Beckmann Distribution sample, tan2theta, alphax == alphay

Percentage Accurate: 55.6% → 89.7%

Time: 6.7s

Alternatives: 5

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: 55.6% 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: 89.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;1 - u0 \leq 0.9998499751091003:\\ \;\;\;\;\left({\alpha}^{2.5} \cdot \left({\alpha}^{1.5} \cdot \left(-{\alpha}^{-2}\right)\right)\right) \cdot \log \left(1 - u0\right)\\ \mathbf{else}:\\ \;\;\;\;\left(u0 \cdot \alpha\right) \cdot \alpha\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha u0)
 :precision binary32
 (if (<= (- 1.0 u0) 0.9998499751091003)
   (*
    (* (pow alpha 2.5) (* (pow alpha 1.5) (- (pow alpha -2.0))))
    (log (- 1.0 u0)))
   (* (* u0 alpha) alpha)))

C

float code(float alpha, float u0) {
	float tmp;
	if ((1.0f - u0) <= 0.9998499751091003f) {
		tmp = (powf(alpha, 2.5f) * (powf(alpha, 1.5f) * -powf(alpha, -2.0f))) * logf((1.0f - u0));
	} else {
		tmp = (u0 * alpha) * alpha;
	}
	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) <= 0.9998499751091003e0) then
        tmp = ((alpha ** 2.5e0) * ((alpha ** 1.5e0) * -(alpha ** (-2.0e0)))) * log((1.0e0 - u0))
    else
        tmp = (u0 * alpha) * alpha
    end if
    code = tmp
end function

Julia

function code(alpha, u0)
	tmp = Float32(0.0)
	if (Float32(Float32(1.0) - u0) <= Float32(0.9998499751091003))
		tmp = Float32(Float32((alpha ^ Float32(2.5)) * Float32((alpha ^ Float32(1.5)) * Float32(-(alpha ^ Float32(-2.0))))) * log(Float32(Float32(1.0) - u0)));
	else
		tmp = Float32(Float32(u0 * alpha) * alpha);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, u0)
	tmp = single(0.0);
	if ((single(1.0) - u0) <= single(0.9998499751091003))
		tmp = ((alpha ^ single(2.5)) * ((alpha ^ single(1.5)) * -(alpha ^ single(-2.0)))) * log((single(1.0) - u0));
	else
		tmp = (u0 * alpha) * alpha;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (-.f32 #s(literal 1 binary32) u0) < 0.999849975

   1. Initial program 87.6%

      \[\left(\left(-\alpha\right) \cdot \alpha\right) \cdot \log \left(1 - u0\right) \]
   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. lift-neg.f32 N/A

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

      3. neg-sub0 N/A

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

      4. flip-- N/A

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

      5. metadata-eval N/A

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

      6. neg-sub0 N/A

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

      7. distribute-lft-neg-out N/A

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

      8. lift-neg.f32 N/A

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

      9. lift-*.f32 N/A

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

      10. +-lft-identity N/A

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

      11. associate-*l/ N/A

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

      12. clear-num N/A

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

      13. lower-/.f32 N/A

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

      14. lower-/.f32 N/A

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

      15. lower-*.f32 87.3

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

   4. Applied rewrites 87.3%

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

   6. Applied rewrites 87.6%

      \[\leadsto \color{blue}{\left({\alpha}^{2.5} \cdot \left({\alpha}^{1.5} \cdot \left(-{\alpha}^{-2}\right)\right)\right)} \cdot \log \left(1 - u0\right) \]

   if 0.999849975 < (-.f32 #s(literal 1 binary32) u0)

   1. Initial program 33.4%

      \[\left(\left(-\alpha\right) \cdot \alpha\right) \cdot \log \left(1 - u0\right) \]
   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 91.8

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

   5. Applied rewrites 91.8%

      \[\leadsto \color{blue}{\left(\alpha \cdot \alpha\right) \cdot u0} \]
   6. Step-by-step derivation

   7. Applied rewrites 92.0%

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

Alternative 2: 89.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;1 - u0 \leq 0.9998499751091003:\\ \;\;\;\;\left(\left(\left(\alpha \cdot \alpha\right) \cdot \frac{-1}{\alpha}\right) \cdot \alpha\right) \cdot \log \left(1 - u0\right)\\ \mathbf{else}:\\ \;\;\;\;\left(u0 \cdot \alpha\right) \cdot \alpha\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha u0)
 :precision binary32
 (if (<= (- 1.0 u0) 0.9998499751091003)
   (* (* (* (* alpha alpha) (/ -1.0 alpha)) alpha) (log (- 1.0 u0)))
   (* (* u0 alpha) alpha)))

C

float code(float alpha, float u0) {
	float tmp;
	if ((1.0f - u0) <= 0.9998499751091003f) {
		tmp = (((alpha * alpha) * (-1.0f / alpha)) * alpha) * logf((1.0f - u0));
	} else {
		tmp = (u0 * alpha) * alpha;
	}
	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) <= 0.9998499751091003e0) then
        tmp = (((alpha * alpha) * ((-1.0e0) / alpha)) * alpha) * log((1.0e0 - u0))
    else
        tmp = (u0 * alpha) * alpha
    end if
    code = tmp
end function

Julia

function code(alpha, u0)
	tmp = Float32(0.0)
	if (Float32(Float32(1.0) - u0) <= Float32(0.9998499751091003))
		tmp = Float32(Float32(Float32(Float32(alpha * alpha) * Float32(Float32(-1.0) / alpha)) * alpha) * log(Float32(Float32(1.0) - u0)));
	else
		tmp = Float32(Float32(u0 * alpha) * alpha);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, u0)
	tmp = single(0.0);
	if ((single(1.0) - u0) <= single(0.9998499751091003))
		tmp = (((alpha * alpha) * (single(-1.0) / alpha)) * alpha) * log((single(1.0) - u0));
	else
		tmp = (u0 * alpha) * alpha;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (-.f32 #s(literal 1 binary32) u0) < 0.999849975

   1. Initial program 87.6%

      \[\left(\left(-\alpha\right) \cdot \alpha\right) \cdot \log \left(1 - u0\right) \]
   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. flip-- N/A

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

      4. metadata-eval N/A

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

      5. neg-sub0 N/A

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

      6. distribute-lft-neg-out N/A

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

      7. lift-neg.f32 N/A

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

      8. lift-*.f32 N/A

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

      9. div-inv N/A

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

      10. lower-*.f32 N/A

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

      11. +-lft-identity N/A

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

      12. lower-/.f32 87.6

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

   4. Applied rewrites 87.6%

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

   if 0.999849975 < (-.f32 #s(literal 1 binary32) u0)

   1. Initial program 33.4%

      \[\left(\left(-\alpha\right) \cdot \alpha\right) \cdot \log \left(1 - u0\right) \]
   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 91.8

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

   5. Applied rewrites 91.8%

      \[\leadsto \color{blue}{\left(\alpha \cdot \alpha\right) \cdot u0} \]
   6. Step-by-step derivation

   7. Applied rewrites 92.0%

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

4. Final simplification 90.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 - u0 \leq 0.9998499751091003:\\ \;\;\;\;\left(\left(\left(\alpha \cdot \alpha\right) \cdot \frac{-1}{\alpha}\right) \cdot \alpha\right) \cdot \log \left(1 - u0\right)\\ \mathbf{else}:\\ \;\;\;\;\left(u0 \cdot \alpha\right) \cdot \alpha\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 89.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;1 - u0 \leq 0.9998499751091003:\\ \;\;\;\;\left(\left(-\alpha\right) \cdot \alpha\right) \cdot \log \left(1 - u0\right)\\ \mathbf{else}:\\ \;\;\;\;\left(u0 \cdot \alpha\right) \cdot \alpha\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha u0)
 :precision binary32
 (if (<= (- 1.0 u0) 0.9998499751091003)
   (* (* (- alpha) alpha) (log (- 1.0 u0)))
   (* (* u0 alpha) alpha)))

C

float code(float alpha, float u0) {
	float tmp;
	if ((1.0f - u0) <= 0.9998499751091003f) {
		tmp = (-alpha * alpha) * logf((1.0f - u0));
	} else {
		tmp = (u0 * alpha) * alpha;
	}
	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) <= 0.9998499751091003e0) then
        tmp = (-alpha * alpha) * log((1.0e0 - u0))
    else
        tmp = (u0 * alpha) * alpha
    end if
    code = tmp
end function

Julia

function code(alpha, u0)
	tmp = Float32(0.0)
	if (Float32(Float32(1.0) - u0) <= Float32(0.9998499751091003))
		tmp = Float32(Float32(Float32(-alpha) * alpha) * log(Float32(Float32(1.0) - u0)));
	else
		tmp = Float32(Float32(u0 * alpha) * alpha);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, u0)
	tmp = single(0.0);
	if ((single(1.0) - u0) <= single(0.9998499751091003))
		tmp = (-alpha * alpha) * log((single(1.0) - u0));
	else
		tmp = (u0 * alpha) * alpha;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (-.f32 #s(literal 1 binary32) u0) < 0.999849975

   1. Initial program 87.6%

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

   if 0.999849975 < (-.f32 #s(literal 1 binary32) u0)

   1. Initial program 33.4%

      \[\left(\left(-\alpha\right) \cdot \alpha\right) \cdot \log \left(1 - u0\right) \]
   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 91.8

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

   5. Applied rewrites 91.8%

      \[\leadsto \color{blue}{\left(\alpha \cdot \alpha\right) \cdot u0} \]
   6. Step-by-step derivation

   7. Applied rewrites 92.0%

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

Alternative 4: 74.8% accurate, 10.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 53.3%

   \[\left(\left(-\alpha\right) \cdot \alpha\right) \cdot \log \left(1 - u0\right) \]
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 76.4

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

5. Applied rewrites 76.4%

   \[\leadsto \color{blue}{\left(\alpha \cdot \alpha\right) \cdot u0} \]
6. Step-by-step derivation

7. Applied rewrites 76.4%

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

Alternative 5: 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 53.3%

   \[\left(\left(-\alpha\right) \cdot \alpha\right) \cdot \log \left(1 - u0\right) \]
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 76.4

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

5. Applied rewrites 76.4%

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

Reproduce

herbie shell --seed 2024325 
(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))))