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

Average Error: 12.4 → 0.6

Time: 38.6s

Precision: binary32

Cost: 10628

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

Math

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

↓

\[\begin{array}{l} t_0 := \frac{sin2phi}{alphay \cdot alphay}\\ \mathbf{if}\;1 - u0 \leq 0.9649999737739563:\\ \;\;\;\;\frac{-\log \left(1 - u0\right)}{\frac{2 \cdot \frac{cos2phi \cdot \frac{4}{alphax \cdot alphax}}{alphax}}{\frac{8}{alphax}} + t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{u0 - \left({u0}^{4} \cdot -0.25 + \left({u0}^{2} \cdot -0.5 - 0.3333333333333333 \cdot {u0}^{3}\right)\right)}{\frac{cos2phi}{alphax \cdot alphax} + t_0}\\ \end{array} \]

FPCore

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

↓

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (let* ((t_0 (/ sin2phi (* alphay alphay))))
   (if (<= (- 1.0 u0) 0.9649999737739563)
     (/
      (- (log (- 1.0 u0)))
      (+
       (/
        (* 2.0 (/ (* cos2phi (/ 4.0 (* alphax alphax))) alphax))
        (/ 8.0 alphax))
       t_0))
     (/
      (-
       u0
       (+
        (* (pow u0 4.0) -0.25)
        (- (* (pow u0 2.0) -0.5) (* 0.3333333333333333 (pow u0 3.0)))))
      (+ (/ cos2phi (* alphax alphax)) t_0)))))

C

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

↓

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float t_0 = sin2phi / (alphay * alphay);
	float tmp;
	if ((1.0f - u0) <= 0.9649999737739563f) {
		tmp = -logf((1.0f - u0)) / (((2.0f * ((cos2phi * (4.0f / (alphax * alphax))) / alphax)) / (8.0f / alphax)) + t_0);
	} else {
		tmp = (u0 - ((powf(u0, 4.0f) * -0.25f) + ((powf(u0, 2.0f) * -0.5f) - (0.3333333333333333f * powf(u0, 3.0f))))) / ((cos2phi / (alphax * alphax)) + t_0);
	}
	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
    code = -log((1.0e0 - u0)) / ((cos2phi / (alphax * alphax)) + (sin2phi / (alphay * alphay)))
end function

↓

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 ((1.0e0 - u0) <= 0.9649999737739563e0) then
        tmp = -log((1.0e0 - u0)) / (((2.0e0 * ((cos2phi * (4.0e0 / (alphax * alphax))) / alphax)) / (8.0e0 / alphax)) + t_0)
    else
        tmp = (u0 - (((u0 ** 4.0e0) * (-0.25e0)) + (((u0 ** 2.0e0) * (-0.5e0)) - (0.3333333333333333e0 * (u0 ** 3.0e0))))) / ((cos2phi / (alphax * alphax)) + t_0)
    end if
    code = tmp
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

↓

function code(alphax, alphay, u0, cos2phi, sin2phi)
	t_0 = Float32(sin2phi / Float32(alphay * alphay))
	tmp = Float32(0.0)
	if (Float32(Float32(1.0) - u0) <= Float32(0.9649999737739563))
		tmp = Float32(Float32(-log(Float32(Float32(1.0) - u0))) / Float32(Float32(Float32(Float32(2.0) * Float32(Float32(cos2phi * Float32(Float32(4.0) / Float32(alphax * alphax))) / alphax)) / Float32(Float32(8.0) / alphax)) + t_0));
	else
		tmp = Float32(Float32(u0 - Float32(Float32((u0 ^ Float32(4.0)) * Float32(-0.25)) + Float32(Float32((u0 ^ Float32(2.0)) * Float32(-0.5)) - Float32(Float32(0.3333333333333333) * (u0 ^ Float32(3.0)))))) / Float32(Float32(cos2phi / Float32(alphax * alphax)) + t_0));
	end
	return tmp
end

MATLAB

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

↓

function tmp_2 = code(alphax, alphay, u0, cos2phi, sin2phi)
	t_0 = sin2phi / (alphay * alphay);
	tmp = single(0.0);
	if ((single(1.0) - u0) <= single(0.9649999737739563))
		tmp = -log((single(1.0) - u0)) / (((single(2.0) * ((cos2phi * (single(4.0) / (alphax * alphax))) / alphax)) / (single(8.0) / alphax)) + t_0);
	else
		tmp = (u0 - (((u0 ^ single(4.0)) * single(-0.25)) + (((u0 ^ single(2.0)) * single(-0.5)) - (single(0.3333333333333333) * (u0 ^ single(3.0)))))) / ((cos2phi / (alphax * alphax)) + t_0);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (-.f32 1 u0) < 0.964999974

   1. Initial program 1.5

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

   2. Applied egg-rr 1.5

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

   3. Applied egg-rr 1.5

      \[\leadsto \frac{-\log \left(1 - u0\right)}{\color{blue}{\frac{2 \cdot \frac{\frac{cos2phi}{alphax} \cdot \frac{4}{alphax}}{alphax}}{\frac{8}{alphax}}} + \frac{sin2phi}{alphay \cdot alphay}} \]

   4. Simplified 1.5

      \[\leadsto \frac{-\log \left(1 - u0\right)}{\color{blue}{\frac{2 \cdot \frac{cos2phi \cdot \frac{4}{alphax \cdot alphax}}{alphax}}{\frac{8}{alphax}}} + \frac{sin2phi}{alphay \cdot alphay}} \]
      Proof

      [Start] 1.5	\[ \frac{-\log \left(1 - u0\right)}{\frac{2 \cdot \frac{\frac{cos2phi}{alphax} \cdot \frac{4}{alphax}}{alphax}}{\frac{8}{alphax}} + \frac{sin2phi}{alphay \cdot alphay}} \]
      rational_best-simplify-55 [=>] 1.5	\[ \frac{-\log \left(1 - u0\right)}{\frac{2 \cdot \frac{\color{blue}{4 \cdot \frac{\frac{cos2phi}{alphax}}{alphax}}}{alphax}}{\frac{8}{alphax}} + \frac{sin2phi}{alphay \cdot alphay}} \]
      rational_best-simplify-53 [=>] 1.5	\[ \frac{-\log \left(1 - u0\right)}{\frac{2 \cdot \frac{4 \cdot \color{blue}{\frac{cos2phi}{alphax \cdot alphax}}}{alphax}}{\frac{8}{alphax}} + \frac{sin2phi}{alphay \cdot alphay}} \]
      rational_best-simplify-55 [=>] 1.5	\[ \frac{-\log \left(1 - u0\right)}{\frac{2 \cdot \frac{\color{blue}{cos2phi \cdot \frac{4}{alphax \cdot alphax}}}{alphax}}{\frac{8}{alphax}} + \frac{sin2phi}{alphay \cdot alphay}} \]

   if 0.964999974 < (-.f32 1 u0)

   1. Initial program 14.5

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

   2. Simplified 14.5

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

      [Start] 14.5	\[ \frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
      rational_best-simplify-13 [=>] 14.5	\[ \frac{\color{blue}{\frac{\log \left(1 - u0\right)}{-1}}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
      rational_best-simplify-53 [=>] 14.5	\[ \color{blue}{\frac{\log \left(1 - u0\right)}{-1 \cdot \left(\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\right)}} \]
      rational_best-simplify-1 [=>] 14.5	\[ \frac{\log \left(1 - u0\right)}{\color{blue}{\left(\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\right) \cdot -1}} \]
      rational_best-simplify-10 [=>] 14.5	\[ \frac{\log \left(1 - u0\right)}{\color{blue}{-\left(\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\right)}} \]
      rational_best-simplify-54 [=>] 14.5	\[ \frac{\log \left(1 - u0\right)}{-\left(\color{blue}{\frac{\frac{cos2phi}{alphax}}{alphax}} + \frac{sin2phi}{alphay \cdot alphay}\right)} \]

   3. Taylor expanded in u0 around 0 0.5

      \[\leadsto \frac{\color{blue}{-1 \cdot u0 + \left(-0.5 \cdot {u0}^{2} + \left(-0.3333333333333333 \cdot {u0}^{3} + -0.25 \cdot {u0}^{4}\right)\right)}}{-\left(\frac{\frac{cos2phi}{alphax}}{alphax} + \frac{sin2phi}{alphay \cdot alphay}\right)} \]

   4. Applied egg-rr 0.5

      \[\leadsto \color{blue}{\frac{{u0}^{3} \cdot 0.3333333333333333 - -0.5 \cdot {u0}^{2}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} + \frac{u0 - -0.25 \cdot {u0}^{4}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}}} \]

   5. Simplified 0.5

      \[\leadsto \color{blue}{\frac{\left(0.3333333333333333 \cdot {u0}^{3} - {u0}^{2} \cdot -0.5\right) + \left(u0 - {u0}^{4} \cdot -0.25\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}}} \]
      Proof

      [Start] 0.5	\[ \frac{{u0}^{3} \cdot 0.3333333333333333 - -0.5 \cdot {u0}^{2}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} + \frac{u0 - -0.25 \cdot {u0}^{4}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
      rational_best-simplify-64 [=>] 0.5	\[ \color{blue}{\frac{\left({u0}^{3} \cdot 0.3333333333333333 - -0.5 \cdot {u0}^{2}\right) + \left(u0 - -0.25 \cdot {u0}^{4}\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}}} \]
      rational_best-simplify-1 [=>] 0.5	\[ \frac{\left(\color{blue}{0.3333333333333333 \cdot {u0}^{3}} - -0.5 \cdot {u0}^{2}\right) + \left(u0 - -0.25 \cdot {u0}^{4}\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
      rational_best-simplify-1 [=>] 0.5	\[ \frac{\left(0.3333333333333333 \cdot {u0}^{3} - \color{blue}{{u0}^{2} \cdot -0.5}\right) + \left(u0 - -0.25 \cdot {u0}^{4}\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
      rational_best-simplify-1 [=>] 0.5	\[ \frac{\left(0.3333333333333333 \cdot {u0}^{3} - {u0}^{2} \cdot -0.5\right) + \left(u0 - \color{blue}{{u0}^{4} \cdot -0.25}\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   6. Applied egg-rr 0.5

      \[\leadsto \color{blue}{\frac{\left(0.3333333333333333 \cdot {u0}^{3} - {u0}^{2} \cdot -0.5\right) + \left(u0 - {u0}^{4} \cdot -0.25\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} + 0} \]

   7. Simplified 0.5

      \[\leadsto \color{blue}{\frac{u0 - \left({u0}^{4} \cdot -0.25 + \left({u0}^{2} \cdot -0.5 - 0.3333333333333333 \cdot {u0}^{3}\right)\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}}} \]
      Proof

      [Start] 0.5	\[ \frac{\left(0.3333333333333333 \cdot {u0}^{3} - {u0}^{2} \cdot -0.5\right) + \left(u0 - {u0}^{4} \cdot -0.25\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} + 0 \]
      rational_best-simplify-3 [<=] 0.5	\[ \color{blue}{0 + \frac{\left(0.3333333333333333 \cdot {u0}^{3} - {u0}^{2} \cdot -0.5\right) + \left(u0 - {u0}^{4} \cdot -0.25\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}}} \]
      rational_best-simplify-6 [=>] 0.5	\[ \color{blue}{\frac{\left(0.3333333333333333 \cdot {u0}^{3} - {u0}^{2} \cdot -0.5\right) + \left(u0 - {u0}^{4} \cdot -0.25\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}}} \]
      rational_best-simplify-64 [<=] 0.5	\[ \color{blue}{\frac{0.3333333333333333 \cdot {u0}^{3} - {u0}^{2} \cdot -0.5}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} + \frac{u0 - {u0}^{4} \cdot -0.25}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}}} \]
      rational_best-simplify-67 [=>] 0.5	\[ \color{blue}{\left(\frac{0.3333333333333333 \cdot {u0}^{3}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} - \frac{{u0}^{2} \cdot -0.5}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}}\right)} + \frac{u0 - {u0}^{4} \cdot -0.25}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
      rational_best-simplify-9 [<=] 0.5	\[ \left(\frac{0.3333333333333333 \cdot {u0}^{3}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} - \frac{{u0}^{2} \cdot -0.5}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}}\right) + \color{blue}{\left(\frac{u0 - {u0}^{4} \cdot -0.25}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} - 0\right)} \]
      metadata-eval [<=] 0.5	\[ \left(\frac{0.3333333333333333 \cdot {u0}^{3}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} - \frac{{u0}^{2} \cdot -0.5}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}}\right) + \left(\frac{u0 - {u0}^{4} \cdot -0.25}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} - \color{blue}{\left(0 - 0\right)}\right) \]
      rational_best-simplify-51 [<=] 0.5	\[ \left(\frac{0.3333333333333333 \cdot {u0}^{3}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} - \frac{{u0}^{2} \cdot -0.5}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}}\right) + \color{blue}{\left(0 - \left(0 - \frac{u0 - {u0}^{4} \cdot -0.25}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}}\right)\right)} \]
      rational_best-simplify-14 [<=] 0.5	\[ \left(\frac{0.3333333333333333 \cdot {u0}^{3}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} - \frac{{u0}^{2} \cdot -0.5}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}}\right) + \left(0 - \color{blue}{\left(-\frac{u0 - {u0}^{4} \cdot -0.25}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}}\right)}\right) \]
      rational_best-simplify-14 [<=] 0.5	\[ \left(\frac{0.3333333333333333 \cdot {u0}^{3}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} - \frac{{u0}^{2} \cdot -0.5}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}}\right) + \color{blue}{\left(-\left(-\frac{u0 - {u0}^{4} \cdot -0.25}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}}\right)\right)} \]

3. Recombined 2 regimes into one program.

4. Final simplification 0.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 - u0 \leq 0.9649999737739563:\\ \;\;\;\;\frac{-\log \left(1 - u0\right)}{\frac{2 \cdot \frac{cos2phi \cdot \frac{4}{alphax \cdot alphax}}{alphax}}{\frac{8}{alphax}} + \frac{sin2phi}{alphay \cdot alphay}}\\ \mathbf{else}:\\ \;\;\;\;\frac{u0 - \left({u0}^{4} \cdot -0.25 + \left({u0}^{2} \cdot -0.5 - 0.3333333333333333 \cdot {u0}^{3}\right)\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}}\\ \end{array} \]

Alternatives

Alternative 1
Error	0.6
Cost	10628

\[\begin{array}{l} t_0 := \frac{sin2phi}{alphay \cdot alphay}\\ \mathbf{if}\;1 - u0 \leq 0.9649999737739563:\\ \;\;\;\;\frac{-\log \left(1 - u0\right)}{\frac{2 \cdot \frac{cos2phi \cdot \frac{4}{alphax \cdot alphax}}{alphax}}{\frac{8}{alphax}} + t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(0.3333333333333333 \cdot {u0}^{3} - {u0}^{2} \cdot -0.5\right) + \left(u0 - {u0}^{4} \cdot -0.25\right)}{\frac{cos2phi}{alphax \cdot alphax} + t_0}\\ \end{array} \]

Alternative 2
Error	0.8
Cost	7428

\[\begin{array}{l} \mathbf{if}\;1 - u0 \leq 0.9850000143051147:\\ \;\;\;\;\frac{-\log \left(1 - u0\right)}{\frac{2 \cdot \frac{cos2phi \cdot \frac{4}{alphax \cdot alphax}}{alphax}}{\frac{8}{alphax}} + \frac{sin2phi}{alphay \cdot alphay}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1 \cdot u0 + \left(-0.5 \cdot {u0}^{2} + -0.3333333333333333 \cdot {u0}^{3}\right)}{-\left(\frac{\frac{cos2phi}{alphax}}{alphax} + \frac{sin2phi}{alphay} \cdot \frac{1}{alphay}\right)}\\ \end{array} \]

Alternative 3
Error	1.2
Cost	7332

\[\begin{array}{l} t_0 := -\log \left(1 - u0\right)\\ \mathbf{if}\;t_0 \leq 0.002199999988079071:\\ \;\;\;\;\frac{u0 \cdot -2 - {u0}^{2}}{\frac{\frac{sin2phi}{alphay}}{-alphay} - \frac{cos2phi}{alphax \cdot alphax}} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0}{\frac{cos2phi \cdot \frac{2}{alphax \cdot alphax}}{alphax \cdot \frac{2}{alphax}} + \frac{sin2phi}{alphay \cdot alphay}}\\ \end{array} \]

Alternative 4
Error	0.8
Cost	7268

\[\begin{array}{l} t_0 := \frac{sin2phi}{alphay \cdot alphay}\\ \mathbf{if}\;1 - u0 \leq 0.9850000143051147:\\ \;\;\;\;\frac{-\log \left(1 - u0\right)}{\frac{2 \cdot \frac{cos2phi \cdot \frac{4}{alphax \cdot alphax}}{alphax}}{\frac{8}{alphax}} + t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{u0 - \left({u0}^{2} \cdot -0.5 + {u0}^{3} \cdot -0.3333333333333333\right)}{\frac{cos2phi}{alphax \cdot alphax} + t_0}\\ \end{array} \]

Alternative 5
Error	1.2
Cost	7236

\[\begin{array}{l} t_0 := \frac{cos2phi}{alphax \cdot alphax}\\ t_1 := -\log \left(1 - u0\right)\\ \mathbf{if}\;t_1 \leq 0.004000000189989805:\\ \;\;\;\;\frac{1}{\frac{\frac{sin2phi}{alphay}}{-alphay} - t_0} \cdot \left(-0.5 \cdot {u0}^{2} - u0\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{t_1}{t_0 + \frac{1}{alphay \cdot alphay} \cdot sin2phi}\\ \end{array} \]

Alternative 6
Error	1.2
Cost	7236

\[\begin{array}{l} t_0 := -\log \left(1 - u0\right)\\ t_1 := \frac{cos2phi}{alphax \cdot alphax}\\ \mathbf{if}\;t_0 \leq 0.004000000189989805:\\ \;\;\;\;\frac{u0 \cdot -2 - {u0}^{2}}{\frac{\frac{sin2phi}{alphay}}{-alphay} - t_1} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0}{t_1 + \frac{1}{alphay \cdot alphay} \cdot sin2phi}\\ \end{array} \]

Alternative 7
Error	1.2
Cost	4228

\[\begin{array}{l} \mathbf{if}\;1 - u0 \leq 0.9976000189781189:\\ \;\;\;\;\frac{-\log \left(1 - u0\right)}{\frac{2 \cdot \frac{cos2phi \cdot \frac{4}{alphax \cdot alphax}}{alphax}}{\frac{8}{alphax}} + \frac{sin2phi}{alphay \cdot alphay}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-u0\right) + -0.5 \cdot {u0}^{2}}{-\left(\frac{\frac{cos2phi}{alphax}}{alphax} + \left(sin2phi \cdot 4\right) \cdot \frac{\frac{1}{alphay}}{\left(alphay + alphay\right) \cdot 2}\right)}\\ \end{array} \]

Alternative 8
Error	1.2
Cost	4164

\[\begin{array}{l} \mathbf{if}\;1 - u0 \leq 0.9976000189781189:\\ \;\;\;\;\frac{-\log \left(1 - u0\right)}{\frac{2 \cdot \frac{cos2phi \cdot \frac{4}{alphax \cdot alphax}}{alphax}}{\frac{8}{alphax}} + \frac{sin2phi}{alphay \cdot alphay}}\\ \mathbf{else}:\\ \;\;\;\;\frac{u0 \cdot -2 - {u0}^{2}}{\frac{\frac{sin2phi}{alphay}}{-alphay} - \frac{cos2phi}{alphax \cdot alphax}} \cdot 0.5\\ \end{array} \]

Alternative 9
Error	3.2
Cost	3908

\[\begin{array}{l} \mathbf{if}\;1 - u0 \leq 0.9998000264167786:\\ \;\;\;\;\frac{-\log \left(1 - u0\right)}{\frac{1}{alphax \cdot alphax} \cdot cos2phi + \frac{sin2phi}{alphay \cdot alphay}}\\ \mathbf{else}:\\ \;\;\;\;\frac{u0}{\frac{cos2phi}{alphax \cdot alphax} + \left(sin2phi \cdot 4\right) \cdot \frac{\frac{1}{alphay}}{\left(alphay + alphay\right) \cdot 2}}\\ \end{array} \]

Alternative 10
Error	3.2
Cost	3908

\[\begin{array}{l} t_0 := \frac{cos2phi}{alphax \cdot alphax}\\ \mathbf{if}\;1 - u0 \leq 0.9998000264167786:\\ \;\;\;\;\frac{-\log \left(1 - u0\right)}{t_0 + \frac{1}{alphay \cdot alphay} \cdot sin2phi}\\ \mathbf{else}:\\ \;\;\;\;\frac{u0}{t_0 + \left(sin2phi \cdot 4\right) \cdot \frac{\frac{1}{alphay}}{\left(alphay + alphay\right) \cdot 2}}\\ \end{array} \]

Alternative 11
Error	1.2
Cost	3908

\[\begin{array}{l} t_0 := \frac{cos2phi}{alphax \cdot alphax}\\ \mathbf{if}\;1 - u0 \leq 0.9959999918937683:\\ \;\;\;\;\frac{-\log \left(1 - u0\right)}{t_0 + \frac{1}{alphay \cdot alphay} \cdot sin2phi}\\ \mathbf{else}:\\ \;\;\;\;\frac{u0 - {u0}^{2} \cdot -0.5}{t_0 + \frac{sin2phi}{alphay \cdot alphay}}\\ \end{array} \]

Alternative 12
Error	3.2
Cost	3844

\[\begin{array}{l} \mathbf{if}\;1 - u0 \leq 0.9998000264167786:\\ \;\;\;\;\frac{\log \left(1 - u0\right)}{-\left(\frac{\frac{cos2phi}{alphax}}{alphax} + \frac{sin2phi}{alphay \cdot alphay}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{u0}{\frac{cos2phi}{alphax \cdot alphax} + \left(sin2phi \cdot 4\right) \cdot \frac{\frac{1}{alphay}}{\left(alphay + alphay\right) \cdot 2}}\\ \end{array} \]

Alternative 13
Error	3.2
Cost	3844

\[\begin{array}{l} t_0 := \frac{cos2phi}{alphax \cdot alphax}\\ \mathbf{if}\;1 - u0 \leq 0.9998000264167786:\\ \;\;\;\;\frac{\log \left(1 - u0\right)}{\left(-\frac{\frac{sin2phi}{alphay}}{alphay}\right) - t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{u0}{t_0 + \left(sin2phi \cdot 4\right) \cdot \frac{\frac{1}{alphay}}{\left(alphay + alphay\right) \cdot 2}}\\ \end{array} \]

Alternative 14
Error	3.2
Cost	3844

\[\begin{array}{l} t_0 := \frac{cos2phi}{alphax \cdot alphax}\\ \mathbf{if}\;1 - u0 \leq 0.9998000264167786:\\ \;\;\;\;\frac{-\log \left(1 - u0\right)}{t_0 + \frac{sin2phi}{alphay \cdot alphay}}\\ \mathbf{else}:\\ \;\;\;\;\frac{u0}{t_0 + \left(sin2phi \cdot 4\right) \cdot \frac{\frac{1}{alphay}}{\left(alphay + alphay\right) \cdot 2}}\\ \end{array} \]

Alternative 15
Error	7.9
Cost	672

\[\frac{u0}{\frac{cos2phi}{alphax \cdot alphax} + \left(sin2phi \cdot 4\right) \cdot \frac{\frac{1}{alphay}}{\left(alphay + alphay\right) \cdot 2}} \]

Alternative 16
Error	7.9
Cost	480

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

Alternative 17
Error	7.9
Cost	416

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

Alternative 18
Error	7.9
Cost	416

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

Reproduce

herbie shell --seed 2023100 
(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)))))