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

Percentage Accurate: 60.5% → 98.5%

Time: 14.2s

Alternatives: 15

Speedup: 12.7×

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: 60.5% 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, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 63.5%

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

   1. neg-sub0 63.5%

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

   2. div-sub 63.5%

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

   3. --rgt-identity 63.5%

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

   4. div-sub 63.5%

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

   5. --rgt-identity 63.5%

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

   6. neg-sub0 63.5%

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

   7. sub-neg 63.5%

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

   8. log1p-def 98.2%

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

3. Simplified 98.2%

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

   1. +-commutative 98.2%

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

   2. associate-/r* 98.2%

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

   3. frac-add 97.9%

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

5. Applied egg-rr 97.9%

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

   1. *-commutative 97.9%

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

   2. fma-def 97.9%

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

   3. *-commutative 97.9%

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

   4. associate-*l* 97.6%

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

7. Simplified 97.6%

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

   1. expm1-log1p-u 96.1%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{-\mathsf{log1p}\left(-u0\right)}{\frac{\mathsf{fma}\left(\frac{sin2phi}{alphay}, alphax \cdot alphax, cos2phi \cdot alphay\right)}{alphax \cdot \left(alphax \cdot alphay\right)}}\right)\right)} \]

   2. expm1-udef 52.6%

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

   3. associate-/r/ 52.6%

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

9. Applied egg-rr 52.6%

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

    1. expm1-def 96.7%

       \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{-\mathsf{log1p}\left(-u0\right)}{\mathsf{fma}\left(\frac{sin2phi}{alphay}, alphax \cdot alphax, cos2phi \cdot alphay\right)} \cdot \left(alphax \cdot \left(alphax \cdot alphay\right)\right)\right)\right)} \]

    2. expm1-log1p 98.2%

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

    3. associate-*r* 98.5%

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

    4. *-commutative 98.5%

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

11. Simplified 98.5%

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

    1. div-inv 98.4%

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

    2. *-commutative 98.4%

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

13. Applied egg-rr 98.4%

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

    1. associate-*r/ 98.5%

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

    2. *-rgt-identity 98.5%

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

    3. fma-udef 98.5%

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

    4. unpow2 98.5%

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

    5. associate-*l/ 98.3%

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

    6. *-commutative 98.3%

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

    7. +-commutative 98.3%

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

    8. *-commutative 98.3%

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

    9. fma-def 98.2%

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

    10. *-commutative 98.2%

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

    11. associate-/l* 98.6%

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

    12. unpow2 98.6%

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

15. Simplified 98.6%

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

16. Final simplification 98.6%

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

Alternative 2: 98.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 63.5%

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

   1. neg-sub0 63.5%

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

   2. div-sub 63.5%

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

   3. --rgt-identity 63.5%

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

   4. div-sub 63.5%

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

   5. --rgt-identity 63.5%

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

   6. neg-sub0 63.5%

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

   7. sub-neg 63.5%

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

   8. log1p-def 98.2%

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

3. Simplified 98.2%

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

   1. +-commutative 98.2%

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

   2. associate-/r* 98.2%

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

   3. frac-add 97.9%

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

5. Applied egg-rr 97.9%

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

   1. *-commutative 97.9%

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

   2. fma-def 97.9%

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

   3. *-commutative 97.9%

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

   4. associate-*l* 97.6%

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

7. Simplified 97.6%

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

   1. expm1-log1p-u 96.1%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{-\mathsf{log1p}\left(-u0\right)}{\frac{\mathsf{fma}\left(\frac{sin2phi}{alphay}, alphax \cdot alphax, cos2phi \cdot alphay\right)}{alphax \cdot \left(alphax \cdot alphay\right)}}\right)\right)} \]

   2. expm1-udef 52.6%

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

   3. associate-/r/ 52.6%

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

9. Applied egg-rr 52.6%

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

    1. expm1-def 96.7%

       \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{-\mathsf{log1p}\left(-u0\right)}{\mathsf{fma}\left(\frac{sin2phi}{alphay}, alphax \cdot alphax, cos2phi \cdot alphay\right)} \cdot \left(alphax \cdot \left(alphax \cdot alphay\right)\right)\right)\right)} \]

    2. expm1-log1p 98.2%

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

    3. associate-*r* 98.5%

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

    4. *-commutative 98.5%

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

11. Simplified 98.5%

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

    1. fma-udef 98.5%

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

13. Applied egg-rr 98.5%

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

14. Final simplification 98.5%

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

Alternative 3: 87.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 0.20000000298023224:\\ \;\;\;\;\frac{u0}{\frac{alphay \cdot \frac{-cos2phi}{alphax} - \frac{sin2phi}{\frac{alphay}{alphax}}}{alphay \cdot \left(-alphax\right)}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(-u0\right) \cdot \frac{-alphay}{\frac{sin2phi}{alphay}}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (if (<= (/ sin2phi (* alphay alphay)) 0.20000000298023224)
   (/
    u0
    (/
     (- (* alphay (/ (- cos2phi) alphax)) (/ sin2phi (/ alphay alphax)))
     (* alphay (- alphax))))
   (* (log1p (- u0)) (/ (- alphay) (/ sin2phi alphay)))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float tmp;
	if ((sin2phi / (alphay * alphay)) <= 0.20000000298023224f) {
		tmp = u0 / (((alphay * (-cos2phi / alphax)) - (sin2phi / (alphay / alphax))) / (alphay * -alphax));
	} 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(0.20000000298023224))
		tmp = Float32(u0 / Float32(Float32(Float32(alphay * Float32(Float32(-cos2phi) / alphax)) - Float32(sin2phi / Float32(alphay / alphax))) / Float32(alphay * Float32(-alphax))));
	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)) < 0.200000003

   1. Initial program 55.1%

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

      1. associate-/r* 55.1%

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

   3. Simplified 55.1%

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

   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. Step-by-step derivation

      1. +-commutative 74.8%

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

      2. associate-/r* 74.8%

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

      3. associate-/r* 74.8%

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

      4. frac-2neg 74.8%

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

      5. frac-add 74.8%

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

      6. distribute-neg-frac 74.8%

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

   8. Applied egg-rr 74.8%

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

      1. +-commutative 74.8%

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

      2. distribute-rgt-neg-out 74.8%

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

      3. unsub-neg 74.8%

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

      4. associate-*l/ 74.8%

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

      5. associate-/l* 74.9%

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

      6. *-commutative 74.9%

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

      7. distribute-lft-neg-out 74.9%

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

      8. distribute-rgt-neg-in 74.9%

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

   10. Simplified 74.9%

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

   if 0.200000003 < (/.f32 sin2phi (*.f32 alphay alphay))

   1. Initial program 69.3%

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

      1. neg-sub0 69.3%

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

      2. div-sub 69.3%

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

      3. --rgt-identity 69.3%

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

      4. div-sub 69.3%

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

      5. --rgt-identity 69.3%

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

      6. neg-sub0 69.3%

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

      7. sub-neg 69.3%

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

      8. log1p-def 97.9%

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

   3. Simplified 97.9%

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

      1. +-commutative 97.9%

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

      2. associate-/r* 97.9%

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

      3. frac-add 97.6%

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

   5. Applied egg-rr 97.6%

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

      1. *-commutative 97.6%

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

      2. fma-def 97.6%

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

      3. *-commutative 97.6%

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

      4. associate-*l* 97.2%

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

   7. Simplified 97.2%

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

      1. expm1-log1p-u 97.2%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{-\mathsf{log1p}\left(-u0\right)}{\frac{\mathsf{fma}\left(\frac{sin2phi}{alphay}, alphax \cdot alphax, cos2phi \cdot alphay\right)}{alphax \cdot \left(alphax \cdot alphay\right)}}\right)\right)} \]

      2. expm1-udef 31.7%

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

      3. associate-/r/ 31.8%

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

   9. Applied egg-rr 31.8%

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

       1. expm1-def 98.2%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{-\mathsf{log1p}\left(-u0\right)}{\mathsf{fma}\left(\frac{sin2phi}{alphay}, alphax \cdot alphax, cos2phi \cdot alphay\right)} \cdot \left(alphax \cdot \left(alphax \cdot alphay\right)\right)\right)\right)} \]

       2. expm1-log1p 98.2%

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

       3. associate-*r* 98.6%

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

       4. *-commutative 98.6%

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

   11. Simplified 98.6%

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

   12. Taylor expanded in sin2phi around inf 68.7%

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

       1. mul-1-neg 68.7%

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

       2. associate-/l* 67.7%

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

       3. unpow2 67.7%

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

   14. Simplified 67.7%

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

   15. Taylor expanded in alphay around 0 68.7%

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

       1. *-commutative 68.7%

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

       2. associate-*r/ 68.7%

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

       3. sub-neg 68.7%

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

       4. log1p-def 96.2%

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

       5. unpow2 96.2%

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

       6. associate-/l* 96.2%

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

   17. Simplified 96.2%

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

4. Final simplification 87.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 0.20000000298023224:\\ \;\;\;\;\frac{u0}{\frac{alphay \cdot \frac{-cos2phi}{alphax} - \frac{sin2phi}{\frac{alphay}{alphax}}}{alphay \cdot \left(-alphax\right)}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(-u0\right) \cdot \frac{-alphay}{\frac{sin2phi}{alphay}}\\ \end{array} \]

Alternative 4: 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 63.5%

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

   1. neg-sub0 63.5%

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

   2. div-sub 63.5%

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

   3. --rgt-identity 63.5%

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

   4. div-sub 63.5%

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

   5. --rgt-identity 63.5%

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

   6. neg-sub0 63.5%

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

   7. sub-neg 63.5%

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

   8. log1p-def 98.2%

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

3. Simplified 98.2%

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

4. Final simplification 98.2%

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

Alternative 5: 98.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 63.5%

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

   1. neg-sub0 63.5%

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

   2. div-sub 63.5%

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

   3. --rgt-identity 63.5%

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

   4. div-sub 63.5%

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

   5. --rgt-identity 63.5%

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

   6. sub-neg 63.5%

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

   7. +-commutative 63.5%

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

   8. neg-sub0 63.5%

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

   9. associate-+l- 63.5%

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

   10. sub0-neg 63.5%

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

   11. neg-mul-1 63.5%

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

   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 98.2%

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

4. Final simplification 98.2%

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

Alternative 6: 83.6% accurate, 4.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 0.20000000298023224:\\ \;\;\;\;\frac{u0}{\frac{alphay \cdot \frac{-cos2phi}{alphax} - \frac{sin2phi}{\frac{alphay}{alphax}}}{alphay \cdot \left(-alphax\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-alphay \cdot alphay}{\left(sin2phi \cdot 0.5 - \frac{sin2phi}{u0}\right) - u0 \cdot \left(sin2phi \cdot 0.25 + sin2phi \cdot -0.3333333333333333\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (if (<= (/ sin2phi (* alphay alphay)) 0.20000000298023224)
   (/
    u0
    (/
     (- (* alphay (/ (- cos2phi) alphax)) (/ sin2phi (/ alphay alphax)))
     (* alphay (- alphax))))
   (/
    (- (* alphay alphay))
    (-
     (- (* sin2phi 0.5) (/ sin2phi u0))
     (* u0 (+ (* sin2phi 0.25) (* sin2phi -0.3333333333333333)))))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float tmp;
	if ((sin2phi / (alphay * alphay)) <= 0.20000000298023224f) {
		tmp = u0 / (((alphay * (-cos2phi / alphax)) - (sin2phi / (alphay / alphax))) / (alphay * -alphax));
	} else {
		tmp = -(alphay * alphay) / (((sin2phi * 0.5f) - (sin2phi / u0)) - (u0 * ((sin2phi * 0.25f) + (sin2phi * -0.3333333333333333f))));
	}
	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)) <= 0.20000000298023224e0) then
        tmp = u0 / (((alphay * (-cos2phi / alphax)) - (sin2phi / (alphay / alphax))) / (alphay * -alphax))
    else
        tmp = -(alphay * alphay) / (((sin2phi * 0.5e0) - (sin2phi / u0)) - (u0 * ((sin2phi * 0.25e0) + (sin2phi * (-0.3333333333333333e0)))))
    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(0.20000000298023224))
		tmp = Float32(u0 / Float32(Float32(Float32(alphay * Float32(Float32(-cos2phi) / alphax)) - Float32(sin2phi / Float32(alphay / alphax))) / Float32(alphay * Float32(-alphax))));
	else
		tmp = Float32(Float32(-Float32(alphay * alphay)) / Float32(Float32(Float32(sin2phi * Float32(0.5)) - Float32(sin2phi / u0)) - Float32(u0 * Float32(Float32(sin2phi * Float32(0.25)) + Float32(sin2phi * Float32(-0.3333333333333333))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = single(0.0);
	if ((sin2phi / (alphay * alphay)) <= single(0.20000000298023224))
		tmp = u0 / (((alphay * (-cos2phi / alphax)) - (sin2phi / (alphay / alphax))) / (alphay * -alphax));
	else
		tmp = -(alphay * alphay) / (((sin2phi * single(0.5)) - (sin2phi / u0)) - (u0 * ((sin2phi * single(0.25)) + (sin2phi * single(-0.3333333333333333)))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 sin2phi (*.f32 alphay alphay)) < 0.200000003

   1. Initial program 55.1%

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

      1. associate-/r* 55.1%

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

   3. Simplified 55.1%

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

   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. Step-by-step derivation

      1. +-commutative 74.8%

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

      2. associate-/r* 74.8%

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

      3. associate-/r* 74.8%

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

      4. frac-2neg 74.8%

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

      5. frac-add 74.8%

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

      6. distribute-neg-frac 74.8%

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

   8. Applied egg-rr 74.8%

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

      1. +-commutative 74.8%

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

      2. distribute-rgt-neg-out 74.8%

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

      3. unsub-neg 74.8%

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

      4. associate-*l/ 74.8%

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

      5. associate-/l* 74.9%

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

      6. *-commutative 74.9%

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

      7. distribute-lft-neg-out 74.9%

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

      8. distribute-rgt-neg-in 74.9%

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

   10. Simplified 74.9%

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

   if 0.200000003 < (/.f32 sin2phi (*.f32 alphay alphay))

   1. Initial program 69.3%

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

      1. neg-sub0 69.3%

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

      2. div-sub 69.3%

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

      3. --rgt-identity 69.3%

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

      4. div-sub 69.3%

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

      5. --rgt-identity 69.3%

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

      6. neg-sub0 69.3%

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

      7. sub-neg 69.3%

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

      8. log1p-def 97.9%

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

   3. Simplified 97.9%

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

      1. +-commutative 97.9%

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

      2. associate-/r* 97.9%

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

      3. frac-add 97.6%

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

   5. Applied egg-rr 97.6%

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

      1. *-commutative 97.6%

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

      2. fma-def 97.6%

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

      3. *-commutative 97.6%

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

      4. associate-*l* 97.2%

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

   7. Simplified 97.2%

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

      1. expm1-log1p-u 97.2%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{-\mathsf{log1p}\left(-u0\right)}{\frac{\mathsf{fma}\left(\frac{sin2phi}{alphay}, alphax \cdot alphax, cos2phi \cdot alphay\right)}{alphax \cdot \left(alphax \cdot alphay\right)}}\right)\right)} \]

      2. expm1-udef 31.7%

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

      3. associate-/r/ 31.8%

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

   9. Applied egg-rr 31.8%

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

       1. expm1-def 98.2%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{-\mathsf{log1p}\left(-u0\right)}{\mathsf{fma}\left(\frac{sin2phi}{alphay}, alphax \cdot alphax, cos2phi \cdot alphay\right)} \cdot \left(alphax \cdot \left(alphax \cdot alphay\right)\right)\right)\right)} \]

       2. expm1-log1p 98.2%

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

       3. associate-*r* 98.6%

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

       4. *-commutative 98.6%

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

   11. Simplified 98.6%

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

   12. Taylor expanded in sin2phi around inf 68.7%

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

       1. mul-1-neg 68.7%

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

       2. associate-/l* 67.7%

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

       3. unpow2 67.7%

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

   14. Simplified 67.7%

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

   15. Taylor expanded in u0 around 0 90.0%

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

4. Final simplification 83.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 0.20000000298023224:\\ \;\;\;\;\frac{u0}{\frac{alphay \cdot \frac{-cos2phi}{alphax} - \frac{sin2phi}{\frac{alphay}{alphax}}}{alphay \cdot \left(-alphax\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-alphay \cdot alphay}{\left(sin2phi \cdot 0.5 - \frac{sin2phi}{u0}\right) - u0 \cdot \left(sin2phi \cdot 0.25 + sin2phi \cdot -0.3333333333333333\right)}\\ \end{array} \]

Alternative 7: 81.8% accurate, 4.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 1.7999999523162842:\\ \;\;\;\;\frac{u0}{\frac{alphay \cdot \frac{-cos2phi}{alphax} - \frac{sin2phi}{\frac{alphay}{alphax}}}{alphay \cdot \left(-alphax\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-alphay \cdot alphay}{sin2phi \cdot 0.5 - \frac{sin2phi}{u0}}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (if (<= (/ sin2phi (* alphay alphay)) 1.7999999523162842)
   (/
    u0
    (/
     (- (* alphay (/ (- cos2phi) alphax)) (/ sin2phi (/ alphay alphax)))
     (* alphay (- alphax))))
   (/ (- (* alphay alphay)) (- (* sin2phi 0.5) (/ sin2phi u0)))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float tmp;
	if ((sin2phi / (alphay * alphay)) <= 1.7999999523162842f) {
		tmp = u0 / (((alphay * (-cos2phi / alphax)) - (sin2phi / (alphay / alphax))) / (alphay * -alphax));
	} else {
		tmp = -(alphay * alphay) / ((sin2phi * 0.5f) - (sin2phi / u0));
	}
	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)) <= 1.7999999523162842e0) then
        tmp = u0 / (((alphay * (-cos2phi / alphax)) - (sin2phi / (alphay / alphax))) / (alphay * -alphax))
    else
        tmp = -(alphay * alphay) / ((sin2phi * 0.5e0) - (sin2phi / u0))
    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(1.7999999523162842))
		tmp = Float32(u0 / Float32(Float32(Float32(alphay * Float32(Float32(-cos2phi) / alphax)) - Float32(sin2phi / Float32(alphay / alphax))) / Float32(alphay * Float32(-alphax))));
	else
		tmp = Float32(Float32(-Float32(alphay * alphay)) / Float32(Float32(sin2phi * Float32(0.5)) - Float32(sin2phi / u0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = single(0.0);
	if ((sin2phi / (alphay * alphay)) <= single(1.7999999523162842))
		tmp = u0 / (((alphay * (-cos2phi / alphax)) - (sin2phi / (alphay / alphax))) / (alphay * -alphax));
	else
		tmp = -(alphay * alphay) / ((sin2phi * single(0.5)) - (sin2phi / u0));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 sin2phi (*.f32 alphay alphay)) < 1.79999995

   1. Initial program 56.3%

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

      1. associate-/r* 56.3%

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

   3. Simplified 56.3%

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

   4. Taylor expanded in u0 around 0 73.7%

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

      1. unpow2 73.7%

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

      2. unpow2 73.7%

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

   6. Simplified 73.7%

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

      1. +-commutative 73.7%

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

      2. associate-/r* 73.7%

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

      3. associate-/r* 73.7%

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

      4. frac-2neg 73.7%

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

      5. frac-add 73.8%

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

      6. distribute-neg-frac 73.8%

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

   8. Applied egg-rr 73.8%

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

      1. +-commutative 73.8%

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

      2. distribute-rgt-neg-out 73.8%

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

      3. unsub-neg 73.8%

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

      4. associate-*l/ 73.8%

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

      5. associate-/l* 73.8%

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

      6. *-commutative 73.8%

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

      7. distribute-lft-neg-out 73.8%

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

      8. distribute-rgt-neg-in 73.8%

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

   10. Simplified 73.8%

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

   if 1.79999995 < (/.f32 sin2phi (*.f32 alphay alphay))

   1. Initial program 69.1%

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

      1. neg-sub0 69.1%

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

      2. div-sub 69.1%

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

      3. --rgt-identity 69.1%

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

      4. div-sub 69.1%

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

      5. --rgt-identity 69.1%

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

      6. neg-sub0 69.1%

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

      7. sub-neg 69.1%

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

      8. log1p-def 97.9%

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

   3. Simplified 97.9%

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

      1. +-commutative 97.9%

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

      2. associate-/r* 97.9%

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

      3. frac-add 97.5%

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

   5. Applied egg-rr 97.5%

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

      1. *-commutative 97.5%

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

      2. fma-def 97.5%

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

      3. *-commutative 97.5%

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

      4. associate-*l* 97.2%

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

   7. Simplified 97.2%

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

      1. expm1-log1p-u 97.2%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{-\mathsf{log1p}\left(-u0\right)}{\frac{\mathsf{fma}\left(\frac{sin2phi}{alphay}, alphax \cdot alphax, cos2phi \cdot alphay\right)}{alphax \cdot \left(alphax \cdot alphay\right)}}\right)\right)} \]

      2. expm1-udef 30.4%

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

      3. associate-/r/ 30.4%

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

   9. Applied egg-rr 30.4%

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

       1. expm1-def 98.3%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{-\mathsf{log1p}\left(-u0\right)}{\mathsf{fma}\left(\frac{sin2phi}{alphay}, alphax \cdot alphax, cos2phi \cdot alphay\right)} \cdot \left(alphax \cdot \left(alphax \cdot alphay\right)\right)\right)\right)} \]

       2. expm1-log1p 98.3%

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

       3. associate-*r* 98.6%

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

       4. *-commutative 98.6%

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

   11. Simplified 98.6%

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

   12. Taylor expanded in sin2phi around inf 69.1%

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

       1. mul-1-neg 69.1%

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

       2. associate-/l* 68.1%

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

       3. unpow2 68.1%

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

   14. Simplified 68.1%

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

   15. Taylor expanded in u0 around 0 87.7%

       \[\leadsto -\frac{alphay \cdot alphay}{\color{blue}{-1 \cdot \frac{sin2phi}{u0} + 0.5 \cdot sin2phi}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 81.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 1.7999999523162842:\\ \;\;\;\;\frac{u0}{\frac{alphay \cdot \frac{-cos2phi}{alphax} - \frac{sin2phi}{\frac{alphay}{alphax}}}{alphay \cdot \left(-alphax\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-alphay \cdot alphay}{sin2phi \cdot 0.5 - \frac{sin2phi}{u0}}\\ \end{array} \]

Alternative 8: 81.8% accurate, 5.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 1.7999999523162842:\\ \;\;\;\;\frac{u0}{\frac{alphay \cdot \frac{cos2phi}{alphax} + alphax \cdot \frac{sin2phi}{alphay}}{alphay \cdot alphax}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-alphay \cdot alphay}{sin2phi \cdot 0.5 - \frac{sin2phi}{u0}}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (if (<= (/ sin2phi (* alphay alphay)) 1.7999999523162842)
   (/
    u0
    (/
     (+ (* alphay (/ cos2phi alphax)) (* alphax (/ sin2phi alphay)))
     (* alphay alphax)))
   (/ (- (* alphay alphay)) (- (* sin2phi 0.5) (/ sin2phi u0)))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float tmp;
	if ((sin2phi / (alphay * alphay)) <= 1.7999999523162842f) {
		tmp = u0 / (((alphay * (cos2phi / alphax)) + (alphax * (sin2phi / alphay))) / (alphay * alphax));
	} else {
		tmp = -(alphay * alphay) / ((sin2phi * 0.5f) - (sin2phi / u0));
	}
	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)) <= 1.7999999523162842e0) then
        tmp = u0 / (((alphay * (cos2phi / alphax)) + (alphax * (sin2phi / alphay))) / (alphay * alphax))
    else
        tmp = -(alphay * alphay) / ((sin2phi * 0.5e0) - (sin2phi / u0))
    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(1.7999999523162842))
		tmp = Float32(u0 / Float32(Float32(Float32(alphay * Float32(cos2phi / alphax)) + Float32(alphax * Float32(sin2phi / alphay))) / Float32(alphay * alphax)));
	else
		tmp = Float32(Float32(-Float32(alphay * alphay)) / Float32(Float32(sin2phi * Float32(0.5)) - Float32(sin2phi / u0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = single(0.0);
	if ((sin2phi / (alphay * alphay)) <= single(1.7999999523162842))
		tmp = u0 / (((alphay * (cos2phi / alphax)) + (alphax * (sin2phi / alphay))) / (alphay * alphax));
	else
		tmp = -(alphay * alphay) / ((sin2phi * single(0.5)) - (sin2phi / u0));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 sin2phi (*.f32 alphay alphay)) < 1.79999995

   1. Initial program 56.3%

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

      1. associate-/r* 56.3%

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

   3. Simplified 56.3%

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

   4. Taylor expanded in u0 around 0 73.7%

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

      1. unpow2 73.7%

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

      2. unpow2 73.7%

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

   6. Simplified 73.7%

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

      1. +-commutative 73.7%

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

      2. associate-/r* 73.7%

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

      3. associate-/r* 73.7%

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

      4. frac-add 73.8%

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

   8. Applied egg-rr 73.8%

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

   if 1.79999995 < (/.f32 sin2phi (*.f32 alphay alphay))

   1. Initial program 69.1%

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

      1. neg-sub0 69.1%

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

      2. div-sub 69.1%

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

      3. --rgt-identity 69.1%

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

      4. div-sub 69.1%

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

      5. --rgt-identity 69.1%

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

      6. neg-sub0 69.1%

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

      7. sub-neg 69.1%

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

      8. log1p-def 97.9%

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

   3. Simplified 97.9%

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

      1. +-commutative 97.9%

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

      2. associate-/r* 97.9%

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

      3. frac-add 97.5%

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

   5. Applied egg-rr 97.5%

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

      1. *-commutative 97.5%

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

      2. fma-def 97.5%

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

      3. *-commutative 97.5%

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

      4. associate-*l* 97.2%

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

   7. Simplified 97.2%

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

      1. expm1-log1p-u 97.2%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{-\mathsf{log1p}\left(-u0\right)}{\frac{\mathsf{fma}\left(\frac{sin2phi}{alphay}, alphax \cdot alphax, cos2phi \cdot alphay\right)}{alphax \cdot \left(alphax \cdot alphay\right)}}\right)\right)} \]

      2. expm1-udef 30.4%

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

      3. associate-/r/ 30.4%

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

   9. Applied egg-rr 30.4%

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

       1. expm1-def 98.3%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{-\mathsf{log1p}\left(-u0\right)}{\mathsf{fma}\left(\frac{sin2phi}{alphay}, alphax \cdot alphax, cos2phi \cdot alphay\right)} \cdot \left(alphax \cdot \left(alphax \cdot alphay\right)\right)\right)\right)} \]

       2. expm1-log1p 98.3%

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

       3. associate-*r* 98.6%

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

       4. *-commutative 98.6%

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

   11. Simplified 98.6%

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

   12. Taylor expanded in sin2phi around inf 69.1%

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

       1. mul-1-neg 69.1%

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

       2. associate-/l* 68.1%

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

       3. unpow2 68.1%

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

   14. Simplified 68.1%

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

   15. Taylor expanded in u0 around 0 87.7%

       \[\leadsto -\frac{alphay \cdot alphay}{\color{blue}{-1 \cdot \frac{sin2phi}{u0} + 0.5 \cdot sin2phi}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 81.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 1.7999999523162842:\\ \;\;\;\;\frac{u0}{\frac{alphay \cdot \frac{cos2phi}{alphax} + alphax \cdot \frac{sin2phi}{alphay}}{alphay \cdot alphax}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-alphay \cdot alphay}{sin2phi \cdot 0.5 - \frac{sin2phi}{u0}}\\ \end{array} \]

Alternative 9: 81.8% accurate, 5.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 1.7999999523162842:\\ \;\;\;\;\frac{u0}{\frac{\frac{alphax}{\frac{alphay}{sin2phi}} + alphay \cdot \frac{cos2phi}{alphax}}{alphay \cdot alphax}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-alphay \cdot alphay}{sin2phi \cdot 0.5 - \frac{sin2phi}{u0}}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (if (<= (/ sin2phi (* alphay alphay)) 1.7999999523162842)
   (/
    u0
    (/
     (+ (/ alphax (/ alphay sin2phi)) (* alphay (/ cos2phi alphax)))
     (* alphay alphax)))
   (/ (- (* alphay alphay)) (- (* sin2phi 0.5) (/ sin2phi u0)))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float tmp;
	if ((sin2phi / (alphay * alphay)) <= 1.7999999523162842f) {
		tmp = u0 / (((alphax / (alphay / sin2phi)) + (alphay * (cos2phi / alphax))) / (alphay * alphax));
	} else {
		tmp = -(alphay * alphay) / ((sin2phi * 0.5f) - (sin2phi / u0));
	}
	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)) <= 1.7999999523162842e0) then
        tmp = u0 / (((alphax / (alphay / sin2phi)) + (alphay * (cos2phi / alphax))) / (alphay * alphax))
    else
        tmp = -(alphay * alphay) / ((sin2phi * 0.5e0) - (sin2phi / u0))
    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(1.7999999523162842))
		tmp = Float32(u0 / Float32(Float32(Float32(alphax / Float32(alphay / sin2phi)) + Float32(alphay * Float32(cos2phi / alphax))) / Float32(alphay * alphax)));
	else
		tmp = Float32(Float32(-Float32(alphay * alphay)) / Float32(Float32(sin2phi * Float32(0.5)) - Float32(sin2phi / u0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = single(0.0);
	if ((sin2phi / (alphay * alphay)) <= single(1.7999999523162842))
		tmp = u0 / (((alphax / (alphay / sin2phi)) + (alphay * (cos2phi / alphax))) / (alphay * alphax));
	else
		tmp = -(alphay * alphay) / ((sin2phi * single(0.5)) - (sin2phi / u0));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 sin2phi (*.f32 alphay alphay)) < 1.79999995

   1. Initial program 56.3%

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

      1. associate-/r* 56.3%

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

   3. Simplified 56.3%

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

   4. Taylor expanded in u0 around 0 73.7%

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

      1. unpow2 73.7%

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

      2. unpow2 73.7%

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

   6. Simplified 73.7%

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

      1. +-commutative 73.7%

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

      2. associate-/r* 73.7%

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

      3. associate-/r* 73.7%

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

      4. frac-add 73.8%

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

   8. Applied egg-rr 73.8%

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

   9. Taylor expanded in sin2phi around 0 73.8%

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

       1. *-commutative 73.8%

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

       2. associate-/l* 73.8%

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

   11. Simplified 73.8%

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

   if 1.79999995 < (/.f32 sin2phi (*.f32 alphay alphay))

   1. Initial program 69.1%

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

      1. neg-sub0 69.1%

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

      2. div-sub 69.1%

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

      3. --rgt-identity 69.1%

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

      4. div-sub 69.1%

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

      5. --rgt-identity 69.1%

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

      6. neg-sub0 69.1%

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

      7. sub-neg 69.1%

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

      8. log1p-def 97.9%

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

   3. Simplified 97.9%

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

      1. +-commutative 97.9%

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

      2. associate-/r* 97.9%

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

      3. frac-add 97.5%

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

   5. Applied egg-rr 97.5%

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

      1. *-commutative 97.5%

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

      2. fma-def 97.5%

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

      3. *-commutative 97.5%

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

      4. associate-*l* 97.2%

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

   7. Simplified 97.2%

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

      1. expm1-log1p-u 97.2%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{-\mathsf{log1p}\left(-u0\right)}{\frac{\mathsf{fma}\left(\frac{sin2phi}{alphay}, alphax \cdot alphax, cos2phi \cdot alphay\right)}{alphax \cdot \left(alphax \cdot alphay\right)}}\right)\right)} \]

      2. expm1-udef 30.4%

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

      3. associate-/r/ 30.4%

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

   9. Applied egg-rr 30.4%

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

       1. expm1-def 98.3%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{-\mathsf{log1p}\left(-u0\right)}{\mathsf{fma}\left(\frac{sin2phi}{alphay}, alphax \cdot alphax, cos2phi \cdot alphay\right)} \cdot \left(alphax \cdot \left(alphax \cdot alphay\right)\right)\right)\right)} \]

       2. expm1-log1p 98.3%

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

       3. associate-*r* 98.6%

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

       4. *-commutative 98.6%

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

   11. Simplified 98.6%

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

   12. Taylor expanded in sin2phi around inf 69.1%

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

       1. mul-1-neg 69.1%

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

       2. associate-/l* 68.1%

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

       3. unpow2 68.1%

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

   14. Simplified 68.1%

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

   15. Taylor expanded in u0 around 0 87.7%

       \[\leadsto -\frac{alphay \cdot alphay}{\color{blue}{-1 \cdot \frac{sin2phi}{u0} + 0.5 \cdot sin2phi}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 81.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 1.7999999523162842:\\ \;\;\;\;\frac{u0}{\frac{\frac{alphax}{\frac{alphay}{sin2phi}} + alphay \cdot \frac{cos2phi}{alphax}}{alphay \cdot alphax}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-alphay \cdot alphay}{sin2phi \cdot 0.5 - \frac{sin2phi}{u0}}\\ \end{array} \]

Alternative 10: 81.8% accurate, 6.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{sin2phi}{alphay \cdot alphay}\\ \mathbf{if}\;t_0 \leq 1.7999999523162842:\\ \;\;\;\;\frac{u0}{t_0 + \frac{cos2phi}{alphax \cdot alphax}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-alphay \cdot alphay}{sin2phi \cdot 0.5 - \frac{sin2phi}{u0}}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (let* ((t_0 (/ sin2phi (* alphay alphay))))
   (if (<= t_0 1.7999999523162842)
     (/ u0 (+ t_0 (/ cos2phi (* alphax alphax))))
     (/ (- (* alphay alphay)) (- (* sin2phi 0.5) (/ sin2phi u0))))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float t_0 = sin2phi / (alphay * alphay);
	float tmp;
	if (t_0 <= 1.7999999523162842f) {
		tmp = u0 / (t_0 + (cos2phi / (alphax * alphax)));
	} else {
		tmp = -(alphay * alphay) / ((sin2phi * 0.5f) - (sin2phi / u0));
	}
	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 <= 1.7999999523162842e0) then
        tmp = u0 / (t_0 + (cos2phi / (alphax * alphax)))
    else
        tmp = -(alphay * alphay) / ((sin2phi * 0.5e0) - (sin2phi / u0))
    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(1.7999999523162842))
		tmp = Float32(u0 / Float32(t_0 + Float32(cos2phi / Float32(alphax * alphax))));
	else
		tmp = Float32(Float32(-Float32(alphay * alphay)) / Float32(Float32(sin2phi * Float32(0.5)) - Float32(sin2phi / u0)));
	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(1.7999999523162842))
		tmp = u0 / (t_0 + (cos2phi / (alphax * alphax)));
	else
		tmp = -(alphay * alphay) / ((sin2phi * single(0.5)) - (sin2phi / u0));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 sin2phi (*.f32 alphay alphay)) < 1.79999995

   1. Initial program 56.3%

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

      1. associate-/r* 56.3%

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

   3. Simplified 56.3%

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

   4. Taylor expanded in u0 around 0 73.7%

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

      1. unpow2 73.7%

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

      2. unpow2 73.7%

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

   6. Simplified 73.7%

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

   if 1.79999995 < (/.f32 sin2phi (*.f32 alphay alphay))

   1. Initial program 69.1%

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

      1. neg-sub0 69.1%

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

      2. div-sub 69.1%

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

      3. --rgt-identity 69.1%

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

      4. div-sub 69.1%

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

      5. --rgt-identity 69.1%

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

      6. neg-sub0 69.1%

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

      7. sub-neg 69.1%

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

      8. log1p-def 97.9%

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

   3. Simplified 97.9%

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

      1. +-commutative 97.9%

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

      2. associate-/r* 97.9%

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

      3. frac-add 97.5%

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

   5. Applied egg-rr 97.5%

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

      1. *-commutative 97.5%

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

      2. fma-def 97.5%

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

      3. *-commutative 97.5%

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

      4. associate-*l* 97.2%

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

   7. Simplified 97.2%

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

      1. expm1-log1p-u 97.2%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{-\mathsf{log1p}\left(-u0\right)}{\frac{\mathsf{fma}\left(\frac{sin2phi}{alphay}, alphax \cdot alphax, cos2phi \cdot alphay\right)}{alphax \cdot \left(alphax \cdot alphay\right)}}\right)\right)} \]

      2. expm1-udef 30.4%

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

      3. associate-/r/ 30.4%

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

   9. Applied egg-rr 30.4%

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

       1. expm1-def 98.3%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{-\mathsf{log1p}\left(-u0\right)}{\mathsf{fma}\left(\frac{sin2phi}{alphay}, alphax \cdot alphax, cos2phi \cdot alphay\right)} \cdot \left(alphax \cdot \left(alphax \cdot alphay\right)\right)\right)\right)} \]

       2. expm1-log1p 98.3%

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

       3. associate-*r* 98.6%

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

       4. *-commutative 98.6%

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

   11. Simplified 98.6%

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

   12. Taylor expanded in sin2phi around inf 69.1%

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

       1. mul-1-neg 69.1%

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

       2. associate-/l* 68.1%

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

       3. unpow2 68.1%

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

   14. Simplified 68.1%

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

   15. Taylor expanded in u0 around 0 87.7%

       \[\leadsto -\frac{alphay \cdot alphay}{\color{blue}{-1 \cdot \frac{sin2phi}{u0} + 0.5 \cdot sin2phi}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 81.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 1.7999999523162842:\\ \;\;\;\;\frac{u0}{\frac{sin2phi}{alphay \cdot alphay} + \frac{cos2phi}{alphax \cdot alphax}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-alphay \cdot alphay}{sin2phi \cdot 0.5 - \frac{sin2phi}{u0}}\\ \end{array} \]

Alternative 11: 66.8% accurate, 8.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 5.499999861132538 \cdot 10^{-16}:\\ \;\;\;\;\frac{u0}{\frac{cos2phi}{alphax \cdot alphax}}\\ \mathbf{else}:\\ \;\;\;\;\left(alphay \cdot alphay\right) \cdot \frac{u0}{sin2phi}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (if (<= (/ sin2phi (* alphay alphay)) 5.499999861132538e-16)
   (/ u0 (/ cos2phi (* alphax alphax)))
   (* (* alphay alphay) (/ u0 sin2phi))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float tmp;
	if ((sin2phi / (alphay * alphay)) <= 5.499999861132538e-16f) {
		tmp = u0 / (cos2phi / (alphax * alphax));
	} 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)) <= 5.499999861132538e-16) then
        tmp = u0 / (cos2phi / (alphax * alphax))
    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(5.499999861132538e-16))
		tmp = Float32(u0 / Float32(cos2phi / Float32(alphax * alphax)));
	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(5.499999861132538e-16))
		tmp = u0 / (cos2phi / (alphax * alphax));
	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)) < 5.49999986e-16

   1. Initial program 57.3%

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

      1. associate-/r* 57.4%

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

   3. Simplified 57.4%

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

   4. Taylor expanded in u0 around 0 72.2%

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

      1. unpow2 72.2%

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

      2. unpow2 72.2%

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

   6. Simplified 72.2%

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

   7. Taylor expanded in cos2phi around inf 55.4%

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

      1. unpow2 55.4%

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

      2. associate-/l* 55.5%

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

   9. Simplified 55.5%

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

   if 5.49999986e-16 < (/.f32 sin2phi (*.f32 alphay alphay))

   1. Initial program 65.1%

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

      1. associate-/r* 65.1%

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

   3. Simplified 65.1%

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

   4. Taylor expanded in u0 around 0 74.6%

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

      1. unpow2 74.6%

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

      2. unpow2 74.6%

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

   6. Simplified 74.6%

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

   7. Taylor expanded in cos2phi around 0 67.9%

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

      1. unpow2 67.9%

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

      2. *-commutative 67.9%

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

      3. *-lft-identity 67.9%

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

      4. times-frac 67.9%

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

      5. /-rgt-identity 67.9%

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

   9. Simplified 67.9%

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

4. Final simplification 65.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 5.499999861132538 \cdot 10^{-16}:\\ \;\;\;\;\frac{u0}{\frac{cos2phi}{alphax \cdot alphax}}\\ \mathbf{else}:\\ \;\;\;\;\left(alphay \cdot alphay\right) \cdot \frac{u0}{sin2phi}\\ \end{array} \]

Alternative 12: 66.8% accurate, 8.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 5.499999861132538 \cdot 10^{-16}:\\ \;\;\;\;\frac{u0}{\frac{cos2phi}{alphax \cdot alphax}}\\ \mathbf{else}:\\ \;\;\;\;\frac{u0 \cdot \left(alphay \cdot alphay\right)}{sin2phi}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (if (<= (/ sin2phi (* alphay alphay)) 5.499999861132538e-16)
   (/ u0 (/ cos2phi (* alphax alphax)))
   (/ (* u0 (* alphay alphay)) sin2phi)))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float tmp;
	if ((sin2phi / (alphay * alphay)) <= 5.499999861132538e-16f) {
		tmp = u0 / (cos2phi / (alphax * alphax));
	} 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)) <= 5.499999861132538e-16) then
        tmp = u0 / (cos2phi / (alphax * alphax))
    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(5.499999861132538e-16))
		tmp = Float32(u0 / Float32(cos2phi / Float32(alphax * alphax)));
	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(5.499999861132538e-16))
		tmp = u0 / (cos2phi / (alphax * alphax));
	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)) < 5.49999986e-16

   1. Initial program 57.3%

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

      1. associate-/r* 57.4%

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

   3. Simplified 57.4%

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

   4. Taylor expanded in u0 around 0 72.2%

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

      1. unpow2 72.2%

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

      2. unpow2 72.2%

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

   6. Simplified 72.2%

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

   7. Taylor expanded in cos2phi around inf 55.4%

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

      1. unpow2 55.4%

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

      2. associate-/l* 55.5%

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

   9. Simplified 55.5%

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

   if 5.49999986e-16 < (/.f32 sin2phi (*.f32 alphay alphay))

   1. Initial program 65.1%

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

      1. associate-/r* 65.1%

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

   3. Simplified 65.1%

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

   4. Taylor expanded in u0 around 0 74.6%

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

      1. unpow2 74.6%

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

      2. unpow2 74.6%

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

   6. Simplified 74.6%

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

   7. Taylor expanded in cos2phi around 0 67.9%

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

      1. unpow2 67.9%

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

   9. Simplified 67.9%

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

4. Final simplification 65.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 5.499999861132538 \cdot 10^{-16}:\\ \;\;\;\;\frac{u0}{\frac{cos2phi}{alphax \cdot alphax}}\\ \mathbf{else}:\\ \;\;\;\;\frac{u0 \cdot \left(alphay \cdot alphay\right)}{sin2phi}\\ \end{array} \]

Alternative 13: 75.6% 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 63.5%

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

   1. associate-/r* 63.5%

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

3. Simplified 63.5%

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

4. Taylor expanded in u0 around 0 74.1%

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

   1. unpow2 74.1%

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

   2. unpow2 74.1%

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

6. Simplified 74.1%

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

7. Final simplification 74.1%

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

Alternative 14: 66.7% accurate, 12.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;sin2phi \leq 1.1000000297155601 \cdot 10^{-19}:\\ \;\;\;\;\left(alphax \cdot alphax\right) \cdot \frac{u0}{cos2phi}\\ \mathbf{else}:\\ \;\;\;\;\left(alphay \cdot alphay\right) \cdot \frac{u0}{sin2phi}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (if (<= sin2phi 1.1000000297155601e-19)
   (* (* alphax alphax) (/ u0 cos2phi))
   (* (* alphay alphay) (/ u0 sin2phi))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float tmp;
	if (sin2phi <= 1.1000000297155601e-19f) {
		tmp = (alphax * alphax) * (u0 / 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 <= 1.1000000297155601e-19) then
        tmp = (alphax * alphax) * (u0 / 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 (sin2phi <= Float32(1.1000000297155601e-19))
		tmp = Float32(Float32(alphax * alphax) * Float32(u0 / 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 <= single(1.1000000297155601e-19))
		tmp = (alphax * alphax) * (u0 / cos2phi);
	else
		tmp = (alphay * alphay) * (u0 / sin2phi);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if sin2phi < 1.10000003e-19

   1. Initial program 54.4%

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

      1. associate-/r* 54.4%

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

   3. Simplified 54.4%

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

   4. Taylor expanded in u0 around 0 74.1%

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

      1. unpow2 74.1%

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

      2. unpow2 74.1%

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

   6. Simplified 74.1%

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

   7. Taylor expanded in cos2phi around inf 53.9%

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

      1. unpow2 53.9%

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

      2. *-commutative 53.9%

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

      3. *-lft-identity 53.9%

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

      4. times-frac 53.9%

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

      5. /-rgt-identity 53.9%

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

   9. Simplified 53.9%

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

   if 1.10000003e-19 < sin2phi

   1. Initial program 66.0%

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

      1. associate-/r* 66.0%

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

   3. Simplified 66.0%

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

   4. Taylor expanded in u0 around 0 74.1%

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

      1. unpow2 74.1%

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

      2. unpow2 74.1%

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

   6. Simplified 74.1%

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

   7. Taylor expanded in cos2phi around 0 67.4%

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

      1. unpow2 67.4%

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

      2. *-commutative 67.4%

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

      3. *-lft-identity 67.4%

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

      4. times-frac 67.4%

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

      5. /-rgt-identity 67.4%

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

   9. Simplified 67.4%

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

4. Final simplification 64.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;sin2phi \leq 1.1000000297155601 \cdot 10^{-19}:\\ \;\;\;\;\left(alphax \cdot alphax\right) \cdot \frac{u0}{cos2phi}\\ \mathbf{else}:\\ \;\;\;\;\left(alphay \cdot alphay\right) \cdot \frac{u0}{sin2phi}\\ \end{array} \]

Alternative 15: 23.4% accurate, 16.6× speedup

Math

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

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(Float32(alphax * alphax) * Float32(u0 / cos2phi))
end

MATLAB

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

Derivation

1. Initial program 63.5%

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

   1. associate-/r* 63.5%

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

3. Simplified 63.5%

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

4. Taylor expanded in u0 around 0 74.1%

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

   1. unpow2 74.1%

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

   2. unpow2 74.1%

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

6. Simplified 74.1%

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

7. Taylor expanded in cos2phi around inf 22.9%

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

   1. unpow2 22.9%

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

   2. *-commutative 22.9%

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

   3. *-lft-identity 22.9%

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

   4. times-frac 22.9%

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

   5. /-rgt-identity 22.9%

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

9. Simplified 22.9%

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

10. Final simplification 22.9%

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

Reproduce

herbie shell --seed 2023202 
(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)))))