cdot \frac{1}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}}\right)\right)}\right)\right) \]

   7. Simplified 92.7%

      \[\leadsto \color{blue}{u0 \cdot \left(\frac{1}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} + u0 \cdot \left(\frac{0.5}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} + u0 \cdot \left(\frac{u0 \cdot 0.25}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} + \frac{0.3333333333333333}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}}\right)\right)\right)} \]

   8. Taylor expanded in sin2phi around inf

      \[\leadsto \color{blue}{\frac{u0 \cdot \left(u0 \cdot \left(\frac{1}{2} \cdot {alphay}^{2} + u0 \cdot \left(\frac{1}{4} \cdot \left({alphay}^{2} \cdot u0\right) + \frac{1}{3} \cdot {alphay}^{2}\right)\right) + {alphay}^{2}\right)}{sin2phi}} \]
   9. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto u0 \cdot \color{blue}{\frac{u0 \cdot \left(\frac{1}{2} \cdot {alphay}^{2} + u0 \cdot \left(\frac{1}{4} \cdot \left({alphay}^{2} \cdot u0\right) + \frac{1}{3} \cdot {alphay}^{2}\right)\right) + {alphay}^{2}}{sin2phi}} \]

      2. *-lowering-*.f32 N/A

         \[\leadsto \mathsf{*.f32}\left(u0, \color{blue}{\left(\frac{u0 \cdot \left(\frac{1}{2} \cdot {alphay}^{2} + u0 \cdot \left(\frac{1}{4} \cdot \left({alphay}^{2} \cdot u0\right) + \frac{1}{3} \cdot {alphay}^{2}\right)\right) + {alphay}^{2}}{sin2phi}\right)}\right) \]

      3. /-lowering-/.f32 N/A

         \[\leadsto \mathsf{*.f32}\left(u0, \mathsf{/.f32}\left(\left(u0 \cdot \left(\frac{1}{2} \cdot {alphay}^{2} + u0 \cdot \left(\frac{1}{4} \cdot \left({alphay}^{2} \cdot u0\right) + \frac{1}{3} \cdot {alphay}^{2}\right)\right) + {alphay}^{2}\right), \color{blue}{sin2phi}\right)\right) \]

   10. Simplified 93.1%

       \[\leadsto \color{blue}{u0 \cdot \frac{alphay \cdot alphay + u0 \cdot \left(u0 \cdot \left(\left(u0 \cdot \left(alphay \cdot alphay\right)\right) \cdot 0.25 + \left(alphay \cdot alphay\right) \cdot 0.3333333333333333\right) + \left(alphay \cdot alphay\right) \cdot 0.5\right)}{sin2phi}} \]

   11. Taylor expanded in u0 around 0

       \[\leadsto \mathsf{*.f32}\left(u0, \mathsf{/.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(alphay, alphay\right), \mathsf{*.f32}\left(u0, \color{blue}{\left(\frac{1}{2} \cdot {alphay}^{2}\right)}\right)\right), sin2phi\right)\right) \]
   12. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \mathsf{*.f32}\left(u0, \mathsf{/.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(alphay, alphay\right), \mathsf{*.f32}\left(u0, \left({alphay}^{2} \cdot \frac{1}{2}\right)\right)\right), sin2phi\right)\right) \]

       2. *-lowering-*.f32 N/A

          \[\leadsto \mathsf{*.f32}\left(u0, \mathsf{/.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(alphay, alphay\right), \mathsf{*.f32}\left(u0, \mathsf{*.f32}\left(\left({alphay}^{2}\right), \frac{1}{2}\right)\right)\right), sin2phi\right)\right) \]

       3. unpow2 N/A

          \[\leadsto \mathsf{*.f32}\left(u0, \mathsf{/.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(alphay, alphay\right), \mathsf{*.f32}\left(u0, \mathsf{*.f32}\left(\left(alphay \cdot alphay\right), \frac{1}{2}\right)\right)\right), sin2phi\right)\right) \]

       4. *-lowering-*.f32 87.6%

          \[\leadsto \mathsf{*.f32}\left(u0, \mathsf{/.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(alphay, alphay\right), \mathsf{*.f32}\left(u0, \mathsf{*.f32}\left(\mathsf{*.f32}\left(alphay, alphay\right), \frac{1}{2}\right)\right)\right), sin2phi\right)\right) \]

   13. Simplified 87.6%

       \[\leadsto u0 \cdot \frac{alphay \cdot alphay + u0 \cdot \color{blue}{\left(\left(alphay \cdot alphay\right) \cdot 0.5\right)}}{sin2phi} \]
3. Recombined 2 regimes into one program.

4. Final simplification 81.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 0.30000001192092896:\\ \;\;\;\;\frac{u0}{\frac{sin2phi}{alphay \cdot alphay} + \frac{\frac{1}{alphax}}{\frac{alphax}{cos2phi}}}\\ \mathbf{else}:\\ \;\;\;\;u0 \cdot \frac{alphay \cdot alphay + u0 \cdot \left(\left(alphay \cdot alphay\right) \cdot 0.5\right)}{sin2phi}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 81.7% accurate, 4.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 0.30000001192092896:\\ \;\;\;\;\frac{u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{\frac{sin2phi}{alphay}}{alphay}}\\ \mathbf{else}:\\ \;\;\;\;u0 \cdot \frac{alphay \cdot alphay + u0 \cdot \left(\left(alphay \cdot alphay\right) \cdot 0.5\right)}{sin2phi}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (if (<= (/ sin2phi (* alphay alphay)) 0.30000001192092896)
   (/ u0 (+ (/ cos2phi (* alphax alphax)) (/ (/ sin2phi alphay) alphay)))
   (* u0 (/ (+ (* alphay alphay) (* u0 (* (* alphay alphay) 0.5))) sin2phi))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float tmp;
	if ((sin2phi / (alphay * alphay)) <= 0.30000001192092896f) {
		tmp = u0 / ((cos2phi / (alphax * alphax)) + ((sin2phi / alphay) / alphay));
	} else {
		tmp = u0 * (((alphay * alphay) + (u0 * ((alphay * alphay) * 0.5f))) / 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)) <= 0.30000001192092896e0) then
        tmp = u0 / ((cos2phi / (alphax * alphax)) + ((sin2phi / alphay) / alphay))
    else
        tmp = u0 * (((alphay * alphay) + (u0 * ((alphay * alphay) * 0.5e0))) / 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(0.30000001192092896))
		tmp = Float32(u0 / Float32(Float32(cos2phi / Float32(alphax * alphax)) + Float32(Float32(sin2phi / alphay) / alphay)));
	else
		tmp = Float32(u0 * Float32(Float32(Float32(alphay * alphay) + Float32(u0 * Float32(Float32(alphay * alphay) * Float32(0.5)))) / sin2phi));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = single(0.0);
	if ((sin2phi / (alphay * alphay)) <= single(0.30000001192092896))
		tmp = u0 / ((cos2phi / (alphax * alphax)) + ((sin2phi / alphay) / alphay));
	else
		tmp = u0 * (((alphay * alphay) + (u0 * ((alphay * alphay) * single(0.5)))) / sin2phi);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 sin2phi (*.f32 alphay alphay)) < 0.300000012

   1. Initial program 55.9%

      \[\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
   2. Step-by-step derivation

      1. distribute-frac-neg N/A

         \[\leadsto \mathsf{neg}\left(\frac{\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}}\right) \]

      2. distribute-neg-frac2 N/A

         \[\leadsto \frac{\log \left(1 - u0\right)}{\color{blue}{\mathsf{neg}\left(\left(\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\right)\right)}} \]

      3. /-lowering-/.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(\log \left(1 - u0\right), \color{blue}{\left(\mathsf{neg}\left(\left(\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\right)\right)\right)}\right) \]

      4. sub-neg N/A

         \[\leadsto \mathsf{/.f32}\left(\log \left(1 + \left(\mathsf{neg}\left(u0\right)\right)\right), \left(\mathsf{neg}\left(\left(\color{blue}{\frac{cos2phi}{alphax \cdot alphax}} + \frac{sin2phi}{alphay \cdot alphay}\right)\right)\right)\right) \]

      5. log1p-define N/A

         \[\leadsto \mathsf{/.f32}\left(\left(\mathsf{log1p}\left(\mathsf{neg}\left(u0\right)\right)\right), \left(\mathsf{neg}\left(\color{blue}{\left(\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\right)}\right)\right)\right) \]

      6. log1p-lowering-log1p.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\left(\mathsf{neg}\left(u0\right)\right)\right), \left(\mathsf{neg}\left(\color{blue}{\left(\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\right)}\right)\right)\right) \]

      7. neg-lowering-neg.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \left(\mathsf{neg}\left(\left(\color{blue}{\frac{cos2phi}{alphax \cdot alphax}} + \frac{sin2phi}{alphay \cdot alphay}\right)\right)\right)\right) \]

      8. distribute-neg-in N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \left(\left(\mathsf{neg}\left(\frac{cos2phi}{alphax \cdot alphax}\right)\right) + \color{blue}{\left(\mathsf{neg}\left(\frac{sin2phi}{alphay \cdot alphay}\right)\right)}\right)\right) \]

      9. unsub-neg N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \left(\left(\mathsf{neg}\left(\frac{cos2phi}{alphax \cdot alphax}\right)\right) - \color{blue}{\frac{sin2phi}{alphay \cdot alphay}}\right)\right) \]

      10. --lowering--.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\left(\mathsf{neg}\left(\frac{cos2phi}{alphax \cdot alphax}\right)\right), \color{blue}{\left(\frac{sin2phi}{alphay \cdot alphay}\right)}\right)\right) \]

      11. neg-sub0 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\left(0 - \frac{cos2phi}{alphax \cdot alphax}\right), \left(\frac{\color{blue}{sin2phi}}{alphay \cdot alphay}\right)\right)\right) \]

      12. --lowering--.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \left(\frac{cos2phi}{alphax \cdot alphax}\right)\right), \left(\frac{\color{blue}{sin2phi}}{alphay \cdot alphay}\right)\right)\right) \]

      13. /-lowering-/.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(cos2phi, \left(alphax \cdot alphax\right)\right)\right), \left(\frac{sin2phi}{alphay \cdot alphay}\right)\right)\right) \]

      14. *-lowering-*.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right)\right), \left(\frac{sin2phi}{alphay \cdot alphay}\right)\right)\right) \]

      15. /-lowering-/.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right)\right), \mathsf{/.f32}\left(sin2phi, \color{blue}{\left(alphay \cdot alphay\right)}\right)\right)\right) \]

      16. *-lowering-*.f32 98.6%

          \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right)\right), \mathsf{/.f32}\left(sin2phi, \mathsf{*.f32}\left(alphay, \color{blue}{alphay}\right)\right)\right)\right) \]

   3. Simplified 98.6%

      \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(-u0\right)}{\left(0 - \frac{cos2phi}{alphax \cdot alphax}\right) - \frac{sin2phi}{alphay \cdot alphay}}} \]
   4. Add Preprocessing

   5. Taylor expanded in u0 around 0

      \[\leadsto \color{blue}{\frac{u0}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}}} \]
   6. Step-by-step derivation

      1. /-lowering-/.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(u0, \color{blue}{\left(\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}\right)}\right) \]

      2. +-lowering-+.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{+.f32}\left(\left(\frac{cos2phi}{{alphax}^{2}}\right), \color{blue}{\left(\frac{sin2phi}{{alphay}^{2}}\right)}\right)\right) \]

      3. /-lowering-/.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{/.f32}\left(cos2phi, \left({alphax}^{2}\right)\right), \left(\frac{\color{blue}{sin2phi}}{{alphay}^{2}}\right)\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{/.f32}\left(cos2phi, \left(alphax \cdot alphax\right)\right), \left(\frac{sin2phi}{{alphay}^{2}}\right)\right)\right) \]

      5. *-lowering-*.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right), \left(\frac{sin2phi}{{alphay}^{2}}\right)\right)\right) \]

      6. /-lowering-/.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right), \mathsf{/.f32}\left(sin2phi, \color{blue}{\left({alphay}^{2}\right)}\right)\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right), \mathsf{/.f32}\left(sin2phi, \left(alphay \cdot \color{blue}{alphay}\right)\right)\right)\right) \]

      8. *-lowering-*.f32 73.7%

         \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right), \mathsf{/.f32}\left(sin2phi, \mathsf{*.f32}\left(alphay, \color{blue}{alphay}\right)\right)\right)\right) \]

   7. Simplified 73.7%

      \[\leadsto \color{blue}{\frac{u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}}} \]
   8. Step-by-step derivation

      1. associate-/r* N/A

         \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right), \left(\frac{\frac{sin2phi}{alphay}}{\color{blue}{alphay}}\right)\right)\right) \]

      2. /-lowering-/.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right), \mathsf{/.f32}\left(\left(\frac{sin2phi}{alphay}\right), \color{blue}{alphay}\right)\right)\right) \]

      3. /-lowering-/.f32 73.7%

         \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(sin2phi, alphay\right), alphay\right)\right)\right) \]

   9. Applied egg-rr 73.7%

      \[\leadsto \frac{u0}{\frac{cos2phi}{alphax \cdot alphax} + \color{blue}{\frac{\frac{sin2phi}{alphay}}{alphay}}} \]

   if 0.300000012 < (/.f32 sin2phi (*.f32 alphay alphay))

   1. Initial program 65.1%

      \[\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
   2. Step-by-step derivation

      1. distribute-frac-neg N/A

         \[\leadsto \mathsf{neg}\left(\frac{\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}}\right) \]

      2. distribute-neg-frac2 N/A

         \[\leadsto \frac{\log \left(1 - u0\right)}{\color{blue}{\mathsf{neg}\left(\left(\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\right)\right)}} \]

      3. /-lowering-/.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(\log \left(1 - u0\right), \color{blue}{\left(\mathsf{neg}\left(\left(\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\right)\right)\right)}\right) \]

      4. sub-neg N/A

         \[\leadsto \mathsf{/.f32}\left(\log \left(1 + \left(\mathsf{neg}\left(u0\right)\right)\right), \left(\mathsf{neg}\left(\left(\color{blue}{\frac{cos2phi}{alphax \cdot alphax}} + \frac{sin2phi}{alphay \cdot alphay}\right)\right)\right)\right) \]

      5. log1p-define N/A

         \[\leadsto \mathsf{/.f32}\left(\left(\mathsf{log1p}\left(\mathsf{neg}\left(u0\right)\right)\right), \left(\mathsf{neg}\left(\color{blue}{\left(\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\right)}\right)\right)\right) \]

      6. log1p-lowering-log1p.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\left(\mathsf{neg}\left(u0\right)\right)\right), \left(\mathsf{neg}\left(\color{blue}{\left(\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\right)}\right)\right)\right) \]

      7. neg-lowering-neg.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \left(\mathsf{neg}\left(\left(\color{blue}{\frac{cos2phi}{alphax \cdot alphax}} + \frac{sin2phi}{alphay \cdot alphay}\right)\right)\right)\right) \]

      8. distribute-neg-in N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \left(\left(\mathsf{neg}\left(\frac{cos2phi}{alphax \cdot alphax}\right)\right) + \color{blue}{\left(\mathsf{neg}\left(\frac{sin2phi}{alphay \cdot alphay}\right)\right)}\right)\right) \]

      9. unsub-neg N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \left(\left(\mathsf{neg}\left(\frac{cos2phi}{alphax \cdot alphax}\right)\right) - \color{blue}{\frac{sin2phi}{alphay \cdot alphay}}\right)\right) \]

      10. --lowering--.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\left(\mathsf{neg}\left(\frac{cos2phi}{alphax \cdot alphax}\right)\right), \color{blue}{\left(\frac{sin2phi}{alphay \cdot alphay}\right)}\right)\right) \]

      11. neg-sub0 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\left(0 - \frac{cos2phi}{alphax \cdot alphax}\right), \left(\frac{\color{blue}{sin2phi}}{alphay \cdot alphay}\right)\right)\right) \]

      12. --lowering--.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \left(\frac{cos2phi}{alphax \cdot alphax}\right)\right), \left(\frac{\color{blue}{sin2phi}}{alphay \cdot alphay}\right)\right)\right) \]

      13. /-lowering-/.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(cos2phi, \left(alphax \cdot alphax\right)\right)\right), \left(\frac{sin2phi}{alphay \cdot alphay}\right)\right)\right) \]

      14. *-lowering-*.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right)\right), \left(\frac{sin2phi}{alphay \cdot alphay}\right)\right)\right) \]

      15. /-lowering-/.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right)\right), \mathsf{/.f32}\left(sin2phi, \color{blue}{\left(alphay \cdot alphay\right)}\right)\right)\right) \]

      16. *-lowering-*.f32 98.2%

          \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right)\right), \mathsf{/.f32}\left(sin2phi, \mathsf{*.f32}\left(alphay, \color{blue}{alphay}\right)\right)\right)\right) \]

   3. Simplified 98.2%

      \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(-u0\right)}{\left(0 - \frac{cos2phi}{alphax \cdot alphax}\right) - \frac{sin2phi}{alphay \cdot alphay}}} \]
   4. Add Preprocessing

   5. Taylor expanded in u0 around 0

      \[\leadsto \color{blue}{u0 \cdot \left(u0 \cdot \left(u0 \cdot \left(\frac{1}{4} \cdot \frac{u0}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}} + \frac{1}{3} \cdot \frac{1}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}}\right) + \frac{1}{2} \cdot \frac{1}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}}\right) + \frac{1}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}}\right)} \]
   6. Step-by-step derivation

      1. *-lowering-*.f32 N/A

         \[\leadsto \mathsf{*.f32}\left(u0, \color{blue}{\left(u0 \cdot \left(u0 \cdot \left(\frac{1}{4} \cdot \frac{u0}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}} + \frac{1}{3} \cdot \frac{1}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}}\right) + \frac{1}{2} \cdot \frac{1}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}}\right) + \frac{1}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}}\right)}\right) \]

      2. +-commutative N/A

         \[\leadsto \mathsf{*.f32}\left(u0, \left(\frac{1}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}} + \color{blue}{u0 \cdot \left(u0 \cdot \left(\frac{1}{4} \cdot \frac{u0}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}} + \frac{1}{3} \cdot \frac{1}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}}\right) + \frac{1}{2} \cdot \frac{1}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}}\right)}\right)\right) \]

      3. +-lowering-+.f32 N/A

         \[\leadsto \mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\left(\frac{1}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}}\right), \color{blue}{\left(u0 \cdot \left(u0 \cdot \left(\frac{1}{4} \cdot \frac{u0}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}} + \frac{1}{3} \cdot \frac{1}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}}\right) + \frac{1}{2} \cdot \frac{1}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}}\right)\right)}\right)\right) \]

   7. Simplified 92.7%

      \[\leadsto \color{blue}{u0 \cdot \left(\frac{1}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} + u0 \cdot \left(\frac{0.5}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} + u0 \cdot \left(\frac{u0 \cdot 0.25}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} + \frac{0.3333333333333333}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}}\right)\right)\right)} \]

   8. Taylor expanded in sin2phi around inf

      \[\leadsto \color{blue}{\frac{u0 \cdot \left(u0 \cdot \left(\frac{1}{2} \cdot {alphay}^{2} + u0 \cdot \left(\frac{1}{4} \cdot \left({alphay}^{2} \cdot u0\right) + \frac{1}{3} \cdot {alphay}^{2}\right)\right) + {alphay}^{2}\right)}{sin2phi}} \]
   9. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto u0 \cdot \color{blue}{\frac{u0 \cdot \left(\frac{1}{2} \cdot {alphay}^{2} + u0 \cdot \left(\frac{1}{4} \cdot \left({alphay}^{2} \cdot u0\right) + \frac{1}{3} \cdot {alphay}^{2}\right)\right) + {alphay}^{2}}{sin2phi}} \]

      2. *-lowering-*.f32 N/A

         \[\leadsto \mathsf{*.f32}\left(u0, \color{blue}{\left(\frac{u0 \cdot \left(\frac{1}{2} \cdot {alphay}^{2} + u0 \cdot \left(\frac{1}{4} \cdot \left({alphay}^{2} \cdot u0\right) + \frac{1}{3} \cdot {alphay}^{2}\right)\right) + {alphay}^{2}}{sin2phi}\right)}\right) \]

      3. /-lowering-/.f32 N/A

         \[\leadsto \mathsf{*.f32}\left(u0, \mathsf{/.f32}\left(\left(u0 \cdot \left(\frac{1}{2} \cdot {alphay}^{2} + u0 \cdot \left(\frac{1}{4} \cdot \left({alphay}^{2} \cdot u0\right) + \frac{1}{3} \cdot {alphay}^{2}\right)\right) + {alphay}^{2}\right), \color{blue}{sin2phi}\right)\right) \]

   10. Simplified 93.1%

       \[\leadsto \color{blue}{u0 \cdot \frac{alphay \cdot alphay + u0 \cdot \left(u0 \cdot \left(\left(u0 \cdot \left(alphay \cdot alphay\right)\right) \cdot 0.25 + \left(alphay \cdot alphay\right) \cdot 0.3333333333333333\right) + \left(alphay \cdot alphay\right) \cdot 0.5\right)}{sin2phi}} \]

   11. Taylor expanded in u0 around 0

       \[\leadsto \mathsf{*.f32}\left(u0, \mathsf{/.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(alphay, alphay\right), \mathsf{*.f32}\left(u0, \color{blue}{\left(\frac{1}{2} \cdot {alphay}^{2}\right)}\right)\right), sin2phi\right)\right) \]
   12. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \mathsf{*.f32}\left(u0, \mathsf{/.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(alphay, alphay\right), \mathsf{*.f32}\left(u0, \left({alphay}^{2} \cdot \frac{1}{2}\right)\right)\right), sin2phi\right)\right) \]

       2. *-lowering-*.f32 N/A

          \[\leadsto \mathsf{*.f32}\left(u0, \mathsf{/.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(alphay, alphay\right), \mathsf{*.f32}\left(u0, \mathsf{*.f32}\left(\left({alphay}^{2}\right), \frac{1}{2}\right)\right)\right), sin2phi\right)\right) \]

       3. unpow2 N/A

          \[\leadsto \mathsf{*.f32}\left(u0, \mathsf{/.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(alphay, alphay\right), \mathsf{*.f32}\left(u0, \mathsf{*.f32}\left(\left(alphay \cdot alphay\right), \frac{1}{2}\right)\right)\right), sin2phi\right)\right) \]

       4. *-lowering-*.f32 87.6%

          \[\leadsto \mathsf{*.f32}\left(u0, \mathsf{/.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(alphay, alphay\right), \mathsf{*.f32}\left(u0, \mathsf{*.f32}\left(\mathsf{*.f32}\left(alphay, alphay\right), \frac{1}{2}\right)\right)\right), sin2phi\right)\right) \]

   13. Simplified 87.6%

       \[\leadsto u0 \cdot \frac{alphay \cdot alphay + u0 \cdot \color{blue}{\left(\left(alphay \cdot alphay\right) \cdot 0.5\right)}}{sin2phi} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 15: 91.2% accurate, 5.0× speedup

Math

\[\begin{array}{l} \\ \frac{u0 \cdot \left(1 - u0 \cdot \left(-0.5 + u0 \cdot -0.3333333333333333\right)\right)}{\frac{\frac{sin2phi}{alphay}}{alphay} + \frac{\frac{cos2phi}{alphax}}{alphax}} \end{array} \]

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(Float32(sin2phi / alphay) / alphay) + Float32(Float32(cos2phi / 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 60.9%

   \[\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
2. Step-by-step derivation

   1. distribute-frac-neg N/A

      \[\leadsto \mathsf{neg}\left(\frac{\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}}\right) \]

   2. distribute-neg-frac2 N/A

      \[\leadsto \frac{\log \left(1 - u0\right)}{\color{blue}{\mathsf{neg}\left(\left(\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\right)\right)}} \]

   3. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\log \left(1 - u0\right), \color{blue}{\left(\mathsf{neg}\left(\left(\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\right)\right)\right)}\right) \]

   4. sub-neg N/A

      \[\leadsto \mathsf{/.f32}\left(\log \left(1 + \left(\mathsf{neg}\left(u0\right)\right)\right), \left(\mathsf{neg}\left(\left(\color{blue}{\frac{cos2phi}{alphax \cdot alphax}} + \frac{sin2phi}{alphay \cdot alphay}\right)\right)\right)\right) \]

   5. log1p-define N/A

      \[\leadsto \mathsf{/.f32}\left(\left(\mathsf{log1p}\left(\mathsf{neg}\left(u0\right)\right)\right), \left(\mathsf{neg}\left(\color{blue}{\left(\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\right)}\right)\right)\right) \]

   6. log1p-lowering-log1p.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\left(\mathsf{neg}\left(u0\right)\right)\right), \left(\mathsf{neg}\left(\color{blue}{\left(\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\right)}\right)\right)\right) \]

   7. neg-lowering-neg.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \left(\mathsf{neg}\left(\left(\color{blue}{\frac{cos2phi}{alphax \cdot alphax}} + \frac{sin2phi}{alphay \cdot alphay}\right)\right)\right)\right) \]

   8. distribute-neg-in N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \left(\left(\mathsf{neg}\left(\frac{cos2phi}{alphax \cdot alphax}\right)\right) + \color{blue}{\left(\mathsf{neg}\left(\frac{sin2phi}{alphay \cdot alphay}\right)\right)}\right)\right) \]

   9. unsub-neg N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \left(\left(\mathsf{neg}\left(\frac{cos2phi}{alphax \cdot alphax}\right)\right) - \color{blue}{\frac{sin2phi}{alphay \cdot alphay}}\right)\right) \]

   10. --lowering--.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\left(\mathsf{neg}\left(\frac{cos2phi}{alphax \cdot alphax}\right)\right), \color{blue}{\left(\frac{sin2phi}{alphay \cdot alphay}\right)}\right)\right) \]

   11. neg-sub0 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\left(0 - \frac{cos2phi}{alphax \cdot alphax}\right), \left(\frac{\color{blue}{sin2phi}}{alphay \cdot alphay}\right)\right)\right) \]

   12. --lowering--.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \left(\frac{cos2phi}{alphax \cdot alphax}\right)\right), \left(\frac{\color{blue}{sin2phi}}{alphay \cdot alphay}\right)\right)\right) \]

   13. /-lowering-/.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(cos2phi, \left(alphax \cdot alphax\right)\right)\right), \left(\frac{sin2phi}{alphay \cdot alphay}\right)\right)\right) \]

   14. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right)\right), \left(\frac{sin2phi}{alphay \cdot alphay}\right)\right)\right) \]

   15. /-lowering-/.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right)\right), \mathsf{/.f32}\left(sin2phi, \color{blue}{\left(alphay \cdot alphay\right)}\right)\right)\right) \]

   16. *-lowering-*.f32 98.4%

       \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right)\right), \mathsf{/.f32}\left(sin2phi, \mathsf{*.f32}\left(alphay, \color{blue}{alphay}\right)\right)\right)\right) \]

3. Simplified 98.4%

   \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(-u0\right)}{\left(0 - \frac{cos2phi}{alphax \cdot alphax}\right) - \frac{sin2phi}{alphay \cdot alphay}}} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. associate-/r* N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \left(\frac{\frac{cos2phi}{alphax}}{alphax}\right)\right), \mathsf{/.f32}\left(sin2phi, \mathsf{*.f32}\left(alphay, alphay\right)\right)\right)\right) \]

   2. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(\left(\frac{cos2phi}{alphax}\right), alphax\right)\right), \mathsf{/.f32}\left(sin2phi, \mathsf{*.f32}\left(alphay, alphay\right)\right)\right)\right) \]

   3. /-lowering-/.f32 98.4%

      \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(\mathsf{/.f32}\left(cos2phi, alphax\right), alphax\right)\right), \mathsf{/.f32}\left(sin2phi, \mathsf{*.f32}\left(alphay, alphay\right)\right)\right)\right) \]

6. Applied egg-rr 98.4%

   \[\leadsto \frac{\mathsf{log1p}\left(-u0\right)}{\left(0 - \color{blue}{\frac{\frac{cos2phi}{alphax}}{alphax}}\right) - \frac{sin2phi}{alphay \cdot alphay}} \]
7. Step-by-step derivation

   1. sub0-neg N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\left(\mathsf{neg}\left(\frac{\frac{cos2phi}{alphax}}{alphax}\right)\right), \mathsf{/.f32}\left(\color{blue}{sin2phi}, \mathsf{*.f32}\left(alphay, alphay\right)\right)\right)\right) \]

   2. distribute-neg-frac2 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\left(\frac{\frac{cos2phi}{alphax}}{\mathsf{neg}\left(alphax\right)}\right), \mathsf{/.f32}\left(\color{blue}{sin2phi}, \mathsf{*.f32}\left(alphay, alphay\right)\right)\right)\right) \]

   3. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{/.f32}\left(\left(\frac{cos2phi}{alphax}\right), \left(\mathsf{neg}\left(alphax\right)\right)\right), \mathsf{/.f32}\left(\color{blue}{sin2phi}, \mathsf{*.f32}\left(alphay, alphay\right)\right)\right)\right) \]

   4. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(cos2phi, alphax\right), \left(\mathsf{neg}\left(alphax\right)\right)\right), \mathsf{/.f32}\left(sin2phi, \mathsf{*.f32}\left(alphay, alphay\right)\right)\right)\right) \]

   5. neg-lowering-neg.f32 98.4%

      \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(cos2phi, alphax\right), \mathsf{neg.f32}\left(alphax\right)\right), \mathsf{/.f32}\left(sin2phi, \mathsf{*.f32}\left(alphay, alphay\right)\right)\right)\right) \]

8. Applied egg-rr 98.4%

   \[\leadsto \frac{\mathsf{log1p}\left(-u0\right)}{\color{blue}{\frac{\frac{cos2phi}{alphax}}{-alphax}} - \frac{sin2phi}{alphay \cdot alphay}} \]
9. Step-by-step derivation

   1. associate-/r* 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 98.4%

      \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(cos2phi, alphax\right), \mathsf{neg.f32}\left(alphax\right)\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(sin2phi, alphay\right), alphay\right)\right)\right) \]

10. Applied egg-rr 98.4%

    \[\leadsto \frac{\mathsf{log1p}\left(-u0\right)}{\frac{\frac{cos2phi}{alphax}}{-alphax} - \color{blue}{\frac{\frac{sin2phi}{alphay}}{alphay}}} \]

11. Taylor expanded in u0 around 0

    \[\leadsto \mathsf{/.f32}\left(\color{blue}{\left(u0 \cdot \left(u0 \cdot \left(\frac{-1}{3} \cdot u0 - \frac{1}{2}\right) - 1\right)\right)}, \mathsf{\_.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(cos2phi, alphax\right), \mathsf{neg.f32}\left(alphax\right)\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(sin2phi, alphay\right), alphay\right)\right)\right) \]
12. Step-by-step derivation

    1. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(u0, \left(u0 \cdot \left(\frac{-1}{3} \cdot u0 - \frac{1}{2}\right) - 1\right)\right), \mathsf{\_.f32}\left(\color{blue}{\mathsf{/.f32}\left(\mathsf{/.f32}\left(cos2phi, alphax\right), \mathsf{neg.f32}\left(alphax\right)\right)}, \mathsf{/.f32}\left(\mathsf{/.f32}\left(sin2phi, alphay\right), alphay\right)\right)\right) \]

    2. sub-neg N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(u0, \left(u0 \cdot \left(\frac{-1}{3} \cdot u0 - \frac{1}{2}\right) + \left(\mathsf{neg}\left(1\right)\right)\right)\right), \mathsf{\_.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(cos2phi, alphax\right), \color{blue}{\mathsf{neg.f32}\left(alphax\right)}\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(sin2phi, alphay\right), alphay\right)\right)\right) \]

    3. metadata-eval N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(u0, \left(u0 \cdot \left(\frac{-1}{3} \cdot u0 - \frac{1}{2}\right) + -1\right)\right), \mathsf{\_.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(cos2phi, alphax\right), \mathsf{neg.f32}\left(alphax\right)\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(sin2phi, alphay\right), alphay\right)\right)\right) \]

    4. +-lowering-+.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\left(u0 \cdot \left(\frac{-1}{3} \cdot u0 - \frac{1}{2}\right)\right), -1\right)\right), \mathsf{\_.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(cos2phi, alphax\right), \color{blue}{\mathsf{neg.f32}\left(alphax\right)}\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(sin2phi, alphay\right), alphay\right)\right)\right) \]

    5. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{*.f32}\left(u0, \left(\frac{-1}{3} \cdot u0 - \frac{1}{2}\right)\right), -1\right)\right), \mathsf{\_.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(cos2phi, alphax\right), \mathsf{neg.f32}\left(\color{blue}{alphax}\right)\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(sin2phi, alphay\right), alphay\right)\right)\right) \]

    6. sub-neg N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{*.f32}\left(u0, \left(\frac{-1}{3} \cdot u0 + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right)\right), -1\right)\right), \mathsf{\_.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(cos2phi, alphax\right), \mathsf{neg.f32}\left(alphax\right)\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(sin2phi, alphay\right), alphay\right)\right)\right) \]

    7. metadata-eval N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{*.f32}\left(u0, \left(\frac{-1}{3} \cdot u0 + \frac{-1}{2}\right)\right), -1\right)\right), \mathsf{\_.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(cos2phi, alphax\right), \mathsf{neg.f32}\left(alphax\right)\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(sin2phi, alphay\right), alphay\right)\right)\right) \]

    8. +-lowering-+.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\left(\frac{-1}{3} \cdot u0\right), \frac{-1}{2}\right)\right), -1\right)\right), \mathsf{\_.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(cos2phi, alphax\right), \mathsf{neg.f32}\left(alphax\right)\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(sin2phi, alphay\right), alphay\right)\right)\right) \]

    9. *-commutative N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\left(u0 \cdot \frac{-1}{3}\right), \frac{-1}{2}\right)\right), -1\right)\right), \mathsf{\_.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(cos2phi, alphax\right), \mathsf{neg.f32}\left(alphax\right)\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(sin2phi, alphay\right), alphay\right)\right)\right) \]

    10. *-lowering-*.f32 91.1%

        \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{*.f32}\left(u0, \frac{-1}{3}\right), \frac{-1}{2}\right)\right), -1\right)\right), \mathsf{\_.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(cos2phi, alphax\right), \mathsf{neg.f32}\left(alphax\right)\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(sin2phi, alphay\right), alphay\right)\right)\right) \]

13. Simplified 91.1%

    \[\leadsto \frac{\color{blue}{u0 \cdot \left(u0 \cdot \left(u0 \cdot -0.3333333333333333 + -0.5\right) + -1\right)}}{\frac{\frac{cos2phi}{alphax}}{-alphax} - \frac{\frac{sin2phi}{alphay}}{alphay}} \]

14. Final simplification 91.1%

    \[\leadsto \frac{u0 \cdot \left(1 - u0 \cdot \left(-0.5 + u0 \cdot -0.3333333333333333\right)\right)}{\frac{\frac{sin2phi}{alphay}}{alphay} + \frac{\frac{cos2phi}{alphax}}{alphax}} \]
15. Add Preprocessing

Alternative 16: 91.2% accurate, 5.0× speedup

Math

\[\begin{array}{l} \\ \frac{u0 \cdot \left(1 - u0 \cdot \left(-0.5 + u0 \cdot -0.3333333333333333\right)\right)}{\frac{sin2phi}{alphay \cdot alphay} + \frac{\frac{cos2phi}{alphax}}{alphax}} \end{array} \]

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(Float32(cos2phi / 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 60.9%

   \[\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
2. Step-by-step derivation

   1. distribute-frac-neg N/A

      \[\leadsto \mathsf{neg}\left(\frac{\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}}\right) \]

   2. distribute-neg-frac2 N/A

      \[\leadsto \frac{\log \left(1 - u0\right)}{\color{blue}{\mathsf{neg}\left(\left(\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\right)\right)}} \]

   3. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\log \left(1 - u0\right), \color{blue}{\left(\mathsf{neg}\left(\left(\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\right)\right)\right)}\right) \]

   4. sub-neg N/A

      \[\leadsto \mathsf{/.f32}\left(\log \left(1 + \left(\mathsf{neg}\left(u0\right)\right)\right), \left(\mathsf{neg}\left(\left(\color{blue}{\frac{cos2phi}{alphax \cdot alphax}} + \frac{sin2phi}{alphay \cdot alphay}\right)\right)\right)\right) \]

   5. log1p-define N/A

      \[\leadsto \mathsf{/.f32}\left(\left(\mathsf{log1p}\left(\mathsf{neg}\left(u0\right)\right)\right), \left(\mathsf{neg}\left(\color{blue}{\left(\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\right)}\right)\right)\right) \]

   6. log1p-lowering-log1p.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\left(\mathsf{neg}\left(u0\right)\right)\right), \left(\mathsf{neg}\left(\color{blue}{\left(\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\right)}\right)\right)\right) \]

   7. neg-lowering-neg.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \left(\mathsf{neg}\left(\left(\color{blue}{\frac{cos2phi}{alphax \cdot alphax}} + \frac{sin2phi}{alphay \cdot alphay}\right)\right)\right)\right) \]

   8. distribute-neg-in N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \left(\left(\mathsf{neg}\left(\frac{cos2phi}{alphax \cdot alphax}\right)\right) + \color{blue}{\left(\mathsf{neg}\left(\frac{sin2phi}{alphay \cdot alphay}\right)\right)}\right)\right) \]

   9. unsub-neg N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \left(\left(\mathsf{neg}\left(\frac{cos2phi}{alphax \cdot alphax}\right)\right) - \color{blue}{\frac{sin2phi}{alphay \cdot alphay}}\right)\right) \]

   10. --lowering--.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\left(\mathsf{neg}\left(\frac{cos2phi}{alphax \cdot alphax}\right)\right), \color{blue}{\left(\frac{sin2phi}{alphay \cdot alphay}\right)}\right)\right) \]

   11. neg-sub0 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\left(0 - \frac{cos2phi}{alphax \cdot alphax}\right), \left(\frac{\color{blue}{sin2phi}}{alphay \cdot alphay}\right)\right)\right) \]

   12. --lowering--.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \left(\frac{cos2phi}{alphax \cdot alphax}\right)\right), \left(\frac{\color{blue}{sin2phi}}{alphay \cdot alphay}\right)\right)\right) \]

   13. /-lowering-/.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(cos2phi, \left(alphax \cdot alphax\right)\right)\right), \left(\frac{sin2phi}{alphay \cdot alphay}\right)\right)\right) \]

   14. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right)\right), \left(\frac{sin2phi}{alphay \cdot alphay}\right)\right)\right) \]

   15. /-lowering-/.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right)\right), \mathsf{/.f32}\left(sin2phi, \color{blue}{\left(alphay \cdot alphay\right)}\right)\right)\right) \]

   16. *-lowering-*.f32 98.4%

       \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right)\right), \mathsf{/.f32}\left(sin2phi, \mathsf{*.f32}\left(alphay, \color{blue}{alphay}\right)\right)\right)\right) \]

3. Simplified 98.4%

   \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(-u0\right)}{\left(0 - \frac{cos2phi}{alphax \cdot alphax}\right) - \frac{sin2phi}{alphay \cdot alphay}}} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. associate-/r* N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \left(\frac{\frac{cos2phi}{alphax}}{alphax}\right)\right), \mathsf{/.f32}\left(sin2phi, \mathsf{*.f32}\left(alphay, alphay\right)\right)\right)\right) \]

   2. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(\left(\frac{cos2phi}{alphax}\right), alphax\right)\right), \mathsf{/.f32}\left(sin2phi, \mathsf{*.f32}\left(alphay, alphay\right)\right)\right)\right) \]

   3. /-lowering-/.f32 98.4%

      \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(\mathsf{/.f32}\left(cos2phi, alphax\right), alphax\right)\right), \mathsf{/.f32}\left(sin2phi, \mathsf{*.f32}\left(alphay, alphay\right)\right)\right)\right) \]

6. Applied egg-rr 98.4%

   \[\leadsto \frac{\mathsf{log1p}\left(-u0\right)}{\left(0 - \color{blue}{\frac{\frac{cos2phi}{alphax}}{alphax}}\right) - \frac{sin2phi}{alphay \cdot alphay}} \]
7. Step-by-step derivation

   1. sub0-neg N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\left(\mathsf{neg}\left(\frac{\frac{cos2phi}{alphax}}{alphax}\right)\right), \mathsf{/.f32}\left(\color{blue}{sin2phi}, \mathsf{*.f32}\left(alphay, alphay\right)\right)\right)\right) \]

   2. distribute-neg-frac2 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\left(\frac{\frac{cos2phi}{alphax}}{\mathsf{neg}\left(alphax\right)}\right), \mathsf{/.f32}\left(\color{blue}{sin2phi}, \mathsf{*.f32}\left(alphay, alphay\right)\right)\right)\right) \]

   3. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{/.f32}\left(\left(\frac{cos2phi}{alphax}\right), \left(\mathsf{neg}\left(alphax\right)\right)\right), \mathsf{/.f32}\left(\color{blue}{sin2phi}, \mathsf{*.f32}\left(alphay, alphay\right)\right)\right)\right) \]

   4. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(cos2phi, alphax\right), \left(\mathsf{neg}\left(alphax\right)\right)\right), \mathsf{/.f32}\left(sin2phi, \mathsf{*.f32}\left(alphay, alphay\right)\right)\right)\right) \]

   5. neg-lowering-neg.f32 98.4%

      \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(cos2phi, alphax\right), \mathsf{neg.f32}\left(alphax\right)\right), \mathsf{/.f32}\left(sin2phi, \mathsf{*.f32}\left(alphay, alphay\right)\right)\right)\right) \]

8. Applied egg-rr 98.4%

   \[\leadsto \frac{\mathsf{log1p}\left(-u0\right)}{\color{blue}{\frac{\frac{cos2phi}{alphax}}{-alphax}} - \frac{sin2phi}{alphay \cdot alphay}} \]

9. Taylor expanded in u0 around 0

   \[\leadsto \mathsf{/.f32}\left(\color{blue}{\left(u0 \cdot \left(u0 \cdot \left(\frac{-1}{3} \cdot u0 - \frac{1}{2}\right) - 1\right)\right)}, \mathsf{\_.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(cos2phi, alphax\right), \mathsf{neg.f32}\left(alphax\right)\right), \mathsf{/.f32}\left(sin2phi, \mathsf{*.f32}\left(alphay, alphay\right)\right)\right)\right) \]
10. Step-by-step derivation

    1. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(u0, \left(u0 \cdot \left(\frac{-1}{3} \cdot u0 - \frac{1}{2}\right) - 1\right)\right), \mathsf{\_.f32}\left(\color{blue}{\mathsf{/.f32}\left(\mathsf{/.f32}\left(cos2phi, alphax\right), \mathsf{neg.f32}\left(alphax\right)\right)}, \mathsf{/.f32}\left(sin2phi, \mathsf{*.f32}\left(alphay, alphay\right)\right)\right)\right) \]

    2. sub-neg N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(u0, \left(u0 \cdot \left(\frac{-1}{3} \cdot u0 - \frac{1}{2}\right) + \left(\mathsf{neg}\left(1\right)\right)\right)\right), \mathsf{\_.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(cos2phi, alphax\right), \color{blue}{\mathsf{neg.f32}\left(alphax\right)}\right), \mathsf{/.f32}\left(sin2phi, \mathsf{*.f32}\left(alphay, alphay\right)\right)\right)\right) \]

    3. metadata-eval N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(u0, \left(u0 \cdot \left(\frac{-1}{3} \cdot u0 - \frac{1}{2}\right) + -1\right)\right), \mathsf{\_.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(cos2phi, alphax\right), \mathsf{neg.f32}\left(alphax\right)\right), \mathsf{/.f32}\left(sin2phi, \mathsf{*.f32}\left(alphay, alphay\right)\right)\right)\right) \]

    4. +-lowering-+.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\left(u0 \cdot \left(\frac{-1}{3} \cdot u0 - \frac{1}{2}\right)\right), -1\right)\right), \mathsf{\_.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(cos2phi, alphax\right), \color{blue}{\mathsf{neg.f32}\left(alphax\right)}\right), \mathsf{/.f32}\left(sin2phi, \mathsf{*.f32}\left(alphay, alphay\right)\right)\right)\right) \]

    5. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{*.f32}\left(u0, \left(\frac{-1}{3} \cdot u0 - \frac{1}{2}\right)\right), -1\right)\right), \mathsf{\_.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(cos2phi, alphax\right), \mathsf{neg.f32}\left(\color{blue}{alphax}\right)\right), \mathsf{/.f32}\left(sin2phi, \mathsf{*.f32}\left(alphay, alphay\right)\right)\right)\right) \]

    6. sub-neg N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{*.f32}\left(u0, \left(\frac{-1}{3} \cdot u0 + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right)\right), -1\right)\right), \mathsf{\_.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(cos2phi, alphax\right), \mathsf{neg.f32}\left(alphax\right)\right), \mathsf{/.f32}\left(sin2phi, \mathsf{*.f32}\left(alphay, alphay\right)\right)\right)\right) \]

    7. metadata-eval N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{*.f32}\left(u0, \left(\frac{-1}{3} \cdot u0 + \frac{-1}{2}\right)\right), -1\right)\right), \mathsf{\_.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(cos2phi, alphax\right), \mathsf{neg.f32}\left(alphax\right)\right), \mathsf{/.f32}\left(sin2phi, \mathsf{*.f32}\left(alphay, alphay\right)\right)\right)\right) \]

    8. +-lowering-+.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\left(\frac{-1}{3} \cdot u0\right), \frac{-1}{2}\right)\right), -1\right)\right), \mathsf{\_.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(cos2phi, alphax\right), \mathsf{neg.f32}\left(alphax\right)\right), \mathsf{/.f32}\left(sin2phi, \mathsf{*.f32}\left(alphay, alphay\right)\right)\right)\right) \]

    9. *-commutative N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\left(u0 \cdot \frac{-1}{3}\right), \frac{-1}{2}\right)\right), -1\right)\right), \mathsf{\_.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(cos2phi, alphax\right), \mathsf{neg.f32}\left(alphax\right)\right), \mathsf{/.f32}\left(sin2phi, \mathsf{*.f32}\left(alphay, alphay\right)\right)\right)\right) \]

    10. *-lowering-*.f32 91.0%

        \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{*.f32}\left(u0, \frac{-1}{3}\right), \frac{-1}{2}\right)\right), -1\right)\right), \mathsf{\_.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(cos2phi, alphax\right), \mathsf{neg.f32}\left(alphax\right)\right), \mathsf{/.f32}\left(sin2phi, \mathsf{*.f32}\left(alphay, alphay\right)\right)\right)\right) \]

11. Simplified 91.0%

    \[\leadsto \frac{\color{blue}{u0 \cdot \left(u0 \cdot \left(u0 \cdot -0.3333333333333333 + -0.5\right) + -1\right)}}{\frac{\frac{cos2phi}{alphax}}{-alphax} - \frac{sin2phi}{alphay \cdot alphay}} \]

12. Final simplification 91.0%

    \[\leadsto \frac{u0 \cdot \left(1 - u0 \cdot \left(-0.5 + u0 \cdot -0.3333333333333333\right)\right)}{\frac{sin2phi}{alphay \cdot alphay} + \frac{\frac{cos2phi}{alphax}}{alphax}} \]
13. Add Preprocessing

Alternative 17: 91.2% accurate, 5.0× speedup

Math

\[\begin{array}{l} \\ \frac{u0 \cdot \left(u0 \cdot \left(0.5 + u0 \cdot 0.3333333333333333\right) + 1\right)}{\frac{sin2phi}{alphay \cdot alphay} + \frac{cos2phi}{alphax \cdot alphax}} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (/
  (* u0 (+ (* u0 (+ 0.5 (* u0 0.3333333333333333))) 1.0))
  (+ (/ sin2phi (* alphay alphay)) (/ cos2phi (* alphax alphax)))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	return (u0 * ((u0 * (0.5f + (u0 * 0.3333333333333333f))) + 1.0f)) / ((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 * (0.5e0 + (u0 * 0.3333333333333333e0))) + 1.0e0)) / ((sin2phi / (alphay * alphay)) + (cos2phi / (alphax * alphax)))
end function

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	return Float32(Float32(u0 * Float32(Float32(u0 * Float32(Float32(0.5) + Float32(u0 * Float32(0.3333333333333333)))) + Float32(1.0))) / Float32(Float32(sin2phi / Float32(alphay * alphay)) + Float32(cos2phi / Float32(alphax * alphax))))
end

MATLAB

function tmp = code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = (u0 * ((u0 * (single(0.5) + (u0 * single(0.3333333333333333)))) + single(1.0))) / ((sin2phi / (alphay * alphay)) + (cos2phi / (alphax * alphax)));
end

Derivation

1. Initial program 60.9%

   \[\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
2. Add Preprocessing

3. Taylor expanded in u0 around 0

   \[\leadsto \mathsf{/.f32}\left(\color{blue}{\left(u0 \cdot \left(1 + u0 \cdot \left(\frac{1}{2} + \frac{1}{3} \cdot u0\right)\right)\right)}, \mathsf{+.f32}\left(\mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right), \mathsf{/.f32}\left(sin2phi, \mathsf{*.f32}\left(alphay, alphay\right)\right)\right)\right) \]
4. Step-by-step derivation

   1. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(u0, \left(1 + u0 \cdot \left(\frac{1}{2} + \frac{1}{3} \cdot u0\right)\right)\right), \mathsf{+.f32}\left(\color{blue}{\mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right)}, \mathsf{/.f32}\left(sin2phi, \mathsf{*.f32}\left(alphay, alphay\right)\right)\right)\right) \]

   2. +-lowering-+.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(1, \left(u0 \cdot \left(\frac{1}{2} + \frac{1}{3} \cdot u0\right)\right)\right)\right), \mathsf{+.f32}\left(\mathsf{/.f32}\left(cos2phi, \color{blue}{\mathsf{*.f32}\left(alphax, alphax\right)}\right), \mathsf{/.f32}\left(sin2phi, \mathsf{*.f32}\left(alphay, alphay\right)\right)\right)\right) \]

   3. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(u0, \left(\frac{1}{2} + \frac{1}{3} \cdot u0\right)\right)\right)\right), \mathsf{+.f32}\left(\mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, \color{blue}{alphax}\right)\right), \mathsf{/.f32}\left(sin2phi, \mathsf{*.f32}\left(alphay, alphay\right)\right)\right)\right) \]

   4. +-lowering-+.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\frac{1}{2}, \left(\frac{1}{3} \cdot u0\right)\right)\right)\right)\right), \mathsf{+.f32}\left(\mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right), \mathsf{/.f32}\left(sin2phi, \mathsf{*.f32}\left(alphay, alphay\right)\right)\right)\right) \]

   5. *-commutative N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\frac{1}{2}, \left(u0 \cdot \frac{1}{3}\right)\right)\right)\right)\right), \mathsf{+.f32}\left(\mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right), \mathsf{/.f32}\left(sin2phi, \mathsf{*.f32}\left(alphay, alphay\right)\right)\right)\right) \]

   6. *-lowering-*.f32 91.0%

      \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\frac{1}{2}, \mathsf{*.f32}\left(u0, \frac{1}{3}\right)\right)\right)\right)\right), \mathsf{+.f32}\left(\mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right), \mathsf{/.f32}\left(sin2phi, \mathsf{*.f32}\left(alphay, alphay\right)\right)\right)\right) \]

5. Simplified 91.0%

   \[\leadsto \frac{\color{blue}{u0 \cdot \left(1 + u0 \cdot \left(0.5 + u0 \cdot 0.3333333333333333\right)\right)}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

6. Final simplification 91.0%

   \[\leadsto \frac{u0 \cdot \left(u0 \cdot \left(0.5 + u0 \cdot 0.3333333333333333\right) + 1\right)}{\frac{sin2phi}{alphay \cdot alphay} + \frac{cos2phi}{alphax \cdot alphax}} \]
7. Add Preprocessing

Alternative 18: 81.6% accurate, 5.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 0.30000001192092896:\\ \;\;\;\;\frac{u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{\frac{sin2phi}{alphay}}{alphay}}\\ \mathbf{else}:\\ \;\;\;\;\frac{alphay \cdot alphay}{sin2phi} \cdot \left(u0 \cdot \left(1 - u0 \cdot -0.5\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (if (<= (/ sin2phi (* alphay alphay)) 0.30000001192092896)
   (/ u0 (+ (/ cos2phi (* alphax alphax)) (/ (/ sin2phi alphay) alphay)))
   (* (/ (* alphay alphay) sin2phi) (* u0 (- 1.0 (* u0 -0.5))))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float tmp;
	if ((sin2phi / (alphay * alphay)) <= 0.30000001192092896f) {
		tmp = u0 / ((cos2phi / (alphax * alphax)) + ((sin2phi / alphay) / alphay));
	} else {
		tmp = ((alphay * alphay) / sin2phi) * (u0 * (1.0f - (u0 * -0.5f)));
	}
	return tmp;
}

Fortran

real(4) function code(alphax, alphay, u0, cos2phi, sin2phi)
    real(4), intent (in) :: alphax
    real(4), intent (in) :: alphay
    real(4), intent (in) :: u0
    real(4), intent (in) :: cos2phi
    real(4), intent (in) :: sin2phi
    real(4) :: tmp
    if ((sin2phi / (alphay * alphay)) <= 0.30000001192092896e0) then
        tmp = u0 / ((cos2phi / (alphax * alphax)) + ((sin2phi / alphay) / alphay))
    else
        tmp = ((alphay * alphay) / sin2phi) * (u0 * (1.0e0 - (u0 * (-0.5e0))))
    end if
    code = tmp
end function

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = Float32(0.0)
	if (Float32(sin2phi / Float32(alphay * alphay)) <= Float32(0.30000001192092896))
		tmp = Float32(u0 / Float32(Float32(cos2phi / Float32(alphax * alphax)) + Float32(Float32(sin2phi / alphay) / alphay)));
	else
		tmp = Float32(Float32(Float32(alphay * alphay) / sin2phi) * Float32(u0 * Float32(Float32(1.0) - Float32(u0 * Float32(-0.5)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = single(0.0);
	if ((sin2phi / (alphay * alphay)) <= single(0.30000001192092896))
		tmp = u0 / ((cos2phi / (alphax * alphax)) + ((sin2phi / alphay) / alphay));
	else
		tmp = ((alphay * alphay) / sin2phi) * (u0 * (single(1.0) - (u0 * single(-0.5))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 sin2phi (*.f32 alphay alphay)) < 0.300000012

   1. Initial program 55.9%

      \[\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
   2. Step-by-step derivation

      1. distribute-frac-neg N/A

         \[\leadsto \mathsf{neg}\left(\frac{\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}}\right) \]

      2. distribute-neg-frac2 N/A

         \[\leadsto \frac{\log \left(1 - u0\right)}{\color{blue}{\mathsf{neg}\left(\left(\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\right)\right)}} \]

      3. /-lowering-/.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(\log \left(1 - u0\right), \color{blue}{\left(\mathsf{neg}\left(\left(\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\right)\right)\right)}\right) \]

      4. sub-neg N/A

         \[\leadsto \mathsf{/.f32}\left(\log \left(1 + \left(\mathsf{neg}\left(u0\right)\right)\right), \left(\mathsf{neg}\left(\left(\color{blue}{\frac{cos2phi}{alphax \cdot alphax}} + \frac{sin2phi}{alphay \cdot alphay}\right)\right)\right)\right) \]

      5. log1p-define N/A

         \[\leadsto \mathsf{/.f32}\left(\left(\mathsf{log1p}\left(\mathsf{neg}\left(u0\right)\right)\right), \left(\mathsf{neg}\left(\color{blue}{\left(\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\right)}\right)\right)\right) \]

      6. log1p-lowering-log1p.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\left(\mathsf{neg}\left(u0\right)\right)\right), \left(\mathsf{neg}\left(\color{blue}{\left(\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\right)}\right)\right)\right) \]

      7. neg-lowering-neg.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \left(\mathsf{neg}\left(\left(\color{blue}{\frac{cos2phi}{alphax \cdot alphax}} + \frac{sin2phi}{alphay \cdot alphay}\right)\right)\right)\right) \]

      8. distribute-neg-in N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \left(\left(\mathsf{neg}\left(\frac{cos2phi}{alphax \cdot alphax}\right)\right) + \color{blue}{\left(\mathsf{neg}\left(\frac{sin2phi}{alphay \cdot alphay}\right)\right)}\right)\right) \]

      9. unsub-neg N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \left(\left(\mathsf{neg}\left(\frac{cos2phi}{alphax \cdot alphax}\right)\right) - \color{blue}{\frac{sin2phi}{alphay \cdot alphay}}\right)\right) \]

      10. --lowering--.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\left(\mathsf{neg}\left(\frac{cos2phi}{alphax \cdot alphax}\right)\right), \color{blue}{\left(\frac{sin2phi}{alphay \cdot alphay}\right)}\right)\right) \]

      11. neg-sub0 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\left(0 - \frac{cos2phi}{alphax \cdot alphax}\right), \left(\frac{\color{blue}{sin2phi}}{alphay \cdot alphay}\right)\right)\right) \]

      12. --lowering--.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \left(\frac{cos2phi}{alphax \cdot alphax}\right)\right), \left(\frac{\color{blue}{sin2phi}}{alphay \cdot alphay}\right)\right)\right) \]

      13. /-lowering-/.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(cos2phi, \left(alphax \cdot alphax\right)\right)\right), \left(\frac{sin2phi}{alphay \cdot alphay}\right)\right)\right) \]

      14. *-lowering-*.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right)\right), \left(\frac{sin2phi}{alphay \cdot alphay}\right)\right)\right) \]

      15. /-lowering-/.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right)\right), \mathsf{/.f32}\left(sin2phi, \color{blue}{\left(alphay \cdot alphay\right)}\right)\right)\right) \]

      16. *-lowering-*.f32 98.6%

          \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right)\right), \mathsf{/.f32}\left(sin2phi, \mathsf{*.f32}\left(alphay, \color{blue}{alphay}\right)\right)\right)\right) \]

   3. Simplified 98.6%

      \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(-u0\right)}{\left(0 - \frac{cos2phi}{alphax \cdot alphax}\right) - \frac{sin2phi}{alphay \cdot alphay}}} \]
   4. Add Preprocessing

   5. Taylor expanded in u0 around 0

      \[\leadsto \color{blue}{\frac{u0}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}}} \]
   6. Step-by-step derivation

      1. /-lowering-/.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(u0, \color{blue}{\left(\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}\right)}\right) \]

      2. +-lowering-+.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{+.f32}\left(\left(\frac{cos2phi}{{alphax}^{2}}\right), \color{blue}{\left(\frac{sin2phi}{{alphay}^{2}}\right)}\right)\right) \]

      3. /-lowering-/.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{/.f32}\left(cos2phi, \left({alphax}^{2}\right)\right), \left(\frac{\color{blue}{sin2phi}}{{alphay}^{2}}\right)\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{/.f32}\left(cos2phi, \left(alphax \cdot alphax\right)\right), \left(\frac{sin2phi}{{alphay}^{2}}\right)\right)\right) \]

      5. *-lowering-*.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right), \left(\frac{sin2phi}{{alphay}^{2}}\right)\right)\right) \]

      6. /-lowering-/.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right), \mathsf{/.f32}\left(sin2phi, \color{blue}{\left({alphay}^{2}\right)}\right)\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right), \mathsf{/.f32}\left(sin2phi, \left(alphay \cdot \color{blue}{alphay}\right)\right)\right)\right) \]

      8. *-lowering-*.f32 73.7%

         \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right), \mathsf{/.f32}\left(sin2phi, \mathsf{*.f32}\left(alphay, \color{blue}{alphay}\right)\right)\right)\right) \]

   7. Simplified 73.7%

      \[\leadsto \color{blue}{\frac{u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}}} \]
   8. Step-by-step derivation

      1. associate-/r* N/A

         \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right), \left(\frac{\frac{sin2phi}{alphay}}{\color{blue}{alphay}}\right)\right)\right) \]

      2. /-lowering-/.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right), \mathsf{/.f32}\left(\left(\frac{sin2phi}{alphay}\right), \color{blue}{alphay}\right)\right)\right) \]

      3. /-lowering-/.f32 73.7%

         \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(sin2phi, alphay\right), alphay\right)\right)\right) \]

   9. Applied egg-rr 73.7%

      \[\leadsto \frac{u0}{\frac{cos2phi}{alphax \cdot alphax} + \color{blue}{\frac{\frac{sin2phi}{alphay}}{alphay}}} \]

   if 0.300000012 < (/.f32 sin2phi (*.f32 alphay alphay))

   1. Initial program 65.1%

      \[\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
   2. Step-by-step derivation

      1. distribute-frac-neg N/A

         \[\leadsto \mathsf{neg}\left(\frac{\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}}\right) \]

      2. distribute-neg-frac2 N/A

         \[\leadsto \frac{\log \left(1 - u0\right)}{\color{blue}{\mathsf{neg}\left(\left(\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\right)\right)}} \]

      3. /-lowering-/.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(\log \left(1 - u0\right), \color{blue}{\left(\mathsf{neg}\left(\left(\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\right)\right)\right)}\right) \]

      4. sub-neg N/A

         \[\leadsto \mathsf{/.f32}\left(\log \left(1 + \left(\mathsf{neg}\left(u0\right)\right)\right), \left(\mathsf{neg}\left(\left(\color{blue}{\frac{cos2phi}{alphax \cdot alphax}} + \frac{sin2phi}{alphay \cdot alphay}\right)\right)\right)\right) \]

      5. log1p-define N/A

         \[\leadsto \mathsf{/.f32}\left(\left(\mathsf{log1p}\left(\mathsf{neg}\left(u0\right)\right)\right), \left(\mathsf{neg}\left(\color{blue}{\left(\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\right)}\right)\right)\right) \]

      6. log1p-lowering-log1p.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\left(\mathsf{neg}\left(u0\right)\right)\right), \left(\mathsf{neg}\left(\color{blue}{\left(\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\right)}\right)\right)\right) \]

      7. neg-lowering-neg.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \left(\mathsf{neg}\left(\left(\color{blue}{\frac{cos2phi}{alphax \cdot alphax}} + \frac{sin2phi}{alphay \cdot alphay}\right)\right)\right)\right) \]

      8. distribute-neg-in N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \left(\left(\mathsf{neg}\left(\frac{cos2phi}{alphax \cdot alphax}\right)\right) + \color{blue}{\left(\mathsf{neg}\left(\frac{sin2phi}{alphay \cdot alphay}\right)\right)}\right)\right) \]

      9. unsub-neg N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \left(\left(\mathsf{neg}\left(\frac{cos2phi}{alphax \cdot alphax}\right)\right) - \color{blue}{\frac{sin2phi}{alphay \cdot alphay}}\right)\right) \]

      10. --lowering--.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\left(\mathsf{neg}\left(\frac{cos2phi}{alphax \cdot alphax}\right)\right), \color{blue}{\left(\frac{sin2phi}{alphay \cdot alphay}\right)}\right)\right) \]

      11. neg-sub0 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\left(0 - \frac{cos2phi}{alphax \cdot alphax}\right), \left(\frac{\color{blue}{sin2phi}}{alphay \cdot alphay}\right)\right)\right) \]

      12. --lowering--.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \left(\frac{cos2phi}{alphax \cdot alphax}\right)\right), \left(\frac{\color{blue}{sin2phi}}{alphay \cdot alphay}\right)\right)\right) \]

      13. /-lowering-/.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(cos2phi, \left(alphax \cdot alphax\right)\right)\right), \left(\frac{sin2phi}{alphay \cdot alphay}\right)\right)\right) \]

      14. *-lowering-*.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right)\right), \left(\frac{sin2phi}{alphay \cdot alphay}\right)\right)\right) \]

      15. /-lowering-/.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right)\right), \mathsf{/.f32}\left(sin2phi, \color{blue}{\left(alphay \cdot alphay\right)}\right)\right)\right) \]

      16. *-lowering-*.f32 98.2%

          \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right)\right), \mathsf{/.f32}\left(sin2phi, \mathsf{*.f32}\left(alphay, \color{blue}{alphay}\right)\right)\right)\right) \]

   3. Simplified 98.2%

      \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(-u0\right)}{\left(0 - \frac{cos2phi}{alphax \cdot alphax}\right) - \frac{sin2phi}{alphay \cdot alphay}}} \]
   4. Add Preprocessing

   5. Taylor expanded in cos2phi around 0

      \[\leadsto \color{blue}{-1 \cdot \frac{{alphay}^{2} \cdot \log \left(1 - u0\right)}{sin2phi}} \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(\frac{{alphay}^{2} \cdot \log \left(1 - u0\right)}{sin2phi}\right) \]

      2. neg-lowering-neg.f32 N/A

         \[\leadsto \mathsf{neg.f32}\left(\left(\frac{{alphay}^{2} \cdot \log \left(1 - u0\right)}{sin2phi}\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{neg.f32}\left(\left(\frac{\log \left(1 - u0\right) \cdot {alphay}^{2}}{sin2phi}\right)\right) \]

      4. associate-/l* N/A

         \[\leadsto \mathsf{neg.f32}\left(\left(\log \left(1 - u0\right) \cdot \frac{{alphay}^{2}}{sin2phi}\right)\right) \]

      5. *-lowering-*.f32 N/A

         \[\leadsto \mathsf{neg.f32}\left(\mathsf{*.f32}\left(\log \left(1 - u0\right), \left(\frac{{alphay}^{2}}{sin2phi}\right)\right)\right) \]

      6. sub-neg N/A

         \[\leadsto \mathsf{neg.f32}\left(\mathsf{*.f32}\left(\log \left(1 + \left(\mathsf{neg}\left(u0\right)\right)\right), \left(\frac{{alphay}^{2}}{sin2phi}\right)\right)\right) \]

      7. mul-1-neg N/A

         \[\leadsto \mathsf{neg.f32}\left(\mathsf{*.f32}\left(\log \left(1 + -1 \cdot u0\right), \left(\frac{{alphay}^{2}}{sin2phi}\right)\right)\right) \]

      8. log1p-define N/A

         \[\leadsto \mathsf{neg.f32}\left(\mathsf{*.f32}\left(\left(\mathsf{log1p}\left(-1 \cdot u0\right)\right), \left(\frac{{alphay}^{2}}{sin2phi}\right)\right)\right) \]

      9. log1p-lowering-log1p.f32 N/A

         \[\leadsto \mathsf{neg.f32}\left(\mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\left(-1 \cdot u0\right)\right), \left(\frac{{alphay}^{2}}{sin2phi}\right)\right)\right) \]

      10. mul-1-neg N/A

          \[\leadsto \mathsf{neg.f32}\left(\mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\left(\mathsf{neg}\left(u0\right)\right)\right), \left(\frac{{alphay}^{2}}{sin2phi}\right)\right)\right) \]

      11. neg-lowering-neg.f32 N/A

          \[\leadsto \mathsf{neg.f32}\left(\mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \left(\frac{{alphay}^{2}}{sin2phi}\right)\right)\right) \]

      12. /-lowering-/.f32 N/A

          \[\leadsto \mathsf{neg.f32}\left(\mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{/.f32}\left(\left({alphay}^{2}\right), sin2phi\right)\right)\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{neg.f32}\left(\mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{/.f32}\left(\left(alphay \cdot alphay\right), sin2phi\right)\right)\right) \]

      14. *-lowering-*.f32 98.4%

          \[\leadsto \mathsf{neg.f32}\left(\mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{/.f32}\left(\mathsf{*.f32}\left(alphay, alphay\right), sin2phi\right)\right)\right) \]

   7. Simplified 98.4%

      \[\leadsto \color{blue}{-\mathsf{log1p}\left(-u0\right) \cdot \frac{alphay \cdot alphay}{sin2phi}} \]

   8. Taylor expanded in u0 around 0

      \[\leadsto \mathsf{neg.f32}\left(\mathsf{*.f32}\left(\color{blue}{\left(u0 \cdot \left(\frac{-1}{2} \cdot u0 - 1\right)\right)}, \mathsf{/.f32}\left(\mathsf{*.f32}\left(alphay, alphay\right), sin2phi\right)\right)\right) \]
   9. Step-by-step derivation

      1. *-lowering-*.f32 N/A

         \[\leadsto \mathsf{neg.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(u0, \left(\frac{-1}{2} \cdot u0 - 1\right)\right), \mathsf{/.f32}\left(\mathsf{*.f32}\left(alphay, alphay\right), sin2phi\right)\right)\right) \]

      2. sub-neg N/A

         \[\leadsto \mathsf{neg.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(u0, \left(\frac{-1}{2} \cdot u0 + \left(\mathsf{neg}\left(1\right)\right)\right)\right), \mathsf{/.f32}\left(\mathsf{*.f32}\left(alphay, alphay\right), sin2phi\right)\right)\right) \]

      3. metadata-eval N/A

         \[\leadsto \mathsf{neg.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(u0, \left(\frac{-1}{2} \cdot u0 + -1\right)\right), \mathsf{/.f32}\left(\mathsf{*.f32}\left(alphay, alphay\right), sin2phi\right)\right)\right) \]

      4. +-lowering-+.f32 N/A

         \[\leadsto \mathsf{neg.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\left(\frac{-1}{2} \cdot u0\right), -1\right)\right), \mathsf{/.f32}\left(\mathsf{*.f32}\left(alphay, alphay\right), sin2phi\right)\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{neg.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\left(u0 \cdot \frac{-1}{2}\right), -1\right)\right), \mathsf{/.f32}\left(\mathsf{*.f32}\left(alphay, alphay\right), sin2phi\right)\right)\right) \]

      6. *-lowering-*.f32 87.2%

         \[\leadsto \mathsf{neg.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{*.f32}\left(u0, \frac{-1}{2}\right), -1\right)\right), \mathsf{/.f32}\left(\mathsf{*.f32}\left(alphay, alphay\right), sin2phi\right)\right)\right) \]

   10. Simplified 87.2%

       \[\leadsto -\color{blue}{\left(u0 \cdot \left(u0 \cdot -0.5 + -1\right)\right)} \cdot \frac{alphay \cdot alphay}{sin2phi} \]
3. Recombined 2 regimes into one program.

4. Final simplification 81.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 0.30000001192092896:\\ \;\;\;\;\frac{u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{\frac{sin2phi}{alphay}}{alphay}}\\ \mathbf{else}:\\ \;\;\;\;\frac{alphay \cdot alphay}{sin2phi} \cdot \left(u0 \cdot \left(1 - u0 \cdot -0.5\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 87.6% accurate, 6.1× speedup

Math

\[\begin{array}{l} \\ \frac{u0 \cdot \left(1 - u0 \cdot -0.5\right)}{\frac{\frac{sin2phi}{alphay}}{alphay} + \frac{\frac{cos2phi}{alphax}}{alphax}} \end{array} \]

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(Float32(sin2phi / alphay) / alphay) + Float32(Float32(cos2phi / 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 60.9%

   \[\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
2. Step-by-step derivation

   1. distribute-frac-neg N/A

      \[\leadsto \mathsf{neg}\left(\frac{\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}}\right) \]

   2. distribute-neg-frac2 N/A

      \[\leadsto \frac{\log \left(1 - u0\right)}{\color{blue}{\mathsf{neg}\left(\left(\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\right)\right)}} \]

   3. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\log \left(1 - u0\right), \color{blue}{\left(\mathsf{neg}\left(\left(\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\right)\right)\right)}\right) \]

   4. sub-neg N/A

      \[\leadsto \mathsf{/.f32}\left(\log \left(1 + \left(\mathsf{neg}\left(u0\right)\right)\right), \left(\mathsf{neg}\left(\left(\color{blue}{\frac{cos2phi}{alphax \cdot alphax}} + \frac{sin2phi}{alphay \cdot alphay}\right)\right)\right)\right) \]

   5. log1p-define N/A

      \[\leadsto \mathsf{/.f32}\left(\left(\mathsf{log1p}\left(\mathsf{neg}\left(u0\right)\right)\right), \left(\mathsf{neg}\left(\color{blue}{\left(\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\right)}\right)\right)\right) \]

   6. log1p-lowering-log1p.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\left(\mathsf{neg}\left(u0\right)\right)\right), \left(\mathsf{neg}\left(\color{blue}{\left(\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\right)}\right)\right)\right) \]

   7. neg-lowering-neg.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \left(\mathsf{neg}\left(\left(\color{blue}{\frac{cos2phi}{alphax \cdot alphax}} + \frac{sin2phi}{alphay \cdot alphay}\right)\right)\right)\right) \]

   8. distribute-neg-in N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \left(\left(\mathsf{neg}\left(\frac{cos2phi}{alphax \cdot alphax}\right)\right) + \color{blue}{\left(\mathsf{neg}\left(\frac{sin2phi}{alphay \cdot alphay}\right)\right)}\right)\right) \]

   9. unsub-neg N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \left(\left(\mathsf{neg}\left(\frac{cos2phi}{alphax \cdot alphax}\right)\right) - \color{blue}{\frac{sin2phi}{alphay \cdot alphay}}\right)\right) \]

   10. --lowering--.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\left(\mathsf{neg}\left(\frac{cos2phi}{alphax \cdot alphax}\right)\right), \color{blue}{\left(\frac{sin2phi}{alphay \cdot alphay}\right)}\right)\right) \]

   11. neg-sub0 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\left(0 - \frac{cos2phi}{alphax \cdot alphax}\right), \left(\frac{\color{blue}{sin2phi}}{alphay \cdot alphay}\right)\right)\right) \]

   12. --lowering--.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \left(\frac{cos2phi}{alphax \cdot alphax}\right)\right), \left(\frac{\color{blue}{sin2phi}}{alphay \cdot alphay}\right)\right)\right) \]

   13. /-lowering-/.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(cos2phi, \left(alphax \cdot alphax\right)\right)\right), \left(\frac{sin2phi}{alphay \cdot alphay}\right)\right)\right) \]

   14. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right)\right), \left(\frac{sin2phi}{alphay \cdot alphay}\right)\right)\right) \]

   15. /-lowering-/.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right)\right), \mathsf{/.f32}\left(sin2phi, \color{blue}{\left(alphay \cdot alphay\right)}\right)\right)\right) \]

   16. *-lowering-*.f32 98.4%

       \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right)\right), \mathsf{/.f32}\left(sin2phi, \mathsf{*.f32}\left(alphay, \color{blue}{alphay}\right)\right)\right)\right) \]

3. Simplified 98.4%

   \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(-u0\right)}{\left(0 - \frac{cos2phi}{alphax \cdot alphax}\right) - \frac{sin2phi}{alphay \cdot alphay}}} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. associate-/r* N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \left(\frac{\frac{cos2phi}{alphax}}{alphax}\right)\right), \mathsf{/.f32}\left(sin2phi, \mathsf{*.f32}\left(alphay, alphay\right)\right)\right)\right) \]

   2. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(\left(\frac{cos2phi}{alphax}\right), alphax\right)\right), \mathsf{/.f32}\left(sin2phi, \mathsf{*.f32}\left(alphay, alphay\right)\right)\right)\right) \]

   3. /-lowering-/.f32 98.4%

      \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(\mathsf{/.f32}\left(cos2phi, alphax\right), alphax\right)\right), \mathsf{/.f32}\left(sin2phi, \mathsf{*.f32}\left(alphay, alphay\right)\right)\right)\right) \]

6. Applied egg-rr 98.4%

   \[\leadsto \frac{\mathsf{log1p}\left(-u0\right)}{\left(0 - \color{blue}{\frac{\frac{cos2phi}{alphax}}{alphax}}\right) - \frac{sin2phi}{alphay \cdot alphay}} \]
7. Step-by-step derivation

   1. sub0-neg N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\left(\mathsf{neg}\left(\frac{\frac{cos2phi}{alphax}}{alphax}\right)\right), \mathsf{/.f32}\left(\color{blue}{sin2phi}, \mathsf{*.f32}\left(alphay, alphay\right)\right)\right)\right) \]

   2. distribute-neg-frac2 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\left(\frac{\frac{cos2phi}{alphax}}{\mathsf{neg}\left(alphax\right)}\right), \mathsf{/.f32}\left(\color{blue}{sin2phi}, \mathsf{*.f32}\left(alphay, alphay\right)\right)\right)\right) \]

   3. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{/.f32}\left(\left(\frac{cos2phi}{alphax}\right), \left(\mathsf{neg}\left(alphax\right)\right)\right), \mathsf{/.f32}\left(\color{blue}{sin2phi}, \mathsf{*.f32}\left(alphay, alphay\right)\right)\right)\right) \]

   4. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(cos2phi, alphax\right), \left(\mathsf{neg}\left(alphax\right)\right)\right), \mathsf{/.f32}\left(sin2phi, \mathsf{*.f32}\left(alphay, alphay\right)\right)\right)\right) \]

   5. neg-lowering-neg.f32 98.4%

      \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(cos2phi, alphax\right), \mathsf{neg.f32}\left(alphax\right)\right), \mathsf{/.f32}\left(sin2phi, \mathsf{*.f32}\left(alphay, alphay\right)\right)\right)\right) \]

8. Applied egg-rr 98.4%

   \[\leadsto \frac{\mathsf{log1p}\left(-u0\right)}{\color{blue}{\frac{\frac{cos2phi}{alphax}}{-alphax}} - \frac{sin2phi}{alphay \cdot alphay}} \]
9. Step-by-step derivation

   1. associate-/r* 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 98.4%

      \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(cos2phi, alphax\right), \mathsf{neg.f32}\left(alphax\right)\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(sin2phi, alphay\right), alphay\right)\right)\right) \]

10. Applied egg-rr 98.4%

    \[\leadsto \frac{\mathsf{log1p}\left(-u0\right)}{\frac{\frac{cos2phi}{alphax}}{-alphax} - \color{blue}{\frac{\frac{sin2phi}{alphay}}{alphay}}} \]

11. Taylor expanded in u0 around 0

    \[\leadsto \mathsf{/.f32}\left(\color{blue}{\left(u0 \cdot \left(\frac{-1}{2} \cdot u0 - 1\right)\right)}, \mathsf{\_.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(cos2phi, alphax\right), \mathsf{neg.f32}\left(alphax\right)\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(sin2phi, alphay\right), alphay\right)\right)\right) \]
12. Step-by-step derivation

    1. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(u0, \left(\frac{-1}{2} \cdot u0 - 1\right)\right), \mathsf{\_.f32}\left(\color{blue}{\mathsf{/.f32}\left(\mathsf{/.f32}\left(cos2phi, alphax\right), \mathsf{neg.f32}\left(alphax\right)\right)}, \mathsf{/.f32}\left(\mathsf{/.f32}\left(sin2phi, alphay\right), alphay\right)\right)\right) \]

    2. sub-neg N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(u0, \left(\frac{-1}{2} \cdot u0 + \left(\mathsf{neg}\left(1\right)\right)\right)\right), \mathsf{\_.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(cos2phi, alphax\right), \color{blue}{\mathsf{neg.f32}\left(alphax\right)}\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(sin2phi, alphay\right), alphay\right)\right)\right) \]

    3. metadata-eval N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(u0, \left(\frac{-1}{2} \cdot u0 + -1\right)\right), \mathsf{\_.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(cos2phi, alphax\right), \mathsf{neg.f32}\left(alphax\right)\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(sin2phi, alphay\right), alphay\right)\right)\right) \]

    4. +-lowering-+.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\left(\frac{-1}{2} \cdot u0\right), -1\right)\right), \mathsf{\_.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(cos2phi, alphax\right), \color{blue}{\mathsf{neg.f32}\left(alphax\right)}\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(sin2phi, alphay\right), alphay\right)\right)\right) \]

    5. *-commutative N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\left(u0 \cdot \frac{-1}{2}\right), -1\right)\right), \mathsf{\_.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(cos2phi, alphax\right), \mathsf{neg.f32}\left(\color{blue}{alphax}\right)\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(sin2phi, alphay\right), alphay\right)\right)\right) \]

    6. *-lowering-*.f32 86.9%

       \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{*.f32}\left(u0, \frac{-1}{2}\right), -1\right)\right), \mathsf{\_.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(cos2phi, alphax\right), \mathsf{neg.f32}\left(\color{blue}{alphax}\right)\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(sin2phi, alphay\right), alphay\right)\right)\right) \]

13. Simplified 86.9%

    \[\leadsto \frac{\color{blue}{u0 \cdot \left(u0 \cdot -0.5 + -1\right)}}{\frac{\frac{cos2phi}{alphax}}{-alphax} - \frac{\frac{sin2phi}{alphay}}{alphay}} \]

14. Final simplification 86.9%

    \[\leadsto \frac{u0 \cdot \left(1 - u0 \cdot -0.5\right)}{\frac{\frac{sin2phi}{alphay}}{alphay} + \frac{\frac{cos2phi}{alphax}}{alphax}} \]
15. Add Preprocessing

Alternative 20: 87.6% accurate, 6.1× speedup

Math

\[\begin{array}{l} \\ \frac{u0 \cdot \left(u0 \cdot 0.5 + 1\right)}{\frac{sin2phi}{alphay \cdot alphay} + \frac{cos2phi}{alphax \cdot alphax}} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (/
  (* u0 (+ (* u0 0.5) 1.0))
  (+ (/ sin2phi (* alphay alphay)) (/ cos2phi (* alphax alphax)))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	return (u0 * ((u0 * 0.5f) + 1.0f)) / ((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 * 0.5e0) + 1.0e0)) / ((sin2phi / (alphay * alphay)) + (cos2phi / (alphax * alphax)))
end function

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	return Float32(Float32(u0 * Float32(Float32(u0 * Float32(0.5)) + Float32(1.0))) / Float32(Float32(sin2phi / Float32(alphay * alphay)) + Float32(cos2phi / Float32(alphax * alphax))))
end

MATLAB

function tmp = code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = (u0 * ((u0 * single(0.5)) + single(1.0))) / ((sin2phi / (alphay * alphay)) + (cos2phi / (alphax * alphax)));
end

Derivation

1. Initial program 60.9%

   \[\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
2. Add Preprocessing

3. Taylor expanded in u0 around 0

   \[\leadsto \mathsf{/.f32}\left(\color{blue}{\left(u0 \cdot \left(1 + \frac{1}{2} \cdot u0\right)\right)}, \mathsf{+.f32}\left(\mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right), \mathsf{/.f32}\left(sin2phi, \mathsf{*.f32}\left(alphay, alphay\right)\right)\right)\right) \]
4. Step-by-step derivation

   1. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(u0, \left(1 + \frac{1}{2} \cdot u0\right)\right), \mathsf{+.f32}\left(\color{blue}{\mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right)}, \mathsf{/.f32}\left(sin2phi, \mathsf{*.f32}\left(alphay, alphay\right)\right)\right)\right) \]

   2. +-lowering-+.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(1, \left(\frac{1}{2} \cdot u0\right)\right)\right), \mathsf{+.f32}\left(\mathsf{/.f32}\left(cos2phi, \color{blue}{\mathsf{*.f32}\left(alphax, alphax\right)}\right), \mathsf{/.f32}\left(sin2phi, \mathsf{*.f32}\left(alphay, alphay\right)\right)\right)\right) \]

   3. *-commutative N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(1, \left(u0 \cdot \frac{1}{2}\right)\right)\right), \mathsf{+.f32}\left(\mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, \color{blue}{alphax}\right)\right), \mathsf{/.f32}\left(sin2phi, \mathsf{*.f32}\left(alphay, alphay\right)\right)\right)\right) \]

   4. *-lowering-*.f32 86.8%

      \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(u0, \frac{1}{2}\right)\right)\right), \mathsf{+.f32}\left(\mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, \color{blue}{alphax}\right)\right), \mathsf{/.f32}\left(sin2phi, \mathsf{*.f32}\left(alphay, alphay\right)\right)\right)\right) \]

5. Simplified 86.8%

   \[\leadsto \frac{\color{blue}{u0 \cdot \left(1 + u0 \cdot 0.5\right)}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

6. Final simplification 86.8%

   \[\leadsto \frac{u0 \cdot \left(u0 \cdot 0.5 + 1\right)}{\frac{sin2phi}{alphay \cdot alphay} + \frac{cos2phi}{alphax \cdot alphax}} \]
7. Add Preprocessing

Alternative 21: 67.3% accurate, 7.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 2.4000000934092797 \cdot 10^{-16}:\\ \;\;\;\;\frac{u0}{\frac{cos2phi}{alphax \cdot alphax}}\\ \mathbf{else}:\\ \;\;\;\;\left(alphay \cdot alphay\right) \cdot \frac{u0}{sin2phi}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (if (<= (/ sin2phi (* alphay alphay)) 2.4000000934092797e-16)
   (/ u0 (/ cos2phi (* alphax alphax)))
   (* (* alphay alphay) (/ u0 sin2phi))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float tmp;
	if ((sin2phi / (alphay * alphay)) <= 2.4000000934092797e-16f) {
		tmp = u0 / (cos2phi / (alphax * alphax));
	} else {
		tmp = (alphay * alphay) * (u0 / sin2phi);
	}
	return tmp;
}

Fortran

real(4) function code(alphax, alphay, u0, cos2phi, sin2phi)
    real(4), intent (in) :: alphax
    real(4), intent (in) :: alphay
    real(4), intent (in) :: u0
    real(4), intent (in) :: cos2phi
    real(4), intent (in) :: sin2phi
    real(4) :: tmp
    if ((sin2phi / (alphay * alphay)) <= 2.4000000934092797e-16) then
        tmp = u0 / (cos2phi / (alphax * alphax))
    else
        tmp = (alphay * alphay) * (u0 / sin2phi)
    end if
    code = tmp
end function

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = Float32(0.0)
	if (Float32(sin2phi / Float32(alphay * alphay)) <= Float32(2.4000000934092797e-16))
		tmp = Float32(u0 / Float32(cos2phi / Float32(alphax * alphax)));
	else
		tmp = Float32(Float32(alphay * alphay) * Float32(u0 / sin2phi));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = single(0.0);
	if ((sin2phi / (alphay * alphay)) <= single(2.4000000934092797e-16))
		tmp = u0 / (cos2phi / (alphax * alphax));
	else
		tmp = (alphay * alphay) * (u0 / sin2phi);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 sin2phi (*.f32 alphay alphay)) < 2.4000001e-16

   1. Initial program 55.7%

      \[\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
   2. Step-by-step derivation

      1. distribute-frac-neg N/A

         \[\leadsto \mathsf{neg}\left(\frac{\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}}\right) \]

      2. distribute-neg-frac2 N/A

         \[\leadsto \frac{\log \left(1 - u0\right)}{\color{blue}{\mathsf{neg}\left(\left(\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\right)\right)}} \]

      3. /-lowering-/.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(\log \left(1 - u0\right), \color{blue}{\left(\mathsf{neg}\left(\left(\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\right)\right)\right)}\right) \]

      4. sub-neg N/A

         \[\leadsto \mathsf{/.f32}\left(\log \left(1 + \left(\mathsf{neg}\left(u0\right)\right)\right), \left(\mathsf{neg}\left(\left(\color{blue}{\frac{cos2phi}{alphax \cdot alphax}} + \frac{sin2phi}{alphay \cdot alphay}\right)\right)\right)\right) \]

      5. log1p-define N/A

         \[\leadsto \mathsf{/.f32}\left(\left(\mathsf{log1p}\left(\mathsf{neg}\left(u0\right)\right)\right), \left(\mathsf{neg}\left(\color{blue}{\left(\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\right)}\right)\right)\right) \]

      6. log1p-lowering-log1p.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\left(\mathsf{neg}\left(u0\right)\right)\right), \left(\mathsf{neg}\left(\color{blue}{\left(\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\right)}\right)\right)\right) \]

      7. neg-lowering-neg.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \left(\mathsf{neg}\left(\left(\color{blue}{\frac{cos2phi}{alphax \cdot alphax}} + \frac{sin2phi}{alphay \cdot alphay}\right)\right)\right)\right) \]

      8. distribute-neg-in N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \left(\left(\mathsf{neg}\left(\frac{cos2phi}{alphax \cdot alphax}\right)\right) + \color{blue}{\left(\mathsf{neg}\left(\frac{sin2phi}{alphay \cdot alphay}\right)\right)}\right)\right) \]

      9. unsub-neg N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \left(\left(\mathsf{neg}\left(\frac{cos2phi}{alphax \cdot alphax}\right)\right) - \color{blue}{\frac{sin2phi}{alphay \cdot alphay}}\right)\right) \]

      10. --lowering--.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\left(\mathsf{neg}\left(\frac{cos2phi}{alphax \cdot alphax}\right)\right), \color{blue}{\left(\frac{sin2phi}{alphay \cdot alphay}\right)}\right)\right) \]

      11. neg-sub0 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\left(0 - \frac{cos2phi}{alphax \cdot alphax}\right), \left(\frac{\color{blue}{sin2phi}}{alphay \cdot alphay}\right)\right)\right) \]

      12. --lowering--.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \left(\frac{cos2phi}{alphax \cdot alphax}\right)\right), \left(\frac{\color{blue}{sin2phi}}{alphay \cdot alphay}\right)\right)\right) \]

      13. /-lowering-/.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(cos2phi, \left(alphax \cdot alphax\right)\right)\right), \left(\frac{sin2phi}{alphay \cdot alphay}\right)\right)\right) \]

      14. *-lowering-*.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right)\right), \left(\frac{sin2phi}{alphay \cdot alphay}\right)\right)\right) \]

      15. /-lowering-/.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right)\right), \mathsf{/.f32}\left(sin2phi, \color{blue}{\left(alphay \cdot alphay\right)}\right)\right)\right) \]

      16. *-lowering-*.f32 98.7%

          \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right)\right), \mathsf{/.f32}\left(sin2phi, \mathsf{*.f32}\left(alphay, \color{blue}{alphay}\right)\right)\right)\right) \]

   3. Simplified 98.7%

      \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(-u0\right)}{\left(0 - \frac{cos2phi}{alphax \cdot alphax}\right) - \frac{sin2phi}{alphay \cdot alphay}}} \]
   4. Add Preprocessing

   5. Taylor expanded in u0 around 0

      \[\leadsto \color{blue}{\frac{u0}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}}} \]
   6. Step-by-step derivation

      1. /-lowering-/.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(u0, \color{blue}{\left(\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}\right)}\right) \]

      2. +-lowering-+.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{+.f32}\left(\left(\frac{cos2phi}{{alphax}^{2}}\right), \color{blue}{\left(\frac{sin2phi}{{alphay}^{2}}\right)}\right)\right) \]

      3. /-lowering-/.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{/.f32}\left(cos2phi, \left({alphax}^{2}\right)\right), \left(\frac{\color{blue}{sin2phi}}{{alphay}^{2}}\right)\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{/.f32}\left(cos2phi, \left(alphax \cdot alphax\right)\right), \left(\frac{sin2phi}{{alphay}^{2}}\right)\right)\right) \]

      5. *-lowering-*.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right), \left(\frac{sin2phi}{{alphay}^{2}}\right)\right)\right) \]

      6. /-lowering-/.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right), \mathsf{/.f32}\left(sin2phi, \color{blue}{\left({alphay}^{2}\right)}\right)\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right), \mathsf{/.f32}\left(sin2phi, \left(alphay \cdot \color{blue}{alphay}\right)\right)\right)\right) \]

      8. *-lowering-*.f32 73.9%

         \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right), \mathsf{/.f32}\left(sin2phi, \mathsf{*.f32}\left(alphay, \color{blue}{alphay}\right)\right)\right)\right) \]

   7. Simplified 73.9%

      \[\leadsto \color{blue}{\frac{u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}}} \]

   8. Taylor expanded in cos2phi around inf

      \[\leadsto \mathsf{/.f32}\left(u0, \color{blue}{\left(\frac{cos2phi}{{alphax}^{2}}\right)}\right) \]
   9. Step-by-step derivation

      1. /-lowering-/.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{/.f32}\left(cos2phi, \color{blue}{\left({alphax}^{2}\right)}\right)\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{/.f32}\left(cos2phi, \left(alphax \cdot \color{blue}{alphax}\right)\right)\right) \]

      3. *-lowering-*.f32 57.0%

         \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, \color{blue}{alphax}\right)\right)\right) \]

   10. Simplified 57.0%

       \[\leadsto \frac{u0}{\color{blue}{\frac{cos2phi}{alphax \cdot alphax}}} \]

   if 2.4000001e-16 < (/.f32 sin2phi (*.f32 alphay alphay))

   1. Initial program 62.9%

      \[\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
   2. Step-by-step derivation

      1. distribute-frac-neg N/A

         \[\leadsto \mathsf{neg}\left(\frac{\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}}\right) \]

      2. distribute-neg-frac2 N/A

         \[\leadsto \frac{\log \left(1 - u0\right)}{\color{blue}{\mathsf{neg}\left(\left(\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\right)\right)}} \]

      3. /-lowering-/.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(\log \left(1 - u0\right), \color{blue}{\left(\mathsf{neg}\left(\left(\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\right)\right)\right)}\right) \]

      4. sub-neg N/A

         \[\leadsto \mathsf{/.f32}\left(\log \left(1 + \left(\mathsf{neg}\left(u0\right)\right)\right), \left(\mathsf{neg}\left(\left(\color{blue}{\frac{cos2phi}{alphax \cdot alphax}} + \frac{sin2phi}{alphay \cdot alphay}\right)\right)\right)\right) \]

      5. log1p-define N/A

         \[\leadsto \mathsf{/.f32}\left(\left(\mathsf{log1p}\left(\mathsf{neg}\left(u0\right)\right)\right), \left(\mathsf{neg}\left(\color{blue}{\left(\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\right)}\right)\right)\right) \]

      6. log1p-lowering-log1p.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\left(\mathsf{neg}\left(u0\right)\right)\right), \left(\mathsf{neg}\left(\color{blue}{\left(\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\right)}\right)\right)\right) \]

      7. neg-lowering-neg.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \left(\mathsf{neg}\left(\left(\color{blue}{\frac{cos2phi}{alphax \cdot alphax}} + \frac{sin2phi}{alphay \cdot alphay}\right)\right)\right)\right) \]

      8. distribute-neg-in N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \left(\left(\mathsf{neg}\left(\frac{cos2phi}{alphax \cdot alphax}\right)\right) + \color{blue}{\left(\mathsf{neg}\left(\frac{sin2phi}{alphay \cdot alphay}\right)\right)}\right)\right) \]

      9. unsub-neg N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \left(\left(\mathsf{neg}\left(\frac{cos2phi}{alphax \cdot alphax}\right)\right) - \color{blue}{\frac{sin2phi}{alphay \cdot alphay}}\right)\right) \]

      10. --lowering--.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\left(\mathsf{neg}\left(\frac{cos2phi}{alphax \cdot alphax}\right)\right), \color{blue}{\left(\frac{sin2phi}{alphay \cdot alphay}\right)}\right)\right) \]

      11. neg-sub0 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\left(0 - \frac{cos2phi}{alphax \cdot alphax}\right), \left(\frac{\color{blue}{sin2phi}}{alphay \cdot alphay}\right)\right)\right) \]

      12. --lowering--.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \left(\frac{cos2phi}{alphax \cdot alphax}\right)\right), \left(\frac{\color{blue}{sin2phi}}{alphay \cdot alphay}\right)\right)\right) \]

      13. /-lowering-/.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(cos2phi, \left(alphax \cdot alphax\right)\right)\right), \left(\frac{sin2phi}{alphay \cdot alphay}\right)\right)\right) \]

      14. *-lowering-*.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right)\right), \left(\frac{sin2phi}{alphay \cdot alphay}\right)\right)\right) \]

      15. /-lowering-/.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right)\right), \mathsf{/.f32}\left(sin2phi, \color{blue}{\left(alphay \cdot alphay\right)}\right)\right)\right) \]

      16. *-lowering-*.f32 98.3%

          \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right)\right), \mathsf{/.f32}\left(sin2phi, \mathsf{*.f32}\left(alphay, \color{blue}{alphay}\right)\right)\right)\right) \]

   3. Simplified 98.3%

      \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(-u0\right)}{\left(0 - \frac{cos2phi}{alphax \cdot alphax}\right) - \frac{sin2phi}{alphay \cdot alphay}}} \]
   4. Add Preprocessing

   5. Taylor expanded in u0 around 0

      \[\leadsto \color{blue}{\frac{u0}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}}} \]
   6. Step-by-step derivation

      1. /-lowering-/.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(u0, \color{blue}{\left(\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}\right)}\right) \]

      2. +-lowering-+.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{+.f32}\left(\left(\frac{cos2phi}{{alphax}^{2}}\right), \color{blue}{\left(\frac{sin2phi}{{alphay}^{2}}\right)}\right)\right) \]

      3. /-lowering-/.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{/.f32}\left(cos2phi, \left({alphax}^{2}\right)\right), \left(\frac{\color{blue}{sin2phi}}{{alphay}^{2}}\right)\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{/.f32}\left(cos2phi, \left(alphax \cdot alphax\right)\right), \left(\frac{sin2phi}{{alphay}^{2}}\right)\right)\right) \]

      5. *-lowering-*.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right), \left(\frac{sin2phi}{{alphay}^{2}}\right)\right)\right) \]

      6. /-lowering-/.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right), \mathsf{/.f32}\left(sin2phi, \color{blue}{\left({alphay}^{2}\right)}\right)\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right), \mathsf{/.f32}\left(sin2phi, \left(alphay \cdot \color{blue}{alphay}\right)\right)\right)\right) \]

      8. *-lowering-*.f32 74.7%

         \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right), \mathsf{/.f32}\left(sin2phi, \mathsf{*.f32}\left(alphay, \color{blue}{alphay}\right)\right)\right)\right) \]

   7. Simplified 74.7%

      \[\leadsto \color{blue}{\frac{u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}}} \]
   8. Step-by-step derivation

      1. clear-num N/A

         \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right), \left(\frac{1}{\color{blue}{\frac{alphay \cdot alphay}{sin2phi}}}\right)\right)\right) \]

      2. /-lowering-/.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right), \mathsf{/.f32}\left(1, \color{blue}{\left(\frac{alphay \cdot alphay}{sin2phi}\right)}\right)\right)\right) \]

      3. associate-/l* N/A

         \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right), \mathsf{/.f32}\left(1, \left(alphay \cdot \color{blue}{\frac{alphay}{sin2phi}}\right)\right)\right)\right) \]

      4. *-lowering-*.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right), \mathsf{/.f32}\left(1, \mathsf{*.f32}\left(alphay, \color{blue}{\left(\frac{alphay}{sin2phi}\right)}\right)\right)\right)\right) \]

      5. /-lowering-/.f32 74.8%

         \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right), \mathsf{/.f32}\left(1, \mathsf{*.f32}\left(alphay, \mathsf{/.f32}\left(alphay, \color{blue}{sin2phi}\right)\right)\right)\right)\right) \]

   9. Applied egg-rr 74.8%

      \[\leadsto \frac{u0}{\frac{cos2phi}{alphax \cdot alphax} + \color{blue}{\frac{1}{alphay \cdot \frac{alphay}{sin2phi}}}} \]

   10. Taylor expanded in cos2phi around 0

       \[\leadsto \color{blue}{\frac{{alphay}^{2} \cdot u0}{sin2phi}} \]
   11. Step-by-step derivation

       1. associate-/l* N/A

          \[\leadsto {alphay}^{2} \cdot \color{blue}{\frac{u0}{sin2phi}} \]

       2. *-lowering-*.f32 N/A

          \[\leadsto \mathsf{*.f32}\left(\left({alphay}^{2}\right), \color{blue}{\left(\frac{u0}{sin2phi}\right)}\right) \]

       3. unpow2 N/A

          \[\leadsto \mathsf{*.f32}\left(\left(alphay \cdot alphay\right), \left(\frac{\color{blue}{u0}}{sin2phi}\right)\right) \]

       4. *-lowering-*.f32 N/A

          \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(alphay, alphay\right), \left(\frac{\color{blue}{u0}}{sin2phi}\right)\right) \]

       5. /-lowering-/.f32 67.5%

          \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(alphay, alphay\right), \mathsf{/.f32}\left(u0, \color{blue}{sin2phi}\right)\right) \]

   12. Simplified 67.5%

       \[\leadsto \color{blue}{\left(alphay \cdot alphay\right) \cdot \frac{u0}{sin2phi}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 22: 75.9% accurate, 8.9× speedup

Math

\[\begin{array}{l} \\ \frac{u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{\frac{sin2phi}{alphay}}{alphay}} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (/ u0 (+ (/ cos2phi (* alphax alphax)) (/ (/ sin2phi alphay) alphay))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	return u0 / ((cos2phi / (alphax * alphax)) + ((sin2phi / alphay) / alphay));
}

Fortran

real(4) function code(alphax, alphay, u0, cos2phi, sin2phi)
    real(4), intent (in) :: alphax
    real(4), intent (in) :: alphay
    real(4), intent (in) :: u0
    real(4), intent (in) :: cos2phi
    real(4), intent (in) :: sin2phi
    code = u0 / ((cos2phi / (alphax * alphax)) + ((sin2phi / alphay) / alphay))
end function

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	return Float32(u0 / Float32(Float32(cos2phi / Float32(alphax * alphax)) + Float32(Float32(sin2phi / alphay) / alphay)))
end

MATLAB

function tmp = code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = u0 / ((cos2phi / (alphax * alphax)) + ((sin2phi / alphay) / alphay));
end

Derivation

1. Initial program 60.9%

   \[\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
2. Step-by-step derivation

   1. distribute-frac-neg N/A

      \[\leadsto \mathsf{neg}\left(\frac{\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}}\right) \]

   2. distribute-neg-frac2 N/A

      \[\leadsto \frac{\log \left(1 - u0\right)}{\color{blue}{\mathsf{neg}\left(\left(\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\right)\right)}} \]

   3. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\log \left(1 - u0\right), \color{blue}{\left(\mathsf{neg}\left(\left(\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\right)\right)\right)}\right) \]

   4. sub-neg N/A

      \[\leadsto \mathsf{/.f32}\left(\log \left(1 + \left(\mathsf{neg}\left(u0\right)\right)\right), \left(\mathsf{neg}\left(\left(\color{blue}{\frac{cos2phi}{alphax \cdot alphax}} + \frac{sin2phi}{alphay \cdot alphay}\right)\right)\right)\right) \]

   5. log1p-define N/A

      \[\leadsto \mathsf{/.f32}\left(\left(\mathsf{log1p}\left(\mathsf{neg}\left(u0\right)\right)\right), \left(\mathsf{neg}\left(\color{blue}{\left(\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\right)}\right)\right)\right) \]

   6. log1p-lowering-log1p.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\left(\mathsf{neg}\left(u0\right)\right)\right), \left(\mathsf{neg}\left(\color{blue}{\left(\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\right)}\right)\right)\right) \]

   7. neg-lowering-neg.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \left(\mathsf{neg}\left(\left(\color{blue}{\frac{cos2phi}{alphax \cdot alphax}} + \frac{sin2phi}{alphay \cdot alphay}\right)\right)\right)\right) \]

   8. distribute-neg-in N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \left(\left(\mathsf{neg}\left(\frac{cos2phi}{alphax \cdot alphax}\right)\right) + \color{blue}{\left(\mathsf{neg}\left(\frac{sin2phi}{alphay \cdot alphay}\right)\right)}\right)\right) \]

   9. unsub-neg N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \left(\left(\mathsf{neg}\left(\frac{cos2phi}{alphax \cdot alphax}\right)\right) - \color{blue}{\frac{sin2phi}{alphay \cdot alphay}}\right)\right) \]

   10. --lowering--.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\left(\mathsf{neg}\left(\frac{cos2phi}{alphax \cdot alphax}\right)\right), \color{blue}{\left(\frac{sin2phi}{alphay \cdot alphay}\right)}\right)\right) \]

   11. neg-sub0 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\left(0 - \frac{cos2phi}{alphax \cdot alphax}\right), \left(\frac{\color{blue}{sin2phi}}{alphay \cdot alphay}\right)\right)\right) \]

   12. --lowering--.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \left(\frac{cos2phi}{alphax \cdot alphax}\right)\right), \left(\frac{\color{blue}{sin2phi}}{alphay \cdot alphay}\right)\right)\right) \]

   13. /-lowering-/.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(cos2phi, \left(alphax \cdot alphax\right)\right)\right), \left(\frac{sin2phi}{alphay \cdot alphay}\right)\right)\right) \]

   14. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right)\right), \left(\frac{sin2phi}{alphay \cdot alphay}\right)\right)\right) \]

   15. /-lowering-/.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right)\right), \mathsf{/.f32}\left(sin2phi, \color{blue}{\left(alphay \cdot alphay\right)}\right)\right)\right) \]

   16. *-lowering-*.f32 98.4%

       \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right)\right), \mathsf{/.f32}\left(sin2phi, \mathsf{*.f32}\left(alphay, \color{blue}{alphay}\right)\right)\right)\right) \]

3. Simplified 98.4%

   \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(-u0\right)}{\left(0 - \frac{cos2phi}{alphax \cdot alphax}\right) - \frac{sin2phi}{alphay \cdot alphay}}} \]
4. Add Preprocessing

5. Taylor expanded in u0 around 0

   \[\leadsto \color{blue}{\frac{u0}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}}} \]
6. Step-by-step derivation

   1. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(u0, \color{blue}{\left(\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}\right)}\right) \]

   2. +-lowering-+.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{+.f32}\left(\left(\frac{cos2phi}{{alphax}^{2}}\right), \color{blue}{\left(\frac{sin2phi}{{alphay}^{2}}\right)}\right)\right) \]

   3. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{/.f32}\left(cos2phi, \left({alphax}^{2}\right)\right), \left(\frac{\color{blue}{sin2phi}}{{alphay}^{2}}\right)\right)\right) \]

   4. unpow2 N/A

      \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{/.f32}\left(cos2phi, \left(alphax \cdot alphax\right)\right), \left(\frac{sin2phi}{{alphay}^{2}}\right)\right)\right) \]

   5. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right), \left(\frac{sin2phi}{{alphay}^{2}}\right)\right)\right) \]

   6. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right), \mathsf{/.f32}\left(sin2phi, \color{blue}{\left({alphay}^{2}\right)}\right)\right)\right) \]

   7. unpow2 N/A

      \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right), \mathsf{/.f32}\left(sin2phi, \left(alphay \cdot \color{blue}{alphay}\right)\right)\right)\right) \]

   8. *-lowering-*.f32 74.5%

      \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right), \mathsf{/.f32}\left(sin2phi, \mathsf{*.f32}\left(alphay, \color{blue}{alphay}\right)\right)\right)\right) \]

7. Simplified 74.5%

   \[\leadsto \color{blue}{\frac{u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}}} \]
8. Step-by-step derivation

   1. associate-/r* N/A

      \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right), \left(\frac{\frac{sin2phi}{alphay}}{\color{blue}{alphay}}\right)\right)\right) \]

   2. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right), \mathsf{/.f32}\left(\left(\frac{sin2phi}{alphay}\right), \color{blue}{alphay}\right)\right)\right) \]

   3. /-lowering-/.f32 74.6%

      \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(sin2phi, alphay\right), alphay\right)\right)\right) \]

9. Applied egg-rr 74.6%

   \[\leadsto \frac{u0}{\frac{cos2phi}{alphax \cdot alphax} + \color{blue}{\frac{\frac{sin2phi}{alphay}}{alphay}}} \]
10. Add Preprocessing

Alternative 23: 76.0% accurate, 8.9× speedup

Math

\[\begin{array}{l} \\ \frac{u0}{\frac{sin2phi}{alphay \cdot alphay} + \frac{cos2phi}{alphax \cdot alphax}} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (/ u0 (+ (/ sin2phi (* alphay alphay)) (/ cos2phi (* alphax alphax)))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	return u0 / ((sin2phi / (alphay * alphay)) + (cos2phi / (alphax * alphax)));
}

Fortran

real(4) function code(alphax, alphay, u0, cos2phi, sin2phi)
    real(4), intent (in) :: alphax
    real(4), intent (in) :: alphay
    real(4), intent (in) :: u0
    real(4), intent (in) :: cos2phi
    real(4), intent (in) :: sin2phi
    code = u0 / ((sin2phi / (alphay * alphay)) + (cos2phi / (alphax * alphax)))
end function

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	return Float32(u0 / Float32(Float32(sin2phi / Float32(alphay * alphay)) + Float32(cos2phi / Float32(alphax * alphax))))
end

MATLAB

function tmp = code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = u0 / ((sin2phi / (alphay * alphay)) + (cos2phi / (alphax * alphax)));
end

Derivation

1. Initial program 60.9%

   \[\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
2. Step-by-step derivation

   1. distribute-frac-neg N/A

      \[\leadsto \mathsf{neg}\left(\frac{\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}}\right) \]

   2. distribute-neg-frac2 N/A

      \[\leadsto \frac{\log \left(1 - u0\right)}{\color{blue}{\mathsf{neg}\left(\left(\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\right)\right)}} \]

   3. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\log \left(1 - u0\right), \color{blue}{\left(\mathsf{neg}\left(\left(\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\right)\right)\right)}\right) \]

   4. sub-neg N/A

      \[\leadsto \mathsf{/.f32}\left(\log \left(1 + \left(\mathsf{neg}\left(u0\right)\right)\right), \left(\mathsf{neg}\left(\left(\color{blue}{\frac{cos2phi}{alphax \cdot alphax}} + \frac{sin2phi}{alphay \cdot alphay}\right)\right)\right)\right) \]

   5. log1p-define N/A

      \[\leadsto \mathsf{/.f32}\left(\left(\mathsf{log1p}\left(\mathsf{neg}\left(u0\right)\right)\right), \left(\mathsf{neg}\left(\color{blue}{\left(\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\right)}\right)\right)\right) \]

   6. log1p-lowering-log1p.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\left(\mathsf{neg}\left(u0\right)\right)\right), \left(\mathsf{neg}\left(\color{blue}{\left(\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\right)}\right)\right)\right) \]

   7. neg-lowering-neg.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \left(\mathsf{neg}\left(\left(\color{blue}{\frac{cos2phi}{alphax \cdot alphax}} + \frac{sin2phi}{alphay \cdot alphay}\right)\right)\right)\right) \]

   8. distribute-neg-in N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \left(\left(\mathsf{neg}\left(\frac{cos2phi}{alphax \cdot alphax}\right)\right) + \color{blue}{\left(\mathsf{neg}\left(\frac{sin2phi}{alphay \cdot alphay}\right)\right)}\right)\right) \]

   9. unsub-neg N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \left(\left(\mathsf{neg}\left(\frac{cos2phi}{alphax \cdot alphax}\right)\right) - \color{blue}{\frac{sin2phi}{alphay \cdot alphay}}\right)\right) \]

   10. --lowering--.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\left(\mathsf{neg}\left(\frac{cos2phi}{alphax \cdot alphax}\right)\right), \color{blue}{\left(\frac{sin2phi}{alphay \cdot alphay}\right)}\right)\right) \]

   11. neg-sub0 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\left(0 - \frac{cos2phi}{alphax \cdot alphax}\right), \left(\frac{\color{blue}{sin2phi}}{alphay \cdot alphay}\right)\right)\right) \]

   12. --lowering--.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \left(\frac{cos2phi}{alphax \cdot alphax}\right)\right), \left(\frac{\color{blue}{sin2phi}}{alphay \cdot alphay}\right)\right)\right) \]

   13. /-lowering-/.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(cos2phi, \left(alphax \cdot alphax\right)\right)\right), \left(\frac{sin2phi}{alphay \cdot alphay}\right)\right)\right) \]

   14. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right)\right), \left(\frac{sin2phi}{alphay \cdot alphay}\right)\right)\right) \]

   15. /-lowering-/.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right)\right), \mathsf{/.f32}\left(sin2phi, \color{blue}{\left(alphay \cdot alphay\right)}\right)\right)\right) \]

   16. *-lowering-*.f32 98.4%

       \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right)\right), \mathsf{/.f32}\left(sin2phi, \mathsf{*.f32}\left(alphay, \color{blue}{alphay}\right)\right)\right)\right) \]

3. Simplified 98.4%

   \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(-u0\right)}{\left(0 - \frac{cos2phi}{alphax \cdot alphax}\right) - \frac{sin2phi}{alphay \cdot alphay}}} \]
4. Add Preprocessing

5. Taylor expanded in u0 around 0

   \[\leadsto \color{blue}{\frac{u0}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}}} \]
6. Step-by-step derivation

   1. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(u0, \color{blue}{\left(\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}\right)}\right) \]

   2. +-lowering-+.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{+.f32}\left(\left(\frac{cos2phi}{{alphax}^{2}}\right), \color{blue}{\left(\frac{sin2phi}{{alphay}^{2}}\right)}\right)\right) \]

   3. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{/.f32}\left(cos2phi, \left({alphax}^{2}\right)\right), \left(\frac{\color{blue}{sin2phi}}{{alphay}^{2}}\right)\right)\right) \]

   4. unpow2 N/A

      \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{/.f32}\left(cos2phi, \left(alphax \cdot alphax\right)\right), \left(\frac{sin2phi}{{alphay}^{2}}\right)\right)\right) \]

   5. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right), \left(\frac{sin2phi}{{alphay}^{2}}\right)\right)\right) \]

   6. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right), \mathsf{/.f32}\left(sin2phi, \color{blue}{\left({alphay}^{2}\right)}\right)\right)\right) \]

   7. unpow2 N/A

      \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right), \mathsf{/.f32}\left(sin2phi, \left(alphay \cdot \color{blue}{alphay}\right)\right)\right)\right) \]

   8. *-lowering-*.f32 74.5%

      \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right), \mathsf{/.f32}\left(sin2phi, \mathsf{*.f32}\left(alphay, \color{blue}{alphay}\right)\right)\right)\right) \]

7. Simplified 74.5%

   \[\leadsto \color{blue}{\frac{u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}}} \]

8. Final simplification 74.5%

   \[\leadsto \frac{u0}{\frac{sin2phi}{alphay \cdot alphay} + \frac{cos2phi}{alphax \cdot alphax}} \]
9. Add Preprocessing

Alternative 24: 67.3% accurate, 9.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;sin2phi \leq 4.9999998413276127 \cdot 10^{-20}:\\ \;\;\;\;alphax \cdot \frac{u0}{\frac{cos2phi}{alphax}}\\ \mathbf{else}:\\ \;\;\;\;\left(alphay \cdot alphay\right) \cdot \frac{u0}{sin2phi}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (if (<= sin2phi 4.9999998413276127e-20)
   (* alphax (/ u0 (/ cos2phi alphax)))
   (* (* alphay alphay) (/ u0 sin2phi))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float tmp;
	if (sin2phi <= 4.9999998413276127e-20f) {
		tmp = alphax * (u0 / (cos2phi / alphax));
	} else {
		tmp = (alphay * alphay) * (u0 / sin2phi);
	}
	return tmp;
}

Fortran

real(4) function code(alphax, alphay, u0, cos2phi, sin2phi)
    real(4), intent (in) :: alphax
    real(4), intent (in) :: alphay
    real(4), intent (in) :: u0
    real(4), intent (in) :: cos2phi
    real(4), intent (in) :: sin2phi
    real(4) :: tmp
    if (sin2phi <= 4.9999998413276127e-20) then
        tmp = alphax * (u0 / (cos2phi / alphax))
    else
        tmp = (alphay * alphay) * (u0 / sin2phi)
    end if
    code = tmp
end function

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = Float32(0.0)
	if (sin2phi <= Float32(4.9999998413276127e-20))
		tmp = Float32(alphax * Float32(u0 / Float32(cos2phi / alphax)));
	else
		tmp = Float32(Float32(alphay * alphay) * Float32(u0 / sin2phi));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = single(0.0);
	if (sin2phi <= single(4.9999998413276127e-20))
		tmp = alphax * (u0 / (cos2phi / alphax));
	else
		tmp = (alphay * alphay) * (u0 / sin2phi);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if sin2phi < 4.99999984e-20

   1. Initial program 55.8%

      \[\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
   2. Step-by-step derivation

      1. distribute-frac-neg N/A

         \[\leadsto \mathsf{neg}\left(\frac{\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}}\right) \]

      2. distribute-neg-frac2 N/A

         \[\leadsto \frac{\log \left(1 - u0\right)}{\color{blue}{\mathsf{neg}\left(\left(\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\right)\right)}} \]

      3. /-lowering-/.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(\log \left(1 - u0\right), \color{blue}{\left(\mathsf{neg}\left(\left(\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\right)\right)\right)}\right) \]

      4. sub-neg N/A

         \[\leadsto \mathsf{/.f32}\left(\log \left(1 + \left(\mathsf{neg}\left(u0\right)\right)\right), \left(\mathsf{neg}\left(\left(\color{blue}{\frac{cos2phi}{alphax \cdot alphax}} + \frac{sin2phi}{alphay \cdot alphay}\right)\right)\right)\right) \]

      5. log1p-define N/A

         \[\leadsto \mathsf{/.f32}\left(\left(\mathsf{log1p}\left(\mathsf{neg}\left(u0\right)\right)\right), \left(\mathsf{neg}\left(\color{blue}{\left(\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\right)}\right)\right)\right) \]

      6. log1p-lowering-log1p.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\left(\mathsf{neg}\left(u0\right)\right)\right), \left(\mathsf{neg}\left(\color{blue}{\left(\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\right)}\right)\right)\right) \]

      7. neg-lowering-neg.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \left(\mathsf{neg}\left(\left(\color{blue}{\frac{cos2phi}{alphax \cdot alphax}} + \frac{sin2phi}{alphay \cdot alphay}\right)\right)\right)\right) \]

      8. distribute-neg-in N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \left(\left(\mathsf{neg}\left(\frac{cos2phi}{alphax \cdot alphax}\right)\right) + \color{blue}{\left(\mathsf{neg}\left(\frac{sin2phi}{alphay \cdot alphay}\right)\right)}\right)\right) \]

      9. unsub-neg N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \left(\left(\mathsf{neg}\left(\frac{cos2phi}{alphax \cdot alphax}\right)\right) - \color{blue}{\frac{sin2phi}{alphay \cdot alphay}}\right)\right) \]

      10. --lowering--.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\left(\mathsf{neg}\left(\frac{cos2phi}{alphax \cdot alphax}\right)\right), \color{blue}{\left(\frac{sin2phi}{alphay \cdot alphay}\right)}\right)\right) \]

      11. neg-sub0 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\left(0 - \frac{cos2phi}{alphax \cdot alphax}\right), \left(\frac{\color{blue}{sin2phi}}{alphay \cdot alphay}\right)\right)\right) \]

      12. --lowering--.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \left(\frac{cos2phi}{alphax \cdot alphax}\right)\right), \left(\frac{\color{blue}{sin2phi}}{alphay \cdot alphay}\right)\right)\right) \]

      13. /-lowering-/.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(cos2phi, \left(alphax \cdot alphax\right)\right)\right), \left(\frac{sin2phi}{alphay \cdot alphay}\right)\right)\right) \]

      14. *-lowering-*.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right)\right), \left(\frac{sin2phi}{alphay \cdot alphay}\right)\right)\right) \]

      15. /-lowering-/.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right)\right), \mathsf{/.f32}\left(sin2phi, \color{blue}{\left(alphay \cdot alphay\right)}\right)\right)\right) \]

      16. *-lowering-*.f32 98.8%

          \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right)\right), \mathsf{/.f32}\left(sin2phi, \mathsf{*.f32}\left(alphay, \color{blue}{alphay}\right)\right)\right)\right) \]

   3. Simplified 98.8%

      \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(-u0\right)}{\left(0 - \frac{cos2phi}{alphax \cdot alphax}\right) - \frac{sin2phi}{alphay \cdot alphay}}} \]
   4. Add Preprocessing

   5. Taylor expanded in u0 around 0

      \[\leadsto \color{blue}{\frac{u0}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}}} \]
   6. Step-by-step derivation

      1. /-lowering-/.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(u0, \color{blue}{\left(\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}\right)}\right) \]

      2. +-lowering-+.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{+.f32}\left(\left(\frac{cos2phi}{{alphax}^{2}}\right), \color{blue}{\left(\frac{sin2phi}{{alphay}^{2}}\right)}\right)\right) \]

      3. /-lowering-/.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{/.f32}\left(cos2phi, \left({alphax}^{2}\right)\right), \left(\frac{\color{blue}{sin2phi}}{{alphay}^{2}}\right)\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{/.f32}\left(cos2phi, \left(alphax \cdot alphax\right)\right), \left(\frac{sin2phi}{{alphay}^{2}}\right)\right)\right) \]

      5. *-lowering-*.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right), \left(\frac{sin2phi}{{alphay}^{2}}\right)\right)\right) \]

      6. /-lowering-/.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right), \mathsf{/.f32}\left(sin2phi, \color{blue}{\left({alphay}^{2}\right)}\right)\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right), \mathsf{/.f32}\left(sin2phi, \left(alphay \cdot \color{blue}{alphay}\right)\right)\right)\right) \]

      8. *-lowering-*.f32 73.3%

         \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right), \mathsf{/.f32}\left(sin2phi, \mathsf{*.f32}\left(alphay, \color{blue}{alphay}\right)\right)\right)\right) \]

   7. Simplified 73.3%

      \[\leadsto \color{blue}{\frac{u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}}} \]

   8. Taylor expanded in cos2phi around inf

      \[\leadsto \color{blue}{\frac{{alphax}^{2} \cdot u0}{cos2phi}} \]
   9. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto {alphax}^{2} \cdot \color{blue}{\frac{u0}{cos2phi}} \]

      2. *-lowering-*.f32 N/A

         \[\leadsto \mathsf{*.f32}\left(\left({alphax}^{2}\right), \color{blue}{\left(\frac{u0}{cos2phi}\right)}\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{*.f32}\left(\left(alphax \cdot alphax\right), \left(\frac{\color{blue}{u0}}{cos2phi}\right)\right) \]

      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 55.3%

         \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(alphax, alphax\right), \mathsf{/.f32}\left(u0, \color{blue}{cos2phi}\right)\right) \]

   10. Simplified 55.3%

       \[\leadsto \color{blue}{\left(alphax \cdot alphax\right) \cdot \frac{u0}{cos2phi}} \]
   11. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \frac{u0}{cos2phi} \cdot \color{blue}{\left(alphax \cdot alphax\right)} \]

       2. div-inv N/A

          \[\leadsto \left(u0 \cdot \frac{1}{cos2phi}\right) \cdot \left(\color{blue}{alphax} \cdot alphax\right) \]

       3. associate-*l* N/A

          \[\leadsto u0 \cdot \color{blue}{\left(\frac{1}{cos2phi} \cdot \left(alphax \cdot alphax\right)\right)} \]

       4. /-rgt-identity N/A

          \[\leadsto u0 \cdot \left(\frac{1}{cos2phi} \cdot \left(\frac{alphax}{1} \cdot alphax\right)\right) \]

       5. /-rgt-identity N/A

          \[\leadsto u0 \cdot \left(\frac{1}{cos2phi} \cdot \left(\frac{alphax}{1} \cdot \frac{alphax}{\color{blue}{1}}\right)\right) \]

       6. clear-num N/A

          \[\leadsto u0 \cdot \left(\frac{1}{cos2phi} \cdot \left(\frac{alphax}{1} \cdot \frac{1}{\color{blue}{\frac{1}{alphax}}}\right)\right) \]

       7. times-frac N/A

          \[\leadsto u0 \cdot \left(\frac{1}{cos2phi} \cdot \frac{alphax \cdot 1}{\color{blue}{1 \cdot \frac{1}{alphax}}}\right) \]

       8. metadata-eval N/A

          \[\leadsto u0 \cdot \left(\frac{1}{cos2phi} \cdot \frac{alphax \cdot \frac{1}{1}}{1 \cdot \frac{1}{alphax}}\right) \]

       9. div-inv N/A

          \[\leadsto u0 \cdot \left(\frac{1}{cos2phi} \cdot \frac{\frac{alphax}{1}}{\color{blue}{1} \cdot \frac{1}{alphax}}\right) \]

       10. /-rgt-identity N/A

           \[\leadsto u0 \cdot \left(\frac{1}{cos2phi} \cdot \frac{alphax}{\color{blue}{1} \cdot \frac{1}{alphax}}\right) \]

       11. div-inv N/A

           \[\leadsto u0 \cdot \left(\frac{1}{cos2phi} \cdot \frac{alphax}{\frac{1}{\color{blue}{alphax}}}\right) \]

       12. times-frac N/A

           \[\leadsto u0 \cdot \frac{1 \cdot alphax}{\color{blue}{cos2phi \cdot \frac{1}{alphax}}} \]

       13. div-inv N/A

           \[\leadsto u0 \cdot \frac{1 \cdot alphax}{\frac{cos2phi}{\color{blue}{alphax}}} \]

       14. associate-*l/ N/A

           \[\leadsto u0 \cdot \left(\frac{1}{\frac{cos2phi}{alphax}} \cdot \color{blue}{alphax}\right) \]

       15. clear-num N/A

           \[\leadsto u0 \cdot \left(\frac{alphax}{cos2phi} \cdot alphax\right) \]

       16. *-lowering-*.f32 N/A

           \[\leadsto \mathsf{*.f32}\left(u0, \color{blue}{\left(\frac{alphax}{cos2phi} \cdot alphax\right)}\right) \]

       17. associate-*l/ N/A

           \[\leadsto \mathsf{*.f32}\left(u0, \left(\frac{alphax \cdot alphax}{\color{blue}{cos2phi}}\right)\right) \]

       18. clear-num N/A

           \[\leadsto \mathsf{*.f32}\left(u0, \left(\frac{1}{\color{blue}{\frac{cos2phi}{alphax \cdot alphax}}}\right)\right) \]

       19. associate-/r* N/A

           \[\leadsto \mathsf{*.f32}\left(u0, \left(\frac{1}{\frac{\frac{cos2phi}{alphax}}{\color{blue}{alphax}}}\right)\right) \]

       20. clear-num N/A

           \[\leadsto \mathsf{*.f32}\left(u0, \left(\frac{alphax}{\color{blue}{\frac{cos2phi}{alphax}}}\right)\right) \]

       21. /-lowering-/.f32 N/A

           \[\leadsto \mathsf{*.f32}\left(u0, \mathsf{/.f32}\left(alphax, \color{blue}{\left(\frac{cos2phi}{alphax}\right)}\right)\right) \]

       22. /-lowering-/.f32 55.3%

           \[\leadsto \mathsf{*.f32}\left(u0, \mathsf{/.f32}\left(alphax, \mathsf{/.f32}\left(cos2phi, \color{blue}{alphax}\right)\right)\right) \]

   12. Applied egg-rr 55.3%

       \[\leadsto \color{blue}{u0 \cdot \frac{alphax}{\frac{cos2phi}{alphax}}} \]
   13. Step-by-step derivation

       1. associate-/r/ N/A

          \[\leadsto u0 \cdot \left(\frac{alphax}{cos2phi} \cdot \color{blue}{alphax}\right) \]

       2. associate-*r* N/A

          \[\leadsto \left(u0 \cdot \frac{alphax}{cos2phi}\right) \cdot \color{blue}{alphax} \]

       3. clear-num N/A

          \[\leadsto \left(u0 \cdot \frac{1}{\frac{cos2phi}{alphax}}\right) \cdot alphax \]

       4. un-div-inv N/A

          \[\leadsto \frac{u0}{\frac{cos2phi}{alphax}} \cdot alphax \]

       5. *-lowering-*.f32 N/A

          \[\leadsto \mathsf{*.f32}\left(\left(\frac{u0}{\frac{cos2phi}{alphax}}\right), \color{blue}{alphax}\right) \]

       6. /-lowering-/.f32 N/A

          \[\leadsto \mathsf{*.f32}\left(\mathsf{/.f32}\left(u0, \left(\frac{cos2phi}{alphax}\right)\right), alphax\right) \]

       7. /-lowering-/.f32 55.3%

          \[\leadsto \mathsf{*.f32}\left(\mathsf{/.f32}\left(u0, \mathsf{/.f32}\left(cos2phi, alphax\right)\right), alphax\right) \]

   14. Applied egg-rr 55.3%

       \[\leadsto \color{blue}{\frac{u0}{\frac{cos2phi}{alphax}} \cdot alphax} \]

   if 4.99999984e-20 < sin2phi

   1. Initial program 62.8%

      \[\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
   2. Step-by-step derivation

      1. distribute-frac-neg N/A

         \[\leadsto \mathsf{neg}\left(\frac{\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}}\right) \]

      2. distribute-neg-frac2 N/A

         \[\leadsto \frac{\log \left(1 - u0\right)}{\color{blue}{\mathsf{neg}\left(\left(\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\right)\right)}} \]

      3. /-lowering-/.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(\log \left(1 - u0\right), \color{blue}{\left(\mathsf{neg}\left(\left(\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\right)\right)\right)}\right) \]

      4. sub-neg N/A

         \[\leadsto \mathsf{/.f32}\left(\log \left(1 + \left(\mathsf{neg}\left(u0\right)\right)\right), \left(\mathsf{neg}\left(\left(\color{blue}{\frac{cos2phi}{alphax \cdot alphax}} + \frac{sin2phi}{alphay \cdot alphay}\right)\right)\right)\right) \]

      5. log1p-define N/A

         \[\leadsto \mathsf{/.f32}\left(\left(\mathsf{log1p}\left(\mathsf{neg}\left(u0\right)\right)\right), \left(\mathsf{neg}\left(\color{blue}{\left(\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\right)}\right)\right)\right) \]

      6. log1p-lowering-log1p.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\left(\mathsf{neg}\left(u0\right)\right)\right), \left(\mathsf{neg}\left(\color{blue}{\left(\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\right)}\right)\right)\right) \]

      7. neg-lowering-neg.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \left(\mathsf{neg}\left(\left(\color{blue}{\frac{cos2phi}{alphax \cdot alphax}} + \frac{sin2phi}{alphay \cdot alphay}\right)\right)\right)\right) \]

      8. distribute-neg-in N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \left(\left(\mathsf{neg}\left(\frac{cos2phi}{alphax \cdot alphax}\right)\right) + \color{blue}{\left(\mathsf{neg}\left(\frac{sin2phi}{alphay \cdot alphay}\right)\right)}\right)\right) \]

      9. unsub-neg N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \left(\left(\mathsf{neg}\left(\frac{cos2phi}{alphax \cdot alphax}\right)\right) - \color{blue}{\frac{sin2phi}{alphay \cdot alphay}}\right)\right) \]

      10. --lowering--.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\left(\mathsf{neg}\left(\frac{cos2phi}{alphax \cdot alphax}\right)\right), \color{blue}{\left(\frac{sin2phi}{alphay \cdot alphay}\right)}\right)\right) \]

      11. neg-sub0 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\left(0 - \frac{cos2phi}{alphax \cdot alphax}\right), \left(\frac{\color{blue}{sin2phi}}{alphay \cdot alphay}\right)\right)\right) \]

      12. --lowering--.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \left(\frac{cos2phi}{alphax \cdot alphax}\right)\right), \left(\frac{\color{blue}{sin2phi}}{alphay \cdot alphay}\right)\right)\right) \]

      13. /-lowering-/.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(cos2phi, \left(alphax \cdot alphax\right)\right)\right), \left(\frac{sin2phi}{alphay \cdot alphay}\right)\right)\right) \]

      14. *-lowering-*.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right)\right), \left(\frac{sin2phi}{alphay \cdot alphay}\right)\right)\right) \]

      15. /-lowering-/.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right)\right), \mathsf{/.f32}\left(sin2phi, \color{blue}{\left(alphay \cdot alphay\right)}\right)\right)\right) \]

      16. *-lowering-*.f32 98.3%

          \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right)\right), \mathsf{/.f32}\left(sin2phi, \mathsf{*.f32}\left(alphay, \color{blue}{alphay}\right)\right)\right)\right) \]

   3. Simplified 98.3%

      \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(-u0\right)}{\left(0 - \frac{cos2phi}{alphax \cdot alphax}\right) - \frac{sin2phi}{alphay \cdot alphay}}} \]
   4. Add Preprocessing

   5. Taylor expanded in u0 around 0

      \[\leadsto \color{blue}{\frac{u0}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}}} \]
   6. Step-by-step derivation

      1. /-lowering-/.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(u0, \color{blue}{\left(\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}\right)}\right) \]

      2. +-lowering-+.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{+.f32}\left(\left(\frac{cos2phi}{{alphax}^{2}}\right), \color{blue}{\left(\frac{sin2phi}{{alphay}^{2}}\right)}\right)\right) \]

      3. /-lowering-/.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{/.f32}\left(cos2phi, \left({alphax}^{2}\right)\right), \left(\frac{\color{blue}{sin2phi}}{{alphay}^{2}}\right)\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{/.f32}\left(cos2phi, \left(alphax \cdot alphax\right)\right), \left(\frac{sin2phi}{{alphay}^{2}}\right)\right)\right) \]

      5. *-lowering-*.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right), \left(\frac{sin2phi}{{alphay}^{2}}\right)\right)\right) \]

      6. /-lowering-/.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right), \mathsf{/.f32}\left(sin2phi, \color{blue}{\left({alphay}^{2}\right)}\right)\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right), \mathsf{/.f32}\left(sin2phi, \left(alphay \cdot \color{blue}{alphay}\right)\right)\right)\right) \]

      8. *-lowering-*.f32 75.0%

         \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right), \mathsf{/.f32}\left(sin2phi, \mathsf{*.f32}\left(alphay, \color{blue}{alphay}\right)\right)\right)\right) \]

   7. Simplified 75.0%

      \[\leadsto \color{blue}{\frac{u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}}} \]
   8. Step-by-step derivation

      1. clear-num N/A

         \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right), \left(\frac{1}{\color{blue}{\frac{alphay \cdot alphay}{sin2phi}}}\right)\right)\right) \]

      2. /-lowering-/.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right), \mathsf{/.f32}\left(1, \color{blue}{\left(\frac{alphay \cdot alphay}{sin2phi}\right)}\right)\right)\right) \]

      3. associate-/l* N/A

         \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right), \mathsf{/.f32}\left(1, \left(alphay \cdot \color{blue}{\frac{alphay}{sin2phi}}\right)\right)\right)\right) \]

      4. *-lowering-*.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right), \mathsf{/.f32}\left(1, \mathsf{*.f32}\left(alphay, \color{blue}{\left(\frac{alphay}{sin2phi}\right)}\right)\right)\right)\right) \]

      5. /-lowering-/.f32 75.0%

         \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right), \mathsf{/.f32}\left(1, \mathsf{*.f32}\left(alphay, \mathsf{/.f32}\left(alphay, \color{blue}{sin2phi}\right)\right)\right)\right)\right) \]

   9. Applied egg-rr 75.0%

      \[\leadsto \frac{u0}{\frac{cos2phi}{alphax \cdot alphax} + \color{blue}{\frac{1}{alphay \cdot \frac{alphay}{sin2phi}}}} \]

   10. Taylor expanded in cos2phi around 0

       \[\leadsto \color{blue}{\frac{{alphay}^{2} \cdot u0}{sin2phi}} \]
   11. Step-by-step derivation

       1. associate-/l* N/A

          \[\leadsto {alphay}^{2} \cdot \color{blue}{\frac{u0}{sin2phi}} \]

       2. *-lowering-*.f32 N/A

          \[\leadsto \mathsf{*.f32}\left(\left({alphay}^{2}\right), \color{blue}{\left(\frac{u0}{sin2phi}\right)}\right) \]

       3. unpow2 N/A

          \[\leadsto \mathsf{*.f32}\left(\left(alphay \cdot alphay\right), \left(\frac{\color{blue}{u0}}{sin2phi}\right)\right) \]

       4. *-lowering-*.f32 N/A

          \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(alphay, alphay\right), \left(\frac{\color{blue}{u0}}{sin2phi}\right)\right) \]

       5. /-lowering-/.f32 67.6%

          \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(alphay, alphay\right), \mathsf{/.f32}\left(u0, \color{blue}{sin2phi}\right)\right) \]

   12. Simplified 67.6%

       \[\leadsto \color{blue}{\left(alphay \cdot alphay\right) \cdot \frac{u0}{sin2phi}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 64.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;sin2phi \leq 4.9999998413276127 \cdot 10^{-20}:\\ \;\;\;\;alphax \cdot \frac{u0}{\frac{cos2phi}{alphax}}\\ \mathbf{else}:\\ \;\;\;\;\left(alphay \cdot alphay\right) \cdot \frac{u0}{sin2phi}\\ \end{array} \]
5. Add Preprocessing

Alternative 25: 67.2% accurate, 9.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;sin2phi \leq 4.9999998413276127 \cdot 10^{-20}:\\ \;\;\;\;u0 \cdot \left(alphax \cdot \frac{alphax}{cos2phi}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(alphay \cdot alphay\right) \cdot \frac{u0}{sin2phi}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (if (<= sin2phi 4.9999998413276127e-20)
   (* u0 (* alphax (/ alphax cos2phi)))
   (* (* alphay alphay) (/ u0 sin2phi))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float tmp;
	if (sin2phi <= 4.9999998413276127e-20f) {
		tmp = u0 * (alphax * (alphax / cos2phi));
	} else {
		tmp = (alphay * alphay) * (u0 / sin2phi);
	}
	return tmp;
}

Fortran

real(4) function code(alphax, alphay, u0, cos2phi, sin2phi)
    real(4), intent (in) :: alphax
    real(4), intent (in) :: alphay
    real(4), intent (in) :: u0
    real(4), intent (in) :: cos2phi
    real(4), intent (in) :: sin2phi
    real(4) :: tmp
    if (sin2phi <= 4.9999998413276127e-20) then
        tmp = u0 * (alphax * (alphax / cos2phi))
    else
        tmp = (alphay * alphay) * (u0 / sin2phi)
    end if
    code = tmp
end function

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = Float32(0.0)
	if (sin2phi <= Float32(4.9999998413276127e-20))
		tmp = Float32(u0 * Float32(alphax * Float32(alphax / cos2phi)));
	else
		tmp = Float32(Float32(alphay * alphay) * Float32(u0 / sin2phi));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = single(0.0);
	if (sin2phi <= single(4.9999998413276127e-20))
		tmp = u0 * (alphax * (alphax / cos2phi));
	else
		tmp = (alphay * alphay) * (u0 / sin2phi);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if sin2phi < 4.99999984e-20

   1. Initial program 55.8%

      \[\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
   2. Step-by-step derivation

      1. distribute-frac-neg N/A

         \[\leadsto \mathsf{neg}\left(\frac{\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}}\right) \]

      2. distribute-neg-frac2 N/A

         \[\leadsto \frac{\log \left(1 - u0\right)}{\color{blue}{\mathsf{neg}\left(\left(\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\right)\right)}} \]

      3. /-lowering-/.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(\log \left(1 - u0\right), \color{blue}{\left(\mathsf{neg}\left(\left(\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\right)\right)\right)}\right) \]

      4. sub-neg N/A

         \[\leadsto \mathsf{/.f32}\left(\log \left(1 + \left(\mathsf{neg}\left(u0\right)\right)\right), \left(\mathsf{neg}\left(\left(\color{blue}{\frac{cos2phi}{alphax \cdot alphax}} + \frac{sin2phi}{alphay \cdot alphay}\right)\right)\right)\right) \]

      5. log1p-define N/A

         \[\leadsto \mathsf{/.f32}\left(\left(\mathsf{log1p}\left(\mathsf{neg}\left(u0\right)\right)\right), \left(\mathsf{neg}\left(\color{blue}{\left(\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\right)}\right)\right)\right) \]

      6. log1p-lowering-log1p.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\left(\mathsf{neg}\left(u0\right)\right)\right), \left(\mathsf{neg}\left(\color{blue}{\left(\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\right)}\right)\right)\right) \]

      7. neg-lowering-neg.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \left(\mathsf{neg}\left(\left(\color{blue}{\frac{cos2phi}{alphax \cdot alphax}} + \frac{sin2phi}{alphay \cdot alphay}\right)\right)\right)\right) \]

      8. distribute-neg-in N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \left(\left(\mathsf{neg}\left(\frac{cos2phi}{alphax \cdot alphax}\right)\right) + \color{blue}{\left(\mathsf{neg}\left(\frac{sin2phi}{alphay \cdot alphay}\right)\right)}\right)\right) \]

      9. unsub-neg N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \left(\left(\mathsf{neg}\left(\frac{cos2phi}{alphax \cdot alphax}\right)\right) - \color{blue}{\frac{sin2phi}{alphay \cdot alphay}}\right)\right) \]

      10. --lowering--.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\left(\mathsf{neg}\left(\frac{cos2phi}{alphax \cdot alphax}\right)\right), \color{blue}{\left(\frac{sin2phi}{alphay \cdot alphay}\right)}\right)\right) \]

      11. neg-sub0 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\left(0 - \frac{cos2phi}{alphax \cdot alphax}\right), \left(\frac{\color{blue}{sin2phi}}{alphay \cdot alphay}\right)\right)\right) \]

      12. --lowering--.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \left(\frac{cos2phi}{alphax \cdot alphax}\right)\right), \left(\frac{\color{blue}{sin2phi}}{alphay \cdot alphay}\right)\right)\right) \]

      13. /-lowering-/.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(cos2phi, \left(alphax \cdot alphax\right)\right)\right), \left(\frac{sin2phi}{alphay \cdot alphay}\right)\right)\right) \]

      14. *-lowering-*.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right)\right), \left(\frac{sin2phi}{alphay \cdot alphay}\right)\right)\right) \]

      15. /-lowering-/.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right)\right), \mathsf{/.f32}\left(sin2phi, \color{blue}{\left(alphay \cdot alphay\right)}\right)\right)\right) \]

      16. *-lowering-*.f32 98.8%

          \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right)\right), \mathsf{/.f32}\left(sin2phi, \mathsf{*.f32}\left(alphay, \color{blue}{alphay}\right)\right)\right)\right) \]

   3. Simplified 98.8%

      \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(-u0\right)}{\left(0 - \frac{cos2phi}{alphax \cdot alphax}\right) - \frac{sin2phi}{alphay \cdot alphay}}} \]
   4. Add Preprocessing

   5. Taylor expanded in u0 around 0

      \[\leadsto \color{blue}{\frac{u0}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}}} \]
   6. Step-by-step derivation

      1. /-lowering-/.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(u0, \color{blue}{\left(\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}\right)}\right) \]

      2. +-lowering-+.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{+.f32}\left(\left(\frac{cos2phi}{{alphax}^{2}}\right), \color{blue}{\left(\frac{sin2phi}{{alphay}^{2}}\right)}\right)\right) \]

      3. /-lowering-/.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{/.f32}\left(cos2phi, \left({alphax}^{2}\right)\right), \left(\frac{\color{blue}{sin2phi}}{{alphay}^{2}}\right)\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{/.f32}\left(cos2phi, \left(alphax \cdot alphax\right)\right), \left(\frac{sin2phi}{{alphay}^{2}}\right)\right)\right) \]

      5. *-lowering-*.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right), \left(\frac{sin2phi}{{alphay}^{2}}\right)\right)\right) \]

      6. /-lowering-/.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right), \mathsf{/.f32}\left(sin2phi, \color{blue}{\left({alphay}^{2}\right)}\right)\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right), \mathsf{/.f32}\left(sin2phi, \left(alphay \cdot \color{blue}{alphay}\right)\right)\right)\right) \]

      8. *-lowering-*.f32 73.3%

         \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right), \mathsf{/.f32}\left(sin2phi, \mathsf{*.f32}\left(alphay, \color{blue}{alphay}\right)\right)\right)\right) \]

   7. Simplified 73.3%

      \[\leadsto \color{blue}{\frac{u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}}} \]

   8. Taylor expanded in cos2phi around inf

      \[\leadsto \color{blue}{\frac{{alphax}^{2} \cdot u0}{cos2phi}} \]
   9. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto {alphax}^{2} \cdot \color{blue}{\frac{u0}{cos2phi}} \]

      2. *-lowering-*.f32 N/A

         \[\leadsto \mathsf{*.f32}\left(\left({alphax}^{2}\right), \color{blue}{\left(\frac{u0}{cos2phi}\right)}\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{*.f32}\left(\left(alphax \cdot alphax\right), \left(\frac{\color{blue}{u0}}{cos2phi}\right)\right) \]

      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 55.3%

         \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(alphax, alphax\right), \mathsf{/.f32}\left(u0, \color{blue}{cos2phi}\right)\right) \]

   10. Simplified 55.3%

       \[\leadsto \color{blue}{\left(alphax \cdot alphax\right) \cdot \frac{u0}{cos2phi}} \]
   11. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \frac{u0}{cos2phi} \cdot \color{blue}{\left(alphax \cdot alphax\right)} \]

       2. div-inv N/A

          \[\leadsto \left(u0 \cdot \frac{1}{cos2phi}\right) \cdot \left(\color{blue}{alphax} \cdot alphax\right) \]

       3. associate-*l* N/A

          \[\leadsto u0 \cdot \color{blue}{\left(\frac{1}{cos2phi} \cdot \left(alphax \cdot alphax\right)\right)} \]

       4. /-rgt-identity N/A

          \[\leadsto u0 \cdot \left(\frac{1}{cos2phi} \cdot \left(\frac{alphax}{1} \cdot alphax\right)\right) \]

       5. /-rgt-identity N/A

          \[\leadsto u0 \cdot \left(\frac{1}{cos2phi} \cdot \left(\frac{alphax}{1} \cdot \frac{alphax}{\color{blue}{1}}\right)\right) \]

       6. clear-num N/A

          \[\leadsto u0 \cdot \left(\frac{1}{cos2phi} \cdot \left(\frac{alphax}{1} \cdot \frac{1}{\color{blue}{\frac{1}{alphax}}}\right)\right) \]

       7. times-frac N/A

          \[\leadsto u0 \cdot \left(\frac{1}{cos2phi} \cdot \frac{alphax \cdot 1}{\color{blue}{1 \cdot \frac{1}{alphax}}}\right) \]

       8. metadata-eval N/A

          \[\leadsto u0 \cdot \left(\frac{1}{cos2phi} \cdot \frac{alphax \cdot \frac{1}{1}}{1 \cdot \frac{1}{alphax}}\right) \]

       9. div-inv N/A

          \[\leadsto u0 \cdot \left(\frac{1}{cos2phi} \cdot \frac{\frac{alphax}{1}}{\color{blue}{1} \cdot \frac{1}{alphax}}\right) \]

       10. /-rgt-identity N/A

           \[\leadsto u0 \cdot \left(\frac{1}{cos2phi} \cdot \frac{alphax}{\color{blue}{1} \cdot \frac{1}{alphax}}\right) \]

       11. div-inv N/A

           \[\leadsto u0 \cdot \left(\frac{1}{cos2phi} \cdot \frac{alphax}{\frac{1}{\color{blue}{alphax}}}\right) \]

       12. times-frac N/A

           \[\leadsto u0 \cdot \frac{1 \cdot alphax}{\color{blue}{cos2phi \cdot \frac{1}{alphax}}} \]

       13. div-inv N/A

           \[\leadsto u0 \cdot \frac{1 \cdot alphax}{\frac{cos2phi}{\color{blue}{alphax}}} \]

       14. associate-*l/ N/A

           \[\leadsto u0 \cdot \left(\frac{1}{\frac{cos2phi}{alphax}} \cdot \color{blue}{alphax}\right) \]

       15. clear-num N/A

           \[\leadsto u0 \cdot \left(\frac{alphax}{cos2phi} \cdot alphax\right) \]

       16. *-lowering-*.f32 N/A

           \[\leadsto \mathsf{*.f32}\left(u0, \color{blue}{\left(\frac{alphax}{cos2phi} \cdot alphax\right)}\right) \]

       17. associate-*l/ N/A

           \[\leadsto \mathsf{*.f32}\left(u0, \left(\frac{alphax \cdot alphax}{\color{blue}{cos2phi}}\right)\right) \]

       18. clear-num N/A

           \[\leadsto \mathsf{*.f32}\left(u0, \left(\frac{1}{\color{blue}{\frac{cos2phi}{alphax \cdot alphax}}}\right)\right) \]

       19. associate-/r* N/A

           \[\leadsto \mathsf{*.f32}\left(u0, \left(\frac{1}{\frac{\frac{cos2phi}{alphax}}{\color{blue}{alphax}}}\right)\right) \]

       20. clear-num N/A

           \[\leadsto \mathsf{*.f32}\left(u0, \left(\frac{alphax}{\color{blue}{\frac{cos2phi}{alphax}}}\right)\right) \]

       21. /-lowering-/.f32 N/A

           \[\leadsto \mathsf{*.f32}\left(u0, \mathsf{/.f32}\left(alphax, \color{blue}{\left(\frac{cos2phi}{alphax}\right)}\right)\right) \]

       22. /-lowering-/.f32 55.3%

           \[\leadsto \mathsf{*.f32}\left(u0, \mathsf{/.f32}\left(alphax, \mathsf{/.f32}\left(cos2phi, \color{blue}{alphax}\right)\right)\right) \]

   12. Applied egg-rr 55.3%

       \[\leadsto \color{blue}{u0 \cdot \frac{alphax}{\frac{cos2phi}{alphax}}} \]
   13. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \frac{alphax}{\frac{cos2phi}{alphax}} \cdot \color{blue}{u0} \]

       2. clear-num N/A

          \[\leadsto \frac{1}{\frac{\frac{cos2phi}{alphax}}{alphax}} \cdot u0 \]

       3. associate-/r* N/A

          \[\leadsto \frac{1}{\frac{cos2phi}{alphax \cdot alphax}} \cdot u0 \]

       4. clear-num N/A

          \[\leadsto \frac{alphax \cdot alphax}{cos2phi} \cdot u0 \]

       5. *-lowering-*.f32 N/A

          \[\leadsto \mathsf{*.f32}\left(\left(\frac{alphax \cdot alphax}{cos2phi}\right), \color{blue}{u0}\right) \]

       6. associate-/l* N/A

          \[\leadsto \mathsf{*.f32}\left(\left(alphax \cdot \frac{alphax}{cos2phi}\right), u0\right) \]

       7. *-lowering-*.f32 N/A

          \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(alphax, \left(\frac{alphax}{cos2phi}\right)\right), u0\right) \]

       8. /-lowering-/.f32 55.3%

          \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(alphax, \mathsf{/.f32}\left(alphax, cos2phi\right)\right), u0\right) \]

   14. Applied egg-rr 55.3%

       \[\leadsto \color{blue}{\left(alphax \cdot \frac{alphax}{cos2phi}\right) \cdot u0} \]

   if 4.99999984e-20 < sin2phi

   1. Initial program 62.8%

      \[\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
   2. Step-by-step derivation

      1. distribute-frac-neg N/A

         \[\leadsto \mathsf{neg}\left(\frac{\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}}\right) \]

      2. distribute-neg-frac2 N/A

         \[\leadsto \frac{\log \left(1 - u0\right)}{\color{blue}{\mathsf{neg}\left(\left(\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\right)\right)}} \]

      3. /-lowering-/.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(\log \left(1 - u0\right), \color{blue}{\left(\mathsf{neg}\left(\left(\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\right)\right)\right)}\right) \]

      4. sub-neg N/A

         \[\leadsto \mathsf{/.f32}\left(\log \left(1 + \left(\mathsf{neg}\left(u0\right)\right)\right), \left(\mathsf{neg}\left(\left(\color{blue}{\frac{cos2phi}{alphax \cdot alphax}} + \frac{sin2phi}{alphay \cdot alphay}\right)\right)\right)\right) \]

      5. log1p-define N/A

         \[\leadsto \mathsf{/.f32}\left(\left(\mathsf{log1p}\left(\mathsf{neg}\left(u0\right)\right)\right), \left(\mathsf{neg}\left(\color{blue}{\left(\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\right)}\right)\right)\right) \]

      6. log1p-lowering-log1p.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\left(\mathsf{neg}\left(u0\right)\right)\right), \left(\mathsf{neg}\left(\color{blue}{\left(\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\right)}\right)\right)\right) \]

      7. neg-lowering-neg.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \left(\mathsf{neg}\left(\left(\color{blue}{\frac{cos2phi}{alphax \cdot alphax}} + \frac{sin2phi}{alphay \cdot alphay}\right)\right)\right)\right) \]

      8. distribute-neg-in N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \left(\left(\mathsf{neg}\left(\frac{cos2phi}{alphax \cdot alphax}\right)\right) + \color{blue}{\left(\mathsf{neg}\left(\frac{sin2phi}{alphay \cdot alphay}\right)\right)}\right)\right) \]

      9. unsub-neg N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \left(\left(\mathsf{neg}\left(\frac{cos2phi}{alphax \cdot alphax}\right)\right) - \color{blue}{\frac{sin2phi}{alphay \cdot alphay}}\right)\right) \]

      10. --lowering--.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\left(\mathsf{neg}\left(\frac{cos2phi}{alphax \cdot alphax}\right)\right), \color{blue}{\left(\frac{sin2phi}{alphay \cdot alphay}\right)}\right)\right) \]

      11. neg-sub0 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\left(0 - \frac{cos2phi}{alphax \cdot alphax}\right), \left(\frac{\color{blue}{sin2phi}}{alphay \cdot alphay}\right)\right)\right) \]

      12. --lowering--.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \left(\frac{cos2phi}{alphax \cdot alphax}\right)\right), \left(\frac{\color{blue}{sin2phi}}{alphay \cdot alphay}\right)\right)\right) \]

      13. /-lowering-/.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(cos2phi, \left(alphax \cdot alphax\right)\right)\right), \left(\frac{sin2phi}{alphay \cdot alphay}\right)\right)\right) \]

      14. *-lowering-*.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right)\right), \left(\frac{sin2phi}{alphay \cdot alphay}\right)\right)\right) \]

      15. /-lowering-/.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right)\right), \mathsf{/.f32}\left(sin2phi, \color{blue}{\left(alphay \cdot alphay\right)}\right)\right)\right) \]

      16. *-lowering-*.f32 98.3%

          \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right)\right), \mathsf{/.f32}\left(sin2phi, \mathsf{*.f32}\left(alphay, \color{blue}{alphay}\right)\right)\right)\right) \]

   3. Simplified 98.3%

      \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(-u0\right)}{\left(0 - \frac{cos2phi}{alphax \cdot alphax}\right) - \frac{sin2phi}{alphay \cdot alphay}}} \]
   4. Add Preprocessing

   5. Taylor expanded in u0 around 0

      \[\leadsto \color{blue}{\frac{u0}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}}} \]
   6. Step-by-step derivation

      1. /-lowering-/.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(u0, \color{blue}{\left(\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}\right)}\right) \]

      2. +-lowering-+.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{+.f32}\left(\left(\frac{cos2phi}{{alphax}^{2}}\right), \color{blue}{\left(\frac{sin2phi}{{alphay}^{2}}\right)}\right)\right) \]

      3. /-lowering-/.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{/.f32}\left(cos2phi, \left({alphax}^{2}\right)\right), \left(\frac{\color{blue}{sin2phi}}{{alphay}^{2}}\right)\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{/.f32}\left(cos2phi, \left(alphax \cdot alphax\right)\right), \left(\frac{sin2phi}{{alphay}^{2}}\right)\right)\right) \]

      5. *-lowering-*.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right), \left(\frac{sin2phi}{{alphay}^{2}}\right)\right)\right) \]

      6. /-lowering-/.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right), \mathsf{/.f32}\left(sin2phi, \color{blue}{\left({alphay}^{2}\right)}\right)\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right), \mathsf{/.f32}\left(sin2phi, \left(alphay \cdot \color{blue}{alphay}\right)\right)\right)\right) \]

      8. *-lowering-*.f32 75.0%

         \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right), \mathsf{/.f32}\left(sin2phi, \mathsf{*.f32}\left(alphay, \color{blue}{alphay}\right)\right)\right)\right) \]

   7. Simplified 75.0%

      \[\leadsto \color{blue}{\frac{u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}}} \]
   8. Step-by-step derivation

      1. clear-num N/A

         \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right), \left(\frac{1}{\color{blue}{\frac{alphay \cdot alphay}{sin2phi}}}\right)\right)\right) \]

      2. /-lowering-/.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right), \mathsf{/.f32}\left(1, \color{blue}{\left(\frac{alphay \cdot alphay}{sin2phi}\right)}\right)\right)\right) \]

      3. associate-/l* N/A

         \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right), \mathsf{/.f32}\left(1, \left(alphay \cdot \color{blue}{\frac{alphay}{sin2phi}}\right)\right)\right)\right) \]

      4. *-lowering-*.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right), \mathsf{/.f32}\left(1, \mathsf{*.f32}\left(alphay, \color{blue}{\left(\frac{alphay}{sin2phi}\right)}\right)\right)\right)\right) \]

      5. /-lowering-/.f32 75.0%

         \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right), \mathsf{/.f32}\left(1, \mathsf{*.f32}\left(alphay, \mathsf{/.f32}\left(alphay, \color{blue}{sin2phi}\right)\right)\right)\right)\right) \]

   9. Applied egg-rr 75.0%

      \[\leadsto \frac{u0}{\frac{cos2phi}{alphax \cdot alphax} + \color{blue}{\frac{1}{alphay \cdot \frac{alphay}{sin2phi}}}} \]

   10. Taylor expanded in cos2phi around 0

       \[\leadsto \color{blue}{\frac{{alphay}^{2} \cdot u0}{sin2phi}} \]
   11. Step-by-step derivation

       1. associate-/l* N/A

          \[\leadsto {alphay}^{2} \cdot \color{blue}{\frac{u0}{sin2phi}} \]

       2. *-lowering-*.f32 N/A

          \[\leadsto \mathsf{*.f32}\left(\left({alphay}^{2}\right), \color{blue}{\left(\frac{u0}{sin2phi}\right)}\right) \]

       3. unpow2 N/A

          \[\leadsto \mathsf{*.f32}\left(\left(alphay \cdot alphay\right), \left(\frac{\color{blue}{u0}}{sin2phi}\right)\right) \]

       4. *-lowering-*.f32 N/A

          \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(alphay, alphay\right), \left(\frac{\color{blue}{u0}}{sin2phi}\right)\right) \]

       5. /-lowering-/.f32 67.6%

          \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(alphay, alphay\right), \mathsf{/.f32}\left(u0, \color{blue}{sin2phi}\right)\right) \]

   12. Simplified 67.6%

       \[\leadsto \color{blue}{\left(alphay \cdot alphay\right) \cdot \frac{u0}{sin2phi}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 64.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;sin2phi \leq 4.9999998413276127 \cdot 10^{-20}:\\ \;\;\;\;u0 \cdot \left(alphax \cdot \frac{alphax}{cos2phi}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(alphay \cdot alphay\right) \cdot \frac{u0}{sin2phi}\\ \end{array} \]
5. Add Preprocessing

Alternative 26: 67.2% accurate, 9.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;sin2phi \leq 4.9999998413276127 \cdot 10^{-20}:\\ \;\;\;\;u0 \cdot \left(alphax \cdot \frac{alphax}{cos2phi}\right)\\ \mathbf{else}:\\ \;\;\;\;u0 \cdot \frac{alphay \cdot alphay}{sin2phi}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (if (<= sin2phi 4.9999998413276127e-20)
   (* u0 (* alphax (/ alphax cos2phi)))
   (* u0 (/ (* alphay alphay) sin2phi))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float tmp;
	if (sin2phi <= 4.9999998413276127e-20f) {
		tmp = u0 * (alphax * (alphax / cos2phi));
	} 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 <= 4.9999998413276127e-20) then
        tmp = u0 * (alphax * (alphax / cos2phi))
    else
        tmp = u0 * ((alphay * alphay) / sin2phi)
    end if
    code = tmp
end function

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = Float32(0.0)
	if (sin2phi <= Float32(4.9999998413276127e-20))
		tmp = Float32(u0 * Float32(alphax * Float32(alphax / cos2phi)));
	else
		tmp = Float32(u0 * Float32(Float32(alphay * alphay) / sin2phi));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = single(0.0);
	if (sin2phi <= single(4.9999998413276127e-20))
		tmp = u0 * (alphax * (alphax / cos2phi));
	else
		tmp = u0 * ((alphay * alphay) / sin2phi);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if sin2phi < 4.99999984e-20

   1. Initial program 55.8%

      \[\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
   2. Step-by-step derivation

      1. distribute-frac-neg N/A

         \[\leadsto \mathsf{neg}\left(\frac{\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}}\right) \]

      2. distribute-neg-frac2 N/A

         \[\leadsto \frac{\log \left(1 - u0\right)}{\color{blue}{\mathsf{neg}\left(\left(\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\right)\right)}} \]

      3. /-lowering-/.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(\log \left(1 - u0\right), \color{blue}{\left(\mathsf{neg}\left(\left(\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\right)\right)\right)}\right) \]

      4. sub-neg N/A

         \[\leadsto \mathsf{/.f32}\left(\log \left(1 + \left(\mathsf{neg}\left(u0\right)\right)\right), \left(\mathsf{neg}\left(\left(\color{blue}{\frac{cos2phi}{alphax \cdot alphax}} + \frac{sin2phi}{alphay \cdot alphay}\right)\right)\right)\right) \]

      5. log1p-define N/A

         \[\leadsto \mathsf{/.f32}\left(\left(\mathsf{log1p}\left(\mathsf{neg}\left(u0\right)\right)\right), \left(\mathsf{neg}\left(\color{blue}{\left(\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\right)}\right)\right)\right) \]

      6. log1p-lowering-log1p.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\left(\mathsf{neg}\left(u0\right)\right)\right), \left(\mathsf{neg}\left(\color{blue}{\left(\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\right)}\right)\right)\right) \]

      7. neg-lowering-neg.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \left(\mathsf{neg}\left(\left(\color{blue}{\frac{cos2phi}{alphax \cdot alphax}} + \frac{sin2phi}{alphay \cdot alphay}\right)\right)\right)\right) \]

      8. distribute-neg-in N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \left(\left(\mathsf{neg}\left(\frac{cos2phi}{alphax \cdot alphax}\right)\right) + \color{blue}{\left(\mathsf{neg}\left(\frac{sin2phi}{alphay \cdot alphay}\right)\right)}\right)\right) \]

      9. unsub-neg N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \left(\left(\mathsf{neg}\left(\frac{cos2phi}{alphax \cdot alphax}\right)\right) - \color{blue}{\frac{sin2phi}{alphay \cdot alphay}}\right)\right) \]

      10. --lowering--.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\left(\mathsf{neg}\left(\frac{cos2phi}{alphax \cdot alphax}\right)\right), \color{blue}{\left(\frac{sin2phi}{alphay \cdot alphay}\right)}\right)\right) \]

      11. neg-sub0 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\left(0 - \frac{cos2phi}{alphax \cdot alphax}\right), \left(\frac{\color{blue}{sin2phi}}{alphay \cdot alphay}\right)\right)\right) \]

      12. --lowering--.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \left(\frac{cos2phi}{alphax \cdot alphax}\right)\right), \left(\frac{\color{blue}{sin2phi}}{alphay \cdot alphay}\right)\right)\right) \]

      13. /-lowering-/.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(cos2phi, \left(alphax \cdot alphax\right)\right)\right), \left(\frac{sin2phi}{alphay \cdot alphay}\right)\right)\right) \]

      14. *-lowering-*.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right)\right), \left(\frac{sin2phi}{alphay \cdot alphay}\right)\right)\right) \]

      15. /-lowering-/.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right)\right), \mathsf{/.f32}\left(sin2phi, \color{blue}{\left(alphay \cdot alphay\right)}\right)\right)\right) \]

      16. *-lowering-*.f32 98.8%

          \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right)\right), \mathsf{/.f32}\left(sin2phi, \mathsf{*.f32}\left(alphay, \color{blue}{alphay}\right)\right)\right)\right) \]

   3. Simplified 98.8%

      \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(-u0\right)}{\left(0 - \frac{cos2phi}{alphax \cdot alphax}\right) - \frac{sin2phi}{alphay \cdot alphay}}} \]
   4. Add Preprocessing

   5. Taylor expanded in u0 around 0

      \[\leadsto \color{blue}{\frac{u0}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}}} \]
   6. Step-by-step derivation

      1. /-lowering-/.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(u0, \color{blue}{\left(\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}\right)}\right) \]

      2. +-lowering-+.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{+.f32}\left(\left(\frac{cos2phi}{{alphax}^{2}}\right), \color{blue}{\left(\frac{sin2phi}{{alphay}^{2}}\right)}\right)\right) \]

      3. /-lowering-/.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{/.f32}\left(cos2phi, \left({alphax}^{2}\right)\right), \left(\frac{\color{blue}{sin2phi}}{{alphay}^{2}}\right)\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{/.f32}\left(cos2phi, \left(alphax \cdot alphax\right)\right), \left(\frac{sin2phi}{{alphay}^{2}}\right)\right)\right) \]

      5. *-lowering-*.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right), \left(\frac{sin2phi}{{alphay}^{2}}\right)\right)\right) \]

      6. /-lowering-/.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right), \mathsf{/.f32}\left(sin2phi, \color{blue}{\left({alphay}^{2}\right)}\right)\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right), \mathsf{/.f32}\left(sin2phi, \left(alphay \cdot \color{blue}{alphay}\right)\right)\right)\right) \]

      8. *-lowering-*.f32 73.3%

         \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right), \mathsf{/.f32}\left(sin2phi, \mathsf{*.f32}\left(alphay, \color{blue}{alphay}\right)\right)\right)\right) \]

   7. Simplified 73.3%

      \[\leadsto \color{blue}{\frac{u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}}} \]

   8. Taylor expanded in cos2phi around inf

      \[\leadsto \color{blue}{\frac{{alphax}^{2} \cdot u0}{cos2phi}} \]
   9. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto {alphax}^{2} \cdot \color{blue}{\frac{u0}{cos2phi}} \]

      2. *-lowering-*.f32 N/A

         \[\leadsto \mathsf{*.f32}\left(\left({alphax}^{2}\right), \color{blue}{\left(\frac{u0}{cos2phi}\right)}\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{*.f32}\left(\left(alphax \cdot alphax\right), \left(\frac{\color{blue}{u0}}{cos2phi}\right)\right) \]

      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 55.3%

         \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(alphax, alphax\right), \mathsf{/.f32}\left(u0, \color{blue}{cos2phi}\right)\right) \]

   10. Simplified 55.3%

       \[\leadsto \color{blue}{\left(alphax \cdot alphax\right) \cdot \frac{u0}{cos2phi}} \]
   11. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \frac{u0}{cos2phi} \cdot \color{blue}{\left(alphax \cdot alphax\right)} \]

       2. div-inv N/A

          \[\leadsto \left(u0 \cdot \frac{1}{cos2phi}\right) \cdot \left(\color{blue}{alphax} \cdot alphax\right) \]

       3. associate-*l* N/A

          \[\leadsto u0 \cdot \color{blue}{\left(\frac{1}{cos2phi} \cdot \left(alphax \cdot alphax\right)\right)} \]

       4. /-rgt-identity N/A

          \[\leadsto u0 \cdot \left(\frac{1}{cos2phi} \cdot \left(\frac{alphax}{1} \cdot alphax\right)\right) \]

       5. /-rgt-identity N/A

          \[\leadsto u0 \cdot \left(\frac{1}{cos2phi} \cdot \left(\frac{alphax}{1} \cdot \frac{alphax}{\color{blue}{1}}\right)\right) \]

       6. clear-num N/A

          \[\leadsto u0 \cdot \left(\frac{1}{cos2phi} \cdot \left(\frac{alphax}{1} \cdot \frac{1}{\color{blue}{\frac{1}{alphax}}}\right)\right) \]

       7. times-frac N/A

          \[\leadsto u0 \cdot \left(\frac{1}{cos2phi} \cdot \frac{alphax \cdot 1}{\color{blue}{1 \cdot \frac{1}{alphax}}}\right) \]

       8. metadata-eval N/A

          \[\leadsto u0 \cdot \left(\frac{1}{cos2phi} \cdot \frac{alphax \cdot \frac{1}{1}}{1 \cdot \frac{1}{alphax}}\right) \]

       9. div-inv N/A

          \[\leadsto u0 \cdot \left(\frac{1}{cos2phi} \cdot \frac{\frac{alphax}{1}}{\color{blue}{1} \cdot \frac{1}{alphax}}\right) \]

       10. /-rgt-identity N/A

           \[\leadsto u0 \cdot \left(\frac{1}{cos2phi} \cdot \frac{alphax}{\color{blue}{1} \cdot \frac{1}{alphax}}\right) \]

       11. div-inv N/A

           \[\leadsto u0 \cdot \left(\frac{1}{cos2phi} \cdot \frac{alphax}{\frac{1}{\color{blue}{alphax}}}\right) \]

       12. times-frac N/A

           \[\leadsto u0 \cdot \frac{1 \cdot alphax}{\color{blue}{cos2phi \cdot \frac{1}{alphax}}} \]

       13. div-inv N/A

           \[\leadsto u0 \cdot \frac{1 \cdot alphax}{\frac{cos2phi}{\color{blue}{alphax}}} \]

       14. associate-*l/ N/A

           \[\leadsto u0 \cdot \left(\frac{1}{\frac{cos2phi}{alphax}} \cdot \color{blue}{alphax}\right) \]

       15. clear-num N/A

           \[\leadsto u0 \cdot \left(\frac{alphax}{cos2phi} \cdot alphax\right) \]

       16. *-lowering-*.f32 N/A

           \[\leadsto \mathsf{*.f32}\left(u0, \color{blue}{\left(\frac{alphax}{cos2phi} \cdot alphax\right)}\right) \]

       17. associate-*l/ N/A

           \[\leadsto \mathsf{*.f32}\left(u0, \left(\frac{alphax \cdot alphax}{\color{blue}{cos2phi}}\right)\right) \]

       18. clear-num N/A

           \[\leadsto \mathsf{*.f32}\left(u0, \left(\frac{1}{\color{blue}{\frac{cos2phi}{alphax \cdot alphax}}}\right)\right) \]

       19. associate-/r* N/A

           \[\leadsto \mathsf{*.f32}\left(u0, \left(\frac{1}{\frac{\frac{cos2phi}{alphax}}{\color{blue}{alphax}}}\right)\right) \]

       20. clear-num N/A

           \[\leadsto \mathsf{*.f32}\left(u0, \left(\frac{alphax}{\color{blue}{\frac{cos2phi}{alphax}}}\right)\right) \]

       21. /-lowering-/.f32 N/A

           \[\leadsto \mathsf{*.f32}\left(u0, \mathsf{/.f32}\left(alphax, \color{blue}{\left(\frac{cos2phi}{alphax}\right)}\right)\right) \]

       22. /-lowering-/.f32 55.3%

           \[\leadsto \mathsf{*.f32}\left(u0, \mathsf{/.f32}\left(alphax, \mathsf{/.f32}\left(cos2phi, \color{blue}{alphax}\right)\right)\right) \]

   12. Applied egg-rr 55.3%

       \[\leadsto \color{blue}{u0 \cdot \frac{alphax}{\frac{cos2phi}{alphax}}} \]
   13. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \frac{alphax}{\frac{cos2phi}{alphax}} \cdot \color{blue}{u0} \]

       2. clear-num N/A

          \[\leadsto \frac{1}{\frac{\frac{cos2phi}{alphax}}{alphax}} \cdot u0 \]

       3. associate-/r* N/A

          \[\leadsto \frac{1}{\frac{cos2phi}{alphax \cdot alphax}} \cdot u0 \]

       4. clear-num N/A

          \[\leadsto \frac{alphax \cdot alphax}{cos2phi} \cdot u0 \]

       5. *-lowering-*.f32 N/A

          \[\leadsto \mathsf{*.f32}\left(\left(\frac{alphax \cdot alphax}{cos2phi}\right), \color{blue}{u0}\right) \]

       6. associate-/l* N/A

          \[\leadsto \mathsf{*.f32}\left(\left(alphax \cdot \frac{alphax}{cos2phi}\right), u0\right) \]

       7. *-lowering-*.f32 N/A

          \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(alphax, \left(\frac{alphax}{cos2phi}\right)\right), u0\right) \]

       8. /-lowering-/.f32 55.3%

          \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(alphax, \mathsf{/.f32}\left(alphax, cos2phi\right)\right), u0\right) \]

   14. Applied egg-rr 55.3%

       \[\leadsto \color{blue}{\left(alphax \cdot \frac{alphax}{cos2phi}\right) \cdot u0} \]

   if 4.99999984e-20 < sin2phi

   1. Initial program 62.8%

      \[\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
   2. Step-by-step derivation

      1. distribute-frac-neg N/A

         \[\leadsto \mathsf{neg}\left(\frac{\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}}\right) \]

      2. distribute-neg-frac2 N/A

         \[\leadsto \frac{\log \left(1 - u0\right)}{\color{blue}{\mathsf{neg}\left(\left(\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\right)\right)}} \]

      3. /-lowering-/.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(\log \left(1 - u0\right), \color{blue}{\left(\mathsf{neg}\left(\left(\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\right)\right)\right)}\right) \]

      4. sub-neg N/A

         \[\leadsto \mathsf{/.f32}\left(\log \left(1 + \left(\mathsf{neg}\left(u0\right)\right)\right), \left(\mathsf{neg}\left(\left(\color{blue}{\frac{cos2phi}{alphax \cdot alphax}} + \frac{sin2phi}{alphay \cdot alphay}\right)\right)\right)\right) \]

      5. log1p-define N/A

         \[\leadsto \mathsf{/.f32}\left(\left(\mathsf{log1p}\left(\mathsf{neg}\left(u0\right)\right)\right), \left(\mathsf{neg}\left(\color{blue}{\left(\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\right)}\right)\right)\right) \]

      6. log1p-lowering-log1p.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\left(\mathsf{neg}\left(u0\right)\right)\right), \left(\mathsf{neg}\left(\color{blue}{\left(\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\right)}\right)\right)\right) \]

      7. neg-lowering-neg.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \left(\mathsf{neg}\left(\left(\color{blue}{\frac{cos2phi}{alphax \cdot alphax}} + \frac{sin2phi}{alphay \cdot alphay}\right)\right)\right)\right) \]

      8. distribute-neg-in N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \left(\left(\mathsf{neg}\left(\frac{cos2phi}{alphax \cdot alphax}\right)\right) + \color{blue}{\left(\mathsf{neg}\left(\frac{sin2phi}{alphay \cdot alphay}\right)\right)}\right)\right) \]

      9. unsub-neg N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \left(\left(\mathsf{neg}\left(\frac{cos2phi}{alphax \cdot alphax}\right)\right) - \color{blue}{\frac{sin2phi}{alphay \cdot alphay}}\right)\right) \]

      10. --lowering--.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\left(\mathsf{neg}\left(\frac{cos2phi}{alphax \cdot alphax}\right)\right), \color{blue}{\left(\frac{sin2phi}{alphay \cdot alphay}\right)}\right)\right) \]

      11. neg-sub0 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\left(0 - \frac{cos2phi}{alphax \cdot alphax}\right), \left(\frac{\color{blue}{sin2phi}}{alphay \cdot alphay}\right)\right)\right) \]

      12. --lowering--.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \left(\frac{cos2phi}{alphax \cdot alphax}\right)\right), \left(\frac{\color{blue}{sin2phi}}{alphay \cdot alphay}\right)\right)\right) \]

      13. /-lowering-/.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(cos2phi, \left(alphax \cdot alphax\right)\right)\right), \left(\frac{sin2phi}{alphay \cdot alphay}\right)\right)\right) \]

      14. *-lowering-*.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right)\right), \left(\frac{sin2phi}{alphay \cdot alphay}\right)\right)\right) \]

      15. /-lowering-/.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right)\right), \mathsf{/.f32}\left(sin2phi, \color{blue}{\left(alphay \cdot alphay\right)}\right)\right)\right) \]

      16. *-lowering-*.f32 98.3%

          \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right)\right), \mathsf{/.f32}\left(sin2phi, \mathsf{*.f32}\left(alphay, \color{blue}{alphay}\right)\right)\right)\right) \]

   3. Simplified 98.3%

      \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(-u0\right)}{\left(0 - \frac{cos2phi}{alphax \cdot alphax}\right) - \frac{sin2phi}{alphay \cdot alphay}}} \]
   4. Add Preprocessing

   5. Taylor expanded in u0 around 0

      \[\leadsto \color{blue}{u0 \cdot \left(u0 \cdot \left(u0 \cdot \left(\frac{1}{4} \cdot \frac{u0}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}} + \frac{1}{3} \cdot \frac{1}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}}\right) + \frac{1}{2} \cdot \frac{1}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}}\right) + \frac{1}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}}\right)} \]
   6. Step-by-step derivation

      1. *-lowering-*.f32 N/A

         \[\leadsto \mathsf{*.f32}\left(u0, \color{blue}{\left(u0 \cdot \left(u0 \cdot \left(\frac{1}{4} \cdot \frac{u0}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}} + \frac{1}{3} \cdot \frac{1}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}}\right) + \frac{1}{2} \cdot \frac{1}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}}\right) + \frac{1}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}}\right)}\right) \]

      2. +-commutative N/A

         \[\leadsto \mathsf{*.f32}\left(u0, \left(\frac{1}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}} + \color{blue}{u0 \cdot \left(u0 \cdot \left(\frac{1}{4} \cdot \frac{u0}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}} + \frac{1}{3} \cdot \frac{1}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}}\right) + \frac{1}{2} \cdot \frac{1}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}}\right)}\right)\right) \]

      3. +-lowering-+.f32 N/A

         \[\leadsto \mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\left(\frac{1}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}}\right), \color{blue}{\left(u0 \cdot \left(u0 \cdot \left(\frac{1}{4} \cdot \frac{u0}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}} + \frac{1}{3} \cdot \frac{1}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}}\right) + \frac{1}{2} \cdot \frac{1}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}}\right)\right)}\right)\right) \]

   7. Simplified 93.0%

      \[\leadsto \color{blue}{u0 \cdot \left(\frac{1}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} + u0 \cdot \left(\frac{0.5}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} + u0 \cdot \left(\frac{u0 \cdot 0.25}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} + \frac{0.3333333333333333}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}}\right)\right)\right)} \]

   8. Taylor expanded in sin2phi around inf

      \[\leadsto \color{blue}{\frac{u0 \cdot \left(u0 \cdot \left(\frac{1}{2} \cdot {alphay}^{2} + u0 \cdot \left(\frac{1}{4} \cdot \left({alphay}^{2} \cdot u0\right) + \frac{1}{3} \cdot {alphay}^{2}\right)\right) + {alphay}^{2}\right)}{sin2phi}} \]
   9. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto u0 \cdot \color{blue}{\frac{u0 \cdot \left(\frac{1}{2} \cdot {alphay}^{2} + u0 \cdot \left(\frac{1}{4} \cdot \left({alphay}^{2} \cdot u0\right) + \frac{1}{3} \cdot {alphay}^{2}\right)\right) + {alphay}^{2}}{sin2phi}} \]

      2. *-lowering-*.f32 N/A

         \[\leadsto \mathsf{*.f32}\left(u0, \color{blue}{\left(\frac{u0 \cdot \left(\frac{1}{2} \cdot {alphay}^{2} + u0 \cdot \left(\frac{1}{4} \cdot \left({alphay}^{2} \cdot u0\right) + \frac{1}{3} \cdot {alphay}^{2}\right)\right) + {alphay}^{2}}{sin2phi}\right)}\right) \]

      3. /-lowering-/.f32 N/A

         \[\leadsto \mathsf{*.f32}\left(u0, \mathsf{/.f32}\left(\left(u0 \cdot \left(\frac{1}{2} \cdot {alphay}^{2} + u0 \cdot \left(\frac{1}{4} \cdot \left({alphay}^{2} \cdot u0\right) + \frac{1}{3} \cdot {alphay}^{2}\right)\right) + {alphay}^{2}\right), \color{blue}{sin2phi}\right)\right) \]

   10. Simplified 82.1%

       \[\leadsto \color{blue}{u0 \cdot \frac{alphay \cdot alphay + u0 \cdot \left(u0 \cdot \left(\left(u0 \cdot \left(alphay \cdot alphay\right)\right) \cdot 0.25 + \left(alphay \cdot alphay\right) \cdot 0.3333333333333333\right) + \left(alphay \cdot alphay\right) \cdot 0.5\right)}{sin2phi}} \]

   11. Taylor expanded in u0 around 0

       \[\leadsto \mathsf{*.f32}\left(u0, \color{blue}{\left(\frac{{alphay}^{2}}{sin2phi}\right)}\right) \]
   12. Step-by-step derivation

       1. /-lowering-/.f32 N/A

          \[\leadsto \mathsf{*.f32}\left(u0, \mathsf{/.f32}\left(\left({alphay}^{2}\right), \color{blue}{sin2phi}\right)\right) \]

       2. unpow2 N/A

          \[\leadsto \mathsf{*.f32}\left(u0, \mathsf{/.f32}\left(\left(alphay \cdot alphay\right), sin2phi\right)\right) \]

       3. *-lowering-*.f32 67.5%

          \[\leadsto \mathsf{*.f32}\left(u0, \mathsf{/.f32}\left(\mathsf{*.f32}\left(alphay, alphay\right), sin2phi\right)\right) \]

   13. Simplified 67.5%

       \[\leadsto u0 \cdot \color{blue}{\frac{alphay \cdot alphay}{sin2phi}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 64.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;sin2phi \leq 4.9999998413276127 \cdot 10^{-20}:\\ \;\;\;\;u0 \cdot \left(alphax \cdot \frac{alphax}{cos2phi}\right)\\ \mathbf{else}:\\ \;\;\;\;u0 \cdot \frac{alphay \cdot alphay}{sin2phi}\\ \end{array} \]
5. Add Preprocessing

Alternative 27: 67.2% accurate, 9.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;sin2phi \leq 4.9999998413276127 \cdot 10^{-20}:\\ \;\;\;\;\left(alphax \cdot alphax\right) \cdot \frac{u0}{cos2phi}\\ \mathbf{else}:\\ \;\;\;\;u0 \cdot \frac{alphay \cdot alphay}{sin2phi}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (if (<= sin2phi 4.9999998413276127e-20)
   (* (* alphax alphax) (/ u0 cos2phi))
   (* u0 (/ (* alphay alphay) sin2phi))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float tmp;
	if (sin2phi <= 4.9999998413276127e-20f) {
		tmp = (alphax * alphax) * (u0 / cos2phi);
	} 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 <= 4.9999998413276127e-20) then
        tmp = (alphax * alphax) * (u0 / cos2phi)
    else
        tmp = u0 * ((alphay * alphay) / sin2phi)
    end if
    code = tmp
end function

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = Float32(0.0)
	if (sin2phi <= Float32(4.9999998413276127e-20))
		tmp = Float32(Float32(alphax * alphax) * Float32(u0 / cos2phi));
	else
		tmp = Float32(u0 * Float32(Float32(alphay * alphay) / sin2phi));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = single(0.0);
	if (sin2phi <= single(4.9999998413276127e-20))
		tmp = (alphax * alphax) * (u0 / cos2phi);
	else
		tmp = u0 * ((alphay * alphay) / sin2phi);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if sin2phi < 4.99999984e-20

   1. Initial program 55.8%

      \[\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
   2. Step-by-step derivation

      1. distribute-frac-neg N/A

         \[\leadsto \mathsf{neg}\left(\frac{\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}}\right) \]

      2. distribute-neg-frac2 N/A

         \[\leadsto \frac{\log \left(1 - u0\right)}{\color{blue}{\mathsf{neg}\left(\left(\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\right)\right)}} \]

      3. /-lowering-/.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(\log \left(1 - u0\right), \color{blue}{\left(\mathsf{neg}\left(\left(\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\right)\right)\right)}\right) \]

      4. sub-neg N/A

         \[\leadsto \mathsf{/.f32}\left(\log \left(1 + \left(\mathsf{neg}\left(u0\right)\right)\right), \left(\mathsf{neg}\left(\left(\color{blue}{\frac{cos2phi}{alphax \cdot alphax}} + \frac{sin2phi}{alphay \cdot alphay}\right)\right)\right)\right) \]

      5. log1p-define N/A

         \[\leadsto \mathsf{/.f32}\left(\left(\mathsf{log1p}\left(\mathsf{neg}\left(u0\right)\right)\right), \left(\mathsf{neg}\left(\color{blue}{\left(\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\right)}\right)\right)\right) \]

      6. log1p-lowering-log1p.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\left(\mathsf{neg}\left(u0\right)\right)\right), \left(\mathsf{neg}\left(\color{blue}{\left(\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\right)}\right)\right)\right) \]

      7. neg-lowering-neg.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \left(\mathsf{neg}\left(\left(\color{blue}{\frac{cos2phi}{alphax \cdot alphax}} + \frac{sin2phi}{alphay \cdot alphay}\right)\right)\right)\right) \]

      8. distribute-neg-in N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \left(\left(\mathsf{neg}\left(\frac{cos2phi}{alphax \cdot alphax}\right)\right) + \color{blue}{\left(\mathsf{neg}\left(\frac{sin2phi}{alphay \cdot alphay}\right)\right)}\right)\right) \]

      9. unsub-neg N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \left(\left(\mathsf{neg}\left(\frac{cos2phi}{alphax \cdot alphax}\right)\right) - \color{blue}{\frac{sin2phi}{alphay \cdot alphay}}\right)\right) \]

      10. --lowering--.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\left(\mathsf{neg}\left(\frac{cos2phi}{alphax \cdot alphax}\right)\right), \color{blue}{\left(\frac{sin2phi}{alphay \cdot alphay}\right)}\right)\right) \]

      11. neg-sub0 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\left(0 - \frac{cos2phi}{alphax \cdot alphax}\right), \left(\frac{\color{blue}{sin2phi}}{alphay \cdot alphay}\right)\right)\right) \]

      12. --lowering--.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \left(\frac{cos2phi}{alphax \cdot alphax}\right)\right), \left(\frac{\color{blue}{sin2phi}}{alphay \cdot alphay}\right)\right)\right) \]

      13. /-lowering-/.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(cos2phi, \left(alphax \cdot alphax\right)\right)\right), \left(\frac{sin2phi}{alphay \cdot alphay}\right)\right)\right) \]

      14. *-lowering-*.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right)\right), \left(\frac{sin2phi}{alphay \cdot alphay}\right)\right)\right) \]

      15. /-lowering-/.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right)\right), \mathsf{/.f32}\left(sin2phi, \color{blue}{\left(alphay \cdot alphay\right)}\right)\right)\right) \]

      16. *-lowering-*.f32 98.8%

          \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right)\right), \mathsf{/.f32}\left(sin2phi, \mathsf{*.f32}\left(alphay, \color{blue}{alphay}\right)\right)\right)\right) \]

   3. Simplified 98.8%

      \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(-u0\right)}{\left(0 - \frac{cos2phi}{alphax \cdot alphax}\right) - \frac{sin2phi}{alphay \cdot alphay}}} \]
   4. Add Preprocessing

   5. Taylor expanded in u0 around 0

      \[\leadsto \color{blue}{\frac{u0}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}}} \]
   6. Step-by-step derivation

      1. /-lowering-/.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(u0, \color{blue}{\left(\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}\right)}\right) \]

      2. +-lowering-+.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{+.f32}\left(\left(\frac{cos2phi}{{alphax}^{2}}\right), \color{blue}{\left(\frac{sin2phi}{{alphay}^{2}}\right)}\right)\right) \]

      3. /-lowering-/.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{/.f32}\left(cos2phi, \left({alphax}^{2}\right)\right), \left(\frac{\color{blue}{sin2phi}}{{alphay}^{2}}\right)\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{/.f32}\left(cos2phi, \left(alphax \cdot alphax\right)\right), \left(\frac{sin2phi}{{alphay}^{2}}\right)\right)\right) \]

      5. *-lowering-*.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right), \left(\frac{sin2phi}{{alphay}^{2}}\right)\right)\right) \]

      6. /-lowering-/.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right), \mathsf{/.f32}\left(sin2phi, \color{blue}{\left({alphay}^{2}\right)}\right)\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right), \mathsf{/.f32}\left(sin2phi, \left(alphay \cdot \color{blue}{alphay}\right)\right)\right)\right) \]

      8. *-lowering-*.f32 73.3%

         \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right), \mathsf{/.f32}\left(sin2phi, \mathsf{*.f32}\left(alphay, \color{blue}{alphay}\right)\right)\right)\right) \]

   7. Simplified 73.3%

      \[\leadsto \color{blue}{\frac{u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}}} \]

   8. Taylor expanded in cos2phi around inf

      \[\leadsto \color{blue}{\frac{{alphax}^{2} \cdot u0}{cos2phi}} \]
   9. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto {alphax}^{2} \cdot \color{blue}{\frac{u0}{cos2phi}} \]

      2. *-lowering-*.f32 N/A

         \[\leadsto \mathsf{*.f32}\left(\left({alphax}^{2}\right), \color{blue}{\left(\frac{u0}{cos2phi}\right)}\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{*.f32}\left(\left(alphax \cdot alphax\right), \left(\frac{\color{blue}{u0}}{cos2phi}\right)\right) \]

      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 55.3%

         \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(alphax, alphax\right), \mathsf{/.f32}\left(u0, \color{blue}{cos2phi}\right)\right) \]

   10. Simplified 55.3%

       \[\leadsto \color{blue}{\left(alphax \cdot alphax\right) \cdot \frac{u0}{cos2phi}} \]

   if 4.99999984e-20 < sin2phi

   1. Initial program 62.8%

      \[\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
   2. Step-by-step derivation

      1. distribute-frac-neg N/A

         \[\leadsto \mathsf{neg}\left(\frac{\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}}\right) \]

      2. distribute-neg-frac2 N/A

         \[\leadsto \frac{\log \left(1 - u0\right)}{\color{blue}{\mathsf{neg}\left(\left(\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\right)\right)}} \]

      3. /-lowering-/.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(\log \left(1 - u0\right), \color{blue}{\left(\mathsf{neg}\left(\left(\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\right)\right)\right)}\right) \]

      4. sub-neg N/A

         \[\leadsto \mathsf{/.f32}\left(\log \left(1 + \left(\mathsf{neg}\left(u0\right)\right)\right), \left(\mathsf{neg}\left(\left(\color{blue}{\frac{cos2phi}{alphax \cdot alphax}} + \frac{sin2phi}{alphay \cdot alphay}\right)\right)\right)\right) \]

      5. log1p-define N/A

         \[\leadsto \mathsf{/.f32}\left(\left(\mathsf{log1p}\left(\mathsf{neg}\left(u0\right)\right)\right), \left(\mathsf{neg}\left(\color{blue}{\left(\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\right)}\right)\right)\right) \]

      6. log1p-lowering-log1p.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\left(\mathsf{neg}\left(u0\right)\right)\right), \left(\mathsf{neg}\left(\color{blue}{\left(\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\right)}\right)\right)\right) \]

      7. neg-lowering-neg.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \left(\mathsf{neg}\left(\left(\color{blue}{\frac{cos2phi}{alphax \cdot alphax}} + \frac{sin2phi}{alphay \cdot alphay}\right)\right)\right)\right) \]

      8. distribute-neg-in N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \left(\left(\mathsf{neg}\left(\frac{cos2phi}{alphax \cdot alphax}\right)\right) + \color{blue}{\left(\mathsf{neg}\left(\frac{sin2phi}{alphay \cdot alphay}\right)\right)}\right)\right) \]

      9. unsub-neg N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \left(\left(\mathsf{neg}\left(\frac{cos2phi}{alphax \cdot alphax}\right)\right) - \color{blue}{\frac{sin2phi}{alphay \cdot alphay}}\right)\right) \]

      10. --lowering--.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\left(\mathsf{neg}\left(\frac{cos2phi}{alphax \cdot alphax}\right)\right), \color{blue}{\left(\frac{sin2phi}{alphay \cdot alphay}\right)}\right)\right) \]

      11. neg-sub0 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\left(0 - \frac{cos2phi}{alphax \cdot alphax}\right), \left(\frac{\color{blue}{sin2phi}}{alphay \cdot alphay}\right)\right)\right) \]

      12. --lowering--.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \left(\frac{cos2phi}{alphax \cdot alphax}\right)\right), \left(\frac{\color{blue}{sin2phi}}{alphay \cdot alphay}\right)\right)\right) \]

      13. /-lowering-/.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(cos2phi, \left(alphax \cdot alphax\right)\right)\right), \left(\frac{sin2phi}{alphay \cdot alphay}\right)\right)\right) \]

      14. *-lowering-*.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right)\right), \left(\frac{sin2phi}{alphay \cdot alphay}\right)\right)\right) \]

      15. /-lowering-/.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right)\right), \mathsf{/.f32}\left(sin2phi, \color{blue}{\left(alphay \cdot alphay\right)}\right)\right)\right) \]

      16. *-lowering-*.f32 98.3%

          \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right)\right), \mathsf{/.f32}\left(sin2phi, \mathsf{*.f32}\left(alphay, \color{blue}{alphay}\right)\right)\right)\right) \]

   3. Simplified 98.3%

      \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(-u0\right)}{\left(0 - \frac{cos2phi}{alphax \cdot alphax}\right) - \frac{sin2phi}{alphay \cdot alphay}}} \]
   4. Add Preprocessing

   5. Taylor expanded in u0 around 0

      \[\leadsto \color{blue}{u0 \cdot \left(u0 \cdot \left(u0 \cdot \left(\frac{1}{4} \cdot \frac{u0}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}} + \frac{1}{3} \cdot \frac{1}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}}\right) + \frac{1}{2} \cdot \frac{1}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}}\right) + \frac{1}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}}\right)} \]
   6. Step-by-step derivation

      1. *-lowering-*.f32 N/A

         \[\leadsto \mathsf{*.f32}\left(u0, \color{blue}{\left(u0 \cdot \left(u0 \cdot \left(\frac{1}{4} \cdot \frac{u0}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}} + \frac{1}{3} \cdot \frac{1}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}}\right) + \frac{1}{2} \cdot \frac{1}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}}\right) + \frac{1}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}}\right)}\right) \]

      2. +-commutative N/A

         \[\leadsto \mathsf{*.f32}\left(u0, \left(\frac{1}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}} + \color{blue}{u0 \cdot \left(u0 \cdot \left(\frac{1}{4} \cdot \frac{u0}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}} + \frac{1}{3} \cdot \frac{1}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}}\right) + \frac{1}{2} \cdot \frac{1}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}}\right)}\right)\right) \]

      3. +-lowering-+.f32 N/A

         \[\leadsto \mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\left(\frac{1}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}}\right), \color{blue}{\left(u0 \cdot \left(u0 \cdot \left(\frac{1}{4} \cdot \frac{u0}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}} + \frac{1}{3} \cdot \frac{1}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}}\right) + \frac{1}{2} \cdot \frac{1}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}}\right)\right)}\right)\right) \]

   7. Simplified 93.0%

      \[\leadsto \color{blue}{u0 \cdot \left(\frac{1}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} + u0 \cdot \left(\frac{0.5}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} + u0 \cdot \left(\frac{u0 \cdot 0.25}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} + \frac{0.3333333333333333}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}}\right)\right)\right)} \]

   8. Taylor expanded in sin2phi around inf

      \[\leadsto \color{blue}{\frac{u0 \cdot \left(u0 \cdot \left(\frac{1}{2} \cdot {alphay}^{2} + u0 \cdot \left(\frac{1}{4} \cdot \left({alphay}^{2} \cdot u0\right) + \frac{1}{3} \cdot {alphay}^{2}\right)\right) + {alphay}^{2}\right)}{sin2phi}} \]
   9. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto u0 \cdot \color{blue}{\frac{u0 \cdot \left(\frac{1}{2} \cdot {alphay}^{2} + u0 \cdot \left(\frac{1}{4} \cdot \left({alphay}^{2} \cdot u0\right) + \frac{1}{3} \cdot {alphay}^{2}\right)\right) + {alphay}^{2}}{sin2phi}} \]

      2. *-lowering-*.f32 N/A

         \[\leadsto \mathsf{*.f32}\left(u0, \color{blue}{\left(\frac{u0 \cdot \left(\frac{1}{2} \cdot {alphay}^{2} + u0 \cdot \left(\frac{1}{4} \cdot \left({alphay}^{2} \cdot u0\right) + \frac{1}{3} \cdot {alphay}^{2}\right)\right) + {alphay}^{2}}{sin2phi}\right)}\right) \]

      3. /-lowering-/.f32 N/A

         \[\leadsto \mathsf{*.f32}\left(u0, \mathsf{/.f32}\left(\left(u0 \cdot \left(\frac{1}{2} \cdot {alphay}^{2} + u0 \cdot \left(\frac{1}{4} \cdot \left({alphay}^{2} \cdot u0\right) + \frac{1}{3} \cdot {alphay}^{2}\right)\right) + {alphay}^{2}\right), \color{blue}{sin2phi}\right)\right) \]

   10. Simplified 82.1%

       \[\leadsto \color{blue}{u0 \cdot \frac{alphay \cdot alphay + u0 \cdot \left(u0 \cdot \left(\left(u0 \cdot \left(alphay \cdot alphay\right)\right) \cdot 0.25 + \left(alphay \cdot alphay\right) \cdot 0.3333333333333333\right) + \left(alphay \cdot alphay\right) \cdot 0.5\right)}{sin2phi}} \]

   11. Taylor expanded in u0 around 0

       \[\leadsto \mathsf{*.f32}\left(u0, \color{blue}{\left(\frac{{alphay}^{2}}{sin2phi}\right)}\right) \]
   12. Step-by-step derivation

       1. /-lowering-/.f32 N/A

          \[\leadsto \mathsf{*.f32}\left(u0, \mathsf{/.f32}\left(\left({alphay}^{2}\right), \color{blue}{sin2phi}\right)\right) \]

       2. unpow2 N/A

          \[\leadsto \mathsf{*.f32}\left(u0, \mathsf{/.f32}\left(\left(alphay \cdot alphay\right), sin2phi\right)\right) \]

       3. *-lowering-*.f32 67.5%

          \[\leadsto \mathsf{*.f32}\left(u0, \mathsf{/.f32}\left(\mathsf{*.f32}\left(alphay, alphay\right), sin2phi\right)\right) \]

   13. Simplified 67.5%

       \[\leadsto u0 \cdot \color{blue}{\frac{alphay \cdot alphay}{sin2phi}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 28: 67.2% accurate, 9.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;sin2phi \leq 4.9999998413276127 \cdot 10^{-20}:\\ \;\;\;\;u0 \cdot \frac{alphax}{\frac{cos2phi}{alphax}}\\ \mathbf{else}:\\ \;\;\;\;u0 \cdot \frac{alphay \cdot alphay}{sin2phi}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (if (<= sin2phi 4.9999998413276127e-20)
   (* u0 (/ alphax (/ cos2phi alphax)))
   (* u0 (/ (* alphay alphay) sin2phi))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float tmp;
	if (sin2phi <= 4.9999998413276127e-20f) {
		tmp = u0 * (alphax / (cos2phi / alphax));
	} else {
		tmp = u0 * ((alphay * alphay) / sin2phi);
	}
	return tmp;
}

Fortran

real(4) function code(alphax, alphay, u0, cos2phi, sin2phi)
    real(4), intent (in) :: alphax
    real(4), intent (in) :: alphay
    real(4), intent (in) :: u0
    real(4), intent (in) :: cos2phi
    real(4), intent (in) :: sin2phi
    real(4) :: tmp
    if (sin2phi <= 4.9999998413276127e-20) then
        tmp = u0 * (alphax / (cos2phi / alphax))
    else
        tmp = u0 * ((alphay * alphay) / sin2phi)
    end if
    code = tmp
end function

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = Float32(0.0)
	if (sin2phi <= Float32(4.9999998413276127e-20))
		tmp = Float32(u0 * Float32(alphax / Float32(cos2phi / alphax)));
	else
		tmp = Float32(u0 * Float32(Float32(alphay * alphay) / sin2phi));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = single(0.0);
	if (sin2phi <= single(4.9999998413276127e-20))
		tmp = u0 * (alphax / (cos2phi / alphax));
	else
		tmp = u0 * ((alphay * alphay) / sin2phi);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if sin2phi < 4.99999984e-20

   1. Initial program 55.8%

      \[\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
   2. Step-by-step derivation

      1. distribute-frac-neg N/A

         \[\leadsto \mathsf{neg}\left(\frac{\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}}\right) \]

      2. distribute-neg-frac2 N/A

         \[\leadsto \frac{\log \left(1 - u0\right)}{\color{blue}{\mathsf{neg}\left(\left(\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\right)\right)}} \]

      3. /-lowering-/.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(\log \left(1 - u0\right), \color{blue}{\left(\mathsf{neg}\left(\left(\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\right)\right)\right)}\right) \]

      4. sub-neg N/A

         \[\leadsto \mathsf{/.f32}\left(\log \left(1 + \left(\mathsf{neg}\left(u0\right)\right)\right), \left(\mathsf{neg}\left(\left(\color{blue}{\frac{cos2phi}{alphax \cdot alphax}} + \frac{sin2phi}{alphay \cdot alphay}\right)\right)\right)\right) \]

      5. log1p-define N/A

         \[\leadsto \mathsf{/.f32}\left(\left(\mathsf{log1p}\left(\mathsf{neg}\left(u0\right)\right)\right), \left(\mathsf{neg}\left(\color{blue}{\left(\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\right)}\right)\right)\right) \]

      6. log1p-lowering-log1p.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\left(\mathsf{neg}\left(u0\right)\right)\right), \left(\mathsf{neg}\left(\color{blue}{\left(\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\right)}\right)\right)\right) \]

      7. neg-lowering-neg.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \left(\mathsf{neg}\left(\left(\color{blue}{\frac{cos2phi}{alphax \cdot alphax}} + \frac{sin2phi}{alphay \cdot alphay}\right)\right)\right)\right) \]

      8. distribute-neg-in N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \left(\left(\mathsf{neg}\left(\frac{cos2phi}{alphax \cdot alphax}\right)\right) + \color{blue}{\left(\mathsf{neg}\left(\frac{sin2phi}{alphay \cdot alphay}\right)\right)}\right)\right) \]

      9. unsub-neg N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \left(\left(\mathsf{neg}\left(\frac{cos2phi}{alphax \cdot alphax}\right)\right) - \color{blue}{\frac{sin2phi}{alphay \cdot alphay}}\right)\right) \]

      10. --lowering--.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\left(\mathsf{neg}\left(\frac{cos2phi}{alphax \cdot alphax}\right)\right), \color{blue}{\left(\frac{sin2phi}{alphay \cdot alphay}\right)}\right)\right) \]

      11. neg-sub0 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\left(0 - \frac{cos2phi}{alphax \cdot alphax}\right), \left(\frac{\color{blue}{sin2phi}}{alphay \cdot alphay}\right)\right)\right) \]

      12. --lowering--.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \left(\frac{cos2phi}{alphax \cdot alphax}\right)\right), \left(\frac{\color{blue}{sin2phi}}{alphay \cdot alphay}\right)\right)\right) \]

      13. /-lowering-/.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(cos2phi, \left(alphax \cdot alphax\right)\right)\right), \left(\frac{sin2phi}{alphay \cdot alphay}\right)\right)\right) \]

      14. *-lowering-*.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right)\right), \left(\frac{sin2phi}{alphay \cdot alphay}\right)\right)\right) \]

      15. /-lowering-/.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right)\right), \mathsf{/.f32}\left(sin2phi, \color{blue}{\left(alphay \cdot alphay\right)}\right)\right)\right) \]

      16. *-lowering-*.f32 98.8%

          \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right)\right), \mathsf{/.f32}\left(sin2phi, \mathsf{*.f32}\left(alphay, \color{blue}{alphay}\right)\right)\right)\right) \]

   3. Simplified 98.8%

      \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(-u0\right)}{\left(0 - \frac{cos2phi}{alphax \cdot alphax}\right) - \frac{sin2phi}{alphay \cdot alphay}}} \]
   4. Add Preprocessing

   5. Taylor expanded in u0 around 0

      \[\leadsto \color{blue}{\frac{u0}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}}} \]
   6. Step-by-step derivation

      1. /-lowering-/.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(u0, \color{blue}{\left(\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}\right)}\right) \]

      2. +-lowering-+.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{+.f32}\left(\left(\frac{cos2phi}{{alphax}^{2}}\right), \color{blue}{\left(\frac{sin2phi}{{alphay}^{2}}\right)}\right)\right) \]

      3. /-lowering-/.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{/.f32}\left(cos2phi, \left({alphax}^{2}\right)\right), \left(\frac{\color{blue}{sin2phi}}{{alphay}^{2}}\right)\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{/.f32}\left(cos2phi, \left(alphax \cdot alphax\right)\right), \left(\frac{sin2phi}{{alphay}^{2}}\right)\right)\right) \]

      5. *-lowering-*.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right), \left(\frac{sin2phi}{{alphay}^{2}}\right)\right)\right) \]

      6. /-lowering-/.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right), \mathsf{/.f32}\left(sin2phi, \color{blue}{\left({alphay}^{2}\right)}\right)\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right), \mathsf{/.f32}\left(sin2phi, \left(alphay \cdot \color{blue}{alphay}\right)\right)\right)\right) \]

      8. *-lowering-*.f32 73.3%

         \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right), \mathsf{/.f32}\left(sin2phi, \mathsf{*.f32}\left(alphay, \color{blue}{alphay}\right)\right)\right)\right) \]

   7. Simplified 73.3%

      \[\leadsto \color{blue}{\frac{u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}}} \]

   8. Taylor expanded in cos2phi around inf

      \[\leadsto \color{blue}{\frac{{alphax}^{2} \cdot u0}{cos2phi}} \]
   9. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto {alphax}^{2} \cdot \color{blue}{\frac{u0}{cos2phi}} \]

      2. *-lowering-*.f32 N/A

         \[\leadsto \mathsf{*.f32}\left(\left({alphax}^{2}\right), \color{blue}{\left(\frac{u0}{cos2phi}\right)}\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{*.f32}\left(\left(alphax \cdot alphax\right), \left(\frac{\color{blue}{u0}}{cos2phi}\right)\right) \]

      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 55.3%

         \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(alphax, alphax\right), \mathsf{/.f32}\left(u0, \color{blue}{cos2phi}\right)\right) \]

   10. Simplified 55.3%

       \[\leadsto \color{blue}{\left(alphax \cdot alphax\right) \cdot \frac{u0}{cos2phi}} \]
   11. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \frac{u0}{cos2phi} \cdot \color{blue}{\left(alphax \cdot alphax\right)} \]

       2. div-inv N/A

          \[\leadsto \left(u0 \cdot \frac{1}{cos2phi}\right) \cdot \left(\color{blue}{alphax} \cdot alphax\right) \]

       3. associate-*l* N/A

          \[\leadsto u0 \cdot \color{blue}{\left(\frac{1}{cos2phi} \cdot \left(alphax \cdot alphax\right)\right)} \]

       4. /-rgt-identity N/A

          \[\leadsto u0 \cdot \left(\frac{1}{cos2phi} \cdot \left(\frac{alphax}{1} \cdot alphax\right)\right) \]

       5. /-rgt-identity N/A

          \[\leadsto u0 \cdot \left(\frac{1}{cos2phi} \cdot \left(\frac{alphax}{1} \cdot \frac{alphax}{\color{blue}{1}}\right)\right) \]

       6. clear-num N/A

          \[\leadsto u0 \cdot \left(\frac{1}{cos2phi} \cdot \left(\frac{alphax}{1} \cdot \frac{1}{\color{blue}{\frac{1}{alphax}}}\right)\right) \]

       7. times-frac N/A

          \[\leadsto u0 \cdot \left(\frac{1}{cos2phi} \cdot \frac{alphax \cdot 1}{\color{blue}{1 \cdot \frac{1}{alphax}}}\right) \]

       8. metadata-eval N/A

          \[\leadsto u0 \cdot \left(\frac{1}{cos2phi} \cdot \frac{alphax \cdot \frac{1}{1}}{1 \cdot \frac{1}{alphax}}\right) \]

       9. div-inv N/A

          \[\leadsto u0 \cdot \left(\frac{1}{cos2phi} \cdot \frac{\frac{alphax}{1}}{\color{blue}{1} \cdot \frac{1}{alphax}}\right) \]

       10. /-rgt-identity N/A

           \[\leadsto u0 \cdot \left(\frac{1}{cos2phi} \cdot \frac{alphax}{\color{blue}{1} \cdot \frac{1}{alphax}}\right) \]

       11. div-inv N/A

           \[\leadsto u0 \cdot \left(\frac{1}{cos2phi} \cdot \frac{alphax}{\frac{1}{\color{blue}{alphax}}}\right) \]

       12. times-frac N/A

           \[\leadsto u0 \cdot \frac{1 \cdot alphax}{\color{blue}{cos2phi \cdot \frac{1}{alphax}}} \]

       13. div-inv N/A

           \[\leadsto u0 \cdot \frac{1 \cdot alphax}{\frac{cos2phi}{\color{blue}{alphax}}} \]

       14. associate-*l/ N/A

           \[\leadsto u0 \cdot \left(\frac{1}{\frac{cos2phi}{alphax}} \cdot \color{blue}{alphax}\right) \]

       15. clear-num N/A

           \[\leadsto u0 \cdot \left(\frac{alphax}{cos2phi} \cdot alphax\right) \]

       16. *-lowering-*.f32 N/A

           \[\leadsto \mathsf{*.f32}\left(u0, \color{blue}{\left(\frac{alphax}{cos2phi} \cdot alphax\right)}\right) \]

       17. associate-*l/ N/A

           \[\leadsto \mathsf{*.f32}\left(u0, \left(\frac{alphax \cdot alphax}{\color{blue}{cos2phi}}\right)\right) \]

       18. clear-num N/A

           \[\leadsto \mathsf{*.f32}\left(u0, \left(\frac{1}{\color{blue}{\frac{cos2phi}{alphax \cdot alphax}}}\right)\right) \]

       19. associate-/r* N/A

           \[\leadsto \mathsf{*.f32}\left(u0, \left(\frac{1}{\frac{\frac{cos2phi}{alphax}}{\color{blue}{alphax}}}\right)\right) \]

       20. clear-num N/A

           \[\leadsto \mathsf{*.f32}\left(u0, \left(\frac{alphax}{\color{blue}{\frac{cos2phi}{alphax}}}\right)\right) \]

       21. /-lowering-/.f32 N/A

           \[\leadsto \mathsf{*.f32}\left(u0, \mathsf{/.f32}\left(alphax, \color{blue}{\left(\frac{cos2phi}{alphax}\right)}\right)\right) \]

       22. /-lowering-/.f32 55.3%

           \[\leadsto \mathsf{*.f32}\left(u0, \mathsf{/.f32}\left(alphax, \mathsf{/.f32}\left(cos2phi, \color{blue}{alphax}\right)\right)\right) \]

   12. Applied egg-rr 55.3%

       \[\leadsto \color{blue}{u0 \cdot \frac{alphax}{\frac{cos2phi}{alphax}}} \]

   if 4.99999984e-20 < sin2phi

   1. Initial program 62.8%

      \[\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
   2. Step-by-step derivation

      1. distribute-frac-neg N/A

         \[\leadsto \mathsf{neg}\left(\frac{\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}}\right) \]

      2. distribute-neg-frac2 N/A

         \[\leadsto \frac{\log \left(1 - u0\right)}{\color{blue}{\mathsf{neg}\left(\left(\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\right)\right)}} \]

      3. /-lowering-/.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(\log \left(1 - u0\right), \color{blue}{\left(\mathsf{neg}\left(\left(\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\right)\right)\right)}\right) \]

      4. sub-neg N/A

         \[\leadsto \mathsf{/.f32}\left(\log \left(1 + \left(\mathsf{neg}\left(u0\right)\right)\right), \left(\mathsf{neg}\left(\left(\color{blue}{\frac{cos2phi}{alphax \cdot alphax}} + \frac{sin2phi}{alphay \cdot alphay}\right)\right)\right)\right) \]

      5. log1p-define N/A

         \[\leadsto \mathsf{/.f32}\left(\left(\mathsf{log1p}\left(\mathsf{neg}\left(u0\right)\right)\right), \left(\mathsf{neg}\left(\color{blue}{\left(\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\right)}\right)\right)\right) \]

      6. log1p-lowering-log1p.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\left(\mathsf{neg}\left(u0\right)\right)\right), \left(\mathsf{neg}\left(\color{blue}{\left(\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\right)}\right)\right)\right) \]

      7. neg-lowering-neg.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \left(\mathsf{neg}\left(\left(\color{blue}{\frac{cos2phi}{alphax \cdot alphax}} + \frac{sin2phi}{alphay \cdot alphay}\right)\right)\right)\right) \]

      8. distribute-neg-in N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \left(\left(\mathsf{neg}\left(\frac{cos2phi}{alphax \cdot alphax}\right)\right) + \color{blue}{\left(\mathsf{neg}\left(\frac{sin2phi}{alphay \cdot alphay}\right)\right)}\right)\right) \]

      9. unsub-neg N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \left(\left(\mathsf{neg}\left(\frac{cos2phi}{alphax \cdot alphax}\right)\right) - \color{blue}{\frac{sin2phi}{alphay \cdot alphay}}\right)\right) \]

      10. --lowering--.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\left(\mathsf{neg}\left(\frac{cos2phi}{alphax \cdot alphax}\right)\right), \color{blue}{\left(\frac{sin2phi}{alphay \cdot alphay}\right)}\right)\right) \]

      11. neg-sub0 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\left(0 - \frac{cos2phi}{alphax \cdot alphax}\right), \left(\frac{\color{blue}{sin2phi}}{alphay \cdot alphay}\right)\right)\right) \]

      12. --lowering--.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \left(\frac{cos2phi}{alphax \cdot alphax}\right)\right), \left(\frac{\color{blue}{sin2phi}}{alphay \cdot alphay}\right)\right)\right) \]

      13. /-lowering-/.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(cos2phi, \left(alphax \cdot alphax\right)\right)\right), \left(\frac{sin2phi}{alphay \cdot alphay}\right)\right)\right) \]

      14. *-lowering-*.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right)\right), \left(\frac{sin2phi}{alphay \cdot alphay}\right)\right)\right) \]

      15. /-lowering-/.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right)\right), \mathsf{/.f32}\left(sin2phi, \color{blue}{\left(alphay \cdot alphay\right)}\right)\right)\right) \]

      16. *-lowering-*.f32 98.3%

          \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right)\right), \mathsf{/.f32}\left(sin2phi, \mathsf{*.f32}\left(alphay, \color{blue}{alphay}\right)\right)\right)\right) \]

   3. Simplified 98.3%

      \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(-u0\right)}{\left(0 - \frac{cos2phi}{alphax \cdot alphax}\right) - \frac{sin2phi}{alphay \cdot alphay}}} \]
   4. Add Preprocessing

   5. Taylor expanded in u0 around 0

      \[\leadsto \color{blue}{u0 \cdot \left(u0 \cdot \left(u0 \cdot \left(\frac{1}{4} \cdot \frac{u0}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}} + \frac{1}{3} \cdot \frac{1}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}}\right) + \frac{1}{2} \cdot \frac{1}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}}\right) + \frac{1}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}}\right)} \]
   6. Step-by-step derivation

      1. *-lowering-*.f32 N/A

         \[\leadsto \mathsf{*.f32}\left(u0, \color{blue}{\left(u0 \cdot \left(u0 \cdot \left(\frac{1}{4} \cdot \frac{u0}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}} + \frac{1}{3} \cdot \frac{1}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}}\right) + \frac{1}{2} \cdot \frac{1}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}}\right) + \frac{1}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}}\right)}\right) \]

      2. +-commutative N/A

         \[\leadsto \mathsf{*.f32}\left(u0, \left(\frac{1}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}} + \color{blue}{u0 \cdot \left(u0 \cdot \left(\frac{1}{4} \cdot \frac{u0}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}} + \frac{1}{3} \cdot \frac{1}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}}\right) + \frac{1}{2} \cdot \frac{1}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}}\right)}\right)\right) \]

      3. +-lowering-+.f32 N/A

         \[\leadsto \mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\left(\frac{1}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}}\right), \color{blue}{\left(u0 \cdot \left(u0 \cdot \left(\frac{1}{4} \cdot \frac{u0}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}} + \frac{1}{3} \cdot \frac{1}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}}\right) + \frac{1}{2} \cdot \frac{1}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}}\right)\right)}\right)\right) \]

   7. Simplified 93.0%

      \[\leadsto \color{blue}{u0 \cdot \left(\frac{1}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} + u0 \cdot \left(\frac{0.5}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} + u0 \cdot \left(\frac{u0 \cdot 0.25}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} + \frac{0.3333333333333333}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}}\right)\right)\right)} \]

   8. Taylor expanded in sin2phi around inf

      \[\leadsto \color{blue}{\frac{u0 \cdot \left(u0 \cdot \left(\frac{1}{2} \cdot {alphay}^{2} + u0 \cdot \left(\frac{1}{4} \cdot \left({alphay}^{2} \cdot u0\right) + \frac{1}{3} \cdot {alphay}^{2}\right)\right) + {alphay}^{2}\right)}{sin2phi}} \]
   9. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto u0 \cdot \color{blue}{\frac{u0 \cdot \left(\frac{1}{2} \cdot {alphay}^{2} + u0 \cdot \left(\frac{1}{4} \cdot \left({alphay}^{2} \cdot u0\right) + \frac{1}{3} \cdot {alphay}^{2}\right)\right) + {alphay}^{2}}{sin2phi}} \]

      2. *-lowering-*.f32 N/A

         \[\leadsto \mathsf{*.f32}\left(u0, \color{blue}{\left(\frac{u0 \cdot \left(\frac{1}{2} \cdot {alphay}^{2} + u0 \cdot \left(\frac{1}{4} \cdot \left({alphay}^{2} \cdot u0\right) + \frac{1}{3} \cdot {alphay}^{2}\right)\right) + {alphay}^{2}}{sin2phi}\right)}\right) \]

      3. /-lowering-/.f32 N/A

         \[\leadsto \mathsf{*.f32}\left(u0, \mathsf{/.f32}\left(\left(u0 \cdot \left(\frac{1}{2} \cdot {alphay}^{2} + u0 \cdot \left(\frac{1}{4} \cdot \left({alphay}^{2} \cdot u0\right) + \frac{1}{3} \cdot {alphay}^{2}\right)\right) + {alphay}^{2}\right), \color{blue}{sin2phi}\right)\right) \]

   10. Simplified 82.1%

       \[\leadsto \color{blue}{u0 \cdot \frac{alphay \cdot alphay + u0 \cdot \left(u0 \cdot \left(\left(u0 \cdot \left(alphay \cdot alphay\right)\right) \cdot 0.25 + \left(alphay \cdot alphay\right) \cdot 0.3333333333333333\right) + \left(alphay \cdot alphay\right) \cdot 0.5\right)}{sin2phi}} \]

   11. Taylor expanded in u0 around 0

       \[\leadsto \mathsf{*.f32}\left(u0, \color{blue}{\left(\frac{{alphay}^{2}}{sin2phi}\right)}\right) \]
   12. Step-by-step derivation

       1. /-lowering-/.f32 N/A

          \[\leadsto \mathsf{*.f32}\left(u0, \mathsf{/.f32}\left(\left({alphay}^{2}\right), \color{blue}{sin2phi}\right)\right) \]

       2. unpow2 N/A

          \[\leadsto \mathsf{*.f32}\left(u0, \mathsf{/.f32}\left(\left(alphay \cdot alphay\right), sin2phi\right)\right) \]

       3. *-lowering-*.f32 67.5%

          \[\leadsto \mathsf{*.f32}\left(u0, \mathsf{/.f32}\left(\mathsf{*.f32}\left(alphay, alphay\right), sin2phi\right)\right) \]

   13. Simplified 67.5%

       \[\leadsto u0 \cdot \color{blue}{\frac{alphay \cdot alphay}{sin2phi}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 29: 23.6% accurate, 16.6× speedup

Math

\[\begin{array}{l} \\ u0 \cdot \frac{alphax}{\frac{cos2phi}{alphax}} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (* u0 (/ alphax (/ cos2phi alphax))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	return u0 * (alphax / (cos2phi / alphax));
}

Fortran

real(4) function code(alphax, alphay, u0, cos2phi, sin2phi)
    real(4), intent (in) :: alphax
    real(4), intent (in) :: alphay
    real(4), intent (in) :: u0
    real(4), intent (in) :: cos2phi
    real(4), intent (in) :: sin2phi
    code = u0 * (alphax / (cos2phi / alphax))
end function

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	return Float32(u0 * Float32(alphax / Float32(cos2phi / alphax)))
end

MATLAB

function tmp = code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = u0 * (alphax / (cos2phi / alphax));
end

Derivation

1. Initial program 60.9%

   \[\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
2. Step-by-step derivation

   1. distribute-frac-neg N/A

      \[\leadsto \mathsf{neg}\left(\frac{\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}}\right) \]

   2. distribute-neg-frac2 N/A

      \[\leadsto \frac{\log \left(1 - u0\right)}{\color{blue}{\mathsf{neg}\left(\left(\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\right)\right)}} \]

   3. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\log \left(1 - u0\right), \color{blue}{\left(\mathsf{neg}\left(\left(\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\right)\right)\right)}\right) \]

   4. sub-neg N/A

      \[\leadsto \mathsf{/.f32}\left(\log \left(1 + \left(\mathsf{neg}\left(u0\right)\right)\right), \left(\mathsf{neg}\left(\left(\color{blue}{\frac{cos2phi}{alphax \cdot alphax}} + \frac{sin2phi}{alphay \cdot alphay}\right)\right)\right)\right) \]

   5. log1p-define N/A

      \[\leadsto \mathsf{/.f32}\left(\left(\mathsf{log1p}\left(\mathsf{neg}\left(u0\right)\right)\right), \left(\mathsf{neg}\left(\color{blue}{\left(\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\right)}\right)\right)\right) \]

   6. log1p-lowering-log1p.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\left(\mathsf{neg}\left(u0\right)\right)\right), \left(\mathsf{neg}\left(\color{blue}{\left(\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\right)}\right)\right)\right) \]

   7. neg-lowering-neg.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \left(\mathsf{neg}\left(\left(\color{blue}{\frac{cos2phi}{alphax \cdot alphax}} + \frac{sin2phi}{alphay \cdot alphay}\right)\right)\right)\right) \]

   8. distribute-neg-in N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \left(\left(\mathsf{neg}\left(\frac{cos2phi}{alphax \cdot alphax}\right)\right) + \color{blue}{\left(\mathsf{neg}\left(\frac{sin2phi}{alphay \cdot alphay}\right)\right)}\right)\right) \]

   9. unsub-neg N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \left(\left(\mathsf{neg}\left(\frac{cos2phi}{alphax \cdot alphax}\right)\right) - \color{blue}{\frac{sin2phi}{alphay \cdot alphay}}\right)\right) \]

   10. --lowering--.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\left(\mathsf{neg}\left(\frac{cos2phi}{alphax \cdot alphax}\right)\right), \color{blue}{\left(\frac{sin2phi}{alphay \cdot alphay}\right)}\right)\right) \]

   11. neg-sub0 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\left(0 - \frac{cos2phi}{alphax \cdot alphax}\right), \left(\frac{\color{blue}{sin2phi}}{alphay \cdot alphay}\right)\right)\right) \]

   12. --lowering--.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \left(\frac{cos2phi}{alphax \cdot alphax}\right)\right), \left(\frac{\color{blue}{sin2phi}}{alphay \cdot alphay}\right)\right)\right) \]

   13. /-lowering-/.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(cos2phi, \left(alphax \cdot alphax\right)\right)\right), \left(\frac{sin2phi}{alphay \cdot alphay}\right)\right)\right) \]

   14. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right)\right), \left(\frac{sin2phi}{alphay \cdot alphay}\right)\right)\right) \]

   15. /-lowering-/.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right)\right), \mathsf{/.f32}\left(sin2phi, \color{blue}{\left(alphay \cdot alphay\right)}\right)\right)\right) \]

   16. *-lowering-*.f32 98.4%

       \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right)\right), \mathsf{/.f32}\left(sin2phi, \mathsf{*.f32}\left(alphay, \color{blue}{alphay}\right)\right)\right)\right) \]

3. Simplified 98.4%

   \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(-u0\right)}{\left(0 - \frac{cos2phi}{alphax \cdot alphax}\right) - \frac{sin2phi}{alphay \cdot alphay}}} \]
4. Add Preprocessing

5. Taylor expanded in u0 around 0

   \[\leadsto \color{blue}{\frac{u0}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}}} \]
6. Step-by-step derivation

   1. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(u0, \color{blue}{\left(\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}\right)}\right) \]

   2. +-lowering-+.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{+.f32}\left(\left(\frac{cos2phi}{{alphax}^{2}}\right), \color{blue}{\left(\frac{sin2phi}{{alphay}^{2}}\right)}\right)\right) \]

   3. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{/.f32}\left(cos2phi, \left({alphax}^{2}\right)\right), \left(\frac{\color{blue}{sin2phi}}{{alphay}^{2}}\right)\right)\right) \]

   4. unpow2 N/A

      \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{/.f32}\left(cos2phi, \left(alphax \cdot alphax\right)\right), \left(\frac{sin2phi}{{alphay}^{2}}\right)\right)\right) \]

   5. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right), \left(\frac{sin2phi}{{alphay}^{2}}\right)\right)\right) \]

   6. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right), \mathsf{/.f32}\left(sin2phi, \color{blue}{\left({alphay}^{2}\right)}\right)\right)\right) \]

   7. unpow2 N/A

      \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right), \mathsf{/.f32}\left(sin2phi, \left(alphay \cdot \color{blue}{alphay}\right)\right)\right)\right) \]

   8. *-lowering-*.f32 74.5%

      \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right), \mathsf{/.f32}\left(sin2phi, \mathsf{*.f32}\left(alphay, \color{blue}{alphay}\right)\right)\right)\right) \]

7. Simplified 74.5%

   \[\leadsto \color{blue}{\frac{u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}}} \]

8. Taylor expanded in cos2phi around inf

   \[\leadsto \color{blue}{\frac{{alphax}^{2} \cdot u0}{cos2phi}} \]
9. Step-by-step derivation

   1. associate-/l* N/A

      \[\leadsto {alphax}^{2} \cdot \color{blue}{\frac{u0}{cos2phi}} \]

   2. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{*.f32}\left(\left({alphax}^{2}\right), \color{blue}{\left(\frac{u0}{cos2phi}\right)}\right) \]

   3. unpow2 N/A

      \[\leadsto \mathsf{*.f32}\left(\left(alphax \cdot alphax\right), \left(\frac{\color{blue}{u0}}{cos2phi}\right)\right) \]

   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 25.0%

      \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(alphax, alphax\right), \mathsf{/.f32}\left(u0, \color{blue}{cos2phi}\right)\right) \]

10. Simplified 25.0%

    \[\leadsto \color{blue}{\left(alphax \cdot alphax\right) \cdot \frac{u0}{cos2phi}} \]
11. Step-by-step derivation

    1. *-commutative N/A

       \[\leadsto \frac{u0}{cos2phi} \cdot \color{blue}{\left(alphax \cdot alphax\right)} \]

    2. div-inv N/A

       \[\leadsto \left(u0 \cdot \frac{1}{cos2phi}\right) \cdot \left(\color{blue}{alphax} \cdot alphax\right) \]

    3. associate-*l* N/A

       \[\leadsto u0 \cdot \color{blue}{\left(\frac{1}{cos2phi} \cdot \left(alphax \cdot alphax\right)\right)} \]

    4. /-rgt-identity N/A

       \[\leadsto u0 \cdot \left(\frac{1}{cos2phi} \cdot \left(\frac{alphax}{1} \cdot alphax\right)\right) \]

    5. /-rgt-identity N/A

       \[\leadsto u0 \cdot \left(\frac{1}{cos2phi} \cdot \left(\frac{alphax}{1} \cdot \frac{alphax}{\color{blue}{1}}\right)\right) \]

    6. clear-num N/A

       \[\leadsto u0 \cdot \left(\frac{1}{cos2phi} \cdot \left(\frac{alphax}{1} \cdot \frac{1}{\color{blue}{\frac{1}{alphax}}}\right)\right) \]

    7. times-frac N/A

       \[\leadsto u0 \cdot \left(\frac{1}{cos2phi} \cdot \frac{alphax \cdot 1}{\color{blue}{1 \cdot \frac{1}{alphax}}}\right) \]

    8. metadata-eval N/A

       \[\leadsto u0 \cdot \left(\frac{1}{cos2phi} \cdot \frac{alphax \cdot \frac{1}{1}}{1 \cdot \frac{1}{alphax}}\right) \]

    9. div-inv N/A

       \[\leadsto u0 \cdot \left(\frac{1}{cos2phi} \cdot \frac{\frac{alphax}{1}}{\color{blue}{1} \cdot \frac{1}{alphax}}\right) \]

    10. /-rgt-identity N/A

        \[\leadsto u0 \cdot \left(\frac{1}{cos2phi} \cdot \frac{alphax}{\color{blue}{1} \cdot \frac{1}{alphax}}\right) \]

    11. div-inv N/A

        \[\leadsto u0 \cdot \left(\frac{1}{cos2phi} \cdot \frac{alphax}{\frac{1}{\color{blue}{alphax}}}\right) \]

    12. times-frac N/A

        \[\leadsto u0 \cdot \frac{1 \cdot alphax}{\color{blue}{cos2phi \cdot \frac{1}{alphax}}} \]

    13. div-inv N/A

        \[\leadsto u0 \cdot \frac{1 \cdot alphax}{\frac{cos2phi}{\color{blue}{alphax}}} \]

    14. associate-*l/ N/A

        \[\leadsto u0 \cdot \left(\frac{1}{\frac{cos2phi}{alphax}} \cdot \color{blue}{alphax}\right) \]

    15. clear-num N/A

        \[\leadsto u0 \cdot \left(\frac{alphax}{cos2phi} \cdot alphax\right) \]

    16. *-lowering-*.f32 N/A

        \[\leadsto \mathsf{*.f32}\left(u0, \color{blue}{\left(\frac{alphax}{cos2phi} \cdot alphax\right)}\right) \]

    17. associate-*l/ N/A

        \[\leadsto \mathsf{*.f32}\left(u0, \left(\frac{alphax \cdot alphax}{\color{blue}{cos2phi}}\right)\right) \]

    18. clear-num N/A

        \[\leadsto \mathsf{*.f32}\left(u0, \left(\frac{1}{\color{blue}{\frac{cos2phi}{alphax \cdot alphax}}}\right)\right) \]

    19. associate-/r* N/A

        \[\leadsto \mathsf{*.f32}\left(u0, \left(\frac{1}{\frac{\frac{cos2phi}{alphax}}{\color{blue}{alphax}}}\right)\right) \]

    20. clear-num N/A

        \[\leadsto \mathsf{*.f32}\left(u0, \left(\frac{alphax}{\color{blue}{\frac{cos2phi}{alphax}}}\right)\right) \]

    21. /-lowering-/.f32 N/A

        \[\leadsto \mathsf{*.f32}\left(u0, \mathsf{/.f32}\left(alphax, \color{blue}{\left(\frac{cos2phi}{alphax}\right)}\right)\right) \]

    22. /-lowering-/.f32 25.0%

        \[\leadsto \mathsf{*.f32}\left(u0, \mathsf{/.f32}\left(alphax, \mathsf{/.f32}\left(cos2phi, \color{blue}{alphax}\right)\right)\right) \]

12. Applied egg-rr 25.0%

    \[\leadsto \color{blue}{u0 \cdot \frac{alphax}{\frac{cos2phi}{alphax}}} \]
13. Add Preprocessing

Reproduce

herbie shell --seed 2024164 
(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)))))