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

Percentage Accurate: 60.2% → 98.3%

Time: 14.0s

Alternatives: 18

Speedup: 8.8×

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: 60.2% 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, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := alphay \cdot \frac{alphay}{sin2phi}\\ \frac{-\mathsf{log1p}\left(-u0\right)}{\mathsf{fma}\left(t_0, \frac{cos2phi}{alphax}, alphax\right)} \cdot \left(alphax \cdot t_0\right) \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (let* ((t_0 (* alphay (/ alphay sin2phi))))
   (*
    (/ (- (log1p (- u0))) (fma t_0 (/ cos2phi alphax) alphax))
    (* alphax t_0))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float t_0 = alphay * (alphay / sin2phi);
	return (-log1pf(-u0) / fmaf(t_0, (cos2phi / alphax), alphax)) * (alphax * t_0);
}

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	t_0 = Float32(alphay * Float32(alphay / sin2phi))
	return Float32(Float32(Float32(-log1p(Float32(-u0))) / fma(t_0, Float32(cos2phi / alphax), alphax)) * Float32(alphax * t_0))
end

Derivation

1. Initial program 61.7%

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

   1. neg-sub0 61.7%

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

   2. div-sub 61.7%

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

   3. --rgt-identity 61.7%

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

   4. div-sub 61.7%

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

   5. --rgt-identity 61.7%

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

   6. neg-sub0 61.7%

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

   7. sub-neg 61.7%

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

   8. log1p-def 98.4%

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

3. Simplified 98.4%

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

   1. clear-num 98.3%

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

   2. associate-/r* 98.2%

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

   3. frac-add 98.0%

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

   4. associate-/l* 97.8%

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

   5. *-commutative 97.8%

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

   6. *-un-lft-identity 97.8%

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

   7. associate-/l* 97.9%

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

5. Applied egg-rr 97.9%

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

   1. associate-/r/ 98.4%

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

   2. *-commutative 98.4%

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

   3. fma-def 98.5%

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

   4. associate-/r/ 98.4%

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

   5. associate-/r/ 98.5%

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

7. Applied egg-rr 98.5%

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

8. Final simplification 98.5%

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

Alternative 2: 98.2% 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 61.7%

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

   1. neg-sub0 61.7%

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

   2. div-sub 61.7%

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

   3. --rgt-identity 61.7%

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

   4. div-sub 61.7%

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

   5. --rgt-identity 61.7%

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

   6. neg-sub0 61.7%

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

   7. sub-neg 61.7%

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

   8. log1p-def 98.4%

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

3. Simplified 98.4%

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

4. Final simplification 98.4%

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

Alternative 3: 92.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;sin2phi \leq 0.014999999664723873:\\ \;\;\;\;\frac{u0 - u0 \cdot \left(u0 \cdot -0.5\right)}{\frac{\frac{cos2phi}{alphax}}{alphax} + sin2phi \cdot \frac{-1}{alphay \cdot \left(-alphay\right)}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(-u0\right) \cdot \left(alphay \cdot \frac{-alphay}{sin2phi}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (if (<= sin2phi 0.014999999664723873)
   (/
    (- u0 (* u0 (* u0 -0.5)))
    (+
     (/ (/ cos2phi alphax) alphax)
     (* sin2phi (/ -1.0 (* alphay (- alphay))))))
   (* (log1p (- u0)) (* alphay (/ (- alphay) sin2phi)))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float tmp;
	if (sin2phi <= 0.014999999664723873f) {
		tmp = (u0 - (u0 * (u0 * -0.5f))) / (((cos2phi / alphax) / alphax) + (sin2phi * (-1.0f / (alphay * -alphay))));
	} else {
		tmp = log1pf(-u0) * (alphay * (-alphay / sin2phi));
	}
	return tmp;
}

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = Float32(0.0)
	if (sin2phi <= Float32(0.014999999664723873))
		tmp = Float32(Float32(u0 - Float32(u0 * Float32(u0 * Float32(-0.5)))) / Float32(Float32(Float32(cos2phi / alphax) / alphax) + Float32(sin2phi * Float32(Float32(-1.0) / Float32(alphay * Float32(-alphay))))));
	else
		tmp = Float32(log1p(Float32(-u0)) * Float32(alphay * Float32(Float32(-alphay) / sin2phi)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if sin2phi < 0.0149999997

   1. Initial program 56.4%

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

      1. associate-/r* 56.4%

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

   3. Simplified 56.4%

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

   4. Taylor expanded in u0 around 0 88.5%

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

      1. +-commutative 88.5%

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

      2. mul-1-neg 88.5%

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

      3. unsub-neg 88.5%

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

      4. unpow2 88.5%

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

      5. associate-*r* 88.5%

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

   6. Simplified 88.5%

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

      1. frac-2neg 88.5%

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

      2. div-inv 88.5%

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

   8. Applied egg-rr 88.5%

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

   if 0.0149999997 < sin2phi

   1. Initial program 66.1%

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

      1. neg-sub0 66.1%

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

      2. div-sub 66.1%

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

      3. --rgt-identity 66.1%

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

      4. div-sub 66.1%

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

      5. --rgt-identity 66.1%

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

      6. sub-neg 66.1%

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

      7. +-commutative 66.1%

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

      8. neg-sub0 66.1%

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

      9. associate-+l- 66.1%

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

      10. sub0-neg 66.1%

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

      11. neg-mul-1 66.1%

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

      12. log-prod -0.0%

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

      13. associate--r+ -0.0%

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

   3. Simplified 98.1%

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

      1. associate-/r* 98.1%

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

      2. frac-2neg 98.1%

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

      3. div-inv 98.1%

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

      4. distribute-rgt-neg-in 98.1%

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

   5. Applied egg-rr 98.1%

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

   6. Taylor expanded in cos2phi around 0 66.8%

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

      1. mul-1-neg 66.8%

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

      2. *-commutative 66.8%

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

      3. sub-neg 66.8%

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

      4. log1p-def 98.5%

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

      5. unpow2 98.5%

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

      6. *-rgt-identity 98.5%

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

      7. associate-*r/ 98.3%

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

      8. unpow2 98.3%

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

      9. associate-*l* 98.3%

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

      10. associate-*r/ 98.5%

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

      11. associate-*l/ 98.5%

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

      12. unpow2 98.5%

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

      13. associate-*l/ 98.4%

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

      14. *-rgt-identity 98.4%

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

      15. *-commutative 98.4%

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

   8. Simplified 98.4%

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

4. Final simplification 93.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;sin2phi \leq 0.014999999664723873:\\ \;\;\;\;\frac{u0 - u0 \cdot \left(u0 \cdot -0.5\right)}{\frac{\frac{cos2phi}{alphax}}{alphax} + sin2phi \cdot \frac{-1}{alphay \cdot \left(-alphay\right)}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(-u0\right) \cdot \left(alphay \cdot \frac{-alphay}{sin2phi}\right)\\ \end{array} \]

Alternative 4: 92.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;sin2phi \leq 0.019999999552965164:\\ \;\;\;\;\frac{u0 - u0 \cdot \left(u0 \cdot -0.5\right)}{\frac{\frac{cos2phi}{alphax}}{alphax} + sin2phi \cdot \frac{-1}{alphay \cdot \left(-alphay\right)}}\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{log1p}\left(-u0\right) \cdot alphay\right) \cdot \frac{-alphay}{sin2phi}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (if (<= sin2phi 0.019999999552965164)
   (/
    (- u0 (* u0 (* u0 -0.5)))
    (+
     (/ (/ cos2phi alphax) alphax)
     (* sin2phi (/ -1.0 (* alphay (- alphay))))))
   (* (* (log1p (- u0)) alphay) (/ (- alphay) sin2phi))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float tmp;
	if (sin2phi <= 0.019999999552965164f) {
		tmp = (u0 - (u0 * (u0 * -0.5f))) / (((cos2phi / alphax) / alphax) + (sin2phi * (-1.0f / (alphay * -alphay))));
	} else {
		tmp = (log1pf(-u0) * alphay) * (-alphay / sin2phi);
	}
	return tmp;
}

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = Float32(0.0)
	if (sin2phi <= Float32(0.019999999552965164))
		tmp = Float32(Float32(u0 - Float32(u0 * Float32(u0 * Float32(-0.5)))) / Float32(Float32(Float32(cos2phi / alphax) / alphax) + Float32(sin2phi * Float32(Float32(-1.0) / Float32(alphay * Float32(-alphay))))));
	else
		tmp = Float32(Float32(log1p(Float32(-u0)) * alphay) * Float32(Float32(-alphay) / sin2phi));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if sin2phi < 0.0199999996

   1. Initial program 56.0%

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

      1. associate-/r* 56.0%

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

   3. Simplified 56.0%

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

   4. Taylor expanded in u0 around 0 88.6%

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

      1. +-commutative 88.6%

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

      2. mul-1-neg 88.6%

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

      3. unsub-neg 88.6%

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

      4. unpow2 88.6%

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

      5. associate-*r* 88.6%

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

   6. Simplified 88.6%

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

      1. frac-2neg 88.6%

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

      2. div-inv 88.6%

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

   8. Applied egg-rr 88.6%

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

   if 0.0199999996 < sin2phi

   1. Initial program 66.6%

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

      1. associate-/r* 66.6%

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

   3. Simplified 66.6%

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

   4. Taylor expanded in cos2phi around 0 67.2%

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

      1. mul-1-neg 67.2%

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

      2. unpow2 67.2%

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

      3. *-commutative 67.2%

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

   6. Simplified 67.2%

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

   7. Taylor expanded in alphay around 0 67.2%

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

      1. unpow2 67.2%

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

      2. sub-neg 67.2%

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

      3. log1p-def 98.5%

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

      4. associate-*l* 98.4%

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

   9. Simplified 98.4%

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

       1. expm1-log1p-u 98.4%

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

       2. expm1-udef 27.6%

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

       3. associate-/l* 27.6%

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

       4. *-commutative 27.6%

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

   11. Applied egg-rr 27.6%

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

       1. expm1-def 97.0%

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

       2. expm1-log1p 97.0%

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

       3. associate-/r/ 98.4%

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

       4. *-commutative 98.4%

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

   13. Simplified 98.4%

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

4. Final simplification 93.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;sin2phi \leq 0.019999999552965164:\\ \;\;\;\;\frac{u0 - u0 \cdot \left(u0 \cdot -0.5\right)}{\frac{\frac{cos2phi}{alphax}}{alphax} + sin2phi \cdot \frac{-1}{alphay \cdot \left(-alphay\right)}}\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{log1p}\left(-u0\right) \cdot alphay\right) \cdot \frac{-alphay}{sin2phi}\\ \end{array} \]

Alternative 5: 87.5% accurate, 5.3× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 61.7%

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

   1. associate-/r* 61.7%

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

3. Simplified 61.7%

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

4. Taylor expanded in u0 around 0 87.5%

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

   1. +-commutative 87.5%

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

   2. mul-1-neg 87.5%

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

   3. unsub-neg 87.5%

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

   4. unpow2 87.5%

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

   5. associate-*r* 87.5%

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

6. Simplified 87.5%

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

   1. frac-2neg 87.5%

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

   2. div-inv 87.5%

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

8. Applied egg-rr 87.5%

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

9. Final simplification 87.5%

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

Alternative 6: 87.5% accurate, 5.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 61.7%

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

   1. associate-/r* 61.7%

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

3. Simplified 61.7%

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

4. Taylor expanded in u0 around 0 87.5%

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

   1. +-commutative 87.5%

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

   2. mul-1-neg 87.5%

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

   3. unsub-neg 87.5%

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

   4. unpow2 87.5%

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

   5. associate-*r* 87.5%

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

6. Simplified 87.5%

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

7. Taylor expanded in u0 around 0 87.5%

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

   1. neg-mul-1 87.5%

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

   2. +-commutative 87.5%

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

   3. *-commutative 87.5%

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

   4. unpow2 87.5%

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

   5. associate-*r* 87.5%

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

   6. neg-mul-1 87.5%

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

   7. *-commutative 87.5%

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

   8. distribute-lft-out 87.3%

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

9. Simplified 87.3%

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

10. Final simplification 87.3%

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

Alternative 7: 68.3% accurate, 6.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 4.9999998413276127 \cdot 10^{-20}:\\ \;\;\;\;\left(alphax \cdot alphax\right) \cdot \frac{u0 + 0.5 \cdot \left(u0 \cdot u0\right)}{cos2phi}\\ \mathbf{else}:\\ \;\;\;\;\frac{alphay \cdot \left(u0 \cdot alphay\right)}{sin2phi}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (if (<= (/ sin2phi (* alphay alphay)) 4.9999998413276127e-20)
   (* (* alphax alphax) (/ (+ u0 (* 0.5 (* u0 u0))) cos2phi))
   (/ (* alphay (* u0 alphay)) sin2phi)))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float tmp;
	if ((sin2phi / (alphay * alphay)) <= 4.9999998413276127e-20f) {
		tmp = (alphax * alphax) * ((u0 + (0.5f * (u0 * u0))) / cos2phi);
	} else {
		tmp = (alphay * (u0 * alphay)) / 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 / (alphay * alphay)) <= 4.9999998413276127e-20) then
        tmp = (alphax * alphax) * ((u0 + (0.5e0 * (u0 * u0))) / cos2phi)
    else
        tmp = (alphay * (u0 * alphay)) / sin2phi
    end if
    code = tmp
end function

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = Float32(0.0)
	if (Float32(sin2phi / Float32(alphay * alphay)) <= Float32(4.9999998413276127e-20))
		tmp = Float32(Float32(alphax * alphax) * Float32(Float32(u0 + Float32(Float32(0.5) * Float32(u0 * u0))) / cos2phi));
	else
		tmp = Float32(Float32(alphay * Float32(u0 * alphay)) / sin2phi);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = single(0.0);
	if ((sin2phi / (alphay * alphay)) <= single(4.9999998413276127e-20))
		tmp = (alphax * alphax) * ((u0 + (single(0.5) * (u0 * u0))) / cos2phi);
	else
		tmp = (alphay * (u0 * alphay)) / sin2phi;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 sin2phi (*.f32 alphay alphay)) < 4.99999984e-20

   1. Initial program 55.1%

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

      1. associate-/r* 55.0%

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

   3. Simplified 55.0%

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

   4. Taylor expanded in u0 around 0 90.4%

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

      1. +-commutative 90.4%

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

      2. mul-1-neg 90.4%

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

      3. unsub-neg 90.4%

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

      4. unpow2 90.4%

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

      5. associate-*r* 90.4%

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

   6. Simplified 90.4%

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

   7. Taylor expanded in cos2phi around inf 76.7%

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

      1. associate-/l* 76.8%

         \[\leadsto \color{blue}{\frac{u0 - -0.5 \cdot {u0}^{2}}{\frac{cos2phi}{{alphax}^{2}}}} \]

      2. associate-/r/ 76.8%

         \[\leadsto \color{blue}{\frac{u0 - -0.5 \cdot {u0}^{2}}{cos2phi} \cdot {alphax}^{2}} \]

      3. cancel-sign-sub-inv 76.8%

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

      4. metadata-eval 76.8%

         \[\leadsto \frac{u0 + \color{blue}{0.5} \cdot {u0}^{2}}{cos2phi} \cdot {alphax}^{2} \]

      5. unpow2 76.8%

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

      6. unpow2 76.8%

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

   9. Simplified 76.8%

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

   if 4.99999984e-20 < (/.f32 sin2phi (*.f32 alphay alphay))

   1. Initial program 63.4%

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

      1. associate-/r* 63.4%

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

   3. Simplified 63.4%

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

   4. Taylor expanded in cos2phi around 0 58.1%

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

      1. mul-1-neg 58.1%

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

      2. unpow2 58.1%

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

      3. *-commutative 58.1%

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

   6. Simplified 58.1%

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

   7. Taylor expanded in alphay around 0 58.1%

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

      1. unpow2 58.1%

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

      2. sub-neg 58.1%

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

      3. log1p-def 86.6%

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

      4. associate-*l* 86.6%

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

   9. Simplified 86.6%

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

   10. Taylor expanded in u0 around 0 67.6%

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

       1. associate-*r* 67.6%

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

       2. neg-mul-1 67.6%

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

   12. Simplified 67.6%

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

4. Final simplification 69.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 4.9999998413276127 \cdot 10^{-20}:\\ \;\;\;\;\left(alphax \cdot alphax\right) \cdot \frac{u0 + 0.5 \cdot \left(u0 \cdot u0\right)}{cos2phi}\\ \mathbf{else}:\\ \;\;\;\;\frac{alphay \cdot \left(u0 \cdot alphay\right)}{sin2phi}\\ \end{array} \]

Alternative 8: 75.9% accurate, 6.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := u0 + 0.5 \cdot \left(u0 \cdot u0\right)\\ \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 4.9999998413276127 \cdot 10^{-20}:\\ \;\;\;\;\left(alphax \cdot alphax\right) \cdot \frac{t_0}{cos2phi}\\ \mathbf{else}:\\ \;\;\;\;\left(alphay \cdot alphay\right) \cdot \frac{t_0}{sin2phi}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (let* ((t_0 (+ u0 (* 0.5 (* u0 u0)))))
   (if (<= (/ sin2phi (* alphay alphay)) 4.9999998413276127e-20)
     (* (* alphax alphax) (/ t_0 cos2phi))
     (* (* alphay alphay) (/ t_0 sin2phi)))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float t_0 = u0 + (0.5f * (u0 * u0));
	float tmp;
	if ((sin2phi / (alphay * alphay)) <= 4.9999998413276127e-20f) {
		tmp = (alphax * alphax) * (t_0 / cos2phi);
	} else {
		tmp = (alphay * alphay) * (t_0 / 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) :: t_0
    real(4) :: tmp
    t_0 = u0 + (0.5e0 * (u0 * u0))
    if ((sin2phi / (alphay * alphay)) <= 4.9999998413276127e-20) then
        tmp = (alphax * alphax) * (t_0 / cos2phi)
    else
        tmp = (alphay * alphay) * (t_0 / sin2phi)
    end if
    code = tmp
end function

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	t_0 = Float32(u0 + Float32(Float32(0.5) * Float32(u0 * u0)))
	tmp = Float32(0.0)
	if (Float32(sin2phi / Float32(alphay * alphay)) <= Float32(4.9999998413276127e-20))
		tmp = Float32(Float32(alphax * alphax) * Float32(t_0 / cos2phi));
	else
		tmp = Float32(Float32(alphay * alphay) * Float32(t_0 / sin2phi));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alphax, alphay, u0, cos2phi, sin2phi)
	t_0 = u0 + (single(0.5) * (u0 * u0));
	tmp = single(0.0);
	if ((sin2phi / (alphay * alphay)) <= single(4.9999998413276127e-20))
		tmp = (alphax * alphax) * (t_0 / cos2phi);
	else
		tmp = (alphay * alphay) * (t_0 / sin2phi);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 sin2phi (*.f32 alphay alphay)) < 4.99999984e-20

   1. Initial program 55.1%

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

      1. associate-/r* 55.0%

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

   3. Simplified 55.0%

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

   4. Taylor expanded in u0 around 0 90.4%

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

      1. +-commutative 90.4%

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

      2. mul-1-neg 90.4%

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

      3. unsub-neg 90.4%

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

      4. unpow2 90.4%

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

      5. associate-*r* 90.4%

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

   6. Simplified 90.4%

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

   7. Taylor expanded in cos2phi around inf 76.7%

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

      1. associate-/l* 76.8%

         \[\leadsto \color{blue}{\frac{u0 - -0.5 \cdot {u0}^{2}}{\frac{cos2phi}{{alphax}^{2}}}} \]

      2. associate-/r/ 76.8%

         \[\leadsto \color{blue}{\frac{u0 - -0.5 \cdot {u0}^{2}}{cos2phi} \cdot {alphax}^{2}} \]

      3. cancel-sign-sub-inv 76.8%

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

      4. metadata-eval 76.8%

         \[\leadsto \frac{u0 + \color{blue}{0.5} \cdot {u0}^{2}}{cos2phi} \cdot {alphax}^{2} \]

      5. unpow2 76.8%

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

      6. unpow2 76.8%

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

   9. Simplified 76.8%

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

   if 4.99999984e-20 < (/.f32 sin2phi (*.f32 alphay alphay))

   1. Initial program 63.4%

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

      1. associate-/r* 63.4%

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

   3. Simplified 63.4%

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

   4. Taylor expanded in u0 around 0 86.7%

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

      1. +-commutative 86.7%

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

      2. mul-1-neg 86.7%

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

      3. unsub-neg 86.7%

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

      4. unpow2 86.7%

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

      5. associate-*r* 86.7%

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

   6. Simplified 86.7%

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

   7. Taylor expanded in cos2phi around 0 77.7%

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

      1. unpow2 77.7%

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

      2. associate-/l* 77.1%

         \[\leadsto \color{blue}{\frac{u0 - -0.5 \cdot {u0}^{2}}{\frac{sin2phi}{alphay \cdot alphay}}} \]

      3. associate-/r/ 77.8%

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

      4. cancel-sign-sub-inv 77.8%

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

      5. metadata-eval 77.8%

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

      6. unpow2 77.8%

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

   9. Simplified 77.8%

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

4. Final simplification 77.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 4.9999998413276127 \cdot 10^{-20}:\\ \;\;\;\;\left(alphax \cdot alphax\right) \cdot \frac{u0 + 0.5 \cdot \left(u0 \cdot u0\right)}{cos2phi}\\ \mathbf{else}:\\ \;\;\;\;\left(alphay \cdot alphay\right) \cdot \frac{u0 + 0.5 \cdot \left(u0 \cdot u0\right)}{sin2phi}\\ \end{array} \]

Alternative 9: 87.6% accurate, 6.1× speedup

Math

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

FPCore

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

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	return (u0 - (u0 * (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 - (u0 * (u0 * (-0.5e0)))) / ((cos2phi / (alphax * alphax)) + (sin2phi / (alphay * alphay)))
end function

Julia

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

Derivation

1. Initial program 61.7%

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

   1. associate-/r* 61.7%

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

3. Simplified 61.7%

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

4. Taylor expanded in u0 around 0 87.5%

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

   1. +-commutative 87.5%

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

   2. mul-1-neg 87.5%

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

   3. unsub-neg 87.5%

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

   4. unpow2 87.5%

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

   5. associate-*r* 87.5%

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

6. Simplified 87.5%

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

7. Taylor expanded in cos2phi around 0 87.5%

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

   1. unpow2 87.5%

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

9. Simplified 87.5%

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

10. Final simplification 87.5%

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

Alternative 10: 81.6% accurate, 6.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;sin2phi \leq 0.014999999664723873:\\ \;\;\;\;\frac{u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay} \cdot \frac{1}{alphay}}\\ \mathbf{else}:\\ \;\;\;\;\left(alphay \cdot alphay\right) \cdot \frac{u0 + 0.5 \cdot \left(u0 \cdot u0\right)}{sin2phi}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (if (<= sin2phi 0.014999999664723873)
   (/
    u0
    (+ (/ cos2phi (* alphax alphax)) (* (/ sin2phi alphay) (/ 1.0 alphay))))
   (* (* alphay alphay) (/ (+ u0 (* 0.5 (* u0 u0))) sin2phi))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float tmp;
	if (sin2phi <= 0.014999999664723873f) {
		tmp = u0 / ((cos2phi / (alphax * alphax)) + ((sin2phi / alphay) * (1.0f / alphay)));
	} else {
		tmp = (alphay * alphay) * ((u0 + (0.5f * (u0 * 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 <= 0.014999999664723873e0) then
        tmp = u0 / ((cos2phi / (alphax * alphax)) + ((sin2phi / alphay) * (1.0e0 / alphay)))
    else
        tmp = (alphay * alphay) * ((u0 + (0.5e0 * (u0 * u0))) / sin2phi)
    end if
    code = tmp
end function

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = Float32(0.0)
	if (sin2phi <= Float32(0.014999999664723873))
		tmp = Float32(u0 / Float32(Float32(cos2phi / Float32(alphax * alphax)) + Float32(Float32(sin2phi / alphay) * Float32(Float32(1.0) / alphay))));
	else
		tmp = Float32(Float32(alphay * alphay) * Float32(Float32(u0 + Float32(Float32(0.5) * Float32(u0 * u0))) / sin2phi));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = single(0.0);
	if (sin2phi <= single(0.014999999664723873))
		tmp = u0 / ((cos2phi / (alphax * alphax)) + ((sin2phi / alphay) * (single(1.0) / alphay)));
	else
		tmp = (alphay * alphay) * ((u0 + (single(0.5) * (u0 * u0))) / sin2phi);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if sin2phi < 0.0149999997

   1. Initial program 56.4%

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

      1. associate-/r* 56.4%

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

   3. Simplified 56.4%

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

   4. Taylor expanded in u0 around 0 74.6%

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

      1. unpow2 74.6%

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

      2. unpow2 74.6%

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

   6. Simplified 74.6%

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

      1. associate-/r* 74.6%

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

      2. div-inv 74.6%

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

   8. Applied egg-rr 74.6%

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

   if 0.0149999997 < sin2phi

   1. Initial program 66.1%

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

      1. associate-/r* 66.1%

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

   3. Simplified 66.1%

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

   4. Taylor expanded in u0 around 0 86.6%

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

      1. +-commutative 86.6%

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

      2. mul-1-neg 86.6%

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

      3. unsub-neg 86.6%

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

      4. unpow2 86.6%

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

      5. associate-*r* 86.6%

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

   6. Simplified 86.6%

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

   7. Taylor expanded in cos2phi around 0 87.5%

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

      1. unpow2 87.5%

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

      2. associate-/l* 86.6%

         \[\leadsto \color{blue}{\frac{u0 - -0.5 \cdot {u0}^{2}}{\frac{sin2phi}{alphay \cdot alphay}}} \]

      3. associate-/r/ 87.5%

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

      4. cancel-sign-sub-inv 87.5%

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

      5. metadata-eval 87.5%

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

      6. unpow2 87.5%

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

   9. Simplified 87.5%

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

4. Final simplification 81.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;sin2phi \leq 0.014999999664723873:\\ \;\;\;\;\frac{u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay} \cdot \frac{1}{alphay}}\\ \mathbf{else}:\\ \;\;\;\;\left(alphay \cdot alphay\right) \cdot \frac{u0 + 0.5 \cdot \left(u0 \cdot u0\right)}{sin2phi}\\ \end{array} \]

Alternative 11: 81.7% accurate, 6.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;sin2phi \leq 0.014999999664723873:\\ \;\;\;\;\frac{u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{1}{alphay \cdot \frac{alphay}{sin2phi}}}\\ \mathbf{else}:\\ \;\;\;\;\left(alphay \cdot alphay\right) \cdot \frac{u0 + 0.5 \cdot \left(u0 \cdot u0\right)}{sin2phi}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (if (<= sin2phi 0.014999999664723873)
   (/
    u0
    (+ (/ cos2phi (* alphax alphax)) (/ 1.0 (* alphay (/ alphay sin2phi)))))
   (* (* alphay alphay) (/ (+ u0 (* 0.5 (* u0 u0))) sin2phi))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float tmp;
	if (sin2phi <= 0.014999999664723873f) {
		tmp = u0 / ((cos2phi / (alphax * alphax)) + (1.0f / (alphay * (alphay / sin2phi))));
	} else {
		tmp = (alphay * alphay) * ((u0 + (0.5f * (u0 * 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 <= 0.014999999664723873e0) then
        tmp = u0 / ((cos2phi / (alphax * alphax)) + (1.0e0 / (alphay * (alphay / sin2phi))))
    else
        tmp = (alphay * alphay) * ((u0 + (0.5e0 * (u0 * u0))) / sin2phi)
    end if
    code = tmp
end function

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = Float32(0.0)
	if (sin2phi <= Float32(0.014999999664723873))
		tmp = Float32(u0 / Float32(Float32(cos2phi / Float32(alphax * alphax)) + Float32(Float32(1.0) / Float32(alphay * Float32(alphay / sin2phi)))));
	else
		tmp = Float32(Float32(alphay * alphay) * Float32(Float32(u0 + Float32(Float32(0.5) * Float32(u0 * u0))) / sin2phi));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = single(0.0);
	if (sin2phi <= single(0.014999999664723873))
		tmp = u0 / ((cos2phi / (alphax * alphax)) + (single(1.0) / (alphay * (alphay / sin2phi))));
	else
		tmp = (alphay * alphay) * ((u0 + (single(0.5) * (u0 * u0))) / sin2phi);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if sin2phi < 0.0149999997

   1. Initial program 56.4%

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

      1. associate-/r* 56.4%

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

   3. Simplified 56.4%

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

   4. Taylor expanded in u0 around 0 74.6%

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

      1. unpow2 74.6%

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

      2. unpow2 74.6%

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

   6. Simplified 74.6%

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

      1. associate-/r* 74.6%

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

      2. div-inv 74.6%

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

   8. Applied egg-rr 74.6%

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

      1. un-div-inv 74.6%

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

      2. associate-/r* 74.6%

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

      3. clear-num 74.6%

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

      4. associate-*l/ 74.6%

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

      5. *-commutative 74.6%

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

   10. Applied egg-rr 74.6%

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

   if 0.0149999997 < sin2phi

   1. Initial program 66.1%

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

      1. associate-/r* 66.1%

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

   3. Simplified 66.1%

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

   4. Taylor expanded in u0 around 0 86.6%

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

      1. +-commutative 86.6%

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

      2. mul-1-neg 86.6%

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

      3. unsub-neg 86.6%

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

      4. unpow2 86.6%

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

      5. associate-*r* 86.6%

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

   6. Simplified 86.6%

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

   7. Taylor expanded in cos2phi around 0 87.5%

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

      1. unpow2 87.5%

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

      2. associate-/l* 86.6%

         \[\leadsto \color{blue}{\frac{u0 - -0.5 \cdot {u0}^{2}}{\frac{sin2phi}{alphay \cdot alphay}}} \]

      3. associate-/r/ 87.5%

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

      4. cancel-sign-sub-inv 87.5%

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

      5. metadata-eval 87.5%

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

      6. unpow2 87.5%

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

   9. Simplified 87.5%

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

4. Final simplification 81.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;sin2phi \leq 0.014999999664723873:\\ \;\;\;\;\frac{u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{1}{alphay \cdot \frac{alphay}{sin2phi}}}\\ \mathbf{else}:\\ \;\;\;\;\left(alphay \cdot alphay\right) \cdot \frac{u0 + 0.5 \cdot \left(u0 \cdot u0\right)}{sin2phi}\\ \end{array} \]

Alternative 12: 81.7% accurate, 7.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;sin2phi \leq 0.014999999664723873:\\ \;\;\;\;\frac{u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}}\\ \mathbf{else}:\\ \;\;\;\;\left(alphay \cdot alphay\right) \cdot \frac{u0 + 0.5 \cdot \left(u0 \cdot u0\right)}{sin2phi}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (if (<= sin2phi 0.014999999664723873)
   (/ u0 (+ (/ cos2phi (* alphax alphax)) (/ sin2phi (* alphay alphay))))
   (* (* alphay alphay) (/ (+ u0 (* 0.5 (* u0 u0))) sin2phi))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float tmp;
	if (sin2phi <= 0.014999999664723873f) {
		tmp = u0 / ((cos2phi / (alphax * alphax)) + (sin2phi / (alphay * alphay)));
	} else {
		tmp = (alphay * alphay) * ((u0 + (0.5f * (u0 * 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 <= 0.014999999664723873e0) then
        tmp = u0 / ((cos2phi / (alphax * alphax)) + (sin2phi / (alphay * alphay)))
    else
        tmp = (alphay * alphay) * ((u0 + (0.5e0 * (u0 * u0))) / sin2phi)
    end if
    code = tmp
end function

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = Float32(0.0)
	if (sin2phi <= Float32(0.014999999664723873))
		tmp = Float32(u0 / Float32(Float32(cos2phi / Float32(alphax * alphax)) + Float32(sin2phi / Float32(alphay * alphay))));
	else
		tmp = Float32(Float32(alphay * alphay) * Float32(Float32(u0 + Float32(Float32(0.5) * Float32(u0 * u0))) / sin2phi));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = single(0.0);
	if (sin2phi <= single(0.014999999664723873))
		tmp = u0 / ((cos2phi / (alphax * alphax)) + (sin2phi / (alphay * alphay)));
	else
		tmp = (alphay * alphay) * ((u0 + (single(0.5) * (u0 * u0))) / sin2phi);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if sin2phi < 0.0149999997

   1. Initial program 56.4%

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

      1. associate-/r* 56.4%

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

   3. Simplified 56.4%

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

   4. Taylor expanded in u0 around 0 74.6%

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

      1. unpow2 74.6%

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

      2. unpow2 74.6%

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

   6. Simplified 74.6%

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

   if 0.0149999997 < sin2phi

   1. Initial program 66.1%

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

      1. associate-/r* 66.1%

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

   3. Simplified 66.1%

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

   4. Taylor expanded in u0 around 0 86.6%

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

      1. +-commutative 86.6%

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

      2. mul-1-neg 86.6%

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

      3. unsub-neg 86.6%

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

      4. unpow2 86.6%

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

      5. associate-*r* 86.6%

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

   6. Simplified 86.6%

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

   7. Taylor expanded in cos2phi around 0 87.5%

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

      1. unpow2 87.5%

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

      2. associate-/l* 86.6%

         \[\leadsto \color{blue}{\frac{u0 - -0.5 \cdot {u0}^{2}}{\frac{sin2phi}{alphay \cdot alphay}}} \]

      3. associate-/r/ 87.5%

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

      4. cancel-sign-sub-inv 87.5%

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

      5. metadata-eval 87.5%

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

      6. unpow2 87.5%

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

   9. Simplified 87.5%

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

4. Final simplification 81.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;sin2phi \leq 0.014999999664723873:\\ \;\;\;\;\frac{u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}}\\ \mathbf{else}:\\ \;\;\;\;\left(alphay \cdot alphay\right) \cdot \frac{u0 + 0.5 \cdot \left(u0 \cdot u0\right)}{sin2phi}\\ \end{array} \]

Alternative 13: 81.7% accurate, 7.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;sin2phi \leq 0.014999999664723873:\\ \;\;\;\;\frac{u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{\frac{sin2phi}{alphay}}{alphay}}\\ \mathbf{else}:\\ \;\;\;\;\left(alphay \cdot alphay\right) \cdot \frac{u0 + 0.5 \cdot \left(u0 \cdot u0\right)}{sin2phi}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (if (<= sin2phi 0.014999999664723873)
   (/ u0 (+ (/ cos2phi (* alphax alphax)) (/ (/ sin2phi alphay) alphay)))
   (* (* alphay alphay) (/ (+ u0 (* 0.5 (* u0 u0))) sin2phi))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float tmp;
	if (sin2phi <= 0.014999999664723873f) {
		tmp = u0 / ((cos2phi / (alphax * alphax)) + ((sin2phi / alphay) / alphay));
	} else {
		tmp = (alphay * alphay) * ((u0 + (0.5f * (u0 * 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 <= 0.014999999664723873e0) then
        tmp = u0 / ((cos2phi / (alphax * alphax)) + ((sin2phi / alphay) / alphay))
    else
        tmp = (alphay * alphay) * ((u0 + (0.5e0 * (u0 * u0))) / sin2phi)
    end if
    code = tmp
end function

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = Float32(0.0)
	if (sin2phi <= Float32(0.014999999664723873))
		tmp = Float32(u0 / Float32(Float32(cos2phi / Float32(alphax * alphax)) + Float32(Float32(sin2phi / alphay) / alphay)));
	else
		tmp = Float32(Float32(alphay * alphay) * Float32(Float32(u0 + Float32(Float32(0.5) * Float32(u0 * u0))) / sin2phi));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = single(0.0);
	if (sin2phi <= single(0.014999999664723873))
		tmp = u0 / ((cos2phi / (alphax * alphax)) + ((sin2phi / alphay) / alphay));
	else
		tmp = (alphay * alphay) * ((u0 + (single(0.5) * (u0 * u0))) / sin2phi);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if sin2phi < 0.0149999997

   1. Initial program 56.4%

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

      1. neg-sub0 56.4%

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

      2. div-sub 56.4%

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

      3. --rgt-identity 56.4%

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

      4. div-sub 56.4%

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

      5. --rgt-identity 56.4%

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

      6. sub-neg 56.4%

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

      7. +-commutative 56.4%

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

      8. neg-sub0 56.4%

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

      9. associate-+l- 56.4%

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

      10. sub0-neg 56.4%

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

      11. neg-mul-1 56.4%

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

      12. log-prod -0.0%

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

      13. associate--r+ -0.0%

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

   3. Simplified 98.7%

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

      1. associate-/r* 98.8%

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

      2. frac-2neg 98.8%

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

      3. div-inv 98.6%

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

      4. distribute-rgt-neg-in 98.6%

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

   5. Applied egg-rr 98.6%

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

   6. Taylor expanded in u0 around 0 74.6%

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

      1. +-commutative 74.6%

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

      2. unpow2 74.6%

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

      3. unpow2 74.6%

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

      4. associate-/r* 74.6%

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

   8. Simplified 74.6%

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

   if 0.0149999997 < sin2phi

   1. Initial program 66.1%

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

      1. associate-/r* 66.1%

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

   3. Simplified 66.1%

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

   4. Taylor expanded in u0 around 0 86.6%

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

      1. +-commutative 86.6%

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

      2. mul-1-neg 86.6%

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

      3. unsub-neg 86.6%

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

      4. unpow2 86.6%

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

      5. associate-*r* 86.6%

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

   6. Simplified 86.6%

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

   7. Taylor expanded in cos2phi around 0 87.5%

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

      1. unpow2 87.5%

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

      2. associate-/l* 86.6%

         \[\leadsto \color{blue}{\frac{u0 - -0.5 \cdot {u0}^{2}}{\frac{sin2phi}{alphay \cdot alphay}}} \]

      3. associate-/r/ 87.5%

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

      4. cancel-sign-sub-inv 87.5%

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

      5. metadata-eval 87.5%

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

      6. unpow2 87.5%

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

   9. Simplified 87.5%

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

4. Final simplification 81.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;sin2phi \leq 0.014999999664723873:\\ \;\;\;\;\frac{u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{\frac{sin2phi}{alphay}}{alphay}}\\ \mathbf{else}:\\ \;\;\;\;\left(alphay \cdot alphay\right) \cdot \frac{u0 + 0.5 \cdot \left(u0 \cdot u0\right)}{sin2phi}\\ \end{array} \]

Alternative 14: 66.4% accurate, 8.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{sin2phi}{alphay \cdot alphay}\\ \mathbf{if}\;t_0 \leq 4.9999998413276127 \cdot 10^{-20}:\\ \;\;\;\;\left(alphax \cdot alphax\right) \cdot \frac{u0}{cos2phi}\\ \mathbf{else}:\\ \;\;\;\;\frac{u0}{t_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (let* ((t_0 (/ sin2phi (* alphay alphay))))
   (if (<= t_0 4.9999998413276127e-20)
     (* (* alphax alphax) (/ u0 cos2phi))
     (/ u0 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 <= 4.9999998413276127e-20f) {
		tmp = (alphax * alphax) * (u0 / cos2phi);
	} else {
		tmp = u0 / t_0;
	}
	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) :: t_0
    real(4) :: tmp
    t_0 = sin2phi / (alphay * alphay)
    if (t_0 <= 4.9999998413276127e-20) then
        tmp = (alphax * alphax) * (u0 / cos2phi)
    else
        tmp = u0 / t_0
    end if
    code = tmp
end function

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	t_0 = Float32(sin2phi / Float32(alphay * alphay))
	tmp = Float32(0.0)
	if (t_0 <= Float32(4.9999998413276127e-20))
		tmp = Float32(Float32(alphax * alphax) * Float32(u0 / cos2phi));
	else
		tmp = Float32(u0 / t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alphax, alphay, u0, cos2phi, sin2phi)
	t_0 = sin2phi / (alphay * alphay);
	tmp = single(0.0);
	if (t_0 <= single(4.9999998413276127e-20))
		tmp = (alphax * alphax) * (u0 / cos2phi);
	else
		tmp = u0 / t_0;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 sin2phi (*.f32 alphay alphay)) < 4.99999984e-20

   1. Initial program 55.1%

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

      1. associate-/r* 55.0%

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

   3. Simplified 55.0%

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

   4. Taylor expanded in u0 around 0 76.6%

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

      1. unpow2 76.6%

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

      2. unpow2 76.6%

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

   6. Simplified 76.6%

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

   7. Taylor expanded in cos2phi around inf 66.1%

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

      1. associate-/l* 66.1%

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

      2. unpow2 66.1%

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

   9. Simplified 66.1%

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

       1. associate-/r/ 66.1%

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

   11. Applied egg-rr 66.1%

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

   if 4.99999984e-20 < (/.f32 sin2phi (*.f32 alphay alphay))

   1. Initial program 63.4%

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

      1. associate-/r* 63.4%

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

   3. Simplified 63.4%

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

   4. Taylor expanded in u0 around 0 74.4%

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

      1. unpow2 74.4%

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

      2. unpow2 74.4%

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

   6. Simplified 74.4%

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

   7. Taylor expanded in cos2phi around 0 67.6%

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

      1. associate-/l* 67.2%

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

      2. unpow2 67.2%

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

   9. Simplified 67.2%

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

4. Final simplification 67.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 4.9999998413276127 \cdot 10^{-20}:\\ \;\;\;\;\left(alphax \cdot alphax\right) \cdot \frac{u0}{cos2phi}\\ \mathbf{else}:\\ \;\;\;\;\frac{u0}{\frac{sin2phi}{alphay \cdot alphay}}\\ \end{array} \]

Alternative 15: 66.4% accurate, 8.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 4.9999998413276127 \cdot 10^{-20}:\\ \;\;\;\;\left(alphax \cdot alphax\right) \cdot \frac{u0}{cos2phi}\\ \mathbf{else}:\\ \;\;\;\;\frac{u0}{\frac{\frac{sin2phi}{alphay}}{alphay}}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (if (<= (/ sin2phi (* alphay alphay)) 4.9999998413276127e-20)
   (* (* alphax alphax) (/ u0 cos2phi))
   (/ u0 (/ (/ sin2phi alphay) alphay))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float tmp;
	if ((sin2phi / (alphay * alphay)) <= 4.9999998413276127e-20f) {
		tmp = (alphax * alphax) * (u0 / cos2phi);
	} else {
		tmp = u0 / ((sin2phi / alphay) / alphay);
	}
	return tmp;
}

Fortran

real(4) function code(alphax, alphay, u0, cos2phi, sin2phi)
    real(4), intent (in) :: alphax
    real(4), intent (in) :: alphay
    real(4), intent (in) :: u0
    real(4), intent (in) :: cos2phi
    real(4), intent (in) :: sin2phi
    real(4) :: tmp
    if ((sin2phi / (alphay * alphay)) <= 4.9999998413276127e-20) then
        tmp = (alphax * alphax) * (u0 / cos2phi)
    else
        tmp = u0 / ((sin2phi / alphay) / alphay)
    end if
    code = tmp
end function

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = Float32(0.0)
	if (Float32(sin2phi / Float32(alphay * alphay)) <= Float32(4.9999998413276127e-20))
		tmp = Float32(Float32(alphax * alphax) * Float32(u0 / cos2phi));
	else
		tmp = Float32(u0 / Float32(Float32(sin2phi / alphay) / alphay));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = single(0.0);
	if ((sin2phi / (alphay * alphay)) <= single(4.9999998413276127e-20))
		tmp = (alphax * alphax) * (u0 / cos2phi);
	else
		tmp = u0 / ((sin2phi / alphay) / alphay);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 sin2phi (*.f32 alphay alphay)) < 4.99999984e-20

   1. Initial program 55.1%

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

      1. associate-/r* 55.0%

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

   3. Simplified 55.0%

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

   4. Taylor expanded in u0 around 0 76.6%

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

      1. unpow2 76.6%

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

      2. unpow2 76.6%

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

   6. Simplified 76.6%

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

   7. Taylor expanded in cos2phi around inf 66.1%

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

      1. associate-/l* 66.1%

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

      2. unpow2 66.1%

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

   9. Simplified 66.1%

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

       1. associate-/r/ 66.1%

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

   11. Applied egg-rr 66.1%

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

   if 4.99999984e-20 < (/.f32 sin2phi (*.f32 alphay alphay))

   1. Initial program 63.4%

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

      1. associate-/r* 63.4%

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

   3. Simplified 63.4%

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

   4. Taylor expanded in u0 around 0 74.4%

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

      1. unpow2 74.4%

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

      2. unpow2 74.4%

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

   6. Simplified 74.4%

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

   7. Taylor expanded in cos2phi around 0 67.6%

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

      1. associate-/l* 67.2%

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

      2. unpow2 67.2%

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

      3. associate-/r* 67.2%

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

   9. Simplified 67.2%

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

4. Final simplification 67.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 4.9999998413276127 \cdot 10^{-20}:\\ \;\;\;\;\left(alphax \cdot alphax\right) \cdot \frac{u0}{cos2phi}\\ \mathbf{else}:\\ \;\;\;\;\frac{u0}{\frac{\frac{sin2phi}{alphay}}{alphay}}\\ \end{array} \]

Alternative 16: 66.4% accurate, 8.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 4.9999998413276127 \cdot 10^{-20}:\\ \;\;\;\;\frac{alphax \cdot \left(u0 \cdot alphax\right)}{cos2phi}\\ \mathbf{else}:\\ \;\;\;\;\frac{u0}{\frac{\frac{sin2phi}{alphay}}{alphay}}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (if (<= (/ sin2phi (* alphay alphay)) 4.9999998413276127e-20)
   (/ (* alphax (* u0 alphax)) cos2phi)
   (/ u0 (/ (/ sin2phi alphay) alphay))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float tmp;
	if ((sin2phi / (alphay * alphay)) <= 4.9999998413276127e-20f) {
		tmp = (alphax * (u0 * alphax)) / cos2phi;
	} else {
		tmp = u0 / ((sin2phi / alphay) / alphay);
	}
	return tmp;
}

Fortran

real(4) function code(alphax, alphay, u0, cos2phi, sin2phi)
    real(4), intent (in) :: alphax
    real(4), intent (in) :: alphay
    real(4), intent (in) :: u0
    real(4), intent (in) :: cos2phi
    real(4), intent (in) :: sin2phi
    real(4) :: tmp
    if ((sin2phi / (alphay * alphay)) <= 4.9999998413276127e-20) then
        tmp = (alphax * (u0 * alphax)) / cos2phi
    else
        tmp = u0 / ((sin2phi / alphay) / alphay)
    end if
    code = tmp
end function

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = Float32(0.0)
	if (Float32(sin2phi / Float32(alphay * alphay)) <= Float32(4.9999998413276127e-20))
		tmp = Float32(Float32(alphax * Float32(u0 * alphax)) / cos2phi);
	else
		tmp = Float32(u0 / Float32(Float32(sin2phi / alphay) / alphay));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = single(0.0);
	if ((sin2phi / (alphay * alphay)) <= single(4.9999998413276127e-20))
		tmp = (alphax * (u0 * alphax)) / cos2phi;
	else
		tmp = u0 / ((sin2phi / alphay) / alphay);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 sin2phi (*.f32 alphay alphay)) < 4.99999984e-20

   1. Initial program 55.1%

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

      1. associate-/r* 55.0%

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

   3. Simplified 55.0%

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

   4. Taylor expanded in u0 around 0 76.6%

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

      1. unpow2 76.6%

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

      2. unpow2 76.6%

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

   6. Simplified 76.6%

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

   7. Taylor expanded in cos2phi around inf 66.1%

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

      1. *-commutative 66.1%

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

      2. unpow2 66.1%

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

      3. associate-*l* 66.2%

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

   9. Simplified 66.2%

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

   if 4.99999984e-20 < (/.f32 sin2phi (*.f32 alphay alphay))

   1. Initial program 63.4%

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

      1. associate-/r* 63.4%

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

   3. Simplified 63.4%

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

   4. Taylor expanded in u0 around 0 74.4%

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

      1. unpow2 74.4%

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

      2. unpow2 74.4%

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

   6. Simplified 74.4%

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

   7. Taylor expanded in cos2phi around 0 67.6%

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

      1. associate-/l* 67.2%

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

      2. unpow2 67.2%

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

      3. associate-/r* 67.2%

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

   9. Simplified 67.2%

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

4. Final simplification 67.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 4.9999998413276127 \cdot 10^{-20}:\\ \;\;\;\;\frac{alphax \cdot \left(u0 \cdot alphax\right)}{cos2phi}\\ \mathbf{else}:\\ \;\;\;\;\frac{u0}{\frac{\frac{sin2phi}{alphay}}{alphay}}\\ \end{array} \]

Alternative 17: 66.7% accurate, 8.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 4.9999998413276127 \cdot 10^{-20}:\\ \;\;\;\;\frac{alphax \cdot \left(u0 \cdot alphax\right)}{cos2phi}\\ \mathbf{else}:\\ \;\;\;\;\frac{alphay \cdot \left(u0 \cdot alphay\right)}{sin2phi}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (if (<= (/ sin2phi (* alphay alphay)) 4.9999998413276127e-20)
   (/ (* alphax (* u0 alphax)) cos2phi)
   (/ (* alphay (* u0 alphay)) sin2phi)))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float tmp;
	if ((sin2phi / (alphay * alphay)) <= 4.9999998413276127e-20f) {
		tmp = (alphax * (u0 * alphax)) / cos2phi;
	} else {
		tmp = (alphay * (u0 * alphay)) / 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 / (alphay * alphay)) <= 4.9999998413276127e-20) then
        tmp = (alphax * (u0 * alphax)) / cos2phi
    else
        tmp = (alphay * (u0 * alphay)) / sin2phi
    end if
    code = tmp
end function

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = Float32(0.0)
	if (Float32(sin2phi / Float32(alphay * alphay)) <= Float32(4.9999998413276127e-20))
		tmp = Float32(Float32(alphax * Float32(u0 * alphax)) / cos2phi);
	else
		tmp = Float32(Float32(alphay * Float32(u0 * alphay)) / sin2phi);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = single(0.0);
	if ((sin2phi / (alphay * alphay)) <= single(4.9999998413276127e-20))
		tmp = (alphax * (u0 * alphax)) / cos2phi;
	else
		tmp = (alphay * (u0 * alphay)) / sin2phi;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 sin2phi (*.f32 alphay alphay)) < 4.99999984e-20

   1. Initial program 55.1%

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

      1. associate-/r* 55.0%

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

   3. Simplified 55.0%

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

   4. Taylor expanded in u0 around 0 76.6%

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

      1. unpow2 76.6%

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

      2. unpow2 76.6%

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

   6. Simplified 76.6%

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

   7. Taylor expanded in cos2phi around inf 66.1%

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

      1. *-commutative 66.1%

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

      2. unpow2 66.1%

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

      3. associate-*l* 66.2%

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

   9. Simplified 66.2%

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

   if 4.99999984e-20 < (/.f32 sin2phi (*.f32 alphay alphay))

   1. Initial program 63.4%

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

      1. associate-/r* 63.4%

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

   3. Simplified 63.4%

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

   4. Taylor expanded in cos2phi around 0 58.1%

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

      1. mul-1-neg 58.1%

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

      2. unpow2 58.1%

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

      3. *-commutative 58.1%

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

   6. Simplified 58.1%

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

   7. Taylor expanded in alphay around 0 58.1%

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

      1. unpow2 58.1%

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

      2. sub-neg 58.1%

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

      3. log1p-def 86.6%

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

      4. associate-*l* 86.6%

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

   9. Simplified 86.6%

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

   10. Taylor expanded in u0 around 0 67.6%

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

       1. associate-*r* 67.6%

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

       2. neg-mul-1 67.6%

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

   12. Simplified 67.6%

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

4. Final simplification 67.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 4.9999998413276127 \cdot 10^{-20}:\\ \;\;\;\;\frac{alphax \cdot \left(u0 \cdot alphax\right)}{cos2phi}\\ \mathbf{else}:\\ \;\;\;\;\frac{alphay \cdot \left(u0 \cdot alphay\right)}{sin2phi}\\ \end{array} \]

Alternative 18: 23.4% accurate, 16.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 61.7%

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

   1. associate-/r* 61.7%

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

3. Simplified 61.7%

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

4. Taylor expanded in u0 around 0 74.9%

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

   1. unpow2 74.9%

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

   2. unpow2 74.9%

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

6. Simplified 74.9%

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

7. Taylor expanded in cos2phi around inf 24.1%

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

   1. associate-/l* 24.1%

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

   2. unpow2 24.1%

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

9. Simplified 24.1%

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

    1. associate-/r/ 24.1%

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

11. Applied egg-rr 24.1%

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

12. Final simplification 24.1%

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

Reproduce

herbie shell --seed 2023214 
(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)))))