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

Percentage Accurate: 59.9% → 98.3%

Time: 14.0s

Alternatives: 8

Speedup: 1.1×

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: 59.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Alternative 1: 98.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	return -log1pf(-u0) / ((cos2phi / (alphax * alphax)) + (1.0f / (alphay / (sin2phi / alphay))));
}

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	return Float32(Float32(-log1p(Float32(-u0))) / Float32(Float32(cos2phi / Float32(alphax * alphax)) + Float32(Float32(1.0) / Float32(alphay / Float32(sin2phi / alphay)))))
end

Derivation

1. Initial program 63.1%

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

   1. sub-neg 63.1%

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

   2. log1p-def 98.6%

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

3. Simplified 98.6%

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

   1. associate-/r* 98.7%

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

   2. div-inv 98.5%

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

5. Applied egg-rr 98.5%

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

   1. div-inv 98.7%

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

   2. clear-num 98.7%

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

7. Applied egg-rr 98.7%

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

8. Final simplification 98.7%

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

Alternative 2: 82.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;sin2phi \leq 0.004999999888241291:\\ \;\;\;\;\frac{u0}{\frac{1}{alphax} \cdot \frac{cos2phi}{alphax} + sin2phi \cdot {alphay}^{-2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-{alphay}^{2}}{sin2phi \cdot 0.5 - \frac{sin2phi}{u0}}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (if (<= sin2phi 0.004999999888241291)
   (/
    u0
    (+ (* (/ 1.0 alphax) (/ cos2phi alphax)) (* sin2phi (pow alphay -2.0))))
   (/ (- (pow alphay 2.0)) (- (* sin2phi 0.5) (/ sin2phi u0)))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float tmp;
	if (sin2phi <= 0.004999999888241291f) {
		tmp = u0 / (((1.0f / alphax) * (cos2phi / alphax)) + (sin2phi * powf(alphay, -2.0f)));
	} else {
		tmp = -powf(alphay, 2.0f) / ((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 <= 0.004999999888241291e0) then
        tmp = u0 / (((1.0e0 / alphax) * (cos2phi / alphax)) + (sin2phi * (alphay ** (-2.0e0))))
    else
        tmp = -(alphay ** 2.0e0) / ((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(0.004999999888241291))
		tmp = Float32(u0 / Float32(Float32(Float32(Float32(1.0) / alphax) * Float32(cos2phi / alphax)) + Float32(sin2phi * (alphay ^ Float32(-2.0)))));
	else
		tmp = Float32(Float32(-(alphay ^ Float32(2.0))) / 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(0.004999999888241291))
		tmp = u0 / (((single(1.0) / alphax) * (cos2phi / alphax)) + (sin2phi * (alphay ^ single(-2.0))));
	else
		tmp = -(alphay ^ single(2.0)) / ((sin2phi * single(0.5)) - (sin2phi / u0));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if sin2phi < 0.00499999989

   1. Initial program 55.5%

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

      1. sub-neg 55.5%

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

      2. log1p-def 98.7%

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

   3. Simplified 98.7%

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

      1. clear-num 98.8%

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

      2. inv-pow 98.8%

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

      3. pow2 98.8%

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

   5. Applied egg-rr 98.8%

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

   6. Taylor expanded in u0 around 0 73.6%

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

      1. unpow2 73.6%

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

      2. associate-/l/ 73.6%

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

      3. *-rgt-identity 73.6%

         \[\leadsto \frac{u0}{\frac{cos2phi}{{alphax}^{2}} + \frac{\frac{\color{blue}{sin2phi \cdot 1}}{alphay}}{alphay}} \]

      4. associate-*r/ 73.6%

         \[\leadsto \frac{u0}{\frac{cos2phi}{{alphax}^{2}} + \frac{\color{blue}{sin2phi \cdot \frac{1}{alphay}}}{alphay}} \]

      5. associate-*r/ 73.6%

         \[\leadsto \frac{u0}{\frac{cos2phi}{{alphax}^{2}} + \color{blue}{sin2phi \cdot \frac{\frac{1}{alphay}}{alphay}}} \]

      6. *-rgt-identity 73.6%

         \[\leadsto \frac{u0}{\frac{cos2phi}{{alphax}^{2}} + sin2phi \cdot \frac{\color{blue}{\frac{1}{alphay} \cdot 1}}{alphay}} \]

      7. associate-*r/ 73.5%

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

      8. unpow-1 73.5%

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

      9. unpow-1 73.5%

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

      10. pow-sqr 73.7%

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

      11. metadata-eval 73.7%

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

   8. Simplified 73.7%

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

      1. *-un-lft-identity 73.7%

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

      2. pow2 73.7%

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

      3. times-frac 73.5%

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

   10. Applied egg-rr 73.5%

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

   if 0.00499999989 < sin2phi

   1. Initial program 68.6%

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

      1. sub-neg 68.6%

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

      2. log1p-def 98.6%

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

   3. Simplified 98.6%

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

   4. Taylor expanded in cos2phi around 0 68.7%

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

      1. mul-1-neg 68.7%

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

      2. associate-/l* 67.3%

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

      3. distribute-neg-frac 67.3%

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

      4. sub-neg 67.3%

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

      5. log1p-def 96.0%

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

   6. Simplified 96.0%

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

   7. Taylor expanded in u0 around 0 89.2%

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

      1. +-commutative 89.2%

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

      2. mul-1-neg 89.2%

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

      3. unsub-neg 89.2%

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

      4. *-commutative 89.2%

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

   9. Simplified 89.2%

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

4. Final simplification 82.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;sin2phi \leq 0.004999999888241291:\\ \;\;\;\;\frac{u0}{\frac{1}{alphax} \cdot \frac{cos2phi}{alphax} + sin2phi \cdot {alphay}^{-2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-{alphay}^{2}}{sin2phi \cdot 0.5 - \frac{sin2phi}{u0}}\\ \end{array} \]

Alternative 3: 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 63.1%

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

   1. sub-neg 63.1%

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

   2. log1p-def 98.6%

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

3. Simplified 98.6%

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

4. Final simplification 98.6%

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

Alternative 4: 98.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 63.1%

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

   1. sub-neg 63.1%

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

   2. log1p-def 98.6%

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

3. Simplified 98.6%

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

   1. clear-num 98.6%

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

   2. inv-pow 98.6%

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

   3. pow2 98.6%

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

5. Applied egg-rr 98.6%

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

   1. unpow-1 98.6%

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

   2. clear-num 98.6%

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

   3. pow2 98.6%

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

   4. associate-/r* 98.7%

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

7. Applied egg-rr 98.7%

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

8. Final simplification 98.7%

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

Alternative 5: 75.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;sin2phi \leq 1.000000031374395 \cdot 10^{-22}:\\ \;\;\;\;\frac{u0 \cdot {alphax}^{2}}{cos2phi}\\ \mathbf{else}:\\ \;\;\;\;\frac{-{alphay}^{2}}{sin2phi \cdot 0.5 - \frac{sin2phi}{u0}}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (if (<= sin2phi 1.000000031374395e-22)
   (/ (* u0 (pow alphax 2.0)) cos2phi)
   (/ (- (pow alphay 2.0)) (- (* sin2phi 0.5) (/ sin2phi u0)))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float tmp;
	if (sin2phi <= 1.000000031374395e-22f) {
		tmp = (u0 * powf(alphax, 2.0f)) / cos2phi;
	} else {
		tmp = -powf(alphay, 2.0f) / ((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 <= 1.000000031374395e-22) then
        tmp = (u0 * (alphax ** 2.0e0)) / cos2phi
    else
        tmp = -(alphay ** 2.0e0) / ((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(1.000000031374395e-22))
		tmp = Float32(Float32(u0 * (alphax ^ Float32(2.0))) / cos2phi);
	else
		tmp = Float32(Float32(-(alphay ^ Float32(2.0))) / 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(1.000000031374395e-22))
		tmp = (u0 * (alphax ^ single(2.0))) / cos2phi;
	else
		tmp = -(alphay ^ single(2.0)) / ((sin2phi * single(0.5)) - (sin2phi / u0));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if sin2phi < 1.00000003e-22

   1. Initial program 54.0%

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

      1. sub-neg 54.0%

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

      2. log1p-def 98.7%

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

   3. Simplified 98.7%

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

      1. clear-num 98.8%

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

      2. inv-pow 98.8%

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

      3. pow2 98.8%

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

   5. Applied egg-rr 98.8%

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

   6. Taylor expanded in u0 around 0 74.7%

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

      1. unpow2 74.7%

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

      2. associate-/l/ 74.8%

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

      3. *-rgt-identity 74.8%

         \[\leadsto \frac{u0}{\frac{cos2phi}{{alphax}^{2}} + \frac{\frac{\color{blue}{sin2phi \cdot 1}}{alphay}}{alphay}} \]

      4. associate-*r/ 74.8%

         \[\leadsto \frac{u0}{\frac{cos2phi}{{alphax}^{2}} + \frac{\color{blue}{sin2phi \cdot \frac{1}{alphay}}}{alphay}} \]

      5. associate-*r/ 74.8%

         \[\leadsto \frac{u0}{\frac{cos2phi}{{alphax}^{2}} + \color{blue}{sin2phi \cdot \frac{\frac{1}{alphay}}{alphay}}} \]

      6. *-rgt-identity 74.8%

         \[\leadsto \frac{u0}{\frac{cos2phi}{{alphax}^{2}} + sin2phi \cdot \frac{\color{blue}{\frac{1}{alphay} \cdot 1}}{alphay}} \]

      7. associate-*r/ 74.7%

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

      8. unpow-1 74.7%

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

      9. unpow-1 74.7%

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

      10. pow-sqr 74.9%

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

      11. metadata-eval 74.9%

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

   8. Simplified 74.9%

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

   9. Taylor expanded in cos2phi around inf 61.4%

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

   if 1.00000003e-22 < sin2phi

   1. Initial program 65.2%

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

      1. sub-neg 65.2%

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

      2. log1p-def 98.6%

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

   3. Simplified 98.6%

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

   4. Taylor expanded in cos2phi around 0 61.6%

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

      1. mul-1-neg 61.6%

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

      2. associate-/l* 60.6%

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

      3. distribute-neg-frac 60.6%

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

      4. sub-neg 60.6%

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

      5. log1p-def 89.4%

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

   6. Simplified 89.4%

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

   7. Taylor expanded in u0 around 0 82.3%

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

      1. +-commutative 82.3%

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

      2. mul-1-neg 82.3%

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

      3. unsub-neg 82.3%

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

      4. *-commutative 82.3%

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

   9. Simplified 82.3%

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

4. Final simplification 78.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;sin2phi \leq 1.000000031374395 \cdot 10^{-22}:\\ \;\;\;\;\frac{u0 \cdot {alphax}^{2}}{cos2phi}\\ \mathbf{else}:\\ \;\;\;\;\frac{-{alphay}^{2}}{sin2phi \cdot 0.5 - \frac{sin2phi}{u0}}\\ \end{array} \]

Alternative 6: 67.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;sin2phi \leq 1.000000031374395 \cdot 10^{-22}:\\ \;\;\;\;u0 \cdot \frac{{alphax}^{2}}{cos2phi}\\ \mathbf{else}:\\ \;\;\;\;u0 \cdot \left(alphay \cdot \frac{alphay}{sin2phi}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (if (<= sin2phi 1.000000031374395e-22)
   (* u0 (/ (pow alphax 2.0) cos2phi))
   (* u0 (* alphay (/ alphay sin2phi)))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float tmp;
	if (sin2phi <= 1.000000031374395e-22f) {
		tmp = u0 * (powf(alphax, 2.0f) / cos2phi);
	} else {
		tmp = u0 * (alphay * (alphay / sin2phi));
	}
	return tmp;
}

Fortran

real(4) function code(alphax, alphay, u0, cos2phi, sin2phi)
    real(4), intent (in) :: alphax
    real(4), intent (in) :: alphay
    real(4), intent (in) :: u0
    real(4), intent (in) :: cos2phi
    real(4), intent (in) :: sin2phi
    real(4) :: tmp
    if (sin2phi <= 1.000000031374395e-22) then
        tmp = u0 * ((alphax ** 2.0e0) / cos2phi)
    else
        tmp = u0 * (alphay * (alphay / sin2phi))
    end if
    code = tmp
end function

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = Float32(0.0)
	if (sin2phi <= Float32(1.000000031374395e-22))
		tmp = Float32(u0 * Float32((alphax ^ Float32(2.0)) / cos2phi));
	else
		tmp = Float32(u0 * Float32(alphay * Float32(alphay / sin2phi)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = single(0.0);
	if (sin2phi <= single(1.000000031374395e-22))
		tmp = u0 * ((alphax ^ single(2.0)) / cos2phi);
	else
		tmp = u0 * (alphay * (alphay / sin2phi));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if sin2phi < 1.00000003e-22

   1. Initial program 54.0%

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

      1. sub-neg 54.0%

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

      2. log1p-def 98.7%

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

   3. Simplified 98.7%

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

      1. clear-num 98.8%

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

      2. inv-pow 98.8%

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

      3. pow2 98.8%

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

   5. Applied egg-rr 98.8%

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

   6. Taylor expanded in u0 around 0 74.7%

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

      1. unpow2 74.7%

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

      2. associate-/l/ 74.8%

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

      3. *-rgt-identity 74.8%

         \[\leadsto \frac{u0}{\frac{cos2phi}{{alphax}^{2}} + \frac{\frac{\color{blue}{sin2phi \cdot 1}}{alphay}}{alphay}} \]

      4. associate-*r/ 74.8%

         \[\leadsto \frac{u0}{\frac{cos2phi}{{alphax}^{2}} + \frac{\color{blue}{sin2phi \cdot \frac{1}{alphay}}}{alphay}} \]

      5. associate-*r/ 74.8%

         \[\leadsto \frac{u0}{\frac{cos2phi}{{alphax}^{2}} + \color{blue}{sin2phi \cdot \frac{\frac{1}{alphay}}{alphay}}} \]

      6. *-rgt-identity 74.8%

         \[\leadsto \frac{u0}{\frac{cos2phi}{{alphax}^{2}} + sin2phi \cdot \frac{\color{blue}{\frac{1}{alphay} \cdot 1}}{alphay}} \]

      7. associate-*r/ 74.7%

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

      8. unpow-1 74.7%

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

      9. unpow-1 74.7%

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

      10. pow-sqr 74.9%

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

      11. metadata-eval 74.9%

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

   8. Simplified 74.9%

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

   9. Taylor expanded in cos2phi around inf 61.4%

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

       1. associate-/l* 61.4%

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

       2. associate-/r/ 61.3%

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

   11. Simplified 61.3%

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

   if 1.00000003e-22 < sin2phi

   1. Initial program 65.2%

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

      1. sub-neg 65.2%

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

      2. log1p-def 98.6%

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

   3. Simplified 98.6%

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

      1. clear-num 98.6%

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

      2. inv-pow 98.6%

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

      3. pow2 98.6%

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

   5. Applied egg-rr 98.6%

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

   6. Taylor expanded in u0 around 0 76.5%

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

      1. unpow2 76.5%

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

      2. associate-/l/ 76.5%

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

      3. *-rgt-identity 76.5%

         \[\leadsto \frac{u0}{\frac{cos2phi}{{alphax}^{2}} + \frac{\frac{\color{blue}{sin2phi \cdot 1}}{alphay}}{alphay}} \]

      4. associate-*r/ 76.5%

         \[\leadsto \frac{u0}{\frac{cos2phi}{{alphax}^{2}} + \frac{\color{blue}{sin2phi \cdot \frac{1}{alphay}}}{alphay}} \]

      5. associate-*r/ 76.5%

         \[\leadsto \frac{u0}{\frac{cos2phi}{{alphax}^{2}} + \color{blue}{sin2phi \cdot \frac{\frac{1}{alphay}}{alphay}}} \]

      6. *-rgt-identity 76.5%

         \[\leadsto \frac{u0}{\frac{cos2phi}{{alphax}^{2}} + sin2phi \cdot \frac{\color{blue}{\frac{1}{alphay} \cdot 1}}{alphay}} \]

      7. associate-*r/ 76.4%

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

      8. unpow-1 76.4%

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

      9. unpow-1 76.4%

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

      10. pow-sqr 76.5%

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

      11. metadata-eval 76.5%

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

   8. Simplified 76.5%

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

   9. Taylor expanded in cos2phi around 0 71.8%

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

       1. *-commutative 71.8%

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

       2. associate-*r/ 71.8%

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

   11. Simplified 71.8%

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

       1. pow2 71.8%

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

       2. *-un-lft-identity 71.8%

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

       3. times-frac 71.9%

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

   13. Applied egg-rr 71.9%

       \[\leadsto u0 \cdot \color{blue}{\left(\frac{alphay}{1} \cdot \frac{alphay}{sin2phi}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 69.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;sin2phi \leq 1.000000031374395 \cdot 10^{-22}:\\ \;\;\;\;u0 \cdot \frac{{alphax}^{2}}{cos2phi}\\ \mathbf{else}:\\ \;\;\;\;u0 \cdot \left(alphay \cdot \frac{alphay}{sin2phi}\right)\\ \end{array} \]

Alternative 7: 67.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;sin2phi \leq 1.000000031374395 \cdot 10^{-22}:\\ \;\;\;\;\frac{u0 \cdot {alphax}^{2}}{cos2phi}\\ \mathbf{else}:\\ \;\;\;\;u0 \cdot \left(alphay \cdot \frac{alphay}{sin2phi}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (if (<= sin2phi 1.000000031374395e-22)
   (/ (* u0 (pow alphax 2.0)) cos2phi)
   (* u0 (* alphay (/ alphay sin2phi)))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float tmp;
	if (sin2phi <= 1.000000031374395e-22f) {
		tmp = (u0 * powf(alphax, 2.0f)) / cos2phi;
	} else {
		tmp = u0 * (alphay * (alphay / sin2phi));
	}
	return tmp;
}

Fortran

real(4) function code(alphax, alphay, u0, cos2phi, sin2phi)
    real(4), intent (in) :: alphax
    real(4), intent (in) :: alphay
    real(4), intent (in) :: u0
    real(4), intent (in) :: cos2phi
    real(4), intent (in) :: sin2phi
    real(4) :: tmp
    if (sin2phi <= 1.000000031374395e-22) then
        tmp = (u0 * (alphax ** 2.0e0)) / cos2phi
    else
        tmp = u0 * (alphay * (alphay / sin2phi))
    end if
    code = tmp
end function

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = Float32(0.0)
	if (sin2phi <= Float32(1.000000031374395e-22))
		tmp = Float32(Float32(u0 * (alphax ^ Float32(2.0))) / cos2phi);
	else
		tmp = Float32(u0 * Float32(alphay * Float32(alphay / sin2phi)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = single(0.0);
	if (sin2phi <= single(1.000000031374395e-22))
		tmp = (u0 * (alphax ^ single(2.0))) / cos2phi;
	else
		tmp = u0 * (alphay * (alphay / sin2phi));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if sin2phi < 1.00000003e-22

   1. Initial program 54.0%

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

      1. sub-neg 54.0%

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

      2. log1p-def 98.7%

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

   3. Simplified 98.7%

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

      1. clear-num 98.8%

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

      2. inv-pow 98.8%

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

      3. pow2 98.8%

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

   5. Applied egg-rr 98.8%

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

   6. Taylor expanded in u0 around 0 74.7%

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

      1. unpow2 74.7%

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

      2. associate-/l/ 74.8%

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

      3. *-rgt-identity 74.8%

         \[\leadsto \frac{u0}{\frac{cos2phi}{{alphax}^{2}} + \frac{\frac{\color{blue}{sin2phi \cdot 1}}{alphay}}{alphay}} \]

      4. associate-*r/ 74.8%

         \[\leadsto \frac{u0}{\frac{cos2phi}{{alphax}^{2}} + \frac{\color{blue}{sin2phi \cdot \frac{1}{alphay}}}{alphay}} \]

      5. associate-*r/ 74.8%

         \[\leadsto \frac{u0}{\frac{cos2phi}{{alphax}^{2}} + \color{blue}{sin2phi \cdot \frac{\frac{1}{alphay}}{alphay}}} \]

      6. *-rgt-identity 74.8%

         \[\leadsto \frac{u0}{\frac{cos2phi}{{alphax}^{2}} + sin2phi \cdot \frac{\color{blue}{\frac{1}{alphay} \cdot 1}}{alphay}} \]

      7. associate-*r/ 74.7%

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

      8. unpow-1 74.7%

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

      9. unpow-1 74.7%

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

      10. pow-sqr 74.9%

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

      11. metadata-eval 74.9%

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

   8. Simplified 74.9%

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

   9. Taylor expanded in cos2phi around inf 61.4%

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

   if 1.00000003e-22 < sin2phi

   1. Initial program 65.2%

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

      1. sub-neg 65.2%

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

      2. log1p-def 98.6%

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

   3. Simplified 98.6%

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

      1. clear-num 98.6%

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

      2. inv-pow 98.6%

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

      3. pow2 98.6%

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

   5. Applied egg-rr 98.6%

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

   6. Taylor expanded in u0 around 0 76.5%

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

      1. unpow2 76.5%

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

      2. associate-/l/ 76.5%

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

      3. *-rgt-identity 76.5%

         \[\leadsto \frac{u0}{\frac{cos2phi}{{alphax}^{2}} + \frac{\frac{\color{blue}{sin2phi \cdot 1}}{alphay}}{alphay}} \]

      4. associate-*r/ 76.5%

         \[\leadsto \frac{u0}{\frac{cos2phi}{{alphax}^{2}} + \frac{\color{blue}{sin2phi \cdot \frac{1}{alphay}}}{alphay}} \]

      5. associate-*r/ 76.5%

         \[\leadsto \frac{u0}{\frac{cos2phi}{{alphax}^{2}} + \color{blue}{sin2phi \cdot \frac{\frac{1}{alphay}}{alphay}}} \]

      6. *-rgt-identity 76.5%

         \[\leadsto \frac{u0}{\frac{cos2phi}{{alphax}^{2}} + sin2phi \cdot \frac{\color{blue}{\frac{1}{alphay} \cdot 1}}{alphay}} \]

      7. associate-*r/ 76.4%

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

      8. unpow-1 76.4%

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

      9. unpow-1 76.4%

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

      10. pow-sqr 76.5%

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

      11. metadata-eval 76.5%

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

   8. Simplified 76.5%

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

   9. Taylor expanded in cos2phi around 0 71.8%

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

       1. *-commutative 71.8%

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

       2. associate-*r/ 71.8%

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

   11. Simplified 71.8%

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

       1. pow2 71.8%

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

       2. *-un-lft-identity 71.8%

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

       3. times-frac 71.9%

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

   13. Applied egg-rr 71.9%

       \[\leadsto u0 \cdot \color{blue}{\left(\frac{alphay}{1} \cdot \frac{alphay}{sin2phi}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 69.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;sin2phi \leq 1.000000031374395 \cdot 10^{-22}:\\ \;\;\;\;\frac{u0 \cdot {alphax}^{2}}{cos2phi}\\ \mathbf{else}:\\ \;\;\;\;u0 \cdot \left(alphay \cdot \frac{alphay}{sin2phi}\right)\\ \end{array} \]

Alternative 8: 59.3% accurate, 16.6× speedup

Math

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

FPCore

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

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	return u0 * (alphay * (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 = u0 * (alphay * (alphay / sin2phi))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 63.1%

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

   1. sub-neg 63.1%

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

   2. log1p-def 98.6%

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

3. Simplified 98.6%

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

   1. clear-num 98.6%

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

   2. inv-pow 98.6%

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

   3. pow2 98.6%

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

5. Applied egg-rr 98.6%

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

6. Taylor expanded in u0 around 0 76.1%

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

   1. unpow2 76.1%

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

   2. associate-/l/ 76.2%

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

   3. *-rgt-identity 76.2%

      \[\leadsto \frac{u0}{\frac{cos2phi}{{alphax}^{2}} + \frac{\frac{\color{blue}{sin2phi \cdot 1}}{alphay}}{alphay}} \]

   4. associate-*r/ 76.1%

      \[\leadsto \frac{u0}{\frac{cos2phi}{{alphax}^{2}} + \frac{\color{blue}{sin2phi \cdot \frac{1}{alphay}}}{alphay}} \]

   5. associate-*r/ 76.2%

      \[\leadsto \frac{u0}{\frac{cos2phi}{{alphax}^{2}} + \color{blue}{sin2phi \cdot \frac{\frac{1}{alphay}}{alphay}}} \]

   6. *-rgt-identity 76.2%

      \[\leadsto \frac{u0}{\frac{cos2phi}{{alphax}^{2}} + sin2phi \cdot \frac{\color{blue}{\frac{1}{alphay} \cdot 1}}{alphay}} \]

   7. associate-*r/ 76.1%

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

   8. unpow-1 76.1%

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

   9. unpow-1 76.1%

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

   10. pow-sqr 76.2%

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

   11. metadata-eval 76.2%

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

8. Simplified 76.2%

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

9. Taylor expanded in cos2phi around 0 62.1%

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

    1. *-commutative 62.1%

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

    2. associate-*r/ 62.1%

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

11. Simplified 62.1%

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

    1. pow2 62.1%

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

    2. *-un-lft-identity 62.1%

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

    3. times-frac 62.2%

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

13. Applied egg-rr 62.2%

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

14. Final simplification 62.2%

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

Reproduce

herbie shell --seed 2023333 
(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)))))