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

Percentage Accurate: 60.2% → 98.3%

Time: 11.7s

Alternatives: 10

Speedup: 12.7×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Initial Program: 60.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Alternative 1: 98.3% accurate, 1.0× speedup

Math

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

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 57.8%

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

   1. neg-sub0 57.8%

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

   2. div-sub 57.8%

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

   3. --rgt-identity 57.8%

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

   4. div-sub 57.8%

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

   5. --rgt-identity 57.8%

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

   6. sub-neg 57.8%

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

   7. +-commutative 57.8%

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

   8. neg-sub0 57.8%

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

   9. associate-+l- 57.8%

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

   10. sub0-neg 57.8%

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

   11. neg-mul-1 57.8%

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

   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 98.5%

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

   1. frac-2neg 98.5%

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

   2. div-inv 98.4%

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

5. Applied egg-rr 98.4%

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

   1. un-div-inv 98.5%

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

   2. frac-2neg 98.5%

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

   3. associate-/r* 98.6%

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

7. Applied egg-rr 98.6%

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

8. Final simplification 98.6%

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

Alternative 2: 98.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 57.8%

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

   1. neg-sub0 57.8%

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

   2. div-sub 57.8%

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

   3. --rgt-identity 57.8%

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

   4. div-sub 57.8%

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

   5. --rgt-identity 57.8%

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

   6. neg-sub0 57.8%

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

   7. sub-neg 57.8%

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

   8. log1p-def 98.5%

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

3. Simplified 98.5%

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

4. Final simplification 98.5%

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

Alternative 3: 85.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Split input into 2 regimes

2. if sin2phi < 0.00999999978

   1. Initial program 51.6%

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

      1. neg-sub0 51.6%

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

      2. div-sub 51.6%

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

      3. --rgt-identity 51.6%

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

      4. div-sub 51.6%

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

      5. --rgt-identity 51.6%

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

      6. sub-neg 51.6%

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

      7. +-commutative 51.6%

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

      8. neg-sub0 51.6%

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

      9. associate-+l- 51.6%

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

      10. sub0-neg 51.6%

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

      11. neg-mul-1 51.6%

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

      12. log-prod -0.0%

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

      13. associate--r+ -0.0%

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

   3. Simplified 98.8%

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

      1. frac-2neg 98.8%

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

      2. div-inv 98.8%

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

   5. Applied egg-rr 98.8%

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

      1. un-div-inv 98.8%

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

      2. frac-2neg 98.8%

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

      3. associate-/r* 98.9%

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

   7. Applied egg-rr 98.9%

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

   8. Taylor expanded in u0 around 0 77.6%

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

      1. +-commutative 77.6%

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

      2. unpow2 77.6%

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

      3. unpow2 77.6%

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

      4. associate-/r* 77.7%

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

   10. Simplified 77.7%

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

   if 0.00999999978 < sin2phi

   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 cos2phi around 0 63.5%

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

      1. mul-1-neg 63.5%

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

      2. unpow2 63.5%

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

      3. *-commutative 63.5%

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

   6. Simplified 63.5%

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

   7. Taylor expanded in alphay around 0 63.5%

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

      1. associate-/l* 63.5%

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

      2. unpow2 63.5%

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

      3. associate-/l* 62.5%

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

      4. sub-neg 62.5%

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

      5. log1p-def 97.0%

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

   9. Simplified 97.0%

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

4. Final simplification 88.4%

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

Alternative 4: 86.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Split input into 2 regimes

2. if sin2phi < 0.00999999978

   1. Initial program 51.6%

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

      1. neg-sub0 51.6%

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

      2. div-sub 51.6%

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

      3. --rgt-identity 51.6%

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

      4. div-sub 51.6%

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

      5. --rgt-identity 51.6%

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

      6. sub-neg 51.6%

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

      7. +-commutative 51.6%

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

      8. neg-sub0 51.6%

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

      9. associate-+l- 51.6%

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

      10. sub0-neg 51.6%

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

      11. neg-mul-1 51.6%

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

      12. log-prod -0.0%

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

      13. associate--r+ -0.0%

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

   3. Simplified 98.8%

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

      1. frac-2neg 98.8%

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

      2. div-inv 98.8%

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

   5. Applied egg-rr 98.8%

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

      1. un-div-inv 98.8%

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

      2. frac-2neg 98.8%

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

      3. associate-/r* 98.9%

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

   7. Applied egg-rr 98.9%

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

   8. Taylor expanded in u0 around 0 77.6%

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

      1. +-commutative 77.6%

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

      2. unpow2 77.6%

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

      3. unpow2 77.6%

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

      4. associate-/r* 77.7%

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

   10. Simplified 77.7%

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

   if 0.00999999978 < sin2phi

   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 cos2phi around 0 63.5%

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

      1. mul-1-neg 63.5%

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

      2. unpow2 63.5%

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

      3. *-commutative 63.5%

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

   6. Simplified 63.5%

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

   7. Taylor expanded in alphay around 0 63.5%

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

      1. sub-neg 63.5%

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

      2. log1p-def 98.8%

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

      3. unpow2 98.8%

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

      4. associate-*l* 98.8%

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

   9. Simplified 98.8%

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

4. Final simplification 89.4%

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

Alternative 5: 81.5% accurate, 7.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Split input into 2 regimes

2. if sin2phi < 0.00999999978

   1. Initial program 51.6%

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

      1. neg-sub0 51.6%

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

      2. div-sub 51.6%

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

      3. --rgt-identity 51.6%

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

      4. div-sub 51.6%

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

      5. --rgt-identity 51.6%

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

      6. sub-neg 51.6%

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

      7. +-commutative 51.6%

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

      8. neg-sub0 51.6%

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

      9. associate-+l- 51.6%

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

      10. sub0-neg 51.6%

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

      11. neg-mul-1 51.6%

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

      12. log-prod -0.0%

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

      13. associate--r+ -0.0%

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

   3. Simplified 98.8%

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

      1. frac-2neg 98.8%

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

      2. div-inv 98.8%

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

   5. Applied egg-rr 98.8%

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

      1. un-div-inv 98.8%

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

      2. frac-2neg 98.8%

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

      3. associate-/r* 98.9%

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

   7. Applied egg-rr 98.9%

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

   8. Taylor expanded in u0 around 0 77.6%

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

      1. +-commutative 77.6%

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

      2. unpow2 77.6%

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

      3. unpow2 77.6%

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

      4. associate-/r* 77.7%

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

   10. Simplified 77.7%

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

   if 0.00999999978 < sin2phi

   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 cos2phi around 0 63.5%

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

      1. mul-1-neg 63.5%

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

      2. unpow2 63.5%

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

      3. *-commutative 63.5%

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

   6. Simplified 63.5%

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

   7. Taylor expanded in u0 around 0 88.1%

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

      1. fma-def 88.1%

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

      2. *-commutative 88.1%

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

      3. unpow2 88.1%

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

      4. unpow2 88.1%

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

      5. mul-1-neg 88.1%

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

      6. unpow2 88.1%

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

   9. Simplified 88.1%

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

   10. Taylor expanded in alphay around 0 88.1%

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

       1. associate-/l* 87.5%

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

       2. associate-/r/ 88.0%

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

       3. unpow2 88.0%

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

       4. *-commutative 88.0%

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

       5. unpow2 88.0%

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

   12. Simplified 88.0%

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

   13. Taylor expanded in alphay around 0 88.1%

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

       1. associate-/l* 87.5%

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

       2. *-commutative 87.5%

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

       3. unpow2 87.5%

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

       4. associate-*r* 87.5%

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

       5. associate-/r/ 88.0%

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

       6. unpow2 88.0%

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

       7. associate-/l* 88.1%

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

       8. *-commutative 88.1%

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

       9. associate-*r* 88.1%

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

       10. associate-/r/ 88.1%

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

       11. *-commutative 88.1%

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

   15. Simplified 88.1%

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

4. Final simplification 83.4%

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

Alternative 6: 81.5% accurate, 7.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

function tmp_2 = code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = single(0.0);
	if (sin2phi <= single(0.009999999776482582))
		tmp = u0 / (((sin2phi / alphay) / alphay) + (cos2phi / (alphax * alphax)));
	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 sin2phi < 0.00999999978

   1. Initial program 51.6%

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

      1. neg-sub0 51.6%

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

      2. div-sub 51.6%

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

      3. --rgt-identity 51.6%

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

      4. div-sub 51.6%

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

      5. --rgt-identity 51.6%

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

      6. sub-neg 51.6%

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

      7. +-commutative 51.6%

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

      8. neg-sub0 51.6%

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

      9. associate-+l- 51.6%

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

      10. sub0-neg 51.6%

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

      11. neg-mul-1 51.6%

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

      12. log-prod -0.0%

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

      13. associate--r+ -0.0%

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

   3. Simplified 98.8%

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

      1. frac-2neg 98.8%

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

      2. div-inv 98.8%

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

   5. Applied egg-rr 98.8%

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

      1. un-div-inv 98.8%

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

      2. frac-2neg 98.8%

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

      3. associate-/r* 98.9%

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

   7. Applied egg-rr 98.9%

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

   8. Taylor expanded in u0 around 0 77.6%

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

      1. +-commutative 77.6%

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

      2. unpow2 77.6%

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

      3. unpow2 77.6%

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

      4. associate-/r* 77.7%

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

   10. Simplified 77.7%

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

   if 0.00999999978 < sin2phi

   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 cos2phi around 0 63.5%

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

      1. mul-1-neg 63.5%

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

      2. unpow2 63.5%

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

      3. *-commutative 63.5%

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

   6. Simplified 63.5%

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

   7. Taylor expanded in u0 around 0 88.1%

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

      1. fma-def 88.1%

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

      2. *-commutative 88.1%

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

      3. unpow2 88.1%

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

      4. unpow2 88.1%

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

      5. mul-1-neg 88.1%

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

      6. unpow2 88.1%

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

   9. Simplified 88.1%

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

   10. Taylor expanded in alphay around 0 88.1%

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

       1. unpow2 88.1%

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

       2. associate-*l* 88.2%

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

       3. *-commutative 88.2%

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

       4. unpow2 88.2%

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

   12. Simplified 88.2%

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

4. Final simplification 83.5%

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

Alternative 7: 76.1% accurate, 8.9× speedup

Math

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

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(cos2phi / Float32(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 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 77.6%

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

   1. unpow2 77.6%

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

   2. unpow2 77.6%

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

6. Simplified 77.6%

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

7. Final simplification 77.6%

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

Alternative 8: 76.1% accurate, 8.9× speedup

Math

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

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(Float32(sin2phi / alphay) / alphay) + Float32(cos2phi / Float32(alphax * alphax))))
end

MATLAB

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

Derivation

1. Initial program 57.8%

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

   1. neg-sub0 57.8%

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

   2. div-sub 57.8%

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

   3. --rgt-identity 57.8%

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

   4. div-sub 57.8%

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

   5. --rgt-identity 57.8%

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

   6. sub-neg 57.8%

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

   7. +-commutative 57.8%

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

   8. neg-sub0 57.8%

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

   9. associate-+l- 57.8%

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

   10. sub0-neg 57.8%

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

   11. neg-mul-1 57.8%

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

   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 98.5%

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

   1. frac-2neg 98.5%

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

   2. div-inv 98.4%

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

5. Applied egg-rr 98.4%

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

   1. un-div-inv 98.5%

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

   2. frac-2neg 98.5%

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

   3. associate-/r* 98.6%

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

7. Applied egg-rr 98.6%

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

8. Taylor expanded in u0 around 0 77.6%

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

   1. +-commutative 77.6%

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

   2. unpow2 77.6%

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

   3. unpow2 77.6%

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

   4. associate-/r* 77.6%

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

10. Simplified 77.6%

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

11. Final simplification 77.6%

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

Alternative 9: 66.4% accurate, 12.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;sin2phi \leq 1.9999999774532045 \cdot 10^{-26}:\\ \;\;\;\;\left(alphax \cdot alphax\right) \cdot \frac{u0}{cos2phi}\\ \mathbf{else}:\\ \;\;\;\;\left(alphay \cdot alphay\right) \cdot \frac{u0}{sin2phi}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (if (<= sin2phi 1.9999999774532045e-26)
   (* (* 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 <= 1.9999999774532045e-26f) {
		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 <= 1.9999999774532045e-26) then
        tmp = (alphax * alphax) * (u0 / cos2phi)
    else
        tmp = (alphay * alphay) * (u0 / sin2phi)
    end if
    code = tmp
end function

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = Float32(0.0)
	if (sin2phi <= Float32(1.9999999774532045e-26))
		tmp = Float32(Float32(alphax * alphax) * Float32(u0 / cos2phi));
	else
		tmp = Float32(Float32(alphay * alphay) * Float32(u0 / sin2phi));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = single(0.0);
	if (sin2phi <= single(1.9999999774532045e-26))
		tmp = (alphax * alphax) * (u0 / cos2phi);
	else
		tmp = (alphay * alphay) * (u0 / sin2phi);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if sin2phi < 1.99999998e-26

   1. Initial program 49.1%

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

      1. associate-/r* 49.2%

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

   3. Simplified 49.2%

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

   4. Taylor expanded in u0 around 0 78.6%

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

      1. unpow2 78.6%

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

      2. unpow2 78.6%

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

   6. Simplified 78.6%

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

   7. Taylor expanded in cos2phi around inf 66.2%

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

      1. associate-/l* 66.2%

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

      2. unpow2 66.2%

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

   9. Simplified 66.2%

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

       1. associate-/r/ 66.3%

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

   11. Applied egg-rr 66.3%

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

   if 1.99999998e-26 < sin2phi

   1. Initial program 59.6%

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

      1. associate-/r* 59.6%

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

   3. Simplified 59.6%

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

   4. Taylor expanded in u0 around 0 77.4%

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

      1. unpow2 77.4%

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

      2. unpow2 77.4%

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

   6. Simplified 77.4%

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

      1. +-commutative 77.4%

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

      2. associate-/r* 77.5%

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

      3. frac-add 77.3%

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

   8. Applied egg-rr 77.3%

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

   9. Taylor expanded in sin2phi around inf 69.2%

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

       1. associate-/l* 68.9%

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

       2. associate-/r/ 69.3%

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

       3. unpow2 69.3%

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

   11. Simplified 69.3%

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

4. Final simplification 68.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;sin2phi \leq 1.9999999774532045 \cdot 10^{-26}:\\ \;\;\;\;\left(alphax \cdot alphax\right) \cdot \frac{u0}{cos2phi}\\ \mathbf{else}:\\ \;\;\;\;\left(alphay \cdot alphay\right) \cdot \frac{u0}{sin2phi}\\ \end{array} \]

Alternative 10: 23.5% accurate, 16.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 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 77.6%

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

   1. unpow2 77.6%

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

   2. unpow2 77.6%

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

6. Simplified 77.6%

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

7. Taylor expanded in cos2phi around inf 23.1%

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

   1. associate-/l* 23.2%

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

   2. unpow2 23.2%

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

9. Simplified 23.2%

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

    1. associate-/r/ 23.2%

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

11. Applied egg-rr 23.2%

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

12. Final simplification 23.2%

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

Reproduce

herbie shell --seed 2023188 
(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)))))