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

Percentage Accurate: 60.6% → 98.5%

Time: 12.3s

Alternatives: 14

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: 60.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Alternative 1: 98.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 59.1%

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

   1. neg-sub0 59.1%

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

   2. div-sub 59.1%

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

   3. --rgt-identity 59.1%

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

   4. div-sub 59.1%

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

   5. --rgt-identity 59.1%

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

   6. neg-sub0 59.1%

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

   7. sub-neg 59.1%

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

   8. log1p-def 98.6%

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

3. Simplified 98.6%

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

   1. +-commutative 98.6%

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

   2. associate-/r* 98.6%

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

   3. associate-/r* 98.5%

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

   4. frac-2neg 98.5%

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

   5. frac-add 98.1%

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

   6. distribute-neg-frac 98.1%

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

5. Applied egg-rr 98.1%

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

   1. +-commutative 98.1%

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

   2. distribute-rgt-neg-out 98.1%

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

   3. unsub-neg 98.1%

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

   4. associate-*l/ 98.2%

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

   5. associate-/l* 98.2%

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

   6. *-commutative 98.2%

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

   7. distribute-lft-neg-out 98.2%

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

   8. distribute-rgt-neg-in 98.2%

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

7. Simplified 98.2%

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

   1. expm1-log1p-u 96.5%

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

   2. expm1-udef 50.7%

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

   3. associate-/r/ 50.7%

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

   4. associate-/r/ 50.7%

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

   5. distribute-rgt-neg-out 50.7%

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

9. Applied egg-rr 50.7%

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

    1. expm1-def 96.7%

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

    2. expm1-log1p 98.5%

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

    3. distribute-frac-neg 98.5%

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

    4. distribute-lft-neg-in 98.5%

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

    5. distribute-rgt-neg-in 98.5%

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

11. Simplified 98.6%

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

12. Taylor expanded in alphax around 0 98.6%

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

    1. associate-*r/ 98.7%

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

14. Simplified 98.7%

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

15. Final simplification 98.7%

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

Alternative 2: 98.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 59.1%

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

   1. neg-sub0 59.1%

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

   2. div-sub 59.1%

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

   3. --rgt-identity 59.1%

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

   4. div-sub 59.1%

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

   5. --rgt-identity 59.1%

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

   6. neg-sub0 59.1%

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

   7. sub-neg 59.1%

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

   8. log1p-def 98.6%

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

3. Simplified 98.6%

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

   1. +-commutative 98.6%

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

   2. associate-/r* 98.6%

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

   3. associate-/r* 98.5%

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

   4. frac-2neg 98.5%

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

   5. frac-add 98.1%

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

   6. distribute-neg-frac 98.1%

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

5. Applied egg-rr 98.1%

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

   1. +-commutative 98.1%

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

   2. distribute-rgt-neg-out 98.1%

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

   3. unsub-neg 98.1%

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

   4. associate-*l/ 98.2%

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

   5. associate-/l* 98.2%

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

   6. *-commutative 98.2%

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

   7. distribute-lft-neg-out 98.2%

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

   8. distribute-rgt-neg-in 98.2%

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

7. Simplified 98.2%

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

   1. expm1-log1p-u 96.5%

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

   2. expm1-udef 50.7%

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

   3. associate-/r/ 50.7%

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

   4. associate-/r/ 50.7%

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

   5. distribute-rgt-neg-out 50.7%

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

9. Applied egg-rr 50.7%

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

    1. expm1-def 96.7%

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

    2. expm1-log1p 98.5%

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

    3. distribute-frac-neg 98.5%

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

    4. distribute-lft-neg-in 98.5%

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

    5. distribute-rgt-neg-in 98.5%

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

11. Simplified 98.6%

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

12. Final simplification 98.6%

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

Alternative 3: 98.4% accurate, 1.0× speedup

Math

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

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(cos2phi / Float32(alphax * alphax)) + Float32(sin2phi / Float32(alphay * alphay))))
end

Derivation

1. Initial program 59.1%

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

   1. neg-sub0 59.1%

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

   2. div-sub 59.1%

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

   3. --rgt-identity 59.1%

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

   4. div-sub 59.1%

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

   5. --rgt-identity 59.1%

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

   6. neg-sub0 59.1%

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

   7. sub-neg 59.1%

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

   8. log1p-def 98.6%

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

3. Simplified 98.6%

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

4. Final simplification 98.6%

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

Alternative 4: 91.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-alphay}{\frac{alphax}{cos2phi}} - alphax \cdot \frac{sin2phi}{alphay}\\ \mathbf{if}\;u0 \leq 0.0026000000070780516:\\ \;\;\;\;\frac{-0.5 \cdot \left(alphay \cdot \left(alphax \cdot \left(u0 \cdot u0\right)\right)\right)}{t_0} - \frac{u0}{\frac{t_0}{alphay \cdot alphax}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\mathsf{log1p}\left(-u0\right)}{\frac{\frac{sin2phi}{alphay}}{alphay}}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (let* ((t_0
         (- (/ (- alphay) (/ alphax cos2phi)) (* alphax (/ sin2phi alphay)))))
   (if (<= u0 0.0026000000070780516)
     (-
      (/ (* -0.5 (* alphay (* alphax (* u0 u0)))) t_0)
      (/ u0 (/ t_0 (* alphay alphax))))
     (/ (- (log1p (- u0))) (/ (/ sin2phi alphay) alphay)))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float t_0 = (-alphay / (alphax / cos2phi)) - (alphax * (sin2phi / alphay));
	float tmp;
	if (u0 <= 0.0026000000070780516f) {
		tmp = ((-0.5f * (alphay * (alphax * (u0 * u0)))) / t_0) - (u0 / (t_0 / (alphay * alphax)));
	} else {
		tmp = -log1pf(-u0) / ((sin2phi / alphay) / alphay);
	}
	return tmp;
}

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	t_0 = Float32(Float32(Float32(-alphay) / Float32(alphax / cos2phi)) - Float32(alphax * Float32(sin2phi / alphay)))
	tmp = Float32(0.0)
	if (u0 <= Float32(0.0026000000070780516))
		tmp = Float32(Float32(Float32(Float32(-0.5) * Float32(alphay * Float32(alphax * Float32(u0 * u0)))) / t_0) - Float32(u0 / Float32(t_0 / Float32(alphay * alphax))));
	else
		tmp = Float32(Float32(-log1p(Float32(-u0))) / Float32(Float32(sin2phi / alphay) / alphay));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if u0 < 0.00260000001

   1. Initial program 48.3%

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

      1. neg-sub0 48.3%

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

      2. div-sub 48.3%

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

      3. --rgt-identity 48.3%

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

      4. div-sub 48.3%

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

      5. --rgt-identity 48.3%

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

      6. neg-sub0 48.3%

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

      7. sub-neg 48.3%

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

      8. log1p-def 98.7%

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

   3. Simplified 98.7%

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

      1. +-commutative 98.7%

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

      2. associate-/r* 98.7%

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

      3. associate-/r* 98.6%

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

      4. frac-2neg 98.6%

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

      5. frac-add 98.3%

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

      6. distribute-neg-frac 98.3%

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

   5. Applied egg-rr 98.3%

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

      1. +-commutative 98.3%

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

      2. distribute-rgt-neg-out 98.3%

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

      3. unsub-neg 98.3%

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

      4. associate-*l/ 98.4%

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

      5. associate-/l* 98.5%

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

      6. *-commutative 98.5%

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

      7. distribute-lft-neg-out 98.5%

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

      8. distribute-rgt-neg-in 98.5%

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

   7. Simplified 98.5%

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

   8. Taylor expanded in u0 around 0 98.0%

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

      1. +-commutative 98.0%

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

      2. mul-1-neg 98.0%

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

      3. unsub-neg 98.0%

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

   10. Simplified 97.8%

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

   if 0.00260000001 < u0

   1. Initial program 91.6%

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

      1. neg-sub0 91.6%

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

      2. div-sub 91.6%

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

      3. --rgt-identity 91.6%

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

      4. div-sub 91.6%

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

      5. --rgt-identity 91.6%

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

      6. sub-neg 91.6%

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

      7. +-commutative 91.6%

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

      8. neg-sub0 91.6%

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

      9. associate-+l- 91.6%

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

      10. sub0-neg 91.6%

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

      11. neg-mul-1 91.6%

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

      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 98.0%

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

      1. div-inv 98.2%

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

   5. Applied egg-rr 98.2%

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

   6. Taylor expanded in cos2phi around 0 77.6%

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

      1. unpow2 77.6%

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

   8. Simplified 77.6%

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

      1. distribute-frac-neg 77.6%

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

      2. associate-/r* 77.6%

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

   10. Applied egg-rr 77.6%

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

4. Final simplification 92.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u0 \leq 0.0026000000070780516:\\ \;\;\;\;\frac{-0.5 \cdot \left(alphay \cdot \left(alphax \cdot \left(u0 \cdot u0\right)\right)\right)}{\frac{-alphay}{\frac{alphax}{cos2phi}} - alphax \cdot \frac{sin2phi}{alphay}} - \frac{u0}{\frac{\frac{-alphay}{\frac{alphax}{cos2phi}} - alphax \cdot \frac{sin2phi}{alphay}}{alphay \cdot alphax}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\mathsf{log1p}\left(-u0\right)}{\frac{\frac{sin2phi}{alphay}}{alphay}}\\ \end{array} \]

Alternative 5: 92.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-alphay}{\frac{alphax}{cos2phi}} - alphax \cdot \frac{sin2phi}{alphay}\\ \mathbf{if}\;sin2phi \leq 8.199999865610152 \cdot 10^{-5}:\\ \;\;\;\;\frac{-0.5 \cdot \left(alphay \cdot \left(alphax \cdot \left(u0 \cdot u0\right)\right)\right)}{t_0} - \frac{u0}{\frac{t_0}{alphay \cdot alphax}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(-u0\right)}{sin2phi} \cdot \left(alphay \cdot \left(-alphay\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (let* ((t_0
         (- (/ (- alphay) (/ alphax cos2phi)) (* alphax (/ sin2phi alphay)))))
   (if (<= sin2phi 8.199999865610152e-5)
     (-
      (/ (* -0.5 (* alphay (* alphax (* u0 u0)))) t_0)
      (/ u0 (/ t_0 (* alphay alphax))))
     (* (/ (log1p (- u0)) sin2phi) (* alphay (- alphay))))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float t_0 = (-alphay / (alphax / cos2phi)) - (alphax * (sin2phi / alphay));
	float tmp;
	if (sin2phi <= 8.199999865610152e-5f) {
		tmp = ((-0.5f * (alphay * (alphax * (u0 * u0)))) / t_0) - (u0 / (t_0 / (alphay * alphax)));
	} else {
		tmp = (log1pf(-u0) / sin2phi) * (alphay * -alphay);
	}
	return tmp;
}

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	t_0 = Float32(Float32(Float32(-alphay) / Float32(alphax / cos2phi)) - Float32(alphax * Float32(sin2phi / alphay)))
	tmp = Float32(0.0)
	if (sin2phi <= Float32(8.199999865610152e-5))
		tmp = Float32(Float32(Float32(Float32(-0.5) * Float32(alphay * Float32(alphax * Float32(u0 * u0)))) / t_0) - Float32(u0 / Float32(t_0 / Float32(alphay * alphax))));
	else
		tmp = Float32(Float32(log1p(Float32(-u0)) / sin2phi) * Float32(alphay * Float32(-alphay)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if sin2phi < 8.19999987e-5

   1. Initial program 53.9%

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

      1. neg-sub0 53.9%

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

      2. div-sub 53.9%

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

      3. --rgt-identity 53.9%

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

      4. div-sub 53.9%

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

      5. --rgt-identity 53.9%

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

      6. neg-sub0 53.9%

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

      7. sub-neg 53.9%

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

      8. log1p-def 98.8%

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

   3. Simplified 98.8%

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

      1. +-commutative 98.8%

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

      2. associate-/r* 98.8%

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

      3. associate-/r* 98.6%

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

      4. frac-2neg 98.6%

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

      5. frac-add 98.3%

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

      6. distribute-neg-frac 98.3%

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

   5. Applied egg-rr 98.3%

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

      1. +-commutative 98.3%

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

      2. distribute-rgt-neg-out 98.3%

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

      3. unsub-neg 98.3%

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

      4. associate-*l/ 98.4%

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

      5. associate-/l* 98.4%

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

      6. *-commutative 98.4%

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

      7. distribute-lft-neg-out 98.4%

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

      8. distribute-rgt-neg-in 98.4%

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

   7. Simplified 98.4%

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

   8. Taylor expanded in u0 around 0 86.5%

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

      1. +-commutative 86.5%

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

      2. mul-1-neg 86.5%

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

      3. unsub-neg 86.5%

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

   10. Simplified 86.5%

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

   if 8.19999987e-5 < sin2phi

   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 98.3%

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

      1. div-inv 98.3%

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

   5. Applied egg-rr 98.3%

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

   6. Taylor expanded in cos2phi around 0 97.8%

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

      1. unpow2 97.8%

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

   8. Simplified 97.8%

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

      1. associate-/r/ 98.5%

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

   10. Applied egg-rr 98.5%

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

4. Final simplification 93.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;sin2phi \leq 8.199999865610152 \cdot 10^{-5}:\\ \;\;\;\;\frac{-0.5 \cdot \left(alphay \cdot \left(alphax \cdot \left(u0 \cdot u0\right)\right)\right)}{\frac{-alphay}{\frac{alphax}{cos2phi}} - alphax \cdot \frac{sin2phi}{alphay}} - \frac{u0}{\frac{\frac{-alphay}{\frac{alphax}{cos2phi}} - alphax \cdot \frac{sin2phi}{alphay}}{alphay \cdot alphax}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(-u0\right)}{sin2phi} \cdot \left(alphay \cdot \left(-alphay\right)\right)\\ \end{array} \]

Alternative 6: 87.4% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-alphay}{\frac{alphax}{cos2phi}} - alphax \cdot \frac{sin2phi}{alphay}\\ \frac{-0.5 \cdot \left(alphay \cdot \left(alphax \cdot \left(u0 \cdot u0\right)\right)\right)}{t_0} - \frac{u0}{\frac{t_0}{alphay \cdot alphax}} \end{array} \end{array} \]

FPCore

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

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float t_0 = (-alphay / (alphax / cos2phi)) - (alphax * (sin2phi / alphay));
	return ((-0.5f * (alphay * (alphax * (u0 * u0)))) / t_0) - (u0 / (t_0 / (alphay * 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
    real(4) :: t_0
    t_0 = (-alphay / (alphax / cos2phi)) - (alphax * (sin2phi / alphay))
    code = (((-0.5e0) * (alphay * (alphax * (u0 * u0)))) / t_0) - (u0 / (t_0 / (alphay * alphax)))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 59.1%

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

   1. neg-sub0 59.1%

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

   2. div-sub 59.1%

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

   3. --rgt-identity 59.1%

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

   4. div-sub 59.1%

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

   5. --rgt-identity 59.1%

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

   6. neg-sub0 59.1%

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

   7. sub-neg 59.1%

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

   8. log1p-def 98.6%

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

3. Simplified 98.6%

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

   1. +-commutative 98.6%

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

   2. associate-/r* 98.6%

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

   3. associate-/r* 98.5%

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

   4. frac-2neg 98.5%

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

   5. frac-add 98.1%

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

   6. distribute-neg-frac 98.1%

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

5. Applied egg-rr 98.1%

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

   1. +-commutative 98.1%

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

   2. distribute-rgt-neg-out 98.1%

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

   3. unsub-neg 98.1%

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

   4. associate-*l/ 98.2%

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

   5. associate-/l* 98.2%

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

   6. *-commutative 98.2%

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

   7. distribute-lft-neg-out 98.2%

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

   8. distribute-rgt-neg-in 98.2%

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

7. Simplified 98.2%

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

8. Taylor expanded in u0 around 0 88.6%

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

   1. +-commutative 88.6%

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

   2. mul-1-neg 88.6%

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

   3. unsub-neg 88.6%

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

10. Simplified 88.3%

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

11. Final simplification 88.3%

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

Alternative 7: 76.2% accurate, 6.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 59.1%

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

   1. neg-sub0 59.1%

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

   2. div-sub 59.1%

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

   3. --rgt-identity 59.1%

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

   4. div-sub 59.1%

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

   5. --rgt-identity 59.1%

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

   6. neg-sub0 59.1%

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

   7. sub-neg 59.1%

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

   8. log1p-def 98.6%

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

3. Simplified 98.6%

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

   1. +-commutative 98.6%

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

   2. associate-/r* 98.6%

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

   3. associate-/r* 98.5%

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

   4. frac-2neg 98.5%

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

   5. frac-add 98.1%

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

   6. distribute-neg-frac 98.1%

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

5. Applied egg-rr 98.1%

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

   1. +-commutative 98.1%

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

   2. distribute-rgt-neg-out 98.1%

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

   3. unsub-neg 98.1%

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

   4. associate-*l/ 98.2%

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

   5. associate-/l* 98.2%

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

   6. *-commutative 98.2%

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

   7. distribute-lft-neg-out 98.2%

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

   8. distribute-rgt-neg-in 98.2%

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

7. Simplified 98.2%

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

   1. expm1-log1p-u 96.5%

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

   2. expm1-udef 50.7%

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

   3. associate-/r/ 50.7%

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

   4. associate-/r/ 50.7%

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

   5. distribute-rgt-neg-out 50.7%

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

9. Applied egg-rr 50.7%

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

    1. expm1-def 96.7%

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

    2. expm1-log1p 98.5%

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

    3. distribute-frac-neg 98.5%

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

    4. distribute-lft-neg-in 98.5%

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

    5. distribute-rgt-neg-in 98.5%

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

11. Simplified 98.6%

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

12. Taylor expanded in u0 around 0 76.8%

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

13. Final simplification 76.8%

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

Alternative 8: 66.7% accurate, 8.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;sin2phi \leq 1.9999999996399175 \cdot 10^{-23}:\\ \;\;\;\;\left(alphax \cdot alphax\right) \cdot \frac{u0}{cos2phi}\\ \mathbf{else}:\\ \;\;\;\;\left(alphay \cdot alphax\right) \cdot \left(\frac{u0}{sin2phi} \cdot \frac{alphay}{alphax}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (if (<= sin2phi 1.9999999996399175e-23)
   (* (* alphax alphax) (/ u0 cos2phi))
   (* (* alphay alphax) (* (/ u0 sin2phi) (/ alphay alphax)))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float tmp;
	if (sin2phi <= 1.9999999996399175e-23f) {
		tmp = (alphax * alphax) * (u0 / cos2phi);
	} else {
		tmp = (alphay * alphax) * ((u0 / sin2phi) * (alphay / alphax));
	}
	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 <= 1.9999999996399175e-23) then
        tmp = (alphax * alphax) * (u0 / cos2phi)
    else
        tmp = (alphay * alphax) * ((u0 / sin2phi) * (alphay / alphax))
    end if
    code = tmp
end function

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = Float32(0.0)
	if (sin2phi <= Float32(1.9999999996399175e-23))
		tmp = Float32(Float32(alphax * alphax) * Float32(u0 / cos2phi));
	else
		tmp = Float32(Float32(alphay * alphax) * Float32(Float32(u0 / sin2phi) * Float32(alphay / alphax)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = single(0.0);
	if (sin2phi <= single(1.9999999996399175e-23))
		tmp = (alphax * alphax) * (u0 / cos2phi);
	else
		tmp = (alphay * alphax) * ((u0 / sin2phi) * (alphay / alphax));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if sin2phi < 2e-23

   1. Initial program 52.9%

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

      1. associate-/r* 52.9%

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

   3. Simplified 52.9%

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

   4. Taylor expanded in u0 around 0 75.3%

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

      1. unpow2 75.3%

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

      2. unpow2 75.3%

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

   6. Simplified 75.3%

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

   7. Taylor expanded in cos2phi around inf 54.0%

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

      1. *-commutative 54.0%

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

      2. *-lft-identity 54.0%

         \[\leadsto \frac{{alphax}^{2} \cdot u0}{\color{blue}{1 \cdot cos2phi}} \]

      3. times-frac 54.0%

         \[\leadsto \color{blue}{\frac{{alphax}^{2}}{1} \cdot \frac{u0}{cos2phi}} \]

      4. /-rgt-identity 54.0%

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

      5. unpow2 54.0%

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

   9. Simplified 54.0%

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

   if 2e-23 < sin2phi

   1. Initial program 60.5%

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

      1. neg-sub0 60.5%

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

      2. div-sub 60.5%

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

      3. --rgt-identity 60.5%

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

      4. div-sub 60.5%

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

      5. --rgt-identity 60.5%

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

      6. neg-sub0 60.5%

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

      7. sub-neg 60.5%

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

      8. log1p-def 98.5%

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

   3. Simplified 98.5%

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

      1. +-commutative 98.5%

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

      2. associate-/r* 98.6%

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

      3. associate-/r* 98.5%

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

      4. frac-2neg 98.5%

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

      5. frac-add 98.0%

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

      6. distribute-neg-frac 98.0%

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

   5. Applied egg-rr 98.0%

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

      1. +-commutative 98.0%

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

      2. distribute-rgt-neg-out 98.0%

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

      3. unsub-neg 98.0%

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

      4. associate-*l/ 98.1%

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

      5. associate-/l* 98.1%

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

      6. *-commutative 98.1%

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

      7. distribute-lft-neg-out 98.1%

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

      8. distribute-rgt-neg-in 98.1%

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

   7. Simplified 98.1%

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

      1. expm1-log1p-u 97.3%

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

      2. expm1-udef 43.1%

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

      3. associate-/r/ 43.1%

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

      4. associate-/r/ 43.1%

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

      5. distribute-rgt-neg-out 43.1%

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

   9. Applied egg-rr 43.1%

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

       1. expm1-def 97.6%

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

       2. expm1-log1p 98.5%

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

       3. distribute-frac-neg 98.5%

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

       4. distribute-lft-neg-in 98.5%

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

       5. distribute-rgt-neg-in 98.5%

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

   11. Simplified 98.6%

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

   12. Taylor expanded in u0 around 0 77.1%

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

       1. associate-*r/ 77.1%

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

       2. associate-/l* 77.1%

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

   14. Simplified 77.1%

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

   15. Taylor expanded in sin2phi around inf 71.1%

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

       1. times-frac 71.1%

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

   17. Simplified 71.1%

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

4. Final simplification 67.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;sin2phi \leq 1.9999999996399175 \cdot 10^{-23}:\\ \;\;\;\;\left(alphax \cdot alphax\right) \cdot \frac{u0}{cos2phi}\\ \mathbf{else}:\\ \;\;\;\;\left(alphay \cdot alphax\right) \cdot \left(\frac{u0}{sin2phi} \cdot \frac{alphay}{alphax}\right)\\ \end{array} \]

Alternative 9: 66.7% accurate, 8.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;sin2phi \leq 1.9999999996399175 \cdot 10^{-23}:\\ \;\;\;\;\left(alphax \cdot alphax\right) \cdot \frac{u0}{cos2phi}\\ \mathbf{else}:\\ \;\;\;\;\left(alphay \cdot alphax\right) \cdot \frac{u0 \cdot alphay}{alphax \cdot sin2phi}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (if (<= sin2phi 1.9999999996399175e-23)
   (* (* alphax alphax) (/ u0 cos2phi))
   (* (* alphay alphax) (/ (* u0 alphay) (* alphax sin2phi)))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float tmp;
	if (sin2phi <= 1.9999999996399175e-23f) {
		tmp = (alphax * alphax) * (u0 / cos2phi);
	} else {
		tmp = (alphay * alphax) * ((u0 * alphay) / (alphax * 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 <= 1.9999999996399175e-23) then
        tmp = (alphax * alphax) * (u0 / cos2phi)
    else
        tmp = (alphay * alphax) * ((u0 * alphay) / (alphax * sin2phi))
    end if
    code = tmp
end function

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = Float32(0.0)
	if (sin2phi <= Float32(1.9999999996399175e-23))
		tmp = Float32(Float32(alphax * alphax) * Float32(u0 / cos2phi));
	else
		tmp = Float32(Float32(alphay * alphax) * Float32(Float32(u0 * alphay) / Float32(alphax * sin2phi)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = single(0.0);
	if (sin2phi <= single(1.9999999996399175e-23))
		tmp = (alphax * alphax) * (u0 / cos2phi);
	else
		tmp = (alphay * alphax) * ((u0 * alphay) / (alphax * sin2phi));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if sin2phi < 2e-23

   1. Initial program 52.9%

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

      1. associate-/r* 52.9%

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

   3. Simplified 52.9%

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

   4. Taylor expanded in u0 around 0 75.3%

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

      1. unpow2 75.3%

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

      2. unpow2 75.3%

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

   6. Simplified 75.3%

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

   7. Taylor expanded in cos2phi around inf 54.0%

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

      1. *-commutative 54.0%

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

      2. *-lft-identity 54.0%

         \[\leadsto \frac{{alphax}^{2} \cdot u0}{\color{blue}{1 \cdot cos2phi}} \]

      3. times-frac 54.0%

         \[\leadsto \color{blue}{\frac{{alphax}^{2}}{1} \cdot \frac{u0}{cos2phi}} \]

      4. /-rgt-identity 54.0%

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

      5. unpow2 54.0%

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

   9. Simplified 54.0%

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

   if 2e-23 < sin2phi

   1. Initial program 60.5%

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

      1. neg-sub0 60.5%

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

      2. div-sub 60.5%

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

      3. --rgt-identity 60.5%

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

      4. div-sub 60.5%

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

      5. --rgt-identity 60.5%

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

      6. neg-sub0 60.5%

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

      7. sub-neg 60.5%

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

      8. log1p-def 98.5%

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

   3. Simplified 98.5%

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

      1. +-commutative 98.5%

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

      2. associate-/r* 98.6%

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

      3. associate-/r* 98.5%

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

      4. frac-2neg 98.5%

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

      5. frac-add 98.0%

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

      6. distribute-neg-frac 98.0%

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

   5. Applied egg-rr 98.0%

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

      1. +-commutative 98.0%

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

      2. distribute-rgt-neg-out 98.0%

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

      3. unsub-neg 98.0%

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

      4. associate-*l/ 98.1%

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

      5. associate-/l* 98.1%

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

      6. *-commutative 98.1%

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

      7. distribute-lft-neg-out 98.1%

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

      8. distribute-rgt-neg-in 98.1%

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

   7. Simplified 98.1%

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

      1. expm1-log1p-u 97.3%

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

      2. expm1-udef 43.1%

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

      3. associate-/r/ 43.1%

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

      4. associate-/r/ 43.1%

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

      5. distribute-rgt-neg-out 43.1%

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

   9. Applied egg-rr 43.1%

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

       1. expm1-def 97.6%

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

       2. expm1-log1p 98.5%

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

       3. distribute-frac-neg 98.5%

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

       4. distribute-lft-neg-in 98.5%

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

       5. distribute-rgt-neg-in 98.5%

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

   11. Simplified 98.6%

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

   12. Taylor expanded in u0 around 0 77.1%

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

       1. associate-*r/ 77.1%

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

       2. associate-/l* 77.1%

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

   14. Simplified 77.1%

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

   15. Taylor expanded in sin2phi around inf 71.1%

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

4. Final simplification 68.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;sin2phi \leq 1.9999999996399175 \cdot 10^{-23}:\\ \;\;\;\;\left(alphax \cdot alphax\right) \cdot \frac{u0}{cos2phi}\\ \mathbf{else}:\\ \;\;\;\;\left(alphay \cdot alphax\right) \cdot \frac{u0 \cdot alphay}{alphax \cdot sin2phi}\\ \end{array} \]

Alternative 10: 76.0% accurate, 8.9× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 59.1%

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

   1. associate-/r* 59.1%

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

3. Simplified 59.1%

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

4. Taylor expanded in u0 around 0 76.6%

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

   1. unpow2 76.6%

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

   2. unpow2 76.6%

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

6. Simplified 76.6%

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

7. Final simplification 76.6%

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

Alternative 11: 66.5% accurate, 12.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;sin2phi \leq 1.9999999996399175 \cdot 10^{-23}:\\ \;\;\;\;\left(alphax \cdot alphax\right) \cdot \frac{u0}{cos2phi}\\ \mathbf{else}:\\ \;\;\;\;\frac{u0}{\frac{sin2phi}{alphay \cdot alphay}}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (if (<= sin2phi 1.9999999996399175e-23)
   (* (* alphax alphax) (/ u0 cos2phi))
   (/ u0 (/ sin2phi (* alphay alphay)))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float tmp;
	if (sin2phi <= 1.9999999996399175e-23f) {
		tmp = (alphax * alphax) * (u0 / cos2phi);
	} else {
		tmp = u0 / (sin2phi / (alphay * alphay));
	}
	return tmp;
}

Fortran

real(4) function code(alphax, alphay, u0, cos2phi, sin2phi)
    real(4), intent (in) :: alphax
    real(4), intent (in) :: alphay
    real(4), intent (in) :: u0
    real(4), intent (in) :: cos2phi
    real(4), intent (in) :: sin2phi
    real(4) :: tmp
    if (sin2phi <= 1.9999999996399175e-23) then
        tmp = (alphax * alphax) * (u0 / cos2phi)
    else
        tmp = u0 / (sin2phi / (alphay * alphay))
    end if
    code = tmp
end function

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = Float32(0.0)
	if (sin2phi <= Float32(1.9999999996399175e-23))
		tmp = Float32(Float32(alphax * alphax) * Float32(u0 / cos2phi));
	else
		tmp = Float32(u0 / Float32(sin2phi / Float32(alphay * alphay)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = single(0.0);
	if (sin2phi <= single(1.9999999996399175e-23))
		tmp = (alphax * alphax) * (u0 / cos2phi);
	else
		tmp = u0 / (sin2phi / (alphay * alphay));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if sin2phi < 2e-23

   1. Initial program 52.9%

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

      1. associate-/r* 52.9%

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

   3. Simplified 52.9%

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

   4. Taylor expanded in u0 around 0 75.3%

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

      1. unpow2 75.3%

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

      2. unpow2 75.3%

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

   6. Simplified 75.3%

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

   7. Taylor expanded in cos2phi around inf 54.0%

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

      1. *-commutative 54.0%

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

      2. *-lft-identity 54.0%

         \[\leadsto \frac{{alphax}^{2} \cdot u0}{\color{blue}{1 \cdot cos2phi}} \]

      3. times-frac 54.0%

         \[\leadsto \color{blue}{\frac{{alphax}^{2}}{1} \cdot \frac{u0}{cos2phi}} \]

      4. /-rgt-identity 54.0%

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

      5. unpow2 54.0%

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

   9. Simplified 54.0%

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

   if 2e-23 < sin2phi

   1. Initial program 60.5%

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

      1. associate-/r* 60.5%

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

   3. Simplified 60.5%

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

   4. Taylor expanded in u0 around 0 76.9%

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

      1. unpow2 76.9%

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

      2. unpow2 76.9%

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

   6. Simplified 76.9%

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

   7. Taylor expanded in cos2phi around 0 71.2%

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

      1. unpow2 71.2%

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

      2. associate-/l* 70.8%

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

   9. Simplified 70.8%

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

4. Final simplification 67.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;sin2phi \leq 1.9999999996399175 \cdot 10^{-23}:\\ \;\;\;\;\left(alphax \cdot alphax\right) \cdot \frac{u0}{cos2phi}\\ \mathbf{else}:\\ \;\;\;\;\frac{u0}{\frac{sin2phi}{alphay \cdot alphay}}\\ \end{array} \]

Alternative 12: 23.6% accurate, 16.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 59.1%

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

   1. associate-/r* 59.1%

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

3. Simplified 59.1%

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

4. Taylor expanded in u0 around 0 76.6%

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

   1. unpow2 76.6%

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

   2. unpow2 76.6%

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

6. Simplified 76.6%

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

7. Taylor expanded in cos2phi around inf 20.4%

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

   1. *-commutative 20.4%

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

   2. *-lft-identity 20.4%

      \[\leadsto \frac{{alphax}^{2} \cdot u0}{\color{blue}{1 \cdot cos2phi}} \]

   3. times-frac 20.4%

      \[\leadsto \color{blue}{\frac{{alphax}^{2}}{1} \cdot \frac{u0}{cos2phi}} \]

   4. /-rgt-identity 20.4%

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

   5. unpow2 20.4%

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

9. Simplified 20.4%

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

10. Taylor expanded in alphax around 0 20.4%

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

    1. *-commutative 20.4%

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

    2. associate-*r/ 20.4%

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

    3. unpow2 20.4%

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

    4. associate-*l* 20.3%

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

12. Simplified 20.3%

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

13. Final simplification 20.3%

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

Alternative 13: 23.6% accurate, 16.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 59.1%

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

   1. associate-/r* 59.1%

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

3. Simplified 59.1%

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

4. Taylor expanded in u0 around 0 76.6%

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

   1. unpow2 76.6%

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

   2. unpow2 76.6%

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

6. Simplified 76.6%

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

7. Taylor expanded in cos2phi around inf 20.4%

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

   1. *-commutative 20.4%

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

   2. *-lft-identity 20.4%

      \[\leadsto \frac{{alphax}^{2} \cdot u0}{\color{blue}{1 \cdot cos2phi}} \]

   3. times-frac 20.4%

      \[\leadsto \color{blue}{\frac{{alphax}^{2}}{1} \cdot \frac{u0}{cos2phi}} \]

   4. /-rgt-identity 20.4%

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

   5. unpow2 20.4%

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

9. Simplified 20.4%

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

10. Taylor expanded in alphax around 0 20.4%

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

    1. *-commutative 20.4%

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

    2. associate-*r/ 20.4%

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

    3. unpow2 20.4%

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

    4. associate-*l* 20.3%

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

12. Simplified 20.3%

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

13. Taylor expanded in alphax around 0 20.3%

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

    1. associate-/l* 20.3%

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

15. Simplified 20.3%

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

16. Final simplification 20.3%

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

Alternative 14: 23.6% 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 59.1%

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

   1. associate-/r* 59.1%

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

3. Simplified 59.1%

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

4. Taylor expanded in u0 around 0 76.6%

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

   1. unpow2 76.6%

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

   2. unpow2 76.6%

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

6. Simplified 76.6%

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

7. Taylor expanded in cos2phi around inf 20.4%

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

   1. *-commutative 20.4%

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

   2. *-lft-identity 20.4%

      \[\leadsto \frac{{alphax}^{2} \cdot u0}{\color{blue}{1 \cdot cos2phi}} \]

   3. times-frac 20.4%

      \[\leadsto \color{blue}{\frac{{alphax}^{2}}{1} \cdot \frac{u0}{cos2phi}} \]

   4. /-rgt-identity 20.4%

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

   5. unpow2 20.4%

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

9. Simplified 20.4%

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

10. Final simplification 20.4%

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

Reproduce

herbie shell --seed 2023189 
(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)))))