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

Percentage Accurate: 60.6% → 98.4%

Time: 11.6s

Alternatives: 12

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: 60.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Alternative 1: 98.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 62.0%

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

   1. neg-sub0 62.0%

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

   2. div-sub 62.0%

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

   3. --rgt-identity 62.0%

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

   4. div-sub 62.0%

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

   5. --rgt-identity 62.0%

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

   6. neg-sub0 62.0%

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

   7. sub-neg 62.0%

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

   8. log1p-def 98.4%

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

3. Simplified 98.4%

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

4. Final simplification 98.4%

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

Alternative 2: 92.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;sin2phi \leq 1.4999999621068127 \cdot 10^{-5}:\\ \;\;\;\;\frac{u0 - u0 \cdot \left(u0 \cdot -0.5\right)}{\frac{sin2phi}{alphay \cdot alphay} + \frac{\frac{cos2phi}{alphax}}{alphax}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(-u0\right) \cdot \left(alphay \cdot \left(-alphay\right)\right)}{sin2phi}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (if (<= sin2phi 1.4999999621068127e-5)
   (/
    (- u0 (* u0 (* u0 -0.5)))
    (+ (/ sin2phi (* alphay alphay)) (/ (/ cos2phi alphax) alphax)))
   (/ (* (log1p (- u0)) (* alphay (- alphay))) sin2phi)))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float tmp;
	if (sin2phi <= 1.4999999621068127e-5f) {
		tmp = (u0 - (u0 * (u0 * -0.5f))) / ((sin2phi / (alphay * alphay)) + ((cos2phi / alphax) / alphax));
	} else {
		tmp = (log1pf(-u0) * (alphay * -alphay)) / sin2phi;
	}
	return tmp;
}

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = Float32(0.0)
	if (sin2phi <= Float32(1.4999999621068127e-5))
		tmp = Float32(Float32(u0 - Float32(u0 * Float32(u0 * Float32(-0.5)))) / Float32(Float32(sin2phi / Float32(alphay * alphay)) + Float32(Float32(cos2phi / alphax) / alphax)));
	else
		tmp = Float32(Float32(log1p(Float32(-u0)) * Float32(alphay * Float32(-alphay))) / sin2phi);
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if sin2phi < 1.49999996e-5

   1. Initial program 55.4%

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

      1. associate-/r* 55.4%

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

   3. Simplified 55.4%

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

   4. Taylor expanded in u0 around 0 87.3%

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

      1. +-commutative 87.3%

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

      2. mul-1-neg 87.3%

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

      3. unsub-neg 87.3%

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

      4. unpow2 87.3%

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

      5. associate-*r* 87.3%

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

   6. Simplified 87.3%

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

   if 1.49999996e-5 < sin2phi

   1. Initial program 67.2%

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

      1. neg-sub0 67.2%

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

      2. div-sub 67.2%

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

      3. --rgt-identity 67.2%

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

      4. div-sub 67.2%

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

      5. --rgt-identity 67.2%

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

      6. sub-neg 67.2%

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

      7. +-commutative 67.2%

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

      8. neg-sub0 67.2%

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

      9. associate-+l- 67.2%

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

      10. sub0-neg 67.2%

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

      11. neg-mul-1 67.2%

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

      12. log-prod -0.0%

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

      13. associate--r+ -0.0%

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

   3. Simplified 98.0%

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

      1. div-inv 98.0%

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

   5. Applied egg-rr 98.0%

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

      1. div-inv 98.0%

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

      2. clear-num 98.0%

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

   7. Applied egg-rr 98.0%

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

   8. Taylor expanded in alphax around inf 67.1%

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

      1. mul-1-neg 67.1%

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

      2. unpow2 67.1%

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

      3. sub-neg 67.1%

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

      4. log1p-def 97.7%

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

   10. Simplified 97.7%

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

4. Final simplification 93.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;sin2phi \leq 1.4999999621068127 \cdot 10^{-5}:\\ \;\;\;\;\frac{u0 - u0 \cdot \left(u0 \cdot -0.5\right)}{\frac{sin2phi}{alphay \cdot alphay} + \frac{\frac{cos2phi}{alphax}}{alphax}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(-u0\right) \cdot \left(alphay \cdot \left(-alphay\right)\right)}{sin2phi}\\ \end{array} \]

Alternative 3: 83.5% accurate, 4.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{sin2phi}{alphay \cdot alphay}\\ \mathbf{if}\;t_0 \leq 0.019999999552965164:\\ \;\;\;\;\frac{u0}{t_0 + \frac{cos2phi}{alphax} \cdot \frac{1}{alphax}}\\ \mathbf{else}:\\ \;\;\;\;\frac{alphay \cdot \left(-alphay\right)}{\left(sin2phi \cdot 0.5 - \frac{sin2phi}{u0}\right) - sin2phi \cdot \left(u0 \cdot -0.08333333333333333\right)}\\ \end{array} \end{array} \]

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) (/ 1.0 alphax))))
     (/
      (* alphay (- alphay))
      (-
       (- (* sin2phi 0.5) (/ sin2phi u0))
       (* sin2phi (* u0 -0.08333333333333333)))))))

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) * (1.0f / alphax)));
	} else {
		tmp = (alphay * -alphay) / (((sin2phi * 0.5f) - (sin2phi / u0)) - (sin2phi * (u0 * -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) :: t_0
    real(4) :: tmp
    t_0 = sin2phi / (alphay * alphay)
    if (t_0 <= 0.019999999552965164e0) then
        tmp = u0 / (t_0 + ((cos2phi / alphax) * (1.0e0 / alphax)))
    else
        tmp = (alphay * -alphay) / (((sin2phi * 0.5e0) - (sin2phi / u0)) - (sin2phi * (u0 * (-0.08333333333333333e0))))
    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(Float32(cos2phi / alphax) * Float32(Float32(1.0) / alphax))));
	else
		tmp = Float32(Float32(alphay * Float32(-alphay)) / Float32(Float32(Float32(sin2phi * Float32(0.5)) - Float32(sin2phi / u0)) - Float32(sin2phi * Float32(u0 * Float32(-0.08333333333333333)))));
	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) * (single(1.0) / alphax)));
	else
		tmp = (alphay * -alphay) / (((sin2phi * single(0.5)) - (sin2phi / u0)) - (sin2phi * (u0 * single(-0.08333333333333333))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

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

   1. Initial program 55.9%

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

      1. associate-/r* 55.9%

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

   3. Simplified 55.9%

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

   4. Taylor expanded in u0 around 0 73.8%

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

      1. unpow2 73.8%

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

      2. unpow2 73.8%

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

   6. Simplified 73.8%

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

      1. associate-/r* 73.7%

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

      2. div-inv 73.8%

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

   8. Applied egg-rr 73.8%

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

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

   1. Initial program 66.9%

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

      1. associate-/r* 66.9%

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

   3. Simplified 66.9%

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

   4. Taylor expanded in cos2phi around 0 67.2%

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

      1. mul-1-neg 67.2%

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

      2. unpow2 67.2%

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

      3. associate-/l* 66.4%

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

      4. distribute-neg-frac 66.4%

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

      5. distribute-rgt-neg-in 66.4%

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

      6. sub-neg 66.4%

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

      7. mul-1-neg 66.4%

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

      8. log1p-def 96.5%

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

      9. mul-1-neg 96.5%

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

   6. Simplified 96.5%

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

   7. Taylor expanded in sin2phi around 0 66.4%

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

   8. Taylor expanded in u0 around 0 90.9%

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

      1. +-commutative 90.9%

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

      2. mul-1-neg 90.9%

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

      3. unsub-neg 90.9%

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

      4. +-commutative 90.9%

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

      5. mul-1-neg 90.9%

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

      6. unsub-neg 90.9%

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

      7. *-commutative 90.9%

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

      8. *-commutative 90.9%

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

      9. distribute-rgt-out 90.9%

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

      10. associate-*l* 90.9%

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

      11. metadata-eval 90.9%

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

   10. Simplified 90.9%

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

4. Final simplification 83.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 0.019999999552965164:\\ \;\;\;\;\frac{u0}{\frac{sin2phi}{alphay \cdot alphay} + \frac{cos2phi}{alphax} \cdot \frac{1}{alphax}}\\ \mathbf{else}:\\ \;\;\;\;\frac{alphay \cdot \left(-alphay\right)}{\left(sin2phi \cdot 0.5 - \frac{sin2phi}{u0}\right) - sin2phi \cdot \left(u0 \cdot -0.08333333333333333\right)}\\ \end{array} \]

Alternative 4: 81.7% accurate, 5.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{sin2phi}{alphay \cdot alphay}\\ \mathbf{if}\;t_0 \leq 0.019999999552965164:\\ \;\;\;\;\frac{u0}{t_0 + cos2phi \cdot \frac{1}{alphax \cdot alphax}}\\ \mathbf{else}:\\ \;\;\;\;\frac{alphay \cdot \left(-alphay\right)}{sin2phi \cdot 0.5 - \frac{sin2phi}{u0}}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (let* ((t_0 (/ sin2phi (* alphay alphay))))
   (if (<= t_0 0.019999999552965164)
     (/ u0 (+ t_0 (* cos2phi (/ 1.0 (* 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 <= 0.019999999552965164f) {
		tmp = u0 / (t_0 + (cos2phi * (1.0f / (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 <= 0.019999999552965164e0) then
        tmp = u0 / (t_0 + (cos2phi * (1.0e0 / (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(0.019999999552965164))
		tmp = Float32(u0 / Float32(t_0 + Float32(cos2phi * Float32(Float32(1.0) / 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(0.019999999552965164))
		tmp = u0 / (t_0 + (cos2phi * (single(1.0) / (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)) < 0.0199999996

   1. Initial program 55.9%

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

      1. associate-/r* 55.9%

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

   3. Simplified 55.9%

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

   4. Taylor expanded in u0 around 0 73.8%

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

      1. unpow2 73.8%

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

      2. unpow2 73.8%

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

   6. Simplified 73.8%

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

      1. div-inv 73.8%

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

   8. Applied egg-rr 73.8%

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

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

   1. Initial program 66.9%

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

      1. associate-/r* 66.9%

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

   3. Simplified 66.9%

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

   4. Taylor expanded in cos2phi around 0 67.2%

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

      1. mul-1-neg 67.2%

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

      2. unpow2 67.2%

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

      3. associate-/l* 66.4%

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

      4. distribute-neg-frac 66.4%

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

      5. distribute-rgt-neg-in 66.4%

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

      6. sub-neg 66.4%

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

      7. mul-1-neg 66.4%

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

      8. log1p-def 96.5%

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

      9. mul-1-neg 96.5%

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

   6. Simplified 96.5%

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

   7. Taylor expanded in sin2phi around 0 66.4%

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

   8. Taylor expanded in u0 around 0 87.9%

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

      1. +-commutative 87.9%

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

      2. mul-1-neg 87.9%

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

      3. unsub-neg 87.9%

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

      4. *-commutative 87.9%

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

   10. Simplified 87.9%

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

4. Final simplification 81.6%

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

Alternative 5: 81.7% accurate, 5.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{sin2phi}{alphay \cdot alphay}\\ \mathbf{if}\;t_0 \leq 0.019999999552965164:\\ \;\;\;\;\frac{u0}{t_0 + \frac{cos2phi}{alphax} \cdot \frac{1}{alphax}}\\ \mathbf{else}:\\ \;\;\;\;\frac{alphay \cdot \left(-alphay\right)}{sin2phi \cdot 0.5 - \frac{sin2phi}{u0}}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (let* ((t_0 (/ sin2phi (* alphay alphay))))
   (if (<= t_0 0.019999999552965164)
     (/ u0 (+ t_0 (* (/ cos2phi alphax) (/ 1.0 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 <= 0.019999999552965164f) {
		tmp = u0 / (t_0 + ((cos2phi / alphax) * (1.0f / 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 <= 0.019999999552965164e0) then
        tmp = u0 / (t_0 + ((cos2phi / alphax) * (1.0e0 / 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(0.019999999552965164))
		tmp = Float32(u0 / Float32(t_0 + Float32(Float32(cos2phi / alphax) * Float32(Float32(1.0) / 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(0.019999999552965164))
		tmp = u0 / (t_0 + ((cos2phi / alphax) * (single(1.0) / 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)) < 0.0199999996

   1. Initial program 55.9%

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

      1. associate-/r* 55.9%

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

   3. Simplified 55.9%

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

   4. Taylor expanded in u0 around 0 73.8%

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

      1. unpow2 73.8%

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

      2. unpow2 73.8%

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

   6. Simplified 73.8%

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

      1. associate-/r* 73.7%

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

      2. div-inv 73.8%

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

   8. Applied egg-rr 73.8%

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

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

   1. Initial program 66.9%

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

      1. associate-/r* 66.9%

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

   3. Simplified 66.9%

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

   4. Taylor expanded in cos2phi around 0 67.2%

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

      1. mul-1-neg 67.2%

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

      2. unpow2 67.2%

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

      3. associate-/l* 66.4%

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

      4. distribute-neg-frac 66.4%

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

      5. distribute-rgt-neg-in 66.4%

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

      6. sub-neg 66.4%

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

      7. mul-1-neg 66.4%

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

      8. log1p-def 96.5%

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

      9. mul-1-neg 96.5%

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

   6. Simplified 96.5%

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

   7. Taylor expanded in sin2phi around 0 66.4%

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

   8. Taylor expanded in u0 around 0 87.9%

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

      1. +-commutative 87.9%

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

      2. mul-1-neg 87.9%

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

      3. unsub-neg 87.9%

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

      4. *-commutative 87.9%

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

   10. Simplified 87.9%

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

4. Final simplification 81.6%

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

Alternative 6: 81.7% accurate, 6.1× speedup

Math

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

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (let* ((t_0 (/ sin2phi (* alphay alphay))))
   (if (<= t_0 0.019999999552965164)
     (/ 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 <= 0.019999999552965164f) {
		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 <= 0.019999999552965164e0) 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(0.019999999552965164))
		tmp = Float32(u0 / Float32(Float32(cos2phi / Float32(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(0.019999999552965164))
		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)) < 0.0199999996

   1. Initial program 55.9%

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

      1. associate-/r* 55.9%

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

   3. Simplified 55.9%

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

   4. Taylor expanded in u0 around 0 73.8%

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

      1. unpow2 73.8%

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

      2. unpow2 73.8%

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

   6. Simplified 73.8%

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

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

   1. Initial program 66.9%

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

      1. associate-/r* 66.9%

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

   3. Simplified 66.9%

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

   4. Taylor expanded in cos2phi around 0 67.2%

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

      1. mul-1-neg 67.2%

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

      2. unpow2 67.2%

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

      3. associate-/l* 66.4%

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

      4. distribute-neg-frac 66.4%

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

      5. distribute-rgt-neg-in 66.4%

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

      6. sub-neg 66.4%

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

      7. mul-1-neg 66.4%

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

      8. log1p-def 96.5%

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

      9. mul-1-neg 96.5%

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

   6. Simplified 96.5%

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

   7. Taylor expanded in sin2phi around 0 66.4%

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

   8. Taylor expanded in u0 around 0 87.9%

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

      1. +-commutative 87.9%

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

      2. mul-1-neg 87.9%

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

      3. unsub-neg 87.9%

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

      4. *-commutative 87.9%

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

   10. Simplified 87.9%

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

4. Final simplification 81.6%

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

Alternative 7: 87.6% accurate, 6.1× speedup

Math

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

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (/
  (- u0 (* 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(Float32(cos2phi / 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 62.0%

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

   1. associate-/r* 62.0%

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

3. Simplified 62.0%

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

4. Taylor expanded in u0 around 0 86.9%

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

   1. +-commutative 86.9%

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

   2. mul-1-neg 86.9%

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

   3. unsub-neg 86.9%

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

   4. unpow2 86.9%

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

   5. associate-*r* 86.9%

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

6. Simplified 86.9%

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

7. Final simplification 86.9%

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

Alternative 8: 74.6% accurate, 8.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;sin2phi \leq 3.8000000086559564 \cdot 10^{-19}:\\ \;\;\;\;u0 \cdot \frac{1}{\frac{cos2phi}{alphax \cdot alphax}}\\ \mathbf{else}:\\ \;\;\;\;\frac{alphay \cdot \left(-alphay\right)}{sin2phi \cdot 0.5 - \frac{sin2phi}{u0}}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (if (<= sin2phi 3.8000000086559564e-19)
   (* u0 (/ 1.0 (/ cos2phi (* alphax alphax))))
   (/ (* alphay (- alphay)) (- (* sin2phi 0.5) (/ sin2phi u0)))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float tmp;
	if (sin2phi <= 3.8000000086559564e-19f) {
		tmp = u0 * (1.0f / (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) :: tmp
    if (sin2phi <= 3.8000000086559564e-19) then
        tmp = u0 * (1.0e0 / (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)
	tmp = Float32(0.0)
	if (sin2phi <= Float32(3.8000000086559564e-19))
		tmp = Float32(u0 * Float32(Float32(1.0) / 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)
	tmp = single(0.0);
	if (sin2phi <= single(3.8000000086559564e-19))
		tmp = u0 * (single(1.0) / (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 sin2phi < 3.80000001e-19

   1. Initial program 56.9%

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

      1. associate-/r* 56.9%

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

   3. Simplified 56.9%

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

   4. Taylor expanded in u0 around 0 73.2%

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

      1. unpow2 73.2%

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

      2. unpow2 73.2%

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

   6. Simplified 73.2%

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

   7. Taylor expanded in cos2phi around inf 61.0%

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

      1. associate-/l* 61.1%

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

      2. unpow2 61.1%

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

   9. Simplified 61.1%

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

       1. div-inv 61.1%

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

   11. Applied egg-rr 61.1%

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

   if 3.80000001e-19 < sin2phi

   1. Initial program 63.6%

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

      1. associate-/r* 63.7%

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

   3. Simplified 63.7%

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

   4. Taylor expanded in cos2phi around 0 59.3%

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

      1. mul-1-neg 59.3%

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

      2. unpow2 59.3%

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

      3. associate-/l* 58.8%

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

      4. distribute-neg-frac 58.8%

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

      5. distribute-rgt-neg-in 58.8%

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

      6. sub-neg 58.8%

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

      7. mul-1-neg 58.8%

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

      8. log1p-def 88.4%

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

      9. mul-1-neg 88.4%

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

   6. Simplified 88.4%

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

   7. Taylor expanded in sin2phi around 0 58.8%

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

   8. Taylor expanded in u0 around 0 80.7%

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

      1. +-commutative 80.7%

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

      2. mul-1-neg 80.7%

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

      3. unsub-neg 80.7%

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

      4. *-commutative 80.7%

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

   10. Simplified 80.7%

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

4. Final simplification 75.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;sin2phi \leq 3.8000000086559564 \cdot 10^{-19}:\\ \;\;\;\;u0 \cdot \frac{1}{\frac{cos2phi}{alphax \cdot alphax}}\\ \mathbf{else}:\\ \;\;\;\;\frac{alphay \cdot \left(-alphay\right)}{sin2phi \cdot 0.5 - \frac{sin2phi}{u0}}\\ \end{array} \]

Alternative 9: 66.9% accurate, 10.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;sin2phi \leq 3.8000000086559564 \cdot 10^{-19}:\\ \;\;\;\;u0 \cdot \frac{1}{\frac{cos2phi}{alphax \cdot alphax}}\\ \mathbf{else}:\\ \;\;\;\;\left(alphay \cdot alphay\right) \cdot \frac{u0}{sin2phi}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (if (<= sin2phi 3.8000000086559564e-19)
   (* u0 (/ 1.0 (/ cos2phi (* alphax alphax))))
   (* (* alphay alphay) (/ u0 sin2phi))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float tmp;
	if (sin2phi <= 3.8000000086559564e-19f) {
		tmp = u0 * (1.0f / (cos2phi / (alphax * alphax)));
	} else {
		tmp = (alphay * alphay) * (u0 / sin2phi);
	}
	return tmp;
}

Fortran

real(4) function code(alphax, alphay, u0, cos2phi, sin2phi)
    real(4), intent (in) :: alphax
    real(4), intent (in) :: alphay
    real(4), intent (in) :: u0
    real(4), intent (in) :: cos2phi
    real(4), intent (in) :: sin2phi
    real(4) :: tmp
    if (sin2phi <= 3.8000000086559564e-19) then
        tmp = u0 * (1.0e0 / (cos2phi / (alphax * alphax)))
    else
        tmp = (alphay * alphay) * (u0 / sin2phi)
    end if
    code = tmp
end function

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = Float32(0.0)
	if (sin2phi <= Float32(3.8000000086559564e-19))
		tmp = Float32(u0 * Float32(Float32(1.0) / Float32(cos2phi / Float32(alphax * alphax))));
	else
		tmp = Float32(Float32(alphay * alphay) * Float32(u0 / sin2phi));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = single(0.0);
	if (sin2phi <= single(3.8000000086559564e-19))
		tmp = u0 * (single(1.0) / (cos2phi / (alphax * alphax)));
	else
		tmp = (alphay * alphay) * (u0 / sin2phi);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if sin2phi < 3.80000001e-19

   1. Initial program 56.9%

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

      1. associate-/r* 56.9%

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

   3. Simplified 56.9%

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

   4. Taylor expanded in u0 around 0 73.2%

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

      1. unpow2 73.2%

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

      2. unpow2 73.2%

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

   6. Simplified 73.2%

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

   7. Taylor expanded in cos2phi around inf 61.0%

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

      1. associate-/l* 61.1%

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

      2. unpow2 61.1%

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

   9. Simplified 61.1%

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

       1. div-inv 61.1%

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

   11. Applied egg-rr 61.1%

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

   if 3.80000001e-19 < sin2phi

   1. Initial program 63.6%

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

      1. associate-/r* 63.7%

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

   3. Simplified 63.7%

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

   4. Taylor expanded in u0 around 0 75.8%

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

      1. unpow2 75.8%

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

      2. unpow2 75.8%

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

   6. Simplified 75.8%

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

      1. associate-/r* 75.7%

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

      2. div-inv 75.7%

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

   8. Applied egg-rr 75.7%

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

   9. Taylor expanded in cos2phi around 0 70.6%

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

       1. associate-/l* 70.2%

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

       2. associate-/r/ 70.5%

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

       3. unpow2 70.5%

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

   11. Simplified 70.5%

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

4. Final simplification 68.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;sin2phi \leq 3.8000000086559564 \cdot 10^{-19}:\\ \;\;\;\;u0 \cdot \frac{1}{\frac{cos2phi}{alphax \cdot alphax}}\\ \mathbf{else}:\\ \;\;\;\;\left(alphay \cdot alphay\right) \cdot \frac{u0}{sin2phi}\\ \end{array} \]

Alternative 10: 66.9% accurate, 12.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;sin2phi \leq 3.8000000086559564 \cdot 10^{-19}:\\ \;\;\;\;\left(alphax \cdot alphax\right) \cdot \frac{u0}{cos2phi}\\ \mathbf{else}:\\ \;\;\;\;\left(alphay \cdot alphay\right) \cdot \frac{u0}{sin2phi}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (if (<= sin2phi 3.8000000086559564e-19)
   (* (* alphax alphax) (/ u0 cos2phi))
   (* (* alphay alphay) (/ u0 sin2phi))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float tmp;
	if (sin2phi <= 3.8000000086559564e-19f) {
		tmp = (alphax * alphax) * (u0 / cos2phi);
	} else {
		tmp = (alphay * alphay) * (u0 / sin2phi);
	}
	return tmp;
}

Fortran

real(4) function code(alphax, alphay, u0, cos2phi, sin2phi)
    real(4), intent (in) :: alphax
    real(4), intent (in) :: alphay
    real(4), intent (in) :: u0
    real(4), intent (in) :: cos2phi
    real(4), intent (in) :: sin2phi
    real(4) :: tmp
    if (sin2phi <= 3.8000000086559564e-19) then
        tmp = (alphax * alphax) * (u0 / cos2phi)
    else
        tmp = (alphay * alphay) * (u0 / sin2phi)
    end if
    code = tmp
end function

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = Float32(0.0)
	if (sin2phi <= Float32(3.8000000086559564e-19))
		tmp = Float32(Float32(alphax * alphax) * Float32(u0 / cos2phi));
	else
		tmp = Float32(Float32(alphay * alphay) * Float32(u0 / sin2phi));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = single(0.0);
	if (sin2phi <= single(3.8000000086559564e-19))
		tmp = (alphax * alphax) * (u0 / cos2phi);
	else
		tmp = (alphay * alphay) * (u0 / sin2phi);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if sin2phi < 3.80000001e-19

   1. Initial program 56.9%

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

      1. associate-/r* 56.9%

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

   3. Simplified 56.9%

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

   4. Taylor expanded in u0 around 0 73.2%

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

      1. unpow2 73.2%

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

      2. unpow2 73.2%

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

   6. Simplified 73.2%

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

   7. Taylor expanded in cos2phi around inf 61.0%

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

      1. *-commutative 61.0%

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

      2. *-lft-identity 61.0%

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

      3. times-frac 61.1%

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

      4. /-rgt-identity 61.1%

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

      5. unpow2 61.1%

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

   9. Simplified 61.1%

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

   if 3.80000001e-19 < sin2phi

   1. Initial program 63.6%

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

      1. associate-/r* 63.7%

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

   3. Simplified 63.7%

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

   4. Taylor expanded in u0 around 0 75.8%

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

      1. unpow2 75.8%

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

      2. unpow2 75.8%

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

   6. Simplified 75.8%

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

      1. associate-/r* 75.7%

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

      2. div-inv 75.7%

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

   8. Applied egg-rr 75.7%

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

   9. Taylor expanded in cos2phi around 0 70.6%

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

       1. associate-/l* 70.2%

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

       2. associate-/r/ 70.5%

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

       3. unpow2 70.5%

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

   11. Simplified 70.5%

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

4. Final simplification 68.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;sin2phi \leq 3.8000000086559564 \cdot 10^{-19}:\\ \;\;\;\;\left(alphax \cdot alphax\right) \cdot \frac{u0}{cos2phi}\\ \mathbf{else}:\\ \;\;\;\;\left(alphay \cdot alphay\right) \cdot \frac{u0}{sin2phi}\\ \end{array} \]

Alternative 11: 23.8% accurate, 16.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 62.0%

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

   1. associate-/r* 62.0%

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

3. Simplified 62.0%

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

4. Taylor expanded in u0 around 0 75.1%

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

   1. unpow2 75.1%

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

   2. unpow2 75.1%

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

6. Simplified 75.1%

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

7. Taylor expanded in cos2phi around inf 23.9%

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

   1. associate-/l* 23.9%

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

   2. unpow2 23.9%

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

9. Simplified 23.9%

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

10. Taylor expanded in u0 around 0 23.9%

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

    1. associate-*l/ 23.9%

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

    2. unpow2 23.9%

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

    3. associate-*r* 23.9%

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

    4. *-commutative 23.9%

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

    5. associate-*l/ 23.9%

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

12. Simplified 23.9%

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

13. Final simplification 23.9%

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

Alternative 12: 23.8% accurate, 16.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 62.0%

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

   1. associate-/r* 62.0%

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

3. Simplified 62.0%

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

4. Taylor expanded in u0 around 0 75.1%

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

   1. unpow2 75.1%

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

   2. unpow2 75.1%

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

6. Simplified 75.1%

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

7. Taylor expanded in cos2phi around inf 23.9%

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

   1. *-commutative 23.9%

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

   2. *-lft-identity 23.9%

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

   3. times-frac 23.9%

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

   4. /-rgt-identity 23.9%

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

   5. unpow2 23.9%

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

9. Simplified 23.9%

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

10. Final simplification 23.9%

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

Reproduce

herbie shell --seed 2023252 
(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)))))