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

Percentage Accurate: 60.4% → 98.4%

Time: 16.9s

Alternatives: 13

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: 60.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Alternative 1: 98.4% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 60.1%

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

   1. distribute-frac-neg 60.1%

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

   2. distribute-neg-frac2 60.1%

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

   3. sub-neg 60.1%

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

   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* 98.0%

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

   2. div-inv 97.7%

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

6. Applied egg-rr 97.7%

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

   1. associate-*r/ 98.0%

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

   2. *-rgt-identity 98.0%

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

8. Simplified 98.0%

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

   1. frac-sub 97.7%

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

10. Applied egg-rr 97.7%

    \[\leadsto \frac{\mathsf{log1p}\left(-u0\right)}{\color{blue}{\frac{\frac{cos2phi}{alphax} \cdot alphay - \left(-alphax\right) \cdot \frac{sin2phi}{alphay}}{\left(-alphax\right) \cdot alphay}}} \]
11. 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. *-commutative 97.7%

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

12. Simplified 97.7%

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

    1. *-un-lft-identity 97.7%

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

    2. associate-/r/ 98.6%

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

    3. associate-*r/ 98.6%

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

    4. +-commutative 98.6%

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

    5. clear-num 98.6%

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

    6. fma-define 98.6%

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

    7. clear-num 98.5%

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

    8. associate-*r/ 98.6%

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

14. Applied egg-rr 98.6%

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

    1. *-lft-identity 98.6%

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

    2. associate-*l/ 98.7%

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

16. Simplified 98.7%

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

17. Final simplification 98.7%

    \[\leadsto \frac{\mathsf{log1p}\left(-u0\right) \cdot \left(alphax \cdot \left(-alphay\right)\right)}{\mathsf{fma}\left(alphax, \frac{sin2phi}{alphay}, alphay \cdot \frac{cos2phi}{alphax}\right)} \]
18. 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 60.1%

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

   1. distribute-frac-neg 60.1%

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

   2. distribute-neg-frac2 60.1%

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

   3. sub-neg 60.1%

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

   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* 98.0%

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

   2. div-inv 97.7%

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

6. Applied egg-rr 97.7%

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

   1. associate-*r/ 98.0%

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

   2. *-rgt-identity 98.0%

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

8. Simplified 98.0%

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

   1. frac-sub 97.7%

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

10. Applied egg-rr 97.7%

    \[\leadsto \frac{\mathsf{log1p}\left(-u0\right)}{\color{blue}{\frac{\frac{cos2phi}{alphax} \cdot alphay - \left(-alphax\right) \cdot \frac{sin2phi}{alphay}}{\left(-alphax\right) \cdot alphay}}} \]
11. 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. *-commutative 97.7%

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

12. Simplified 97.7%

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

    1. associate-/r/ 98.6%

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

    2. associate-*r/ 98.6%

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

    3. +-commutative 98.6%

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

    4. clear-num 98.6%

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

    5. fma-define 98.6%

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

    6. clear-num 98.5%

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

    7. associate-*r/ 98.6%

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

14. Applied egg-rr 98.6%

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

15. Final simplification 98.6%

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

Alternative 3: 98.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 60.1%

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

   1. distribute-frac-neg 60.1%

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

   2. distribute-neg-frac2 60.1%

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

   3. sub-neg 60.1%

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

   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* 98.0%

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

   2. div-inv 97.7%

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

6. Applied egg-rr 97.7%

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

   1. associate-*r/ 98.0%

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

   2. *-rgt-identity 98.0%

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

8. Simplified 98.0%

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

Alternative 4: 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 / alphax) / Float32(-alphax)) - Float32(sin2phi / Float32(alphay * alphay))))
end

Derivation

1. Initial program 60.1%

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

   1. distribute-frac-neg 60.1%

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

   2. distribute-neg-frac2 60.1%

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

   3. sub-neg 60.1%

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

   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. Add Preprocessing

Alternative 5: 93.2% accurate, 1.5× speedup

Math

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

FPCore

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

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 = (alphay * alphax) / t_0;
	return u0 * (t_1 + (u0 * ((0.5f * t_1) + (u0 * ((0.25f * ((alphax * (u0 * alphay)) / t_0)) + (t_1 * 0.3333333333333333f))))));
}

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 = (alphay * alphax) / t_0
    code = u0 * (t_1 + (u0 * ((0.5e0 * t_1) + (u0 * ((0.25e0 * ((alphax * (u0 * alphay)) / t_0)) + (t_1 * 0.3333333333333333e0))))))
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(alphay * alphax) / t_0)
	return Float32(u0 * Float32(t_1 + Float32(u0 * Float32(Float32(Float32(0.5) * t_1) + Float32(u0 * Float32(Float32(Float32(0.25) * Float32(Float32(alphax * Float32(u0 * alphay)) / t_0)) + Float32(t_1 * Float32(0.3333333333333333))))))))
end

MATLAB

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

Derivation

1. Initial program 60.1%

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

   1. distribute-frac-neg 60.1%

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

   2. distribute-neg-frac2 60.1%

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

   3. sub-neg 60.1%

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

   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* 98.0%

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

   2. div-inv 97.7%

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

6. Applied egg-rr 97.7%

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

   1. associate-*r/ 98.0%

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

   2. *-rgt-identity 98.0%

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

8. Simplified 98.0%

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

   1. frac-sub 97.7%

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

10. Applied egg-rr 97.7%

    \[\leadsto \frac{\mathsf{log1p}\left(-u0\right)}{\color{blue}{\frac{\frac{cos2phi}{alphax} \cdot alphay - \left(-alphax\right) \cdot \frac{sin2phi}{alphay}}{\left(-alphax\right) \cdot alphay}}} \]
11. 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. *-commutative 97.7%

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

12. Simplified 97.7%

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

13. Taylor expanded in u0 around 0 94.2%

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

14. Final simplification 94.2%

    \[\leadsto u0 \cdot \left(\frac{alphay \cdot alphax}{\frac{alphax \cdot sin2phi}{alphay} + \frac{alphay \cdot cos2phi}{alphax}} + u0 \cdot \left(0.5 \cdot \frac{alphay \cdot alphax}{\frac{alphax \cdot sin2phi}{alphay} + \frac{alphay \cdot cos2phi}{alphax}} + u0 \cdot \left(0.25 \cdot \frac{alphax \cdot \left(u0 \cdot alphay\right)}{\frac{alphax \cdot sin2phi}{alphay} + \frac{alphay \cdot cos2phi}{alphax}} + \frac{alphay \cdot alphax}{\frac{alphax \cdot sin2phi}{alphay} + \frac{alphay \cdot cos2phi}{alphax}} \cdot 0.3333333333333333\right)\right)\right) \]
15. 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 60.1%

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

3. Taylor expanded in u0 around 0 93.4%

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

   1. *-commutative 93.4%

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

5. Simplified 93.4%

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

6. Final simplification 93.4%

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

Alternative 7: 91.0% 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 60.1%

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

3. Taylor expanded in u0 around 0 92.0%

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

   1. *-commutative 92.0%

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

5. Simplified 92.0%

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

6. Final simplification 92.0%

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

Alternative 8: 87.2% accurate, 5.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 60.1%

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

   1. distribute-frac-neg 60.1%

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

   2. distribute-neg-frac2 60.1%

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

   3. neg-mul-1 60.1%

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

   4. associate-/r* 60.1%

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

   5. remove-double-neg 60.1%

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

   6. distribute-frac-neg 60.1%

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

   7. distribute-neg-frac2 60.1%

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

   8. metadata-eval 60.1%

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

   9. /-rgt-identity 60.1%

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

   10. sub-neg 60.1%

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

   11. log1p-define 97.9%

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

3. Simplified 97.9%

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

   1. associate-/r* 98.0%

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

   2. div-inv 97.7%

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

6. Applied egg-rr 97.7%

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

   1. *-commutative 97.7%

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

   2. associate-*r/ 97.8%

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

8. Simplified 97.8%

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

   1. associate-*l/ 98.0%

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

   2. *-un-lft-identity 98.0%

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

   3. clear-num 97.9%

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

10. Applied egg-rr 97.9%

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

11. Taylor expanded in u0 around 0 88.4%

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

    1. *-commutative 88.4%

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

13. Simplified 88.4%

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

    1. distribute-rgt-in 88.6%

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

    2. *-un-lft-identity 88.6%

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

15. Applied egg-rr 88.6%

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

16. Final simplification 88.6%

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

Alternative 9: 87.2% accurate, 6.1× speedup

Math

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

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(Float32(sin2phi / 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 60.1%

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

   1. distribute-frac-neg 60.1%

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

   2. distribute-neg-frac2 60.1%

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

   3. neg-mul-1 60.1%

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

   4. associate-/r* 60.1%

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

   5. remove-double-neg 60.1%

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

   6. distribute-frac-neg 60.1%

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

   7. distribute-neg-frac2 60.1%

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

   8. metadata-eval 60.1%

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

   9. /-rgt-identity 60.1%

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

   10. sub-neg 60.1%

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

   11. log1p-define 97.9%

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

3. Simplified 97.9%

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

   1. associate-/r* 98.0%

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

   2. div-inv 97.7%

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

6. Applied egg-rr 97.7%

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

   1. *-commutative 97.7%

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

   2. associate-*r/ 97.8%

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

8. Simplified 97.8%

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

   1. associate-*l/ 98.0%

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

   2. *-un-lft-identity 98.0%

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

   3. clear-num 97.9%

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

10. Applied egg-rr 97.9%

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

11. Taylor expanded in u0 around 0 88.4%

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

    1. *-commutative 88.4%

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

13. Simplified 88.4%

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

14. Taylor expanded in alphay around 0 88.5%

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

15. Final simplification 88.5%

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

Alternative 10: 87.2% 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 60.1%

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

3. Taylor expanded in u0 around 0 88.4%

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

   1. *-commutative 88.4%

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

5. Simplified 88.4%

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

6. Final simplification 88.4%

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

Alternative 11: 75.8% 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 60.1%

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

   1. distribute-frac-neg 60.1%

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

   2. distribute-neg-frac2 60.1%

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

   3. sub-neg 60.1%

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

   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* 98.0%

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

   2. div-inv 97.7%

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

6. Applied egg-rr 97.7%

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

   1. associate-*r/ 98.0%

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

   2. *-rgt-identity 98.0%

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

8. Simplified 98.0%

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

   1. frac-sub 97.7%

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

10. Applied egg-rr 97.7%

    \[\leadsto \frac{\mathsf{log1p}\left(-u0\right)}{\color{blue}{\frac{\frac{cos2phi}{alphax} \cdot alphay - \left(-alphax\right) \cdot \frac{sin2phi}{alphay}}{\left(-alphax\right) \cdot alphay}}} \]
11. 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. *-commutative 97.7%

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

12. Simplified 97.7%

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

13. Taylor expanded in u0 around 0 78.2%

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

14. Final simplification 78.2%

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

Alternative 12: 75.7% 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 60.1%

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

3. Taylor expanded in u0 around 0 77.7%

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

   1. associate-/r* 98.0%

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

   2. div-inv 97.7%

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

5. Applied egg-rr 77.7%

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

   1. associate-*r/ 98.0%

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

   2. *-rgt-identity 98.0%

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

7. Simplified 77.7%

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

8. Final simplification 77.7%

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

Alternative 13: 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 60.1%

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

3. Taylor expanded in u0 around 0 77.7%

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

4. Final simplification 77.7%

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

Reproduce

herbie shell --seed 2024135 
(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)))))