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

Percentage Accurate: 61.0% → 98.3%

Time: 14.2s

Alternatives: 12

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: 61.0% 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 60.6%

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

   1. neg-sub0 60.6%

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

   2. div-sub 60.6%

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

   3. --rgt-identity 60.6%

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

   4. div-sub 60.6%

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

   5. --rgt-identity 60.6%

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

   6. sub-neg 60.6%

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

   7. +-commutative 60.6%

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

   8. neg-sub0 60.6%

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

   9. associate-+l- 60.6%

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

   10. sub0-neg 60.6%

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

   11. neg-mul-1 60.6%

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

   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 98.7%

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

   1. associate-/r* 98.8%

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

   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 98.8%

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

7. Applied egg-rr 98.8%

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

8. Final simplification 98.8%

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

Alternative 2: 98.3% accurate, 1.0× speedup

Math

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

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 60.6%

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

   1. neg-sub0 60.6%

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

   2. div-sub 60.6%

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

   3. --rgt-identity 60.6%

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

   4. div-sub 60.6%

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

   5. --rgt-identity 60.6%

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

   6. neg-sub0 60.6%

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

   7. sub-neg 60.6%

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

   8. log1p-def 98.7%

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

3. Simplified 98.7%

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

4. Final simplification 98.7%

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

Alternative 3: 92.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;sin2phi \leq 4.999999873689376 \cdot 10^{-5}:\\ \;\;\;\;\frac{u0 - u0 \cdot \left(u0 \cdot -0.5\right)}{\frac{\frac{cos2phi}{alphax}}{alphax} + \frac{sin2phi}{alphay \cdot alphay}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(-u0\right)}{sin2phi} \cdot \left(-alphay \cdot alphay\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (if (<= sin2phi 4.999999873689376e-5)
   (/
    (- u0 (* u0 (* u0 -0.5)))
    (+ (/ (/ cos2phi alphax) alphax) (/ sin2phi (* alphay alphay))))
   (* (/ (log1p (- u0)) sin2phi) (- (* alphay alphay)))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float tmp;
	if (sin2phi <= 4.999999873689376e-5f) {
		tmp = (u0 - (u0 * (u0 * -0.5f))) / (((cos2phi / alphax) / alphax) + (sin2phi / (alphay * alphay)));
	} else {
		tmp = (log1pf(-u0) / sin2phi) * -(alphay * alphay);
	}
	return tmp;
}

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = Float32(0.0)
	if (sin2phi <= Float32(4.999999873689376e-5))
		tmp = Float32(Float32(u0 - Float32(u0 * Float32(u0 * Float32(-0.5)))) / Float32(Float32(Float32(cos2phi / alphax) / alphax) + Float32(sin2phi / Float32(alphay * alphay))));
	else
		tmp = Float32(Float32(log1p(Float32(-u0)) / sin2phi) * Float32(-Float32(alphay * alphay)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if sin2phi < 4.99999987e-5

   1. Initial program 53.1%

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

      1. associate-/r* 53.2%

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

   3. Simplified 53.2%

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

   4. Taylor expanded in u0 around 0 88.7%

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

      1. +-commutative 41.5%

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

      2. mul-1-neg 41.5%

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

      3. unsub-neg 41.5%

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

      4. unpow2 41.5%

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

      5. associate-*r* 41.5%

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

   6. Simplified 88.7%

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

   if 4.99999987e-5 < sin2phi

   1. Initial program 65.8%

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

      1. associate-/r* 65.8%

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

   3. Simplified 65.8%

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

   4. Taylor expanded in cos2phi around 0 65.8%

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

      1. mul-1-neg 65.8%

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

      2. unpow2 65.8%

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

      3. *-commutative 65.8%

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

   6. Simplified 65.8%

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

   7. Taylor expanded in alphay around 0 65.8%

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

      1. *-commutative 65.8%

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

      2. sub-neg 65.8%

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

      3. log1p-def 97.6%

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

      4. unpow2 97.6%

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

      5. unpow2 97.6%

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

      6. *-commutative 97.6%

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

      7. *-lft-identity 97.6%

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

      8. times-frac 97.7%

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

      9. /-rgt-identity 97.7%

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

      10. unpow2 97.7%

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

   9. Simplified 97.7%

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

4. Final simplification 94.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;sin2phi \leq 4.999999873689376 \cdot 10^{-5}:\\ \;\;\;\;\frac{u0 - u0 \cdot \left(u0 \cdot -0.5\right)}{\frac{\frac{cos2phi}{alphax}}{alphax} + \frac{sin2phi}{alphay \cdot alphay}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(-u0\right)}{sin2phi} \cdot \left(-alphay \cdot alphay\right)\\ \end{array} \]

Alternative 4: 87.2% accurate, 5.8× speedup

Math

\[\begin{array}{l} \\ \frac{u0 \cdot \left(u0 \cdot \left(--0.5\right) - -1\right)}{\frac{\frac{cos2phi}{alphax}}{alphax} + \frac{\frac{sin2phi}{alphay}}{alphay}} \end{array} \]

FPCore

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

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	return (u0 * ((u0 * -(-0.5f)) - -1.0f)) / (((cos2phi / alphax) / alphax) + ((sin2phi / alphay) / alphay));
}

Fortran

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

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	return Float32(Float32(u0 * Float32(Float32(u0 * Float32(-Float32(-0.5))) - Float32(-1.0))) / Float32(Float32(Float32(cos2phi / alphax) / alphax) + Float32(Float32(sin2phi / alphay) / alphay)))
end

MATLAB

function tmp = code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = (u0 * ((u0 * -single(-0.5)) - single(-1.0))) / (((cos2phi / alphax) / alphax) + ((sin2phi / alphay) / alphay));
end

Derivation

1. Initial program 60.6%

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

   1. neg-sub0 60.6%

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

   2. div-sub 60.6%

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

   3. --rgt-identity 60.6%

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

   4. div-sub 60.6%

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

   5. --rgt-identity 60.6%

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

   6. sub-neg 60.6%

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

   7. +-commutative 60.6%

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

   8. neg-sub0 60.6%

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

   9. associate-+l- 60.6%

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

   10. sub0-neg 60.6%

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

   11. neg-mul-1 60.6%

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

   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 98.7%

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

   1. associate-/r* 98.8%

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

   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 98.8%

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

7. Applied egg-rr 98.8%

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

8. Taylor expanded in u0 around 0 88.9%

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

   1. unpow2 88.9%

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

   2. associate-*l* 88.9%

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

   3. distribute-rgt-out 88.7%

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

   4. *-commutative 88.7%

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

10. Simplified 88.7%

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

11. Final simplification 88.7%

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

Alternative 5: 87.3% accurate, 6.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 60.6%

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

   1. associate-/r* 60.6%

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

3. Simplified 60.6%

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

4. Taylor expanded in u0 around 0 88.8%

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

   1. +-commutative 68.8%

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

   2. mul-1-neg 68.8%

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

   3. unsub-neg 68.8%

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

   4. unpow2 68.8%

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

   5. associate-*r* 68.8%

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

6. Simplified 88.8%

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

7. Final simplification 88.8%

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

Alternative 6: 81.2% accurate, 7.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;sin2phi \leq 4.999999873689376 \cdot 10^{-5}:\\ \;\;\;\;\frac{u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-1 + u0 \cdot -0.5\right) \cdot \left(alphay \cdot \left(\left(-u0\right) \cdot alphay\right)\right)}{sin2phi}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (if (<= sin2phi 4.999999873689376e-5)
   (/ u0 (+ (/ cos2phi (* alphax alphax)) (/ sin2phi (* alphay alphay))))
   (/ (* (+ -1.0 (* u0 -0.5)) (* alphay (* (- u0) alphay))) sin2phi)))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float tmp;
	if (sin2phi <= 4.999999873689376e-5f) {
		tmp = u0 / ((cos2phi / (alphax * alphax)) + (sin2phi / (alphay * alphay)));
	} else {
		tmp = ((-1.0f + (u0 * -0.5f)) * (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 <= 4.999999873689376e-5) then
        tmp = u0 / ((cos2phi / (alphax * alphax)) + (sin2phi / (alphay * alphay)))
    else
        tmp = (((-1.0e0) + (u0 * (-0.5e0))) * (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(4.999999873689376e-5))
		tmp = Float32(u0 / Float32(Float32(cos2phi / Float32(alphax * alphax)) + Float32(sin2phi / Float32(alphay * alphay))));
	else
		tmp = Float32(Float32(Float32(Float32(-1.0) + Float32(u0 * Float32(-0.5))) * Float32(alphay * Float32(Float32(-u0) * alphay))) / sin2phi);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = single(0.0);
	if (sin2phi <= single(4.999999873689376e-5))
		tmp = u0 / ((cos2phi / (alphax * alphax)) + (sin2phi / (alphay * alphay)));
	else
		tmp = ((single(-1.0) + (u0 * single(-0.5))) * (alphay * (-u0 * alphay))) / sin2phi;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if sin2phi < 4.99999987e-5

   1. Initial program 53.1%

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

      1. associate-/r* 53.2%

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

   3. Simplified 53.2%

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

   4. Taylor expanded in u0 around 0 76.2%

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

      1. unpow2 76.2%

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

      2. unpow2 76.2%

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

   6. Simplified 76.2%

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

   if 4.99999987e-5 < sin2phi

   1. Initial program 65.8%

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

      1. associate-/r* 65.8%

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

   3. Simplified 65.8%

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

   4. Taylor expanded in cos2phi around 0 65.8%

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

      1. mul-1-neg 65.8%

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

      2. unpow2 65.8%

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

      3. *-commutative 65.8%

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

   6. Simplified 65.8%

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

   7. Taylor expanded in u0 around 0 88.0%

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

      1. mul-1-neg 88.0%

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

      2. unsub-neg 88.0%

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

      3. associate-*r/ 88.0%

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

      4. associate-*r* 88.0%

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

      5. *-rgt-identity 88.0%

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

      6. times-frac 88.0%

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

      7. unpow2 88.0%

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

      8. /-rgt-identity 88.0%

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

      9. unpow2 88.0%

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

      10. *-commutative 88.0%

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

      11. *-lft-identity 88.0%

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

      12. times-frac 88.0%

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

      13. /-rgt-identity 88.0%

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

      14. unpow2 88.0%

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

   9. Simplified 88.0%

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

   10. Taylor expanded in u0 around 0 88.0%

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

       1. fma-def 88.0%

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

       2. mul-1-neg 88.0%

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

       3. unpow2 88.0%

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

       4. *-commutative 88.0%

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

       5. associate-*r/ 88.0%

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

       6. fma-neg 88.0%

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

       7. associate-*r/ 88.0%

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

       8. associate-*r/ 88.0%

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

       9. *-commutative 88.0%

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

       10. unpow2 88.0%

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

       11. div-sub 88.0%

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

   12. Simplified 87.9%

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

   13. Taylor expanded in u0 around 0 87.9%

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

       1. *-commutative 87.9%

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

       2. unpow2 87.9%

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

       3. associate-*l* 88.0%

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

       4. *-commutative 88.0%

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

   15. Simplified 88.0%

       \[\leadsto -\frac{\color{blue}{\left(alphay \cdot \left(u0 \cdot alphay\right)\right)} \cdot \left(u0 \cdot -0.5 + -1\right)}{sin2phi} \]
3. Recombined 2 regimes into one program.

4. Final simplification 83.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;sin2phi \leq 4.999999873689376 \cdot 10^{-5}:\\ \;\;\;\;\frac{u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-1 + u0 \cdot -0.5\right) \cdot \left(alphay \cdot \left(\left(-u0\right) \cdot alphay\right)\right)}{sin2phi}\\ \end{array} \]

Alternative 7: 81.2% accurate, 7.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;sin2phi \leq 4.999999873689376 \cdot 10^{-5}:\\ \;\;\;\;\frac{u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(alphay \cdot alphay\right) \cdot \left(u0 - u0 \cdot \left(u0 \cdot -0.5\right)\right)}{sin2phi}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (if (<= sin2phi 4.999999873689376e-5)
   (/ u0 (+ (/ cos2phi (* alphax alphax)) (/ sin2phi (* alphay alphay))))
   (/ (* (* alphay alphay) (- u0 (* u0 (* u0 -0.5)))) sin2phi)))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float tmp;
	if (sin2phi <= 4.999999873689376e-5f) {
		tmp = u0 / ((cos2phi / (alphax * alphax)) + (sin2phi / (alphay * alphay)));
	} else {
		tmp = ((alphay * alphay) * (u0 - (u0 * (u0 * -0.5f)))) / sin2phi;
	}
	return tmp;
}

Fortran

real(4) function code(alphax, alphay, u0, cos2phi, sin2phi)
    real(4), intent (in) :: alphax
    real(4), intent (in) :: alphay
    real(4), intent (in) :: u0
    real(4), intent (in) :: cos2phi
    real(4), intent (in) :: sin2phi
    real(4) :: tmp
    if (sin2phi <= 4.999999873689376e-5) then
        tmp = u0 / ((cos2phi / (alphax * alphax)) + (sin2phi / (alphay * alphay)))
    else
        tmp = ((alphay * alphay) * (u0 - (u0 * (u0 * (-0.5e0))))) / sin2phi
    end if
    code = tmp
end function

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = Float32(0.0)
	if (sin2phi <= Float32(4.999999873689376e-5))
		tmp = Float32(u0 / Float32(Float32(cos2phi / Float32(alphax * alphax)) + Float32(sin2phi / Float32(alphay * alphay))));
	else
		tmp = Float32(Float32(Float32(alphay * alphay) * Float32(u0 - Float32(u0 * Float32(u0 * Float32(-0.5))))) / sin2phi);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = single(0.0);
	if (sin2phi <= single(4.999999873689376e-5))
		tmp = u0 / ((cos2phi / (alphax * alphax)) + (sin2phi / (alphay * alphay)));
	else
		tmp = ((alphay * alphay) * (u0 - (u0 * (u0 * single(-0.5))))) / sin2phi;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if sin2phi < 4.99999987e-5

   1. Initial program 53.1%

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

      1. associate-/r* 53.2%

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

   3. Simplified 53.2%

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

   4. Taylor expanded in u0 around 0 76.2%

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

      1. unpow2 76.2%

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

      2. unpow2 76.2%

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

   6. Simplified 76.2%

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

   if 4.99999987e-5 < sin2phi

   1. Initial program 65.8%

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

      1. associate-/r* 65.8%

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

   3. Simplified 65.8%

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

   4. Taylor expanded in cos2phi around 0 65.8%

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

      1. mul-1-neg 65.8%

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

      2. unpow2 65.8%

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

      3. *-commutative 65.8%

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

   6. Simplified 65.8%

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

   7. Taylor expanded in u0 around 0 88.2%

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

      1. +-commutative 88.2%

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

      2. mul-1-neg 88.2%

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

      3. unsub-neg 88.2%

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

      4. unpow2 88.2%

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

      5. associate-*r* 88.2%

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

   9. Simplified 88.2%

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

4. Final simplification 83.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;sin2phi \leq 4.999999873689376 \cdot 10^{-5}:\\ \;\;\;\;\frac{u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(alphay \cdot alphay\right) \cdot \left(u0 - u0 \cdot \left(u0 \cdot -0.5\right)\right)}{sin2phi}\\ \end{array} \]

Alternative 8: 75.6% 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 60.6%

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

   1. associate-/r* 60.6%

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

3. Simplified 60.6%

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

4. Taylor expanded in u0 around 0 77.5%

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

   1. unpow2 77.5%

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

   2. unpow2 77.5%

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

6. Simplified 77.5%

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

7. Final simplification 77.5%

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

Alternative 9: 66.9% accurate, 12.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;sin2phi \leq 6.000000068087077 \cdot 10^{-18}:\\ \;\;\;\;\left(alphax \cdot alphax\right) \cdot \frac{u0}{cos2phi}\\ \mathbf{else}:\\ \;\;\;\;alphay \cdot \left(alphay \cdot \frac{u0}{sin2phi}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (if (<= sin2phi 6.000000068087077e-18)
   (* (* 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 <= 6.000000068087077e-18f) {
		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 <= 6.000000068087077e-18) 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(6.000000068087077e-18))
		tmp = Float32(Float32(alphax * alphax) * Float32(u0 / cos2phi));
	else
		tmp = Float32(alphay * Float32(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(6.000000068087077e-18))
		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 < 6.00000007e-18

   1. Initial program 56.7%

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

      1. associate-/r* 56.8%

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

   3. Simplified 56.8%

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

   4. Taylor expanded in u0 around 0 72.4%

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

      1. unpow2 72.4%

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

      2. unpow2 72.4%

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

   6. Simplified 72.4%

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

      1. associate-/r* 72.2%

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

      2. div-inv 72.2%

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

   8. Applied egg-rr 72.2%

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

   9. Taylor expanded in cos2phi around inf 53.9%

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

       1. *-commutative 53.9%

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

       2. *-lft-identity 53.9%

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

       3. times-frac 54.0%

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

       4. /-rgt-identity 54.0%

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

       5. unpow2 54.0%

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

   11. Simplified 54.0%

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

   if 6.00000007e-18 < sin2phi

   1. Initial program 62.0%

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

      1. associate-/r* 62.0%

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

   3. Simplified 62.0%

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

   4. Taylor expanded in u0 around 0 79.3%

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

      1. unpow2 79.3%

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

      2. unpow2 79.3%

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

   6. Simplified 79.3%

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

   7. Taylor expanded in cos2phi around 0 74.3%

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

      1. *-commutative 74.3%

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

      2. *-lft-identity 74.3%

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

      3. times-frac 74.3%

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

      4. /-rgt-identity 74.3%

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

      5. unpow2 74.3%

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

   9. Simplified 74.3%

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

   10. Taylor expanded in alphay around 0 74.3%

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

       1. unpow2 74.3%

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

       2. *-commutative 74.3%

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

       3. associate-*r/ 74.3%

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

       4. associate-*l* 74.3%

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

   12. Simplified 74.3%

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

4. Final simplification 68.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;sin2phi \leq 6.000000068087077 \cdot 10^{-18}:\\ \;\;\;\;\left(alphax \cdot alphax\right) \cdot \frac{u0}{cos2phi}\\ \mathbf{else}:\\ \;\;\;\;alphay \cdot \left(alphay \cdot \frac{u0}{sin2phi}\right)\\ \end{array} \]

Alternative 10: 66.9% accurate, 12.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;sin2phi \leq 6.000000068087077 \cdot 10^{-18}:\\ \;\;\;\;\left(alphax \cdot alphax\right) \cdot \frac{u0}{cos2phi}\\ \mathbf{else}:\\ \;\;\;\;\left(alphay \cdot alphay\right) \cdot \frac{u0}{sin2phi}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (if (<= sin2phi 6.000000068087077e-18)
   (* (* 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 <= 6.000000068087077e-18f) {
		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 <= 6.000000068087077e-18) 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(6.000000068087077e-18))
		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(6.000000068087077e-18))
		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 < 6.00000007e-18

   1. Initial program 56.7%

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

      1. associate-/r* 56.8%

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

   3. Simplified 56.8%

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

   4. Taylor expanded in u0 around 0 72.4%

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

      1. unpow2 72.4%

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

      2. unpow2 72.4%

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

   6. Simplified 72.4%

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

      1. associate-/r* 72.2%

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

      2. div-inv 72.2%

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

   8. Applied egg-rr 72.2%

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

   9. Taylor expanded in cos2phi around inf 53.9%

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

       1. *-commutative 53.9%

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

       2. *-lft-identity 53.9%

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

       3. times-frac 54.0%

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

       4. /-rgt-identity 54.0%

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

       5. unpow2 54.0%

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

   11. Simplified 54.0%

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

   if 6.00000007e-18 < sin2phi

   1. Initial program 62.0%

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

      1. associate-/r* 62.0%

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

   3. Simplified 62.0%

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

   4. Taylor expanded in u0 around 0 79.3%

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

      1. unpow2 79.3%

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

      2. unpow2 79.3%

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

   6. Simplified 79.3%

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

   7. Taylor expanded in cos2phi around 0 74.3%

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

      1. *-commutative 74.3%

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

      2. *-lft-identity 74.3%

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

      3. times-frac 74.3%

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

      4. /-rgt-identity 74.3%

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

      5. unpow2 74.3%

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

   9. Simplified 74.3%

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

4. Final simplification 68.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;sin2phi \leq 6.000000068087077 \cdot 10^{-18}:\\ \;\;\;\;\left(alphax \cdot alphax\right) \cdot \frac{u0}{cos2phi}\\ \mathbf{else}:\\ \;\;\;\;\left(alphay \cdot alphay\right) \cdot \frac{u0}{sin2phi}\\ \end{array} \]

Alternative 11: 66.9% accurate, 12.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;sin2phi \leq 6.000000068087077 \cdot 10^{-18}:\\ \;\;\;\;\frac{u0}{\frac{cos2phi}{alphax \cdot alphax}}\\ \mathbf{else}:\\ \;\;\;\;\left(alphay \cdot alphay\right) \cdot \frac{u0}{sin2phi}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (if (<= sin2phi 6.000000068087077e-18)
   (/ u0 (/ cos2phi (* alphax alphax)))
   (* (* alphay alphay) (/ u0 sin2phi))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float tmp;
	if (sin2phi <= 6.000000068087077e-18f) {
		tmp = u0 / (cos2phi / (alphax * alphax));
	} 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 <= 6.000000068087077e-18) then
        tmp = u0 / (cos2phi / (alphax * alphax))
    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(6.000000068087077e-18))
		tmp = Float32(u0 / Float32(cos2phi / Float32(alphax * alphax)));
	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(6.000000068087077e-18))
		tmp = u0 / (cos2phi / (alphax * alphax));
	else
		tmp = (alphay * alphay) * (u0 / sin2phi);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if sin2phi < 6.00000007e-18

   1. Initial program 56.7%

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

      1. associate-/r* 56.8%

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

   3. Simplified 56.8%

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

   4. Taylor expanded in u0 around 0 72.4%

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

      1. unpow2 72.4%

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

      2. unpow2 72.4%

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

   6. Simplified 72.4%

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

   7. Taylor expanded in cos2phi around inf 53.9%

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

      1. associate-/l* 54.0%

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

      2. unpow2 54.0%

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

   9. Simplified 54.0%

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

   if 6.00000007e-18 < sin2phi

   1. Initial program 62.0%

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

      1. associate-/r* 62.0%

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

   3. Simplified 62.0%

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

   4. Taylor expanded in u0 around 0 79.3%

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

      1. unpow2 79.3%

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

      2. unpow2 79.3%

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

   6. Simplified 79.3%

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

   7. Taylor expanded in cos2phi around 0 74.3%

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

      1. *-commutative 74.3%

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

      2. *-lft-identity 74.3%

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

      3. times-frac 74.3%

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

      4. /-rgt-identity 74.3%

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

      5. unpow2 74.3%

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

   9. Simplified 74.3%

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

4. Final simplification 68.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;sin2phi \leq 6.000000068087077 \cdot 10^{-18}:\\ \;\;\;\;\frac{u0}{\frac{cos2phi}{alphax \cdot alphax}}\\ \mathbf{else}:\\ \;\;\;\;\left(alphay \cdot alphay\right) \cdot \frac{u0}{sin2phi}\\ \end{array} \]

Alternative 12: 59.6% accurate, 16.6× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 60.6%

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

   1. associate-/r* 60.6%

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

3. Simplified 60.6%

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

4. Taylor expanded in u0 around 0 77.5%

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

   1. unpow2 77.5%

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

   2. unpow2 77.5%

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

6. Simplified 77.5%

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

7. Taylor expanded in cos2phi around 0 61.1%

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

   1. *-commutative 61.1%

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

   2. *-lft-identity 61.1%

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

   3. times-frac 61.1%

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

   4. /-rgt-identity 61.1%

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

   5. unpow2 61.1%

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

9. Simplified 61.1%

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

10. Taylor expanded in alphay around 0 61.1%

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

    1. unpow2 61.1%

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

    2. *-commutative 61.1%

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

    3. associate-*r/ 61.1%

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

    4. associate-*l* 61.1%

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

12. Simplified 61.1%

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

13. Final simplification 61.1%

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

Reproduce

herbie shell --seed 2023257 
(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)))))