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

Percentage Accurate: 61.0% → 98.4%

Time: 17.8s

Alternatives: 10

Speedup: 3.1×

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: 98.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \frac{\mathsf{log1p}\left(-u0\right)}{{alphax}^{-2} \cdot \left(-cos2phi\right) - \frac{sin2phi}{alphay \cdot alphay}} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (/
  (log1p (- u0))
  (- (* (pow alphax -2.0) (- cos2phi)) (/ sin2phi (* alphay alphay)))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	return log1pf(-u0) / ((powf(alphax, -2.0f) * -cos2phi) - (sin2phi / (alphay * alphay)));
}

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	return Float32(log1p(Float32(-u0)) / Float32(Float32((alphax ^ Float32(-2.0)) * Float32(-cos2phi)) - Float32(sin2phi / Float32(alphay * alphay))))
end

Derivation

1. Initial program 60.5%

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

   1. distribute-frac-neg 60.5%

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

   2. distribute-neg-frac2 60.5%

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

   3. sub-neg 60.5%

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

   4. log1p-define 98.1%

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

   5. neg-sub0 98.1%

      \[\leadsto \frac{\mathsf{log1p}\left(-u0\right)}{\color{blue}{0 - \left(\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\right)}} \]

   6. associate--r+ 98.1%

      \[\leadsto \frac{\mathsf{log1p}\left(-u0\right)}{\color{blue}{\left(0 - \frac{cos2phi}{alphax \cdot alphax}\right) - \frac{sin2phi}{alphay \cdot alphay}}} \]

   7. neg-sub0 98.1%

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

   8. associate-/r* 98.2%

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

   9. distribute-neg-frac2 98.2%

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

3. Simplified 98.2%

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

   1. distribute-frac-neg2 98.2%

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

   2. associate-/r* 98.1%

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

   3. neg-sub0 98.1%

      \[\leadsto \frac{\mathsf{log1p}\left(-u0\right)}{\color{blue}{\left(0 - \frac{cos2phi}{alphax \cdot alphax}\right)} - \frac{sin2phi}{alphay \cdot alphay}} \]

   4. div-inv 98.2%

      \[\leadsto \frac{\mathsf{log1p}\left(-u0\right)}{\left(0 - \color{blue}{cos2phi \cdot \frac{1}{alphax \cdot alphax}}\right) - \frac{sin2phi}{alphay \cdot alphay}} \]

   5. pow2 98.2%

      \[\leadsto \frac{\mathsf{log1p}\left(-u0\right)}{\left(0 - cos2phi \cdot \frac{1}{\color{blue}{{alphax}^{2}}}\right) - \frac{sin2phi}{alphay \cdot alphay}} \]

   6. pow-flip 98.3%

      \[\leadsto \frac{\mathsf{log1p}\left(-u0\right)}{\left(0 - cos2phi \cdot \color{blue}{{alphax}^{\left(-2\right)}}\right) - \frac{sin2phi}{alphay \cdot alphay}} \]

   7. metadata-eval 98.3%

      \[\leadsto \frac{\mathsf{log1p}\left(-u0\right)}{\left(0 - cos2phi \cdot {alphax}^{\color{blue}{-2}}\right) - \frac{sin2phi}{alphay \cdot alphay}} \]

6. Applied egg-rr 98.3%

   \[\leadsto \frac{\mathsf{log1p}\left(-u0\right)}{\color{blue}{\left(0 - cos2phi \cdot {alphax}^{-2}\right)} - \frac{sin2phi}{alphay \cdot alphay}} \]
7. Step-by-step derivation

   1. neg-sub0 98.3%

      \[\leadsto \frac{\mathsf{log1p}\left(-u0\right)}{\color{blue}{\left(-cos2phi \cdot {alphax}^{-2}\right)} - \frac{sin2phi}{alphay \cdot alphay}} \]

   2. distribute-lft-neg-in 98.3%

      \[\leadsto \frac{\mathsf{log1p}\left(-u0\right)}{\color{blue}{\left(-cos2phi\right) \cdot {alphax}^{-2}} - \frac{sin2phi}{alphay \cdot alphay}} \]

   3. *-commutative 98.3%

      \[\leadsto \frac{\mathsf{log1p}\left(-u0\right)}{\color{blue}{{alphax}^{-2} \cdot \left(-cos2phi\right)} - \frac{sin2phi}{alphay \cdot alphay}} \]

8. Simplified 98.3%

   \[\leadsto \frac{\mathsf{log1p}\left(-u0\right)}{\color{blue}{{alphax}^{-2} \cdot \left(-cos2phi\right)} - \frac{sin2phi}{alphay \cdot alphay}} \]

9. Final simplification 98.3%

   \[\leadsto \frac{\mathsf{log1p}\left(-u0\right)}{{alphax}^{-2} \cdot \left(-cos2phi\right) - \frac{sin2phi}{alphay \cdot alphay}} \]
10. Add Preprocessing

Alternative 2: 81.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{sin2phi}{alphay \cdot alphay}\\ \mathbf{if}\;t\_0 \leq 2.0000000072549875 \cdot 10^{-15}:\\ \;\;\;\;{alphax}^{2} \cdot \left(u0 \cdot \left(0.5 \cdot \frac{u0}{cos2phi} + \frac{1}{cos2phi}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(-u0\right)}{\frac{\frac{cos2phi}{alphax}}{alphax} - t\_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (let* ((t_0 (/ sin2phi (* alphay alphay))))
   (if (<= t_0 2.0000000072549875e-15)
     (* (pow alphax 2.0) (* u0 (+ (* 0.5 (/ u0 cos2phi)) (/ 1.0 cos2phi))))
     (/ (log1p (- u0)) (- (/ (/ cos2phi alphax) alphax) t_0)))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float t_0 = sin2phi / (alphay * alphay);
	float tmp;
	if (t_0 <= 2.0000000072549875e-15f) {
		tmp = powf(alphax, 2.0f) * (u0 * ((0.5f * (u0 / cos2phi)) + (1.0f / cos2phi)));
	} else {
		tmp = log1pf(-u0) / (((cos2phi / alphax) / alphax) - t_0);
	}
	return tmp;
}

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	t_0 = Float32(sin2phi / Float32(alphay * alphay))
	tmp = Float32(0.0)
	if (t_0 <= Float32(2.0000000072549875e-15))
		tmp = Float32((alphax ^ Float32(2.0)) * Float32(u0 * Float32(Float32(Float32(0.5) * Float32(u0 / cos2phi)) + Float32(Float32(1.0) / cos2phi))));
	else
		tmp = Float32(log1p(Float32(-u0)) / Float32(Float32(Float32(cos2phi / alphax) / alphax) - t_0));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

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

   1. Initial program 50.3%

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

      1. distribute-frac-neg 50.3%

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

      2. distribute-neg-frac2 50.3%

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

      3. sub-neg 50.3%

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

      4. log1p-define 98.6%

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

      5. neg-sub0 98.6%

         \[\leadsto \frac{\mathsf{log1p}\left(-u0\right)}{\color{blue}{0 - \left(\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\right)}} \]

      6. associate--r+ 98.6%

         \[\leadsto \frac{\mathsf{log1p}\left(-u0\right)}{\color{blue}{\left(0 - \frac{cos2phi}{alphax \cdot alphax}\right) - \frac{sin2phi}{alphay \cdot alphay}}} \]

      7. neg-sub0 98.6%

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

      8. associate-/r* 98.8%

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

      9. distribute-neg-frac2 98.8%

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

   3. Simplified 98.8%

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

   5. Taylor expanded in cos2phi around inf 41.7%

      \[\leadsto \color{blue}{-1 \cdot \frac{{alphax}^{2} \cdot \log \left(1 - u0\right)}{cos2phi}} \]
   6. Step-by-step derivation

      1. mul-1-neg 41.7%

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

      2. associate-/l* 41.7%

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

      3. distribute-rgt-neg-in 41.7%

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

      4. distribute-neg-frac2 41.7%

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

      5. sub-neg 41.7%

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

      6. log1p-define 78.2%

         \[\leadsto {alphax}^{2} \cdot \frac{\color{blue}{\mathsf{log1p}\left(-u0\right)}}{-cos2phi} \]

   7. Simplified 78.2%

      \[\leadsto \color{blue}{{alphax}^{2} \cdot \frac{\mathsf{log1p}\left(-u0\right)}{-cos2phi}} \]

   8. Taylor expanded in u0 around 0 71.9%

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

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

   1. Initial program 63.2%

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

      1. distribute-frac-neg 63.2%

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

      2. distribute-neg-frac2 63.2%

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

      3. sub-neg 63.2%

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

      4. log1p-define 98.0%

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

      5. neg-sub0 98.0%

         \[\leadsto \frac{\mathsf{log1p}\left(-u0\right)}{\color{blue}{0 - \left(\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\right)}} \]

      6. associate--r+ 98.0%

         \[\leadsto \frac{\mathsf{log1p}\left(-u0\right)}{\color{blue}{\left(0 - \frac{cos2phi}{alphax \cdot alphax}\right) - \frac{sin2phi}{alphay \cdot alphay}}} \]

      7. neg-sub0 98.0%

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

      8. associate-/r* 98.0%

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

      9. distribute-neg-frac2 98.0%

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

   3. Simplified 98.0%

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

      1. add-sqr-sqrt -0.0%

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

      2. sqrt-unprod 88.5%

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

      3. sqr-neg 88.5%

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

      4. sqrt-prod 88.5%

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

      5. add-sqr-sqrt 88.5%

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

      6. div-inv 88.5%

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

   6. Applied egg-rr 88.5%

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

      1. associate-*r/ 88.5%

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

      2. *-rgt-identity 88.5%

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

   8. Simplified 88.5%

      \[\leadsto \frac{\mathsf{log1p}\left(-u0\right)}{\color{blue}{\frac{\frac{cos2phi}{alphax}}{alphax}} - \frac{sin2phi}{alphay \cdot alphay}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 85.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 2.0000000072549875 \cdot 10^{-15}:\\ \;\;\;\;{alphax}^{2} \cdot \left(u0 \cdot \left(0.5 \cdot \frac{u0}{cos2phi} + \frac{1}{cos2phi}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(-u0\right)}{\frac{\frac{cos2phi}{alphax}}{alphax} - \frac{sin2phi}{alphay \cdot alphay}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 78.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{alphax \cdot sin2phi}{alphay}\\ t_1 := \frac{cos2phi \cdot alphay}{alphax}\\ t_2 := t\_1 - t\_0\\ t_3 := \frac{alphax \cdot alphay}{t\_2}\\ \mathbf{if}\;sin2phi \leq 2.00000006274879 \cdot 10^{-22}:\\ \;\;\;\;{alphax}^{2} \cdot \left(u0 \cdot \left(0.5 \cdot \frac{u0}{cos2phi} + \frac{1}{cos2phi}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;u0 \cdot \left(\frac{alphax \cdot alphay}{t\_0 - t\_1} + u0 \cdot \left(t\_3 \cdot -0.5 + u0 \cdot \left(t\_3 \cdot -0.3333333333333333 + -0.25 \cdot \frac{alphax \cdot \left(u0 \cdot alphay\right)}{t\_2}\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (let* ((t_0 (/ (* alphax sin2phi) alphay))
        (t_1 (/ (* cos2phi alphay) alphax))
        (t_2 (- t_1 t_0))
        (t_3 (/ (* alphax alphay) t_2)))
   (if (<= sin2phi 2.00000006274879e-22)
     (* (pow alphax 2.0) (* u0 (+ (* 0.5 (/ u0 cos2phi)) (/ 1.0 cos2phi))))
     (*
      u0
      (+
       (/ (* alphax alphay) (- t_0 t_1))
       (*
        u0
        (+
         (* t_3 -0.5)
         (*
          u0
          (+
           (* t_3 -0.3333333333333333)
           (* -0.25 (/ (* alphax (* u0 alphay)) t_2)))))))))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float t_0 = (alphax * sin2phi) / alphay;
	float t_1 = (cos2phi * alphay) / alphax;
	float t_2 = t_1 - t_0;
	float t_3 = (alphax * alphay) / t_2;
	float tmp;
	if (sin2phi <= 2.00000006274879e-22f) {
		tmp = powf(alphax, 2.0f) * (u0 * ((0.5f * (u0 / cos2phi)) + (1.0f / cos2phi)));
	} else {
		tmp = u0 * (((alphax * alphay) / (t_0 - t_1)) + (u0 * ((t_3 * -0.5f) + (u0 * ((t_3 * -0.3333333333333333f) + (-0.25f * ((alphax * (u0 * alphay)) / t_2)))))));
	}
	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) :: t_1
    real(4) :: t_2
    real(4) :: t_3
    real(4) :: tmp
    t_0 = (alphax * sin2phi) / alphay
    t_1 = (cos2phi * alphay) / alphax
    t_2 = t_1 - t_0
    t_3 = (alphax * alphay) / t_2
    if (sin2phi <= 2.00000006274879e-22) then
        tmp = (alphax ** 2.0e0) * (u0 * ((0.5e0 * (u0 / cos2phi)) + (1.0e0 / cos2phi)))
    else
        tmp = u0 * (((alphax * alphay) / (t_0 - t_1)) + (u0 * ((t_3 * (-0.5e0)) + (u0 * ((t_3 * (-0.3333333333333333e0)) + ((-0.25e0) * ((alphax * (u0 * alphay)) / t_2)))))))
    end if
    code = tmp
end function

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	t_0 = Float32(Float32(alphax * sin2phi) / alphay)
	t_1 = Float32(Float32(cos2phi * alphay) / alphax)
	t_2 = Float32(t_1 - t_0)
	t_3 = Float32(Float32(alphax * alphay) / t_2)
	tmp = Float32(0.0)
	if (sin2phi <= Float32(2.00000006274879e-22))
		tmp = Float32((alphax ^ Float32(2.0)) * Float32(u0 * Float32(Float32(Float32(0.5) * Float32(u0 / cos2phi)) + Float32(Float32(1.0) / cos2phi))));
	else
		tmp = Float32(u0 * Float32(Float32(Float32(alphax * alphay) / Float32(t_0 - t_1)) + Float32(u0 * Float32(Float32(t_3 * Float32(-0.5)) + Float32(u0 * Float32(Float32(t_3 * Float32(-0.3333333333333333)) + Float32(Float32(-0.25) * Float32(Float32(alphax * Float32(u0 * alphay)) / t_2))))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alphax, alphay, u0, cos2phi, sin2phi)
	t_0 = (alphax * sin2phi) / alphay;
	t_1 = (cos2phi * alphay) / alphax;
	t_2 = t_1 - t_0;
	t_3 = (alphax * alphay) / t_2;
	tmp = single(0.0);
	if (sin2phi <= single(2.00000006274879e-22))
		tmp = (alphax ^ single(2.0)) * (u0 * ((single(0.5) * (u0 / cos2phi)) + (single(1.0) / cos2phi)));
	else
		tmp = u0 * (((alphax * alphay) / (t_0 - t_1)) + (u0 * ((t_3 * single(-0.5)) + (u0 * ((t_3 * single(-0.3333333333333333)) + (single(-0.25) * ((alphax * (u0 * alphay)) / t_2)))))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if sin2phi < 2.00000006e-22

   1. Initial program 51.5%

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

      1. distribute-frac-neg 51.5%

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

      2. distribute-neg-frac2 51.5%

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

      3. sub-neg 51.5%

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

      4. log1p-define 98.7%

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

      5. neg-sub0 98.7%

         \[\leadsto \frac{\mathsf{log1p}\left(-u0\right)}{\color{blue}{0 - \left(\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\right)}} \]

      6. associate--r+ 98.7%

         \[\leadsto \frac{\mathsf{log1p}\left(-u0\right)}{\color{blue}{\left(0 - \frac{cos2phi}{alphax \cdot alphax}\right) - \frac{sin2phi}{alphay \cdot alphay}}} \]

      7. neg-sub0 98.7%

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

      8. associate-/r* 98.8%

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

      9. distribute-neg-frac2 98.8%

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

   3. Simplified 98.8%

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

   5. Taylor expanded in cos2phi around inf 42.3%

      \[\leadsto \color{blue}{-1 \cdot \frac{{alphax}^{2} \cdot \log \left(1 - u0\right)}{cos2phi}} \]
   6. Step-by-step derivation

      1. mul-1-neg 42.3%

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

      2. associate-/l* 42.2%

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

      3. distribute-rgt-neg-in 42.2%

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

      4. distribute-neg-frac2 42.2%

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

      5. sub-neg 42.2%

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

      6. log1p-define 79.7%

         \[\leadsto {alphax}^{2} \cdot \frac{\color{blue}{\mathsf{log1p}\left(-u0\right)}}{-cos2phi} \]

   7. Simplified 79.7%

      \[\leadsto \color{blue}{{alphax}^{2} \cdot \frac{\mathsf{log1p}\left(-u0\right)}{-cos2phi}} \]

   8. Taylor expanded in u0 around 0 73.3%

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

   if 2.00000006e-22 < sin2phi

   1. Initial program 62.7%

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

      1. distribute-frac-neg 62.7%

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

      2. distribute-neg-frac2 62.7%

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

      3. sub-neg 62.7%

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

      4. log1p-define 98.0%

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

      5. neg-sub0 98.0%

         \[\leadsto \frac{\mathsf{log1p}\left(-u0\right)}{\color{blue}{0 - \left(\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\right)}} \]

      6. associate--r+ 98.0%

         \[\leadsto \frac{\mathsf{log1p}\left(-u0\right)}{\color{blue}{\left(0 - \frac{cos2phi}{alphax \cdot alphax}\right) - \frac{sin2phi}{alphay \cdot alphay}}} \]

      7. neg-sub0 98.0%

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

      8. associate-/r* 98.0%

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

      9. distribute-neg-frac2 98.0%

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

   3. Simplified 98.0%

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

      1. frac-2neg 98.0%

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

      2. frac-2neg 98.0%

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

      3. add-sqr-sqrt -0.0%

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

      4. sqrt-unprod 87.3%

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

      5. sqr-neg 87.3%

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

      6. sqrt-prod 87.3%

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

      7. add-sqr-sqrt 87.3%

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

      8. associate-/r* 87.3%

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

      9. frac-sub 87.1%

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

   6. Applied egg-rr 87.1%

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

      1. associate-*l/ 87.1%

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

   8. Simplified 87.1%

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

   9. Taylor expanded in u0 around 0 82.9%

      \[\leadsto \color{blue}{u0 \cdot \left(-1 \cdot \frac{alphax \cdot alphay}{\frac{alphay \cdot cos2phi}{alphax} - \frac{alphax \cdot sin2phi}{alphay}} + u0 \cdot \left(-0.5 \cdot \frac{alphax \cdot alphay}{\frac{alphay \cdot cos2phi}{alphax} - \frac{alphax \cdot sin2phi}{alphay}} + u0 \cdot \left(-0.3333333333333333 \cdot \frac{alphax \cdot alphay}{\frac{alphay \cdot cos2phi}{alphax} - \frac{alphax \cdot sin2phi}{alphay}} + -0.25 \cdot \frac{alphax \cdot \left(alphay \cdot u0\right)}{\frac{alphay \cdot cos2phi}{alphax} - \frac{alphax \cdot sin2phi}{alphay}}\right)\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 81.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;sin2phi \leq 2.00000006274879 \cdot 10^{-22}:\\ \;\;\;\;{alphax}^{2} \cdot \left(u0 \cdot \left(0.5 \cdot \frac{u0}{cos2phi} + \frac{1}{cos2phi}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;u0 \cdot \left(\frac{alphax \cdot alphay}{\frac{alphax \cdot sin2phi}{alphay} - \frac{cos2phi \cdot alphay}{alphax}} + u0 \cdot \left(\frac{alphax \cdot alphay}{\frac{cos2phi \cdot alphay}{alphax} - \frac{alphax \cdot sin2phi}{alphay}} \cdot -0.5 + u0 \cdot \left(\frac{alphax \cdot alphay}{\frac{cos2phi \cdot alphay}{alphax} - \frac{alphax \cdot sin2phi}{alphay}} \cdot -0.3333333333333333 + -0.25 \cdot \frac{alphax \cdot \left(u0 \cdot alphay\right)}{\frac{cos2phi \cdot alphay}{alphax} - \frac{alphax \cdot sin2phi}{alphay}}\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 60.5%

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

   1. distribute-frac-neg 60.5%

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

   2. distribute-neg-frac2 60.5%

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

   3. sub-neg 60.5%

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

   4. log1p-define 98.1%

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

   5. neg-sub0 98.1%

      \[\leadsto \frac{\mathsf{log1p}\left(-u0\right)}{\color{blue}{0 - \left(\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\right)}} \]

   6. associate--r+ 98.1%

      \[\leadsto \frac{\mathsf{log1p}\left(-u0\right)}{\color{blue}{\left(0 - \frac{cos2phi}{alphax \cdot alphax}\right) - \frac{sin2phi}{alphay \cdot alphay}}} \]

   7. neg-sub0 98.1%

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

   8. associate-/r* 98.2%

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

   9. distribute-neg-frac2 98.2%

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

3. Simplified 98.2%

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

5. Final simplification 98.2%

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

Alternative 5: 76.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{alphax \cdot sin2phi}{alphay}\\ t_1 := \frac{cos2phi \cdot alphay}{alphax}\\ t_2 := t\_1 - t\_0\\ t_3 := \frac{alphax \cdot alphay}{t\_2}\\ \mathbf{if}\;sin2phi \leq 2.00000006274879 \cdot 10^{-22}:\\ \;\;\;\;{alphax}^{2} \cdot \frac{u0}{cos2phi}\\ \mathbf{else}:\\ \;\;\;\;u0 \cdot \left(\frac{alphax \cdot alphay}{t\_0 - t\_1} + u0 \cdot \left(t\_3 \cdot -0.5 + u0 \cdot \left(t\_3 \cdot -0.3333333333333333 + -0.25 \cdot \frac{alphax \cdot \left(u0 \cdot alphay\right)}{t\_2}\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (let* ((t_0 (/ (* alphax sin2phi) alphay))
        (t_1 (/ (* cos2phi alphay) alphax))
        (t_2 (- t_1 t_0))
        (t_3 (/ (* alphax alphay) t_2)))
   (if (<= sin2phi 2.00000006274879e-22)
     (* (pow alphax 2.0) (/ u0 cos2phi))
     (*
      u0
      (+
       (/ (* alphax alphay) (- t_0 t_1))
       (*
        u0
        (+
         (* t_3 -0.5)
         (*
          u0
          (+
           (* t_3 -0.3333333333333333)
           (* -0.25 (/ (* alphax (* u0 alphay)) t_2)))))))))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float t_0 = (alphax * sin2phi) / alphay;
	float t_1 = (cos2phi * alphay) / alphax;
	float t_2 = t_1 - t_0;
	float t_3 = (alphax * alphay) / t_2;
	float tmp;
	if (sin2phi <= 2.00000006274879e-22f) {
		tmp = powf(alphax, 2.0f) * (u0 / cos2phi);
	} else {
		tmp = u0 * (((alphax * alphay) / (t_0 - t_1)) + (u0 * ((t_3 * -0.5f) + (u0 * ((t_3 * -0.3333333333333333f) + (-0.25f * ((alphax * (u0 * alphay)) / t_2)))))));
	}
	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) :: t_1
    real(4) :: t_2
    real(4) :: t_3
    real(4) :: tmp
    t_0 = (alphax * sin2phi) / alphay
    t_1 = (cos2phi * alphay) / alphax
    t_2 = t_1 - t_0
    t_3 = (alphax * alphay) / t_2
    if (sin2phi <= 2.00000006274879e-22) then
        tmp = (alphax ** 2.0e0) * (u0 / cos2phi)
    else
        tmp = u0 * (((alphax * alphay) / (t_0 - t_1)) + (u0 * ((t_3 * (-0.5e0)) + (u0 * ((t_3 * (-0.3333333333333333e0)) + ((-0.25e0) * ((alphax * (u0 * alphay)) / t_2)))))))
    end if
    code = tmp
end function

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	t_0 = Float32(Float32(alphax * sin2phi) / alphay)
	t_1 = Float32(Float32(cos2phi * alphay) / alphax)
	t_2 = Float32(t_1 - t_0)
	t_3 = Float32(Float32(alphax * alphay) / t_2)
	tmp = Float32(0.0)
	if (sin2phi <= Float32(2.00000006274879e-22))
		tmp = Float32((alphax ^ Float32(2.0)) * Float32(u0 / cos2phi));
	else
		tmp = Float32(u0 * Float32(Float32(Float32(alphax * alphay) / Float32(t_0 - t_1)) + Float32(u0 * Float32(Float32(t_3 * Float32(-0.5)) + Float32(u0 * Float32(Float32(t_3 * Float32(-0.3333333333333333)) + Float32(Float32(-0.25) * Float32(Float32(alphax * Float32(u0 * alphay)) / t_2))))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alphax, alphay, u0, cos2phi, sin2phi)
	t_0 = (alphax * sin2phi) / alphay;
	t_1 = (cos2phi * alphay) / alphax;
	t_2 = t_1 - t_0;
	t_3 = (alphax * alphay) / t_2;
	tmp = single(0.0);
	if (sin2phi <= single(2.00000006274879e-22))
		tmp = (alphax ^ single(2.0)) * (u0 / cos2phi);
	else
		tmp = u0 * (((alphax * alphay) / (t_0 - t_1)) + (u0 * ((t_3 * single(-0.5)) + (u0 * ((t_3 * single(-0.3333333333333333)) + (single(-0.25) * ((alphax * (u0 * alphay)) / t_2)))))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if sin2phi < 2.00000006e-22

   1. Initial program 51.5%

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

      1. distribute-frac-neg 51.5%

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

      2. distribute-neg-frac2 51.5%

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

      3. sub-neg 51.5%

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

      4. log1p-define 98.7%

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

      5. neg-sub0 98.7%

         \[\leadsto \frac{\mathsf{log1p}\left(-u0\right)}{\color{blue}{0 - \left(\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\right)}} \]

      6. associate--r+ 98.7%

         \[\leadsto \frac{\mathsf{log1p}\left(-u0\right)}{\color{blue}{\left(0 - \frac{cos2phi}{alphax \cdot alphax}\right) - \frac{sin2phi}{alphay \cdot alphay}}} \]

      7. neg-sub0 98.7%

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

      8. associate-/r* 98.8%

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

      9. distribute-neg-frac2 98.8%

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

   3. Simplified 98.8%

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

   5. Taylor expanded in cos2phi around inf 42.3%

      \[\leadsto \color{blue}{-1 \cdot \frac{{alphax}^{2} \cdot \log \left(1 - u0\right)}{cos2phi}} \]
   6. Step-by-step derivation

      1. mul-1-neg 42.3%

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

      2. associate-/l* 42.2%

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

      3. distribute-rgt-neg-in 42.2%

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

      4. distribute-neg-frac2 42.2%

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

      5. sub-neg 42.2%

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

      6. log1p-define 79.7%

         \[\leadsto {alphax}^{2} \cdot \frac{\color{blue}{\mathsf{log1p}\left(-u0\right)}}{-cos2phi} \]

   7. Simplified 79.7%

      \[\leadsto \color{blue}{{alphax}^{2} \cdot \frac{\mathsf{log1p}\left(-u0\right)}{-cos2phi}} \]

   8. Taylor expanded in u0 around 0 65.2%

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

      1. associate-/l* 65.2%

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

   10. Simplified 65.2%

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

   if 2.00000006e-22 < sin2phi

   1. Initial program 62.7%

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

      1. distribute-frac-neg 62.7%

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

      2. distribute-neg-frac2 62.7%

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

      3. sub-neg 62.7%

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

      4. log1p-define 98.0%

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

      5. neg-sub0 98.0%

         \[\leadsto \frac{\mathsf{log1p}\left(-u0\right)}{\color{blue}{0 - \left(\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\right)}} \]

      6. associate--r+ 98.0%

         \[\leadsto \frac{\mathsf{log1p}\left(-u0\right)}{\color{blue}{\left(0 - \frac{cos2phi}{alphax \cdot alphax}\right) - \frac{sin2phi}{alphay \cdot alphay}}} \]

      7. neg-sub0 98.0%

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

      8. associate-/r* 98.0%

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

      9. distribute-neg-frac2 98.0%

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

   3. Simplified 98.0%

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

      1. frac-2neg 98.0%

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

      2. frac-2neg 98.0%

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

      3. add-sqr-sqrt -0.0%

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

      4. sqrt-unprod 87.3%

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

      5. sqr-neg 87.3%

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

      6. sqrt-prod 87.3%

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

      7. add-sqr-sqrt 87.3%

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

      8. associate-/r* 87.3%

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

      9. frac-sub 87.1%

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

   6. Applied egg-rr 87.1%

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

      1. associate-*l/ 87.1%

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

   8. Simplified 87.1%

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

   9. Taylor expanded in u0 around 0 82.9%

      \[\leadsto \color{blue}{u0 \cdot \left(-1 \cdot \frac{alphax \cdot alphay}{\frac{alphay \cdot cos2phi}{alphax} - \frac{alphax \cdot sin2phi}{alphay}} + u0 \cdot \left(-0.5 \cdot \frac{alphax \cdot alphay}{\frac{alphay \cdot cos2phi}{alphax} - \frac{alphax \cdot sin2phi}{alphay}} + u0 \cdot \left(-0.3333333333333333 \cdot \frac{alphax \cdot alphay}{\frac{alphay \cdot cos2phi}{alphax} - \frac{alphax \cdot sin2phi}{alphay}} + -0.25 \cdot \frac{alphax \cdot \left(alphay \cdot u0\right)}{\frac{alphay \cdot cos2phi}{alphax} - \frac{alphax \cdot sin2phi}{alphay}}\right)\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 79.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;sin2phi \leq 2.00000006274879 \cdot 10^{-22}:\\ \;\;\;\;{alphax}^{2} \cdot \frac{u0}{cos2phi}\\ \mathbf{else}:\\ \;\;\;\;u0 \cdot \left(\frac{alphax \cdot alphay}{\frac{alphax \cdot sin2phi}{alphay} - \frac{cos2phi \cdot alphay}{alphax}} + u0 \cdot \left(\frac{alphax \cdot alphay}{\frac{cos2phi \cdot alphay}{alphax} - \frac{alphax \cdot sin2phi}{alphay}} \cdot -0.5 + u0 \cdot \left(\frac{alphax \cdot alphay}{\frac{cos2phi \cdot alphay}{alphax} - \frac{alphax \cdot sin2phi}{alphay}} \cdot -0.3333333333333333 + -0.25 \cdot \frac{alphax \cdot \left(u0 \cdot alphay\right)}{\frac{cos2phi \cdot alphay}{alphax} - \frac{alphax \cdot sin2phi}{alphay}}\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 68.0% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{cos2phi \cdot alphay}{alphax}\\ t_1 := \frac{alphax \cdot sin2phi}{alphay}\\ t_2 := t\_0 - t\_1\\ t_3 := \frac{alphax \cdot alphay}{t\_2}\\ u0 \cdot \left(\frac{alphax \cdot alphay}{t\_1 - t\_0} + u0 \cdot \left(t\_3 \cdot -0.5 + u0 \cdot \left(t\_3 \cdot -0.3333333333333333 + -0.25 \cdot \frac{alphax \cdot \left(u0 \cdot alphay\right)}{t\_2}\right)\right)\right) \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (let* ((t_0 (/ (* cos2phi alphay) alphax))
        (t_1 (/ (* alphax sin2phi) alphay))
        (t_2 (- t_0 t_1))
        (t_3 (/ (* alphax alphay) t_2)))
   (*
    u0
    (+
     (/ (* alphax alphay) (- t_1 t_0))
     (*
      u0
      (+
       (* t_3 -0.5)
       (*
        u0
        (+
         (* t_3 -0.3333333333333333)
         (* -0.25 (/ (* alphax (* u0 alphay)) t_2))))))))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float t_0 = (cos2phi * alphay) / alphax;
	float t_1 = (alphax * sin2phi) / alphay;
	float t_2 = t_0 - t_1;
	float t_3 = (alphax * alphay) / t_2;
	return u0 * (((alphax * alphay) / (t_1 - t_0)) + (u0 * ((t_3 * -0.5f) + (u0 * ((t_3 * -0.3333333333333333f) + (-0.25f * ((alphax * (u0 * alphay)) / t_2)))))));
}

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) :: t_1
    real(4) :: t_2
    real(4) :: t_3
    t_0 = (cos2phi * alphay) / alphax
    t_1 = (alphax * sin2phi) / alphay
    t_2 = t_0 - t_1
    t_3 = (alphax * alphay) / t_2
    code = u0 * (((alphax * alphay) / (t_1 - t_0)) + (u0 * ((t_3 * (-0.5e0)) + (u0 * ((t_3 * (-0.3333333333333333e0)) + ((-0.25e0) * ((alphax * (u0 * alphay)) / t_2)))))))
end function

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	t_0 = Float32(Float32(cos2phi * alphay) / alphax)
	t_1 = Float32(Float32(alphax * sin2phi) / alphay)
	t_2 = Float32(t_0 - t_1)
	t_3 = Float32(Float32(alphax * alphay) / t_2)
	return Float32(u0 * Float32(Float32(Float32(alphax * alphay) / Float32(t_1 - t_0)) + Float32(u0 * Float32(Float32(t_3 * Float32(-0.5)) + Float32(u0 * Float32(Float32(t_3 * Float32(-0.3333333333333333)) + Float32(Float32(-0.25) * Float32(Float32(alphax * Float32(u0 * alphay)) / t_2))))))))
end

MATLAB

function tmp = code(alphax, alphay, u0, cos2phi, sin2phi)
	t_0 = (cos2phi * alphay) / alphax;
	t_1 = (alphax * sin2phi) / alphay;
	t_2 = t_0 - t_1;
	t_3 = (alphax * alphay) / t_2;
	tmp = u0 * (((alphax * alphay) / (t_1 - t_0)) + (u0 * ((t_3 * single(-0.5)) + (u0 * ((t_3 * single(-0.3333333333333333)) + (single(-0.25) * ((alphax * (u0 * alphay)) / t_2)))))));
end

Derivation

1. Initial program 60.5%

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

   1. distribute-frac-neg 60.5%

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

   2. distribute-neg-frac2 60.5%

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

   3. sub-neg 60.5%

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

   4. log1p-define 98.1%

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

   5. neg-sub0 98.1%

      \[\leadsto \frac{\mathsf{log1p}\left(-u0\right)}{\color{blue}{0 - \left(\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\right)}} \]

   6. associate--r+ 98.1%

      \[\leadsto \frac{\mathsf{log1p}\left(-u0\right)}{\color{blue}{\left(0 - \frac{cos2phi}{alphax \cdot alphax}\right) - \frac{sin2phi}{alphay \cdot alphay}}} \]

   7. neg-sub0 98.1%

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

   8. associate-/r* 98.2%

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

   9. distribute-neg-frac2 98.2%

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

3. Simplified 98.2%

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

   1. frac-2neg 98.2%

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

   2. frac-2neg 98.2%

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

   3. add-sqr-sqrt -0.0%

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

   4. sqrt-unprod 73.0%

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

   5. sqr-neg 73.0%

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

   6. sqrt-prod 73.0%

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

   7. add-sqr-sqrt 73.0%

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

   8. associate-/r* 72.9%

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

   9. frac-sub 72.7%

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

6. Applied egg-rr 72.7%

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

   1. associate-*l/ 72.7%

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

8. Simplified 72.7%

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

9. Taylor expanded in u0 around 0 69.2%

   \[\leadsto \color{blue}{u0 \cdot \left(-1 \cdot \frac{alphax \cdot alphay}{\frac{alphay \cdot cos2phi}{alphax} - \frac{alphax \cdot sin2phi}{alphay}} + u0 \cdot \left(-0.5 \cdot \frac{alphax \cdot alphay}{\frac{alphay \cdot cos2phi}{alphax} - \frac{alphax \cdot sin2phi}{alphay}} + u0 \cdot \left(-0.3333333333333333 \cdot \frac{alphax \cdot alphay}{\frac{alphay \cdot cos2phi}{alphax} - \frac{alphax \cdot sin2phi}{alphay}} + -0.25 \cdot \frac{alphax \cdot \left(alphay \cdot u0\right)}{\frac{alphay \cdot cos2phi}{alphax} - \frac{alphax \cdot sin2phi}{alphay}}\right)\right)\right)} \]

10. Final simplification 69.2%

    \[\leadsto u0 \cdot \left(\frac{alphax \cdot alphay}{\frac{alphax \cdot sin2phi}{alphay} - \frac{cos2phi \cdot alphay}{alphax}} + u0 \cdot \left(\frac{alphax \cdot alphay}{\frac{cos2phi \cdot alphay}{alphax} - \frac{alphax \cdot sin2phi}{alphay}} \cdot -0.5 + u0 \cdot \left(\frac{alphax \cdot alphay}{\frac{cos2phi \cdot alphay}{alphax} - \frac{alphax \cdot sin2phi}{alphay}} \cdot -0.3333333333333333 + -0.25 \cdot \frac{alphax \cdot \left(u0 \cdot alphay\right)}{\frac{cos2phi \cdot alphay}{alphax} - \frac{alphax \cdot sin2phi}{alphay}}\right)\right)\right) \]
11. Add Preprocessing

Alternative 7: 66.7% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{cos2phi \cdot alphay}{alphax}\\ t_1 := \frac{alphax \cdot sin2phi}{alphay}\\ t_2 := t\_0 - t\_1\\ u0 \cdot \left(\frac{alphax \cdot alphay}{t\_1 - t\_0} + u0 \cdot \left(\frac{alphax \cdot alphay}{t\_2} \cdot -0.5 + -0.3333333333333333 \cdot \frac{alphax \cdot \left(u0 \cdot alphay\right)}{t\_2}\right)\right) \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (let* ((t_0 (/ (* cos2phi alphay) alphax))
        (t_1 (/ (* alphax sin2phi) alphay))
        (t_2 (- t_0 t_1)))
   (*
    u0
    (+
     (/ (* alphax alphay) (- t_1 t_0))
     (*
      u0
      (+
       (* (/ (* alphax alphay) t_2) -0.5)
       (* -0.3333333333333333 (/ (* alphax (* u0 alphay)) t_2))))))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float t_0 = (cos2phi * alphay) / alphax;
	float t_1 = (alphax * sin2phi) / alphay;
	float t_2 = t_0 - t_1;
	return u0 * (((alphax * alphay) / (t_1 - t_0)) + (u0 * ((((alphax * alphay) / t_2) * -0.5f) + (-0.3333333333333333f * ((alphax * (u0 * alphay)) / t_2)))));
}

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) :: t_1
    real(4) :: t_2
    t_0 = (cos2phi * alphay) / alphax
    t_1 = (alphax * sin2phi) / alphay
    t_2 = t_0 - t_1
    code = u0 * (((alphax * alphay) / (t_1 - t_0)) + (u0 * ((((alphax * alphay) / t_2) * (-0.5e0)) + ((-0.3333333333333333e0) * ((alphax * (u0 * alphay)) / t_2)))))
end function

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	t_0 = Float32(Float32(cos2phi * alphay) / alphax)
	t_1 = Float32(Float32(alphax * sin2phi) / alphay)
	t_2 = Float32(t_0 - t_1)
	return Float32(u0 * Float32(Float32(Float32(alphax * alphay) / Float32(t_1 - t_0)) + Float32(u0 * Float32(Float32(Float32(Float32(alphax * alphay) / t_2) * Float32(-0.5)) + Float32(Float32(-0.3333333333333333) * Float32(Float32(alphax * Float32(u0 * alphay)) / t_2))))))
end

MATLAB

function tmp = code(alphax, alphay, u0, cos2phi, sin2phi)
	t_0 = (cos2phi * alphay) / alphax;
	t_1 = (alphax * sin2phi) / alphay;
	t_2 = t_0 - t_1;
	tmp = u0 * (((alphax * alphay) / (t_1 - t_0)) + (u0 * ((((alphax * alphay) / t_2) * single(-0.5)) + (single(-0.3333333333333333) * ((alphax * (u0 * alphay)) / t_2)))));
end

Derivation

1. Initial program 60.5%

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

   1. distribute-frac-neg 60.5%

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

   2. distribute-neg-frac2 60.5%

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

   3. sub-neg 60.5%

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

   4. log1p-define 98.1%

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

   5. neg-sub0 98.1%

      \[\leadsto \frac{\mathsf{log1p}\left(-u0\right)}{\color{blue}{0 - \left(\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\right)}} \]

   6. associate--r+ 98.1%

      \[\leadsto \frac{\mathsf{log1p}\left(-u0\right)}{\color{blue}{\left(0 - \frac{cos2phi}{alphax \cdot alphax}\right) - \frac{sin2phi}{alphay \cdot alphay}}} \]

   7. neg-sub0 98.1%

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

   8. associate-/r* 98.2%

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

   9. distribute-neg-frac2 98.2%

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

3. Simplified 98.2%

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

   1. frac-2neg 98.2%

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

   2. frac-2neg 98.2%

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

   3. add-sqr-sqrt -0.0%

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

   4. sqrt-unprod 73.0%

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

   5. sqr-neg 73.0%

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

   6. sqrt-prod 73.0%

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

   7. add-sqr-sqrt 73.0%

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

   8. associate-/r* 72.9%

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

   9. frac-sub 72.7%

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

6. Applied egg-rr 72.7%

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

   1. associate-*l/ 72.7%

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

8. Simplified 72.7%

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

9. Taylor expanded in u0 around 0 67.7%

   \[\leadsto \color{blue}{u0 \cdot \left(-1 \cdot \frac{alphax \cdot alphay}{\frac{alphay \cdot cos2phi}{alphax} - \frac{alphax \cdot sin2phi}{alphay}} + u0 \cdot \left(-0.5 \cdot \frac{alphax \cdot alphay}{\frac{alphay \cdot cos2phi}{alphax} - \frac{alphax \cdot sin2phi}{alphay}} + -0.3333333333333333 \cdot \frac{alphax \cdot \left(alphay \cdot u0\right)}{\frac{alphay \cdot cos2phi}{alphax} - \frac{alphax \cdot sin2phi}{alphay}}\right)\right)} \]

10. Final simplification 67.7%

    \[\leadsto u0 \cdot \left(\frac{alphax \cdot alphay}{\frac{alphax \cdot sin2phi}{alphay} - \frac{cos2phi \cdot alphay}{alphax}} + u0 \cdot \left(\frac{alphax \cdot alphay}{\frac{cos2phi \cdot alphay}{alphax} - \frac{alphax \cdot sin2phi}{alphay}} \cdot -0.5 + -0.3333333333333333 \cdot \frac{alphax \cdot \left(u0 \cdot alphay\right)}{\frac{cos2phi \cdot alphay}{alphax} - \frac{alphax \cdot sin2phi}{alphay}}\right)\right) \]
11. Add Preprocessing

Alternative 8: 64.1% accurate, 3.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := cos2phi \cdot \frac{alphay}{alphax}\\ t_1 := \frac{alphax \cdot sin2phi}{alphay}\\ u0 \cdot \left(-0.5 \cdot \left(alphax \cdot \frac{u0 \cdot alphay}{t\_0 - t\_1}\right) + alphax \cdot \frac{alphay}{t\_1 - t\_0}\right) \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (let* ((t_0 (* cos2phi (/ alphay alphax)))
        (t_1 (/ (* alphax sin2phi) alphay)))
   (*
    u0
    (+
     (* -0.5 (* alphax (/ (* u0 alphay) (- t_0 t_1))))
     (* alphax (/ alphay (- t_1 t_0)))))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float t_0 = cos2phi * (alphay / alphax);
	float t_1 = (alphax * sin2phi) / alphay;
	return u0 * ((-0.5f * (alphax * ((u0 * alphay) / (t_0 - t_1)))) + (alphax * (alphay / (t_1 - t_0))));
}

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) :: t_1
    t_0 = cos2phi * (alphay / alphax)
    t_1 = (alphax * sin2phi) / alphay
    code = u0 * (((-0.5e0) * (alphax * ((u0 * alphay) / (t_0 - t_1)))) + (alphax * (alphay / (t_1 - t_0))))
end function

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	t_0 = Float32(cos2phi * Float32(alphay / alphax))
	t_1 = Float32(Float32(alphax * sin2phi) / alphay)
	return Float32(u0 * Float32(Float32(Float32(-0.5) * Float32(alphax * Float32(Float32(u0 * alphay) / Float32(t_0 - t_1)))) + Float32(alphax * Float32(alphay / Float32(t_1 - t_0)))))
end

MATLAB

function tmp = code(alphax, alphay, u0, cos2phi, sin2phi)
	t_0 = cos2phi * (alphay / alphax);
	t_1 = (alphax * sin2phi) / alphay;
	tmp = u0 * ((single(-0.5) * (alphax * ((u0 * alphay) / (t_0 - t_1)))) + (alphax * (alphay / (t_1 - t_0))));
end

Derivation

1. Initial program 60.5%

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

   1. distribute-frac-neg 60.5%

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

   2. distribute-neg-frac2 60.5%

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

   3. sub-neg 60.5%

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

   4. log1p-define 98.1%

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

   5. neg-sub0 98.1%

      \[\leadsto \frac{\mathsf{log1p}\left(-u0\right)}{\color{blue}{0 - \left(\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\right)}} \]

   6. associate--r+ 98.1%

      \[\leadsto \frac{\mathsf{log1p}\left(-u0\right)}{\color{blue}{\left(0 - \frac{cos2phi}{alphax \cdot alphax}\right) - \frac{sin2phi}{alphay \cdot alphay}}} \]

   7. neg-sub0 98.1%

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

   8. associate-/r* 98.2%

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

   9. distribute-neg-frac2 98.2%

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

3. Simplified 98.2%

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

   1. frac-2neg 98.2%

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

   2. frac-2neg 98.2%

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

   3. add-sqr-sqrt -0.0%

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

   4. sqrt-unprod 73.0%

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

   5. sqr-neg 73.0%

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

   6. sqrt-prod 73.0%

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

   7. add-sqr-sqrt 73.0%

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

   8. associate-/r* 72.9%

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

   9. frac-sub 72.7%

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

6. Applied egg-rr 72.7%

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

   1. associate-*l/ 72.7%

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

8. Simplified 72.7%

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

9. Taylor expanded in u0 around 0 65.1%

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

    1. +-commutative 65.1%

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

    2. mul-1-neg 65.1%

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

    3. unsub-neg 65.1%

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

    4. associate-/l* 65.1%

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

    5. *-commutative 65.1%

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

    6. *-commutative 65.1%

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

    7. associate-*r/ 65.1%

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

    8. associate-/l* 65.1%

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

11. Simplified 65.1%

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

12. Final simplification 65.1%

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

Alternative 9: 56.2% accurate, 6.8× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 60.5%

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

   1. distribute-frac-neg 60.5%

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

   2. distribute-neg-frac2 60.5%

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

   3. sub-neg 60.5%

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

   4. log1p-define 98.1%

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

   5. neg-sub0 98.1%

      \[\leadsto \frac{\mathsf{log1p}\left(-u0\right)}{\color{blue}{0 - \left(\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\right)}} \]

   6. associate--r+ 98.1%

      \[\leadsto \frac{\mathsf{log1p}\left(-u0\right)}{\color{blue}{\left(0 - \frac{cos2phi}{alphax \cdot alphax}\right) - \frac{sin2phi}{alphay \cdot alphay}}} \]

   7. neg-sub0 98.1%

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

   8. associate-/r* 98.2%

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

   9. distribute-neg-frac2 98.2%

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

3. Simplified 98.2%

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

   1. frac-2neg 98.2%

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

   2. frac-2neg 98.2%

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

   3. add-sqr-sqrt -0.0%

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

   4. sqrt-unprod 73.0%

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

   5. sqr-neg 73.0%

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

   6. sqrt-prod 73.0%

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

   7. add-sqr-sqrt 73.0%

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

   8. associate-/r* 72.9%

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

   9. frac-sub 72.7%

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

6. Applied egg-rr 72.7%

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

   1. associate-*l/ 72.7%

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

8. Simplified 72.7%

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

9. Taylor expanded in u0 around 0 56.9%

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

    1. mul-1-neg 56.9%

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

    2. *-commutative 56.9%

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

    3. *-commutative 56.9%

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

    4. associate-*r/ 56.9%

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

11. Simplified 56.9%

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

12. Final simplification 56.9%

    \[\leadsto \frac{alphax \cdot \left(u0 \cdot alphay\right)}{alphax \cdot \frac{sin2phi}{alphay} - \frac{cos2phi \cdot alphay}{alphax}} \]
13. Add Preprocessing

Alternative 10: 56.3% accurate, 6.8× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 60.5%

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

   1. distribute-frac-neg 60.5%

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

   2. distribute-neg-frac2 60.5%

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

   3. sub-neg 60.5%

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

   4. log1p-define 98.1%

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

   5. neg-sub0 98.1%

      \[\leadsto \frac{\mathsf{log1p}\left(-u0\right)}{\color{blue}{0 - \left(\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\right)}} \]

   6. associate--r+ 98.1%

      \[\leadsto \frac{\mathsf{log1p}\left(-u0\right)}{\color{blue}{\left(0 - \frac{cos2phi}{alphax \cdot alphax}\right) - \frac{sin2phi}{alphay \cdot alphay}}} \]

   7. neg-sub0 98.1%

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

   8. associate-/r* 98.2%

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

   9. distribute-neg-frac2 98.2%

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

3. Simplified 98.2%

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

   1. frac-2neg 98.2%

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

   2. frac-2neg 98.2%

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

   3. add-sqr-sqrt -0.0%

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

   4. sqrt-unprod 73.0%

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

   5. sqr-neg 73.0%

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

   6. sqrt-prod 73.0%

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

   7. add-sqr-sqrt 73.0%

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

   8. associate-/r* 72.9%

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

   9. frac-sub 72.7%

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

6. Applied egg-rr 72.7%

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

   1. associate-*l/ 72.7%

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

8. Simplified 72.7%

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

9. Taylor expanded in u0 around 0 56.9%

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

    1. associate-*r/ 56.9%

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

    2. mul-1-neg 56.9%

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

    3. *-commutative 56.9%

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

    4. *-commutative 56.9%

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

    5. associate-*r/ 56.9%

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

11. Simplified 56.9%

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

12. Final simplification 56.9%

    \[\leadsto \frac{alphax \cdot \left(u0 \cdot alphay\right)}{\frac{alphax \cdot sin2phi}{alphay} - cos2phi \cdot \frac{alphay}{alphax}} \]
13. Add Preprocessing

Reproduce

herbie shell --seed 2024096 
(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)))))