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

Percentage Accurate: 60.7% → 98.5%

Time: 15.0s

Alternatives: 8

Speedup: 8.9×

Specification

\[\left(\left(\left(\left(0.0001 \leq alphax \land alphax \leq 1\right) \land \left(0.0001 \leq alphay \land alphay \leq 1\right)\right) \land \left(2.328306437 \cdot 10^{-10} \leq u0 \land u0 \leq 1\right)\right) \land \left(0 \leq cos2phi \land cos2phi \leq 1\right)\right) \land 0 \leq sin2phi\]

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Initial Program: 60.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Alternative 1: 98.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 62.1%

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

   1. sub-neg 62.1%

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

   2. 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. associate-/r* 98.5%

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

   2. associate-/r* 98.5%

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

   3. frac-2neg 98.5%

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

   4. frac-add 98.4%

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

   5. distribute-neg-frac 98.4%

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

5. Applied egg-rr 98.4%

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

   1. +-commutative 98.4%

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

   2. *-commutative 98.4%

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

   3. distribute-lft-neg-out 98.4%

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

   4. unsub-neg 98.4%

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

   5. associate-*r/ 98.5%

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

   6. associate-/l* 98.4%

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

   7. associate-*r/ 98.4%

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

   8. associate-/l* 98.5%

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

7. Simplified 98.5%

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

   1. associate-/r/ 98.7%

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

   2. associate-/r/ 98.7%

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

   3. associate-/r/ 98.7%

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

9. Applied egg-rr 98.7%

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

10. Final simplification 98.7%

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

Alternative 2: 98.3% 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 62.1%

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

   1. sub-neg 62.1%

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

   2. 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. Final simplification 98.7%

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

Alternative 3: 75.8% accurate, 6.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 62.1%

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

   1. sub-neg 62.1%

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

   2. 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. associate-/r* 98.5%

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

   2. associate-/r* 98.5%

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

   3. frac-2neg 98.5%

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

   4. frac-add 98.4%

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

   5. distribute-neg-frac 98.4%

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

5. Applied egg-rr 98.4%

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

   1. +-commutative 98.4%

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

   2. *-commutative 98.4%

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

   3. distribute-lft-neg-out 98.4%

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

   4. unsub-neg 98.4%

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

   5. associate-*r/ 98.5%

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

   6. associate-/l* 98.4%

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

   7. associate-*r/ 98.4%

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

   8. associate-/l* 98.5%

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

7. Simplified 98.5%

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

8. Taylor expanded in u0 around 0 75.0%

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

   1. mul-1-neg 75.0%

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

   2. mul-1-neg 75.0%

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

   3. associate-/l* 75.0%

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

   4. associate-*l/ 75.0%

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

   5. *-commutative 75.0%

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

10. Simplified 75.0%

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

11. Final simplification 75.0%

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

Alternative 4: 75.8% accurate, 6.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 62.1%

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

   1. sub-neg 62.1%

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

   2. 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. associate-/r* 98.5%

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

   2. associate-/r* 98.5%

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

   3. frac-2neg 98.5%

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

   4. frac-add 98.4%

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

   5. distribute-neg-frac 98.4%

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

5. Applied egg-rr 98.4%

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

   1. +-commutative 98.4%

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

   2. *-commutative 98.4%

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

   3. distribute-lft-neg-out 98.4%

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

   4. unsub-neg 98.4%

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

   5. associate-*r/ 98.5%

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

   6. associate-/l* 98.4%

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

   7. associate-*r/ 98.4%

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

   8. associate-/l* 98.5%

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

7. Simplified 98.5%

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

8. Taylor expanded in u0 around 0 75.0%

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

   1. associate-*r/ 75.0%

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

   2. mul-1-neg 75.0%

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

   3. mul-1-neg 75.0%

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

   4. associate-/l* 75.0%

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

   5. distribute-neg-frac 75.0%

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

   6. *-commutative 75.0%

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

10. Simplified 75.0%

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

11. Final simplification 75.0%

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

Alternative 5: 75.8% accurate, 6.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 62.1%

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

   1. sub-neg 62.1%

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

   2. 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. associate-/r* 98.5%

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

   2. associate-/r* 98.5%

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

   3. frac-2neg 98.5%

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

   4. frac-add 98.4%

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

   5. distribute-neg-frac 98.4%

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

5. Applied egg-rr 98.4%

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

   1. +-commutative 98.4%

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

   2. *-commutative 98.4%

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

   3. distribute-lft-neg-out 98.4%

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

   4. unsub-neg 98.4%

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

   5. associate-*r/ 98.5%

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

   6. associate-/l* 98.4%

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

   7. associate-*r/ 98.4%

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

   8. associate-/l* 98.5%

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

7. Simplified 98.5%

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

8. Taylor expanded in u0 around 0 75.0%

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

9. Final simplification 75.0%

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

Alternative 6: 75.6% accurate, 7.7× speedup

Math

\[\begin{array}{l} \\ \frac{u0}{\frac{cos2phi}{alphax} \cdot \frac{1}{alphax} + \frac{\frac{sin2phi}{alphay}}{alphay}} \end{array} \]

FPCore

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

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	return u0 / (((cos2phi / alphax) * (1.0f / 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) * (1.0e0 / alphax)) + ((sin2phi / alphay) / alphay))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 62.1%

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

   1. associate-/r* 62.1%

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

3. Simplified 62.1%

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

4. Taylor expanded in u0 around 0 74.9%

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

   1. mul-1-neg 74.9%

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

6. Simplified 74.9%

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

   1. associate-/r* 74.9%

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

   2. div-inv 74.9%

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

8. Applied egg-rr 74.9%

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

9. Final simplification 74.9%

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

Alternative 7: 75.6% accurate, 8.9× speedup

Math

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

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(Float32(sin2phi / 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 62.1%

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

   1. associate-/r* 62.1%

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

3. Simplified 62.1%

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

4. Taylor expanded in u0 around 0 74.9%

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

   1. mul-1-neg 74.9%

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

6. Simplified 74.9%

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

7. Final simplification 74.9%

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

Alternative 8: 75.6% accurate, 8.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 62.1%

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

   1. sqr-neg 62.1%

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

   2. sub-neg 62.1%

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

   3. log1p-def 98.7%

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

   4. sqr-neg 98.7%

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

   5. associate-/r* 98.5%

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

3. Simplified 98.5%

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

   1. clear-num 98.4%

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

   2. associate-/r/ 98.4%

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

   3. pow2 98.4%

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

   4. pow-flip 98.5%

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

   5. metadata-eval 98.5%

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

5. Applied egg-rr 98.5%

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

   1. *-commutative 98.5%

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

   2. metadata-eval 98.5%

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

   3. pow-flip 98.4%

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

   4. pow2 98.4%

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

   5. div-inv 98.5%

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

   6. associate-/r* 98.5%

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

7. Applied egg-rr 98.5%

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

8. Taylor expanded in u0 around 0 74.9%

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

   1. mul-1-neg 74.9%

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

10. Simplified 74.9%

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

11. Final simplification 74.9%

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

Reproduce

herbie shell --seed 2023310 
(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)))))