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

Percentage Accurate: 60.5% → 98.3%

Time: 15.7s

Alternatives: 18

Speedup: 8.9×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Initial Program: 60.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Alternative 1: 98.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 57.9%

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

   1. neg-sub0 57.9%

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

   2. div-sub 57.9%

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

   3. --rgt-identity 57.9%

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

   4. div-sub 57.9%

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

   5. --rgt-identity 57.9%

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

   6. neg-sub0 57.9%

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

   7. sub-neg 57.9%

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

   8. log1p-def 98.0%

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

3. Simplified 98.0%

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

   1. +-commutative 98.0%

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

   2. associate-/r* 98.0%

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

   3. frac-2neg 98.0%

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

   4. frac-add 97.6%

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

   5. distribute-rgt-neg-in 97.6%

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

   6. distribute-rgt-neg-in 97.6%

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

5. Applied egg-rr 97.6%

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

   1. *-commutative 97.6%

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

   2. associate-*l* 97.6%

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

   3. *-commutative 97.6%

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

   4. distribute-rgt-neg-out 97.6%

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

   5. unsub-neg 97.6%

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

   6. associate-*r/ 97.4%

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

   7. distribute-rgt-neg-out 97.4%

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

   8. distribute-lft-neg-in 97.4%

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

   9. distribute-rgt-neg-out 97.4%

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

   10. *-commutative 97.4%

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

   11. associate-/l* 97.6%

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

   12. *-commutative 97.6%

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

   13. distribute-lft-neg-out 97.6%

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

   14. distribute-rgt-neg-in 97.6%

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

7. Simplified 97.6%

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

   1. div-inv 97.4%

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

   2. associate-/r* 97.4%

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

9. Applied egg-rr 97.4%

   \[\leadsto \color{blue}{\left(-\mathsf{log1p}\left(-u0\right)\right) \cdot \frac{1}{\frac{\frac{-sin2phi}{\frac{\frac{alphay}{alphax}}{alphax}} - cos2phi \cdot alphay}{alphax \cdot \left(alphax \cdot \left(-alphay\right)\right)}}} \]
10. Step-by-step derivation

    1. associate-*r/ 97.5%

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

    2. *-rgt-identity 97.5%

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

    3. distribute-neg-frac 97.5%

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

    4. associate-/r/ 98.4%

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

    5. distribute-lft-neg-in 98.4%

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

    6. distribute-frac-neg 98.4%

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

    7. *-commutative 98.4%

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

    8. distribute-rgt-neg-out 98.4%

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

    9. distribute-lft-neg-in 98.4%

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

11. Simplified 98.3%

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

12. Final simplification 98.3%

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

Alternative 2: 92.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{sin2phi}{alphay \cdot alphay}\\ \mathbf{if}\;t_0 \leq 100000:\\ \;\;\;\;\frac{u0 - u0 \cdot \left(u0 \cdot -0.5\right)}{\frac{\frac{cos2phi}{alphax}}{alphax} + t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{alphay \cdot \left(alphay \cdot \left(-\mathsf{log1p}\left(-u0\right)\right)\right)}{sin2phi}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (let* ((t_0 (/ sin2phi (* alphay alphay))))
   (if (<= t_0 100000.0)
     (/ (- u0 (* u0 (* u0 -0.5))) (+ (/ (/ cos2phi alphax) alphax) t_0))
     (/ (* alphay (* alphay (- (log1p (- u0))))) sin2phi))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float t_0 = sin2phi / (alphay * alphay);
	float tmp;
	if (t_0 <= 100000.0f) {
		tmp = (u0 - (u0 * (u0 * -0.5f))) / (((cos2phi / alphax) / alphax) + t_0);
	} else {
		tmp = (alphay * (alphay * -log1pf(-u0))) / sin2phi;
	}
	return tmp;
}

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	t_0 = Float32(sin2phi / Float32(alphay * alphay))
	tmp = Float32(0.0)
	if (t_0 <= Float32(100000.0))
		tmp = Float32(Float32(u0 - Float32(u0 * Float32(u0 * Float32(-0.5)))) / Float32(Float32(Float32(cos2phi / alphax) / alphax) + t_0));
	else
		tmp = Float32(Float32(alphay * Float32(alphay * Float32(-log1p(Float32(-u0))))) / sin2phi);
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 sin2phi (*.f32 alphay alphay)) < 1e5

   1. Initial program 54.0%

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

      1. associate-/r* 54.0%

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

   3. Simplified 54.0%

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

   4. Taylor expanded in u0 around 0 88.0%

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

      1. +-commutative 46.3%

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

      2. mul-1-neg 46.3%

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

      3. unsub-neg 46.3%

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

      4. unpow2 46.3%

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

      5. associate-*r* 46.3%

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

   6. Simplified 88.0%

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

   if 1e5 < (/.f32 sin2phi (*.f32 alphay alphay))

   1. Initial program 61.6%

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

      1. associate-/r* 61.6%

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

   3. Simplified 61.6%

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

   4. Taylor expanded in cos2phi around 0 63.2%

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

      1. mul-1-neg 63.2%

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

      2. unpow2 63.2%

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

      3. *-commutative 63.2%

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

   6. Simplified 63.2%

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

   7. Taylor expanded in alphay around 0 63.2%

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

      1. unpow2 63.2%

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

      2. sub-neg 63.2%

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

      3. log1p-def 98.7%

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

   9. Simplified 98.7%

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

   10. Taylor expanded in alphay around 0 63.2%

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

       1. unpow2 63.2%

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

       2. sub-neg 63.2%

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

       3. log1p-def 98.7%

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

       4. associate-*r* 98.8%

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

   12. Simplified 98.8%

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

4. Final simplification 93.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 100000:\\ \;\;\;\;\frac{u0 - u0 \cdot \left(u0 \cdot -0.5\right)}{\frac{\frac{cos2phi}{alphax}}{alphax} + \frac{sin2phi}{alphay \cdot alphay}}\\ \mathbf{else}:\\ \;\;\;\;\frac{alphay \cdot \left(alphay \cdot \left(-\mathsf{log1p}\left(-u0\right)\right)\right)}{sin2phi}\\ \end{array} \]

Alternative 3: 92.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{sin2phi}{alphay \cdot alphay}\\ \mathbf{if}\;t_0 \leq 100000:\\ \;\;\;\;\frac{u0 - u0 \cdot \left(u0 \cdot -0.5\right)}{\frac{\frac{cos2phi}{alphax}}{alphax} + t_0}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(-u0\right) \cdot \frac{-alphay \cdot alphay}{sin2phi}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (let* ((t_0 (/ sin2phi (* alphay alphay))))
   (if (<= t_0 100000.0)
     (/ (- u0 (* u0 (* u0 -0.5))) (+ (/ (/ cos2phi alphax) alphax) t_0))
     (* (log1p (- u0)) (/ (- (* alphay alphay)) sin2phi)))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float t_0 = sin2phi / (alphay * alphay);
	float tmp;
	if (t_0 <= 100000.0f) {
		tmp = (u0 - (u0 * (u0 * -0.5f))) / (((cos2phi / alphax) / alphax) + t_0);
	} else {
		tmp = log1pf(-u0) * (-(alphay * alphay) / sin2phi);
	}
	return tmp;
}

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	t_0 = Float32(sin2phi / Float32(alphay * alphay))
	tmp = Float32(0.0)
	if (t_0 <= Float32(100000.0))
		tmp = Float32(Float32(u0 - Float32(u0 * Float32(u0 * Float32(-0.5)))) / Float32(Float32(Float32(cos2phi / alphax) / alphax) + t_0));
	else
		tmp = Float32(log1p(Float32(-u0)) * Float32(Float32(-Float32(alphay * alphay)) / sin2phi));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 sin2phi (*.f32 alphay alphay)) < 1e5

   1. Initial program 54.0%

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

      1. associate-/r* 54.0%

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

   3. Simplified 54.0%

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

   4. Taylor expanded in u0 around 0 88.0%

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

      1. +-commutative 46.3%

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

      2. mul-1-neg 46.3%

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

      3. unsub-neg 46.3%

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

      4. unpow2 46.3%

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

      5. associate-*r* 46.3%

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

   6. Simplified 88.0%

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

   if 1e5 < (/.f32 sin2phi (*.f32 alphay alphay))

   1. Initial program 61.6%

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

      1. neg-sub0 61.6%

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

      2. div-sub 61.6%

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

      3. --rgt-identity 61.6%

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

      4. div-sub 61.6%

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

      5. --rgt-identity 61.6%

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

      6. neg-sub0 61.6%

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

      7. sub-neg 61.6%

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

      8. log1p-def 97.4%

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

   3. Simplified 97.4%

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

      1. +-commutative 97.4%

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

      2. associate-/r* 97.4%

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

      3. frac-2neg 97.4%

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

      4. frac-add 97.0%

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

      5. distribute-rgt-neg-in 97.0%

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

      6. distribute-rgt-neg-in 97.0%

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

   5. Applied egg-rr 97.0%

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

      1. *-commutative 97.0%

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

      2. associate-*l* 96.9%

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

      3. *-commutative 96.9%

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

      4. distribute-rgt-neg-out 96.9%

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

      5. unsub-neg 96.9%

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

      6. associate-*r/ 96.7%

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

      7. distribute-rgt-neg-out 96.7%

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

      8. distribute-lft-neg-in 96.7%

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

      9. distribute-rgt-neg-out 96.7%

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

      10. *-commutative 96.7%

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

      11. associate-/l* 96.9%

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

      12. *-commutative 96.9%

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

      13. distribute-lft-neg-out 96.9%

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

      14. distribute-rgt-neg-in 96.9%

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

   7. Simplified 96.9%

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

      1. div-inv 96.8%

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

      2. associate-/r* 96.8%

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

   9. Applied egg-rr 96.8%

      \[\leadsto \color{blue}{\left(-\mathsf{log1p}\left(-u0\right)\right) \cdot \frac{1}{\frac{\frac{-sin2phi}{\frac{\frac{alphay}{alphax}}{alphax}} - cos2phi \cdot alphay}{alphax \cdot \left(alphax \cdot \left(-alphay\right)\right)}}} \]
   10. Step-by-step derivation

       1. associate-/r/ 98.3%

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

       2. distribute-frac-neg 98.3%

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

       3. associate-/l/ 98.4%

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

   11. Simplified 98.4%

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

   12. Taylor expanded in sin2phi around inf 98.9%

       \[\leadsto \left(-\mathsf{log1p}\left(-u0\right)\right) \cdot \color{blue}{\frac{{alphay}^{2}}{sin2phi}} \]
   13. Step-by-step derivation

       1. unpow2 98.9%

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

   14. Simplified 98.9%

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

4. Final simplification 93.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 100000:\\ \;\;\;\;\frac{u0 - u0 \cdot \left(u0 \cdot -0.5\right)}{\frac{\frac{cos2phi}{alphax}}{alphax} + \frac{sin2phi}{alphay \cdot alphay}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(-u0\right) \cdot \frac{-alphay \cdot alphay}{sin2phi}\\ \end{array} \]

Alternative 4: 98.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 57.9%

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

   1. neg-sub0 57.9%

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

   2. div-sub 57.9%

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

   3. --rgt-identity 57.9%

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

   4. div-sub 57.9%

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

   5. --rgt-identity 57.9%

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

   6. neg-sub0 57.9%

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

   7. sub-neg 57.9%

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

   8. log1p-def 98.0%

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

3. Simplified 98.0%

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

4. Final simplification 98.0%

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

Alternative 5: 98.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 57.9%

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

   1. neg-sub0 57.9%

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

   2. div-sub 57.9%

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

   3. --rgt-identity 57.9%

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

   4. div-sub 57.9%

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

   5. --rgt-identity 57.9%

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

   6. sub-neg 57.9%

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

   7. +-commutative 57.9%

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

   8. neg-sub0 57.9%

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

   9. associate-+l- 57.9%

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

   10. sub0-neg 57.9%

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

   11. neg-mul-1 57.9%

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

   12. log-prod -0.0%

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

   13. associate--r+ -0.0%

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

3. Simplified 98.1%

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

4. Final simplification 98.1%

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

Alternative 6: 87.6% accurate, 4.5× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 57.9%

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

   1. neg-sub0 57.9%

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

   2. div-sub 57.9%

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

   3. --rgt-identity 57.9%

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

   4. div-sub 57.9%

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

   5. --rgt-identity 57.9%

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

   6. neg-sub0 57.9%

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

   7. sub-neg 57.9%

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

   8. log1p-def 98.0%

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

3. Simplified 98.0%

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

   1. +-commutative 98.0%

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

   2. associate-/r* 98.0%

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

   3. frac-2neg 98.0%

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

   4. frac-add 97.6%

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

   5. distribute-rgt-neg-in 97.6%

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

   6. distribute-rgt-neg-in 97.6%

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

5. Applied egg-rr 97.6%

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

   1. *-commutative 97.6%

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

   2. associate-*l* 97.6%

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

   3. *-commutative 97.6%

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

   4. distribute-rgt-neg-out 97.6%

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

   5. unsub-neg 97.6%

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

   6. associate-*r/ 97.4%

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

   7. distribute-rgt-neg-out 97.4%

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

   8. distribute-lft-neg-in 97.4%

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

   9. distribute-rgt-neg-out 97.4%

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

   10. *-commutative 97.4%

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

   11. associate-/l* 97.6%

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

   12. *-commutative 97.6%

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

   13. distribute-lft-neg-out 97.6%

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

   14. distribute-rgt-neg-in 97.6%

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

7. Simplified 97.6%

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

   1. div-inv 97.4%

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

   2. associate-/r* 97.4%

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

9. Applied egg-rr 97.4%

   \[\leadsto \color{blue}{\left(-\mathsf{log1p}\left(-u0\right)\right) \cdot \frac{1}{\frac{\frac{-sin2phi}{\frac{\frac{alphay}{alphax}}{alphax}} - cos2phi \cdot alphay}{alphax \cdot \left(alphax \cdot \left(-alphay\right)\right)}}} \]
10. Step-by-step derivation

    1. associate-*r/ 97.5%

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

    2. *-rgt-identity 97.5%

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

    3. distribute-neg-frac 97.5%

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

    4. associate-/r/ 98.4%

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

    5. distribute-lft-neg-in 98.4%

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

    6. distribute-frac-neg 98.4%

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

    7. *-commutative 98.4%

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

    8. distribute-rgt-neg-out 98.4%

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

    9. distribute-lft-neg-in 98.4%

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

11. Simplified 98.3%

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

12. Taylor expanded in u0 around 0 87.9%

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

    1. +-commutative 87.9%

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

    2. neg-mul-1 87.9%

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

    3. unsub-neg 87.9%

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

    4. *-commutative 87.9%

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

    5. unpow2 87.9%

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

    6. associate-*l* 87.9%

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

14. Simplified 87.9%

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

15. Taylor expanded in alphax around 0 87.9%

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

    1. mul-1-neg 87.9%

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

    2. *-commutative 87.9%

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

    3. unpow2 87.9%

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

17. Simplified 87.9%

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

18. Final simplification 87.9%

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

Alternative 7: 87.6% accurate, 4.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 57.9%

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

   1. neg-sub0 57.9%

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

   2. div-sub 57.9%

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

   3. --rgt-identity 57.9%

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

   4. div-sub 57.9%

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

   5. --rgt-identity 57.9%

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

   6. neg-sub0 57.9%

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

   7. sub-neg 57.9%

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

   8. log1p-def 98.0%

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

3. Simplified 98.0%

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

   1. +-commutative 98.0%

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

   2. associate-/r* 98.0%

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

   3. frac-2neg 98.0%

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

   4. frac-add 97.6%

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

   5. distribute-rgt-neg-in 97.6%

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

   6. distribute-rgt-neg-in 97.6%

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

5. Applied egg-rr 97.6%

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

   1. *-commutative 97.6%

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

   2. associate-*l* 97.6%

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

   3. *-commutative 97.6%

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

   4. distribute-rgt-neg-out 97.6%

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

   5. unsub-neg 97.6%

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

   6. associate-*r/ 97.4%

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

   7. distribute-rgt-neg-out 97.4%

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

   8. distribute-lft-neg-in 97.4%

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

   9. distribute-rgt-neg-out 97.4%

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

   10. *-commutative 97.4%

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

   11. associate-/l* 97.6%

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

   12. *-commutative 97.6%

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

   13. distribute-lft-neg-out 97.6%

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

   14. distribute-rgt-neg-in 97.6%

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

7. Simplified 97.6%

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

   1. div-inv 97.4%

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

   2. associate-/r* 97.4%

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

9. Applied egg-rr 97.4%

   \[\leadsto \color{blue}{\left(-\mathsf{log1p}\left(-u0\right)\right) \cdot \frac{1}{\frac{\frac{-sin2phi}{\frac{\frac{alphay}{alphax}}{alphax}} - cos2phi \cdot alphay}{alphax \cdot \left(alphax \cdot \left(-alphay\right)\right)}}} \]
10. Step-by-step derivation

    1. associate-*r/ 97.5%

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

    2. *-rgt-identity 97.5%

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

    3. distribute-neg-frac 97.5%

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

    4. associate-/r/ 98.4%

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

    5. distribute-lft-neg-in 98.4%

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

    6. distribute-frac-neg 98.4%

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

    7. *-commutative 98.4%

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

    8. distribute-rgt-neg-out 98.4%

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

    9. distribute-lft-neg-in 98.4%

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

11. Simplified 98.3%

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

12. Taylor expanded in u0 around 0 87.9%

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

    1. +-commutative 87.9%

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

    2. neg-mul-1 87.9%

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

    3. unsub-neg 87.9%

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

    4. *-commutative 87.9%

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

    5. unpow2 87.9%

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

    6. associate-*l* 87.9%

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

14. Simplified 87.9%

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

    1. *-un-lft-identity 87.9%

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

    2. distribute-rgt-neg-in 87.9%

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

    3. *-commutative 87.9%

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

    4. *-commutative 87.9%

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

16. Applied egg-rr 87.9%

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

    1. *-lft-identity 87.9%

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

    2. associate-/l* 88.0%

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

    3. distribute-rgt-neg-in 88.0%

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

    4. *-commutative 88.0%

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

18. Simplified 88.0%

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

19. Final simplification 88.0%

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

Alternative 8: 81.5% accurate, 4.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 18000:\\ \;\;\;\;\frac{u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay} \cdot \frac{1}{alphay}}\\ \mathbf{else}:\\ \;\;\;\;\left(alphay \cdot alphay\right) \cdot \frac{u0}{sin2phi} - -0.5 \cdot \frac{\left(alphay \cdot u0\right) \cdot \left(alphay \cdot u0\right)}{sin2phi}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (if (<= (/ sin2phi (* alphay alphay)) 18000.0)
   (/
    u0
    (+ (/ cos2phi (* alphax alphax)) (* (/ sin2phi alphay) (/ 1.0 alphay))))
   (-
    (* (* alphay alphay) (/ u0 sin2phi))
    (* -0.5 (/ (* (* alphay u0) (* alphay u0)) sin2phi)))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float tmp;
	if ((sin2phi / (alphay * alphay)) <= 18000.0f) {
		tmp = u0 / ((cos2phi / (alphax * alphax)) + ((sin2phi / alphay) * (1.0f / alphay)));
	} else {
		tmp = ((alphay * alphay) * (u0 / sin2phi)) - (-0.5f * (((alphay * u0) * (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)) <= 18000.0e0) then
        tmp = u0 / ((cos2phi / (alphax * alphax)) + ((sin2phi / alphay) * (1.0e0 / alphay)))
    else
        tmp = ((alphay * alphay) * (u0 / sin2phi)) - ((-0.5e0) * (((alphay * u0) * (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(18000.0))
		tmp = Float32(u0 / Float32(Float32(cos2phi / Float32(alphax * alphax)) + Float32(Float32(sin2phi / alphay) * Float32(Float32(1.0) / alphay))));
	else
		tmp = Float32(Float32(Float32(alphay * alphay) * Float32(u0 / sin2phi)) - Float32(Float32(-0.5) * Float32(Float32(Float32(alphay * u0) * Float32(alphay * 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(18000.0))
		tmp = u0 / ((cos2phi / (alphax * alphax)) + ((sin2phi / alphay) * (single(1.0) / alphay)));
	else
		tmp = ((alphay * alphay) * (u0 / sin2phi)) - (single(-0.5) * (((alphay * u0) * (alphay * u0)) / sin2phi));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 sin2phi (*.f32 alphay alphay)) < 18000

   1. Initial program 54.1%

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

      1. associate-/r* 54.2%

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

   3. Simplified 54.2%

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

   4. Taylor expanded in u0 around 0 75.8%

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

      1. unpow2 75.8%

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

      2. unpow2 75.8%

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

   6. Simplified 75.8%

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

      1. associate-/r* 75.8%

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

      2. div-inv 75.9%

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

   8. Applied egg-rr 75.9%

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

   if 18000 < (/.f32 sin2phi (*.f32 alphay alphay))

   1. Initial program 61.3%

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

      1. associate-/r* 61.3%

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

   3. Simplified 61.3%

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

   4. Taylor expanded in cos2phi around 0 62.7%

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

      1. mul-1-neg 62.7%

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

      2. unpow2 62.7%

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

      3. *-commutative 62.7%

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

   6. Simplified 62.7%

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

   7. Taylor expanded in alphay around 0 62.7%

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

      1. unpow2 62.7%

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

      2. sub-neg 62.7%

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

      3. log1p-def 98.4%

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

   9. Simplified 98.4%

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

   10. Taylor expanded in u0 around 0 88.4%

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

       1. mul-1-neg 88.4%

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

       2. unsub-neg 88.4%

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

       3. *-commutative 88.4%

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

       4. unpow2 88.4%

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

       5. unpow2 88.4%

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

       6. unswap-sqr 88.4%

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

       7. *-commutative 88.4%

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

       8. *-lft-identity 88.4%

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

       9. times-frac 88.4%

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

       10. /-rgt-identity 88.4%

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

       11. unpow2 88.4%

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

   12. Simplified 88.4%

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

4. Final simplification 82.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 18000:\\ \;\;\;\;\frac{u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay} \cdot \frac{1}{alphay}}\\ \mathbf{else}:\\ \;\;\;\;\left(alphay \cdot alphay\right) \cdot \frac{u0}{sin2phi} - -0.5 \cdot \frac{\left(alphay \cdot u0\right) \cdot \left(alphay \cdot u0\right)}{sin2phi}\\ \end{array} \]

Alternative 9: 81.5% accurate, 5.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 18000:\\ \;\;\;\;\frac{u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay} \cdot \frac{1}{alphay}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(alphay \cdot alphay\right) \cdot \left(u0 - u0 \cdot \left(u0 \cdot -0.5\right)\right)}{sin2phi}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (if (<= (/ sin2phi (* alphay alphay)) 18000.0)
   (/
    u0
    (+ (/ cos2phi (* alphax alphax)) (* (/ sin2phi alphay) (/ 1.0 alphay))))
   (/ (* (* alphay alphay) (- u0 (* u0 (* u0 -0.5)))) sin2phi)))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float tmp;
	if ((sin2phi / (alphay * alphay)) <= 18000.0f) {
		tmp = u0 / ((cos2phi / (alphax * alphax)) + ((sin2phi / alphay) * (1.0f / alphay)));
	} else {
		tmp = ((alphay * alphay) * (u0 - (u0 * (u0 * -0.5f)))) / sin2phi;
	}
	return tmp;
}

Fortran

real(4) function code(alphax, alphay, u0, cos2phi, sin2phi)
    real(4), intent (in) :: alphax
    real(4), intent (in) :: alphay
    real(4), intent (in) :: u0
    real(4), intent (in) :: cos2phi
    real(4), intent (in) :: sin2phi
    real(4) :: tmp
    if ((sin2phi / (alphay * alphay)) <= 18000.0e0) then
        tmp = u0 / ((cos2phi / (alphax * alphax)) + ((sin2phi / alphay) * (1.0e0 / alphay)))
    else
        tmp = ((alphay * alphay) * (u0 - (u0 * (u0 * (-0.5e0))))) / sin2phi
    end if
    code = tmp
end function

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = Float32(0.0)
	if (Float32(sin2phi / Float32(alphay * alphay)) <= Float32(18000.0))
		tmp = Float32(u0 / Float32(Float32(cos2phi / Float32(alphax * alphax)) + Float32(Float32(sin2phi / alphay) * Float32(Float32(1.0) / alphay))));
	else
		tmp = Float32(Float32(Float32(alphay * alphay) * Float32(u0 - Float32(u0 * Float32(u0 * Float32(-0.5))))) / sin2phi);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = single(0.0);
	if ((sin2phi / (alphay * alphay)) <= single(18000.0))
		tmp = u0 / ((cos2phi / (alphax * alphax)) + ((sin2phi / alphay) * (single(1.0) / alphay)));
	else
		tmp = ((alphay * alphay) * (u0 - (u0 * (u0 * single(-0.5))))) / sin2phi;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 sin2phi (*.f32 alphay alphay)) < 18000

   1. Initial program 54.1%

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

      1. associate-/r* 54.2%

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

   3. Simplified 54.2%

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

   4. Taylor expanded in u0 around 0 75.8%

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

      1. unpow2 75.8%

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

      2. unpow2 75.8%

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

   6. Simplified 75.8%

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

      1. associate-/r* 75.8%

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

      2. div-inv 75.9%

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

   8. Applied egg-rr 75.9%

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

   if 18000 < (/.f32 sin2phi (*.f32 alphay alphay))

   1. Initial program 61.3%

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

      1. associate-/r* 61.3%

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

   3. Simplified 61.3%

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

   4. Taylor expanded in cos2phi around 0 62.7%

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

      1. mul-1-neg 62.7%

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

      2. unpow2 62.7%

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

      3. *-commutative 62.7%

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

   6. Simplified 62.7%

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

   7. Taylor expanded in u0 around 0 88.2%

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

      1. +-commutative 88.2%

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

      2. mul-1-neg 88.2%

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

      3. unsub-neg 88.2%

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

      4. unpow2 88.2%

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

      5. associate-*r* 88.2%

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

   9. Simplified 88.2%

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

4. Final simplification 82.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 18000:\\ \;\;\;\;\frac{u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay} \cdot \frac{1}{alphay}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(alphay \cdot alphay\right) \cdot \left(u0 - u0 \cdot \left(u0 \cdot -0.5\right)\right)}{sin2phi}\\ \end{array} \]

Alternative 10: 81.1% accurate, 6.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{sin2phi}{alphay \cdot alphay}\\ \mathbf{if}\;t_0 \leq 18000:\\ \;\;\;\;\frac{u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{\frac{sin2phi}{alphay}}{alphay}}\\ \mathbf{else}:\\ \;\;\;\;\frac{u0 + 0.5 \cdot \left(u0 \cdot u0\right)}{t_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (let* ((t_0 (/ sin2phi (* alphay alphay))))
   (if (<= t_0 18000.0)
     (/ u0 (+ (/ cos2phi (* alphax alphax)) (/ (/ sin2phi alphay) alphay)))
     (/ (+ u0 (* 0.5 (* u0 u0))) t_0))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float t_0 = sin2phi / (alphay * alphay);
	float tmp;
	if (t_0 <= 18000.0f) {
		tmp = u0 / ((cos2phi / (alphax * alphax)) + ((sin2phi / alphay) / alphay));
	} else {
		tmp = (u0 + (0.5f * (u0 * u0))) / t_0;
	}
	return tmp;
}

Fortran

real(4) function code(alphax, alphay, u0, cos2phi, sin2phi)
    real(4), intent (in) :: alphax
    real(4), intent (in) :: alphay
    real(4), intent (in) :: u0
    real(4), intent (in) :: cos2phi
    real(4), intent (in) :: sin2phi
    real(4) :: t_0
    real(4) :: tmp
    t_0 = sin2phi / (alphay * alphay)
    if (t_0 <= 18000.0e0) then
        tmp = u0 / ((cos2phi / (alphax * alphax)) + ((sin2phi / alphay) / alphay))
    else
        tmp = (u0 + (0.5e0 * (u0 * u0))) / t_0
    end if
    code = tmp
end function

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	t_0 = Float32(sin2phi / Float32(alphay * alphay))
	tmp = Float32(0.0)
	if (t_0 <= Float32(18000.0))
		tmp = Float32(u0 / Float32(Float32(cos2phi / Float32(alphax * alphax)) + Float32(Float32(sin2phi / alphay) / alphay)));
	else
		tmp = Float32(Float32(u0 + Float32(Float32(0.5) * Float32(u0 * u0))) / t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alphax, alphay, u0, cos2phi, sin2phi)
	t_0 = sin2phi / (alphay * alphay);
	tmp = single(0.0);
	if (t_0 <= single(18000.0))
		tmp = u0 / ((cos2phi / (alphax * alphax)) + ((sin2phi / alphay) / alphay));
	else
		tmp = (u0 + (single(0.5) * (u0 * u0))) / t_0;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 sin2phi (*.f32 alphay alphay)) < 18000

   1. Initial program 54.1%

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

      1. associate-/r* 54.2%

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

   3. Simplified 54.2%

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

   4. Taylor expanded in u0 around 0 75.8%

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

      1. unpow2 75.8%

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

      2. unpow2 75.8%

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

   6. Simplified 75.8%

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

      1. associate-/r* 75.8%

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

      2. div-inv 75.9%

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

   8. Applied egg-rr 75.9%

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

      1. un-div-inv 75.8%

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

   10. Applied egg-rr 75.8%

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

   if 18000 < (/.f32 sin2phi (*.f32 alphay alphay))

   1. Initial program 61.3%

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

      1. neg-sub0 61.3%

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

      2. div-sub 61.3%

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

      3. --rgt-identity 61.3%

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

      4. div-sub 61.3%

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

      5. --rgt-identity 61.3%

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

      6. neg-sub0 61.3%

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

      7. sub-neg 61.3%

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

      8. log1p-def 97.5%

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

   3. Simplified 97.5%

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

      1. +-commutative 97.5%

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

      2. associate-/r* 97.4%

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

      3. frac-2neg 97.4%

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

      4. frac-add 97.0%

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

      5. distribute-rgt-neg-in 97.0%

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

      6. distribute-rgt-neg-in 97.0%

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

   5. Applied egg-rr 97.0%

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

      1. *-commutative 97.0%

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

      2. associate-*l* 97.0%

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

      3. *-commutative 97.0%

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

      4. distribute-rgt-neg-out 97.0%

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

      5. unsub-neg 97.0%

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

      6. associate-*r/ 96.7%

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

      7. distribute-rgt-neg-out 96.7%

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

      8. distribute-lft-neg-in 96.7%

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

      9. distribute-rgt-neg-out 96.7%

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

      10. *-commutative 96.7%

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

      11. associate-/l* 97.0%

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

      12. *-commutative 97.0%

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

      13. distribute-lft-neg-out 97.0%

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

      14. distribute-rgt-neg-in 97.0%

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

   7. Simplified 97.0%

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

      1. div-inv 96.9%

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

      2. associate-/r* 96.9%

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

   9. Applied egg-rr 96.9%

      \[\leadsto \color{blue}{\left(-\mathsf{log1p}\left(-u0\right)\right) \cdot \frac{1}{\frac{\frac{-sin2phi}{\frac{\frac{alphay}{alphax}}{alphax}} - cos2phi \cdot alphay}{alphax \cdot \left(alphax \cdot \left(-alphay\right)\right)}}} \]
   10. Step-by-step derivation

       1. associate-*r/ 96.9%

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

       2. *-rgt-identity 96.9%

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

       3. distribute-neg-frac 96.9%

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

       4. associate-/r/ 98.6%

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

       5. distribute-lft-neg-in 98.6%

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

       6. distribute-frac-neg 98.6%

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

       7. *-commutative 98.6%

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

       8. distribute-rgt-neg-out 98.6%

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

       9. distribute-lft-neg-in 98.6%

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

   11. Simplified 98.7%

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

   12. Taylor expanded in u0 around 0 88.7%

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

       1. +-commutative 88.7%

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

       2. neg-mul-1 88.7%

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

       3. unsub-neg 88.7%

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

       4. *-commutative 88.7%

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

       5. unpow2 88.7%

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

       6. associate-*l* 88.7%

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

   14. Simplified 88.7%

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

   15. Taylor expanded in alphax around inf 88.2%

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

       1. associate-/l* 87.4%

          \[\leadsto \color{blue}{\frac{u0 - -0.5 \cdot {u0}^{2}}{\frac{sin2phi}{{alphay}^{2}}}} \]

       2. cancel-sign-sub-inv 87.4%

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

       3. metadata-eval 87.4%

          \[\leadsto \frac{u0 + \color{blue}{0.5} \cdot {u0}^{2}}{\frac{sin2phi}{{alphay}^{2}}} \]

       4. unpow2 87.4%

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

       5. unpow2 87.4%

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

   17. Simplified 87.4%

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

4. Final simplification 82.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 18000:\\ \;\;\;\;\frac{u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{\frac{sin2phi}{alphay}}{alphay}}\\ \mathbf{else}:\\ \;\;\;\;\frac{u0 + 0.5 \cdot \left(u0 \cdot u0\right)}{\frac{sin2phi}{alphay \cdot alphay}}\\ \end{array} \]

Alternative 11: 81.5% accurate, 6.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 18000:\\ \;\;\;\;\frac{u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{\frac{sin2phi}{alphay}}{alphay}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(alphay \cdot alphay\right) \cdot \left(u0 - u0 \cdot \left(u0 \cdot -0.5\right)\right)}{sin2phi}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (if (<= (/ sin2phi (* alphay alphay)) 18000.0)
   (/ u0 (+ (/ cos2phi (* alphax alphax)) (/ (/ sin2phi alphay) alphay)))
   (/ (* (* alphay alphay) (- u0 (* u0 (* u0 -0.5)))) sin2phi)))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float tmp;
	if ((sin2phi / (alphay * alphay)) <= 18000.0f) {
		tmp = u0 / ((cos2phi / (alphax * alphax)) + ((sin2phi / alphay) / alphay));
	} else {
		tmp = ((alphay * alphay) * (u0 - (u0 * (u0 * -0.5f)))) / sin2phi;
	}
	return tmp;
}

Fortran

real(4) function code(alphax, alphay, u0, cos2phi, sin2phi)
    real(4), intent (in) :: alphax
    real(4), intent (in) :: alphay
    real(4), intent (in) :: u0
    real(4), intent (in) :: cos2phi
    real(4), intent (in) :: sin2phi
    real(4) :: tmp
    if ((sin2phi / (alphay * alphay)) <= 18000.0e0) then
        tmp = u0 / ((cos2phi / (alphax * alphax)) + ((sin2phi / alphay) / alphay))
    else
        tmp = ((alphay * alphay) * (u0 - (u0 * (u0 * (-0.5e0))))) / sin2phi
    end if
    code = tmp
end function

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = Float32(0.0)
	if (Float32(sin2phi / Float32(alphay * alphay)) <= Float32(18000.0))
		tmp = Float32(u0 / Float32(Float32(cos2phi / Float32(alphax * alphax)) + Float32(Float32(sin2phi / alphay) / alphay)));
	else
		tmp = Float32(Float32(Float32(alphay * alphay) * Float32(u0 - Float32(u0 * Float32(u0 * Float32(-0.5))))) / sin2phi);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = single(0.0);
	if ((sin2phi / (alphay * alphay)) <= single(18000.0))
		tmp = u0 / ((cos2phi / (alphax * alphax)) + ((sin2phi / alphay) / alphay));
	else
		tmp = ((alphay * alphay) * (u0 - (u0 * (u0 * single(-0.5))))) / sin2phi;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 sin2phi (*.f32 alphay alphay)) < 18000

   1. Initial program 54.1%

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

      1. associate-/r* 54.2%

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

   3. Simplified 54.2%

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

   4. Taylor expanded in u0 around 0 75.8%

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

      1. unpow2 75.8%

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

      2. unpow2 75.8%

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

   6. Simplified 75.8%

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

      1. associate-/r* 75.8%

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

      2. div-inv 75.9%

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

   8. Applied egg-rr 75.9%

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

      1. un-div-inv 75.8%

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

   10. Applied egg-rr 75.8%

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

   if 18000 < (/.f32 sin2phi (*.f32 alphay alphay))

   1. Initial program 61.3%

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

      1. associate-/r* 61.3%

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

   3. Simplified 61.3%

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

   4. Taylor expanded in cos2phi around 0 62.7%

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

      1. mul-1-neg 62.7%

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

      2. unpow2 62.7%

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

      3. *-commutative 62.7%

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

   6. Simplified 62.7%

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

   7. Taylor expanded in u0 around 0 88.2%

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

      1. +-commutative 88.2%

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

      2. mul-1-neg 88.2%

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

      3. unsub-neg 88.2%

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

      4. unpow2 88.2%

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

      5. associate-*r* 88.2%

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

   9. Simplified 88.2%

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

4. Final simplification 82.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 18000:\\ \;\;\;\;\frac{u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{\frac{sin2phi}{alphay}}{alphay}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(alphay \cdot alphay\right) \cdot \left(u0 - u0 \cdot \left(u0 \cdot -0.5\right)\right)}{sin2phi}\\ \end{array} \]

Alternative 12: 87.5% accurate, 6.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 57.9%

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

   1. associate-/r* 57.9%

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

3. Simplified 57.9%

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

4. Taylor expanded in u0 around 0 87.8%

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

   1. +-commutative 68.0%

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

   2. mul-1-neg 68.0%

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

   3. unsub-neg 68.0%

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

   4. unpow2 68.0%

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

   5. associate-*r* 68.0%

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

6. Simplified 87.8%

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

7. Final simplification 87.8%

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

Alternative 13: 67.0% accurate, 8.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 1.9999999920083944 \cdot 10^{-11}:\\ \;\;\;\;alphax \cdot \frac{alphax \cdot u0}{cos2phi}\\ \mathbf{else}:\\ \;\;\;\;u0 \cdot \left(alphay \cdot \frac{alphay}{sin2phi}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (if (<= (/ sin2phi (* alphay alphay)) 1.9999999920083944e-11)
   (* 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)) <= 1.9999999920083944e-11f) {
		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)) <= 1.9999999920083944e-11) 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(1.9999999920083944e-11))
		tmp = Float32(alphax * Float32(Float32(alphax * u0) / cos2phi));
	else
		tmp = Float32(u0 * Float32(alphay * Float32(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(1.9999999920083944e-11))
		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)) < 1.99999999e-11

   1. Initial program 55.0%

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

      1. associate-/r* 55.2%

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

   3. Simplified 55.2%

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

   4. Taylor expanded in u0 around 0 75.0%

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

      1. unpow2 75.0%

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

      2. unpow2 75.0%

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

   6. Simplified 75.0%

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

   7. Taylor expanded in cos2phi around inf 54.5%

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

      1. unpow2 54.5%

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

      2. associate-/l* 54.3%

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

      3. associate-/r/ 54.4%

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

   9. Simplified 54.4%

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

   10. Taylor expanded in u0 around 0 54.5%

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

       1. associate-*l/ 54.4%

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

       2. *-commutative 54.4%

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

       3. unpow2 54.4%

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

       4. associate-*l* 54.4%

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

   12. Simplified 54.4%

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

   13. Taylor expanded in alphax around 0 54.5%

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

   if 1.99999999e-11 < (/.f32 sin2phi (*.f32 alphay alphay))

   1. Initial program 59.0%

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

      1. associate-/r* 59.0%

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

   3. Simplified 59.0%

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

   4. Taylor expanded in u0 around 0 76.7%

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

      1. unpow2 76.7%

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

      2. unpow2 76.7%

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

   6. Simplified 76.7%

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

      1. div-inv 76.7%

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

      2. +-commutative 76.7%

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

   8. Applied egg-rr 76.7%

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

   9. Taylor expanded in sin2phi around inf 73.1%

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

       1. unpow2 73.1%

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

       2. *-lft-identity 73.1%

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

       3. times-frac 73.3%

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

       4. /-rgt-identity 73.3%

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

   11. Simplified 73.3%

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

4. Final simplification 68.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 1.9999999920083944 \cdot 10^{-11}:\\ \;\;\;\;alphax \cdot \frac{alphax \cdot u0}{cos2phi}\\ \mathbf{else}:\\ \;\;\;\;u0 \cdot \left(alphay \cdot \frac{alphay}{sin2phi}\right)\\ \end{array} \]

Alternative 14: 67.0% accurate, 8.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 1.9999999920083944 \cdot 10^{-11}:\\ \;\;\;\;\frac{alphax \cdot \left(alphax \cdot u0\right)}{cos2phi}\\ \mathbf{else}:\\ \;\;\;\;u0 \cdot \left(alphay \cdot \frac{alphay}{sin2phi}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (if (<= (/ sin2phi (* alphay alphay)) 1.9999999920083944e-11)
   (/ (* 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)) <= 1.9999999920083944e-11f) {
		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)) <= 1.9999999920083944e-11) 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(1.9999999920083944e-11))
		tmp = Float32(Float32(alphax * Float32(alphax * u0)) / cos2phi);
	else
		tmp = Float32(u0 * Float32(alphay * Float32(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(1.9999999920083944e-11))
		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)) < 1.99999999e-11

   1. Initial program 55.0%

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

      1. associate-/r* 55.2%

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

   3. Simplified 55.2%

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

   4. Taylor expanded in cos2phi around inf 41.0%

      \[\leadsto \color{blue}{-1 \cdot \frac{\log \left(1 - u0\right) \cdot {alphax}^{2}}{cos2phi}} \]
   5. Step-by-step derivation

      1. associate-*r/ 41.0%

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

      2. unpow2 41.0%

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

      3. *-commutative 41.0%

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

      4. associate-*r* 41.0%

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

      5. neg-mul-1 41.0%

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

      6. distribute-rgt-neg-out 41.0%

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

      7. sub-neg 41.0%

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

      8. mul-1-neg 41.0%

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

      9. log1p-def 68.8%

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

      10. mul-1-neg 68.8%

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

   6. Simplified 68.8%

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

   7. Taylor expanded in u0 around 0 54.5%

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

      1. *-commutative 54.5%

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

      2. unpow2 54.5%

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

      3. associate-*l* 54.5%

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

   9. Simplified 54.5%

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

   if 1.99999999e-11 < (/.f32 sin2phi (*.f32 alphay alphay))

   1. Initial program 59.0%

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

      1. associate-/r* 59.0%

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

   3. Simplified 59.0%

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

   4. Taylor expanded in u0 around 0 76.7%

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

      1. unpow2 76.7%

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

      2. unpow2 76.7%

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

   6. Simplified 76.7%

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

      1. div-inv 76.7%

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

      2. +-commutative 76.7%

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

   8. Applied egg-rr 76.7%

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

   9. Taylor expanded in sin2phi around inf 73.1%

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

       1. unpow2 73.1%

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

       2. *-lft-identity 73.1%

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

       3. times-frac 73.3%

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

       4. /-rgt-identity 73.3%

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

   11. Simplified 73.3%

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

4. Final simplification 68.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 1.9999999920083944 \cdot 10^{-11}:\\ \;\;\;\;\frac{alphax \cdot \left(alphax \cdot u0\right)}{cos2phi}\\ \mathbf{else}:\\ \;\;\;\;u0 \cdot \left(alphay \cdot \frac{alphay}{sin2phi}\right)\\ \end{array} \]

Alternative 15: 76.0% accurate, 8.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 57.9%

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

   1. associate-/r* 57.9%

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

3. Simplified 57.9%

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

4. Taylor expanded in u0 around 0 76.3%

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

   1. unpow2 76.3%

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

   2. unpow2 76.3%

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

6. Simplified 76.3%

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

7. Final simplification 76.3%

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

Alternative 16: 76.0% accurate, 8.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 57.9%

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

   1. associate-/r* 57.9%

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

3. Simplified 57.9%

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

4. Taylor expanded in u0 around 0 76.3%

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

   1. unpow2 76.3%

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

   2. unpow2 76.3%

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

6. Simplified 76.3%

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

   1. associate-/r* 76.3%

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

   2. div-inv 76.3%

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

8. Applied egg-rr 76.3%

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

   1. un-div-inv 76.3%

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

10. Applied egg-rr 76.3%

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

11. Final simplification 76.3%

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

Alternative 17: 24.0% 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 57.9%

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

   1. associate-/r* 57.9%

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

3. Simplified 57.9%

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

4. Taylor expanded in u0 around 0 76.3%

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

   1. unpow2 76.3%

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

   2. unpow2 76.3%

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

6. Simplified 76.3%

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

7. Taylor expanded in cos2phi around inf 22.4%

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

   1. unpow2 22.4%

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

   2. associate-/l* 22.4%

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

   3. associate-/r/ 22.4%

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

9. Simplified 22.4%

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

10. Taylor expanded in u0 around 0 22.4%

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

    1. associate-*l/ 22.4%

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

    2. *-commutative 22.4%

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

    3. unpow2 22.4%

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

    4. associate-*l* 22.4%

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

12. Simplified 22.4%

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

13. Final simplification 22.4%

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

Alternative 18: 24.0% accurate, 16.6× speedup

Math

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

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(Float32(alphax * u0) / cos2phi))
end

MATLAB

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

Derivation

1. Initial program 57.9%

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

   1. associate-/r* 57.9%

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

3. Simplified 57.9%

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

4. Taylor expanded in u0 around 0 76.3%

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

   1. unpow2 76.3%

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

   2. unpow2 76.3%

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

6. Simplified 76.3%

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

7. Taylor expanded in cos2phi around inf 22.4%

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

   1. unpow2 22.4%

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

   2. associate-/l* 22.4%

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

   3. associate-/r/ 22.4%

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

9. Simplified 22.4%

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

10. Taylor expanded in u0 around 0 22.4%

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

    1. associate-*l/ 22.4%

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

    2. *-commutative 22.4%

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

    3. unpow2 22.4%

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

    4. associate-*l* 22.4%

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

12. Simplified 22.4%

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

13. Taylor expanded in alphax around 0 22.4%

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

14. Final simplification 22.4%

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

Reproduce

herbie shell --seed 2023222 
(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)))))