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

Percentage Accurate: 60.7% → 98.4%

Time: 20.6s

Alternatives: 12

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: 60.7% 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

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

FPCore

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

C

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

Julia

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

Derivation

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 98.1%

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

   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* 98.1%

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

   9. distribute-neg-frac2 98.1%

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

3. Simplified 98.1%

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

   1. distribute-frac-neg2 98.1%

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

   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. div-inv 98.1%

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

   4. distribute-lft-neg-in 98.1%

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

   5. pow2 98.1%

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

   6. pow-flip 98.1%

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

   7. metadata-eval 98.1%

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

6. Applied egg-rr 98.1%

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

   1. distribute-lft-neg-out 98.1%

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

   2. metadata-eval 98.1%

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

   3. pow-flip 98.1%

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

   4. pow2 98.1%

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

   5. div-inv 98.1%

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

   6. associate-/r* 98.1%

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

   7. distribute-frac-neg2 98.1%

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

   8. associate-/r* 98.1%

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

   9. frac-sub 97.7%

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

8. Applied egg-rr 97.7%

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

   1. cancel-sign-sub 97.7%

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

   2. *-commutative 97.7%

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

   3. distribute-lft-neg-out 97.7%

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

10. Simplified 97.7%

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

    1. associate-/r/ 98.3%

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

    2. +-commutative 98.3%

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

    3. fma-define 98.3%

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

    4. *-commutative 98.3%

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

    5. distribute-rgt-neg-in 98.3%

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

12. Applied egg-rr 98.3%

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

13. Taylor expanded in alphay around 0 98.6%

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

14. Final simplification 98.6%

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

Alternative 2: 98.4% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

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 98.1%

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

   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* 98.1%

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

   9. distribute-neg-frac2 98.1%

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

3. Simplified 98.1%

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

   1. distribute-frac-neg2 98.1%

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

   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. div-inv 98.1%

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

   4. distribute-lft-neg-in 98.1%

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

   5. pow2 98.1%

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

   6. pow-flip 98.1%

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

   7. metadata-eval 98.1%

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

6. Applied egg-rr 98.1%

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

   1. distribute-lft-neg-out 98.1%

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

   2. metadata-eval 98.1%

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

   3. pow-flip 98.1%

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

   4. pow2 98.1%

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

   5. div-inv 98.1%

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

   6. associate-/r* 98.1%

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

   7. distribute-frac-neg2 98.1%

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

   8. associate-/r* 98.1%

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

   9. frac-sub 97.7%

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

8. Applied egg-rr 97.7%

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

   1. cancel-sign-sub 97.7%

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

   2. *-commutative 97.7%

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

   3. distribute-lft-neg-out 97.7%

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

10. Simplified 97.7%

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

    1. associate-/r/ 98.3%

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

    2. +-commutative 98.3%

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

    3. fma-define 98.3%

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

    4. *-commutative 98.3%

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

    5. distribute-rgt-neg-in 98.3%

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

12. Applied egg-rr 98.3%

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

13. Final simplification 98.3%

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

Alternative 3: 98.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

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 98.1%

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

   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* 98.1%

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

   9. distribute-neg-frac2 98.1%

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

3. Simplified 98.1%

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

5. Final simplification 98.1%

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

Alternative 4: 93.1% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{alphax \cdot sin2phi}{alphay} + \frac{alphay \cdot cos2phi}{alphax}\\ t_1 := \frac{1}{t\_0}\\ \left(alphax \cdot alphay\right) \cdot \left(u0 \cdot \left(t\_1 + u0 \cdot \left(t\_1 \cdot 0.5 + u0 \cdot \left(0.3333333333333333 \cdot t\_1 - -0.25 \cdot \frac{u0}{t\_0}\right)\right)\right)\right) \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (let* ((t_0 (+ (/ (* alphax sin2phi) alphay) (/ (* alphay cos2phi) alphax)))
        (t_1 (/ 1.0 t_0)))
   (*
    (* alphax alphay)
    (*
     u0
     (+
      t_1
      (*
       u0
       (+
        (* t_1 0.5)
        (* u0 (- (* 0.3333333333333333 t_1) (* -0.25 (/ u0 t_0)))))))))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float t_0 = ((alphax * sin2phi) / alphay) + ((alphay * cos2phi) / alphax);
	float t_1 = 1.0f / t_0;
	return (alphax * alphay) * (u0 * (t_1 + (u0 * ((t_1 * 0.5f) + (u0 * ((0.3333333333333333f * t_1) - (-0.25f * (u0 / 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 = ((alphax * sin2phi) / alphay) + ((alphay * cos2phi) / alphax)
    t_1 = 1.0e0 / t_0
    code = (alphax * alphay) * (u0 * (t_1 + (u0 * ((t_1 * 0.5e0) + (u0 * ((0.3333333333333333e0 * t_1) - ((-0.25e0) * (u0 / t_0))))))))
end function

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	t_0 = Float32(Float32(Float32(alphax * sin2phi) / alphay) + Float32(Float32(alphay * cos2phi) / alphax))
	t_1 = Float32(Float32(1.0) / t_0)
	return Float32(Float32(alphax * alphay) * Float32(u0 * Float32(t_1 + Float32(u0 * Float32(Float32(t_1 * Float32(0.5)) + Float32(u0 * Float32(Float32(Float32(0.3333333333333333) * t_1) - Float32(Float32(-0.25) * Float32(u0 / t_0)))))))))
end

MATLAB

function tmp = code(alphax, alphay, u0, cos2phi, sin2phi)
	t_0 = ((alphax * sin2phi) / alphay) + ((alphay * cos2phi) / alphax);
	t_1 = single(1.0) / t_0;
	tmp = (alphax * alphay) * (u0 * (t_1 + (u0 * ((t_1 * single(0.5)) + (u0 * ((single(0.3333333333333333) * t_1) - (single(-0.25) * (u0 / t_0))))))));
end

Derivation

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 98.1%

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

   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* 98.1%

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

   9. distribute-neg-frac2 98.1%

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

3. Simplified 98.1%

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

   1. distribute-frac-neg2 98.1%

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

   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. div-inv 98.1%

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

   4. distribute-lft-neg-in 98.1%

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

   5. pow2 98.1%

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

   6. pow-flip 98.1%

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

   7. metadata-eval 98.1%

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

6. Applied egg-rr 98.1%

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

   1. distribute-lft-neg-out 98.1%

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

   2. metadata-eval 98.1%

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

   3. pow-flip 98.1%

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

   4. pow2 98.1%

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

   5. div-inv 98.1%

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

   6. associate-/r* 98.1%

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

   7. distribute-frac-neg2 98.1%

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

   8. associate-/r* 98.1%

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

   9. frac-sub 97.7%

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

8. Applied egg-rr 97.7%

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

   1. cancel-sign-sub 97.7%

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

   2. *-commutative 97.7%

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

   3. distribute-lft-neg-out 97.7%

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

10. Simplified 97.7%

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

    1. associate-/r/ 98.3%

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

    2. +-commutative 98.3%

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

    3. fma-define 98.3%

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

    4. *-commutative 98.3%

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

    5. distribute-rgt-neg-in 98.3%

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

12. Applied egg-rr 98.3%

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

13. Taylor expanded in u0 around 0 93.5%

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

14. Final simplification 93.5%

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

Alternative 5: 80.7% accurate, 4.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\frac{sin2phi}{alphay}}{alphay}\\ \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 0.0005000000237487257:\\ \;\;\;\;\frac{u0}{\frac{cos2phi}{alphax \cdot alphax} + t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{u0 \cdot \left(u0 \cdot -0.5 + -1\right)}{\frac{\frac{cos2phi}{alphax}}{alphax} - t\_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (let* ((t_0 (/ (/ sin2phi alphay) alphay)))
   (if (<= (/ sin2phi (* alphay alphay)) 0.0005000000237487257)
     (/ u0 (+ (/ cos2phi (* alphax alphax)) t_0))
     (/ (* u0 (+ (* u0 -0.5) -1.0)) (- (/ (/ 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 ((sin2phi / (alphay * alphay)) <= 0.0005000000237487257f) {
		tmp = u0 / ((cos2phi / (alphax * alphax)) + t_0);
	} else {
		tmp = (u0 * ((u0 * -0.5f) + -1.0f)) / (((cos2phi / alphax) / alphax) - t_0);
	}
	return tmp;
}

Fortran

real(4) function code(alphax, alphay, u0, cos2phi, sin2phi)
    real(4), intent (in) :: alphax
    real(4), intent (in) :: alphay
    real(4), intent (in) :: u0
    real(4), intent (in) :: cos2phi
    real(4), intent (in) :: sin2phi
    real(4) :: t_0
    real(4) :: tmp
    t_0 = (sin2phi / alphay) / alphay
    if ((sin2phi / (alphay * alphay)) <= 0.0005000000237487257e0) then
        tmp = u0 / ((cos2phi / (alphax * alphax)) + t_0)
    else
        tmp = (u0 * ((u0 * (-0.5e0)) + (-1.0e0))) / (((cos2phi / alphax) / alphax) - t_0)
    end if
    code = tmp
end function

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	t_0 = Float32(Float32(sin2phi / alphay) / alphay)
	tmp = Float32(0.0)
	if (Float32(sin2phi / Float32(alphay * alphay)) <= Float32(0.0005000000237487257))
		tmp = Float32(u0 / Float32(Float32(cos2phi / Float32(alphax * alphax)) + t_0));
	else
		tmp = Float32(Float32(u0 * Float32(Float32(u0 * Float32(-0.5)) + Float32(-1.0))) / Float32(Float32(Float32(cos2phi / alphax) / alphax) - t_0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alphax, alphay, u0, cos2phi, sin2phi)
	t_0 = (sin2phi / alphay) / alphay;
	tmp = single(0.0);
	if ((sin2phi / (alphay * alphay)) <= single(0.0005000000237487257))
		tmp = u0 / ((cos2phi / (alphax * alphax)) + t_0);
	else
		tmp = (u0 * ((u0 * single(-0.5)) + single(-1.0))) / (((cos2phi / alphax) / alphax) - t_0);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 sin2phi (*.f32 alphay alphay)) < 5.00000024e-4

   1. Initial program 54.5%

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

   3. Taylor expanded in u0 around 0 75.2%

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

      1. mul-1-neg 75.2%

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

   5. Simplified 75.2%

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

      1. associate-/r* 75.2%

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

      2. div-inv 75.2%

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

   7. Applied egg-rr 75.2%

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

      1. associate-*r/ 75.2%

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

      2. *-rgt-identity 75.2%

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

   9. Simplified 75.2%

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

   if 5.00000024e-4 < (/.f32 sin2phi (*.f32 alphay alphay))

   1. Initial program 67.7%

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

      1. distribute-frac-neg 67.7%

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

      2. distribute-neg-frac2 67.7%

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

      3. sub-neg 67.7%

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

      4. log1p-define 97.8%

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

      5. neg-sub0 97.8%

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

      6. associate--r+ 97.8%

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

      7. neg-sub0 97.8%

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

      8. associate-/r* 97.8%

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

      9. distribute-neg-frac2 97.8%

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

   3. Simplified 97.8%

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

      1. div-inv 97.8%

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

      2. fma-neg 97.8%

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

      3. add-sqr-sqrt -0.0%

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

      4. sqrt-unprod 96.0%

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

      5. sqr-neg 96.0%

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

      6. sqrt-prod 96.0%

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

      7. add-sqr-sqrt 96.0%

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

      8. div-inv 95.9%

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

      9. distribute-rgt-neg-in 95.9%

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

      10. pow2 95.9%

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

      11. pow-flip 95.9%

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

      12. metadata-eval 95.9%

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

   6. Applied egg-rr 95.9%

      \[\leadsto \frac{\mathsf{log1p}\left(-u0\right)}{\color{blue}{\mathsf{fma}\left(\frac{cos2phi}{alphax}, \frac{1}{alphax}, sin2phi \cdot \left(-{alphay}^{-2}\right)\right)}} \]
   7. Step-by-step derivation

      1. fma-undefine 95.9%

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

      2. distribute-rgt-neg-out 95.9%

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

      3. unsub-neg 95.9%

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

      4. associate-*r/ 95.9%

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

      5. *-rgt-identity 95.9%

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

   8. Simplified 95.9%

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

      1. metadata-eval 95.9%

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

      2. pow-flip 95.9%

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

      3. pow2 95.9%

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

      4. div-inv 96.0%

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

      5. associate-/r* 95.9%

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

   10. Applied egg-rr 95.9%

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

   11. Taylor expanded in u0 around 0 86.6%

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

4. Final simplification 82.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 0.0005000000237487257:\\ \;\;\;\;\frac{u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{\frac{sin2phi}{alphay}}{alphay}}\\ \mathbf{else}:\\ \;\;\;\;\frac{u0 \cdot \left(u0 \cdot -0.5 + -1\right)}{\frac{\frac{cos2phi}{alphax}}{alphax} - \frac{\frac{sin2phi}{alphay}}{alphay}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 92.9% 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 62.7%

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

3. Taylor expanded in u0 around 0 93.1%

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

4. Final simplification 93.1%

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

Alternative 7: 92.9% accurate, 4.3× speedup

Math

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

FPCore

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

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))))))) / ((cos2phi / (alphax * alphax)) + ((sin2phi / alphay) / alphay));
}

Fortran

real(4) function code(alphax, alphay, u0, cos2phi, sin2phi)
    real(4), intent (in) :: alphax
    real(4), intent (in) :: alphay
    real(4), intent (in) :: u0
    real(4), intent (in) :: cos2phi
    real(4), intent (in) :: sin2phi
    code = (u0 * (1.0e0 + (u0 * (0.5e0 + (u0 * (0.3333333333333333e0 - (u0 * (-0.25e0)))))))) / ((cos2phi / (alphax * alphax)) + ((sin2phi / alphay) / alphay))
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(cos2phi / Float32(alphax * alphax)) + Float32(Float32(sin2phi / alphay) / alphay)))
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)))))))) / ((cos2phi / (alphax * alphax)) + ((sin2phi / alphay) / alphay));
end

Derivation

1. Initial program 62.7%

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

3. Taylor expanded in u0 around 0 93.1%

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

   1. associate-/r* 76.9%

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

   2. div-inv 76.8%

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

5. Applied egg-rr 93.0%

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

   1. associate-*r/ 76.9%

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

   2. *-rgt-identity 76.9%

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

7. Simplified 93.1%

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

8. Final simplification 93.1%

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

Alternative 8: 91.1% accurate, 5.0× speedup

Math

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

FPCore

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

Derivation

1. Initial program 62.7%

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

3. Taylor expanded in u0 around 0 91.4%

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

4. Final simplification 91.4%

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

Alternative 9: 87.3% accurate, 6.1× speedup

Math

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

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 62.7%

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

3. Taylor expanded in u0 around 0 88.0%

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

4. Final simplification 88.0%

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

Alternative 10: 75.9% accurate, 6.8× speedup

Math

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

FPCore

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

C

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

MATLAB

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

Derivation

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 98.1%

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

   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* 98.1%

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

   9. distribute-neg-frac2 98.1%

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

3. Simplified 98.1%

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

   1. distribute-frac-neg2 98.1%

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

   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. div-inv 98.1%

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

   4. distribute-lft-neg-in 98.1%

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

   5. pow2 98.1%

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

   6. pow-flip 98.1%

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

   7. metadata-eval 98.1%

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

6. Applied egg-rr 98.1%

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

   1. distribute-lft-neg-out 98.1%

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

   2. metadata-eval 98.1%

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

   3. pow-flip 98.1%

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

   4. pow2 98.1%

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

   5. div-inv 98.1%

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

   6. associate-/r* 98.1%

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

   7. distribute-frac-neg2 98.1%

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

   8. associate-/r* 98.1%

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

   9. frac-sub 97.7%

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

8. Applied egg-rr 97.7%

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

   1. cancel-sign-sub 97.7%

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

   2. *-commutative 97.7%

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

   3. distribute-lft-neg-out 97.7%

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

10. Simplified 97.7%

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

11. Taylor expanded in u0 around 0 77.2%

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

12. Final simplification 77.2%

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

Alternative 11: 75.7% 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 62.7%

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

3. Taylor expanded in u0 around 0 76.9%

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

   1. mul-1-neg 76.9%

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

5. Simplified 76.9%

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

6. Final simplification 76.9%

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

Alternative 12: 59.0% accurate, 16.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 62.7%

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

3. Taylor expanded in u0 around 0 76.9%

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

   1. mul-1-neg 76.9%

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

5. Simplified 76.9%

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

6. Taylor expanded in cos2phi around 0 60.9%

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

   1. associate-/l* 61.0%

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

8. Simplified 61.0%

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

   1. pow2 61.0%

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

10. Applied egg-rr 61.0%

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

11. Final simplification 61.0%

    \[\leadsto \left(alphay \cdot alphay\right) \cdot \frac{u0}{sin2phi} \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2024058 
(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)))))