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

Percentage Accurate: 60.9% → 98.5%

Time: 14.7s

Alternatives: 17

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: 60.9% 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: 98.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 59.8%

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

   1. neg-sub0 59.8%

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

   2. div-sub 59.8%

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

   3. --rgt-identity 59.8%

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

   4. div-sub 59.8%

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

   5. --rgt-identity 59.8%

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

   6. neg-sub0 59.8%

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

   7. sub-neg 59.8%

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

   8. log1p-def 98.3%

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

3. Simplified 98.3%

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

   1. +-commutative 98.3%

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

   2. associate-/r* 98.1%

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

   3. associate-/r* 98.1%

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

   4. frac-2neg 98.1%

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

   5. frac-add 98.1%

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

   6. distribute-neg-frac 98.1%

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

5. Applied egg-rr 98.1%

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

   1. +-commutative 98.1%

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

   2. distribute-rgt-neg-out 98.1%

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

   3. unsub-neg 98.1%

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

   4. associate-*l/ 98.1%

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

   5. associate-/l* 98.1%

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

   6. *-commutative 98.1%

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

   7. distribute-lft-neg-out 98.1%

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

   8. distribute-rgt-neg-in 98.1%

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

7. Simplified 98.1%

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

   1. div-inv 97.8%

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

   2. associate-/r/ 97.9%

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

   3. distribute-rgt-neg-out 97.9%

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

9. Applied egg-rr 97.9%

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

    1. associate-/r/ 98.1%

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

    2. associate-*l/ 98.4%

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

    3. *-commutative 98.4%

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

    4. distribute-rgt-neg-in 98.4%

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

11. Simplified 98.4%

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

    1. associate-*l/ 98.6%

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

    2. *-un-lft-identity 98.6%

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

    3. associate-/l* 98.5%

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

13. Applied egg-rr 98.5%

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

    1. associate-/l* 98.4%

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

    2. associate-/r/ 98.7%

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

    3. associate-/r/ 98.7%

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

    4. *-commutative 98.7%

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

15. Simplified 98.7%

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

16. Final simplification 98.7%

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

Alternative 2: 98.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{-\mathsf{log1p}\left(-u0\right)}{\frac{cos2phi}{alphax} \cdot \frac{1}{alphax} + \frac{sin2phi}{alphay \cdot alphay}} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (/
  (- (log1p (- u0)))
  (+ (* (/ cos2phi alphax) (/ 1.0 alphax)) (/ sin2phi (* alphay alphay)))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	return -log1pf(-u0) / (((cos2phi / alphax) * (1.0f / alphax)) + (sin2phi / (alphay * alphay)));
}

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	return Float32(Float32(-log1p(Float32(-u0))) / Float32(Float32(Float32(cos2phi / alphax) * Float32(Float32(1.0) / alphax)) + Float32(sin2phi / Float32(alphay * alphay))))
end

Derivation

1. Initial program 59.8%

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

   1. neg-sub0 59.8%

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

   2. div-sub 59.8%

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

   3. --rgt-identity 59.8%

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

   4. div-sub 59.8%

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

   5. --rgt-identity 59.8%

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

   6. sub-neg 59.8%

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

   7. +-commutative 59.8%

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

   8. neg-sub0 59.8%

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

   9. associate-+l- 59.8%

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

   10. sub0-neg 59.8%

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

   11. neg-mul-1 59.8%

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

   12. log-prod -0.0%

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

   13. associate--r+ -0.0%

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

3. Simplified 98.3%

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

   1. div-inv 98.3%

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

5. Applied egg-rr 98.3%

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

6. Final simplification 98.3%

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

Alternative 3: 92.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 0.550000011920929:\\ \;\;\;\;\left(-0.5 \cdot \left(u0 \cdot u0\right) - u0\right) \cdot \left(alphay \cdot \frac{alphax}{alphay \cdot \frac{-cos2phi}{alphax} - sin2phi \cdot \frac{alphax}{alphay}}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(-u0\right) \cdot \frac{-alphay}{\frac{sin2phi}{alphay}}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (if (<= (/ sin2phi (* alphay alphay)) 0.550000011920929)
   (*
    (- (* -0.5 (* u0 u0)) u0)
    (*
     alphay
     (/
      alphax
      (- (* alphay (/ (- cos2phi) alphax)) (* sin2phi (/ alphax alphay))))))
   (* (log1p (- u0)) (/ (- alphay) (/ sin2phi alphay)))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float tmp;
	if ((sin2phi / (alphay * alphay)) <= 0.550000011920929f) {
		tmp = ((-0.5f * (u0 * u0)) - u0) * (alphay * (alphax / ((alphay * (-cos2phi / alphax)) - (sin2phi * (alphax / alphay)))));
	} else {
		tmp = log1pf(-u0) * (-alphay / (sin2phi / alphay));
	}
	return tmp;
}

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = Float32(0.0)
	if (Float32(sin2phi / Float32(alphay * alphay)) <= Float32(0.550000011920929))
		tmp = Float32(Float32(Float32(Float32(-0.5) * Float32(u0 * u0)) - u0) * Float32(alphay * Float32(alphax / Float32(Float32(alphay * Float32(Float32(-cos2phi) / alphax)) - Float32(sin2phi * Float32(alphax / alphay))))));
	else
		tmp = Float32(log1p(Float32(-u0)) * Float32(Float32(-alphay) / Float32(sin2phi / alphay)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 sin2phi (*.f32 alphay alphay)) < 0.550000012

   1. Initial program 56.0%

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

      1. neg-sub0 56.0%

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

      2. div-sub 56.0%

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

      3. --rgt-identity 56.0%

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

      4. div-sub 56.0%

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

      5. --rgt-identity 56.0%

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

      6. neg-sub0 56.0%

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

      7. sub-neg 56.0%

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

      8. log1p-def 98.6%

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

   3. Simplified 98.6%

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

      1. +-commutative 98.6%

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

      2. associate-/r* 98.6%

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

      3. associate-/r* 98.6%

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

      4. frac-2neg 98.6%

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

      5. frac-add 98.6%

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

      6. distribute-neg-frac 98.6%

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

   5. Applied egg-rr 98.6%

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

      1. +-commutative 98.6%

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

      2. distribute-rgt-neg-out 98.6%

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

      3. unsub-neg 98.6%

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

      4. associate-*l/ 98.6%

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

      5. associate-/l* 98.6%

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

      6. *-commutative 98.6%

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

      7. distribute-lft-neg-out 98.6%

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

      8. distribute-rgt-neg-in 98.6%

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

   7. Simplified 98.6%

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

      1. div-inv 98.3%

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

      2. associate-/r/ 98.4%

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

      3. distribute-rgt-neg-out 98.4%

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

   9. Applied egg-rr 98.4%

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

       1. associate-/r/ 98.3%

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

       2. associate-*l/ 98.2%

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

       3. *-commutative 98.2%

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

       4. distribute-rgt-neg-in 98.2%

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

   11. Simplified 98.2%

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

       1. associate-*l/ 98.3%

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

       2. *-un-lft-identity 98.3%

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

       3. associate-/l* 98.3%

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

   13. Applied egg-rr 98.3%

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

       1. associate-/l* 98.3%

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

       2. associate-/r/ 98.5%

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

       3. associate-/r/ 98.5%

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

       4. *-commutative 98.5%

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

   15. Simplified 98.5%

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

   16. Taylor expanded in u0 around 0 87.9%

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

       1. +-commutative 87.9%

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

       2. neg-mul-1 87.9%

          \[\leadsto \left(-\left(-0.5 \cdot {u0}^{2} + \color{blue}{\left(-u0\right)}\right)\right) \cdot \left(\frac{alphax}{alphay \cdot \frac{-cos2phi}{alphax} - sin2phi \cdot \frac{alphax}{alphay}} \cdot \left(-alphay\right)\right) \]

       3. unsub-neg 87.9%

          \[\leadsto \left(-\color{blue}{\left(-0.5 \cdot {u0}^{2} - u0\right)}\right) \cdot \left(\frac{alphax}{alphay \cdot \frac{-cos2phi}{alphax} - sin2phi \cdot \frac{alphax}{alphay}} \cdot \left(-alphay\right)\right) \]

       4. unpow2 87.9%

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

   18. Simplified 87.9%

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

   if 0.550000012 < (/.f32 sin2phi (*.f32 alphay alphay))

   1. Initial program 63.0%

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

      1. neg-sub0 63.0%

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

      2. div-sub 63.0%

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

      3. --rgt-identity 63.0%

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

      4. div-sub 63.0%

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

      5. --rgt-identity 63.0%

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

      6. neg-sub0 63.0%

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

      7. sub-neg 63.0%

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

      8. log1p-def 98.0%

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

   3. Simplified 98.0%

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

      1. +-commutative 98.0%

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

      2. associate-/r* 97.8%

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

      3. associate-/r* 97.7%

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

      4. frac-2neg 97.7%

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

      5. frac-add 97.6%

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

      6. distribute-neg-frac 97.6%

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

   5. Applied egg-rr 97.6%

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

      1. +-commutative 97.6%

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

      2. distribute-rgt-neg-out 97.6%

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

      3. unsub-neg 97.6%

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

      4. associate-*l/ 97.7%

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

      5. associate-/l* 97.6%

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

      6. *-commutative 97.6%

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

      7. distribute-lft-neg-out 97.6%

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

      8. distribute-rgt-neg-in 97.6%

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

   7. Simplified 97.6%

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

      1. div-inv 97.4%

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

      2. associate-/r/ 97.4%

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

      3. distribute-rgt-neg-out 97.4%

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

   9. Applied egg-rr 97.4%

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

       1. associate-/r/ 97.9%

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

       2. associate-*l/ 98.6%

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

       3. *-commutative 98.6%

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

       4. distribute-rgt-neg-in 98.6%

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

   11. Simplified 98.6%

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

   12. Taylor expanded in alphay around 0 97.0%

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

       1. unpow2 97.0%

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

       2. associate-/l* 96.4%

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

   14. Simplified 96.4%

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

4. Final simplification 92.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 0.550000011920929:\\ \;\;\;\;\left(-0.5 \cdot \left(u0 \cdot u0\right) - u0\right) \cdot \left(alphay \cdot \frac{alphax}{alphay \cdot \frac{-cos2phi}{alphax} - sin2phi \cdot \frac{alphax}{alphay}}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(-u0\right) \cdot \frac{-alphay}{\frac{sin2phi}{alphay}}\\ \end{array} \]

Alternative 4: 92.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 0.550000011920929:\\ \;\;\;\;\left(-0.5 \cdot \left(u0 \cdot u0\right) - u0\right) \cdot \left(alphay \cdot \frac{alphax}{alphay \cdot \frac{-cos2phi}{alphax} - sin2phi \cdot \frac{alphax}{alphay}}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(-u0\right) \cdot \frac{alphay \cdot \left(-alphay\right)}{sin2phi}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (if (<= (/ sin2phi (* alphay alphay)) 0.550000011920929)
   (*
    (- (* -0.5 (* u0 u0)) u0)
    (*
     alphay
     (/
      alphax
      (- (* alphay (/ (- cos2phi) alphax)) (* sin2phi (/ alphax alphay))))))
   (* (log1p (- u0)) (/ (* alphay (- alphay)) sin2phi))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float tmp;
	if ((sin2phi / (alphay * alphay)) <= 0.550000011920929f) {
		tmp = ((-0.5f * (u0 * u0)) - u0) * (alphay * (alphax / ((alphay * (-cos2phi / alphax)) - (sin2phi * (alphax / alphay)))));
	} else {
		tmp = log1pf(-u0) * ((alphay * -alphay) / sin2phi);
	}
	return tmp;
}

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = Float32(0.0)
	if (Float32(sin2phi / Float32(alphay * alphay)) <= Float32(0.550000011920929))
		tmp = Float32(Float32(Float32(Float32(-0.5) * Float32(u0 * u0)) - u0) * Float32(alphay * Float32(alphax / Float32(Float32(alphay * Float32(Float32(-cos2phi) / alphax)) - Float32(sin2phi * Float32(alphax / alphay))))));
	else
		tmp = Float32(log1p(Float32(-u0)) * Float32(Float32(alphay * Float32(-alphay)) / sin2phi));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 sin2phi (*.f32 alphay alphay)) < 0.550000012

   1. Initial program 56.0%

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

      1. neg-sub0 56.0%

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

      2. div-sub 56.0%

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

      3. --rgt-identity 56.0%

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

      4. div-sub 56.0%

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

      5. --rgt-identity 56.0%

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

      6. neg-sub0 56.0%

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

      7. sub-neg 56.0%

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

      8. log1p-def 98.6%

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

   3. Simplified 98.6%

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

      1. +-commutative 98.6%

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

      2. associate-/r* 98.6%

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

      3. associate-/r* 98.6%

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

      4. frac-2neg 98.6%

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

      5. frac-add 98.6%

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

      6. distribute-neg-frac 98.6%

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

   5. Applied egg-rr 98.6%

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

      1. +-commutative 98.6%

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

      2. distribute-rgt-neg-out 98.6%

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

      3. unsub-neg 98.6%

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

      4. associate-*l/ 98.6%

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

      5. associate-/l* 98.6%

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

      6. *-commutative 98.6%

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

      7. distribute-lft-neg-out 98.6%

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

      8. distribute-rgt-neg-in 98.6%

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

   7. Simplified 98.6%

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

      1. div-inv 98.3%

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

      2. associate-/r/ 98.4%

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

      3. distribute-rgt-neg-out 98.4%

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

   9. Applied egg-rr 98.4%

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

       1. associate-/r/ 98.3%

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

       2. associate-*l/ 98.2%

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

       3. *-commutative 98.2%

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

       4. distribute-rgt-neg-in 98.2%

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

   11. Simplified 98.2%

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

       1. associate-*l/ 98.3%

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

       2. *-un-lft-identity 98.3%

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

       3. associate-/l* 98.3%

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

   13. Applied egg-rr 98.3%

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

       1. associate-/l* 98.3%

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

       2. associate-/r/ 98.5%

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

       3. associate-/r/ 98.5%

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

       4. *-commutative 98.5%

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

   15. Simplified 98.5%

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

   16. Taylor expanded in u0 around 0 87.9%

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

       1. +-commutative 87.9%

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

       2. neg-mul-1 87.9%

          \[\leadsto \left(-\left(-0.5 \cdot {u0}^{2} + \color{blue}{\left(-u0\right)}\right)\right) \cdot \left(\frac{alphax}{alphay \cdot \frac{-cos2phi}{alphax} - sin2phi \cdot \frac{alphax}{alphay}} \cdot \left(-alphay\right)\right) \]

       3. unsub-neg 87.9%

          \[\leadsto \left(-\color{blue}{\left(-0.5 \cdot {u0}^{2} - u0\right)}\right) \cdot \left(\frac{alphax}{alphay \cdot \frac{-cos2phi}{alphax} - sin2phi \cdot \frac{alphax}{alphay}} \cdot \left(-alphay\right)\right) \]

       4. unpow2 87.9%

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

   18. Simplified 87.9%

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

   if 0.550000012 < (/.f32 sin2phi (*.f32 alphay alphay))

   1. Initial program 63.0%

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

      1. neg-sub0 63.0%

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

      2. div-sub 63.0%

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

      3. --rgt-identity 63.0%

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

      4. div-sub 63.0%

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

      5. --rgt-identity 63.0%

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

      6. neg-sub0 63.0%

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

      7. sub-neg 63.0%

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

      8. log1p-def 98.0%

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

   3. Simplified 98.0%

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

      1. +-commutative 98.0%

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

      2. associate-/r* 97.8%

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

      3. associate-/r* 97.7%

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

      4. frac-2neg 97.7%

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

      5. frac-add 97.6%

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

      6. distribute-neg-frac 97.6%

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

   5. Applied egg-rr 97.6%

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

      1. +-commutative 97.6%

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

      2. distribute-rgt-neg-out 97.6%

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

      3. unsub-neg 97.6%

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

      4. associate-*l/ 97.7%

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

      5. associate-/l* 97.6%

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

      6. *-commutative 97.6%

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

      7. distribute-lft-neg-out 97.6%

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

      8. distribute-rgt-neg-in 97.6%

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

   7. Simplified 97.6%

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

      1. div-inv 97.4%

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

      2. associate-/r/ 97.4%

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

      3. distribute-rgt-neg-out 97.4%

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

   9. Applied egg-rr 97.4%

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

       1. associate-/r/ 97.9%

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

       2. associate-*l/ 98.6%

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

       3. *-commutative 98.6%

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

       4. distribute-rgt-neg-in 98.6%

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

   11. Simplified 98.6%

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

   12. Taylor expanded in alphay around 0 97.0%

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

       1. unpow2 97.0%

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

   14. Simplified 97.0%

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

4. Final simplification 92.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 0.550000011920929:\\ \;\;\;\;\left(-0.5 \cdot \left(u0 \cdot u0\right) - u0\right) \cdot \left(alphay \cdot \frac{alphax}{alphay \cdot \frac{-cos2phi}{alphax} - sin2phi \cdot \frac{alphax}{alphay}}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(-u0\right) \cdot \frac{alphay \cdot \left(-alphay\right)}{sin2phi}\\ \end{array} \]

Alternative 5: 92.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 0.550000011920929:\\ \;\;\;\;\left(-0.5 \cdot \left(u0 \cdot u0\right) - u0\right) \cdot \left(alphay \cdot \frac{alphax}{alphay \cdot \frac{-cos2phi}{alphax} - sin2phi \cdot \frac{alphax}{alphay}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(-u0\right)}{sin2phi} \cdot \left(alphay \cdot \left(-alphay\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (if (<= (/ sin2phi (* alphay alphay)) 0.550000011920929)
   (*
    (- (* -0.5 (* u0 u0)) u0)
    (*
     alphay
     (/
      alphax
      (- (* alphay (/ (- cos2phi) alphax)) (* sin2phi (/ alphax alphay))))))
   (* (/ (log1p (- u0)) sin2phi) (* alphay (- alphay)))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float tmp;
	if ((sin2phi / (alphay * alphay)) <= 0.550000011920929f) {
		tmp = ((-0.5f * (u0 * u0)) - u0) * (alphay * (alphax / ((alphay * (-cos2phi / alphax)) - (sin2phi * (alphax / alphay)))));
	} else {
		tmp = (log1pf(-u0) / sin2phi) * (alphay * -alphay);
	}
	return tmp;
}

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = Float32(0.0)
	if (Float32(sin2phi / Float32(alphay * alphay)) <= Float32(0.550000011920929))
		tmp = Float32(Float32(Float32(Float32(-0.5) * Float32(u0 * u0)) - u0) * Float32(alphay * Float32(alphax / Float32(Float32(alphay * Float32(Float32(-cos2phi) / alphax)) - Float32(sin2phi * Float32(alphax / alphay))))));
	else
		tmp = Float32(Float32(log1p(Float32(-u0)) / sin2phi) * Float32(alphay * Float32(-alphay)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 sin2phi (*.f32 alphay alphay)) < 0.550000012

   1. Initial program 56.0%

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

      1. neg-sub0 56.0%

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

      2. div-sub 56.0%

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

      3. --rgt-identity 56.0%

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

      4. div-sub 56.0%

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

      5. --rgt-identity 56.0%

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

      6. neg-sub0 56.0%

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

      7. sub-neg 56.0%

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

      8. log1p-def 98.6%

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

   3. Simplified 98.6%

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

      1. +-commutative 98.6%

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

      2. associate-/r* 98.6%

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

      3. associate-/r* 98.6%

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

      4. frac-2neg 98.6%

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

      5. frac-add 98.6%

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

      6. distribute-neg-frac 98.6%

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

   5. Applied egg-rr 98.6%

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

      1. +-commutative 98.6%

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

      2. distribute-rgt-neg-out 98.6%

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

      3. unsub-neg 98.6%

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

      4. associate-*l/ 98.6%

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

      5. associate-/l* 98.6%

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

      6. *-commutative 98.6%

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

      7. distribute-lft-neg-out 98.6%

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

      8. distribute-rgt-neg-in 98.6%

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

   7. Simplified 98.6%

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

      1. div-inv 98.3%

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

      2. associate-/r/ 98.4%

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

      3. distribute-rgt-neg-out 98.4%

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

   9. Applied egg-rr 98.4%

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

       1. associate-/r/ 98.3%

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

       2. associate-*l/ 98.2%

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

       3. *-commutative 98.2%

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

       4. distribute-rgt-neg-in 98.2%

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

   11. Simplified 98.2%

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

       1. associate-*l/ 98.3%

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

       2. *-un-lft-identity 98.3%

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

       3. associate-/l* 98.3%

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

   13. Applied egg-rr 98.3%

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

       1. associate-/l* 98.3%

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

       2. associate-/r/ 98.5%

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

       3. associate-/r/ 98.5%

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

       4. *-commutative 98.5%

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

   15. Simplified 98.5%

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

   16. Taylor expanded in u0 around 0 87.9%

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

       1. +-commutative 87.9%

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

       2. neg-mul-1 87.9%

          \[\leadsto \left(-\left(-0.5 \cdot {u0}^{2} + \color{blue}{\left(-u0\right)}\right)\right) \cdot \left(\frac{alphax}{alphay \cdot \frac{-cos2phi}{alphax} - sin2phi \cdot \frac{alphax}{alphay}} \cdot \left(-alphay\right)\right) \]

       3. unsub-neg 87.9%

          \[\leadsto \left(-\color{blue}{\left(-0.5 \cdot {u0}^{2} - u0\right)}\right) \cdot \left(\frac{alphax}{alphay \cdot \frac{-cos2phi}{alphax} - sin2phi \cdot \frac{alphax}{alphay}} \cdot \left(-alphay\right)\right) \]

       4. unpow2 87.9%

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

   18. Simplified 87.9%

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

   if 0.550000012 < (/.f32 sin2phi (*.f32 alphay alphay))

   1. Initial program 63.0%

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

      1. neg-sub0 63.0%

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

      2. div-sub 63.0%

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

      3. --rgt-identity 63.0%

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

      4. div-sub 63.0%

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

      5. --rgt-identity 63.0%

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

      6. neg-sub0 63.0%

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

      7. sub-neg 63.0%

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

      8. log1p-def 98.0%

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

   3. Simplified 98.0%

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

      1. +-commutative 98.0%

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

      2. associate-/r* 97.8%

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

      3. associate-/r* 97.7%

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

      4. frac-2neg 97.7%

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

      5. frac-add 97.6%

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

      6. distribute-neg-frac 97.6%

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

   5. Applied egg-rr 97.6%

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

      1. +-commutative 97.6%

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

      2. distribute-rgt-neg-out 97.6%

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

      3. unsub-neg 97.6%

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

      4. associate-*l/ 97.7%

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

      5. associate-/l* 97.6%

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

      6. *-commutative 97.6%

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

      7. distribute-lft-neg-out 97.6%

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

      8. distribute-rgt-neg-in 97.6%

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

   7. Simplified 97.6%

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

      1. div-inv 97.4%

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

      2. associate-/r/ 97.4%

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

      3. distribute-rgt-neg-out 97.4%

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

   9. Applied egg-rr 97.4%

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

       1. associate-/r/ 97.9%

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

       2. associate-*l/ 98.6%

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

       3. *-commutative 98.6%

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

       4. distribute-rgt-neg-in 98.6%

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

   11. Simplified 98.6%

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

       1. associate-*l/ 98.8%

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

       2. *-un-lft-identity 98.8%

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

       3. associate-/l* 98.7%

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

   13. Applied egg-rr 98.7%

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

       1. associate-/l* 98.5%

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

       2. associate-/r/ 99.0%

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

       3. associate-/r/ 98.9%

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

       4. *-commutative 98.9%

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

   15. Simplified 98.9%

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

   16. Taylor expanded in alphax around inf 62.8%

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

       1. associate-*r/ 62.8%

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

       2. mul-1-neg 62.8%

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

       3. sub-neg 62.8%

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

       4. log1p-def 97.1%

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

       5. *-commutative 97.1%

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

       6. distribute-lft-neg-in 97.1%

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

       7. associate-*l/ 97.1%

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

       8. *-commutative 97.1%

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

       9. unpow2 97.1%

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

   18. Simplified 97.1%

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

4. Final simplification 92.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 0.550000011920929:\\ \;\;\;\;\left(-0.5 \cdot \left(u0 \cdot u0\right) - u0\right) \cdot \left(alphay \cdot \frac{alphax}{alphay \cdot \frac{-cos2phi}{alphax} - sin2phi \cdot \frac{alphax}{alphay}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(-u0\right)}{sin2phi} \cdot \left(alphay \cdot \left(-alphay\right)\right)\\ \end{array} \]

Alternative 6: 92.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 0.550000011920929:\\ \;\;\;\;\left(-0.5 \cdot \left(u0 \cdot u0\right) - u0\right) \cdot \left(alphay \cdot \frac{alphax}{alphay \cdot \frac{-cos2phi}{alphax} - sin2phi \cdot \frac{alphax}{alphay}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{alphay \cdot \left(\mathsf{log1p}\left(-u0\right) \cdot \left(-alphay\right)\right)}{sin2phi}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (if (<= (/ sin2phi (* alphay alphay)) 0.550000011920929)
   (*
    (- (* -0.5 (* u0 u0)) u0)
    (*
     alphay
     (/
      alphax
      (- (* alphay (/ (- cos2phi) alphax)) (* sin2phi (/ alphax alphay))))))
   (/ (* alphay (* (log1p (- u0)) (- alphay))) sin2phi)))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float tmp;
	if ((sin2phi / (alphay * alphay)) <= 0.550000011920929f) {
		tmp = ((-0.5f * (u0 * u0)) - u0) * (alphay * (alphax / ((alphay * (-cos2phi / alphax)) - (sin2phi * (alphax / alphay)))));
	} else {
		tmp = (alphay * (log1pf(-u0) * -alphay)) / sin2phi;
	}
	return tmp;
}

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = Float32(0.0)
	if (Float32(sin2phi / Float32(alphay * alphay)) <= Float32(0.550000011920929))
		tmp = Float32(Float32(Float32(Float32(-0.5) * Float32(u0 * u0)) - u0) * Float32(alphay * Float32(alphax / Float32(Float32(alphay * Float32(Float32(-cos2phi) / alphax)) - Float32(sin2phi * Float32(alphax / alphay))))));
	else
		tmp = Float32(Float32(alphay * Float32(log1p(Float32(-u0)) * Float32(-alphay))) / sin2phi);
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 sin2phi (*.f32 alphay alphay)) < 0.550000012

   1. Initial program 56.0%

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

      1. neg-sub0 56.0%

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

      2. div-sub 56.0%

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

      3. --rgt-identity 56.0%

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

      4. div-sub 56.0%

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

      5. --rgt-identity 56.0%

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

      6. neg-sub0 56.0%

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

      7. sub-neg 56.0%

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

      8. log1p-def 98.6%

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

   3. Simplified 98.6%

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

      1. +-commutative 98.6%

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

      2. associate-/r* 98.6%

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

      3. associate-/r* 98.6%

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

      4. frac-2neg 98.6%

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

      5. frac-add 98.6%

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

      6. distribute-neg-frac 98.6%

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

   5. Applied egg-rr 98.6%

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

      1. +-commutative 98.6%

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

      2. distribute-rgt-neg-out 98.6%

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

      3. unsub-neg 98.6%

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

      4. associate-*l/ 98.6%

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

      5. associate-/l* 98.6%

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

      6. *-commutative 98.6%

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

      7. distribute-lft-neg-out 98.6%

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

      8. distribute-rgt-neg-in 98.6%

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

   7. Simplified 98.6%

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

      1. div-inv 98.3%

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

      2. associate-/r/ 98.4%

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

      3. distribute-rgt-neg-out 98.4%

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

   9. Applied egg-rr 98.4%

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

       1. associate-/r/ 98.3%

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

       2. associate-*l/ 98.2%

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

       3. *-commutative 98.2%

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

       4. distribute-rgt-neg-in 98.2%

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

   11. Simplified 98.2%

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

       1. associate-*l/ 98.3%

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

       2. *-un-lft-identity 98.3%

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

       3. associate-/l* 98.3%

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

   13. Applied egg-rr 98.3%

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

       1. associate-/l* 98.3%

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

       2. associate-/r/ 98.5%

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

       3. associate-/r/ 98.5%

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

       4. *-commutative 98.5%

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

   15. Simplified 98.5%

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

   16. Taylor expanded in u0 around 0 87.9%

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

       1. +-commutative 87.9%

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

       2. neg-mul-1 87.9%

          \[\leadsto \left(-\left(-0.5 \cdot {u0}^{2} + \color{blue}{\left(-u0\right)}\right)\right) \cdot \left(\frac{alphax}{alphay \cdot \frac{-cos2phi}{alphax} - sin2phi \cdot \frac{alphax}{alphay}} \cdot \left(-alphay\right)\right) \]

       3. unsub-neg 87.9%

          \[\leadsto \left(-\color{blue}{\left(-0.5 \cdot {u0}^{2} - u0\right)}\right) \cdot \left(\frac{alphax}{alphay \cdot \frac{-cos2phi}{alphax} - sin2phi \cdot \frac{alphax}{alphay}} \cdot \left(-alphay\right)\right) \]

       4. unpow2 87.9%

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

   18. Simplified 87.9%

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

   if 0.550000012 < (/.f32 sin2phi (*.f32 alphay alphay))

   1. Initial program 63.0%

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

      1. neg-sub0 63.0%

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

      2. div-sub 63.0%

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

      3. --rgt-identity 63.0%

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

      4. div-sub 63.0%

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

      5. --rgt-identity 63.0%

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

      6. neg-sub0 63.0%

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

      7. sub-neg 63.0%

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

      8. log1p-def 98.0%

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

   3. Simplified 98.0%

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

      1. +-commutative 98.0%

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

      2. associate-/r* 97.8%

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

      3. associate-/r* 97.7%

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

      4. frac-2neg 97.7%

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

      5. frac-add 97.6%

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

      6. distribute-neg-frac 97.6%

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

   5. Applied egg-rr 97.6%

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

      1. +-commutative 97.6%

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

      2. distribute-rgt-neg-out 97.6%

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

      3. unsub-neg 97.6%

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

      4. associate-*l/ 97.7%

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

      5. associate-/l* 97.6%

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

      6. *-commutative 97.6%

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

      7. distribute-lft-neg-out 97.6%

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

      8. distribute-rgt-neg-in 97.6%

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

   7. Simplified 97.6%

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

      1. div-inv 97.4%

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

      2. associate-/r/ 97.4%

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

      3. distribute-rgt-neg-out 97.4%

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

   9. Applied egg-rr 97.4%

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

       1. associate-/r/ 97.9%

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

       2. associate-*l/ 98.6%

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

       3. *-commutative 98.6%

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

       4. distribute-rgt-neg-in 98.6%

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

   11. Simplified 98.6%

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

   12. Taylor expanded in alphay around 0 62.8%

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

       1. associate-*r/ 62.8%

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

       2. *-commutative 62.8%

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

       3. associate-*r* 62.8%

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

       4. sub-neg 62.8%

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

       5. log1p-def 97.1%

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

       6. neg-mul-1 97.1%

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

       7. unpow2 97.1%

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

   14. Simplified 97.1%

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

   15. Taylor expanded in alphay around 0 62.8%

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

       1. mul-1-neg 62.8%

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

       2. distribute-rgt-neg-in 62.8%

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

       3. sub-neg 62.8%

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

       4. log1p-def 97.1%

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

       5. unpow2 97.1%

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

       6. associate-*l* 97.2%

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

   17. Simplified 97.2%

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

4. Final simplification 92.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 0.550000011920929:\\ \;\;\;\;\left(-0.5 \cdot \left(u0 \cdot u0\right) - u0\right) \cdot \left(alphay \cdot \frac{alphax}{alphay \cdot \frac{-cos2phi}{alphax} - sin2phi \cdot \frac{alphax}{alphay}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{alphay \cdot \left(\mathsf{log1p}\left(-u0\right) \cdot \left(-alphay\right)\right)}{sin2phi}\\ \end{array} \]

Alternative 7: 98.4% accurate, 1.0× speedup

Math

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

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(Float32(-log1p(Float32(-u0))) / Float32(Float32(sin2phi / Float32(alphay * alphay)) + Float32(cos2phi / Float32(alphax * alphax))))
end

Derivation

1. Initial program 59.8%

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

   1. neg-sub0 59.8%

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

   2. div-sub 59.8%

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

   3. --rgt-identity 59.8%

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

   4. div-sub 59.8%

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

   5. --rgt-identity 59.8%

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

   6. neg-sub0 59.8%

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

   7. sub-neg 59.8%

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

   8. log1p-def 98.3%

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

3. Simplified 98.3%

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

4. Final simplification 98.3%

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

Alternative 8: 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(Float32(-log1p(Float32(-u0))) / Float32(Float32(sin2phi / Float32(alphay * alphay)) + Float32(Float32(cos2phi / alphax) / alphax)))
end

Derivation

1. Initial program 59.8%

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

   1. neg-sub0 59.8%

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

   2. div-sub 59.8%

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

   3. --rgt-identity 59.8%

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

   4. div-sub 59.8%

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

   5. --rgt-identity 59.8%

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

   6. sub-neg 59.8%

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

   7. +-commutative 59.8%

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

   8. neg-sub0 59.8%

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

   9. associate-+l- 59.8%

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

   10. sub0-neg 59.8%

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

   11. neg-mul-1 59.8%

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

   12. log-prod -0.0%

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

   13. associate--r+ -0.0%

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

3. Simplified 98.3%

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

4. Final simplification 98.3%

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

Alternative 9: 87.5% accurate, 4.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 59.8%

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

   1. neg-sub0 59.8%

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

   2. div-sub 59.8%

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

   3. --rgt-identity 59.8%

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

   4. div-sub 59.8%

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

   5. --rgt-identity 59.8%

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

   6. neg-sub0 59.8%

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

   7. sub-neg 59.8%

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

   8. log1p-def 98.3%

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

3. Simplified 98.3%

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

   1. +-commutative 98.3%

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

   2. associate-/r* 98.1%

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

   3. associate-/r* 98.1%

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

   4. frac-2neg 98.1%

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

   5. frac-add 98.1%

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

   6. distribute-neg-frac 98.1%

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

5. Applied egg-rr 98.1%

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

   1. +-commutative 98.1%

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

   2. distribute-rgt-neg-out 98.1%

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

   3. unsub-neg 98.1%

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

   4. associate-*l/ 98.1%

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

   5. associate-/l* 98.1%

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

   6. *-commutative 98.1%

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

   7. distribute-lft-neg-out 98.1%

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

   8. distribute-rgt-neg-in 98.1%

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

7. Simplified 98.1%

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

   1. div-inv 97.8%

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

   2. associate-/r/ 97.9%

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

   3. distribute-rgt-neg-out 97.9%

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

9. Applied egg-rr 97.9%

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

    1. associate-/r/ 98.1%

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

    2. associate-*l/ 98.4%

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

    3. *-commutative 98.4%

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

    4. distribute-rgt-neg-in 98.4%

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

11. Simplified 98.4%

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

    1. associate-*l/ 98.6%

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

    2. *-un-lft-identity 98.6%

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

    3. associate-/l* 98.5%

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

13. Applied egg-rr 98.5%

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

    1. associate-/l* 98.4%

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

    2. associate-/r/ 98.7%

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

    3. associate-/r/ 98.7%

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

    4. *-commutative 98.7%

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

15. Simplified 98.7%

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

16. Taylor expanded in u0 around 0 88.4%

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

    1. +-commutative 88.4%

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

    2. neg-mul-1 88.4%

       \[\leadsto \left(-\left(-0.5 \cdot {u0}^{2} + \color{blue}{\left(-u0\right)}\right)\right) \cdot \left(\frac{alphax}{alphay \cdot \frac{-cos2phi}{alphax} - sin2phi \cdot \frac{alphax}{alphay}} \cdot \left(-alphay\right)\right) \]

    3. unsub-neg 88.4%

       \[\leadsto \left(-\color{blue}{\left(-0.5 \cdot {u0}^{2} - u0\right)}\right) \cdot \left(\frac{alphax}{alphay \cdot \frac{-cos2phi}{alphax} - sin2phi \cdot \frac{alphax}{alphay}} \cdot \left(-alphay\right)\right) \]

    4. unpow2 88.4%

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

18. Simplified 88.4%

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

19. Final simplification 88.4%

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

Alternative 10: 81.4% accurate, 5.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{sin2phi}{alphay \cdot alphay}\\ \mathbf{if}\;t_0 \leq 0.550000011920929:\\ \;\;\;\;\frac{u0}{t_0 + cos2phi \cdot \frac{1}{alphax \cdot alphax}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(alphay \cdot alphay\right) \cdot \left(u0 + \left(u0 \cdot u0\right) \cdot 0.5\right)}{sin2phi}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (let* ((t_0 (/ sin2phi (* alphay alphay))))
   (if (<= t_0 0.550000011920929)
     (/ u0 (+ t_0 (* cos2phi (/ 1.0 (* alphax alphax)))))
     (/ (* (* alphay alphay) (+ u0 (* (* u0 u0) 0.5))) sin2phi))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float t_0 = sin2phi / (alphay * alphay);
	float tmp;
	if (t_0 <= 0.550000011920929f) {
		tmp = u0 / (t_0 + (cos2phi * (1.0f / (alphax * alphax))));
	} else {
		tmp = ((alphay * alphay) * (u0 + ((u0 * u0) * 0.5f))) / sin2phi;
	}
	return tmp;
}

Fortran

real(4) function code(alphax, alphay, u0, cos2phi, sin2phi)
    real(4), intent (in) :: alphax
    real(4), intent (in) :: alphay
    real(4), intent (in) :: u0
    real(4), intent (in) :: cos2phi
    real(4), intent (in) :: sin2phi
    real(4) :: t_0
    real(4) :: tmp
    t_0 = sin2phi / (alphay * alphay)
    if (t_0 <= 0.550000011920929e0) then
        tmp = u0 / (t_0 + (cos2phi * (1.0e0 / (alphax * alphax))))
    else
        tmp = ((alphay * alphay) * (u0 + ((u0 * u0) * 0.5e0))) / sin2phi
    end if
    code = tmp
end function

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	t_0 = Float32(sin2phi / Float32(alphay * alphay))
	tmp = Float32(0.0)
	if (t_0 <= Float32(0.550000011920929))
		tmp = Float32(u0 / Float32(t_0 + Float32(cos2phi * Float32(Float32(1.0) / Float32(alphax * alphax)))));
	else
		tmp = Float32(Float32(Float32(alphay * alphay) * Float32(u0 + Float32(Float32(u0 * u0) * Float32(0.5)))) / sin2phi);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alphax, alphay, u0, cos2phi, sin2phi)
	t_0 = sin2phi / (alphay * alphay);
	tmp = single(0.0);
	if (t_0 <= single(0.550000011920929))
		tmp = u0 / (t_0 + (cos2phi * (single(1.0) / (alphax * alphax))));
	else
		tmp = ((alphay * alphay) * (u0 + ((u0 * u0) * single(0.5)))) / sin2phi;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 sin2phi (*.f32 alphay alphay)) < 0.550000012

   1. Initial program 56.0%

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

      1. associate-/r* 56.0%

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

   3. Simplified 56.0%

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

   4. Taylor expanded in u0 around 0 75.1%

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

      1. unpow2 75.1%

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

      2. unpow2 75.1%

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

   6. Simplified 75.1%

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

      1. div-inv 75.1%

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

   8. Applied egg-rr 75.1%

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

   if 0.550000012 < (/.f32 sin2phi (*.f32 alphay alphay))

   1. Initial program 63.0%

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

      1. neg-sub0 63.0%

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

      2. div-sub 63.0%

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

      3. --rgt-identity 63.0%

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

      4. div-sub 63.0%

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

      5. --rgt-identity 63.0%

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

      6. neg-sub0 63.0%

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

      7. sub-neg 63.0%

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

      8. log1p-def 98.0%

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

   3. Simplified 98.0%

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

      1. +-commutative 98.0%

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

      2. associate-/r* 97.8%

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

      3. associate-/r* 97.7%

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

      4. frac-2neg 97.7%

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

      5. frac-add 97.6%

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

      6. distribute-neg-frac 97.6%

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

   5. Applied egg-rr 97.6%

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

      1. +-commutative 97.6%

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

      2. distribute-rgt-neg-out 97.6%

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

      3. unsub-neg 97.6%

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

      4. associate-*l/ 97.7%

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

      5. associate-/l* 97.6%

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

      6. *-commutative 97.6%

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

      7. distribute-lft-neg-out 97.6%

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

      8. distribute-rgt-neg-in 97.6%

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

   7. Simplified 97.6%

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

      1. div-inv 97.4%

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

      2. associate-/r/ 97.4%

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

      3. distribute-rgt-neg-out 97.4%

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

   9. Applied egg-rr 97.4%

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

       1. associate-/r/ 97.9%

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

       2. associate-*l/ 98.6%

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

       3. *-commutative 98.6%

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

       4. distribute-rgt-neg-in 98.6%

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

   11. Simplified 98.6%

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

   12. Taylor expanded in alphay around 0 62.8%

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

       1. associate-*r/ 62.8%

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

       2. *-commutative 62.8%

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

       3. associate-*r* 62.8%

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

       4. sub-neg 62.8%

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

       5. log1p-def 97.1%

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

       6. neg-mul-1 97.1%

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

       7. unpow2 97.1%

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

   14. Simplified 97.1%

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

   15. Taylor expanded in u0 around 0 87.9%

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

       1. associate-*r* 87.9%

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

       2. distribute-rgt-out 88.0%

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

       3. unpow2 88.0%

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

       4. *-commutative 88.0%

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

       5. unpow2 88.0%

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

   17. Simplified 88.0%

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

4. Final simplification 82.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 0.550000011920929:\\ \;\;\;\;\frac{u0}{\frac{sin2phi}{alphay \cdot alphay} + cos2phi \cdot \frac{1}{alphax \cdot alphax}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(alphay \cdot alphay\right) \cdot \left(u0 + \left(u0 \cdot u0\right) \cdot 0.5\right)}{sin2phi}\\ \end{array} \]

Alternative 11: 81.4% accurate, 6.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{sin2phi}{alphay \cdot alphay}\\ \mathbf{if}\;t_0 \leq 0.550000011920929:\\ \;\;\;\;\frac{u0}{t_0 + \frac{cos2phi}{alphax \cdot alphax}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(alphay \cdot alphay\right) \cdot \left(u0 + \left(u0 \cdot u0\right) \cdot 0.5\right)}{sin2phi}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (let* ((t_0 (/ sin2phi (* alphay alphay))))
   (if (<= t_0 0.550000011920929)
     (/ u0 (+ t_0 (/ cos2phi (* alphax alphax))))
     (/ (* (* alphay alphay) (+ u0 (* (* u0 u0) 0.5))) sin2phi))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float t_0 = sin2phi / (alphay * alphay);
	float tmp;
	if (t_0 <= 0.550000011920929f) {
		tmp = u0 / (t_0 + (cos2phi / (alphax * alphax)));
	} else {
		tmp = ((alphay * alphay) * (u0 + ((u0 * u0) * 0.5f))) / sin2phi;
	}
	return tmp;
}

Fortran

real(4) function code(alphax, alphay, u0, cos2phi, sin2phi)
    real(4), intent (in) :: alphax
    real(4), intent (in) :: alphay
    real(4), intent (in) :: u0
    real(4), intent (in) :: cos2phi
    real(4), intent (in) :: sin2phi
    real(4) :: t_0
    real(4) :: tmp
    t_0 = sin2phi / (alphay * alphay)
    if (t_0 <= 0.550000011920929e0) then
        tmp = u0 / (t_0 + (cos2phi / (alphax * alphax)))
    else
        tmp = ((alphay * alphay) * (u0 + ((u0 * u0) * 0.5e0))) / sin2phi
    end if
    code = tmp
end function

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	t_0 = Float32(sin2phi / Float32(alphay * alphay))
	tmp = Float32(0.0)
	if (t_0 <= Float32(0.550000011920929))
		tmp = Float32(u0 / Float32(t_0 + Float32(cos2phi / Float32(alphax * alphax))));
	else
		tmp = Float32(Float32(Float32(alphay * alphay) * Float32(u0 + Float32(Float32(u0 * u0) * Float32(0.5)))) / sin2phi);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alphax, alphay, u0, cos2phi, sin2phi)
	t_0 = sin2phi / (alphay * alphay);
	tmp = single(0.0);
	if (t_0 <= single(0.550000011920929))
		tmp = u0 / (t_0 + (cos2phi / (alphax * alphax)));
	else
		tmp = ((alphay * alphay) * (u0 + ((u0 * u0) * single(0.5)))) / sin2phi;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 sin2phi (*.f32 alphay alphay)) < 0.550000012

   1. Initial program 56.0%

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

      1. associate-/r* 56.0%

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

   3. Simplified 56.0%

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

   4. Taylor expanded in u0 around 0 75.1%

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

      1. unpow2 75.1%

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

      2. unpow2 75.1%

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

   6. Simplified 75.1%

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

   if 0.550000012 < (/.f32 sin2phi (*.f32 alphay alphay))

   1. Initial program 63.0%

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

      1. neg-sub0 63.0%

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

      2. div-sub 63.0%

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

      3. --rgt-identity 63.0%

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

      4. div-sub 63.0%

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

      5. --rgt-identity 63.0%

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

      6. neg-sub0 63.0%

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

      7. sub-neg 63.0%

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

      8. log1p-def 98.0%

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

   3. Simplified 98.0%

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

      1. +-commutative 98.0%

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

      2. associate-/r* 97.8%

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

      3. associate-/r* 97.7%

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

      4. frac-2neg 97.7%

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

      5. frac-add 97.6%

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

      6. distribute-neg-frac 97.6%

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

   5. Applied egg-rr 97.6%

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

      1. +-commutative 97.6%

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

      2. distribute-rgt-neg-out 97.6%

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

      3. unsub-neg 97.6%

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

      4. associate-*l/ 97.7%

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

      5. associate-/l* 97.6%

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

      6. *-commutative 97.6%

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

      7. distribute-lft-neg-out 97.6%

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

      8. distribute-rgt-neg-in 97.6%

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

   7. Simplified 97.6%

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

      1. div-inv 97.4%

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

      2. associate-/r/ 97.4%

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

      3. distribute-rgt-neg-out 97.4%

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

   9. Applied egg-rr 97.4%

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

       1. associate-/r/ 97.9%

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

       2. associate-*l/ 98.6%

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

       3. *-commutative 98.6%

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

       4. distribute-rgt-neg-in 98.6%

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

   11. Simplified 98.6%

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

   12. Taylor expanded in alphay around 0 62.8%

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

       1. associate-*r/ 62.8%

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

       2. *-commutative 62.8%

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

       3. associate-*r* 62.8%

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

       4. sub-neg 62.8%

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

       5. log1p-def 97.1%

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

       6. neg-mul-1 97.1%

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

       7. unpow2 97.1%

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

   14. Simplified 97.1%

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

   15. Taylor expanded in u0 around 0 87.9%

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

       1. associate-*r* 87.9%

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

       2. distribute-rgt-out 88.0%

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

       3. unpow2 88.0%

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

       4. *-commutative 88.0%

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

       5. unpow2 88.0%

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

   17. Simplified 88.0%

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

4. Final simplification 82.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 0.550000011920929:\\ \;\;\;\;\frac{u0}{\frac{sin2phi}{alphay \cdot alphay} + \frac{cos2phi}{alphax \cdot alphax}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(alphay \cdot alphay\right) \cdot \left(u0 + \left(u0 \cdot u0\right) \cdot 0.5\right)}{sin2phi}\\ \end{array} \]

Alternative 12: 67.4% accurate, 8.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 2.0000000072549875 \cdot 10^{-15}:\\ \;\;\;\;alphax \cdot \frac{u0 \cdot alphax}{cos2phi}\\ \mathbf{else}:\\ \;\;\;\;\left(alphay \cdot alphay\right) \cdot \frac{u0}{sin2phi}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (if (<= (/ sin2phi (* alphay alphay)) 2.0000000072549875e-15)
   (* alphax (/ (* u0 alphax) cos2phi))
   (* (* alphay alphay) (/ u0 sin2phi))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float tmp;
	if ((sin2phi / (alphay * alphay)) <= 2.0000000072549875e-15f) {
		tmp = alphax * ((u0 * alphax) / cos2phi);
	} else {
		tmp = (alphay * alphay) * (u0 / sin2phi);
	}
	return tmp;
}

Fortran

real(4) function code(alphax, alphay, u0, cos2phi, sin2phi)
    real(4), intent (in) :: alphax
    real(4), intent (in) :: alphay
    real(4), intent (in) :: u0
    real(4), intent (in) :: cos2phi
    real(4), intent (in) :: sin2phi
    real(4) :: tmp
    if ((sin2phi / (alphay * alphay)) <= 2.0000000072549875e-15) then
        tmp = alphax * ((u0 * alphax) / cos2phi)
    else
        tmp = (alphay * alphay) * (u0 / sin2phi)
    end if
    code = tmp
end function

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = Float32(0.0)
	if (Float32(sin2phi / Float32(alphay * alphay)) <= Float32(2.0000000072549875e-15))
		tmp = Float32(alphax * Float32(Float32(u0 * alphax) / cos2phi));
	else
		tmp = Float32(Float32(alphay * alphay) * Float32(u0 / sin2phi));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = single(0.0);
	if ((sin2phi / (alphay * alphay)) <= single(2.0000000072549875e-15))
		tmp = alphax * ((u0 * alphax) / cos2phi);
	else
		tmp = (alphay * alphay) * (u0 / sin2phi);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

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

   1. Initial program 57.7%

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

      1. associate-/r* 57.7%

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

   3. Simplified 57.7%

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

   4. Taylor expanded in u0 around 0 74.8%

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

      1. unpow2 74.8%

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

      2. unpow2 74.8%

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

   6. Simplified 74.8%

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

   7. Taylor expanded in cos2phi around inf 58.6%

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

      1. pow2 58.6%

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

      2. *-commutative 58.6%

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

      3. associate-*r/ 58.5%

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

      4. associate-*l* 58.5%

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

   9. Applied egg-rr 58.5%

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

   10. Taylor expanded in alphax around 0 58.6%

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

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

   1. Initial program 60.4%

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

      1. associate-/r* 60.4%

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

   3. Simplified 60.4%

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

   4. Taylor expanded in u0 around 0 76.7%

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

      1. unpow2 76.7%

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

      2. unpow2 76.7%

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

   6. Simplified 76.7%

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

   7. Taylor expanded in cos2phi around 0 70.4%

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

      1. unpow2 70.4%

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

      2. *-commutative 70.4%

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

      3. *-lft-identity 70.4%

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

      4. times-frac 70.3%

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

      5. /-rgt-identity 70.3%

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

   9. Simplified 70.3%

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

4. Final simplification 67.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 2.0000000072549875 \cdot 10^{-15}:\\ \;\;\;\;alphax \cdot \frac{u0 \cdot alphax}{cos2phi}\\ \mathbf{else}:\\ \;\;\;\;\left(alphay \cdot alphay\right) \cdot \frac{u0}{sin2phi}\\ \end{array} \]

Alternative 13: 67.4% accurate, 8.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 2.0000000072549875 \cdot 10^{-15}:\\ \;\;\;\;alphax \cdot \frac{u0 \cdot alphax}{cos2phi}\\ \mathbf{else}:\\ \;\;\;\;\frac{u0 \cdot \left(alphay \cdot alphay\right)}{sin2phi}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (if (<= (/ sin2phi (* alphay alphay)) 2.0000000072549875e-15)
   (* alphax (/ (* u0 alphax) cos2phi))
   (/ (* u0 (* alphay alphay)) sin2phi)))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float tmp;
	if ((sin2phi / (alphay * alphay)) <= 2.0000000072549875e-15f) {
		tmp = alphax * ((u0 * alphax) / cos2phi);
	} else {
		tmp = (u0 * (alphay * alphay)) / sin2phi;
	}
	return tmp;
}

Fortran

real(4) function code(alphax, alphay, u0, cos2phi, sin2phi)
    real(4), intent (in) :: alphax
    real(4), intent (in) :: alphay
    real(4), intent (in) :: u0
    real(4), intent (in) :: cos2phi
    real(4), intent (in) :: sin2phi
    real(4) :: tmp
    if ((sin2phi / (alphay * alphay)) <= 2.0000000072549875e-15) then
        tmp = alphax * ((u0 * alphax) / cos2phi)
    else
        tmp = (u0 * (alphay * alphay)) / sin2phi
    end if
    code = tmp
end function

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = Float32(0.0)
	if (Float32(sin2phi / Float32(alphay * alphay)) <= Float32(2.0000000072549875e-15))
		tmp = Float32(alphax * Float32(Float32(u0 * alphax) / cos2phi));
	else
		tmp = Float32(Float32(u0 * Float32(alphay * alphay)) / sin2phi);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = single(0.0);
	if ((sin2phi / (alphay * alphay)) <= single(2.0000000072549875e-15))
		tmp = alphax * ((u0 * alphax) / cos2phi);
	else
		tmp = (u0 * (alphay * alphay)) / sin2phi;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

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

   1. Initial program 57.7%

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

      1. associate-/r* 57.7%

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

   3. Simplified 57.7%

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

   4. Taylor expanded in u0 around 0 74.8%

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

      1. unpow2 74.8%

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

      2. unpow2 74.8%

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

   6. Simplified 74.8%

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

   7. Taylor expanded in cos2phi around inf 58.6%

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

      1. pow2 58.6%

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

      2. *-commutative 58.6%

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

      3. associate-*r/ 58.5%

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

      4. associate-*l* 58.5%

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

   9. Applied egg-rr 58.5%

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

   10. Taylor expanded in alphax around 0 58.6%

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

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

   1. Initial program 60.4%

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

      1. associate-/r* 60.4%

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

   3. Simplified 60.4%

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

   4. Taylor expanded in u0 around 0 76.7%

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

      1. unpow2 76.7%

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

      2. unpow2 76.7%

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

   6. Simplified 76.7%

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

      1. +-commutative 76.7%

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

      2. associate-/r* 76.6%

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

      3. frac-add 76.6%

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

   8. Applied egg-rr 76.6%

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

   9. Taylor expanded in sin2phi around inf 70.4%

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

       1. *-commutative 70.4%

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

       2. unpow2 70.4%

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

   11. Simplified 70.4%

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

4. Final simplification 67.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 2.0000000072549875 \cdot 10^{-15}:\\ \;\;\;\;alphax \cdot \frac{u0 \cdot alphax}{cos2phi}\\ \mathbf{else}:\\ \;\;\;\;\frac{u0 \cdot \left(alphay \cdot alphay\right)}{sin2phi}\\ \end{array} \]

Alternative 14: 75.8% 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 59.8%

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

   1. associate-/r* 59.8%

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

3. Simplified 59.8%

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

4. Taylor expanded in u0 around 0 76.2%

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

   1. unpow2 76.2%

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

   2. unpow2 76.2%

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

6. Simplified 76.2%

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

7. Final simplification 76.2%

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

Alternative 15: 23.9% accurate, 16.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 59.8%

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

   1. associate-/r* 59.8%

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

3. Simplified 59.8%

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

4. Taylor expanded in u0 around 0 76.2%

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

   1. unpow2 76.2%

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

   2. unpow2 76.2%

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

6. Simplified 76.2%

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

7. Taylor expanded in cos2phi around inf 24.8%

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

   1. pow2 24.8%

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

   2. *-commutative 24.8%

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

   3. associate-*r/ 24.8%

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

   4. associate-*l* 24.8%

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

9. Applied egg-rr 24.8%

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

10. Final simplification 24.8%

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

Alternative 16: 23.9% accurate, 16.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 59.8%

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

   1. associate-/r* 59.8%

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

3. Simplified 59.8%

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

4. Taylor expanded in u0 around 0 76.2%

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

   1. unpow2 76.2%

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

   2. unpow2 76.2%

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

6. Simplified 76.2%

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

7. Taylor expanded in cos2phi around inf 24.8%

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

   1. pow2 24.8%

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

   2. *-commutative 24.8%

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

   3. associate-*r/ 24.8%

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

   4. associate-*l* 24.8%

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

9. Applied egg-rr 24.8%

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

10. Taylor expanded in alphax around 0 24.8%

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

11. Final simplification 24.8%

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

Alternative 17: 23.9% accurate, 16.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 59.8%

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

   1. associate-/r* 59.8%

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

3. Simplified 59.8%

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

4. Taylor expanded in u0 around 0 76.2%

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

   1. unpow2 76.2%

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

   2. unpow2 76.2%

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

6. Simplified 76.2%

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

7. Taylor expanded in cos2phi around inf 24.8%

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

   1. *-commutative 24.8%

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

   2. *-lft-identity 24.8%

      \[\leadsto \frac{{alphax}^{2} \cdot u0}{\color{blue}{1 \cdot cos2phi}} \]

   3. times-frac 24.8%

      \[\leadsto \color{blue}{\frac{{alphax}^{2}}{1} \cdot \frac{u0}{cos2phi}} \]

   4. /-rgt-identity 24.8%

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

   5. unpow2 24.8%

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

9. Simplified 24.8%

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

10. Taylor expanded in alphax around 0 24.8%

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

    1. unpow2 24.8%

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

    2. associate-*r/ 24.8%

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

12. Simplified 24.8%

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

13. Final simplification 24.8%

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

Reproduce

herbie shell --seed 2023228 
(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)))))