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

Percentage Accurate: 60.6% → 98.3%

Time: 14.1s

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: 60.6% 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

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 62.9%

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

   1. neg-sub0 62.9%

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

   2. div-sub 62.9%

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

   3. --rgt-identity 62.9%

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

   4. div-sub 62.9%

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

   5. --rgt-identity 62.9%

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

   6. sub-neg 62.9%

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

   7. +-commutative 62.9%

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

   8. neg-sub0 62.9%

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

   9. associate-+l- 62.9%

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

   10. sub0-neg 62.9%

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

   11. neg-mul-1 62.9%

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

   12. log-prod -0.0%

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

   13. associate--r+ -0.0%

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

3. Simplified 98.2%

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

   1. associate-/r* 98.3%

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

   2. div-inv 98.1%

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

5. Applied egg-rr 98.1%

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

   1. un-div-inv 98.3%

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

7. Applied egg-rr 98.3%

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

8. Final simplification 98.3%

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

Alternative 2: 98.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 62.9%

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

   1. neg-sub0 62.9%

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

   2. div-sub 62.9%

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

   3. --rgt-identity 62.9%

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

   4. div-sub 62.9%

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

   5. --rgt-identity 62.9%

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

   6. neg-sub0 62.9%

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

   7. sub-neg 62.9%

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

   8. log1p-def 98.2%

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

3. Simplified 98.2%

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

4. Final simplification 98.2%

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

Alternative 3: 98.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 62.9%

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

   1. neg-sub0 62.9%

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

   2. div-sub 62.9%

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

   3. --rgt-identity 62.9%

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

   4. div-sub 62.9%

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

   5. --rgt-identity 62.9%

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

   6. sub-neg 62.9%

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

   7. +-commutative 62.9%

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

   8. neg-sub0 62.9%

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

   9. associate-+l- 62.9%

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

   10. sub0-neg 62.9%

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

   11. neg-mul-1 62.9%

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

   12. log-prod -0.0%

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

   13. associate--r+ -0.0%

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

3. Simplified 98.2%

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

4. Final simplification 98.2%

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

Alternative 4: 91.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u0 \leq 0.014000000432133675:\\ \;\;\;\;\frac{u0 - u0 \cdot \left(u0 \cdot -0.5\right)}{\frac{sin2phi}{alphay \cdot alphay} + \frac{cos2phi}{alphax \cdot alphax}}\\ \mathbf{else}:\\ \;\;\;\;\frac{alphay \cdot \left(-alphay\right)}{\frac{sin2phi}{\mathsf{log1p}\left(-u0\right)}}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (if (<= u0 0.014000000432133675)
   (/
    (- u0 (* u0 (* u0 -0.5)))
    (+ (/ sin2phi (* alphay alphay)) (/ cos2phi (* alphax alphax))))
   (/ (* alphay (- alphay)) (/ sin2phi (log1p (- u0))))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float tmp;
	if (u0 <= 0.014000000432133675f) {
		tmp = (u0 - (u0 * (u0 * -0.5f))) / ((sin2phi / (alphay * alphay)) + (cos2phi / (alphax * alphax)));
	} else {
		tmp = (alphay * -alphay) / (sin2phi / log1pf(-u0));
	}
	return tmp;
}

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = Float32(0.0)
	if (u0 <= Float32(0.014000000432133675))
		tmp = Float32(Float32(u0 - Float32(u0 * Float32(u0 * Float32(-0.5)))) / Float32(Float32(sin2phi / Float32(alphay * alphay)) + Float32(cos2phi / Float32(alphax * alphax))));
	else
		tmp = Float32(Float32(alphay * Float32(-alphay)) / Float32(sin2phi / log1p(Float32(-u0))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if u0 < 0.0140000004

   1. Initial program 55.2%

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

      1. associate-/r* 55.2%

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

   3. Simplified 55.2%

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

   4. Taylor expanded in u0 around 0 95.8%

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

      1. +-commutative 95.8%

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

      2. fma-def 95.8%

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

      3. unpow2 95.8%

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

      4. mul-1-neg 95.8%

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

   6. Simplified 95.8%

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

      1. div-inv 95.6%

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

      2. associate-/r* 95.6%

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

      3. associate-/r* 95.5%

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

   8. Applied egg-rr 95.5%

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

      1. associate-*r/ 95.8%

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

      2. *-lft-identity 95.8%

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

      3. *-rgt-identity 95.8%

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

      4. unpow2 95.8%

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

      5. fma-neg 95.8%

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

      6. *-commutative 95.8%

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

      7. unpow2 95.8%

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

      8. associate-*l* 95.8%

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

      9. *-lft-identity 95.8%

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

      10. associate-/l/ 95.8%

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

   10. Simplified 95.8%

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

   if 0.0140000004 < u0

   1. Initial program 93.2%

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

      1. associate-/r* 93.4%

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

   3. Simplified 93.4%

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

   4. Taylor expanded in cos2phi around 0 71.2%

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

      1. mul-1-neg 71.2%

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

      2. unpow2 71.2%

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

      3. associate-/l* 69.8%

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

      4. distribute-neg-frac 69.8%

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

      5. distribute-rgt-neg-in 69.8%

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

      6. sub-neg 69.8%

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

      7. mul-1-neg 69.8%

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

      8. log1p-def 71.5%

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

      9. mul-1-neg 71.5%

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

   6. Simplified 71.5%

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

4. Final simplification 90.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u0 \leq 0.014000000432133675:\\ \;\;\;\;\frac{u0 - u0 \cdot \left(u0 \cdot -0.5\right)}{\frac{sin2phi}{alphay \cdot alphay} + \frac{cos2phi}{alphax \cdot alphax}}\\ \mathbf{else}:\\ \;\;\;\;\frac{alphay \cdot \left(-alphay\right)}{\frac{sin2phi}{\mathsf{log1p}\left(-u0\right)}}\\ \end{array} \]

Alternative 5: 92.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;sin2phi \leq 90:\\ \;\;\;\;\frac{u0 - \left(u0 \cdot u0\right) \cdot -0.5}{\frac{\frac{cos2phi}{alphax}}{alphax} + sin2phi \cdot \frac{1}{alphay \cdot alphay}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(-u0\right) \cdot \left(alphay \cdot \left(-alphay\right)\right)}{sin2phi}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (if (<= sin2phi 90.0)
   (/
    (- u0 (* (* u0 u0) -0.5))
    (+ (/ (/ cos2phi alphax) alphax) (* sin2phi (/ 1.0 (* alphay alphay)))))
   (/ (* (log1p (- u0)) (* alphay (- alphay))) sin2phi)))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float tmp;
	if (sin2phi <= 90.0f) {
		tmp = (u0 - ((u0 * u0) * -0.5f)) / (((cos2phi / alphax) / alphax) + (sin2phi * (1.0f / (alphay * alphay))));
	} else {
		tmp = (log1pf(-u0) * (alphay * -alphay)) / sin2phi;
	}
	return tmp;
}

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = Float32(0.0)
	if (sin2phi <= Float32(90.0))
		tmp = Float32(Float32(u0 - Float32(Float32(u0 * u0) * Float32(-0.5))) / Float32(Float32(Float32(cos2phi / alphax) / alphax) + Float32(sin2phi * Float32(Float32(1.0) / Float32(alphay * alphay)))));
	else
		tmp = Float32(Float32(log1p(Float32(-u0)) * Float32(alphay * Float32(-alphay))) / sin2phi);
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if sin2phi < 90

   1. Initial program 58.1%

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

      1. neg-sub0 58.1%

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

      2. div-sub 58.1%

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

      3. --rgt-identity 58.1%

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

      4. div-sub 58.1%

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

      5. --rgt-identity 58.1%

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

      6. sub-neg 58.1%

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

      7. +-commutative 58.1%

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

      8. neg-sub0 58.1%

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

      9. associate-+l- 58.1%

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

      10. sub0-neg 58.1%

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

      11. neg-mul-1 58.1%

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

      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. associate-/r* 98.7%

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

      2. div-inv 98.6%

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

   5. Applied egg-rr 98.6%

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

      1. un-div-inv 98.7%

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

   7. Applied egg-rr 98.7%

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

      1. associate-/r* 98.6%

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

      2. div-inv 98.7%

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

   9. Applied egg-rr 98.7%

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

   10. Taylor expanded in u0 around 0 84.5%

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

       1. neg-mul-1 84.5%

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

       2. +-commutative 84.5%

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

       3. sub-neg 84.5%

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

       4. *-commutative 84.5%

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

       5. unpow2 84.5%

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

   12. Simplified 84.5%

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

   if 90 < sin2phi

   1. Initial program 67.8%

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

      1. neg-sub0 67.8%

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

      2. div-sub 67.8%

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

      3. --rgt-identity 67.8%

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

      4. div-sub 67.8%

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

      5. --rgt-identity 67.8%

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

      6. sub-neg 67.8%

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

      7. +-commutative 67.8%

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

      8. neg-sub0 67.8%

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

      9. associate-+l- 67.8%

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

      10. sub0-neg 67.8%

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

      11. neg-mul-1 67.8%

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

      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 97.8%

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

      1. associate-/r* 97.9%

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

      2. div-inv 97.7%

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

   5. Applied egg-rr 97.7%

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

      1. un-div-inv 97.9%

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

   7. Applied egg-rr 97.9%

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

   8. Taylor expanded in cos2phi around 0 68.9%

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

      1. associate-*r/ 68.9%

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

      2. *-commutative 68.9%

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

      3. associate-*r* 68.9%

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

      4. sub-neg 68.9%

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

      5. log1p-def 98.9%

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

      6. neg-mul-1 98.9%

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

      7. unpow2 98.9%

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

   10. Simplified 98.9%

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

4. Final simplification 91.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;sin2phi \leq 90:\\ \;\;\;\;\frac{u0 - \left(u0 \cdot u0\right) \cdot -0.5}{\frac{\frac{cos2phi}{alphax}}{alphax} + sin2phi \cdot \frac{1}{alphay \cdot alphay}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(-u0\right) \cdot \left(alphay \cdot \left(-alphay\right)\right)}{sin2phi}\\ \end{array} \]

Alternative 6: 81.1% accurate, 5.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{sin2phi}{alphay \cdot alphay}\\ \mathbf{if}\;t_0 \leq 0.019999999552965164:\\ \;\;\;\;\frac{u0}{\frac{\frac{cos2phi}{alphax}}{alphax} + t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{alphay \cdot \left(-alphay\right)}{\frac{sin2phi}{\left(u0 \cdot u0\right) \cdot -0.5 - u0}}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (let* ((t_0 (/ sin2phi (* alphay alphay))))
   (if (<= t_0 0.019999999552965164)
     (/ u0 (+ (/ (/ cos2phi alphax) alphax) t_0))
     (/ (* alphay (- alphay)) (/ sin2phi (- (* (* u0 u0) -0.5) u0))))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float t_0 = sin2phi / (alphay * alphay);
	float tmp;
	if (t_0 <= 0.019999999552965164f) {
		tmp = u0 / (((cos2phi / alphax) / alphax) + t_0);
	} else {
		tmp = (alphay * -alphay) / (sin2phi / (((u0 * u0) * -0.5f) - u0));
	}
	return tmp;
}

Fortran

real(4) function code(alphax, alphay, u0, cos2phi, sin2phi)
    real(4), intent (in) :: alphax
    real(4), intent (in) :: alphay
    real(4), intent (in) :: u0
    real(4), intent (in) :: cos2phi
    real(4), intent (in) :: sin2phi
    real(4) :: t_0
    real(4) :: tmp
    t_0 = sin2phi / (alphay * alphay)
    if (t_0 <= 0.019999999552965164e0) then
        tmp = u0 / (((cos2phi / alphax) / alphax) + t_0)
    else
        tmp = (alphay * -alphay) / (sin2phi / (((u0 * u0) * (-0.5e0)) - u0))
    end if
    code = tmp
end function

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	t_0 = Float32(sin2phi / Float32(alphay * alphay))
	tmp = Float32(0.0)
	if (t_0 <= Float32(0.019999999552965164))
		tmp = Float32(u0 / Float32(Float32(Float32(cos2phi / alphax) / alphax) + t_0));
	else
		tmp = Float32(Float32(alphay * Float32(-alphay)) / Float32(sin2phi / Float32(Float32(Float32(u0 * u0) * Float32(-0.5)) - u0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alphax, alphay, u0, cos2phi, sin2phi)
	t_0 = sin2phi / (alphay * alphay);
	tmp = single(0.0);
	if (t_0 <= single(0.019999999552965164))
		tmp = u0 / (((cos2phi / alphax) / alphax) + t_0);
	else
		tmp = (alphay * -alphay) / (sin2phi / (((u0 * u0) * single(-0.5)) - u0));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 sin2phi (*.f32 alphay alphay)) < 0.0199999996

   1. Initial program 58.6%

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

      1. associate-/r* 58.6%

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

   3. Simplified 58.6%

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

   4. Taylor expanded in u0 around 0 70.5%

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

      1. unpow2 70.5%

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

      2. unpow2 70.5%

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

   6. Simplified 70.5%

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

      1. associate-/r* 70.6%

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

      2. div-inv 70.5%

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

   8. Applied egg-rr 70.5%

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

      1. div-inv 70.6%

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

   10. Applied egg-rr 70.6%

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

   if 0.0199999996 < (/.f32 sin2phi (*.f32 alphay alphay))

   1. Initial program 66.3%

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

      1. associate-/r* 66.3%

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

   3. Simplified 66.3%

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

   4. Taylor expanded in u0 around 0 88.6%

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

      1. +-commutative 88.6%

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

      2. fma-def 88.6%

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

      3. unpow2 88.6%

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

      4. mul-1-neg 88.6%

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

   6. Simplified 88.6%

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

   7. Taylor expanded in cos2phi around 0 88.1%

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

      1. mul-1-neg 88.1%

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

      2. associate-/l* 86.7%

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

      3. unpow2 86.7%

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

      4. *-commutative 86.7%

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

      5. unpow2 86.7%

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

   9. Simplified 86.7%

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

4. Final simplification 79.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 0.019999999552965164:\\ \;\;\;\;\frac{u0}{\frac{\frac{cos2phi}{alphax}}{alphax} + \frac{sin2phi}{alphay \cdot alphay}}\\ \mathbf{else}:\\ \;\;\;\;\frac{alphay \cdot \left(-alphay\right)}{\frac{sin2phi}{\left(u0 \cdot u0\right) \cdot -0.5 - u0}}\\ \end{array} \]

Alternative 7: 81.6% accurate, 6.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{sin2phi}{alphay \cdot alphay}\\ \mathbf{if}\;t_0 \leq 0.019999999552965164:\\ \;\;\;\;\frac{u0}{\frac{\frac{cos2phi}{alphax}}{alphax} + t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(alphay \cdot alphay\right) \cdot \left(u0 - u0 \cdot \left(u0 \cdot -0.5\right)\right)}{sin2phi}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (let* ((t_0 (/ sin2phi (* alphay alphay))))
   (if (<= t_0 0.019999999552965164)
     (/ u0 (+ (/ (/ cos2phi alphax) alphax) t_0))
     (/ (* (* alphay alphay) (- u0 (* u0 (* u0 -0.5)))) sin2phi))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float t_0 = sin2phi / (alphay * alphay);
	float tmp;
	if (t_0 <= 0.019999999552965164f) {
		tmp = u0 / (((cos2phi / alphax) / alphax) + t_0);
	} else {
		tmp = ((alphay * alphay) * (u0 - (u0 * (u0 * -0.5f)))) / sin2phi;
	}
	return tmp;
}

Fortran

real(4) function code(alphax, alphay, u0, cos2phi, sin2phi)
    real(4), intent (in) :: alphax
    real(4), intent (in) :: alphay
    real(4), intent (in) :: u0
    real(4), intent (in) :: cos2phi
    real(4), intent (in) :: sin2phi
    real(4) :: t_0
    real(4) :: tmp
    t_0 = sin2phi / (alphay * alphay)
    if (t_0 <= 0.019999999552965164e0) then
        tmp = u0 / (((cos2phi / alphax) / alphax) + t_0)
    else
        tmp = ((alphay * alphay) * (u0 - (u0 * (u0 * (-0.5e0))))) / sin2phi
    end if
    code = tmp
end function

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	t_0 = Float32(sin2phi / Float32(alphay * alphay))
	tmp = Float32(0.0)
	if (t_0 <= Float32(0.019999999552965164))
		tmp = Float32(u0 / Float32(Float32(Float32(cos2phi / alphax) / alphax) + t_0));
	else
		tmp = Float32(Float32(Float32(alphay * alphay) * Float32(u0 - Float32(u0 * Float32(u0 * Float32(-0.5))))) / sin2phi);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alphax, alphay, u0, cos2phi, sin2phi)
	t_0 = sin2phi / (alphay * alphay);
	tmp = single(0.0);
	if (t_0 <= single(0.019999999552965164))
		tmp = u0 / (((cos2phi / alphax) / alphax) + t_0);
	else
		tmp = ((alphay * alphay) * (u0 - (u0 * (u0 * single(-0.5))))) / sin2phi;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 sin2phi (*.f32 alphay alphay)) < 0.0199999996

   1. Initial program 58.6%

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

      1. associate-/r* 58.6%

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

   3. Simplified 58.6%

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

   4. Taylor expanded in u0 around 0 70.5%

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

      1. unpow2 70.5%

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

      2. unpow2 70.5%

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

   6. Simplified 70.5%

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

      1. associate-/r* 70.6%

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

      2. div-inv 70.5%

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

   8. Applied egg-rr 70.5%

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

      1. div-inv 70.6%

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

   10. Applied egg-rr 70.6%

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

   if 0.0199999996 < (/.f32 sin2phi (*.f32 alphay alphay))

   1. Initial program 66.3%

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

      1. associate-/r* 66.3%

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

   3. Simplified 66.3%

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

   4. Taylor expanded in u0 around 0 88.6%

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

      1. +-commutative 88.6%

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

      2. fma-def 88.6%

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

      3. unpow2 88.6%

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

      4. mul-1-neg 88.6%

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

   6. Simplified 88.6%

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

   7. Taylor expanded in cos2phi around 0 88.1%

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

      1. associate-*r/ 88.1%

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

      2. *-commutative 88.1%

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

      3. associate-*r* 88.1%

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

      4. neg-mul-1 88.1%

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

      5. *-commutative 88.1%

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

      6. unpow2 88.1%

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

      7. associate-*l* 88.1%

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

      8. unpow2 88.2%

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

   9. Simplified 88.2%

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

4. Final simplification 80.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 0.019999999552965164:\\ \;\;\;\;\frac{u0}{\frac{\frac{cos2phi}{alphax}}{alphax} + \frac{sin2phi}{alphay \cdot alphay}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(alphay \cdot alphay\right) \cdot \left(u0 - u0 \cdot \left(u0 \cdot -0.5\right)\right)}{sin2phi}\\ \end{array} \]

Alternative 8: 87.5% 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 62.9%

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

   1. associate-/r* 62.9%

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

3. Simplified 62.9%

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

4. Taylor expanded in u0 around 0 86.2%

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

   1. +-commutative 86.2%

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

   2. fma-def 86.2%

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

   3. unpow2 86.2%

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

   4. mul-1-neg 86.2%

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

6. Simplified 86.2%

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

   1. div-inv 86.1%

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

   2. associate-/r* 86.1%

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

   3. associate-/r* 86.0%

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

8. Applied egg-rr 86.0%

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

   1. associate-*r/ 86.2%

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

   2. *-lft-identity 86.2%

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

   3. *-rgt-identity 86.2%

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

   4. unpow2 86.2%

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

   5. fma-neg 86.2%

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

   6. *-commutative 86.2%

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

   7. unpow2 86.2%

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

   8. associate-*l* 86.2%

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

   9. *-lft-identity 86.2%

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

   10. associate-/l/ 86.2%

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

10. Simplified 86.2%

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

11. Final simplification 86.2%

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

Alternative 9: 66.6% accurate, 8.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 2.0000000072549875 \cdot 10^{-15}:\\ \;\;\;\;\left(alphax \cdot alphax\right) \cdot \frac{u0}{cos2phi}\\ \mathbf{else}:\\ \;\;\;\;\left(alphay \cdot alphay\right) \cdot \frac{u0}{sin2phi}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (if (<= (/ sin2phi (* alphay alphay)) 2.0000000072549875e-15)
   (* (* alphax alphax) (/ u0 cos2phi))
   (* (* alphay alphay) (/ u0 sin2phi))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float tmp;
	if ((sin2phi / (alphay * alphay)) <= 2.0000000072549875e-15f) {
		tmp = (alphax * alphax) * (u0 / cos2phi);
	} else {
		tmp = (alphay * alphay) * (u0 / sin2phi);
	}
	return tmp;
}

Fortran

real(4) function code(alphax, alphay, u0, cos2phi, sin2phi)
    real(4), intent (in) :: alphax
    real(4), intent (in) :: alphay
    real(4), intent (in) :: u0
    real(4), intent (in) :: cos2phi
    real(4), intent (in) :: sin2phi
    real(4) :: tmp
    if ((sin2phi / (alphay * alphay)) <= 2.0000000072549875e-15) then
        tmp = (alphax * alphax) * (u0 / cos2phi)
    else
        tmp = (alphay * alphay) * (u0 / sin2phi)
    end if
    code = tmp
end function

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = Float32(0.0)
	if (Float32(sin2phi / Float32(alphay * alphay)) <= Float32(2.0000000072549875e-15))
		tmp = Float32(Float32(alphax * alphax) * Float32(u0 / cos2phi));
	else
		tmp = Float32(Float32(alphay * alphay) * Float32(u0 / sin2phi));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = single(0.0);
	if ((sin2phi / (alphay * alphay)) <= single(2.0000000072549875e-15))
		tmp = (alphax * alphax) * (u0 / cos2phi);
	else
		tmp = (alphay * alphay) * (u0 / sin2phi);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 sin2phi (*.f32 alphay alphay)) < 2.00000001e-15

   1. Initial program 57.4%

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

      1. associate-/r* 57.4%

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

   3. Simplified 57.4%

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

   4. Taylor expanded in u0 around 0 70.5%

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

      1. unpow2 70.5%

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

      2. unpow2 70.5%

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

   6. Simplified 70.5%

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

      1. associate-/r* 70.6%

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

      2. div-inv 70.5%

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

   8. Applied egg-rr 70.5%

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

   9. Taylor expanded in cos2phi around inf 60.6%

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

       1. *-commutative 60.6%

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

       2. *-lft-identity 60.6%

          \[\leadsto \frac{{alphax}^{2} \cdot u0}{\color{blue}{1 \cdot cos2phi}} \]

       3. times-frac 60.4%

          \[\leadsto \color{blue}{\frac{{alphax}^{2}}{1} \cdot \frac{u0}{cos2phi}} \]

       4. /-rgt-identity 60.4%

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

       5. unpow2 60.4%

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

   11. Simplified 60.4%

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

   if 2.00000001e-15 < (/.f32 sin2phi (*.f32 alphay alphay))

   1. Initial program 64.5%

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

      1. associate-/r* 64.5%

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

   3. Simplified 64.5%

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

   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. Taylor expanded in cos2phi around 0 70.9%

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

      1. unpow2 70.9%

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

      2. *-commutative 70.9%

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

      3. *-lft-identity 70.9%

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

      4. times-frac 70.9%

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

      5. /-rgt-identity 70.9%

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

   9. Simplified 70.9%

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

4. Final simplification 68.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 2.0000000072549875 \cdot 10^{-15}:\\ \;\;\;\;\left(alphax \cdot alphax\right) \cdot \frac{u0}{cos2phi}\\ \mathbf{else}:\\ \;\;\;\;\left(alphay \cdot alphay\right) \cdot \frac{u0}{sin2phi}\\ \end{array} \]

Alternative 10: 66.6% accurate, 8.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 2.0000000072549875 \cdot 10^{-15}:\\ \;\;\;\;\frac{u0 \cdot \left(alphax \cdot alphax\right)}{cos2phi}\\ \mathbf{else}:\\ \;\;\;\;\left(alphay \cdot alphay\right) \cdot \frac{u0}{sin2phi}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (if (<= (/ sin2phi (* alphay alphay)) 2.0000000072549875e-15)
   (/ (* u0 (* alphax alphax)) cos2phi)
   (* (* alphay alphay) (/ u0 sin2phi))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float tmp;
	if ((sin2phi / (alphay * alphay)) <= 2.0000000072549875e-15f) {
		tmp = (u0 * (alphax * alphax)) / cos2phi;
	} else {
		tmp = (alphay * alphay) * (u0 / sin2phi);
	}
	return tmp;
}

Fortran

real(4) function code(alphax, alphay, u0, cos2phi, sin2phi)
    real(4), intent (in) :: alphax
    real(4), intent (in) :: alphay
    real(4), intent (in) :: u0
    real(4), intent (in) :: cos2phi
    real(4), intent (in) :: sin2phi
    real(4) :: tmp
    if ((sin2phi / (alphay * alphay)) <= 2.0000000072549875e-15) then
        tmp = (u0 * (alphax * alphax)) / cos2phi
    else
        tmp = (alphay * alphay) * (u0 / sin2phi)
    end if
    code = tmp
end function

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = Float32(0.0)
	if (Float32(sin2phi / Float32(alphay * alphay)) <= Float32(2.0000000072549875e-15))
		tmp = Float32(Float32(u0 * Float32(alphax * alphax)) / cos2phi);
	else
		tmp = Float32(Float32(alphay * alphay) * Float32(u0 / sin2phi));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = single(0.0);
	if ((sin2phi / (alphay * alphay)) <= single(2.0000000072549875e-15))
		tmp = (u0 * (alphax * alphax)) / cos2phi;
	else
		tmp = (alphay * alphay) * (u0 / sin2phi);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 sin2phi (*.f32 alphay alphay)) < 2.00000001e-15

   1. Initial program 57.4%

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

      1. associate-/r* 57.4%

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

   3. Simplified 57.4%

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

   4. Taylor expanded in u0 around 0 70.5%

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

      1. unpow2 70.5%

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

      2. unpow2 70.5%

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

   6. Simplified 70.5%

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

      1. associate-/r* 70.6%

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

      2. div-inv 70.5%

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

   8. Applied egg-rr 70.5%

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

   9. Taylor expanded in cos2phi around inf 60.6%

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

       1. unpow2 60.6%

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

   11. Simplified 60.6%

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

   if 2.00000001e-15 < (/.f32 sin2phi (*.f32 alphay alphay))

   1. Initial program 64.5%

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

      1. associate-/r* 64.5%

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

   3. Simplified 64.5%

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

   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. Taylor expanded in cos2phi around 0 70.9%

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

      1. unpow2 70.9%

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

      2. *-commutative 70.9%

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

      3. *-lft-identity 70.9%

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

      4. times-frac 70.9%

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

      5. /-rgt-identity 70.9%

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

   9. Simplified 70.9%

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

4. Final simplification 68.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 2.0000000072549875 \cdot 10^{-15}:\\ \;\;\;\;\frac{u0 \cdot \left(alphax \cdot alphax\right)}{cos2phi}\\ \mathbf{else}:\\ \;\;\;\;\left(alphay \cdot alphay\right) \cdot \frac{u0}{sin2phi}\\ \end{array} \]

Alternative 11: 66.6% accurate, 8.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 2.0000000072549875 \cdot 10^{-15}:\\ \;\;\;\;\frac{u0 \cdot \left(alphax \cdot alphax\right)}{cos2phi}\\ \mathbf{else}:\\ \;\;\;\;\frac{u0 \cdot \left(alphay \cdot alphay\right)}{sin2phi}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (if (<= (/ sin2phi (* alphay alphay)) 2.0000000072549875e-15)
   (/ (* u0 (* alphax alphax)) cos2phi)
   (/ (* u0 (* alphay alphay)) sin2phi)))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float tmp;
	if ((sin2phi / (alphay * alphay)) <= 2.0000000072549875e-15f) {
		tmp = (u0 * (alphax * alphax)) / cos2phi;
	} else {
		tmp = (u0 * (alphay * alphay)) / sin2phi;
	}
	return tmp;
}

Fortran

real(4) function code(alphax, alphay, u0, cos2phi, sin2phi)
    real(4), intent (in) :: alphax
    real(4), intent (in) :: alphay
    real(4), intent (in) :: u0
    real(4), intent (in) :: cos2phi
    real(4), intent (in) :: sin2phi
    real(4) :: tmp
    if ((sin2phi / (alphay * alphay)) <= 2.0000000072549875e-15) then
        tmp = (u0 * (alphax * alphax)) / cos2phi
    else
        tmp = (u0 * (alphay * alphay)) / sin2phi
    end if
    code = tmp
end function

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = Float32(0.0)
	if (Float32(sin2phi / Float32(alphay * alphay)) <= Float32(2.0000000072549875e-15))
		tmp = Float32(Float32(u0 * Float32(alphax * alphax)) / cos2phi);
	else
		tmp = Float32(Float32(u0 * Float32(alphay * alphay)) / sin2phi);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = single(0.0);
	if ((sin2phi / (alphay * alphay)) <= single(2.0000000072549875e-15))
		tmp = (u0 * (alphax * alphax)) / cos2phi;
	else
		tmp = (u0 * (alphay * alphay)) / sin2phi;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 sin2phi (*.f32 alphay alphay)) < 2.00000001e-15

   1. Initial program 57.4%

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

      1. associate-/r* 57.4%

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

   3. Simplified 57.4%

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

   4. Taylor expanded in u0 around 0 70.5%

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

      1. unpow2 70.5%

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

      2. unpow2 70.5%

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

   6. Simplified 70.5%

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

      1. associate-/r* 70.6%

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

      2. div-inv 70.5%

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

   8. Applied egg-rr 70.5%

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

   9. Taylor expanded in cos2phi around inf 60.6%

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

       1. unpow2 60.6%

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

   11. Simplified 60.6%

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

   if 2.00000001e-15 < (/.f32 sin2phi (*.f32 alphay alphay))

   1. Initial program 64.5%

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

      1. associate-/r* 64.5%

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

   3. Simplified 64.5%

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

   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. Taylor expanded in cos2phi around 0 70.9%

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

      1. unpow2 70.9%

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

      2. *-commutative 70.9%

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

      3. *-lft-identity 70.9%

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

      4. times-frac 70.9%

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

      5. /-rgt-identity 70.9%

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

   9. Simplified 70.9%

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

       1. associate-*r/ 70.9%

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

   11. Applied egg-rr 70.9%

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

4. Final simplification 68.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 2.0000000072549875 \cdot 10^{-15}:\\ \;\;\;\;\frac{u0 \cdot \left(alphax \cdot alphax\right)}{cos2phi}\\ \mathbf{else}:\\ \;\;\;\;\frac{u0 \cdot \left(alphay \cdot alphay\right)}{sin2phi}\\ \end{array} \]

Alternative 12: 75.9% accurate, 8.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 62.9%

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

   1. associate-/r* 62.9%

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

3. Simplified 62.9%

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

4. Taylor expanded in u0 around 0 74.4%

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

   1. unpow2 74.4%

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

   2. unpow2 74.4%

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

6. Simplified 74.4%

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

7. Final simplification 74.4%

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

Alternative 13: 75.9% accurate, 8.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 62.9%

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

   1. associate-/r* 62.9%

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

3. Simplified 62.9%

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

4. Taylor expanded in u0 around 0 74.4%

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

   1. unpow2 74.4%

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

   2. unpow2 74.4%

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

6. Simplified 74.4%

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

   1. associate-/r* 74.4%

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

   2. div-inv 74.4%

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

8. Applied egg-rr 74.4%

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

   1. div-inv 74.4%

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

10. Applied egg-rr 74.4%

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

11. Final simplification 74.4%

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

Alternative 14: 66.5% accurate, 12.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;sin2phi \leq 3.99999987306209 \cdot 10^{-19}:\\ \;\;\;\;\left(alphax \cdot alphax\right) \cdot \frac{u0}{cos2phi}\\ \mathbf{else}:\\ \;\;\;\;alphay \cdot \left(u0 \cdot \frac{alphay}{sin2phi}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (if (<= sin2phi 3.99999987306209e-19)
   (* (* alphax alphax) (/ u0 cos2phi))
   (* alphay (* u0 (/ alphay sin2phi)))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float tmp;
	if (sin2phi <= 3.99999987306209e-19f) {
		tmp = (alphax * alphax) * (u0 / cos2phi);
	} else {
		tmp = alphay * (u0 * (alphay / sin2phi));
	}
	return tmp;
}

Fortran

real(4) function code(alphax, alphay, u0, cos2phi, sin2phi)
    real(4), intent (in) :: alphax
    real(4), intent (in) :: alphay
    real(4), intent (in) :: u0
    real(4), intent (in) :: cos2phi
    real(4), intent (in) :: sin2phi
    real(4) :: tmp
    if (sin2phi <= 3.99999987306209e-19) then
        tmp = (alphax * alphax) * (u0 / cos2phi)
    else
        tmp = alphay * (u0 * (alphay / sin2phi))
    end if
    code = tmp
end function

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = Float32(0.0)
	if (sin2phi <= Float32(3.99999987306209e-19))
		tmp = Float32(Float32(alphax * alphax) * Float32(u0 / cos2phi));
	else
		tmp = Float32(alphay * Float32(u0 * Float32(alphay / sin2phi)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = single(0.0);
	if (sin2phi <= single(3.99999987306209e-19))
		tmp = (alphax * alphax) * (u0 / cos2phi);
	else
		tmp = alphay * (u0 * (alphay / sin2phi));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if sin2phi < 3.99999987e-19

   1. Initial program 57.8%

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

      1. associate-/r* 57.9%

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

   3. Simplified 57.9%

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

   4. Taylor expanded in u0 around 0 70.2%

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

      1. unpow2 70.2%

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

      2. unpow2 70.2%

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

   6. Simplified 70.2%

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

      1. associate-/r* 70.4%

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

      2. div-inv 70.3%

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

   8. Applied egg-rr 70.3%

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

   9. Taylor expanded in cos2phi around inf 57.9%

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

       1. *-commutative 57.9%

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

       2. *-lft-identity 57.9%

          \[\leadsto \frac{{alphax}^{2} \cdot u0}{\color{blue}{1 \cdot cos2phi}} \]

       3. times-frac 57.7%

          \[\leadsto \color{blue}{\frac{{alphax}^{2}}{1} \cdot \frac{u0}{cos2phi}} \]

       4. /-rgt-identity 57.7%

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

       5. unpow2 57.7%

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

   11. Simplified 57.7%

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

   if 3.99999987e-19 < sin2phi

   1. Initial program 64.3%

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

      1. associate-/r* 64.3%

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

   3. Simplified 64.3%

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

   4. Taylor expanded in u0 around 0 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 0 70.5%

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

      1. unpow2 70.5%

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

      2. *-commutative 70.5%

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

      3. *-lft-identity 70.5%

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

      4. times-frac 70.5%

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

      5. /-rgt-identity 70.5%

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

   9. Simplified 70.5%

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

   10. Taylor expanded in alphay around 0 70.5%

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

       1. *-commutative 70.5%

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

       2. unpow2 70.5%

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

       3. associate-*l* 70.4%

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

       4. associate-*r/ 70.3%

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

       5. associate-/l* 69.5%

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

   12. Simplified 69.5%

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

       1. associate-/r/ 70.4%

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

   14. Applied egg-rr 70.4%

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

4. Final simplification 67.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;sin2phi \leq 3.99999987306209 \cdot 10^{-19}:\\ \;\;\;\;\left(alphax \cdot alphax\right) \cdot \frac{u0}{cos2phi}\\ \mathbf{else}:\\ \;\;\;\;alphay \cdot \left(u0 \cdot \frac{alphay}{sin2phi}\right)\\ \end{array} \]

Alternative 15: 58.5% accurate, 16.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 62.9%

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

   1. associate-/r* 62.9%

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

3. Simplified 62.9%

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

4. Taylor expanded in u0 around 0 74.4%

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

   1. unpow2 74.4%

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

   2. unpow2 74.4%

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

6. Simplified 74.4%

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

7. Taylor expanded in cos2phi around 0 59.7%

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

   1. unpow2 59.7%

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

   2. *-commutative 59.7%

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

   3. *-lft-identity 59.7%

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

   4. times-frac 59.7%

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

   5. /-rgt-identity 59.7%

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

9. Simplified 59.7%

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

10. Taylor expanded in alphay around 0 59.7%

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

    1. *-commutative 59.7%

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

    2. unpow2 59.7%

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

    3. associate-*l* 59.7%

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

    4. associate-*r/ 59.6%

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

    5. associate-/l* 58.9%

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

12. Simplified 58.9%

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

    1. associate-/r/ 59.6%

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

14. Applied egg-rr 59.6%

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

15. Final simplification 59.6%

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

Reproduce

herbie shell --seed 2023238 
(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)))))