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

Percentage Accurate: 60.7% → 98.3%

Time: 13.9s

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: 60.7% 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: 98.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \frac{-\sqrt[3]{{\left(\mathsf{log1p}\left(-u0\right)\right)}^{3}}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (/
  (- (cbrt (pow (log1p (- u0)) 3.0)))
  (+ (/ cos2phi (* alphax alphax)) (/ sin2phi (* alphay alphay)))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	return -cbrtf(powf(log1pf(-u0), 3.0f)) / ((cos2phi / (alphax * alphax)) + (sin2phi / (alphay * alphay)));
}

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	return Float32(Float32(-cbrt((log1p(Float32(-u0)) ^ Float32(3.0)))) / 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. distribute-frac-neg 62.1%

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

   2. distribute-neg-frac2 62.1%

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

   3. neg-mul-1 62.1%

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

   4. associate-/r* 62.1%

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

   5. metadata-eval 62.1%

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

   6. distribute-neg-frac2 62.1%

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

   7. /-rgt-identity 62.1%

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

   8. sub-neg 62.1%

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

   9. log1p-define 98.2%

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

3. Simplified 98.2%

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

   1. neg-sub0 98.2%

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

   2. flip3-- 98.0%

      \[\leadsto \frac{-\mathsf{log1p}\left(\color{blue}{\frac{{0}^{3} - {u0}^{3}}{0 \cdot 0 + \left(u0 \cdot u0 + 0 \cdot u0\right)}}\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   3. metadata-eval 98.0%

      \[\leadsto \frac{-\mathsf{log1p}\left(\frac{\color{blue}{0} - {u0}^{3}}{0 \cdot 0 + \left(u0 \cdot u0 + 0 \cdot u0\right)}\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   4. metadata-eval 98.0%

      \[\leadsto \frac{-\mathsf{log1p}\left(\frac{0 - {u0}^{3}}{\color{blue}{0} + \left(u0 \cdot u0 + 0 \cdot u0\right)}\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   5. pow2 98.0%

      \[\leadsto \frac{-\mathsf{log1p}\left(\frac{0 - {u0}^{3}}{0 + \left(\color{blue}{{u0}^{2}} + 0 \cdot u0\right)}\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

6. Applied egg-rr 98.0%

   \[\leadsto \frac{-\mathsf{log1p}\left(\color{blue}{\frac{0 - {u0}^{3}}{0 + \left({u0}^{2} + 0 \cdot u0\right)}}\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
7. Step-by-step derivation

   1. sub0-neg 98.0%

      \[\leadsto \frac{-\mathsf{log1p}\left(\frac{\color{blue}{-{u0}^{3}}}{0 + \left({u0}^{2} + 0 \cdot u0\right)}\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   2. +-lft-identity 98.0%

      \[\leadsto \frac{-\mathsf{log1p}\left(\frac{-{u0}^{3}}{\color{blue}{{u0}^{2} + 0 \cdot u0}}\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   3. mul0-lft 98.0%

      \[\leadsto \frac{-\mathsf{log1p}\left(\frac{-{u0}^{3}}{{u0}^{2} + \color{blue}{0}}\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   4. +-rgt-identity 98.0%

      \[\leadsto \frac{-\mathsf{log1p}\left(\frac{-{u0}^{3}}{\color{blue}{{u0}^{2}}}\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

8. Simplified 98.0%

   \[\leadsto \frac{-\mathsf{log1p}\left(\color{blue}{\frac{-{u0}^{3}}{{u0}^{2}}}\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
9. Step-by-step derivation

   1. distribute-frac-neg 98.0%

      \[\leadsto \frac{-\mathsf{log1p}\left(\color{blue}{-\frac{{u0}^{3}}{{u0}^{2}}}\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   2. pow-div 98.2%

      \[\leadsto \frac{-\mathsf{log1p}\left(-\color{blue}{{u0}^{\left(3 - 2\right)}}\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   3. metadata-eval 98.2%

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

   4. pow1 98.2%

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

   5. add-cbrt-cube 98.2%

      \[\leadsto \frac{-\color{blue}{\sqrt[3]{\left(\mathsf{log1p}\left(-u0\right) \cdot \mathsf{log1p}\left(-u0\right)\right) \cdot \mathsf{log1p}\left(-u0\right)}}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   6. pow3 98.2%

      \[\leadsto \frac{-\sqrt[3]{\color{blue}{{\left(\mathsf{log1p}\left(-u0\right)\right)}^{3}}}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

10. Applied egg-rr 98.2%

    \[\leadsto \frac{-\color{blue}{\sqrt[3]{{\left(\mathsf{log1p}\left(-u0\right)\right)}^{3}}}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
11. Add Preprocessing

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. distribute-frac-neg 62.1%

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

   2. distribute-neg-frac2 62.1%

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

   3. neg-mul-1 62.1%

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

   4. associate-/r* 62.1%

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

   5. metadata-eval 62.1%

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

   6. distribute-neg-frac2 62.1%

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

   7. /-rgt-identity 62.1%

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

   8. sub-neg 62.1%

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

   9. log1p-define 98.2%

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

3. Simplified 98.2%

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

Alternative 3: 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 62.1%

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

   1. distribute-frac-neg 62.1%

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

   2. distribute-neg-frac2 62.1%

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

   3. sub-neg 62.1%

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

   4. log1p-define 98.2%

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

   5. neg-sub0 98.2%

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

   6. associate--r+ 98.2%

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

   7. neg-sub0 98.2%

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

   8. associate-/r* 98.2%

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

   9. distribute-neg-frac2 98.2%

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

3. Simplified 98.2%

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

Alternative 4: 94.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u0 \leq 0.05999999865889549:\\ \;\;\;\;\frac{u0 \cdot \left(1 + u0 \cdot \left(0.5 + u0 \cdot \left(0.3333333333333333 + u0 \cdot 0.25\right)\right)\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(-u0\right) \cdot \left(alphay \cdot \left(-alphay\right)\right)}{sin2phi}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (if (<= u0 0.05999999865889549)
   (/
    (* u0 (+ 1.0 (* u0 (+ 0.5 (* u0 (+ 0.3333333333333333 (* u0 0.25)))))))
    (+ (/ cos2phi (* alphax alphax)) (/ sin2phi (* alphay alphay))))
   (/ (* (log1p (- u0)) (* alphay (- alphay))) sin2phi)))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float tmp;
	if (u0 <= 0.05999999865889549f) {
		tmp = (u0 * (1.0f + (u0 * (0.5f + (u0 * (0.3333333333333333f + (u0 * 0.25f))))))) / ((cos2phi / (alphax * alphax)) + (sin2phi / (alphay * alphay)));
	} else {
		tmp = (log1pf(-u0) * (alphay * -alphay)) / sin2phi;
	}
	return tmp;
}

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = Float32(0.0)
	if (u0 <= Float32(0.05999999865889549))
		tmp = 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(cos2phi / Float32(alphax * alphax)) + Float32(sin2phi / Float32(alphay * alphay))));
	else
		tmp = Float32(Float32(log1p(Float32(-u0)) * Float32(alphay * Float32(-alphay))) / sin2phi);
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if u0 < 0.0599999987

   1. Initial program 56.8%

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

   3. Taylor expanded in u0 around 0 97.5%

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

      1. *-commutative 97.5%

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

   5. Simplified 97.5%

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

   if 0.0599999987 < u0

   1. Initial program 96.1%

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

      1. distribute-frac-neg 96.1%

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

      2. distribute-neg-frac2 96.1%

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

      3. neg-mul-1 96.1%

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

      4. associate-/r* 96.1%

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

      5. metadata-eval 96.1%

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

      6. distribute-neg-frac2 96.1%

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

      7. /-rgt-identity 96.1%

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

      8. sub-neg 96.1%

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

      9. log1p-define 97.5%

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

   3. Simplified 97.5%

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

      1. neg-sub0 97.5%

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

      2. flip3-- 96.6%

         \[\leadsto \frac{-\mathsf{log1p}\left(\color{blue}{\frac{{0}^{3} - {u0}^{3}}{0 \cdot 0 + \left(u0 \cdot u0 + 0 \cdot u0\right)}}\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      3. metadata-eval 96.6%

         \[\leadsto \frac{-\mathsf{log1p}\left(\frac{\color{blue}{0} - {u0}^{3}}{0 \cdot 0 + \left(u0 \cdot u0 + 0 \cdot u0\right)}\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      4. metadata-eval 96.6%

         \[\leadsto \frac{-\mathsf{log1p}\left(\frac{0 - {u0}^{3}}{\color{blue}{0} + \left(u0 \cdot u0 + 0 \cdot u0\right)}\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      5. pow2 96.6%

         \[\leadsto \frac{-\mathsf{log1p}\left(\frac{0 - {u0}^{3}}{0 + \left(\color{blue}{{u0}^{2}} + 0 \cdot u0\right)}\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   6. Applied egg-rr 96.6%

      \[\leadsto \frac{-\mathsf{log1p}\left(\color{blue}{\frac{0 - {u0}^{3}}{0 + \left({u0}^{2} + 0 \cdot u0\right)}}\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
   7. Step-by-step derivation

      1. sub0-neg 96.6%

         \[\leadsto \frac{-\mathsf{log1p}\left(\frac{\color{blue}{-{u0}^{3}}}{0 + \left({u0}^{2} + 0 \cdot u0\right)}\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      2. +-lft-identity 96.6%

         \[\leadsto \frac{-\mathsf{log1p}\left(\frac{-{u0}^{3}}{\color{blue}{{u0}^{2} + 0 \cdot u0}}\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      3. mul0-lft 96.6%

         \[\leadsto \frac{-\mathsf{log1p}\left(\frac{-{u0}^{3}}{{u0}^{2} + \color{blue}{0}}\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      4. +-rgt-identity 96.6%

         \[\leadsto \frac{-\mathsf{log1p}\left(\frac{-{u0}^{3}}{\color{blue}{{u0}^{2}}}\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   8. Simplified 96.6%

      \[\leadsto \frac{-\mathsf{log1p}\left(\color{blue}{\frac{-{u0}^{3}}{{u0}^{2}}}\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   9. Taylor expanded in cos2phi around 0 84.7%

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

       1. neg-mul-1 84.7%

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

       2. sub-neg 84.7%

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

       3. associate-*r/ 84.7%

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

       4. associate-*r* 84.7%

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

       5. mul-1-neg 84.7%

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

       6. sub-neg 84.7%

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

       7. log1p-define 85.9%

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

   11. Simplified 85.9%

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

       1. pow2 85.9%

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

   13. Applied egg-rr 85.9%

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

4. Final simplification 95.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u0 \leq 0.05999999865889549:\\ \;\;\;\;\frac{u0 \cdot \left(1 + u0 \cdot \left(0.5 + u0 \cdot \left(0.3333333333333333 + u0 \cdot 0.25\right)\right)\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(-u0\right) \cdot \left(alphay \cdot \left(-alphay\right)\right)}{sin2phi}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 92.7% accurate, 4.3× speedup

Math

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

FPCore

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

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))))))) / ((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 * (1.0e0 + (u0 * (0.5e0 + (u0 * (0.3333333333333333e0 + (u0 * 0.25e0))))))) / ((cos2phi / (alphax * alphax)) + (sin2phi / (alphay * alphay)))
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(cos2phi / Float32(alphax * alphax)) + Float32(sin2phi / Float32(alphay * alphay))))
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)))))))) / ((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. Add Preprocessing

3. Taylor expanded in u0 around 0 92.0%

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

   1. *-commutative 92.0%

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

5. Simplified 92.0%

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

Alternative 6: 90.9% accurate, 5.0× speedup

Math

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

FPCore

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

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	return (u0 * (1.0f + (u0 * (0.5f + (u0 * 0.3333333333333333f))))) / ((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 * (1.0e0 + (u0 * (0.5e0 + (u0 * 0.3333333333333333e0))))) / ((cos2phi / (alphax * alphax)) + (sin2phi / (alphay * alphay)))
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(cos2phi / Float32(alphax * alphax)) + Float32(sin2phi / Float32(alphay * alphay))))
end

MATLAB

function tmp = code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = (u0 * (single(1.0) + (u0 * (single(0.5) + (u0 * single(0.3333333333333333)))))) / ((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. Add Preprocessing

3. Taylor expanded in u0 around 0 89.8%

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

   1. *-commutative 89.8%

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

5. Simplified 89.8%

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

Alternative 7: 87.1% accurate, 6.1× speedup

Math

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

FPCore

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

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	return (u0 * (1.0f + (u0 * 0.5f))) / ((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 * (1.0e0 + (u0 * 0.5e0))) / ((cos2phi / (alphax * alphax)) + (sin2phi / (alphay * alphay)))
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(cos2phi / Float32(alphax * alphax)) + Float32(sin2phi / Float32(alphay * alphay))))
end

MATLAB

function tmp = code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = (u0 * (single(1.0) + (u0 * single(0.5)))) / ((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. Add Preprocessing

3. Taylor expanded in u0 around 0 85.8%

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

   1. *-commutative 85.8%

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

5. Simplified 85.8%

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

Alternative 8: 75.9% 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. Add Preprocessing

3. Taylor expanded in u0 around 0 75.0%

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

   1. associate-/r* 75.0%

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

   2. div-inv 74.9%

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

5. Applied egg-rr 74.9%

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

   1. associate-*r/ 75.0%

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

   2. *-rgt-identity 75.0%

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

7. Simplified 75.0%

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

Alternative 9: 75.9% 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 62.1%

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

3. Taylor expanded in u0 around 0 75.0%

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

Alternative 10: 59.3% accurate, 16.6× speedup

Math

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

FPCore

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

C

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

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 * alphay)) / sin2phi
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 62.1%

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

3. Taylor expanded in u0 around 0 75.0%

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

4. Taylor expanded in cos2phi around 0 58.9%

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

   1. pow2 75.7%

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

6. Applied egg-rr 58.9%

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

7. Final simplification 58.9%

   \[\leadsto \frac{u0 \cdot \left(alphay \cdot alphay\right)}{sin2phi} \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024177 
(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)))))