Beckmann Distribution sample, tan2theta, alphax == alphay

Percentage Accurate: 55.8% → 89.9%

Time: 5.5s

Alternatives: 4

Speedup: 10.5×

Specification

\[\left(0.0001 \leq \alpha \land \alpha \leq 1\right) \land \left(2.328306437 \cdot 10^{-10} \leq u0 \land u0 \leq 1\right)\]

Math

\[\begin{array}{l} \\ \left(\left(-\alpha\right) \cdot \alpha\right) \cdot \log \left(1 - u0\right) \end{array} \]

FPCore

(FPCore (alpha u0)
 :precision binary32
 (* (* (- alpha) alpha) (log (- 1.0 u0))))

C

float code(float alpha, float u0) {
	return (-alpha * alpha) * logf((1.0f - u0));
}

Fortran

real(4) function code(alpha, u0)
    real(4), intent (in) :: alpha
    real(4), intent (in) :: u0
    code = (-alpha * alpha) * log((1.0e0 - u0))
end function

Julia

function code(alpha, u0)
	return Float32(Float32(Float32(-alpha) * alpha) * log(Float32(Float32(1.0) - u0)))
end

MATLAB

function tmp = code(alpha, u0)
	tmp = (-alpha * alpha) * log((single(1.0) - u0));
end

Initial Program: 55.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(\left(-\alpha\right) \cdot \alpha\right) \cdot \log \left(1 - u0\right) \end{array} \]

FPCore

(FPCore (alpha u0)
 :precision binary32
 (* (* (- alpha) alpha) (log (- 1.0 u0))))

C

float code(float alpha, float u0) {
	return (-alpha * alpha) * logf((1.0f - u0));
}

Fortran

real(4) function code(alpha, u0)
    real(4), intent (in) :: alpha
    real(4), intent (in) :: u0
    code = (-alpha * alpha) * log((1.0e0 - u0))
end function

Julia

function code(alpha, u0)
	return Float32(Float32(Float32(-alpha) * alpha) * log(Float32(Float32(1.0) - u0)))
end

MATLAB

function tmp = code(alpha, u0)
	tmp = (-alpha * alpha) * log((single(1.0) - u0));
end

Alternative 1: 89.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;1 - u0 \leq 0.9998940229415894:\\ \;\;\;\;\left(\frac{1}{\frac{-1}{\alpha}} \cdot \alpha\right) \cdot \log \left(1 - u0\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\alpha \cdot \alpha\right) \cdot u0\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha u0)
 :precision binary32
 (if (<= (- 1.0 u0) 0.9998940229415894)
   (* (* (/ 1.0 (/ -1.0 alpha)) alpha) (log (- 1.0 u0)))
   (* (* alpha alpha) u0)))

C

float code(float alpha, float u0) {
	float tmp;
	if ((1.0f - u0) <= 0.9998940229415894f) {
		tmp = ((1.0f / (-1.0f / alpha)) * alpha) * logf((1.0f - u0));
	} else {
		tmp = (alpha * alpha) * u0;
	}
	return tmp;
}

Fortran

real(4) function code(alpha, u0)
    real(4), intent (in) :: alpha
    real(4), intent (in) :: u0
    real(4) :: tmp
    if ((1.0e0 - u0) <= 0.9998940229415894e0) then
        tmp = ((1.0e0 / ((-1.0e0) / alpha)) * alpha) * log((1.0e0 - u0))
    else
        tmp = (alpha * alpha) * u0
    end if
    code = tmp
end function

Julia

function code(alpha, u0)
	tmp = Float32(0.0)
	if (Float32(Float32(1.0) - u0) <= Float32(0.9998940229415894))
		tmp = Float32(Float32(Float32(Float32(1.0) / Float32(Float32(-1.0) / alpha)) * alpha) * log(Float32(Float32(1.0) - u0)));
	else
		tmp = Float32(Float32(alpha * alpha) * u0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, u0)
	tmp = single(0.0);
	if ((single(1.0) - u0) <= single(0.9998940229415894))
		tmp = ((single(1.0) / (single(-1.0) / alpha)) * alpha) * log((single(1.0) - u0));
	else
		tmp = (alpha * alpha) * u0;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (-.f32 #s(literal 1 binary32) u0) < 0.999894023

   1. Initial program 84.5%

      \[\left(\left(-\alpha\right) \cdot \alpha\right) \cdot \log \left(1 - u0\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-neg.f32 N/A

         \[\leadsto \left(\color{blue}{\left(\mathsf{neg}\left(\alpha\right)\right)} \cdot \alpha\right) \cdot \log \left(1 - u0\right) \]

      2. neg-sub0 N/A

         \[\leadsto \left(\color{blue}{\left(0 - \alpha\right)} \cdot \alpha\right) \cdot \log \left(1 - u0\right) \]

      3. flip3-- N/A

         \[\leadsto \left(\color{blue}{\frac{{0}^{3} - {\alpha}^{3}}{0 \cdot 0 + \left(\alpha \cdot \alpha + 0 \cdot \alpha\right)}} \cdot \alpha\right) \cdot \log \left(1 - u0\right) \]

      4. clear-num N/A

         \[\leadsto \left(\color{blue}{\frac{1}{\frac{0 \cdot 0 + \left(\alpha \cdot \alpha + 0 \cdot \alpha\right)}{{0}^{3} - {\alpha}^{3}}}} \cdot \alpha\right) \cdot \log \left(1 - u0\right) \]

      5. lower-/.f32 N/A

         \[\leadsto \left(\color{blue}{\frac{1}{\frac{0 \cdot 0 + \left(\alpha \cdot \alpha + 0 \cdot \alpha\right)}{{0}^{3} - {\alpha}^{3}}}} \cdot \alpha\right) \cdot \log \left(1 - u0\right) \]

      6. metadata-eval N/A

         \[\leadsto \left(\frac{1}{\frac{\color{blue}{0} + \left(\alpha \cdot \alpha + 0 \cdot \alpha\right)}{{0}^{3} - {\alpha}^{3}}} \cdot \alpha\right) \cdot \log \left(1 - u0\right) \]

      7. +-lft-identity N/A

         \[\leadsto \left(\frac{1}{\frac{\color{blue}{\alpha \cdot \alpha + 0 \cdot \alpha}}{{0}^{3} - {\alpha}^{3}}} \cdot \alpha\right) \cdot \log \left(1 - u0\right) \]

      8. mul0-lft N/A

         \[\leadsto \left(\frac{1}{\frac{\alpha \cdot \alpha + \color{blue}{0}}{{0}^{3} - {\alpha}^{3}}} \cdot \alpha\right) \cdot \log \left(1 - u0\right) \]

      9. +-rgt-identity N/A

         \[\leadsto \left(\frac{1}{\frac{\color{blue}{\alpha \cdot \alpha}}{{0}^{3} - {\alpha}^{3}}} \cdot \alpha\right) \cdot \log \left(1 - u0\right) \]

      10. clear-num N/A

          \[\leadsto \left(\frac{1}{\color{blue}{\frac{1}{\frac{{0}^{3} - {\alpha}^{3}}{\alpha \cdot \alpha}}}} \cdot \alpha\right) \cdot \log \left(1 - u0\right) \]

      11. +-rgt-identity N/A

          \[\leadsto \left(\frac{1}{\frac{1}{\frac{{0}^{3} - {\alpha}^{3}}{\color{blue}{\alpha \cdot \alpha + 0}}}} \cdot \alpha\right) \cdot \log \left(1 - u0\right) \]

      12. mul0-lft N/A

          \[\leadsto \left(\frac{1}{\frac{1}{\frac{{0}^{3} - {\alpha}^{3}}{\alpha \cdot \alpha + \color{blue}{0 \cdot \alpha}}}} \cdot \alpha\right) \cdot \log \left(1 - u0\right) \]

      13. +-lft-identity N/A

          \[\leadsto \left(\frac{1}{\frac{1}{\frac{{0}^{3} - {\alpha}^{3}}{\color{blue}{0 + \left(\alpha \cdot \alpha + 0 \cdot \alpha\right)}}}} \cdot \alpha\right) \cdot \log \left(1 - u0\right) \]

      14. metadata-eval N/A

          \[\leadsto \left(\frac{1}{\frac{1}{\frac{{0}^{3} - {\alpha}^{3}}{\color{blue}{0 \cdot 0} + \left(\alpha \cdot \alpha + 0 \cdot \alpha\right)}}} \cdot \alpha\right) \cdot \log \left(1 - u0\right) \]

      15. flip3-- N/A

          \[\leadsto \left(\frac{1}{\frac{1}{\color{blue}{0 - \alpha}}} \cdot \alpha\right) \cdot \log \left(1 - u0\right) \]

      16. neg-sub0 N/A

          \[\leadsto \left(\frac{1}{\frac{1}{\color{blue}{\mathsf{neg}\left(\alpha\right)}}} \cdot \alpha\right) \cdot \log \left(1 - u0\right) \]

      17. lift-neg.f32 N/A

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

      18. lower-/.f32 84.6

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

   4. Applied rewrites 84.6%

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

   if 0.999894023 < (-.f32 #s(literal 1 binary32) u0)

   1. Initial program 37.0%

      \[\left(\left(-\alpha\right) \cdot \alpha\right) \cdot \log \left(1 - u0\right) \]
   2. Add Preprocessing

   3. Taylor expanded in u0 around 0

      \[\leadsto \color{blue}{{\alpha}^{2} \cdot u0} \]
   4. Step-by-step derivation

      1. lower-*.f32 N/A

         \[\leadsto \color{blue}{{\alpha}^{2} \cdot u0} \]

      2. unpow2 N/A

         \[\leadsto \color{blue}{\left(\alpha \cdot \alpha\right)} \cdot u0 \]

      3. lower-*.f32 90.6

         \[\leadsto \color{blue}{\left(\alpha \cdot \alpha\right)} \cdot u0 \]

   5. Applied rewrites 90.6%

      \[\leadsto \color{blue}{\left(\alpha \cdot \alpha\right) \cdot u0} \]
3. Recombined 2 regimes into one program.

4. Final simplification 88.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 - u0 \leq 0.9998940229415894:\\ \;\;\;\;\left(\frac{1}{\frac{-1}{\alpha}} \cdot \alpha\right) \cdot \log \left(1 - u0\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\alpha \cdot \alpha\right) \cdot u0\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 89.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u0 \leq 0.00010599999950500205:\\ \;\;\;\;\left(\alpha \cdot \alpha\right) \cdot u0\\ \mathbf{else}:\\ \;\;\;\;\frac{\alpha}{\frac{-1}{\alpha}} \cdot \log \left(1 - u0\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha u0)
 :precision binary32
 (if (<= u0 0.00010599999950500205)
   (* (* alpha alpha) u0)
   (* (/ alpha (/ -1.0 alpha)) (log (- 1.0 u0)))))

C

float code(float alpha, float u0) {
	float tmp;
	if (u0 <= 0.00010599999950500205f) {
		tmp = (alpha * alpha) * u0;
	} else {
		tmp = (alpha / (-1.0f / alpha)) * logf((1.0f - u0));
	}
	return tmp;
}

Fortran

real(4) function code(alpha, u0)
    real(4), intent (in) :: alpha
    real(4), intent (in) :: u0
    real(4) :: tmp
    if (u0 <= 0.00010599999950500205e0) then
        tmp = (alpha * alpha) * u0
    else
        tmp = (alpha / ((-1.0e0) / alpha)) * log((1.0e0 - u0))
    end if
    code = tmp
end function

Julia

function code(alpha, u0)
	tmp = Float32(0.0)
	if (u0 <= Float32(0.00010599999950500205))
		tmp = Float32(Float32(alpha * alpha) * u0);
	else
		tmp = Float32(Float32(alpha / Float32(Float32(-1.0) / alpha)) * log(Float32(Float32(1.0) - u0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, u0)
	tmp = single(0.0);
	if (u0 <= single(0.00010599999950500205))
		tmp = (alpha * alpha) * u0;
	else
		tmp = (alpha / (single(-1.0) / alpha)) * log((single(1.0) - u0));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if u0 < 1.06e-4

   1. Initial program 37.0%

      \[\left(\left(-\alpha\right) \cdot \alpha\right) \cdot \log \left(1 - u0\right) \]
   2. Add Preprocessing

   3. Taylor expanded in u0 around 0

      \[\leadsto \color{blue}{{\alpha}^{2} \cdot u0} \]
   4. Step-by-step derivation

      1. lower-*.f32 N/A

         \[\leadsto \color{blue}{{\alpha}^{2} \cdot u0} \]

      2. unpow2 N/A

         \[\leadsto \color{blue}{\left(\alpha \cdot \alpha\right)} \cdot u0 \]

      3. lower-*.f32 90.6

         \[\leadsto \color{blue}{\left(\alpha \cdot \alpha\right)} \cdot u0 \]

   5. Applied rewrites 90.6%

      \[\leadsto \color{blue}{\left(\alpha \cdot \alpha\right) \cdot u0} \]

   if 1.06e-4 < u0

   1. Initial program 84.5%

      \[\left(\left(-\alpha\right) \cdot \alpha\right) \cdot \log \left(1 - u0\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f32 N/A

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

      2. *-commutative N/A

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

      3. lift-neg.f32 N/A

         \[\leadsto \left(\alpha \cdot \color{blue}{\left(\mathsf{neg}\left(\alpha\right)\right)}\right) \cdot \log \left(1 - u0\right) \]

      4. neg-sub0 N/A

         \[\leadsto \left(\alpha \cdot \color{blue}{\left(0 - \alpha\right)}\right) \cdot \log \left(1 - u0\right) \]

      5. flip3-- N/A

         \[\leadsto \left(\alpha \cdot \color{blue}{\frac{{0}^{3} - {\alpha}^{3}}{0 \cdot 0 + \left(\alpha \cdot \alpha + 0 \cdot \alpha\right)}}\right) \cdot \log \left(1 - u0\right) \]

      6. clear-num N/A

         \[\leadsto \left(\alpha \cdot \color{blue}{\frac{1}{\frac{0 \cdot 0 + \left(\alpha \cdot \alpha + 0 \cdot \alpha\right)}{{0}^{3} - {\alpha}^{3}}}}\right) \cdot \log \left(1 - u0\right) \]

      7. un-div-inv N/A

         \[\leadsto \color{blue}{\frac{\alpha}{\frac{0 \cdot 0 + \left(\alpha \cdot \alpha + 0 \cdot \alpha\right)}{{0}^{3} - {\alpha}^{3}}}} \cdot \log \left(1 - u0\right) \]

      8. lower-/.f32 N/A

         \[\leadsto \color{blue}{\frac{\alpha}{\frac{0 \cdot 0 + \left(\alpha \cdot \alpha + 0 \cdot \alpha\right)}{{0}^{3} - {\alpha}^{3}}}} \cdot \log \left(1 - u0\right) \]

      9. metadata-eval N/A

         \[\leadsto \frac{\alpha}{\frac{\color{blue}{0} + \left(\alpha \cdot \alpha + 0 \cdot \alpha\right)}{{0}^{3} - {\alpha}^{3}}} \cdot \log \left(1 - u0\right) \]

      10. +-lft-identity N/A

          \[\leadsto \frac{\alpha}{\frac{\color{blue}{\alpha \cdot \alpha + 0 \cdot \alpha}}{{0}^{3} - {\alpha}^{3}}} \cdot \log \left(1 - u0\right) \]

      11. mul0-lft N/A

          \[\leadsto \frac{\alpha}{\frac{\alpha \cdot \alpha + \color{blue}{0}}{{0}^{3} - {\alpha}^{3}}} \cdot \log \left(1 - u0\right) \]

      12. +-rgt-identity N/A

          \[\leadsto \frac{\alpha}{\frac{\color{blue}{\alpha \cdot \alpha}}{{0}^{3} - {\alpha}^{3}}} \cdot \log \left(1 - u0\right) \]

      13. clear-num N/A

          \[\leadsto \frac{\alpha}{\color{blue}{\frac{1}{\frac{{0}^{3} - {\alpha}^{3}}{\alpha \cdot \alpha}}}} \cdot \log \left(1 - u0\right) \]

      14. +-rgt-identity N/A

          \[\leadsto \frac{\alpha}{\frac{1}{\frac{{0}^{3} - {\alpha}^{3}}{\color{blue}{\alpha \cdot \alpha + 0}}}} \cdot \log \left(1 - u0\right) \]

      15. mul0-lft N/A

          \[\leadsto \frac{\alpha}{\frac{1}{\frac{{0}^{3} - {\alpha}^{3}}{\alpha \cdot \alpha + \color{blue}{0 \cdot \alpha}}}} \cdot \log \left(1 - u0\right) \]

      16. +-lft-identity N/A

          \[\leadsto \frac{\alpha}{\frac{1}{\frac{{0}^{3} - {\alpha}^{3}}{\color{blue}{0 + \left(\alpha \cdot \alpha + 0 \cdot \alpha\right)}}}} \cdot \log \left(1 - u0\right) \]

      17. metadata-eval N/A

          \[\leadsto \frac{\alpha}{\frac{1}{\frac{{0}^{3} - {\alpha}^{3}}{\color{blue}{0 \cdot 0} + \left(\alpha \cdot \alpha + 0 \cdot \alpha\right)}}} \cdot \log \left(1 - u0\right) \]

      18. flip3-- N/A

          \[\leadsto \frac{\alpha}{\frac{1}{\color{blue}{0 - \alpha}}} \cdot \log \left(1 - u0\right) \]

      19. neg-sub0 N/A

          \[\leadsto \frac{\alpha}{\frac{1}{\color{blue}{\mathsf{neg}\left(\alpha\right)}}} \cdot \log \left(1 - u0\right) \]

      20. lift-neg.f32 N/A

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

      21. lower-/.f32 84.5

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

   4. Applied rewrites 84.5%

      \[\leadsto \color{blue}{\frac{\alpha}{\frac{1}{-\alpha}}} \cdot \log \left(1 - u0\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 88.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u0 \leq 0.00010599999950500205:\\ \;\;\;\;\left(\alpha \cdot \alpha\right) \cdot u0\\ \mathbf{else}:\\ \;\;\;\;\frac{\alpha}{\frac{-1}{\alpha}} \cdot \log \left(1 - u0\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 89.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;1 - u0 \leq 0.9998940229415894:\\ \;\;\;\;\left(\left(-\alpha\right) \cdot \alpha\right) \cdot \log \left(1 - u0\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\alpha \cdot \alpha\right) \cdot u0\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha u0)
 :precision binary32
 (if (<= (- 1.0 u0) 0.9998940229415894)
   (* (* (- alpha) alpha) (log (- 1.0 u0)))
   (* (* alpha alpha) u0)))

C

float code(float alpha, float u0) {
	float tmp;
	if ((1.0f - u0) <= 0.9998940229415894f) {
		tmp = (-alpha * alpha) * logf((1.0f - u0));
	} else {
		tmp = (alpha * alpha) * u0;
	}
	return tmp;
}

Fortran

real(4) function code(alpha, u0)
    real(4), intent (in) :: alpha
    real(4), intent (in) :: u0
    real(4) :: tmp
    if ((1.0e0 - u0) <= 0.9998940229415894e0) then
        tmp = (-alpha * alpha) * log((1.0e0 - u0))
    else
        tmp = (alpha * alpha) * u0
    end if
    code = tmp
end function

Julia

function code(alpha, u0)
	tmp = Float32(0.0)
	if (Float32(Float32(1.0) - u0) <= Float32(0.9998940229415894))
		tmp = Float32(Float32(Float32(-alpha) * alpha) * log(Float32(Float32(1.0) - u0)));
	else
		tmp = Float32(Float32(alpha * alpha) * u0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, u0)
	tmp = single(0.0);
	if ((single(1.0) - u0) <= single(0.9998940229415894))
		tmp = (-alpha * alpha) * log((single(1.0) - u0));
	else
		tmp = (alpha * alpha) * u0;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (-.f32 #s(literal 1 binary32) u0) < 0.999894023

   1. Initial program 84.5%

      \[\left(\left(-\alpha\right) \cdot \alpha\right) \cdot \log \left(1 - u0\right) \]
   2. Add Preprocessing

   if 0.999894023 < (-.f32 #s(literal 1 binary32) u0)

   1. Initial program 37.0%

      \[\left(\left(-\alpha\right) \cdot \alpha\right) \cdot \log \left(1 - u0\right) \]
   2. Add Preprocessing

   3. Taylor expanded in u0 around 0

      \[\leadsto \color{blue}{{\alpha}^{2} \cdot u0} \]
   4. Step-by-step derivation

      1. lower-*.f32 N/A

         \[\leadsto \color{blue}{{\alpha}^{2} \cdot u0} \]

      2. unpow2 N/A

         \[\leadsto \color{blue}{\left(\alpha \cdot \alpha\right)} \cdot u0 \]

      3. lower-*.f32 90.6

         \[\leadsto \color{blue}{\left(\alpha \cdot \alpha\right)} \cdot u0 \]

   5. Applied rewrites 90.6%

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

Alternative 4: 74.7% accurate, 10.5× speedup

Math

\[\begin{array}{l} \\ \left(\alpha \cdot \alpha\right) \cdot u0 \end{array} \]

FPCore

(FPCore (alpha u0) :precision binary32 (* (* alpha alpha) u0))

C

float code(float alpha, float u0) {
	return (alpha * alpha) * u0;
}

Fortran

real(4) function code(alpha, u0)
    real(4), intent (in) :: alpha
    real(4), intent (in) :: u0
    code = (alpha * alpha) * u0
end function

Julia

function code(alpha, u0)
	return Float32(Float32(alpha * alpha) * u0)
end

MATLAB

function tmp = code(alpha, u0)
	tmp = (alpha * alpha) * u0;
end

Derivation

1. Initial program 56.8%

   \[\left(\left(-\alpha\right) \cdot \alpha\right) \cdot \log \left(1 - u0\right) \]
2. Add Preprocessing

3. Taylor expanded in u0 around 0

   \[\leadsto \color{blue}{{\alpha}^{2} \cdot u0} \]
4. Step-by-step derivation

   1. lower-*.f32 N/A

      \[\leadsto \color{blue}{{\alpha}^{2} \cdot u0} \]

   2. unpow2 N/A

      \[\leadsto \color{blue}{\left(\alpha \cdot \alpha\right)} \cdot u0 \]

   3. lower-*.f32 74.7

      \[\leadsto \color{blue}{\left(\alpha \cdot \alpha\right)} \cdot u0 \]

5. Applied rewrites 74.7%

   \[\leadsto \color{blue}{\left(\alpha \cdot \alpha\right) \cdot u0} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024319 
(FPCore (alpha u0)
  :name "Beckmann Distribution sample, tan2theta, alphax == alphay"
  :precision binary32
  :pre (and (and (<= 0.0001 alpha) (<= alpha 1.0)) (and (<= 2.328306437e-10 u0) (<= u0 1.0)))
  (* (* (- alpha) alpha) (log (- 1.0 u0))))