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

Percentage Accurate: 60.4% → 98.3%

Time: 10.6s

Alternatives: 9

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: 60.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Alternative 1: 98.3% accurate, 1.0× speedup

Math

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

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 62.8%

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

   1. neg-sub0 62.8%

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

   2. div-sub 62.8%

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

   3. --rgt-identity 62.8%

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

   4. div-sub 62.8%

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

   5. --rgt-identity 62.8%

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

   6. neg-sub0 62.8%

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

   7. sub-neg 62.8%

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

   8. log1p-def 98.7%

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

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 2: 92.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;sin2phi \leq 0.00019999999494757503:\\ \;\;\;\;\frac{u0 - u0 \cdot \left(u0 \cdot -0.5\right)}{\frac{sin2phi}{alphay \cdot alphay} + \frac{\frac{cos2phi}{alphax}}{alphax}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(-u0\right)}{sin2phi} \cdot \left(alphay \cdot \left(-alphay\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (if (<= sin2phi 0.00019999999494757503)
   (/
    (- u0 (* u0 (* u0 -0.5)))
    (+ (/ sin2phi (* alphay alphay)) (/ (/ cos2phi alphax) alphax)))
   (* (/ (log1p (- u0)) sin2phi) (* alphay (- alphay)))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float tmp;
	if (sin2phi <= 0.00019999999494757503f) {
		tmp = (u0 - (u0 * (u0 * -0.5f))) / ((sin2phi / (alphay * alphay)) + ((cos2phi / alphax) / alphax));
	} else {
		tmp = (log1pf(-u0) / sin2phi) * (alphay * -alphay);
	}
	return tmp;
}

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = Float32(0.0)
	if (sin2phi <= Float32(0.00019999999494757503))
		tmp = Float32(Float32(u0 - Float32(u0 * Float32(u0 * Float32(-0.5)))) / Float32(Float32(sin2phi / Float32(alphay * alphay)) + Float32(Float32(cos2phi / alphax) / alphax)));
	else
		tmp = Float32(Float32(log1p(Float32(-u0)) / sin2phi) * Float32(alphay * Float32(-alphay)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if sin2phi < 1.99999995e-4

   1. Initial program 56.2%

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

      1. associate-/r* 56.3%

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

   3. Simplified 56.3%

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

   4. Taylor expanded in u0 around 0 86.7%

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

      1. +-commutative 40.8%

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

      2. mul-1-neg 40.8%

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

      3. unsub-neg 40.8%

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

      4. unpow2 40.8%

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

      5. associate-*r* 40.8%

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

   6. Simplified 86.7%

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

   if 1.99999995e-4 < sin2phi

   1. Initial program 68.2%

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

      1. associate-/r* 68.2%

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

   3. Simplified 68.2%

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

   4. Taylor expanded in cos2phi around 0 68.5%

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

      1. mul-1-neg 68.5%

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

      2. unpow2 68.5%

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

      3. *-commutative 68.5%

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

   6. Simplified 68.5%

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

   7. Taylor expanded in alphay around 0 68.5%

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

      1. associate-/l* 68.0%

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

      2. sub-neg 68.0%

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

      3. log1p-def 97.2%

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

      4. associate-/l* 98.5%

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

      5. *-commutative 98.5%

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

      6. associate-*l/ 98.4%

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

      7. unpow2 98.4%

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

   9. Simplified 98.4%

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

4. Final simplification 93.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;sin2phi \leq 0.00019999999494757503:\\ \;\;\;\;\frac{u0 - u0 \cdot \left(u0 \cdot -0.5\right)}{\frac{sin2phi}{alphay \cdot alphay} + \frac{\frac{cos2phi}{alphax}}{alphax}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(-u0\right)}{sin2phi} \cdot \left(alphay \cdot \left(-alphay\right)\right)\\ \end{array} \]

Alternative 3: 81.6% accurate, 6.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{sin2phi}{alphay \cdot alphay}\\ \mathbf{if}\;t_0 \leq 0.05999999865889549:\\ \;\;\;\;\frac{u0}{\frac{cos2phi}{alphax \cdot alphax} + t_0}\\ \mathbf{else}:\\ \;\;\;\;\left(u0 - u0 \cdot \left(u0 \cdot -0.5\right)\right) \cdot \left(alphay \cdot \frac{alphay}{sin2phi}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (let* ((t_0 (/ sin2phi (* alphay alphay))))
   (if (<= t_0 0.05999999865889549)
     (/ u0 (+ (/ cos2phi (* alphax alphax)) t_0))
     (* (- u0 (* u0 (* u0 -0.5))) (* alphay (/ alphay sin2phi))))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float t_0 = sin2phi / (alphay * alphay);
	float tmp;
	if (t_0 <= 0.05999999865889549f) {
		tmp = u0 / ((cos2phi / (alphax * alphax)) + t_0);
	} else {
		tmp = (u0 - (u0 * (u0 * -0.5f))) * (alphay * (alphay / sin2phi));
	}
	return tmp;
}

Fortran

real(4) function code(alphax, alphay, u0, cos2phi, sin2phi)
    real(4), intent (in) :: alphax
    real(4), intent (in) :: alphay
    real(4), intent (in) :: u0
    real(4), intent (in) :: cos2phi
    real(4), intent (in) :: sin2phi
    real(4) :: t_0
    real(4) :: tmp
    t_0 = sin2phi / (alphay * alphay)
    if (t_0 <= 0.05999999865889549e0) then
        tmp = u0 / ((cos2phi / (alphax * alphax)) + t_0)
    else
        tmp = (u0 - (u0 * (u0 * (-0.5e0)))) * (alphay * (alphay / sin2phi))
    end if
    code = tmp
end function

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	t_0 = Float32(sin2phi / Float32(alphay * alphay))
	tmp = Float32(0.0)
	if (t_0 <= Float32(0.05999999865889549))
		tmp = Float32(u0 / Float32(Float32(cos2phi / Float32(alphax * alphax)) + t_0));
	else
		tmp = Float32(Float32(u0 - Float32(u0 * Float32(u0 * Float32(-0.5)))) * Float32(alphay * Float32(alphay / sin2phi)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alphax, alphay, u0, cos2phi, sin2phi)
	t_0 = sin2phi / (alphay * alphay);
	tmp = single(0.0);
	if (t_0 <= single(0.05999999865889549))
		tmp = u0 / ((cos2phi / (alphax * alphax)) + t_0);
	else
		tmp = (u0 - (u0 * (u0 * single(-0.5)))) * (alphay * (alphay / sin2phi));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 sin2phi (*.f32 alphay alphay)) < 0.0599999987

   1. Initial program 57.2%

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

      1. associate-/r* 57.4%

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

   3. Simplified 57.4%

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

   4. Taylor expanded in u0 around 0 73.4%

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

      1. unpow2 73.4%

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

      2. unpow2 73.4%

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

   6. Simplified 73.4%

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

   if 0.0599999987 < (/.f32 sin2phi (*.f32 alphay alphay))

   1. Initial program 67.0%

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

      1. associate-/r* 67.0%

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

   3. Simplified 67.0%

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

   4. Taylor expanded in cos2phi around 0 67.3%

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

      1. mul-1-neg 67.3%

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

      2. unpow2 67.3%

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

      3. *-commutative 67.3%

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

   6. Simplified 67.3%

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

   7. Taylor expanded in u0 around 0 88.6%

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

      1. +-commutative 88.6%

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

      2. mul-1-neg 88.6%

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

      3. unsub-neg 88.6%

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

      4. unpow2 88.6%

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

      5. associate-*r* 88.6%

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

   9. Simplified 88.6%

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

   10. Taylor expanded in u0 around 0 88.7%

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

       1. mul-1-neg 88.7%

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

       2. unsub-neg 88.7%

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

       3. associate-/l* 88.6%

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

       4. associate-*r/ 88.5%

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

       5. *-commutative 88.5%

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

       6. unpow2 88.5%

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

       7. associate-*r* 88.5%

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

       8. associate-/l* 88.3%

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

       9. div-sub 88.2%

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

       10. associate-/l* 88.6%

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

       11. *-commutative 88.6%

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

       12. associate-/l* 87.2%

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

       13. associate-/r/ 88.3%

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

       14. unpow2 88.3%

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

       15. associate-*r/ 88.4%

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

   12. Simplified 88.4%

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

4. Final simplification 82.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 0.05999999865889549:\\ \;\;\;\;\frac{u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}}\\ \mathbf{else}:\\ \;\;\;\;\left(u0 - u0 \cdot \left(u0 \cdot -0.5\right)\right) \cdot \left(alphay \cdot \frac{alphay}{sin2phi}\right)\\ \end{array} \]

Alternative 4: 81.6% accurate, 6.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{sin2phi}{alphay \cdot alphay}\\ \mathbf{if}\;t_0 \leq 0.10000000149011612:\\ \;\;\;\;\frac{u0}{\frac{cos2phi}{alphax \cdot alphax} + t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{alphay \cdot \left(alphay \cdot \left(u0 - u0 \cdot \left(u0 \cdot -0.5\right)\right)\right)}{sin2phi}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (let* ((t_0 (/ sin2phi (* alphay alphay))))
   (if (<= t_0 0.10000000149011612)
     (/ u0 (+ (/ cos2phi (* alphax alphax)) t_0))
     (/ (* alphay (* alphay (- u0 (* u0 (* u0 -0.5))))) sin2phi))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float t_0 = sin2phi / (alphay * alphay);
	float tmp;
	if (t_0 <= 0.10000000149011612f) {
		tmp = u0 / ((cos2phi / (alphax * alphax)) + t_0);
	} else {
		tmp = (alphay * (alphay * (u0 - (u0 * (u0 * -0.5f))))) / sin2phi;
	}
	return tmp;
}

Fortran

real(4) function code(alphax, alphay, u0, cos2phi, sin2phi)
    real(4), intent (in) :: alphax
    real(4), intent (in) :: alphay
    real(4), intent (in) :: u0
    real(4), intent (in) :: cos2phi
    real(4), intent (in) :: sin2phi
    real(4) :: t_0
    real(4) :: tmp
    t_0 = sin2phi / (alphay * alphay)
    if (t_0 <= 0.10000000149011612e0) then
        tmp = u0 / ((cos2phi / (alphax * alphax)) + t_0)
    else
        tmp = (alphay * (alphay * (u0 - (u0 * (u0 * (-0.5e0)))))) / sin2phi
    end if
    code = tmp
end function

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	t_0 = Float32(sin2phi / Float32(alphay * alphay))
	tmp = Float32(0.0)
	if (t_0 <= Float32(0.10000000149011612))
		tmp = Float32(u0 / Float32(Float32(cos2phi / Float32(alphax * alphax)) + t_0));
	else
		tmp = Float32(Float32(alphay * Float32(alphay * Float32(u0 - Float32(u0 * Float32(u0 * Float32(-0.5)))))) / sin2phi);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alphax, alphay, u0, cos2phi, sin2phi)
	t_0 = sin2phi / (alphay * alphay);
	tmp = single(0.0);
	if (t_0 <= single(0.10000000149011612))
		tmp = u0 / ((cos2phi / (alphax * alphax)) + t_0);
	else
		tmp = (alphay * (alphay * (u0 - (u0 * (u0 * single(-0.5)))))) / sin2phi;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 sin2phi (*.f32 alphay alphay)) < 0.100000001

   1. Initial program 57.0%

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

      1. associate-/r* 57.1%

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

   3. Simplified 57.1%

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

   4. Taylor expanded in u0 around 0 73.6%

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

      1. unpow2 73.6%

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

      2. unpow2 73.6%

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

   6. Simplified 73.6%

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

   if 0.100000001 < (/.f32 sin2phi (*.f32 alphay alphay))

   1. Initial program 67.3%

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

      1. associate-/r* 67.3%

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

   3. Simplified 67.3%

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

   4. Taylor expanded in cos2phi around 0 67.6%

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

      1. mul-1-neg 67.6%

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

      2. unpow2 67.6%

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

      3. *-commutative 67.6%

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

   6. Simplified 67.6%

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

   7. Taylor expanded in u0 around 0 88.5%

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

      1. +-commutative 88.5%

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

      2. mul-1-neg 88.5%

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

      3. unsub-neg 88.5%

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

      4. unpow2 88.5%

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

      5. associate-*r* 88.5%

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

   9. Simplified 88.5%

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

   10. Taylor expanded in u0 around 0 88.4%

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

       1. associate-*r* 88.4%

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

       2. *-commutative 88.4%

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

       3. unpow2 88.4%

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

       4. associate-*r* 88.4%

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

       5. associate-*r* 88.4%

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

       6. neg-mul-1 88.4%

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

       7. distribute-rgt-in 88.5%

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

       8. sub-neg 88.5%

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

       9. unpow2 88.5%

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

       10. associate-*l* 88.4%

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

   12. Simplified 88.4%

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

4. Final simplification 82.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 0.10000000149011612:\\ \;\;\;\;\frac{u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}}\\ \mathbf{else}:\\ \;\;\;\;\frac{alphay \cdot \left(alphay \cdot \left(u0 - u0 \cdot \left(u0 \cdot -0.5\right)\right)\right)}{sin2phi}\\ \end{array} \]

Alternative 5: 81.6% accurate, 6.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{sin2phi}{alphay \cdot alphay}\\ \mathbf{if}\;t_0 \leq 0.05999999865889549:\\ \;\;\;\;\frac{u0}{\frac{cos2phi}{alphax \cdot alphax} + t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(u0 - u0 \cdot \left(u0 \cdot -0.5\right)\right) \cdot \left(alphay \cdot alphay\right)}{sin2phi}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (let* ((t_0 (/ sin2phi (* alphay alphay))))
   (if (<= t_0 0.05999999865889549)
     (/ u0 (+ (/ cos2phi (* alphax alphax)) t_0))
     (/ (* (- u0 (* u0 (* u0 -0.5))) (* alphay alphay)) sin2phi))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float t_0 = sin2phi / (alphay * alphay);
	float tmp;
	if (t_0 <= 0.05999999865889549f) {
		tmp = u0 / ((cos2phi / (alphax * alphax)) + t_0);
	} else {
		tmp = ((u0 - (u0 * (u0 * -0.5f))) * (alphay * alphay)) / sin2phi;
	}
	return tmp;
}

Fortran

real(4) function code(alphax, alphay, u0, cos2phi, sin2phi)
    real(4), intent (in) :: alphax
    real(4), intent (in) :: alphay
    real(4), intent (in) :: u0
    real(4), intent (in) :: cos2phi
    real(4), intent (in) :: sin2phi
    real(4) :: t_0
    real(4) :: tmp
    t_0 = sin2phi / (alphay * alphay)
    if (t_0 <= 0.05999999865889549e0) then
        tmp = u0 / ((cos2phi / (alphax * alphax)) + t_0)
    else
        tmp = ((u0 - (u0 * (u0 * (-0.5e0)))) * (alphay * alphay)) / sin2phi
    end if
    code = tmp
end function

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	t_0 = Float32(sin2phi / Float32(alphay * alphay))
	tmp = Float32(0.0)
	if (t_0 <= Float32(0.05999999865889549))
		tmp = Float32(u0 / Float32(Float32(cos2phi / Float32(alphax * alphax)) + t_0));
	else
		tmp = Float32(Float32(Float32(u0 - Float32(u0 * Float32(u0 * Float32(-0.5)))) * Float32(alphay * alphay)) / sin2phi);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alphax, alphay, u0, cos2phi, sin2phi)
	t_0 = sin2phi / (alphay * alphay);
	tmp = single(0.0);
	if (t_0 <= single(0.05999999865889549))
		tmp = u0 / ((cos2phi / (alphax * alphax)) + t_0);
	else
		tmp = ((u0 - (u0 * (u0 * single(-0.5)))) * (alphay * alphay)) / sin2phi;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 sin2phi (*.f32 alphay alphay)) < 0.0599999987

   1. Initial program 57.2%

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

      1. associate-/r* 57.4%

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

   3. Simplified 57.4%

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

   4. Taylor expanded in u0 around 0 73.4%

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

      1. unpow2 73.4%

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

      2. unpow2 73.4%

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

   6. Simplified 73.4%

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

   if 0.0599999987 < (/.f32 sin2phi (*.f32 alphay alphay))

   1. Initial program 67.0%

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

      1. associate-/r* 67.0%

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

   3. Simplified 67.0%

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

   4. Taylor expanded in cos2phi around 0 67.3%

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

      1. mul-1-neg 67.3%

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

      2. unpow2 67.3%

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

      3. *-commutative 67.3%

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

   6. Simplified 67.3%

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

   7. Taylor expanded in u0 around 0 88.6%

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

      1. +-commutative 88.6%

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

      2. mul-1-neg 88.6%

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

      3. unsub-neg 88.6%

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

      4. unpow2 88.6%

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

      5. associate-*r* 88.6%

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

   9. Simplified 88.6%

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

4. Final simplification 82.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 0.05999999865889549:\\ \;\;\;\;\frac{u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(u0 - u0 \cdot \left(u0 \cdot -0.5\right)\right) \cdot \left(alphay \cdot alphay\right)}{sin2phi}\\ \end{array} \]

Alternative 6: 87.7% accurate, 6.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 62.8%

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

   1. associate-/r* 62.9%

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

3. Simplified 62.9%

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

4. Taylor expanded in u0 around 0 87.4%

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

   1. +-commutative 66.8%

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

   2. mul-1-neg 66.8%

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

   3. unsub-neg 66.8%

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

   4. unpow2 66.8%

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

   5. associate-*r* 66.8%

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

6. Simplified 87.4%

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

7. Final simplification 87.4%

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

Alternative 7: 76.0% 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 62.8%

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

   1. associate-/r* 62.9%

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

3. Simplified 62.9%

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

4. Taylor expanded in u0 around 0 76.2%

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

   1. unpow2 76.2%

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

   2. unpow2 76.2%

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

6. Simplified 76.2%

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

7. Final simplification 76.2%

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

Alternative 8: 66.6% accurate, 12.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;sin2phi \leq 3.499999920483765 \cdot 10^{-24}:\\ \;\;\;\;alphax \cdot \left(alphax \cdot \frac{u0}{cos2phi}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{u0}{sin2phi} \cdot \left(alphay \cdot alphay\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (if (<= sin2phi 3.499999920483765e-24)
   (* alphax (* alphax (/ u0 cos2phi)))
   (* (/ u0 sin2phi) (* alphay alphay))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float tmp;
	if (sin2phi <= 3.499999920483765e-24f) {
		tmp = alphax * (alphax * (u0 / cos2phi));
	} else {
		tmp = (u0 / sin2phi) * (alphay * alphay);
	}
	return tmp;
}

Fortran

real(4) function code(alphax, alphay, u0, cos2phi, sin2phi)
    real(4), intent (in) :: alphax
    real(4), intent (in) :: alphay
    real(4), intent (in) :: u0
    real(4), intent (in) :: cos2phi
    real(4), intent (in) :: sin2phi
    real(4) :: tmp
    if (sin2phi <= 3.499999920483765e-24) then
        tmp = alphax * (alphax * (u0 / cos2phi))
    else
        tmp = (u0 / sin2phi) * (alphay * alphay)
    end if
    code = tmp
end function

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = Float32(0.0)
	if (sin2phi <= Float32(3.499999920483765e-24))
		tmp = Float32(alphax * Float32(alphax * Float32(u0 / cos2phi)));
	else
		tmp = Float32(Float32(u0 / sin2phi) * Float32(alphay * alphay));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = single(0.0);
	if (sin2phi <= single(3.499999920483765e-24))
		tmp = alphax * (alphax * (u0 / cos2phi));
	else
		tmp = (u0 / sin2phi) * (alphay * alphay);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if sin2phi < 3.49999992e-24

   1. Initial program 55.6%

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

      1. associate-/r* 55.8%

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

   3. Simplified 55.8%

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

   4. Taylor expanded in u0 around 0 72.7%

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

      1. unpow2 72.7%

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

      2. unpow2 72.7%

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

   6. Simplified 72.7%

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

   7. Taylor expanded in cos2phi around inf 56.7%

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

      1. associate-/l* 56.8%

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

      2. unpow2 56.8%

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

   9. Simplified 56.8%

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

   10. Taylor expanded in u0 around 0 56.7%

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

       1. associate-*l/ 56.8%

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

       2. unpow2 56.8%

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

       3. associate-*r* 56.9%

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

   12. Simplified 56.9%

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

   if 3.49999992e-24 < sin2phi

   1. Initial program 64.9%

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

      1. associate-/r* 64.9%

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

   3. Simplified 64.9%

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

   4. Taylor expanded in u0 around 0 77.3%

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

      1. unpow2 77.3%

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

      2. unpow2 77.3%

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

   6. Simplified 77.3%

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

      1. clear-num 77.2%

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

      2. clear-num 77.2%

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

      3. frac-add 73.2%

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

      4. *-un-lft-identity 73.2%

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

      5. associate-/l* 73.2%

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

      6. associate-/l* 73.1%

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

      7. div-inv 73.2%

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

      8. clear-num 73.2%

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

      9. associate-/l* 73.2%

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

      10. div-inv 73.2%

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

      11. clear-num 73.2%

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

      12. associate-/l* 73.1%

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

   8. Applied egg-rr 73.1%

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

   9. Taylor expanded in alphay around 0 71.1%

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

       1. associate-/l* 71.0%

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

       2. associate-/r/ 71.1%

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

       3. unpow2 71.1%

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

   11. Simplified 71.1%

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

4. Final simplification 68.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;sin2phi \leq 3.499999920483765 \cdot 10^{-24}:\\ \;\;\;\;alphax \cdot \left(alphax \cdot \frac{u0}{cos2phi}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{u0}{sin2phi} \cdot \left(alphay \cdot alphay\right)\\ \end{array} \]

Alternative 9: 23.6% accurate, 16.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 62.8%

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

   1. associate-/r* 62.9%

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

3. Simplified 62.9%

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

4. Taylor expanded in u0 around 0 76.2%

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

   1. unpow2 76.2%

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

   2. unpow2 76.2%

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

6. Simplified 76.2%

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

7. Taylor expanded in cos2phi around inf 23.6%

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

   1. associate-/l* 23.6%

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

   2. unpow2 23.6%

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

9. Simplified 23.6%

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

10. Taylor expanded in u0 around 0 23.6%

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

    1. associate-*l/ 23.6%

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

    2. unpow2 23.6%

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

    3. associate-*r* 23.6%

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

12. Simplified 23.6%

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

13. Final simplification 23.6%

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

Reproduce

herbie shell --seed 2023181 
(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)))))