Beckmann Distribution sample, tan2theta, alphax == alphay

Percentage Accurate: 55.8% → 99.0%

Time: 8.4s

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: 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: 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 56.9%

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

   1. associate-*l* 56.8%

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

   2. distribute-lft-neg-out 56.8%

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

   3. distribute-rgt-neg-in 56.8%

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

   4. distribute-rgt-neg-in 56.8%

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

   5. distribute-rgt-neg-in 56.8%

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

   6. distribute-lft-neg-out 56.8%

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

   7. sub-neg 56.8%

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

   8. log1p-def 99.2%

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

3. Simplified 99.2%

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

4. Final simplification 99.2%

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

Alternative 2: 87.1% accurate, 8.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 56.9%

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

   1. associate-*l* 56.8%

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

   2. distribute-lft-neg-out 56.8%

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

   3. distribute-rgt-neg-in 56.8%

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

   4. distribute-rgt-neg-in 56.8%

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

   5. distribute-rgt-neg-in 56.8%

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

   6. distribute-lft-neg-out 56.8%

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

   7. sub-neg 56.8%

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

   8. log1p-def 99.2%

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

3. Simplified 99.2%

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

4. Taylor expanded in u0 around 0 87.6%

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

   1. +-commutative 87.6%

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

   2. fma-def 87.6%

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

   3. *-commutative 87.6%

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

   4. associate-*l* 87.6%

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

6. Simplified 87.6%

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

7. Taylor expanded in alpha around -inf 87.6%

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

   1. mul-1-neg 87.6%

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

   2. neg-mul-1 87.6%

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

   3. distribute-rgt-in 87.6%

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

   4. +-commutative 87.6%

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

   5. *-commutative 87.6%

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

   6. unpow2 87.6%

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

   7. associate-*r* 87.6%

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

   8. distribute-rgt-in 87.6%

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

   9. neg-mul-1 87.6%

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

   10. *-commutative 87.6%

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

   11. +-commutative 87.6%

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

   12. distribute-lft-out 87.3%

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

9. Simplified 87.3%

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

    1. associate-*r* 87.3%

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

    2. +-commutative 87.3%

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

    3. distribute-rgt-in 87.6%

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

11. Applied egg-rr 87.6%

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

12. Final simplification 87.6%

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

Alternative 3: 86.9% accurate, 9.0× speedup

Math

\[\begin{array}{l} \\ \alpha \cdot \left(\alpha \cdot \left(u0 \cdot \left(u0 \cdot \left(--0.5\right) - -1\right)\right)\right) \end{array} \]

FPCore

(FPCore (alpha u0)
 :precision binary32
 (* alpha (* alpha (* u0 (- (* u0 (- -0.5)) -1.0)))))

C

float code(float alpha, float u0) {
	return alpha * (alpha * (u0 * ((u0 * -(-0.5f)) - -1.0f)));
}

Fortran

real(4) function code(alpha, u0)
    real(4), intent (in) :: alpha
    real(4), intent (in) :: u0
    code = alpha * (alpha * (u0 * ((u0 * -(-0.5e0)) - (-1.0e0))))
end function

Julia

function code(alpha, u0)
	return Float32(alpha * Float32(alpha * Float32(u0 * Float32(Float32(u0 * Float32(-Float32(-0.5))) - Float32(-1.0)))))
end

MATLAB

function tmp = code(alpha, u0)
	tmp = alpha * (alpha * (u0 * ((u0 * -single(-0.5)) - single(-1.0))));
end

Derivation

1. Initial program 56.9%

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

   1. associate-*l* 56.8%

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

   2. distribute-lft-neg-out 56.8%

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

   3. distribute-rgt-neg-in 56.8%

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

   4. distribute-rgt-neg-in 56.8%

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

   5. distribute-rgt-neg-in 56.8%

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

   6. distribute-lft-neg-out 56.8%

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

   7. sub-neg 56.8%

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

   8. log1p-def 99.2%

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

3. Simplified 99.2%

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

4. Taylor expanded in u0 around 0 87.6%

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

   1. +-commutative 87.6%

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

   2. fma-def 87.6%

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

   3. *-commutative 87.6%

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

   4. associate-*l* 87.6%

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

6. Simplified 87.6%

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

7. Taylor expanded in alpha around -inf 87.6%

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

   1. mul-1-neg 87.6%

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

   2. neg-mul-1 87.6%

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

   3. distribute-rgt-in 87.6%

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

   4. +-commutative 87.6%

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

   5. *-commutative 87.6%

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

   6. unpow2 87.6%

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

   7. associate-*r* 87.6%

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

   8. distribute-rgt-in 87.6%

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

   9. neg-mul-1 87.6%

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

   10. *-commutative 87.6%

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

   11. +-commutative 87.6%

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

   12. distribute-lft-out 87.3%

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

9. Simplified 87.3%

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

10. Final simplification 87.3%

    \[\leadsto \alpha \cdot \left(\alpha \cdot \left(u0 \cdot \left(u0 \cdot \left(--0.5\right) - -1\right)\right)\right) \]

Alternative 4: 87.1% accurate, 9.8× speedup

Math

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

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(Float32(alpha * 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 56.9%

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

2. Taylor expanded in u0 around 0 87.5%

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

   1. unpow2 87.5%

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

   2. associate-*r* 87.5%

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

   3. distribute-rgt-out 87.2%

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

4. Simplified 87.2%

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

   1. distribute-rgt-in 87.5%

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

   2. mul-1-neg 87.5%

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

   3. *-commutative 87.5%

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

6. Applied egg-rr 87.5%

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

7. Final simplification 87.5%

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

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 56.9%

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

   1. associate-*l* 56.8%

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

   2. distribute-lft-neg-out 56.8%

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

   3. distribute-rgt-neg-in 56.8%

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

   4. distribute-rgt-neg-in 56.8%

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

   5. distribute-rgt-neg-in 56.8%

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

   6. distribute-lft-neg-out 56.8%

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

   7. sub-neg 56.8%

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

   8. log1p-def 99.2%

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

3. Simplified 99.2%

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

4. Taylor expanded in u0 around 0 74.8%

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

5. Final simplification 74.8%

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

Reproduce

herbie shell --seed 2023333 
(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))))