Beckmann Distribution sample, tan2theta, alphax == alphay

Percentage Accurate: 56.3% → 99.0%

Time: 7.4s

Alternatives: 9

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: 56.3% 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 54.2%

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

   1. *-commutative 54.2%

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

   2. sub-neg 54.2%

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

   3. log1p-def 99.1%

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

3. Simplified 99.1%

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

4. Final simplification 99.1%

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

Alternative 2: 99.0% accurate, 1.0× speedup

Math

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

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(Float32(-alpha) * Float32(alpha * log1p(Float32(-u0))))
end

Derivation

1. Initial program 54.2%

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

   1. associate-*l* 54.1%

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

   2. sub-neg 54.1%

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

   3. log1p-def 99.1%

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

3. Simplified 99.1%

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

4. Final simplification 99.1%

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

Alternative 3: 91.1% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \left(u0 \cdot u0\right)\\ t_1 := \left(u0 \cdot u0\right) \cdot \left(u0 \cdot 0.3333333333333333\right)\\ \left(\alpha \cdot \alpha\right) \cdot \left(u0 + \frac{t_0 \cdot t_0 - t_1 \cdot t_1}{t_0 - t_1}\right) \end{array} \end{array} \]

FPCore

(FPCore (alpha u0)
 :precision binary32
 (let* ((t_0 (* 0.5 (* u0 u0))) (t_1 (* (* u0 u0) (* u0 0.3333333333333333))))
   (* (* alpha alpha) (+ u0 (/ (- (* t_0 t_0) (* t_1 t_1)) (- t_0 t_1))))))

C

float code(float alpha, float u0) {
	float t_0 = 0.5f * (u0 * u0);
	float t_1 = (u0 * u0) * (u0 * 0.3333333333333333f);
	return (alpha * alpha) * (u0 + (((t_0 * t_0) - (t_1 * t_1)) / (t_0 - t_1)));
}

Fortran

real(4) function code(alpha, u0)
    real(4), intent (in) :: alpha
    real(4), intent (in) :: u0
    real(4) :: t_0
    real(4) :: t_1
    t_0 = 0.5e0 * (u0 * u0)
    t_1 = (u0 * u0) * (u0 * 0.3333333333333333e0)
    code = (alpha * alpha) * (u0 + (((t_0 * t_0) - (t_1 * t_1)) / (t_0 - t_1)))
end function

Julia

function code(alpha, u0)
	t_0 = Float32(Float32(0.5) * Float32(u0 * u0))
	t_1 = Float32(Float32(u0 * u0) * Float32(u0 * Float32(0.3333333333333333)))
	return Float32(Float32(alpha * alpha) * Float32(u0 + Float32(Float32(Float32(t_0 * t_0) - Float32(t_1 * t_1)) / Float32(t_0 - t_1))))
end

MATLAB

function tmp = code(alpha, u0)
	t_0 = single(0.5) * (u0 * u0);
	t_1 = (u0 * u0) * (u0 * single(0.3333333333333333));
	tmp = (alpha * alpha) * (u0 + (((t_0 * t_0) - (t_1 * t_1)) / (t_0 - t_1)));
end

Derivation

1. Initial program 54.2%

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

   1. associate-*l* 54.1%

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

   2. sub-neg 54.1%

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

   3. log1p-def 99.1%

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

3. Simplified 99.1%

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

4. Taylor expanded in u0 around 0 93.5%

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

   1. associate-+r+ 93.5%

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

   2. +-commutative 93.5%

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

   3. +-commutative 93.5%

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

   4. associate-*r* 93.5%

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

   5. associate-*r* 93.5%

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

   6. distribute-rgt-out 93.5%

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

   7. associate-*r* 93.5%

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

   8. distribute-rgt-out 93.5%

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

   9. distribute-lft-out 93.6%

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

   10. unpow2 93.6%

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

6. Simplified 93.6%

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

7. Taylor expanded in u0 around 0 91.8%

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

   1. associate-+r+ 91.8%

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

   2. unpow2 91.8%

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

   3. *-commutative 91.8%

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

   4. associate-+r+ 91.8%

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

   5. *-commutative 91.8%

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

   6. unpow3 91.8%

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

   7. associate-*r* 91.8%

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

   8. distribute-lft-in 91.8%

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

9. Simplified 91.8%

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

    1. distribute-rgt-in 91.8%

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

    2. flip-+ 91.8%

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

11. Applied egg-rr 91.8%

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

12. Final simplification 91.8%

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

Alternative 4: 91.1% accurate, 5.7× speedup

Math

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

FPCore

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

C

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

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)) + ((u0 * u0) * (u0 * 0.3333333333333333e0))))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 54.2%

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

   1. associate-*l* 54.1%

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

   2. sub-neg 54.1%

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

   3. log1p-def 99.1%

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

3. Simplified 99.1%

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

4. Taylor expanded in u0 around 0 93.5%

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

   1. associate-+r+ 93.5%

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

   2. +-commutative 93.5%

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

   3. +-commutative 93.5%

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

   4. associate-*r* 93.5%

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

   5. associate-*r* 93.5%

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

   6. distribute-rgt-out 93.5%

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

   7. associate-*r* 93.5%

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

   8. distribute-rgt-out 93.5%

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

   9. distribute-lft-out 93.6%

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

   10. unpow2 93.6%

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

6. Simplified 93.6%

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

7. Taylor expanded in u0 around 0 91.8%

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

   1. associate-+r+ 91.8%

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

   2. unpow2 91.8%

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

   3. *-commutative 91.8%

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

   4. associate-+r+ 91.8%

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

   5. *-commutative 91.8%

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

   6. unpow3 91.8%

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

   7. associate-*r* 91.8%

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

   8. distribute-lft-in 91.8%

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

9. Simplified 91.8%

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

    1. +-commutative 91.8%

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

    2. distribute-rgt-in 91.8%

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

11. Applied egg-rr 91.8%

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

12. Final simplification 91.8%

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

Alternative 5: 91.1% accurate, 7.2× speedup

Math

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

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(Float32(u0 * 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 54.2%

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

   1. associate-*l* 54.1%

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

   2. sub-neg 54.1%

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

   3. log1p-def 99.1%

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

3. Simplified 99.1%

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

4. Taylor expanded in u0 around 0 91.8%

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

   1. *-commutative 91.8%

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

   2. +-commutative 91.8%

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

   3. associate-*r* 91.8%

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

   4. associate-*r* 91.8%

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

   5. distribute-rgt-out 91.8%

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

   6. distribute-lft-out 91.8%

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

   7. unpow2 91.8%

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

   8. cube-mult 91.8%

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

   9. unpow2 91.8%

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

   10. associate-*r* 91.8%

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

   11. distribute-rgt-out 91.8%

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

   12. unpow2 91.8%

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

6. Simplified 91.8%

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

7. Final simplification 91.8%

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

Alternative 6: 86.9% 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 54.2%

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

   1. associate-*l* 54.1%

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

   2. sub-neg 54.1%

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

   3. log1p-def 99.1%

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

3. Simplified 99.1%

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

4. Taylor expanded in u0 around 0 87.8%

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

   1. +-commutative 87.8%

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

   2. associate-*r* 87.8%

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

   3. distribute-rgt-out 87.9%

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

   4. unpow2 87.9%

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

   5. unpow2 87.9%

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

6. Simplified 87.9%

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

7. Final simplification 87.9%

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

Alternative 7: 86.9% accurate, 9.8× speedup

Math

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

FPCore

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

C

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

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)))))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 54.2%

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

   1. associate-*l* 54.1%

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

   2. sub-neg 54.1%

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

   3. log1p-def 99.1%

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

3. Simplified 99.1%

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

4. Taylor expanded in u0 around 0 87.8%

   \[\leadsto \left(-\alpha\right) \cdot \color{blue}{\left(-1 \cdot \left(u0 \cdot \alpha\right) + -0.5 \cdot \left({u0}^{2} \cdot \alpha\right)\right)} \]
5. Step-by-step derivation

   1. +-commutative 87.8%

      \[\leadsto \left(-\alpha\right) \cdot \color{blue}{\left(-0.5 \cdot \left({u0}^{2} \cdot \alpha\right) + -1 \cdot \left(u0 \cdot \alpha\right)\right)} \]

   2. mul-1-neg 87.8%

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

   3. unsub-neg 87.8%

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

   4. associate-*r* 87.8%

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

   5. distribute-rgt-out-- 87.9%

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

   6. *-commutative 87.9%

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

   7. unpow2 87.9%

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

   8. associate-*l* 87.9%

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

6. Simplified 87.9%

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

7. Final simplification 87.9%

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

Alternative 8: 74.1% 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 54.2%

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

   1. associate-*l* 54.1%

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

   2. sub-neg 54.1%

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

   3. log1p-def 99.1%

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

3. Simplified 99.1%

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

4. Taylor expanded in u0 around 0 76.1%

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

   1. *-commutative 76.1%

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

   2. unpow2 76.1%

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

   3. associate-*l* 76.1%

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

6. Simplified 76.1%

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

7. Final simplification 76.1%

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

Alternative 9: 74.1% accurate, 21.6× speedup

Math

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

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(u0 * Float32(alpha * alpha))
end

MATLAB

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

Derivation

1. Initial program 54.2%

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

   1. associate-*l* 54.1%

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

   2. sub-neg 54.1%

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

   3. log1p-def 99.1%

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

3. Simplified 99.1%

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

4. Taylor expanded in u0 around 0 76.1%

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

   1. *-commutative 76.1%

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

   2. unpow2 76.1%

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

6. Simplified 76.1%

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

7. Final simplification 76.1%

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

Reproduce

herbie shell --seed 2023238 
(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))))