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

Percentage Accurate: 60.4% → 98.3%

Time: 16.9s

Alternatives: 8

Speedup: 9.7×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Initial Program: 60.4% 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

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

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 59.8%

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

   1. sub-neg 59.8%

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

   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. Add Preprocessing
5. Step-by-step derivation

   1. clear-num 98.5%

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

   2. associate-/r/ 98.4%

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

   3. pow2 98.4%

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

   4. pow-flip 98.5%

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

   5. metadata-eval 98.5%

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

6. Applied egg-rr 98.5%

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

   1. *-commutative 98.5%

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

   2. metadata-eval 98.5%

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

   3. pow-flip 98.4%

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

   4. pow2 98.4%

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

   5. div-inv 98.6%

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

   6. associate-/r* 98.6%

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

8. Applied egg-rr 98.6%

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

9. Final simplification 98.6%

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

Alternative 2: 82.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;1 - u0 \leq 0.9982500076293945:\\ \;\;\;\;\frac{alphay \cdot \left(-alphay\right)}{\frac{sin2phi}{\log \left(1 - u0\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{u0}{\frac{cos2phi}{alphax \cdot alphax} + sin2phi \cdot {alphay}^{-2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (if (<= (- 1.0 u0) 0.9982500076293945)
   (/ (* alphay (- alphay)) (/ sin2phi (log (- 1.0 u0))))
   (/ 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) <= 0.9982500076293945f) {
		tmp = (alphay * -alphay) / (sin2phi / logf((1.0f - u0)));
	} 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) <= 0.9982500076293945e0) then
        tmp = (alphay * -alphay) / (sin2phi / log((1.0e0 - u0)))
    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(0.9982500076293945))
		tmp = Float32(Float32(alphay * Float32(-alphay)) / Float32(sin2phi / log(Float32(Float32(1.0) - u0))));
	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(0.9982500076293945))
		tmp = (alphay * -alphay) / (sin2phi / log((single(1.0) - u0)));
	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) < 0.99825001

   1. Initial program 91.9%

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

   3. Taylor expanded in cos2phi around 0 79.4%

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

      1. mul-1-neg 79.4%

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

      2. associate-/l* 79.3%

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

   5. Simplified 79.3%

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

      1. pow2 40.9%

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

   7. Applied egg-rr 79.3%

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

   if 0.99825001 < (-.f32 1 u0)

   1. Initial program 48.2%

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

   3. Taylor expanded in u0 around 0 87.6%

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

      1. mul-1-neg 87.6%

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

   5. Simplified 87.6%

      \[\leadsto \frac{-\color{blue}{\left(-u0\right)}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
   6. 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. associate-/r/ 98.5%

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

      3. pow2 98.5%

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

      4. pow-flip 98.6%

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

      5. metadata-eval 98.6%

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

   7. Applied egg-rr 87.6%

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

4. Final simplification 85.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 - u0 \leq 0.9982500076293945:\\ \;\;\;\;\frac{alphay \cdot \left(-alphay\right)}{\frac{sin2phi}{\log \left(1 - u0\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{u0}{\frac{cos2phi}{alphax \cdot alphax} + sin2phi \cdot {alphay}^{-2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 82.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;1 - u0 \leq 0.9982500076293945:\\ \;\;\;\;\frac{alphay \cdot \left(-alphay\right)}{\frac{sin2phi}{\log \left(1 - u0\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (if (<= (- 1.0 u0) 0.9982500076293945)
   (/ (* alphay (- alphay)) (/ sin2phi (log (- 1.0 u0))))
   (/ u0 (+ (/ cos2phi (* alphax alphax)) (/ sin2phi (* alphay alphay))))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float tmp;
	if ((1.0f - u0) <= 0.9982500076293945f) {
		tmp = (alphay * -alphay) / (sin2phi / logf((1.0f - u0)));
	} else {
		tmp = u0 / ((cos2phi / (alphax * alphax)) + (sin2phi / (alphay * alphay)));
	}
	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) <= 0.9982500076293945e0) then
        tmp = (alphay * -alphay) / (sin2phi / log((1.0e0 - u0)))
    else
        tmp = u0 / ((cos2phi / (alphax * alphax)) + (sin2phi / (alphay * alphay)))
    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(0.9982500076293945))
		tmp = Float32(Float32(alphay * Float32(-alphay)) / Float32(sin2phi / log(Float32(Float32(1.0) - u0))));
	else
		tmp = Float32(u0 / Float32(Float32(cos2phi / Float32(alphax * alphax)) + Float32(sin2phi / Float32(alphay * alphay))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = single(0.0);
	if ((single(1.0) - u0) <= single(0.9982500076293945))
		tmp = (alphay * -alphay) / (sin2phi / log((single(1.0) - u0)));
	else
		tmp = u0 / ((cos2phi / (alphax * alphax)) + (sin2phi / (alphay * alphay)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (-.f32 1 u0) < 0.99825001

   1. Initial program 91.9%

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

   3. Taylor expanded in cos2phi around 0 79.4%

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

      1. mul-1-neg 79.4%

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

      2. associate-/l* 79.3%

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

   5. Simplified 79.3%

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

      1. pow2 40.9%

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

   7. Applied egg-rr 79.3%

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

   if 0.99825001 < (-.f32 1 u0)

   1. Initial program 48.2%

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

   3. Taylor expanded in u0 around 0 87.6%

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

      1. mul-1-neg 87.6%

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

   5. Simplified 87.6%

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

4. Final simplification 85.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 - u0 \leq 0.9982500076293945:\\ \;\;\;\;\frac{alphay \cdot \left(-alphay\right)}{\frac{sin2phi}{\log \left(1 - u0\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 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 59.8%

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

   1. sub-neg 59.8%

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

   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. Add Preprocessing

5. Final simplification 98.6%

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

Alternative 5: 81.6% accurate, 5.3× speedup

Math

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

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (let* ((t_0 (/ sin2phi (* alphay alphay))))
   (if (<= t_0 10.0)
     (/ u0 (+ (/ cos2phi (* alphax alphax)) t_0))
     (/ (* alphay (- alphay)) (- (* sin2phi 0.5) (/ sin2phi u0))))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float t_0 = sin2phi / (alphay * alphay);
	float tmp;
	if (t_0 <= 10.0f) {
		tmp = u0 / ((cos2phi / (alphax * alphax)) + t_0);
	} else {
		tmp = (alphay * -alphay) / ((sin2phi * 0.5f) - (sin2phi / u0));
	}
	return tmp;
}

Fortran

real(4) function code(alphax, alphay, u0, cos2phi, sin2phi)
    real(4), intent (in) :: alphax
    real(4), intent (in) :: alphay
    real(4), intent (in) :: u0
    real(4), intent (in) :: cos2phi
    real(4), intent (in) :: sin2phi
    real(4) :: t_0
    real(4) :: tmp
    t_0 = sin2phi / (alphay * alphay)
    if (t_0 <= 10.0e0) then
        tmp = u0 / ((cos2phi / (alphax * alphax)) + t_0)
    else
        tmp = (alphay * -alphay) / ((sin2phi * 0.5e0) - (sin2phi / u0))
    end if
    code = tmp
end function

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	t_0 = Float32(sin2phi / Float32(alphay * alphay))
	tmp = Float32(0.0)
	if (t_0 <= Float32(10.0))
		tmp = Float32(u0 / Float32(Float32(cos2phi / Float32(alphax * alphax)) + t_0));
	else
		tmp = Float32(Float32(alphay * Float32(-alphay)) / Float32(Float32(sin2phi * Float32(0.5)) - Float32(sin2phi / u0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alphax, alphay, u0, cos2phi, sin2phi)
	t_0 = sin2phi / (alphay * alphay);
	tmp = single(0.0);
	if (t_0 <= single(10.0))
		tmp = u0 / ((cos2phi / (alphax * alphax)) + t_0);
	else
		tmp = (alphay * -alphay) / ((sin2phi * single(0.5)) - (sin2phi / u0));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 sin2phi (*.f32 alphay alphay)) < 10

   1. Initial program 53.0%

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

   3. Taylor expanded in u0 around 0 77.8%

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

      1. mul-1-neg 77.8%

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

   5. Simplified 77.8%

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

   if 10 < (/.f32 sin2phi (*.f32 alphay alphay))

   1. Initial program 65.3%

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

   3. Taylor expanded in cos2phi around 0 65.4%

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

      1. mul-1-neg 65.4%

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

      2. associate-/l* 65.2%

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

   5. Simplified 65.2%

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

   6. Taylor expanded in u0 around 0 87.1%

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

      1. +-commutative 87.1%

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

      2. mul-1-neg 87.1%

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

      3. unsub-neg 87.1%

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

      4. *-commutative 87.1%

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

   8. Simplified 87.1%

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

      1. pow2 74.7%

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

   10. Applied egg-rr 87.1%

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

4. Final simplification 83.0%

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

Alternative 6: 66.8% accurate, 9.7× speedup

Math

\[\begin{array}{l} \\ \frac{alphay \cdot \left(-alphay\right)}{sin2phi \cdot \left(0.5 + \frac{-1}{u0}\right)} \end{array} \]

FPCore

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

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	return (alphay * -alphay) / (sin2phi * (0.5f + (-1.0f / 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 * (0.5e0 + ((-1.0e0) / u0)))
end function

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	return Float32(Float32(alphay * Float32(-alphay)) / Float32(sin2phi * Float32(Float32(0.5) + Float32(Float32(-1.0) / u0))))
end

MATLAB

function tmp = code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = (alphay * -alphay) / (sin2phi * (single(0.5) + (single(-1.0) / u0)));
end

Derivation

1. Initial program 59.8%

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

3. Taylor expanded in cos2phi around 0 50.6%

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

   1. mul-1-neg 50.6%

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

   2. associate-/l* 50.4%

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

5. Simplified 50.4%

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

6. Taylor expanded in u0 around 0 68.9%

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

   1. +-commutative 68.9%

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

   2. mul-1-neg 68.9%

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

   3. unsub-neg 68.9%

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

   4. *-commutative 68.9%

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

8. Simplified 68.9%

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

   1. pow2 59.8%

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

10. Applied egg-rr 68.9%

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

11. Taylor expanded in sin2phi around 0 68.8%

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

12. Final simplification 68.8%

    \[\leadsto \frac{alphay \cdot \left(-alphay\right)}{sin2phi \cdot \left(0.5 + \frac{-1}{u0}\right)} \]
13. Add Preprocessing

Alternative 7: 66.8% accurate, 9.7× speedup

Math

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

FPCore

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

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	return (alphay * -alphay) / ((sin2phi * 0.5f) - (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 * 0.5e0) - (sin2phi / u0))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 59.8%

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

3. Taylor expanded in cos2phi around 0 50.6%

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

   1. mul-1-neg 50.6%

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

   2. associate-/l* 50.4%

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

5. Simplified 50.4%

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

6. Taylor expanded in u0 around 0 68.9%

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

   1. +-commutative 68.9%

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

   2. mul-1-neg 68.9%

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

   3. unsub-neg 68.9%

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

   4. *-commutative 68.9%

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

8. Simplified 68.9%

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

   1. pow2 59.8%

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

10. Applied egg-rr 68.9%

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

11. Final simplification 68.9%

    \[\leadsto \frac{alphay \cdot \left(-alphay\right)}{sin2phi \cdot 0.5 - \frac{sin2phi}{u0}} \]
12. Add Preprocessing

Alternative 8: 58.0% 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 59.8%

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

3. Taylor expanded in u0 around 0 76.3%

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

   1. mul-1-neg 76.3%

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

5. Simplified 76.3%

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

6. Taylor expanded in cos2phi around 0 60.1%

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

   1. associate-/l* 59.8%

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

8. Simplified 59.8%

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

   1. pow2 59.8%

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

10. Applied egg-rr 59.8%

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

11. Final simplification 59.8%

    \[\leadsto \frac{alphay \cdot alphay}{\frac{sin2phi}{u0}} \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2024021 
(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)))))