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

Percentage Accurate: 60.5% → 98.2%

Time: 22.8s

Alternatives: 23

Speedup: 9.6×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Initial Program: 60.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Alternative 1: 98.2% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 61.4%

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

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

5. Final simplification 98.4%

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

Alternative 2: 98.0% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 61.4%

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

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

   4. associate-*r* N/A

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

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

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

4. Applied egg-rr 98.2%

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

5. Final simplification 98.2%

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

Alternative 3: 95.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;1 - u0 \leq 0.8999999761581421:\\ \;\;\;\;\frac{\mathsf{log1p}\left(-u0\right) \cdot \left(alphay \cdot alphay\right)}{-sin2phi}\\ \mathbf{else}:\\ \;\;\;\;\frac{u0 + \left(0.5 + u0 \cdot \left(0.3333333333333333 + u0 \cdot 0.25\right)\right) \cdot \left(u0 \cdot u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (if (<= (- 1.0 u0) 0.8999999761581421)
   (/ (* (log1p (- u0)) (* alphay alphay)) (- sin2phi))
   (/
    (+ u0 (* (+ 0.5 (* u0 (+ 0.3333333333333333 (* u0 0.25)))) (* u0 u0)))
    (+ (/ cos2phi (* alphax alphax)) (/ sin2phi (* alphay alphay))))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float tmp;
	if ((1.0f - u0) <= 0.8999999761581421f) {
		tmp = (log1pf(-u0) * (alphay * alphay)) / -sin2phi;
	} else {
		tmp = (u0 + ((0.5f + (u0 * (0.3333333333333333f + (u0 * 0.25f)))) * (u0 * u0))) / ((cos2phi / (alphax * alphax)) + (sin2phi / (alphay * alphay)));
	}
	return tmp;
}

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = Float32(0.0)
	if (Float32(Float32(1.0) - u0) <= Float32(0.8999999761581421))
		tmp = Float32(Float32(log1p(Float32(-u0)) * Float32(alphay * alphay)) / Float32(-sin2phi));
	else
		tmp = Float32(Float32(u0 + Float32(Float32(Float32(0.5) + Float32(u0 * Float32(Float32(0.3333333333333333) + Float32(u0 * Float32(0.25))))) * Float32(u0 * u0))) / Float32(Float32(cos2phi / Float32(alphax * alphax)) + Float32(sin2phi / Float32(alphay * alphay))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (-.f32 #s(literal 1 binary32) u0) < 0.899999976

   1. Initial program 94.6%

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

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

      1. *-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. *-commutative N/A

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

      8. div-inv N/A

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

      9. associate-*l* N/A

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

   6. Applied egg-rr 95.2%

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

   7. Taylor expanded in sin2phi around inf

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

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

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

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

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

      3. unpow2 N/A

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

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

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

      5. sub-neg N/A

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

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

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

      7. neg-lowering-neg.f32 80.5%

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

   9. Simplified 80.5%

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

   if 0.899999976 < (-.f32 #s(literal 1 binary32) u0)

   1. Initial program 56.4%

      \[\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
   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 97.7%

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

   5. Simplified 97.7%

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. *-lft-identity N/A

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

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

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. sqr-neg N/A

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

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

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

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

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

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

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

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

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

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

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

      13. sqr-neg N/A

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

      14. *-lowering-*.f32 98.0%

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

   7. Applied egg-rr 98.0%

      \[\leadsto \frac{\color{blue}{\left(0.5 + u0 \cdot \left(0.3333333333333333 + u0 \cdot 0.25\right)\right) \cdot \left(u0 \cdot u0\right) + u0}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 95.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 - u0 \leq 0.8999999761581421:\\ \;\;\;\;\frac{\mathsf{log1p}\left(-u0\right) \cdot \left(alphay \cdot alphay\right)}{-sin2phi}\\ \mathbf{else}:\\ \;\;\;\;\frac{u0 + \left(0.5 + u0 \cdot \left(0.3333333333333333 + u0 \cdot 0.25\right)\right) \cdot \left(u0 \cdot u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 61.4%

   \[\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
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 98.1%

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

4. Applied egg-rr 98.1%

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

5. Final simplification 98.1%

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

Alternative 5: 98.3% accurate, 1.0× speedup

Math

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

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 / Float32(alphax * alphax))) - Float32(Float32(sin2phi / alphay) / alphay)))
end

Derivation

1. Initial program 61.4%

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

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

   1. *-commutative N/A

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

   2. *-commutative N/A

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

   3. associate-*r* N/A

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

   4. associate-*r* N/A

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

   5. *-commutative N/A

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

   6. associate-*l* N/A

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

   7. *-commutative N/A

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

   8. div-inv N/A

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

   9. associate-*l* N/A

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

6. Applied egg-rr 98.1%

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

7. Final simplification 98.1%

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

Alternative 6: 95.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u0 \leq 0.10000000149011612:\\ \;\;\;\;\frac{u0 + \left(0.5 + u0 \cdot \left(0.3333333333333333 + u0 \cdot 0.25\right)\right) \cdot \left(u0 \cdot u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}}\\ \mathbf{else}:\\ \;\;\;\;-\mathsf{log1p}\left(-u0\right) \cdot \frac{alphay \cdot alphay}{sin2phi}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (if (<= u0 0.10000000149011612)
   (/
    (+ u0 (* (+ 0.5 (* u0 (+ 0.3333333333333333 (* u0 0.25)))) (* u0 u0)))
    (+ (/ cos2phi (* alphax alphax)) (/ sin2phi (* alphay alphay))))
   (- (* (log1p (- u0)) (/ (* alphay alphay) sin2phi)))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float tmp;
	if (u0 <= 0.10000000149011612f) {
		tmp = (u0 + ((0.5f + (u0 * (0.3333333333333333f + (u0 * 0.25f)))) * (u0 * u0))) / ((cos2phi / (alphax * alphax)) + (sin2phi / (alphay * alphay)));
	} else {
		tmp = -(log1pf(-u0) * ((alphay * alphay) / sin2phi));
	}
	return tmp;
}

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = Float32(0.0)
	if (u0 <= Float32(0.10000000149011612))
		tmp = Float32(Float32(u0 + Float32(Float32(Float32(0.5) + Float32(u0 * Float32(Float32(0.3333333333333333) + Float32(u0 * Float32(0.25))))) * Float32(u0 * u0))) / Float32(Float32(cos2phi / Float32(alphax * alphax)) + Float32(sin2phi / Float32(alphay * alphay))));
	else
		tmp = Float32(-Float32(log1p(Float32(-u0)) * Float32(Float32(alphay * alphay) / sin2phi)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if u0 < 0.100000001

   1. Initial program 56.4%

      \[\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
   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 97.7%

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

   5. Simplified 97.7%

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. *-lft-identity N/A

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

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

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. sqr-neg N/A

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

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

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

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

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

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

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

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

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

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

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

      13. sqr-neg N/A

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

      14. *-lowering-*.f32 98.0%

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

   7. Applied egg-rr 98.0%

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

   if 0.100000001 < u0

   1. Initial program 94.6%

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

   3. Taylor expanded in cos2phi around 0

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

      1. mul-1-neg N/A

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

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

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

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

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

      6. sub-neg N/A

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

      7. mul-1-neg N/A

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

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

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

      9. mul-1-neg N/A

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

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

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

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

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

      12. unpow2 N/A

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

      13. *-lowering-*.f32 80.2%

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

   5. Simplified 80.2%

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

4. Final simplification 95.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u0 \leq 0.10000000149011612:\\ \;\;\;\;\frac{u0 + \left(0.5 + u0 \cdot \left(0.3333333333333333 + u0 \cdot 0.25\right)\right) \cdot \left(u0 \cdot u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}}\\ \mathbf{else}:\\ \;\;\;\;-\mathsf{log1p}\left(-u0\right) \cdot \frac{alphay \cdot alphay}{sin2phi}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 93.3% accurate, 1.2× speedup

Math

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

FPCore

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

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float t_0 = (cos2phi * (alphay * alphay)) + (sin2phi * (alphax * alphax));
	float t_1 = (alphax * alphax) * ((alphay * alphay) / t_0);
	return u0 * (t_1 + (u0 * ((0.5f * t_1) + (u0 * ((0.25f * (((alphax * alphax) * (u0 * (alphay * alphay))) / 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 = (cos2phi * (alphay * alphay)) + (sin2phi * (alphax * alphax))
    t_1 = (alphax * alphax) * ((alphay * alphay) / t_0)
    code = u0 * (t_1 + (u0 * ((0.5e0 * t_1) + (u0 * ((0.25e0 * (((alphax * alphax) * (u0 * (alphay * alphay))) / t_0)) + (0.3333333333333333e0 * t_1))))))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 61.4%

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

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

5. Taylor expanded in u0 around 0

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

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

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

   2. +-commutative N/A

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

7. Simplified 92.2%

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

8. Final simplification 92.2%

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

Alternative 8: 93.2% accurate, 3.3× speedup

Math

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

FPCore

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

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)))))) * (alphay * (alphay * (alphax * alphax)))) / ((sin2phi * (alphax * alphax)) + (alphay * (cos2phi * alphay))));
}

Fortran

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

Julia

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

Derivation

1. Initial program 61.4%

   \[\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
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 91.6%

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

5. Simplified 91.6%

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

   1. associate-/l* N/A

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

   2. *-commutative N/A

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

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

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

7. Applied egg-rr 91.6%

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

   1. associate-/r* N/A

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

   2. frac-add N/A

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

   3. associate-/r/ N/A

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

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

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

9. Applied egg-rr 92.1%

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

    1. associate-*l/ N/A

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

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

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

11. Applied egg-rr 92.2%

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

12. Final simplification 92.2%

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

Alternative 9: 93.2% accurate, 3.3× speedup

Math

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

FPCore

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

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)))))) * ((alphay * (alphay * (alphax * alphax))) / ((sin2phi * (alphax * alphax)) + (alphay * (cos2phi * alphay)))));
}

Fortran

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

Julia

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

Derivation

1. Initial program 61.4%

   \[\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
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 91.6%

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

5. Simplified 91.6%

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

   1. associate-/l* N/A

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

   2. *-commutative N/A

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

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

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

7. Applied egg-rr 91.6%

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

   1. associate-/r* N/A

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

   2. frac-add N/A

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

   3. associate-/r/ N/A

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

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

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

9. Applied egg-rr 92.1%

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

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

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

11. Applied egg-rr 92.2%

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

12. Final simplification 92.2%

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

Alternative 10: 93.2% accurate, 3.5× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 61.4%

   \[\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
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 91.6%

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

5. Simplified 91.6%

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

   1. associate-/l* N/A

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

   2. *-commutative N/A

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

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

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

7. Applied egg-rr 91.6%

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

   1. associate-*l/ N/A

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

   2. frac-add N/A

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

   3. associate-/r/ N/A

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

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

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

9. Applied egg-rr 92.1%

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

10. Final simplification 92.1%

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

Alternative 11: 93.2% accurate, 3.5× speedup

Math

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

FPCore

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

Julia

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

Derivation

1. Initial program 61.4%

   \[\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
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 91.6%

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

5. Simplified 91.6%

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

   1. associate-/r* N/A

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

   2. frac-add N/A

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

   3. associate-/r/ N/A

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

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

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

7. Applied egg-rr 92.0%

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

Alternative 12: 93.2% accurate, 4.3× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 61.4%

   \[\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
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 91.6%

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

5. Simplified 91.6%

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

   1. +-commutative N/A

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

   2. distribute-rgt-in N/A

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

   3. *-lft-identity N/A

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

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

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

   5. *-commutative N/A

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

   6. associate-*l* N/A

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

   7. sqr-neg N/A

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

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

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

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

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

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

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

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

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

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

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

   13. sqr-neg N/A

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

   14. *-lowering-*.f32 91.9%

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

7. Applied egg-rr 91.9%

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

8. Final simplification 91.9%

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

Alternative 13: 93.0% accurate, 4.3× speedup

Math

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

FPCore

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

MATLAB

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

Derivation

1. Initial program 61.4%

   \[\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
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 91.6%

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

5. Simplified 91.6%

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

   1. associate-/l* N/A

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

   2. *-commutative N/A

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

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

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

7. Applied egg-rr 91.6%

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

   1. associate-/r* N/A

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

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

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

   3. /-lowering-/.f32 91.7%

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

9. Applied egg-rr 91.7%

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

10. Final simplification 91.7%

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

Alternative 14: 93.0% accurate, 4.3× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 61.4%

   \[\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
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 91.6%

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

5. Simplified 91.6%

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

   1. *-commutative N/A

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

   2. associate-/l* N/A

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

   3. +-commutative N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

   11. +-commutative N/A

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

7. Applied egg-rr 91.6%

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

8. Final simplification 91.6%

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

Alternative 15: 83.6% accurate, 4.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{sin2phi}{alphay \cdot alphay}\\ \mathbf{if}\;t\_0 \leq 0.6000000238418579:\\ \;\;\;\;\frac{u0}{t\_0 + \frac{\frac{1}{alphax}}{\frac{alphax}{cos2phi}}}\\ \mathbf{else}:\\ \;\;\;\;u0 \cdot \frac{\left(alphay \cdot alphay\right) \cdot \left(1 + u0 \cdot \left(0.5 + u0 \cdot 0.3333333333333333\right)\right)}{sin2phi}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (let* ((t_0 (/ sin2phi (* alphay alphay))))
   (if (<= t_0 0.6000000238418579)
     (/ u0 (+ t_0 (/ (/ 1.0 alphax) (/ alphax cos2phi))))
     (*
      u0
      (/
       (* (* alphay alphay) (+ 1.0 (* u0 (+ 0.5 (* u0 0.3333333333333333)))))
       sin2phi)))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float t_0 = sin2phi / (alphay * alphay);
	float tmp;
	if (t_0 <= 0.6000000238418579f) {
		tmp = u0 / (t_0 + ((1.0f / alphax) / (alphax / cos2phi)));
	} else {
		tmp = u0 * (((alphay * alphay) * (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) :: t_0
    real(4) :: tmp
    t_0 = sin2phi / (alphay * alphay)
    if (t_0 <= 0.6000000238418579e0) then
        tmp = u0 / (t_0 + ((1.0e0 / alphax) / (alphax / cos2phi)))
    else
        tmp = u0 * (((alphay * alphay) * (1.0e0 + (u0 * (0.5e0 + (u0 * 0.3333333333333333e0))))) / sin2phi)
    end if
    code = tmp
end function

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	t_0 = Float32(sin2phi / Float32(alphay * alphay))
	tmp = Float32(0.0)
	if (t_0 <= Float32(0.6000000238418579))
		tmp = Float32(u0 / Float32(t_0 + Float32(Float32(Float32(1.0) / alphax) / Float32(alphax / cos2phi))));
	else
		tmp = Float32(u0 * Float32(Float32(Float32(alphay * alphay) * Float32(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)
	t_0 = sin2phi / (alphay * alphay);
	tmp = single(0.0);
	if (t_0 <= single(0.6000000238418579))
		tmp = u0 / (t_0 + ((single(1.0) / alphax) / (alphax / cos2phi)));
	else
		tmp = u0 * (((alphay * alphay) * (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 (/.f32 sin2phi (*.f32 alphay alphay)) < 0.600000024

   1. Initial program 57.3%

      \[\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
   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 73.2%

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

   5. Simplified 73.2%

      \[\leadsto \color{blue}{\frac{u0}{\frac{sin2phi}{alphay \cdot alphay} + \frac{cos2phi}{alphax \cdot alphax}}} \]
   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 73.3%

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

   7. Applied egg-rr 73.3%

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

   if 0.600000024 < (/.f32 sin2phi (*.f32 alphay alphay))

   1. Initial program 64.5%

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

   3. Taylor expanded in u0 around 0

      \[\leadsto \mathsf{/.f32}\left(\color{blue}{\left(u0 \cdot \left(1 + u0 \cdot \left(\frac{1}{2} + u0 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right)\right)\right)\right)}, \mathsf{+.f32}\left(\mathsf{/.f32}\left(cos2phi, \mathsf{*.f32}\left(alphax, alphax\right)\right), \mathsf{/.f32}\left(sin2phi, \mathsf{*.f32}\left(alphay, alphay\right)\right)\right)\right) \]
   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 92.1%

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

   5. Simplified 92.1%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

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

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

   7. Applied egg-rr 92.1%

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

   8. Taylor expanded in u0 around 0

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

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

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

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

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

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

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

      4. *-commutative N/A

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

      5. *-lowering-*.f32 90.6%

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

   10. Simplified 90.6%

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

   11. Taylor expanded in sin2phi around inf

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

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

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

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

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

       3. unpow2 N/A

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

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

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

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

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

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

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

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

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

       8. *-commutative N/A

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

       9. *-lowering-*.f32 90.2%

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

   13. Simplified 90.2%

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

4. Final simplification 82.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 0.6000000238418579:\\ \;\;\;\;\frac{u0}{\frac{sin2phi}{alphay \cdot alphay} + \frac{\frac{1}{alphax}}{\frac{alphax}{cos2phi}}}\\ \mathbf{else}:\\ \;\;\;\;u0 \cdot \frac{\left(alphay \cdot alphay\right) \cdot \left(1 + u0 \cdot \left(0.5 + u0 \cdot 0.3333333333333333\right)\right)}{sin2phi}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 91.1% accurate, 5.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 61.4%

   \[\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
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 91.6%

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

5. Simplified 91.6%

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

   1. associate-/l* N/A

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

   2. *-commutative N/A

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

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

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

7. Applied egg-rr 91.6%

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

8. Taylor expanded in u0 around 0

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

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

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

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

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

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

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

   4. *-commutative N/A

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

   5. *-lowering-*.f32 90.2%

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

10. Simplified 90.2%

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

    1. associate-/r* N/A

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

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

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

    3. /-lowering-/.f32 90.3%

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

12. Applied egg-rr 90.3%

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

13. Final simplification 90.3%

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

Alternative 17: 91.2% accurate, 5.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 61.4%

   \[\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
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 91.6%

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

5. Simplified 91.6%

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

   1. associate-/l* N/A

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

   2. *-commutative N/A

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

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

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

7. Applied egg-rr 91.6%

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

8. Taylor expanded in u0 around 0

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

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

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

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

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

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

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

   4. *-commutative N/A

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

   5. *-lowering-*.f32 90.2%

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

10. Simplified 90.2%

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

    1. associate-*l/ N/A

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

    2. associate-/l* N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

    9. +-commutative N/A

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

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

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

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

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

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

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

    13. associate-/l/ N/A

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

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

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

    15. *-lowering-*.f32 90.2%

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

12. Applied egg-rr 90.2%

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

Alternative 18: 87.5% accurate, 6.1× speedup

Math

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

FPCore

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

C

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

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)))) * (1.0e0 + (u0 * 0.5e0))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 61.4%

   \[\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
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 87.0%

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

6. Final simplification 87.0%

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

Alternative 19: 76.0% accurate, 8.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 61.4%

   \[\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
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 75.4%

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

5. Simplified 75.4%

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

   1. +-commutative N/A

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

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

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

   3. +-commutative N/A

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

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

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

   5. associate-/r* N/A

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

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

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

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

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

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

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

   9. *-lowering-*.f32 75.5%

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

7. Applied egg-rr 75.5%

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

8. Final simplification 75.5%

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

Alternative 20: 76.0% accurate, 8.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 61.4%

   \[\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
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 75.4%

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

5. Simplified 75.4%

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

6. Final simplification 75.4%

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

Alternative 21: 66.7% accurate, 9.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;sin2phi \leq 5.0000000843119176 \cdot 10^{-17}:\\ \;\;\;\;alphax \cdot \left(u0 \cdot \frac{alphax}{cos2phi}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{u0 \cdot \left(alphay \cdot alphay\right)}{sin2phi}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (if (<= sin2phi 5.0000000843119176e-17)
   (* alphax (* u0 (/ alphax cos2phi)))
   (/ (* u0 (* alphay alphay)) sin2phi)))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float tmp;
	if (sin2phi <= 5.0000000843119176e-17f) {
		tmp = alphax * (u0 * (alphax / cos2phi));
	} else {
		tmp = (u0 * (alphay * alphay)) / sin2phi;
	}
	return tmp;
}

Fortran

real(4) function code(alphax, alphay, u0, cos2phi, sin2phi)
    real(4), intent (in) :: alphax
    real(4), intent (in) :: alphay
    real(4), intent (in) :: u0
    real(4), intent (in) :: cos2phi
    real(4), intent (in) :: sin2phi
    real(4) :: tmp
    if (sin2phi <= 5.0000000843119176e-17) then
        tmp = alphax * (u0 * (alphax / cos2phi))
    else
        tmp = (u0 * (alphay * alphay)) / sin2phi
    end if
    code = tmp
end function

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = Float32(0.0)
	if (sin2phi <= Float32(5.0000000843119176e-17))
		tmp = Float32(alphax * Float32(u0 * Float32(alphax / cos2phi)));
	else
		tmp = Float32(Float32(u0 * Float32(alphay * alphay)) / sin2phi);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = single(0.0);
	if (sin2phi <= single(5.0000000843119176e-17))
		tmp = alphax * (u0 * (alphax / cos2phi));
	else
		tmp = (u0 * (alphay * alphay)) / sin2phi;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if sin2phi < 5.00000008e-17

   1. Initial program 56.7%

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

   3. Taylor expanded in u0 around 0

      \[\leadsto \color{blue}{\frac{u0}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}}} \]
   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 73.8%

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

   5. Simplified 73.8%

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

   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 58.8%

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

   8. Simplified 58.8%

      \[\leadsto \frac{u0}{\color{blue}{\frac{cos2phi}{alphax \cdot alphax}}} \]
   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 58.9%

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

   10. Applied egg-rr 58.9%

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

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

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

       2. associate-*l/ N/A

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

       3. associate-/l* N/A

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

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

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

       5. /-lowering-/.f32 59.0%

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

   12. Applied egg-rr 59.0%

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

   if 5.00000008e-17 < sin2phi

   1. Initial program 63.3%

      \[\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
   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 76.1%

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

   5. Simplified 76.1%

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

   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. *-lowering-*.f32 N/A

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

      3. unpow2 N/A

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

      4. *-lowering-*.f32 71.6%

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

   8. Simplified 71.6%

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

4. Final simplification 67.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;sin2phi \leq 5.0000000843119176 \cdot 10^{-17}:\\ \;\;\;\;alphax \cdot \left(u0 \cdot \frac{alphax}{cos2phi}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{u0 \cdot \left(alphay \cdot alphay\right)}{sin2phi}\\ \end{array} \]
5. Add Preprocessing

Alternative 22: 66.4% accurate, 9.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;sin2phi \leq 5.0000000843119176 \cdot 10^{-17}:\\ \;\;\;\;alphax \cdot \left(u0 \cdot \frac{alphax}{cos2phi}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{u0}{\frac{sin2phi}{alphay \cdot alphay}}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (if (<= sin2phi 5.0000000843119176e-17)
   (* alphax (* u0 (/ alphax cos2phi)))
   (/ u0 (/ sin2phi (* alphay alphay)))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float tmp;
	if (sin2phi <= 5.0000000843119176e-17f) {
		tmp = alphax * (u0 * (alphax / cos2phi));
	} else {
		tmp = u0 / (sin2phi / (alphay * 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 <= 5.0000000843119176e-17) then
        tmp = alphax * (u0 * (alphax / cos2phi))
    else
        tmp = u0 / (sin2phi / (alphay * alphay))
    end if
    code = tmp
end function

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = Float32(0.0)
	if (sin2phi <= Float32(5.0000000843119176e-17))
		tmp = Float32(alphax * Float32(u0 * Float32(alphax / cos2phi)));
	else
		tmp = Float32(u0 / Float32(sin2phi / Float32(alphay * alphay)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = single(0.0);
	if (sin2phi <= single(5.0000000843119176e-17))
		tmp = alphax * (u0 * (alphax / cos2phi));
	else
		tmp = u0 / (sin2phi / (alphay * alphay));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if sin2phi < 5.00000008e-17

   1. Initial program 56.7%

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

   3. Taylor expanded in u0 around 0

      \[\leadsto \color{blue}{\frac{u0}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}}} \]
   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 73.8%

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

   5. Simplified 73.8%

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

   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 58.8%

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

   8. Simplified 58.8%

      \[\leadsto \frac{u0}{\color{blue}{\frac{cos2phi}{alphax \cdot alphax}}} \]
   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 58.9%

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

   10. Applied egg-rr 58.9%

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

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

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

       2. associate-*l/ N/A

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

       3. associate-/l* N/A

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

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

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

       5. /-lowering-/.f32 59.0%

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

   12. Applied egg-rr 59.0%

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

   if 5.00000008e-17 < sin2phi

   1. Initial program 63.3%

      \[\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
   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 76.1%

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

   5. Simplified 76.1%

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

   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 71.1%

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

   8. Simplified 71.1%

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

4. Final simplification 67.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;sin2phi \leq 5.0000000843119176 \cdot 10^{-17}:\\ \;\;\;\;alphax \cdot \left(u0 \cdot \frac{alphax}{cos2phi}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{u0}{\frac{sin2phi}{alphay \cdot alphay}}\\ \end{array} \]
5. Add Preprocessing

Alternative 23: 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 61.4%

   \[\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
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 75.4%

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

5. Simplified 75.4%

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

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 25.4%

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

8. Simplified 25.4%

   \[\leadsto \frac{u0}{\color{blue}{\frac{cos2phi}{alphax \cdot alphax}}} \]
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 25.4%

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

10. Applied egg-rr 25.4%

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

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

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

    2. associate-*l/ N/A

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

    3. associate-/l* N/A

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

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

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

    5. /-lowering-/.f32 25.4%

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

12. Applied egg-rr 25.4%

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

13. Final simplification 25.4%

    \[\leadsto alphax \cdot \left(u0 \cdot \frac{alphax}{cos2phi}\right) \]
14. Add Preprocessing

Reproduce

herbie shell --seed 2024191 
(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)))))