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

Percentage Accurate: 60.5% → 98.3%

Time: 21.8s

Alternatives: 23

Speedup: 8.9×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Initial Program: 60.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Alternative 1: 98.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 60.2%

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

   1. sub-neg N/A

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

   2. log1p-define N/A

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

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

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

   4. neg-lowering-neg.f32 98.0%

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

4. Applied egg-rr 98.0%

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

   1. associate-/r* N/A

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

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

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

   3. /-lowering-/.f32 98.1%

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

6. Applied egg-rr 98.1%

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

7. Final simplification 98.1%

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

Alternative 2: 98.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 60.2%

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

   1. sub-neg N/A

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

   2. log1p-define N/A

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

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

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

   4. neg-lowering-neg.f32 98.0%

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

4. Applied egg-rr 98.0%

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

5. Final simplification 98.0%

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

Alternative 3: 95.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;sin2phi \leq 0.30000001192092896:\\ \;\;\;\;\frac{u0 - u0 \cdot \left(u0 \cdot \left(-0.5 + u0 \cdot \left(u0 \cdot -0.25 + -0.3333333333333333\right)\right)\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{\frac{sin2phi}{alphay}}{alphay}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(-u0\right)}{0 - \frac{sin2phi}{alphay \cdot alphay}}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (if (<= sin2phi 0.30000001192092896)
   (/
    (- u0 (* u0 (* u0 (+ -0.5 (* u0 (+ (* u0 -0.25) -0.3333333333333333))))))
    (+ (/ cos2phi (* alphax alphax)) (/ (/ sin2phi alphay) alphay)))
   (/ (log1p (- u0)) (- 0.0 (/ sin2phi (* alphay alphay))))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float tmp;
	if (sin2phi <= 0.30000001192092896f) {
		tmp = (u0 - (u0 * (u0 * (-0.5f + (u0 * ((u0 * -0.25f) + -0.3333333333333333f)))))) / ((cos2phi / (alphax * alphax)) + ((sin2phi / alphay) / alphay));
	} else {
		tmp = log1pf(-u0) / (0.0f - (sin2phi / (alphay * alphay)));
	}
	return tmp;
}

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = Float32(0.0)
	if (sin2phi <= Float32(0.30000001192092896))
		tmp = Float32(Float32(u0 - Float32(u0 * Float32(u0 * Float32(Float32(-0.5) + Float32(u0 * Float32(Float32(u0 * Float32(-0.25)) + Float32(-0.3333333333333333))))))) / Float32(Float32(cos2phi / Float32(alphax * alphax)) + Float32(Float32(sin2phi / alphay) / alphay)));
	else
		tmp = Float32(log1p(Float32(-u0)) / Float32(Float32(0.0) - Float32(sin2phi / Float32(alphay * alphay))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if sin2phi < 0.300000012

   1. Initial program 54.1%

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

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

   3. Simplified 98.7%

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

      1. associate-/r* N/A

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

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

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

      3. /-lowering-/.f32 98.8%

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

   6. Applied egg-rr 98.8%

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

   7. Taylor expanded in u0 around 0

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

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

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

      2. sub-neg N/A

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

      3. metadata-eval N/A

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

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

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

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

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

      6. sub-neg N/A

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

      7. metadata-eval N/A

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

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

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

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

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

      10. sub-neg N/A

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

      11. metadata-eval N/A

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

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

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

      13. *-commutative N/A

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

      14. *-lowering-*.f32 92.7%

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

   9. Simplified 92.7%

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

       1. distribute-rgt-in N/A

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

       2. neg-mul-1 N/A

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

       3. unsub-neg N/A

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

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

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

       5. *-commutative N/A

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

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

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

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

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

       8. +-commutative N/A

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

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

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

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

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

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

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

       12. *-lowering-*.f32 93.1%

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

   11. Applied egg-rr 93.1%

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

   if 0.300000012 < sin2phi

   1. Initial program 66.9%

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

      1. sub-neg N/A

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

      2. log1p-define N/A

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

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

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

      4. neg-lowering-neg.f32 97.3%

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

   4. Applied egg-rr 97.3%

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

   5. Taylor expanded in cos2phi around 0

      \[\leadsto \mathsf{/.f32}\left(\mathsf{neg.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right)\right), \color{blue}{\left(\frac{sin2phi}{{alphay}^{2}}\right)}\right) \]
   6. Step-by-step derivation

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

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

      2. unpow2 N/A

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

      3. *-lowering-*.f32 97.1%

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

   7. Simplified 97.1%

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

4. Final simplification 95.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;sin2phi \leq 0.30000001192092896:\\ \;\;\;\;\frac{u0 - u0 \cdot \left(u0 \cdot \left(-0.5 + u0 \cdot \left(u0 \cdot -0.25 + -0.3333333333333333\right)\right)\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{\frac{sin2phi}{alphay}}{alphay}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(-u0\right)}{0 - \frac{sin2phi}{alphay \cdot alphay}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 60.2%

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

   1. distribute-frac-neg N/A

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

   2. distribute-neg-frac2 N/A

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

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

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

   4. sub-neg N/A

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

   5. log1p-define N/A

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

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

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

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

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

   8. distribute-neg-in N/A

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

   9. unsub-neg N/A

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

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

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

   11. neg-sub0 N/A

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

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

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

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

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

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

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

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

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

   16. *-lowering-*.f32 98.0%

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

3. Simplified 98.0%

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

   1. associate--l- N/A

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

   2. neg-sub0 N/A

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

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

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

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

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

   5. associate-/r* N/A

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

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

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

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

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

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

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

   9. *-lowering-*.f32 98.0%

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

6. Applied egg-rr 98.0%

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

7. Final simplification 98.0%

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

Alternative 5: 95.7% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (if (<= sin2phi 5.0)
   (/
    (- u0 (* u0 (* u0 (+ -0.5 (* u0 (+ (* u0 -0.25) -0.3333333333333333))))))
    (+ (/ cos2phi (* alphax alphax)) (/ (/ sin2phi alphay) alphay)))
   (/ (* (log1p (- u0)) (* alphay alphay)) (- sin2phi))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float tmp;
	if (sin2phi <= 5.0f) {
		tmp = (u0 - (u0 * (u0 * (-0.5f + (u0 * ((u0 * -0.25f) + -0.3333333333333333f)))))) / ((cos2phi / (alphax * alphax)) + ((sin2phi / alphay) / alphay));
	} else {
		tmp = (log1pf(-u0) * (alphay * alphay)) / -sin2phi;
	}
	return tmp;
}

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = Float32(0.0)
	if (sin2phi <= Float32(5.0))
		tmp = Float32(Float32(u0 - Float32(u0 * Float32(u0 * Float32(Float32(-0.5) + Float32(u0 * Float32(Float32(u0 * Float32(-0.25)) + Float32(-0.3333333333333333))))))) / Float32(Float32(cos2phi / Float32(alphax * alphax)) + Float32(Float32(sin2phi / alphay) / alphay)));
	else
		tmp = Float32(Float32(log1p(Float32(-u0)) * Float32(alphay * alphay)) / Float32(-sin2phi));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if sin2phi < 5

   1. Initial program 54.0%

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

      1. distribute-frac-neg N/A

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

      2. distribute-neg-frac2 N/A

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

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

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

      4. sub-neg N/A

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

      5. log1p-define N/A

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

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

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

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

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

      8. distribute-neg-in N/A

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

      9. unsub-neg N/A

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

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

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

      11. neg-sub0 N/A

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

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

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

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

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

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

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

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

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

      16. *-lowering-*.f32 98.7%

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

   3. Simplified 98.7%

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

      1. associate-/r* N/A

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

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

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

      3. /-lowering-/.f32 98.8%

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

   6. Applied egg-rr 98.8%

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

   7. Taylor expanded in u0 around 0

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

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

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

      2. sub-neg N/A

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

      3. metadata-eval N/A

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

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

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

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

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

      6. sub-neg N/A

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

      7. metadata-eval N/A

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

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

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

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

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

      10. sub-neg N/A

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

      11. metadata-eval N/A

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

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

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

      13. *-commutative N/A

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

      14. *-lowering-*.f32 92.9%

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

   9. Simplified 92.9%

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

       1. distribute-rgt-in N/A

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

       2. neg-mul-1 N/A

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

       3. unsub-neg N/A

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

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

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

       5. *-commutative N/A

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

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

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

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

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

       8. +-commutative N/A

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

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

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

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

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

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

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

       12. *-lowering-*.f32 93.3%

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

   11. Applied egg-rr 93.3%

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

   if 5 < sin2phi

   1. Initial program 67.6%

      \[\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
   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 97.2%

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

   3. Simplified 97.2%

      \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(-u0\right)}{\left(0 - \frac{cos2phi}{alphax \cdot alphax}\right) - \frac{sin2phi}{alphay \cdot alphay}}} \]
   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. distribute-neg-frac2 N/A

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

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

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

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

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

      5. unpow2 N/A

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

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

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

      7. sub-neg N/A

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

      8. mul-1-neg N/A

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

      9. log1p-define N/A

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

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

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

      11. mul-1-neg N/A

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

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

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

      13. neg-lowering-neg.f32 98.8%

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

   7. Simplified 98.8%

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

4. Final simplification 95.8%

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

Alternative 6: 89.9% accurate, 3.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 500:\\ \;\;\;\;\frac{u0 - u0 \cdot \left(u0 \cdot -0.5\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{\frac{sin2phi}{alphay}}{alphay}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(u0 \cdot \left(alphay \cdot alphay\right)\right) \cdot \left(1 - u0 \cdot \left(-0.5 + u0 \cdot \left(u0 \cdot -0.25 + -0.3333333333333333\right)\right)\right)}{sin2phi}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (if (<= (/ sin2phi (* alphay alphay)) 500.0)
   (/
    (- u0 (* u0 (* u0 -0.5)))
    (+ (/ cos2phi (* alphax alphax)) (/ (/ sin2phi alphay) alphay)))
   (/
    (*
     (* u0 (* alphay alphay))
     (- 1.0 (* u0 (+ -0.5 (* u0 (+ (* u0 -0.25) -0.3333333333333333))))))
    sin2phi)))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float tmp;
	if ((sin2phi / (alphay * alphay)) <= 500.0f) {
		tmp = (u0 - (u0 * (u0 * -0.5f))) / ((cos2phi / (alphax * alphax)) + ((sin2phi / alphay) / alphay));
	} else {
		tmp = ((u0 * (alphay * alphay)) * (1.0f - (u0 * (-0.5f + (u0 * ((u0 * -0.25f) + -0.3333333333333333f)))))) / 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)) <= 500.0e0) then
        tmp = (u0 - (u0 * (u0 * (-0.5e0)))) / ((cos2phi / (alphax * alphax)) + ((sin2phi / alphay) / alphay))
    else
        tmp = ((u0 * (alphay * alphay)) * (1.0e0 - (u0 * ((-0.5e0) + (u0 * ((u0 * (-0.25e0)) + (-0.3333333333333333e0))))))) / 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(500.0))
		tmp = Float32(Float32(u0 - Float32(u0 * Float32(u0 * Float32(-0.5)))) / Float32(Float32(cos2phi / Float32(alphax * alphax)) + Float32(Float32(sin2phi / alphay) / alphay)));
	else
		tmp = Float32(Float32(Float32(u0 * Float32(alphay * alphay)) * Float32(Float32(1.0) - Float32(u0 * Float32(Float32(-0.5) + Float32(u0 * Float32(Float32(u0 * Float32(-0.25)) + Float32(-0.3333333333333333))))))) / sin2phi);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = single(0.0);
	if ((sin2phi / (alphay * alphay)) <= single(500.0))
		tmp = (u0 - (u0 * (u0 * single(-0.5)))) / ((cos2phi / (alphax * alphax)) + ((sin2phi / alphay) / alphay));
	else
		tmp = ((u0 * (alphay * alphay)) * (single(1.0) - (u0 * (single(-0.5) + (u0 * ((u0 * single(-0.25)) + single(-0.3333333333333333))))))) / sin2phi;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 sin2phi (*.f32 alphay alphay)) < 500

   1. Initial program 53.7%

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

      1. associate-/r* N/A

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

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

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

      3. /-lowering-/.f32 98.8%

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

   6. Applied egg-rr 98.8%

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

   7. Taylor expanded in u0 around 0

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

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

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

      2. sub-neg N/A

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

      3. metadata-eval N/A

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

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

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

      5. *-commutative N/A

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

      6. *-lowering-*.f32 87.9%

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

   9. Simplified 87.9%

      \[\leadsto \frac{\color{blue}{u0 \cdot \left(u0 \cdot -0.5 + -1\right)}}{\left(0 - \frac{cos2phi}{alphax \cdot alphax}\right) - \frac{\frac{sin2phi}{alphay}}{alphay}} \]
   10. Step-by-step derivation

       1. distribute-rgt-in N/A

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

       2. neg-mul-1 N/A

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

       3. unsub-neg N/A

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

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

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

       5. *-commutative N/A

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

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

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

       7. *-lowering-*.f32 88.1%

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

   11. Applied egg-rr 88.1%

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

   if 500 < (/.f32 sin2phi (*.f32 alphay alphay))

   1. Initial program 67.1%

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

      1. distribute-frac-neg N/A

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

      2. distribute-neg-frac2 N/A

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

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

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

      4. sub-neg N/A

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

      5. log1p-define N/A

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

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

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

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

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

      8. distribute-neg-in N/A

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

      9. unsub-neg N/A

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

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

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

      11. neg-sub0 N/A

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

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

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

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

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

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

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

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

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

      16. *-lowering-*.f32 97.3%

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

   3. Simplified 97.3%

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

      1. associate-/r* N/A

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

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

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

      3. /-lowering-/.f32 97.4%

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

   6. Applied egg-rr 97.4%

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

   7. Taylor expanded in u0 around 0

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

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

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

      2. sub-neg N/A

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

      3. metadata-eval N/A

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

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

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

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

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

      6. sub-neg N/A

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

      7. metadata-eval N/A

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

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

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

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

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

      10. sub-neg N/A

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

      11. metadata-eval N/A

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

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

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

      13. *-commutative N/A

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

      14. *-lowering-*.f32 89.9%

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

   9. Simplified 89.9%

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

   10. Taylor expanded in cos2phi around 0

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

       1. associate-*r/ N/A

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

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

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

   12. Simplified 91.0%

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

4. Final simplification 89.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 500:\\ \;\;\;\;\frac{u0 - u0 \cdot \left(u0 \cdot -0.5\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{\frac{sin2phi}{alphay}}{alphay}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(u0 \cdot \left(alphay \cdot alphay\right)\right) \cdot \left(1 - u0 \cdot \left(-0.5 + u0 \cdot \left(u0 \cdot -0.25 + -0.3333333333333333\right)\right)\right)}{sin2phi}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 92.8% accurate, 4.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 60.2%

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

   1. distribute-frac-neg N/A

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

   2. distribute-neg-frac2 N/A

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

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

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

   4. sub-neg N/A

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

   5. log1p-define N/A

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

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

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

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

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

   8. distribute-neg-in N/A

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

   9. unsub-neg N/A

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

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

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

   11. neg-sub0 N/A

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

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

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

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

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

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

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

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

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

   16. *-lowering-*.f32 98.0%

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

3. Simplified 98.0%

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

   1. associate-/r* N/A

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

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

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

   3. /-lowering-/.f32 98.1%

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

6. Applied egg-rr 98.1%

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

7. Taylor expanded in u0 around 0

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

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

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

   2. sub-neg N/A

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

   3. metadata-eval N/A

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

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

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

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

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

   6. sub-neg N/A

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

   7. metadata-eval N/A

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

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

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

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

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

   10. sub-neg N/A

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

   11. metadata-eval N/A

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

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

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

   13. *-commutative N/A

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

   14. *-lowering-*.f32 91.4%

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

9. Simplified 91.4%

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

    1. distribute-rgt-in N/A

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

    2. neg-mul-1 N/A

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

    3. unsub-neg N/A

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

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

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

    5. *-commutative N/A

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

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

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

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

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

    8. +-commutative N/A

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

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

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

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

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

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

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

    12. *-lowering-*.f32 91.7%

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

11. Applied egg-rr 91.7%

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

12. Final simplification 91.7%

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

Alternative 8: 92.7% accurate, 4.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 60.2%

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

3. Taylor expanded in u0 around 0

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

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

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

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

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

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

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

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

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

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

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

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

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

   7. *-commutative N/A

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

   8. *-lowering-*.f32 91.4%

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

5. Simplified 91.4%

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

6. Final simplification 91.4%

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

Alternative 9: 90.8% accurate, 5.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 60.2%

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

   1. sub-neg N/A

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

   2. log1p-define N/A

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

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

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

   4. neg-lowering-neg.f32 98.0%

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

4. Applied egg-rr 98.0%

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

   1. associate-/r* N/A

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

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

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

   3. /-lowering-/.f32 98.1%

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

6. Applied egg-rr 98.1%

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

7. Taylor expanded in u0 around 0

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

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

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

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

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

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

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

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

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

   5. *-commutative N/A

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

   6. *-lowering-*.f32 89.4%

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

9. Simplified 89.4%

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

10. Final simplification 89.4%

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

Alternative 10: 90.8% accurate, 5.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 60.2%

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

3. Taylor expanded in u0 around 0

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

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

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

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

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

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

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

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

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

   5. *-commutative N/A

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

   6. *-lowering-*.f32 89.4%

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

5. Simplified 89.4%

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

6. Final simplification 89.4%

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

Alternative 11: 81.2% accurate, 5.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{sin2phi}{alphay \cdot alphay}\\ \mathbf{if}\;t\_0 \leq 500:\\ \;\;\;\;\frac{u0}{t\_0 + \frac{\frac{cos2phi}{alphax}}{alphax}}\\ \mathbf{else}:\\ \;\;\;\;\frac{alphay \cdot \left(\left(u0 \cdot alphay\right) \cdot \left(1 - u0 \cdot -0.5\right)\right)}{sin2phi}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (let* ((t_0 (/ sin2phi (* alphay alphay))))
   (if (<= t_0 500.0)
     (/ u0 (+ t_0 (/ (/ cos2phi alphax) alphax)))
     (/ (* alphay (* (* u0 alphay) (- 1.0 (* u0 -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 <= 500.0f) {
		tmp = u0 / (t_0 + ((cos2phi / alphax) / alphax));
	} else {
		tmp = (alphay * ((u0 * alphay) * (1.0f - (u0 * -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 <= 500.0e0) then
        tmp = u0 / (t_0 + ((cos2phi / alphax) / alphax))
    else
        tmp = (alphay * ((u0 * alphay) * (1.0e0 - (u0 * (-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(500.0))
		tmp = Float32(u0 / Float32(t_0 + Float32(Float32(cos2phi / alphax) / alphax)));
	else
		tmp = Float32(Float32(alphay * Float32(Float32(u0 * alphay) * Float32(Float32(1.0) - Float32(u0 * 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(500.0))
		tmp = u0 / (t_0 + ((cos2phi / alphax) / alphax));
	else
		tmp = (alphay * ((u0 * alphay) * (single(1.0) - (u0 * single(-0.5))))) / sin2phi;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 sin2phi (*.f32 alphay alphay)) < 500

   1. Initial program 53.7%

      \[\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
   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 75.1%

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

   7. Simplified 75.1%

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

      1. associate-/r* N/A

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

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

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

      3. /-lowering-/.f32 75.2%

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

   9. Applied egg-rr 75.2%

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

   if 500 < (/.f32 sin2phi (*.f32 alphay alphay))

   1. Initial program 67.1%

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

      1. distribute-frac-neg N/A

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

      2. distribute-neg-frac2 N/A

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

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

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

      4. sub-neg N/A

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

      5. log1p-define N/A

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

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

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

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

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

      8. distribute-neg-in N/A

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

      9. unsub-neg N/A

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

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

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

      11. neg-sub0 N/A

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

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

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

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

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

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

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

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

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

      16. *-lowering-*.f32 97.3%

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

   3. Simplified 97.3%

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

      1. associate-/r* N/A

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

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

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

      3. /-lowering-/.f32 97.4%

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

   6. Applied egg-rr 97.4%

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

   7. Taylor expanded in u0 around 0

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

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

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

      2. sub-neg N/A

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

      3. metadata-eval N/A

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

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

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

      5. *-commutative N/A

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

      6. *-lowering-*.f32 83.3%

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

   9. Simplified 83.3%

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

   10. Taylor expanded in cos2phi around 0

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

       1. mul-1-neg N/A

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

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

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

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

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

       4. associate-*r* N/A

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

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

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

       6. *-commutative N/A

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

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

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

       8. unpow2 N/A

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

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

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

       10. sub-neg N/A

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

       11. metadata-eval N/A

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

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

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

       13. *-commutative N/A

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

       14. *-lowering-*.f32 84.3%

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

   12. Simplified 84.3%

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

       1. *-commutative N/A

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

       2. associate-*r* N/A

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

       3. associate-*r* N/A

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

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

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

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

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

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

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

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

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

       8. *-lowering-*.f32 84.4%

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

   14. Applied egg-rr 84.4%

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

4. Final simplification 79.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 500:\\ \;\;\;\;\frac{u0}{\frac{sin2phi}{alphay \cdot alphay} + \frac{\frac{cos2phi}{alphax}}{alphax}}\\ \mathbf{else}:\\ \;\;\;\;\frac{alphay \cdot \left(\left(u0 \cdot alphay\right) \cdot \left(1 - u0 \cdot -0.5\right)\right)}{sin2phi}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 81.2% accurate, 5.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{sin2phi}{alphay \cdot alphay}\\ \mathbf{if}\;t\_0 \leq 500:\\ \;\;\;\;\frac{u0}{t\_0 + \frac{\frac{cos2phi}{alphax}}{alphax}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(u0 \cdot alphay\right) \cdot \left(alphay \cdot \left(1 - u0 \cdot -0.5\right)\right)}{sin2phi}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (let* ((t_0 (/ sin2phi (* alphay alphay))))
   (if (<= t_0 500.0)
     (/ u0 (+ t_0 (/ (/ cos2phi alphax) alphax)))
     (/ (* (* u0 alphay) (* alphay (- 1.0 (* u0 -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 <= 500.0f) {
		tmp = u0 / (t_0 + ((cos2phi / alphax) / alphax));
	} else {
		tmp = ((u0 * alphay) * (alphay * (1.0f - (u0 * -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 <= 500.0e0) then
        tmp = u0 / (t_0 + ((cos2phi / alphax) / alphax))
    else
        tmp = ((u0 * alphay) * (alphay * (1.0e0 - (u0 * (-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(500.0))
		tmp = Float32(u0 / Float32(t_0 + Float32(Float32(cos2phi / alphax) / alphax)));
	else
		tmp = Float32(Float32(Float32(u0 * alphay) * Float32(alphay * Float32(Float32(1.0) - Float32(u0 * 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(500.0))
		tmp = u0 / (t_0 + ((cos2phi / alphax) / alphax));
	else
		tmp = ((u0 * alphay) * (alphay * (single(1.0) - (u0 * single(-0.5))))) / sin2phi;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 sin2phi (*.f32 alphay alphay)) < 500

   1. Initial program 53.7%

      \[\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
   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 75.1%

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

   7. Simplified 75.1%

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

      1. associate-/r* N/A

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

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

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

      3. /-lowering-/.f32 75.2%

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

   9. Applied egg-rr 75.2%

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

   if 500 < (/.f32 sin2phi (*.f32 alphay alphay))

   1. Initial program 67.1%

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

      1. distribute-frac-neg N/A

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

      2. distribute-neg-frac2 N/A

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

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

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

      4. sub-neg N/A

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

      5. log1p-define N/A

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

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

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

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

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

      8. distribute-neg-in N/A

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

      9. unsub-neg N/A

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

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

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

      11. neg-sub0 N/A

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

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

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

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

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

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

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

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

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

      16. *-lowering-*.f32 97.3%

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

   3. Simplified 97.3%

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

      1. associate-/r* N/A

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

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

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

      3. /-lowering-/.f32 97.4%

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

   6. Applied egg-rr 97.4%

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

   7. Taylor expanded in u0 around 0

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

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

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

      2. sub-neg N/A

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

      3. metadata-eval N/A

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

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

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

      5. *-commutative N/A

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

      6. *-lowering-*.f32 83.3%

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

   9. Simplified 83.3%

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

   10. Taylor expanded in cos2phi around 0

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

       1. mul-1-neg N/A

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

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

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

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

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

       4. associate-*r* N/A

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

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

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

       6. *-commutative N/A

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

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

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

       8. unpow2 N/A

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

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

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

       10. sub-neg N/A

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

       11. metadata-eval N/A

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

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

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

       13. *-commutative N/A

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

       14. *-lowering-*.f32 84.3%

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

   12. Simplified 84.3%

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

       1. associate-*r* N/A

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

       2. associate-*l* N/A

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

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

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

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

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

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

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

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

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

       7. *-lowering-*.f32 84.3%

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

   14. Applied egg-rr 84.3%

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

4. Final simplification 79.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 500:\\ \;\;\;\;\frac{u0}{\frac{sin2phi}{alphay \cdot alphay} + \frac{\frac{cos2phi}{alphax}}{alphax}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(u0 \cdot alphay\right) \cdot \left(alphay \cdot \left(1 - u0 \cdot -0.5\right)\right)}{sin2phi}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 81.2% accurate, 5.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{sin2phi}{alphay \cdot alphay}\\ \mathbf{if}\;t\_0 \leq 500:\\ \;\;\;\;\frac{u0}{t\_0 + \frac{\frac{cos2phi}{alphax}}{alphax}}\\ \mathbf{else}:\\ \;\;\;\;\left(u0 \cdot alphay\right) \cdot \left(alphay \cdot \frac{1 - u0 \cdot -0.5}{sin2phi}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (let* ((t_0 (/ sin2phi (* alphay alphay))))
   (if (<= t_0 500.0)
     (/ u0 (+ t_0 (/ (/ cos2phi alphax) alphax)))
     (* (* u0 alphay) (* alphay (/ (- 1.0 (* u0 -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 <= 500.0f) {
		tmp = u0 / (t_0 + ((cos2phi / alphax) / alphax));
	} else {
		tmp = (u0 * alphay) * (alphay * ((1.0f - (u0 * -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 <= 500.0e0) then
        tmp = u0 / (t_0 + ((cos2phi / alphax) / alphax))
    else
        tmp = (u0 * alphay) * (alphay * ((1.0e0 - (u0 * (-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(500.0))
		tmp = Float32(u0 / Float32(t_0 + Float32(Float32(cos2phi / alphax) / alphax)));
	else
		tmp = Float32(Float32(u0 * alphay) * Float32(alphay * Float32(Float32(Float32(1.0) - Float32(u0 * 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(500.0))
		tmp = u0 / (t_0 + ((cos2phi / alphax) / alphax));
	else
		tmp = (u0 * alphay) * (alphay * ((single(1.0) - (u0 * single(-0.5))) / sin2phi));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 sin2phi (*.f32 alphay alphay)) < 500

   1. Initial program 53.7%

      \[\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
   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 75.1%

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

   7. Simplified 75.1%

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

      1. associate-/r* N/A

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

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

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

      3. /-lowering-/.f32 75.2%

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

   9. Applied egg-rr 75.2%

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

   if 500 < (/.f32 sin2phi (*.f32 alphay alphay))

   1. Initial program 67.1%

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

      1. distribute-frac-neg N/A

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

      2. distribute-neg-frac2 N/A

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

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

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

      4. sub-neg N/A

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

      5. log1p-define N/A

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

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

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

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

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

      8. distribute-neg-in N/A

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

      9. unsub-neg N/A

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

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

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

      11. neg-sub0 N/A

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

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

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

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

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

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

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

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

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

      16. *-lowering-*.f32 97.3%

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

   3. Simplified 97.3%

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

      1. associate-/r* N/A

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

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

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

      3. /-lowering-/.f32 97.4%

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

   6. Applied egg-rr 97.4%

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

   7. Taylor expanded in u0 around 0

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

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

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

      2. sub-neg N/A

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

      3. metadata-eval N/A

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

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

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

      5. *-commutative N/A

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

      6. *-lowering-*.f32 83.3%

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

   9. Simplified 83.3%

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

   10. Taylor expanded in cos2phi around 0

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

       1. mul-1-neg N/A

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

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

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

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

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

       4. associate-*r* N/A

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

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

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

       6. *-commutative N/A

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

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

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

       8. unpow2 N/A

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

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

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

       10. sub-neg N/A

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

       11. metadata-eval N/A

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

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

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

       13. *-commutative N/A

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

       14. *-lowering-*.f32 84.3%

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

   12. Simplified 84.3%

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

       1. associate-/l* N/A

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

       2. associate-*r* N/A

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

       3. associate-*l* N/A

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

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

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

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

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

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

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

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

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

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

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

       9. *-lowering-*.f32 84.2%

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

   14. Applied egg-rr 84.2%

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

4. Final simplification 79.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 500:\\ \;\;\;\;\frac{u0}{\frac{sin2phi}{alphay \cdot alphay} + \frac{\frac{cos2phi}{alphax}}{alphax}}\\ \mathbf{else}:\\ \;\;\;\;\left(u0 \cdot alphay\right) \cdot \left(alphay \cdot \frac{1 - u0 \cdot -0.5}{sin2phi}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 81.2% accurate, 5.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{sin2phi}{alphay \cdot alphay}\\ \mathbf{if}\;t\_0 \leq 500:\\ \;\;\;\;\frac{u0}{t\_0 + \frac{\frac{cos2phi}{alphax}}{alphax}}\\ \mathbf{else}:\\ \;\;\;\;u0 \cdot \left(\left(alphay \cdot alphay\right) \cdot \frac{1 - u0 \cdot -0.5}{sin2phi}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (let* ((t_0 (/ sin2phi (* alphay alphay))))
   (if (<= t_0 500.0)
     (/ u0 (+ t_0 (/ (/ cos2phi alphax) alphax)))
     (* u0 (* (* alphay alphay) (/ (- 1.0 (* u0 -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 <= 500.0f) {
		tmp = u0 / (t_0 + ((cos2phi / alphax) / alphax));
	} else {
		tmp = u0 * ((alphay * alphay) * ((1.0f - (u0 * -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 <= 500.0e0) then
        tmp = u0 / (t_0 + ((cos2phi / alphax) / alphax))
    else
        tmp = u0 * ((alphay * alphay) * ((1.0e0 - (u0 * (-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(500.0))
		tmp = Float32(u0 / Float32(t_0 + Float32(Float32(cos2phi / alphax) / alphax)));
	else
		tmp = Float32(u0 * Float32(Float32(alphay * alphay) * Float32(Float32(Float32(1.0) - Float32(u0 * 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(500.0))
		tmp = u0 / (t_0 + ((cos2phi / alphax) / alphax));
	else
		tmp = u0 * ((alphay * alphay) * ((single(1.0) - (u0 * single(-0.5))) / sin2phi));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 sin2phi (*.f32 alphay alphay)) < 500

   1. Initial program 53.7%

      \[\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
   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 75.1%

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

   7. Simplified 75.1%

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

      1. associate-/r* N/A

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

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

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

      3. /-lowering-/.f32 75.2%

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

   9. Applied egg-rr 75.2%

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

   if 500 < (/.f32 sin2phi (*.f32 alphay alphay))

   1. Initial program 67.1%

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

      1. distribute-frac-neg N/A

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

      2. distribute-neg-frac2 N/A

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

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

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

      4. sub-neg N/A

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

      5. log1p-define N/A

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

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

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

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

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

      8. distribute-neg-in N/A

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

      9. unsub-neg N/A

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

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

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

      11. neg-sub0 N/A

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

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

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

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

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

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

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

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

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

      16. *-lowering-*.f32 97.3%

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

   3. Simplified 97.3%

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

      1. associate-/r* N/A

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

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

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

      3. /-lowering-/.f32 97.4%

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

   6. Applied egg-rr 97.4%

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

   7. Taylor expanded in u0 around 0

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

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

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

      2. sub-neg N/A

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

      3. metadata-eval N/A

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

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

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

      5. *-commutative N/A

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

      6. *-lowering-*.f32 83.3%

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

   9. Simplified 83.3%

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

   10. Taylor expanded in cos2phi around 0

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

       1. mul-1-neg N/A

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

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

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

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

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

       4. associate-*r* N/A

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

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

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

       6. *-commutative N/A

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

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

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

       8. unpow2 N/A

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

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

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

       10. sub-neg N/A

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

       11. metadata-eval N/A

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

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

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

       13. *-commutative N/A

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

       14. *-lowering-*.f32 84.3%

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

   12. Simplified 84.3%

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

       1. associate-/l* N/A

          \[\leadsto \mathsf{neg.f32}\left(\left(\left(u0 \cdot \left(alphay \cdot alphay\right)\right) \cdot \frac{u0 \cdot \frac{-1}{2} + -1}{sin2phi}\right)\right) \]

       2. associate-*l* N/A

          \[\leadsto \mathsf{neg.f32}\left(\left(u0 \cdot \left(\left(alphay \cdot alphay\right) \cdot \frac{u0 \cdot \frac{-1}{2} + -1}{sin2phi}\right)\right)\right) \]

       3. *-lowering-*.f32 N/A

          \[\leadsto \mathsf{neg.f32}\left(\mathsf{*.f32}\left(u0, \left(\left(alphay \cdot alphay\right) \cdot \frac{u0 \cdot \frac{-1}{2} + -1}{sin2phi}\right)\right)\right) \]

       4. *-lowering-*.f32 N/A

          \[\leadsto \mathsf{neg.f32}\left(\mathsf{*.f32}\left(u0, \mathsf{*.f32}\left(\left(alphay \cdot alphay\right), \left(\frac{u0 \cdot \frac{-1}{2} + -1}{sin2phi}\right)\right)\right)\right) \]

       5. *-lowering-*.f32 N/A

          \[\leadsto \mathsf{neg.f32}\left(\mathsf{*.f32}\left(u0, \mathsf{*.f32}\left(\mathsf{*.f32}\left(alphay, alphay\right), \left(\frac{u0 \cdot \frac{-1}{2} + -1}{sin2phi}\right)\right)\right)\right) \]

       6. /-lowering-/.f32 N/A

          \[\leadsto \mathsf{neg.f32}\left(\mathsf{*.f32}\left(u0, \mathsf{*.f32}\left(\mathsf{*.f32}\left(alphay, alphay\right), \mathsf{/.f32}\left(\left(u0 \cdot \frac{-1}{2} + -1\right), sin2phi\right)\right)\right)\right) \]

       7. +-lowering-+.f32 N/A

          \[\leadsto \mathsf{neg.f32}\left(\mathsf{*.f32}\left(u0, \mathsf{*.f32}\left(\mathsf{*.f32}\left(alphay, alphay\right), \mathsf{/.f32}\left(\mathsf{+.f32}\left(\left(u0 \cdot \frac{-1}{2}\right), -1\right), sin2phi\right)\right)\right)\right) \]

       8. *-lowering-*.f32 84.2%

          \[\leadsto \mathsf{neg.f32}\left(\mathsf{*.f32}\left(u0, \mathsf{*.f32}\left(\mathsf{*.f32}\left(alphay, alphay\right), \mathsf{/.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(u0, \frac{-1}{2}\right), -1\right), sin2phi\right)\right)\right)\right) \]

   14. Applied egg-rr 84.2%

       \[\leadsto -\color{blue}{u0 \cdot \left(\left(alphay \cdot alphay\right) \cdot \frac{u0 \cdot -0.5 + -1}{sin2phi}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 79.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 500:\\ \;\;\;\;\frac{u0}{\frac{sin2phi}{alphay \cdot alphay} + \frac{\frac{cos2phi}{alphax}}{alphax}}\\ \mathbf{else}:\\ \;\;\;\;u0 \cdot \left(\left(alphay \cdot alphay\right) \cdot \frac{1 - u0 \cdot -0.5}{sin2phi}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 87.1% accurate, 6.1× speedup

Math

\[\begin{array}{l} \\ \frac{u0 - u0 \cdot \left(u0 \cdot -0.5\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{\frac{sin2phi}{alphay}}{alphay}} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (/
  (- u0 (* u0 (* u0 -0.5)))
  (+ (/ cos2phi (* alphax alphax)) (/ (/ sin2phi alphay) alphay))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	return (u0 - (u0 * (u0 * -0.5f))) / ((cos2phi / (alphax * alphax)) + ((sin2phi / alphay) / alphay));
}

Fortran

real(4) function code(alphax, alphay, u0, cos2phi, sin2phi)
    real(4), intent (in) :: alphax
    real(4), intent (in) :: alphay
    real(4), intent (in) :: u0
    real(4), intent (in) :: cos2phi
    real(4), intent (in) :: sin2phi
    code = (u0 - (u0 * (u0 * (-0.5e0)))) / ((cos2phi / (alphax * alphax)) + ((sin2phi / alphay) / alphay))
end function

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	return Float32(Float32(u0 - Float32(u0 * Float32(u0 * Float32(-0.5)))) / Float32(Float32(cos2phi / Float32(alphax * alphax)) + Float32(Float32(sin2phi / alphay) / alphay)))
end

MATLAB

function tmp = code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = (u0 - (u0 * (u0 * single(-0.5)))) / ((cos2phi / (alphax * alphax)) + ((sin2phi / alphay) / alphay));
end

Derivation

1. Initial program 60.2%

   \[\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
2. Step-by-step derivation

   1. distribute-frac-neg N/A

      \[\leadsto \mathsf{neg}\left(\frac{\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}}\right) \]

   2. distribute-neg-frac2 N/A

      \[\leadsto \frac{\log \left(1 - u0\right)}{\color{blue}{\mathsf{neg}\left(\left(\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\right)\right)}} \]

   3. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\log \left(1 - u0\right), \color{blue}{\left(\mathsf{neg}\left(\left(\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\right)\right)\right)}\right) \]

   4. sub-neg N/A

      \[\leadsto \mathsf{/.f32}\left(\log \left(1 + \left(\mathsf{neg}\left(u0\right)\right)\right), \left(\mathsf{neg}\left(\left(\color{blue}{\frac{cos2phi}{alphax \cdot alphax}} + \frac{sin2phi}{alphay \cdot alphay}\right)\right)\right)\right) \]

   5. log1p-define N/A

      \[\leadsto \mathsf{/.f32}\left(\left(\mathsf{log1p}\left(\mathsf{neg}\left(u0\right)\right)\right), \left(\mathsf{neg}\left(\color{blue}{\left(\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\right)}\right)\right)\right) \]

   6. log1p-lowering-log1p.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\left(\mathsf{neg}\left(u0\right)\right)\right), \left(\mathsf{neg}\left(\color{blue}{\left(\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\right)}\right)\right)\right) \]

   7. neg-lowering-neg.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \left(\mathsf{neg}\left(\left(\color{blue}{\frac{cos2phi}{alphax \cdot alphax}} + \frac{sin2phi}{alphay \cdot alphay}\right)\right)\right)\right) \]

   8. distribute-neg-in N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \left(\left(\mathsf{neg}\left(\frac{cos2phi}{alphax \cdot alphax}\right)\right) + \color{blue}{\left(\mathsf{neg}\left(\frac{sin2phi}{alphay \cdot alphay}\right)\right)}\right)\right) \]

   9. unsub-neg N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \left(\left(\mathsf{neg}\left(\frac{cos2phi}{alphax \cdot alphax}\right)\right) - \color{blue}{\frac{sin2phi}{alphay \cdot alphay}}\right)\right) \]

   10. --lowering--.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\left(\mathsf{neg}\left(\frac{cos2phi}{alphax \cdot alphax}\right)\right), \color{blue}{\left(\frac{sin2phi}{alphay \cdot alphay}\right)}\right)\right) \]

   11. neg-sub0 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\left(0 - \frac{cos2phi}{alphax \cdot alphax}\right), \left(\frac{\color{blue}{sin2phi}}{alphay \cdot alphay}\right)\right)\right) \]

   12. --lowering--.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \left(\frac{cos2phi}{alphax \cdot alphax}\right)\right), \left(\frac{\color{blue}{sin2phi}}{alphay \cdot alphay}\right)\right)\right) \]

   13. /-lowering-/.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(cos2phi, \left(alphax \cdot alphax\right)\right)\right), \left(\frac{sin2phi}{alphay \cdot alphay}\right)\right)\right) \]

   14. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right)\right), \left(\frac{sin2phi}{alphay \cdot alphay}\right)\right)\right) \]

   15. /-lowering-/.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right)\right), \mathsf{/.f32}\left(sin2phi, \color{blue}{\left(alphay \cdot alphay\right)}\right)\right)\right) \]

   16. *-lowering-*.f32 98.0%

       \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right)\right), \mathsf{/.f32}\left(sin2phi, \mathsf{*.f32}\left(alphay, \color{blue}{alphay}\right)\right)\right)\right) \]

3. Simplified 98.0%

   \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(-u0\right)}{\left(0 - \frac{cos2phi}{alphax \cdot alphax}\right) - \frac{sin2phi}{alphay \cdot alphay}}} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. associate-/r* N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right)\right), \left(\frac{\frac{sin2phi}{alphay}}{\color{blue}{alphay}}\right)\right)\right) \]

   2. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right)\right), \mathsf{/.f32}\left(\left(\frac{sin2phi}{alphay}\right), \color{blue}{alphay}\right)\right)\right) \]

   3. /-lowering-/.f32 98.1%

      \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right)\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(sin2phi, alphay\right), alphay\right)\right)\right) \]

6. Applied egg-rr 98.1%

   \[\leadsto \frac{\mathsf{log1p}\left(-u0\right)}{\left(0 - \frac{cos2phi}{alphax \cdot alphax}\right) - \color{blue}{\frac{\frac{sin2phi}{alphay}}{alphay}}} \]

7. Taylor expanded in u0 around 0

   \[\leadsto \mathsf{/.f32}\left(\color{blue}{\left(u0 \cdot \left(\frac{-1}{2} \cdot u0 - 1\right)\right)}, \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right)\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(sin2phi, alphay\right), alphay\right)\right)\right) \]
8. Step-by-step derivation

   1. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(u0, \left(\frac{-1}{2} \cdot u0 - 1\right)\right), \mathsf{\_.f32}\left(\color{blue}{\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right)\right)}, \mathsf{/.f32}\left(\mathsf{/.f32}\left(sin2phi, alphay\right), alphay\right)\right)\right) \]

   2. sub-neg N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(u0, \left(\frac{-1}{2} \cdot u0 + \left(\mathsf{neg}\left(1\right)\right)\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \color{blue}{\mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right)}\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(sin2phi, alphay\right), alphay\right)\right)\right) \]

   3. metadata-eval N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(u0, \left(\frac{-1}{2} \cdot u0 + -1\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(cos2phi, \color{blue}{\mathsf{*.f32}\left(alphax, alphax\right)}\right)\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(sin2phi, alphay\right), alphay\right)\right)\right) \]

   4. +-lowering-+.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\left(\frac{-1}{2} \cdot u0\right), -1\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \color{blue}{\mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right)}\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(sin2phi, alphay\right), alphay\right)\right)\right) \]

   5. *-commutative N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\left(u0 \cdot \frac{-1}{2}\right), -1\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(\color{blue}{cos2phi}, \mathsf{*.f32}\left(alphax, alphax\right)\right)\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(sin2phi, alphay\right), alphay\right)\right)\right) \]

   6. *-lowering-*.f32 85.7%

      \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{*.f32}\left(u0, \frac{-1}{2}\right), -1\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(\color{blue}{cos2phi}, \mathsf{*.f32}\left(alphax, alphax\right)\right)\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(sin2phi, alphay\right), alphay\right)\right)\right) \]

9. Simplified 85.7%

   \[\leadsto \frac{\color{blue}{u0 \cdot \left(u0 \cdot -0.5 + -1\right)}}{\left(0 - \frac{cos2phi}{alphax \cdot alphax}\right) - \frac{\frac{sin2phi}{alphay}}{alphay}} \]
10. Step-by-step derivation

    1. distribute-rgt-in N/A

       \[\leadsto \mathsf{/.f32}\left(\left(\left(u0 \cdot \frac{-1}{2}\right) \cdot u0 + -1 \cdot u0\right), \mathsf{\_.f32}\left(\color{blue}{\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right)\right)}, \mathsf{/.f32}\left(\mathsf{/.f32}\left(sin2phi, alphay\right), alphay\right)\right)\right) \]

    2. neg-mul-1 N/A

       \[\leadsto \mathsf{/.f32}\left(\left(\left(u0 \cdot \frac{-1}{2}\right) \cdot u0 + \left(\mathsf{neg}\left(u0\right)\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \color{blue}{\mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right)}\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(sin2phi, alphay\right), alphay\right)\right)\right) \]

    3. unsub-neg N/A

       \[\leadsto \mathsf{/.f32}\left(\left(\left(u0 \cdot \frac{-1}{2}\right) \cdot u0 - u0\right), \mathsf{\_.f32}\left(\color{blue}{\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right)\right)}, \mathsf{/.f32}\left(\mathsf{/.f32}\left(sin2phi, alphay\right), alphay\right)\right)\right) \]

    4. --lowering--.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{\_.f32}\left(\left(\left(u0 \cdot \frac{-1}{2}\right) \cdot u0\right), u0\right), \mathsf{\_.f32}\left(\color{blue}{\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right)\right)}, \mathsf{/.f32}\left(\mathsf{/.f32}\left(sin2phi, alphay\right), alphay\right)\right)\right) \]

    5. *-commutative N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{\_.f32}\left(\left(u0 \cdot \left(u0 \cdot \frac{-1}{2}\right)\right), u0\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(\color{blue}{0}, \mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right)\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(sin2phi, alphay\right), alphay\right)\right)\right) \]

    6. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{\_.f32}\left(\mathsf{*.f32}\left(u0, \left(u0 \cdot \frac{-1}{2}\right)\right), u0\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(\color{blue}{0}, \mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right)\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(sin2phi, alphay\right), alphay\right)\right)\right) \]

    7. *-lowering-*.f32 85.8%

       \[\leadsto \mathsf{/.f32}\left(\mathsf{\_.f32}\left(\mathsf{*.f32}\left(u0, \mathsf{*.f32}\left(u0, \frac{-1}{2}\right)\right), u0\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right)\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(sin2phi, alphay\right), alphay\right)\right)\right) \]

11. Applied egg-rr 85.8%

    \[\leadsto \frac{\color{blue}{u0 \cdot \left(u0 \cdot -0.5\right) - u0}}{\left(0 - \frac{cos2phi}{alphax \cdot alphax}\right) - \frac{\frac{sin2phi}{alphay}}{alphay}} \]

12. Final simplification 85.8%

    \[\leadsto \frac{u0 - u0 \cdot \left(u0 \cdot -0.5\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{\frac{sin2phi}{alphay}}{alphay}} \]
13. Add Preprocessing

Alternative 16: 87.0% accurate, 6.1× speedup

Math

\[\begin{array}{l} \\ \frac{u0 \cdot \left(u0 \cdot 0.5 + 1\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{\frac{sin2phi}{alphay}}{alphay}} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (/
  (* u0 (+ (* u0 0.5) 1.0))
  (+ (/ cos2phi (* alphax alphax)) (/ (/ sin2phi alphay) alphay))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	return (u0 * ((u0 * 0.5f) + 1.0f)) / ((cos2phi / (alphax * alphax)) + ((sin2phi / alphay) / alphay));
}

Fortran

real(4) function code(alphax, alphay, u0, cos2phi, sin2phi)
    real(4), intent (in) :: alphax
    real(4), intent (in) :: alphay
    real(4), intent (in) :: u0
    real(4), intent (in) :: cos2phi
    real(4), intent (in) :: sin2phi
    code = (u0 * ((u0 * 0.5e0) + 1.0e0)) / ((cos2phi / (alphax * alphax)) + ((sin2phi / alphay) / alphay))
end function

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	return Float32(Float32(u0 * Float32(Float32(u0 * Float32(0.5)) + Float32(1.0))) / Float32(Float32(cos2phi / Float32(alphax * alphax)) + Float32(Float32(sin2phi / alphay) / alphay)))
end

MATLAB

function tmp = code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = (u0 * ((u0 * single(0.5)) + single(1.0))) / ((cos2phi / (alphax * alphax)) + ((sin2phi / alphay) / alphay));
end

Derivation

1. Initial program 60.2%

   \[\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. sub-neg N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{neg.f32}\left(\log \left(1 + \left(\mathsf{neg}\left(u0\right)\right)\right)\right), \mathsf{+.f32}\left(\mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right), \mathsf{/.f32}\left(sin2phi, \mathsf{*.f32}\left(alphay, alphay\right)\right)\right)\right) \]

   2. log1p-define N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{neg.f32}\left(\left(\mathsf{log1p}\left(\mathsf{neg}\left(u0\right)\right)\right)\right), \mathsf{+.f32}\left(\mathsf{/.f32}\left(\color{blue}{cos2phi}, \mathsf{*.f32}\left(alphax, alphax\right)\right), \mathsf{/.f32}\left(sin2phi, \mathsf{*.f32}\left(alphay, alphay\right)\right)\right)\right) \]

   3. log1p-lowering-log1p.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{neg.f32}\left(\mathsf{log1p.f32}\left(\left(\mathsf{neg}\left(u0\right)\right)\right)\right), \mathsf{+.f32}\left(\mathsf{/.f32}\left(\color{blue}{cos2phi}, \mathsf{*.f32}\left(alphax, alphax\right)\right), \mathsf{/.f32}\left(sin2phi, \mathsf{*.f32}\left(alphay, alphay\right)\right)\right)\right) \]

   4. neg-lowering-neg.f32 98.0%

      \[\leadsto \mathsf{/.f32}\left(\mathsf{neg.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right)\right), \mathsf{+.f32}\left(\mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right), \mathsf{/.f32}\left(sin2phi, \mathsf{*.f32}\left(alphay, alphay\right)\right)\right)\right) \]

4. Applied egg-rr 98.0%

   \[\leadsto \frac{-\color{blue}{\mathsf{log1p}\left(-u0\right)}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
5. Step-by-step derivation

   1. associate-/r* N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{neg.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right)\right), \mathsf{+.f32}\left(\mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right), \left(\frac{\frac{sin2phi}{alphay}}{\color{blue}{alphay}}\right)\right)\right) \]

   2. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{neg.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right)\right), \mathsf{+.f32}\left(\mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right), \mathsf{/.f32}\left(\left(\frac{sin2phi}{alphay}\right), \color{blue}{alphay}\right)\right)\right) \]

   3. /-lowering-/.f32 98.1%

      \[\leadsto \mathsf{/.f32}\left(\mathsf{neg.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right)\right), \mathsf{+.f32}\left(\mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(sin2phi, alphay\right), alphay\right)\right)\right) \]

6. Applied egg-rr 98.1%

   \[\leadsto \frac{-\mathsf{log1p}\left(-u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \color{blue}{\frac{\frac{sin2phi}{alphay}}{alphay}}} \]

7. Taylor expanded in u0 around 0

   \[\leadsto \mathsf{/.f32}\left(\color{blue}{\left(u0 \cdot \left(1 + \frac{1}{2} \cdot u0\right)\right)}, \mathsf{+.f32}\left(\mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(sin2phi, alphay\right), alphay\right)\right)\right) \]
8. Step-by-step derivation

   1. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(u0, \left(1 + \frac{1}{2} \cdot u0\right)\right), \mathsf{+.f32}\left(\color{blue}{\mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right)}, \mathsf{/.f32}\left(\mathsf{/.f32}\left(sin2phi, alphay\right), alphay\right)\right)\right) \]

   2. +-lowering-+.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(1, \left(\frac{1}{2} \cdot u0\right)\right)\right), \mathsf{+.f32}\left(\mathsf{/.f32}\left(cos2phi, \color{blue}{\mathsf{*.f32}\left(alphax, alphax\right)}\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(sin2phi, alphay\right), alphay\right)\right)\right) \]

   3. *-commutative N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(1, \left(u0 \cdot \frac{1}{2}\right)\right)\right), \mathsf{+.f32}\left(\mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, \color{blue}{alphax}\right)\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(sin2phi, alphay\right), alphay\right)\right)\right) \]

   4. *-lowering-*.f32 85.7%

      \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(u0, \frac{1}{2}\right)\right)\right), \mathsf{+.f32}\left(\mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, \color{blue}{alphax}\right)\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(sin2phi, alphay\right), alphay\right)\right)\right) \]

9. Simplified 85.7%

   \[\leadsto \frac{\color{blue}{u0 \cdot \left(1 + u0 \cdot 0.5\right)}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{\frac{sin2phi}{alphay}}{alphay}} \]

10. Final simplification 85.7%

    \[\leadsto \frac{u0 \cdot \left(u0 \cdot 0.5 + 1\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{\frac{sin2phi}{alphay}}{alphay}} \]
11. Add Preprocessing

Alternative 17: 87.0% accurate, 6.1× speedup

Math

\[\begin{array}{l} \\ \frac{u0 \cdot \left(u0 \cdot 0.5 + 1\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (/
  (* u0 (+ (* u0 0.5) 1.0))
  (+ (/ cos2phi (* alphax alphax)) (/ sin2phi (* alphay alphay)))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	return (u0 * ((u0 * 0.5f) + 1.0f)) / ((cos2phi / (alphax * alphax)) + (sin2phi / (alphay * alphay)));
}

Fortran

real(4) function code(alphax, alphay, u0, cos2phi, sin2phi)
    real(4), intent (in) :: alphax
    real(4), intent (in) :: alphay
    real(4), intent (in) :: u0
    real(4), intent (in) :: cos2phi
    real(4), intent (in) :: sin2phi
    code = (u0 * ((u0 * 0.5e0) + 1.0e0)) / ((cos2phi / (alphax * alphax)) + (sin2phi / (alphay * alphay)))
end function

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	return Float32(Float32(u0 * Float32(Float32(u0 * Float32(0.5)) + Float32(1.0))) / Float32(Float32(cos2phi / Float32(alphax * alphax)) + Float32(sin2phi / Float32(alphay * alphay))))
end

MATLAB

function tmp = code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = (u0 * ((u0 * single(0.5)) + single(1.0))) / ((cos2phi / (alphax * alphax)) + (sin2phi / (alphay * alphay)));
end

Derivation

1. Initial program 60.2%

   \[\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
2. Add Preprocessing

3. Taylor expanded in u0 around 0

   \[\leadsto \mathsf{/.f32}\left(\color{blue}{\left(u0 \cdot \left(1 + \frac{1}{2} \cdot u0\right)\right)}, \mathsf{+.f32}\left(\mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right), \mathsf{/.f32}\left(sin2phi, \mathsf{*.f32}\left(alphay, alphay\right)\right)\right)\right) \]
4. Step-by-step derivation

   1. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(u0, \left(1 + \frac{1}{2} \cdot u0\right)\right), \mathsf{+.f32}\left(\color{blue}{\mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right)}, \mathsf{/.f32}\left(sin2phi, \mathsf{*.f32}\left(alphay, alphay\right)\right)\right)\right) \]

   2. +-lowering-+.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(1, \left(\frac{1}{2} \cdot u0\right)\right)\right), \mathsf{+.f32}\left(\mathsf{/.f32}\left(cos2phi, \color{blue}{\mathsf{*.f32}\left(alphax, alphax\right)}\right), \mathsf{/.f32}\left(sin2phi, \mathsf{*.f32}\left(alphay, alphay\right)\right)\right)\right) \]

   3. *-commutative N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(1, \left(u0 \cdot \frac{1}{2}\right)\right)\right), \mathsf{+.f32}\left(\mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, \color{blue}{alphax}\right)\right), \mathsf{/.f32}\left(sin2phi, \mathsf{*.f32}\left(alphay, alphay\right)\right)\right)\right) \]

   4. *-lowering-*.f32 85.6%

      \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(u0, \frac{1}{2}\right)\right)\right), \mathsf{+.f32}\left(\mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, \color{blue}{alphax}\right)\right), \mathsf{/.f32}\left(sin2phi, \mathsf{*.f32}\left(alphay, alphay\right)\right)\right)\right) \]

5. Simplified 85.6%

   \[\leadsto \frac{\color{blue}{u0 \cdot \left(1 + u0 \cdot 0.5\right)}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

6. Final simplification 85.6%

   \[\leadsto \frac{u0 \cdot \left(u0 \cdot 0.5 + 1\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
7. Add Preprocessing

Alternative 18: 66.2% accurate, 7.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 8.000000134899068 \cdot 10^{-16}:\\ \;\;\;\;alphax \cdot \left(u0 \cdot \frac{alphax}{cos2phi}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{u0 \cdot \left(alphay \cdot alphay\right)}{sin2phi}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (if (<= (/ sin2phi (* alphay alphay)) 8.000000134899068e-16)
   (* alphax (* u0 (/ alphax cos2phi)))
   (/ (* u0 (* alphay alphay)) sin2phi)))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float tmp;
	if ((sin2phi / (alphay * alphay)) <= 8.000000134899068e-16f) {
		tmp = alphax * (u0 * (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 / (alphay * alphay)) <= 8.000000134899068e-16) then
        tmp = alphax * (u0 * (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 (Float32(sin2phi / Float32(alphay * alphay)) <= Float32(8.000000134899068e-16))
		tmp = Float32(alphax * Float32(u0 * Float32(alphax / cos2phi)));
	else
		tmp = Float32(Float32(u0 * Float32(alphay * alphay)) / sin2phi);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = single(0.0);
	if ((sin2phi / (alphay * alphay)) <= single(8.000000134899068e-16))
		tmp = alphax * (u0 * (alphax / cos2phi));
	else
		tmp = (u0 * (alphay * alphay)) / sin2phi;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 sin2phi (*.f32 alphay alphay)) < 8.00000013e-16

   1. Initial program 59.3%

      \[\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
   2. Step-by-step derivation

      1. distribute-frac-neg N/A

         \[\leadsto \mathsf{neg}\left(\frac{\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}}\right) \]

      2. distribute-neg-frac2 N/A

         \[\leadsto \frac{\log \left(1 - u0\right)}{\color{blue}{\mathsf{neg}\left(\left(\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\right)\right)}} \]

      3. /-lowering-/.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(\log \left(1 - u0\right), \color{blue}{\left(\mathsf{neg}\left(\left(\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\right)\right)\right)}\right) \]

      4. sub-neg N/A

         \[\leadsto \mathsf{/.f32}\left(\log \left(1 + \left(\mathsf{neg}\left(u0\right)\right)\right), \left(\mathsf{neg}\left(\left(\color{blue}{\frac{cos2phi}{alphax \cdot alphax}} + \frac{sin2phi}{alphay \cdot alphay}\right)\right)\right)\right) \]

      5. log1p-define N/A

         \[\leadsto \mathsf{/.f32}\left(\left(\mathsf{log1p}\left(\mathsf{neg}\left(u0\right)\right)\right), \left(\mathsf{neg}\left(\color{blue}{\left(\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\right)}\right)\right)\right) \]

      6. log1p-lowering-log1p.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\left(\mathsf{neg}\left(u0\right)\right)\right), \left(\mathsf{neg}\left(\color{blue}{\left(\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\right)}\right)\right)\right) \]

      7. neg-lowering-neg.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \left(\mathsf{neg}\left(\left(\color{blue}{\frac{cos2phi}{alphax \cdot alphax}} + \frac{sin2phi}{alphay \cdot alphay}\right)\right)\right)\right) \]

      8. distribute-neg-in N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \left(\left(\mathsf{neg}\left(\frac{cos2phi}{alphax \cdot alphax}\right)\right) + \color{blue}{\left(\mathsf{neg}\left(\frac{sin2phi}{alphay \cdot alphay}\right)\right)}\right)\right) \]

      9. unsub-neg N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \left(\left(\mathsf{neg}\left(\frac{cos2phi}{alphax \cdot alphax}\right)\right) - \color{blue}{\frac{sin2phi}{alphay \cdot alphay}}\right)\right) \]

      10. --lowering--.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\left(\mathsf{neg}\left(\frac{cos2phi}{alphax \cdot alphax}\right)\right), \color{blue}{\left(\frac{sin2phi}{alphay \cdot alphay}\right)}\right)\right) \]

      11. neg-sub0 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\left(0 - \frac{cos2phi}{alphax \cdot alphax}\right), \left(\frac{\color{blue}{sin2phi}}{alphay \cdot alphay}\right)\right)\right) \]

      12. --lowering--.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \left(\frac{cos2phi}{alphax \cdot alphax}\right)\right), \left(\frac{\color{blue}{sin2phi}}{alphay \cdot alphay}\right)\right)\right) \]

      13. /-lowering-/.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(cos2phi, \left(alphax \cdot alphax\right)\right)\right), \left(\frac{sin2phi}{alphay \cdot alphay}\right)\right)\right) \]

      14. *-lowering-*.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right)\right), \left(\frac{sin2phi}{alphay \cdot alphay}\right)\right)\right) \]

      15. /-lowering-/.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right)\right), \mathsf{/.f32}\left(sin2phi, \color{blue}{\left(alphay \cdot alphay\right)}\right)\right)\right) \]

      16. *-lowering-*.f32 98.9%

          \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right)\right), \mathsf{/.f32}\left(sin2phi, \mathsf{*.f32}\left(alphay, \color{blue}{alphay}\right)\right)\right)\right) \]

   3. Simplified 98.9%

      \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(-u0\right)}{\left(0 - \frac{cos2phi}{alphax \cdot alphax}\right) - \frac{sin2phi}{alphay \cdot alphay}}} \]
   4. Add Preprocessing

   5. Taylor expanded in 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 70.8%

         \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right), \mathsf{/.f32}\left(sin2phi, \mathsf{*.f32}\left(alphay, \color{blue}{alphay}\right)\right)\right)\right) \]

   7. Simplified 70.8%

      \[\leadsto \color{blue}{\frac{u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}}} \]

   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 56.5%

         \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, \color{blue}{alphax}\right)\right)\right) \]

   10. Simplified 56.5%

       \[\leadsto \frac{u0}{\color{blue}{\frac{cos2phi}{alphax \cdot alphax}}} \]
   11. Step-by-step derivation

       1. associate-/r/ N/A

          \[\leadsto \frac{u0}{cos2phi} \cdot \color{blue}{\left(alphax \cdot alphax\right)} \]

       2. associate-*r* N/A

          \[\leadsto \left(\frac{u0}{cos2phi} \cdot alphax\right) \cdot \color{blue}{alphax} \]

       3. *-lowering-*.f32 N/A

          \[\leadsto \mathsf{*.f32}\left(\left(\frac{u0}{cos2phi} \cdot alphax\right), \color{blue}{alphax}\right) \]

       4. *-lowering-*.f32 N/A

          \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(\left(\frac{u0}{cos2phi}\right), alphax\right), alphax\right) \]

       5. /-lowering-/.f32 56.5%

          \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{/.f32}\left(u0, cos2phi\right), alphax\right), alphax\right) \]

   12. Applied egg-rr 56.5%

       \[\leadsto \color{blue}{\left(\frac{u0}{cos2phi} \cdot alphax\right) \cdot alphax} \]
   13. Step-by-step derivation

       1. associate-*l/ N/A

          \[\leadsto \mathsf{*.f32}\left(\left(\frac{u0 \cdot alphax}{cos2phi}\right), alphax\right) \]

       2. associate-/l* N/A

          \[\leadsto \mathsf{*.f32}\left(\left(u0 \cdot \frac{alphax}{cos2phi}\right), alphax\right) \]

       3. *-lowering-*.f32 N/A

          \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(u0, \left(\frac{alphax}{cos2phi}\right)\right), alphax\right) \]

       4. /-lowering-/.f32 56.6%

          \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(u0, \mathsf{/.f32}\left(alphax, cos2phi\right)\right), alphax\right) \]

   14. Applied egg-rr 56.6%

       \[\leadsto \color{blue}{\left(u0 \cdot \frac{alphax}{cos2phi}\right)} \cdot alphax \]

   if 8.00000013e-16 < (/.f32 sin2phi (*.f32 alphay alphay))

   1. Initial program 60.5%

      \[\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
   2. Step-by-step derivation

      1. distribute-frac-neg N/A

         \[\leadsto \mathsf{neg}\left(\frac{\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}}\right) \]

      2. distribute-neg-frac2 N/A

         \[\leadsto \frac{\log \left(1 - u0\right)}{\color{blue}{\mathsf{neg}\left(\left(\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\right)\right)}} \]

      3. /-lowering-/.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(\log \left(1 - u0\right), \color{blue}{\left(\mathsf{neg}\left(\left(\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\right)\right)\right)}\right) \]

      4. sub-neg N/A

         \[\leadsto \mathsf{/.f32}\left(\log \left(1 + \left(\mathsf{neg}\left(u0\right)\right)\right), \left(\mathsf{neg}\left(\left(\color{blue}{\frac{cos2phi}{alphax \cdot alphax}} + \frac{sin2phi}{alphay \cdot alphay}\right)\right)\right)\right) \]

      5. log1p-define N/A

         \[\leadsto \mathsf{/.f32}\left(\left(\mathsf{log1p}\left(\mathsf{neg}\left(u0\right)\right)\right), \left(\mathsf{neg}\left(\color{blue}{\left(\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\right)}\right)\right)\right) \]

      6. log1p-lowering-log1p.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\left(\mathsf{neg}\left(u0\right)\right)\right), \left(\mathsf{neg}\left(\color{blue}{\left(\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\right)}\right)\right)\right) \]

      7. neg-lowering-neg.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \left(\mathsf{neg}\left(\left(\color{blue}{\frac{cos2phi}{alphax \cdot alphax}} + \frac{sin2phi}{alphay \cdot alphay}\right)\right)\right)\right) \]

      8. distribute-neg-in N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \left(\left(\mathsf{neg}\left(\frac{cos2phi}{alphax \cdot alphax}\right)\right) + \color{blue}{\left(\mathsf{neg}\left(\frac{sin2phi}{alphay \cdot alphay}\right)\right)}\right)\right) \]

      9. unsub-neg N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \left(\left(\mathsf{neg}\left(\frac{cos2phi}{alphax \cdot alphax}\right)\right) - \color{blue}{\frac{sin2phi}{alphay \cdot alphay}}\right)\right) \]

      10. --lowering--.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\left(\mathsf{neg}\left(\frac{cos2phi}{alphax \cdot alphax}\right)\right), \color{blue}{\left(\frac{sin2phi}{alphay \cdot alphay}\right)}\right)\right) \]

      11. neg-sub0 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\left(0 - \frac{cos2phi}{alphax \cdot alphax}\right), \left(\frac{\color{blue}{sin2phi}}{alphay \cdot alphay}\right)\right)\right) \]

      12. --lowering--.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \left(\frac{cos2phi}{alphax \cdot alphax}\right)\right), \left(\frac{\color{blue}{sin2phi}}{alphay \cdot alphay}\right)\right)\right) \]

      13. /-lowering-/.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(cos2phi, \left(alphax \cdot alphax\right)\right)\right), \left(\frac{sin2phi}{alphay \cdot alphay}\right)\right)\right) \]

      14. *-lowering-*.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right)\right), \left(\frac{sin2phi}{alphay \cdot alphay}\right)\right)\right) \]

      15. /-lowering-/.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right)\right), \mathsf{/.f32}\left(sin2phi, \color{blue}{\left(alphay \cdot alphay\right)}\right)\right)\right) \]

      16. *-lowering-*.f32 97.7%

          \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right)\right), \mathsf{/.f32}\left(sin2phi, \mathsf{*.f32}\left(alphay, \color{blue}{alphay}\right)\right)\right)\right) \]

   3. Simplified 97.7%

      \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(-u0\right)}{\left(0 - \frac{cos2phi}{alphax \cdot alphax}\right) - \frac{sin2phi}{alphay \cdot alphay}}} \]
   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 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. Taylor expanded in cos2phi around 0

      \[\leadsto \color{blue}{\frac{{alphay}^{2} \cdot u0}{sin2phi}} \]
   9. Step-by-step derivation

      1. /-lowering-/.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(\left({alphay}^{2} \cdot u0\right), \color{blue}{sin2phi}\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{/.f32}\left(\left(u0 \cdot {alphay}^{2}\right), sin2phi\right) \]

      3. *-lowering-*.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(u0, \left({alphay}^{2}\right)\right), sin2phi\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(u0, \left(alphay \cdot alphay\right)\right), sin2phi\right) \]

      5. *-lowering-*.f32 67.9%

         \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(u0, \mathsf{*.f32}\left(alphay, alphay\right)\right), sin2phi\right) \]

   10. Simplified 67.9%

       \[\leadsto \color{blue}{\frac{u0 \cdot \left(alphay \cdot alphay\right)}{sin2phi}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 65.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 8.000000134899068 \cdot 10^{-16}:\\ \;\;\;\;alphax \cdot \left(u0 \cdot \frac{alphax}{cos2phi}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{u0 \cdot \left(alphay \cdot alphay\right)}{sin2phi}\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 66.0% accurate, 7.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{sin2phi}{alphay \cdot alphay}\\ \mathbf{if}\;t\_0 \leq 8.000000134899068 \cdot 10^{-16}:\\ \;\;\;\;alphax \cdot \left(u0 \cdot \frac{alphax}{cos2phi}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{u0}{t\_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (let* ((t_0 (/ sin2phi (* alphay alphay))))
   (if (<= t_0 8.000000134899068e-16)
     (* alphax (* u0 (/ alphax cos2phi)))
     (/ u0 t_0))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float t_0 = sin2phi / (alphay * alphay);
	float tmp;
	if (t_0 <= 8.000000134899068e-16f) {
		tmp = alphax * (u0 * (alphax / cos2phi));
	} else {
		tmp = u0 / t_0;
	}
	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 <= 8.000000134899068e-16) then
        tmp = alphax * (u0 * (alphax / cos2phi))
    else
        tmp = u0 / t_0
    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(8.000000134899068e-16))
		tmp = Float32(alphax * Float32(u0 * Float32(alphax / cos2phi)));
	else
		tmp = Float32(u0 / t_0);
	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(8.000000134899068e-16))
		tmp = alphax * (u0 * (alphax / cos2phi));
	else
		tmp = u0 / t_0;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 sin2phi (*.f32 alphay alphay)) < 8.00000013e-16

   1. Initial program 59.3%

      \[\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
   2. Step-by-step derivation

      1. distribute-frac-neg N/A

         \[\leadsto \mathsf{neg}\left(\frac{\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}}\right) \]

      2. distribute-neg-frac2 N/A

         \[\leadsto \frac{\log \left(1 - u0\right)}{\color{blue}{\mathsf{neg}\left(\left(\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\right)\right)}} \]

      3. /-lowering-/.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(\log \left(1 - u0\right), \color{blue}{\left(\mathsf{neg}\left(\left(\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\right)\right)\right)}\right) \]

      4. sub-neg N/A

         \[\leadsto \mathsf{/.f32}\left(\log \left(1 + \left(\mathsf{neg}\left(u0\right)\right)\right), \left(\mathsf{neg}\left(\left(\color{blue}{\frac{cos2phi}{alphax \cdot alphax}} + \frac{sin2phi}{alphay \cdot alphay}\right)\right)\right)\right) \]

      5. log1p-define N/A

         \[\leadsto \mathsf{/.f32}\left(\left(\mathsf{log1p}\left(\mathsf{neg}\left(u0\right)\right)\right), \left(\mathsf{neg}\left(\color{blue}{\left(\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\right)}\right)\right)\right) \]

      6. log1p-lowering-log1p.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\left(\mathsf{neg}\left(u0\right)\right)\right), \left(\mathsf{neg}\left(\color{blue}{\left(\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\right)}\right)\right)\right) \]

      7. neg-lowering-neg.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \left(\mathsf{neg}\left(\left(\color{blue}{\frac{cos2phi}{alphax \cdot alphax}} + \frac{sin2phi}{alphay \cdot alphay}\right)\right)\right)\right) \]

      8. distribute-neg-in N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \left(\left(\mathsf{neg}\left(\frac{cos2phi}{alphax \cdot alphax}\right)\right) + \color{blue}{\left(\mathsf{neg}\left(\frac{sin2phi}{alphay \cdot alphay}\right)\right)}\right)\right) \]

      9. unsub-neg N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \left(\left(\mathsf{neg}\left(\frac{cos2phi}{alphax \cdot alphax}\right)\right) - \color{blue}{\frac{sin2phi}{alphay \cdot alphay}}\right)\right) \]

      10. --lowering--.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\left(\mathsf{neg}\left(\frac{cos2phi}{alphax \cdot alphax}\right)\right), \color{blue}{\left(\frac{sin2phi}{alphay \cdot alphay}\right)}\right)\right) \]

      11. neg-sub0 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\left(0 - \frac{cos2phi}{alphax \cdot alphax}\right), \left(\frac{\color{blue}{sin2phi}}{alphay \cdot alphay}\right)\right)\right) \]

      12. --lowering--.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \left(\frac{cos2phi}{alphax \cdot alphax}\right)\right), \left(\frac{\color{blue}{sin2phi}}{alphay \cdot alphay}\right)\right)\right) \]

      13. /-lowering-/.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(cos2phi, \left(alphax \cdot alphax\right)\right)\right), \left(\frac{sin2phi}{alphay \cdot alphay}\right)\right)\right) \]

      14. *-lowering-*.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right)\right), \left(\frac{sin2phi}{alphay \cdot alphay}\right)\right)\right) \]

      15. /-lowering-/.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right)\right), \mathsf{/.f32}\left(sin2phi, \color{blue}{\left(alphay \cdot alphay\right)}\right)\right)\right) \]

      16. *-lowering-*.f32 98.9%

          \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right)\right), \mathsf{/.f32}\left(sin2phi, \mathsf{*.f32}\left(alphay, \color{blue}{alphay}\right)\right)\right)\right) \]

   3. Simplified 98.9%

      \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(-u0\right)}{\left(0 - \frac{cos2phi}{alphax \cdot alphax}\right) - \frac{sin2phi}{alphay \cdot alphay}}} \]
   4. Add Preprocessing

   5. Taylor expanded in 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 70.8%

         \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right), \mathsf{/.f32}\left(sin2phi, \mathsf{*.f32}\left(alphay, \color{blue}{alphay}\right)\right)\right)\right) \]

   7. Simplified 70.8%

      \[\leadsto \color{blue}{\frac{u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}}} \]

   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 56.5%

         \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, \color{blue}{alphax}\right)\right)\right) \]

   10. Simplified 56.5%

       \[\leadsto \frac{u0}{\color{blue}{\frac{cos2phi}{alphax \cdot alphax}}} \]
   11. Step-by-step derivation

       1. associate-/r/ N/A

          \[\leadsto \frac{u0}{cos2phi} \cdot \color{blue}{\left(alphax \cdot alphax\right)} \]

       2. associate-*r* N/A

          \[\leadsto \left(\frac{u0}{cos2phi} \cdot alphax\right) \cdot \color{blue}{alphax} \]

       3. *-lowering-*.f32 N/A

          \[\leadsto \mathsf{*.f32}\left(\left(\frac{u0}{cos2phi} \cdot alphax\right), \color{blue}{alphax}\right) \]

       4. *-lowering-*.f32 N/A

          \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(\left(\frac{u0}{cos2phi}\right), alphax\right), alphax\right) \]

       5. /-lowering-/.f32 56.5%

          \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{/.f32}\left(u0, cos2phi\right), alphax\right), alphax\right) \]

   12. Applied egg-rr 56.5%

       \[\leadsto \color{blue}{\left(\frac{u0}{cos2phi} \cdot alphax\right) \cdot alphax} \]
   13. Step-by-step derivation

       1. associate-*l/ N/A

          \[\leadsto \mathsf{*.f32}\left(\left(\frac{u0 \cdot alphax}{cos2phi}\right), alphax\right) \]

       2. associate-/l* N/A

          \[\leadsto \mathsf{*.f32}\left(\left(u0 \cdot \frac{alphax}{cos2phi}\right), alphax\right) \]

       3. *-lowering-*.f32 N/A

          \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(u0, \left(\frac{alphax}{cos2phi}\right)\right), alphax\right) \]

       4. /-lowering-/.f32 56.6%

          \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(u0, \mathsf{/.f32}\left(alphax, cos2phi\right)\right), alphax\right) \]

   14. Applied egg-rr 56.6%

       \[\leadsto \color{blue}{\left(u0 \cdot \frac{alphax}{cos2phi}\right)} \cdot alphax \]

   if 8.00000013e-16 < (/.f32 sin2phi (*.f32 alphay alphay))

   1. Initial program 60.5%

      \[\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
   2. Step-by-step derivation

      1. distribute-frac-neg N/A

         \[\leadsto \mathsf{neg}\left(\frac{\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}}\right) \]

      2. distribute-neg-frac2 N/A

         \[\leadsto \frac{\log \left(1 - u0\right)}{\color{blue}{\mathsf{neg}\left(\left(\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\right)\right)}} \]

      3. /-lowering-/.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(\log \left(1 - u0\right), \color{blue}{\left(\mathsf{neg}\left(\left(\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\right)\right)\right)}\right) \]

      4. sub-neg N/A

         \[\leadsto \mathsf{/.f32}\left(\log \left(1 + \left(\mathsf{neg}\left(u0\right)\right)\right), \left(\mathsf{neg}\left(\left(\color{blue}{\frac{cos2phi}{alphax \cdot alphax}} + \frac{sin2phi}{alphay \cdot alphay}\right)\right)\right)\right) \]

      5. log1p-define N/A

         \[\leadsto \mathsf{/.f32}\left(\left(\mathsf{log1p}\left(\mathsf{neg}\left(u0\right)\right)\right), \left(\mathsf{neg}\left(\color{blue}{\left(\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\right)}\right)\right)\right) \]

      6. log1p-lowering-log1p.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\left(\mathsf{neg}\left(u0\right)\right)\right), \left(\mathsf{neg}\left(\color{blue}{\left(\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\right)}\right)\right)\right) \]

      7. neg-lowering-neg.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \left(\mathsf{neg}\left(\left(\color{blue}{\frac{cos2phi}{alphax \cdot alphax}} + \frac{sin2phi}{alphay \cdot alphay}\right)\right)\right)\right) \]

      8. distribute-neg-in N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \left(\left(\mathsf{neg}\left(\frac{cos2phi}{alphax \cdot alphax}\right)\right) + \color{blue}{\left(\mathsf{neg}\left(\frac{sin2phi}{alphay \cdot alphay}\right)\right)}\right)\right) \]

      9. unsub-neg N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \left(\left(\mathsf{neg}\left(\frac{cos2phi}{alphax \cdot alphax}\right)\right) - \color{blue}{\frac{sin2phi}{alphay \cdot alphay}}\right)\right) \]

      10. --lowering--.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\left(\mathsf{neg}\left(\frac{cos2phi}{alphax \cdot alphax}\right)\right), \color{blue}{\left(\frac{sin2phi}{alphay \cdot alphay}\right)}\right)\right) \]

      11. neg-sub0 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\left(0 - \frac{cos2phi}{alphax \cdot alphax}\right), \left(\frac{\color{blue}{sin2phi}}{alphay \cdot alphay}\right)\right)\right) \]

      12. --lowering--.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \left(\frac{cos2phi}{alphax \cdot alphax}\right)\right), \left(\frac{\color{blue}{sin2phi}}{alphay \cdot alphay}\right)\right)\right) \]

      13. /-lowering-/.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(cos2phi, \left(alphax \cdot alphax\right)\right)\right), \left(\frac{sin2phi}{alphay \cdot alphay}\right)\right)\right) \]

      14. *-lowering-*.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right)\right), \left(\frac{sin2phi}{alphay \cdot alphay}\right)\right)\right) \]

      15. /-lowering-/.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right)\right), \mathsf{/.f32}\left(sin2phi, \color{blue}{\left(alphay \cdot alphay\right)}\right)\right)\right) \]

      16. *-lowering-*.f32 97.7%

          \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right)\right), \mathsf{/.f32}\left(sin2phi, \mathsf{*.f32}\left(alphay, \color{blue}{alphay}\right)\right)\right)\right) \]

   3. Simplified 97.7%

      \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(-u0\right)}{\left(0 - \frac{cos2phi}{alphax \cdot alphax}\right) - \frac{sin2phi}{alphay \cdot alphay}}} \]
   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 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. Taylor expanded in cos2phi around 0

      \[\leadsto \mathsf{/.f32}\left(u0, \color{blue}{\left(\frac{sin2phi}{{alphay}^{2}}\right)}\right) \]
   9. Step-by-step derivation

      1. /-lowering-/.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{/.f32}\left(sin2phi, \color{blue}{\left({alphay}^{2}\right)}\right)\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{/.f32}\left(sin2phi, \left(alphay \cdot \color{blue}{alphay}\right)\right)\right) \]

      3. *-lowering-*.f32 67.3%

         \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{/.f32}\left(sin2phi, \mathsf{*.f32}\left(alphay, \color{blue}{alphay}\right)\right)\right) \]

   10. Simplified 67.3%

       \[\leadsto \frac{u0}{\color{blue}{\frac{sin2phi}{alphay \cdot alphay}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 64.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 8.000000134899068 \cdot 10^{-16}:\\ \;\;\;\;alphax \cdot \left(u0 \cdot \frac{alphax}{cos2phi}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{u0}{\frac{sin2phi}{alphay \cdot alphay}}\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 75.4% accurate, 8.9× speedup

Math

\[\begin{array}{l} \\ \frac{u0}{\frac{sin2phi}{alphay \cdot alphay} + \frac{\frac{cos2phi}{alphax}}{alphax}} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (/ u0 (+ (/ sin2phi (* alphay alphay)) (/ (/ cos2phi alphax) alphax))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	return u0 / ((sin2phi / (alphay * alphay)) + ((cos2phi / alphax) / alphax));
}

Fortran

real(4) function code(alphax, alphay, u0, cos2phi, sin2phi)
    real(4), intent (in) :: alphax
    real(4), intent (in) :: alphay
    real(4), intent (in) :: u0
    real(4), intent (in) :: cos2phi
    real(4), intent (in) :: sin2phi
    code = u0 / ((sin2phi / (alphay * alphay)) + ((cos2phi / alphax) / alphax))
end function

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	return Float32(u0 / Float32(Float32(sin2phi / Float32(alphay * alphay)) + Float32(Float32(cos2phi / alphax) / alphax)))
end

MATLAB

function tmp = code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = u0 / ((sin2phi / (alphay * alphay)) + ((cos2phi / alphax) / alphax));
end

Derivation

1. Initial program 60.2%

   \[\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
2. Step-by-step derivation

   1. distribute-frac-neg N/A

      \[\leadsto \mathsf{neg}\left(\frac{\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}}\right) \]

   2. distribute-neg-frac2 N/A

      \[\leadsto \frac{\log \left(1 - u0\right)}{\color{blue}{\mathsf{neg}\left(\left(\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\right)\right)}} \]

   3. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\log \left(1 - u0\right), \color{blue}{\left(\mathsf{neg}\left(\left(\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\right)\right)\right)}\right) \]

   4. sub-neg N/A

      \[\leadsto \mathsf{/.f32}\left(\log \left(1 + \left(\mathsf{neg}\left(u0\right)\right)\right), \left(\mathsf{neg}\left(\left(\color{blue}{\frac{cos2phi}{alphax \cdot alphax}} + \frac{sin2phi}{alphay \cdot alphay}\right)\right)\right)\right) \]

   5. log1p-define N/A

      \[\leadsto \mathsf{/.f32}\left(\left(\mathsf{log1p}\left(\mathsf{neg}\left(u0\right)\right)\right), \left(\mathsf{neg}\left(\color{blue}{\left(\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\right)}\right)\right)\right) \]

   6. log1p-lowering-log1p.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\left(\mathsf{neg}\left(u0\right)\right)\right), \left(\mathsf{neg}\left(\color{blue}{\left(\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\right)}\right)\right)\right) \]

   7. neg-lowering-neg.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \left(\mathsf{neg}\left(\left(\color{blue}{\frac{cos2phi}{alphax \cdot alphax}} + \frac{sin2phi}{alphay \cdot alphay}\right)\right)\right)\right) \]

   8. distribute-neg-in N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \left(\left(\mathsf{neg}\left(\frac{cos2phi}{alphax \cdot alphax}\right)\right) + \color{blue}{\left(\mathsf{neg}\left(\frac{sin2phi}{alphay \cdot alphay}\right)\right)}\right)\right) \]

   9. unsub-neg N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \left(\left(\mathsf{neg}\left(\frac{cos2phi}{alphax \cdot alphax}\right)\right) - \color{blue}{\frac{sin2phi}{alphay \cdot alphay}}\right)\right) \]

   10. --lowering--.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\left(\mathsf{neg}\left(\frac{cos2phi}{alphax \cdot alphax}\right)\right), \color{blue}{\left(\frac{sin2phi}{alphay \cdot alphay}\right)}\right)\right) \]

   11. neg-sub0 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\left(0 - \frac{cos2phi}{alphax \cdot alphax}\right), \left(\frac{\color{blue}{sin2phi}}{alphay \cdot alphay}\right)\right)\right) \]

   12. --lowering--.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \left(\frac{cos2phi}{alphax \cdot alphax}\right)\right), \left(\frac{\color{blue}{sin2phi}}{alphay \cdot alphay}\right)\right)\right) \]

   13. /-lowering-/.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(cos2phi, \left(alphax \cdot alphax\right)\right)\right), \left(\frac{sin2phi}{alphay \cdot alphay}\right)\right)\right) \]

   14. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right)\right), \left(\frac{sin2phi}{alphay \cdot alphay}\right)\right)\right) \]

   15. /-lowering-/.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right)\right), \mathsf{/.f32}\left(sin2phi, \color{blue}{\left(alphay \cdot alphay\right)}\right)\right)\right) \]

   16. *-lowering-*.f32 98.0%

       \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right)\right), \mathsf{/.f32}\left(sin2phi, \mathsf{*.f32}\left(alphay, \color{blue}{alphay}\right)\right)\right)\right) \]

3. Simplified 98.0%

   \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(-u0\right)}{\left(0 - \frac{cos2phi}{alphax \cdot alphax}\right) - \frac{sin2phi}{alphay \cdot alphay}}} \]
4. Add Preprocessing

5. Taylor expanded in 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. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{/.f32}\left(\left(\frac{cos2phi}{alphax}\right), alphax\right), \mathsf{/.f32}\left(\color{blue}{sin2phi}, \mathsf{*.f32}\left(alphay, alphay\right)\right)\right)\right) \]

   3. /-lowering-/.f32 73.8%

      \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(cos2phi, alphax\right), alphax\right), \mathsf{/.f32}\left(sin2phi, \mathsf{*.f32}\left(alphay, alphay\right)\right)\right)\right) \]

9. Applied egg-rr 73.8%

   \[\leadsto \frac{u0}{\color{blue}{\frac{\frac{cos2phi}{alphax}}{alphax}} + \frac{sin2phi}{alphay \cdot alphay}} \]

10. Final simplification 73.8%

    \[\leadsto \frac{u0}{\frac{sin2phi}{alphay \cdot alphay} + \frac{\frac{cos2phi}{alphax}}{alphax}} \]
11. Add Preprocessing

Alternative 21: 75.4% 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 60.2%

   \[\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
2. Step-by-step derivation

   1. distribute-frac-neg N/A

      \[\leadsto \mathsf{neg}\left(\frac{\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}}\right) \]

   2. distribute-neg-frac2 N/A

      \[\leadsto \frac{\log \left(1 - u0\right)}{\color{blue}{\mathsf{neg}\left(\left(\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\right)\right)}} \]

   3. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\log \left(1 - u0\right), \color{blue}{\left(\mathsf{neg}\left(\left(\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\right)\right)\right)}\right) \]

   4. sub-neg N/A

      \[\leadsto \mathsf{/.f32}\left(\log \left(1 + \left(\mathsf{neg}\left(u0\right)\right)\right), \left(\mathsf{neg}\left(\left(\color{blue}{\frac{cos2phi}{alphax \cdot alphax}} + \frac{sin2phi}{alphay \cdot alphay}\right)\right)\right)\right) \]

   5. log1p-define N/A

      \[\leadsto \mathsf{/.f32}\left(\left(\mathsf{log1p}\left(\mathsf{neg}\left(u0\right)\right)\right), \left(\mathsf{neg}\left(\color{blue}{\left(\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\right)}\right)\right)\right) \]

   6. log1p-lowering-log1p.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\left(\mathsf{neg}\left(u0\right)\right)\right), \left(\mathsf{neg}\left(\color{blue}{\left(\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\right)}\right)\right)\right) \]

   7. neg-lowering-neg.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \left(\mathsf{neg}\left(\left(\color{blue}{\frac{cos2phi}{alphax \cdot alphax}} + \frac{sin2phi}{alphay \cdot alphay}\right)\right)\right)\right) \]

   8. distribute-neg-in N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \left(\left(\mathsf{neg}\left(\frac{cos2phi}{alphax \cdot alphax}\right)\right) + \color{blue}{\left(\mathsf{neg}\left(\frac{sin2phi}{alphay \cdot alphay}\right)\right)}\right)\right) \]

   9. unsub-neg N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \left(\left(\mathsf{neg}\left(\frac{cos2phi}{alphax \cdot alphax}\right)\right) - \color{blue}{\frac{sin2phi}{alphay \cdot alphay}}\right)\right) \]

   10. --lowering--.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\left(\mathsf{neg}\left(\frac{cos2phi}{alphax \cdot alphax}\right)\right), \color{blue}{\left(\frac{sin2phi}{alphay \cdot alphay}\right)}\right)\right) \]

   11. neg-sub0 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\left(0 - \frac{cos2phi}{alphax \cdot alphax}\right), \left(\frac{\color{blue}{sin2phi}}{alphay \cdot alphay}\right)\right)\right) \]

   12. --lowering--.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \left(\frac{cos2phi}{alphax \cdot alphax}\right)\right), \left(\frac{\color{blue}{sin2phi}}{alphay \cdot alphay}\right)\right)\right) \]

   13. /-lowering-/.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(cos2phi, \left(alphax \cdot alphax\right)\right)\right), \left(\frac{sin2phi}{alphay \cdot alphay}\right)\right)\right) \]

   14. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right)\right), \left(\frac{sin2phi}{alphay \cdot alphay}\right)\right)\right) \]

   15. /-lowering-/.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right)\right), \mathsf{/.f32}\left(sin2phi, \color{blue}{\left(alphay \cdot alphay\right)}\right)\right)\right) \]

   16. *-lowering-*.f32 98.0%

       \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right)\right), \mathsf{/.f32}\left(sin2phi, \mathsf{*.f32}\left(alphay, \color{blue}{alphay}\right)\right)\right)\right) \]

3. Simplified 98.0%

   \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(-u0\right)}{\left(0 - \frac{cos2phi}{alphax \cdot alphax}\right) - \frac{sin2phi}{alphay \cdot alphay}}} \]
4. Add Preprocessing

5. Taylor expanded in 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%

   \[\leadsto \frac{u0}{\frac{cos2phi}{alphax \cdot alphax} + \color{blue}{\frac{\frac{sin2phi}{alphay}}{alphay}}} \]
10. Add Preprocessing

Alternative 22: 75.4% accurate, 8.9× speedup

Math

\[\begin{array}{l} \\ \frac{u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (/ u0 (+ (/ cos2phi (* alphax alphax)) (/ sin2phi (* alphay alphay)))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	return u0 / ((cos2phi / (alphax * alphax)) + (sin2phi / (alphay * alphay)));
}

Fortran

real(4) function code(alphax, alphay, u0, cos2phi, sin2phi)
    real(4), intent (in) :: alphax
    real(4), intent (in) :: alphay
    real(4), intent (in) :: u0
    real(4), intent (in) :: cos2phi
    real(4), intent (in) :: sin2phi
    code = u0 / ((cos2phi / (alphax * alphax)) + (sin2phi / (alphay * alphay)))
end function

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	return Float32(u0 / Float32(Float32(cos2phi / Float32(alphax * alphax)) + Float32(sin2phi / Float32(alphay * alphay))))
end

MATLAB

function tmp = code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = u0 / ((cos2phi / (alphax * alphax)) + (sin2phi / (alphay * alphay)));
end

Derivation

1. Initial program 60.2%

   \[\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
2. Step-by-step derivation

   1. distribute-frac-neg N/A

      \[\leadsto \mathsf{neg}\left(\frac{\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}}\right) \]

   2. distribute-neg-frac2 N/A

      \[\leadsto \frac{\log \left(1 - u0\right)}{\color{blue}{\mathsf{neg}\left(\left(\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\right)\right)}} \]

   3. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\log \left(1 - u0\right), \color{blue}{\left(\mathsf{neg}\left(\left(\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\right)\right)\right)}\right) \]

   4. sub-neg N/A

      \[\leadsto \mathsf{/.f32}\left(\log \left(1 + \left(\mathsf{neg}\left(u0\right)\right)\right), \left(\mathsf{neg}\left(\left(\color{blue}{\frac{cos2phi}{alphax \cdot alphax}} + \frac{sin2phi}{alphay \cdot alphay}\right)\right)\right)\right) \]

   5. log1p-define N/A

      \[\leadsto \mathsf{/.f32}\left(\left(\mathsf{log1p}\left(\mathsf{neg}\left(u0\right)\right)\right), \left(\mathsf{neg}\left(\color{blue}{\left(\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\right)}\right)\right)\right) \]

   6. log1p-lowering-log1p.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\left(\mathsf{neg}\left(u0\right)\right)\right), \left(\mathsf{neg}\left(\color{blue}{\left(\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\right)}\right)\right)\right) \]

   7. neg-lowering-neg.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \left(\mathsf{neg}\left(\left(\color{blue}{\frac{cos2phi}{alphax \cdot alphax}} + \frac{sin2phi}{alphay \cdot alphay}\right)\right)\right)\right) \]

   8. distribute-neg-in N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \left(\left(\mathsf{neg}\left(\frac{cos2phi}{alphax \cdot alphax}\right)\right) + \color{blue}{\left(\mathsf{neg}\left(\frac{sin2phi}{alphay \cdot alphay}\right)\right)}\right)\right) \]

   9. unsub-neg N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \left(\left(\mathsf{neg}\left(\frac{cos2phi}{alphax \cdot alphax}\right)\right) - \color{blue}{\frac{sin2phi}{alphay \cdot alphay}}\right)\right) \]

   10. --lowering--.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\left(\mathsf{neg}\left(\frac{cos2phi}{alphax \cdot alphax}\right)\right), \color{blue}{\left(\frac{sin2phi}{alphay \cdot alphay}\right)}\right)\right) \]

   11. neg-sub0 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\left(0 - \frac{cos2phi}{alphax \cdot alphax}\right), \left(\frac{\color{blue}{sin2phi}}{alphay \cdot alphay}\right)\right)\right) \]

   12. --lowering--.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \left(\frac{cos2phi}{alphax \cdot alphax}\right)\right), \left(\frac{\color{blue}{sin2phi}}{alphay \cdot alphay}\right)\right)\right) \]

   13. /-lowering-/.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(cos2phi, \left(alphax \cdot alphax\right)\right)\right), \left(\frac{sin2phi}{alphay \cdot alphay}\right)\right)\right) \]

   14. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right)\right), \left(\frac{sin2phi}{alphay \cdot alphay}\right)\right)\right) \]

   15. /-lowering-/.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right)\right), \mathsf{/.f32}\left(sin2phi, \color{blue}{\left(alphay \cdot alphay\right)}\right)\right)\right) \]

   16. *-lowering-*.f32 98.0%

       \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right)\right), \mathsf{/.f32}\left(sin2phi, \mathsf{*.f32}\left(alphay, \color{blue}{alphay}\right)\right)\right)\right) \]

3. Simplified 98.0%

   \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(-u0\right)}{\left(0 - \frac{cos2phi}{alphax \cdot alphax}\right) - \frac{sin2phi}{alphay \cdot alphay}}} \]
4. Add Preprocessing

5. Taylor expanded in 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. Add Preprocessing

Alternative 23: 23.9% accurate, 16.6× speedup

Math

\[\begin{array}{l} \\ alphax \cdot \left(u0 \cdot \frac{alphax}{cos2phi}\right) \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (* alphax (* u0 (/ alphax cos2phi))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	return alphax * (u0 * (alphax / cos2phi));
}

Fortran

real(4) function code(alphax, alphay, u0, cos2phi, sin2phi)
    real(4), intent (in) :: alphax
    real(4), intent (in) :: alphay
    real(4), intent (in) :: u0
    real(4), intent (in) :: cos2phi
    real(4), intent (in) :: sin2phi
    code = alphax * (u0 * (alphax / cos2phi))
end function

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	return Float32(alphax * Float32(u0 * Float32(alphax / cos2phi)))
end

MATLAB

function tmp = code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = alphax * (u0 * (alphax / cos2phi));
end

Derivation

1. Initial program 60.2%

   \[\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
2. Step-by-step derivation

   1. distribute-frac-neg N/A

      \[\leadsto \mathsf{neg}\left(\frac{\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}}\right) \]

   2. distribute-neg-frac2 N/A

      \[\leadsto \frac{\log \left(1 - u0\right)}{\color{blue}{\mathsf{neg}\left(\left(\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\right)\right)}} \]

   3. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\log \left(1 - u0\right), \color{blue}{\left(\mathsf{neg}\left(\left(\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\right)\right)\right)}\right) \]

   4. sub-neg N/A

      \[\leadsto \mathsf{/.f32}\left(\log \left(1 + \left(\mathsf{neg}\left(u0\right)\right)\right), \left(\mathsf{neg}\left(\left(\color{blue}{\frac{cos2phi}{alphax \cdot alphax}} + \frac{sin2phi}{alphay \cdot alphay}\right)\right)\right)\right) \]

   5. log1p-define N/A

      \[\leadsto \mathsf{/.f32}\left(\left(\mathsf{log1p}\left(\mathsf{neg}\left(u0\right)\right)\right), \left(\mathsf{neg}\left(\color{blue}{\left(\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\right)}\right)\right)\right) \]

   6. log1p-lowering-log1p.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\left(\mathsf{neg}\left(u0\right)\right)\right), \left(\mathsf{neg}\left(\color{blue}{\left(\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\right)}\right)\right)\right) \]

   7. neg-lowering-neg.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \left(\mathsf{neg}\left(\left(\color{blue}{\frac{cos2phi}{alphax \cdot alphax}} + \frac{sin2phi}{alphay \cdot alphay}\right)\right)\right)\right) \]

   8. distribute-neg-in N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \left(\left(\mathsf{neg}\left(\frac{cos2phi}{alphax \cdot alphax}\right)\right) + \color{blue}{\left(\mathsf{neg}\left(\frac{sin2phi}{alphay \cdot alphay}\right)\right)}\right)\right) \]

   9. unsub-neg N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \left(\left(\mathsf{neg}\left(\frac{cos2phi}{alphax \cdot alphax}\right)\right) - \color{blue}{\frac{sin2phi}{alphay \cdot alphay}}\right)\right) \]

   10. --lowering--.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\left(\mathsf{neg}\left(\frac{cos2phi}{alphax \cdot alphax}\right)\right), \color{blue}{\left(\frac{sin2phi}{alphay \cdot alphay}\right)}\right)\right) \]

   11. neg-sub0 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\left(0 - \frac{cos2phi}{alphax \cdot alphax}\right), \left(\frac{\color{blue}{sin2phi}}{alphay \cdot alphay}\right)\right)\right) \]

   12. --lowering--.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \left(\frac{cos2phi}{alphax \cdot alphax}\right)\right), \left(\frac{\color{blue}{sin2phi}}{alphay \cdot alphay}\right)\right)\right) \]

   13. /-lowering-/.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(cos2phi, \left(alphax \cdot alphax\right)\right)\right), \left(\frac{sin2phi}{alphay \cdot alphay}\right)\right)\right) \]

   14. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right)\right), \left(\frac{sin2phi}{alphay \cdot alphay}\right)\right)\right) \]

   15. /-lowering-/.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right)\right), \mathsf{/.f32}\left(sin2phi, \color{blue}{\left(alphay \cdot alphay\right)}\right)\right)\right) \]

   16. *-lowering-*.f32 98.0%

       \[\leadsto \mathsf{/.f32}\left(\mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right)\right), \mathsf{/.f32}\left(sin2phi, \mathsf{*.f32}\left(alphay, \color{blue}{alphay}\right)\right)\right)\right) \]

3. Simplified 98.0%

   \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(-u0\right)}{\left(0 - \frac{cos2phi}{alphax \cdot alphax}\right) - \frac{sin2phi}{alphay \cdot alphay}}} \]
4. Add Preprocessing

5. Taylor expanded in 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. 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 24.6%

      \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, \color{blue}{alphax}\right)\right)\right) \]

10. Simplified 24.6%

    \[\leadsto \frac{u0}{\color{blue}{\frac{cos2phi}{alphax \cdot alphax}}} \]
11. Step-by-step derivation

    1. associate-/r/ N/A

       \[\leadsto \frac{u0}{cos2phi} \cdot \color{blue}{\left(alphax \cdot alphax\right)} \]

    2. associate-*r* N/A

       \[\leadsto \left(\frac{u0}{cos2phi} \cdot alphax\right) \cdot \color{blue}{alphax} \]

    3. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{*.f32}\left(\left(\frac{u0}{cos2phi} \cdot alphax\right), \color{blue}{alphax}\right) \]

    4. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(\left(\frac{u0}{cos2phi}\right), alphax\right), alphax\right) \]

    5. /-lowering-/.f32 24.5%

       \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{/.f32}\left(u0, cos2phi\right), alphax\right), alphax\right) \]

12. Applied egg-rr 24.5%

    \[\leadsto \color{blue}{\left(\frac{u0}{cos2phi} \cdot alphax\right) \cdot alphax} \]
13. Step-by-step derivation

    1. associate-*l/ N/A

       \[\leadsto \mathsf{*.f32}\left(\left(\frac{u0 \cdot alphax}{cos2phi}\right), alphax\right) \]

    2. associate-/l* N/A

       \[\leadsto \mathsf{*.f32}\left(\left(u0 \cdot \frac{alphax}{cos2phi}\right), alphax\right) \]

    3. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(u0, \left(\frac{alphax}{cos2phi}\right)\right), alphax\right) \]

    4. /-lowering-/.f32 24.6%

       \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(u0, \mathsf{/.f32}\left(alphax, cos2phi\right)\right), alphax\right) \]

14. Applied egg-rr 24.6%

    \[\leadsto \color{blue}{\left(u0 \cdot \frac{alphax}{cos2phi}\right)} \cdot alphax \]

15. Final simplification 24.6%

    \[\leadsto alphax \cdot \left(u0 \cdot \frac{alphax}{cos2phi}\right) \]
16. Add Preprocessing

Reproduce

herbie shell --seed 2024158 
(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)))))