Beckmann Distribution sample, tan2theta, alphax == alphay

Percentage Accurate: 56.0% → 99.0%

Time: 7.1s

Alternatives: 5

Speedup: 21.6×

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: 56.0% 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: 99.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(\alpha \cdot \left(-\alpha\right)\right) \cdot \mathsf{log1p}\left(-u0\right) \end{array} \]

FPCore

(FPCore (alpha u0) :precision binary32 (* (* alpha (- alpha)) (log1p (- u0))))

C

float code(float alpha, float u0) {
	return (alpha * -alpha) * log1pf(-u0);
}

Julia

function code(alpha, u0)
	return Float32(Float32(alpha * Float32(-alpha)) * log1p(Float32(-u0)))
end

Derivation

1. Initial program 60.9%

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

   1. *-commutative 60.9%

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

   2. sub-neg 60.9%

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

   3. log1p-def 99.0%

      \[\leadsto \left(\alpha \cdot \left(-\alpha\right)\right) \cdot \color{blue}{\mathsf{log1p}\left(-u0\right)} \]

3. Simplified 99.0%

   \[\leadsto \color{blue}{\left(\alpha \cdot \left(-\alpha\right)\right) \cdot \mathsf{log1p}\left(-u0\right)} \]

4. Final simplification 99.0%

   \[\leadsto \left(\alpha \cdot \left(-\alpha\right)\right) \cdot \mathsf{log1p}\left(-u0\right) \]

Alternative 2: 99.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \alpha \cdot \left(\alpha \cdot \left(-\mathsf{log1p}\left(-u0\right)\right)\right) \end{array} \]

FPCore

(FPCore (alpha u0) :precision binary32 (* alpha (* alpha (- (log1p (- u0))))))

C

float code(float alpha, float u0) {
	return alpha * (alpha * -log1pf(-u0));
}

Julia

function code(alpha, u0)
	return Float32(alpha * Float32(alpha * Float32(-log1p(Float32(-u0)))))
end

Derivation

1. Initial program 60.9%

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

   1. associate-*l* 61.0%

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

   2. sub-neg 61.0%

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

   3. log1p-def 98.9%

      \[\leadsto \left(-\alpha\right) \cdot \left(\alpha \cdot \color{blue}{\mathsf{log1p}\left(-u0\right)}\right) \]

3. Simplified 98.9%

   \[\leadsto \color{blue}{\left(-\alpha\right) \cdot \left(\alpha \cdot \mathsf{log1p}\left(-u0\right)\right)} \]

4. Final simplification 98.9%

   \[\leadsto \alpha \cdot \left(\alpha \cdot \left(-\mathsf{log1p}\left(-u0\right)\right)\right) \]

Alternative 3: 91.3% accurate, 7.2× speedup

Math

\[\begin{array}{l} \\ \left(\alpha \cdot \alpha\right) \cdot \left(u0 + u0 \cdot \left(u0 \cdot \left(0.5 + u0 \cdot 0.3333333333333333\right)\right)\right) \end{array} \]

FPCore

(FPCore (alpha u0)
 :precision binary32
 (* (* alpha alpha) (+ u0 (* u0 (* u0 (+ 0.5 (* u0 0.3333333333333333)))))))

C

float code(float alpha, float u0) {
	return (alpha * alpha) * (u0 + (u0 * (u0 * (0.5f + (u0 * 0.3333333333333333f)))));
}

Fortran

real(4) function code(alpha, u0)
    real(4), intent (in) :: alpha
    real(4), intent (in) :: u0
    code = (alpha * alpha) * (u0 + (u0 * (u0 * (0.5e0 + (u0 * 0.3333333333333333e0)))))
end function

Julia

function code(alpha, u0)
	return Float32(Float32(alpha * alpha) * Float32(u0 + Float32(u0 * Float32(u0 * Float32(Float32(0.5) + Float32(u0 * Float32(0.3333333333333333)))))))
end

MATLAB

function tmp = code(alpha, u0)
	tmp = (alpha * alpha) * (u0 + (u0 * (u0 * (single(0.5) + (u0 * single(0.3333333333333333))))));
end

Derivation

1. Initial program 60.9%

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

   1. associate-*l* 61.0%

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

   2. sub-neg 61.0%

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

   3. log1p-def 98.9%

      \[\leadsto \left(-\alpha\right) \cdot \left(\alpha \cdot \color{blue}{\mathsf{log1p}\left(-u0\right)}\right) \]

3. Simplified 98.9%

   \[\leadsto \color{blue}{\left(-\alpha\right) \cdot \left(\alpha \cdot \mathsf{log1p}\left(-u0\right)\right)} \]

4. Taylor expanded in u0 around 0 92.1%

   \[\leadsto \color{blue}{0.25 \cdot \left({u0}^{4} \cdot {\alpha}^{2}\right) + \left(u0 \cdot {\alpha}^{2} + \left(0.3333333333333333 \cdot \left({u0}^{3} \cdot {\alpha}^{2}\right) + 0.5 \cdot \left({u0}^{2} \cdot {\alpha}^{2}\right)\right)\right)} \]
5. Step-by-step derivation

   1. associate-+r+ 92.1%

      \[\leadsto \color{blue}{\left(0.25 \cdot \left({u0}^{4} \cdot {\alpha}^{2}\right) + u0 \cdot {\alpha}^{2}\right) + \left(0.3333333333333333 \cdot \left({u0}^{3} \cdot {\alpha}^{2}\right) + 0.5 \cdot \left({u0}^{2} \cdot {\alpha}^{2}\right)\right)} \]

   2. +-commutative 92.1%

      \[\leadsto \color{blue}{\left(0.3333333333333333 \cdot \left({u0}^{3} \cdot {\alpha}^{2}\right) + 0.5 \cdot \left({u0}^{2} \cdot {\alpha}^{2}\right)\right) + \left(0.25 \cdot \left({u0}^{4} \cdot {\alpha}^{2}\right) + u0 \cdot {\alpha}^{2}\right)} \]

   3. +-commutative 92.1%

      \[\leadsto \color{blue}{\left(0.5 \cdot \left({u0}^{2} \cdot {\alpha}^{2}\right) + 0.3333333333333333 \cdot \left({u0}^{3} \cdot {\alpha}^{2}\right)\right)} + \left(0.25 \cdot \left({u0}^{4} \cdot {\alpha}^{2}\right) + u0 \cdot {\alpha}^{2}\right) \]

   4. associate-*r* 92.1%

      \[\leadsto \left(\color{blue}{\left(0.5 \cdot {u0}^{2}\right) \cdot {\alpha}^{2}} + 0.3333333333333333 \cdot \left({u0}^{3} \cdot {\alpha}^{2}\right)\right) + \left(0.25 \cdot \left({u0}^{4} \cdot {\alpha}^{2}\right) + u0 \cdot {\alpha}^{2}\right) \]

   5. associate-*r* 92.1%

      \[\leadsto \left(\left(0.5 \cdot {u0}^{2}\right) \cdot {\alpha}^{2} + \color{blue}{\left(0.3333333333333333 \cdot {u0}^{3}\right) \cdot {\alpha}^{2}}\right) + \left(0.25 \cdot \left({u0}^{4} \cdot {\alpha}^{2}\right) + u0 \cdot {\alpha}^{2}\right) \]

   6. distribute-rgt-out 92.1%

      \[\leadsto \color{blue}{{\alpha}^{2} \cdot \left(0.5 \cdot {u0}^{2} + 0.3333333333333333 \cdot {u0}^{3}\right)} + \left(0.25 \cdot \left({u0}^{4} \cdot {\alpha}^{2}\right) + u0 \cdot {\alpha}^{2}\right) \]

   7. associate-*r* 92.1%

      \[\leadsto {\alpha}^{2} \cdot \left(0.5 \cdot {u0}^{2} + 0.3333333333333333 \cdot {u0}^{3}\right) + \left(\color{blue}{\left(0.25 \cdot {u0}^{4}\right) \cdot {\alpha}^{2}} + u0 \cdot {\alpha}^{2}\right) \]

   8. distribute-rgt-out 92.1%

      \[\leadsto {\alpha}^{2} \cdot \left(0.5 \cdot {u0}^{2} + 0.3333333333333333 \cdot {u0}^{3}\right) + \color{blue}{{\alpha}^{2} \cdot \left(0.25 \cdot {u0}^{4} + u0\right)} \]

   9. unpow2 92.1%

      \[\leadsto \color{blue}{\left(\alpha \cdot \alpha\right)} \cdot \left(0.5 \cdot {u0}^{2} + 0.3333333333333333 \cdot {u0}^{3}\right) + {\alpha}^{2} \cdot \left(0.25 \cdot {u0}^{4} + u0\right) \]

   10. unpow2 92.1%

       \[\leadsto \left(\alpha \cdot \alpha\right) \cdot \left(0.5 \cdot \color{blue}{\left(u0 \cdot u0\right)} + 0.3333333333333333 \cdot {u0}^{3}\right) + {\alpha}^{2} \cdot \left(0.25 \cdot {u0}^{4} + u0\right) \]

   11. unpow2 92.1%

       \[\leadsto \left(\alpha \cdot \alpha\right) \cdot \left(0.5 \cdot \left(u0 \cdot u0\right) + 0.3333333333333333 \cdot {u0}^{3}\right) + \color{blue}{\left(\alpha \cdot \alpha\right)} \cdot \left(0.25 \cdot {u0}^{4} + u0\right) \]

6. Simplified 92.1%

   \[\leadsto \color{blue}{\left(\alpha \cdot \alpha\right) \cdot \left(0.5 \cdot \left(u0 \cdot u0\right) + 0.3333333333333333 \cdot {u0}^{3}\right) + \left(\alpha \cdot \alpha\right) \cdot \left(0.25 \cdot {u0}^{4} + u0\right)} \]

7. Taylor expanded in u0 around 0 90.0%

   \[\leadsto \color{blue}{u0 \cdot {\alpha}^{2} + \left(0.3333333333333333 \cdot \left({u0}^{3} \cdot {\alpha}^{2}\right) + 0.5 \cdot \left({u0}^{2} \cdot {\alpha}^{2}\right)\right)} \]
8. Step-by-step derivation

   1. unpow2 90.0%

      \[\leadsto u0 \cdot \color{blue}{\left(\alpha \cdot \alpha\right)} + \left(0.3333333333333333 \cdot \left({u0}^{3} \cdot {\alpha}^{2}\right) + 0.5 \cdot \left({u0}^{2} \cdot {\alpha}^{2}\right)\right) \]

   2. +-commutative 90.0%

      \[\leadsto u0 \cdot \left(\alpha \cdot \alpha\right) + \color{blue}{\left(0.5 \cdot \left({u0}^{2} \cdot {\alpha}^{2}\right) + 0.3333333333333333 \cdot \left({u0}^{3} \cdot {\alpha}^{2}\right)\right)} \]

   3. unpow2 90.0%

      \[\leadsto u0 \cdot \left(\alpha \cdot \alpha\right) + \left(0.5 \cdot \left(\color{blue}{\left(u0 \cdot u0\right)} \cdot {\alpha}^{2}\right) + 0.3333333333333333 \cdot \left({u0}^{3} \cdot {\alpha}^{2}\right)\right) \]

   4. unpow2 90.0%

      \[\leadsto u0 \cdot \left(\alpha \cdot \alpha\right) + \left(0.5 \cdot \left(\left(u0 \cdot u0\right) \cdot \color{blue}{\left(\alpha \cdot \alpha\right)}\right) + 0.3333333333333333 \cdot \left({u0}^{3} \cdot {\alpha}^{2}\right)\right) \]

   5. unpow2 90.0%

      \[\leadsto u0 \cdot \left(\alpha \cdot \alpha\right) + \left(0.5 \cdot \left(\left(u0 \cdot u0\right) \cdot \left(\alpha \cdot \alpha\right)\right) + 0.3333333333333333 \cdot \left({u0}^{3} \cdot \color{blue}{\left(\alpha \cdot \alpha\right)}\right)\right) \]

   6. associate-*r* 90.0%

      \[\leadsto u0 \cdot \left(\alpha \cdot \alpha\right) + \left(0.5 \cdot \left(\left(u0 \cdot u0\right) \cdot \left(\alpha \cdot \alpha\right)\right) + \color{blue}{\left(0.3333333333333333 \cdot {u0}^{3}\right) \cdot \left(\alpha \cdot \alpha\right)}\right) \]

   7. cube-mult 90.0%

      \[\leadsto u0 \cdot \left(\alpha \cdot \alpha\right) + \left(0.5 \cdot \left(\left(u0 \cdot u0\right) \cdot \left(\alpha \cdot \alpha\right)\right) + \left(0.3333333333333333 \cdot \color{blue}{\left(u0 \cdot \left(u0 \cdot u0\right)\right)}\right) \cdot \left(\alpha \cdot \alpha\right)\right) \]

   8. associate-*r* 90.0%

      \[\leadsto u0 \cdot \left(\alpha \cdot \alpha\right) + \left(0.5 \cdot \left(\left(u0 \cdot u0\right) \cdot \left(\alpha \cdot \alpha\right)\right) + \color{blue}{\left(\left(0.3333333333333333 \cdot u0\right) \cdot \left(u0 \cdot u0\right)\right)} \cdot \left(\alpha \cdot \alpha\right)\right) \]

   9. *-commutative 90.0%

      \[\leadsto u0 \cdot \left(\alpha \cdot \alpha\right) + \left(0.5 \cdot \left(\left(u0 \cdot u0\right) \cdot \left(\alpha \cdot \alpha\right)\right) + \left(\color{blue}{\left(u0 \cdot 0.3333333333333333\right)} \cdot \left(u0 \cdot u0\right)\right) \cdot \left(\alpha \cdot \alpha\right)\right) \]

   10. associate-*l* 90.0%

       \[\leadsto u0 \cdot \left(\alpha \cdot \alpha\right) + \left(0.5 \cdot \left(\left(u0 \cdot u0\right) \cdot \left(\alpha \cdot \alpha\right)\right) + \color{blue}{\left(u0 \cdot 0.3333333333333333\right) \cdot \left(\left(u0 \cdot u0\right) \cdot \left(\alpha \cdot \alpha\right)\right)}\right) \]

   11. distribute-rgt-in 90.0%

       \[\leadsto u0 \cdot \left(\alpha \cdot \alpha\right) + \color{blue}{\left(\left(u0 \cdot u0\right) \cdot \left(\alpha \cdot \alpha\right)\right) \cdot \left(0.5 + u0 \cdot 0.3333333333333333\right)} \]

   12. *-commutative 90.0%

       \[\leadsto u0 \cdot \left(\alpha \cdot \alpha\right) + \color{blue}{\left(0.5 + u0 \cdot 0.3333333333333333\right) \cdot \left(\left(u0 \cdot u0\right) \cdot \left(\alpha \cdot \alpha\right)\right)} \]

   13. associate-*l* 90.0%

       \[\leadsto u0 \cdot \left(\alpha \cdot \alpha\right) + \color{blue}{\left(\left(0.5 + u0 \cdot 0.3333333333333333\right) \cdot \left(u0 \cdot u0\right)\right) \cdot \left(\alpha \cdot \alpha\right)} \]

   14. *-commutative 90.0%

       \[\leadsto u0 \cdot \left(\alpha \cdot \alpha\right) + \color{blue}{\left(\left(u0 \cdot u0\right) \cdot \left(0.5 + u0 \cdot 0.3333333333333333\right)\right)} \cdot \left(\alpha \cdot \alpha\right) \]

9. Simplified 90.1%

   \[\leadsto \color{blue}{\left(\alpha \cdot \alpha\right) \cdot \left(u0 + u0 \cdot \left(u0 \cdot \left(0.5 + u0 \cdot 0.3333333333333333\right)\right)\right)} \]

10. Final simplification 90.1%

    \[\leadsto \left(\alpha \cdot \alpha\right) \cdot \left(u0 + u0 \cdot \left(u0 \cdot \left(0.5 + u0 \cdot 0.3333333333333333\right)\right)\right) \]

Alternative 4: 87.2% accurate, 9.8× speedup

Math

\[\begin{array}{l} \\ \left(\alpha \cdot \alpha\right) \cdot \left(u0 + 0.5 \cdot \left(u0 \cdot u0\right)\right) \end{array} \]

FPCore

(FPCore (alpha u0)
 :precision binary32
 (* (* alpha alpha) (+ u0 (* 0.5 (* u0 u0)))))

C

float code(float alpha, float u0) {
	return (alpha * alpha) * (u0 + (0.5f * (u0 * u0)));
}

Fortran

real(4) function code(alpha, u0)
    real(4), intent (in) :: alpha
    real(4), intent (in) :: u0
    code = (alpha * alpha) * (u0 + (0.5e0 * (u0 * u0)))
end function

Julia

function code(alpha, u0)
	return Float32(Float32(alpha * alpha) * Float32(u0 + Float32(Float32(0.5) * Float32(u0 * u0))))
end

MATLAB

function tmp = code(alpha, u0)
	tmp = (alpha * alpha) * (u0 + (single(0.5) * (u0 * u0)));
end

Derivation

1. Initial program 60.9%

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

   1. associate-*l* 61.0%

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

   2. sub-neg 61.0%

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

   3. log1p-def 98.9%

      \[\leadsto \left(-\alpha\right) \cdot \left(\alpha \cdot \color{blue}{\mathsf{log1p}\left(-u0\right)}\right) \]

3. Simplified 98.9%

   \[\leadsto \color{blue}{\left(-\alpha\right) \cdot \left(\alpha \cdot \mathsf{log1p}\left(-u0\right)\right)} \]

4. Taylor expanded in u0 around 0 85.5%

   \[\leadsto \color{blue}{u0 \cdot {\alpha}^{2} + 0.5 \cdot \left({u0}^{2} \cdot {\alpha}^{2}\right)} \]
5. Step-by-step derivation

   1. +-commutative 85.5%

      \[\leadsto \color{blue}{0.5 \cdot \left({u0}^{2} \cdot {\alpha}^{2}\right) + u0 \cdot {\alpha}^{2}} \]

   2. associate-*r* 85.5%

      \[\leadsto \color{blue}{\left(0.5 \cdot {u0}^{2}\right) \cdot {\alpha}^{2}} + u0 \cdot {\alpha}^{2} \]

   3. distribute-rgt-out 85.5%

      \[\leadsto \color{blue}{{\alpha}^{2} \cdot \left(0.5 \cdot {u0}^{2} + u0\right)} \]

   4. unpow2 85.5%

      \[\leadsto \color{blue}{\left(\alpha \cdot \alpha\right)} \cdot \left(0.5 \cdot {u0}^{2} + u0\right) \]

   5. unpow2 85.5%

      \[\leadsto \left(\alpha \cdot \alpha\right) \cdot \left(0.5 \cdot \color{blue}{\left(u0 \cdot u0\right)} + u0\right) \]

6. Simplified 85.5%

   \[\leadsto \color{blue}{\left(\alpha \cdot \alpha\right) \cdot \left(0.5 \cdot \left(u0 \cdot u0\right) + u0\right)} \]

7. Final simplification 85.5%

   \[\leadsto \left(\alpha \cdot \alpha\right) \cdot \left(u0 + 0.5 \cdot \left(u0 \cdot u0\right)\right) \]

Alternative 5: 74.5% accurate, 21.6× speedup

Math

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

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(alpha * Float32(alpha * u0))
end

MATLAB

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

Derivation

1. Initial program 60.9%

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

   1. associate-*l* 61.0%

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

   2. sub-neg 61.0%

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

   3. log1p-def 98.9%

      \[\leadsto \left(-\alpha\right) \cdot \left(\alpha \cdot \color{blue}{\mathsf{log1p}\left(-u0\right)}\right) \]

3. Simplified 98.9%

   \[\leadsto \color{blue}{\left(-\alpha\right) \cdot \left(\alpha \cdot \mathsf{log1p}\left(-u0\right)\right)} \]

4. Taylor expanded in u0 around 0 71.0%

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

   1. *-commutative 71.0%

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

   2. unpow2 71.0%

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

   3. associate-*l* 71.0%

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

6. Simplified 71.0%

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

7. Final simplification 71.0%

   \[\leadsto \alpha \cdot \left(\alpha \cdot u0\right) \]

Reproduce

herbie shell --seed 2023174 
(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))))