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

Percentage Accurate: 61.3% → 98.3%

Time: 22.9s

Alternatives: 19

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: 61.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Alternative 1: 98.3% accurate, 1.0× speedup

Math

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

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 60.0%

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

   1. sub-neg 60.0%

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

   2. 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. Step-by-step derivation

   1. *-un-lft-identity 98.8%

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

   2. *-commutative 98.8%

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

   3. associate-/r* 98.9%

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

5. Applied egg-rr 98.9%

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

6. Final simplification 98.9%

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

Alternative 2: 91.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 1.2000000424450263 \cdot 10^{-6}:\\ \;\;\;\;\frac{u0 - u0 \cdot \left(u0 \cdot -0.5\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay} \cdot \frac{1}{alphay}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(-u0\right) \cdot \frac{alphay}{\frac{sin2phi}{-alphay}}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (if (<= (/ sin2phi (* alphay alphay)) 1.2000000424450263e-6)
   (/
    (- u0 (* u0 (* u0 -0.5)))
    (+ (/ cos2phi (* alphax alphax)) (* (/ sin2phi alphay) (/ 1.0 alphay))))
   (* (log1p (- u0)) (/ alphay (/ sin2phi (- alphay))))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float tmp;
	if ((sin2phi / (alphay * alphay)) <= 1.2000000424450263e-6f) {
		tmp = (u0 - (u0 * (u0 * -0.5f))) / ((cos2phi / (alphax * alphax)) + ((sin2phi / alphay) * (1.0f / alphay)));
	} else {
		tmp = log1pf(-u0) * (alphay / (sin2phi / -alphay));
	}
	return tmp;
}

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = Float32(0.0)
	if (Float32(sin2phi / Float32(alphay * alphay)) <= Float32(1.2000000424450263e-6))
		tmp = Float32(Float32(u0 - Float32(u0 * Float32(u0 * Float32(-0.5)))) / Float32(Float32(cos2phi / Float32(alphax * alphax)) + Float32(Float32(sin2phi / alphay) * Float32(Float32(1.0) / alphay))));
	else
		tmp = Float32(log1p(Float32(-u0)) * Float32(alphay / Float32(sin2phi / Float32(-alphay))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 sin2phi (*.f32 alphay alphay)) < 1.2e-6

   1. Initial program 52.4%

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

      1. associate-/r* 77.1%

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

      2. div-inv 77.2%

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

   3. Applied egg-rr 52.4%

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

   4. Taylor expanded in u0 around 0 89.8%

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

      1. +-commutative 89.7%

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

      2. mul-1-neg 89.7%

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

      3. unsub-neg 89.7%

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

      4. *-commutative 89.7%

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

      5. unpow2 89.7%

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

      6. associate-*l* 89.7%

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

   6. Simplified 89.8%

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

   if 1.2e-6 < (/.f32 sin2phi (*.f32 alphay alphay))

   1. Initial program 64.7%

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

      1. sub-neg 64.7%

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

      2. 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. Step-by-step derivation

      1. frac-2neg 98.8%

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

      2. div-inv 98.7%

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

      3. distribute-rgt-neg-in 98.7%

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

   5. Applied egg-rr 98.7%

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

      1. un-div-inv 98.8%

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

      2. distribute-rgt-neg-out 98.8%

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

      3. frac-2neg 98.8%

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

      4. associate-/r* 98.7%

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

   7. Applied egg-rr 98.7%

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

   8. Taylor expanded in cos2phi around 0 64.6%

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

      1. mul-1-neg 64.6%

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

      2. distribute-neg-frac 64.6%

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

      3. distribute-lft-neg-out 64.6%

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

      4. unpow2 64.6%

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

      5. distribute-rgt-neg-out 64.6%

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

      6. sub-neg 64.6%

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

      7. log1p-def 97.9%

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

      8. associate-*l/ 97.9%

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

      9. *-commutative 97.9%

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

      10. associate-/l* 97.7%

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

   10. Simplified 97.7%

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

4. Final simplification 94.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 1.2000000424450263 \cdot 10^{-6}:\\ \;\;\;\;\frac{u0 - u0 \cdot \left(u0 \cdot -0.5\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay} \cdot \frac{1}{alphay}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(-u0\right) \cdot \frac{alphay}{\frac{sin2phi}{-alphay}}\\ \end{array} \]

Alternative 3: 91.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{sin2phi}{alphay \cdot alphay}\\ \mathbf{if}\;t_0 \leq 1.2000000424450263 \cdot 10^{-6}:\\ \;\;\;\;\frac{u0 - u0 \cdot \left(u0 \cdot -0.5\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay} \cdot \frac{1}{alphay}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\mathsf{log1p}\left(-u0\right)}{t_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (let* ((t_0 (/ sin2phi (* alphay alphay))))
   (if (<= t_0 1.2000000424450263e-6)
     (/
      (- u0 (* u0 (* u0 -0.5)))
      (+ (/ cos2phi (* alphax alphax)) (* (/ sin2phi alphay) (/ 1.0 alphay))))
     (/ (- (log1p (- u0))) t_0))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float t_0 = sin2phi / (alphay * alphay);
	float tmp;
	if (t_0 <= 1.2000000424450263e-6f) {
		tmp = (u0 - (u0 * (u0 * -0.5f))) / ((cos2phi / (alphax * alphax)) + ((sin2phi / alphay) * (1.0f / alphay)));
	} else {
		tmp = -log1pf(-u0) / t_0;
	}
	return tmp;
}

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	t_0 = Float32(sin2phi / Float32(alphay * alphay))
	tmp = Float32(0.0)
	if (t_0 <= Float32(1.2000000424450263e-6))
		tmp = Float32(Float32(u0 - Float32(u0 * Float32(u0 * Float32(-0.5)))) / Float32(Float32(cos2phi / Float32(alphax * alphax)) + Float32(Float32(sin2phi / alphay) * Float32(Float32(1.0) / alphay))));
	else
		tmp = Float32(Float32(-log1p(Float32(-u0))) / t_0);
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 sin2phi (*.f32 alphay alphay)) < 1.2e-6

   1. Initial program 52.4%

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

      1. associate-/r* 77.1%

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

      2. div-inv 77.2%

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

   3. Applied egg-rr 52.4%

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

   4. Taylor expanded in u0 around 0 89.8%

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

      1. +-commutative 89.7%

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

      2. mul-1-neg 89.7%

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

      3. unsub-neg 89.7%

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

      4. *-commutative 89.7%

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

      5. unpow2 89.7%

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

      6. associate-*l* 89.7%

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

   6. Simplified 89.8%

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

   if 1.2e-6 < (/.f32 sin2phi (*.f32 alphay alphay))

   1. Initial program 64.7%

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

      1. sub-neg 64.7%

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

      2. 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. Step-by-step derivation

      1. *-un-lft-identity 98.8%

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

      2. *-commutative 98.8%

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

      3. associate-/r* 98.8%

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

   5. Applied egg-rr 98.8%

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

   6. Taylor expanded in cos2phi around 0 97.7%

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

      1. unpow2 97.7%

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

   8. Simplified 97.7%

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

4. Final simplification 94.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 1.2000000424450263 \cdot 10^{-6}:\\ \;\;\;\;\frac{u0 - u0 \cdot \left(u0 \cdot -0.5\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay} \cdot \frac{1}{alphay}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\mathsf{log1p}\left(-u0\right)}{\frac{sin2phi}{alphay \cdot alphay}}\\ \end{array} \]

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 60.0%

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

   1. sub-neg 60.0%

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

   2. 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. Final simplification 98.8%

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

Alternative 5: 89.0% accurate, 3.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 0.00019999999494757503:\\ \;\;\;\;\frac{u0 - u0 \cdot \left(u0 \cdot -0.5\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay} \cdot \frac{1}{alphay}}\\ \mathbf{else}:\\ \;\;\;\;\frac{alphay \cdot \left(-alphay\right)}{\left(\left(sin2phi \cdot 0.5 - \frac{sin2phi}{u0}\right) - u0 \cdot \left(u0 \cdot \left(0.5 \cdot \left(sin2phi \cdot -0.08333333333333333\right)\right)\right)\right) - u0 \cdot \left(sin2phi \cdot -0.08333333333333333\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (if (<= (/ sin2phi (* alphay alphay)) 0.00019999999494757503)
   (/
    (- u0 (* u0 (* u0 -0.5)))
    (+ (/ cos2phi (* alphax alphax)) (* (/ sin2phi alphay) (/ 1.0 alphay))))
   (/
    (* alphay (- alphay))
    (-
     (-
      (- (* sin2phi 0.5) (/ sin2phi u0))
      (* u0 (* u0 (* 0.5 (* sin2phi -0.08333333333333333)))))
     (* u0 (* sin2phi -0.08333333333333333))))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float tmp;
	if ((sin2phi / (alphay * alphay)) <= 0.00019999999494757503f) {
		tmp = (u0 - (u0 * (u0 * -0.5f))) / ((cos2phi / (alphax * alphax)) + ((sin2phi / alphay) * (1.0f / alphay)));
	} else {
		tmp = (alphay * -alphay) / ((((sin2phi * 0.5f) - (sin2phi / u0)) - (u0 * (u0 * (0.5f * (sin2phi * -0.08333333333333333f))))) - (u0 * (sin2phi * -0.08333333333333333f)));
	}
	return tmp;
}

Fortran

real(4) function code(alphax, alphay, u0, cos2phi, sin2phi)
    real(4), intent (in) :: alphax
    real(4), intent (in) :: alphay
    real(4), intent (in) :: u0
    real(4), intent (in) :: cos2phi
    real(4), intent (in) :: sin2phi
    real(4) :: tmp
    if ((sin2phi / (alphay * alphay)) <= 0.00019999999494757503e0) then
        tmp = (u0 - (u0 * (u0 * (-0.5e0)))) / ((cos2phi / (alphax * alphax)) + ((sin2phi / alphay) * (1.0e0 / alphay)))
    else
        tmp = (alphay * -alphay) / ((((sin2phi * 0.5e0) - (sin2phi / u0)) - (u0 * (u0 * (0.5e0 * (sin2phi * (-0.08333333333333333e0)))))) - (u0 * (sin2phi * (-0.08333333333333333e0))))
    end if
    code = tmp
end function

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = Float32(0.0)
	if (Float32(sin2phi / Float32(alphay * alphay)) <= Float32(0.00019999999494757503))
		tmp = Float32(Float32(u0 - Float32(u0 * Float32(u0 * Float32(-0.5)))) / Float32(Float32(cos2phi / Float32(alphax * alphax)) + Float32(Float32(sin2phi / alphay) * Float32(Float32(1.0) / alphay))));
	else
		tmp = Float32(Float32(alphay * Float32(-alphay)) / Float32(Float32(Float32(Float32(sin2phi * Float32(0.5)) - Float32(sin2phi / u0)) - Float32(u0 * Float32(u0 * Float32(Float32(0.5) * Float32(sin2phi * Float32(-0.08333333333333333)))))) - Float32(u0 * Float32(sin2phi * Float32(-0.08333333333333333)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = single(0.0);
	if ((sin2phi / (alphay * alphay)) <= single(0.00019999999494757503))
		tmp = (u0 - (u0 * (u0 * single(-0.5)))) / ((cos2phi / (alphax * alphax)) + ((sin2phi / alphay) * (single(1.0) / alphay)));
	else
		tmp = (alphay * -alphay) / ((((sin2phi * single(0.5)) - (sin2phi / u0)) - (u0 * (u0 * (single(0.5) * (sin2phi * single(-0.08333333333333333)))))) - (u0 * (sin2phi * single(-0.08333333333333333))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 sin2phi (*.f32 alphay alphay)) < 1.99999995e-4

   1. Initial program 52.4%

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

      1. associate-/r* 77.0%

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

      2. div-inv 77.1%

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

   3. Applied egg-rr 52.4%

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

   4. Taylor expanded in u0 around 0 89.5%

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

      1. +-commutative 89.4%

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

      2. mul-1-neg 89.4%

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

      3. unsub-neg 89.4%

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

      4. *-commutative 89.4%

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

      5. unpow2 89.4%

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

      6. associate-*l* 89.4%

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

   6. Simplified 89.5%

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

   if 1.99999995e-4 < (/.f32 sin2phi (*.f32 alphay alphay))

   1. Initial program 65.0%

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

   2. Taylor expanded in cos2phi around 0 64.9%

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

      1. mul-1-neg 64.9%

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

      2. unpow2 64.9%

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

      3. associate-/l* 64.8%

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

      4. distribute-neg-frac 64.8%

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

      5. distribute-rgt-neg-out 64.8%

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

      6. sub-neg 64.8%

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

      7. mul-1-neg 64.8%

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

      8. log1p-def 97.6%

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

      9. mul-1-neg 97.6%

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

   4. Simplified 97.6%

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

   5. Taylor expanded in u0 around 0 94.6%

      \[\leadsto \frac{alphay \cdot \left(-alphay\right)}{\color{blue}{-1 \cdot \left(u0 \cdot \left(-0.3333333333333333 \cdot sin2phi + 0.25 \cdot sin2phi\right)\right) + \left(-1 \cdot \left({u0}^{2} \cdot \left(-0.5 \cdot \left(-0.3333333333333333 \cdot sin2phi + 0.25 \cdot sin2phi\right) + \left(-0.25 \cdot sin2phi + 0.16666666666666666 \cdot sin2phi\right)\right)\right) + \left(-1 \cdot \frac{sin2phi}{u0} + 0.5 \cdot sin2phi\right)\right)}} \]
   6. Step-by-step derivation

      1. +-commutative 94.6%

         \[\leadsto \frac{alphay \cdot \left(-alphay\right)}{\color{blue}{\left(-1 \cdot \left({u0}^{2} \cdot \left(-0.5 \cdot \left(-0.3333333333333333 \cdot sin2phi + 0.25 \cdot sin2phi\right) + \left(-0.25 \cdot sin2phi + 0.16666666666666666 \cdot sin2phi\right)\right)\right) + \left(-1 \cdot \frac{sin2phi}{u0} + 0.5 \cdot sin2phi\right)\right) + -1 \cdot \left(u0 \cdot \left(-0.3333333333333333 \cdot sin2phi + 0.25 \cdot sin2phi\right)\right)}} \]

      2. mul-1-neg 94.6%

         \[\leadsto \frac{alphay \cdot \left(-alphay\right)}{\left(-1 \cdot \left({u0}^{2} \cdot \left(-0.5 \cdot \left(-0.3333333333333333 \cdot sin2phi + 0.25 \cdot sin2phi\right) + \left(-0.25 \cdot sin2phi + 0.16666666666666666 \cdot sin2phi\right)\right)\right) + \left(-1 \cdot \frac{sin2phi}{u0} + 0.5 \cdot sin2phi\right)\right) + \color{blue}{\left(-u0 \cdot \left(-0.3333333333333333 \cdot sin2phi + 0.25 \cdot sin2phi\right)\right)}} \]

      3. unsub-neg 94.6%

         \[\leadsto \frac{alphay \cdot \left(-alphay\right)}{\color{blue}{\left(-1 \cdot \left({u0}^{2} \cdot \left(-0.5 \cdot \left(-0.3333333333333333 \cdot sin2phi + 0.25 \cdot sin2phi\right) + \left(-0.25 \cdot sin2phi + 0.16666666666666666 \cdot sin2phi\right)\right)\right) + \left(-1 \cdot \frac{sin2phi}{u0} + 0.5 \cdot sin2phi\right)\right) - u0 \cdot \left(-0.3333333333333333 \cdot sin2phi + 0.25 \cdot sin2phi\right)}} \]

   7. Simplified 94.6%

      \[\leadsto \frac{alphay \cdot \left(-alphay\right)}{\color{blue}{\left(\left(sin2phi \cdot 0.5 - \frac{sin2phi}{u0}\right) - u0 \cdot \left(u0 \cdot \left(0.5 \cdot \left(sin2phi \cdot -0.08333333333333333\right)\right)\right)\right) - u0 \cdot \left(sin2phi \cdot -0.08333333333333333\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 92.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 0.00019999999494757503:\\ \;\;\;\;\frac{u0 - u0 \cdot \left(u0 \cdot -0.5\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay} \cdot \frac{1}{alphay}}\\ \mathbf{else}:\\ \;\;\;\;\frac{alphay \cdot \left(-alphay\right)}{\left(\left(sin2phi \cdot 0.5 - \frac{sin2phi}{u0}\right) - u0 \cdot \left(u0 \cdot \left(0.5 \cdot \left(sin2phi \cdot -0.08333333333333333\right)\right)\right)\right) - u0 \cdot \left(sin2phi \cdot -0.08333333333333333\right)}\\ \end{array} \]

Alternative 6: 82.8% accurate, 4.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 0.00019999999494757503:\\ \;\;\;\;u0 \cdot \frac{1}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay} \cdot \frac{1}{alphay}}\\ \mathbf{else}:\\ \;\;\;\;\frac{alphay \cdot \left(-alphay\right)}{\left(sin2phi \cdot 0.5 - \frac{sin2phi}{u0}\right) - u0 \cdot \left(sin2phi \cdot -0.08333333333333333\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (if (<= (/ sin2phi (* alphay alphay)) 0.00019999999494757503)
   (*
    u0
    (/
     1.0
     (+ (/ cos2phi (* alphax alphax)) (* (/ sin2phi alphay) (/ 1.0 alphay)))))
   (/
    (* alphay (- alphay))
    (-
     (- (* sin2phi 0.5) (/ sin2phi u0))
     (* u0 (* sin2phi -0.08333333333333333))))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float tmp;
	if ((sin2phi / (alphay * alphay)) <= 0.00019999999494757503f) {
		tmp = u0 * (1.0f / ((cos2phi / (alphax * alphax)) + ((sin2phi / alphay) * (1.0f / alphay))));
	} else {
		tmp = (alphay * -alphay) / (((sin2phi * 0.5f) - (sin2phi / u0)) - (u0 * (sin2phi * -0.08333333333333333f)));
	}
	return tmp;
}

Fortran

real(4) function code(alphax, alphay, u0, cos2phi, sin2phi)
    real(4), intent (in) :: alphax
    real(4), intent (in) :: alphay
    real(4), intent (in) :: u0
    real(4), intent (in) :: cos2phi
    real(4), intent (in) :: sin2phi
    real(4) :: tmp
    if ((sin2phi / (alphay * alphay)) <= 0.00019999999494757503e0) then
        tmp = u0 * (1.0e0 / ((cos2phi / (alphax * alphax)) + ((sin2phi / alphay) * (1.0e0 / alphay))))
    else
        tmp = (alphay * -alphay) / (((sin2phi * 0.5e0) - (sin2phi / u0)) - (u0 * (sin2phi * (-0.08333333333333333e0))))
    end if
    code = tmp
end function

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = Float32(0.0)
	if (Float32(sin2phi / Float32(alphay * alphay)) <= Float32(0.00019999999494757503))
		tmp = Float32(u0 * Float32(Float32(1.0) / Float32(Float32(cos2phi / Float32(alphax * alphax)) + Float32(Float32(sin2phi / alphay) * Float32(Float32(1.0) / alphay)))));
	else
		tmp = Float32(Float32(alphay * Float32(-alphay)) / Float32(Float32(Float32(sin2phi * Float32(0.5)) - Float32(sin2phi / u0)) - Float32(u0 * Float32(sin2phi * Float32(-0.08333333333333333)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = single(0.0);
	if ((sin2phi / (alphay * alphay)) <= single(0.00019999999494757503))
		tmp = u0 * (single(1.0) / ((cos2phi / (alphax * alphax)) + ((sin2phi / alphay) * (single(1.0) / alphay))));
	else
		tmp = (alphay * -alphay) / (((sin2phi * single(0.5)) - (sin2phi / u0)) - (u0 * (sin2phi * single(-0.08333333333333333))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 sin2phi (*.f32 alphay alphay)) < 1.99999995e-4

   1. Initial program 52.4%

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

   2. Taylor expanded in u0 around 0 76.9%

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

      1. +-commutative 76.9%

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

      2. unpow2 76.9%

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

      3. unpow2 76.9%

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

   4. Simplified 76.9%

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

      1. frac-2neg 76.9%

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

      2. div-inv 76.9%

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

      3. distribute-rgt-neg-in 76.9%

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

   6. Applied egg-rr 76.9%

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

      1. div-inv 77.0%

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

      2. un-div-inv 77.0%

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

      3. distribute-rgt-neg-out 77.0%

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

      4. frac-2neg 77.0%

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

   8. Applied egg-rr 77.0%

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

      1. associate-/r* 77.0%

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

      2. div-inv 77.1%

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

   10. Applied egg-rr 77.2%

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

   if 1.99999995e-4 < (/.f32 sin2phi (*.f32 alphay alphay))

   1. Initial program 65.0%

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

   2. Taylor expanded in cos2phi around 0 64.9%

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

      1. mul-1-neg 64.9%

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

      2. unpow2 64.9%

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

      3. associate-/l* 64.8%

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

      4. distribute-neg-frac 64.8%

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

      5. distribute-rgt-neg-out 64.8%

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

      6. sub-neg 64.8%

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

      7. mul-1-neg 64.8%

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

      8. log1p-def 97.6%

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

      9. mul-1-neg 97.6%

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

   4. Simplified 97.6%

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

   5. Taylor expanded in u0 around 0 93.1%

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

      1. +-commutative 93.1%

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

      2. mul-1-neg 93.1%

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

      3. unsub-neg 93.1%

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

      4. +-commutative 93.1%

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

      5. mul-1-neg 93.1%

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

      6. unsub-neg 93.1%

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

      7. *-commutative 93.1%

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

      8. distribute-rgt-out 93.1%

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

      9. metadata-eval 93.1%

         \[\leadsto \frac{alphay \cdot \left(-alphay\right)}{\left(sin2phi \cdot 0.5 - \frac{sin2phi}{u0}\right) - u0 \cdot \left(sin2phi \cdot \color{blue}{-0.08333333333333333}\right)} \]

   7. Simplified 93.1%

      \[\leadsto \frac{alphay \cdot \left(-alphay\right)}{\color{blue}{\left(sin2phi \cdot 0.5 - \frac{sin2phi}{u0}\right) - u0 \cdot \left(sin2phi \cdot -0.08333333333333333\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 86.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 0.00019999999494757503:\\ \;\;\;\;u0 \cdot \frac{1}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay} \cdot \frac{1}{alphay}}\\ \mathbf{else}:\\ \;\;\;\;\frac{alphay \cdot \left(-alphay\right)}{\left(sin2phi \cdot 0.5 - \frac{sin2phi}{u0}\right) - u0 \cdot \left(sin2phi \cdot -0.08333333333333333\right)}\\ \end{array} \]

Alternative 7: 80.8% accurate, 5.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 0.00019999999494757503:\\ \;\;\;\;u0 \cdot \frac{1}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay} \cdot \frac{1}{alphay}}\\ \mathbf{else}:\\ \;\;\;\;\frac{alphay \cdot \left(-alphay\right)}{sin2phi \cdot 0.5 - \frac{sin2phi}{u0}}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (if (<= (/ sin2phi (* alphay alphay)) 0.00019999999494757503)
   (*
    u0
    (/
     1.0
     (+ (/ cos2phi (* alphax alphax)) (* (/ sin2phi alphay) (/ 1.0 alphay)))))
   (/ (* alphay (- alphay)) (- (* sin2phi 0.5) (/ sin2phi u0)))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float tmp;
	if ((sin2phi / (alphay * alphay)) <= 0.00019999999494757503f) {
		tmp = u0 * (1.0f / ((cos2phi / (alphax * alphax)) + ((sin2phi / alphay) * (1.0f / alphay))));
	} else {
		tmp = (alphay * -alphay) / ((sin2phi * 0.5f) - (sin2phi / u0));
	}
	return tmp;
}

Fortran

real(4) function code(alphax, alphay, u0, cos2phi, sin2phi)
    real(4), intent (in) :: alphax
    real(4), intent (in) :: alphay
    real(4), intent (in) :: u0
    real(4), intent (in) :: cos2phi
    real(4), intent (in) :: sin2phi
    real(4) :: tmp
    if ((sin2phi / (alphay * alphay)) <= 0.00019999999494757503e0) then
        tmp = u0 * (1.0e0 / ((cos2phi / (alphax * alphax)) + ((sin2phi / alphay) * (1.0e0 / alphay))))
    else
        tmp = (alphay * -alphay) / ((sin2phi * 0.5e0) - (sin2phi / u0))
    end if
    code = tmp
end function

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = Float32(0.0)
	if (Float32(sin2phi / Float32(alphay * alphay)) <= Float32(0.00019999999494757503))
		tmp = Float32(u0 * Float32(Float32(1.0) / Float32(Float32(cos2phi / Float32(alphax * alphax)) + Float32(Float32(sin2phi / alphay) * Float32(Float32(1.0) / alphay)))));
	else
		tmp = Float32(Float32(alphay * Float32(-alphay)) / Float32(Float32(sin2phi * Float32(0.5)) - Float32(sin2phi / u0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = single(0.0);
	if ((sin2phi / (alphay * alphay)) <= single(0.00019999999494757503))
		tmp = u0 * (single(1.0) / ((cos2phi / (alphax * alphax)) + ((sin2phi / alphay) * (single(1.0) / alphay))));
	else
		tmp = (alphay * -alphay) / ((sin2phi * single(0.5)) - (sin2phi / u0));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 sin2phi (*.f32 alphay alphay)) < 1.99999995e-4

   1. Initial program 52.4%

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

   2. Taylor expanded in u0 around 0 76.9%

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

      1. +-commutative 76.9%

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

      2. unpow2 76.9%

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

      3. unpow2 76.9%

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

   4. Simplified 76.9%

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

      1. frac-2neg 76.9%

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

      2. div-inv 76.9%

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

      3. distribute-rgt-neg-in 76.9%

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

   6. Applied egg-rr 76.9%

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

      1. div-inv 77.0%

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

      2. un-div-inv 77.0%

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

      3. distribute-rgt-neg-out 77.0%

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

      4. frac-2neg 77.0%

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

   8. Applied egg-rr 77.0%

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

      1. associate-/r* 77.0%

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

      2. div-inv 77.1%

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

   10. Applied egg-rr 77.2%

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

   if 1.99999995e-4 < (/.f32 sin2phi (*.f32 alphay alphay))

   1. Initial program 65.0%

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

   2. Taylor expanded in cos2phi around 0 64.9%

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

      1. mul-1-neg 64.9%

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

      2. unpow2 64.9%

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

      3. associate-/l* 64.8%

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

      4. distribute-neg-frac 64.8%

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

      5. distribute-rgt-neg-out 64.8%

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

      6. sub-neg 64.8%

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

      7. mul-1-neg 64.8%

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

      8. log1p-def 97.6%

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

      9. mul-1-neg 97.6%

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

   4. Simplified 97.6%

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

   5. Taylor expanded in u0 around 0 89.0%

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

      1. +-commutative 89.0%

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

      2. mul-1-neg 89.0%

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

      3. unsub-neg 89.0%

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

      4. *-commutative 89.0%

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

   7. Simplified 89.0%

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

4. Final simplification 84.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 0.00019999999494757503:\\ \;\;\;\;u0 \cdot \frac{1}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay} \cdot \frac{1}{alphay}}\\ \mathbf{else}:\\ \;\;\;\;\frac{alphay \cdot \left(-alphay\right)}{sin2phi \cdot 0.5 - \frac{sin2phi}{u0}}\\ \end{array} \]

Alternative 8: 80.8% accurate, 5.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{sin2phi}{alphay \cdot alphay}\\ \mathbf{if}\;t_0 \leq 0.00019999999494757503:\\ \;\;\;\;u0 \cdot \frac{1}{t_0 + \frac{cos2phi}{alphax \cdot alphax}}\\ \mathbf{else}:\\ \;\;\;\;\frac{alphay \cdot \left(-alphay\right)}{sin2phi \cdot 0.5 - \frac{sin2phi}{u0}}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (let* ((t_0 (/ sin2phi (* alphay alphay))))
   (if (<= t_0 0.00019999999494757503)
     (* u0 (/ 1.0 (+ t_0 (/ cos2phi (* alphax alphax)))))
     (/ (* alphay (- alphay)) (- (* sin2phi 0.5) (/ sin2phi u0))))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float t_0 = sin2phi / (alphay * alphay);
	float tmp;
	if (t_0 <= 0.00019999999494757503f) {
		tmp = u0 * (1.0f / (t_0 + (cos2phi / (alphax * alphax))));
	} else {
		tmp = (alphay * -alphay) / ((sin2phi * 0.5f) - (sin2phi / u0));
	}
	return tmp;
}

Fortran

real(4) function code(alphax, alphay, u0, cos2phi, sin2phi)
    real(4), intent (in) :: alphax
    real(4), intent (in) :: alphay
    real(4), intent (in) :: u0
    real(4), intent (in) :: cos2phi
    real(4), intent (in) :: sin2phi
    real(4) :: t_0
    real(4) :: tmp
    t_0 = sin2phi / (alphay * alphay)
    if (t_0 <= 0.00019999999494757503e0) then
        tmp = u0 * (1.0e0 / (t_0 + (cos2phi / (alphax * alphax))))
    else
        tmp = (alphay * -alphay) / ((sin2phi * 0.5e0) - (sin2phi / u0))
    end if
    code = tmp
end function

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	t_0 = Float32(sin2phi / Float32(alphay * alphay))
	tmp = Float32(0.0)
	if (t_0 <= Float32(0.00019999999494757503))
		tmp = Float32(u0 * Float32(Float32(1.0) / Float32(t_0 + Float32(cos2phi / Float32(alphax * alphax)))));
	else
		tmp = Float32(Float32(alphay * Float32(-alphay)) / Float32(Float32(sin2phi * Float32(0.5)) - Float32(sin2phi / u0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alphax, alphay, u0, cos2phi, sin2phi)
	t_0 = sin2phi / (alphay * alphay);
	tmp = single(0.0);
	if (t_0 <= single(0.00019999999494757503))
		tmp = u0 * (single(1.0) / (t_0 + (cos2phi / (alphax * alphax))));
	else
		tmp = (alphay * -alphay) / ((sin2phi * single(0.5)) - (sin2phi / u0));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 sin2phi (*.f32 alphay alphay)) < 1.99999995e-4

   1. Initial program 52.4%

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

   2. Taylor expanded in u0 around 0 76.9%

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

      1. +-commutative 76.9%

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

      2. unpow2 76.9%

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

      3. unpow2 76.9%

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

   4. Simplified 76.9%

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

      1. frac-2neg 76.9%

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

      2. div-inv 76.9%

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

      3. distribute-rgt-neg-in 76.9%

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

   6. Applied egg-rr 76.9%

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

      1. div-inv 77.0%

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

      2. un-div-inv 77.0%

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

      3. distribute-rgt-neg-out 77.0%

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

      4. frac-2neg 77.0%

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

   8. Applied egg-rr 77.0%

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

   if 1.99999995e-4 < (/.f32 sin2phi (*.f32 alphay alphay))

   1. Initial program 65.0%

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

   2. Taylor expanded in cos2phi around 0 64.9%

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

      1. mul-1-neg 64.9%

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

      2. unpow2 64.9%

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

      3. associate-/l* 64.8%

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

      4. distribute-neg-frac 64.8%

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

      5. distribute-rgt-neg-out 64.8%

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

      6. sub-neg 64.8%

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

      7. mul-1-neg 64.8%

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

      8. log1p-def 97.6%

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

      9. mul-1-neg 97.6%

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

   4. Simplified 97.6%

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

   5. Taylor expanded in u0 around 0 89.0%

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

      1. +-commutative 89.0%

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

      2. mul-1-neg 89.0%

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

      3. unsub-neg 89.0%

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

      4. *-commutative 89.0%

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

   7. Simplified 89.0%

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

4. Final simplification 84.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 0.00019999999494757503:\\ \;\;\;\;u0 \cdot \frac{1}{\frac{sin2phi}{alphay \cdot alphay} + \frac{cos2phi}{alphax \cdot alphax}}\\ \mathbf{else}:\\ \;\;\;\;\frac{alphay \cdot \left(-alphay\right)}{sin2phi \cdot 0.5 - \frac{sin2phi}{u0}}\\ \end{array} \]

Alternative 9: 80.8% accurate, 5.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 0.00019999999494757503:\\ \;\;\;\;\frac{u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay} \cdot \frac{1}{alphay}}\\ \mathbf{else}:\\ \;\;\;\;\frac{alphay \cdot \left(-alphay\right)}{sin2phi \cdot 0.5 - \frac{sin2phi}{u0}}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (if (<= (/ sin2phi (* alphay alphay)) 0.00019999999494757503)
   (/
    u0
    (+ (/ cos2phi (* alphax alphax)) (* (/ sin2phi alphay) (/ 1.0 alphay))))
   (/ (* alphay (- alphay)) (- (* sin2phi 0.5) (/ sin2phi u0)))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float tmp;
	if ((sin2phi / (alphay * alphay)) <= 0.00019999999494757503f) {
		tmp = u0 / ((cos2phi / (alphax * alphax)) + ((sin2phi / alphay) * (1.0f / alphay)));
	} else {
		tmp = (alphay * -alphay) / ((sin2phi * 0.5f) - (sin2phi / u0));
	}
	return tmp;
}

Fortran

real(4) function code(alphax, alphay, u0, cos2phi, sin2phi)
    real(4), intent (in) :: alphax
    real(4), intent (in) :: alphay
    real(4), intent (in) :: u0
    real(4), intent (in) :: cos2phi
    real(4), intent (in) :: sin2phi
    real(4) :: tmp
    if ((sin2phi / (alphay * alphay)) <= 0.00019999999494757503e0) then
        tmp = u0 / ((cos2phi / (alphax * alphax)) + ((sin2phi / alphay) * (1.0e0 / alphay)))
    else
        tmp = (alphay * -alphay) / ((sin2phi * 0.5e0) - (sin2phi / u0))
    end if
    code = tmp
end function

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = Float32(0.0)
	if (Float32(sin2phi / Float32(alphay * alphay)) <= Float32(0.00019999999494757503))
		tmp = Float32(u0 / Float32(Float32(cos2phi / Float32(alphax * alphax)) + Float32(Float32(sin2phi / alphay) * Float32(Float32(1.0) / alphay))));
	else
		tmp = Float32(Float32(alphay * Float32(-alphay)) / Float32(Float32(sin2phi * Float32(0.5)) - Float32(sin2phi / u0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = single(0.0);
	if ((sin2phi / (alphay * alphay)) <= single(0.00019999999494757503))
		tmp = u0 / ((cos2phi / (alphax * alphax)) + ((sin2phi / alphay) * (single(1.0) / alphay)));
	else
		tmp = (alphay * -alphay) / ((sin2phi * single(0.5)) - (sin2phi / u0));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 sin2phi (*.f32 alphay alphay)) < 1.99999995e-4

   1. Initial program 52.4%

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

   2. Taylor expanded in u0 around 0 76.9%

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

      1. +-commutative 76.9%

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

      2. unpow2 76.9%

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

      3. unpow2 76.9%

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

   4. Simplified 76.9%

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

      1. associate-/r* 77.0%

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

      2. div-inv 77.1%

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

   6. Applied egg-rr 77.1%

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

   if 1.99999995e-4 < (/.f32 sin2phi (*.f32 alphay alphay))

   1. Initial program 65.0%

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

   2. Taylor expanded in cos2phi around 0 64.9%

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

      1. mul-1-neg 64.9%

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

      2. unpow2 64.9%

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

      3. associate-/l* 64.8%

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

      4. distribute-neg-frac 64.8%

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

      5. distribute-rgt-neg-out 64.8%

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

      6. sub-neg 64.8%

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

      7. mul-1-neg 64.8%

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

      8. log1p-def 97.6%

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

      9. mul-1-neg 97.6%

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

   4. Simplified 97.6%

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

   5. Taylor expanded in u0 around 0 89.0%

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

      1. +-commutative 89.0%

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

      2. mul-1-neg 89.0%

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

      3. unsub-neg 89.0%

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

      4. *-commutative 89.0%

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

   7. Simplified 89.0%

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

4. Final simplification 84.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 0.00019999999494757503:\\ \;\;\;\;\frac{u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay} \cdot \frac{1}{alphay}}\\ \mathbf{else}:\\ \;\;\;\;\frac{alphay \cdot \left(-alphay\right)}{sin2phi \cdot 0.5 - \frac{sin2phi}{u0}}\\ \end{array} \]

Alternative 10: 80.8% accurate, 6.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{sin2phi}{alphay \cdot alphay}\\ \mathbf{if}\;t_0 \leq 0.00019999999494757503:\\ \;\;\;\;\frac{u0}{t_0 + \frac{cos2phi}{alphax \cdot alphax}}\\ \mathbf{else}:\\ \;\;\;\;\frac{alphay \cdot \left(-alphay\right)}{sin2phi \cdot 0.5 - \frac{sin2phi}{u0}}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (let* ((t_0 (/ sin2phi (* alphay alphay))))
   (if (<= t_0 0.00019999999494757503)
     (/ u0 (+ t_0 (/ cos2phi (* alphax alphax))))
     (/ (* alphay (- alphay)) (- (* sin2phi 0.5) (/ sin2phi u0))))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float t_0 = sin2phi / (alphay * alphay);
	float tmp;
	if (t_0 <= 0.00019999999494757503f) {
		tmp = u0 / (t_0 + (cos2phi / (alphax * alphax)));
	} else {
		tmp = (alphay * -alphay) / ((sin2phi * 0.5f) - (sin2phi / u0));
	}
	return tmp;
}

Fortran

real(4) function code(alphax, alphay, u0, cos2phi, sin2phi)
    real(4), intent (in) :: alphax
    real(4), intent (in) :: alphay
    real(4), intent (in) :: u0
    real(4), intent (in) :: cos2phi
    real(4), intent (in) :: sin2phi
    real(4) :: t_0
    real(4) :: tmp
    t_0 = sin2phi / (alphay * alphay)
    if (t_0 <= 0.00019999999494757503e0) then
        tmp = u0 / (t_0 + (cos2phi / (alphax * alphax)))
    else
        tmp = (alphay * -alphay) / ((sin2phi * 0.5e0) - (sin2phi / u0))
    end if
    code = tmp
end function

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	t_0 = Float32(sin2phi / Float32(alphay * alphay))
	tmp = Float32(0.0)
	if (t_0 <= Float32(0.00019999999494757503))
		tmp = Float32(u0 / Float32(t_0 + Float32(cos2phi / Float32(alphax * alphax))));
	else
		tmp = Float32(Float32(alphay * Float32(-alphay)) / Float32(Float32(sin2phi * Float32(0.5)) - Float32(sin2phi / u0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alphax, alphay, u0, cos2phi, sin2phi)
	t_0 = sin2phi / (alphay * alphay);
	tmp = single(0.0);
	if (t_0 <= single(0.00019999999494757503))
		tmp = u0 / (t_0 + (cos2phi / (alphax * alphax)));
	else
		tmp = (alphay * -alphay) / ((sin2phi * single(0.5)) - (sin2phi / u0));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 sin2phi (*.f32 alphay alphay)) < 1.99999995e-4

   1. Initial program 52.4%

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

   2. Taylor expanded in u0 around 0 76.9%

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

      1. +-commutative 76.9%

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

      2. unpow2 76.9%

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

      3. unpow2 76.9%

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

   4. Simplified 76.9%

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

   if 1.99999995e-4 < (/.f32 sin2phi (*.f32 alphay alphay))

   1. Initial program 65.0%

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

   2. Taylor expanded in cos2phi around 0 64.9%

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

      1. mul-1-neg 64.9%

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

      2. unpow2 64.9%

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

      3. associate-/l* 64.8%

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

      4. distribute-neg-frac 64.8%

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

      5. distribute-rgt-neg-out 64.8%

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

      6. sub-neg 64.8%

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

      7. mul-1-neg 64.8%

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

      8. log1p-def 97.6%

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

      9. mul-1-neg 97.6%

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

   4. Simplified 97.6%

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

   5. Taylor expanded in u0 around 0 89.0%

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

      1. +-commutative 89.0%

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

      2. mul-1-neg 89.0%

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

      3. unsub-neg 89.0%

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

      4. *-commutative 89.0%

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

   7. Simplified 89.0%

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

4. Final simplification 84.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 0.00019999999494757503:\\ \;\;\;\;\frac{u0}{\frac{sin2phi}{alphay \cdot alphay} + \frac{cos2phi}{alphax \cdot alphax}}\\ \mathbf{else}:\\ \;\;\;\;\frac{alphay \cdot \left(-alphay\right)}{sin2phi \cdot 0.5 - \frac{sin2phi}{u0}}\\ \end{array} \]

Alternative 11: 80.8% accurate, 6.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{sin2phi}{alphay \cdot alphay}\\ \mathbf{if}\;t_0 \leq 0.00019999999494757503:\\ \;\;\;\;\frac{u0}{\frac{\frac{cos2phi}{alphax}}{alphax} + t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{alphay \cdot \left(-alphay\right)}{sin2phi \cdot 0.5 - \frac{sin2phi}{u0}}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (let* ((t_0 (/ sin2phi (* alphay alphay))))
   (if (<= t_0 0.00019999999494757503)
     (/ u0 (+ (/ (/ cos2phi alphax) alphax) t_0))
     (/ (* alphay (- alphay)) (- (* sin2phi 0.5) (/ sin2phi u0))))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float t_0 = sin2phi / (alphay * alphay);
	float tmp;
	if (t_0 <= 0.00019999999494757503f) {
		tmp = u0 / (((cos2phi / alphax) / alphax) + t_0);
	} else {
		tmp = (alphay * -alphay) / ((sin2phi * 0.5f) - (sin2phi / u0));
	}
	return tmp;
}

Fortran

real(4) function code(alphax, alphay, u0, cos2phi, sin2phi)
    real(4), intent (in) :: alphax
    real(4), intent (in) :: alphay
    real(4), intent (in) :: u0
    real(4), intent (in) :: cos2phi
    real(4), intent (in) :: sin2phi
    real(4) :: t_0
    real(4) :: tmp
    t_0 = sin2phi / (alphay * alphay)
    if (t_0 <= 0.00019999999494757503e0) then
        tmp = u0 / (((cos2phi / alphax) / alphax) + t_0)
    else
        tmp = (alphay * -alphay) / ((sin2phi * 0.5e0) - (sin2phi / u0))
    end if
    code = tmp
end function

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	t_0 = Float32(sin2phi / Float32(alphay * alphay))
	tmp = Float32(0.0)
	if (t_0 <= Float32(0.00019999999494757503))
		tmp = Float32(u0 / Float32(Float32(Float32(cos2phi / alphax) / alphax) + t_0));
	else
		tmp = Float32(Float32(alphay * Float32(-alphay)) / Float32(Float32(sin2phi * Float32(0.5)) - Float32(sin2phi / u0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alphax, alphay, u0, cos2phi, sin2phi)
	t_0 = sin2phi / (alphay * alphay);
	tmp = single(0.0);
	if (t_0 <= single(0.00019999999494757503))
		tmp = u0 / (((cos2phi / alphax) / alphax) + t_0);
	else
		tmp = (alphay * -alphay) / ((sin2phi * single(0.5)) - (sin2phi / u0));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 sin2phi (*.f32 alphay alphay)) < 1.99999995e-4

   1. Initial program 52.4%

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

   2. Taylor expanded in u0 around 0 76.9%

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

      1. +-commutative 76.9%

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

      2. unpow2 76.9%

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

      3. unpow2 76.9%

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

   4. Simplified 76.9%

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

      1. frac-2neg 76.9%

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

      2. div-inv 76.9%

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

      3. distribute-rgt-neg-in 76.9%

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

   6. Applied egg-rr 76.9%

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

      1. un-div-inv 76.9%

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

      2. distribute-rgt-neg-out 76.9%

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

      3. frac-2neg 76.9%

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

      4. associate-/r* 76.9%

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

   8. Applied egg-rr 76.9%

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

   if 1.99999995e-4 < (/.f32 sin2phi (*.f32 alphay alphay))

   1. Initial program 65.0%

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

   2. Taylor expanded in cos2phi around 0 64.9%

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

      1. mul-1-neg 64.9%

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

      2. unpow2 64.9%

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

      3. associate-/l* 64.8%

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

      4. distribute-neg-frac 64.8%

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

      5. distribute-rgt-neg-out 64.8%

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

      6. sub-neg 64.8%

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

      7. mul-1-neg 64.8%

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

      8. log1p-def 97.6%

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

      9. mul-1-neg 97.6%

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

   4. Simplified 97.6%

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

   5. Taylor expanded in u0 around 0 89.0%

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

      1. +-commutative 89.0%

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

      2. mul-1-neg 89.0%

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

      3. unsub-neg 89.0%

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

      4. *-commutative 89.0%

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

   7. Simplified 89.0%

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

4. Final simplification 84.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 0.00019999999494757503:\\ \;\;\;\;\frac{u0}{\frac{\frac{cos2phi}{alphax}}{alphax} + \frac{sin2phi}{alphay \cdot alphay}}\\ \mathbf{else}:\\ \;\;\;\;\frac{alphay \cdot \left(-alphay\right)}{sin2phi \cdot 0.5 - \frac{sin2phi}{u0}}\\ \end{array} \]

Alternative 12: 87.0% accurate, 6.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 60.0%

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

2. Taylor expanded in u0 around 0 88.8%

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

   1. +-commutative 88.8%

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

   2. mul-1-neg 88.8%

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

   3. unsub-neg 88.8%

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

   4. *-commutative 88.8%

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

   5. unpow2 88.8%

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

   6. associate-*l* 88.8%

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

4. Simplified 88.8%

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

5. Final simplification 88.8%

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

Alternative 13: 68.7% accurate, 8.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;sin2phi \leq 6.000000068087077 \cdot 10^{-19}:\\ \;\;\;\;\frac{-alphax \cdot alphax}{cos2phi \cdot 0.5 - \frac{cos2phi}{u0}}\\ \mathbf{else}:\\ \;\;\;\;\frac{u0 \cdot \left(alphay \cdot alphay\right)}{sin2phi}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (if (<= sin2phi 6.000000068087077e-19)
   (/ (- (* alphax alphax)) (- (* cos2phi 0.5) (/ cos2phi u0)))
   (/ (* u0 (* alphay alphay)) sin2phi)))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float tmp;
	if (sin2phi <= 6.000000068087077e-19f) {
		tmp = -(alphax * alphax) / ((cos2phi * 0.5f) - (cos2phi / u0));
	} 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 <= 6.000000068087077e-19) then
        tmp = -(alphax * alphax) / ((cos2phi * 0.5e0) - (cos2phi / u0))
    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(6.000000068087077e-19))
		tmp = Float32(Float32(-Float32(alphax * alphax)) / Float32(Float32(cos2phi * Float32(0.5)) - Float32(cos2phi / u0)));
	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(6.000000068087077e-19))
		tmp = -(alphax * alphax) / ((cos2phi * single(0.5)) - (cos2phi / u0));
	else
		tmp = (u0 * (alphay * alphay)) / sin2phi;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if sin2phi < 6.00000007e-19

   1. Initial program 54.4%

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

   2. Taylor expanded in cos2phi around inf 42.8%

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

      1. mul-1-neg 42.8%

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

      2. unpow2 42.8%

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

      3. associate-/l* 42.7%

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

      4. distribute-neg-frac 42.7%

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

      5. distribute-rgt-neg-out 42.7%

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

      6. sub-neg 42.7%

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

      7. mul-1-neg 42.7%

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

      8. log1p-def 66.8%

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

      9. mul-1-neg 66.8%

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

   4. Simplified 66.8%

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

   5. Taylor expanded in u0 around 0 62.4%

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

      1. +-commutative 62.4%

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

      2. mul-1-neg 62.4%

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

      3. unsub-neg 62.4%

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

      4. *-commutative 62.4%

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

   7. Simplified 62.4%

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

   if 6.00000007e-19 < sin2phi

   1. Initial program 61.8%

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

   2. Taylor expanded in u0 around 0 77.6%

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

      1. +-commutative 77.6%

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

      2. unpow2 77.6%

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

      3. unpow2 77.6%

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

   4. Simplified 77.6%

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

   5. Taylor expanded in sin2phi around inf 72.0%

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

      1. unpow2 72.0%

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

      2. *-lft-identity 72.0%

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

      3. times-frac 71.9%

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

      4. /-rgt-identity 71.9%

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

   7. Simplified 71.9%

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

      1. associate-*r/ 72.0%

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

   9. Applied egg-rr 72.0%

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

4. Final simplification 69.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;sin2phi \leq 6.000000068087077 \cdot 10^{-19}:\\ \;\;\;\;\frac{-alphax \cdot alphax}{cos2phi \cdot 0.5 - \frac{cos2phi}{u0}}\\ \mathbf{else}:\\ \;\;\;\;\frac{u0 \cdot \left(alphay \cdot alphay\right)}{sin2phi}\\ \end{array} \]

Alternative 14: 76.4% accurate, 8.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;sin2phi \leq 6.000000068087077 \cdot 10^{-19}:\\ \;\;\;\;\frac{-alphax \cdot alphax}{cos2phi \cdot 0.5 - \frac{cos2phi}{u0}}\\ \mathbf{else}:\\ \;\;\;\;\frac{alphay \cdot \left(-alphay\right)}{sin2phi \cdot 0.5 - \frac{sin2phi}{u0}}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (if (<= sin2phi 6.000000068087077e-19)
   (/ (- (* alphax alphax)) (- (* cos2phi 0.5) (/ cos2phi u0)))
   (/ (* alphay (- alphay)) (- (* sin2phi 0.5) (/ sin2phi u0)))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float tmp;
	if (sin2phi <= 6.000000068087077e-19f) {
		tmp = -(alphax * alphax) / ((cos2phi * 0.5f) - (cos2phi / u0));
	} else {
		tmp = (alphay * -alphay) / ((sin2phi * 0.5f) - (sin2phi / u0));
	}
	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 <= 6.000000068087077e-19) then
        tmp = -(alphax * alphax) / ((cos2phi * 0.5e0) - (cos2phi / u0))
    else
        tmp = (alphay * -alphay) / ((sin2phi * 0.5e0) - (sin2phi / u0))
    end if
    code = tmp
end function

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = Float32(0.0)
	if (sin2phi <= Float32(6.000000068087077e-19))
		tmp = Float32(Float32(-Float32(alphax * alphax)) / Float32(Float32(cos2phi * Float32(0.5)) - Float32(cos2phi / u0)));
	else
		tmp = Float32(Float32(alphay * Float32(-alphay)) / Float32(Float32(sin2phi * Float32(0.5)) - Float32(sin2phi / u0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = single(0.0);
	if (sin2phi <= single(6.000000068087077e-19))
		tmp = -(alphax * alphax) / ((cos2phi * single(0.5)) - (cos2phi / u0));
	else
		tmp = (alphay * -alphay) / ((sin2phi * single(0.5)) - (sin2phi / u0));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if sin2phi < 6.00000007e-19

   1. Initial program 54.4%

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

   2. Taylor expanded in cos2phi around inf 42.8%

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

      1. mul-1-neg 42.8%

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

      2. unpow2 42.8%

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

      3. associate-/l* 42.7%

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

      4. distribute-neg-frac 42.7%

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

      5. distribute-rgt-neg-out 42.7%

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

      6. sub-neg 42.7%

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

      7. mul-1-neg 42.7%

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

      8. log1p-def 66.8%

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

      9. mul-1-neg 66.8%

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

   4. Simplified 66.8%

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

   5. Taylor expanded in u0 around 0 62.4%

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

      1. +-commutative 62.4%

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

      2. mul-1-neg 62.4%

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

      3. unsub-neg 62.4%

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

      4. *-commutative 62.4%

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

   7. Simplified 62.4%

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

   if 6.00000007e-19 < sin2phi

   1. Initial program 61.8%

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

   2. Taylor expanded in cos2phi around 0 58.4%

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

      1. mul-1-neg 58.4%

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

      2. unpow2 58.4%

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

      3. associate-/l* 58.3%

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

      4. distribute-neg-frac 58.3%

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

      5. distribute-rgt-neg-out 58.3%

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

      6. sub-neg 58.3%

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

      7. mul-1-neg 58.3%

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

      8. log1p-def 90.1%

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

      9. mul-1-neg 90.1%

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

   4. Simplified 90.1%

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

   5. Taylor expanded in u0 around 0 82.8%

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

      1. +-commutative 82.8%

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

      2. mul-1-neg 82.8%

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

      3. unsub-neg 82.8%

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

      4. *-commutative 82.8%

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

   7. Simplified 82.8%

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

4. Final simplification 77.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;sin2phi \leq 6.000000068087077 \cdot 10^{-19}:\\ \;\;\;\;\frac{-alphax \cdot alphax}{cos2phi \cdot 0.5 - \frac{cos2phi}{u0}}\\ \mathbf{else}:\\ \;\;\;\;\frac{alphay \cdot \left(-alphay\right)}{sin2phi \cdot 0.5 - \frac{sin2phi}{u0}}\\ \end{array} \]

Alternative 15: 66.2% accurate, 12.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;sin2phi \leq 6.000000068087077 \cdot 10^{-19}:\\ \;\;\;\;u0 \cdot \frac{alphax \cdot alphax}{cos2phi}\\ \mathbf{else}:\\ \;\;\;\;alphay \cdot \frac{u0 \cdot alphay}{sin2phi}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (if (<= sin2phi 6.000000068087077e-19)
   (* u0 (/ (* alphax alphax) cos2phi))
   (* alphay (/ (* u0 alphay) sin2phi))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float tmp;
	if (sin2phi <= 6.000000068087077e-19f) {
		tmp = u0 * ((alphax * alphax) / cos2phi);
	} else {
		tmp = alphay * ((u0 * alphay) / sin2phi);
	}
	return tmp;
}

Fortran

real(4) function code(alphax, alphay, u0, cos2phi, sin2phi)
    real(4), intent (in) :: alphax
    real(4), intent (in) :: alphay
    real(4), intent (in) :: u0
    real(4), intent (in) :: cos2phi
    real(4), intent (in) :: sin2phi
    real(4) :: tmp
    if (sin2phi <= 6.000000068087077e-19) then
        tmp = u0 * ((alphax * alphax) / cos2phi)
    else
        tmp = alphay * ((u0 * alphay) / sin2phi)
    end if
    code = tmp
end function

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = Float32(0.0)
	if (sin2phi <= Float32(6.000000068087077e-19))
		tmp = Float32(u0 * Float32(Float32(alphax * alphax) / cos2phi));
	else
		tmp = Float32(alphay * Float32(Float32(u0 * alphay) / sin2phi));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = single(0.0);
	if (sin2phi <= single(6.000000068087077e-19))
		tmp = u0 * ((alphax * alphax) / cos2phi);
	else
		tmp = alphay * ((u0 * alphay) / sin2phi);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if sin2phi < 6.00000007e-19

   1. Initial program 54.4%

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

   2. Taylor expanded in u0 around 0 75.4%

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

      1. +-commutative 75.4%

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

      2. unpow2 75.4%

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

      3. unpow2 75.4%

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

   4. Simplified 75.4%

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

      1. frac-2neg 75.4%

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

      2. div-inv 75.4%

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

      3. distribute-rgt-neg-in 75.4%

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

   6. Applied egg-rr 75.4%

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

      1. div-inv 75.4%

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

      2. un-div-inv 75.4%

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

      3. distribute-rgt-neg-out 75.4%

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

      4. frac-2neg 75.4%

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

   8. Applied egg-rr 75.4%

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

   9. Taylor expanded in sin2phi around 0 53.5%

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

       1. unpow2 53.5%

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

   11. Simplified 53.5%

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

   if 6.00000007e-19 < sin2phi

   1. Initial program 61.8%

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

   2. Taylor expanded in u0 around 0 77.6%

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

      1. +-commutative 77.6%

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

      2. unpow2 77.6%

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

      3. unpow2 77.6%

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

   4. Simplified 77.6%

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

   5. Taylor expanded in sin2phi around inf 72.0%

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

      1. unpow2 72.0%

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

      2. *-lft-identity 72.0%

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

      3. times-frac 71.9%

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

      4. /-rgt-identity 71.9%

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

   7. Simplified 71.9%

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

   8. Taylor expanded in alphay around 0 72.0%

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

      1. associate-*r/ 71.9%

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

      2. unpow2 71.9%

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

      3. associate-*l* 71.8%

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

   10. Simplified 71.8%

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

       1. associate-*r/ 71.9%

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

       2. *-commutative 71.9%

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

   12. Applied egg-rr 71.9%

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

4. Final simplification 67.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;sin2phi \leq 6.000000068087077 \cdot 10^{-19}:\\ \;\;\;\;u0 \cdot \frac{alphax \cdot alphax}{cos2phi}\\ \mathbf{else}:\\ \;\;\;\;alphay \cdot \frac{u0 \cdot alphay}{sin2phi}\\ \end{array} \]

Alternative 16: 66.2% accurate, 12.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;sin2phi \leq 6.000000068087077 \cdot 10^{-19}:\\ \;\;\;\;\frac{u0 \cdot \left(alphax \cdot alphax\right)}{cos2phi}\\ \mathbf{else}:\\ \;\;\;\;alphay \cdot \frac{u0 \cdot alphay}{sin2phi}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (if (<= sin2phi 6.000000068087077e-19)
   (/ (* u0 (* alphax alphax)) cos2phi)
   (* alphay (/ (* u0 alphay) sin2phi))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float tmp;
	if (sin2phi <= 6.000000068087077e-19f) {
		tmp = (u0 * (alphax * alphax)) / cos2phi;
	} else {
		tmp = alphay * ((u0 * alphay) / sin2phi);
	}
	return tmp;
}

Fortran

real(4) function code(alphax, alphay, u0, cos2phi, sin2phi)
    real(4), intent (in) :: alphax
    real(4), intent (in) :: alphay
    real(4), intent (in) :: u0
    real(4), intent (in) :: cos2phi
    real(4), intent (in) :: sin2phi
    real(4) :: tmp
    if (sin2phi <= 6.000000068087077e-19) then
        tmp = (u0 * (alphax * alphax)) / cos2phi
    else
        tmp = alphay * ((u0 * alphay) / sin2phi)
    end if
    code = tmp
end function

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = Float32(0.0)
	if (sin2phi <= Float32(6.000000068087077e-19))
		tmp = Float32(Float32(u0 * Float32(alphax * alphax)) / cos2phi);
	else
		tmp = Float32(alphay * Float32(Float32(u0 * alphay) / sin2phi));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = single(0.0);
	if (sin2phi <= single(6.000000068087077e-19))
		tmp = (u0 * (alphax * alphax)) / cos2phi;
	else
		tmp = alphay * ((u0 * alphay) / sin2phi);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if sin2phi < 6.00000007e-19

   1. Initial program 54.4%

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

   2. Taylor expanded in u0 around 0 75.4%

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

      1. +-commutative 75.4%

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

      2. unpow2 75.4%

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

      3. unpow2 75.4%

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

   4. Simplified 75.4%

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

      1. frac-2neg 75.4%

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

      2. div-inv 75.4%

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

      3. distribute-rgt-neg-in 75.4%

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

   6. Applied egg-rr 75.4%

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

      1. div-inv 75.4%

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

      2. un-div-inv 75.4%

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

      3. distribute-rgt-neg-out 75.4%

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

      4. frac-2neg 75.4%

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

   8. Applied egg-rr 75.4%

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

   9. Taylor expanded in sin2phi around 0 53.7%

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

       1. unpow2 53.7%

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

       2. *-commutative 53.7%

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

   11. Simplified 53.7%

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

   if 6.00000007e-19 < sin2phi

   1. Initial program 61.8%

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

   2. Taylor expanded in u0 around 0 77.6%

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

      1. +-commutative 77.6%

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

      2. unpow2 77.6%

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

      3. unpow2 77.6%

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

   4. Simplified 77.6%

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

   5. Taylor expanded in sin2phi around inf 72.0%

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

      1. unpow2 72.0%

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

      2. *-lft-identity 72.0%

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

      3. times-frac 71.9%

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

      4. /-rgt-identity 71.9%

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

   7. Simplified 71.9%

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

   8. Taylor expanded in alphay around 0 72.0%

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

      1. associate-*r/ 71.9%

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

      2. unpow2 71.9%

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

      3. associate-*l* 71.8%

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

   10. Simplified 71.8%

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

       1. associate-*r/ 71.9%

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

       2. *-commutative 71.9%

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

   12. Applied egg-rr 71.9%

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

4. Final simplification 67.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;sin2phi \leq 6.000000068087077 \cdot 10^{-19}:\\ \;\;\;\;\frac{u0 \cdot \left(alphax \cdot alphax\right)}{cos2phi}\\ \mathbf{else}:\\ \;\;\;\;alphay \cdot \frac{u0 \cdot alphay}{sin2phi}\\ \end{array} \]

Alternative 17: 66.2% accurate, 12.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;sin2phi \leq 6.000000068087077 \cdot 10^{-19}:\\ \;\;\;\;\frac{u0 \cdot \left(alphax \cdot alphax\right)}{cos2phi}\\ \mathbf{else}:\\ \;\;\;\;\frac{u0 \cdot \left(alphay \cdot alphay\right)}{sin2phi}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (if (<= sin2phi 6.000000068087077e-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 <= 6.000000068087077e-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 <= 6.000000068087077e-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(6.000000068087077e-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(6.000000068087077e-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 < 6.00000007e-19

   1. Initial program 54.4%

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

   2. Taylor expanded in u0 around 0 75.4%

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

      1. +-commutative 75.4%

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

      2. unpow2 75.4%

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

      3. unpow2 75.4%

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

   4. Simplified 75.4%

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

      1. frac-2neg 75.4%

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

      2. div-inv 75.4%

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

      3. distribute-rgt-neg-in 75.4%

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

   6. Applied egg-rr 75.4%

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

      1. div-inv 75.4%

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

      2. un-div-inv 75.4%

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

      3. distribute-rgt-neg-out 75.4%

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

      4. frac-2neg 75.4%

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

   8. Applied egg-rr 75.4%

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

   9. Taylor expanded in sin2phi around 0 53.7%

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

       1. unpow2 53.7%

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

       2. *-commutative 53.7%

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

   11. Simplified 53.7%

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

   if 6.00000007e-19 < sin2phi

   1. Initial program 61.8%

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

   2. Taylor expanded in u0 around 0 77.6%

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

      1. +-commutative 77.6%

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

      2. unpow2 77.6%

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

      3. unpow2 77.6%

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

   4. Simplified 77.6%

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

   5. Taylor expanded in sin2phi around inf 72.0%

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

      1. unpow2 72.0%

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

      2. *-lft-identity 72.0%

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

      3. times-frac 71.9%

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

      4. /-rgt-identity 71.9%

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

   7. Simplified 71.9%

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

      1. associate-*r/ 72.0%

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

   9. Applied egg-rr 72.0%

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

4. Final simplification 67.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;sin2phi \leq 6.000000068087077 \cdot 10^{-19}:\\ \;\;\;\;\frac{u0 \cdot \left(alphax \cdot alphax\right)}{cos2phi}\\ \mathbf{else}:\\ \;\;\;\;\frac{u0 \cdot \left(alphay \cdot alphay\right)}{sin2phi}\\ \end{array} \]

Alternative 18: 57.9% 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 60.0%

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

2. Taylor expanded in u0 around 0 77.1%

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

   1. +-commutative 77.1%

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

   2. unpow2 77.1%

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

   3. unpow2 77.1%

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

4. Simplified 77.1%

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

5. Taylor expanded in sin2phi around inf 61.0%

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

   1. unpow2 61.0%

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

   2. *-lft-identity 61.0%

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

   3. times-frac 61.0%

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

   4. /-rgt-identity 61.0%

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

7. Simplified 61.0%

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

8. Taylor expanded in alphay around 0 61.0%

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

   1. associate-*r/ 61.0%

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

   2. unpow2 61.0%

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

   3. associate-*l* 61.0%

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

10. Simplified 61.0%

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

11. Final simplification 61.0%

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

Alternative 19: 57.9% accurate, 16.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 60.0%

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

2. Taylor expanded in u0 around 0 77.1%

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

   1. +-commutative 77.1%

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

   2. unpow2 77.1%

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

   3. unpow2 77.1%

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

4. Simplified 77.1%

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

5. Taylor expanded in sin2phi around inf 61.0%

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

   1. unpow2 61.0%

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

   2. *-lft-identity 61.0%

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

   3. times-frac 61.0%

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

   4. /-rgt-identity 61.0%

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

7. Simplified 61.0%

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

8. Taylor expanded in alphay around 0 61.0%

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

   1. associate-*r/ 61.0%

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

   2. unpow2 61.0%

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

   3. associate-*l* 61.0%

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

10. Simplified 61.0%

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

    1. associate-*r/ 61.0%

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

    2. *-commutative 61.0%

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

12. Applied egg-rr 61.0%

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

13. Final simplification 61.0%

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

Reproduce

herbie shell --seed 2023275 
(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)))))