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

Percentage Accurate: 61.0% → 98.4%

Time: 14.5s

Alternatives: 10

Speedup: 1.0×

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: 61.0% 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: 98.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \frac{\mathsf{log1p}\left(-u0\right)}{{alphax}^{-2} \cdot \left(-cos2phi\right) - \frac{sin2phi}{alphay \cdot alphay}} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (/
  (log1p (- u0))
  (- (* (pow alphax -2.0) (- cos2phi)) (/ sin2phi (* alphay alphay)))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	return log1pf(-u0) / ((powf(alphax, -2.0f) * -cos2phi) - (sin2phi / (alphay * alphay)));
}

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	return Float32(log1p(Float32(-u0)) / Float32(Float32((alphax ^ Float32(-2.0)) * Float32(-cos2phi)) - Float32(sin2phi / Float32(alphay * alphay))))
end

Derivation

1. Initial program 60.5%

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

   1. distribute-frac-neg 60.5%

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

   2. distribute-neg-frac2 60.5%

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

   3. sub-neg 60.5%

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

   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. distribute-neg-frac2 98.2%

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

   9. distribute-rgt-neg-out 98.2%

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

3. Simplified 98.2%

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

   1. frac-2neg 98.2%

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

   2. div-inv 98.2%

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

   3. distribute-rgt-neg-out 98.2%

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

   4. remove-double-neg 98.2%

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

   5. pow2 98.2%

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

6. Applied egg-rr 98.2%

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

   1. *-commutative 98.2%

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

   2. pow-flip 98.3%

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

   3. metadata-eval 98.3%

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

8. Applied egg-rr 98.3%

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

Alternative 2: 81.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{sin2phi}{alphay \cdot alphay}\\ \mathbf{if}\;t\_0 \leq 2.0000000072549875 \cdot 10^{-15}:\\ \;\;\;\;{alphax}^{2} \cdot \left(u0 \cdot \left(0.5 \cdot \frac{u0}{cos2phi} + \frac{1}{cos2phi}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(-u0\right)}{\frac{\frac{cos2phi}{alphax}}{alphax} - t\_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (let* ((t_0 (/ sin2phi (* alphay alphay))))
   (if (<= t_0 2.0000000072549875e-15)
     (* (pow alphax 2.0) (* u0 (+ (* 0.5 (/ u0 cos2phi)) (/ 1.0 cos2phi))))
     (/ (log1p (- u0)) (- (/ (/ cos2phi alphax) alphax) t_0)))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float t_0 = sin2phi / (alphay * alphay);
	float tmp;
	if (t_0 <= 2.0000000072549875e-15f) {
		tmp = powf(alphax, 2.0f) * (u0 * ((0.5f * (u0 / cos2phi)) + (1.0f / cos2phi)));
	} else {
		tmp = log1pf(-u0) / (((cos2phi / alphax) / alphax) - t_0);
	}
	return tmp;
}

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	t_0 = Float32(sin2phi / Float32(alphay * alphay))
	tmp = Float32(0.0)
	if (t_0 <= Float32(2.0000000072549875e-15))
		tmp = Float32((alphax ^ Float32(2.0)) * Float32(u0 * Float32(Float32(Float32(0.5) * Float32(u0 / cos2phi)) + Float32(Float32(1.0) / cos2phi))));
	else
		tmp = Float32(log1p(Float32(-u0)) / Float32(Float32(Float32(cos2phi / alphax) / alphax) - t_0));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 sin2phi (*.f32 alphay alphay)) < 2.00000001e-15

   1. Initial program 50.3%

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

      1. distribute-frac-neg 50.3%

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

      2. distribute-neg-frac2 50.3%

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

      3. sub-neg 50.3%

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

      4. log1p-define 98.6%

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

      5. neg-sub0 98.6%

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

      6. associate--r+ 98.6%

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

      7. neg-sub0 98.6%

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

      8. distribute-neg-frac2 98.6%

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

      9. distribute-rgt-neg-out 98.6%

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

   3. Simplified 98.6%

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

   5. Taylor expanded in cos2phi around inf 41.7%

      \[\leadsto \color{blue}{-1 \cdot \frac{{alphax}^{2} \cdot \log \left(1 - u0\right)}{cos2phi}} \]
   6. Step-by-step derivation

      1. mul-1-neg 41.7%

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

      2. associate-/l* 41.7%

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

      3. distribute-rgt-neg-in 41.7%

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

      4. distribute-neg-frac2 41.7%

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

      5. sub-neg 41.7%

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

      6. log1p-define 78.2%

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

   7. Simplified 78.2%

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

   8. Taylor expanded in u0 around 0 71.9%

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

   if 2.00000001e-15 < (/.f32 sin2phi (*.f32 alphay alphay))

   1. Initial program 63.2%

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

      1. distribute-frac-neg 63.2%

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

      2. distribute-neg-frac2 63.2%

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

      3. sub-neg 63.2%

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

      4. log1p-define 98.1%

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

      5. neg-sub0 98.1%

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

      6. associate--r+ 98.1%

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

      7. neg-sub0 98.1%

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

      8. distribute-neg-frac2 98.1%

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

      9. distribute-rgt-neg-out 98.1%

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

   3. Simplified 98.1%

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

      1. associate-/r* 98.0%

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

      2. add-sqr-sqrt -0.0%

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

      3. sqrt-unprod 88.6%

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

      4. sqr-neg 88.6%

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

      5. sqrt-unprod 88.6%

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

      6. add-sqr-sqrt 88.6%

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

   6. Applied egg-rr 88.6%

      \[\leadsto \frac{\mathsf{log1p}\left(-u0\right)}{\color{blue}{\frac{\frac{cos2phi}{alphax}}{alphax}} - \frac{sin2phi}{alphay \cdot alphay}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 75.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;sin2phi \leq 2.00000006274879 \cdot 10^{-22}:\\ \;\;\;\;{alphax}^{2} \cdot \left(u0 \cdot \left(0.5 \cdot \frac{u0}{cos2phi} + \frac{1}{cos2phi}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;u0 \cdot \left({alphay}^{2} \cdot \left(0.5 \cdot \frac{u0}{sin2phi} + \frac{1}{sin2phi}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (if (<= sin2phi 2.00000006274879e-22)
   (* (pow alphax 2.0) (* u0 (+ (* 0.5 (/ u0 cos2phi)) (/ 1.0 cos2phi))))
   (* u0 (* (pow alphay 2.0) (+ (* 0.5 (/ u0 sin2phi)) (/ 1.0 sin2phi))))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float tmp;
	if (sin2phi <= 2.00000006274879e-22f) {
		tmp = powf(alphax, 2.0f) * (u0 * ((0.5f * (u0 / cos2phi)) + (1.0f / cos2phi)));
	} else {
		tmp = u0 * (powf(alphay, 2.0f) * ((0.5f * (u0 / sin2phi)) + (1.0f / sin2phi)));
	}
	return tmp;
}

Fortran

real(4) function code(alphax, alphay, u0, cos2phi, sin2phi)
    real(4), intent (in) :: alphax
    real(4), intent (in) :: alphay
    real(4), intent (in) :: u0
    real(4), intent (in) :: cos2phi
    real(4), intent (in) :: sin2phi
    real(4) :: tmp
    if (sin2phi <= 2.00000006274879e-22) then
        tmp = (alphax ** 2.0e0) * (u0 * ((0.5e0 * (u0 / cos2phi)) + (1.0e0 / cos2phi)))
    else
        tmp = u0 * ((alphay ** 2.0e0) * ((0.5e0 * (u0 / sin2phi)) + (1.0e0 / sin2phi)))
    end if
    code = tmp
end function

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = Float32(0.0)
	if (sin2phi <= Float32(2.00000006274879e-22))
		tmp = Float32((alphax ^ Float32(2.0)) * Float32(u0 * Float32(Float32(Float32(0.5) * Float32(u0 / cos2phi)) + Float32(Float32(1.0) / cos2phi))));
	else
		tmp = Float32(u0 * Float32((alphay ^ Float32(2.0)) * Float32(Float32(Float32(0.5) * Float32(u0 / sin2phi)) + Float32(Float32(1.0) / sin2phi))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = single(0.0);
	if (sin2phi <= single(2.00000006274879e-22))
		tmp = (alphax ^ single(2.0)) * (u0 * ((single(0.5) * (u0 / cos2phi)) + (single(1.0) / cos2phi)));
	else
		tmp = u0 * ((alphay ^ single(2.0)) * ((single(0.5) * (u0 / sin2phi)) + (single(1.0) / sin2phi)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if sin2phi < 2.00000006e-22

   1. Initial program 51.5%

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

      1. distribute-frac-neg 51.5%

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

      2. distribute-neg-frac2 51.5%

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

      3. sub-neg 51.5%

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

      4. log1p-define 98.7%

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

      5. neg-sub0 98.7%

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

      6. associate--r+ 98.7%

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

      7. neg-sub0 98.7%

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

      8. distribute-neg-frac2 98.7%

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

      9. distribute-rgt-neg-out 98.7%

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

   3. Simplified 98.7%

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

   5. Taylor expanded in cos2phi around inf 42.3%

      \[\leadsto \color{blue}{-1 \cdot \frac{{alphax}^{2} \cdot \log \left(1 - u0\right)}{cos2phi}} \]
   6. Step-by-step derivation

      1. mul-1-neg 42.3%

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

      2. associate-/l* 42.2%

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

      3. distribute-rgt-neg-in 42.2%

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

      4. distribute-neg-frac2 42.2%

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

      5. sub-neg 42.2%

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

      6. log1p-define 79.7%

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

   7. Simplified 79.7%

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

   8. Taylor expanded in u0 around 0 73.3%

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

   if 2.00000006e-22 < sin2phi

   1. Initial program 62.7%

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

      1. distribute-frac-neg 62.7%

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

      2. distribute-neg-frac2 62.7%

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

      3. sub-neg 62.7%

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

      4. log1p-define 98.0%

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

      5. neg-sub0 98.0%

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

      6. associate--r+ 98.0%

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

      7. neg-sub0 98.0%

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

      8. distribute-neg-frac2 98.0%

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

      9. distribute-rgt-neg-out 98.0%

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

   3. Simplified 98.0%

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

   5. Taylor expanded in u0 around 0 86.7%

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

      1. sub-neg 86.7%

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

   7. Simplified 86.7%

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

   8. Taylor expanded in alphay around 0 79.5%

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

Alternative 4: 75.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;sin2phi \leq 2.00000006274879 \cdot 10^{-22}:\\ \;\;\;\;u0 \cdot \left({alphax}^{2} \cdot \left(0.5 \cdot \frac{u0}{cos2phi} + \frac{1}{cos2phi}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;u0 \cdot \left({alphay}^{2} \cdot \left(0.5 \cdot \frac{u0}{sin2phi} + \frac{1}{sin2phi}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (if (<= sin2phi 2.00000006274879e-22)
   (* u0 (* (pow alphax 2.0) (+ (* 0.5 (/ u0 cos2phi)) (/ 1.0 cos2phi))))
   (* u0 (* (pow alphay 2.0) (+ (* 0.5 (/ u0 sin2phi)) (/ 1.0 sin2phi))))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float tmp;
	if (sin2phi <= 2.00000006274879e-22f) {
		tmp = u0 * (powf(alphax, 2.0f) * ((0.5f * (u0 / cos2phi)) + (1.0f / cos2phi)));
	} else {
		tmp = u0 * (powf(alphay, 2.0f) * ((0.5f * (u0 / sin2phi)) + (1.0f / sin2phi)));
	}
	return tmp;
}

Fortran

real(4) function code(alphax, alphay, u0, cos2phi, sin2phi)
    real(4), intent (in) :: alphax
    real(4), intent (in) :: alphay
    real(4), intent (in) :: u0
    real(4), intent (in) :: cos2phi
    real(4), intent (in) :: sin2phi
    real(4) :: tmp
    if (sin2phi <= 2.00000006274879e-22) then
        tmp = u0 * ((alphax ** 2.0e0) * ((0.5e0 * (u0 / cos2phi)) + (1.0e0 / cos2phi)))
    else
        tmp = u0 * ((alphay ** 2.0e0) * ((0.5e0 * (u0 / sin2phi)) + (1.0e0 / sin2phi)))
    end if
    code = tmp
end function

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = Float32(0.0)
	if (sin2phi <= Float32(2.00000006274879e-22))
		tmp = Float32(u0 * Float32((alphax ^ Float32(2.0)) * Float32(Float32(Float32(0.5) * Float32(u0 / cos2phi)) + Float32(Float32(1.0) / cos2phi))));
	else
		tmp = Float32(u0 * Float32((alphay ^ Float32(2.0)) * Float32(Float32(Float32(0.5) * Float32(u0 / sin2phi)) + Float32(Float32(1.0) / sin2phi))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = single(0.0);
	if (sin2phi <= single(2.00000006274879e-22))
		tmp = u0 * ((alphax ^ single(2.0)) * ((single(0.5) * (u0 / cos2phi)) + (single(1.0) / cos2phi)));
	else
		tmp = u0 * ((alphay ^ single(2.0)) * ((single(0.5) * (u0 / sin2phi)) + (single(1.0) / sin2phi)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if sin2phi < 2.00000006e-22

   1. Initial program 51.5%

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

      1. distribute-frac-neg 51.5%

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

      2. distribute-neg-frac2 51.5%

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

      3. sub-neg 51.5%

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

      4. log1p-define 98.7%

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

      5. neg-sub0 98.7%

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

      6. associate--r+ 98.7%

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

      7. neg-sub0 98.7%

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

      8. distribute-neg-frac2 98.7%

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

      9. distribute-rgt-neg-out 98.7%

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

   3. Simplified 98.7%

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

   5. Taylor expanded in u0 around 0 87.7%

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

      1. sub-neg 87.7%

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

   7. Simplified 87.7%

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

   8. Taylor expanded in alphax around 0 73.2%

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

   if 2.00000006e-22 < sin2phi

   1. Initial program 62.7%

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

      1. distribute-frac-neg 62.7%

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

      2. distribute-neg-frac2 62.7%

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

      3. sub-neg 62.7%

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

      4. log1p-define 98.0%

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

      5. neg-sub0 98.0%

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

      6. associate--r+ 98.0%

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

      7. neg-sub0 98.0%

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

      8. distribute-neg-frac2 98.0%

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

      9. distribute-rgt-neg-out 98.0%

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

   3. Simplified 98.0%

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

   5. Taylor expanded in u0 around 0 86.7%

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

      1. sub-neg 86.7%

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

   7. Simplified 86.7%

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

   8. Taylor expanded in alphay around 0 79.5%

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

Alternative 5: 68.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;sin2phi \leq 2.00000006274879 \cdot 10^{-22}:\\ \;\;\;\;u0 \cdot \left({alphax}^{2} \cdot \left(0.5 \cdot \frac{u0}{cos2phi} + \frac{1}{cos2phi}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{alphay}^{2} \cdot \frac{u0}{sin2phi}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (if (<= sin2phi 2.00000006274879e-22)
   (* u0 (* (pow alphax 2.0) (+ (* 0.5 (/ u0 cos2phi)) (/ 1.0 cos2phi))))
   (* (pow alphay 2.0) (/ u0 sin2phi))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float tmp;
	if (sin2phi <= 2.00000006274879e-22f) {
		tmp = u0 * (powf(alphax, 2.0f) * ((0.5f * (u0 / cos2phi)) + (1.0f / cos2phi)));
	} else {
		tmp = powf(alphay, 2.0f) * (u0 / sin2phi);
	}
	return tmp;
}

Fortran

real(4) function code(alphax, alphay, u0, cos2phi, sin2phi)
    real(4), intent (in) :: alphax
    real(4), intent (in) :: alphay
    real(4), intent (in) :: u0
    real(4), intent (in) :: cos2phi
    real(4), intent (in) :: sin2phi
    real(4) :: tmp
    if (sin2phi <= 2.00000006274879e-22) then
        tmp = u0 * ((alphax ** 2.0e0) * ((0.5e0 * (u0 / cos2phi)) + (1.0e0 / cos2phi)))
    else
        tmp = (alphay ** 2.0e0) * (u0 / sin2phi)
    end if
    code = tmp
end function

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = Float32(0.0)
	if (sin2phi <= Float32(2.00000006274879e-22))
		tmp = Float32(u0 * Float32((alphax ^ Float32(2.0)) * Float32(Float32(Float32(0.5) * Float32(u0 / cos2phi)) + Float32(Float32(1.0) / cos2phi))));
	else
		tmp = Float32((alphay ^ Float32(2.0)) * Float32(u0 / sin2phi));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = single(0.0);
	if (sin2phi <= single(2.00000006274879e-22))
		tmp = u0 * ((alphax ^ single(2.0)) * ((single(0.5) * (u0 / cos2phi)) + (single(1.0) / cos2phi)));
	else
		tmp = (alphay ^ single(2.0)) * (u0 / sin2phi);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if sin2phi < 2.00000006e-22

   1. Initial program 51.5%

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

      1. distribute-frac-neg 51.5%

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

      2. distribute-neg-frac2 51.5%

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

      3. sub-neg 51.5%

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

      4. log1p-define 98.7%

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

      5. neg-sub0 98.7%

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

      6. associate--r+ 98.7%

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

      7. neg-sub0 98.7%

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

      8. distribute-neg-frac2 98.7%

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

      9. distribute-rgt-neg-out 98.7%

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

   3. Simplified 98.7%

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

   5. Taylor expanded in u0 around 0 87.7%

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

      1. sub-neg 87.7%

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

   7. Simplified 87.7%

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

   8. Taylor expanded in alphax around 0 73.2%

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

   if 2.00000006e-22 < sin2phi

   1. Initial program 62.7%

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

      1. distribute-frac-neg 62.7%

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

      2. distribute-neg-frac2 62.7%

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

      3. sub-neg 62.7%

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

      4. log1p-define 98.0%

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

      5. neg-sub0 98.0%

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

      6. associate--r+ 98.0%

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

      7. neg-sub0 98.0%

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

      8. distribute-neg-frac2 98.0%

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

      9. distribute-rgt-neg-out 98.0%

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

   3. Simplified 98.0%

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

   6. Applied egg-rr 70.4%

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

   7. Taylor expanded in cos2phi around 0 42.2%

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

      1. associate-/r* 42.2%

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

      2. log1p-define 65.1%

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

   9. Simplified 65.1%

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

   10. Taylor expanded in u0 around 0 69.7%

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

       1. associate-/l* 69.8%

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

   12. Simplified 69.8%

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

Alternative 6: 98.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 60.5%

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

   1. distribute-frac-neg 60.5%

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

   2. distribute-neg-frac2 60.5%

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

   3. sub-neg 60.5%

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

   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. distribute-neg-frac2 98.2%

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

   9. distribute-rgt-neg-out 98.2%

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

3. Simplified 98.2%

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

   1. associate-/r* 98.2%

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

   2. frac-2neg 98.2%

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

   3. add-sqr-sqrt -0.0%

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

   4. sqrt-unprod 73.0%

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

   5. sqr-neg 73.0%

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

   6. sqrt-unprod 73.0%

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

   7. add-sqr-sqrt 73.0%

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

   8. add-sqr-sqrt -0.0%

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

   9. sqrt-unprod 98.2%

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

   10. sqr-neg 98.2%

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

   11. sqrt-unprod 98.0%

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

   12. add-sqr-sqrt 98.2%

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

6. Applied egg-rr 98.2%

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

7. Final simplification 98.2%

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

Alternative 7: 98.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 60.5%

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

   1. distribute-frac-neg 60.5%

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

   2. distribute-neg-frac2 60.5%

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

   3. sub-neg 60.5%

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

   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. distribute-neg-frac2 98.2%

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

   9. distribute-rgt-neg-out 98.2%

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

3. Simplified 98.2%

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

5. Final simplification 98.2%

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

Alternative 8: 66.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;sin2phi \leq 2.00000006274879 \cdot 10^{-22}:\\ \;\;\;\;\frac{1}{\frac{cos2phi}{u0 \cdot {alphax}^{2}}}\\ \mathbf{else}:\\ \;\;\;\;{alphay}^{2} \cdot \frac{u0}{sin2phi}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (if (<= sin2phi 2.00000006274879e-22)
   (/ 1.0 (/ cos2phi (* u0 (pow alphax 2.0))))
   (* (pow alphay 2.0) (/ u0 sin2phi))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float tmp;
	if (sin2phi <= 2.00000006274879e-22f) {
		tmp = 1.0f / (cos2phi / (u0 * powf(alphax, 2.0f)));
	} else {
		tmp = powf(alphay, 2.0f) * (u0 / sin2phi);
	}
	return tmp;
}

Fortran

real(4) function code(alphax, alphay, u0, cos2phi, sin2phi)
    real(4), intent (in) :: alphax
    real(4), intent (in) :: alphay
    real(4), intent (in) :: u0
    real(4), intent (in) :: cos2phi
    real(4), intent (in) :: sin2phi
    real(4) :: tmp
    if (sin2phi <= 2.00000006274879e-22) then
        tmp = 1.0e0 / (cos2phi / (u0 * (alphax ** 2.0e0)))
    else
        tmp = (alphay ** 2.0e0) * (u0 / sin2phi)
    end if
    code = tmp
end function

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = Float32(0.0)
	if (sin2phi <= Float32(2.00000006274879e-22))
		tmp = Float32(Float32(1.0) / Float32(cos2phi / Float32(u0 * (alphax ^ Float32(2.0)))));
	else
		tmp = Float32((alphay ^ Float32(2.0)) * Float32(u0 / sin2phi));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = single(0.0);
	if (sin2phi <= single(2.00000006274879e-22))
		tmp = single(1.0) / (cos2phi / (u0 * (alphax ^ single(2.0))));
	else
		tmp = (alphay ^ single(2.0)) * (u0 / sin2phi);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if sin2phi < 2.00000006e-22

   1. Initial program 51.5%

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

      1. distribute-frac-neg 51.5%

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

      2. distribute-neg-frac2 51.5%

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

      3. sub-neg 51.5%

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

      4. log1p-define 98.7%

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

      5. neg-sub0 98.7%

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

      6. associate--r+ 98.7%

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

      7. neg-sub0 98.7%

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

      8. distribute-neg-frac2 98.7%

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

      9. distribute-rgt-neg-out 98.7%

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

   3. Simplified 98.7%

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

   5. Taylor expanded in cos2phi around inf 42.3%

      \[\leadsto \color{blue}{-1 \cdot \frac{{alphax}^{2} \cdot \log \left(1 - u0\right)}{cos2phi}} \]
   6. Step-by-step derivation

      1. mul-1-neg 42.3%

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

      2. associate-/l* 42.2%

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

      3. distribute-rgt-neg-in 42.2%

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

      4. distribute-neg-frac2 42.2%

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

      5. sub-neg 42.2%

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

      6. log1p-define 79.7%

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

   7. Simplified 79.7%

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

   8. Taylor expanded in u0 around 0 65.2%

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

      1. clear-num 65.3%

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

      2. *-commutative 65.3%

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

   10. Applied egg-rr 65.3%

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

   if 2.00000006e-22 < sin2phi

   1. Initial program 62.7%

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

      1. distribute-frac-neg 62.7%

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

      2. distribute-neg-frac2 62.7%

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

      3. sub-neg 62.7%

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

      4. log1p-define 98.0%

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

      5. neg-sub0 98.0%

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

      6. associate--r+ 98.0%

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

      7. neg-sub0 98.0%

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

      8. distribute-neg-frac2 98.0%

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

      9. distribute-rgt-neg-out 98.0%

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

   3. Simplified 98.0%

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

   6. Applied egg-rr 70.4%

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

   7. Taylor expanded in cos2phi around 0 42.2%

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

      1. associate-/r* 42.2%

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

      2. log1p-define 65.1%

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

   9. Simplified 65.1%

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

   10. Taylor expanded in u0 around 0 69.7%

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

       1. associate-/l* 69.8%

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

   12. Simplified 69.8%

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

Alternative 9: 66.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;sin2phi \leq 2.00000006274879 \cdot 10^{-22}:\\ \;\;\;\;{alphax}^{2} \cdot \frac{u0}{cos2phi}\\ \mathbf{else}:\\ \;\;\;\;{alphay}^{2} \cdot \frac{u0}{sin2phi}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (if (<= sin2phi 2.00000006274879e-22)
   (* (pow alphax 2.0) (/ u0 cos2phi))
   (* (pow alphay 2.0) (/ u0 sin2phi))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float tmp;
	if (sin2phi <= 2.00000006274879e-22f) {
		tmp = powf(alphax, 2.0f) * (u0 / cos2phi);
	} else {
		tmp = powf(alphay, 2.0f) * (u0 / sin2phi);
	}
	return tmp;
}

Fortran

real(4) function code(alphax, alphay, u0, cos2phi, sin2phi)
    real(4), intent (in) :: alphax
    real(4), intent (in) :: alphay
    real(4), intent (in) :: u0
    real(4), intent (in) :: cos2phi
    real(4), intent (in) :: sin2phi
    real(4) :: tmp
    if (sin2phi <= 2.00000006274879e-22) then
        tmp = (alphax ** 2.0e0) * (u0 / cos2phi)
    else
        tmp = (alphay ** 2.0e0) * (u0 / sin2phi)
    end if
    code = tmp
end function

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = Float32(0.0)
	if (sin2phi <= Float32(2.00000006274879e-22))
		tmp = Float32((alphax ^ Float32(2.0)) * Float32(u0 / cos2phi));
	else
		tmp = Float32((alphay ^ Float32(2.0)) * Float32(u0 / sin2phi));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = single(0.0);
	if (sin2phi <= single(2.00000006274879e-22))
		tmp = (alphax ^ single(2.0)) * (u0 / cos2phi);
	else
		tmp = (alphay ^ single(2.0)) * (u0 / sin2phi);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if sin2phi < 2.00000006e-22

   1. Initial program 51.5%

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

      1. distribute-frac-neg 51.5%

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

      2. distribute-neg-frac2 51.5%

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

      3. sub-neg 51.5%

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

      4. log1p-define 98.7%

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

      5. neg-sub0 98.7%

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

      6. associate--r+ 98.7%

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

      7. neg-sub0 98.7%

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

      8. distribute-neg-frac2 98.7%

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

      9. distribute-rgt-neg-out 98.7%

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

   3. Simplified 98.7%

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

   5. Taylor expanded in cos2phi around inf 42.3%

      \[\leadsto \color{blue}{-1 \cdot \frac{{alphax}^{2} \cdot \log \left(1 - u0\right)}{cos2phi}} \]
   6. Step-by-step derivation

      1. mul-1-neg 42.3%

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

      2. associate-/l* 42.2%

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

      3. distribute-rgt-neg-in 42.2%

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

      4. distribute-neg-frac2 42.2%

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

      5. sub-neg 42.2%

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

      6. log1p-define 79.7%

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

   7. Simplified 79.7%

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

   8. Taylor expanded in u0 around 0 65.2%

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

      1. associate-/l* 65.2%

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

   10. Simplified 65.2%

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

   if 2.00000006e-22 < sin2phi

   1. Initial program 62.7%

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

      1. distribute-frac-neg 62.7%

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

      2. distribute-neg-frac2 62.7%

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

      3. sub-neg 62.7%

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

      4. log1p-define 98.0%

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

      5. neg-sub0 98.0%

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

      6. associate--r+ 98.0%

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

      7. neg-sub0 98.0%

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

      8. distribute-neg-frac2 98.0%

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

      9. distribute-rgt-neg-out 98.0%

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

   3. Simplified 98.0%

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

   6. Applied egg-rr 70.4%

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

   7. Taylor expanded in cos2phi around 0 42.2%

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

      1. associate-/r* 42.2%

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

      2. log1p-define 65.1%

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

   9. Simplified 65.1%

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

   10. Taylor expanded in u0 around 0 69.7%

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

       1. associate-/l* 69.8%

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

   12. Simplified 69.8%

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

Alternative 10: 23.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ {alphax}^{2} \cdot \frac{u0}{cos2phi} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (* (pow alphax 2.0) (/ u0 cos2phi)))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	return powf(alphax, 2.0f) * (u0 / cos2phi);
}

Fortran

real(4) function code(alphax, alphay, u0, cos2phi, sin2phi)
    real(4), intent (in) :: alphax
    real(4), intent (in) :: alphay
    real(4), intent (in) :: u0
    real(4), intent (in) :: cos2phi
    real(4), intent (in) :: sin2phi
    code = (alphax ** 2.0e0) * (u0 / cos2phi)
end function

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	return Float32((alphax ^ Float32(2.0)) * Float32(u0 / cos2phi))
end

MATLAB

function tmp = code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = (alphax ^ single(2.0)) * (u0 / cos2phi);
end

Derivation

1. Initial program 60.5%

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

   1. distribute-frac-neg 60.5%

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

   2. distribute-neg-frac2 60.5%

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

   3. sub-neg 60.5%

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

   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. distribute-neg-frac2 98.2%

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

   9. distribute-rgt-neg-out 98.2%

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

3. Simplified 98.2%

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

5. Taylor expanded in cos2phi around inf 21.6%

   \[\leadsto \color{blue}{-1 \cdot \frac{{alphax}^{2} \cdot \log \left(1 - u0\right)}{cos2phi}} \]
6. Step-by-step derivation

   1. mul-1-neg 21.6%

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

   2. associate-/l* 21.6%

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

   3. distribute-rgt-neg-in 21.6%

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

   4. distribute-neg-frac2 21.6%

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

   5. sub-neg 21.6%

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

   6. log1p-define 26.7%

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

7. Simplified 26.7%

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

8. Taylor expanded in u0 around 0 23.0%

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

   1. associate-/l* 23.0%

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

10. Simplified 23.0%

    \[\leadsto \color{blue}{{alphax}^{2} \cdot \frac{u0}{cos2phi}} \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2024096 
(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)))))