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

Percentage Accurate: 60.9% → 98.3%

Time: 14.7s

Alternatives: 17

Speedup: 12.7×

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: 60.9% 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

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 61.4%

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

   1. neg-sub0 61.4%

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

   2. div-sub 61.4%

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

   3. --rgt-identity 61.4%

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

   4. div-sub 61.4%

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

   5. --rgt-identity 61.4%

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

   6. sub-neg 61.4%

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

   7. +-commutative 61.4%

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

   8. neg-sub0 61.4%

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

   9. associate-+l- 61.4%

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

   10. sub0-neg 61.4%

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

   11. neg-mul-1 61.4%

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

   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. clear-num 98.6%

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

   2. associate-/r/ 98.5%

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

   3. pow2 98.5%

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

   4. pow-flip 98.7%

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

   5. metadata-eval 98.7%

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

5. Applied egg-rr 98.7%

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

6. Final simplification 98.7%

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

Alternative 2: 87.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := alphax \cdot \frac{alphax}{cos2phi}\\ \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 1.9999999920083944 \cdot 10^{-11}:\\ \;\;\;\;\frac{u0}{\frac{alphay + \frac{sin2phi}{alphay} \cdot t_0}{alphay \cdot t_0}}\\ \mathbf{else}:\\ \;\;\;\;\left(alphay \cdot alphay\right) \cdot \frac{-\mathsf{log1p}\left(-u0\right)}{sin2phi}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (let* ((t_0 (* alphax (/ alphax cos2phi))))
   (if (<= (/ sin2phi (* alphay alphay)) 1.9999999920083944e-11)
     (/ u0 (/ (+ alphay (* (/ sin2phi alphay) t_0)) (* alphay t_0)))
     (* (* alphay alphay) (/ (- (log1p (- u0))) sin2phi)))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float t_0 = alphax * (alphax / cos2phi);
	float tmp;
	if ((sin2phi / (alphay * alphay)) <= 1.9999999920083944e-11f) {
		tmp = u0 / ((alphay + ((sin2phi / alphay) * t_0)) / (alphay * t_0));
	} else {
		tmp = (alphay * alphay) * (-log1pf(-u0) / sin2phi);
	}
	return tmp;
}

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	t_0 = Float32(alphax * Float32(alphax / cos2phi))
	tmp = Float32(0.0)
	if (Float32(sin2phi / Float32(alphay * alphay)) <= Float32(1.9999999920083944e-11))
		tmp = Float32(u0 / Float32(Float32(alphay + Float32(Float32(sin2phi / alphay) * t_0)) / Float32(alphay * t_0)));
	else
		tmp = Float32(Float32(alphay * alphay) * Float32(Float32(-log1p(Float32(-u0))) / sin2phi));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 sin2phi (*.f32 alphay alphay)) < 1.99999999e-11

   1. Initial program 59.0%

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

      1. neg-sub0 59.0%

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

      2. div-sub 59.0%

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

      3. --rgt-identity 59.0%

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

      4. div-sub 59.0%

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

      5. --rgt-identity 59.0%

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

      6. neg-sub0 59.0%

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

      7. sub-neg 59.0%

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

      8. log1p-def 98.8%

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

   3. Simplified 98.8%

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

   4. Taylor expanded in u0 around 0 73.5%

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

      1. unpow2 73.5%

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

      2. unpow2 73.5%

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

   6. Simplified 73.5%

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

      1. associate-/r* 73.5%

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

      2. clear-num 73.5%

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

      3. frac-add 73.5%

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

      4. *-un-lft-identity 73.5%

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

      5. associate-/l* 73.4%

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

      6. div-inv 73.4%

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

      7. clear-num 73.5%

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

      8. associate-/l* 73.4%

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

      9. div-inv 73.5%

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

      10. clear-num 73.5%

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

   8. Applied egg-rr 73.5%

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

   if 1.99999999e-11 < (/.f32 sin2phi (*.f32 alphay alphay))

   1. Initial program 62.5%

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

      1. neg-sub0 62.5%

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

      2. div-sub 62.5%

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

      3. --rgt-identity 62.5%

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

      4. div-sub 62.5%

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

      5. --rgt-identity 62.5%

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

      6. neg-sub0 62.5%

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

      7. sub-neg 62.5%

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

      8. log1p-def 98.6%

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

   3. Simplified 98.6%

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

   4. Taylor expanded in cos2phi around 0 62.0%

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

      1. mul-1-neg 62.0%

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

      2. unpow2 62.0%

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

      3. *-commutative 62.0%

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

   6. Simplified 62.0%

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

   7. Taylor expanded in alphay around 0 62.0%

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

      1. associate-/l* 61.6%

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

      2. sub-neg 61.6%

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

      3. neg-mul-1 61.6%

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

      4. log1p-def 95.8%

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

      5. neg-mul-1 95.8%

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

      6. associate-/l* 96.3%

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

      7. *-commutative 96.3%

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

      8. associate-*l/ 96.3%

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

      9. unpow2 96.3%

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

   9. Simplified 96.3%

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

4. Final simplification 89.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 1.9999999920083944 \cdot 10^{-11}:\\ \;\;\;\;\frac{u0}{\frac{alphay + \frac{sin2phi}{alphay} \cdot \left(alphax \cdot \frac{alphax}{cos2phi}\right)}{alphay \cdot \left(alphax \cdot \frac{alphax}{cos2phi}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\left(alphay \cdot alphay\right) \cdot \frac{-\mathsf{log1p}\left(-u0\right)}{sin2phi}\\ \end{array} \]

Alternative 3: 87.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := alphax \cdot \frac{alphax}{cos2phi}\\ \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 1.9999999920083944 \cdot 10^{-11}:\\ \;\;\;\;\frac{u0}{\frac{alphay + \frac{sin2phi}{alphay} \cdot t_0}{alphay \cdot t_0}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(-u0\right) \cdot \frac{-alphay \cdot alphay}{sin2phi}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (let* ((t_0 (* alphax (/ alphax cos2phi))))
   (if (<= (/ sin2phi (* alphay alphay)) 1.9999999920083944e-11)
     (/ u0 (/ (+ alphay (* (/ sin2phi alphay) t_0)) (* alphay t_0)))
     (* (log1p (- u0)) (/ (- (* alphay alphay)) sin2phi)))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float t_0 = alphax * (alphax / cos2phi);
	float tmp;
	if ((sin2phi / (alphay * alphay)) <= 1.9999999920083944e-11f) {
		tmp = u0 / ((alphay + ((sin2phi / alphay) * t_0)) / (alphay * t_0));
	} else {
		tmp = log1pf(-u0) * (-(alphay * alphay) / sin2phi);
	}
	return tmp;
}

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	t_0 = Float32(alphax * Float32(alphax / cos2phi))
	tmp = Float32(0.0)
	if (Float32(sin2phi / Float32(alphay * alphay)) <= Float32(1.9999999920083944e-11))
		tmp = Float32(u0 / Float32(Float32(alphay + Float32(Float32(sin2phi / alphay) * t_0)) / Float32(alphay * t_0)));
	else
		tmp = Float32(log1p(Float32(-u0)) * Float32(Float32(-Float32(alphay * alphay)) / sin2phi));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 sin2phi (*.f32 alphay alphay)) < 1.99999999e-11

   1. Initial program 59.0%

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

      1. neg-sub0 59.0%

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

      2. div-sub 59.0%

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

      3. --rgt-identity 59.0%

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

      4. div-sub 59.0%

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

      5. --rgt-identity 59.0%

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

      6. neg-sub0 59.0%

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

      7. sub-neg 59.0%

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

      8. log1p-def 98.8%

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

   3. Simplified 98.8%

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

   4. Taylor expanded in u0 around 0 73.5%

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

      1. unpow2 73.5%

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

      2. unpow2 73.5%

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

   6. Simplified 73.5%

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

      1. associate-/r* 73.5%

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

      2. clear-num 73.5%

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

      3. frac-add 73.5%

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

      4. *-un-lft-identity 73.5%

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

      5. associate-/l* 73.4%

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

      6. div-inv 73.4%

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

      7. clear-num 73.5%

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

      8. associate-/l* 73.4%

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

      9. div-inv 73.5%

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

      10. clear-num 73.5%

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

   8. Applied egg-rr 73.5%

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

   if 1.99999999e-11 < (/.f32 sin2phi (*.f32 alphay alphay))

   1. Initial program 62.5%

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

      1. neg-sub0 62.5%

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

      2. div-sub 62.5%

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

      3. --rgt-identity 62.5%

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

      4. div-sub 62.5%

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

      5. --rgt-identity 62.5%

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

      6. sub-neg 62.5%

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

      7. +-commutative 62.5%

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

      8. neg-sub0 62.5%

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

      9. associate-+l- 62.5%

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

      10. sub0-neg 62.5%

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

      11. neg-mul-1 62.5%

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

      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 98.6%

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

      1. associate-/r* 98.6%

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

      2. div-inv 98.3%

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

   5. Applied egg-rr 98.3%

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

      1. un-div-inv 98.6%

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

   7. Applied egg-rr 98.6%

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

      1. div-inv 98.6%

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

      2. associate-/r* 98.6%

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

      3. +-commutative 98.6%

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

      4. associate-/l/ 98.6%

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

   9. Applied egg-rr 98.6%

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

   10. Taylor expanded in sin2phi around inf 96.4%

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

       1. unpow2 96.4%

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

   12. Simplified 96.4%

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

4. Final simplification 89.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 1.9999999920083944 \cdot 10^{-11}:\\ \;\;\;\;\frac{u0}{\frac{alphay + \frac{sin2phi}{alphay} \cdot \left(alphax \cdot \frac{alphax}{cos2phi}\right)}{alphay \cdot \left(alphax \cdot \frac{alphax}{cos2phi}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(-u0\right) \cdot \frac{-alphay \cdot alphay}{sin2phi}\\ \end{array} \]

Alternative 4: 98.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 61.4%

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

   1. neg-sub0 61.4%

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

   2. div-sub 61.4%

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

   3. --rgt-identity 61.4%

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

   4. div-sub 61.4%

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

   5. --rgt-identity 61.4%

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

   6. neg-sub0 61.4%

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

   7. sub-neg 61.4%

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

   8. log1p-def 98.6%

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

3. Simplified 98.6%

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

4. Final simplification 98.6%

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

Alternative 5: 98.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 61.4%

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

   1. neg-sub0 61.4%

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

   2. div-sub 61.4%

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

   3. --rgt-identity 61.4%

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

   4. div-sub 61.4%

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

   5. --rgt-identity 61.4%

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

   6. sub-neg 61.4%

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

   7. +-commutative 61.4%

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

   8. neg-sub0 61.4%

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

   9. associate-+l- 61.4%

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

   10. sub0-neg 61.4%

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

   11. neg-mul-1 61.4%

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

   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. Final simplification 98.7%

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

Alternative 6: 98.3% accurate, 1.0× speedup

Math

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

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(Float32(cos2phi / alphax) / alphax) + Float32(Float32(sin2phi / alphay) / alphay)))
end

Derivation

1. Initial program 61.4%

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

   1. neg-sub0 61.4%

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

   2. div-sub 61.4%

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

   3. --rgt-identity 61.4%

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

   4. div-sub 61.4%

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

   5. --rgt-identity 61.4%

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

   6. sub-neg 61.4%

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

   7. +-commutative 61.4%

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

   8. neg-sub0 61.4%

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

   9. associate-+l- 61.4%

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

   10. sub0-neg 61.4%

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

   11. neg-mul-1 61.4%

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

   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* 98.7%

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

   2. div-inv 98.4%

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

5. Applied egg-rr 98.4%

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

   1. un-div-inv 98.7%

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

7. Applied egg-rr 98.7%

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

8. Final simplification 98.7%

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

Alternative 7: 81.3% accurate, 5.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := alphax \cdot \frac{alphax}{cos2phi}\\ \mathbf{if}\;sin2phi \leq 1.0000000116860974 \cdot 10^{-7}:\\ \;\;\;\;\frac{u0}{\frac{alphay + \frac{sin2phi}{alphay} \cdot t_0}{alphay \cdot t_0}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(alphay \cdot alphay\right) \cdot \left(u0 - \left(u0 \cdot u0\right) \cdot -0.5\right)}{sin2phi}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (let* ((t_0 (* alphax (/ alphax cos2phi))))
   (if (<= sin2phi 1.0000000116860974e-7)
     (/ u0 (/ (+ alphay (* (/ sin2phi alphay) t_0)) (* alphay t_0)))
     (/ (* (* alphay alphay) (- u0 (* (* u0 u0) -0.5))) sin2phi))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float t_0 = alphax * (alphax / cos2phi);
	float tmp;
	if (sin2phi <= 1.0000000116860974e-7f) {
		tmp = u0 / ((alphay + ((sin2phi / alphay) * t_0)) / (alphay * t_0));
	} else {
		tmp = ((alphay * alphay) * (u0 - ((u0 * u0) * -0.5f))) / 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 = alphax * (alphax / cos2phi)
    if (sin2phi <= 1.0000000116860974e-7) then
        tmp = u0 / ((alphay + ((sin2phi / alphay) * t_0)) / (alphay * t_0))
    else
        tmp = ((alphay * alphay) * (u0 - ((u0 * u0) * (-0.5e0)))) / sin2phi
    end if
    code = tmp
end function

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	t_0 = Float32(alphax * Float32(alphax / cos2phi))
	tmp = Float32(0.0)
	if (sin2phi <= Float32(1.0000000116860974e-7))
		tmp = Float32(u0 / Float32(Float32(alphay + Float32(Float32(sin2phi / alphay) * t_0)) / Float32(alphay * t_0)));
	else
		tmp = Float32(Float32(Float32(alphay * alphay) * Float32(u0 - Float32(Float32(u0 * u0) * Float32(-0.5)))) / sin2phi);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alphax, alphay, u0, cos2phi, sin2phi)
	t_0 = alphax * (alphax / cos2phi);
	tmp = single(0.0);
	if (sin2phi <= single(1.0000000116860974e-7))
		tmp = u0 / ((alphay + ((sin2phi / alphay) * t_0)) / (alphay * t_0));
	else
		tmp = ((alphay * alphay) * (u0 - ((u0 * u0) * single(-0.5)))) / sin2phi;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if sin2phi < 1.00000001e-7

   1. Initial program 56.5%

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

      1. neg-sub0 56.5%

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

      2. div-sub 56.5%

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

      3. --rgt-identity 56.5%

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

      4. div-sub 56.5%

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

      5. --rgt-identity 56.5%

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

      6. neg-sub0 56.5%

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

      7. sub-neg 56.5%

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

      8. log1p-def 98.7%

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

   3. Simplified 98.7%

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

   4. Taylor expanded in u0 around 0 75.1%

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

      1. unpow2 75.1%

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

      2. unpow2 75.1%

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

   6. Simplified 75.1%

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

      1. associate-/r* 75.1%

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

      2. clear-num 75.1%

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

      3. frac-add 75.1%

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

      4. *-un-lft-identity 75.1%

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

      5. associate-/l* 75.1%

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

      6. div-inv 75.1%

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

      7. clear-num 75.1%

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

      8. associate-/l* 75.1%

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

      9. div-inv 75.1%

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

      10. clear-num 75.2%

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

   8. Applied egg-rr 75.2%

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

   if 1.00000001e-7 < sin2phi

   1. Initial program 64.2%

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

      1. neg-sub0 64.2%

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

      2. div-sub 64.2%

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

      3. --rgt-identity 64.2%

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

      4. div-sub 64.2%

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

      5. --rgt-identity 64.2%

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

      6. neg-sub0 64.2%

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

      7. sub-neg 64.2%

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

      8. log1p-def 98.6%

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

   3. Simplified 98.6%

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

   4. Taylor expanded in cos2phi around 0 64.2%

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

      1. mul-1-neg 64.2%

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

      2. unpow2 64.2%

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

      3. *-commutative 64.2%

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

   6. Simplified 64.2%

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

   7. Taylor expanded in u0 around 0 87.7%

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

      1. +-commutative 87.7%

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

      2. neg-mul-1 87.7%

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

      3. unsub-neg 87.7%

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

      4. *-commutative 87.7%

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

      5. unpow2 87.7%

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

   9. Simplified 87.7%

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

4. Final simplification 83.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;sin2phi \leq 1.0000000116860974 \cdot 10^{-7}:\\ \;\;\;\;\frac{u0}{\frac{alphay + \frac{sin2phi}{alphay} \cdot \left(alphax \cdot \frac{alphax}{cos2phi}\right)}{alphay \cdot \left(alphax \cdot \frac{alphax}{cos2phi}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(alphay \cdot alphay\right) \cdot \left(u0 - \left(u0 \cdot u0\right) \cdot -0.5\right)}{sin2phi}\\ \end{array} \]

Alternative 8: 81.3% accurate, 5.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;sin2phi \leq 1.0000000116860974 \cdot 10^{-7}:\\ \;\;\;\;\frac{u0}{\frac{\frac{sin2phi}{\frac{alphay}{alphax \cdot alphax}} + cos2phi \cdot alphay}{alphay \cdot \left(alphax \cdot alphax\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(alphay \cdot alphay\right) \cdot \left(u0 - \left(u0 \cdot u0\right) \cdot -0.5\right)}{sin2phi}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (if (<= sin2phi 1.0000000116860974e-7)
   (/
    u0
    (/
     (+ (/ sin2phi (/ alphay (* alphax alphax))) (* cos2phi alphay))
     (* alphay (* alphax alphax))))
   (/ (* (* alphay alphay) (- u0 (* (* u0 u0) -0.5))) sin2phi)))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float tmp;
	if (sin2phi <= 1.0000000116860974e-7f) {
		tmp = u0 / (((sin2phi / (alphay / (alphax * alphax))) + (cos2phi * alphay)) / (alphay * (alphax * alphax)));
	} else {
		tmp = ((alphay * alphay) * (u0 - ((u0 * u0) * -0.5f))) / 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 <= 1.0000000116860974e-7) then
        tmp = u0 / (((sin2phi / (alphay / (alphax * alphax))) + (cos2phi * alphay)) / (alphay * (alphax * alphax)))
    else
        tmp = ((alphay * alphay) * (u0 - ((u0 * u0) * (-0.5e0)))) / sin2phi
    end if
    code = tmp
end function

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = Float32(0.0)
	if (sin2phi <= Float32(1.0000000116860974e-7))
		tmp = Float32(u0 / Float32(Float32(Float32(sin2phi / Float32(alphay / Float32(alphax * alphax))) + Float32(cos2phi * alphay)) / Float32(alphay * Float32(alphax * alphax))));
	else
		tmp = Float32(Float32(Float32(alphay * alphay) * Float32(u0 - Float32(Float32(u0 * u0) * Float32(-0.5)))) / sin2phi);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = single(0.0);
	if (sin2phi <= single(1.0000000116860974e-7))
		tmp = u0 / (((sin2phi / (alphay / (alphax * alphax))) + (cos2phi * alphay)) / (alphay * (alphax * alphax)));
	else
		tmp = ((alphay * alphay) * (u0 - ((u0 * u0) * single(-0.5)))) / sin2phi;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if sin2phi < 1.00000001e-7

   1. Initial program 56.5%

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

      1. neg-sub0 56.5%

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

      2. div-sub 56.5%

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

      3. --rgt-identity 56.5%

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

      4. div-sub 56.5%

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

      5. --rgt-identity 56.5%

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

      6. neg-sub0 56.5%

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

      7. sub-neg 56.5%

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

      8. log1p-def 98.7%

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

   3. Simplified 98.7%

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

      1. associate-/r* 98.8%

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

      2. frac-add 98.2%

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

      3. fma-def 98.3%

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

   5. Applied egg-rr 98.3%

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

      1. associate-*l* 98.0%

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

      2. associate-*r/ 98.1%

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

      3. associate-/l* 98.1%

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

      4. associate-*l* 98.2%

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

   7. Simplified 98.2%

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

   8. Taylor expanded in u0 around 0 75.0%

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

      1. associate-/l* 75.1%

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

      2. associate-/l* 75.1%

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

      3. unpow2 75.1%

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

      4. *-commutative 75.1%

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

      5. unpow2 75.1%

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

   10. Simplified 75.1%

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

   if 1.00000001e-7 < sin2phi

   1. Initial program 64.2%

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

      1. neg-sub0 64.2%

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

      2. div-sub 64.2%

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

      3. --rgt-identity 64.2%

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

      4. div-sub 64.2%

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

      5. --rgt-identity 64.2%

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

      6. neg-sub0 64.2%

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

      7. sub-neg 64.2%

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

      8. log1p-def 98.6%

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

   3. Simplified 98.6%

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

   4. Taylor expanded in cos2phi around 0 64.2%

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

      1. mul-1-neg 64.2%

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

      2. unpow2 64.2%

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

      3. *-commutative 64.2%

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

   6. Simplified 64.2%

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

   7. Taylor expanded in u0 around 0 87.7%

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

      1. +-commutative 87.7%

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

      2. neg-mul-1 87.7%

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

      3. unsub-neg 87.7%

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

      4. *-commutative 87.7%

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

      5. unpow2 87.7%

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

   9. Simplified 87.7%

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

4. Final simplification 83.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;sin2phi \leq 1.0000000116860974 \cdot 10^{-7}:\\ \;\;\;\;\frac{u0}{\frac{\frac{sin2phi}{\frac{alphay}{alphax \cdot alphax}} + cos2phi \cdot alphay}{alphay \cdot \left(alphax \cdot alphax\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(alphay \cdot alphay\right) \cdot \left(u0 - \left(u0 \cdot u0\right) \cdot -0.5\right)}{sin2phi}\\ \end{array} \]

Alternative 9: 81.4% accurate, 7.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;sin2phi \leq 1.0000000116860974 \cdot 10^{-7}:\\ \;\;\;\;\frac{u0}{\frac{sin2phi}{alphay \cdot alphay} - \frac{\frac{cos2phi}{alphax}}{-alphax}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(alphay \cdot alphay\right) \cdot \left(u0 - \left(u0 \cdot u0\right) \cdot -0.5\right)}{sin2phi}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (if (<= sin2phi 1.0000000116860974e-7)
   (/ u0 (- (/ sin2phi (* alphay alphay)) (/ (/ cos2phi alphax) (- alphax))))
   (/ (* (* alphay alphay) (- u0 (* (* u0 u0) -0.5))) sin2phi)))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float tmp;
	if (sin2phi <= 1.0000000116860974e-7f) {
		tmp = u0 / ((sin2phi / (alphay * alphay)) - ((cos2phi / alphax) / -alphax));
	} else {
		tmp = ((alphay * alphay) * (u0 - ((u0 * u0) * -0.5f))) / 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 <= 1.0000000116860974e-7) then
        tmp = u0 / ((sin2phi / (alphay * alphay)) - ((cos2phi / alphax) / -alphax))
    else
        tmp = ((alphay * alphay) * (u0 - ((u0 * u0) * (-0.5e0)))) / sin2phi
    end if
    code = tmp
end function

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = Float32(0.0)
	if (sin2phi <= Float32(1.0000000116860974e-7))
		tmp = Float32(u0 / Float32(Float32(sin2phi / Float32(alphay * alphay)) - Float32(Float32(cos2phi / alphax) / Float32(-alphax))));
	else
		tmp = Float32(Float32(Float32(alphay * alphay) * Float32(u0 - Float32(Float32(u0 * u0) * Float32(-0.5)))) / sin2phi);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = single(0.0);
	if (sin2phi <= single(1.0000000116860974e-7))
		tmp = u0 / ((sin2phi / (alphay * alphay)) - ((cos2phi / alphax) / -alphax));
	else
		tmp = ((alphay * alphay) * (u0 - ((u0 * u0) * single(-0.5)))) / sin2phi;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if sin2phi < 1.00000001e-7

   1. Initial program 56.5%

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

      1. neg-sub0 56.5%

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

      2. div-sub 56.5%

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

      3. --rgt-identity 56.5%

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

      4. div-sub 56.5%

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

      5. --rgt-identity 56.5%

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

      6. neg-sub0 56.5%

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

      7. sub-neg 56.5%

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

      8. log1p-def 98.7%

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

   3. Simplified 98.7%

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

   4. Taylor expanded in u0 around 0 75.1%

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

      1. unpow2 75.1%

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

      2. unpow2 75.1%

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

   6. Simplified 75.1%

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

      1. associate-/r* 75.1%

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

      2. frac-2neg 75.1%

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

      3. distribute-frac-neg 75.1%

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

   8. Applied egg-rr 75.1%

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

   if 1.00000001e-7 < sin2phi

   1. Initial program 64.2%

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

      1. neg-sub0 64.2%

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

      2. div-sub 64.2%

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

      3. --rgt-identity 64.2%

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

      4. div-sub 64.2%

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

      5. --rgt-identity 64.2%

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

      6. neg-sub0 64.2%

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

      7. sub-neg 64.2%

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

      8. log1p-def 98.6%

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

   3. Simplified 98.6%

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

   4. Taylor expanded in cos2phi around 0 64.2%

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

      1. mul-1-neg 64.2%

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

      2. unpow2 64.2%

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

      3. *-commutative 64.2%

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

   6. Simplified 64.2%

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

   7. Taylor expanded in u0 around 0 87.7%

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

      1. +-commutative 87.7%

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

      2. neg-mul-1 87.7%

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

      3. unsub-neg 87.7%

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

      4. *-commutative 87.7%

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

      5. unpow2 87.7%

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

   9. Simplified 87.7%

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

4. Final simplification 83.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;sin2phi \leq 1.0000000116860974 \cdot 10^{-7}:\\ \;\;\;\;\frac{u0}{\frac{sin2phi}{alphay \cdot alphay} - \frac{\frac{cos2phi}{alphax}}{-alphax}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(alphay \cdot alphay\right) \cdot \left(u0 - \left(u0 \cdot u0\right) \cdot -0.5\right)}{sin2phi}\\ \end{array} \]

Alternative 10: 81.4% accurate, 7.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;sin2phi \leq 1.0000000116860974 \cdot 10^{-7}:\\ \;\;\;\;\frac{u0}{\frac{sin2phi}{alphay \cdot alphay} + \frac{cos2phi}{alphax \cdot alphax}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(alphay \cdot alphay\right) \cdot \left(u0 - \left(u0 \cdot u0\right) \cdot -0.5\right)}{sin2phi}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (if (<= sin2phi 1.0000000116860974e-7)
   (/ u0 (+ (/ sin2phi (* alphay alphay)) (/ cos2phi (* alphax alphax))))
   (/ (* (* alphay alphay) (- u0 (* (* u0 u0) -0.5))) sin2phi)))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float tmp;
	if (sin2phi <= 1.0000000116860974e-7f) {
		tmp = u0 / ((sin2phi / (alphay * alphay)) + (cos2phi / (alphax * alphax)));
	} else {
		tmp = ((alphay * alphay) * (u0 - ((u0 * u0) * -0.5f))) / 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 <= 1.0000000116860974e-7) then
        tmp = u0 / ((sin2phi / (alphay * alphay)) + (cos2phi / (alphax * alphax)))
    else
        tmp = ((alphay * alphay) * (u0 - ((u0 * u0) * (-0.5e0)))) / sin2phi
    end if
    code = tmp
end function

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = Float32(0.0)
	if (sin2phi <= Float32(1.0000000116860974e-7))
		tmp = Float32(u0 / Float32(Float32(sin2phi / Float32(alphay * alphay)) + Float32(cos2phi / Float32(alphax * alphax))));
	else
		tmp = Float32(Float32(Float32(alphay * alphay) * Float32(u0 - Float32(Float32(u0 * u0) * Float32(-0.5)))) / sin2phi);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = single(0.0);
	if (sin2phi <= single(1.0000000116860974e-7))
		tmp = u0 / ((sin2phi / (alphay * alphay)) + (cos2phi / (alphax * alphax)));
	else
		tmp = ((alphay * alphay) * (u0 - ((u0 * u0) * single(-0.5)))) / sin2phi;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if sin2phi < 1.00000001e-7

   1. Initial program 56.5%

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

      1. neg-sub0 56.5%

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

      2. div-sub 56.5%

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

      3. --rgt-identity 56.5%

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

      4. div-sub 56.5%

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

      5. --rgt-identity 56.5%

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

      6. neg-sub0 56.5%

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

      7. sub-neg 56.5%

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

      8. log1p-def 98.7%

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

   3. Simplified 98.7%

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

   4. Taylor expanded in u0 around 0 75.1%

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

      1. unpow2 75.1%

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

      2. unpow2 75.1%

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

   6. Simplified 75.1%

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

   if 1.00000001e-7 < sin2phi

   1. Initial program 64.2%

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

      1. neg-sub0 64.2%

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

      2. div-sub 64.2%

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

      3. --rgt-identity 64.2%

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

      4. div-sub 64.2%

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

      5. --rgt-identity 64.2%

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

      6. neg-sub0 64.2%

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

      7. sub-neg 64.2%

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

      8. log1p-def 98.6%

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

   3. Simplified 98.6%

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

   4. Taylor expanded in cos2phi around 0 64.2%

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

      1. mul-1-neg 64.2%

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

      2. unpow2 64.2%

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

      3. *-commutative 64.2%

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

   6. Simplified 64.2%

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

   7. Taylor expanded in u0 around 0 87.7%

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

      1. +-commutative 87.7%

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

      2. neg-mul-1 87.7%

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

      3. unsub-neg 87.7%

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

      4. *-commutative 87.7%

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

      5. unpow2 87.7%

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

   9. Simplified 87.7%

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

4. Final simplification 83.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;sin2phi \leq 1.0000000116860974 \cdot 10^{-7}:\\ \;\;\;\;\frac{u0}{\frac{sin2phi}{alphay \cdot alphay} + \frac{cos2phi}{alphax \cdot alphax}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(alphay \cdot alphay\right) \cdot \left(u0 - \left(u0 \cdot u0\right) \cdot -0.5\right)}{sin2phi}\\ \end{array} \]

Alternative 11: 75.6% accurate, 8.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 61.4%

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

   1. neg-sub0 61.4%

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

   2. div-sub 61.4%

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

   3. --rgt-identity 61.4%

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

   4. div-sub 61.4%

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

   5. --rgt-identity 61.4%

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

   6. neg-sub0 61.4%

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

   7. sub-neg 61.4%

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

   8. log1p-def 98.6%

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

3. Simplified 98.6%

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

4. Taylor expanded in u0 around 0 76.4%

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

   1. unpow2 76.4%

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

   2. unpow2 76.4%

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

6. Simplified 76.4%

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

7. Final simplification 76.4%

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

Alternative 12: 67.3% accurate, 12.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;sin2phi \leq 5.000000229068525 \cdot 10^{-19}:\\ \;\;\;\;\left(alphax \cdot alphax\right) \cdot \frac{u0}{cos2phi}\\ \mathbf{else}:\\ \;\;\;\;\left(alphay \cdot alphay\right) \cdot \frac{u0}{sin2phi}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (if (<= sin2phi 5.000000229068525e-19)
   (* (* alphax alphax) (/ u0 cos2phi))
   (* (* alphay alphay) (/ u0 sin2phi))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float tmp;
	if (sin2phi <= 5.000000229068525e-19f) {
		tmp = (alphax * alphax) * (u0 / cos2phi);
	} else {
		tmp = (alphay * alphay) * (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 <= 5.000000229068525e-19) then
        tmp = (alphax * alphax) * (u0 / cos2phi)
    else
        tmp = (alphay * alphay) * (u0 / sin2phi)
    end if
    code = tmp
end function

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = Float32(0.0)
	if (sin2phi <= Float32(5.000000229068525e-19))
		tmp = Float32(Float32(alphax * alphax) * Float32(u0 / cos2phi));
	else
		tmp = Float32(Float32(alphay * alphay) * Float32(u0 / sin2phi));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = single(0.0);
	if (sin2phi <= single(5.000000229068525e-19))
		tmp = (alphax * alphax) * (u0 / cos2phi);
	else
		tmp = (alphay * alphay) * (u0 / sin2phi);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if sin2phi < 5.00000023e-19

   1. Initial program 57.8%

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

      1. neg-sub0 57.8%

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

      2. div-sub 57.8%

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

      3. --rgt-identity 57.8%

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

      4. div-sub 57.8%

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

      5. --rgt-identity 57.8%

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

      6. neg-sub0 57.8%

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

      7. sub-neg 57.8%

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

      8. log1p-def 98.7%

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

   3. Simplified 98.7%

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

   4. Taylor expanded in u0 around 0 74.7%

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

      1. unpow2 74.7%

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

      2. unpow2 74.7%

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

   6. Simplified 74.7%

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

   7. Taylor expanded in cos2phi around inf 59.6%

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

      1. associate-/l* 59.5%

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

      2. associate-/r/ 59.5%

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

      3. unpow2 59.5%

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

   9. Simplified 59.5%

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

   if 5.00000023e-19 < sin2phi

   1. Initial program 62.7%

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

      1. neg-sub0 62.7%

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

      2. div-sub 62.7%

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

      3. --rgt-identity 62.7%

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

      4. div-sub 62.7%

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

      5. --rgt-identity 62.7%

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

      6. neg-sub0 62.7%

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

      7. sub-neg 62.7%

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

      8. log1p-def 98.6%

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

   3. Simplified 98.6%

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

   4. Taylor expanded in u0 around 0 77.0%

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

      1. unpow2 77.0%

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

      2. unpow2 77.0%

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

   6. Simplified 77.0%

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

   7. Taylor expanded in cos2phi around 0 74.4%

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

      1. unpow2 74.4%

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

      2. *-commutative 74.4%

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

      3. *-lft-identity 74.4%

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

      4. times-frac 74.4%

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

      5. /-rgt-identity 74.4%

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

   9. Simplified 74.4%

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

4. Final simplification 70.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;sin2phi \leq 5.000000229068525 \cdot 10^{-19}:\\ \;\;\;\;\left(alphax \cdot alphax\right) \cdot \frac{u0}{cos2phi}\\ \mathbf{else}:\\ \;\;\;\;\left(alphay \cdot alphay\right) \cdot \frac{u0}{sin2phi}\\ \end{array} \]

Alternative 13: 67.3% accurate, 12.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;sin2phi \leq 5.000000229068525 \cdot 10^{-19}:\\ \;\;\;\;\frac{u0}{\frac{cos2phi}{alphax \cdot alphax}}\\ \mathbf{else}:\\ \;\;\;\;\left(alphay \cdot alphay\right) \cdot \frac{u0}{sin2phi}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (if (<= sin2phi 5.000000229068525e-19)
   (/ u0 (/ cos2phi (* alphax alphax)))
   (* (* alphay alphay) (/ u0 sin2phi))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float tmp;
	if (sin2phi <= 5.000000229068525e-19f) {
		tmp = u0 / (cos2phi / (alphax * alphax));
	} else {
		tmp = (alphay * alphay) * (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 <= 5.000000229068525e-19) then
        tmp = u0 / (cos2phi / (alphax * alphax))
    else
        tmp = (alphay * alphay) * (u0 / sin2phi)
    end if
    code = tmp
end function

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = Float32(0.0)
	if (sin2phi <= Float32(5.000000229068525e-19))
		tmp = Float32(u0 / Float32(cos2phi / Float32(alphax * alphax)));
	else
		tmp = Float32(Float32(alphay * alphay) * Float32(u0 / sin2phi));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = single(0.0);
	if (sin2phi <= single(5.000000229068525e-19))
		tmp = u0 / (cos2phi / (alphax * alphax));
	else
		tmp = (alphay * alphay) * (u0 / sin2phi);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if sin2phi < 5.00000023e-19

   1. Initial program 57.8%

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

      1. neg-sub0 57.8%

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

      2. div-sub 57.8%

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

      3. --rgt-identity 57.8%

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

      4. div-sub 57.8%

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

      5. --rgt-identity 57.8%

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

      6. neg-sub0 57.8%

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

      7. sub-neg 57.8%

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

      8. log1p-def 98.7%

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

   3. Simplified 98.7%

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

   4. Taylor expanded in u0 around 0 74.7%

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

      1. unpow2 74.7%

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

      2. unpow2 74.7%

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

   6. Simplified 74.7%

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

      1. frac-2neg 74.7%

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

      2. div-inv 74.7%

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

      3. distribute-rgt-neg-in 74.7%

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

   8. Applied egg-rr 74.7%

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

   9. Taylor expanded in cos2phi around inf 59.6%

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

       1. associate-/l* 59.5%

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

       2. unpow2 59.5%

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

   11. Simplified 59.5%

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

   if 5.00000023e-19 < sin2phi

   1. Initial program 62.7%

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

      1. neg-sub0 62.7%

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

      2. div-sub 62.7%

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

      3. --rgt-identity 62.7%

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

      4. div-sub 62.7%

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

      5. --rgt-identity 62.7%

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

      6. neg-sub0 62.7%

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

      7. sub-neg 62.7%

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

      8. log1p-def 98.6%

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

   3. Simplified 98.6%

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

   4. Taylor expanded in u0 around 0 77.0%

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

      1. unpow2 77.0%

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

      2. unpow2 77.0%

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

   6. Simplified 77.0%

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

   7. Taylor expanded in cos2phi around 0 74.4%

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

      1. unpow2 74.4%

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

      2. *-commutative 74.4%

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

      3. *-lft-identity 74.4%

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

      4. times-frac 74.4%

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

      5. /-rgt-identity 74.4%

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

   9. Simplified 74.4%

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

4. Final simplification 70.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;sin2phi \leq 5.000000229068525 \cdot 10^{-19}:\\ \;\;\;\;\frac{u0}{\frac{cos2phi}{alphax \cdot alphax}}\\ \mathbf{else}:\\ \;\;\;\;\left(alphay \cdot alphay\right) \cdot \frac{u0}{sin2phi}\\ \end{array} \]

Alternative 14: 67.3% accurate, 12.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;sin2phi \leq 5.000000229068525 \cdot 10^{-19}:\\ \;\;\;\;\frac{u0 \cdot \left(alphax \cdot alphax\right)}{cos2phi}\\ \mathbf{else}:\\ \;\;\;\;\left(alphay \cdot alphay\right) \cdot \frac{u0}{sin2phi}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (if (<= sin2phi 5.000000229068525e-19)
   (/ (* u0 (* alphax alphax)) cos2phi)
   (* (* alphay alphay) (/ u0 sin2phi))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float tmp;
	if (sin2phi <= 5.000000229068525e-19f) {
		tmp = (u0 * (alphax * alphax)) / cos2phi;
	} else {
		tmp = (alphay * alphay) * (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 <= 5.000000229068525e-19) then
        tmp = (u0 * (alphax * alphax)) / cos2phi
    else
        tmp = (alphay * alphay) * (u0 / sin2phi)
    end if
    code = tmp
end function

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = Float32(0.0)
	if (sin2phi <= Float32(5.000000229068525e-19))
		tmp = Float32(Float32(u0 * Float32(alphax * alphax)) / cos2phi);
	else
		tmp = Float32(Float32(alphay * alphay) * Float32(u0 / sin2phi));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = single(0.0);
	if (sin2phi <= single(5.000000229068525e-19))
		tmp = (u0 * (alphax * alphax)) / cos2phi;
	else
		tmp = (alphay * alphay) * (u0 / sin2phi);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if sin2phi < 5.00000023e-19

   1. Initial program 57.8%

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

      1. neg-sub0 57.8%

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

      2. div-sub 57.8%

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

      3. --rgt-identity 57.8%

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

      4. div-sub 57.8%

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

      5. --rgt-identity 57.8%

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

      6. neg-sub0 57.8%

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

      7. sub-neg 57.8%

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

      8. log1p-def 98.7%

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

   3. Simplified 98.7%

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

   4. Taylor expanded in u0 around 0 74.7%

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

      1. unpow2 74.7%

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

      2. unpow2 74.7%

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

   6. Simplified 74.7%

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

   7. Taylor expanded in cos2phi around inf 59.6%

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

      1. unpow2 59.6%

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

   9. Simplified 59.6%

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

   if 5.00000023e-19 < sin2phi

   1. Initial program 62.7%

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

      1. neg-sub0 62.7%

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

      2. div-sub 62.7%

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

      3. --rgt-identity 62.7%

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

      4. div-sub 62.7%

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

      5. --rgt-identity 62.7%

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

      6. neg-sub0 62.7%

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

      7. sub-neg 62.7%

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

      8. log1p-def 98.6%

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

   3. Simplified 98.6%

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

   4. Taylor expanded in u0 around 0 77.0%

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

      1. unpow2 77.0%

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

      2. unpow2 77.0%

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

   6. Simplified 77.0%

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

   7. Taylor expanded in cos2phi around 0 74.4%

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

      1. unpow2 74.4%

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

      2. *-commutative 74.4%

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

      3. *-lft-identity 74.4%

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

      4. times-frac 74.4%

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

      5. /-rgt-identity 74.4%

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

   9. Simplified 74.4%

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

4. Final simplification 70.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;sin2phi \leq 5.000000229068525 \cdot 10^{-19}:\\ \;\;\;\;\frac{u0 \cdot \left(alphax \cdot alphax\right)}{cos2phi}\\ \mathbf{else}:\\ \;\;\;\;\left(alphay \cdot alphay\right) \cdot \frac{u0}{sin2phi}\\ \end{array} \]

Alternative 15: 67.3% accurate, 12.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;sin2phi \leq 5.000000229068525 \cdot 10^{-19}:\\ \;\;\;\;\frac{u0 \cdot \left(alphax \cdot alphax\right)}{cos2phi}\\ \mathbf{else}:\\ \;\;\;\;\frac{u0 \cdot \left(alphay \cdot alphay\right)}{sin2phi}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (if (<= sin2phi 5.000000229068525e-19)
   (/ (* u0 (* alphax alphax)) cos2phi)
   (/ (* u0 (* alphay alphay)) sin2phi)))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float tmp;
	if (sin2phi <= 5.000000229068525e-19f) {
		tmp = (u0 * (alphax * alphax)) / cos2phi;
	} else {
		tmp = (u0 * (alphay * 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 <= 5.000000229068525e-19) then
        tmp = (u0 * (alphax * alphax)) / cos2phi
    else
        tmp = (u0 * (alphay * alphay)) / sin2phi
    end if
    code = tmp
end function

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = Float32(0.0)
	if (sin2phi <= Float32(5.000000229068525e-19))
		tmp = Float32(Float32(u0 * Float32(alphax * alphax)) / cos2phi);
	else
		tmp = Float32(Float32(u0 * Float32(alphay * alphay)) / sin2phi);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = single(0.0);
	if (sin2phi <= single(5.000000229068525e-19))
		tmp = (u0 * (alphax * alphax)) / cos2phi;
	else
		tmp = (u0 * (alphay * alphay)) / sin2phi;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if sin2phi < 5.00000023e-19

   1. Initial program 57.8%

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

      1. neg-sub0 57.8%

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

      2. div-sub 57.8%

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

      3. --rgt-identity 57.8%

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

      4. div-sub 57.8%

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

      5. --rgt-identity 57.8%

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

      6. neg-sub0 57.8%

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

      7. sub-neg 57.8%

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

      8. log1p-def 98.7%

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

   3. Simplified 98.7%

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

   4. Taylor expanded in u0 around 0 74.7%

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

      1. unpow2 74.7%

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

      2. unpow2 74.7%

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

   6. Simplified 74.7%

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

   7. Taylor expanded in cos2phi around inf 59.6%

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

      1. unpow2 59.6%

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

   9. Simplified 59.6%

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

   if 5.00000023e-19 < sin2phi

   1. Initial program 62.7%

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

      1. neg-sub0 62.7%

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

      2. div-sub 62.7%

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

      3. --rgt-identity 62.7%

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

      4. div-sub 62.7%

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

      5. --rgt-identity 62.7%

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

      6. neg-sub0 62.7%

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

      7. sub-neg 62.7%

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

      8. log1p-def 98.6%

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

   3. Simplified 98.6%

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

   4. Taylor expanded in u0 around 0 77.0%

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

      1. unpow2 77.0%

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

      2. unpow2 77.0%

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

   6. Simplified 77.0%

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

   7. Taylor expanded in cos2phi around 0 74.4%

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

      1. unpow2 74.4%

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

      2. *-commutative 74.4%

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

      3. *-lft-identity 74.4%

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

      4. times-frac 74.4%

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

      5. /-rgt-identity 74.4%

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

   9. Simplified 74.4%

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

       1. associate-*r/ 74.4%

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

   11. Applied egg-rr 74.4%

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

4. Final simplification 70.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;sin2phi \leq 5.000000229068525 \cdot 10^{-19}:\\ \;\;\;\;\frac{u0 \cdot \left(alphax \cdot alphax\right)}{cos2phi}\\ \mathbf{else}:\\ \;\;\;\;\frac{u0 \cdot \left(alphay \cdot alphay\right)}{sin2phi}\\ \end{array} \]

Alternative 16: 59.3% accurate, 16.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 61.4%

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

   1. neg-sub0 61.4%

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

   2. div-sub 61.4%

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

   3. --rgt-identity 61.4%

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

   4. div-sub 61.4%

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

   5. --rgt-identity 61.4%

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

   6. neg-sub0 61.4%

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

   7. sub-neg 61.4%

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

   8. log1p-def 98.6%

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

3. Simplified 98.6%

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

4. Taylor expanded in u0 around 0 76.4%

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

   1. unpow2 76.4%

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

   2. unpow2 76.4%

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

6. Simplified 76.4%

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

7. Taylor expanded in cos2phi around 0 61.0%

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

   1. unpow2 61.0%

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

   2. *-commutative 61.0%

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

   3. *-lft-identity 61.0%

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

   4. times-frac 61.0%

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

   5. /-rgt-identity 61.0%

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

9. Simplified 61.0%

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

10. Taylor expanded in alphay around 0 61.0%

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

    1. *-commutative 61.0%

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

    2. associate-*r/ 61.0%

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

    3. unpow2 61.0%

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

    4. associate-*l* 61.0%

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

12. Simplified 61.0%

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

13. Final simplification 61.0%

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

Alternative 17: 59.3% accurate, 16.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 61.4%

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

   1. neg-sub0 61.4%

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

   2. div-sub 61.4%

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

   3. --rgt-identity 61.4%

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

   4. div-sub 61.4%

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

   5. --rgt-identity 61.4%

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

   6. neg-sub0 61.4%

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

   7. sub-neg 61.4%

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

   8. log1p-def 98.6%

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

3. Simplified 98.6%

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

4. Taylor expanded in u0 around 0 76.4%

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

   1. unpow2 76.4%

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

   2. unpow2 76.4%

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

6. Simplified 76.4%

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

7. Taylor expanded in cos2phi around 0 61.0%

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

   1. unpow2 61.0%

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

   2. *-commutative 61.0%

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

   3. *-lft-identity 61.0%

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

   4. times-frac 61.0%

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

   5. /-rgt-identity 61.0%

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

9. Simplified 61.0%

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

10. Final simplification 61.0%

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

Reproduce

herbie shell --seed 2023230 
(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)))))