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

Percentage Accurate: 60.6% → 98.3%

Time: 13.6s

Alternatives: 8

Speedup: 8.9×

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: 98.3% 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 61.6%

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

   1. sub-neg 61.6%

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

   2. log1p-def 98.8%

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

3. Simplified 98.8%

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

4. Final simplification 98.8%

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

Alternative 2: 82.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;1 - u0 \leq 0.9994999766349792:\\ \;\;\;\;\frac{\left(alphay \cdot alphay\right) \cdot \left(-\log \left(1 - u0\right)\right)}{sin2phi}\\ \mathbf{else}:\\ \;\;\;\;\frac{u0}{\frac{cos2phi}{alphax \cdot alphax} + sin2phi \cdot {alphay}^{-2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (if (<= (- 1.0 u0) 0.9994999766349792)
   (/ (* (* alphay alphay) (- (log (- 1.0 u0)))) sin2phi)
   (/ u0 (+ (/ cos2phi (* alphax alphax)) (* sin2phi (pow alphay -2.0))))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float tmp;
	if ((1.0f - u0) <= 0.9994999766349792f) {
		tmp = ((alphay * alphay) * -logf((1.0f - u0))) / sin2phi;
	} else {
		tmp = u0 / ((cos2phi / (alphax * alphax)) + (sin2phi * powf(alphay, -2.0f)));
	}
	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 ((1.0e0 - u0) <= 0.9994999766349792e0) then
        tmp = ((alphay * alphay) * -log((1.0e0 - u0))) / sin2phi
    else
        tmp = u0 / ((cos2phi / (alphax * alphax)) + (sin2phi * (alphay ** (-2.0e0))))
    end if
    code = tmp
end function

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = Float32(0.0)
	if (Float32(Float32(1.0) - u0) <= Float32(0.9994999766349792))
		tmp = Float32(Float32(Float32(alphay * alphay) * Float32(-log(Float32(Float32(1.0) - u0)))) / sin2phi);
	else
		tmp = Float32(u0 / Float32(Float32(cos2phi / Float32(alphax * alphax)) + Float32(sin2phi * (alphay ^ Float32(-2.0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = single(0.0);
	if ((single(1.0) - u0) <= single(0.9994999766349792))
		tmp = ((alphay * alphay) * -log((single(1.0) - u0))) / sin2phi;
	else
		tmp = u0 / ((cos2phi / (alphax * alphax)) + (sin2phi * (alphay ^ single(-2.0))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (-.f32 1 u0) < 0.999499977

   1. Initial program 90.4%

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

      1. associate-/r* 90.4%

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

   3. Simplified 90.4%

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

   4. Taylor expanded in cos2phi around 0 65.6%

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

      1. unpow2 38.5%

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

   6. Applied egg-rr 65.5%

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

   if 0.999499977 < (-.f32 1 u0)

   1. Initial program 46.5%

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

      1. associate-/r* 46.5%

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

   3. Simplified 46.5%

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

   4. Taylor expanded in u0 around 0 90.4%

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

      1. mul-1-neg 90.4%

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

   6. Simplified 90.4%

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

      1. associate-/r* 90.6%

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

      2. clear-num 90.5%

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

      3. associate-/r/ 90.5%

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

      4. pow2 90.5%

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

      5. pow-flip 90.6%

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

      6. metadata-eval 90.6%

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

   8. Applied egg-rr 90.6%

      \[\leadsto \frac{-\left(-u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \color{blue}{{alphay}^{-2} \cdot sin2phi}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 82.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 - u0 \leq 0.9994999766349792:\\ \;\;\;\;\frac{\left(alphay \cdot alphay\right) \cdot \left(-\log \left(1 - u0\right)\right)}{sin2phi}\\ \mathbf{else}:\\ \;\;\;\;\frac{u0}{\frac{cos2phi}{alphax \cdot alphax} + sin2phi \cdot {alphay}^{-2}}\\ \end{array} \]

Alternative 3: 82.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;1 - u0 \leq 0.9994999766349792:\\ \;\;\;\;\frac{\left(alphay \cdot alphay\right) \cdot \left(-\log \left(1 - u0\right)\right)}{sin2phi}\\ \mathbf{else}:\\ \;\;\;\;\frac{alphax \cdot \left(\left(-u0\right) \cdot alphay\right)}{sin2phi \cdot \frac{-alphax}{alphay} - cos2phi \cdot \frac{alphay}{alphax}}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (if (<= (- 1.0 u0) 0.9994999766349792)
   (/ (* (* alphay alphay) (- (log (- 1.0 u0)))) sin2phi)
   (/
    (* alphax (* (- u0) alphay))
    (- (* sin2phi (/ (- alphax) alphay)) (* cos2phi (/ alphay alphax))))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float tmp;
	if ((1.0f - u0) <= 0.9994999766349792f) {
		tmp = ((alphay * alphay) * -logf((1.0f - u0))) / sin2phi;
	} else {
		tmp = (alphax * (-u0 * alphay)) / ((sin2phi * (-alphax / alphay)) - (cos2phi * (alphay / alphax)));
	}
	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 ((1.0e0 - u0) <= 0.9994999766349792e0) then
        tmp = ((alphay * alphay) * -log((1.0e0 - u0))) / sin2phi
    else
        tmp = (alphax * (-u0 * alphay)) / ((sin2phi * (-alphax / alphay)) - (cos2phi * (alphay / alphax)))
    end if
    code = tmp
end function

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = Float32(0.0)
	if (Float32(Float32(1.0) - u0) <= Float32(0.9994999766349792))
		tmp = Float32(Float32(Float32(alphay * alphay) * Float32(-log(Float32(Float32(1.0) - u0)))) / sin2phi);
	else
		tmp = Float32(Float32(alphax * Float32(Float32(-u0) * alphay)) / Float32(Float32(sin2phi * Float32(Float32(-alphax) / alphay)) - Float32(cos2phi * Float32(alphay / alphax))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = single(0.0);
	if ((single(1.0) - u0) <= single(0.9994999766349792))
		tmp = ((alphay * alphay) * -log((single(1.0) - u0))) / sin2phi;
	else
		tmp = (alphax * (-u0 * alphay)) / ((sin2phi * (-alphax / alphay)) - (cos2phi * (alphay / alphax)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (-.f32 1 u0) < 0.999499977

   1. Initial program 90.4%

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

      1. associate-/r* 90.4%

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

   3. Simplified 90.4%

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

   4. Taylor expanded in cos2phi around 0 65.6%

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

      1. unpow2 38.5%

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

   6. Applied egg-rr 65.5%

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

   if 0.999499977 < (-.f32 1 u0)

   1. Initial program 46.5%

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

      1. sub-neg 46.5%

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

      2. log1p-def 99.0%

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

   3. Simplified 99.0%

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

      1. associate-/r* 98.8%

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

      2. associate-/r* 98.8%

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

      3. frac-2neg 98.8%

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

      4. frac-add 98.6%

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

      5. distribute-neg-frac 98.6%

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

   5. Applied egg-rr 98.6%

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

      1. +-commutative 98.6%

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

      2. *-commutative 98.6%

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

      3. distribute-lft-neg-out 98.6%

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

      4. unsub-neg 98.6%

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

      5. associate-*r/ 98.5%

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

      6. associate-/l* 98.5%

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

      7. associate-*r/ 98.6%

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

      8. associate-/l* 98.6%

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

   7. Simplified 98.6%

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

      1. div-inv 98.2%

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

      2. associate-/r/ 98.3%

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

      3. associate-/r/ 98.3%

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

   9. Applied egg-rr 98.3%

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

   10. Taylor expanded in u0 around 0 90.5%

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

       1. associate-*r/ 90.5%

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

       2. associate-*r* 90.3%

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

       3. associate-*r* 90.3%

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

       4. neg-mul-1 90.3%

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

       5. *-commutative 90.3%

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

       6. distribute-rgt-neg-in 90.3%

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

       7. mul-1-neg 90.3%

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

       8. associate-*l/ 90.5%

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

       9. distribute-rgt-neg-out 90.5%

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

       10. *-commutative 90.5%

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

       11. associate-*r/ 90.4%

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

   12. Simplified 90.4%

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

   13. Taylor expanded in u0 around 0 90.5%

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

       1. associate-*r/ 90.5%

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

       2. associate-*r* 90.5%

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

       3. neg-mul-1 90.5%

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

       4. *-commutative 90.5%

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

       5. associate-*l/ 90.6%

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

       6. neg-mul-1 90.6%

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

       7. distribute-rgt-neg-in 90.6%

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

       8. *-commutative 90.6%

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

       9. associate-*r/ 90.6%

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

   15. Simplified 90.6%

       \[\leadsto \color{blue}{\frac{\left(-alphax\right) \cdot \left(u0 \cdot alphay\right)}{\frac{alphax}{alphay} \cdot \left(-sin2phi\right) - cos2phi \cdot \frac{alphay}{alphax}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 82.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 - u0 \leq 0.9994999766349792:\\ \;\;\;\;\frac{\left(alphay \cdot alphay\right) \cdot \left(-\log \left(1 - u0\right)\right)}{sin2phi}\\ \mathbf{else}:\\ \;\;\;\;\frac{alphax \cdot \left(\left(-u0\right) \cdot alphay\right)}{sin2phi \cdot \frac{-alphax}{alphay} - cos2phi \cdot \frac{alphay}{alphax}}\\ \end{array} \]

Alternative 4: 76.0% accurate, 6.1× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 61.6%

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

   1. sub-neg 61.6%

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

   2. log1p-def 98.8%

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

3. Simplified 98.8%

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

   1. associate-/r* 98.7%

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

   2. associate-/r* 98.6%

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

   3. frac-2neg 98.6%

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

   4. frac-add 98.4%

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

   5. distribute-neg-frac 98.4%

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

5. Applied egg-rr 98.4%

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

   1. +-commutative 98.4%

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

   2. *-commutative 98.4%

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

   3. distribute-lft-neg-out 98.4%

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

   4. unsub-neg 98.4%

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

   5. associate-*r/ 98.4%

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

   6. associate-/l* 98.4%

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

   7. associate-*r/ 98.4%

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

   8. associate-/l* 98.5%

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

7. Simplified 98.5%

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

   1. div-inv 98.2%

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

   2. associate-/r/ 98.3%

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

   3. associate-/r/ 98.2%

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

9. Applied egg-rr 98.2%

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

10. Taylor expanded in u0 around 0 75.8%

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

    1. mul-1-neg 75.8%

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

    2. mul-1-neg 75.8%

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

    3. associate-/l* 75.9%

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

    4. *-commutative 75.9%

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

    5. associate-*r/ 75.9%

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

12. Simplified 75.9%

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

13. Final simplification 75.9%

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

Alternative 5: 76.0% accurate, 6.1× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 61.6%

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

   1. sub-neg 61.6%

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

   2. log1p-def 98.8%

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

3. Simplified 98.8%

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

   1. associate-/r* 98.7%

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

   2. associate-/r* 98.6%

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

   3. frac-2neg 98.6%

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

   4. frac-add 98.4%

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

   5. distribute-neg-frac 98.4%

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

5. Applied egg-rr 98.4%

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

   1. +-commutative 98.4%

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

   2. *-commutative 98.4%

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

   3. distribute-lft-neg-out 98.4%

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

   4. unsub-neg 98.4%

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

   5. associate-*r/ 98.4%

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

   6. associate-/l* 98.4%

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

   7. associate-*r/ 98.4%

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

   8. associate-/l* 98.5%

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

7. Simplified 98.5%

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

   1. div-inv 98.2%

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

   2. associate-/r/ 98.3%

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

   3. associate-/r/ 98.2%

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

9. Applied egg-rr 98.2%

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

10. Taylor expanded in u0 around 0 75.8%

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

    1. associate-*r/ 75.8%

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

    2. associate-*r* 75.7%

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

    3. associate-*r* 75.7%

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

    4. neg-mul-1 75.7%

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

    5. *-commutative 75.7%

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

    6. distribute-rgt-neg-in 75.7%

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

    7. mul-1-neg 75.7%

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

    8. associate-*l/ 75.8%

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

    9. distribute-rgt-neg-out 75.8%

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

    10. *-commutative 75.8%

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

    11. associate-*r/ 75.8%

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

12. Simplified 75.8%

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

13. Taylor expanded in u0 around 0 75.8%

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

    1. associate-*r/ 75.8%

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

    2. associate-*r* 75.8%

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

    3. neg-mul-1 75.8%

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

    4. *-commutative 75.8%

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

    5. associate-*l/ 75.9%

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

    6. neg-mul-1 75.9%

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

    7. distribute-rgt-neg-in 75.9%

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

    8. *-commutative 75.9%

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

    9. associate-*r/ 75.9%

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

15. Simplified 75.9%

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

16. Final simplification 75.9%

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

Alternative 6: 75.8% accurate, 7.7× speedup

Math

\[\begin{array}{l} \\ \frac{u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay} \cdot \frac{1}{alphay}} \end{array} \]

FPCore

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

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	return u0 / ((cos2phi / (alphax * alphax)) + ((sin2phi / alphay) * (1.0f / 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 = u0 / ((cos2phi / (alphax * alphax)) + ((sin2phi / alphay) * (1.0e0 / alphay)))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 61.6%

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

   1. associate-/r* 61.6%

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

3. Simplified 61.6%

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

4. Taylor expanded in u0 around 0 75.7%

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

   1. mul-1-neg 75.7%

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

6. Simplified 75.7%

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

   1. div-inv 75.7%

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

8. Applied egg-rr 75.7%

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

9. Final simplification 75.7%

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

Alternative 7: 75.8% accurate, 8.9× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 61.6%

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

   1. associate-/r* 61.6%

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

3. Simplified 61.6%

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

4. Taylor expanded in u0 around 0 75.7%

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

   1. mul-1-neg 75.7%

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

6. Simplified 75.7%

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

7. Final simplification 75.7%

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

Alternative 8: 58.4% accurate, 16.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 61.6%

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

   1. associate-/r* 61.6%

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

3. Simplified 61.6%

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

4. Taylor expanded in u0 around 0 75.7%

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

   1. mul-1-neg 75.7%

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

6. Simplified 75.7%

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

7. Taylor expanded in cos2phi around 0 59.1%

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

   1. associate-/l* 58.0%

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

9. Simplified 58.0%

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

    1. unpow2 58.0%

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

11. Applied egg-rr 58.0%

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

12. Final simplification 58.0%

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

Reproduce

herbie shell --seed 2023306 
(FPCore (alphax alphay u0 cos2phi sin2phi)
  :name "Beckmann Distribution sample, tan2theta, alphax != alphay, u1 <= 0.5"
  :precision binary32
  :pre (and (and (and (and (and (<= 0.0001 alphax) (<= alphax 1.0)) (and (<= 0.0001 alphay) (<= alphay 1.0))) (and (<= 2.328306437e-10 u0) (<= u0 1.0))) (and (<= 0.0 cos2phi) (<= cos2phi 1.0))) (<= 0.0 sin2phi))
  (/ (- (log (- 1.0 u0))) (+ (/ cos2phi (* alphax alphax)) (/ sin2phi (* alphay alphay)))))