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

Percentage Accurate: 60.2% → 98.3%

Time: 21.1s

Alternatives: 16

Speedup: 8.9×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Initial Program: 60.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Alternative 1: 98.3% accurate, 1.0× speedup

Math

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

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 / Float32(-alphax)) / alphax) - Float32(sin2phi / Float32(alphay * alphay))))
end

Derivation

1. Initial program 61.4%

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

   1. distribute-frac-neg 61.4%

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

   2. distribute-neg-frac2 61.4%

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

   3. sub-neg 61.4%

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

   4. log1p-define 98.3%

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

   5. neg-sub0 98.3%

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

   6. associate--r+ 98.3%

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

   7. neg-sub0 98.3%

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

   8. associate-/r* 98.3%

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

   9. distribute-neg-frac2 98.3%

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

3. Simplified 98.3%

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

5. Final simplification 98.3%

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

Alternative 2: 93.2% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

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) + ((alphax * sin2phi) / alphay)
    t_1 = (alphax * alphay) / t_0
    code = u0 * (t_1 - (u0 * ((u0 * (((-0.25e0) * ((alphax * (u0 * alphay)) / t_0)) + (t_1 * (-0.3333333333333333e0)))) + (t_1 * (-0.5e0)))))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 61.4%

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

   1. distribute-frac-neg 61.4%

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

   2. distribute-neg-frac2 61.4%

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

   3. sub-neg 61.4%

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

   4. log1p-define 98.3%

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

   5. neg-sub0 98.3%

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

   6. associate--r+ 98.3%

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

   7. neg-sub0 98.3%

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

   8. associate-/r* 98.3%

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

   9. distribute-neg-frac2 98.3%

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

3. Simplified 98.3%

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

   1. associate-/r* 98.3%

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

   2. div-inv 98.2%

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

6. Applied egg-rr 98.2%

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

   1. associate-*r/ 98.3%

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

   2. *-rgt-identity 98.3%

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

8. Simplified 98.3%

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

   1. frac-2neg 98.3%

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

   2. frac-sub 97.9%

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

   3. add-sqr-sqrt -0.0%

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

   4. sqrt-unprod 38.2%

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

   5. sqr-neg 38.2%

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

   6. sqrt-prod 38.2%

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

   7. add-sqr-sqrt 38.2%

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

   8. add-sqr-sqrt -0.0%

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

   9. sqrt-unprod 97.9%

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

   10. sqr-neg 97.9%

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

   11. sqrt-prod 97.6%

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

   12. add-sqr-sqrt 97.9%

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

10. Applied egg-rr 97.9%

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

11. Taylor expanded in u0 around 0 94.3%

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

12. Final simplification 94.3%

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

Alternative 3: 92.7% accurate, 4.0× speedup

Math

\[\begin{array}{l} \\ \frac{u0 \cdot \left(-1 + u0 \cdot \left(u0 \cdot \left(u0 \cdot -0.25 - 0.3333333333333333\right) - 0.5\right)\right)}{\frac{-1}{alphay \cdot \frac{alphay}{sin2phi}} - \frac{\frac{cos2phi}{alphax}}{alphax}} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (/
  (* u0 (+ -1.0 (* u0 (- (* u0 (- (* u0 -0.25) 0.3333333333333333)) 0.5))))
  (- (/ -1.0 (* alphay (/ alphay sin2phi))) (/ (/ cos2phi alphax) alphax))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	return (u0 * (-1.0f + (u0 * ((u0 * ((u0 * -0.25f) - 0.3333333333333333f)) - 0.5f)))) / ((-1.0f / (alphay * (alphay / sin2phi))) - ((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 * ((-1.0e0) + (u0 * ((u0 * ((u0 * (-0.25e0)) - 0.3333333333333333e0)) - 0.5e0)))) / (((-1.0e0) / (alphay * (alphay / sin2phi))) - ((cos2phi / alphax) / alphax))
end function

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	return Float32(Float32(u0 * Float32(Float32(-1.0) + Float32(u0 * Float32(Float32(u0 * Float32(Float32(u0 * Float32(-0.25)) - Float32(0.3333333333333333))) - Float32(0.5))))) / Float32(Float32(Float32(-1.0) / Float32(alphay * Float32(alphay / sin2phi))) - Float32(Float32(cos2phi / alphax) / alphax)))
end

MATLAB

function tmp = code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = (u0 * (single(-1.0) + (u0 * ((u0 * ((u0 * single(-0.25)) - single(0.3333333333333333))) - single(0.5))))) / ((single(-1.0) / (alphay * (alphay / sin2phi))) - ((cos2phi / alphax) / alphax));
end

Derivation

1. Initial program 61.4%

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

   1. distribute-frac-neg 61.4%

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

   2. distribute-neg-frac2 61.4%

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

   3. sub-neg 61.4%

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

   4. log1p-define 98.3%

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

   5. neg-sub0 98.3%

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

   6. associate--r+ 98.3%

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

   7. neg-sub0 98.3%

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

   8. associate-/r* 98.3%

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

   9. distribute-neg-frac2 98.3%

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

3. Simplified 98.3%

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

   1. associate-/r* 98.3%

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

   2. div-inv 98.2%

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

6. Applied egg-rr 98.2%

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

   1. associate-*r/ 98.3%

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

   2. *-rgt-identity 98.3%

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

8. Simplified 98.3%

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

9. Taylor expanded in u0 around 0 93.8%

   \[\leadsto \frac{\color{blue}{u0 \cdot \left(u0 \cdot \left(u0 \cdot \left(-0.25 \cdot u0 - 0.3333333333333333\right) - 0.5\right) - 1\right)}}{\frac{\frac{cos2phi}{alphax}}{-alphax} - \frac{\frac{sin2phi}{alphay}}{alphay}} \]
10. Step-by-step derivation

    1. clear-num 93.8%

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

    2. inv-pow 93.8%

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

11. Applied egg-rr 93.8%

    \[\leadsto \frac{u0 \cdot \left(u0 \cdot \left(u0 \cdot \left(-0.25 \cdot u0 - 0.3333333333333333\right) - 0.5\right) - 1\right)}{\frac{\frac{cos2phi}{alphax}}{-alphax} - \color{blue}{{\left(\frac{alphay}{\frac{sin2phi}{alphay}}\right)}^{-1}}} \]
12. Step-by-step derivation

    1. unpow-1 93.8%

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

    2. associate-/r/ 93.9%

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

13. Simplified 93.9%

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

14. Final simplification 93.9%

    \[\leadsto \frac{u0 \cdot \left(-1 + u0 \cdot \left(u0 \cdot \left(u0 \cdot -0.25 - 0.3333333333333333\right) - 0.5\right)\right)}{\frac{-1}{alphay \cdot \frac{alphay}{sin2phi}} - \frac{\frac{cos2phi}{alphax}}{alphax}} \]
15. Add Preprocessing

Alternative 4: 81.0% accurate, 4.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 20:\\ \;\;\;\;\frac{u0}{\frac{cos2phi}{alphax \cdot alphax} + sin2phi \cdot \frac{\frac{1}{alphay}}{alphay}}\\ \mathbf{else}:\\ \;\;\;\;\frac{u0 \cdot \left(-1 + u0 \cdot -0.5\right)}{\frac{\frac{cos2phi}{alphax}}{alphax} - \frac{\frac{sin2phi}{alphay}}{alphay}}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (if (<= (/ sin2phi (* alphay alphay)) 20.0)
   (/
    u0
    (+ (/ cos2phi (* alphax alphax)) (* sin2phi (/ (/ 1.0 alphay) alphay))))
   (/
    (* u0 (+ -1.0 (* u0 -0.5)))
    (- (/ (/ cos2phi alphax) alphax) (/ (/ sin2phi alphay) alphay)))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float tmp;
	if ((sin2phi / (alphay * alphay)) <= 20.0f) {
		tmp = u0 / ((cos2phi / (alphax * alphax)) + (sin2phi * ((1.0f / alphay) / alphay)));
	} else {
		tmp = (u0 * (-1.0f + (u0 * -0.5f))) / (((cos2phi / alphax) / alphax) - ((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 / (alphay * alphay)) <= 20.0e0) then
        tmp = u0 / ((cos2phi / (alphax * alphax)) + (sin2phi * ((1.0e0 / alphay) / alphay)))
    else
        tmp = (u0 * ((-1.0e0) + (u0 * (-0.5e0)))) / (((cos2phi / alphax) / alphax) - ((sin2phi / alphay) / alphay))
    end if
    code = tmp
end function

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = Float32(0.0)
	if (Float32(sin2phi / Float32(alphay * alphay)) <= Float32(20.0))
		tmp = Float32(u0 / Float32(Float32(cos2phi / Float32(alphax * alphax)) + Float32(sin2phi * Float32(Float32(Float32(1.0) / alphay) / alphay))));
	else
		tmp = Float32(Float32(u0 * Float32(Float32(-1.0) + Float32(u0 * Float32(-0.5)))) / Float32(Float32(Float32(cos2phi / alphax) / alphax) - Float32(Float32(sin2phi / alphay) / alphay)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = single(0.0);
	if ((sin2phi / (alphay * alphay)) <= single(20.0))
		tmp = u0 / ((cos2phi / (alphax * alphax)) + (sin2phi * ((single(1.0) / alphay) / alphay)));
	else
		tmp = (u0 * (single(-1.0) + (u0 * single(-0.5)))) / (((cos2phi / alphax) / alphax) - ((sin2phi / alphay) / alphay));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 sin2phi (*.f32 alphay alphay)) < 20

   1. Initial program 51.6%

      \[\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
   2. Add Preprocessing

   3. Taylor expanded in u0 around 0 77.0%

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

      1. mul-1-neg 77.0%

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

   5. Simplified 77.0%

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

      1. associate-/r* 98.8%

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

      2. div-inv 98.8%

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

   7. Applied egg-rr 77.0%

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

      1. associate-*r/ 98.8%

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

      2. *-rgt-identity 98.8%

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

   9. Simplified 77.0%

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

       1. div-inv 76.9%

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

       2. *-un-lft-identity 76.9%

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

       3. times-frac 77.1%

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

   11. Applied egg-rr 77.1%

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

   if 20 < (/.f32 sin2phi (*.f32 alphay alphay))

   1. Initial program 69.7%

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

      1. distribute-frac-neg 69.7%

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

      2. distribute-neg-frac2 69.7%

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

      3. sub-neg 69.7%

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

      4. log1p-define 97.9%

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

      5. neg-sub0 97.9%

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

      6. associate--r+ 97.9%

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

      7. neg-sub0 97.9%

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

      8. associate-/r* 97.9%

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

      9. distribute-neg-frac2 97.9%

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

   3. Simplified 97.9%

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

      1. associate-/r* 97.8%

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

      2. div-inv 97.6%

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

   6. Applied egg-rr 97.6%

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

      1. associate-*r/ 97.8%

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

      2. *-rgt-identity 97.8%

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

   8. Simplified 97.8%

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

   9. Taylor expanded in u0 around 0 86.2%

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

       1. add-sqr-sqrt -0.0%

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

       2. sqrt-unprod 85.2%

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

       3. sqr-neg 85.2%

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

       4. sqrt-prod 85.2%

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

       5. add-sqr-sqrt 85.2%

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

       6. div-inv 85.2%

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

   11. Applied egg-rr 85.2%

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

       1. associate-*r/ 85.2%

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

       2. *-rgt-identity 85.2%

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

   13. Simplified 85.2%

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

4. Final simplification 81.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 20:\\ \;\;\;\;\frac{u0}{\frac{cos2phi}{alphax \cdot alphax} + sin2phi \cdot \frac{\frac{1}{alphay}}{alphay}}\\ \mathbf{else}:\\ \;\;\;\;\frac{u0 \cdot \left(-1 + u0 \cdot -0.5\right)}{\frac{\frac{cos2phi}{alphax}}{alphax} - \frac{\frac{sin2phi}{alphay}}{alphay}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 92.8% accurate, 4.3× speedup

Math

\[\begin{array}{l} \\ \frac{u0 \cdot \left(1 + u0 \cdot \left(0.5 + u0 \cdot \left(0.3333333333333333 - u0 \cdot -0.25\right)\right)\right)}{\frac{\frac{sin2phi}{alphay}}{alphay} + \frac{\frac{cos2phi}{alphax}}{alphax}} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (/
  (* u0 (+ 1.0 (* u0 (+ 0.5 (* u0 (- 0.3333333333333333 (* u0 -0.25)))))))
  (+ (/ (/ sin2phi alphay) alphay) (/ (/ cos2phi alphax) alphax))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	return (u0 * (1.0f + (u0 * (0.5f + (u0 * (0.3333333333333333f - (u0 * -0.25f))))))) / (((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 * (1.0e0 + (u0 * (0.5e0 + (u0 * (0.3333333333333333e0 - (u0 * (-0.25e0)))))))) / (((sin2phi / alphay) / alphay) + ((cos2phi / alphax) / alphax))
end function

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	return Float32(Float32(u0 * Float32(Float32(1.0) + Float32(u0 * Float32(Float32(0.5) + Float32(u0 * Float32(Float32(0.3333333333333333) - Float32(u0 * Float32(-0.25)))))))) / Float32(Float32(Float32(sin2phi / alphay) / alphay) + Float32(Float32(cos2phi / alphax) / alphax)))
end

MATLAB

function tmp = code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = (u0 * (single(1.0) + (u0 * (single(0.5) + (u0 * (single(0.3333333333333333) - (u0 * single(-0.25)))))))) / (((sin2phi / alphay) / alphay) + ((cos2phi / alphax) / alphax));
end

Derivation

1. Initial program 61.4%

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

   1. distribute-frac-neg 61.4%

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

   2. distribute-neg-frac2 61.4%

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

   3. sub-neg 61.4%

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

   4. log1p-define 98.3%

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

   5. neg-sub0 98.3%

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

   6. associate--r+ 98.3%

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

   7. neg-sub0 98.3%

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

   8. associate-/r* 98.3%

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

   9. distribute-neg-frac2 98.3%

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

3. Simplified 98.3%

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

   1. associate-/r* 98.3%

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

   2. div-inv 98.2%

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

6. Applied egg-rr 98.2%

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

   1. associate-*r/ 98.3%

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

   2. *-rgt-identity 98.3%

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

8. Simplified 98.3%

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

9. Taylor expanded in u0 around 0 93.8%

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

10. Final simplification 93.8%

    \[\leadsto \frac{u0 \cdot \left(1 + u0 \cdot \left(0.5 + u0 \cdot \left(0.3333333333333333 - u0 \cdot -0.25\right)\right)\right)}{\frac{\frac{sin2phi}{alphay}}{alphay} + \frac{\frac{cos2phi}{alphax}}{alphax}} \]
11. Add Preprocessing

Alternative 6: 92.8% accurate, 4.3× speedup

Math

\[\begin{array}{l} \\ \frac{u0 \cdot \left(1 + u0 \cdot \left(0.5 + u0 \cdot \left(0.3333333333333333 - u0 \cdot -0.25\right)\right)\right)}{\frac{sin2phi}{alphay \cdot alphay} + \frac{cos2phi}{alphax \cdot alphax}} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (/
  (* u0 (+ 1.0 (* u0 (+ 0.5 (* u0 (- 0.3333333333333333 (* u0 -0.25)))))))
  (+ (/ sin2phi (* alphay alphay)) (/ cos2phi (* alphax alphax)))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	return (u0 * (1.0f + (u0 * (0.5f + (u0 * (0.3333333333333333f - (u0 * -0.25f))))))) / ((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 * (1.0e0 + (u0 * (0.5e0 + (u0 * (0.3333333333333333e0 - (u0 * (-0.25e0)))))))) / ((sin2phi / (alphay * alphay)) + (cos2phi / (alphax * alphax)))
end function

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	return Float32(Float32(u0 * Float32(Float32(1.0) + Float32(u0 * Float32(Float32(0.5) + Float32(u0 * Float32(Float32(0.3333333333333333) - Float32(u0 * Float32(-0.25)))))))) / Float32(Float32(sin2phi / Float32(alphay * alphay)) + Float32(cos2phi / Float32(alphax * alphax))))
end

MATLAB

function tmp = code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = (u0 * (single(1.0) + (u0 * (single(0.5) + (u0 * (single(0.3333333333333333) - (u0 * single(-0.25)))))))) / ((sin2phi / (alphay * alphay)) + (cos2phi / (alphax * alphax)));
end

Derivation

1. Initial program 61.4%

   \[\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
2. Add Preprocessing

3. Taylor expanded in u0 around 0 93.7%

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

4. Final simplification 93.7%

   \[\leadsto \frac{u0 \cdot \left(1 + u0 \cdot \left(0.5 + u0 \cdot \left(0.3333333333333333 - u0 \cdot -0.25\right)\right)\right)}{\frac{sin2phi}{alphay \cdot alphay} + \frac{cos2phi}{alphax \cdot alphax}} \]
5. Add Preprocessing

Alternative 7: 91.0% accurate, 5.0× speedup

Math

\[\begin{array}{l} \\ \frac{u0 \cdot \left(1 - u0 \cdot \left(u0 \cdot -0.3333333333333333 - 0.5\right)\right)}{\frac{\frac{sin2phi}{alphay}}{alphay} + \frac{\frac{cos2phi}{alphax}}{alphax}} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (/
  (* u0 (- 1.0 (* u0 (- (* u0 -0.3333333333333333) 0.5))))
  (+ (/ (/ sin2phi alphay) alphay) (/ (/ cos2phi alphax) alphax))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	return (u0 * (1.0f - (u0 * ((u0 * -0.3333333333333333f) - 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 * (1.0e0 - (u0 * ((u0 * (-0.3333333333333333e0)) - 0.5e0)))) / (((sin2phi / alphay) / alphay) + ((cos2phi / alphax) / alphax))
end function

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	return Float32(Float32(u0 * Float32(Float32(1.0) - Float32(u0 * Float32(Float32(u0 * Float32(-0.3333333333333333)) - Float32(0.5))))) / Float32(Float32(Float32(sin2phi / alphay) / alphay) + Float32(Float32(cos2phi / alphax) / alphax)))
end

MATLAB

function tmp = code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = (u0 * (single(1.0) - (u0 * ((u0 * single(-0.3333333333333333)) - single(0.5))))) / (((sin2phi / alphay) / alphay) + ((cos2phi / alphax) / alphax));
end

Derivation

1. Initial program 61.4%

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

   1. distribute-frac-neg 61.4%

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

   2. distribute-neg-frac2 61.4%

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

   3. sub-neg 61.4%

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

   4. log1p-define 98.3%

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

   5. neg-sub0 98.3%

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

   6. associate--r+ 98.3%

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

   7. neg-sub0 98.3%

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

   8. associate-/r* 98.3%

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

   9. distribute-neg-frac2 98.3%

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

3. Simplified 98.3%

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

   1. associate-/r* 98.3%

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

   2. div-inv 98.2%

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

6. Applied egg-rr 98.2%

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

   1. associate-*r/ 98.3%

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

   2. *-rgt-identity 98.3%

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

8. Simplified 98.3%

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

9. Taylor expanded in u0 around 0 92.0%

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

10. Final simplification 92.0%

    \[\leadsto \frac{u0 \cdot \left(1 - u0 \cdot \left(u0 \cdot -0.3333333333333333 - 0.5\right)\right)}{\frac{\frac{sin2phi}{alphay}}{alphay} + \frac{\frac{cos2phi}{alphax}}{alphax}} \]
11. Add Preprocessing

Alternative 8: 91.0% accurate, 5.0× speedup

Math

\[\begin{array}{l} \\ \frac{u0 \cdot \left(1 - u0 \cdot \left(u0 \cdot -0.3333333333333333 - 0.5\right)\right)}{\frac{sin2phi}{alphay \cdot alphay} + \frac{cos2phi}{alphax \cdot alphax}} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (/
  (* u0 (- 1.0 (* u0 (- (* u0 -0.3333333333333333) 0.5))))
  (+ (/ sin2phi (* alphay alphay)) (/ cos2phi (* alphax alphax)))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	return (u0 * (1.0f - (u0 * ((u0 * -0.3333333333333333f) - 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 * (1.0e0 - (u0 * ((u0 * (-0.3333333333333333e0)) - 0.5e0)))) / ((sin2phi / (alphay * alphay)) + (cos2phi / (alphax * alphax)))
end function

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	return Float32(Float32(u0 * Float32(Float32(1.0) - Float32(u0 * Float32(Float32(u0 * Float32(-0.3333333333333333)) - Float32(0.5))))) / Float32(Float32(sin2phi / Float32(alphay * alphay)) + Float32(cos2phi / Float32(alphax * alphax))))
end

MATLAB

function tmp = code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = (u0 * (single(1.0) - (u0 * ((u0 * single(-0.3333333333333333)) - single(0.5))))) / ((sin2phi / (alphay * alphay)) + (cos2phi / (alphax * alphax)));
end

Derivation

1. Initial program 61.4%

   \[\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
2. Add Preprocessing

3. Taylor expanded in u0 around 0 91.9%

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

4. Final simplification 91.9%

   \[\leadsto \frac{u0 \cdot \left(1 - u0 \cdot \left(u0 \cdot -0.3333333333333333 - 0.5\right)\right)}{\frac{sin2phi}{alphay \cdot alphay} + \frac{cos2phi}{alphax \cdot alphax}} \]
5. Add Preprocessing

Alternative 9: 87.3% accurate, 6.1× speedup

Math

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

FPCore

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

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	return (u0 * (1.0f - (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 * (1.0e0 - (u0 * (-0.5e0)))) / (((sin2phi / alphay) / alphay) + ((cos2phi / alphax) / alphax))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 61.4%

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

   1. distribute-frac-neg 61.4%

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

   2. distribute-neg-frac2 61.4%

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

   3. sub-neg 61.4%

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

   4. log1p-define 98.3%

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

   5. neg-sub0 98.3%

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

   6. associate--r+ 98.3%

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

   7. neg-sub0 98.3%

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

   8. associate-/r* 98.3%

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

   9. distribute-neg-frac2 98.3%

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

3. Simplified 98.3%

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

   1. associate-/r* 98.3%

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

   2. div-inv 98.2%

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

6. Applied egg-rr 98.2%

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

   1. associate-*r/ 98.3%

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

   2. *-rgt-identity 98.3%

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

8. Simplified 98.3%

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

9. Taylor expanded in u0 around 0 87.9%

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

10. Final simplification 87.9%

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

Alternative 10: 87.3% accurate, 6.1× speedup

Math

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

FPCore

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

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	return (u0 * (1.0f - (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 * (1.0e0 - (u0 * (-0.5e0)))) / ((sin2phi / (alphay * alphay)) + (cos2phi / (alphax * alphax)))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 61.4%

   \[\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
2. Add Preprocessing

3. Taylor expanded in u0 around 0 87.9%

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

4. Final simplification 87.9%

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

Alternative 11: 76.2% accurate, 6.8× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 61.4%

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

   1. distribute-frac-neg 61.4%

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

   2. distribute-neg-frac2 61.4%

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

   3. sub-neg 61.4%

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

   4. log1p-define 98.3%

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

   5. neg-sub0 98.3%

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

   6. associate--r+ 98.3%

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

   7. neg-sub0 98.3%

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

   8. associate-/r* 98.3%

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

   9. distribute-neg-frac2 98.3%

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

3. Simplified 98.3%

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

   1. associate-/r* 98.3%

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

   2. div-inv 98.2%

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

6. Applied egg-rr 98.2%

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

   1. associate-*r/ 98.3%

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

   2. *-rgt-identity 98.3%

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

8. Simplified 98.3%

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

   1. frac-2neg 98.3%

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

   2. frac-sub 97.9%

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

   3. add-sqr-sqrt -0.0%

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

   4. sqrt-unprod 38.2%

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

   5. sqr-neg 38.2%

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

   6. sqrt-prod 38.2%

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

   7. add-sqr-sqrt 38.2%

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

   8. add-sqr-sqrt -0.0%

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

   9. sqrt-unprod 97.9%

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

   10. sqr-neg 97.9%

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

   11. sqrt-prod 97.6%

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

   12. add-sqr-sqrt 97.9%

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

10. Applied egg-rr 97.9%

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

11. Taylor expanded in u0 around 0 75.9%

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

    1. mul-1-neg 75.9%

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

    2. associate-/l* 75.8%

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

    3. distribute-rgt-neg-in 75.8%

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

    4. *-commutative 75.8%

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

    5. mul-1-neg 75.8%

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

    6. associate-/l* 75.9%

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

    7. *-commutative 75.9%

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

    8. distribute-rgt-neg-in 75.9%

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

    9. associate-*r/ 75.7%

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

13. Simplified 75.7%

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

14. Final simplification 75.7%

    \[\leadsto alphax \cdot \frac{u0 \cdot alphay}{alphax \cdot \frac{sin2phi}{alphay} + \frac{cos2phi}{alphax} \cdot alphay} \]
15. Add Preprocessing

Alternative 12: 76.0% accurate, 7.7× speedup

Math

\[\begin{array}{l} \\ \frac{u0}{\frac{1}{alphay \cdot \frac{alphay}{sin2phi}} + \frac{cos2phi}{alphax \cdot alphax}} \end{array} \]

FPCore

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

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	return u0 / ((1.0f / (alphay * (alphay / sin2phi))) + (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 / ((1.0e0 / (alphay * (alphay / sin2phi))) + (cos2phi / (alphax * alphax)))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 61.4%

   \[\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
2. Add Preprocessing

3. Taylor expanded in u0 around 0 75.6%

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

   1. mul-1-neg 75.6%

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

5. Simplified 75.6%

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

   1. associate-/r* 98.3%

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

   2. div-inv 98.2%

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

7. Applied egg-rr 75.6%

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

   1. associate-*r/ 98.3%

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

   2. *-rgt-identity 98.3%

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

9. Simplified 75.6%

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

    1. clear-num 93.8%

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

    2. inv-pow 93.8%

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

11. Applied egg-rr 75.6%

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

    1. unpow-1 93.8%

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

    2. associate-/r/ 93.9%

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

13. Simplified 75.6%

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

14. Final simplification 75.6%

    \[\leadsto \frac{u0}{\frac{1}{alphay \cdot \frac{alphay}{sin2phi}} + \frac{cos2phi}{alphax \cdot alphax}} \]
15. Add Preprocessing

Alternative 13: 76.0% accurate, 8.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 61.4%

   \[\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
2. Add Preprocessing

3. Taylor expanded in u0 around 0 75.6%

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

   1. mul-1-neg 75.6%

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

5. Simplified 75.6%

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

   1. associate-/r* 98.3%

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

   2. div-inv 98.2%

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

7. Applied egg-rr 75.6%

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

   1. associate-*r/ 98.3%

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

   2. *-rgt-identity 98.3%

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

9. Simplified 75.6%

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

10. Final simplification 75.6%

    \[\leadsto \frac{u0}{\frac{\frac{sin2phi}{alphay}}{alphay} + \frac{cos2phi}{alphax \cdot alphax}} \]
11. Add Preprocessing

Alternative 14: 76.0% accurate, 8.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 61.4%

   \[\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
2. Add Preprocessing

3. Taylor expanded in u0 around 0 75.6%

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

   1. mul-1-neg 75.6%

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

5. Simplified 75.6%

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

6. Final simplification 75.6%

   \[\leadsto \frac{u0}{\frac{sin2phi}{alphay \cdot alphay} + \frac{cos2phi}{alphax \cdot alphax}} \]
7. Add Preprocessing

Alternative 15: 76.0% accurate, 8.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 61.4%

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

   1. distribute-frac-neg 61.4%

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

   2. distribute-neg-frac2 61.4%

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

   3. sub-neg 61.4%

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

   4. log1p-define 98.3%

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

   5. neg-sub0 98.3%

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

   6. associate--r+ 98.3%

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

   7. neg-sub0 98.3%

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

   8. associate-/r* 98.3%

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

   9. distribute-neg-frac2 98.3%

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

3. Simplified 98.3%

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

   1. associate-/r* 98.3%

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

   2. div-inv 98.2%

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

6. Applied egg-rr 98.2%

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

   1. associate-*r/ 98.3%

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

   2. *-rgt-identity 98.3%

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

8. Simplified 98.3%

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

9. Taylor expanded in u0 around 0 75.6%

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

    1. mul-1-neg 75.6%

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

11. Simplified 75.6%

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

12. Final simplification 75.6%

    \[\leadsto \frac{u0}{\frac{\frac{sin2phi}{alphay}}{alphay} + \frac{\frac{cos2phi}{alphax}}{alphax}} \]
13. Add Preprocessing

Alternative 16: 55.7% accurate, 8.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 61.4%

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

   1. distribute-frac-neg 61.4%

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

   2. distribute-neg-frac2 61.4%

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

   3. sub-neg 61.4%

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

   4. log1p-define 98.3%

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

   5. neg-sub0 98.3%

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

   6. associate--r+ 98.3%

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

   7. neg-sub0 98.3%

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

   8. associate-/r* 98.3%

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

   9. distribute-neg-frac2 98.3%

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

3. Simplified 98.3%

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

   1. associate-/r* 98.3%

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

   2. div-inv 98.2%

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

6. Applied egg-rr 98.2%

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

   1. associate-*r/ 98.3%

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

   2. *-rgt-identity 98.3%

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

8. Simplified 98.3%

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

9. Taylor expanded in u0 around 0 75.6%

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

    1. mul-1-neg 75.6%

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

11. Simplified 75.6%

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

    1. add-sqr-sqrt -0.0%

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

    2. sqrt-unprod 62.8%

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

    3. sqr-neg 62.8%

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

    4. sqrt-prod 62.8%

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

    5. add-sqr-sqrt 62.8%

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

    6. div-inv 62.8%

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

13. Applied egg-rr 55.2%

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

    1. associate-*r/ 62.8%

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

    2. *-rgt-identity 62.8%

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

15. Simplified 55.2%

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

16. Final simplification 55.2%

    \[\leadsto \frac{u0}{\frac{\frac{sin2phi}{alphay}}{alphay} - \frac{\frac{cos2phi}{alphax}}{alphax}} \]
17. Add Preprocessing

Reproduce

herbie shell --seed 2024087 
(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)))))