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

Percentage Accurate: 60.8% → 98.3%

Time: 14.7s

Alternatives: 10

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: 60.8% 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: 98.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 62.3%

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

   1. distribute-frac-neg 62.3%

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

   2. distribute-neg-frac2 62.3%

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

   3. sub-neg 62.3%

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

   4. log1p-define 98.4%

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

   5. neg-sub0 98.4%

      \[\leadsto \frac{\mathsf{log1p}\left(-u0\right)}{\color{blue}{0 - \left(\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\right)}} \]

   6. associate--r+ 98.4%

      \[\leadsto \frac{\mathsf{log1p}\left(-u0\right)}{\color{blue}{\left(0 - \frac{cos2phi}{alphax \cdot alphax}\right) - \frac{sin2phi}{alphay \cdot alphay}}} \]

   7. neg-sub0 98.4%

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

   8. associate-/r* 98.5%

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

   9. distribute-neg-frac2 98.5%

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

3. Simplified 98.5%

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

   1. distribute-frac-neg2 98.5%

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

   2. associate-/r* 98.4%

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

   3. neg-sub0 98.4%

      \[\leadsto \frac{\mathsf{log1p}\left(-u0\right)}{\color{blue}{\left(0 - \frac{cos2phi}{alphax \cdot alphax}\right)} - \frac{sin2phi}{alphay \cdot alphay}} \]

   4. div-inv 98.4%

      \[\leadsto \frac{\mathsf{log1p}\left(-u0\right)}{\left(0 - \color{blue}{cos2phi \cdot \frac{1}{alphax \cdot alphax}}\right) - \frac{sin2phi}{alphay \cdot alphay}} \]

   5. pow2 98.4%

      \[\leadsto \frac{\mathsf{log1p}\left(-u0\right)}{\left(0 - cos2phi \cdot \frac{1}{\color{blue}{{alphax}^{2}}}\right) - \frac{sin2phi}{alphay \cdot alphay}} \]

   6. pow-flip 98.5%

      \[\leadsto \frac{\mathsf{log1p}\left(-u0\right)}{\left(0 - cos2phi \cdot \color{blue}{{alphax}^{\left(-2\right)}}\right) - \frac{sin2phi}{alphay \cdot alphay}} \]

   7. metadata-eval 98.5%

      \[\leadsto \frac{\mathsf{log1p}\left(-u0\right)}{\left(0 - cos2phi \cdot {alphax}^{\color{blue}{-2}}\right) - \frac{sin2phi}{alphay \cdot alphay}} \]

6. Applied egg-rr 98.5%

   \[\leadsto \frac{\mathsf{log1p}\left(-u0\right)}{\color{blue}{\left(0 - cos2phi \cdot {alphax}^{-2}\right)} - \frac{sin2phi}{alphay \cdot alphay}} \]
7. Step-by-step derivation

   1. neg-sub0 98.5%

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

   2. distribute-lft-neg-in 98.5%

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

   3. *-commutative 98.5%

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

8. Simplified 98.5%

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

   1. sqr-pow 98.3%

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

   2. metadata-eval 98.3%

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

   3. inv-pow 98.3%

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

   4. metadata-eval 98.3%

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

   5. inv-pow 98.3%

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

10. Applied egg-rr 98.3%

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

    1. associate-*l/ 98.5%

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

    2. *-lft-identity 98.5%

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

12. Simplified 98.5%

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

13. Final simplification 98.5%

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

Alternative 2: 98.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 62.3%

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

   1. distribute-frac-neg 62.3%

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

   2. distribute-neg-frac2 62.3%

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

   3. sub-neg 62.3%

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

   4. log1p-define 98.4%

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

   5. neg-sub0 98.4%

      \[\leadsto \frac{\mathsf{log1p}\left(-u0\right)}{\color{blue}{0 - \left(\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\right)}} \]

   6. associate--r+ 98.4%

      \[\leadsto \frac{\mathsf{log1p}\left(-u0\right)}{\color{blue}{\left(0 - \frac{cos2phi}{alphax \cdot alphax}\right) - \frac{sin2phi}{alphay \cdot alphay}}} \]

   7. neg-sub0 98.4%

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

   8. associate-/r* 98.5%

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

   9. distribute-neg-frac2 98.5%

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

3. Simplified 98.5%

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

Alternative 3: 92.9% accurate, 4.3× speedup

Math

\[\begin{array}{l} \\ \frac{u0 \cdot \left(1 + u0 \cdot \left(0.5 + u0 \cdot \left(0.3333333333333333 - u0 \cdot -0.25\right)\right)\right)}{\frac{sin2phi}{alphay \cdot alphay} + \frac{\frac{cos2phi}{alphax}}{alphax}} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (/
  (* u0 (+ 1.0 (* u0 (+ 0.5 (* u0 (- 0.3333333333333333 (* u0 -0.25)))))))
  (+ (/ sin2phi (* alphay alphay)) (/ (/ cos2phi alphax) alphax))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	return (u0 * (1.0f + (u0 * (0.5f + (u0 * (0.3333333333333333f - (u0 * -0.25f))))))) / ((sin2phi / (alphay * alphay)) + ((cos2phi / alphax) / alphax));
}

Fortran

real(4) function code(alphax, alphay, u0, cos2phi, sin2phi)
    real(4), intent (in) :: alphax
    real(4), intent (in) :: alphay
    real(4), intent (in) :: u0
    real(4), intent (in) :: cos2phi
    real(4), intent (in) :: sin2phi
    code = (u0 * (1.0e0 + (u0 * (0.5e0 + (u0 * (0.3333333333333333e0 - (u0 * (-0.25e0)))))))) / ((sin2phi / (alphay * alphay)) + ((cos2phi / alphax) / alphax))
end function

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	return Float32(Float32(u0 * Float32(Float32(1.0) + Float32(u0 * Float32(Float32(0.5) + Float32(u0 * Float32(Float32(0.3333333333333333) - Float32(u0 * Float32(-0.25)))))))) / Float32(Float32(sin2phi / Float32(alphay * alphay)) + Float32(Float32(cos2phi / alphax) / alphax)))
end

MATLAB

function tmp = code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = (u0 * (single(1.0) + (u0 * (single(0.5) + (u0 * (single(0.3333333333333333) - (u0 * single(-0.25)))))))) / ((sin2phi / (alphay * alphay)) + ((cos2phi / alphax) / alphax));
end

Derivation

1. Initial program 62.3%

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

   1. distribute-frac-neg 62.3%

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

   2. distribute-neg-frac2 62.3%

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

   3. sub-neg 62.3%

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

   4. log1p-define 98.4%

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

   5. neg-sub0 98.4%

      \[\leadsto \frac{\mathsf{log1p}\left(-u0\right)}{\color{blue}{0 - \left(\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\right)}} \]

   6. associate--r+ 98.4%

      \[\leadsto \frac{\mathsf{log1p}\left(-u0\right)}{\color{blue}{\left(0 - \frac{cos2phi}{alphax \cdot alphax}\right) - \frac{sin2phi}{alphay \cdot alphay}}} \]

   7. neg-sub0 98.4%

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

   8. associate-/r* 98.5%

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

   9. distribute-neg-frac2 98.5%

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

3. Simplified 98.5%

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

5. Taylor expanded in u0 around 0 92.9%

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

6. Final simplification 92.9%

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

Alternative 4: 92.9% accurate, 4.3× speedup

Math

\[\begin{array}{l} \\ \frac{u0 \cdot \left(1 + u0 \cdot \left(0.5 + u0 \cdot \left(0.3333333333333333 + u0 \cdot 0.25\right)\right)\right)}{\frac{sin2phi}{alphay \cdot alphay} + \frac{cos2phi}{alphax \cdot alphax}} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (/
  (* u0 (+ 1.0 (* u0 (+ 0.5 (* u0 (+ 0.3333333333333333 (* u0 0.25)))))))
  (+ (/ sin2phi (* alphay alphay)) (/ cos2phi (* alphax alphax)))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	return (u0 * (1.0f + (u0 * (0.5f + (u0 * (0.3333333333333333f + (u0 * 0.25f))))))) / ((sin2phi / (alphay * alphay)) + (cos2phi / (alphax * alphax)));
}

Fortran

real(4) function code(alphax, alphay, u0, cos2phi, sin2phi)
    real(4), intent (in) :: alphax
    real(4), intent (in) :: alphay
    real(4), intent (in) :: u0
    real(4), intent (in) :: cos2phi
    real(4), intent (in) :: sin2phi
    code = (u0 * (1.0e0 + (u0 * (0.5e0 + (u0 * (0.3333333333333333e0 + (u0 * 0.25e0))))))) / ((sin2phi / (alphay * alphay)) + (cos2phi / (alphax * alphax)))
end function

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	return Float32(Float32(u0 * Float32(Float32(1.0) + Float32(u0 * Float32(Float32(0.5) + Float32(u0 * Float32(Float32(0.3333333333333333) + Float32(u0 * Float32(0.25)))))))) / Float32(Float32(sin2phi / Float32(alphay * alphay)) + Float32(cos2phi / Float32(alphax * alphax))))
end

MATLAB

function tmp = code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = (u0 * (single(1.0) + (u0 * (single(0.5) + (u0 * (single(0.3333333333333333) + (u0 * single(0.25)))))))) / ((sin2phi / (alphay * alphay)) + (cos2phi / (alphax * alphax)));
end

Derivation

1. Initial program 62.3%

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

3. Taylor expanded in u0 around 0 92.9%

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

   1. *-commutative 92.9%

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

5. Simplified 92.9%

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

6. Final simplification 92.9%

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

Alternative 5: 91.0% accurate, 5.0× speedup

Math

\[\begin{array}{l} \\ \frac{u0 \cdot \left(1 + u0 \cdot \left(0.5 - u0 \cdot -0.3333333333333333\right)\right)}{\frac{sin2phi}{alphay \cdot alphay} + \frac{\frac{cos2phi}{alphax}}{alphax}} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (/
  (* u0 (+ 1.0 (* u0 (- 0.5 (* u0 -0.3333333333333333)))))
  (+ (/ sin2phi (* alphay alphay)) (/ (/ cos2phi alphax) alphax))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	return (u0 * (1.0f + (u0 * (0.5f - (u0 * -0.3333333333333333f))))) / ((sin2phi / (alphay * alphay)) + ((cos2phi / alphax) / alphax));
}

Fortran

real(4) function code(alphax, alphay, u0, cos2phi, sin2phi)
    real(4), intent (in) :: alphax
    real(4), intent (in) :: alphay
    real(4), intent (in) :: u0
    real(4), intent (in) :: cos2phi
    real(4), intent (in) :: sin2phi
    code = (u0 * (1.0e0 + (u0 * (0.5e0 - (u0 * (-0.3333333333333333e0)))))) / ((sin2phi / (alphay * alphay)) + ((cos2phi / alphax) / alphax))
end function

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	return Float32(Float32(u0 * Float32(Float32(1.0) + Float32(u0 * Float32(Float32(0.5) - Float32(u0 * Float32(-0.3333333333333333)))))) / Float32(Float32(sin2phi / Float32(alphay * alphay)) + Float32(Float32(cos2phi / alphax) / alphax)))
end

MATLAB

function tmp = code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = (u0 * (single(1.0) + (u0 * (single(0.5) - (u0 * single(-0.3333333333333333)))))) / ((sin2phi / (alphay * alphay)) + ((cos2phi / alphax) / alphax));
end

Derivation

1. Initial program 62.3%

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

   1. distribute-frac-neg 62.3%

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

   2. distribute-neg-frac2 62.3%

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

   3. sub-neg 62.3%

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

   4. log1p-define 98.4%

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

   5. neg-sub0 98.4%

      \[\leadsto \frac{\mathsf{log1p}\left(-u0\right)}{\color{blue}{0 - \left(\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\right)}} \]

   6. associate--r+ 98.4%

      \[\leadsto \frac{\mathsf{log1p}\left(-u0\right)}{\color{blue}{\left(0 - \frac{cos2phi}{alphax \cdot alphax}\right) - \frac{sin2phi}{alphay \cdot alphay}}} \]

   7. neg-sub0 98.4%

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

   8. associate-/r* 98.5%

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

   9. distribute-neg-frac2 98.5%

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

3. Simplified 98.5%

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

5. Taylor expanded in u0 around 0 91.1%

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

6. Final simplification 91.1%

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

Alternative 6: 91.0% accurate, 5.0× speedup

Math

\[\begin{array}{l} \\ \frac{u0 \cdot \left(1 + u0 \cdot \left(0.5 + u0 \cdot 0.3333333333333333\right)\right)}{\frac{sin2phi}{alphay \cdot alphay} + \frac{cos2phi}{alphax \cdot alphax}} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (/
  (* u0 (+ 1.0 (* u0 (+ 0.5 (* u0 0.3333333333333333)))))
  (+ (/ sin2phi (* alphay alphay)) (/ cos2phi (* alphax alphax)))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	return (u0 * (1.0f + (u0 * (0.5f + (u0 * 0.3333333333333333f))))) / ((sin2phi / (alphay * alphay)) + (cos2phi / (alphax * alphax)));
}

Fortran

real(4) function code(alphax, alphay, u0, cos2phi, sin2phi)
    real(4), intent (in) :: alphax
    real(4), intent (in) :: alphay
    real(4), intent (in) :: u0
    real(4), intent (in) :: cos2phi
    real(4), intent (in) :: sin2phi
    code = (u0 * (1.0e0 + (u0 * (0.5e0 + (u0 * 0.3333333333333333e0))))) / ((sin2phi / (alphay * alphay)) + (cos2phi / (alphax * alphax)))
end function

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	return Float32(Float32(u0 * Float32(Float32(1.0) + Float32(u0 * Float32(Float32(0.5) + Float32(u0 * Float32(0.3333333333333333)))))) / Float32(Float32(sin2phi / Float32(alphay * alphay)) + Float32(cos2phi / Float32(alphax * alphax))))
end

MATLAB

function tmp = code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = (u0 * (single(1.0) + (u0 * (single(0.5) + (u0 * single(0.3333333333333333)))))) / ((sin2phi / (alphay * alphay)) + (cos2phi / (alphax * alphax)));
end

Derivation

1. Initial program 62.3%

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

3. Taylor expanded in u0 around 0 91.0%

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

   1. *-commutative 91.0%

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

5. Simplified 91.0%

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

6. Final simplification 91.0%

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

Alternative 7: 87.1% accurate, 6.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 62.3%

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

   1. distribute-frac-neg 62.3%

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

   2. distribute-neg-frac2 62.3%

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

   3. sub-neg 62.3%

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

   4. log1p-define 98.4%

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

   5. neg-sub0 98.4%

      \[\leadsto \frac{\mathsf{log1p}\left(-u0\right)}{\color{blue}{0 - \left(\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\right)}} \]

   6. associate--r+ 98.4%

      \[\leadsto \frac{\mathsf{log1p}\left(-u0\right)}{\color{blue}{\left(0 - \frac{cos2phi}{alphax \cdot alphax}\right) - \frac{sin2phi}{alphay \cdot alphay}}} \]

   7. neg-sub0 98.4%

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

   8. associate-/r* 98.5%

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

   9. distribute-neg-frac2 98.5%

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

3. Simplified 98.5%

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

5. Taylor expanded in u0 around 0 86.9%

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

6. Final simplification 86.9%

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

Alternative 8: 87.1% accurate, 6.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 62.3%

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

3. Taylor expanded in u0 around 0 86.8%

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

   1. *-commutative 86.8%

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

5. Simplified 86.8%

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

6. Final simplification 86.8%

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

Alternative 9: 75.5% accurate, 7.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 62.3%

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

   1. distribute-frac-neg 62.3%

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

   2. distribute-neg-frac2 62.3%

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

   3. sub-neg 62.3%

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

   4. log1p-define 98.4%

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

   5. neg-sub0 98.4%

      \[\leadsto \frac{\mathsf{log1p}\left(-u0\right)}{\color{blue}{0 - \left(\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\right)}} \]

   6. associate--r+ 98.4%

      \[\leadsto \frac{\mathsf{log1p}\left(-u0\right)}{\color{blue}{\left(0 - \frac{cos2phi}{alphax \cdot alphax}\right) - \frac{sin2phi}{alphay \cdot alphay}}} \]

   7. neg-sub0 98.4%

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

   8. associate-/r* 98.5%

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

   9. distribute-neg-frac2 98.5%

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

3. Simplified 98.5%

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

   1. distribute-frac-neg2 98.5%

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

   2. associate-/r* 98.4%

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

   3. neg-sub0 98.4%

      \[\leadsto \frac{\mathsf{log1p}\left(-u0\right)}{\color{blue}{\left(0 - \frac{cos2phi}{alphax \cdot alphax}\right)} - \frac{sin2phi}{alphay \cdot alphay}} \]

   4. div-inv 98.4%

      \[\leadsto \frac{\mathsf{log1p}\left(-u0\right)}{\left(0 - \color{blue}{cos2phi \cdot \frac{1}{alphax \cdot alphax}}\right) - \frac{sin2phi}{alphay \cdot alphay}} \]

   5. pow2 98.4%

      \[\leadsto \frac{\mathsf{log1p}\left(-u0\right)}{\left(0 - cos2phi \cdot \frac{1}{\color{blue}{{alphax}^{2}}}\right) - \frac{sin2phi}{alphay \cdot alphay}} \]

   6. pow-flip 98.5%

      \[\leadsto \frac{\mathsf{log1p}\left(-u0\right)}{\left(0 - cos2phi \cdot \color{blue}{{alphax}^{\left(-2\right)}}\right) - \frac{sin2phi}{alphay \cdot alphay}} \]

   7. metadata-eval 98.5%

      \[\leadsto \frac{\mathsf{log1p}\left(-u0\right)}{\left(0 - cos2phi \cdot {alphax}^{\color{blue}{-2}}\right) - \frac{sin2phi}{alphay \cdot alphay}} \]

6. Applied egg-rr 98.5%

   \[\leadsto \frac{\mathsf{log1p}\left(-u0\right)}{\color{blue}{\left(0 - cos2phi \cdot {alphax}^{-2}\right)} - \frac{sin2phi}{alphay \cdot alphay}} \]
7. Step-by-step derivation

   1. neg-sub0 98.5%

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

   2. distribute-lft-neg-in 98.5%

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

   3. *-commutative 98.5%

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

8. Simplified 98.5%

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

   1. sqr-pow 98.3%

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

   2. metadata-eval 98.3%

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

   3. inv-pow 98.3%

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

   4. metadata-eval 98.3%

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

   5. inv-pow 98.3%

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

10. Applied egg-rr 98.3%

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

    1. associate-*l/ 98.5%

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

    2. *-lft-identity 98.5%

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

12. Simplified 98.5%

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

13. Taylor expanded in u0 around 0 75.0%

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

    1. mul-1-neg 75.0%

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

15. Simplified 75.0%

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

16. Final simplification 75.0%

    \[\leadsto \frac{u0}{\frac{sin2phi}{alphay \cdot alphay} + \frac{\frac{1}{alphax}}{alphax} \cdot cos2phi} \]
17. Add Preprocessing

Alternative 10: 75.5% accurate, 8.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 62.3%

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

3. Taylor expanded in u0 around 0 74.9%

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

4. Final simplification 74.9%

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

Reproduce

herbie shell --seed 2024165 
(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)))))