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

Percentage Accurate: 61.3% → 90.3%

Time: 12.2s

Alternatives: 8

Speedup: 3.5×

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: 61.3% 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: 90.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;1 - u0 \leq 0.9998840093612671:\\ \;\;\;\;\frac{\log \left(1 - u0\right)}{cos2phi \cdot \frac{-1}{alphax \cdot alphax} - \frac{sin2phi}{alphay \cdot alphay}}\\ \mathbf{else}:\\ \;\;\;\;\frac{u0}{\frac{cos2phi}{alphax \cdot alphax} - \frac{-1}{\frac{alphay}{sin2phi} \cdot alphay}}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (if (<= (- 1.0 u0) 0.9998840093612671)
   (/
    (log (- 1.0 u0))
    (- (* cos2phi (/ -1.0 (* alphax alphax))) (/ sin2phi (* alphay alphay))))
   (/
    u0
    (- (/ cos2phi (* alphax alphax)) (/ -1.0 (* (/ alphay sin2phi) alphay))))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float tmp;
	if ((1.0f - u0) <= 0.9998840093612671f) {
		tmp = logf((1.0f - u0)) / ((cos2phi * (-1.0f / (alphax * alphax))) - (sin2phi / (alphay * alphay)));
	} else {
		tmp = u0 / ((cos2phi / (alphax * alphax)) - (-1.0f / ((alphay / sin2phi) * 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) <= 0.9998840093612671e0) then
        tmp = log((1.0e0 - u0)) / ((cos2phi * ((-1.0e0) / (alphax * alphax))) - (sin2phi / (alphay * alphay)))
    else
        tmp = u0 / ((cos2phi / (alphax * alphax)) - ((-1.0e0) / ((alphay / sin2phi) * 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(0.9998840093612671))
		tmp = Float32(log(Float32(Float32(1.0) - u0)) / Float32(Float32(cos2phi * Float32(Float32(-1.0) / Float32(alphax * alphax))) - Float32(sin2phi / Float32(alphay * alphay))));
	else
		tmp = Float32(u0 / Float32(Float32(cos2phi / Float32(alphax * alphax)) - Float32(Float32(-1.0) / Float32(Float32(alphay / sin2phi) * 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(0.9998840093612671))
		tmp = log((single(1.0) - u0)) / ((cos2phi * (single(-1.0) / (alphax * alphax))) - (sin2phi / (alphay * alphay)));
	else
		tmp = u0 / ((cos2phi / (alphax * alphax)) - (single(-1.0) / ((alphay / sin2phi) * alphay)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (-.f32 #s(literal 1 binary32) u0) < 0.999884009

   1. Initial program 87.1%

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

      1. lift-/.f32 N/A

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

      2. clear-num N/A

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

      3. lower-/.f32 N/A

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

      4. lower-/.f32 87.0

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

   4. Applied rewrites 87.0%

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

      1. lift-/.f32 N/A

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

      2. lift-/.f32 N/A

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

      3. frac-2neg N/A

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

      4. associate-/r/ N/A

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

      5. lower-*.f32 N/A

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

      6. lower-/.f32 N/A

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

      7. lift-*.f32 N/A

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

      8. distribute-lft-neg-in N/A

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

      9. lower-*.f32 N/A

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

      10. lower-neg.f32 N/A

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

      11. lower-neg.f32 87.1

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

   6. Applied rewrites 87.1%

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

   if 0.999884009 < (-.f32 #s(literal 1 binary32) u0)

   1. Initial program 42.9%

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

   3. Taylor expanded in u0 around 0

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

      1. lower-/.f32 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f32 N/A

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

      4. lower-/.f32 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f32 N/A

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

      7. lower-/.f32 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f32 91.8

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

   5. Applied rewrites 91.8%

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

   7. Applied rewrites 91.7%

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

   9. Applied rewrites 91.9%

      \[\leadsto \frac{u0}{\frac{-1}{\left(-alphay\right) \cdot \frac{alphay}{sin2phi}} + \frac{\color{blue}{cos2phi}}{alphax \cdot alphax}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 90.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 - u0 \leq 0.9998840093612671:\\ \;\;\;\;\frac{\log \left(1 - u0\right)}{cos2phi \cdot \frac{-1}{alphax \cdot alphax} - \frac{sin2phi}{alphay \cdot alphay}}\\ \mathbf{else}:\\ \;\;\;\;\frac{u0}{\frac{cos2phi}{alphax \cdot alphax} - \frac{-1}{\frac{alphay}{sin2phi} \cdot alphay}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 90.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{cos2phi}{alphax \cdot alphax}\\ \mathbf{if}\;1 - u0 \leq 0.9998840093612671:\\ \;\;\;\;\frac{-\log \left(1 - u0\right)}{t\_0 + \frac{sin2phi}{alphay \cdot alphay}}\\ \mathbf{else}:\\ \;\;\;\;\frac{u0}{t\_0 - \frac{-1}{\frac{alphay}{sin2phi} \cdot alphay}}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (let* ((t_0 (/ cos2phi (* alphax alphax))))
   (if (<= (- 1.0 u0) 0.9998840093612671)
     (/ (- (log (- 1.0 u0))) (+ t_0 (/ sin2phi (* alphay alphay))))
     (/ u0 (- t_0 (/ -1.0 (* (/ alphay sin2phi) alphay)))))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float t_0 = cos2phi / (alphax * alphax);
	float tmp;
	if ((1.0f - u0) <= 0.9998840093612671f) {
		tmp = -logf((1.0f - u0)) / (t_0 + (sin2phi / (alphay * alphay)));
	} else {
		tmp = u0 / (t_0 - (-1.0f / ((alphay / sin2phi) * 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) :: t_0
    real(4) :: tmp
    t_0 = cos2phi / (alphax * alphax)
    if ((1.0e0 - u0) <= 0.9998840093612671e0) then
        tmp = -log((1.0e0 - u0)) / (t_0 + (sin2phi / (alphay * alphay)))
    else
        tmp = u0 / (t_0 - ((-1.0e0) / ((alphay / sin2phi) * alphay)))
    end if
    code = tmp
end function

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	t_0 = Float32(cos2phi / Float32(alphax * alphax))
	tmp = Float32(0.0)
	if (Float32(Float32(1.0) - u0) <= Float32(0.9998840093612671))
		tmp = Float32(Float32(-log(Float32(Float32(1.0) - u0))) / Float32(t_0 + Float32(sin2phi / Float32(alphay * alphay))));
	else
		tmp = Float32(u0 / Float32(t_0 - Float32(Float32(-1.0) / Float32(Float32(alphay / sin2phi) * alphay))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alphax, alphay, u0, cos2phi, sin2phi)
	t_0 = cos2phi / (alphax * alphax);
	tmp = single(0.0);
	if ((single(1.0) - u0) <= single(0.9998840093612671))
		tmp = -log((single(1.0) - u0)) / (t_0 + (sin2phi / (alphay * alphay)));
	else
		tmp = u0 / (t_0 - (single(-1.0) / ((alphay / sin2phi) * alphay)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (-.f32 #s(literal 1 binary32) u0) < 0.999884009

   1. Initial program 87.1%

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

   if 0.999884009 < (-.f32 #s(literal 1 binary32) u0)

   1. Initial program 42.9%

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

   3. Taylor expanded in u0 around 0

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

      1. lower-/.f32 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f32 N/A

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

      4. lower-/.f32 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f32 N/A

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

      7. lower-/.f32 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f32 91.8

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

   5. Applied rewrites 91.8%

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

   7. Applied rewrites 91.7%

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

   9. Applied rewrites 91.9%

      \[\leadsto \frac{u0}{\frac{-1}{\left(-alphay\right) \cdot \frac{alphay}{sin2phi}} + \frac{\color{blue}{cos2phi}}{alphax \cdot alphax}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 90.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 - u0 \leq 0.9998840093612671:\\ \;\;\;\;\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}}\\ \mathbf{else}:\\ \;\;\;\;\frac{u0}{\frac{cos2phi}{alphax \cdot alphax} - \frac{-1}{\frac{alphay}{sin2phi} \cdot alphay}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 82.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;1 - u0 \leq 0.9995200037956238:\\ \;\;\;\;\frac{-\log \left(1 - u0\right)}{\frac{sin2phi}{alphay \cdot alphay}}\\ \mathbf{else}:\\ \;\;\;\;\frac{u0}{\frac{cos2phi}{alphax \cdot alphax} - \frac{-1}{\frac{alphay}{sin2phi} \cdot alphay}}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (if (<= (- 1.0 u0) 0.9995200037956238)
   (/ (- (log (- 1.0 u0))) (/ sin2phi (* alphay alphay)))
   (/
    u0
    (- (/ cos2phi (* alphax alphax)) (/ -1.0 (* (/ alphay sin2phi) alphay))))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float tmp;
	if ((1.0f - u0) <= 0.9995200037956238f) {
		tmp = -logf((1.0f - u0)) / (sin2phi / (alphay * alphay));
	} else {
		tmp = u0 / ((cos2phi / (alphax * alphax)) - (-1.0f / ((alphay / sin2phi) * 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) <= 0.9995200037956238e0) then
        tmp = -log((1.0e0 - u0)) / (sin2phi / (alphay * alphay))
    else
        tmp = u0 / ((cos2phi / (alphax * alphax)) - ((-1.0e0) / ((alphay / sin2phi) * 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(0.9995200037956238))
		tmp = Float32(Float32(-log(Float32(Float32(1.0) - u0))) / Float32(sin2phi / Float32(alphay * alphay)));
	else
		tmp = Float32(u0 / Float32(Float32(cos2phi / Float32(alphax * alphax)) - Float32(Float32(-1.0) / Float32(Float32(alphay / sin2phi) * 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(0.9995200037956238))
		tmp = -log((single(1.0) - u0)) / (sin2phi / (alphay * alphay));
	else
		tmp = u0 / ((cos2phi / (alphax * alphax)) - (single(-1.0) / ((alphay / sin2phi) * alphay)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (-.f32 #s(literal 1 binary32) u0) < 0.999520004

   1. Initial program 89.3%

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

      1. lift-/.f32 N/A

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

      2. clear-num N/A

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

      3. inv-pow N/A

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

      4. div-inv N/A

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

      5. unpow-prod-down N/A

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

      6. inv-pow N/A

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

      7. lower-*.f32 N/A

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

      8. lift-*.f32 N/A

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

      9. pow2 N/A

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

      10. pow-flip N/A

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

      11. lower-pow.f32 N/A

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

      12. metadata-eval N/A

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

      13. lower-pow.f32 N/A

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

      14. lower-/.f32 89.2

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

   4. Applied rewrites 89.2%

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

   5. Taylor expanded in alphax around inf

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

      1. lower-/.f32 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f32 67.0

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

   7. Applied rewrites 67.0%

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

   if 0.999520004 < (-.f32 #s(literal 1 binary32) u0)

   1. Initial program 45.6%

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

   3. Taylor expanded in u0 around 0

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

      1. lower-/.f32 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f32 N/A

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

      4. lower-/.f32 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f32 N/A

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

      7. lower-/.f32 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f32 89.7

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

   5. Applied rewrites 89.7%

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

   7. Applied rewrites 89.7%

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

   9. Applied rewrites 89.9%

      \[\leadsto \frac{u0}{\frac{-1}{\left(-alphay\right) \cdot \frac{alphay}{sin2phi}} + \frac{\color{blue}{cos2phi}}{alphax \cdot alphax}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 82.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 - u0 \leq 0.9995200037956238:\\ \;\;\;\;\frac{-\log \left(1 - u0\right)}{\frac{sin2phi}{alphay \cdot alphay}}\\ \mathbf{else}:\\ \;\;\;\;\frac{u0}{\frac{cos2phi}{alphax \cdot alphax} - \frac{-1}{\frac{alphay}{sin2phi} \cdot alphay}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 75.4% accurate, 2.6× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 60.7%

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

3. Taylor expanded in u0 around 0

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

   1. lower-/.f32 N/A

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

   2. +-commutative N/A

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

   3. lower-+.f32 N/A

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

   4. lower-/.f32 N/A

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

   5. unpow2 N/A

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

   6. lower-*.f32 N/A

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

   7. lower-/.f32 N/A

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

   8. unpow2 N/A

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

   9. lower-*.f32 75.1

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

5. Applied rewrites 75.1%

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

7. Applied rewrites 75.1%

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

9. Applied rewrites 75.2%

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

10. Final simplification 75.2%

    \[\leadsto \frac{u0}{\frac{cos2phi}{alphax \cdot alphax} - \frac{-1}{\frac{alphay}{sin2phi} \cdot alphay}} \]
11. Add Preprocessing

Alternative 5: 75.4% accurate, 2.9× speedup

Math

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

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(Float32(cos2phi / 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 60.7%

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

3. Taylor expanded in u0 around 0

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

   1. lower-/.f32 N/A

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

   2. +-commutative N/A

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

   3. lower-+.f32 N/A

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

   4. lower-/.f32 N/A

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

   5. unpow2 N/A

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

   6. lower-*.f32 N/A

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

   7. lower-/.f32 N/A

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

   8. unpow2 N/A

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

   9. lower-*.f32 75.1

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

5. Applied rewrites 75.1%

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

7. Applied rewrites 75.1%

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

8. Final simplification 75.1%

   \[\leadsto \frac{u0}{\frac{\frac{cos2phi}{alphax}}{alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
9. Add Preprocessing

Alternative 6: 75.4% accurate, 3.2× 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 60.7%

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

3. Taylor expanded in u0 around 0

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

   1. lower-/.f32 N/A

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

   2. +-commutative N/A

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

   3. lower-+.f32 N/A

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

   4. lower-/.f32 N/A

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

   5. unpow2 N/A

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

   6. lower-*.f32 N/A

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

   7. lower-/.f32 N/A

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

   8. unpow2 N/A

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

   9. lower-*.f32 75.1

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

5. Applied rewrites 75.1%

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

6. Final simplification 75.1%

   \[\leadsto \frac{u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
7. Add Preprocessing

Alternative 7: 66.6% accurate, 3.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{sin2phi}{alphay \cdot alphay} \leq 1.499999973677505 \cdot 10^{-14}:\\ \;\;\;\;\left(\frac{u0}{cos2phi} \cdot alphax\right) \cdot alphax\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(alphay \cdot alphay\right) \cdot u0}{sin2phi}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (if (<= (/ sin2phi (* alphay alphay)) 1.499999973677505e-14)
   (* (* (/ u0 cos2phi) alphax) alphax)
   (/ (* (* alphay alphay) u0) sin2phi)))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float tmp;
	if ((sin2phi / (alphay * alphay)) <= 1.499999973677505e-14f) {
		tmp = ((u0 / cos2phi) * alphax) * alphax;
	} else {
		tmp = ((alphay * alphay) * u0) / sin2phi;
	}
	return tmp;
}

Fortran

real(4) function code(alphax, alphay, u0, cos2phi, sin2phi)
    real(4), intent (in) :: alphax
    real(4), intent (in) :: alphay
    real(4), intent (in) :: u0
    real(4), intent (in) :: cos2phi
    real(4), intent (in) :: sin2phi
    real(4) :: tmp
    if ((sin2phi / (alphay * alphay)) <= 1.499999973677505e-14) then
        tmp = ((u0 / cos2phi) * alphax) * alphax
    else
        tmp = ((alphay * alphay) * u0) / sin2phi
    end if
    code = tmp
end function

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = Float32(0.0)
	if (Float32(sin2phi / Float32(alphay * alphay)) <= Float32(1.499999973677505e-14))
		tmp = Float32(Float32(Float32(u0 / cos2phi) * alphax) * alphax);
	else
		tmp = Float32(Float32(Float32(alphay * alphay) * u0) / sin2phi);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = single(0.0);
	if ((sin2phi / (alphay * alphay)) <= single(1.499999973677505e-14))
		tmp = ((u0 / cos2phi) * alphax) * alphax;
	else
		tmp = ((alphay * alphay) * u0) / sin2phi;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 sin2phi (*.f32 alphay alphay)) < 1.49999997e-14

   1. Initial program 61.0%

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

   3. Taylor expanded in u0 around 0

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

      1. lower-/.f32 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f32 N/A

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

      4. lower-/.f32 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f32 N/A

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

      7. lower-/.f32 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f32 70.4

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

   5. Applied rewrites 70.4%

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

   6. Taylor expanded in alphax around 0

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

   8. Applied rewrites 53.6%

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

   10. Applied rewrites 53.7%

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

   12. Applied rewrites 53.7%

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

   if 1.49999997e-14 < (/.f32 sin2phi (*.f32 alphay alphay))

   1. Initial program 60.5%

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

   3. Taylor expanded in u0 around 0

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

      1. lower-/.f32 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f32 N/A

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

      4. lower-/.f32 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f32 N/A

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

      7. lower-/.f32 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f32 76.7

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

   5. Applied rewrites 76.7%

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

   6. Taylor expanded in alphax around inf

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

   8. Applied rewrites 71.7%

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

Alternative 8: 23.5% accurate, 6.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 60.7%

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

3. Taylor expanded in u0 around 0

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

   1. lower-/.f32 N/A

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

   2. +-commutative N/A

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

   3. lower-+.f32 N/A

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

   4. lower-/.f32 N/A

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

   5. unpow2 N/A

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

   6. lower-*.f32 N/A

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

   7. lower-/.f32 N/A

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

   8. unpow2 N/A

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

   9. lower-*.f32 75.1

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

5. Applied rewrites 75.1%

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

6. Taylor expanded in alphax around 0

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

8. Applied rewrites 22.9%

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

10. Applied rewrites 22.9%

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

11. Final simplification 22.9%

    \[\leadsto \frac{u0}{cos2phi} \cdot \left(alphax \cdot alphax\right) \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2024255 
(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)))))