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

Percentage Accurate: 60.9% → 98.2%

Time: 15.4s

Alternatives: 15

Speedup: 12.7×

Specification

\[\left(\left(\left(\left(0.0001 \leq alphax \land alphax \leq 1\right) \land \left(0.0001 \leq alphay \land alphay \leq 1\right)\right) \land \left(2.328306437 \cdot 10^{-10} \leq u0 \land u0 \leq 1\right)\right) \land \left(0 \leq cos2phi \land cos2phi \leq 1\right)\right) \land 0 \leq sin2phi\]

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Initial Program: 60.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Alternative 1: 98.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{-\mathsf{log1p}\left(-u0\right)}{\frac{sin2phi}{alphay \cdot alphay} - \frac{cos2phi}{\frac{alphax}{\frac{-1}{alphax}}}} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (/
  (- (log1p (- u0)))
  (- (/ sin2phi (* alphay alphay)) (/ cos2phi (/ alphax (/ -1.0 alphax))))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	return -log1pf(-u0) / ((sin2phi / (alphay * alphay)) - (cos2phi / (alphax / (-1.0f / 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 / Float32(Float32(-1.0) / alphax)))))
end

Derivation

1. Initial program 61.3%

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

   1. sub-neg 61.3%

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

   2. log1p-def 98.1%

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

3. Simplified 98.1%

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

   1. frac-2neg 98.1%

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

   2. div-inv 98.1%

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

   3. distribute-rgt-neg-in 98.1%

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

5. Applied egg-rr 98.1%

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

6. Taylor expanded in alphax around 0 98.1%

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

   1. unpow2 98.1%

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

8. Simplified 98.1%

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

   1. clear-num 98.1%

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

   2. inv-pow 98.1%

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

10. Applied egg-rr 98.1%

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

    1. unpow-1 98.1%

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

    2. associate-/l* 98.1%

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

12. Simplified 98.1%

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

    1. un-div-inv 98.1%

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

14. Applied egg-rr 98.1%

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

15. Final simplification 98.1%

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

Alternative 2: 98.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 61.3%

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

   1. sub-neg 61.3%

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

   2. log1p-def 98.1%

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

3. Simplified 98.1%

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

4. Final simplification 98.1%

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

Alternative 3: 92.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;sin2phi \leq 0.004000000189989805:\\ \;\;\;\;\frac{u0 - u0 \cdot \left(u0 \cdot -0.5\right)}{\frac{sin2phi}{alphay \cdot alphay} + \frac{cos2phi}{alphax \cdot alphax}}\\ \mathbf{else}:\\ \;\;\;\;\frac{alphay \cdot \left(\mathsf{log1p}\left(-u0\right) \cdot \left(-alphay\right)\right)}{sin2phi}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (if (<= sin2phi 0.004000000189989805)
   (/
    (- u0 (* u0 (* u0 -0.5)))
    (+ (/ sin2phi (* alphay alphay)) (/ cos2phi (* alphax alphax))))
   (/ (* alphay (* (log1p (- u0)) (- alphay))) sin2phi)))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float tmp;
	if (sin2phi <= 0.004000000189989805f) {
		tmp = (u0 - (u0 * (u0 * -0.5f))) / ((sin2phi / (alphay * alphay)) + (cos2phi / (alphax * alphax)));
	} else {
		tmp = (alphay * (log1pf(-u0) * -alphay)) / sin2phi;
	}
	return tmp;
}

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = Float32(0.0)
	if (sin2phi <= Float32(0.004000000189989805))
		tmp = Float32(Float32(u0 - Float32(u0 * Float32(u0 * Float32(-0.5)))) / Float32(Float32(sin2phi / Float32(alphay * alphay)) + Float32(cos2phi / Float32(alphax * alphax))));
	else
		tmp = Float32(Float32(alphay * Float32(log1p(Float32(-u0)) * Float32(-alphay))) / sin2phi);
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if sin2phi < 0.00400000019

   1. Initial program 55.6%

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

   2. Taylor expanded in u0 around 0 88.4%

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

      1. +-commutative 41.9%

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

      2. mul-1-neg 41.9%

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

      3. unsub-neg 41.9%

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

      4. *-commutative 41.9%

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

      5. unpow2 41.9%

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

      6. associate-*l* 41.9%

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

   4. Simplified 88.4%

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

   if 0.00400000019 < sin2phi

   1. Initial program 66.9%

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

   2. Taylor expanded in cos2phi around 0 68.4%

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

      1. mul-1-neg 68.4%

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

      2. unpow2 68.4%

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

      3. *-commutative 68.4%

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

   4. Simplified 68.4%

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

   5. Taylor expanded in alphay around 0 68.4%

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

      1. sub-neg 68.4%

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

      2. log1p-def 99.0%

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

      3. unpow2 99.0%

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

      4. associate-*l* 99.1%

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

   7. Simplified 99.1%

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

4. Final simplification 93.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;sin2phi \leq 0.004000000189989805:\\ \;\;\;\;\frac{u0 - u0 \cdot \left(u0 \cdot -0.5\right)}{\frac{sin2phi}{alphay \cdot alphay} + \frac{cos2phi}{alphax \cdot alphax}}\\ \mathbf{else}:\\ \;\;\;\;\frac{alphay \cdot \left(\mathsf{log1p}\left(-u0\right) \cdot \left(-alphay\right)\right)}{sin2phi}\\ \end{array} \]

Alternative 4: 80.9% accurate, 5.5× speedup

Math

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

FPCore

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

C

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

Fortran

real(4) function code(alphax, alphay, u0, cos2phi, sin2phi)
    real(4), intent (in) :: alphax
    real(4), intent (in) :: alphay
    real(4), intent (in) :: u0
    real(4), intent (in) :: cos2phi
    real(4), intent (in) :: sin2phi
    real(4) :: tmp
    if ((sin2phi / (alphay * alphay)) <= 0.019999999552965164e0) then
        tmp = u0 / ((cos2phi / (alphax * alphax)) + ((sin2phi / alphay) * (1.0e0 / alphay)))
    else
        tmp = ((alphay * alphay) * (u0 - (u0 * (u0 * (-0.5e0))))) / sin2phi
    end if
    code = tmp
end function

Julia

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

MATLAB

function tmp_2 = code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = single(0.0);
	if ((sin2phi / (alphay * alphay)) <= single(0.019999999552965164))
		tmp = u0 / ((cos2phi / (alphax * alphax)) + ((sin2phi / alphay) * (single(1.0) / alphay)));
	else
		tmp = ((alphay * alphay) * (u0 - (u0 * (u0 * single(-0.5))))) / sin2phi;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 sin2phi (*.f32 alphay alphay)) < 0.0199999996

   1. Initial program 55.4%

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

   2. Taylor expanded in u0 around 0 74.7%

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

      1. +-commutative 74.7%

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

      2. unpow2 74.7%

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

      3. unpow2 74.7%

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

   4. Simplified 74.7%

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

      1. associate-/r* 74.7%

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

      2. div-inv 74.8%

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

   6. Applied egg-rr 74.8%

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

   if 0.0199999996 < (/.f32 sin2phi (*.f32 alphay alphay))

   1. Initial program 66.5%

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

   2. Taylor expanded in cos2phi around 0 67.7%

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

      1. mul-1-neg 67.7%

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

      2. unpow2 67.7%

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

      3. *-commutative 67.7%

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

   4. Simplified 67.7%

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

   5. Taylor expanded in u0 around 0 87.3%

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

      1. +-commutative 87.3%

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

      2. mul-1-neg 87.3%

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

      3. unsub-neg 87.3%

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

      4. *-commutative 87.3%

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

      5. unpow2 87.3%

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

      6. associate-*l* 87.3%

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

   7. Simplified 87.3%

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

4. Final simplification 81.5%

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

Alternative 5: 80.9% accurate, 6.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{sin2phi}{alphay \cdot alphay}\\ \mathbf{if}\;t_0 \leq 0.019999999552965164:\\ \;\;\;\;\frac{u0}{t_0 + \frac{cos2phi}{alphax \cdot alphax}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(alphay \cdot alphay\right) \cdot \left(u0 - u0 \cdot \left(u0 \cdot -0.5\right)\right)}{sin2phi}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (let* ((t_0 (/ sin2phi (* alphay alphay))))
   (if (<= t_0 0.019999999552965164)
     (/ u0 (+ t_0 (/ cos2phi (* alphax alphax))))
     (/ (* (* alphay alphay) (- u0 (* u0 (* u0 -0.5)))) sin2phi))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float t_0 = sin2phi / (alphay * alphay);
	float tmp;
	if (t_0 <= 0.019999999552965164f) {
		tmp = u0 / (t_0 + (cos2phi / (alphax * alphax)));
	} else {
		tmp = ((alphay * alphay) * (u0 - (u0 * (u0 * -0.5f)))) / sin2phi;
	}
	return tmp;
}

Fortran

real(4) function code(alphax, alphay, u0, cos2phi, sin2phi)
    real(4), intent (in) :: alphax
    real(4), intent (in) :: alphay
    real(4), intent (in) :: u0
    real(4), intent (in) :: cos2phi
    real(4), intent (in) :: sin2phi
    real(4) :: t_0
    real(4) :: tmp
    t_0 = sin2phi / (alphay * alphay)
    if (t_0 <= 0.019999999552965164e0) then
        tmp = u0 / (t_0 + (cos2phi / (alphax * alphax)))
    else
        tmp = ((alphay * alphay) * (u0 - (u0 * (u0 * (-0.5e0))))) / sin2phi
    end if
    code = tmp
end function

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	t_0 = Float32(sin2phi / Float32(alphay * alphay))
	tmp = Float32(0.0)
	if (t_0 <= Float32(0.019999999552965164))
		tmp = Float32(u0 / Float32(t_0 + Float32(cos2phi / Float32(alphax * alphax))));
	else
		tmp = Float32(Float32(Float32(alphay * alphay) * Float32(u0 - Float32(u0 * Float32(u0 * Float32(-0.5))))) / sin2phi);
	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(0.019999999552965164))
		tmp = u0 / (t_0 + (cos2phi / (alphax * alphax)));
	else
		tmp = ((alphay * alphay) * (u0 - (u0 * (u0 * single(-0.5))))) / sin2phi;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 sin2phi (*.f32 alphay alphay)) < 0.0199999996

   1. Initial program 55.4%

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

   2. Taylor expanded in u0 around 0 74.7%

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

      1. +-commutative 74.7%

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

      2. unpow2 74.7%

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

      3. unpow2 74.7%

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

   4. Simplified 74.7%

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

   if 0.0199999996 < (/.f32 sin2phi (*.f32 alphay alphay))

   1. Initial program 66.5%

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

   2. Taylor expanded in cos2phi around 0 67.7%

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

      1. mul-1-neg 67.7%

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

      2. unpow2 67.7%

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

      3. *-commutative 67.7%

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

   4. Simplified 67.7%

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

   5. Taylor expanded in u0 around 0 87.3%

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

      1. +-commutative 87.3%

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

      2. mul-1-neg 87.3%

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

      3. unsub-neg 87.3%

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

      4. *-commutative 87.3%

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

      5. unpow2 87.3%

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

      6. associate-*l* 87.3%

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

   7. Simplified 87.3%

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

4. Final simplification 81.5%

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

Alternative 6: 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 61.3%

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

2. Taylor expanded in u0 around 0 87.2%

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

   1. +-commutative 64.7%

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

   2. mul-1-neg 64.7%

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

   3. unsub-neg 64.7%

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

   4. *-commutative 64.7%

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

   5. unpow2 64.7%

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

   6. associate-*l* 64.7%

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

4. Simplified 87.2%

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

5. Final simplification 87.2%

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

Alternative 7: 68.7% accurate, 6.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 1.3999999839026675 \cdot 10^{-15}:\\ \;\;\;\;\frac{\frac{alphax \cdot alphax}{cos2phi}}{-0.5 + \frac{1}{u0}}\\ \mathbf{else}:\\ \;\;\;\;\left(alphay \cdot alphay\right) \cdot \frac{u0}{sin2phi}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (if (<= (/ sin2phi (* alphay alphay)) 1.3999999839026675e-15)
   (/ (/ (* alphax alphax) cos2phi) (+ -0.5 (/ 1.0 u0)))
   (* (* alphay alphay) (/ u0 sin2phi))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float tmp;
	if ((sin2phi / (alphay * alphay)) <= 1.3999999839026675e-15f) {
		tmp = ((alphax * alphax) / cos2phi) / (-0.5f + (1.0f / u0));
	} else {
		tmp = (alphay * alphay) * (u0 / sin2phi);
	}
	return tmp;
}

Fortran

real(4) function code(alphax, alphay, u0, cos2phi, sin2phi)
    real(4), intent (in) :: alphax
    real(4), intent (in) :: alphay
    real(4), intent (in) :: u0
    real(4), intent (in) :: cos2phi
    real(4), intent (in) :: sin2phi
    real(4) :: tmp
    if ((sin2phi / (alphay * alphay)) <= 1.3999999839026675e-15) then
        tmp = ((alphax * alphax) / cos2phi) / ((-0.5e0) + (1.0e0 / u0))
    else
        tmp = (alphay * alphay) * (u0 / sin2phi)
    end if
    code = tmp
end function

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = Float32(0.0)
	if (Float32(sin2phi / Float32(alphay * alphay)) <= Float32(1.3999999839026675e-15))
		tmp = Float32(Float32(Float32(alphax * alphax) / cos2phi) / Float32(Float32(-0.5) + Float32(Float32(1.0) / u0)));
	else
		tmp = Float32(Float32(alphay * alphay) * Float32(u0 / sin2phi));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = single(0.0);
	if ((sin2phi / (alphay * alphay)) <= single(1.3999999839026675e-15))
		tmp = ((alphax * alphax) / cos2phi) / (single(-0.5) + (single(1.0) / u0));
	else
		tmp = (alphay * alphay) * (u0 / sin2phi);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 sin2phi (*.f32 alphay alphay)) < 1.39999998e-15

   1. Initial program 55.7%

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

   2. Taylor expanded in cos2phi around inf 42.7%

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

      1. mul-1-neg 42.7%

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

      2. unpow2 42.7%

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

      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 72.3%

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

      9. mul-1-neg 72.3%

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

   4. Simplified 72.3%

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

   5. Taylor expanded in u0 around 0 68.5%

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

      1. +-commutative 68.5%

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

      2. mul-1-neg 68.5%

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

      3. unsub-neg 68.5%

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

      4. *-commutative 68.5%

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

   7. Simplified 68.5%

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

   8. Taylor expanded in cos2phi around -inf 68.4%

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

      1. associate-/r* 68.5%

         \[\leadsto \color{blue}{\frac{\frac{{alphax}^{2}}{cos2phi}}{\frac{1}{u0} - 0.5}} \]

      2. unpow2 68.5%

         \[\leadsto \frac{\frac{\color{blue}{alphax \cdot alphax}}{cos2phi}}{\frac{1}{u0} - 0.5} \]

      3. sub-neg 68.5%

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

      4. metadata-eval 68.5%

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

   10. Simplified 68.5%

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

   if 1.39999998e-15 < (/.f32 sin2phi (*.f32 alphay alphay))

   1. Initial program 63.3%

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

   2. Taylor expanded in u0 around 0 75.3%

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

      1. +-commutative 75.3%

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

      2. unpow2 75.3%

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

      3. unpow2 75.3%

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

   4. Simplified 75.3%

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

   5. Taylor expanded in sin2phi around inf 68.9%

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

      1. unpow2 68.9%

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

   7. Simplified 68.9%

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

      1. associate-/r/ 69.2%

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

   9. Applied egg-rr 69.2%

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

4. Final simplification 69.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 1.3999999839026675 \cdot 10^{-15}:\\ \;\;\;\;\frac{\frac{alphax \cdot alphax}{cos2phi}}{-0.5 + \frac{1}{u0}}\\ \mathbf{else}:\\ \;\;\;\;\left(alphay \cdot alphay\right) \cdot \frac{u0}{sin2phi}\\ \end{array} \]

Alternative 8: 68.7% accurate, 8.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;sin2phi \leq 9.000000102130615 \cdot 10^{-19}:\\ \;\;\;\;\frac{alphax}{\frac{cos2phi \cdot 0.5 - \frac{cos2phi}{u0}}{-alphax}}\\ \mathbf{else}:\\ \;\;\;\;alphay \cdot \frac{u0 \cdot alphay}{sin2phi}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (if (<= sin2phi 9.000000102130615e-19)
   (/ alphax (/ (- (* cos2phi 0.5) (/ cos2phi u0)) (- alphax)))
   (* alphay (/ (* u0 alphay) sin2phi))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float tmp;
	if (sin2phi <= 9.000000102130615e-19f) {
		tmp = alphax / (((cos2phi * 0.5f) - (cos2phi / u0)) / -alphax);
	} 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 <= 9.000000102130615e-19) then
        tmp = alphax / (((cos2phi * 0.5e0) - (cos2phi / u0)) / -alphax)
    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(9.000000102130615e-19))
		tmp = Float32(alphax / Float32(Float32(Float32(cos2phi * Float32(0.5)) - Float32(cos2phi / u0)) / Float32(-alphax)));
	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(9.000000102130615e-19))
		tmp = alphax / (((cos2phi * single(0.5)) - (cos2phi / u0)) / -alphax);
	else
		tmp = alphay * ((u0 * alphay) / sin2phi);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if sin2phi < 9.0000001e-19

   1. Initial program 58.3%

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

   2. Taylor expanded in cos2phi around inf 41.4%

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

      1. mul-1-neg 41.4%

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

      2. unpow2 41.4%

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

      3. associate-/l* 41.4%

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

      4. distribute-neg-frac 41.4%

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

      5. distribute-rgt-neg-out 41.4%

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

      6. sub-neg 41.4%

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

      7. mul-1-neg 41.4%

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

      8. log1p-def 67.9%

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

      9. mul-1-neg 67.9%

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

   4. Simplified 67.9%

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

   5. Taylor expanded in u0 around 0 64.4%

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

      1. +-commutative 64.4%

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

      2. mul-1-neg 64.4%

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

      3. unsub-neg 64.4%

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

      4. *-commutative 64.4%

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

   7. Simplified 64.4%

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

   8. Taylor expanded in alphax around 0 64.4%

      \[\leadsto \color{blue}{-1 \cdot \frac{{alphax}^{2}}{0.5 \cdot cos2phi - \frac{cos2phi}{u0}}} \]
   9. Step-by-step derivation

      1. associate-*r/ 64.4%

         \[\leadsto \color{blue}{\frac{-1 \cdot {alphax}^{2}}{0.5 \cdot cos2phi - \frac{cos2phi}{u0}}} \]

      2. mul-1-neg 64.4%

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

      3. unpow2 64.4%

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

      4. distribute-rgt-neg-out 64.4%

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

      5. *-commutative 64.4%

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

      6. associate-/l* 64.4%

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

   10. Simplified 64.4%

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

   if 9.0000001e-19 < sin2phi

   1. Initial program 62.6%

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

   2. Taylor expanded in u0 around 0 76.2%

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

      1. +-commutative 76.2%

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

      2. unpow2 76.2%

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

      3. unpow2 76.2%

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

   4. Simplified 76.2%

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

   5. Taylor expanded in sin2phi around inf 70.7%

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

      1. unpow2 70.7%

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

   7. Simplified 70.7%

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

   8. Taylor expanded in u0 around 0 71.0%

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

      1. associate-*r/ 71.1%

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

      2. unpow2 71.1%

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

      3. associate-*l* 71.0%

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

   10. Simplified 71.0%

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

       1. associate-*r/ 71.1%

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

       2. *-commutative 71.1%

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

   12. Applied egg-rr 71.1%

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

4. Final simplification 69.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;sin2phi \leq 9.000000102130615 \cdot 10^{-19}:\\ \;\;\;\;\frac{alphax}{\frac{cos2phi \cdot 0.5 - \frac{cos2phi}{u0}}{-alphax}}\\ \mathbf{else}:\\ \;\;\;\;alphay \cdot \frac{u0 \cdot alphay}{sin2phi}\\ \end{array} \]

Alternative 9: 68.7% accurate, 8.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;sin2phi \leq 9.000000102130615 \cdot 10^{-19}:\\ \;\;\;\;\frac{-alphax \cdot alphax}{cos2phi \cdot 0.5 - \frac{cos2phi}{u0}}\\ \mathbf{else}:\\ \;\;\;\;alphay \cdot \frac{u0 \cdot alphay}{sin2phi}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (if (<= sin2phi 9.000000102130615e-19)
   (/ (- (* alphax alphax)) (- (* cos2phi 0.5) (/ cos2phi u0)))
   (* alphay (/ (* u0 alphay) sin2phi))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float tmp;
	if (sin2phi <= 9.000000102130615e-19f) {
		tmp = -(alphax * alphax) / ((cos2phi * 0.5f) - (cos2phi / u0));
	} 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 <= 9.000000102130615e-19) then
        tmp = -(alphax * alphax) / ((cos2phi * 0.5e0) - (cos2phi / u0))
    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(9.000000102130615e-19))
		tmp = Float32(Float32(-Float32(alphax * alphax)) / Float32(Float32(cos2phi * Float32(0.5)) - Float32(cos2phi / u0)));
	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(9.000000102130615e-19))
		tmp = -(alphax * alphax) / ((cos2phi * single(0.5)) - (cos2phi / u0));
	else
		tmp = alphay * ((u0 * alphay) / sin2phi);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if sin2phi < 9.0000001e-19

   1. Initial program 58.3%

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

   2. Taylor expanded in cos2phi around inf 41.4%

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

      1. mul-1-neg 41.4%

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

      2. unpow2 41.4%

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

      3. associate-/l* 41.4%

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

      4. distribute-neg-frac 41.4%

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

      5. distribute-rgt-neg-out 41.4%

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

      6. sub-neg 41.4%

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

      7. mul-1-neg 41.4%

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

      8. log1p-def 67.9%

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

      9. mul-1-neg 67.9%

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

   4. Simplified 67.9%

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

   5. Taylor expanded in u0 around 0 64.4%

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

      1. +-commutative 64.4%

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

      2. mul-1-neg 64.4%

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

      3. unsub-neg 64.4%

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

      4. *-commutative 64.4%

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

   7. Simplified 64.4%

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

   if 9.0000001e-19 < sin2phi

   1. Initial program 62.6%

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

   2. Taylor expanded in u0 around 0 76.2%

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

      1. +-commutative 76.2%

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

      2. unpow2 76.2%

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

      3. unpow2 76.2%

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

   4. Simplified 76.2%

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

   5. Taylor expanded in sin2phi around inf 70.7%

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

      1. unpow2 70.7%

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

   7. Simplified 70.7%

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

   8. Taylor expanded in u0 around 0 71.0%

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

      1. associate-*r/ 71.1%

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

      2. unpow2 71.1%

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

      3. associate-*l* 71.0%

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

   10. Simplified 71.0%

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

       1. associate-*r/ 71.1%

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

       2. *-commutative 71.1%

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

   12. Applied egg-rr 71.1%

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

4. Final simplification 69.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;sin2phi \leq 9.000000102130615 \cdot 10^{-19}:\\ \;\;\;\;\frac{-alphax \cdot alphax}{cos2phi \cdot 0.5 - \frac{cos2phi}{u0}}\\ \mathbf{else}:\\ \;\;\;\;alphay \cdot \frac{u0 \cdot alphay}{sin2phi}\\ \end{array} \]

Alternative 10: 66.4% accurate, 8.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 1.3999999839026675 \cdot 10^{-15}:\\ \;\;\;\;u0 \cdot \frac{alphax \cdot alphax}{cos2phi}\\ \mathbf{else}:\\ \;\;\;\;\left(alphay \cdot alphay\right) \cdot \frac{u0}{sin2phi}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (if (<= (/ sin2phi (* alphay alphay)) 1.3999999839026675e-15)
   (* u0 (/ (* alphax alphax) cos2phi))
   (* (* alphay alphay) (/ u0 sin2phi))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float tmp;
	if ((sin2phi / (alphay * alphay)) <= 1.3999999839026675e-15f) {
		tmp = u0 * ((alphax * alphax) / cos2phi);
	} else {
		tmp = (alphay * alphay) * (u0 / sin2phi);
	}
	return tmp;
}

Fortran

real(4) function code(alphax, alphay, u0, cos2phi, sin2phi)
    real(4), intent (in) :: alphax
    real(4), intent (in) :: alphay
    real(4), intent (in) :: u0
    real(4), intent (in) :: cos2phi
    real(4), intent (in) :: sin2phi
    real(4) :: tmp
    if ((sin2phi / (alphay * alphay)) <= 1.3999999839026675e-15) then
        tmp = u0 * ((alphax * alphax) / cos2phi)
    else
        tmp = (alphay * alphay) * (u0 / sin2phi)
    end if
    code = tmp
end function

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = Float32(0.0)
	if (Float32(sin2phi / Float32(alphay * alphay)) <= Float32(1.3999999839026675e-15))
		tmp = Float32(u0 * Float32(Float32(alphax * alphax) / cos2phi));
	else
		tmp = Float32(Float32(alphay * alphay) * Float32(u0 / sin2phi));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = single(0.0);
	if ((sin2phi / (alphay * alphay)) <= single(1.3999999839026675e-15))
		tmp = u0 * ((alphax * alphax) / cos2phi);
	else
		tmp = (alphay * alphay) * (u0 / sin2phi);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 sin2phi (*.f32 alphay alphay)) < 1.39999998e-15

   1. Initial program 55.7%

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

   2. Taylor expanded in cos2phi around inf 42.7%

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

      1. mul-1-neg 42.7%

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

      2. unpow2 42.7%

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

      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 72.3%

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

      9. mul-1-neg 72.3%

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

   4. Simplified 72.3%

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

   5. Taylor expanded in u0 around 0 68.5%

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

      1. +-commutative 68.5%

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

      2. mul-1-neg 68.5%

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

      3. unsub-neg 68.5%

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

      4. *-commutative 68.5%

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

   7. Simplified 68.5%

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

   8. Taylor expanded in u0 around 0 58.0%

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

      1. associate-/l* 58.1%

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

      2. associate-/r/ 58.1%

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

      3. unpow2 58.1%

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

   10. Simplified 58.1%

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

   if 1.39999998e-15 < (/.f32 sin2phi (*.f32 alphay alphay))

   1. Initial program 63.3%

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

   2. Taylor expanded in u0 around 0 75.3%

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

      1. +-commutative 75.3%

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

      2. unpow2 75.3%

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

      3. unpow2 75.3%

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

   4. Simplified 75.3%

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

   5. Taylor expanded in sin2phi around inf 68.9%

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

      1. unpow2 68.9%

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

   7. Simplified 68.9%

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

      1. associate-/r/ 69.2%

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

   9. Applied egg-rr 69.2%

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

4. Final simplification 66.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 1.3999999839026675 \cdot 10^{-15}:\\ \;\;\;\;u0 \cdot \frac{alphax \cdot alphax}{cos2phi}\\ \mathbf{else}:\\ \;\;\;\;\left(alphay \cdot alphay\right) \cdot \frac{u0}{sin2phi}\\ \end{array} \]

Alternative 11: 66.4% accurate, 8.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 1.3999999839026675 \cdot 10^{-15}:\\ \;\;\;\;\frac{alphax \cdot alphax}{\frac{cos2phi}{u0}}\\ \mathbf{else}:\\ \;\;\;\;\left(alphay \cdot alphay\right) \cdot \frac{u0}{sin2phi}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (if (<= (/ sin2phi (* alphay alphay)) 1.3999999839026675e-15)
   (/ (* alphax alphax) (/ cos2phi u0))
   (* (* alphay alphay) (/ u0 sin2phi))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float tmp;
	if ((sin2phi / (alphay * alphay)) <= 1.3999999839026675e-15f) {
		tmp = (alphax * alphax) / (cos2phi / u0);
	} else {
		tmp = (alphay * alphay) * (u0 / sin2phi);
	}
	return tmp;
}

Fortran

real(4) function code(alphax, alphay, u0, cos2phi, sin2phi)
    real(4), intent (in) :: alphax
    real(4), intent (in) :: alphay
    real(4), intent (in) :: u0
    real(4), intent (in) :: cos2phi
    real(4), intent (in) :: sin2phi
    real(4) :: tmp
    if ((sin2phi / (alphay * alphay)) <= 1.3999999839026675e-15) then
        tmp = (alphax * alphax) / (cos2phi / u0)
    else
        tmp = (alphay * alphay) * (u0 / sin2phi)
    end if
    code = tmp
end function

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = Float32(0.0)
	if (Float32(sin2phi / Float32(alphay * alphay)) <= Float32(1.3999999839026675e-15))
		tmp = Float32(Float32(alphax * alphax) / Float32(cos2phi / u0));
	else
		tmp = Float32(Float32(alphay * alphay) * Float32(u0 / sin2phi));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = single(0.0);
	if ((sin2phi / (alphay * alphay)) <= single(1.3999999839026675e-15))
		tmp = (alphax * alphax) / (cos2phi / u0);
	else
		tmp = (alphay * alphay) * (u0 / sin2phi);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 sin2phi (*.f32 alphay alphay)) < 1.39999998e-15

   1. Initial program 55.7%

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

   2. Taylor expanded in u0 around 0 75.0%

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

      1. +-commutative 75.0%

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

      2. unpow2 75.0%

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

      3. unpow2 75.0%

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

   4. Simplified 75.0%

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

   5. Taylor expanded in sin2phi around 0 58.0%

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

      1. associate-/l* 58.1%

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

      2. unpow2 58.1%

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

   7. Simplified 58.1%

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

   if 1.39999998e-15 < (/.f32 sin2phi (*.f32 alphay alphay))

   1. Initial program 63.3%

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

   2. Taylor expanded in u0 around 0 75.3%

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

      1. +-commutative 75.3%

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

      2. unpow2 75.3%

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

      3. unpow2 75.3%

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

   4. Simplified 75.3%

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

   5. Taylor expanded in sin2phi around inf 68.9%

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

      1. unpow2 68.9%

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

   7. Simplified 68.9%

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

      1. associate-/r/ 69.2%

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

   9. Applied egg-rr 69.2%

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

4. Final simplification 66.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 1.3999999839026675 \cdot 10^{-15}:\\ \;\;\;\;\frac{alphax \cdot alphax}{\frac{cos2phi}{u0}}\\ \mathbf{else}:\\ \;\;\;\;\left(alphay \cdot alphay\right) \cdot \frac{u0}{sin2phi}\\ \end{array} \]

Alternative 12: 75.4% accurate, 8.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 61.3%

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

2. Taylor expanded in u0 around 0 75.2%

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

   1. +-commutative 75.2%

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

   2. unpow2 75.2%

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

   3. unpow2 75.2%

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

4. Simplified 75.2%

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

5. Final simplification 75.2%

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

Alternative 13: 66.4% accurate, 12.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;sin2phi \leq 9.000000102130615 \cdot 10^{-19}:\\ \;\;\;\;u0 \cdot \left(alphax \cdot \frac{alphax}{cos2phi}\right)\\ \mathbf{else}:\\ \;\;\;\;alphay \cdot \frac{u0 \cdot alphay}{sin2phi}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (if (<= sin2phi 9.000000102130615e-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 <= 9.000000102130615e-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 <= 9.000000102130615e-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(9.000000102130615e-19))
		tmp = Float32(u0 * Float32(alphax * Float32(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(9.000000102130615e-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 < 9.0000001e-19

   1. Initial program 58.3%

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

   2. Taylor expanded in u0 around 0 73.0%

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

      1. +-commutative 73.0%

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

      2. unpow2 73.0%

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

      3. unpow2 73.0%

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

   4. Simplified 73.0%

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

      1. *-un-lft-identity 73.0%

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

      2. *-commutative 73.0%

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

      3. associate-/r* 72.9%

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

   6. Applied egg-rr 72.9%

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

   7. Taylor expanded in sin2phi around 0 54.3%

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

      1. associate-/l* 54.5%

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

      2. associate-/r/ 54.4%

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

      3. unpow2 54.4%

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

      4. associate-*l/ 54.4%

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

      5. *-commutative 54.4%

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

   9. Simplified 54.4%

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

   if 9.0000001e-19 < sin2phi

   1. Initial program 62.6%

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

   2. Taylor expanded in u0 around 0 76.2%

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

      1. +-commutative 76.2%

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

      2. unpow2 76.2%

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

      3. unpow2 76.2%

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

   4. Simplified 76.2%

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

   5. Taylor expanded in sin2phi around inf 70.7%

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

      1. unpow2 70.7%

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

   7. Simplified 70.7%

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

   8. Taylor expanded in u0 around 0 71.0%

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

      1. associate-*r/ 71.1%

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

      2. unpow2 71.1%

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

      3. associate-*l* 71.0%

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

   10. Simplified 71.0%

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

       1. associate-*r/ 71.1%

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

       2. *-commutative 71.1%

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

   12. Applied egg-rr 71.1%

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

4. Final simplification 66.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;sin2phi \leq 9.000000102130615 \cdot 10^{-19}:\\ \;\;\;\;u0 \cdot \left(alphax \cdot \frac{alphax}{cos2phi}\right)\\ \mathbf{else}:\\ \;\;\;\;alphay \cdot \frac{u0 \cdot alphay}{sin2phi}\\ \end{array} \]

Alternative 14: 58.3% accurate, 16.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 61.3%

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

2. Taylor expanded in u0 around 0 75.2%

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

   1. +-commutative 75.2%

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

   2. unpow2 75.2%

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

   3. unpow2 75.2%

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

4. Simplified 75.2%

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

5. Taylor expanded in sin2phi around inf 56.7%

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

   1. unpow2 56.7%

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

7. Simplified 56.7%

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

8. Taylor expanded in u0 around 0 56.9%

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

   1. associate-*r/ 57.0%

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

   2. unpow2 57.0%

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

   3. associate-*l* 56.9%

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

10. Simplified 56.9%

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

11. Final simplification 56.9%

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

Alternative 15: 58.3% 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 61.3%

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

2. Taylor expanded in u0 around 0 75.2%

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

   1. +-commutative 75.2%

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

   2. unpow2 75.2%

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

   3. unpow2 75.2%

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

4. Simplified 75.2%

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

5. Taylor expanded in sin2phi around inf 56.7%

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

   1. unpow2 56.7%

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

7. Simplified 56.7%

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

8. Taylor expanded in u0 around 0 56.9%

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

   1. associate-*r/ 57.0%

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

   2. unpow2 57.0%

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

   3. associate-*l* 56.9%

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

10. Simplified 56.9%

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

    1. associate-*r/ 57.0%

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

    2. *-commutative 57.0%

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

12. Applied egg-rr 57.0%

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

13. Final simplification 57.0%

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

Reproduce

herbie shell --seed 2023306 
(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)))))