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

Percentage Accurate: 60.2% → 98.3%

Time: 13.0s

Alternatives: 16

Speedup: 12.7×

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: 60.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Alternative 1: 98.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 66.3%

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

   1. neg-sub0 66.3%

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

   2. div-sub 66.3%

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

   3. --rgt-identity 66.3%

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

   4. div-sub 66.3%

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

   5. --rgt-identity 66.3%

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

   6. neg-sub0 66.3%

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

   7. sub-neg 66.3%

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

   8. log1p-def 98.7%

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

3. Simplified 98.7%

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

4. Final simplification 98.7%

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

Alternative 2: 92.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{alphay}{\frac{alphax}{-cos2phi}} - \frac{alphax}{\frac{alphay}{sin2phi}}\\ \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 30:\\ \;\;\;\;\frac{-0.5}{\frac{t_0}{alphay \cdot \left(alphax \cdot \left(u0 \cdot u0\right)\right)}} - \frac{alphax \cdot alphay}{\frac{t_0}{u0}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(-u0\right) \cdot \left(\frac{alphay}{sin2phi} \cdot \left(-alphay\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (let* ((t_0
         (- (/ alphay (/ alphax (- cos2phi))) (/ alphax (/ alphay sin2phi)))))
   (if (<= (/ sin2phi (* alphay alphay)) 30.0)
     (-
      (/ -0.5 (/ t_0 (* alphay (* alphax (* u0 u0)))))
      (/ (* alphax alphay) (/ t_0 u0)))
     (* (log1p (- u0)) (* (/ alphay sin2phi) (- alphay))))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float t_0 = (alphay / (alphax / -cos2phi)) - (alphax / (alphay / sin2phi));
	float tmp;
	if ((sin2phi / (alphay * alphay)) <= 30.0f) {
		tmp = (-0.5f / (t_0 / (alphay * (alphax * (u0 * u0))))) - ((alphax * alphay) / (t_0 / u0));
	} else {
		tmp = log1pf(-u0) * ((alphay / sin2phi) * -alphay);
	}
	return tmp;
}

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	t_0 = Float32(Float32(alphay / Float32(alphax / Float32(-cos2phi))) - Float32(alphax / Float32(alphay / sin2phi)))
	tmp = Float32(0.0)
	if (Float32(sin2phi / Float32(alphay * alphay)) <= Float32(30.0))
		tmp = Float32(Float32(Float32(-0.5) / Float32(t_0 / Float32(alphay * Float32(alphax * Float32(u0 * u0))))) - Float32(Float32(alphax * alphay) / Float32(t_0 / u0)));
	else
		tmp = Float32(log1p(Float32(-u0)) * Float32(Float32(alphay / sin2phi) * Float32(-alphay)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 sin2phi (*.f32 alphay alphay)) < 30

   1. Initial program 60.9%

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

      1. neg-sub0 60.9%

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

      2. div-sub 60.9%

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

      3. --rgt-identity 60.9%

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

      4. div-sub 60.9%

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

      5. --rgt-identity 60.9%

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

      6. neg-sub0 60.9%

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

      7. sub-neg 60.9%

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

      8. log1p-def 98.6%

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

   3. Simplified 98.6%

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

      1. +-commutative 98.6%

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

      2. associate-/r* 98.8%

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

      3. associate-/r* 98.6%

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

      4. frac-2neg 98.6%

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

      5. frac-add 98.2%

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

      6. distribute-neg-frac 98.2%

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

   5. Applied egg-rr 98.2%

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

      1. +-commutative 98.2%

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

      2. distribute-rgt-neg-out 98.2%

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

      3. unsub-neg 98.2%

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

      4. associate-*l/ 98.3%

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

      5. associate-/l* 98.2%

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

      6. *-commutative 98.2%

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

      7. distribute-lft-neg-out 98.2%

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

      8. distribute-rgt-neg-in 98.2%

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

   7. Simplified 98.2%

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

   8. Taylor expanded in u0 around 0 83.8%

      \[\leadsto \color{blue}{-1 \cdot \frac{u0 \cdot \left(alphay \cdot alphax\right)}{-1 \cdot \frac{cos2phi \cdot alphay}{alphax} - \frac{sin2phi \cdot alphax}{alphay}} + -0.5 \cdot \frac{{u0}^{2} \cdot \left(alphay \cdot alphax\right)}{-1 \cdot \frac{cos2phi \cdot alphay}{alphax} - \frac{sin2phi \cdot alphax}{alphay}}} \]
   9. Step-by-step derivation

      1. +-commutative 83.8%

         \[\leadsto \color{blue}{-0.5 \cdot \frac{{u0}^{2} \cdot \left(alphay \cdot alphax\right)}{-1 \cdot \frac{cos2phi \cdot alphay}{alphax} - \frac{sin2phi \cdot alphax}{alphay}} + -1 \cdot \frac{u0 \cdot \left(alphay \cdot alphax\right)}{-1 \cdot \frac{cos2phi \cdot alphay}{alphax} - \frac{sin2phi \cdot alphax}{alphay}}} \]

      2. mul-1-neg 83.8%

         \[\leadsto -0.5 \cdot \frac{{u0}^{2} \cdot \left(alphay \cdot alphax\right)}{-1 \cdot \frac{cos2phi \cdot alphay}{alphax} - \frac{sin2phi \cdot alphax}{alphay}} + \color{blue}{\left(-\frac{u0 \cdot \left(alphay \cdot alphax\right)}{-1 \cdot \frac{cos2phi \cdot alphay}{alphax} - \frac{sin2phi \cdot alphax}{alphay}}\right)} \]

      3. unsub-neg 83.8%

         \[\leadsto \color{blue}{-0.5 \cdot \frac{{u0}^{2} \cdot \left(alphay \cdot alphax\right)}{-1 \cdot \frac{cos2phi \cdot alphay}{alphax} - \frac{sin2phi \cdot alphax}{alphay}} - \frac{u0 \cdot \left(alphay \cdot alphax\right)}{-1 \cdot \frac{cos2phi \cdot alphay}{alphax} - \frac{sin2phi \cdot alphax}{alphay}}} \]

   10. Simplified 83.7%

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

   if 30 < (/.f32 sin2phi (*.f32 alphay alphay))

   1. Initial program 70.7%

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

      1. associate-/r* 70.7%

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

   3. Simplified 70.7%

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

   4. Taylor expanded in cos2phi around 0 70.6%

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

      1. mul-1-neg 70.6%

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

      2. unpow2 70.6%

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

      3. *-commutative 70.6%

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

   6. Simplified 70.6%

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

   7. Taylor expanded in alphay around 0 70.6%

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

      1. sub-neg 70.6%

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

      2. log1p-def 98.2%

         \[\leadsto -\frac{{alphay}^{2} \cdot \color{blue}{\mathsf{log1p}\left(-u0\right)}}{sin2phi} \]

      3. unpow2 98.2%

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

      4. associate-*l* 98.2%

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

   9. Simplified 98.2%

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

   10. Taylor expanded in alphay around 0 70.6%

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

       1. unpow2 70.6%

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

       2. associate-*r* 70.5%

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

       3. sub-neg 70.5%

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

       4. log1p-def 98.2%

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

       5. *-commutative 98.2%

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

       6. associate-*l/ 98.1%

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

       7. *-commutative 98.1%

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

       8. log1p-def 70.5%

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

       9. sub-neg 70.5%

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

       10. associate-*r* 70.5%

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

       11. sub-neg 70.5%

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

       12. log1p-def 98.0%

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

   12. Simplified 98.0%

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

4. Final simplification 91.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 30:\\ \;\;\;\;\frac{-0.5}{\frac{\frac{alphay}{\frac{alphax}{-cos2phi}} - \frac{alphax}{\frac{alphay}{sin2phi}}}{alphay \cdot \left(alphax \cdot \left(u0 \cdot u0\right)\right)}} - \frac{alphax \cdot alphay}{\frac{\frac{alphay}{\frac{alphax}{-cos2phi}} - \frac{alphax}{\frac{alphay}{sin2phi}}}{u0}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(-u0\right) \cdot \left(\frac{alphay}{sin2phi} \cdot \left(-alphay\right)\right)\\ \end{array} \]

Alternative 3: 92.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{alphay}{\frac{alphax}{-cos2phi}} - \frac{alphax}{\frac{alphay}{sin2phi}}\\ \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 30:\\ \;\;\;\;\frac{-0.5}{\frac{t_0}{alphay \cdot \left(alphax \cdot \left(u0 \cdot u0\right)\right)}} - \frac{alphax \cdot alphay}{\frac{t_0}{u0}}\\ \mathbf{else}:\\ \;\;\;\;\frac{alphay \cdot \left(\mathsf{log1p}\left(-u0\right) \cdot \left(-alphay\right)\right)}{sin2phi}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (let* ((t_0
         (- (/ alphay (/ alphax (- cos2phi))) (/ alphax (/ alphay sin2phi)))))
   (if (<= (/ sin2phi (* alphay alphay)) 30.0)
     (-
      (/ -0.5 (/ t_0 (* alphay (* alphax (* u0 u0)))))
      (/ (* alphax alphay) (/ t_0 u0)))
     (/ (* alphay (* (log1p (- u0)) (- alphay))) sin2phi))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float t_0 = (alphay / (alphax / -cos2phi)) - (alphax / (alphay / sin2phi));
	float tmp;
	if ((sin2phi / (alphay * alphay)) <= 30.0f) {
		tmp = (-0.5f / (t_0 / (alphay * (alphax * (u0 * u0))))) - ((alphax * alphay) / (t_0 / u0));
	} else {
		tmp = (alphay * (log1pf(-u0) * -alphay)) / sin2phi;
	}
	return tmp;
}

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	t_0 = Float32(Float32(alphay / Float32(alphax / Float32(-cos2phi))) - Float32(alphax / Float32(alphay / sin2phi)))
	tmp = Float32(0.0)
	if (Float32(sin2phi / Float32(alphay * alphay)) <= Float32(30.0))
		tmp = Float32(Float32(Float32(-0.5) / Float32(t_0 / Float32(alphay * Float32(alphax * Float32(u0 * u0))))) - Float32(Float32(alphax * alphay) / Float32(t_0 / u0)));
	else
		tmp = Float32(Float32(alphay * Float32(log1p(Float32(-u0)) * Float32(-alphay))) / sin2phi);
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 sin2phi (*.f32 alphay alphay)) < 30

   1. Initial program 60.9%

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

      1. neg-sub0 60.9%

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

      2. div-sub 60.9%

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

      3. --rgt-identity 60.9%

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

      4. div-sub 60.9%

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

      5. --rgt-identity 60.9%

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

      6. neg-sub0 60.9%

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

      7. sub-neg 60.9%

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

      8. log1p-def 98.6%

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

   3. Simplified 98.6%

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

      1. +-commutative 98.6%

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

      2. associate-/r* 98.8%

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

      3. associate-/r* 98.6%

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

      4. frac-2neg 98.6%

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

      5. frac-add 98.2%

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

      6. distribute-neg-frac 98.2%

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

   5. Applied egg-rr 98.2%

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

      1. +-commutative 98.2%

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

      2. distribute-rgt-neg-out 98.2%

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

      3. unsub-neg 98.2%

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

      4. associate-*l/ 98.3%

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

      5. associate-/l* 98.2%

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

      6. *-commutative 98.2%

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

      7. distribute-lft-neg-out 98.2%

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

      8. distribute-rgt-neg-in 98.2%

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

   7. Simplified 98.2%

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

   8. Taylor expanded in u0 around 0 83.8%

      \[\leadsto \color{blue}{-1 \cdot \frac{u0 \cdot \left(alphay \cdot alphax\right)}{-1 \cdot \frac{cos2phi \cdot alphay}{alphax} - \frac{sin2phi \cdot alphax}{alphay}} + -0.5 \cdot \frac{{u0}^{2} \cdot \left(alphay \cdot alphax\right)}{-1 \cdot \frac{cos2phi \cdot alphay}{alphax} - \frac{sin2phi \cdot alphax}{alphay}}} \]
   9. Step-by-step derivation

      1. +-commutative 83.8%

         \[\leadsto \color{blue}{-0.5 \cdot \frac{{u0}^{2} \cdot \left(alphay \cdot alphax\right)}{-1 \cdot \frac{cos2phi \cdot alphay}{alphax} - \frac{sin2phi \cdot alphax}{alphay}} + -1 \cdot \frac{u0 \cdot \left(alphay \cdot alphax\right)}{-1 \cdot \frac{cos2phi \cdot alphay}{alphax} - \frac{sin2phi \cdot alphax}{alphay}}} \]

      2. mul-1-neg 83.8%

         \[\leadsto -0.5 \cdot \frac{{u0}^{2} \cdot \left(alphay \cdot alphax\right)}{-1 \cdot \frac{cos2phi \cdot alphay}{alphax} - \frac{sin2phi \cdot alphax}{alphay}} + \color{blue}{\left(-\frac{u0 \cdot \left(alphay \cdot alphax\right)}{-1 \cdot \frac{cos2phi \cdot alphay}{alphax} - \frac{sin2phi \cdot alphax}{alphay}}\right)} \]

      3. unsub-neg 83.8%

         \[\leadsto \color{blue}{-0.5 \cdot \frac{{u0}^{2} \cdot \left(alphay \cdot alphax\right)}{-1 \cdot \frac{cos2phi \cdot alphay}{alphax} - \frac{sin2phi \cdot alphax}{alphay}} - \frac{u0 \cdot \left(alphay \cdot alphax\right)}{-1 \cdot \frac{cos2phi \cdot alphay}{alphax} - \frac{sin2phi \cdot alphax}{alphay}}} \]

   10. Simplified 83.7%

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

   if 30 < (/.f32 sin2phi (*.f32 alphay alphay))

   1. Initial program 70.7%

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

      1. associate-/r* 70.7%

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

   3. Simplified 70.7%

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

   4. Taylor expanded in cos2phi around 0 70.6%

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

      1. mul-1-neg 70.6%

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

      2. unpow2 70.6%

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

      3. *-commutative 70.6%

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

   6. Simplified 70.6%

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

   7. Taylor expanded in alphay around 0 70.6%

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

      1. sub-neg 70.6%

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

      2. log1p-def 98.2%

         \[\leadsto -\frac{{alphay}^{2} \cdot \color{blue}{\mathsf{log1p}\left(-u0\right)}}{sin2phi} \]

      3. unpow2 98.2%

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

      4. associate-*l* 98.2%

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

   9. Simplified 98.2%

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

4. Final simplification 91.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 30:\\ \;\;\;\;\frac{-0.5}{\frac{\frac{alphay}{\frac{alphax}{-cos2phi}} - \frac{alphax}{\frac{alphay}{sin2phi}}}{alphay \cdot \left(alphax \cdot \left(u0 \cdot u0\right)\right)}} - \frac{alphax \cdot alphay}{\frac{\frac{alphay}{\frac{alphax}{-cos2phi}} - \frac{alphax}{\frac{alphay}{sin2phi}}}{u0}}\\ \mathbf{else}:\\ \;\;\;\;\frac{alphay \cdot \left(\mathsf{log1p}\left(-u0\right) \cdot \left(-alphay\right)\right)}{sin2phi}\\ \end{array} \]

Alternative 4: 89.7% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{alphay}{\frac{alphax}{-cos2phi}} - \frac{alphax}{\frac{alphay}{sin2phi}}\\ \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 30:\\ \;\;\;\;\frac{-0.5}{\frac{t_0}{alphay \cdot \left(alphax \cdot \left(u0 \cdot u0\right)\right)}} - \frac{alphax \cdot alphay}{\frac{t_0}{u0}}\\ \mathbf{else}:\\ \;\;\;\;\frac{alphay \cdot \left(-alphay\right)}{\left(\left(sin2phi \cdot 0.5 - \frac{sin2phi}{u0}\right) - u0 \cdot \left(sin2phi \cdot -0.08333333333333333\right)\right) - u0 \cdot \left(u0 \cdot \left(sin2phi \cdot -0.08333333333333333 + sin2phi \cdot 0.041666666666666664\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (let* ((t_0
         (- (/ alphay (/ alphax (- cos2phi))) (/ alphax (/ alphay sin2phi)))))
   (if (<= (/ sin2phi (* alphay alphay)) 30.0)
     (-
      (/ -0.5 (/ t_0 (* alphay (* alphax (* u0 u0)))))
      (/ (* alphax alphay) (/ t_0 u0)))
     (/
      (* alphay (- alphay))
      (-
       (-
        (- (* sin2phi 0.5) (/ sin2phi u0))
        (* u0 (* sin2phi -0.08333333333333333)))
       (*
        u0
        (*
         u0
         (+
          (* sin2phi -0.08333333333333333)
          (* sin2phi 0.041666666666666664)))))))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float t_0 = (alphay / (alphax / -cos2phi)) - (alphax / (alphay / sin2phi));
	float tmp;
	if ((sin2phi / (alphay * alphay)) <= 30.0f) {
		tmp = (-0.5f / (t_0 / (alphay * (alphax * (u0 * u0))))) - ((alphax * alphay) / (t_0 / u0));
	} else {
		tmp = (alphay * -alphay) / ((((sin2phi * 0.5f) - (sin2phi / u0)) - (u0 * (sin2phi * -0.08333333333333333f))) - (u0 * (u0 * ((sin2phi * -0.08333333333333333f) + (sin2phi * 0.041666666666666664f)))));
	}
	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 = (alphay / (alphax / -cos2phi)) - (alphax / (alphay / sin2phi))
    if ((sin2phi / (alphay * alphay)) <= 30.0e0) then
        tmp = ((-0.5e0) / (t_0 / (alphay * (alphax * (u0 * u0))))) - ((alphax * alphay) / (t_0 / u0))
    else
        tmp = (alphay * -alphay) / ((((sin2phi * 0.5e0) - (sin2phi / u0)) - (u0 * (sin2phi * (-0.08333333333333333e0)))) - (u0 * (u0 * ((sin2phi * (-0.08333333333333333e0)) + (sin2phi * 0.041666666666666664e0)))))
    end if
    code = tmp
end function

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	t_0 = Float32(Float32(alphay / Float32(alphax / Float32(-cos2phi))) - Float32(alphax / Float32(alphay / sin2phi)))
	tmp = Float32(0.0)
	if (Float32(sin2phi / Float32(alphay * alphay)) <= Float32(30.0))
		tmp = Float32(Float32(Float32(-0.5) / Float32(t_0 / Float32(alphay * Float32(alphax * Float32(u0 * u0))))) - Float32(Float32(alphax * alphay) / Float32(t_0 / u0)));
	else
		tmp = Float32(Float32(alphay * Float32(-alphay)) / Float32(Float32(Float32(Float32(sin2phi * Float32(0.5)) - Float32(sin2phi / u0)) - Float32(u0 * Float32(sin2phi * Float32(-0.08333333333333333)))) - Float32(u0 * Float32(u0 * Float32(Float32(sin2phi * Float32(-0.08333333333333333)) + Float32(sin2phi * Float32(0.041666666666666664)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alphax, alphay, u0, cos2phi, sin2phi)
	t_0 = (alphay / (alphax / -cos2phi)) - (alphax / (alphay / sin2phi));
	tmp = single(0.0);
	if ((sin2phi / (alphay * alphay)) <= single(30.0))
		tmp = (single(-0.5) / (t_0 / (alphay * (alphax * (u0 * u0))))) - ((alphax * alphay) / (t_0 / u0));
	else
		tmp = (alphay * -alphay) / ((((sin2phi * single(0.5)) - (sin2phi / u0)) - (u0 * (sin2phi * single(-0.08333333333333333)))) - (u0 * (u0 * ((sin2phi * single(-0.08333333333333333)) + (sin2phi * single(0.041666666666666664))))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 sin2phi (*.f32 alphay alphay)) < 30

   1. Initial program 60.9%

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

      1. neg-sub0 60.9%

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

      2. div-sub 60.9%

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

      3. --rgt-identity 60.9%

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

      4. div-sub 60.9%

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

      5. --rgt-identity 60.9%

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

      6. neg-sub0 60.9%

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

      7. sub-neg 60.9%

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

      8. log1p-def 98.6%

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

   3. Simplified 98.6%

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

      1. +-commutative 98.6%

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

      2. associate-/r* 98.8%

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

      3. associate-/r* 98.6%

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

      4. frac-2neg 98.6%

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

      5. frac-add 98.2%

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

      6. distribute-neg-frac 98.2%

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

   5. Applied egg-rr 98.2%

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

      1. +-commutative 98.2%

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

      2. distribute-rgt-neg-out 98.2%

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

      3. unsub-neg 98.2%

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

      4. associate-*l/ 98.3%

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

      5. associate-/l* 98.2%

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

      6. *-commutative 98.2%

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

      7. distribute-lft-neg-out 98.2%

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

      8. distribute-rgt-neg-in 98.2%

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

   7. Simplified 98.2%

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

   8. Taylor expanded in u0 around 0 83.8%

      \[\leadsto \color{blue}{-1 \cdot \frac{u0 \cdot \left(alphay \cdot alphax\right)}{-1 \cdot \frac{cos2phi \cdot alphay}{alphax} - \frac{sin2phi \cdot alphax}{alphay}} + -0.5 \cdot \frac{{u0}^{2} \cdot \left(alphay \cdot alphax\right)}{-1 \cdot \frac{cos2phi \cdot alphay}{alphax} - \frac{sin2phi \cdot alphax}{alphay}}} \]
   9. Step-by-step derivation

      1. +-commutative 83.8%

         \[\leadsto \color{blue}{-0.5 \cdot \frac{{u0}^{2} \cdot \left(alphay \cdot alphax\right)}{-1 \cdot \frac{cos2phi \cdot alphay}{alphax} - \frac{sin2phi \cdot alphax}{alphay}} + -1 \cdot \frac{u0 \cdot \left(alphay \cdot alphax\right)}{-1 \cdot \frac{cos2phi \cdot alphay}{alphax} - \frac{sin2phi \cdot alphax}{alphay}}} \]

      2. mul-1-neg 83.8%

         \[\leadsto -0.5 \cdot \frac{{u0}^{2} \cdot \left(alphay \cdot alphax\right)}{-1 \cdot \frac{cos2phi \cdot alphay}{alphax} - \frac{sin2phi \cdot alphax}{alphay}} + \color{blue}{\left(-\frac{u0 \cdot \left(alphay \cdot alphax\right)}{-1 \cdot \frac{cos2phi \cdot alphay}{alphax} - \frac{sin2phi \cdot alphax}{alphay}}\right)} \]

      3. unsub-neg 83.8%

         \[\leadsto \color{blue}{-0.5 \cdot \frac{{u0}^{2} \cdot \left(alphay \cdot alphax\right)}{-1 \cdot \frac{cos2phi \cdot alphay}{alphax} - \frac{sin2phi \cdot alphax}{alphay}} - \frac{u0 \cdot \left(alphay \cdot alphax\right)}{-1 \cdot \frac{cos2phi \cdot alphay}{alphax} - \frac{sin2phi \cdot alphax}{alphay}}} \]

   10. Simplified 83.7%

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

   if 30 < (/.f32 sin2phi (*.f32 alphay alphay))

   1. Initial program 70.7%

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

      1. associate-/r* 70.7%

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

   3. Simplified 70.7%

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

   4. Taylor expanded in cos2phi around 0 70.6%

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

      1. mul-1-neg 70.6%

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

      2. unpow2 70.6%

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

      3. associate-/l* 69.8%

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

      4. distribute-neg-frac 69.8%

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

      5. distribute-rgt-neg-in 69.8%

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

      6. sub-neg 69.8%

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

      7. mul-1-neg 69.8%

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

      8. log1p-def 96.7%

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

      9. mul-1-neg 96.7%

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

   6. Simplified 96.7%

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

   7. Taylor expanded in u0 around 0 92.1%

      \[\leadsto \frac{alphay \cdot \left(-alphay\right)}{\color{blue}{-1 \cdot \left({u0}^{2} \cdot \left(-0.25 \cdot sin2phi + \left(0.16666666666666666 \cdot sin2phi + -0.5 \cdot \left(0.25 \cdot sin2phi + -0.3333333333333333 \cdot sin2phi\right)\right)\right)\right) + \left(-1 \cdot \left(u0 \cdot \left(0.25 \cdot sin2phi + -0.3333333333333333 \cdot sin2phi\right)\right) + \left(-1 \cdot \frac{sin2phi}{u0} + 0.5 \cdot sin2phi\right)\right)}} \]
   8. Step-by-step derivation

      1. +-commutative 92.1%

         \[\leadsto \frac{alphay \cdot \left(-alphay\right)}{\color{blue}{\left(-1 \cdot \left(u0 \cdot \left(0.25 \cdot sin2phi + -0.3333333333333333 \cdot sin2phi\right)\right) + \left(-1 \cdot \frac{sin2phi}{u0} + 0.5 \cdot sin2phi\right)\right) + -1 \cdot \left({u0}^{2} \cdot \left(-0.25 \cdot sin2phi + \left(0.16666666666666666 \cdot sin2phi + -0.5 \cdot \left(0.25 \cdot sin2phi + -0.3333333333333333 \cdot sin2phi\right)\right)\right)\right)}} \]

      2. mul-1-neg 92.1%

         \[\leadsto \frac{alphay \cdot \left(-alphay\right)}{\left(-1 \cdot \left(u0 \cdot \left(0.25 \cdot sin2phi + -0.3333333333333333 \cdot sin2phi\right)\right) + \left(-1 \cdot \frac{sin2phi}{u0} + 0.5 \cdot sin2phi\right)\right) + \color{blue}{\left(-{u0}^{2} \cdot \left(-0.25 \cdot sin2phi + \left(0.16666666666666666 \cdot sin2phi + -0.5 \cdot \left(0.25 \cdot sin2phi + -0.3333333333333333 \cdot sin2phi\right)\right)\right)\right)}} \]

      3. unsub-neg 92.1%

         \[\leadsto \frac{alphay \cdot \left(-alphay\right)}{\color{blue}{\left(-1 \cdot \left(u0 \cdot \left(0.25 \cdot sin2phi + -0.3333333333333333 \cdot sin2phi\right)\right) + \left(-1 \cdot \frac{sin2phi}{u0} + 0.5 \cdot sin2phi\right)\right) - {u0}^{2} \cdot \left(-0.25 \cdot sin2phi + \left(0.16666666666666666 \cdot sin2phi + -0.5 \cdot \left(0.25 \cdot sin2phi + -0.3333333333333333 \cdot sin2phi\right)\right)\right)}} \]

   9. Simplified 92.1%

      \[\leadsto \frac{alphay \cdot \left(-alphay\right)}{\color{blue}{\left(\left(sin2phi \cdot 0.5 - \frac{sin2phi}{u0}\right) - u0 \cdot \left(sin2phi \cdot -0.08333333333333333\right)\right) - u0 \cdot \left(u0 \cdot \left(sin2phi \cdot -0.08333333333333333 + sin2phi \cdot 0.041666666666666664\right)\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 88.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 30:\\ \;\;\;\;\frac{-0.5}{\frac{\frac{alphay}{\frac{alphax}{-cos2phi}} - \frac{alphax}{\frac{alphay}{sin2phi}}}{alphay \cdot \left(alphax \cdot \left(u0 \cdot u0\right)\right)}} - \frac{alphax \cdot alphay}{\frac{\frac{alphay}{\frac{alphax}{-cos2phi}} - \frac{alphax}{\frac{alphay}{sin2phi}}}{u0}}\\ \mathbf{else}:\\ \;\;\;\;\frac{alphay \cdot \left(-alphay\right)}{\left(\left(sin2phi \cdot 0.5 - \frac{sin2phi}{u0}\right) - u0 \cdot \left(sin2phi \cdot -0.08333333333333333\right)\right) - u0 \cdot \left(u0 \cdot \left(sin2phi \cdot -0.08333333333333333 + sin2phi \cdot 0.041666666666666664\right)\right)}\\ \end{array} \]

Alternative 5: 84.1% accurate, 3.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 1.2000000424450263 \cdot 10^{-6}:\\ \;\;\;\;\frac{u0 \cdot \left(alphax \cdot \left(-alphay\right)\right)}{\frac{alphax \cdot \left(-sin2phi\right)}{alphay} - \frac{cos2phi \cdot alphay}{alphax}}\\ \mathbf{else}:\\ \;\;\;\;\frac{alphay \cdot \left(-alphay\right)}{\left(\left(sin2phi \cdot 0.5 - \frac{sin2phi}{u0}\right) - u0 \cdot \left(sin2phi \cdot -0.08333333333333333\right)\right) - u0 \cdot \left(u0 \cdot \left(sin2phi \cdot -0.08333333333333333 + sin2phi \cdot 0.041666666666666664\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (if (<= (/ sin2phi (* alphay alphay)) 1.2000000424450263e-6)
   (/
    (* u0 (* alphax (- alphay)))
    (- (/ (* alphax (- sin2phi)) alphay) (/ (* cos2phi alphay) alphax)))
   (/
    (* alphay (- alphay))
    (-
     (-
      (- (* sin2phi 0.5) (/ sin2phi u0))
      (* u0 (* sin2phi -0.08333333333333333)))
     (*
      u0
      (*
       u0
       (+
        (* sin2phi -0.08333333333333333)
        (* sin2phi 0.041666666666666664))))))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float tmp;
	if ((sin2phi / (alphay * alphay)) <= 1.2000000424450263e-6f) {
		tmp = (u0 * (alphax * -alphay)) / (((alphax * -sin2phi) / alphay) - ((cos2phi * alphay) / alphax));
	} else {
		tmp = (alphay * -alphay) / ((((sin2phi * 0.5f) - (sin2phi / u0)) - (u0 * (sin2phi * -0.08333333333333333f))) - (u0 * (u0 * ((sin2phi * -0.08333333333333333f) + (sin2phi * 0.041666666666666664f)))));
	}
	return tmp;
}

Fortran

real(4) function code(alphax, alphay, u0, cos2phi, sin2phi)
    real(4), intent (in) :: alphax
    real(4), intent (in) :: alphay
    real(4), intent (in) :: u0
    real(4), intent (in) :: cos2phi
    real(4), intent (in) :: sin2phi
    real(4) :: tmp
    if ((sin2phi / (alphay * alphay)) <= 1.2000000424450263e-6) then
        tmp = (u0 * (alphax * -alphay)) / (((alphax * -sin2phi) / alphay) - ((cos2phi * alphay) / alphax))
    else
        tmp = (alphay * -alphay) / ((((sin2phi * 0.5e0) - (sin2phi / u0)) - (u0 * (sin2phi * (-0.08333333333333333e0)))) - (u0 * (u0 * ((sin2phi * (-0.08333333333333333e0)) + (sin2phi * 0.041666666666666664e0)))))
    end if
    code = tmp
end function

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = Float32(0.0)
	if (Float32(sin2phi / Float32(alphay * alphay)) <= Float32(1.2000000424450263e-6))
		tmp = Float32(Float32(u0 * Float32(alphax * Float32(-alphay))) / Float32(Float32(Float32(alphax * Float32(-sin2phi)) / alphay) - Float32(Float32(cos2phi * alphay) / alphax)));
	else
		tmp = Float32(Float32(alphay * Float32(-alphay)) / Float32(Float32(Float32(Float32(sin2phi * Float32(0.5)) - Float32(sin2phi / u0)) - Float32(u0 * Float32(sin2phi * Float32(-0.08333333333333333)))) - Float32(u0 * Float32(u0 * Float32(Float32(sin2phi * Float32(-0.08333333333333333)) + Float32(sin2phi * Float32(0.041666666666666664)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = single(0.0);
	if ((sin2phi / (alphay * alphay)) <= single(1.2000000424450263e-6))
		tmp = (u0 * (alphax * -alphay)) / (((alphax * -sin2phi) / alphay) - ((cos2phi * alphay) / alphax));
	else
		tmp = (alphay * -alphay) / ((((sin2phi * single(0.5)) - (sin2phi / u0)) - (u0 * (sin2phi * single(-0.08333333333333333)))) - (u0 * (u0 * ((sin2phi * single(-0.08333333333333333)) + (sin2phi * single(0.041666666666666664))))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 sin2phi (*.f32 alphay alphay)) < 1.2e-6

   1. Initial program 61.2%

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

      1. neg-sub0 61.2%

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

      2. div-sub 61.2%

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

      3. --rgt-identity 61.2%

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

      4. div-sub 61.2%

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

      5. --rgt-identity 61.2%

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

      6. neg-sub0 61.2%

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

      7. sub-neg 61.2%

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

      8. log1p-def 98.7%

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

   3. Simplified 98.7%

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

      1. +-commutative 98.7%

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

      2. associate-/r* 98.7%

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

      3. associate-/r* 98.5%

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

      4. frac-2neg 98.5%

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

      5. frac-add 98.1%

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

      6. distribute-neg-frac 98.1%

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

   5. Applied egg-rr 98.1%

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

      1. +-commutative 98.1%

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

      2. distribute-rgt-neg-out 98.1%

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

      3. unsub-neg 98.1%

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

      4. associate-*l/ 98.2%

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

      5. associate-/l* 98.1%

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

      6. *-commutative 98.1%

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

      7. distribute-lft-neg-out 98.1%

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

      8. distribute-rgt-neg-in 98.1%

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

   7. Simplified 98.1%

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

   8. Taylor expanded in u0 around 0 69.8%

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

   if 1.2e-6 < (/.f32 sin2phi (*.f32 alphay alphay))

   1. Initial program 69.1%

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

      1. associate-/r* 69.1%

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

   3. Simplified 69.1%

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

   4. Taylor expanded in cos2phi around 0 67.9%

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

      1. mul-1-neg 67.9%

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

      2. unpow2 67.9%

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

      3. associate-/l* 67.3%

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

      4. distribute-neg-frac 67.3%

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

      5. distribute-rgt-neg-in 67.3%

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

      6. sub-neg 67.3%

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

      7. mul-1-neg 67.3%

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

      8. log1p-def 94.1%

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

      9. mul-1-neg 94.1%

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

   6. Simplified 94.1%

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

   7. Taylor expanded in u0 around 0 90.0%

      \[\leadsto \frac{alphay \cdot \left(-alphay\right)}{\color{blue}{-1 \cdot \left({u0}^{2} \cdot \left(-0.25 \cdot sin2phi + \left(0.16666666666666666 \cdot sin2phi + -0.5 \cdot \left(0.25 \cdot sin2phi + -0.3333333333333333 \cdot sin2phi\right)\right)\right)\right) + \left(-1 \cdot \left(u0 \cdot \left(0.25 \cdot sin2phi + -0.3333333333333333 \cdot sin2phi\right)\right) + \left(-1 \cdot \frac{sin2phi}{u0} + 0.5 \cdot sin2phi\right)\right)}} \]
   8. Step-by-step derivation

      1. +-commutative 90.0%

         \[\leadsto \frac{alphay \cdot \left(-alphay\right)}{\color{blue}{\left(-1 \cdot \left(u0 \cdot \left(0.25 \cdot sin2phi + -0.3333333333333333 \cdot sin2phi\right)\right) + \left(-1 \cdot \frac{sin2phi}{u0} + 0.5 \cdot sin2phi\right)\right) + -1 \cdot \left({u0}^{2} \cdot \left(-0.25 \cdot sin2phi + \left(0.16666666666666666 \cdot sin2phi + -0.5 \cdot \left(0.25 \cdot sin2phi + -0.3333333333333333 \cdot sin2phi\right)\right)\right)\right)}} \]

      2. mul-1-neg 90.0%

         \[\leadsto \frac{alphay \cdot \left(-alphay\right)}{\left(-1 \cdot \left(u0 \cdot \left(0.25 \cdot sin2phi + -0.3333333333333333 \cdot sin2phi\right)\right) + \left(-1 \cdot \frac{sin2phi}{u0} + 0.5 \cdot sin2phi\right)\right) + \color{blue}{\left(-{u0}^{2} \cdot \left(-0.25 \cdot sin2phi + \left(0.16666666666666666 \cdot sin2phi + -0.5 \cdot \left(0.25 \cdot sin2phi + -0.3333333333333333 \cdot sin2phi\right)\right)\right)\right)}} \]

      3. unsub-neg 90.0%

         \[\leadsto \frac{alphay \cdot \left(-alphay\right)}{\color{blue}{\left(-1 \cdot \left(u0 \cdot \left(0.25 \cdot sin2phi + -0.3333333333333333 \cdot sin2phi\right)\right) + \left(-1 \cdot \frac{sin2phi}{u0} + 0.5 \cdot sin2phi\right)\right) - {u0}^{2} \cdot \left(-0.25 \cdot sin2phi + \left(0.16666666666666666 \cdot sin2phi + -0.5 \cdot \left(0.25 \cdot sin2phi + -0.3333333333333333 \cdot sin2phi\right)\right)\right)}} \]

   9. Simplified 90.0%

      \[\leadsto \frac{alphay \cdot \left(-alphay\right)}{\color{blue}{\left(\left(sin2phi \cdot 0.5 - \frac{sin2phi}{u0}\right) - u0 \cdot \left(sin2phi \cdot -0.08333333333333333\right)\right) - u0 \cdot \left(u0 \cdot \left(sin2phi \cdot -0.08333333333333333 + sin2phi \cdot 0.041666666666666664\right)\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 82.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 1.2000000424450263 \cdot 10^{-6}:\\ \;\;\;\;\frac{u0 \cdot \left(alphax \cdot \left(-alphay\right)\right)}{\frac{alphax \cdot \left(-sin2phi\right)}{alphay} - \frac{cos2phi \cdot alphay}{alphax}}\\ \mathbf{else}:\\ \;\;\;\;\frac{alphay \cdot \left(-alphay\right)}{\left(\left(sin2phi \cdot 0.5 - \frac{sin2phi}{u0}\right) - u0 \cdot \left(sin2phi \cdot -0.08333333333333333\right)\right) - u0 \cdot \left(u0 \cdot \left(sin2phi \cdot -0.08333333333333333 + sin2phi \cdot 0.041666666666666664\right)\right)}\\ \end{array} \]

Alternative 6: 83.2% accurate, 4.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 1.2000000424450263 \cdot 10^{-6}:\\ \;\;\;\;\frac{u0 \cdot \left(alphax \cdot \left(-alphay\right)\right)}{\frac{alphax \cdot \left(-sin2phi\right)}{alphay} - \frac{cos2phi \cdot alphay}{alphax}}\\ \mathbf{else}:\\ \;\;\;\;\frac{alphay \cdot \left(-alphay\right)}{\left(sin2phi \cdot 0.5 - \frac{sin2phi}{u0}\right) - u0 \cdot \left(sin2phi \cdot -0.08333333333333333\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (if (<= (/ sin2phi (* alphay alphay)) 1.2000000424450263e-6)
   (/
    (* u0 (* alphax (- alphay)))
    (- (/ (* alphax (- sin2phi)) alphay) (/ (* cos2phi alphay) alphax)))
   (/
    (* alphay (- alphay))
    (-
     (- (* sin2phi 0.5) (/ sin2phi u0))
     (* u0 (* sin2phi -0.08333333333333333))))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float tmp;
	if ((sin2phi / (alphay * alphay)) <= 1.2000000424450263e-6f) {
		tmp = (u0 * (alphax * -alphay)) / (((alphax * -sin2phi) / alphay) - ((cos2phi * alphay) / alphax));
	} else {
		tmp = (alphay * -alphay) / (((sin2phi * 0.5f) - (sin2phi / u0)) - (u0 * (sin2phi * -0.08333333333333333f)));
	}
	return tmp;
}

Fortran

real(4) function code(alphax, alphay, u0, cos2phi, sin2phi)
    real(4), intent (in) :: alphax
    real(4), intent (in) :: alphay
    real(4), intent (in) :: u0
    real(4), intent (in) :: cos2phi
    real(4), intent (in) :: sin2phi
    real(4) :: tmp
    if ((sin2phi / (alphay * alphay)) <= 1.2000000424450263e-6) then
        tmp = (u0 * (alphax * -alphay)) / (((alphax * -sin2phi) / alphay) - ((cos2phi * alphay) / alphax))
    else
        tmp = (alphay * -alphay) / (((sin2phi * 0.5e0) - (sin2phi / u0)) - (u0 * (sin2phi * (-0.08333333333333333e0))))
    end if
    code = tmp
end function

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = Float32(0.0)
	if (Float32(sin2phi / Float32(alphay * alphay)) <= Float32(1.2000000424450263e-6))
		tmp = Float32(Float32(u0 * Float32(alphax * Float32(-alphay))) / Float32(Float32(Float32(alphax * Float32(-sin2phi)) / alphay) - Float32(Float32(cos2phi * alphay) / alphax)));
	else
		tmp = Float32(Float32(alphay * Float32(-alphay)) / Float32(Float32(Float32(sin2phi * Float32(0.5)) - Float32(sin2phi / u0)) - Float32(u0 * Float32(sin2phi * Float32(-0.08333333333333333)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = single(0.0);
	if ((sin2phi / (alphay * alphay)) <= single(1.2000000424450263e-6))
		tmp = (u0 * (alphax * -alphay)) / (((alphax * -sin2phi) / alphay) - ((cos2phi * alphay) / alphax));
	else
		tmp = (alphay * -alphay) / (((sin2phi * single(0.5)) - (sin2phi / u0)) - (u0 * (sin2phi * single(-0.08333333333333333))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 sin2phi (*.f32 alphay alphay)) < 1.2e-6

   1. Initial program 61.2%

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

      1. neg-sub0 61.2%

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

      2. div-sub 61.2%

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

      3. --rgt-identity 61.2%

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

      4. div-sub 61.2%

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

      5. --rgt-identity 61.2%

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

      6. neg-sub0 61.2%

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

      7. sub-neg 61.2%

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

      8. log1p-def 98.7%

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

   3. Simplified 98.7%

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

      1. +-commutative 98.7%

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

      2. associate-/r* 98.7%

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

      3. associate-/r* 98.5%

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

      4. frac-2neg 98.5%

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

      5. frac-add 98.1%

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

      6. distribute-neg-frac 98.1%

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

   5. Applied egg-rr 98.1%

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

      1. +-commutative 98.1%

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

      2. distribute-rgt-neg-out 98.1%

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

      3. unsub-neg 98.1%

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

      4. associate-*l/ 98.2%

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

      5. associate-/l* 98.1%

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

      6. *-commutative 98.1%

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

      7. distribute-lft-neg-out 98.1%

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

      8. distribute-rgt-neg-in 98.1%

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

   7. Simplified 98.1%

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

   8. Taylor expanded in u0 around 0 69.8%

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

   if 1.2e-6 < (/.f32 sin2phi (*.f32 alphay alphay))

   1. Initial program 69.1%

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

      1. associate-/r* 69.1%

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

   3. Simplified 69.1%

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

   4. Taylor expanded in cos2phi around 0 67.9%

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

      1. mul-1-neg 67.9%

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

      2. unpow2 67.9%

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

      3. associate-/l* 67.3%

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

      4. distribute-neg-frac 67.3%

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

      5. distribute-rgt-neg-in 67.3%

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

      6. sub-neg 67.3%

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

      7. mul-1-neg 67.3%

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

      8. log1p-def 94.1%

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

      9. mul-1-neg 94.1%

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

   6. Simplified 94.1%

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

   7. Taylor expanded in u0 around 0 88.0%

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

      1. +-commutative 88.0%

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

      2. mul-1-neg 88.0%

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

      3. unsub-neg 88.0%

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

      4. +-commutative 88.0%

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

      5. mul-1-neg 88.0%

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

      6. unsub-neg 88.0%

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

      7. *-commutative 88.0%

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

      8. distribute-rgt-out 88.0%

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

      9. metadata-eval 88.0%

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

   9. Simplified 88.0%

      \[\leadsto \frac{alphay \cdot \left(-alphay\right)}{\color{blue}{\left(sin2phi \cdot 0.5 - \frac{sin2phi}{u0}\right) - u0 \cdot \left(sin2phi \cdot -0.08333333333333333\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 81.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 1.2000000424450263 \cdot 10^{-6}:\\ \;\;\;\;\frac{u0 \cdot \left(alphax \cdot \left(-alphay\right)\right)}{\frac{alphax \cdot \left(-sin2phi\right)}{alphay} - \frac{cos2phi \cdot alphay}{alphax}}\\ \mathbf{else}:\\ \;\;\;\;\frac{alphay \cdot \left(-alphay\right)}{\left(sin2phi \cdot 0.5 - \frac{sin2phi}{u0}\right) - u0 \cdot \left(sin2phi \cdot -0.08333333333333333\right)}\\ \end{array} \]

Alternative 7: 83.3% accurate, 4.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 1.2000000424450263 \cdot 10^{-6}:\\ \;\;\;\;u0 \cdot \frac{1}{\frac{cos2phi}{alphax \cdot alphax} + \frac{\frac{sin2phi}{alphay}}{alphay}}\\ \mathbf{else}:\\ \;\;\;\;\frac{alphay \cdot \left(-alphay\right)}{\left(sin2phi \cdot 0.5 - \frac{sin2phi}{u0}\right) - u0 \cdot \left(sin2phi \cdot -0.08333333333333333\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (if (<= (/ sin2phi (* alphay alphay)) 1.2000000424450263e-6)
   (*
    u0
    (/ 1.0 (+ (/ cos2phi (* alphax alphax)) (/ (/ sin2phi alphay) alphay))))
   (/
    (* alphay (- alphay))
    (-
     (- (* sin2phi 0.5) (/ sin2phi u0))
     (* u0 (* sin2phi -0.08333333333333333))))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float tmp;
	if ((sin2phi / (alphay * alphay)) <= 1.2000000424450263e-6f) {
		tmp = u0 * (1.0f / ((cos2phi / (alphax * alphax)) + ((sin2phi / alphay) / alphay)));
	} else {
		tmp = (alphay * -alphay) / (((sin2phi * 0.5f) - (sin2phi / u0)) - (u0 * (sin2phi * -0.08333333333333333f)));
	}
	return tmp;
}

Fortran

real(4) function code(alphax, alphay, u0, cos2phi, sin2phi)
    real(4), intent (in) :: alphax
    real(4), intent (in) :: alphay
    real(4), intent (in) :: u0
    real(4), intent (in) :: cos2phi
    real(4), intent (in) :: sin2phi
    real(4) :: tmp
    if ((sin2phi / (alphay * alphay)) <= 1.2000000424450263e-6) then
        tmp = u0 * (1.0e0 / ((cos2phi / (alphax * alphax)) + ((sin2phi / alphay) / alphay)))
    else
        tmp = (alphay * -alphay) / (((sin2phi * 0.5e0) - (sin2phi / u0)) - (u0 * (sin2phi * (-0.08333333333333333e0))))
    end if
    code = tmp
end function

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = Float32(0.0)
	if (Float32(sin2phi / Float32(alphay * alphay)) <= Float32(1.2000000424450263e-6))
		tmp = Float32(u0 * Float32(Float32(1.0) / Float32(Float32(cos2phi / Float32(alphax * alphax)) + Float32(Float32(sin2phi / alphay) / alphay))));
	else
		tmp = Float32(Float32(alphay * Float32(-alphay)) / Float32(Float32(Float32(sin2phi * Float32(0.5)) - Float32(sin2phi / u0)) - Float32(u0 * Float32(sin2phi * Float32(-0.08333333333333333)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = single(0.0);
	if ((sin2phi / (alphay * alphay)) <= single(1.2000000424450263e-6))
		tmp = u0 * (single(1.0) / ((cos2phi / (alphax * alphax)) + ((sin2phi / alphay) / alphay)));
	else
		tmp = (alphay * -alphay) / (((sin2phi * single(0.5)) - (sin2phi / u0)) - (u0 * (sin2phi * single(-0.08333333333333333))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 sin2phi (*.f32 alphay alphay)) < 1.2e-6

   1. Initial program 61.2%

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

      1. associate-/r* 61.2%

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

   3. Simplified 61.2%

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

   4. Taylor expanded in u0 around 0 69.8%

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

      1. unpow2 69.8%

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

      2. unpow2 69.8%

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

   6. Simplified 69.8%

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

      1. associate-/r* 69.8%

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

      2. div-inv 69.8%

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

   8. Applied egg-rr 69.8%

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

      1. div-inv 69.8%

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

      2. un-div-inv 69.8%

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

   10. Applied egg-rr 69.8%

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

   if 1.2e-6 < (/.f32 sin2phi (*.f32 alphay alphay))

   1. Initial program 69.1%

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

      1. associate-/r* 69.1%

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

   3. Simplified 69.1%

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

   4. Taylor expanded in cos2phi around 0 67.9%

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

      1. mul-1-neg 67.9%

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

      2. unpow2 67.9%

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

      3. associate-/l* 67.3%

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

      4. distribute-neg-frac 67.3%

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

      5. distribute-rgt-neg-in 67.3%

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

      6. sub-neg 67.3%

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

      7. mul-1-neg 67.3%

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

      8. log1p-def 94.1%

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

      9. mul-1-neg 94.1%

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

   6. Simplified 94.1%

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

   7. Taylor expanded in u0 around 0 88.0%

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

      1. +-commutative 88.0%

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

      2. mul-1-neg 88.0%

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

      3. unsub-neg 88.0%

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

      4. +-commutative 88.0%

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

      5. mul-1-neg 88.0%

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

      6. unsub-neg 88.0%

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

      7. *-commutative 88.0%

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

      8. distribute-rgt-out 88.0%

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

      9. metadata-eval 88.0%

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

   9. Simplified 88.0%

      \[\leadsto \frac{alphay \cdot \left(-alphay\right)}{\color{blue}{\left(sin2phi \cdot 0.5 - \frac{sin2phi}{u0}\right) - u0 \cdot \left(sin2phi \cdot -0.08333333333333333\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 81.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 1.2000000424450263 \cdot 10^{-6}:\\ \;\;\;\;u0 \cdot \frac{1}{\frac{cos2phi}{alphax \cdot alphax} + \frac{\frac{sin2phi}{alphay}}{alphay}}\\ \mathbf{else}:\\ \;\;\;\;\frac{alphay \cdot \left(-alphay\right)}{\left(sin2phi \cdot 0.5 - \frac{sin2phi}{u0}\right) - u0 \cdot \left(sin2phi \cdot -0.08333333333333333\right)}\\ \end{array} \]

Alternative 8: 81.9% accurate, 5.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 1.2000000424450263 \cdot 10^{-6}:\\ \;\;\;\;u0 \cdot \frac{1}{\frac{cos2phi}{alphax \cdot alphax} + \frac{\frac{sin2phi}{alphay}}{alphay}}\\ \mathbf{else}:\\ \;\;\;\;\frac{alphay}{sin2phi} \cdot \frac{alphay}{-0.5 + \frac{1}{u0}}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (if (<= (/ sin2phi (* alphay alphay)) 1.2000000424450263e-6)
   (*
    u0
    (/ 1.0 (+ (/ cos2phi (* alphax alphax)) (/ (/ sin2phi alphay) alphay))))
   (* (/ alphay sin2phi) (/ alphay (+ -0.5 (/ 1.0 u0))))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float tmp;
	if ((sin2phi / (alphay * alphay)) <= 1.2000000424450263e-6f) {
		tmp = u0 * (1.0f / ((cos2phi / (alphax * alphax)) + ((sin2phi / alphay) / alphay)));
	} else {
		tmp = (alphay / sin2phi) * (alphay / (-0.5f + (1.0f / u0)));
	}
	return tmp;
}

Fortran

real(4) function code(alphax, alphay, u0, cos2phi, sin2phi)
    real(4), intent (in) :: alphax
    real(4), intent (in) :: alphay
    real(4), intent (in) :: u0
    real(4), intent (in) :: cos2phi
    real(4), intent (in) :: sin2phi
    real(4) :: tmp
    if ((sin2phi / (alphay * alphay)) <= 1.2000000424450263e-6) then
        tmp = u0 * (1.0e0 / ((cos2phi / (alphax * alphax)) + ((sin2phi / alphay) / alphay)))
    else
        tmp = (alphay / sin2phi) * (alphay / ((-0.5e0) + (1.0e0 / u0)))
    end if
    code = tmp
end function

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = Float32(0.0)
	if (Float32(sin2phi / Float32(alphay * alphay)) <= Float32(1.2000000424450263e-6))
		tmp = Float32(u0 * Float32(Float32(1.0) / Float32(Float32(cos2phi / Float32(alphax * alphax)) + Float32(Float32(sin2phi / alphay) / alphay))));
	else
		tmp = Float32(Float32(alphay / sin2phi) * Float32(alphay / Float32(Float32(-0.5) + Float32(Float32(1.0) / u0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = single(0.0);
	if ((sin2phi / (alphay * alphay)) <= single(1.2000000424450263e-6))
		tmp = u0 * (single(1.0) / ((cos2phi / (alphax * alphax)) + ((sin2phi / alphay) / alphay)));
	else
		tmp = (alphay / sin2phi) * (alphay / (single(-0.5) + (single(1.0) / u0)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 sin2phi (*.f32 alphay alphay)) < 1.2e-6

   1. Initial program 61.2%

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

      1. associate-/r* 61.2%

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

   3. Simplified 61.2%

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

   4. Taylor expanded in u0 around 0 69.8%

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

      1. unpow2 69.8%

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

      2. unpow2 69.8%

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

   6. Simplified 69.8%

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

      1. associate-/r* 69.8%

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

      2. div-inv 69.8%

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

   8. Applied egg-rr 69.8%

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

      1. div-inv 69.8%

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

      2. un-div-inv 69.8%

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

   10. Applied egg-rr 69.8%

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

   if 1.2e-6 < (/.f32 sin2phi (*.f32 alphay alphay))

   1. Initial program 69.1%

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

      1. associate-/r* 69.1%

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

   3. Simplified 69.1%

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

   4. Taylor expanded in cos2phi around 0 67.9%

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

      1. mul-1-neg 67.9%

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

      2. unpow2 67.9%

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

      3. associate-/l* 67.3%

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

      4. distribute-neg-frac 67.3%

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

      5. distribute-rgt-neg-in 67.3%

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

      6. sub-neg 67.3%

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

      7. mul-1-neg 67.3%

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

      8. log1p-def 94.1%

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

      9. mul-1-neg 94.1%

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

   6. Simplified 94.1%

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

   7. Taylor expanded in u0 around 0 83.5%

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

      1. +-commutative 83.5%

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

      2. mul-1-neg 83.5%

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

      3. unsub-neg 83.5%

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

      4. *-commutative 83.5%

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

   9. Simplified 83.5%

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

   10. Taylor expanded in sin2phi around -inf 83.5%

       \[\leadsto \color{blue}{\frac{{alphay}^{2}}{\left(\frac{1}{u0} - 0.5\right) \cdot sin2phi}} \]
   11. Step-by-step derivation

       1. unpow2 83.5%

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

       2. *-commutative 83.5%

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

       3. times-frac 84.5%

          \[\leadsto \color{blue}{\frac{alphay}{sin2phi} \cdot \frac{alphay}{\frac{1}{u0} - 0.5}} \]

       4. sub-neg 84.5%

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

       5. metadata-eval 84.5%

          \[\leadsto \frac{alphay}{sin2phi} \cdot \frac{alphay}{\frac{1}{u0} + \color{blue}{-0.5}} \]

   12. Simplified 84.5%

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

4. Final simplification 79.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 1.2000000424450263 \cdot 10^{-6}:\\ \;\;\;\;u0 \cdot \frac{1}{\frac{cos2phi}{alphax \cdot alphax} + \frac{\frac{sin2phi}{alphay}}{alphay}}\\ \mathbf{else}:\\ \;\;\;\;\frac{alphay}{sin2phi} \cdot \frac{alphay}{-0.5 + \frac{1}{u0}}\\ \end{array} \]

Alternative 9: 81.9% accurate, 6.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{sin2phi}{alphay \cdot alphay}\\ \mathbf{if}\;t_0 \leq 1.2000000424450263 \cdot 10^{-6}:\\ \;\;\;\;\frac{u0}{\frac{cos2phi}{alphax \cdot alphax} + t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{alphay}{sin2phi} \cdot \frac{alphay}{-0.5 + \frac{1}{u0}}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (let* ((t_0 (/ sin2phi (* alphay alphay))))
   (if (<= t_0 1.2000000424450263e-6)
     (/ u0 (+ (/ cos2phi (* alphax alphax)) t_0))
     (* (/ alphay sin2phi) (/ alphay (+ -0.5 (/ 1.0 u0)))))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float t_0 = sin2phi / (alphay * alphay);
	float tmp;
	if (t_0 <= 1.2000000424450263e-6f) {
		tmp = u0 / ((cos2phi / (alphax * alphax)) + t_0);
	} else {
		tmp = (alphay / sin2phi) * (alphay / (-0.5f + (1.0f / u0)));
	}
	return tmp;
}

Fortran

real(4) function code(alphax, alphay, u0, cos2phi, sin2phi)
    real(4), intent (in) :: alphax
    real(4), intent (in) :: alphay
    real(4), intent (in) :: u0
    real(4), intent (in) :: cos2phi
    real(4), intent (in) :: sin2phi
    real(4) :: t_0
    real(4) :: tmp
    t_0 = sin2phi / (alphay * alphay)
    if (t_0 <= 1.2000000424450263e-6) then
        tmp = u0 / ((cos2phi / (alphax * alphax)) + t_0)
    else
        tmp = (alphay / sin2phi) * (alphay / ((-0.5e0) + (1.0e0 / u0)))
    end if
    code = tmp
end function

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	t_0 = Float32(sin2phi / Float32(alphay * alphay))
	tmp = Float32(0.0)
	if (t_0 <= Float32(1.2000000424450263e-6))
		tmp = Float32(u0 / Float32(Float32(cos2phi / Float32(alphax * alphax)) + t_0));
	else
		tmp = Float32(Float32(alphay / sin2phi) * Float32(alphay / Float32(Float32(-0.5) + Float32(Float32(1.0) / u0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alphax, alphay, u0, cos2phi, sin2phi)
	t_0 = sin2phi / (alphay * alphay);
	tmp = single(0.0);
	if (t_0 <= single(1.2000000424450263e-6))
		tmp = u0 / ((cos2phi / (alphax * alphax)) + t_0);
	else
		tmp = (alphay / sin2phi) * (alphay / (single(-0.5) + (single(1.0) / u0)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 sin2phi (*.f32 alphay alphay)) < 1.2e-6

   1. Initial program 61.2%

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

      1. associate-/r* 61.2%

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

   3. Simplified 61.2%

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

   4. Taylor expanded in u0 around 0 69.8%

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

      1. unpow2 69.8%

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

      2. unpow2 69.8%

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

   6. Simplified 69.8%

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

   if 1.2e-6 < (/.f32 sin2phi (*.f32 alphay alphay))

   1. Initial program 69.1%

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

      1. associate-/r* 69.1%

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

   3. Simplified 69.1%

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

   4. Taylor expanded in cos2phi around 0 67.9%

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

      1. mul-1-neg 67.9%

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

      2. unpow2 67.9%

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

      3. associate-/l* 67.3%

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

      4. distribute-neg-frac 67.3%

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

      5. distribute-rgt-neg-in 67.3%

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

      6. sub-neg 67.3%

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

      7. mul-1-neg 67.3%

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

      8. log1p-def 94.1%

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

      9. mul-1-neg 94.1%

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

   6. Simplified 94.1%

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

   7. Taylor expanded in u0 around 0 83.5%

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

      1. +-commutative 83.5%

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

      2. mul-1-neg 83.5%

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

      3. unsub-neg 83.5%

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

      4. *-commutative 83.5%

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

   9. Simplified 83.5%

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

   10. Taylor expanded in sin2phi around -inf 83.5%

       \[\leadsto \color{blue}{\frac{{alphay}^{2}}{\left(\frac{1}{u0} - 0.5\right) \cdot sin2phi}} \]
   11. Step-by-step derivation

       1. unpow2 83.5%

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

       2. *-commutative 83.5%

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

       3. times-frac 84.5%

          \[\leadsto \color{blue}{\frac{alphay}{sin2phi} \cdot \frac{alphay}{\frac{1}{u0} - 0.5}} \]

       4. sub-neg 84.5%

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

       5. metadata-eval 84.5%

          \[\leadsto \frac{alphay}{sin2phi} \cdot \frac{alphay}{\frac{1}{u0} + \color{blue}{-0.5}} \]

   12. Simplified 84.5%

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

4. Final simplification 79.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 1.2000000424450263 \cdot 10^{-6}:\\ \;\;\;\;\frac{u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}}\\ \mathbf{else}:\\ \;\;\;\;\frac{alphay}{sin2phi} \cdot \frac{alphay}{-0.5 + \frac{1}{u0}}\\ \end{array} \]

Alternative 10: 81.9% accurate, 6.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 1.2000000424450263 \cdot 10^{-6}:\\ \;\;\;\;\frac{u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{\frac{sin2phi}{alphay}}{alphay}}\\ \mathbf{else}:\\ \;\;\;\;\frac{alphay}{sin2phi} \cdot \frac{alphay}{-0.5 + \frac{1}{u0}}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (if (<= (/ sin2phi (* alphay alphay)) 1.2000000424450263e-6)
   (/ u0 (+ (/ cos2phi (* alphax alphax)) (/ (/ sin2phi alphay) alphay)))
   (* (/ alphay sin2phi) (/ alphay (+ -0.5 (/ 1.0 u0))))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float tmp;
	if ((sin2phi / (alphay * alphay)) <= 1.2000000424450263e-6f) {
		tmp = u0 / ((cos2phi / (alphax * alphax)) + ((sin2phi / alphay) / alphay));
	} else {
		tmp = (alphay / sin2phi) * (alphay / (-0.5f + (1.0f / u0)));
	}
	return tmp;
}

Fortran

real(4) function code(alphax, alphay, u0, cos2phi, sin2phi)
    real(4), intent (in) :: alphax
    real(4), intent (in) :: alphay
    real(4), intent (in) :: u0
    real(4), intent (in) :: cos2phi
    real(4), intent (in) :: sin2phi
    real(4) :: tmp
    if ((sin2phi / (alphay * alphay)) <= 1.2000000424450263e-6) then
        tmp = u0 / ((cos2phi / (alphax * alphax)) + ((sin2phi / alphay) / alphay))
    else
        tmp = (alphay / sin2phi) * (alphay / ((-0.5e0) + (1.0e0 / u0)))
    end if
    code = tmp
end function

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = Float32(0.0)
	if (Float32(sin2phi / Float32(alphay * alphay)) <= Float32(1.2000000424450263e-6))
		tmp = Float32(u0 / Float32(Float32(cos2phi / Float32(alphax * alphax)) + Float32(Float32(sin2phi / alphay) / alphay)));
	else
		tmp = Float32(Float32(alphay / sin2phi) * Float32(alphay / Float32(Float32(-0.5) + Float32(Float32(1.0) / u0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = single(0.0);
	if ((sin2phi / (alphay * alphay)) <= single(1.2000000424450263e-6))
		tmp = u0 / ((cos2phi / (alphax * alphax)) + ((sin2phi / alphay) / alphay));
	else
		tmp = (alphay / sin2phi) * (alphay / (single(-0.5) + (single(1.0) / u0)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 sin2phi (*.f32 alphay alphay)) < 1.2e-6

   1. Initial program 61.2%

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

      1. associate-/r* 61.2%

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

   3. Simplified 61.2%

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

   4. Taylor expanded in u0 around 0 69.8%

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

      1. unpow2 69.8%

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

      2. unpow2 69.8%

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

   6. Simplified 69.8%

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

      1. associate-/r* 69.8%

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

      2. div-inv 69.8%

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

   8. Applied egg-rr 69.8%

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

      1. un-div-inv 69.8%

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

   10. Applied egg-rr 69.8%

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

   if 1.2e-6 < (/.f32 sin2phi (*.f32 alphay alphay))

   1. Initial program 69.1%

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

      1. associate-/r* 69.1%

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

   3. Simplified 69.1%

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

   4. Taylor expanded in cos2phi around 0 67.9%

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

      1. mul-1-neg 67.9%

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

      2. unpow2 67.9%

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

      3. associate-/l* 67.3%

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

      4. distribute-neg-frac 67.3%

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

      5. distribute-rgt-neg-in 67.3%

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

      6. sub-neg 67.3%

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

      7. mul-1-neg 67.3%

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

      8. log1p-def 94.1%

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

      9. mul-1-neg 94.1%

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

   6. Simplified 94.1%

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

   7. Taylor expanded in u0 around 0 83.5%

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

      1. +-commutative 83.5%

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

      2. mul-1-neg 83.5%

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

      3. unsub-neg 83.5%

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

      4. *-commutative 83.5%

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

   9. Simplified 83.5%

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

   10. Taylor expanded in sin2phi around -inf 83.5%

       \[\leadsto \color{blue}{\frac{{alphay}^{2}}{\left(\frac{1}{u0} - 0.5\right) \cdot sin2phi}} \]
   11. Step-by-step derivation

       1. unpow2 83.5%

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

       2. *-commutative 83.5%

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

       3. times-frac 84.5%

          \[\leadsto \color{blue}{\frac{alphay}{sin2phi} \cdot \frac{alphay}{\frac{1}{u0} - 0.5}} \]

       4. sub-neg 84.5%

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

       5. metadata-eval 84.5%

          \[\leadsto \frac{alphay}{sin2phi} \cdot \frac{alphay}{\frac{1}{u0} + \color{blue}{-0.5}} \]

   12. Simplified 84.5%

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

4. Final simplification 79.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 1.2000000424450263 \cdot 10^{-6}:\\ \;\;\;\;\frac{u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{\frac{sin2phi}{alphay}}{alphay}}\\ \mathbf{else}:\\ \;\;\;\;\frac{alphay}{sin2phi} \cdot \frac{alphay}{-0.5 + \frac{1}{u0}}\\ \end{array} \]

Alternative 11: 75.1% accurate, 6.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 5.0000000843119176 \cdot 10^{-17}:\\ \;\;\;\;\left(alphax \cdot alphax\right) \cdot \frac{u0}{cos2phi}\\ \mathbf{else}:\\ \;\;\;\;\frac{alphay}{sin2phi} \cdot \frac{alphay}{-0.5 + \frac{1}{u0}}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (if (<= (/ sin2phi (* alphay alphay)) 5.0000000843119176e-17)
   (* (* alphax alphax) (/ u0 cos2phi))
   (* (/ alphay sin2phi) (/ alphay (+ -0.5 (/ 1.0 u0))))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float tmp;
	if ((sin2phi / (alphay * alphay)) <= 5.0000000843119176e-17f) {
		tmp = (alphax * alphax) * (u0 / cos2phi);
	} else {
		tmp = (alphay / sin2phi) * (alphay / (-0.5f + (1.0f / u0)));
	}
	return tmp;
}

Fortran

real(4) function code(alphax, alphay, u0, cos2phi, sin2phi)
    real(4), intent (in) :: alphax
    real(4), intent (in) :: alphay
    real(4), intent (in) :: u0
    real(4), intent (in) :: cos2phi
    real(4), intent (in) :: sin2phi
    real(4) :: tmp
    if ((sin2phi / (alphay * alphay)) <= 5.0000000843119176e-17) then
        tmp = (alphax * alphax) * (u0 / cos2phi)
    else
        tmp = (alphay / sin2phi) * (alphay / ((-0.5e0) + (1.0e0 / u0)))
    end if
    code = tmp
end function

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = Float32(0.0)
	if (Float32(sin2phi / Float32(alphay * alphay)) <= Float32(5.0000000843119176e-17))
		tmp = Float32(Float32(alphax * alphax) * Float32(u0 / cos2phi));
	else
		tmp = Float32(Float32(alphay / sin2phi) * Float32(alphay / Float32(Float32(-0.5) + Float32(Float32(1.0) / u0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = single(0.0);
	if ((sin2phi / (alphay * alphay)) <= single(5.0000000843119176e-17))
		tmp = (alphax * alphax) * (u0 / cos2phi);
	else
		tmp = (alphay / sin2phi) * (alphay / (single(-0.5) + (single(1.0) / u0)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 sin2phi (*.f32 alphay alphay)) < 5.00000008e-17

   1. Initial program 61.4%

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

      1. associate-/r* 61.4%

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

   3. Simplified 61.4%

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

   4. Taylor expanded in u0 around 0 70.4%

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

      1. unpow2 70.4%

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

      2. unpow2 70.4%

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

   6. Simplified 70.4%

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

   7. Taylor expanded in cos2phi around inf 52.5%

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

      1. associate-/l* 52.4%

         \[\leadsto \color{blue}{\frac{u0}{\frac{cos2phi}{{alphax}^{2}}}} \]

      2. unpow2 52.4%

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

   9. Simplified 52.4%

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

       1. associate-/r/ 52.5%

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

   11. Applied egg-rr 52.5%

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

   if 5.00000008e-17 < (/.f32 sin2phi (*.f32 alphay alphay))

   1. Initial program 67.9%

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

      1. associate-/r* 67.9%

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

   3. Simplified 67.9%

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

   4. Taylor expanded in cos2phi around 0 64.3%

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

      1. mul-1-neg 64.3%

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

      2. unpow2 64.3%

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

      3. associate-/l* 63.8%

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

      4. distribute-neg-frac 63.8%

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

      5. distribute-rgt-neg-in 63.8%

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

      6. sub-neg 63.8%

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

      7. mul-1-neg 63.8%

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

      8. log1p-def 90.2%

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

      9. mul-1-neg 90.2%

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

   6. Simplified 90.2%

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

   7. Taylor expanded in u0 around 0 79.5%

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

      1. +-commutative 79.5%

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

      2. mul-1-neg 79.5%

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

      3. unsub-neg 79.5%

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

      4. *-commutative 79.5%

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

   9. Simplified 79.5%

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

   10. Taylor expanded in sin2phi around -inf 79.5%

       \[\leadsto \color{blue}{\frac{{alphay}^{2}}{\left(\frac{1}{u0} - 0.5\right) \cdot sin2phi}} \]
   11. Step-by-step derivation

       1. unpow2 79.5%

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

       2. *-commutative 79.5%

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

       3. times-frac 80.4%

          \[\leadsto \color{blue}{\frac{alphay}{sin2phi} \cdot \frac{alphay}{\frac{1}{u0} - 0.5}} \]

       4. sub-neg 80.4%

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

       5. metadata-eval 80.4%

          \[\leadsto \frac{alphay}{sin2phi} \cdot \frac{alphay}{\frac{1}{u0} + \color{blue}{-0.5}} \]

   12. Simplified 80.4%

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

4. Final simplification 73.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 5.0000000843119176 \cdot 10^{-17}:\\ \;\;\;\;\left(alphax \cdot alphax\right) \cdot \frac{u0}{cos2phi}\\ \mathbf{else}:\\ \;\;\;\;\frac{alphay}{sin2phi} \cdot \frac{alphay}{-0.5 + \frac{1}{u0}}\\ \end{array} \]

Alternative 12: 66.8% accurate, 8.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Split input into 2 regimes

2. if (/.f32 sin2phi (*.f32 alphay alphay)) < 5.00000008e-17

   1. Initial program 61.4%

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

      1. associate-/r* 61.4%

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

   3. Simplified 61.4%

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

   4. Taylor expanded in u0 around 0 70.4%

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

      1. unpow2 70.4%

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

      2. unpow2 70.4%

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

   6. Simplified 70.4%

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

   7. Taylor expanded in cos2phi around inf 52.5%

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

      1. associate-/l* 52.4%

         \[\leadsto \color{blue}{\frac{u0}{\frac{cos2phi}{{alphax}^{2}}}} \]

      2. unpow2 52.4%

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

   9. Simplified 52.4%

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

       1. associate-/r/ 52.5%

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

   11. Applied egg-rr 52.5%

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

   if 5.00000008e-17 < (/.f32 sin2phi (*.f32 alphay alphay))

   1. Initial program 67.9%

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

      1. associate-/r* 67.9%

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

   3. Simplified 67.9%

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

   4. Taylor expanded in u0 around 0 72.7%

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

      1. unpow2 72.7%

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

      2. unpow2 72.7%

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

   6. Simplified 72.7%

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

      1. associate-/r* 72.6%

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

      2. div-inv 72.6%

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

   8. Applied egg-rr 72.6%

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

   9. Taylor expanded in cos2phi around 0 68.4%

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

       1. *-commutative 68.4%

          \[\leadsto \frac{\color{blue}{{alphay}^{2} \cdot u0}}{sin2phi} \]

       2. *-lft-identity 68.4%

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

       3. times-frac 68.3%

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

       4. /-rgt-identity 68.3%

          \[\leadsto \color{blue}{{alphay}^{2}} \cdot \frac{u0}{sin2phi} \]

       5. unpow2 68.3%

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

   11. Simplified 68.3%

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

4. Final simplification 64.3%

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

Alternative 13: 66.8% accurate, 8.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Split input into 2 regimes

2. if (/.f32 sin2phi (*.f32 alphay alphay)) < 5.00000008e-17

   1. Initial program 61.4%

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

      1. associate-/r* 61.4%

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

   3. Simplified 61.4%

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

   4. Taylor expanded in u0 around 0 70.4%

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

      1. unpow2 70.4%

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

      2. unpow2 70.4%

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

   6. Simplified 70.4%

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

   7. Taylor expanded in cos2phi around inf 52.5%

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

      1. associate-/l* 52.4%

         \[\leadsto \color{blue}{\frac{u0}{\frac{cos2phi}{{alphax}^{2}}}} \]

      2. unpow2 52.4%

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

   9. Simplified 52.4%

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

       1. associate-/r/ 52.5%

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

   11. Applied egg-rr 52.5%

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

   if 5.00000008e-17 < (/.f32 sin2phi (*.f32 alphay alphay))

   1. Initial program 67.9%

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

      1. associate-/r* 67.9%

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

   3. Simplified 67.9%

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

   4. Taylor expanded in cos2phi around 0 64.3%

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

      1. mul-1-neg 64.3%

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

      2. unpow2 64.3%

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

      3. associate-/l* 63.8%

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

      4. distribute-neg-frac 63.8%

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

      5. distribute-rgt-neg-in 63.8%

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

      6. sub-neg 63.8%

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

      7. mul-1-neg 63.8%

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

      8. log1p-def 90.2%

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

      9. mul-1-neg 90.2%

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

   6. Simplified 90.2%

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

   7. Taylor expanded in u0 around 0 79.5%

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

      1. +-commutative 79.5%

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

      2. mul-1-neg 79.5%

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

      3. unsub-neg 79.5%

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

      4. *-commutative 79.5%

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

   9. Simplified 79.5%

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

   10. Taylor expanded in u0 around 0 68.4%

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

       1. *-commutative 68.4%

          \[\leadsto \frac{\color{blue}{{alphay}^{2} \cdot u0}}{sin2phi} \]

       2. unpow2 68.4%

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

       3. associate-*l* 68.4%

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

       4. *-commutative 68.4%

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

   12. Simplified 68.4%

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

4. Final simplification 64.4%

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

Alternative 14: 66.8% accurate, 8.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Split input into 2 regimes

2. if (/.f32 sin2phi (*.f32 alphay alphay)) < 5.00000008e-17

   1. Initial program 61.4%

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

      1. associate-/r* 61.4%

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

   3. Simplified 61.4%

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

   4. Taylor expanded in u0 around 0 70.4%

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

      1. unpow2 70.4%

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

      2. unpow2 70.4%

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

   6. Simplified 70.4%

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

   7. Taylor expanded in cos2phi around inf 52.5%

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

      1. associate-/l* 52.4%

         \[\leadsto \color{blue}{\frac{u0}{\frac{cos2phi}{{alphax}^{2}}}} \]

      2. unpow2 52.4%

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

   9. Simplified 52.4%

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

       1. associate-/r/ 52.5%

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

   11. Applied egg-rr 52.5%

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

   if 5.00000008e-17 < (/.f32 sin2phi (*.f32 alphay alphay))

   1. Initial program 67.9%

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

      1. associate-/r* 67.9%

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

   3. Simplified 67.9%

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

   4. Taylor expanded in cos2phi around 0 64.3%

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

      1. mul-1-neg 64.3%

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

      2. unpow2 64.3%

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

      3. *-commutative 64.3%

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

   6. Simplified 64.3%

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

   7. Taylor expanded in u0 around 0 68.4%

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

      1. associate-*r* 68.4%

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

      2. neg-mul-1 68.4%

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

      3. unpow2 68.4%

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

   9. Simplified 68.4%

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

4. Final simplification 64.4%

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

Alternative 15: 66.6% accurate, 12.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;sin2phi \leq 2.5000000784359874 \cdot 10^{-23}:\\ \;\;\;\;alphax \cdot \frac{u0 \cdot alphax}{cos2phi}\\ \mathbf{else}:\\ \;\;\;\;\left(alphay \cdot alphay\right) \cdot \frac{u0}{sin2phi}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (if (<= sin2phi 2.5000000784359874e-23)
   (* alphax (/ (* u0 alphax) cos2phi))
   (* (* alphay alphay) (/ u0 sin2phi))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float tmp;
	if (sin2phi <= 2.5000000784359874e-23f) {
		tmp = alphax * ((u0 * alphax) / cos2phi);
	} else {
		tmp = (alphay * alphay) * (u0 / sin2phi);
	}
	return tmp;
}

Fortran

real(4) function code(alphax, alphay, u0, cos2phi, sin2phi)
    real(4), intent (in) :: alphax
    real(4), intent (in) :: alphay
    real(4), intent (in) :: u0
    real(4), intent (in) :: cos2phi
    real(4), intent (in) :: sin2phi
    real(4) :: tmp
    if (sin2phi <= 2.5000000784359874e-23) then
        tmp = alphax * ((u0 * alphax) / cos2phi)
    else
        tmp = (alphay * alphay) * (u0 / sin2phi)
    end if
    code = tmp
end function

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = Float32(0.0)
	if (sin2phi <= Float32(2.5000000784359874e-23))
		tmp = Float32(alphax * Float32(Float32(u0 * alphax) / cos2phi));
	else
		tmp = Float32(Float32(alphay * alphay) * Float32(u0 / sin2phi));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = single(0.0);
	if (sin2phi <= single(2.5000000784359874e-23))
		tmp = alphax * ((u0 * alphax) / cos2phi);
	else
		tmp = (alphay * alphay) * (u0 / sin2phi);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if sin2phi < 2.50000008e-23

   1. Initial program 61.2%

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

      1. associate-/r* 61.1%

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

   3. Simplified 61.1%

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

   4. Taylor expanded in u0 around 0 70.1%

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

      1. unpow2 70.1%

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

      2. unpow2 70.1%

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

   6. Simplified 70.1%

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

   7. Taylor expanded in cos2phi around inf 53.6%

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

      1. associate-/l* 53.5%

         \[\leadsto \color{blue}{\frac{u0}{\frac{cos2phi}{{alphax}^{2}}}} \]

      2. unpow2 53.5%

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

   9. Simplified 53.5%

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

   10. Taylor expanded in u0 around 0 53.6%

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

       1. associate-*l/ 53.7%

          \[\leadsto \color{blue}{\frac{u0}{cos2phi} \cdot {alphax}^{2}} \]

       2. *-commutative 53.7%

          \[\leadsto \color{blue}{{alphax}^{2} \cdot \frac{u0}{cos2phi}} \]

       3. unpow2 53.7%

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

       4. associate-*l* 53.6%

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

   12. Simplified 53.6%

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

   13. Taylor expanded in alphax around 0 53.6%

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

   if 2.50000008e-23 < sin2phi

   1. Initial program 67.5%

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

      1. associate-/r* 67.6%

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

   3. Simplified 67.6%

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

   4. Taylor expanded in u0 around 0 72.6%

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

      1. unpow2 72.6%

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

      2. unpow2 72.6%

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

   6. Simplified 72.6%

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

      1. associate-/r* 72.6%

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

      2. div-inv 72.6%

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

   8. Applied egg-rr 72.6%

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

   9. Taylor expanded in cos2phi around 0 66.6%

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

       1. *-commutative 66.6%

          \[\leadsto \frac{\color{blue}{{alphay}^{2} \cdot u0}}{sin2phi} \]

       2. *-lft-identity 66.6%

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

       3. times-frac 66.5%

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

       4. /-rgt-identity 66.5%

          \[\leadsto \color{blue}{{alphay}^{2}} \cdot \frac{u0}{sin2phi} \]

       5. unpow2 66.5%

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

   11. Simplified 66.5%

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

4. Final simplification 63.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;sin2phi \leq 2.5000000784359874 \cdot 10^{-23}:\\ \;\;\;\;alphax \cdot \frac{u0 \cdot alphax}{cos2phi}\\ \mathbf{else}:\\ \;\;\;\;\left(alphay \cdot alphay\right) \cdot \frac{u0}{sin2phi}\\ \end{array} \]

Alternative 16: 23.8% accurate, 16.6× speedup

Math

\[\begin{array}{l} \\ alphax \cdot \left(alphax \cdot \frac{u0}{cos2phi}\right) \end{array} \]

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 66.3%

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

   1. associate-/r* 66.3%

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

3. Simplified 66.3%

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

4. Taylor expanded in u0 around 0 72.1%

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

   1. unpow2 72.1%

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

   2. unpow2 72.1%

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

6. Simplified 72.1%

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

7. Taylor expanded in cos2phi around inf 21.2%

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

   1. associate-/l* 21.2%

      \[\leadsto \color{blue}{\frac{u0}{\frac{cos2phi}{{alphax}^{2}}}} \]

   2. unpow2 21.2%

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

9. Simplified 21.2%

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

10. Taylor expanded in u0 around 0 21.2%

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

    1. associate-*l/ 21.2%

       \[\leadsto \color{blue}{\frac{u0}{cos2phi} \cdot {alphax}^{2}} \]

    2. *-commutative 21.2%

       \[\leadsto \color{blue}{{alphax}^{2} \cdot \frac{u0}{cos2phi}} \]

    3. unpow2 21.2%

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

    4. associate-*l* 21.2%

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

12. Simplified 21.2%

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

13. Final simplification 21.2%

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

Reproduce

herbie shell --seed 2023182 
(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)))))