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

Percentage Accurate: 60.3% → 98.4%

Time: 20.4s

Alternatives: 28

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: 60.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Alternative 1: 98.4% accurate, 1.0× speedup

Math

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

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 59.4%

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

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

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

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

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

   3. sub-neg N/A

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

   4. accelerator-lowering-log1p.f32 N/A

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

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

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

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

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

   7. associate-/r* N/A

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

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

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

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

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

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

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

   11. *-lowering-*.f32 98.5%

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

4. Applied egg-rr 98.5%

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

5. Final simplification 98.5%

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

Alternative 2: 98.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 59.4%

   \[\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
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. accelerator-lowering-log1p.f32 N/A

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

   3. neg-lowering-neg.f32 98.5%

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

4. Applied egg-rr 98.5%

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

5. Final simplification 98.5%

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

Alternative 3: 93.4% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 59.4%

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

   1. frac-add N/A

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

   2. associate-/r/ N/A

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

   3. associate-*r* N/A

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

   4. associate-*r* N/A

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

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

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

4. Applied egg-rr 97.7%

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

5. Taylor expanded in u0 around 0

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

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

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

   2. +-commutative N/A

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

7. Simplified 92.8%

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

8. Final simplification 92.8%

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

Alternative 4: 93.2% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 59.4%

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

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

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

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

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

   3. sub-neg N/A

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

   4. accelerator-lowering-log1p.f32 N/A

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

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

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

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

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

   7. associate-/r* N/A

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

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

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

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

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

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

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

   11. *-lowering-*.f32 98.5%

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

4. Applied egg-rr 98.5%

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

5. Taylor expanded in u0 around 0

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

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

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

   2. +-commutative N/A

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

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

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

7. Simplified 92.7%

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

8. Final simplification 92.7%

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

Alternative 5: 93.1% accurate, 4.0× speedup

Math

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

FPCore

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

C

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

Fortran

real(4) function code(alphax, alphay, u0, cos2phi, sin2phi)
    real(4), intent (in) :: alphax
    real(4), intent (in) :: alphay
    real(4), intent (in) :: u0
    real(4), intent (in) :: cos2phi
    real(4), intent (in) :: sin2phi
    code = (u0 * ((u0 * ((u0 * ((u0 * (-0.25e0)) + (-0.3333333333333333e0))) + (-0.5e0))) + (-1.0e0))) * ((-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(u0 * Float32(Float32(u0 * Float32(-0.25)) + Float32(-0.3333333333333333))) + Float32(-0.5))) + Float32(-1.0))) * Float32(Float32(-1.0) / Float32(Float32(Float32(cos2phi / alphax) / alphax) + Float32(sin2phi / Float32(alphay * alphay)))))
end

MATLAB

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

Derivation

1. Initial program 59.4%

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

   1. frac-2neg N/A

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

   2. div-inv N/A

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

   3. remove-double-neg N/A

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

   4. frac-2neg N/A

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

   5. metadata-eval N/A

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

   6. remove-double-neg N/A

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

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

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

   8. sub-neg N/A

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

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

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

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

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

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

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

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

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

   13. associate-/r* N/A

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

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

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

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

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

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

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

   17. *-lowering-*.f32 98.4%

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

4. Applied egg-rr 98.4%

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

5. Taylor expanded in u0 around 0

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

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

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

   2. sub-neg N/A

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

   3. metadata-eval N/A

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

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

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

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

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

   6. sub-neg N/A

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

   7. metadata-eval N/A

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

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

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

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

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

   10. sub-neg N/A

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

   11. metadata-eval N/A

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

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

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

   13. *-commutative N/A

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

   14. *-lowering-*.f32 92.6%

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

7. Simplified 92.6%

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

Alternative 6: 93.2% accurate, 4.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 59.4%

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

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

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

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

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

   3. sub-neg N/A

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

   4. accelerator-lowering-log1p.f32 N/A

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

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

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

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

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

   7. associate-/r* N/A

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

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

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

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

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

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

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

   11. *-lowering-*.f32 98.5%

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

4. Applied egg-rr 98.5%

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

5. Taylor expanded in u0 around 0

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

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

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

   2. sub-neg N/A

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

   3. metadata-eval N/A

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

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

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

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

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

   6. sub-neg N/A

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

   7. metadata-eval N/A

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

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

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

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

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

   10. sub-neg N/A

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

   11. metadata-eval N/A

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

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

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

   13. *-commutative N/A

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

   14. *-lowering-*.f32 92.6%

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

7. Simplified 92.6%

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

8. Final simplification 92.6%

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

Alternative 7: 93.2% accurate, 4.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 59.4%

   \[\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
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 92.5%

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

5. Simplified 92.5%

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

6. Final simplification 92.5%

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

Alternative 8: 91.4% accurate, 4.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 59.4%

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

3. Taylor expanded in u0 around 0

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

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

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

   2. +-commutative N/A

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

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

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

5. Simplified 90.9%

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

   1. +-commutative N/A

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

   2. distribute-lft-in N/A

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

   3. *-commutative N/A

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

   4. div-inv N/A

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

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

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

7. Applied egg-rr 90.9%

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

   1. div-inv N/A

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

   2. div-inv N/A

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

   3. distribute-rgt-out N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

9. Applied egg-rr 90.9%

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

    1. associate-/r* N/A

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

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

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

    3. /-lowering-/.f32 91.0%

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

11. Applied egg-rr 91.0%

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

12. Final simplification 91.0%

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

Alternative 9: 91.4% accurate, 4.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 59.4%

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

3. Taylor expanded in u0 around 0

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

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

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

   2. +-commutative N/A

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

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

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

5. Simplified 90.9%

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

   1. +-commutative N/A

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

   2. distribute-lft-in N/A

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

   3. *-commutative N/A

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

   4. div-inv N/A

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

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

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

7. Applied egg-rr 90.9%

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

   1. div-inv N/A

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

   2. div-inv N/A

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

   3. distribute-rgt-out N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

9. Applied egg-rr 90.9%

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

10. Final simplification 90.9%

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

Alternative 10: 81.6% accurate, 4.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{sin2phi}{alphay \cdot alphay}\\ \mathbf{if}\;t\_0 \leq 2000:\\ \;\;\;\;\frac{u0}{t\_0 + \frac{\frac{1}{alphax}}{\frac{alphax}{cos2phi}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{u0 \cdot \left(alphay \cdot alphay + 0.5 \cdot \left(u0 \cdot \left(alphay \cdot alphay\right)\right)\right)}{sin2phi}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (let* ((t_0 (/ sin2phi (* alphay alphay))))
   (if (<= t_0 2000.0)
     (/ u0 (+ t_0 (/ (/ 1.0 alphax) (/ alphax cos2phi))))
     (/
      (* u0 (+ (* alphay alphay) (* 0.5 (* u0 (* alphay alphay)))))
      sin2phi))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float t_0 = sin2phi / (alphay * alphay);
	float tmp;
	if (t_0 <= 2000.0f) {
		tmp = u0 / (t_0 + ((1.0f / alphax) / (alphax / cos2phi)));
	} else {
		tmp = (u0 * ((alphay * alphay) + (0.5f * (u0 * (alphay * alphay))))) / sin2phi;
	}
	return tmp;
}

Fortran

real(4) function code(alphax, alphay, u0, cos2phi, sin2phi)
    real(4), intent (in) :: alphax
    real(4), intent (in) :: alphay
    real(4), intent (in) :: u0
    real(4), intent (in) :: cos2phi
    real(4), intent (in) :: sin2phi
    real(4) :: t_0
    real(4) :: tmp
    t_0 = sin2phi / (alphay * alphay)
    if (t_0 <= 2000.0e0) then
        tmp = u0 / (t_0 + ((1.0e0 / alphax) / (alphax / cos2phi)))
    else
        tmp = (u0 * ((alphay * alphay) + (0.5e0 * (u0 * (alphay * alphay))))) / sin2phi
    end if
    code = tmp
end function

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	t_0 = Float32(sin2phi / Float32(alphay * alphay))
	tmp = Float32(0.0)
	if (t_0 <= Float32(2000.0))
		tmp = Float32(u0 / Float32(t_0 + Float32(Float32(Float32(1.0) / alphax) / Float32(alphax / cos2phi))));
	else
		tmp = Float32(Float32(u0 * Float32(Float32(alphay * alphay) + Float32(Float32(0.5) * Float32(u0 * Float32(alphay * alphay))))) / sin2phi);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alphax, alphay, u0, cos2phi, sin2phi)
	t_0 = sin2phi / (alphay * alphay);
	tmp = single(0.0);
	if (t_0 <= single(2000.0))
		tmp = u0 / (t_0 + ((single(1.0) / alphax) / (alphax / cos2phi)));
	else
		tmp = (u0 * ((alphay * alphay) + (single(0.5) * (u0 * (alphay * alphay))))) / sin2phi;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 sin2phi (*.f32 alphay alphay)) < 2e3

   1. Initial program 54.1%

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

   3. Taylor expanded in u0 around 0

      \[\leadsto \color{blue}{\frac{u0}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}}} \]
   4. 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. +-commutative N/A

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

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

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

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

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

      5. unpow2 N/A

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

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

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

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

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

      8. unpow2 N/A

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

      9. *-lowering-*.f32 75.3%

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

   5. Simplified 75.3%

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

      1. associate-/r* N/A

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

      2. div-inv N/A

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

      3. clear-num N/A

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

      4. associate-*l/ N/A

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

      5. div-inv N/A

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

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

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

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

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

      8. /-lowering-/.f32 75.3%

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

   7. Applied egg-rr 75.3%

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

   if 2e3 < (/.f32 sin2phi (*.f32 alphay alphay))

   1. Initial program 65.1%

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

   3. Taylor expanded in u0 around 0

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

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

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

      2. +-commutative N/A

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

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

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

   5. Simplified 91.4%

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-commutative N/A

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

      4. div-inv N/A

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

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

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

   7. Applied egg-rr 91.2%

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

   8. Taylor expanded in sin2phi around inf

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

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

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

      2. distribute-lft-out N/A

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

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

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

      4. unpow2 N/A

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

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

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

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

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

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

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

      8. unpow2 N/A

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

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

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

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

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

      11. *-commutative N/A

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

      12. *-lowering-*.f32 91.5%

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

   10. Simplified 91.5%

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

   11. Taylor expanded in u0 around 0

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

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

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

       2. +-commutative N/A

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

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

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

       4. unpow2 N/A

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

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

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

       6. *-commutative N/A

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

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

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

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

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

       9. unpow2 N/A

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

       10. *-lowering-*.f32 87.8%

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

   13. Simplified 87.8%

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

4. Final simplification 81.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 2000:\\ \;\;\;\;\frac{u0}{\frac{sin2phi}{alphay \cdot alphay} + \frac{\frac{1}{alphax}}{\frac{alphax}{cos2phi}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{u0 \cdot \left(alphay \cdot alphay + 0.5 \cdot \left(u0 \cdot \left(alphay \cdot alphay\right)\right)\right)}{sin2phi}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 81.6% accurate, 4.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{sin2phi}{alphay \cdot alphay}\\ \mathbf{if}\;t\_0 \leq 2000:\\ \;\;\;\;\frac{u0}{t\_0 + \frac{\frac{1}{alphax}}{\frac{alphax}{cos2phi}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(alphay \cdot alphay\right) \cdot \left(u0 + 0.5 \cdot \left(u0 \cdot u0\right)\right)}{sin2phi}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (let* ((t_0 (/ sin2phi (* alphay alphay))))
   (if (<= t_0 2000.0)
     (/ u0 (+ t_0 (/ (/ 1.0 alphax) (/ alphax cos2phi))))
     (/ (* (* alphay alphay) (+ u0 (* 0.5 (* u0 u0)))) 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 <= 2000.0f) {
		tmp = u0 / (t_0 + ((1.0f / alphax) / (alphax / cos2phi)));
	} else {
		tmp = ((alphay * alphay) * (u0 + (0.5f * (u0 * u0)))) / sin2phi;
	}
	return tmp;
}

Fortran

real(4) function code(alphax, alphay, u0, cos2phi, sin2phi)
    real(4), intent (in) :: alphax
    real(4), intent (in) :: alphay
    real(4), intent (in) :: u0
    real(4), intent (in) :: cos2phi
    real(4), intent (in) :: sin2phi
    real(4) :: t_0
    real(4) :: tmp
    t_0 = sin2phi / (alphay * alphay)
    if (t_0 <= 2000.0e0) then
        tmp = u0 / (t_0 + ((1.0e0 / alphax) / (alphax / cos2phi)))
    else
        tmp = ((alphay * alphay) * (u0 + (0.5e0 * (u0 * u0)))) / 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(2000.0))
		tmp = Float32(u0 / Float32(t_0 + Float32(Float32(Float32(1.0) / alphax) / Float32(alphax / cos2phi))));
	else
		tmp = Float32(Float32(Float32(alphay * alphay) * Float32(u0 + Float32(Float32(0.5) * Float32(u0 * u0)))) / 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(2000.0))
		tmp = u0 / (t_0 + ((single(1.0) / alphax) / (alphax / cos2phi)));
	else
		tmp = ((alphay * alphay) * (u0 + (single(0.5) * (u0 * u0)))) / sin2phi;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 sin2phi (*.f32 alphay alphay)) < 2e3

   1. Initial program 54.1%

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

   3. Taylor expanded in u0 around 0

      \[\leadsto \color{blue}{\frac{u0}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}}} \]
   4. 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. +-commutative N/A

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

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

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

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

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

      5. unpow2 N/A

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

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

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

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

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

      8. unpow2 N/A

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

      9. *-lowering-*.f32 75.3%

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

   5. Simplified 75.3%

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

      1. associate-/r* N/A

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

      2. div-inv N/A

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

      3. clear-num N/A

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

      4. associate-*l/ N/A

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

      5. div-inv N/A

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

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

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

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

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

      8. /-lowering-/.f32 75.3%

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

   7. Applied egg-rr 75.3%

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

   if 2e3 < (/.f32 sin2phi (*.f32 alphay alphay))

   1. Initial program 65.1%

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

   3. Taylor expanded in u0 around 0

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

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

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

      2. +-commutative N/A

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

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

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

   5. Simplified 91.4%

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-commutative N/A

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

      4. div-inv N/A

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

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

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

   7. Applied egg-rr 91.2%

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

   8. Taylor expanded in sin2phi around inf

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

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

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

      2. distribute-lft-out N/A

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

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

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

      4. unpow2 N/A

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

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

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

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

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

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

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

      8. unpow2 N/A

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

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

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

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

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

      11. *-commutative N/A

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

      12. *-lowering-*.f32 91.5%

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

   10. Simplified 91.5%

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

   11. Taylor expanded in u0 around 0

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

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

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

       2. unpow2 N/A

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

       3. *-lowering-*.f32 87.7%

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

   13. Simplified 87.7%

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

Alternative 12: 91.4% accurate, 4.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 59.4%

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

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

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

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

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

   3. sub-neg N/A

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

   4. accelerator-lowering-log1p.f32 N/A

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

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

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

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

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

   7. associate-/r* N/A

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

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

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

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

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

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

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

   11. *-lowering-*.f32 98.5%

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

4. Applied egg-rr 98.5%

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

5. Taylor expanded in u0 around 0

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

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

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

   2. sub-neg N/A

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

   3. metadata-eval N/A

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

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

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

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

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

   6. sub-neg N/A

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

   7. metadata-eval N/A

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

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

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

   9. *-commutative N/A

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

   10. *-lowering-*.f32 90.8%

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

7. Simplified 90.8%

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

8. Final simplification 90.8%

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

Alternative 13: 91.4% accurate, 5.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 59.4%

   \[\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
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 90.7%

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

5. Simplified 90.7%

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

6. Final simplification 90.7%

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

Alternative 14: 83.3% accurate, 5.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;sin2phi \leq 1.5:\\ \;\;\;\;\frac{u0}{\frac{cos2phi + \frac{sin2phi \cdot \left(alphax \cdot alphax\right)}{alphay \cdot alphay}}{alphax \cdot alphax}}\\ \mathbf{else}:\\ \;\;\;\;\left(u0 + \left(0.5 + u0 \cdot 0.3333333333333333\right) \cdot \left(u0 \cdot u0\right)\right) \cdot \frac{alphay \cdot alphay}{sin2phi}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (if (<= sin2phi 1.5)
   (/
    u0
    (/
     (+ cos2phi (/ (* sin2phi (* alphax alphax)) (* alphay alphay)))
     (* alphax alphax)))
   (*
    (+ u0 (* (+ 0.5 (* u0 0.3333333333333333)) (* u0 u0)))
    (/ (* alphay alphay) sin2phi))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float tmp;
	if (sin2phi <= 1.5f) {
		tmp = u0 / ((cos2phi + ((sin2phi * (alphax * alphax)) / (alphay * alphay))) / (alphax * alphax));
	} else {
		tmp = (u0 + ((0.5f + (u0 * 0.3333333333333333f)) * (u0 * u0))) * ((alphay * alphay) / sin2phi);
	}
	return tmp;
}

Fortran

real(4) function code(alphax, alphay, u0, cos2phi, sin2phi)
    real(4), intent (in) :: alphax
    real(4), intent (in) :: alphay
    real(4), intent (in) :: u0
    real(4), intent (in) :: cos2phi
    real(4), intent (in) :: sin2phi
    real(4) :: tmp
    if (sin2phi <= 1.5e0) then
        tmp = u0 / ((cos2phi + ((sin2phi * (alphax * alphax)) / (alphay * alphay))) / (alphax * alphax))
    else
        tmp = (u0 + ((0.5e0 + (u0 * 0.3333333333333333e0)) * (u0 * u0))) * ((alphay * alphay) / sin2phi)
    end if
    code = tmp
end function

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = Float32(0.0)
	if (sin2phi <= Float32(1.5))
		tmp = Float32(u0 / Float32(Float32(cos2phi + Float32(Float32(sin2phi * Float32(alphax * alphax)) / Float32(alphay * alphay))) / Float32(alphax * alphax)));
	else
		tmp = Float32(Float32(u0 + Float32(Float32(Float32(0.5) + Float32(u0 * Float32(0.3333333333333333))) * Float32(u0 * u0))) * Float32(Float32(alphay * alphay) / sin2phi));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = single(0.0);
	if (sin2phi <= single(1.5))
		tmp = u0 / ((cos2phi + ((sin2phi * (alphax * alphax)) / (alphay * alphay))) / (alphax * alphax));
	else
		tmp = (u0 + ((single(0.5) + (u0 * single(0.3333333333333333))) * (u0 * u0))) * ((alphay * alphay) / sin2phi);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if sin2phi < 1.5

   1. Initial program 53.6%

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

   3. Taylor expanded in u0 around 0

      \[\leadsto \color{blue}{\frac{u0}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}}} \]
   4. 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. +-commutative N/A

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

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

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

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

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

      5. unpow2 N/A

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

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

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

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

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

      8. unpow2 N/A

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

      9. *-lowering-*.f32 75.5%

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

   5. Simplified 75.5%

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

   6. Taylor expanded in alphax around 0

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

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

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

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

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

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

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

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

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

      5. unpow2 N/A

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

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

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

      7. unpow2 N/A

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

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

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

      9. unpow2 N/A

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

      10. *-lowering-*.f32 75.6%

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

   8. Simplified 75.6%

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

   if 1.5 < sin2phi

   1. Initial program 66.1%

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

   3. Taylor expanded in u0 around 0

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

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

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

      2. +-commutative N/A

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

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

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

   5. Simplified 91.4%

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-commutative N/A

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

      4. div-inv N/A

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

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

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

   7. Applied egg-rr 91.3%

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

      1. div-inv N/A

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

      2. div-inv N/A

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

      3. distribute-rgt-out N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

   9. Applied egg-rr 91.4%

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

   10. Taylor expanded in sin2phi around inf

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

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

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

       2. unpow2 N/A

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

       3. *-lowering-*.f32 92.1%

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

   12. Simplified 92.1%

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

4. Final simplification 83.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;sin2phi \leq 1.5:\\ \;\;\;\;\frac{u0}{\frac{cos2phi + \frac{sin2phi \cdot \left(alphax \cdot alphax\right)}{alphay \cdot alphay}}{alphax \cdot alphax}}\\ \mathbf{else}:\\ \;\;\;\;\left(u0 + \left(0.5 + u0 \cdot 0.3333333333333333\right) \cdot \left(u0 \cdot u0\right)\right) \cdot \frac{alphay \cdot alphay}{sin2phi}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 83.3% accurate, 5.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;sin2phi \leq 1.5:\\ \;\;\;\;\frac{u0}{\frac{sin2phi}{alphay \cdot alphay} + \frac{\frac{1}{alphax}}{\frac{alphax}{cos2phi}}}\\ \mathbf{else}:\\ \;\;\;\;\left(u0 + \left(0.5 + u0 \cdot 0.3333333333333333\right) \cdot \left(u0 \cdot u0\right)\right) \cdot \frac{alphay \cdot alphay}{sin2phi}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (if (<= sin2phi 1.5)
   (/
    u0
    (+ (/ sin2phi (* alphay alphay)) (/ (/ 1.0 alphax) (/ alphax cos2phi))))
   (*
    (+ u0 (* (+ 0.5 (* u0 0.3333333333333333)) (* u0 u0)))
    (/ (* alphay alphay) sin2phi))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float tmp;
	if (sin2phi <= 1.5f) {
		tmp = u0 / ((sin2phi / (alphay * alphay)) + ((1.0f / alphax) / (alphax / cos2phi)));
	} else {
		tmp = (u0 + ((0.5f + (u0 * 0.3333333333333333f)) * (u0 * u0))) * ((alphay * alphay) / sin2phi);
	}
	return tmp;
}

Fortran

real(4) function code(alphax, alphay, u0, cos2phi, sin2phi)
    real(4), intent (in) :: alphax
    real(4), intent (in) :: alphay
    real(4), intent (in) :: u0
    real(4), intent (in) :: cos2phi
    real(4), intent (in) :: sin2phi
    real(4) :: tmp
    if (sin2phi <= 1.5e0) then
        tmp = u0 / ((sin2phi / (alphay * alphay)) + ((1.0e0 / alphax) / (alphax / cos2phi)))
    else
        tmp = (u0 + ((0.5e0 + (u0 * 0.3333333333333333e0)) * (u0 * u0))) * ((alphay * alphay) / sin2phi)
    end if
    code = tmp
end function

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = Float32(0.0)
	if (sin2phi <= Float32(1.5))
		tmp = Float32(u0 / Float32(Float32(sin2phi / Float32(alphay * alphay)) + Float32(Float32(Float32(1.0) / alphax) / Float32(alphax / cos2phi))));
	else
		tmp = Float32(Float32(u0 + Float32(Float32(Float32(0.5) + Float32(u0 * Float32(0.3333333333333333))) * Float32(u0 * u0))) * Float32(Float32(alphay * alphay) / sin2phi));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = single(0.0);
	if (sin2phi <= single(1.5))
		tmp = u0 / ((sin2phi / (alphay * alphay)) + ((single(1.0) / alphax) / (alphax / cos2phi)));
	else
		tmp = (u0 + ((single(0.5) + (u0 * single(0.3333333333333333))) * (u0 * u0))) * ((alphay * alphay) / sin2phi);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if sin2phi < 1.5

   1. Initial program 53.6%

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

   3. Taylor expanded in u0 around 0

      \[\leadsto \color{blue}{\frac{u0}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}}} \]
   4. 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. +-commutative N/A

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

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

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

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

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

      5. unpow2 N/A

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

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

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

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

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

      8. unpow2 N/A

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

      9. *-lowering-*.f32 75.5%

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

   5. Simplified 75.5%

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

      1. associate-/r* N/A

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

      2. div-inv N/A

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

      3. clear-num N/A

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

      4. associate-*l/ N/A

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

      5. div-inv N/A

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

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

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

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

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

      8. /-lowering-/.f32 75.6%

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

   7. Applied egg-rr 75.6%

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

   if 1.5 < sin2phi

   1. Initial program 66.1%

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

   3. Taylor expanded in u0 around 0

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

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

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

      2. +-commutative N/A

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

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

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

   5. Simplified 91.4%

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-commutative N/A

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

      4. div-inv N/A

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

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

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

   7. Applied egg-rr 91.3%

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

      1. div-inv N/A

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

      2. div-inv N/A

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

      3. distribute-rgt-out N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

   9. Applied egg-rr 91.4%

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

   10. Taylor expanded in sin2phi around inf

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

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

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

       2. unpow2 N/A

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

       3. *-lowering-*.f32 92.1%

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

   12. Simplified 92.1%

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

4. Final simplification 83.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;sin2phi \leq 1.5:\\ \;\;\;\;\frac{u0}{\frac{sin2phi}{alphay \cdot alphay} + \frac{\frac{1}{alphax}}{\frac{alphax}{cos2phi}}}\\ \mathbf{else}:\\ \;\;\;\;\left(u0 + \left(0.5 + u0 \cdot 0.3333333333333333\right) \cdot \left(u0 \cdot u0\right)\right) \cdot \frac{alphay \cdot alphay}{sin2phi}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 83.2% accurate, 5.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;sin2phi \leq 1.5:\\ \;\;\;\;\frac{u0}{\frac{sin2phi}{alphay \cdot alphay} + \frac{\frac{1}{alphax}}{\frac{alphax}{cos2phi}}}\\ \mathbf{else}:\\ \;\;\;\;\left(u0 \cdot \left(1 + u0 \cdot \left(0.5 + u0 \cdot 0.3333333333333333\right)\right)\right) \cdot \frac{alphay}{\frac{sin2phi}{alphay}}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (if (<= sin2phi 1.5)
   (/
    u0
    (+ (/ sin2phi (* alphay alphay)) (/ (/ 1.0 alphax) (/ alphax cos2phi))))
   (*
    (* u0 (+ 1.0 (* u0 (+ 0.5 (* u0 0.3333333333333333)))))
    (/ alphay (/ sin2phi alphay)))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float tmp;
	if (sin2phi <= 1.5f) {
		tmp = u0 / ((sin2phi / (alphay * alphay)) + ((1.0f / alphax) / (alphax / cos2phi)));
	} else {
		tmp = (u0 * (1.0f + (u0 * (0.5f + (u0 * 0.3333333333333333f))))) * (alphay / (sin2phi / alphay));
	}
	return tmp;
}

Fortran

real(4) function code(alphax, alphay, u0, cos2phi, sin2phi)
    real(4), intent (in) :: alphax
    real(4), intent (in) :: alphay
    real(4), intent (in) :: u0
    real(4), intent (in) :: cos2phi
    real(4), intent (in) :: sin2phi
    real(4) :: tmp
    if (sin2phi <= 1.5e0) then
        tmp = u0 / ((sin2phi / (alphay * alphay)) + ((1.0e0 / alphax) / (alphax / cos2phi)))
    else
        tmp = (u0 * (1.0e0 + (u0 * (0.5e0 + (u0 * 0.3333333333333333e0))))) * (alphay / (sin2phi / alphay))
    end if
    code = tmp
end function

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = Float32(0.0)
	if (sin2phi <= Float32(1.5))
		tmp = Float32(u0 / Float32(Float32(sin2phi / Float32(alphay * alphay)) + Float32(Float32(Float32(1.0) / alphax) / Float32(alphax / cos2phi))));
	else
		tmp = Float32(Float32(u0 * Float32(Float32(1.0) + Float32(u0 * Float32(Float32(0.5) + Float32(u0 * Float32(0.3333333333333333)))))) * Float32(alphay / Float32(sin2phi / alphay)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = single(0.0);
	if (sin2phi <= single(1.5))
		tmp = u0 / ((sin2phi / (alphay * alphay)) + ((single(1.0) / alphax) / (alphax / cos2phi)));
	else
		tmp = (u0 * (single(1.0) + (u0 * (single(0.5) + (u0 * single(0.3333333333333333)))))) * (alphay / (sin2phi / alphay));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if sin2phi < 1.5

   1. Initial program 53.6%

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

   3. Taylor expanded in u0 around 0

      \[\leadsto \color{blue}{\frac{u0}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}}} \]
   4. 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. +-commutative N/A

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

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

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

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

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

      5. unpow2 N/A

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

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

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

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

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

      8. unpow2 N/A

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

      9. *-lowering-*.f32 75.5%

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

   5. Simplified 75.5%

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

      1. associate-/r* N/A

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

      2. div-inv N/A

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

      3. clear-num N/A

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

      4. associate-*l/ N/A

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

      5. div-inv N/A

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

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

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

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

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

      8. /-lowering-/.f32 75.6%

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

   7. Applied egg-rr 75.6%

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

   if 1.5 < sin2phi

   1. Initial program 66.1%

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

   3. Taylor expanded in u0 around 0

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

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

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

      2. +-commutative N/A

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

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

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

   5. Simplified 91.4%

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-commutative N/A

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

      4. div-inv N/A

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

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

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

   7. Applied egg-rr 91.3%

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

   8. Taylor expanded in sin2phi around inf

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

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

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

      2. distribute-lft-out N/A

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

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

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

      4. unpow2 N/A

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

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

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

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

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

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

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

      8. unpow2 N/A

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

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

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

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

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

      11. *-commutative N/A

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

      12. *-lowering-*.f32 91.8%

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

   10. Simplified 91.8%

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

       1. *-commutative N/A

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

       2. associate-/l* N/A

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

       3. clear-num N/A

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

       4. associate-/r* N/A

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

       5. clear-num N/A

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

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

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

       7. *-commutative N/A

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

       8. associate-*r* N/A

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

       9. distribute-rgt1-in N/A

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

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

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

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

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

       12. *-commutative N/A

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

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

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

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

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

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

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

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

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

       17. /-lowering-/.f32 91.9%

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

   12. Applied egg-rr 91.9%

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

4. Final simplification 83.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;sin2phi \leq 1.5:\\ \;\;\;\;\frac{u0}{\frac{sin2phi}{alphay \cdot alphay} + \frac{\frac{1}{alphax}}{\frac{alphax}{cos2phi}}}\\ \mathbf{else}:\\ \;\;\;\;\left(u0 \cdot \left(1 + u0 \cdot \left(0.5 + u0 \cdot 0.3333333333333333\right)\right)\right) \cdot \frac{alphay}{\frac{sin2phi}{alphay}}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 83.2% accurate, 5.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;sin2phi \leq 1.5:\\ \;\;\;\;\frac{u0}{\frac{sin2phi}{alphay \cdot alphay} + \frac{\frac{1}{alphax}}{\frac{alphax}{cos2phi}}}\\ \mathbf{else}:\\ \;\;\;\;alphay \cdot \left(alphay \cdot \frac{u0 \cdot \left(1 + u0 \cdot \left(0.5 + u0 \cdot 0.3333333333333333\right)\right)}{sin2phi}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (if (<= sin2phi 1.5)
   (/
    u0
    (+ (/ sin2phi (* alphay alphay)) (/ (/ 1.0 alphax) (/ alphax cos2phi))))
   (*
    alphay
    (*
     alphay
     (/ (* u0 (+ 1.0 (* u0 (+ 0.5 (* u0 0.3333333333333333))))) sin2phi)))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float tmp;
	if (sin2phi <= 1.5f) {
		tmp = u0 / ((sin2phi / (alphay * alphay)) + ((1.0f / alphax) / (alphax / cos2phi)));
	} else {
		tmp = alphay * (alphay * ((u0 * (1.0f + (u0 * (0.5f + (u0 * 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 <= 1.5e0) then
        tmp = u0 / ((sin2phi / (alphay * alphay)) + ((1.0e0 / alphax) / (alphax / cos2phi)))
    else
        tmp = alphay * (alphay * ((u0 * (1.0e0 + (u0 * (0.5e0 + (u0 * 0.3333333333333333e0))))) / sin2phi))
    end if
    code = tmp
end function

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = Float32(0.0)
	if (sin2phi <= Float32(1.5))
		tmp = Float32(u0 / Float32(Float32(sin2phi / Float32(alphay * alphay)) + Float32(Float32(Float32(1.0) / alphax) / Float32(alphax / cos2phi))));
	else
		tmp = Float32(alphay * Float32(alphay * Float32(Float32(u0 * Float32(Float32(1.0) + Float32(u0 * Float32(Float32(0.5) + Float32(u0 * Float32(0.3333333333333333)))))) / sin2phi)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = single(0.0);
	if (sin2phi <= single(1.5))
		tmp = u0 / ((sin2phi / (alphay * alphay)) + ((single(1.0) / alphax) / (alphax / cos2phi)));
	else
		tmp = alphay * (alphay * ((u0 * (single(1.0) + (u0 * (single(0.5) + (u0 * single(0.3333333333333333)))))) / sin2phi));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if sin2phi < 1.5

   1. Initial program 53.6%

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

   3. Taylor expanded in u0 around 0

      \[\leadsto \color{blue}{\frac{u0}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}}} \]
   4. 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. +-commutative N/A

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

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

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

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

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

      5. unpow2 N/A

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

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

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

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

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

      8. unpow2 N/A

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

      9. *-lowering-*.f32 75.5%

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

   5. Simplified 75.5%

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

      1. associate-/r* N/A

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

      2. div-inv N/A

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

      3. clear-num N/A

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

      4. associate-*l/ N/A

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

      5. div-inv N/A

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

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

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

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

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

      8. /-lowering-/.f32 75.6%

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

   7. Applied egg-rr 75.6%

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

   if 1.5 < sin2phi

   1. Initial program 66.1%

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

   3. Taylor expanded in u0 around 0

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

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

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

      2. +-commutative N/A

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

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

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

   5. Simplified 91.4%

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-commutative N/A

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

      4. div-inv N/A

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

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

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

   7. Applied egg-rr 91.3%

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

   8. Taylor expanded in sin2phi around inf

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

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

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

      2. distribute-lft-out N/A

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

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

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

      4. unpow2 N/A

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

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

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

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

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

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

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

      8. unpow2 N/A

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

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

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

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

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

      11. *-commutative N/A

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

      12. *-lowering-*.f32 91.8%

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

   10. Simplified 91.8%

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

       1. associate-/l* N/A

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

       2. associate-*l* N/A

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

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

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

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

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

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

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

       6. *-commutative N/A

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

       7. associate-*r* N/A

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

       8. distribute-rgt1-in N/A

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

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

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

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

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

       11. *-commutative N/A

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

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

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

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

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

       14. *-lowering-*.f32 91.7%

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

   12. Applied egg-rr 91.7%

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

4. Final simplification 83.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;sin2phi \leq 1.5:\\ \;\;\;\;\frac{u0}{\frac{sin2phi}{alphay \cdot alphay} + \frac{\frac{1}{alphax}}{\frac{alphax}{cos2phi}}}\\ \mathbf{else}:\\ \;\;\;\;alphay \cdot \left(alphay \cdot \frac{u0 \cdot \left(1 + u0 \cdot \left(0.5 + u0 \cdot 0.3333333333333333\right)\right)}{sin2phi}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 81.6% accurate, 5.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{sin2phi}{alphay \cdot alphay}\\ \mathbf{if}\;t\_0 \leq 2000:\\ \;\;\;\;\frac{u0}{\frac{\frac{cos2phi}{alphax}}{alphax} + t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(alphay \cdot alphay\right) \cdot \left(u0 + 0.5 \cdot \left(u0 \cdot u0\right)\right)}{sin2phi}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (let* ((t_0 (/ sin2phi (* alphay alphay))))
   (if (<= t_0 2000.0)
     (/ u0 (+ (/ (/ cos2phi alphax) alphax) t_0))
     (/ (* (* alphay alphay) (+ u0 (* 0.5 (* u0 u0)))) 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 <= 2000.0f) {
		tmp = u0 / (((cos2phi / alphax) / alphax) + t_0);
	} else {
		tmp = ((alphay * alphay) * (u0 + (0.5f * (u0 * u0)))) / sin2phi;
	}
	return tmp;
}

Fortran

real(4) function code(alphax, alphay, u0, cos2phi, sin2phi)
    real(4), intent (in) :: alphax
    real(4), intent (in) :: alphay
    real(4), intent (in) :: u0
    real(4), intent (in) :: cos2phi
    real(4), intent (in) :: sin2phi
    real(4) :: t_0
    real(4) :: tmp
    t_0 = sin2phi / (alphay * alphay)
    if (t_0 <= 2000.0e0) then
        tmp = u0 / (((cos2phi / alphax) / alphax) + t_0)
    else
        tmp = ((alphay * alphay) * (u0 + (0.5e0 * (u0 * u0)))) / 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(2000.0))
		tmp = Float32(u0 / Float32(Float32(Float32(cos2phi / alphax) / alphax) + t_0));
	else
		tmp = Float32(Float32(Float32(alphay * alphay) * Float32(u0 + Float32(Float32(0.5) * Float32(u0 * u0)))) / 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(2000.0))
		tmp = u0 / (((cos2phi / alphax) / alphax) + t_0);
	else
		tmp = ((alphay * alphay) * (u0 + (single(0.5) * (u0 * u0)))) / sin2phi;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 sin2phi (*.f32 alphay alphay)) < 2e3

   1. Initial program 54.1%

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

   3. Taylor expanded in u0 around 0

      \[\leadsto \color{blue}{\frac{u0}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}}} \]
   4. 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. +-commutative N/A

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

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

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

      4. /-lowering-/.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{/.f32}\left(sin2phi, \left({alphay}^{2}\right)\right), \left(\frac{\color{blue}{cos2phi}}{{alphax}^{2}}\right)\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{/.f32}\left(sin2phi, \left(alphay \cdot alphay\right)\right), \left(\frac{cos2phi}{{alphax}^{2}}\right)\right)\right) \]

      6. *-lowering-*.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{/.f32}\left(sin2phi, \mathsf{*.f32}\left(alphay, alphay\right)\right), \left(\frac{cos2phi}{{alphax}^{2}}\right)\right)\right) \]

      7. /-lowering-/.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{/.f32}\left(sin2phi, \mathsf{*.f32}\left(alphay, alphay\right)\right), \mathsf{/.f32}\left(cos2phi, \color{blue}{\left({alphax}^{2}\right)}\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{/.f32}\left(sin2phi, \mathsf{*.f32}\left(alphay, alphay\right)\right), \mathsf{/.f32}\left(cos2phi, \left(alphax \cdot \color{blue}{alphax}\right)\right)\right)\right) \]

      9. *-lowering-*.f32 75.3%

         \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{/.f32}\left(sin2phi, \mathsf{*.f32}\left(alphay, alphay\right)\right), \mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, \color{blue}{alphax}\right)\right)\right)\right) \]

   5. Simplified 75.3%

      \[\leadsto \color{blue}{\frac{u0}{\frac{sin2phi}{alphay \cdot alphay} + \frac{cos2phi}{alphax \cdot alphax}}} \]
   6. Step-by-step derivation

      1. associate-/r* N/A

         \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{/.f32}\left(sin2phi, \mathsf{*.f32}\left(alphay, alphay\right)\right), \left(\frac{\frac{cos2phi}{alphax}}{\color{blue}{alphax}}\right)\right)\right) \]

      2. /-lowering-/.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{/.f32}\left(sin2phi, \mathsf{*.f32}\left(alphay, alphay\right)\right), \mathsf{/.f32}\left(\left(\frac{cos2phi}{alphax}\right), \color{blue}{alphax}\right)\right)\right) \]

      3. /-lowering-/.f32 75.3%

         \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{/.f32}\left(sin2phi, \mathsf{*.f32}\left(alphay, alphay\right)\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(cos2phi, alphax\right), alphax\right)\right)\right) \]

   7. Applied egg-rr 75.3%

      \[\leadsto \frac{u0}{\frac{sin2phi}{alphay \cdot alphay} + \color{blue}{\frac{\frac{cos2phi}{alphax}}{alphax}}} \]

   if 2e3 < (/.f32 sin2phi (*.f32 alphay alphay))

   1. Initial program 65.1%

      \[\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
   2. Add Preprocessing

   3. Taylor expanded in u0 around 0

      \[\leadsto \color{blue}{u0 \cdot \left(u0 \cdot \left(\frac{1}{3} \cdot \frac{u0}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}} + \frac{1}{2} \cdot \frac{1}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}}\right) + \frac{1}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}}\right)} \]
   4. Step-by-step derivation

      1. *-lowering-*.f32 N/A

         \[\leadsto \mathsf{*.f32}\left(u0, \color{blue}{\left(u0 \cdot \left(\frac{1}{3} \cdot \frac{u0}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}} + \frac{1}{2} \cdot \frac{1}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}}\right) + \frac{1}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}}\right)}\right) \]

      2. +-commutative N/A

         \[\leadsto \mathsf{*.f32}\left(u0, \left(\frac{1}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}} + \color{blue}{u0 \cdot \left(\frac{1}{3} \cdot \frac{u0}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}} + \frac{1}{2} \cdot \frac{1}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}}\right)}\right)\right) \]

      3. +-lowering-+.f32 N/A

         \[\leadsto \mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\left(\frac{1}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}}\right), \color{blue}{\left(u0 \cdot \left(\frac{1}{3} \cdot \frac{u0}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}} + \frac{1}{2} \cdot \frac{1}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}}\right)\right)}\right)\right) \]

   5. Simplified 91.4%

      \[\leadsto \color{blue}{u0 \cdot \left(\frac{1}{\frac{sin2phi}{alphay \cdot alphay} + \frac{cos2phi}{alphax \cdot alphax}} + \frac{u0}{\frac{sin2phi}{alphay \cdot alphay} + \frac{cos2phi}{alphax \cdot alphax}} \cdot \left(0.5 + u0 \cdot 0.3333333333333333\right)\right)} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto u0 \cdot \left(\frac{u0}{\frac{sin2phi}{alphay \cdot alphay} + \frac{cos2phi}{alphax \cdot alphax}} \cdot \left(\frac{1}{2} + u0 \cdot \frac{1}{3}\right) + \color{blue}{\frac{1}{\frac{sin2phi}{alphay \cdot alphay} + \frac{cos2phi}{alphax \cdot alphax}}}\right) \]

      2. distribute-lft-in N/A

         \[\leadsto u0 \cdot \left(\frac{u0}{\frac{sin2phi}{alphay \cdot alphay} + \frac{cos2phi}{alphax \cdot alphax}} \cdot \left(\frac{1}{2} + u0 \cdot \frac{1}{3}\right)\right) + \color{blue}{u0 \cdot \frac{1}{\frac{sin2phi}{alphay \cdot alphay} + \frac{cos2phi}{alphax \cdot alphax}}} \]

      3. *-commutative N/A

         \[\leadsto \left(\frac{u0}{\frac{sin2phi}{alphay \cdot alphay} + \frac{cos2phi}{alphax \cdot alphax}} \cdot \left(\frac{1}{2} + u0 \cdot \frac{1}{3}\right)\right) \cdot u0 + \color{blue}{u0} \cdot \frac{1}{\frac{sin2phi}{alphay \cdot alphay} + \frac{cos2phi}{alphax \cdot alphax}} \]

      4. div-inv N/A

         \[\leadsto \left(\frac{u0}{\frac{sin2phi}{alphay \cdot alphay} + \frac{cos2phi}{alphax \cdot alphax}} \cdot \left(\frac{1}{2} + u0 \cdot \frac{1}{3}\right)\right) \cdot u0 + \frac{u0}{\color{blue}{\frac{sin2phi}{alphay \cdot alphay} + \frac{cos2phi}{alphax \cdot alphax}}} \]

      5. +-lowering-+.f32 N/A

         \[\leadsto \mathsf{+.f32}\left(\left(\left(\frac{u0}{\frac{sin2phi}{alphay \cdot alphay} + \frac{cos2phi}{alphax \cdot alphax}} \cdot \left(\frac{1}{2} + u0 \cdot \frac{1}{3}\right)\right) \cdot u0\right), \color{blue}{\left(\frac{u0}{\frac{sin2phi}{alphay \cdot alphay} + \frac{cos2phi}{alphax \cdot alphax}}\right)}\right) \]

   7. Applied egg-rr 91.2%

      \[\leadsto \color{blue}{\frac{u0 \cdot \left(u0 \cdot \left(0.5 + u0 \cdot 0.3333333333333333\right)\right)}{\frac{sin2phi}{alphay \cdot alphay} + \frac{cos2phi}{alphax \cdot alphax}} + \frac{u0}{\frac{sin2phi}{alphay \cdot alphay} + \frac{cos2phi}{alphax \cdot alphax}}} \]

   8. Taylor expanded in sin2phi around inf

      \[\leadsto \color{blue}{\frac{{alphay}^{2} \cdot u0 + {alphay}^{2} \cdot \left({u0}^{2} \cdot \left(\frac{1}{2} + \frac{1}{3} \cdot u0\right)\right)}{sin2phi}} \]
   9. Step-by-step derivation

      1. /-lowering-/.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(\left({alphay}^{2} \cdot u0 + {alphay}^{2} \cdot \left({u0}^{2} \cdot \left(\frac{1}{2} + \frac{1}{3} \cdot u0\right)\right)\right), \color{blue}{sin2phi}\right) \]

      2. distribute-lft-out N/A

         \[\leadsto \mathsf{/.f32}\left(\left({alphay}^{2} \cdot \left(u0 + {u0}^{2} \cdot \left(\frac{1}{2} + \frac{1}{3} \cdot u0\right)\right)\right), sin2phi\right) \]

      3. *-lowering-*.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(\left({alphay}^{2}\right), \left(u0 + {u0}^{2} \cdot \left(\frac{1}{2} + \frac{1}{3} \cdot u0\right)\right)\right), sin2phi\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(\left(alphay \cdot alphay\right), \left(u0 + {u0}^{2} \cdot \left(\frac{1}{2} + \frac{1}{3} \cdot u0\right)\right)\right), sin2phi\right) \]

      5. *-lowering-*.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(alphay, alphay\right), \left(u0 + {u0}^{2} \cdot \left(\frac{1}{2} + \frac{1}{3} \cdot u0\right)\right)\right), sin2phi\right) \]

      6. +-lowering-+.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(alphay, alphay\right), \mathsf{+.f32}\left(u0, \left({u0}^{2} \cdot \left(\frac{1}{2} + \frac{1}{3} \cdot u0\right)\right)\right)\right), sin2phi\right) \]

      7. *-lowering-*.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(alphay, alphay\right), \mathsf{+.f32}\left(u0, \mathsf{*.f32}\left(\left({u0}^{2}\right), \left(\frac{1}{2} + \frac{1}{3} \cdot u0\right)\right)\right)\right), sin2phi\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(alphay, alphay\right), \mathsf{+.f32}\left(u0, \mathsf{*.f32}\left(\left(u0 \cdot u0\right), \left(\frac{1}{2} + \frac{1}{3} \cdot u0\right)\right)\right)\right), sin2phi\right) \]

      9. *-lowering-*.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(alphay, alphay\right), \mathsf{+.f32}\left(u0, \mathsf{*.f32}\left(\mathsf{*.f32}\left(u0, u0\right), \left(\frac{1}{2} + \frac{1}{3} \cdot u0\right)\right)\right)\right), sin2phi\right) \]

      10. +-lowering-+.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(alphay, alphay\right), \mathsf{+.f32}\left(u0, \mathsf{*.f32}\left(\mathsf{*.f32}\left(u0, u0\right), \mathsf{+.f32}\left(\frac{1}{2}, \left(\frac{1}{3} \cdot u0\right)\right)\right)\right)\right), sin2phi\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(alphay, alphay\right), \mathsf{+.f32}\left(u0, \mathsf{*.f32}\left(\mathsf{*.f32}\left(u0, u0\right), \mathsf{+.f32}\left(\frac{1}{2}, \left(u0 \cdot \frac{1}{3}\right)\right)\right)\right)\right), sin2phi\right) \]

      12. *-lowering-*.f32 91.5%

          \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(alphay, alphay\right), \mathsf{+.f32}\left(u0, \mathsf{*.f32}\left(\mathsf{*.f32}\left(u0, u0\right), \mathsf{+.f32}\left(\frac{1}{2}, \mathsf{*.f32}\left(u0, \frac{1}{3}\right)\right)\right)\right)\right), sin2phi\right) \]

   10. Simplified 91.5%

       \[\leadsto \color{blue}{\frac{\left(alphay \cdot alphay\right) \cdot \left(u0 + \left(u0 \cdot u0\right) \cdot \left(0.5 + u0 \cdot 0.3333333333333333\right)\right)}{sin2phi}} \]

   11. Taylor expanded in u0 around 0

       \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(alphay, alphay\right), \mathsf{+.f32}\left(u0, \color{blue}{\left(\frac{1}{2} \cdot {u0}^{2}\right)}\right)\right), sin2phi\right) \]
   12. Step-by-step derivation

       1. *-lowering-*.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(alphay, alphay\right), \mathsf{+.f32}\left(u0, \mathsf{*.f32}\left(\frac{1}{2}, \left({u0}^{2}\right)\right)\right)\right), sin2phi\right) \]

       2. unpow2 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(alphay, alphay\right), \mathsf{+.f32}\left(u0, \mathsf{*.f32}\left(\frac{1}{2}, \left(u0 \cdot u0\right)\right)\right)\right), sin2phi\right) \]

       3. *-lowering-*.f32 87.7%

          \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(alphay, alphay\right), \mathsf{+.f32}\left(u0, \mathsf{*.f32}\left(\frac{1}{2}, \mathsf{*.f32}\left(u0, u0\right)\right)\right)\right), sin2phi\right) \]

   13. Simplified 87.7%

       \[\leadsto \frac{\left(alphay \cdot alphay\right) \cdot \left(u0 + \color{blue}{0.5 \cdot \left(u0 \cdot u0\right)}\right)}{sin2phi} \]
3. Recombined 2 regimes into one program.

4. Final simplification 81.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 2000:\\ \;\;\;\;\frac{u0}{\frac{\frac{cos2phi}{alphax}}{alphax} + \frac{sin2phi}{alphay \cdot alphay}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(alphay \cdot alphay\right) \cdot \left(u0 + 0.5 \cdot \left(u0 \cdot u0\right)\right)}{sin2phi}\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 81.6% accurate, 5.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{sin2phi}{alphay \cdot alphay}\\ \mathbf{if}\;t\_0 \leq 2000:\\ \;\;\;\;\frac{u0}{\frac{\frac{cos2phi}{alphax}}{alphax} + t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(alphay \cdot alphay\right) \cdot \left(u0 \cdot \left(1 + u0 \cdot 0.5\right)\right)}{sin2phi}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (let* ((t_0 (/ sin2phi (* alphay alphay))))
   (if (<= t_0 2000.0)
     (/ u0 (+ (/ (/ cos2phi alphax) alphax) t_0))
     (/ (* (* alphay alphay) (* u0 (+ 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 <= 2000.0f) {
		tmp = u0 / (((cos2phi / alphax) / alphax) + t_0);
	} else {
		tmp = ((alphay * alphay) * (u0 * (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 <= 2000.0e0) then
        tmp = u0 / (((cos2phi / alphax) / alphax) + t_0)
    else
        tmp = ((alphay * alphay) * (u0 * (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(2000.0))
		tmp = Float32(u0 / Float32(Float32(Float32(cos2phi / alphax) / alphax) + t_0));
	else
		tmp = Float32(Float32(Float32(alphay * alphay) * Float32(u0 * 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(2000.0))
		tmp = u0 / (((cos2phi / alphax) / alphax) + t_0);
	else
		tmp = ((alphay * alphay) * (u0 * (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)) < 2e3

   1. Initial program 54.1%

      \[\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
   2. Add Preprocessing

   3. Taylor expanded in u0 around 0

      \[\leadsto \color{blue}{\frac{u0}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}}} \]
   4. 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. +-commutative N/A

         \[\leadsto \mathsf{/.f32}\left(u0, \left(\frac{sin2phi}{{alphay}^{2}} + \color{blue}{\frac{cos2phi}{{alphax}^{2}}}\right)\right) \]

      3. +-lowering-+.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{+.f32}\left(\left(\frac{sin2phi}{{alphay}^{2}}\right), \color{blue}{\left(\frac{cos2phi}{{alphax}^{2}}\right)}\right)\right) \]

      4. /-lowering-/.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{/.f32}\left(sin2phi, \left({alphay}^{2}\right)\right), \left(\frac{\color{blue}{cos2phi}}{{alphax}^{2}}\right)\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{/.f32}\left(sin2phi, \left(alphay \cdot alphay\right)\right), \left(\frac{cos2phi}{{alphax}^{2}}\right)\right)\right) \]

      6. *-lowering-*.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{/.f32}\left(sin2phi, \mathsf{*.f32}\left(alphay, alphay\right)\right), \left(\frac{cos2phi}{{alphax}^{2}}\right)\right)\right) \]

      7. /-lowering-/.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{/.f32}\left(sin2phi, \mathsf{*.f32}\left(alphay, alphay\right)\right), \mathsf{/.f32}\left(cos2phi, \color{blue}{\left({alphax}^{2}\right)}\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{/.f32}\left(sin2phi, \mathsf{*.f32}\left(alphay, alphay\right)\right), \mathsf{/.f32}\left(cos2phi, \left(alphax \cdot \color{blue}{alphax}\right)\right)\right)\right) \]

      9. *-lowering-*.f32 75.3%

         \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{/.f32}\left(sin2phi, \mathsf{*.f32}\left(alphay, alphay\right)\right), \mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, \color{blue}{alphax}\right)\right)\right)\right) \]

   5. Simplified 75.3%

      \[\leadsto \color{blue}{\frac{u0}{\frac{sin2phi}{alphay \cdot alphay} + \frac{cos2phi}{alphax \cdot alphax}}} \]
   6. Step-by-step derivation

      1. associate-/r* N/A

         \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{/.f32}\left(sin2phi, \mathsf{*.f32}\left(alphay, alphay\right)\right), \left(\frac{\frac{cos2phi}{alphax}}{\color{blue}{alphax}}\right)\right)\right) \]

      2. /-lowering-/.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{/.f32}\left(sin2phi, \mathsf{*.f32}\left(alphay, alphay\right)\right), \mathsf{/.f32}\left(\left(\frac{cos2phi}{alphax}\right), \color{blue}{alphax}\right)\right)\right) \]

      3. /-lowering-/.f32 75.3%

         \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{/.f32}\left(sin2phi, \mathsf{*.f32}\left(alphay, alphay\right)\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(cos2phi, alphax\right), alphax\right)\right)\right) \]

   7. Applied egg-rr 75.3%

      \[\leadsto \frac{u0}{\frac{sin2phi}{alphay \cdot alphay} + \color{blue}{\frac{\frac{cos2phi}{alphax}}{alphax}}} \]

   if 2e3 < (/.f32 sin2phi (*.f32 alphay alphay))

   1. Initial program 65.1%

      \[\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
   2. Add Preprocessing

   3. Taylor expanded in u0 around 0

      \[\leadsto \color{blue}{u0 \cdot \left(u0 \cdot \left(\frac{1}{3} \cdot \frac{u0}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}} + \frac{1}{2} \cdot \frac{1}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}}\right) + \frac{1}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}}\right)} \]
   4. Step-by-step derivation

      1. *-lowering-*.f32 N/A

         \[\leadsto \mathsf{*.f32}\left(u0, \color{blue}{\left(u0 \cdot \left(\frac{1}{3} \cdot \frac{u0}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}} + \frac{1}{2} \cdot \frac{1}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}}\right) + \frac{1}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}}\right)}\right) \]

      2. +-commutative N/A

         \[\leadsto \mathsf{*.f32}\left(u0, \left(\frac{1}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}} + \color{blue}{u0 \cdot \left(\frac{1}{3} \cdot \frac{u0}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}} + \frac{1}{2} \cdot \frac{1}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}}\right)}\right)\right) \]

      3. +-lowering-+.f32 N/A

         \[\leadsto \mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\left(\frac{1}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}}\right), \color{blue}{\left(u0 \cdot \left(\frac{1}{3} \cdot \frac{u0}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}} + \frac{1}{2} \cdot \frac{1}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}}\right)\right)}\right)\right) \]

   5. Simplified 91.4%

      \[\leadsto \color{blue}{u0 \cdot \left(\frac{1}{\frac{sin2phi}{alphay \cdot alphay} + \frac{cos2phi}{alphax \cdot alphax}} + \frac{u0}{\frac{sin2phi}{alphay \cdot alphay} + \frac{cos2phi}{alphax \cdot alphax}} \cdot \left(0.5 + u0 \cdot 0.3333333333333333\right)\right)} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto u0 \cdot \left(\frac{u0}{\frac{sin2phi}{alphay \cdot alphay} + \frac{cos2phi}{alphax \cdot alphax}} \cdot \left(\frac{1}{2} + u0 \cdot \frac{1}{3}\right) + \color{blue}{\frac{1}{\frac{sin2phi}{alphay \cdot alphay} + \frac{cos2phi}{alphax \cdot alphax}}}\right) \]

      2. distribute-lft-in N/A

         \[\leadsto u0 \cdot \left(\frac{u0}{\frac{sin2phi}{alphay \cdot alphay} + \frac{cos2phi}{alphax \cdot alphax}} \cdot \left(\frac{1}{2} + u0 \cdot \frac{1}{3}\right)\right) + \color{blue}{u0 \cdot \frac{1}{\frac{sin2phi}{alphay \cdot alphay} + \frac{cos2phi}{alphax \cdot alphax}}} \]

      3. *-commutative N/A

         \[\leadsto \left(\frac{u0}{\frac{sin2phi}{alphay \cdot alphay} + \frac{cos2phi}{alphax \cdot alphax}} \cdot \left(\frac{1}{2} + u0 \cdot \frac{1}{3}\right)\right) \cdot u0 + \color{blue}{u0} \cdot \frac{1}{\frac{sin2phi}{alphay \cdot alphay} + \frac{cos2phi}{alphax \cdot alphax}} \]

      4. div-inv N/A

         \[\leadsto \left(\frac{u0}{\frac{sin2phi}{alphay \cdot alphay} + \frac{cos2phi}{alphax \cdot alphax}} \cdot \left(\frac{1}{2} + u0 \cdot \frac{1}{3}\right)\right) \cdot u0 + \frac{u0}{\color{blue}{\frac{sin2phi}{alphay \cdot alphay} + \frac{cos2phi}{alphax \cdot alphax}}} \]

      5. +-lowering-+.f32 N/A

         \[\leadsto \mathsf{+.f32}\left(\left(\left(\frac{u0}{\frac{sin2phi}{alphay \cdot alphay} + \frac{cos2phi}{alphax \cdot alphax}} \cdot \left(\frac{1}{2} + u0 \cdot \frac{1}{3}\right)\right) \cdot u0\right), \color{blue}{\left(\frac{u0}{\frac{sin2phi}{alphay \cdot alphay} + \frac{cos2phi}{alphax \cdot alphax}}\right)}\right) \]

   7. Applied egg-rr 91.2%

      \[\leadsto \color{blue}{\frac{u0 \cdot \left(u0 \cdot \left(0.5 + u0 \cdot 0.3333333333333333\right)\right)}{\frac{sin2phi}{alphay \cdot alphay} + \frac{cos2phi}{alphax \cdot alphax}} + \frac{u0}{\frac{sin2phi}{alphay \cdot alphay} + \frac{cos2phi}{alphax \cdot alphax}}} \]

   8. Taylor expanded in sin2phi around inf

      \[\leadsto \color{blue}{\frac{{alphay}^{2} \cdot u0 + {alphay}^{2} \cdot \left({u0}^{2} \cdot \left(\frac{1}{2} + \frac{1}{3} \cdot u0\right)\right)}{sin2phi}} \]
   9. Step-by-step derivation

      1. /-lowering-/.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(\left({alphay}^{2} \cdot u0 + {alphay}^{2} \cdot \left({u0}^{2} \cdot \left(\frac{1}{2} + \frac{1}{3} \cdot u0\right)\right)\right), \color{blue}{sin2phi}\right) \]

      2. distribute-lft-out N/A

         \[\leadsto \mathsf{/.f32}\left(\left({alphay}^{2} \cdot \left(u0 + {u0}^{2} \cdot \left(\frac{1}{2} + \frac{1}{3} \cdot u0\right)\right)\right), sin2phi\right) \]

      3. *-lowering-*.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(\left({alphay}^{2}\right), \left(u0 + {u0}^{2} \cdot \left(\frac{1}{2} + \frac{1}{3} \cdot u0\right)\right)\right), sin2phi\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(\left(alphay \cdot alphay\right), \left(u0 + {u0}^{2} \cdot \left(\frac{1}{2} + \frac{1}{3} \cdot u0\right)\right)\right), sin2phi\right) \]

      5. *-lowering-*.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(alphay, alphay\right), \left(u0 + {u0}^{2} \cdot \left(\frac{1}{2} + \frac{1}{3} \cdot u0\right)\right)\right), sin2phi\right) \]

      6. +-lowering-+.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(alphay, alphay\right), \mathsf{+.f32}\left(u0, \left({u0}^{2} \cdot \left(\frac{1}{2} + \frac{1}{3} \cdot u0\right)\right)\right)\right), sin2phi\right) \]

      7. *-lowering-*.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(alphay, alphay\right), \mathsf{+.f32}\left(u0, \mathsf{*.f32}\left(\left({u0}^{2}\right), \left(\frac{1}{2} + \frac{1}{3} \cdot u0\right)\right)\right)\right), sin2phi\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(alphay, alphay\right), \mathsf{+.f32}\left(u0, \mathsf{*.f32}\left(\left(u0 \cdot u0\right), \left(\frac{1}{2} + \frac{1}{3} \cdot u0\right)\right)\right)\right), sin2phi\right) \]

      9. *-lowering-*.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(alphay, alphay\right), \mathsf{+.f32}\left(u0, \mathsf{*.f32}\left(\mathsf{*.f32}\left(u0, u0\right), \left(\frac{1}{2} + \frac{1}{3} \cdot u0\right)\right)\right)\right), sin2phi\right) \]

      10. +-lowering-+.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(alphay, alphay\right), \mathsf{+.f32}\left(u0, \mathsf{*.f32}\left(\mathsf{*.f32}\left(u0, u0\right), \mathsf{+.f32}\left(\frac{1}{2}, \left(\frac{1}{3} \cdot u0\right)\right)\right)\right)\right), sin2phi\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(alphay, alphay\right), \mathsf{+.f32}\left(u0, \mathsf{*.f32}\left(\mathsf{*.f32}\left(u0, u0\right), \mathsf{+.f32}\left(\frac{1}{2}, \left(u0 \cdot \frac{1}{3}\right)\right)\right)\right)\right), sin2phi\right) \]

      12. *-lowering-*.f32 91.5%

          \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(alphay, alphay\right), \mathsf{+.f32}\left(u0, \mathsf{*.f32}\left(\mathsf{*.f32}\left(u0, u0\right), \mathsf{+.f32}\left(\frac{1}{2}, \mathsf{*.f32}\left(u0, \frac{1}{3}\right)\right)\right)\right)\right), sin2phi\right) \]

   10. Simplified 91.5%

       \[\leadsto \color{blue}{\frac{\left(alphay \cdot alphay\right) \cdot \left(u0 + \left(u0 \cdot u0\right) \cdot \left(0.5 + u0 \cdot 0.3333333333333333\right)\right)}{sin2phi}} \]

   11. Taylor expanded in u0 around 0

       \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(alphay, alphay\right), \color{blue}{\left(u0 \cdot \left(1 + \frac{1}{2} \cdot u0\right)\right)}\right), sin2phi\right) \]
   12. Step-by-step derivation

       1. *-lowering-*.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(alphay, alphay\right), \mathsf{*.f32}\left(u0, \left(1 + \frac{1}{2} \cdot u0\right)\right)\right), sin2phi\right) \]

       2. +-lowering-+.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(alphay, alphay\right), \mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(1, \left(\frac{1}{2} \cdot u0\right)\right)\right)\right), sin2phi\right) \]

       3. *-commutative N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(alphay, alphay\right), \mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(1, \left(u0 \cdot \frac{1}{2}\right)\right)\right)\right), sin2phi\right) \]

       4. *-lowering-*.f32 87.6%

          \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(alphay, alphay\right), \mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(u0, \frac{1}{2}\right)\right)\right)\right), sin2phi\right) \]

   13. Simplified 87.6%

       \[\leadsto \frac{\left(alphay \cdot alphay\right) \cdot \color{blue}{\left(u0 \cdot \left(1 + u0 \cdot 0.5\right)\right)}}{sin2phi} \]
3. Recombined 2 regimes into one program.

4. Final simplification 81.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 2000:\\ \;\;\;\;\frac{u0}{\frac{\frac{cos2phi}{alphax}}{alphax} + \frac{sin2phi}{alphay \cdot alphay}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(alphay \cdot alphay\right) \cdot \left(u0 \cdot \left(1 + u0 \cdot 0.5\right)\right)}{sin2phi}\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 81.3% accurate, 5.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{sin2phi}{alphay \cdot alphay}\\ \mathbf{if}\;t\_0 \leq 1900:\\ \;\;\;\;\frac{u0}{\frac{\frac{cos2phi}{alphax}}{alphax} + t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{u0 \cdot \left(1 + u0 \cdot 0.5\right)}{t\_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (let* ((t_0 (/ sin2phi (* alphay alphay))))
   (if (<= t_0 1900.0)
     (/ u0 (+ (/ (/ cos2phi alphax) alphax) t_0))
     (/ (* u0 (+ 1.0 (* u0 0.5))) 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 <= 1900.0f) {
		tmp = u0 / (((cos2phi / alphax) / alphax) + t_0);
	} else {
		tmp = (u0 * (1.0f + (u0 * 0.5f))) / 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 <= 1900.0e0) then
        tmp = u0 / (((cos2phi / alphax) / alphax) + t_0)
    else
        tmp = (u0 * (1.0e0 + (u0 * 0.5e0))) / 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(1900.0))
		tmp = Float32(u0 / Float32(Float32(Float32(cos2phi / alphax) / alphax) + t_0));
	else
		tmp = Float32(Float32(u0 * Float32(Float32(1.0) + Float32(u0 * Float32(0.5)))) / 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(1900.0))
		tmp = u0 / (((cos2phi / alphax) / alphax) + t_0);
	else
		tmp = (u0 * (single(1.0) + (u0 * single(0.5)))) / t_0;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 sin2phi (*.f32 alphay alphay)) < 1900

   1. Initial program 54.5%

      \[\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
   2. Add Preprocessing

   3. Taylor expanded in u0 around 0

      \[\leadsto \color{blue}{\frac{u0}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}}} \]
   4. 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. +-commutative N/A

         \[\leadsto \mathsf{/.f32}\left(u0, \left(\frac{sin2phi}{{alphay}^{2}} + \color{blue}{\frac{cos2phi}{{alphax}^{2}}}\right)\right) \]

      3. +-lowering-+.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{+.f32}\left(\left(\frac{sin2phi}{{alphay}^{2}}\right), \color{blue}{\left(\frac{cos2phi}{{alphax}^{2}}\right)}\right)\right) \]

      4. /-lowering-/.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{/.f32}\left(sin2phi, \left({alphay}^{2}\right)\right), \left(\frac{\color{blue}{cos2phi}}{{alphax}^{2}}\right)\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{/.f32}\left(sin2phi, \left(alphay \cdot alphay\right)\right), \left(\frac{cos2phi}{{alphax}^{2}}\right)\right)\right) \]

      6. *-lowering-*.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{/.f32}\left(sin2phi, \mathsf{*.f32}\left(alphay, alphay\right)\right), \left(\frac{cos2phi}{{alphax}^{2}}\right)\right)\right) \]

      7. /-lowering-/.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{/.f32}\left(sin2phi, \mathsf{*.f32}\left(alphay, alphay\right)\right), \mathsf{/.f32}\left(cos2phi, \color{blue}{\left({alphax}^{2}\right)}\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{/.f32}\left(sin2phi, \mathsf{*.f32}\left(alphay, alphay\right)\right), \mathsf{/.f32}\left(cos2phi, \left(alphax \cdot \color{blue}{alphax}\right)\right)\right)\right) \]

      9. *-lowering-*.f32 75.1%

         \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{/.f32}\left(sin2phi, \mathsf{*.f32}\left(alphay, alphay\right)\right), \mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, \color{blue}{alphax}\right)\right)\right)\right) \]

   5. Simplified 75.1%

      \[\leadsto \color{blue}{\frac{u0}{\frac{sin2phi}{alphay \cdot alphay} + \frac{cos2phi}{alphax \cdot alphax}}} \]
   6. Step-by-step derivation

      1. associate-/r* N/A

         \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{/.f32}\left(sin2phi, \mathsf{*.f32}\left(alphay, alphay\right)\right), \left(\frac{\frac{cos2phi}{alphax}}{\color{blue}{alphax}}\right)\right)\right) \]

      2. /-lowering-/.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{/.f32}\left(sin2phi, \mathsf{*.f32}\left(alphay, alphay\right)\right), \mathsf{/.f32}\left(\left(\frac{cos2phi}{alphax}\right), \color{blue}{alphax}\right)\right)\right) \]

      3. /-lowering-/.f32 75.1%

         \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{/.f32}\left(sin2phi, \mathsf{*.f32}\left(alphay, alphay\right)\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(cos2phi, alphax\right), alphax\right)\right)\right) \]

   7. Applied egg-rr 75.1%

      \[\leadsto \frac{u0}{\frac{sin2phi}{alphay \cdot alphay} + \color{blue}{\frac{\frac{cos2phi}{alphax}}{alphax}}} \]

   if 1900 < (/.f32 sin2phi (*.f32 alphay alphay))

   1. Initial program 64.7%

      \[\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
   2. Add Preprocessing

   3. Taylor expanded in cos2phi around 0

      \[\leadsto \mathsf{/.f32}\left(\mathsf{neg.f32}\left(\mathsf{log.f32}\left(\mathsf{\_.f32}\left(1, u0\right)\right)\right), \color{blue}{\left(\frac{sin2phi}{{alphay}^{2}}\right)}\right) \]
   4. Step-by-step derivation

      1. /-lowering-/.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{neg.f32}\left(\mathsf{log.f32}\left(\mathsf{\_.f32}\left(1, u0\right)\right)\right), \mathsf{/.f32}\left(sin2phi, \color{blue}{\left({alphay}^{2}\right)}\right)\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{neg.f32}\left(\mathsf{log.f32}\left(\mathsf{\_.f32}\left(1, u0\right)\right)\right), \mathsf{/.f32}\left(sin2phi, \left(alphay \cdot \color{blue}{alphay}\right)\right)\right) \]

      3. *-lowering-*.f32 64.6%

         \[\leadsto \mathsf{/.f32}\left(\mathsf{neg.f32}\left(\mathsf{log.f32}\left(\mathsf{\_.f32}\left(1, u0\right)\right)\right), \mathsf{/.f32}\left(sin2phi, \mathsf{*.f32}\left(alphay, \color{blue}{alphay}\right)\right)\right) \]

   5. Simplified 64.6%

      \[\leadsto \frac{-\log \left(1 - u0\right)}{\color{blue}{\frac{sin2phi}{alphay \cdot alphay}}} \]

   6. Taylor expanded in u0 around 0

      \[\leadsto \mathsf{/.f32}\left(\color{blue}{\left(u0 \cdot \left(1 + \frac{1}{2} \cdot u0\right)\right)}, \mathsf{/.f32}\left(sin2phi, \mathsf{*.f32}\left(alphay, alphay\right)\right)\right) \]
   7. Step-by-step derivation

      1. *-lowering-*.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(u0, \left(1 + \frac{1}{2} \cdot u0\right)\right), \mathsf{/.f32}\left(\color{blue}{sin2phi}, \mathsf{*.f32}\left(alphay, alphay\right)\right)\right) \]

      2. +-lowering-+.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(1, \left(\frac{1}{2} \cdot u0\right)\right)\right), \mathsf{/.f32}\left(sin2phi, \mathsf{*.f32}\left(alphay, alphay\right)\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(1, \left(u0 \cdot \frac{1}{2}\right)\right)\right), \mathsf{/.f32}\left(sin2phi, \mathsf{*.f32}\left(alphay, alphay\right)\right)\right) \]

      4. *-lowering-*.f32 87.1%

         \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(u0, \frac{1}{2}\right)\right)\right), \mathsf{/.f32}\left(sin2phi, \mathsf{*.f32}\left(alphay, alphay\right)\right)\right) \]

   8. Simplified 87.1%

      \[\leadsto \frac{\color{blue}{u0 \cdot \left(1 + u0 \cdot 0.5\right)}}{\frac{sin2phi}{alphay \cdot alphay}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 80.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 1900:\\ \;\;\;\;\frac{u0}{\frac{\frac{cos2phi}{alphax}}{alphax} + \frac{sin2phi}{alphay \cdot alphay}}\\ \mathbf{else}:\\ \;\;\;\;\frac{u0 \cdot \left(1 + u0 \cdot 0.5\right)}{\frac{sin2phi}{alphay \cdot alphay}}\\ \end{array} \]
5. Add Preprocessing

Alternative 21: 87.7% accurate, 5.5× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\frac{sin2phi}{alphay \cdot alphay} + \frac{cos2phi}{alphax \cdot alphax}} \cdot \left(u0 + 0.5 \cdot \left(u0 \cdot u0\right)\right) \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (*
  (/ 1.0 (+ (/ sin2phi (* alphay alphay)) (/ cos2phi (* alphax alphax))))
  (+ u0 (* 0.5 (* u0 u0)))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	return (1.0f / ((sin2phi / (alphay * alphay)) + (cos2phi / (alphax * alphax)))) * (u0 + (0.5f * (u0 * u0)));
}

Fortran

real(4) function code(alphax, alphay, u0, cos2phi, sin2phi)
    real(4), intent (in) :: alphax
    real(4), intent (in) :: alphay
    real(4), intent (in) :: u0
    real(4), intent (in) :: cos2phi
    real(4), intent (in) :: sin2phi
    code = (1.0e0 / ((sin2phi / (alphay * alphay)) + (cos2phi / (alphax * alphax)))) * (u0 + (0.5e0 * (u0 * u0)))
end function

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	return Float32(Float32(Float32(1.0) / Float32(Float32(sin2phi / Float32(alphay * alphay)) + Float32(cos2phi / Float32(alphax * alphax)))) * Float32(u0 + Float32(Float32(0.5) * Float32(u0 * u0))))
end

MATLAB

function tmp = code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = (single(1.0) / ((sin2phi / (alphay * alphay)) + (cos2phi / (alphax * alphax)))) * (u0 + (single(0.5) * (u0 * u0)));
end

Derivation

1. Initial program 59.4%

   \[\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
2. Add Preprocessing

3. Taylor expanded in u0 around 0

   \[\leadsto \color{blue}{u0 \cdot \left(u0 \cdot \left(\frac{1}{3} \cdot \frac{u0}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}} + \frac{1}{2} \cdot \frac{1}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}}\right) + \frac{1}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}}\right)} \]
4. Step-by-step derivation

   1. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{*.f32}\left(u0, \color{blue}{\left(u0 \cdot \left(\frac{1}{3} \cdot \frac{u0}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}} + \frac{1}{2} \cdot \frac{1}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}}\right) + \frac{1}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}}\right)}\right) \]

   2. +-commutative N/A

      \[\leadsto \mathsf{*.f32}\left(u0, \left(\frac{1}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}} + \color{blue}{u0 \cdot \left(\frac{1}{3} \cdot \frac{u0}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}} + \frac{1}{2} \cdot \frac{1}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}}\right)}\right)\right) \]

   3. +-lowering-+.f32 N/A

      \[\leadsto \mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\left(\frac{1}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}}\right), \color{blue}{\left(u0 \cdot \left(\frac{1}{3} \cdot \frac{u0}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}} + \frac{1}{2} \cdot \frac{1}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}}\right)\right)}\right)\right) \]

5. Simplified 90.9%

   \[\leadsto \color{blue}{u0 \cdot \left(\frac{1}{\frac{sin2phi}{alphay \cdot alphay} + \frac{cos2phi}{alphax \cdot alphax}} + \frac{u0}{\frac{sin2phi}{alphay \cdot alphay} + \frac{cos2phi}{alphax \cdot alphax}} \cdot \left(0.5 + u0 \cdot 0.3333333333333333\right)\right)} \]
6. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto u0 \cdot \left(\frac{u0}{\frac{sin2phi}{alphay \cdot alphay} + \frac{cos2phi}{alphax \cdot alphax}} \cdot \left(\frac{1}{2} + u0 \cdot \frac{1}{3}\right) + \color{blue}{\frac{1}{\frac{sin2phi}{alphay \cdot alphay} + \frac{cos2phi}{alphax \cdot alphax}}}\right) \]

   2. distribute-lft-in N/A

      \[\leadsto u0 \cdot \left(\frac{u0}{\frac{sin2phi}{alphay \cdot alphay} + \frac{cos2phi}{alphax \cdot alphax}} \cdot \left(\frac{1}{2} + u0 \cdot \frac{1}{3}\right)\right) + \color{blue}{u0 \cdot \frac{1}{\frac{sin2phi}{alphay \cdot alphay} + \frac{cos2phi}{alphax \cdot alphax}}} \]

   3. *-commutative N/A

      \[\leadsto \left(\frac{u0}{\frac{sin2phi}{alphay \cdot alphay} + \frac{cos2phi}{alphax \cdot alphax}} \cdot \left(\frac{1}{2} + u0 \cdot \frac{1}{3}\right)\right) \cdot u0 + \color{blue}{u0} \cdot \frac{1}{\frac{sin2phi}{alphay \cdot alphay} + \frac{cos2phi}{alphax \cdot alphax}} \]

   4. div-inv N/A

      \[\leadsto \left(\frac{u0}{\frac{sin2phi}{alphay \cdot alphay} + \frac{cos2phi}{alphax \cdot alphax}} \cdot \left(\frac{1}{2} + u0 \cdot \frac{1}{3}\right)\right) \cdot u0 + \frac{u0}{\color{blue}{\frac{sin2phi}{alphay \cdot alphay} + \frac{cos2phi}{alphax \cdot alphax}}} \]

   5. +-lowering-+.f32 N/A

      \[\leadsto \mathsf{+.f32}\left(\left(\left(\frac{u0}{\frac{sin2phi}{alphay \cdot alphay} + \frac{cos2phi}{alphax \cdot alphax}} \cdot \left(\frac{1}{2} + u0 \cdot \frac{1}{3}\right)\right) \cdot u0\right), \color{blue}{\left(\frac{u0}{\frac{sin2phi}{alphay \cdot alphay} + \frac{cos2phi}{alphax \cdot alphax}}\right)}\right) \]

7. Applied egg-rr 90.9%

   \[\leadsto \color{blue}{\frac{u0 \cdot \left(u0 \cdot \left(0.5 + u0 \cdot 0.3333333333333333\right)\right)}{\frac{sin2phi}{alphay \cdot alphay} + \frac{cos2phi}{alphax \cdot alphax}} + \frac{u0}{\frac{sin2phi}{alphay \cdot alphay} + \frac{cos2phi}{alphax \cdot alphax}}} \]
8. Step-by-step derivation

   1. div-inv N/A

      \[\leadsto \left(u0 \cdot \left(u0 \cdot \left(\frac{1}{2} + u0 \cdot \frac{1}{3}\right)\right)\right) \cdot \frac{1}{\frac{sin2phi}{alphay \cdot alphay} + \frac{cos2phi}{alphax \cdot alphax}} + \frac{\color{blue}{u0}}{\frac{sin2phi}{alphay \cdot alphay} + \frac{cos2phi}{alphax \cdot alphax}} \]

   2. div-inv N/A

      \[\leadsto \left(u0 \cdot \left(u0 \cdot \left(\frac{1}{2} + u0 \cdot \frac{1}{3}\right)\right)\right) \cdot \frac{1}{\frac{sin2phi}{alphay \cdot alphay} + \frac{cos2phi}{alphax \cdot alphax}} + u0 \cdot \color{blue}{\frac{1}{\frac{sin2phi}{alphay \cdot alphay} + \frac{cos2phi}{alphax \cdot alphax}}} \]

   3. distribute-rgt-out N/A

      \[\leadsto \frac{1}{\frac{sin2phi}{alphay \cdot alphay} + \frac{cos2phi}{alphax \cdot alphax}} \cdot \color{blue}{\left(u0 \cdot \left(u0 \cdot \left(\frac{1}{2} + u0 \cdot \frac{1}{3}\right)\right) + u0\right)} \]

   4. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{*.f32}\left(\left(\frac{1}{\frac{sin2phi}{alphay \cdot alphay} + \frac{cos2phi}{alphax \cdot alphax}}\right), \color{blue}{\left(u0 \cdot \left(u0 \cdot \left(\frac{1}{2} + u0 \cdot \frac{1}{3}\right)\right) + u0\right)}\right) \]

   5. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{/.f32}\left(1, \left(\frac{sin2phi}{alphay \cdot alphay} + \frac{cos2phi}{alphax \cdot alphax}\right)\right), \left(\color{blue}{u0 \cdot \left(u0 \cdot \left(\frac{1}{2} + u0 \cdot \frac{1}{3}\right)\right)} + u0\right)\right) \]

   6. +-lowering-+.f32 N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{/.f32}\left(1, \mathsf{+.f32}\left(\left(\frac{sin2phi}{alphay \cdot alphay}\right), \left(\frac{cos2phi}{alphax \cdot alphax}\right)\right)\right), \left(u0 \cdot \color{blue}{\left(u0 \cdot \left(\frac{1}{2} + u0 \cdot \frac{1}{3}\right)\right)} + u0\right)\right) \]

   7. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{/.f32}\left(1, \mathsf{+.f32}\left(\mathsf{/.f32}\left(sin2phi, \left(alphay \cdot alphay\right)\right), \left(\frac{cos2phi}{alphax \cdot alphax}\right)\right)\right), \left(u0 \cdot \left(\color{blue}{u0} \cdot \left(\frac{1}{2} + u0 \cdot \frac{1}{3}\right)\right) + u0\right)\right) \]

   8. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{/.f32}\left(1, \mathsf{+.f32}\left(\mathsf{/.f32}\left(sin2phi, \mathsf{*.f32}\left(alphay, alphay\right)\right), \left(\frac{cos2phi}{alphax \cdot alphax}\right)\right)\right), \left(u0 \cdot \left(u0 \cdot \left(\frac{1}{2} + u0 \cdot \frac{1}{3}\right)\right) + u0\right)\right) \]

   9. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{/.f32}\left(1, \mathsf{+.f32}\left(\mathsf{/.f32}\left(sin2phi, \mathsf{*.f32}\left(alphay, alphay\right)\right), \mathsf{/.f32}\left(cos2phi, \left(alphax \cdot alphax\right)\right)\right)\right), \left(u0 \cdot \left(u0 \cdot \color{blue}{\left(\frac{1}{2} + u0 \cdot \frac{1}{3}\right)}\right) + u0\right)\right) \]

   10. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{/.f32}\left(1, \mathsf{+.f32}\left(\mathsf{/.f32}\left(sin2phi, \mathsf{*.f32}\left(alphay, alphay\right)\right), \mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right)\right)\right), \left(u0 \cdot \left(u0 \cdot \left(\frac{1}{2} + \color{blue}{u0 \cdot \frac{1}{3}}\right)\right) + u0\right)\right) \]

   11. +-lowering-+.f32 N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{/.f32}\left(1, \mathsf{+.f32}\left(\mathsf{/.f32}\left(sin2phi, \mathsf{*.f32}\left(alphay, alphay\right)\right), \mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right)\right)\right), \mathsf{+.f32}\left(\left(u0 \cdot \left(u0 \cdot \left(\frac{1}{2} + u0 \cdot \frac{1}{3}\right)\right)\right), \color{blue}{u0}\right)\right) \]

9. Applied egg-rr 90.9%

   \[\leadsto \color{blue}{\frac{1}{\frac{sin2phi}{alphay \cdot alphay} + \frac{cos2phi}{alphax \cdot alphax}} \cdot \left(\left(0.5 + u0 \cdot 0.3333333333333333\right) \cdot \left(u0 \cdot u0\right) + u0\right)} \]

10. Taylor expanded in u0 around 0

    \[\leadsto \mathsf{*.f32}\left(\mathsf{/.f32}\left(1, \mathsf{+.f32}\left(\mathsf{/.f32}\left(sin2phi, \mathsf{*.f32}\left(alphay, alphay\right)\right), \mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right)\right)\right), \mathsf{+.f32}\left(\color{blue}{\left(\frac{1}{2} \cdot {u0}^{2}\right)}, u0\right)\right) \]
11. Step-by-step derivation

    1. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{/.f32}\left(1, \mathsf{+.f32}\left(\mathsf{/.f32}\left(sin2phi, \mathsf{*.f32}\left(alphay, alphay\right)\right), \mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right)\right)\right), \mathsf{+.f32}\left(\mathsf{*.f32}\left(\frac{1}{2}, \left({u0}^{2}\right)\right), u0\right)\right) \]

    2. unpow2 N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{/.f32}\left(1, \mathsf{+.f32}\left(\mathsf{/.f32}\left(sin2phi, \mathsf{*.f32}\left(alphay, alphay\right)\right), \mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right)\right)\right), \mathsf{+.f32}\left(\mathsf{*.f32}\left(\frac{1}{2}, \left(u0 \cdot u0\right)\right), u0\right)\right) \]

    3. *-lowering-*.f32 87.3%

       \[\leadsto \mathsf{*.f32}\left(\mathsf{/.f32}\left(1, \mathsf{+.f32}\left(\mathsf{/.f32}\left(sin2phi, \mathsf{*.f32}\left(alphay, alphay\right)\right), \mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right)\right)\right), \mathsf{+.f32}\left(\mathsf{*.f32}\left(\frac{1}{2}, \mathsf{*.f32}\left(u0, u0\right)\right), u0\right)\right) \]

12. Simplified 87.3%

    \[\leadsto \frac{1}{\frac{sin2phi}{alphay \cdot alphay} + \frac{cos2phi}{alphax \cdot alphax}} \cdot \left(\color{blue}{0.5 \cdot \left(u0 \cdot u0\right)} + u0\right) \]

13. Final simplification 87.3%

    \[\leadsto \frac{1}{\frac{sin2phi}{alphay \cdot alphay} + \frac{cos2phi}{alphax \cdot alphax}} \cdot \left(u0 + 0.5 \cdot \left(u0 \cdot u0\right)\right) \]
14. Add Preprocessing

Alternative 22: 87.7% accurate, 6.1× speedup

Math

\[\begin{array}{l} \\ \left(1 + u0 \cdot 0.5\right) \cdot \frac{u0}{\frac{sin2phi}{alphay \cdot alphay} + \frac{cos2phi}{alphax \cdot alphax}} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (*
  (+ 1.0 (* u0 0.5))
  (/ u0 (+ (/ sin2phi (* alphay alphay)) (/ cos2phi (* alphax alphax))))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	return (1.0f + (u0 * 0.5f)) * (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 = (1.0e0 + (u0 * 0.5e0)) * (u0 / ((sin2phi / (alphay * alphay)) + (cos2phi / (alphax * alphax))))
end function

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	return Float32(Float32(Float32(1.0) + Float32(u0 * Float32(0.5))) * Float32(u0 / Float32(Float32(sin2phi / Float32(alphay * alphay)) + Float32(cos2phi / Float32(alphax * alphax)))))
end

MATLAB

function tmp = code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = (single(1.0) + (u0 * single(0.5))) * (u0 / ((sin2phi / (alphay * alphay)) + (cos2phi / (alphax * alphax))));
end

Derivation

1. Initial program 59.4%

   \[\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
2. Add Preprocessing

3. Taylor expanded in u0 around 0

   \[\leadsto \color{blue}{u0 \cdot \left(\frac{1}{2} \cdot \frac{u0}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}} + \frac{1}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}}\right)} \]
4. Step-by-step derivation

   1. distribute-lft-in N/A

      \[\leadsto u0 \cdot \left(\frac{1}{2} \cdot \frac{u0}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}}\right) + \color{blue}{u0 \cdot \frac{1}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}}} \]

   2. associate-*r* N/A

      \[\leadsto \left(u0 \cdot \frac{1}{2}\right) \cdot \frac{u0}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}} + \color{blue}{u0} \cdot \frac{1}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}} \]

   3. *-commutative N/A

      \[\leadsto \left(\frac{1}{2} \cdot u0\right) \cdot \frac{u0}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}} + u0 \cdot \frac{1}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}} \]

   4. associate-*r/ N/A

      \[\leadsto \left(\frac{1}{2} \cdot u0\right) \cdot \frac{u0}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}} + \frac{u0 \cdot 1}{\color{blue}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}}} \]

   5. *-rgt-identity N/A

      \[\leadsto \left(\frac{1}{2} \cdot u0\right) \cdot \frac{u0}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}} + \frac{u0}{\color{blue}{\frac{cos2phi}{{alphax}^{2}}} + \frac{sin2phi}{{alphay}^{2}}} \]

   6. distribute-lft1-in N/A

      \[\leadsto \left(\frac{1}{2} \cdot u0 + 1\right) \cdot \color{blue}{\frac{u0}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}}} \]

   7. +-commutative N/A

      \[\leadsto \left(1 + \frac{1}{2} \cdot u0\right) \cdot \frac{\color{blue}{u0}}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}} \]

   8. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{*.f32}\left(\left(1 + \frac{1}{2} \cdot u0\right), \color{blue}{\left(\frac{u0}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}}\right)}\right) \]

   9. +-lowering-+.f32 N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{+.f32}\left(1, \left(\frac{1}{2} \cdot u0\right)\right), \left(\frac{\color{blue}{u0}}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}}\right)\right) \]

   10. *-commutative N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{+.f32}\left(1, \left(u0 \cdot \frac{1}{2}\right)\right), \left(\frac{u0}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}}\right)\right) \]

   11. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{+.f32}\left(1, \mathsf{*.f32}\left(u0, \frac{1}{2}\right)\right), \left(\frac{u0}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}}\right)\right) \]

   12. /-lowering-/.f32 N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{+.f32}\left(1, \mathsf{*.f32}\left(u0, \frac{1}{2}\right)\right), \mathsf{/.f32}\left(u0, \color{blue}{\left(\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}\right)}\right)\right) \]

   13. +-commutative N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{+.f32}\left(1, \mathsf{*.f32}\left(u0, \frac{1}{2}\right)\right), \mathsf{/.f32}\left(u0, \left(\frac{sin2phi}{{alphay}^{2}} + \color{blue}{\frac{cos2phi}{{alphax}^{2}}}\right)\right)\right) \]

   14. +-lowering-+.f32 N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{+.f32}\left(1, \mathsf{*.f32}\left(u0, \frac{1}{2}\right)\right), \mathsf{/.f32}\left(u0, \mathsf{+.f32}\left(\left(\frac{sin2phi}{{alphay}^{2}}\right), \color{blue}{\left(\frac{cos2phi}{{alphax}^{2}}\right)}\right)\right)\right) \]

5. Simplified 87.3%

   \[\leadsto \color{blue}{\left(1 + u0 \cdot 0.5\right) \cdot \frac{u0}{\frac{sin2phi}{alphay \cdot alphay} + \frac{cos2phi}{alphax \cdot alphax}}} \]
6. Add Preprocessing

Alternative 23: 67.0% accurate, 7.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 9.9999998245167 \cdot 10^{-14}:\\ \;\;\;\;u0 \cdot \frac{alphax}{\frac{cos2phi}{alphax}}\\ \mathbf{else}:\\ \;\;\;\;\frac{u0 \cdot \left(alphay \cdot alphay\right)}{sin2phi}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (if (<= (/ sin2phi (* alphay alphay)) 9.9999998245167e-14)
   (* u0 (/ alphax (/ cos2phi alphax)))
   (/ (* u0 (* alphay alphay)) sin2phi)))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float tmp;
	if ((sin2phi / (alphay * alphay)) <= 9.9999998245167e-14f) {
		tmp = u0 * (alphax / (cos2phi / alphax));
	} else {
		tmp = (u0 * (alphay * alphay)) / sin2phi;
	}
	return tmp;
}

Fortran

real(4) function code(alphax, alphay, u0, cos2phi, sin2phi)
    real(4), intent (in) :: alphax
    real(4), intent (in) :: alphay
    real(4), intent (in) :: u0
    real(4), intent (in) :: cos2phi
    real(4), intent (in) :: sin2phi
    real(4) :: tmp
    if ((sin2phi / (alphay * alphay)) <= 9.9999998245167e-14) then
        tmp = u0 * (alphax / (cos2phi / alphax))
    else
        tmp = (u0 * (alphay * alphay)) / sin2phi
    end if
    code = tmp
end function

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = Float32(0.0)
	if (Float32(sin2phi / Float32(alphay * alphay)) <= Float32(9.9999998245167e-14))
		tmp = Float32(u0 * Float32(alphax / Float32(cos2phi / alphax)));
	else
		tmp = Float32(Float32(u0 * Float32(alphay * alphay)) / sin2phi);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = single(0.0);
	if ((sin2phi / (alphay * alphay)) <= single(9.9999998245167e-14))
		tmp = u0 * (alphax / (cos2phi / alphax));
	else
		tmp = (u0 * (alphay * alphay)) / sin2phi;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 sin2phi (*.f32 alphay alphay)) < 9.99999982e-14

   1. Initial program 54.0%

      \[\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
   2. Add Preprocessing

   3. Taylor expanded in u0 around 0

      \[\leadsto \color{blue}{\frac{u0}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}}} \]
   4. 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. +-commutative N/A

         \[\leadsto \mathsf{/.f32}\left(u0, \left(\frac{sin2phi}{{alphay}^{2}} + \color{blue}{\frac{cos2phi}{{alphax}^{2}}}\right)\right) \]

      3. +-lowering-+.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{+.f32}\left(\left(\frac{sin2phi}{{alphay}^{2}}\right), \color{blue}{\left(\frac{cos2phi}{{alphax}^{2}}\right)}\right)\right) \]

      4. /-lowering-/.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{/.f32}\left(sin2phi, \left({alphay}^{2}\right)\right), \left(\frac{\color{blue}{cos2phi}}{{alphax}^{2}}\right)\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{/.f32}\left(sin2phi, \left(alphay \cdot alphay\right)\right), \left(\frac{cos2phi}{{alphax}^{2}}\right)\right)\right) \]

      6. *-lowering-*.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{/.f32}\left(sin2phi, \mathsf{*.f32}\left(alphay, alphay\right)\right), \left(\frac{cos2phi}{{alphax}^{2}}\right)\right)\right) \]

      7. /-lowering-/.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{/.f32}\left(sin2phi, \mathsf{*.f32}\left(alphay, alphay\right)\right), \mathsf{/.f32}\left(cos2phi, \color{blue}{\left({alphax}^{2}\right)}\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{/.f32}\left(sin2phi, \mathsf{*.f32}\left(alphay, alphay\right)\right), \mathsf{/.f32}\left(cos2phi, \left(alphax \cdot \color{blue}{alphax}\right)\right)\right)\right) \]

      9. *-lowering-*.f32 74.7%

         \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{/.f32}\left(sin2phi, \mathsf{*.f32}\left(alphay, alphay\right)\right), \mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, \color{blue}{alphax}\right)\right)\right)\right) \]

   5. Simplified 74.7%

      \[\leadsto \color{blue}{\frac{u0}{\frac{sin2phi}{alphay \cdot alphay} + \frac{cos2phi}{alphax \cdot alphax}}} \]

   6. Taylor expanded in sin2phi around 0

      \[\leadsto \mathsf{/.f32}\left(u0, \color{blue}{\left(\frac{cos2phi}{{alphax}^{2}}\right)}\right) \]
   7. 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 58.1%

         \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, \color{blue}{alphax}\right)\right)\right) \]

   8. Simplified 58.1%

      \[\leadsto \frac{u0}{\color{blue}{\frac{cos2phi}{alphax \cdot alphax}}} \]
   9. Step-by-step derivation

      1. clear-num N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{\frac{cos2phi}{alphax \cdot alphax}}{u0}}} \]

      2. associate-/r/ N/A

         \[\leadsto \frac{1}{\frac{cos2phi}{alphax \cdot alphax}} \cdot \color{blue}{u0} \]

      3. clear-num N/A

         \[\leadsto \frac{alphax \cdot alphax}{cos2phi} \cdot u0 \]

      4. *-lowering-*.f32 N/A

         \[\leadsto \mathsf{*.f32}\left(\left(\frac{alphax \cdot alphax}{cos2phi}\right), \color{blue}{u0}\right) \]

      5. associate-*l/ N/A

         \[\leadsto \mathsf{*.f32}\left(\left(\frac{alphax}{cos2phi} \cdot alphax\right), u0\right) \]

      6. associate-/r/ N/A

         \[\leadsto \mathsf{*.f32}\left(\left(\frac{alphax}{\frac{cos2phi}{alphax}}\right), u0\right) \]

      7. /-lowering-/.f32 N/A

         \[\leadsto \mathsf{*.f32}\left(\mathsf{/.f32}\left(alphax, \left(\frac{cos2phi}{alphax}\right)\right), u0\right) \]

      8. /-lowering-/.f32 58.3%

         \[\leadsto \mathsf{*.f32}\left(\mathsf{/.f32}\left(alphax, \mathsf{/.f32}\left(cos2phi, alphax\right)\right), u0\right) \]

   10. Applied egg-rr 58.3%

       \[\leadsto \color{blue}{\frac{alphax}{\frac{cos2phi}{alphax}} \cdot u0} \]

   if 9.99999982e-14 < (/.f32 sin2phi (*.f32 alphay alphay))

   1. Initial program 61.9%

      \[\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
   2. Add Preprocessing

   3. Taylor expanded in u0 around 0

      \[\leadsto \color{blue}{\frac{u0}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}}} \]
   4. 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. +-commutative N/A

         \[\leadsto \mathsf{/.f32}\left(u0, \left(\frac{sin2phi}{{alphay}^{2}} + \color{blue}{\frac{cos2phi}{{alphax}^{2}}}\right)\right) \]

      3. +-lowering-+.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{+.f32}\left(\left(\frac{sin2phi}{{alphay}^{2}}\right), \color{blue}{\left(\frac{cos2phi}{{alphax}^{2}}\right)}\right)\right) \]

      4. /-lowering-/.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{/.f32}\left(sin2phi, \left({alphay}^{2}\right)\right), \left(\frac{\color{blue}{cos2phi}}{{alphax}^{2}}\right)\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{/.f32}\left(sin2phi, \left(alphay \cdot alphay\right)\right), \left(\frac{cos2phi}{{alphax}^{2}}\right)\right)\right) \]

      6. *-lowering-*.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{/.f32}\left(sin2phi, \mathsf{*.f32}\left(alphay, alphay\right)\right), \left(\frac{cos2phi}{{alphax}^{2}}\right)\right)\right) \]

      7. /-lowering-/.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{/.f32}\left(sin2phi, \mathsf{*.f32}\left(alphay, alphay\right)\right), \mathsf{/.f32}\left(cos2phi, \color{blue}{\left({alphax}^{2}\right)}\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{/.f32}\left(sin2phi, \mathsf{*.f32}\left(alphay, alphay\right)\right), \mathsf{/.f32}\left(cos2phi, \left(alphax \cdot \color{blue}{alphax}\right)\right)\right)\right) \]

      9. *-lowering-*.f32 75.6%

         \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{/.f32}\left(sin2phi, \mathsf{*.f32}\left(alphay, alphay\right)\right), \mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, \color{blue}{alphax}\right)\right)\right)\right) \]

   5. Simplified 75.6%

      \[\leadsto \color{blue}{\frac{u0}{\frac{sin2phi}{alphay \cdot alphay} + \frac{cos2phi}{alphax \cdot alphax}}} \]

   6. Taylor expanded in sin2phi around inf

      \[\leadsto \color{blue}{\frac{{alphay}^{2} \cdot u0}{sin2phi}} \]
   7. 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 69.5%

         \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(u0, \mathsf{*.f32}\left(alphay, alphay\right)\right), sin2phi\right) \]

   8. Simplified 69.5%

      \[\leadsto \color{blue}{\frac{u0 \cdot \left(alphay \cdot alphay\right)}{sin2phi}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 65.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 9.9999998245167 \cdot 10^{-14}:\\ \;\;\;\;u0 \cdot \frac{alphax}{\frac{cos2phi}{alphax}}\\ \mathbf{else}:\\ \;\;\;\;\frac{u0 \cdot \left(alphay \cdot alphay\right)}{sin2phi}\\ \end{array} \]
5. Add Preprocessing

Alternative 24: 66.8% accurate, 7.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{sin2phi}{alphay \cdot alphay}\\ \mathbf{if}\;t\_0 \leq 9.9999998245167 \cdot 10^{-14}:\\ \;\;\;\;u0 \cdot \frac{alphax}{\frac{cos2phi}{alphax}}\\ \mathbf{else}:\\ \;\;\;\;\frac{u0}{t\_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (let* ((t_0 (/ sin2phi (* alphay alphay))))
   (if (<= t_0 9.9999998245167e-14)
     (* u0 (/ alphax (/ cos2phi alphax)))
     (/ 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 <= 9.9999998245167e-14f) {
		tmp = u0 * (alphax / (cos2phi / alphax));
	} 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 <= 9.9999998245167e-14) then
        tmp = u0 * (alphax / (cos2phi / alphax))
    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(9.9999998245167e-14))
		tmp = Float32(u0 * Float32(alphax / Float32(cos2phi / alphax)));
	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(9.9999998245167e-14))
		tmp = u0 * (alphax / (cos2phi / alphax));
	else
		tmp = u0 / t_0;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 sin2phi (*.f32 alphay alphay)) < 9.99999982e-14

   1. Initial program 54.0%

      \[\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
   2. Add Preprocessing

   3. Taylor expanded in u0 around 0

      \[\leadsto \color{blue}{\frac{u0}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}}} \]
   4. 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. +-commutative N/A

         \[\leadsto \mathsf{/.f32}\left(u0, \left(\frac{sin2phi}{{alphay}^{2}} + \color{blue}{\frac{cos2phi}{{alphax}^{2}}}\right)\right) \]

      3. +-lowering-+.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{+.f32}\left(\left(\frac{sin2phi}{{alphay}^{2}}\right), \color{blue}{\left(\frac{cos2phi}{{alphax}^{2}}\right)}\right)\right) \]

      4. /-lowering-/.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{/.f32}\left(sin2phi, \left({alphay}^{2}\right)\right), \left(\frac{\color{blue}{cos2phi}}{{alphax}^{2}}\right)\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{/.f32}\left(sin2phi, \left(alphay \cdot alphay\right)\right), \left(\frac{cos2phi}{{alphax}^{2}}\right)\right)\right) \]

      6. *-lowering-*.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{/.f32}\left(sin2phi, \mathsf{*.f32}\left(alphay, alphay\right)\right), \left(\frac{cos2phi}{{alphax}^{2}}\right)\right)\right) \]

      7. /-lowering-/.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{/.f32}\left(sin2phi, \mathsf{*.f32}\left(alphay, alphay\right)\right), \mathsf{/.f32}\left(cos2phi, \color{blue}{\left({alphax}^{2}\right)}\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{/.f32}\left(sin2phi, \mathsf{*.f32}\left(alphay, alphay\right)\right), \mathsf{/.f32}\left(cos2phi, \left(alphax \cdot \color{blue}{alphax}\right)\right)\right)\right) \]

      9. *-lowering-*.f32 74.7%

         \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{/.f32}\left(sin2phi, \mathsf{*.f32}\left(alphay, alphay\right)\right), \mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, \color{blue}{alphax}\right)\right)\right)\right) \]

   5. Simplified 74.7%

      \[\leadsto \color{blue}{\frac{u0}{\frac{sin2phi}{alphay \cdot alphay} + \frac{cos2phi}{alphax \cdot alphax}}} \]

   6. Taylor expanded in sin2phi around 0

      \[\leadsto \mathsf{/.f32}\left(u0, \color{blue}{\left(\frac{cos2phi}{{alphax}^{2}}\right)}\right) \]
   7. 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 58.1%

         \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, \color{blue}{alphax}\right)\right)\right) \]

   8. Simplified 58.1%

      \[\leadsto \frac{u0}{\color{blue}{\frac{cos2phi}{alphax \cdot alphax}}} \]
   9. Step-by-step derivation

      1. clear-num N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{\frac{cos2phi}{alphax \cdot alphax}}{u0}}} \]

      2. associate-/r/ N/A

         \[\leadsto \frac{1}{\frac{cos2phi}{alphax \cdot alphax}} \cdot \color{blue}{u0} \]

      3. clear-num N/A

         \[\leadsto \frac{alphax \cdot alphax}{cos2phi} \cdot u0 \]

      4. *-lowering-*.f32 N/A

         \[\leadsto \mathsf{*.f32}\left(\left(\frac{alphax \cdot alphax}{cos2phi}\right), \color{blue}{u0}\right) \]

      5. associate-*l/ N/A

         \[\leadsto \mathsf{*.f32}\left(\left(\frac{alphax}{cos2phi} \cdot alphax\right), u0\right) \]

      6. associate-/r/ N/A

         \[\leadsto \mathsf{*.f32}\left(\left(\frac{alphax}{\frac{cos2phi}{alphax}}\right), u0\right) \]

      7. /-lowering-/.f32 N/A

         \[\leadsto \mathsf{*.f32}\left(\mathsf{/.f32}\left(alphax, \left(\frac{cos2phi}{alphax}\right)\right), u0\right) \]

      8. /-lowering-/.f32 58.3%

         \[\leadsto \mathsf{*.f32}\left(\mathsf{/.f32}\left(alphax, \mathsf{/.f32}\left(cos2phi, alphax\right)\right), u0\right) \]

   10. Applied egg-rr 58.3%

       \[\leadsto \color{blue}{\frac{alphax}{\frac{cos2phi}{alphax}} \cdot u0} \]

   if 9.99999982e-14 < (/.f32 sin2phi (*.f32 alphay alphay))

   1. Initial program 61.9%

      \[\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
   2. Add Preprocessing

   3. Taylor expanded in u0 around 0

      \[\leadsto \color{blue}{\frac{u0}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}}} \]
   4. 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. +-commutative N/A

         \[\leadsto \mathsf{/.f32}\left(u0, \left(\frac{sin2phi}{{alphay}^{2}} + \color{blue}{\frac{cos2phi}{{alphax}^{2}}}\right)\right) \]

      3. +-lowering-+.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{+.f32}\left(\left(\frac{sin2phi}{{alphay}^{2}}\right), \color{blue}{\left(\frac{cos2phi}{{alphax}^{2}}\right)}\right)\right) \]

      4. /-lowering-/.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{/.f32}\left(sin2phi, \left({alphay}^{2}\right)\right), \left(\frac{\color{blue}{cos2phi}}{{alphax}^{2}}\right)\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{/.f32}\left(sin2phi, \left(alphay \cdot alphay\right)\right), \left(\frac{cos2phi}{{alphax}^{2}}\right)\right)\right) \]

      6. *-lowering-*.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{/.f32}\left(sin2phi, \mathsf{*.f32}\left(alphay, alphay\right)\right), \left(\frac{cos2phi}{{alphax}^{2}}\right)\right)\right) \]

      7. /-lowering-/.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{/.f32}\left(sin2phi, \mathsf{*.f32}\left(alphay, alphay\right)\right), \mathsf{/.f32}\left(cos2phi, \color{blue}{\left({alphax}^{2}\right)}\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{/.f32}\left(sin2phi, \mathsf{*.f32}\left(alphay, alphay\right)\right), \mathsf{/.f32}\left(cos2phi, \left(alphax \cdot \color{blue}{alphax}\right)\right)\right)\right) \]

      9. *-lowering-*.f32 75.6%

         \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{/.f32}\left(sin2phi, \mathsf{*.f32}\left(alphay, alphay\right)\right), \mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, \color{blue}{alphax}\right)\right)\right)\right) \]

   5. Simplified 75.6%

      \[\leadsto \color{blue}{\frac{u0}{\frac{sin2phi}{alphay \cdot alphay} + \frac{cos2phi}{alphax \cdot alphax}}} \]

   6. Taylor expanded in sin2phi around inf

      \[\leadsto \mathsf{/.f32}\left(u0, \color{blue}{\left(\frac{sin2phi}{{alphay}^{2}}\right)}\right) \]
   7. 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 69.1%

         \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{/.f32}\left(sin2phi, \mathsf{*.f32}\left(alphay, \color{blue}{alphay}\right)\right)\right) \]

   8. Simplified 69.1%

      \[\leadsto \frac{u0}{\color{blue}{\frac{sin2phi}{alphay \cdot alphay}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 65.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 9.9999998245167 \cdot 10^{-14}:\\ \;\;\;\;u0 \cdot \frac{alphax}{\frac{cos2phi}{alphax}}\\ \mathbf{else}:\\ \;\;\;\;\frac{u0}{\frac{sin2phi}{alphay \cdot alphay}}\\ \end{array} \]
5. Add Preprocessing

Alternative 25: 76.2% accurate, 8.9× speedup

Math

\[\begin{array}{l} \\ \frac{u0}{\frac{\frac{cos2phi}{alphax}}{alphax} + \frac{sin2phi}{alphay \cdot alphay}} \end{array} \]

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(Float32(cos2phi / 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 59.4%

   \[\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
2. Add Preprocessing

3. Taylor expanded in u0 around 0

   \[\leadsto \color{blue}{\frac{u0}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}}} \]
4. 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. +-commutative N/A

      \[\leadsto \mathsf{/.f32}\left(u0, \left(\frac{sin2phi}{{alphay}^{2}} + \color{blue}{\frac{cos2phi}{{alphax}^{2}}}\right)\right) \]

   3. +-lowering-+.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{+.f32}\left(\left(\frac{sin2phi}{{alphay}^{2}}\right), \color{blue}{\left(\frac{cos2phi}{{alphax}^{2}}\right)}\right)\right) \]

   4. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{/.f32}\left(sin2phi, \left({alphay}^{2}\right)\right), \left(\frac{\color{blue}{cos2phi}}{{alphax}^{2}}\right)\right)\right) \]

   5. unpow2 N/A

      \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{/.f32}\left(sin2phi, \left(alphay \cdot alphay\right)\right), \left(\frac{cos2phi}{{alphax}^{2}}\right)\right)\right) \]

   6. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{/.f32}\left(sin2phi, \mathsf{*.f32}\left(alphay, alphay\right)\right), \left(\frac{cos2phi}{{alphax}^{2}}\right)\right)\right) \]

   7. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{/.f32}\left(sin2phi, \mathsf{*.f32}\left(alphay, alphay\right)\right), \mathsf{/.f32}\left(cos2phi, \color{blue}{\left({alphax}^{2}\right)}\right)\right)\right) \]

   8. unpow2 N/A

      \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{/.f32}\left(sin2phi, \mathsf{*.f32}\left(alphay, alphay\right)\right), \mathsf{/.f32}\left(cos2phi, \left(alphax \cdot \color{blue}{alphax}\right)\right)\right)\right) \]

   9. *-lowering-*.f32 75.3%

      \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{/.f32}\left(sin2phi, \mathsf{*.f32}\left(alphay, alphay\right)\right), \mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, \color{blue}{alphax}\right)\right)\right)\right) \]

5. Simplified 75.3%

   \[\leadsto \color{blue}{\frac{u0}{\frac{sin2phi}{alphay \cdot alphay} + \frac{cos2phi}{alphax \cdot alphax}}} \]
6. Step-by-step derivation

   1. associate-/r* N/A

      \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{/.f32}\left(sin2phi, \mathsf{*.f32}\left(alphay, alphay\right)\right), \left(\frac{\frac{cos2phi}{alphax}}{\color{blue}{alphax}}\right)\right)\right) \]

   2. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{/.f32}\left(sin2phi, \mathsf{*.f32}\left(alphay, alphay\right)\right), \mathsf{/.f32}\left(\left(\frac{cos2phi}{alphax}\right), \color{blue}{alphax}\right)\right)\right) \]

   3. /-lowering-/.f32 75.3%

      \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{/.f32}\left(sin2phi, \mathsf{*.f32}\left(alphay, alphay\right)\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(cos2phi, alphax\right), alphax\right)\right)\right) \]

7. Applied egg-rr 75.3%

   \[\leadsto \frac{u0}{\frac{sin2phi}{alphay \cdot alphay} + \color{blue}{\frac{\frac{cos2phi}{alphax}}{alphax}}} \]

8. Final simplification 75.3%

   \[\leadsto \frac{u0}{\frac{\frac{cos2phi}{alphax}}{alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
9. Add Preprocessing

Alternative 26: 76.2% accurate, 8.9× speedup

Math

\[\begin{array}{l} \\ \frac{u0}{\frac{sin2phi}{alphay \cdot alphay} + \frac{cos2phi}{alphax \cdot alphax}} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (/ u0 (+ (/ sin2phi (* alphay alphay)) (/ cos2phi (* alphax alphax)))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	return u0 / ((sin2phi / (alphay * alphay)) + (cos2phi / (alphax * alphax)));
}

Fortran

real(4) function code(alphax, alphay, u0, cos2phi, sin2phi)
    real(4), intent (in) :: alphax
    real(4), intent (in) :: alphay
    real(4), intent (in) :: u0
    real(4), intent (in) :: cos2phi
    real(4), intent (in) :: sin2phi
    code = u0 / ((sin2phi / (alphay * alphay)) + (cos2phi / (alphax * alphax)))
end function

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	return Float32(u0 / Float32(Float32(sin2phi / Float32(alphay * alphay)) + Float32(cos2phi / Float32(alphax * alphax))))
end

MATLAB

function tmp = code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = u0 / ((sin2phi / (alphay * alphay)) + (cos2phi / (alphax * alphax)));
end

Derivation

1. Initial program 59.4%

   \[\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
2. Add Preprocessing

3. Taylor expanded in u0 around 0

   \[\leadsto \color{blue}{\frac{u0}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}}} \]
4. 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. +-commutative N/A

      \[\leadsto \mathsf{/.f32}\left(u0, \left(\frac{sin2phi}{{alphay}^{2}} + \color{blue}{\frac{cos2phi}{{alphax}^{2}}}\right)\right) \]

   3. +-lowering-+.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{+.f32}\left(\left(\frac{sin2phi}{{alphay}^{2}}\right), \color{blue}{\left(\frac{cos2phi}{{alphax}^{2}}\right)}\right)\right) \]

   4. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{/.f32}\left(sin2phi, \left({alphay}^{2}\right)\right), \left(\frac{\color{blue}{cos2phi}}{{alphax}^{2}}\right)\right)\right) \]

   5. unpow2 N/A

      \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{/.f32}\left(sin2phi, \left(alphay \cdot alphay\right)\right), \left(\frac{cos2phi}{{alphax}^{2}}\right)\right)\right) \]

   6. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{/.f32}\left(sin2phi, \mathsf{*.f32}\left(alphay, alphay\right)\right), \left(\frac{cos2phi}{{alphax}^{2}}\right)\right)\right) \]

   7. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{/.f32}\left(sin2phi, \mathsf{*.f32}\left(alphay, alphay\right)\right), \mathsf{/.f32}\left(cos2phi, \color{blue}{\left({alphax}^{2}\right)}\right)\right)\right) \]

   8. unpow2 N/A

      \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{/.f32}\left(sin2phi, \mathsf{*.f32}\left(alphay, alphay\right)\right), \mathsf{/.f32}\left(cos2phi, \left(alphax \cdot \color{blue}{alphax}\right)\right)\right)\right) \]

   9. *-lowering-*.f32 75.3%

      \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{/.f32}\left(sin2phi, \mathsf{*.f32}\left(alphay, alphay\right)\right), \mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, \color{blue}{alphax}\right)\right)\right)\right) \]

5. Simplified 75.3%

   \[\leadsto \color{blue}{\frac{u0}{\frac{sin2phi}{alphay \cdot alphay} + \frac{cos2phi}{alphax \cdot alphax}}} \]
6. Add Preprocessing

Alternative 27: 24.0% accurate, 16.6× speedup

Math

\[\begin{array}{l} \\ u0 \cdot \frac{alphax}{\frac{cos2phi}{alphax}} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (* u0 (/ alphax (/ cos2phi alphax))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	return u0 * (alphax / (cos2phi / alphax));
}

Fortran

real(4) function code(alphax, alphay, u0, cos2phi, sin2phi)
    real(4), intent (in) :: alphax
    real(4), intent (in) :: alphay
    real(4), intent (in) :: u0
    real(4), intent (in) :: cos2phi
    real(4), intent (in) :: sin2phi
    code = u0 * (alphax / (cos2phi / alphax))
end function

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	return Float32(u0 * Float32(alphax / Float32(cos2phi / alphax)))
end

MATLAB

function tmp = code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = u0 * (alphax / (cos2phi / alphax));
end

Derivation

1. Initial program 59.4%

   \[\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
2. Add Preprocessing

3. Taylor expanded in u0 around 0

   \[\leadsto \color{blue}{\frac{u0}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}}} \]
4. 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. +-commutative N/A

      \[\leadsto \mathsf{/.f32}\left(u0, \left(\frac{sin2phi}{{alphay}^{2}} + \color{blue}{\frac{cos2phi}{{alphax}^{2}}}\right)\right) \]

   3. +-lowering-+.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{+.f32}\left(\left(\frac{sin2phi}{{alphay}^{2}}\right), \color{blue}{\left(\frac{cos2phi}{{alphax}^{2}}\right)}\right)\right) \]

   4. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{/.f32}\left(sin2phi, \left({alphay}^{2}\right)\right), \left(\frac{\color{blue}{cos2phi}}{{alphax}^{2}}\right)\right)\right) \]

   5. unpow2 N/A

      \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{/.f32}\left(sin2phi, \left(alphay \cdot alphay\right)\right), \left(\frac{cos2phi}{{alphax}^{2}}\right)\right)\right) \]

   6. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{/.f32}\left(sin2phi, \mathsf{*.f32}\left(alphay, alphay\right)\right), \left(\frac{cos2phi}{{alphax}^{2}}\right)\right)\right) \]

   7. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{/.f32}\left(sin2phi, \mathsf{*.f32}\left(alphay, alphay\right)\right), \mathsf{/.f32}\left(cos2phi, \color{blue}{\left({alphax}^{2}\right)}\right)\right)\right) \]

   8. unpow2 N/A

      \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{/.f32}\left(sin2phi, \mathsf{*.f32}\left(alphay, alphay\right)\right), \mathsf{/.f32}\left(cos2phi, \left(alphax \cdot \color{blue}{alphax}\right)\right)\right)\right) \]

   9. *-lowering-*.f32 75.3%

      \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{/.f32}\left(sin2phi, \mathsf{*.f32}\left(alphay, alphay\right)\right), \mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, \color{blue}{alphax}\right)\right)\right)\right) \]

5. Simplified 75.3%

   \[\leadsto \color{blue}{\frac{u0}{\frac{sin2phi}{alphay \cdot alphay} + \frac{cos2phi}{alphax \cdot alphax}}} \]

6. Taylor expanded in sin2phi around 0

   \[\leadsto \mathsf{/.f32}\left(u0, \color{blue}{\left(\frac{cos2phi}{{alphax}^{2}}\right)}\right) \]
7. 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 27.3%

      \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, \color{blue}{alphax}\right)\right)\right) \]

8. Simplified 27.3%

   \[\leadsto \frac{u0}{\color{blue}{\frac{cos2phi}{alphax \cdot alphax}}} \]
9. Step-by-step derivation

   1. clear-num N/A

      \[\leadsto \frac{1}{\color{blue}{\frac{\frac{cos2phi}{alphax \cdot alphax}}{u0}}} \]

   2. associate-/r/ N/A

      \[\leadsto \frac{1}{\frac{cos2phi}{alphax \cdot alphax}} \cdot \color{blue}{u0} \]

   3. clear-num N/A

      \[\leadsto \frac{alphax \cdot alphax}{cos2phi} \cdot u0 \]

   4. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{*.f32}\left(\left(\frac{alphax \cdot alphax}{cos2phi}\right), \color{blue}{u0}\right) \]

   5. associate-*l/ N/A

      \[\leadsto \mathsf{*.f32}\left(\left(\frac{alphax}{cos2phi} \cdot alphax\right), u0\right) \]

   6. associate-/r/ N/A

      \[\leadsto \mathsf{*.f32}\left(\left(\frac{alphax}{\frac{cos2phi}{alphax}}\right), u0\right) \]

   7. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{/.f32}\left(alphax, \left(\frac{cos2phi}{alphax}\right)\right), u0\right) \]

   8. /-lowering-/.f32 27.4%

      \[\leadsto \mathsf{*.f32}\left(\mathsf{/.f32}\left(alphax, \mathsf{/.f32}\left(cos2phi, alphax\right)\right), u0\right) \]

10. Applied egg-rr 27.4%

    \[\leadsto \color{blue}{\frac{alphax}{\frac{cos2phi}{alphax}} \cdot u0} \]

11. Final simplification 27.4%

    \[\leadsto u0 \cdot \frac{alphax}{\frac{cos2phi}{alphax}} \]
12. Add Preprocessing

Alternative 28: 24.0% 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 59.4%

   \[\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
2. Add Preprocessing

3. Taylor expanded in u0 around 0

   \[\leadsto \color{blue}{\frac{u0}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}}} \]
4. 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. +-commutative N/A

      \[\leadsto \mathsf{/.f32}\left(u0, \left(\frac{sin2phi}{{alphay}^{2}} + \color{blue}{\frac{cos2phi}{{alphax}^{2}}}\right)\right) \]

   3. +-lowering-+.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{+.f32}\left(\left(\frac{sin2phi}{{alphay}^{2}}\right), \color{blue}{\left(\frac{cos2phi}{{alphax}^{2}}\right)}\right)\right) \]

   4. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{/.f32}\left(sin2phi, \left({alphay}^{2}\right)\right), \left(\frac{\color{blue}{cos2phi}}{{alphax}^{2}}\right)\right)\right) \]

   5. unpow2 N/A

      \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{/.f32}\left(sin2phi, \left(alphay \cdot alphay\right)\right), \left(\frac{cos2phi}{{alphax}^{2}}\right)\right)\right) \]

   6. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{/.f32}\left(sin2phi, \mathsf{*.f32}\left(alphay, alphay\right)\right), \left(\frac{cos2phi}{{alphax}^{2}}\right)\right)\right) \]

   7. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{/.f32}\left(sin2phi, \mathsf{*.f32}\left(alphay, alphay\right)\right), \mathsf{/.f32}\left(cos2phi, \color{blue}{\left({alphax}^{2}\right)}\right)\right)\right) \]

   8. unpow2 N/A

      \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{/.f32}\left(sin2phi, \mathsf{*.f32}\left(alphay, alphay\right)\right), \mathsf{/.f32}\left(cos2phi, \left(alphax \cdot \color{blue}{alphax}\right)\right)\right)\right) \]

   9. *-lowering-*.f32 75.3%

      \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{/.f32}\left(sin2phi, \mathsf{*.f32}\left(alphay, alphay\right)\right), \mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, \color{blue}{alphax}\right)\right)\right)\right) \]

5. Simplified 75.3%

   \[\leadsto \color{blue}{\frac{u0}{\frac{sin2phi}{alphay \cdot alphay} + \frac{cos2phi}{alphax \cdot alphax}}} \]

6. Taylor expanded in sin2phi around 0

   \[\leadsto \mathsf{/.f32}\left(u0, \color{blue}{\left(\frac{cos2phi}{{alphax}^{2}}\right)}\right) \]
7. 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 27.3%

      \[\leadsto \mathsf{/.f32}\left(u0, \mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, \color{blue}{alphax}\right)\right)\right) \]

8. Simplified 27.3%

   \[\leadsto \frac{u0}{\color{blue}{\frac{cos2phi}{alphax \cdot alphax}}} \]
9. 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 27.3%

      \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{/.f32}\left(u0, cos2phi\right), alphax\right), alphax\right) \]

10. Applied egg-rr 27.3%

    \[\leadsto \color{blue}{\left(\frac{u0}{cos2phi} \cdot alphax\right) \cdot alphax} \]
11. 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 27.3%

       \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(u0, \mathsf{/.f32}\left(alphax, cos2phi\right)\right), alphax\right) \]

12. Applied egg-rr 27.3%

    \[\leadsto \color{blue}{\left(u0 \cdot \frac{alphax}{cos2phi}\right)} \cdot alphax \]

13. Final simplification 27.3%

    \[\leadsto alphax \cdot \left(u0 \cdot \frac{alphax}{cos2phi}\right) \]
14. Add Preprocessing

Reproduce

herbie shell --seed 2024192 
(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)))))