Beckmann Distribution sample, tan2theta, alphax == alphay

Percentage Accurate: 55.7% → 99.0%

Time: 7.6s

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: 55.7% 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

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

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

Derivation

1. Initial program 56.3%

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

   1. associate-*l* 56.3%

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

   2. sub-neg 56.3%

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

   3. log1p-def 99.0%

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

3. Simplified 99.0%

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

4. Final simplification 99.0%

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

Alternative 2: 91.5% 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 56.3%

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

   1. associate-*l* 56.3%

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

   2. sub-neg 56.3%

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

   3. log1p-def 99.0%

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

3. Simplified 99.0%

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

4. Taylor expanded in u0 around 0 91.9%

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

   1. *-commutative 91.9%

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

   2. +-commutative 91.9%

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

   3. associate-*r* 91.9%

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

   4. associate-*r* 91.9%

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

   5. distribute-rgt-out 91.9%

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

   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 3: 91.5% accurate, 7.2× speedup

Math

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

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(alpha * Float32(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 56.3%

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

   1. associate-*l* 56.3%

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

   2. sub-neg 56.3%

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

   3. log1p-def 99.0%

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

3. Simplified 99.0%

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

4. Taylor expanded in u0 around 0 91.8%

   \[\leadsto \left(-\alpha\right) \cdot \color{blue}{\left(-0.3333333333333333 \cdot \left({u0}^{3} \cdot \alpha\right) + \left(-1 \cdot \left(u0 \cdot \alpha\right) + -0.5 \cdot \left({u0}^{2} \cdot \alpha\right)\right)\right)} \]
5. Step-by-step derivation

   1. associate-+r+ 91.8%

      \[\leadsto \left(-\alpha\right) \cdot \color{blue}{\left(\left(-0.3333333333333333 \cdot \left({u0}^{3} \cdot \alpha\right) + -1 \cdot \left(u0 \cdot \alpha\right)\right) + -0.5 \cdot \left({u0}^{2} \cdot \alpha\right)\right)} \]

   2. +-commutative 91.8%

      \[\leadsto \left(-\alpha\right) \cdot \color{blue}{\left(-0.5 \cdot \left({u0}^{2} \cdot \alpha\right) + \left(-0.3333333333333333 \cdot \left({u0}^{3} \cdot \alpha\right) + -1 \cdot \left(u0 \cdot \alpha\right)\right)\right)} \]

   3. associate-*r* 91.8%

      \[\leadsto \left(-\alpha\right) \cdot \left(\color{blue}{\left(-0.5 \cdot {u0}^{2}\right) \cdot \alpha} + \left(-0.3333333333333333 \cdot \left({u0}^{3} \cdot \alpha\right) + -1 \cdot \left(u0 \cdot \alpha\right)\right)\right) \]

   4. *-commutative 91.8%

      \[\leadsto \left(-\alpha\right) \cdot \left(\color{blue}{\alpha \cdot \left(-0.5 \cdot {u0}^{2}\right)} + \left(-0.3333333333333333 \cdot \left({u0}^{3} \cdot \alpha\right) + -1 \cdot \left(u0 \cdot \alpha\right)\right)\right) \]

   5. associate-*r* 91.8%

      \[\leadsto \left(-\alpha\right) \cdot \left(\alpha \cdot \left(-0.5 \cdot {u0}^{2}\right) + \left(\color{blue}{\left(-0.3333333333333333 \cdot {u0}^{3}\right) \cdot \alpha} + -1 \cdot \left(u0 \cdot \alpha\right)\right)\right) \]

   6. associate-*r* 91.8%

      \[\leadsto \left(-\alpha\right) \cdot \left(\alpha \cdot \left(-0.5 \cdot {u0}^{2}\right) + \left(\left(-0.3333333333333333 \cdot {u0}^{3}\right) \cdot \alpha + \color{blue}{\left(-1 \cdot u0\right) \cdot \alpha}\right)\right) \]

   7. distribute-rgt-out 91.9%

      \[\leadsto \left(-\alpha\right) \cdot \left(\alpha \cdot \left(-0.5 \cdot {u0}^{2}\right) + \color{blue}{\alpha \cdot \left(-0.3333333333333333 \cdot {u0}^{3} + -1 \cdot u0\right)}\right) \]

   8. distribute-lft-out 91.9%

      \[\leadsto \left(-\alpha\right) \cdot \color{blue}{\left(\alpha \cdot \left(-0.5 \cdot {u0}^{2} + \left(-0.3333333333333333 \cdot {u0}^{3} + -1 \cdot u0\right)\right)\right)} \]

   9. associate-+l+ 92.0%

      \[\leadsto \left(-\alpha\right) \cdot \left(\alpha \cdot \color{blue}{\left(\left(-0.5 \cdot {u0}^{2} + -0.3333333333333333 \cdot {u0}^{3}\right) + -1 \cdot u0\right)}\right) \]

   10. mul-1-neg 92.0%

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

   11. unsub-neg 92.0%

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

   12. *-commutative 92.0%

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

   13. *-commutative 92.0%

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

   14. unpow3 92.0%

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

   15. unpow2 92.0%

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

   16. associate-*l* 92.0%

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

   17. distribute-lft-out 92.0%

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

   18. unpow2 92.0%

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

6. Simplified 92.0%

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

7. Final simplification 92.0%

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

Alternative 4: 87.3% accurate, 9.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 56.3%

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

   1. associate-*l* 56.3%

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

   2. sub-neg 56.3%

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

   3. log1p-def 99.0%

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

3. Simplified 99.0%

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

4. Taylor expanded in u0 around 0 88.2%

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

   1. associate-*r* 88.2%

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

   2. distribute-rgt-out 88.1%

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

   3. unpow2 88.1%

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

   4. *-commutative 88.1%

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

   5. unpow2 88.1%

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

6. Simplified 88.1%

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

   1. distribute-rgt-in 88.2%

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

   2. associate-*l* 88.2%

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

8. Applied egg-rr 88.2%

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

   1. +-commutative 88.2%

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

   2. associate-*r* 88.2%

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

   3. fma-def 88.2%

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

   4. add-sqr-sqrt 87.6%

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

   5. sqrt-prod 88.2%

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

   6. associate-*r* 88.2%

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

   7. associate-*r* 88.3%

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

   8. swap-sqr 87.9%

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

   9. sqr-neg 87.9%

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

   10. distribute-rgt-neg-out 87.9%

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

   11. distribute-rgt-neg-out 87.9%

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

   12. sqr-neg 87.9%

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

   13. swap-sqr 88.3%

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

   14. *-commutative 88.3%

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

   15. *-commutative 88.3%

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

   16. sqrt-unprod 87.6%

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

   17. add-sqr-sqrt 88.3%

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

10. Applied egg-rr 88.3%

    \[\leadsto \color{blue}{\mathsf{fma}\left(\left(u0 \cdot \left(u0 \cdot 0.5\right)\right) \cdot \alpha, \alpha, \alpha \cdot \left(u0 \cdot \alpha\right)\right)} \]
11. Step-by-step derivation

    1. fma-udef 88.3%

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

    2. associate-*l* 88.3%

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

    3. associate-*l* 88.3%

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

    4. associate-*l* 88.3%

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

    5. *-commutative 88.3%

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

    6. associate-*l* 88.2%

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

12. Applied egg-rr 88.2%

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

    1. associate-*r* 88.2%

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

    2. associate-*r* 88.3%

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

    3. distribute-rgt-out 88.2%

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

    4. associate-*r* 88.2%

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

    5. *-commutative 88.2%

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

    6. associate-*l* 88.2%

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

    7. +-commutative 88.2%

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

    8. *-rgt-identity 88.2%

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

    9. distribute-lft-out 88.0%

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

    10. *-commutative 88.0%

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

    11. *-commutative 88.0%

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

14. Simplified 88.0%

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

15. Final simplification 88.0%

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

Alternative 5: 87.3% accurate, 9.8× speedup

Math

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

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(Float32(alpha * alpha) * Float32(u0 * Float32(Float32(u0 * 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 56.3%

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

   1. associate-*l* 56.3%

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

   2. sub-neg 56.3%

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

   3. log1p-def 99.0%

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

3. Simplified 99.0%

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

4. Taylor expanded in u0 around 0 88.2%

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

   1. associate-*r* 88.2%

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

   2. distribute-rgt-out 88.1%

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

   3. unpow2 88.1%

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

   4. unpow2 88.1%

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

   5. associate-*r* 88.1%

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

   6. distribute-rgt1-in 88.1%

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

6. Simplified 88.1%

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

7. Final simplification 88.1%

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

Alternative 6: 87.5% accurate, 9.8× speedup

Math

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

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(Float32(u0 * 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 56.3%

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

   1. associate-*l* 56.3%

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

   2. sub-neg 56.3%

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

   3. log1p-def 99.0%

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

3. Simplified 99.0%

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

4. Taylor expanded in u0 around 0 88.2%

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

   1. associate-*r* 88.2%

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

   2. distribute-rgt-out 88.1%

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

   3. unpow2 88.1%

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

   4. *-commutative 88.1%

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

   5. unpow2 88.1%

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

6. Simplified 88.1%

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

7. Final simplification 88.1%

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

Alternative 7: 87.5% 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 56.3%

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

   1. associate-*l* 56.3%

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

   2. sub-neg 56.3%

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

   3. log1p-def 99.0%

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

3. Simplified 99.0%

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

4. Taylor expanded in u0 around 0 88.2%

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

   1. +-commutative 88.2%

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

   2. mul-1-neg 88.2%

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

   3. unsub-neg 88.2%

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

   4. associate-*r* 88.2%

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

   5. distribute-rgt-out-- 88.3%

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

   6. *-commutative 88.3%

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

   7. unpow2 88.3%

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

   8. associate-*l* 88.3%

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

6. Simplified 88.3%

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

7. Final simplification 88.3%

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

Alternative 8: 74.7% 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 56.3%

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

   1. associate-*l* 56.3%

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

   2. sub-neg 56.3%

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

   3. log1p-def 99.0%

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

3. Simplified 99.0%

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

4. Taylor expanded in u0 around 0 74.9%

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

   1. unpow2 74.9%

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

6. Simplified 74.9%

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

7. Final simplification 74.9%

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

Alternative 9: 74.7% 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 56.3%

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

   1. associate-*l* 56.3%

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

   2. sub-neg 56.3%

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

   3. log1p-def 99.0%

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

3. Simplified 99.0%

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

4. Taylor expanded in u0 around 0 74.9%

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

   1. unpow2 74.9%

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

6. Simplified 74.9%

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

   1. add-cbrt-cube 67.7%

      \[\leadsto \color{blue}{\sqrt[3]{\left(\left(u0 \cdot \left(\alpha \cdot \alpha\right)\right) \cdot \left(u0 \cdot \left(\alpha \cdot \alpha\right)\right)\right) \cdot \left(u0 \cdot \left(\alpha \cdot \alpha\right)\right)}} \]

8. Applied egg-rr 67.7%

   \[\leadsto \color{blue}{\sqrt[3]{\left(\left(u0 \cdot \left(\alpha \cdot \alpha\right)\right) \cdot \left(u0 \cdot \left(\alpha \cdot \alpha\right)\right)\right) \cdot \left(u0 \cdot \left(\alpha \cdot \alpha\right)\right)}} \]
9. Step-by-step derivation

   1. add-cbrt-cube 74.9%

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

   2. associate-*r* 75.0%

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

10. Applied egg-rr 75.0%

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

11. Final simplification 75.0%

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

Reproduce

herbie shell --seed 2023257 
(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))))