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

Percentage Accurate: 61.1% → 98.4%

Time: 19.9s

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: 61.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Alternative 1: 98.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \frac{-\mathsf{log1p}\left(-u0\right)}{{alphax}^{-2} \cdot cos2phi + \frac{\frac{sin2phi}{alphay}}{alphay}} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (/
  (- (log1p (- u0)))
  (+ (* (pow alphax -2.0) cos2phi) (/ (/ sin2phi alphay) alphay))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	return -log1pf(-u0) / ((powf(alphax, -2.0f) * cos2phi) + ((sin2phi / alphay) / alphay));
}

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	return Float32(Float32(-log1p(Float32(-u0))) / Float32(Float32((alphax ^ Float32(-2.0)) * cos2phi) + Float32(Float32(sin2phi / alphay) / alphay)))
end

Derivation

1. Initial program 58.9%

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

   1. sqr-neg 58.9%

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

   2. sub-neg 58.9%

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

   3. log1p-def 98.1%

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

   4. sqr-neg 98.1%

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

   5. associate-/r* 98.0%

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

3. Simplified 98.0%

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

   1. div-inv 98.0%

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

   2. *-commutative 98.0%

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

   3. pow2 98.0%

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

   4. pow-flip 98.1%

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

   5. metadata-eval 98.1%

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

5. Applied egg-rr 98.1%

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

6. Final simplification 98.1%

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

Alternative 2: 83.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;1 - u0 \leq 0.9990000128746033:\\ \;\;\;\;\frac{alphay \cdot \left(-alphay\right)}{\frac{sin2phi}{\log \left(1 - u0\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (if (<= (- 1.0 u0) 0.9990000128746033)
   (/ (* 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) <= 0.9990000128746033f) {
		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) <= 0.9990000128746033e0) 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(0.9990000128746033))
		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(0.9990000128746033))
		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) < 0.999000013

   1. Initial program 89.9%

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

   2. Taylor expanded in cos2phi around 0 72.1%

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

      1. associate-/l* 72.3%

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

      2. associate-*r/ 72.3%

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

      3. neg-mul-1 72.3%

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

   4. Simplified 72.3%

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

      1. unpow2 72.3%

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

   6. Applied egg-rr 72.3%

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

   if 0.999000013 < (-.f32 1 u0)

   1. Initial program 44.3%

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

   2. Taylor expanded in u0 around 0 89.0%

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

      1. mul-1-neg 89.0%

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

   4. Simplified 89.0%

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

4. Final simplification 83.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 - u0 \leq 0.9990000128746033:\\ \;\;\;\;\frac{alphay \cdot \left(-alphay\right)}{\frac{sin2phi}{\log \left(1 - u0\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}}\\ \end{array} \]

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 58.9%

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

   1. sub-neg 58.9%

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

   2. log1p-def 98.1%

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

3. Simplified 98.1%

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

4. Final simplification 98.1%

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

Alternative 4: 98.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 58.9%

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

   1. sqr-neg 58.9%

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

   2. sub-neg 58.9%

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

   3. log1p-def 98.1%

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

   4. sqr-neg 98.1%

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

   5. associate-/r* 98.0%

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

3. Simplified 98.0%

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

   1. add-sqr-sqrt 97.9%

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

   2. times-frac 97.8%

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

5. Applied egg-rr 97.8%

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

   1. unpow2 97.8%

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

7. Simplified 97.8%

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

   1. unpow2 97.8%

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

   2. times-frac 97.9%

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

   3. add-sqr-sqrt 98.0%

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

   4. associate-/r* 98.1%

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

9. Applied egg-rr 98.1%

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

10. Final simplification 98.1%

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

Alternative 5: 75.6% accurate, 8.9× speedup

Math

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

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(sin2phi / Float32(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 58.9%

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

2. Taylor expanded in u0 around 0 75.0%

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

   1. mul-1-neg 75.0%

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

4. Simplified 75.0%

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

5. Final simplification 75.0%

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

Alternative 6: 59.0% accurate, 12.9× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 58.9%

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

2. Taylor expanded in u0 around 0 75.0%

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

   1. mul-1-neg 75.0%

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

4. Simplified 75.0%

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

5. Taylor expanded in cos2phi around 0 57.3%

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

   1. associate-/l* 56.9%

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

   2. *-un-lft-identity 56.9%

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

   3. div-inv 56.8%

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

   4. times-frac 57.2%

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

7. Applied egg-rr 57.2%

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

   1. div-inv 57.2%

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

   2. unpow2 57.2%

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

   3. associate-*l* 57.3%

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

   4. remove-double-div 57.4%

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

9. Applied egg-rr 57.4%

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

10. Final simplification 57.4%

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

Alternative 7: 59.0% accurate, 16.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 58.9%

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

2. Taylor expanded in u0 around 0 75.0%

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

   1. mul-1-neg 75.0%

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

4. Simplified 75.0%

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

5. Taylor expanded in cos2phi around 0 57.3%

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

   1. associate-/l* 56.9%

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

   2. unpow2 56.9%

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

   3. *-un-lft-identity 56.9%

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

   4. times-frac 56.9%

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

   5. /-rgt-identity 56.9%

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

7. Applied egg-rr 56.9%

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

   1. associate-/r/ 57.2%

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

9. Applied egg-rr 57.2%

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

10. Final simplification 57.2%

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

Alternative 8: 59.0% accurate, 16.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 58.9%

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

2. Taylor expanded in u0 around 0 75.0%

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

   1. mul-1-neg 75.0%

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

4. Simplified 75.0%

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

5. Taylor expanded in cos2phi around 0 57.3%

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

   1. associate-/l* 56.9%

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

   2. unpow2 56.9%

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

   3. *-un-lft-identity 56.9%

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

   4. times-frac 56.9%

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

   5. /-rgt-identity 56.9%

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

7. Applied egg-rr 56.9%

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

   1. div-inv 56.8%

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

   2. *-commutative 56.8%

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

   3. clear-num 57.3%

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

9. Applied egg-rr 57.3%

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

10. Final simplification 57.3%

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

Reproduce

herbie shell --seed 2023336 
(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)))))