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

Percentage Accurate: 60.8% → 98.4%

Time: 14.5s

Alternatives: 16

Speedup: 12.7×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Initial Program: 60.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Alternative 1: 98.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 58.3%

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

   1. neg-sub0 58.3%

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

   2. div-sub 58.3%

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

   3. --rgt-identity 58.3%

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

   4. div-sub 58.3%

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

   5. --rgt-identity 58.3%

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

   6. sub-neg 58.3%

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

   7. +-commutative 58.3%

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

   8. neg-sub0 58.3%

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

   9. associate-+l- 58.3%

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

   10. sub0-neg 58.3%

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

   11. neg-mul-1 58.3%

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

   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 98.7%

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

4. Final simplification 98.7%

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

Alternative 2: 86.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{sin2phi}{alphay \cdot alphay}\\ \mathbf{if}\;t_0 \leq 1.9999999920083944 \cdot 10^{-11}:\\ \;\;\;\;\frac{u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{\frac{sin2phi}{alphay}}{alphay}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\mathsf{log1p}\left(-u0\right)}{t_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (let* ((t_0 (/ sin2phi (* alphay alphay))))
   (if (<= t_0 1.9999999920083944e-11)
     (/ u0 (+ (/ cos2phi (* alphax alphax)) (/ (/ sin2phi alphay) alphay)))
     (/ (- (log1p (- 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 <= 1.9999999920083944e-11f) {
		tmp = u0 / ((cos2phi / (alphax * alphax)) + ((sin2phi / alphay) / alphay));
	} else {
		tmp = -log1pf(-u0) / t_0;
	}
	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(1.9999999920083944e-11))
		tmp = Float32(u0 / Float32(Float32(cos2phi / Float32(alphax * alphax)) + Float32(Float32(sin2phi / alphay) / alphay)));
	else
		tmp = Float32(Float32(-log1p(Float32(-u0))) / t_0);
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

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

   1. Initial program 52.6%

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

      1. neg-sub0 52.6%

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

      2. div-sub 52.6%

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

      3. --rgt-identity 52.6%

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

      4. div-sub 52.6%

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

      5. --rgt-identity 52.6%

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

      6. neg-sub0 52.6%

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

      7. sub-neg 52.6%

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

      8. log1p-def 98.8%

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

   3. Simplified 98.8%

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

      1. associate-/r* 98.8%

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

      2. div-inv 98.6%

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

      3. div-inv 97.9%

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

      4. associate-*l* 98.4%

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

   5. Applied egg-rr 98.4%

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

      1. un-div-inv 98.7%

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

   7. Applied egg-rr 98.7%

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

   8. Taylor expanded in u0 around 0 77.5%

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

      1. unpow2 77.5%

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

      2. unpow2 77.5%

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

      3. associate-/r* 77.6%

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

   10. Simplified 77.6%

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

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

   1. Initial program 61.0%

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

      1. neg-sub0 61.0%

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

      2. div-sub 61.0%

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

      3. --rgt-identity 61.0%

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

      4. div-sub 61.0%

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

      5. --rgt-identity 61.0%

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

      6. neg-sub0 61.0%

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

      7. sub-neg 61.0%

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

      8. log1p-def 98.6%

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

   3. Simplified 98.6%

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

      1. associate-/r* 98.6%

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

      2. div-inv 98.6%

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

   5. Applied egg-rr 98.6%

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

   6. Taylor expanded in cos2phi around 0 93.9%

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

      1. unpow2 93.9%

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

   8. Simplified 93.9%

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

4. Final simplification 88.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 1.9999999920083944 \cdot 10^{-11}:\\ \;\;\;\;\frac{u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{\frac{sin2phi}{alphay}}{alphay}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\mathsf{log1p}\left(-u0\right)}{\frac{sin2phi}{alphay \cdot alphay}}\\ \end{array} \]

Alternative 3: 87.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 1.9999999920083944 \cdot 10^{-11}:\\ \;\;\;\;\frac{u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{\frac{sin2phi}{alphay}}{alphay}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(-u0\right) \cdot \left(alphay \cdot \left(-alphay\right)\right)}{sin2phi}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (if (<= (/ sin2phi (* alphay alphay)) 1.9999999920083944e-11)
   (/ u0 (+ (/ cos2phi (* alphax alphax)) (/ (/ sin2phi alphay) alphay)))
   (/ (* (log1p (- u0)) (* alphay (- alphay))) sin2phi)))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float tmp;
	if ((sin2phi / (alphay * alphay)) <= 1.9999999920083944e-11f) {
		tmp = u0 / ((cos2phi / (alphax * alphax)) + ((sin2phi / alphay) / alphay));
	} else {
		tmp = (log1pf(-u0) * (alphay * -alphay)) / sin2phi;
	}
	return tmp;
}

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = Float32(0.0)
	if (Float32(sin2phi / Float32(alphay * alphay)) <= Float32(1.9999999920083944e-11))
		tmp = Float32(u0 / Float32(Float32(cos2phi / Float32(alphax * alphax)) + Float32(Float32(sin2phi / alphay) / alphay)));
	else
		tmp = Float32(Float32(log1p(Float32(-u0)) * Float32(alphay * Float32(-alphay))) / sin2phi);
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

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

   1. Initial program 52.6%

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

      1. neg-sub0 52.6%

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

      2. div-sub 52.6%

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

      3. --rgt-identity 52.6%

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

      4. div-sub 52.6%

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

      5. --rgt-identity 52.6%

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

      6. neg-sub0 52.6%

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

      7. sub-neg 52.6%

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

      8. log1p-def 98.8%

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

   3. Simplified 98.8%

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

      1. associate-/r* 98.8%

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

      2. div-inv 98.6%

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

      3. div-inv 97.9%

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

      4. associate-*l* 98.4%

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

   5. Applied egg-rr 98.4%

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

      1. un-div-inv 98.7%

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

   7. Applied egg-rr 98.7%

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

   8. Taylor expanded in u0 around 0 77.5%

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

      1. unpow2 77.5%

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

      2. unpow2 77.5%

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

      3. associate-/r* 77.6%

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

   10. Simplified 77.6%

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

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

   1. Initial program 61.0%

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

      1. neg-sub0 61.0%

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

      2. div-sub 61.0%

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

      3. --rgt-identity 61.0%

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

      4. div-sub 61.0%

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

      5. --rgt-identity 61.0%

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

      6. neg-sub0 61.0%

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

      7. sub-neg 61.0%

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

      8. log1p-def 98.6%

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

   3. Simplified 98.6%

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

      1. associate-/r* 98.6%

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

      2. div-inv 98.6%

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

   5. Applied egg-rr 98.6%

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

   6. Taylor expanded in cos2phi around 0 59.6%

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

      1. associate-*r/ 59.6%

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

      2. *-commutative 59.6%

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

      3. sub-neg 59.6%

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

      4. log1p-def 94.2%

         \[\leadsto \frac{-1 \cdot \left(\color{blue}{\mathsf{log1p}\left(-u0\right)} \cdot {alphay}^{2}\right)}{sin2phi} \]

      5. associate-*r* 94.2%

         \[\leadsto \frac{\color{blue}{\left(-1 \cdot \mathsf{log1p}\left(-u0\right)\right) \cdot {alphay}^{2}}}{sin2phi} \]

      6. neg-mul-1 94.2%

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

      7. unpow2 94.2%

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

   8. Simplified 94.2%

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

4. Final simplification 88.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 1.9999999920083944 \cdot 10^{-11}:\\ \;\;\;\;\frac{u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{\frac{sin2phi}{alphay}}{alphay}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(-u0\right) \cdot \left(alphay \cdot \left(-alphay\right)\right)}{sin2phi}\\ \end{array} \]

Alternative 4: 98.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 58.3%

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

   1. neg-sub0 58.3%

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

   2. div-sub 58.3%

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

   3. --rgt-identity 58.3%

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

   4. div-sub 58.3%

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

   5. --rgt-identity 58.3%

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

   6. neg-sub0 58.3%

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

   7. sub-neg 58.3%

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

   8. log1p-def 98.7%

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

3. Simplified 98.7%

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

4. Final simplification 98.7%

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

Alternative 5: 75.8% accurate, 5.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 58.3%

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

   1. neg-sub0 58.3%

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

   2. div-sub 58.3%

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

   3. --rgt-identity 58.3%

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

   4. div-sub 58.3%

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

   5. --rgt-identity 58.3%

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

   6. neg-sub0 58.3%

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

   7. sub-neg 58.3%

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

   8. log1p-def 98.7%

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

3. Simplified 98.7%

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

   1. clear-num 98.6%

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

   2. associate-/r* 98.6%

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

   3. frac-add 98.3%

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

   4. associate-/l* 98.2%

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

   5. *-commutative 98.2%

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

   6. *-un-lft-identity 98.2%

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

   7. associate-/l* 98.1%

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

5. Applied egg-rr 98.1%

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

6. Taylor expanded in u0 around 0 78.5%

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

   1. times-frac 78.5%

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

   2. +-commutative 78.5%

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

   3. times-frac 78.7%

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

   4. unpow2 78.7%

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

   5. associate-/l* 78.5%

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

   6. unpow2 78.5%

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

8. Simplified 78.5%

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

9. Taylor expanded in alphay around 0 78.7%

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

    1. unpow2 78.7%

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

    2. associate-*l/ 78.8%

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

    3. *-commutative 78.8%

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

    4. associate-*l/ 78.8%

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

11. Simplified 78.8%

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

12. Final simplification 78.8%

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

Alternative 6: 75.8% accurate, 5.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 58.3%

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

   1. neg-sub0 58.3%

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

   2. div-sub 58.3%

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

   3. --rgt-identity 58.3%

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

   4. div-sub 58.3%

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

   5. --rgt-identity 58.3%

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

   6. neg-sub0 58.3%

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

   7. sub-neg 58.3%

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

   8. log1p-def 98.7%

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

3. Simplified 98.7%

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

   1. clear-num 98.6%

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

   2. associate-/r* 98.6%

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

   3. frac-add 98.3%

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

   4. associate-/l* 98.2%

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

   5. *-commutative 98.2%

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

   6. *-un-lft-identity 98.2%

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

   7. associate-/l* 98.1%

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

5. Applied egg-rr 98.1%

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

6. Taylor expanded in u0 around 0 78.5%

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

   1. times-frac 78.5%

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

   2. +-commutative 78.5%

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

   3. times-frac 78.7%

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

   4. unpow2 78.7%

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

   5. associate-/l* 78.5%

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

   6. unpow2 78.5%

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

8. Simplified 78.5%

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

9. Taylor expanded in alphay around 0 78.7%

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

    1. unpow2 78.7%

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

    2. *-commutative 78.7%

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

    3. associate-*r* 78.8%

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

11. Simplified 78.8%

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

12. Final simplification 78.8%

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

Alternative 7: 66.6% accurate, 7.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 1.9999999920083944 \cdot 10^{-12}:\\ \;\;\;\;u0 \cdot \frac{1}{\frac{cos2phi}{alphax \cdot alphax}}\\ \mathbf{else}:\\ \;\;\;\;\frac{u0 \cdot \left(alphay \cdot alphay\right)}{sin2phi}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (if (<= (/ sin2phi (* alphay alphay)) 1.9999999920083944e-12)
   (* u0 (/ 1.0 (/ cos2phi (* alphax alphax))))
   (/ (* 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-12f) {
		tmp = u0 * (1.0f / (cos2phi / (alphax * alphax)));
	} else {
		tmp = (u0 * (alphay * alphay)) / sin2phi;
	}
	return tmp;
}

Fortran

real(4) function code(alphax, alphay, u0, cos2phi, sin2phi)
    real(4), intent (in) :: alphax
    real(4), intent (in) :: alphay
    real(4), intent (in) :: u0
    real(4), intent (in) :: cos2phi
    real(4), intent (in) :: sin2phi
    real(4) :: tmp
    if ((sin2phi / (alphay * alphay)) <= 1.9999999920083944e-12) then
        tmp = u0 * (1.0e0 / (cos2phi / (alphax * alphax)))
    else
        tmp = (u0 * (alphay * alphay)) / sin2phi
    end if
    code = tmp
end function

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = Float32(0.0)
	if (Float32(sin2phi / Float32(alphay * alphay)) <= Float32(1.9999999920083944e-12))
		tmp = Float32(u0 * Float32(Float32(1.0) / Float32(cos2phi / Float32(alphax * alphax))));
	else
		tmp = Float32(Float32(u0 * Float32(alphay * alphay)) / sin2phi);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = single(0.0);
	if ((sin2phi / (alphay * alphay)) <= single(1.9999999920083944e-12))
		tmp = u0 * (single(1.0) / (cos2phi / (alphax * alphax)));
	else
		tmp = (u0 * (alphay * alphay)) / sin2phi;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 sin2phi (*.f32 alphay alphay)) < 1.99999999e-12

   1. Initial program 53.7%

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

   2. Taylor expanded in u0 around 0 77.0%

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

      1. unpow2 77.0%

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

      2. unpow2 77.0%

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

   4. Simplified 77.0%

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

   5. Taylor expanded in cos2phi around inf 58.9%

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

      1. associate-/l* 59.0%

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

      2. unpow2 59.0%

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

   7. Simplified 59.0%

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

      1. div-inv 59.1%

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

   9. Applied egg-rr 59.1%

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

   if 1.99999999e-12 < (/.f32 sin2phi (*.f32 alphay alphay))

   1. Initial program 60.4%

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

   2. Taylor expanded in u0 around 0 79.2%

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

      1. unpow2 79.2%

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

      2. unpow2 79.2%

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

   4. Simplified 79.2%

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

   5. Taylor expanded in cos2phi around 0 76.4%

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

      1. expm1-log1p-u 76.4%

         \[\leadsto \frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(u0 \cdot {alphay}^{2}\right)\right)}}{sin2phi} \]

      2. expm1-udef 31.3%

         \[\leadsto \frac{\color{blue}{e^{\mathsf{log1p}\left(u0 \cdot {alphay}^{2}\right)} - 1}}{sin2phi} \]

      3. pow2 31.3%

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

   7. Applied egg-rr 31.3%

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

      1. expm1-def 76.4%

         \[\leadsto \frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(u0 \cdot \left(alphay \cdot alphay\right)\right)\right)}}{sin2phi} \]

      2. expm1-log1p 76.4%

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

   9. Simplified 76.4%

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

4. Final simplification 70.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 1.9999999920083944 \cdot 10^{-12}:\\ \;\;\;\;u0 \cdot \frac{1}{\frac{cos2phi}{alphax \cdot alphax}}\\ \mathbf{else}:\\ \;\;\;\;\frac{u0 \cdot \left(alphay \cdot alphay\right)}{sin2phi}\\ \end{array} \]

Alternative 8: 66.6% accurate, 7.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 1.9999999920083944 \cdot 10^{-12}:\\ \;\;\;\;u0 \cdot \frac{1}{\frac{cos2phi}{alphax \cdot alphax}}\\ \mathbf{else}:\\ \;\;\;\;\left(u0 \cdot \left(alphay \cdot alphay\right)\right) \cdot \frac{1}{sin2phi}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (if (<= (/ sin2phi (* alphay alphay)) 1.9999999920083944e-12)
   (* u0 (/ 1.0 (/ cos2phi (* alphax alphax))))
   (* (* u0 (* alphay alphay)) (/ 1.0 sin2phi))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float tmp;
	if ((sin2phi / (alphay * alphay)) <= 1.9999999920083944e-12f) {
		tmp = u0 * (1.0f / (cos2phi / (alphax * alphax)));
	} else {
		tmp = (u0 * (alphay * alphay)) * (1.0f / 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-12) then
        tmp = u0 * (1.0e0 / (cos2phi / (alphax * alphax)))
    else
        tmp = (u0 * (alphay * alphay)) * (1.0e0 / 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-12))
		tmp = Float32(u0 * Float32(Float32(1.0) / Float32(cos2phi / Float32(alphax * alphax))));
	else
		tmp = Float32(Float32(u0 * Float32(alphay * alphay)) * Float32(Float32(1.0) / 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-12))
		tmp = u0 * (single(1.0) / (cos2phi / (alphax * alphax)));
	else
		tmp = (u0 * (alphay * alphay)) * (single(1.0) / sin2phi);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 sin2phi (*.f32 alphay alphay)) < 1.99999999e-12

   1. Initial program 53.7%

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

   2. Taylor expanded in u0 around 0 77.0%

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

      1. unpow2 77.0%

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

      2. unpow2 77.0%

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

   4. Simplified 77.0%

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

   5. Taylor expanded in cos2phi around inf 58.9%

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

      1. associate-/l* 59.0%

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

      2. unpow2 59.0%

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

   7. Simplified 59.0%

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

      1. div-inv 59.1%

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

   9. Applied egg-rr 59.1%

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

   if 1.99999999e-12 < (/.f32 sin2phi (*.f32 alphay alphay))

   1. Initial program 60.4%

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

   2. Taylor expanded in u0 around 0 79.2%

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

      1. unpow2 79.2%

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

      2. unpow2 79.2%

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

   4. Simplified 79.2%

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

   5. Taylor expanded in cos2phi around 0 76.4%

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

      1. div-inv 76.4%

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

      2. pow2 76.4%

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

   7. Applied egg-rr 76.4%

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

4. Final simplification 70.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 1.9999999920083944 \cdot 10^{-12}:\\ \;\;\;\;u0 \cdot \frac{1}{\frac{cos2phi}{alphax \cdot alphax}}\\ \mathbf{else}:\\ \;\;\;\;\left(u0 \cdot \left(alphay \cdot alphay\right)\right) \cdot \frac{1}{sin2phi}\\ \end{array} \]

Alternative 9: 66.6% accurate, 8.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 1.9999999920083944 \cdot 10^{-12}:\\ \;\;\;\;\frac{u0}{\frac{cos2phi}{alphax \cdot alphax}}\\ \mathbf{else}:\\ \;\;\;\;u0 \cdot \left(alphay \cdot \frac{alphay}{sin2phi}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (if (<= (/ sin2phi (* alphay alphay)) 1.9999999920083944e-12)
   (/ u0 (/ cos2phi (* alphax alphax)))
   (* 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-12f) {
		tmp = u0 / (cos2phi / (alphax * alphax));
	} else {
		tmp = u0 * (alphay * (alphay / sin2phi));
	}
	return tmp;
}

Fortran

real(4) function code(alphax, alphay, u0, cos2phi, sin2phi)
    real(4), intent (in) :: alphax
    real(4), intent (in) :: alphay
    real(4), intent (in) :: u0
    real(4), intent (in) :: cos2phi
    real(4), intent (in) :: sin2phi
    real(4) :: tmp
    if ((sin2phi / (alphay * alphay)) <= 1.9999999920083944e-12) then
        tmp = u0 / (cos2phi / (alphax * alphax))
    else
        tmp = u0 * (alphay * (alphay / sin2phi))
    end if
    code = tmp
end function

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = Float32(0.0)
	if (Float32(sin2phi / Float32(alphay * alphay)) <= Float32(1.9999999920083944e-12))
		tmp = Float32(u0 / Float32(cos2phi / Float32(alphax * alphax)));
	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-12))
		tmp = u0 / (cos2phi / (alphax * alphax));
	else
		tmp = u0 * (alphay * (alphay / sin2phi));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 sin2phi (*.f32 alphay alphay)) < 1.99999999e-12

   1. Initial program 53.7%

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

   2. Taylor expanded in u0 around 0 77.0%

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

      1. unpow2 77.0%

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

      2. unpow2 77.0%

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

   4. Simplified 77.0%

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

   5. Taylor expanded in cos2phi around inf 58.9%

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

      1. associate-/l* 59.0%

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

      2. unpow2 59.0%

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

   7. Simplified 59.0%

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

   if 1.99999999e-12 < (/.f32 sin2phi (*.f32 alphay alphay))

   1. Initial program 60.4%

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

   2. Taylor expanded in u0 around 0 79.2%

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

      1. unpow2 79.2%

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

      2. unpow2 79.2%

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

   4. Simplified 79.2%

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

      1. div-inv 79.2%

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

      2. +-commutative 79.2%

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

   6. Applied egg-rr 79.2%

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

   7. Taylor expanded in sin2phi around inf 76.3%

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

      1. unpow2 76.3%

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

      2. associate-*l/ 76.3%

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

   9. Simplified 76.3%

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

4. Final simplification 70.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 1.9999999920083944 \cdot 10^{-12}:\\ \;\;\;\;\frac{u0}{\frac{cos2phi}{alphax \cdot alphax}}\\ \mathbf{else}:\\ \;\;\;\;u0 \cdot \left(alphay \cdot \frac{alphay}{sin2phi}\right)\\ \end{array} \]

Alternative 10: 66.6% accurate, 8.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 1.9999999920083944 \cdot 10^{-12}:\\ \;\;\;\;\frac{u0}{\frac{cos2phi}{alphax \cdot alphax}}\\ \mathbf{else}:\\ \;\;\;\;\frac{u0 \cdot \left(alphay \cdot alphay\right)}{sin2phi}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (if (<= (/ sin2phi (* alphay alphay)) 1.9999999920083944e-12)
   (/ u0 (/ cos2phi (* alphax alphax)))
   (/ (* 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-12f) {
		tmp = u0 / (cos2phi / (alphax * alphax));
	} else {
		tmp = (u0 * (alphay * alphay)) / sin2phi;
	}
	return tmp;
}

Fortran

real(4) function code(alphax, alphay, u0, cos2phi, sin2phi)
    real(4), intent (in) :: alphax
    real(4), intent (in) :: alphay
    real(4), intent (in) :: u0
    real(4), intent (in) :: cos2phi
    real(4), intent (in) :: sin2phi
    real(4) :: tmp
    if ((sin2phi / (alphay * alphay)) <= 1.9999999920083944e-12) then
        tmp = u0 / (cos2phi / (alphax * alphax))
    else
        tmp = (u0 * (alphay * alphay)) / sin2phi
    end if
    code = tmp
end function

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = Float32(0.0)
	if (Float32(sin2phi / Float32(alphay * alphay)) <= Float32(1.9999999920083944e-12))
		tmp = Float32(u0 / Float32(cos2phi / Float32(alphax * alphax)));
	else
		tmp = Float32(Float32(u0 * Float32(alphay * alphay)) / sin2phi);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = single(0.0);
	if ((sin2phi / (alphay * alphay)) <= single(1.9999999920083944e-12))
		tmp = u0 / (cos2phi / (alphax * alphax));
	else
		tmp = (u0 * (alphay * alphay)) / sin2phi;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 sin2phi (*.f32 alphay alphay)) < 1.99999999e-12

   1. Initial program 53.7%

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

   2. Taylor expanded in u0 around 0 77.0%

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

      1. unpow2 77.0%

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

      2. unpow2 77.0%

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

   4. Simplified 77.0%

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

   5. Taylor expanded in cos2phi around inf 58.9%

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

      1. associate-/l* 59.0%

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

      2. unpow2 59.0%

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

   7. Simplified 59.0%

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

   if 1.99999999e-12 < (/.f32 sin2phi (*.f32 alphay alphay))

   1. Initial program 60.4%

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

   2. Taylor expanded in u0 around 0 79.2%

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

      1. unpow2 79.2%

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

      2. unpow2 79.2%

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

   4. Simplified 79.2%

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

   5. Taylor expanded in cos2phi around 0 76.4%

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

      1. expm1-log1p-u 76.4%

         \[\leadsto \frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(u0 \cdot {alphay}^{2}\right)\right)}}{sin2phi} \]

      2. expm1-udef 31.3%

         \[\leadsto \frac{\color{blue}{e^{\mathsf{log1p}\left(u0 \cdot {alphay}^{2}\right)} - 1}}{sin2phi} \]

      3. pow2 31.3%

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

   7. Applied egg-rr 31.3%

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

      1. expm1-def 76.4%

         \[\leadsto \frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(u0 \cdot \left(alphay \cdot alphay\right)\right)\right)}}{sin2phi} \]

      2. expm1-log1p 76.4%

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

   9. Simplified 76.4%

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

4. Final simplification 70.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 1.9999999920083944 \cdot 10^{-12}:\\ \;\;\;\;\frac{u0}{\frac{cos2phi}{alphax \cdot alphax}}\\ \mathbf{else}:\\ \;\;\;\;\frac{u0 \cdot \left(alphay \cdot alphay\right)}{sin2phi}\\ \end{array} \]

Alternative 11: 75.8% 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 58.3%

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

2. Taylor expanded in u0 around 0 78.5%

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

   1. unpow2 78.5%

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

   2. unpow2 78.5%

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

4. Simplified 78.5%

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

5. Final simplification 78.5%

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

Alternative 12: 75.8% 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 58.3%

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

   1. neg-sub0 58.3%

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

   2. div-sub 58.3%

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

   3. --rgt-identity 58.3%

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

   4. div-sub 58.3%

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

   5. --rgt-identity 58.3%

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

   6. neg-sub0 58.3%

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

   7. sub-neg 58.3%

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

   8. log1p-def 98.7%

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

3. Simplified 98.7%

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

   1. associate-/r* 98.7%

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

   2. div-inv 98.6%

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

   3. div-inv 98.4%

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

   4. associate-*l* 98.5%

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

5. Applied egg-rr 98.5%

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

   1. un-div-inv 98.6%

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

7. Applied egg-rr 98.6%

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

8. Taylor expanded in u0 around 0 78.5%

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

   1. unpow2 78.5%

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

   2. unpow2 78.5%

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

   3. associate-/r* 78.5%

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

10. Simplified 78.5%

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

11. Final simplification 78.5%

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

Alternative 13: 66.7% accurate, 12.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = Float32(0.0)
	if (sin2phi <= Float32(5.0000000843119176e-17))
		tmp = Float32(alphax * Float32(Float32(u0 * alphax) / 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 <= single(5.0000000843119176e-17))
		tmp = alphax * ((u0 * alphax) / cos2phi);
	else
		tmp = u0 * (alphay * (alphay / sin2phi));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if sin2phi < 5.00000008e-17

   1. Initial program 54.0%

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

   2. Taylor expanded in u0 around 0 76.7%

      \[\leadsto \color{blue}{\frac{u0}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}}} \]
   3. 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%

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

   4. Simplified 76.7%

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

   5. Taylor expanded in cos2phi around inf 59.1%

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

      1. associate-/l* 59.2%

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

      2. unpow2 59.2%

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

   7. Simplified 59.2%

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

   8. Taylor expanded in u0 around 0 59.1%

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

      1. unpow2 59.1%

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

      2. associate-*l/ 59.2%

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

      3. *-commutative 59.2%

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

      4. associate-*l* 59.1%

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

   10. Simplified 59.1%

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

   11. Taylor expanded in alphax around 0 59.2%

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

   if 5.00000008e-17 < sin2phi

   1. Initial program 60.1%

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

   2. Taylor expanded in u0 around 0 79.3%

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

      1. unpow2 79.3%

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

      2. unpow2 79.3%

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

   4. Simplified 79.3%

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

      1. div-inv 79.3%

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

      2. +-commutative 79.3%

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

   6. Applied egg-rr 79.3%

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

   7. Taylor expanded in sin2phi around inf 75.1%

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

      1. unpow2 75.1%

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

      2. associate-*l/ 75.1%

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

   9. Simplified 75.1%

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

4. Final simplification 70.3%

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

Alternative 14: 66.7% accurate, 12.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = Float32(0.0)
	if (sin2phi <= Float32(5.0000000843119176e-17))
		tmp = Float32(Float32(alphax * alphax) * Float32(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 <= single(5.0000000843119176e-17))
		tmp = (alphax * alphax) * (u0 / cos2phi);
	else
		tmp = u0 * (alphay * (alphay / sin2phi));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if sin2phi < 5.00000008e-17

   1. Initial program 54.0%

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

   2. Taylor expanded in u0 around 0 76.7%

      \[\leadsto \color{blue}{\frac{u0}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}}} \]
   3. 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%

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

   4. Simplified 76.7%

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

   5. Taylor expanded in cos2phi around inf 59.1%

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

      1. associate-/l* 59.2%

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

      2. unpow2 59.2%

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

   7. Simplified 59.2%

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

      1. associate-/r/ 59.2%

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

   9. Applied egg-rr 59.2%

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

   if 5.00000008e-17 < sin2phi

   1. Initial program 60.1%

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

   2. Taylor expanded in u0 around 0 79.3%

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

      1. unpow2 79.3%

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

      2. unpow2 79.3%

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

   4. Simplified 79.3%

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

      1. div-inv 79.3%

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

      2. +-commutative 79.3%

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

   6. Applied egg-rr 79.3%

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

   7. Taylor expanded in sin2phi around inf 75.1%

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

      1. unpow2 75.1%

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

      2. associate-*l/ 75.1%

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

   9. Simplified 75.1%

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

4. Final simplification 70.3%

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

Alternative 15: 23.5% 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 58.3%

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

2. Taylor expanded in u0 around 0 78.5%

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

   1. unpow2 78.5%

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

   2. unpow2 78.5%

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

4. Simplified 78.5%

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

5. Taylor expanded in cos2phi around inf 25.3%

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

   1. associate-/l* 25.4%

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

   2. unpow2 25.4%

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

7. Simplified 25.4%

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

8. Taylor expanded in u0 around 0 25.3%

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

   1. unpow2 25.3%

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

   2. associate-*l/ 25.4%

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

   3. *-commutative 25.4%

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

   4. associate-*l* 25.3%

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

10. Simplified 25.3%

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

11. Final simplification 25.3%

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

Alternative 16: 23.5% accurate, 16.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 58.3%

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

2. Taylor expanded in u0 around 0 78.5%

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

   1. unpow2 78.5%

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

   2. unpow2 78.5%

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

4. Simplified 78.5%

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

5. Taylor expanded in cos2phi around inf 25.3%

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

   1. associate-/l* 25.4%

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

   2. unpow2 25.4%

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

7. Simplified 25.4%

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

8. Taylor expanded in u0 around 0 25.3%

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

   1. unpow2 25.3%

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

   2. associate-*l/ 25.4%

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

   3. *-commutative 25.4%

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

   4. associate-*l* 25.3%

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

10. Simplified 25.3%

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

11. Taylor expanded in alphax around 0 25.4%

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

12. Final simplification 25.4%

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

Reproduce

herbie shell --seed 2023194 
(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)))))