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

Percentage Accurate: 60.9% → 98.3%

Time: 14.6s

Alternatives: 15

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: 60.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Alternative 1: 98.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{-\mathsf{log1p}\left(-u0\right)}{\frac{cos2phi}{alphax} \cdot \frac{1}{alphax} + \frac{sin2phi}{alphay \cdot alphay}} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (/
  (- (log1p (- u0)))
  (+ (* (/ cos2phi alphax) (/ 1.0 alphax)) (/ sin2phi (* alphay alphay)))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	return -log1pf(-u0) / (((cos2phi / alphax) * (1.0f / alphax)) + (sin2phi / (alphay * alphay)));
}

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	return Float32(Float32(-log1p(Float32(-u0))) / Float32(Float32(Float32(cos2phi / alphax) * Float32(Float32(1.0) / alphax)) + Float32(sin2phi / Float32(alphay * alphay))))
end

Derivation

1. Initial program 59.2%

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

   1. neg-sub0 59.2%

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

   2. div-sub 59.2%

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

   3. --rgt-identity 59.2%

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

   4. div-sub 59.2%

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

   5. --rgt-identity 59.2%

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

   6. sub-neg 59.2%

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

   7. +-commutative 59.2%

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

   8. neg-sub0 59.2%

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

   9. associate-+l- 59.2%

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

   10. sub0-neg 59.2%

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

   11. neg-mul-1 59.2%

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

   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 98.8%

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

   1. div-inv 98.8%

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

5. Applied egg-rr 98.8%

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

6. Final simplification 98.8%

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

Alternative 2: 98.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 59.2%

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

   1. neg-sub0 59.2%

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

   2. div-sub 59.2%

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

   3. --rgt-identity 59.2%

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

   4. div-sub 59.2%

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

   5. --rgt-identity 59.2%

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

   6. neg-sub0 59.2%

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

   7. sub-neg 59.2%

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

   8. log1p-def 98.8%

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

3. Simplified 98.8%

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

4. Final simplification 98.8%

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

Alternative 3: 98.4% accurate, 1.0× speedup

Math

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

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(Float32(cos2phi / alphax) / alphax)))
end

Derivation

1. Initial program 59.2%

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

   1. neg-sub0 59.2%

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

   2. div-sub 59.2%

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

   3. --rgt-identity 59.2%

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

   4. div-sub 59.2%

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

   5. --rgt-identity 59.2%

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

   6. sub-neg 59.2%

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

   7. +-commutative 59.2%

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

   8. neg-sub0 59.2%

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

   9. associate-+l- 59.2%

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

   10. sub0-neg 59.2%

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

   11. neg-mul-1 59.2%

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

   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 98.8%

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

4. Final simplification 98.8%

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

Alternative 4: 92.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;sin2phi \leq 0.009999999776482582:\\ \;\;\;\;\frac{u0 - u0 \cdot \left(u0 \cdot -0.5\right)}{\frac{sin2phi}{alphay \cdot alphay} + \frac{\frac{cos2phi}{alphax}}{alphax}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(-u0\right) \cdot \left(alphay \cdot \frac{-alphay}{sin2phi}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (if (<= sin2phi 0.009999999776482582)
   (/
    (- u0 (* u0 (* u0 -0.5)))
    (+ (/ sin2phi (* alphay alphay)) (/ (/ cos2phi alphax) alphax)))
   (* (log1p (- u0)) (* alphay (/ (- alphay) sin2phi)))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float tmp;
	if (sin2phi <= 0.009999999776482582f) {
		tmp = (u0 - (u0 * (u0 * -0.5f))) / ((sin2phi / (alphay * alphay)) + ((cos2phi / alphax) / alphax));
	} else {
		tmp = log1pf(-u0) * (alphay * (-alphay / sin2phi));
	}
	return tmp;
}

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = Float32(0.0)
	if (sin2phi <= Float32(0.009999999776482582))
		tmp = Float32(Float32(u0 - Float32(u0 * Float32(u0 * Float32(-0.5)))) / Float32(Float32(sin2phi / Float32(alphay * alphay)) + Float32(Float32(cos2phi / alphax) / alphax)));
	else
		tmp = Float32(log1p(Float32(-u0)) * Float32(alphay * Float32(Float32(-alphay) / sin2phi)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if sin2phi < 0.00999999978

   1. Initial program 55.6%

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

      1. associate-/r* 55.5%

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

   3. Simplified 55.5%

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

   4. Taylor expanded in u0 around 0 88.9%

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

      1. +-commutative 88.9%

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

      2. mul-1-neg 88.9%

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

      3. unsub-neg 88.9%

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

      4. unpow2 88.9%

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

      5. associate-*r* 88.9%

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

   6. Simplified 88.9%

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

   if 0.00999999978 < sin2phi

   1. Initial program 63.4%

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

      1. associate-/r* 63.4%

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

   3. Simplified 63.4%

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

   4. Taylor expanded in cos2phi around 0 62.8%

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

      1. mul-1-neg 62.8%

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

      2. unpow2 62.8%

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

      3. *-commutative 62.8%

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

   6. Simplified 62.8%

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

   7. Taylor expanded in alphay around 0 62.8%

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

      1. *-commutative 62.8%

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

      2. sub-neg 62.8%

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

      3. log1p-def 97.8%

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

      4. unpow2 97.8%

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

      5. *-rgt-identity 97.8%

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

      6. associate-*r/ 97.7%

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

      7. unpow2 97.7%

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

      8. associate-*l* 97.6%

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

      9. associate-*r/ 97.9%

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

      10. associate-*l/ 97.9%

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

      11. unpow2 97.9%

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

      12. associate-*l/ 98.0%

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

      13. *-rgt-identity 98.0%

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

      14. *-commutative 98.0%

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

   9. Simplified 98.0%

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

4. Final simplification 93.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;sin2phi \leq 0.009999999776482582:\\ \;\;\;\;\frac{u0 - u0 \cdot \left(u0 \cdot -0.5\right)}{\frac{sin2phi}{alphay \cdot alphay} + \frac{\frac{cos2phi}{alphax}}{alphax}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(-u0\right) \cdot \left(alphay \cdot \frac{-alphay}{sin2phi}\right)\\ \end{array} \]

Alternative 5: 92.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{sin2phi}{alphay \cdot alphay}\\ \mathbf{if}\;sin2phi \leq 0.009999999776482582:\\ \;\;\;\;\frac{u0 - u0 \cdot \left(u0 \cdot -0.5\right)}{t_0 + \frac{\frac{cos2phi}{alphax}}{alphax}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\mathsf{log1p}\left(-u0\right)}{t_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (let* ((t_0 (/ sin2phi (* alphay alphay))))
   (if (<= sin2phi 0.009999999776482582)
     (/ (- u0 (* u0 (* u0 -0.5))) (+ t_0 (/ (/ cos2phi alphax) alphax)))
     (/ (- (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 (sin2phi <= 0.009999999776482582f) {
		tmp = (u0 - (u0 * (u0 * -0.5f))) / (t_0 + ((cos2phi / alphax) / alphax));
	} 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 (sin2phi <= Float32(0.009999999776482582))
		tmp = Float32(Float32(u0 - Float32(u0 * Float32(u0 * Float32(-0.5)))) / Float32(t_0 + Float32(Float32(cos2phi / alphax) / alphax)));
	else
		tmp = Float32(Float32(-log1p(Float32(-u0))) / t_0);
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if sin2phi < 0.00999999978

   1. Initial program 55.6%

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

      1. associate-/r* 55.5%

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

   3. Simplified 55.5%

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

   4. Taylor expanded in u0 around 0 88.9%

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

      1. +-commutative 88.9%

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

      2. mul-1-neg 88.9%

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

      3. unsub-neg 88.9%

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

      4. unpow2 88.9%

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

      5. associate-*r* 88.9%

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

   6. Simplified 88.9%

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

   if 0.00999999978 < sin2phi

   1. Initial program 63.4%

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

      1. neg-sub0 63.4%

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

      2. div-sub 63.4%

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

      3. --rgt-identity 63.4%

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

      4. div-sub 63.4%

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

      5. --rgt-identity 63.4%

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

      6. sub-neg 63.4%

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

      7. +-commutative 63.4%

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

      8. neg-sub0 63.4%

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

      9. associate-+l- 63.4%

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

      10. sub0-neg 63.4%

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

      11. neg-mul-1 63.4%

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

      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 99.0%

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

      1. div-inv 99.0%

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

   5. Applied egg-rr 99.0%

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

   6. Taylor expanded in cos2phi around 0 98.0%

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

      1. unpow2 98.0%

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

   8. Simplified 98.0%

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

4. Final simplification 93.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;sin2phi \leq 0.009999999776482582:\\ \;\;\;\;\frac{u0 - u0 \cdot \left(u0 \cdot -0.5\right)}{\frac{sin2phi}{alphay \cdot alphay} + \frac{\frac{cos2phi}{alphax}}{alphax}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\mathsf{log1p}\left(-u0\right)}{\frac{sin2phi}{alphay \cdot alphay}}\\ \end{array} \]

Alternative 6: 87.2% accurate, 6.1× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 59.2%

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

   1. neg-sub0 59.2%

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

   2. div-sub 59.2%

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

   3. --rgt-identity 59.2%

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

   4. div-sub 59.2%

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

   5. --rgt-identity 59.2%

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

   6. neg-sub0 59.2%

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

   7. sub-neg 59.2%

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

   8. log1p-def 98.8%

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

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. +-commutative 75.6%

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

   2. associate-/r* 75.6%

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

   3. associate-/r* 75.6%

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

   4. frac-add 75.5%

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

5. Applied egg-rr 98.5%

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

6. Taylor expanded in u0 around 0 87.4%

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

   1. +-commutative 87.4%

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

   2. mul-1-neg 87.4%

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

   3. unsub-neg 87.4%

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

   4. *-commutative 87.4%

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

   5. unpow2 87.4%

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

   6. associate-*l* 87.4%

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

8. Simplified 87.4%

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

9. Taylor expanded in sin2phi around 0 87.5%

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

    1. unpow2 87.5%

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

    2. unpow2 87.5%

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

11. Simplified 87.5%

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

12. Final simplification 87.5%

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

Alternative 7: 87.2% accurate, 6.1× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 59.2%

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

   1. associate-/r* 59.2%

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

3. Simplified 59.2%

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

4. Taylor expanded in u0 around 0 87.6%

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

   1. +-commutative 87.6%

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

   2. mul-1-neg 87.6%

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

   3. unsub-neg 87.6%

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

   4. unpow2 87.6%

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

   5. associate-*r* 87.6%

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

6. Simplified 87.6%

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

7. Final simplification 87.6%

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

Alternative 8: 81.0% accurate, 6.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;sin2phi \leq 0.009999999776482582:\\ \;\;\;\;\frac{u0}{\frac{cos2phi}{alphax} \cdot \frac{1}{alphax} + \frac{sin2phi}{alphay \cdot alphay}}\\ \mathbf{else}:\\ \;\;\;\;\left(alphay \cdot alphay\right) \cdot \frac{u0 + 0.5 \cdot \left(u0 \cdot u0\right)}{sin2phi}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (if (<= sin2phi 0.009999999776482582)
   (/
    u0
    (+ (* (/ cos2phi alphax) (/ 1.0 alphax)) (/ sin2phi (* alphay alphay))))
   (* (* alphay alphay) (/ (+ u0 (* 0.5 (* u0 u0))) sin2phi))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float tmp;
	if (sin2phi <= 0.009999999776482582f) {
		tmp = u0 / (((cos2phi / alphax) * (1.0f / alphax)) + (sin2phi / (alphay * alphay)));
	} else {
		tmp = (alphay * alphay) * ((u0 + (0.5f * (u0 * u0))) / sin2phi);
	}
	return tmp;
}

Fortran

real(4) function code(alphax, alphay, u0, cos2phi, sin2phi)
    real(4), intent (in) :: alphax
    real(4), intent (in) :: alphay
    real(4), intent (in) :: u0
    real(4), intent (in) :: cos2phi
    real(4), intent (in) :: sin2phi
    real(4) :: tmp
    if (sin2phi <= 0.009999999776482582e0) then
        tmp = u0 / (((cos2phi / alphax) * (1.0e0 / alphax)) + (sin2phi / (alphay * alphay)))
    else
        tmp = (alphay * alphay) * ((u0 + (0.5e0 * (u0 * u0))) / sin2phi)
    end if
    code = tmp
end function

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = Float32(0.0)
	if (sin2phi <= Float32(0.009999999776482582))
		tmp = Float32(u0 / Float32(Float32(Float32(cos2phi / alphax) * Float32(Float32(1.0) / alphax)) + Float32(sin2phi / Float32(alphay * alphay))));
	else
		tmp = Float32(Float32(alphay * alphay) * Float32(Float32(u0 + Float32(Float32(0.5) * Float32(u0 * u0))) / sin2phi));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = single(0.0);
	if (sin2phi <= single(0.009999999776482582))
		tmp = u0 / (((cos2phi / alphax) * (single(1.0) / alphax)) + (sin2phi / (alphay * alphay)));
	else
		tmp = (alphay * alphay) * ((u0 + (single(0.5) * (u0 * u0))) / sin2phi);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if sin2phi < 0.00999999978

   1. Initial program 55.6%

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

      1. associate-/r* 55.5%

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

   3. Simplified 55.5%

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

   4. Taylor expanded in u0 around 0 75.5%

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

      1. unpow2 75.5%

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

      2. unpow2 75.5%

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

   6. Simplified 75.5%

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

      1. associate-/r* 75.5%

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

      2. un-div-inv 75.5%

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

   8. Applied egg-rr 75.5%

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

   if 0.00999999978 < sin2phi

   1. Initial program 63.4%

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

      1. neg-sub0 63.4%

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

      2. div-sub 63.4%

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

      3. --rgt-identity 63.4%

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

      4. div-sub 63.4%

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

      5. --rgt-identity 63.4%

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

      6. neg-sub0 63.4%

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

      7. sub-neg 63.4%

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

      8. log1p-def 99.0%

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

   3. Simplified 99.0%

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

      1. +-commutative 75.8%

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

      2. associate-/r* 75.8%

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

      3. associate-/r* 75.8%

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

      4. frac-add 75.7%

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

   5. Applied egg-rr 98.5%

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

   6. Taylor expanded in u0 around 0 85.9%

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

      1. +-commutative 85.9%

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

      2. mul-1-neg 85.9%

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

      3. unsub-neg 85.9%

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

      4. *-commutative 85.9%

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

      5. unpow2 85.9%

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

      6. associate-*l* 85.9%

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

   8. Simplified 85.9%

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

   9. Taylor expanded in sin2phi around inf 85.8%

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

       1. associate-/l* 85.9%

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

       2. associate-/r/ 86.0%

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

       3. cancel-sign-sub-inv 86.0%

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

       4. metadata-eval 86.0%

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

       5. unpow2 86.0%

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

       6. unpow2 86.0%

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

   11. Simplified 86.0%

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

4. Final simplification 80.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;sin2phi \leq 0.009999999776482582:\\ \;\;\;\;\frac{u0}{\frac{cos2phi}{alphax} \cdot \frac{1}{alphax} + \frac{sin2phi}{alphay \cdot alphay}}\\ \mathbf{else}:\\ \;\;\;\;\left(alphay \cdot alphay\right) \cdot \frac{u0 + 0.5 \cdot \left(u0 \cdot u0\right)}{sin2phi}\\ \end{array} \]

Alternative 9: 73.7% accurate, 7.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;sin2phi \leq 2.00000006274879 \cdot 10^{-22}:\\ \;\;\;\;\left(alphax \cdot alphax\right) \cdot \frac{u0}{cos2phi}\\ \mathbf{else}:\\ \;\;\;\;\left(alphay \cdot alphay\right) \cdot \frac{u0 + 0.5 \cdot \left(u0 \cdot u0\right)}{sin2phi}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (if (<= sin2phi 2.00000006274879e-22)
   (* (* alphax alphax) (/ u0 cos2phi))
   (* (* alphay alphay) (/ (+ u0 (* 0.5 (* u0 u0))) sin2phi))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float tmp;
	if (sin2phi <= 2.00000006274879e-22f) {
		tmp = (alphax * alphax) * (u0 / cos2phi);
	} else {
		tmp = (alphay * alphay) * ((u0 + (0.5f * (u0 * u0))) / sin2phi);
	}
	return tmp;
}

Fortran

real(4) function code(alphax, alphay, u0, cos2phi, sin2phi)
    real(4), intent (in) :: alphax
    real(4), intent (in) :: alphay
    real(4), intent (in) :: u0
    real(4), intent (in) :: cos2phi
    real(4), intent (in) :: sin2phi
    real(4) :: tmp
    if (sin2phi <= 2.00000006274879e-22) then
        tmp = (alphax * alphax) * (u0 / cos2phi)
    else
        tmp = (alphay * alphay) * ((u0 + (0.5e0 * (u0 * u0))) / sin2phi)
    end if
    code = tmp
end function

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = Float32(0.0)
	if (sin2phi <= Float32(2.00000006274879e-22))
		tmp = Float32(Float32(alphax * alphax) * Float32(u0 / cos2phi));
	else
		tmp = Float32(Float32(alphay * alphay) * Float32(Float32(u0 + Float32(Float32(0.5) * Float32(u0 * u0))) / sin2phi));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = single(0.0);
	if (sin2phi <= single(2.00000006274879e-22))
		tmp = (alphax * alphax) * (u0 / cos2phi);
	else
		tmp = (alphay * alphay) * ((u0 + (single(0.5) * (u0 * u0))) / sin2phi);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if sin2phi < 2.00000006e-22

   1. Initial program 57.2%

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

      1. associate-/r* 57.2%

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

   3. Simplified 57.2%

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

   4. Taylor expanded in u0 around 0 73.8%

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

      1. unpow2 73.8%

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

      2. unpow2 73.8%

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

   6. Simplified 73.8%

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

   7. Taylor expanded in cos2phi around inf 55.7%

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

      1. unpow2 55.7%

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

      2. *-commutative 55.7%

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

      3. *-lft-identity 55.7%

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

      4. times-frac 55.8%

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

      5. /-rgt-identity 55.8%

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

   9. Simplified 55.8%

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

   if 2.00000006e-22 < sin2phi

   1. Initial program 59.9%

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

      1. neg-sub0 59.9%

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

      2. div-sub 59.9%

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

      3. --rgt-identity 59.9%

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

      4. div-sub 59.9%

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

      5. --rgt-identity 59.9%

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

      6. neg-sub0 59.9%

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

      7. sub-neg 59.9%

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

      8. log1p-def 98.8%

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

   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. +-commutative 76.2%

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

      2. associate-/r* 76.2%

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

      3. associate-/r* 76.2%

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

      4. frac-add 76.1%

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

   5. Applied egg-rr 98.5%

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

   6. Taylor expanded in u0 around 0 87.5%

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

      1. +-commutative 87.5%

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

      2. mul-1-neg 87.5%

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

      3. unsub-neg 87.5%

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

      4. *-commutative 87.5%

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

      5. unpow2 87.5%

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

      6. associate-*l* 87.5%

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

   8. Simplified 87.5%

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

   9. Taylor expanded in sin2phi around inf 77.4%

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

       1. associate-/l* 77.4%

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

       2. associate-/r/ 77.5%

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

       3. cancel-sign-sub-inv 77.5%

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

       4. metadata-eval 77.5%

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

       5. unpow2 77.5%

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

       6. unpow2 77.5%

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

   11. Simplified 77.5%

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

4. Final simplification 72.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;sin2phi \leq 2.00000006274879 \cdot 10^{-22}:\\ \;\;\;\;\left(alphax \cdot alphax\right) \cdot \frac{u0}{cos2phi}\\ \mathbf{else}:\\ \;\;\;\;\left(alphay \cdot alphay\right) \cdot \frac{u0 + 0.5 \cdot \left(u0 \cdot u0\right)}{sin2phi}\\ \end{array} \]

Alternative 10: 81.0% accurate, 7.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;sin2phi \leq 0.009999999776482582:\\ \;\;\;\;\frac{u0}{\frac{sin2phi}{alphay \cdot alphay} + \frac{cos2phi}{alphax \cdot alphax}}\\ \mathbf{else}:\\ \;\;\;\;\left(alphay \cdot alphay\right) \cdot \frac{u0 + 0.5 \cdot \left(u0 \cdot u0\right)}{sin2phi}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (if (<= sin2phi 0.009999999776482582)
   (/ u0 (+ (/ sin2phi (* alphay alphay)) (/ cos2phi (* alphax alphax))))
   (* (* alphay alphay) (/ (+ u0 (* 0.5 (* u0 u0))) sin2phi))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float tmp;
	if (sin2phi <= 0.009999999776482582f) {
		tmp = u0 / ((sin2phi / (alphay * alphay)) + (cos2phi / (alphax * alphax)));
	} else {
		tmp = (alphay * alphay) * ((u0 + (0.5f * (u0 * u0))) / sin2phi);
	}
	return tmp;
}

Fortran

real(4) function code(alphax, alphay, u0, cos2phi, sin2phi)
    real(4), intent (in) :: alphax
    real(4), intent (in) :: alphay
    real(4), intent (in) :: u0
    real(4), intent (in) :: cos2phi
    real(4), intent (in) :: sin2phi
    real(4) :: tmp
    if (sin2phi <= 0.009999999776482582e0) then
        tmp = u0 / ((sin2phi / (alphay * alphay)) + (cos2phi / (alphax * alphax)))
    else
        tmp = (alphay * alphay) * ((u0 + (0.5e0 * (u0 * u0))) / sin2phi)
    end if
    code = tmp
end function

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = Float32(0.0)
	if (sin2phi <= Float32(0.009999999776482582))
		tmp = Float32(u0 / Float32(Float32(sin2phi / Float32(alphay * alphay)) + Float32(cos2phi / Float32(alphax * alphax))));
	else
		tmp = Float32(Float32(alphay * alphay) * Float32(Float32(u0 + Float32(Float32(0.5) * Float32(u0 * u0))) / sin2phi));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = single(0.0);
	if (sin2phi <= single(0.009999999776482582))
		tmp = u0 / ((sin2phi / (alphay * alphay)) + (cos2phi / (alphax * alphax)));
	else
		tmp = (alphay * alphay) * ((u0 + (single(0.5) * (u0 * u0))) / sin2phi);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if sin2phi < 0.00999999978

   1. Initial program 55.6%

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

      1. associate-/r* 55.5%

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

   3. Simplified 55.5%

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

   4. Taylor expanded in u0 around 0 75.5%

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

      1. unpow2 75.5%

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

      2. unpow2 75.5%

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

   6. Simplified 75.5%

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

   if 0.00999999978 < sin2phi

   1. Initial program 63.4%

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

      1. neg-sub0 63.4%

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

      2. div-sub 63.4%

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

      3. --rgt-identity 63.4%

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

      4. div-sub 63.4%

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

      5. --rgt-identity 63.4%

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

      6. neg-sub0 63.4%

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

      7. sub-neg 63.4%

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

      8. log1p-def 99.0%

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

   3. Simplified 99.0%

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

      1. +-commutative 75.8%

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

      2. associate-/r* 75.8%

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

      3. associate-/r* 75.8%

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

      4. frac-add 75.7%

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

   5. Applied egg-rr 98.5%

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

   6. Taylor expanded in u0 around 0 85.9%

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

      1. +-commutative 85.9%

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

      2. mul-1-neg 85.9%

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

      3. unsub-neg 85.9%

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

      4. *-commutative 85.9%

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

      5. unpow2 85.9%

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

      6. associate-*l* 85.9%

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

   8. Simplified 85.9%

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

   9. Taylor expanded in sin2phi around inf 85.8%

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

       1. associate-/l* 85.9%

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

       2. associate-/r/ 86.0%

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

       3. cancel-sign-sub-inv 86.0%

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

       4. metadata-eval 86.0%

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

       5. unpow2 86.0%

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

       6. unpow2 86.0%

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

   11. Simplified 86.0%

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

4. Final simplification 80.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;sin2phi \leq 0.009999999776482582:\\ \;\;\;\;\frac{u0}{\frac{sin2phi}{alphay \cdot alphay} + \frac{cos2phi}{alphax \cdot alphax}}\\ \mathbf{else}:\\ \;\;\;\;\left(alphay \cdot alphay\right) \cdot \frac{u0 + 0.5 \cdot \left(u0 \cdot u0\right)}{sin2phi}\\ \end{array} \]

Alternative 11: 81.0% accurate, 7.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;sin2phi \leq 0.009999999776482582:\\ \;\;\;\;\frac{u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{\frac{sin2phi}{alphay}}{alphay}}\\ \mathbf{else}:\\ \;\;\;\;\left(alphay \cdot alphay\right) \cdot \frac{u0 + 0.5 \cdot \left(u0 \cdot u0\right)}{sin2phi}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (if (<= sin2phi 0.009999999776482582)
   (/ u0 (+ (/ cos2phi (* alphax alphax)) (/ (/ sin2phi alphay) alphay)))
   (* (* alphay alphay) (/ (+ u0 (* 0.5 (* u0 u0))) sin2phi))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float tmp;
	if (sin2phi <= 0.009999999776482582f) {
		tmp = u0 / ((cos2phi / (alphax * alphax)) + ((sin2phi / alphay) / alphay));
	} else {
		tmp = (alphay * alphay) * ((u0 + (0.5f * (u0 * u0))) / sin2phi);
	}
	return tmp;
}

Fortran

real(4) function code(alphax, alphay, u0, cos2phi, sin2phi)
    real(4), intent (in) :: alphax
    real(4), intent (in) :: alphay
    real(4), intent (in) :: u0
    real(4), intent (in) :: cos2phi
    real(4), intent (in) :: sin2phi
    real(4) :: tmp
    if (sin2phi <= 0.009999999776482582e0) then
        tmp = u0 / ((cos2phi / (alphax * alphax)) + ((sin2phi / alphay) / alphay))
    else
        tmp = (alphay * alphay) * ((u0 + (0.5e0 * (u0 * u0))) / sin2phi)
    end if
    code = tmp
end function

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = Float32(0.0)
	if (sin2phi <= Float32(0.009999999776482582))
		tmp = Float32(u0 / Float32(Float32(cos2phi / Float32(alphax * alphax)) + Float32(Float32(sin2phi / alphay) / alphay)));
	else
		tmp = Float32(Float32(alphay * alphay) * Float32(Float32(u0 + Float32(Float32(0.5) * Float32(u0 * u0))) / sin2phi));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = single(0.0);
	if (sin2phi <= single(0.009999999776482582))
		tmp = u0 / ((cos2phi / (alphax * alphax)) + ((sin2phi / alphay) / alphay));
	else
		tmp = (alphay * alphay) * ((u0 + (single(0.5) * (u0 * u0))) / sin2phi);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if sin2phi < 0.00999999978

   1. Initial program 55.6%

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

      1. neg-sub0 55.6%

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

      2. div-sub 55.6%

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

      3. --rgt-identity 55.6%

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

      4. div-sub 55.6%

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

      5. --rgt-identity 55.6%

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

      6. sub-neg 55.6%

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

      7. +-commutative 55.6%

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

      8. neg-sub0 55.6%

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

      9. associate-+l- 55.6%

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

      10. sub0-neg 55.6%

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

      11. neg-mul-1 55.6%

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

      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 98.6%

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

      1. div-inv 98.7%

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

   5. Applied egg-rr 98.7%

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

      1. un-div-inv 98.6%

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

      2. associate-/r* 98.6%

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

      3. clear-num 98.6%

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

   7. Applied egg-rr 98.6%

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

      1. inv-pow 98.6%

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

      2. associate-/l* 98.6%

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

   9. Applied egg-rr 98.6%

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

       1. unpow-1 98.6%

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

       2. associate-/r/ 98.6%

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

   11. Simplified 98.6%

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

   12. Taylor expanded in u0 around 0 75.5%

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

       1. +-commutative 75.5%

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

       2. unpow2 75.5%

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

       3. unpow2 75.5%

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

       4. associate-/r* 75.5%

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

   14. Simplified 75.5%

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

   if 0.00999999978 < sin2phi

   1. Initial program 63.4%

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

      1. neg-sub0 63.4%

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

      2. div-sub 63.4%

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

      3. --rgt-identity 63.4%

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

      4. div-sub 63.4%

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

      5. --rgt-identity 63.4%

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

      6. neg-sub0 63.4%

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

      7. sub-neg 63.4%

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

      8. log1p-def 99.0%

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

   3. Simplified 99.0%

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

      1. +-commutative 75.8%

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

      2. associate-/r* 75.8%

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

      3. associate-/r* 75.8%

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

      4. frac-add 75.7%

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

   5. Applied egg-rr 98.5%

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

   6. Taylor expanded in u0 around 0 85.9%

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

      1. +-commutative 85.9%

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

      2. mul-1-neg 85.9%

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

      3. unsub-neg 85.9%

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

      4. *-commutative 85.9%

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

      5. unpow2 85.9%

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

      6. associate-*l* 85.9%

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

   8. Simplified 85.9%

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

   9. Taylor expanded in sin2phi around inf 85.8%

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

       1. associate-/l* 85.9%

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

       2. associate-/r/ 86.0%

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

       3. cancel-sign-sub-inv 86.0%

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

       4. metadata-eval 86.0%

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

       5. unpow2 86.0%

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

       6. unpow2 86.0%

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

   11. Simplified 86.0%

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

4. Final simplification 80.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;sin2phi \leq 0.009999999776482582:\\ \;\;\;\;\frac{u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{\frac{sin2phi}{alphay}}{alphay}}\\ \mathbf{else}:\\ \;\;\;\;\left(alphay \cdot alphay\right) \cdot \frac{u0 + 0.5 \cdot \left(u0 \cdot u0\right)}{sin2phi}\\ \end{array} \]

Alternative 12: 81.0% accurate, 7.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;sin2phi \leq 0.009999999776482582:\\ \;\;\;\;\frac{u0}{\frac{sin2phi}{alphay \cdot alphay} + \frac{\frac{cos2phi}{alphax}}{alphax}}\\ \mathbf{else}:\\ \;\;\;\;\left(alphay \cdot alphay\right) \cdot \frac{u0 + 0.5 \cdot \left(u0 \cdot u0\right)}{sin2phi}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (if (<= sin2phi 0.009999999776482582)
   (/ u0 (+ (/ sin2phi (* alphay alphay)) (/ (/ cos2phi alphax) alphax)))
   (* (* alphay alphay) (/ (+ u0 (* 0.5 (* u0 u0))) sin2phi))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float tmp;
	if (sin2phi <= 0.009999999776482582f) {
		tmp = u0 / ((sin2phi / (alphay * alphay)) + ((cos2phi / alphax) / alphax));
	} else {
		tmp = (alphay * alphay) * ((u0 + (0.5f * (u0 * u0))) / sin2phi);
	}
	return tmp;
}

Fortran

real(4) function code(alphax, alphay, u0, cos2phi, sin2phi)
    real(4), intent (in) :: alphax
    real(4), intent (in) :: alphay
    real(4), intent (in) :: u0
    real(4), intent (in) :: cos2phi
    real(4), intent (in) :: sin2phi
    real(4) :: tmp
    if (sin2phi <= 0.009999999776482582e0) then
        tmp = u0 / ((sin2phi / (alphay * alphay)) + ((cos2phi / alphax) / alphax))
    else
        tmp = (alphay * alphay) * ((u0 + (0.5e0 * (u0 * u0))) / sin2phi)
    end if
    code = tmp
end function

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = Float32(0.0)
	if (sin2phi <= Float32(0.009999999776482582))
		tmp = Float32(u0 / Float32(Float32(sin2phi / Float32(alphay * alphay)) + Float32(Float32(cos2phi / alphax) / alphax)));
	else
		tmp = Float32(Float32(alphay * alphay) * Float32(Float32(u0 + Float32(Float32(0.5) * Float32(u0 * u0))) / sin2phi));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = single(0.0);
	if (sin2phi <= single(0.009999999776482582))
		tmp = u0 / ((sin2phi / (alphay * alphay)) + ((cos2phi / alphax) / alphax));
	else
		tmp = (alphay * alphay) * ((u0 + (single(0.5) * (u0 * u0))) / sin2phi);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if sin2phi < 0.00999999978

   1. Initial program 55.6%

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

      1. associate-/r* 55.5%

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

   3. Simplified 55.5%

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

   4. Taylor expanded in u0 around 0 75.5%

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

      1. mul-1-neg 75.5%

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

   6. Simplified 75.5%

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

   if 0.00999999978 < sin2phi

   1. Initial program 63.4%

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

      1. neg-sub0 63.4%

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

      2. div-sub 63.4%

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

      3. --rgt-identity 63.4%

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

      4. div-sub 63.4%

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

      5. --rgt-identity 63.4%

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

      6. neg-sub0 63.4%

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

      7. sub-neg 63.4%

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

      8. log1p-def 99.0%

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

   3. Simplified 99.0%

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

      1. +-commutative 75.8%

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

      2. associate-/r* 75.8%

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

      3. associate-/r* 75.8%

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

      4. frac-add 75.7%

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

   5. Applied egg-rr 98.5%

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

   6. Taylor expanded in u0 around 0 85.9%

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

      1. +-commutative 85.9%

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

      2. mul-1-neg 85.9%

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

      3. unsub-neg 85.9%

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

      4. *-commutative 85.9%

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

      5. unpow2 85.9%

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

      6. associate-*l* 85.9%

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

   8. Simplified 85.9%

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

   9. Taylor expanded in sin2phi around inf 85.8%

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

       1. associate-/l* 85.9%

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

       2. associate-/r/ 86.0%

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

       3. cancel-sign-sub-inv 86.0%

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

       4. metadata-eval 86.0%

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

       5. unpow2 86.0%

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

       6. unpow2 86.0%

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

   11. Simplified 86.0%

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

4. Final simplification 80.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;sin2phi \leq 0.009999999776482582:\\ \;\;\;\;\frac{u0}{\frac{sin2phi}{alphay \cdot alphay} + \frac{\frac{cos2phi}{alphax}}{alphax}}\\ \mathbf{else}:\\ \;\;\;\;\left(alphay \cdot alphay\right) \cdot \frac{u0 + 0.5 \cdot \left(u0 \cdot u0\right)}{sin2phi}\\ \end{array} \]

Alternative 13: 66.5% accurate, 12.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;sin2phi \leq 2.00000006274879 \cdot 10^{-22}:\\ \;\;\;\;\left(alphax \cdot alphax\right) \cdot \frac{u0}{cos2phi}\\ \mathbf{else}:\\ \;\;\;\;\left(alphay \cdot alphay\right) \cdot \frac{u0}{sin2phi}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (if (<= sin2phi 2.00000006274879e-22)
   (* (* alphax alphax) (/ u0 cos2phi))
   (* (* alphay alphay) (/ u0 sin2phi))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float tmp;
	if (sin2phi <= 2.00000006274879e-22f) {
		tmp = (alphax * alphax) * (u0 / cos2phi);
	} else {
		tmp = (alphay * alphay) * (u0 / sin2phi);
	}
	return tmp;
}

Fortran

real(4) function code(alphax, alphay, u0, cos2phi, sin2phi)
    real(4), intent (in) :: alphax
    real(4), intent (in) :: alphay
    real(4), intent (in) :: u0
    real(4), intent (in) :: cos2phi
    real(4), intent (in) :: sin2phi
    real(4) :: tmp
    if (sin2phi <= 2.00000006274879e-22) then
        tmp = (alphax * alphax) * (u0 / cos2phi)
    else
        tmp = (alphay * alphay) * (u0 / sin2phi)
    end if
    code = tmp
end function

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = Float32(0.0)
	if (sin2phi <= Float32(2.00000006274879e-22))
		tmp = Float32(Float32(alphax * alphax) * Float32(u0 / cos2phi));
	else
		tmp = Float32(Float32(alphay * alphay) * Float32(u0 / sin2phi));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = single(0.0);
	if (sin2phi <= single(2.00000006274879e-22))
		tmp = (alphax * alphax) * (u0 / cos2phi);
	else
		tmp = (alphay * alphay) * (u0 / sin2phi);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if sin2phi < 2.00000006e-22

   1. Initial program 57.2%

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

      1. associate-/r* 57.2%

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

   3. Simplified 57.2%

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

   4. Taylor expanded in u0 around 0 73.8%

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

      1. unpow2 73.8%

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

      2. unpow2 73.8%

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

   6. Simplified 73.8%

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

   7. Taylor expanded in cos2phi around inf 55.7%

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

      1. unpow2 55.7%

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

      2. *-commutative 55.7%

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

      3. *-lft-identity 55.7%

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

      4. times-frac 55.8%

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

      5. /-rgt-identity 55.8%

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

   9. Simplified 55.8%

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

   if 2.00000006e-22 < sin2phi

   1. Initial program 59.9%

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

      1. associate-/r* 59.9%

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

   3. Simplified 59.9%

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

   4. Taylor expanded in u0 around 0 76.2%

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

      1. unpow2 76.2%

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

      2. unpow2 76.2%

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

   6. Simplified 76.2%

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

      1. +-commutative 76.2%

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

      2. associate-/r* 76.2%

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

      3. associate-/r* 76.2%

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

      4. frac-add 76.1%

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

   8. Applied egg-rr 76.1%

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

   9. Taylor expanded in alphay around 0 76.1%

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

       1. associate-/l* 76.1%

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

   11. Simplified 76.1%

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

   12. Taylor expanded in sin2phi around inf 67.8%

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

       1. associate-/l* 67.8%

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

       2. associate-/r/ 67.8%

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

       3. unpow2 67.8%

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

   14. Simplified 67.8%

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

4. Final simplification 65.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;sin2phi \leq 2.00000006274879 \cdot 10^{-22}:\\ \;\;\;\;\left(alphax \cdot alphax\right) \cdot \frac{u0}{cos2phi}\\ \mathbf{else}:\\ \;\;\;\;\left(alphay \cdot alphay\right) \cdot \frac{u0}{sin2phi}\\ \end{array} \]

Alternative 14: 66.5% accurate, 12.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;sin2phi \leq 2.00000006274879 \cdot 10^{-22}:\\ \;\;\;\;\left(alphax \cdot alphax\right) \cdot \frac{u0}{cos2phi}\\ \mathbf{else}:\\ \;\;\;\;\frac{u0 \cdot \left(alphay \cdot alphay\right)}{sin2phi}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (if (<= sin2phi 2.00000006274879e-22)
   (* (* 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 <= 2.00000006274879e-22f) {
		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 <= 2.00000006274879e-22) 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(2.00000006274879e-22))
		tmp = Float32(Float32(alphax * alphax) * Float32(u0 / cos2phi));
	else
		tmp = Float32(Float32(u0 * Float32(alphay * alphay)) / sin2phi);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = single(0.0);
	if (sin2phi <= single(2.00000006274879e-22))
		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 < 2.00000006e-22

   1. Initial program 57.2%

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

      1. associate-/r* 57.2%

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

   3. Simplified 57.2%

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

   4. Taylor expanded in u0 around 0 73.8%

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

      1. unpow2 73.8%

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

      2. unpow2 73.8%

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

   6. Simplified 73.8%

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

   7. Taylor expanded in cos2phi around inf 55.7%

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

      1. unpow2 55.7%

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

      2. *-commutative 55.7%

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

      3. *-lft-identity 55.7%

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

      4. times-frac 55.8%

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

      5. /-rgt-identity 55.8%

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

   9. Simplified 55.8%

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

   if 2.00000006e-22 < sin2phi

   1. Initial program 59.9%

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

      1. associate-/r* 59.9%

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

   3. Simplified 59.9%

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

   4. Taylor expanded in cos2phi around 0 54.7%

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

      1. mul-1-neg 54.7%

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

      2. unpow2 54.7%

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

      3. *-commutative 54.7%

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

   6. Simplified 54.7%

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

   7. Taylor expanded in u0 around 0 67.8%

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

      1. associate-*r/ 67.8%

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

      2. mul-1-neg 67.8%

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

      3. *-commutative 67.8%

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

      4. distribute-rgt-neg-in 67.8%

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

      5. unpow2 67.8%

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

   9. Simplified 67.8%

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

4. Final simplification 65.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;sin2phi \leq 2.00000006274879 \cdot 10^{-22}:\\ \;\;\;\;\left(alphax \cdot alphax\right) \cdot \frac{u0}{cos2phi}\\ \mathbf{else}:\\ \;\;\;\;\frac{u0 \cdot \left(alphay \cdot alphay\right)}{sin2phi}\\ \end{array} \]

Alternative 15: 24.0% accurate, 16.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 59.2%

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

   1. associate-/r* 59.2%

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

3. Simplified 59.2%

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

4. Taylor expanded in u0 around 0 75.6%

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

   1. unpow2 75.6%

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

   2. unpow2 75.6%

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

6. Simplified 75.6%

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

7. Taylor expanded in cos2phi around inf 24.2%

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

   1. unpow2 24.2%

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

   2. *-commutative 24.2%

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

   3. *-lft-identity 24.2%

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

   4. times-frac 24.2%

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

   5. /-rgt-identity 24.2%

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

9. Simplified 24.2%

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

10. Final simplification 24.2%

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

Reproduce

herbie shell --seed 2023213 
(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)))))