Beckmann Distribution sample, tan2theta, alphax == alphay

Percentage Accurate: 56.0% → 99.0%

Time: 12.3s

Alternatives: 22

Speedup: 10.5×

Specification

\[\left(0.0001 \leq \alpha \land \alpha \leq 1\right) \land \left(2.328306437 \cdot 10^{-10} \leq u0 \land u0 \leq 1\right)\]

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Initial Program: 56.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Alternative 1: 99.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 57.3%

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

   1. sub-neg N/A

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

   2. accelerator-lowering-log1p.f32 N/A

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

   3. neg-lowering-neg.f32 98.9

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

4. Applied egg-rr 98.9%

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

5. Final simplification 98.9%

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

Alternative 2: 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 57.3%

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

   1. sub-neg N/A

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

   2. accelerator-lowering-log1p.f32 N/A

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

   3. neg-lowering-neg.f32 98.9

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

4. Applied egg-rr 98.9%

   \[\leadsto \left(\left(-\alpha\right) \cdot \alpha\right) \cdot \color{blue}{\mathsf{log1p}\left(-u0\right)} \]
5. Step-by-step derivation

   1. neg-sub0 N/A

      \[\leadsto \left(\color{blue}{\left(0 - \alpha\right)} \cdot \alpha\right) \cdot \mathsf{log1p}\left(\mathsf{neg}\left(u0\right)\right) \]

   2. sub-neg N/A

      \[\leadsto \left(\color{blue}{\left(0 + \left(\mathsf{neg}\left(\alpha\right)\right)\right)} \cdot \alpha\right) \cdot \mathsf{log1p}\left(\mathsf{neg}\left(u0\right)\right) \]

   3. flip3-+ N/A

      \[\leadsto \left(\color{blue}{\frac{{0}^{3} + {\left(\mathsf{neg}\left(\alpha\right)\right)}^{3}}{0 \cdot 0 + \left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \left(\mathsf{neg}\left(\alpha\right)\right) - 0 \cdot \left(\mathsf{neg}\left(\alpha\right)\right)\right)}} \cdot \alpha\right) \cdot \mathsf{log1p}\left(\mathsf{neg}\left(u0\right)\right) \]

   4. /-lowering-/.f32 N/A

      \[\leadsto \left(\color{blue}{\frac{{0}^{3} + {\left(\mathsf{neg}\left(\alpha\right)\right)}^{3}}{0 \cdot 0 + \left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \left(\mathsf{neg}\left(\alpha\right)\right) - 0 \cdot \left(\mathsf{neg}\left(\alpha\right)\right)\right)}} \cdot \alpha\right) \cdot \mathsf{log1p}\left(\mathsf{neg}\left(u0\right)\right) \]

   5. metadata-eval N/A

      \[\leadsto \left(\frac{\color{blue}{0} + {\left(\mathsf{neg}\left(\alpha\right)\right)}^{3}}{0 \cdot 0 + \left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \left(\mathsf{neg}\left(\alpha\right)\right) - 0 \cdot \left(\mathsf{neg}\left(\alpha\right)\right)\right)} \cdot \alpha\right) \cdot \mathsf{log1p}\left(\mathsf{neg}\left(u0\right)\right) \]

   6. +-lowering-+.f32 N/A

      \[\leadsto \left(\frac{\color{blue}{0 + {\left(\mathsf{neg}\left(\alpha\right)\right)}^{3}}}{0 \cdot 0 + \left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \left(\mathsf{neg}\left(\alpha\right)\right) - 0 \cdot \left(\mathsf{neg}\left(\alpha\right)\right)\right)} \cdot \alpha\right) \cdot \mathsf{log1p}\left(\mathsf{neg}\left(u0\right)\right) \]

   7. pow-lowering-pow.f32 N/A

      \[\leadsto \left(\frac{0 + \color{blue}{{\left(\mathsf{neg}\left(\alpha\right)\right)}^{3}}}{0 \cdot 0 + \left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \left(\mathsf{neg}\left(\alpha\right)\right) - 0 \cdot \left(\mathsf{neg}\left(\alpha\right)\right)\right)} \cdot \alpha\right) \cdot \mathsf{log1p}\left(\mathsf{neg}\left(u0\right)\right) \]

   8. neg-lowering-neg.f32 N/A

      \[\leadsto \left(\frac{0 + {\color{blue}{\left(\mathsf{neg}\left(\alpha\right)\right)}}^{3}}{0 \cdot 0 + \left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \left(\mathsf{neg}\left(\alpha\right)\right) - 0 \cdot \left(\mathsf{neg}\left(\alpha\right)\right)\right)} \cdot \alpha\right) \cdot \mathsf{log1p}\left(\mathsf{neg}\left(u0\right)\right) \]

   9. metadata-eval N/A

      \[\leadsto \left(\frac{0 + {\left(\mathsf{neg}\left(\alpha\right)\right)}^{3}}{\color{blue}{0} + \left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \left(\mathsf{neg}\left(\alpha\right)\right) - 0 \cdot \left(\mathsf{neg}\left(\alpha\right)\right)\right)} \cdot \alpha\right) \cdot \mathsf{log1p}\left(\mathsf{neg}\left(u0\right)\right) \]

   10. +-lowering-+.f32 N/A

       \[\leadsto \left(\frac{0 + {\left(\mathsf{neg}\left(\alpha\right)\right)}^{3}}{\color{blue}{0 + \left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \left(\mathsf{neg}\left(\alpha\right)\right) - 0 \cdot \left(\mathsf{neg}\left(\alpha\right)\right)\right)}} \cdot \alpha\right) \cdot \mathsf{log1p}\left(\mathsf{neg}\left(u0\right)\right) \]

   11. --lowering--.f32 N/A

       \[\leadsto \left(\frac{0 + {\left(\mathsf{neg}\left(\alpha\right)\right)}^{3}}{0 + \color{blue}{\left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \left(\mathsf{neg}\left(\alpha\right)\right) - 0 \cdot \left(\mathsf{neg}\left(\alpha\right)\right)\right)}} \cdot \alpha\right) \cdot \mathsf{log1p}\left(\mathsf{neg}\left(u0\right)\right) \]

   12. *-lowering-*.f32 N/A

       \[\leadsto \left(\frac{0 + {\left(\mathsf{neg}\left(\alpha\right)\right)}^{3}}{0 + \left(\color{blue}{\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \left(\mathsf{neg}\left(\alpha\right)\right)} - 0 \cdot \left(\mathsf{neg}\left(\alpha\right)\right)\right)} \cdot \alpha\right) \cdot \mathsf{log1p}\left(\mathsf{neg}\left(u0\right)\right) \]

   13. neg-lowering-neg.f32 N/A

       \[\leadsto \left(\frac{0 + {\left(\mathsf{neg}\left(\alpha\right)\right)}^{3}}{0 + \left(\color{blue}{\left(\mathsf{neg}\left(\alpha\right)\right)} \cdot \left(\mathsf{neg}\left(\alpha\right)\right) - 0 \cdot \left(\mathsf{neg}\left(\alpha\right)\right)\right)} \cdot \alpha\right) \cdot \mathsf{log1p}\left(\mathsf{neg}\left(u0\right)\right) \]

   14. neg-lowering-neg.f32 N/A

       \[\leadsto \left(\frac{0 + {\left(\mathsf{neg}\left(\alpha\right)\right)}^{3}}{0 + \left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \color{blue}{\left(\mathsf{neg}\left(\alpha\right)\right)} - 0 \cdot \left(\mathsf{neg}\left(\alpha\right)\right)\right)} \cdot \alpha\right) \cdot \mathsf{log1p}\left(\mathsf{neg}\left(u0\right)\right) \]

   15. *-lowering-*.f32 N/A

       \[\leadsto \left(\frac{0 + {\left(\mathsf{neg}\left(\alpha\right)\right)}^{3}}{0 + \left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \left(\mathsf{neg}\left(\alpha\right)\right) - \color{blue}{0 \cdot \left(\mathsf{neg}\left(\alpha\right)\right)}\right)} \cdot \alpha\right) \cdot \mathsf{log1p}\left(\mathsf{neg}\left(u0\right)\right) \]

   16. neg-lowering-neg.f32 98.6

       \[\leadsto \left(\frac{0 + {\left(-\alpha\right)}^{3}}{0 + \left(\left(-\alpha\right) \cdot \left(-\alpha\right) - 0 \cdot \color{blue}{\left(-\alpha\right)}\right)} \cdot \alpha\right) \cdot \mathsf{log1p}\left(-u0\right) \]

6. Applied egg-rr 98.6%

   \[\leadsto \left(\color{blue}{\frac{0 + {\left(-\alpha\right)}^{3}}{0 + \left(\left(-\alpha\right) \cdot \left(-\alpha\right) - 0 \cdot \left(-\alpha\right)\right)}} \cdot \alpha\right) \cdot \mathsf{log1p}\left(-u0\right) \]
7. Step-by-step derivation

   1. +-lft-identity N/A

      \[\leadsto \left(\frac{\color{blue}{{\left(\mathsf{neg}\left(\alpha\right)\right)}^{3}}}{0 + \left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \left(\mathsf{neg}\left(\alpha\right)\right) - 0 \cdot \left(\mathsf{neg}\left(\alpha\right)\right)\right)} \cdot \alpha\right) \cdot \mathsf{log1p}\left(\mathsf{neg}\left(u0\right)\right) \]

   2. +-lft-identity N/A

      \[\leadsto \left(\frac{{\left(\mathsf{neg}\left(\alpha\right)\right)}^{3}}{\color{blue}{\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \left(\mathsf{neg}\left(\alpha\right)\right) - 0 \cdot \left(\mathsf{neg}\left(\alpha\right)\right)}} \cdot \alpha\right) \cdot \mathsf{log1p}\left(\mathsf{neg}\left(u0\right)\right) \]

   3. distribute-rgt-out-- N/A

      \[\leadsto \left(\frac{{\left(\mathsf{neg}\left(\alpha\right)\right)}^{3}}{\color{blue}{\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \left(\left(\mathsf{neg}\left(\alpha\right)\right) - 0\right)}} \cdot \alpha\right) \cdot \mathsf{log1p}\left(\mathsf{neg}\left(u0\right)\right) \]

   4. --rgt-identity N/A

      \[\leadsto \left(\frac{{\left(\mathsf{neg}\left(\alpha\right)\right)}^{3}}{\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \color{blue}{\left(\mathsf{neg}\left(\alpha\right)\right)}} \cdot \alpha\right) \cdot \mathsf{log1p}\left(\mathsf{neg}\left(u0\right)\right) \]

   5. pow2 N/A

      \[\leadsto \left(\frac{{\left(\mathsf{neg}\left(\alpha\right)\right)}^{3}}{\color{blue}{{\left(\mathsf{neg}\left(\alpha\right)\right)}^{2}}} \cdot \alpha\right) \cdot \mathsf{log1p}\left(\mathsf{neg}\left(u0\right)\right) \]

   6. pow-div N/A

      \[\leadsto \left(\color{blue}{{\left(\mathsf{neg}\left(\alpha\right)\right)}^{\left(3 - 2\right)}} \cdot \alpha\right) \cdot \mathsf{log1p}\left(\mathsf{neg}\left(u0\right)\right) \]

   7. metadata-eval N/A

      \[\leadsto \left({\left(\mathsf{neg}\left(\alpha\right)\right)}^{\color{blue}{1}} \cdot \alpha\right) \cdot \mathsf{log1p}\left(\mathsf{neg}\left(u0\right)\right) \]

   8. unpow1 N/A

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

   9. remove-double-div N/A

      \[\leadsto \left(\color{blue}{\frac{1}{\frac{1}{\mathsf{neg}\left(\alpha\right)}}} \cdot \alpha\right) \cdot \mathsf{log1p}\left(\mathsf{neg}\left(u0\right)\right) \]

   10. /-lowering-/.f32 N/A

       \[\leadsto \left(\color{blue}{\frac{1}{\frac{1}{\mathsf{neg}\left(\alpha\right)}}} \cdot \alpha\right) \cdot \mathsf{log1p}\left(\mathsf{neg}\left(u0\right)\right) \]

   11. inv-pow N/A

       \[\leadsto \left(\frac{1}{\color{blue}{{\left(\mathsf{neg}\left(\alpha\right)\right)}^{-1}}} \cdot \alpha\right) \cdot \mathsf{log1p}\left(\mathsf{neg}\left(u0\right)\right) \]

   12. metadata-eval N/A

       \[\leadsto \left(\frac{1}{{\left(\mathsf{neg}\left(\alpha\right)\right)}^{\color{blue}{\left(2 - 3\right)}}} \cdot \alpha\right) \cdot \mathsf{log1p}\left(\mathsf{neg}\left(u0\right)\right) \]

   13. pow-div N/A

       \[\leadsto \left(\frac{1}{\color{blue}{\frac{{\left(\mathsf{neg}\left(\alpha\right)\right)}^{2}}{{\left(\mathsf{neg}\left(\alpha\right)\right)}^{3}}}} \cdot \alpha\right) \cdot \mathsf{log1p}\left(\mathsf{neg}\left(u0\right)\right) \]

   14. pow2 N/A

       \[\leadsto \left(\frac{1}{\frac{\color{blue}{\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \left(\mathsf{neg}\left(\alpha\right)\right)}}{{\left(\mathsf{neg}\left(\alpha\right)\right)}^{3}}} \cdot \alpha\right) \cdot \mathsf{log1p}\left(\mathsf{neg}\left(u0\right)\right) \]

   15. sqr-neg N/A

       \[\leadsto \left(\frac{1}{\frac{\color{blue}{\alpha \cdot \alpha}}{{\left(\mathsf{neg}\left(\alpha\right)\right)}^{3}}} \cdot \alpha\right) \cdot \mathsf{log1p}\left(\mathsf{neg}\left(u0\right)\right) \]

   16. pow2 N/A

       \[\leadsto \left(\frac{1}{\frac{\color{blue}{{\alpha}^{2}}}{{\left(\mathsf{neg}\left(\alpha\right)\right)}^{3}}} \cdot \alpha\right) \cdot \mathsf{log1p}\left(\mathsf{neg}\left(u0\right)\right) \]

   17. sqr-pow N/A

       \[\leadsto \left(\frac{1}{\frac{{\alpha}^{2}}{\color{blue}{{\left(\mathsf{neg}\left(\alpha\right)\right)}^{\left(\frac{3}{2}\right)} \cdot {\left(\mathsf{neg}\left(\alpha\right)\right)}^{\left(\frac{3}{2}\right)}}}} \cdot \alpha\right) \cdot \mathsf{log1p}\left(\mathsf{neg}\left(u0\right)\right) \]

   18. pow-prod-down N/A

       \[\leadsto \left(\frac{1}{\frac{{\alpha}^{2}}{\color{blue}{{\left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \left(\mathsf{neg}\left(\alpha\right)\right)\right)}^{\left(\frac{3}{2}\right)}}}} \cdot \alpha\right) \cdot \mathsf{log1p}\left(\mathsf{neg}\left(u0\right)\right) \]

   19. sqr-neg N/A

       \[\leadsto \left(\frac{1}{\frac{{\alpha}^{2}}{{\color{blue}{\left(\alpha \cdot \alpha\right)}}^{\left(\frac{3}{2}\right)}}} \cdot \alpha\right) \cdot \mathsf{log1p}\left(\mathsf{neg}\left(u0\right)\right) \]

   20. unpow-prod-down N/A

       \[\leadsto \left(\frac{1}{\frac{{\alpha}^{2}}{\color{blue}{{\alpha}^{\left(\frac{3}{2}\right)} \cdot {\alpha}^{\left(\frac{3}{2}\right)}}}} \cdot \alpha\right) \cdot \mathsf{log1p}\left(\mathsf{neg}\left(u0\right)\right) \]

   21. sqr-pow N/A

       \[\leadsto \left(\frac{1}{\frac{{\alpha}^{2}}{\color{blue}{{\alpha}^{3}}}} \cdot \alpha\right) \cdot \mathsf{log1p}\left(\mathsf{neg}\left(u0\right)\right) \]

   22. pow-div N/A

       \[\leadsto \left(\frac{1}{\color{blue}{{\alpha}^{\left(2 - 3\right)}}} \cdot \alpha\right) \cdot \mathsf{log1p}\left(\mathsf{neg}\left(u0\right)\right) \]

   23. metadata-eval N/A

       \[\leadsto \left(\frac{1}{{\alpha}^{\color{blue}{-1}}} \cdot \alpha\right) \cdot \mathsf{log1p}\left(\mathsf{neg}\left(u0\right)\right) \]

   24. inv-pow N/A

       \[\leadsto \left(\frac{1}{\color{blue}{\frac{1}{\alpha}}} \cdot \alpha\right) \cdot \mathsf{log1p}\left(\mathsf{neg}\left(u0\right)\right) \]

   25. frac-2neg N/A

       \[\leadsto \left(\frac{1}{\color{blue}{\frac{\mathsf{neg}\left(1\right)}{\mathsf{neg}\left(\alpha\right)}}} \cdot \alpha\right) \cdot \mathsf{log1p}\left(\mathsf{neg}\left(u0\right)\right) \]

   26. metadata-eval N/A

       \[\leadsto \left(\frac{1}{\frac{\color{blue}{-1}}{\mathsf{neg}\left(\alpha\right)}} \cdot \alpha\right) \cdot \mathsf{log1p}\left(\mathsf{neg}\left(u0\right)\right) \]

   27. /-lowering-/.f32 N/A

       \[\leadsto \left(\frac{1}{\color{blue}{\frac{-1}{\mathsf{neg}\left(\alpha\right)}}} \cdot \alpha\right) \cdot \mathsf{log1p}\left(\mathsf{neg}\left(u0\right)\right) \]

8. Applied egg-rr 98.8%

   \[\leadsto \left(\color{blue}{\frac{1}{\frac{-1}{\alpha}}} \cdot \alpha\right) \cdot \mathsf{log1p}\left(-u0\right) \]
9. Step-by-step derivation

   1. *-commutative N/A

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

   2. associate-/r/ N/A

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

   3. metadata-eval N/A

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

   4. neg-mul-1 N/A

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

   5. associate-*r* N/A

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

   6. *-lowering-*.f32 N/A

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

   7. *-lowering-*.f32 N/A

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

   8. accelerator-lowering-log1p.f32 N/A

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

   9. neg-lowering-neg.f32 N/A

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

   10. neg-lowering-neg.f32 98.9

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

10. Applied egg-rr 98.9%

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

11. Final simplification 98.9%

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

Alternative 3: 94.2% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, -0.25, -0.3333333333333333\right), -0.5\right)\\ \left(\alpha \cdot \left(-\alpha\right)\right) \cdot \frac{u0 \cdot \mathsf{fma}\left(\mathsf{fma}\left(u0, -0.3333333333333333, -0.5\right), t\_0 \cdot \left(u0 \cdot u0\right), -1\right)}{\mathsf{fma}\left(u0, t\_0, 1\right)} \end{array} \end{array} \]

FPCore

(FPCore (alpha u0)
 :precision binary32
 (let* ((t_0 (fma u0 (fma u0 -0.25 -0.3333333333333333) -0.5)))
   (*
    (* alpha (- alpha))
    (/
     (* u0 (fma (fma u0 -0.3333333333333333 -0.5) (* t_0 (* u0 u0)) -1.0))
     (fma u0 t_0 1.0)))))

C

float code(float alpha, float u0) {
	float t_0 = fmaf(u0, fmaf(u0, -0.25f, -0.3333333333333333f), -0.5f);
	return (alpha * -alpha) * ((u0 * fmaf(fmaf(u0, -0.3333333333333333f, -0.5f), (t_0 * (u0 * u0)), -1.0f)) / fmaf(u0, t_0, 1.0f));
}

Julia

function code(alpha, u0)
	t_0 = fma(u0, fma(u0, Float32(-0.25), Float32(-0.3333333333333333)), Float32(-0.5))
	return Float32(Float32(alpha * Float32(-alpha)) * Float32(Float32(u0 * fma(fma(u0, Float32(-0.3333333333333333), Float32(-0.5)), Float32(t_0 * Float32(u0 * u0)), Float32(-1.0))) / fma(u0, t_0, Float32(1.0))))
end

Derivation

1. Initial program 57.3%

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

3. Taylor expanded in u0 around 0

   \[\leadsto \left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right) \cdot \color{blue}{\left(u0 \cdot \left(u0 \cdot \left(u0 \cdot \left(\frac{-1}{4} \cdot u0 - \frac{1}{3}\right) - \frac{1}{2}\right) - 1\right)\right)} \]
4. Step-by-step derivation

   1. *-lowering-*.f32 N/A

      \[\leadsto \left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right) \cdot \color{blue}{\left(u0 \cdot \left(u0 \cdot \left(u0 \cdot \left(\frac{-1}{4} \cdot u0 - \frac{1}{3}\right) - \frac{1}{2}\right) - 1\right)\right)} \]

   2. sub-neg N/A

      \[\leadsto \left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right) \cdot \left(u0 \cdot \color{blue}{\left(u0 \cdot \left(u0 \cdot \left(\frac{-1}{4} \cdot u0 - \frac{1}{3}\right) - \frac{1}{2}\right) + \left(\mathsf{neg}\left(1\right)\right)\right)}\right) \]

   3. metadata-eval N/A

      \[\leadsto \left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right) \cdot \left(u0 \cdot \left(u0 \cdot \left(u0 \cdot \left(\frac{-1}{4} \cdot u0 - \frac{1}{3}\right) - \frac{1}{2}\right) + \color{blue}{-1}\right)\right) \]

   4. accelerator-lowering-fma.f32 N/A

      \[\leadsto \left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right) \cdot \left(u0 \cdot \color{blue}{\mathsf{fma}\left(u0, u0 \cdot \left(\frac{-1}{4} \cdot u0 - \frac{1}{3}\right) - \frac{1}{2}, -1\right)}\right) \]

   5. sub-neg N/A

      \[\leadsto \left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right) \cdot \left(u0 \cdot \mathsf{fma}\left(u0, \color{blue}{u0 \cdot \left(\frac{-1}{4} \cdot u0 - \frac{1}{3}\right) + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}, -1\right)\right) \]

   6. metadata-eval N/A

      \[\leadsto \left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right) \cdot \left(u0 \cdot \mathsf{fma}\left(u0, u0 \cdot \left(\frac{-1}{4} \cdot u0 - \frac{1}{3}\right) + \color{blue}{\frac{-1}{2}}, -1\right)\right) \]

   7. accelerator-lowering-fma.f32 N/A

      \[\leadsto \left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right) \cdot \left(u0 \cdot \mathsf{fma}\left(u0, \color{blue}{\mathsf{fma}\left(u0, \frac{-1}{4} \cdot u0 - \frac{1}{3}, \frac{-1}{2}\right)}, -1\right)\right) \]

   8. sub-neg N/A

      \[\leadsto \left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right) \cdot \left(u0 \cdot \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, \color{blue}{\frac{-1}{4} \cdot u0 + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)}, \frac{-1}{2}\right), -1\right)\right) \]

   9. *-commutative N/A

      \[\leadsto \left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right) \cdot \left(u0 \cdot \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, \color{blue}{u0 \cdot \frac{-1}{4}} + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right), \frac{-1}{2}\right), -1\right)\right) \]

   10. metadata-eval N/A

       \[\leadsto \left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right) \cdot \left(u0 \cdot \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, u0 \cdot \frac{-1}{4} + \color{blue}{\frac{-1}{3}}, \frac{-1}{2}\right), -1\right)\right) \]

   11. accelerator-lowering-fma.f32 93.5

       \[\leadsto \left(\left(-\alpha\right) \cdot \alpha\right) \cdot \left(u0 \cdot \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, \color{blue}{\mathsf{fma}\left(u0, -0.25, -0.3333333333333333\right)}, -0.5\right), -1\right)\right) \]

5. Simplified 93.5%

   \[\leadsto \left(\left(-\alpha\right) \cdot \alpha\right) \cdot \color{blue}{\left(u0 \cdot \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, -0.25, -0.3333333333333333\right), -0.5\right), -1\right)\right)} \]
6. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right) \cdot \color{blue}{\left(\left(u0 \cdot \left(u0 \cdot \left(u0 \cdot \frac{-1}{4} + \frac{-1}{3}\right) + \frac{-1}{2}\right) + -1\right) \cdot u0\right)} \]

   2. flip-+ N/A

      \[\leadsto \left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right) \cdot \left(\color{blue}{\frac{\left(u0 \cdot \left(u0 \cdot \left(u0 \cdot \frac{-1}{4} + \frac{-1}{3}\right) + \frac{-1}{2}\right)\right) \cdot \left(u0 \cdot \left(u0 \cdot \left(u0 \cdot \frac{-1}{4} + \frac{-1}{3}\right) + \frac{-1}{2}\right)\right) - -1 \cdot -1}{u0 \cdot \left(u0 \cdot \left(u0 \cdot \frac{-1}{4} + \frac{-1}{3}\right) + \frac{-1}{2}\right) - -1}} \cdot u0\right) \]

   3. associate-*l/ N/A

      \[\leadsto \left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right) \cdot \color{blue}{\frac{\left(\left(u0 \cdot \left(u0 \cdot \left(u0 \cdot \frac{-1}{4} + \frac{-1}{3}\right) + \frac{-1}{2}\right)\right) \cdot \left(u0 \cdot \left(u0 \cdot \left(u0 \cdot \frac{-1}{4} + \frac{-1}{3}\right) + \frac{-1}{2}\right)\right) - -1 \cdot -1\right) \cdot u0}{u0 \cdot \left(u0 \cdot \left(u0 \cdot \frac{-1}{4} + \frac{-1}{3}\right) + \frac{-1}{2}\right) - -1}} \]

   4. /-lowering-/.f32 N/A

      \[\leadsto \left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right) \cdot \color{blue}{\frac{\left(\left(u0 \cdot \left(u0 \cdot \left(u0 \cdot \frac{-1}{4} + \frac{-1}{3}\right) + \frac{-1}{2}\right)\right) \cdot \left(u0 \cdot \left(u0 \cdot \left(u0 \cdot \frac{-1}{4} + \frac{-1}{3}\right) + \frac{-1}{2}\right)\right) - -1 \cdot -1\right) \cdot u0}{u0 \cdot \left(u0 \cdot \left(u0 \cdot \frac{-1}{4} + \frac{-1}{3}\right) + \frac{-1}{2}\right) - -1}} \]

7. Applied egg-rr 93.5%

   \[\leadsto \left(\left(-\alpha\right) \cdot \alpha\right) \cdot \color{blue}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(u0, \mathsf{fma}\left(u0, -0.25, -0.3333333333333333\right), -0.5\right), \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, -0.25, -0.3333333333333333\right), -0.5\right) \cdot \left(u0 \cdot u0\right), -1\right) \cdot u0}{\mathsf{fma}\left(u0, \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, -0.25, -0.3333333333333333\right), -0.5\right), 1\right)}} \]

8. Taylor expanded in u0 around 0

   \[\leadsto \left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right) \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(u0, \color{blue}{\frac{-1}{3}}, \frac{-1}{2}\right), \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, \frac{-1}{4}, \frac{-1}{3}\right), \frac{-1}{2}\right) \cdot \left(u0 \cdot u0\right), -1\right) \cdot u0}{\mathsf{fma}\left(u0, \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, \frac{-1}{4}, \frac{-1}{3}\right), \frac{-1}{2}\right), 1\right)} \]
9. Step-by-step derivation

10. Simplified 94.4%

    \[\leadsto \left(\left(-\alpha\right) \cdot \alpha\right) \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(u0, \color{blue}{-0.3333333333333333}, -0.5\right), \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, -0.25, -0.3333333333333333\right), -0.5\right) \cdot \left(u0 \cdot u0\right), -1\right) \cdot u0}{\mathsf{fma}\left(u0, \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, -0.25, -0.3333333333333333\right), -0.5\right), 1\right)} \]

11. Final simplification 94.4%

    \[\leadsto \left(\alpha \cdot \left(-\alpha\right)\right) \cdot \frac{u0 \cdot \mathsf{fma}\left(\mathsf{fma}\left(u0, -0.3333333333333333, -0.5\right), \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, -0.25, -0.3333333333333333\right), -0.5\right) \cdot \left(u0 \cdot u0\right), -1\right)}{\mathsf{fma}\left(u0, \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, -0.25, -0.3333333333333333\right), -0.5\right), 1\right)} \]
12. Add Preprocessing

Alternative 4: 93.5% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\left(\alpha \cdot \left(-\alpha\right)\right) \cdot u0, -1, \left(u0 \cdot \left(\alpha \cdot \alpha\right)\right) \cdot \left(\left(-u0\right) \cdot \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, -0.25, -0.3333333333333333\right), -0.5\right)\right)\right) \end{array} \]

FPCore

(FPCore (alpha u0)
 :precision binary32
 (fma
  (* (* alpha (- alpha)) u0)
  -1.0
  (*
   (* u0 (* alpha alpha))
   (* (- u0) (fma u0 (fma u0 -0.25 -0.3333333333333333) -0.5)))))

C

float code(float alpha, float u0) {
	return fmaf(((alpha * -alpha) * u0), -1.0f, ((u0 * (alpha * alpha)) * (-u0 * fmaf(u0, fmaf(u0, -0.25f, -0.3333333333333333f), -0.5f))));
}

Julia

function code(alpha, u0)
	return fma(Float32(Float32(alpha * Float32(-alpha)) * u0), Float32(-1.0), Float32(Float32(u0 * Float32(alpha * alpha)) * Float32(Float32(-u0) * fma(u0, fma(u0, Float32(-0.25), Float32(-0.3333333333333333)), Float32(-0.5)))))
end

Derivation

1. Initial program 57.3%

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

3. Taylor expanded in u0 around 0

   \[\leadsto \left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right) \cdot \color{blue}{\left(u0 \cdot \left(u0 \cdot \left(u0 \cdot \left(\frac{-1}{4} \cdot u0 - \frac{1}{3}\right) - \frac{1}{2}\right) - 1\right)\right)} \]
4. Step-by-step derivation

   1. *-lowering-*.f32 N/A

      \[\leadsto \left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right) \cdot \color{blue}{\left(u0 \cdot \left(u0 \cdot \left(u0 \cdot \left(\frac{-1}{4} \cdot u0 - \frac{1}{3}\right) - \frac{1}{2}\right) - 1\right)\right)} \]

   2. sub-neg N/A

      \[\leadsto \left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right) \cdot \left(u0 \cdot \color{blue}{\left(u0 \cdot \left(u0 \cdot \left(\frac{-1}{4} \cdot u0 - \frac{1}{3}\right) - \frac{1}{2}\right) + \left(\mathsf{neg}\left(1\right)\right)\right)}\right) \]

   3. metadata-eval N/A

      \[\leadsto \left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right) \cdot \left(u0 \cdot \left(u0 \cdot \left(u0 \cdot \left(\frac{-1}{4} \cdot u0 - \frac{1}{3}\right) - \frac{1}{2}\right) + \color{blue}{-1}\right)\right) \]

   4. accelerator-lowering-fma.f32 N/A

      \[\leadsto \left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right) \cdot \left(u0 \cdot \color{blue}{\mathsf{fma}\left(u0, u0 \cdot \left(\frac{-1}{4} \cdot u0 - \frac{1}{3}\right) - \frac{1}{2}, -1\right)}\right) \]

   5. sub-neg N/A

      \[\leadsto \left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right) \cdot \left(u0 \cdot \mathsf{fma}\left(u0, \color{blue}{u0 \cdot \left(\frac{-1}{4} \cdot u0 - \frac{1}{3}\right) + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}, -1\right)\right) \]

   6. metadata-eval N/A

      \[\leadsto \left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right) \cdot \left(u0 \cdot \mathsf{fma}\left(u0, u0 \cdot \left(\frac{-1}{4} \cdot u0 - \frac{1}{3}\right) + \color{blue}{\frac{-1}{2}}, -1\right)\right) \]

   7. accelerator-lowering-fma.f32 N/A

      \[\leadsto \left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right) \cdot \left(u0 \cdot \mathsf{fma}\left(u0, \color{blue}{\mathsf{fma}\left(u0, \frac{-1}{4} \cdot u0 - \frac{1}{3}, \frac{-1}{2}\right)}, -1\right)\right) \]

   8. sub-neg N/A

      \[\leadsto \left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right) \cdot \left(u0 \cdot \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, \color{blue}{\frac{-1}{4} \cdot u0 + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)}, \frac{-1}{2}\right), -1\right)\right) \]

   9. *-commutative N/A

      \[\leadsto \left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right) \cdot \left(u0 \cdot \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, \color{blue}{u0 \cdot \frac{-1}{4}} + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right), \frac{-1}{2}\right), -1\right)\right) \]

   10. metadata-eval N/A

       \[\leadsto \left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right) \cdot \left(u0 \cdot \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, u0 \cdot \frac{-1}{4} + \color{blue}{\frac{-1}{3}}, \frac{-1}{2}\right), -1\right)\right) \]

   11. accelerator-lowering-fma.f32 93.5

       \[\leadsto \left(\left(-\alpha\right) \cdot \alpha\right) \cdot \left(u0 \cdot \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, \color{blue}{\mathsf{fma}\left(u0, -0.25, -0.3333333333333333\right)}, -0.5\right), -1\right)\right) \]

5. Simplified 93.5%

   \[\leadsto \left(\left(-\alpha\right) \cdot \alpha\right) \cdot \color{blue}{\left(u0 \cdot \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, -0.25, -0.3333333333333333\right), -0.5\right), -1\right)\right)} \]
6. Step-by-step derivation

   1. associate-*r* N/A

      \[\leadsto \color{blue}{\left(\left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right) \cdot u0\right) \cdot \left(u0 \cdot \left(u0 \cdot \left(u0 \cdot \frac{-1}{4} + \frac{-1}{3}\right) + \frac{-1}{2}\right) + -1\right)} \]

   2. +-commutative N/A

      \[\leadsto \left(\left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right) \cdot u0\right) \cdot \color{blue}{\left(-1 + u0 \cdot \left(u0 \cdot \left(u0 \cdot \frac{-1}{4} + \frac{-1}{3}\right) + \frac{-1}{2}\right)\right)} \]

   3. distribute-lft-in N/A

      \[\leadsto \color{blue}{\left(\left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right) \cdot u0\right) \cdot -1 + \left(\left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right) \cdot u0\right) \cdot \left(u0 \cdot \left(u0 \cdot \left(u0 \cdot \frac{-1}{4} + \frac{-1}{3}\right) + \frac{-1}{2}\right)\right)} \]

   4. accelerator-lowering-fma.f32 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right) \cdot u0, -1, \left(\left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right) \cdot u0\right) \cdot \left(u0 \cdot \left(u0 \cdot \left(u0 \cdot \frac{-1}{4} + \frac{-1}{3}\right) + \frac{-1}{2}\right)\right)\right)} \]

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

      \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\mathsf{neg}\left(\alpha \cdot \alpha\right)\right)} \cdot u0, -1, \left(\left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right) \cdot u0\right) \cdot \left(u0 \cdot \left(u0 \cdot \left(u0 \cdot \frac{-1}{4} + \frac{-1}{3}\right) + \frac{-1}{2}\right)\right)\right) \]

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

      \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{neg}\left(\left(\alpha \cdot \alpha\right) \cdot u0\right)}, -1, \left(\left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right) \cdot u0\right) \cdot \left(u0 \cdot \left(u0 \cdot \left(u0 \cdot \frac{-1}{4} + \frac{-1}{3}\right) + \frac{-1}{2}\right)\right)\right) \]

   7. *-commutative N/A

      \[\leadsto \mathsf{fma}\left(\mathsf{neg}\left(\color{blue}{u0 \cdot \left(\alpha \cdot \alpha\right)}\right), -1, \left(\left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right) \cdot u0\right) \cdot \left(u0 \cdot \left(u0 \cdot \left(u0 \cdot \frac{-1}{4} + \frac{-1}{3}\right) + \frac{-1}{2}\right)\right)\right) \]

   8. neg-lowering-neg.f32 N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{neg}\left(u0 \cdot \left(\alpha \cdot \alpha\right)\right)}, -1, \left(\left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right) \cdot u0\right) \cdot \left(u0 \cdot \left(u0 \cdot \left(u0 \cdot \frac{-1}{4} + \frac{-1}{3}\right) + \frac{-1}{2}\right)\right)\right) \]

   9. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{fma}\left(\mathsf{neg}\left(\color{blue}{u0 \cdot \left(\alpha \cdot \alpha\right)}\right), -1, \left(\left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right) \cdot u0\right) \cdot \left(u0 \cdot \left(u0 \cdot \left(u0 \cdot \frac{-1}{4} + \frac{-1}{3}\right) + \frac{-1}{2}\right)\right)\right) \]

   10. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{neg}\left(u0 \cdot \color{blue}{\left(\alpha \cdot \alpha\right)}\right), -1, \left(\left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right) \cdot u0\right) \cdot \left(u0 \cdot \left(u0 \cdot \left(u0 \cdot \frac{-1}{4} + \frac{-1}{3}\right) + \frac{-1}{2}\right)\right)\right) \]

   11. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{neg}\left(u0 \cdot \left(\alpha \cdot \alpha\right)\right), -1, \color{blue}{\left(\left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right) \cdot u0\right) \cdot \left(u0 \cdot \left(u0 \cdot \left(u0 \cdot \frac{-1}{4} + \frac{-1}{3}\right) + \frac{-1}{2}\right)\right)}\right) \]

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

       \[\leadsto \mathsf{fma}\left(\mathsf{neg}\left(u0 \cdot \left(\alpha \cdot \alpha\right)\right), -1, \left(\color{blue}{\left(\mathsf{neg}\left(\alpha \cdot \alpha\right)\right)} \cdot u0\right) \cdot \left(u0 \cdot \left(u0 \cdot \left(u0 \cdot \frac{-1}{4} + \frac{-1}{3}\right) + \frac{-1}{2}\right)\right)\right) \]

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

       \[\leadsto \mathsf{fma}\left(\mathsf{neg}\left(u0 \cdot \left(\alpha \cdot \alpha\right)\right), -1, \color{blue}{\left(\mathsf{neg}\left(\left(\alpha \cdot \alpha\right) \cdot u0\right)\right)} \cdot \left(u0 \cdot \left(u0 \cdot \left(u0 \cdot \frac{-1}{4} + \frac{-1}{3}\right) + \frac{-1}{2}\right)\right)\right) \]

   14. *-commutative N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{neg}\left(u0 \cdot \left(\alpha \cdot \alpha\right)\right), -1, \left(\mathsf{neg}\left(\color{blue}{u0 \cdot \left(\alpha \cdot \alpha\right)}\right)\right) \cdot \left(u0 \cdot \left(u0 \cdot \left(u0 \cdot \frac{-1}{4} + \frac{-1}{3}\right) + \frac{-1}{2}\right)\right)\right) \]

   15. neg-lowering-neg.f32 N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{neg}\left(u0 \cdot \left(\alpha \cdot \alpha\right)\right), -1, \color{blue}{\left(\mathsf{neg}\left(u0 \cdot \left(\alpha \cdot \alpha\right)\right)\right)} \cdot \left(u0 \cdot \left(u0 \cdot \left(u0 \cdot \frac{-1}{4} + \frac{-1}{3}\right) + \frac{-1}{2}\right)\right)\right) \]

   16. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{neg}\left(u0 \cdot \left(\alpha \cdot \alpha\right)\right), -1, \left(\mathsf{neg}\left(\color{blue}{u0 \cdot \left(\alpha \cdot \alpha\right)}\right)\right) \cdot \left(u0 \cdot \left(u0 \cdot \left(u0 \cdot \frac{-1}{4} + \frac{-1}{3}\right) + \frac{-1}{2}\right)\right)\right) \]

   17. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{neg}\left(u0 \cdot \left(\alpha \cdot \alpha\right)\right), -1, \left(\mathsf{neg}\left(u0 \cdot \color{blue}{\left(\alpha \cdot \alpha\right)}\right)\right) \cdot \left(u0 \cdot \left(u0 \cdot \left(u0 \cdot \frac{-1}{4} + \frac{-1}{3}\right) + \frac{-1}{2}\right)\right)\right) \]

7. Applied egg-rr 93.8%

   \[\leadsto \color{blue}{\mathsf{fma}\left(-u0 \cdot \left(\alpha \cdot \alpha\right), -1, \left(-u0 \cdot \left(\alpha \cdot \alpha\right)\right) \cdot \left(u0 \cdot \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, -0.25, -0.3333333333333333\right), -0.5\right)\right)\right)} \]

8. Final simplification 93.8%

   \[\leadsto \mathsf{fma}\left(\left(\alpha \cdot \left(-\alpha\right)\right) \cdot u0, -1, \left(u0 \cdot \left(\alpha \cdot \alpha\right)\right) \cdot \left(\left(-u0\right) \cdot \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, -0.25, -0.3333333333333333\right), -0.5\right)\right)\right) \]
9. Add Preprocessing

Alternative 5: 93.5% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\mathsf{fma}\left(u0, \mathsf{fma}\left(u0, -0.25, -0.3333333333333333\right), -0.5\right) \cdot \left(u0 \cdot u0\right), \alpha \cdot \left(-\alpha\right), u0 \cdot \left(\alpha \cdot \alpha\right)\right) \end{array} \]

FPCore

(FPCore (alpha u0)
 :precision binary32
 (fma
  (* (fma u0 (fma u0 -0.25 -0.3333333333333333) -0.5) (* u0 u0))
  (* alpha (- alpha))
  (* u0 (* alpha alpha))))

C

float code(float alpha, float u0) {
	return fmaf((fmaf(u0, fmaf(u0, -0.25f, -0.3333333333333333f), -0.5f) * (u0 * u0)), (alpha * -alpha), (u0 * (alpha * alpha)));
}

Julia

function code(alpha, u0)
	return fma(Float32(fma(u0, fma(u0, Float32(-0.25), Float32(-0.3333333333333333)), Float32(-0.5)) * Float32(u0 * u0)), Float32(alpha * Float32(-alpha)), Float32(u0 * Float32(alpha * alpha)))
end

Derivation

1. Initial program 57.3%

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

3. Taylor expanded in u0 around 0

   \[\leadsto \left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right) \cdot \color{blue}{\left(u0 \cdot \left(u0 \cdot \left(u0 \cdot \left(\frac{-1}{4} \cdot u0 - \frac{1}{3}\right) - \frac{1}{2}\right) - 1\right)\right)} \]
4. Step-by-step derivation

   1. *-lowering-*.f32 N/A

      \[\leadsto \left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right) \cdot \color{blue}{\left(u0 \cdot \left(u0 \cdot \left(u0 \cdot \left(\frac{-1}{4} \cdot u0 - \frac{1}{3}\right) - \frac{1}{2}\right) - 1\right)\right)} \]

   2. sub-neg N/A

      \[\leadsto \left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right) \cdot \left(u0 \cdot \color{blue}{\left(u0 \cdot \left(u0 \cdot \left(\frac{-1}{4} \cdot u0 - \frac{1}{3}\right) - \frac{1}{2}\right) + \left(\mathsf{neg}\left(1\right)\right)\right)}\right) \]

   3. metadata-eval N/A

      \[\leadsto \left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right) \cdot \left(u0 \cdot \left(u0 \cdot \left(u0 \cdot \left(\frac{-1}{4} \cdot u0 - \frac{1}{3}\right) - \frac{1}{2}\right) + \color{blue}{-1}\right)\right) \]

   4. accelerator-lowering-fma.f32 N/A

      \[\leadsto \left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right) \cdot \left(u0 \cdot \color{blue}{\mathsf{fma}\left(u0, u0 \cdot \left(\frac{-1}{4} \cdot u0 - \frac{1}{3}\right) - \frac{1}{2}, -1\right)}\right) \]

   5. sub-neg N/A

      \[\leadsto \left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right) \cdot \left(u0 \cdot \mathsf{fma}\left(u0, \color{blue}{u0 \cdot \left(\frac{-1}{4} \cdot u0 - \frac{1}{3}\right) + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}, -1\right)\right) \]

   6. metadata-eval N/A

      \[\leadsto \left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right) \cdot \left(u0 \cdot \mathsf{fma}\left(u0, u0 \cdot \left(\frac{-1}{4} \cdot u0 - \frac{1}{3}\right) + \color{blue}{\frac{-1}{2}}, -1\right)\right) \]

   7. accelerator-lowering-fma.f32 N/A

      \[\leadsto \left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right) \cdot \left(u0 \cdot \mathsf{fma}\left(u0, \color{blue}{\mathsf{fma}\left(u0, \frac{-1}{4} \cdot u0 - \frac{1}{3}, \frac{-1}{2}\right)}, -1\right)\right) \]

   8. sub-neg N/A

      \[\leadsto \left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right) \cdot \left(u0 \cdot \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, \color{blue}{\frac{-1}{4} \cdot u0 + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)}, \frac{-1}{2}\right), -1\right)\right) \]

   9. *-commutative N/A

      \[\leadsto \left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right) \cdot \left(u0 \cdot \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, \color{blue}{u0 \cdot \frac{-1}{4}} + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right), \frac{-1}{2}\right), -1\right)\right) \]

   10. metadata-eval N/A

       \[\leadsto \left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right) \cdot \left(u0 \cdot \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, u0 \cdot \frac{-1}{4} + \color{blue}{\frac{-1}{3}}, \frac{-1}{2}\right), -1\right)\right) \]

   11. accelerator-lowering-fma.f32 93.5

       \[\leadsto \left(\left(-\alpha\right) \cdot \alpha\right) \cdot \left(u0 \cdot \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, \color{blue}{\mathsf{fma}\left(u0, -0.25, -0.3333333333333333\right)}, -0.5\right), -1\right)\right) \]

5. Simplified 93.5%

   \[\leadsto \left(\left(-\alpha\right) \cdot \alpha\right) \cdot \color{blue}{\left(u0 \cdot \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, -0.25, -0.3333333333333333\right), -0.5\right), -1\right)\right)} \]
6. Step-by-step derivation

   1. distribute-rgt-in N/A

      \[\leadsto \left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right) \cdot \color{blue}{\left(\left(u0 \cdot \left(u0 \cdot \left(u0 \cdot \frac{-1}{4} + \frac{-1}{3}\right) + \frac{-1}{2}\right)\right) \cdot u0 + -1 \cdot u0\right)} \]

   2. neg-mul-1 N/A

      \[\leadsto \left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right) \cdot \left(\left(u0 \cdot \left(u0 \cdot \left(u0 \cdot \frac{-1}{4} + \frac{-1}{3}\right) + \frac{-1}{2}\right)\right) \cdot u0 + \color{blue}{\left(\mathsf{neg}\left(u0\right)\right)}\right) \]

   3. distribute-rgt-in N/A

      \[\leadsto \color{blue}{\left(\left(u0 \cdot \left(u0 \cdot \left(u0 \cdot \frac{-1}{4} + \frac{-1}{3}\right) + \frac{-1}{2}\right)\right) \cdot u0\right) \cdot \left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right) + \left(\mathsf{neg}\left(u0\right)\right) \cdot \left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right)} \]

   4. accelerator-lowering-fma.f32 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(u0 \cdot \left(u0 \cdot \left(u0 \cdot \frac{-1}{4} + \frac{-1}{3}\right) + \frac{-1}{2}\right)\right) \cdot u0, \left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha, \left(\mathsf{neg}\left(u0\right)\right) \cdot \left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right)\right)} \]

   5. *-commutative N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\left(u0 \cdot \left(u0 \cdot \frac{-1}{4} + \frac{-1}{3}\right) + \frac{-1}{2}\right) \cdot u0\right)} \cdot u0, \left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha, \left(\mathsf{neg}\left(u0\right)\right) \cdot \left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right)\right) \]

   6. associate-*l* N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{\left(u0 \cdot \left(u0 \cdot \frac{-1}{4} + \frac{-1}{3}\right) + \frac{-1}{2}\right) \cdot \left(u0 \cdot u0\right)}, \left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha, \left(\mathsf{neg}\left(u0\right)\right) \cdot \left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right)\right) \]

   7. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{\left(u0 \cdot \left(u0 \cdot \frac{-1}{4} + \frac{-1}{3}\right) + \frac{-1}{2}\right) \cdot \left(u0 \cdot u0\right)}, \left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha, \left(\mathsf{neg}\left(u0\right)\right) \cdot \left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right)\right) \]

   8. accelerator-lowering-fma.f32 N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(u0, u0 \cdot \frac{-1}{4} + \frac{-1}{3}, \frac{-1}{2}\right)} \cdot \left(u0 \cdot u0\right), \left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha, \left(\mathsf{neg}\left(u0\right)\right) \cdot \left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right)\right) \]

   9. accelerator-lowering-fma.f32 N/A

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(u0, \color{blue}{\mathsf{fma}\left(u0, \frac{-1}{4}, \frac{-1}{3}\right)}, \frac{-1}{2}\right) \cdot \left(u0 \cdot u0\right), \left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha, \left(\mathsf{neg}\left(u0\right)\right) \cdot \left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right)\right) \]

   10. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(u0, \mathsf{fma}\left(u0, \frac{-1}{4}, \frac{-1}{3}\right), \frac{-1}{2}\right) \cdot \color{blue}{\left(u0 \cdot u0\right)}, \left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha, \left(\mathsf{neg}\left(u0\right)\right) \cdot \left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right)\right) \]

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

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(u0, \mathsf{fma}\left(u0, \frac{-1}{4}, \frac{-1}{3}\right), \frac{-1}{2}\right) \cdot \left(u0 \cdot u0\right), \color{blue}{\mathsf{neg}\left(\alpha \cdot \alpha\right)}, \left(\mathsf{neg}\left(u0\right)\right) \cdot \left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right)\right) \]

   12. neg-lowering-neg.f32 N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(u0, \mathsf{fma}\left(u0, \frac{-1}{4}, \frac{-1}{3}\right), \frac{-1}{2}\right) \cdot \left(u0 \cdot u0\right), \color{blue}{\mathsf{neg}\left(\alpha \cdot \alpha\right)}, \left(\mathsf{neg}\left(u0\right)\right) \cdot \left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right)\right) \]

   13. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(u0, \mathsf{fma}\left(u0, \frac{-1}{4}, \frac{-1}{3}\right), \frac{-1}{2}\right) \cdot \left(u0 \cdot u0\right), \mathsf{neg}\left(\color{blue}{\alpha \cdot \alpha}\right), \left(\mathsf{neg}\left(u0\right)\right) \cdot \left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right)\right) \]

   14. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(u0, \mathsf{fma}\left(u0, \frac{-1}{4}, \frac{-1}{3}\right), \frac{-1}{2}\right) \cdot \left(u0 \cdot u0\right), \mathsf{neg}\left(\alpha \cdot \alpha\right), \color{blue}{\left(\mathsf{neg}\left(u0\right)\right) \cdot \left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right)}\right) \]

   15. neg-lowering-neg.f32 N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(u0, \mathsf{fma}\left(u0, \frac{-1}{4}, \frac{-1}{3}\right), \frac{-1}{2}\right) \cdot \left(u0 \cdot u0\right), \mathsf{neg}\left(\alpha \cdot \alpha\right), \color{blue}{\left(\mathsf{neg}\left(u0\right)\right)} \cdot \left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right)\right) \]

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

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(u0, \mathsf{fma}\left(u0, \frac{-1}{4}, \frac{-1}{3}\right), \frac{-1}{2}\right) \cdot \left(u0 \cdot u0\right), \mathsf{neg}\left(\alpha \cdot \alpha\right), \left(\mathsf{neg}\left(u0\right)\right) \cdot \color{blue}{\left(\mathsf{neg}\left(\alpha \cdot \alpha\right)\right)}\right) \]

   17. neg-lowering-neg.f32 N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(u0, \mathsf{fma}\left(u0, \frac{-1}{4}, \frac{-1}{3}\right), \frac{-1}{2}\right) \cdot \left(u0 \cdot u0\right), \mathsf{neg}\left(\alpha \cdot \alpha\right), \left(\mathsf{neg}\left(u0\right)\right) \cdot \color{blue}{\left(\mathsf{neg}\left(\alpha \cdot \alpha\right)\right)}\right) \]

   18. *-lowering-*.f32 93.8

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(u0, \mathsf{fma}\left(u0, -0.25, -0.3333333333333333\right), -0.5\right) \cdot \left(u0 \cdot u0\right), -\alpha \cdot \alpha, \left(-u0\right) \cdot \left(-\color{blue}{\alpha \cdot \alpha}\right)\right) \]

7. Applied egg-rr 93.8%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(u0, \mathsf{fma}\left(u0, -0.25, -0.3333333333333333\right), -0.5\right) \cdot \left(u0 \cdot u0\right), -\alpha \cdot \alpha, \left(-u0\right) \cdot \left(-\alpha \cdot \alpha\right)\right)} \]

8. Final simplification 93.8%

   \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(u0, \mathsf{fma}\left(u0, -0.25, -0.3333333333333333\right), -0.5\right) \cdot \left(u0 \cdot u0\right), \alpha \cdot \left(-\alpha\right), u0 \cdot \left(\alpha \cdot \alpha\right)\right) \]
9. Add Preprocessing

Alternative 6: 93.6% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ u0 \cdot \mathsf{fma}\left(u0, \left(\alpha \cdot \alpha\right) \cdot \mathsf{fma}\left(u0, u0 \cdot 0.25, \mathsf{fma}\left(u0, 0.3333333333333333, 0.5\right)\right), \alpha \cdot \alpha\right) \end{array} \]

FPCore

(FPCore (alpha u0)
 :precision binary32
 (*
  u0
  (fma
   u0
   (* (* alpha alpha) (fma u0 (* u0 0.25) (fma u0 0.3333333333333333 0.5)))
   (* alpha alpha))))

C

float code(float alpha, float u0) {
	return u0 * fmaf(u0, ((alpha * alpha) * fmaf(u0, (u0 * 0.25f), fmaf(u0, 0.3333333333333333f, 0.5f))), (alpha * alpha));
}

Julia

function code(alpha, u0)
	return Float32(u0 * fma(u0, Float32(Float32(alpha * alpha) * fma(u0, Float32(u0 * Float32(0.25)), fma(u0, Float32(0.3333333333333333), Float32(0.5)))), Float32(alpha * alpha)))
end

Derivation

1. Initial program 57.3%

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

3. Taylor expanded in u0 around 0

   \[\leadsto \color{blue}{u0 \cdot \left(u0 \cdot \left(\frac{1}{2} \cdot {\alpha}^{2} + u0 \cdot \left(\frac{1}{4} \cdot \left({\alpha}^{2} \cdot u0\right) + \frac{1}{3} \cdot {\alpha}^{2}\right)\right) + {\alpha}^{2}\right)} \]
4. Step-by-step derivation

   1. *-lowering-*.f32 N/A

      \[\leadsto \color{blue}{u0 \cdot \left(u0 \cdot \left(\frac{1}{2} \cdot {\alpha}^{2} + u0 \cdot \left(\frac{1}{4} \cdot \left({\alpha}^{2} \cdot u0\right) + \frac{1}{3} \cdot {\alpha}^{2}\right)\right) + {\alpha}^{2}\right)} \]

   2. accelerator-lowering-fma.f32 N/A

      \[\leadsto u0 \cdot \color{blue}{\mathsf{fma}\left(u0, \frac{1}{2} \cdot {\alpha}^{2} + u0 \cdot \left(\frac{1}{4} \cdot \left({\alpha}^{2} \cdot u0\right) + \frac{1}{3} \cdot {\alpha}^{2}\right), {\alpha}^{2}\right)} \]

5. Simplified 93.7%

   \[\leadsto \color{blue}{u0 \cdot \mathsf{fma}\left(u0, \left(\alpha \cdot \alpha\right) \cdot \mathsf{fma}\left(u0, u0 \cdot 0.25, \mathsf{fma}\left(u0, 0.3333333333333333, 0.5\right)\right), \alpha \cdot \alpha\right)} \]
6. Add Preprocessing

Alternative 7: 93.5% accurate, 3.1× speedup

Math

\[\begin{array}{l} \\ \left(\alpha \cdot \left(-\alpha\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(u0, \mathsf{fma}\left(u0, -0.25, -0.3333333333333333\right), -0.5\right), u0 \cdot u0, -u0\right) \end{array} \]

FPCore

(FPCore (alpha u0)
 :precision binary32
 (*
  (* alpha (- alpha))
  (fma (fma u0 (fma u0 -0.25 -0.3333333333333333) -0.5) (* u0 u0) (- u0))))

C

float code(float alpha, float u0) {
	return (alpha * -alpha) * fmaf(fmaf(u0, fmaf(u0, -0.25f, -0.3333333333333333f), -0.5f), (u0 * u0), -u0);
}

Julia

function code(alpha, u0)
	return Float32(Float32(alpha * Float32(-alpha)) * fma(fma(u0, fma(u0, Float32(-0.25), Float32(-0.3333333333333333)), Float32(-0.5)), Float32(u0 * u0), Float32(-u0)))
end

Derivation

1. Initial program 57.3%

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

3. Taylor expanded in u0 around 0

   \[\leadsto \left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right) \cdot \color{blue}{\left(u0 \cdot \left(u0 \cdot \left(u0 \cdot \left(\frac{-1}{4} \cdot u0 - \frac{1}{3}\right) - \frac{1}{2}\right) - 1\right)\right)} \]
4. Step-by-step derivation

   1. *-lowering-*.f32 N/A

      \[\leadsto \left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right) \cdot \color{blue}{\left(u0 \cdot \left(u0 \cdot \left(u0 \cdot \left(\frac{-1}{4} \cdot u0 - \frac{1}{3}\right) - \frac{1}{2}\right) - 1\right)\right)} \]

   2. sub-neg N/A

      \[\leadsto \left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right) \cdot \left(u0 \cdot \color{blue}{\left(u0 \cdot \left(u0 \cdot \left(\frac{-1}{4} \cdot u0 - \frac{1}{3}\right) - \frac{1}{2}\right) + \left(\mathsf{neg}\left(1\right)\right)\right)}\right) \]

   3. metadata-eval N/A

      \[\leadsto \left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right) \cdot \left(u0 \cdot \left(u0 \cdot \left(u0 \cdot \left(\frac{-1}{4} \cdot u0 - \frac{1}{3}\right) - \frac{1}{2}\right) + \color{blue}{-1}\right)\right) \]

   4. accelerator-lowering-fma.f32 N/A

      \[\leadsto \left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right) \cdot \left(u0 \cdot \color{blue}{\mathsf{fma}\left(u0, u0 \cdot \left(\frac{-1}{4} \cdot u0 - \frac{1}{3}\right) - \frac{1}{2}, -1\right)}\right) \]

   5. sub-neg N/A

      \[\leadsto \left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right) \cdot \left(u0 \cdot \mathsf{fma}\left(u0, \color{blue}{u0 \cdot \left(\frac{-1}{4} \cdot u0 - \frac{1}{3}\right) + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}, -1\right)\right) \]

   6. metadata-eval N/A

      \[\leadsto \left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right) \cdot \left(u0 \cdot \mathsf{fma}\left(u0, u0 \cdot \left(\frac{-1}{4} \cdot u0 - \frac{1}{3}\right) + \color{blue}{\frac{-1}{2}}, -1\right)\right) \]

   7. accelerator-lowering-fma.f32 N/A

      \[\leadsto \left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right) \cdot \left(u0 \cdot \mathsf{fma}\left(u0, \color{blue}{\mathsf{fma}\left(u0, \frac{-1}{4} \cdot u0 - \frac{1}{3}, \frac{-1}{2}\right)}, -1\right)\right) \]

   8. sub-neg N/A

      \[\leadsto \left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right) \cdot \left(u0 \cdot \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, \color{blue}{\frac{-1}{4} \cdot u0 + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)}, \frac{-1}{2}\right), -1\right)\right) \]

   9. *-commutative N/A

      \[\leadsto \left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right) \cdot \left(u0 \cdot \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, \color{blue}{u0 \cdot \frac{-1}{4}} + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right), \frac{-1}{2}\right), -1\right)\right) \]

   10. metadata-eval N/A

       \[\leadsto \left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right) \cdot \left(u0 \cdot \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, u0 \cdot \frac{-1}{4} + \color{blue}{\frac{-1}{3}}, \frac{-1}{2}\right), -1\right)\right) \]

   11. accelerator-lowering-fma.f32 93.5

       \[\leadsto \left(\left(-\alpha\right) \cdot \alpha\right) \cdot \left(u0 \cdot \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, \color{blue}{\mathsf{fma}\left(u0, -0.25, -0.3333333333333333\right)}, -0.5\right), -1\right)\right) \]

5. Simplified 93.5%

   \[\leadsto \left(\left(-\alpha\right) \cdot \alpha\right) \cdot \color{blue}{\left(u0 \cdot \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, -0.25, -0.3333333333333333\right), -0.5\right), -1\right)\right)} \]
6. Step-by-step derivation

   1. distribute-rgt-in N/A

      \[\leadsto \left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right) \cdot \color{blue}{\left(\left(u0 \cdot \left(u0 \cdot \left(u0 \cdot \frac{-1}{4} + \frac{-1}{3}\right) + \frac{-1}{2}\right)\right) \cdot u0 + -1 \cdot u0\right)} \]

   2. neg-mul-1 N/A

      \[\leadsto \left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right) \cdot \left(\left(u0 \cdot \left(u0 \cdot \left(u0 \cdot \frac{-1}{4} + \frac{-1}{3}\right) + \frac{-1}{2}\right)\right) \cdot u0 + \color{blue}{\left(\mathsf{neg}\left(u0\right)\right)}\right) \]

   3. *-commutative N/A

      \[\leadsto \left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right) \cdot \left(\color{blue}{\left(\left(u0 \cdot \left(u0 \cdot \frac{-1}{4} + \frac{-1}{3}\right) + \frac{-1}{2}\right) \cdot u0\right)} \cdot u0 + \left(\mathsf{neg}\left(u0\right)\right)\right) \]

   4. associate-*l* N/A

      \[\leadsto \left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right) \cdot \left(\color{blue}{\left(u0 \cdot \left(u0 \cdot \frac{-1}{4} + \frac{-1}{3}\right) + \frac{-1}{2}\right) \cdot \left(u0 \cdot u0\right)} + \left(\mathsf{neg}\left(u0\right)\right)\right) \]

   5. accelerator-lowering-fma.f32 N/A

      \[\leadsto \left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right) \cdot \color{blue}{\mathsf{fma}\left(u0 \cdot \left(u0 \cdot \frac{-1}{4} + \frac{-1}{3}\right) + \frac{-1}{2}, u0 \cdot u0, \mathsf{neg}\left(u0\right)\right)} \]

   6. accelerator-lowering-fma.f32 N/A

      \[\leadsto \left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right) \cdot \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(u0, u0 \cdot \frac{-1}{4} + \frac{-1}{3}, \frac{-1}{2}\right)}, u0 \cdot u0, \mathsf{neg}\left(u0\right)\right) \]

   7. accelerator-lowering-fma.f32 N/A

      \[\leadsto \left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(u0, \color{blue}{\mathsf{fma}\left(u0, \frac{-1}{4}, \frac{-1}{3}\right)}, \frac{-1}{2}\right), u0 \cdot u0, \mathsf{neg}\left(u0\right)\right) \]

   8. *-lowering-*.f32 N/A

      \[\leadsto \left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(u0, \mathsf{fma}\left(u0, \frac{-1}{4}, \frac{-1}{3}\right), \frac{-1}{2}\right), \color{blue}{u0 \cdot u0}, \mathsf{neg}\left(u0\right)\right) \]

   9. neg-lowering-neg.f32 93.7

      \[\leadsto \left(\left(-\alpha\right) \cdot \alpha\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(u0, \mathsf{fma}\left(u0, -0.25, -0.3333333333333333\right), -0.5\right), u0 \cdot u0, \color{blue}{-u0}\right) \]

7. Applied egg-rr 93.7%

   \[\leadsto \left(\left(-\alpha\right) \cdot \alpha\right) \cdot \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(u0, \mathsf{fma}\left(u0, -0.25, -0.3333333333333333\right), -0.5\right), u0 \cdot u0, -u0\right)} \]

8. Final simplification 93.7%

   \[\leadsto \left(\alpha \cdot \left(-\alpha\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(u0, \mathsf{fma}\left(u0, -0.25, -0.3333333333333333\right), -0.5\right), u0 \cdot u0, -u0\right) \]
9. Add Preprocessing

Alternative 8: 93.5% accurate, 3.1× speedup

Math

\[\begin{array}{l} \\ \left(\alpha \cdot \left(-\alpha\right)\right) \cdot \mathsf{fma}\left(u0 \cdot \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, -0.25, -0.3333333333333333\right), -0.5\right), u0, -u0\right) \end{array} \]

FPCore

(FPCore (alpha u0)
 :precision binary32
 (*
  (* alpha (- alpha))
  (fma (* u0 (fma u0 (fma u0 -0.25 -0.3333333333333333) -0.5)) u0 (- u0))))

C

float code(float alpha, float u0) {
	return (alpha * -alpha) * fmaf((u0 * fmaf(u0, fmaf(u0, -0.25f, -0.3333333333333333f), -0.5f)), u0, -u0);
}

Julia

function code(alpha, u0)
	return Float32(Float32(alpha * Float32(-alpha)) * fma(Float32(u0 * fma(u0, fma(u0, Float32(-0.25), Float32(-0.3333333333333333)), Float32(-0.5))), u0, Float32(-u0)))
end

Derivation

1. Initial program 57.3%

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

3. Taylor expanded in u0 around 0

   \[\leadsto \left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right) \cdot \color{blue}{\left(u0 \cdot \left(u0 \cdot \left(u0 \cdot \left(\frac{-1}{4} \cdot u0 - \frac{1}{3}\right) - \frac{1}{2}\right) - 1\right)\right)} \]
4. Step-by-step derivation

   1. *-lowering-*.f32 N/A

      \[\leadsto \left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right) \cdot \color{blue}{\left(u0 \cdot \left(u0 \cdot \left(u0 \cdot \left(\frac{-1}{4} \cdot u0 - \frac{1}{3}\right) - \frac{1}{2}\right) - 1\right)\right)} \]

   2. sub-neg N/A

      \[\leadsto \left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right) \cdot \left(u0 \cdot \color{blue}{\left(u0 \cdot \left(u0 \cdot \left(\frac{-1}{4} \cdot u0 - \frac{1}{3}\right) - \frac{1}{2}\right) + \left(\mathsf{neg}\left(1\right)\right)\right)}\right) \]

   3. metadata-eval N/A

      \[\leadsto \left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right) \cdot \left(u0 \cdot \left(u0 \cdot \left(u0 \cdot \left(\frac{-1}{4} \cdot u0 - \frac{1}{3}\right) - \frac{1}{2}\right) + \color{blue}{-1}\right)\right) \]

   4. accelerator-lowering-fma.f32 N/A

      \[\leadsto \left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right) \cdot \left(u0 \cdot \color{blue}{\mathsf{fma}\left(u0, u0 \cdot \left(\frac{-1}{4} \cdot u0 - \frac{1}{3}\right) - \frac{1}{2}, -1\right)}\right) \]

   5. sub-neg N/A

      \[\leadsto \left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right) \cdot \left(u0 \cdot \mathsf{fma}\left(u0, \color{blue}{u0 \cdot \left(\frac{-1}{4} \cdot u0 - \frac{1}{3}\right) + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}, -1\right)\right) \]

   6. metadata-eval N/A

      \[\leadsto \left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right) \cdot \left(u0 \cdot \mathsf{fma}\left(u0, u0 \cdot \left(\frac{-1}{4} \cdot u0 - \frac{1}{3}\right) + \color{blue}{\frac{-1}{2}}, -1\right)\right) \]

   7. accelerator-lowering-fma.f32 N/A

      \[\leadsto \left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right) \cdot \left(u0 \cdot \mathsf{fma}\left(u0, \color{blue}{\mathsf{fma}\left(u0, \frac{-1}{4} \cdot u0 - \frac{1}{3}, \frac{-1}{2}\right)}, -1\right)\right) \]

   8. sub-neg N/A

      \[\leadsto \left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right) \cdot \left(u0 \cdot \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, \color{blue}{\frac{-1}{4} \cdot u0 + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)}, \frac{-1}{2}\right), -1\right)\right) \]

   9. *-commutative N/A

      \[\leadsto \left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right) \cdot \left(u0 \cdot \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, \color{blue}{u0 \cdot \frac{-1}{4}} + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right), \frac{-1}{2}\right), -1\right)\right) \]

   10. metadata-eval N/A

       \[\leadsto \left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right) \cdot \left(u0 \cdot \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, u0 \cdot \frac{-1}{4} + \color{blue}{\frac{-1}{3}}, \frac{-1}{2}\right), -1\right)\right) \]

   11. accelerator-lowering-fma.f32 93.5

       \[\leadsto \left(\left(-\alpha\right) \cdot \alpha\right) \cdot \left(u0 \cdot \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, \color{blue}{\mathsf{fma}\left(u0, -0.25, -0.3333333333333333\right)}, -0.5\right), -1\right)\right) \]

5. Simplified 93.5%

   \[\leadsto \left(\left(-\alpha\right) \cdot \alpha\right) \cdot \color{blue}{\left(u0 \cdot \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, -0.25, -0.3333333333333333\right), -0.5\right), -1\right)\right)} \]
6. Step-by-step derivation

   1. distribute-rgt-in N/A

      \[\leadsto \left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right) \cdot \color{blue}{\left(\left(u0 \cdot \left(u0 \cdot \left(u0 \cdot \frac{-1}{4} + \frac{-1}{3}\right) + \frac{-1}{2}\right)\right) \cdot u0 + -1 \cdot u0\right)} \]

   2. neg-mul-1 N/A

      \[\leadsto \left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right) \cdot \left(\left(u0 \cdot \left(u0 \cdot \left(u0 \cdot \frac{-1}{4} + \frac{-1}{3}\right) + \frac{-1}{2}\right)\right) \cdot u0 + \color{blue}{\left(\mathsf{neg}\left(u0\right)\right)}\right) \]

   3. accelerator-lowering-fma.f32 N/A

      \[\leadsto \left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right) \cdot \color{blue}{\mathsf{fma}\left(u0 \cdot \left(u0 \cdot \left(u0 \cdot \frac{-1}{4} + \frac{-1}{3}\right) + \frac{-1}{2}\right), u0, \mathsf{neg}\left(u0\right)\right)} \]

   4. *-lowering-*.f32 N/A

      \[\leadsto \left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right) \cdot \mathsf{fma}\left(\color{blue}{u0 \cdot \left(u0 \cdot \left(u0 \cdot \frac{-1}{4} + \frac{-1}{3}\right) + \frac{-1}{2}\right)}, u0, \mathsf{neg}\left(u0\right)\right) \]

   5. accelerator-lowering-fma.f32 N/A

      \[\leadsto \left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right) \cdot \mathsf{fma}\left(u0 \cdot \color{blue}{\mathsf{fma}\left(u0, u0 \cdot \frac{-1}{4} + \frac{-1}{3}, \frac{-1}{2}\right)}, u0, \mathsf{neg}\left(u0\right)\right) \]

   6. accelerator-lowering-fma.f32 N/A

      \[\leadsto \left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right) \cdot \mathsf{fma}\left(u0 \cdot \mathsf{fma}\left(u0, \color{blue}{\mathsf{fma}\left(u0, \frac{-1}{4}, \frac{-1}{3}\right)}, \frac{-1}{2}\right), u0, \mathsf{neg}\left(u0\right)\right) \]

   7. neg-lowering-neg.f32 93.7

      \[\leadsto \left(\left(-\alpha\right) \cdot \alpha\right) \cdot \mathsf{fma}\left(u0 \cdot \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, -0.25, -0.3333333333333333\right), -0.5\right), u0, \color{blue}{-u0}\right) \]

7. Applied egg-rr 93.7%

   \[\leadsto \left(\left(-\alpha\right) \cdot \alpha\right) \cdot \color{blue}{\mathsf{fma}\left(u0 \cdot \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, -0.25, -0.3333333333333333\right), -0.5\right), u0, -u0\right)} \]

8. Final simplification 93.7%

   \[\leadsto \left(\alpha \cdot \left(-\alpha\right)\right) \cdot \mathsf{fma}\left(u0 \cdot \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, -0.25, -0.3333333333333333\right), -0.5\right), u0, -u0\right) \]
9. Add Preprocessing

Alternative 9: 93.5% accurate, 3.2× speedup

Math

\[\begin{array}{l} \\ \left(\alpha \cdot \alpha\right) \cdot \left(u0 - u0 \cdot \left(u0 \cdot \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, -0.25, -0.3333333333333333\right), -0.5\right)\right)\right) \end{array} \]

FPCore

(FPCore (alpha u0)
 :precision binary32
 (*
  (* alpha alpha)
  (- u0 (* u0 (* u0 (fma u0 (fma u0 -0.25 -0.3333333333333333) -0.5))))))

C

float code(float alpha, float u0) {
	return (alpha * alpha) * (u0 - (u0 * (u0 * fmaf(u0, fmaf(u0, -0.25f, -0.3333333333333333f), -0.5f))));
}

Julia

function code(alpha, u0)
	return Float32(Float32(alpha * alpha) * Float32(u0 - Float32(u0 * Float32(u0 * fma(u0, fma(u0, Float32(-0.25), Float32(-0.3333333333333333)), Float32(-0.5))))))
end

Derivation

1. Initial program 57.3%

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

3. Taylor expanded in u0 around 0

   \[\leadsto \left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right) \cdot \color{blue}{\left(u0 \cdot \left(u0 \cdot \left(u0 \cdot \left(\frac{-1}{4} \cdot u0 - \frac{1}{3}\right) - \frac{1}{2}\right) - 1\right)\right)} \]
4. Step-by-step derivation

   1. *-lowering-*.f32 N/A

      \[\leadsto \left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right) \cdot \color{blue}{\left(u0 \cdot \left(u0 \cdot \left(u0 \cdot \left(\frac{-1}{4} \cdot u0 - \frac{1}{3}\right) - \frac{1}{2}\right) - 1\right)\right)} \]

   2. sub-neg N/A

      \[\leadsto \left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right) \cdot \left(u0 \cdot \color{blue}{\left(u0 \cdot \left(u0 \cdot \left(\frac{-1}{4} \cdot u0 - \frac{1}{3}\right) - \frac{1}{2}\right) + \left(\mathsf{neg}\left(1\right)\right)\right)}\right) \]

   3. metadata-eval N/A

      \[\leadsto \left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right) \cdot \left(u0 \cdot \left(u0 \cdot \left(u0 \cdot \left(\frac{-1}{4} \cdot u0 - \frac{1}{3}\right) - \frac{1}{2}\right) + \color{blue}{-1}\right)\right) \]

   4. accelerator-lowering-fma.f32 N/A

      \[\leadsto \left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right) \cdot \left(u0 \cdot \color{blue}{\mathsf{fma}\left(u0, u0 \cdot \left(\frac{-1}{4} \cdot u0 - \frac{1}{3}\right) - \frac{1}{2}, -1\right)}\right) \]

   5. sub-neg N/A

      \[\leadsto \left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right) \cdot \left(u0 \cdot \mathsf{fma}\left(u0, \color{blue}{u0 \cdot \left(\frac{-1}{4} \cdot u0 - \frac{1}{3}\right) + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}, -1\right)\right) \]

   6. metadata-eval N/A

      \[\leadsto \left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right) \cdot \left(u0 \cdot \mathsf{fma}\left(u0, u0 \cdot \left(\frac{-1}{4} \cdot u0 - \frac{1}{3}\right) + \color{blue}{\frac{-1}{2}}, -1\right)\right) \]

   7. accelerator-lowering-fma.f32 N/A

      \[\leadsto \left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right) \cdot \left(u0 \cdot \mathsf{fma}\left(u0, \color{blue}{\mathsf{fma}\left(u0, \frac{-1}{4} \cdot u0 - \frac{1}{3}, \frac{-1}{2}\right)}, -1\right)\right) \]

   8. sub-neg N/A

      \[\leadsto \left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right) \cdot \left(u0 \cdot \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, \color{blue}{\frac{-1}{4} \cdot u0 + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)}, \frac{-1}{2}\right), -1\right)\right) \]

   9. *-commutative N/A

      \[\leadsto \left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right) \cdot \left(u0 \cdot \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, \color{blue}{u0 \cdot \frac{-1}{4}} + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right), \frac{-1}{2}\right), -1\right)\right) \]

   10. metadata-eval N/A

       \[\leadsto \left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right) \cdot \left(u0 \cdot \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, u0 \cdot \frac{-1}{4} + \color{blue}{\frac{-1}{3}}, \frac{-1}{2}\right), -1\right)\right) \]

   11. accelerator-lowering-fma.f32 93.5

       \[\leadsto \left(\left(-\alpha\right) \cdot \alpha\right) \cdot \left(u0 \cdot \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, \color{blue}{\mathsf{fma}\left(u0, -0.25, -0.3333333333333333\right)}, -0.5\right), -1\right)\right) \]

5. Simplified 93.5%

   \[\leadsto \left(\left(-\alpha\right) \cdot \alpha\right) \cdot \color{blue}{\left(u0 \cdot \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, -0.25, -0.3333333333333333\right), -0.5\right), -1\right)\right)} \]
6. Step-by-step derivation

   1. distribute-rgt-in N/A

      \[\leadsto \left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right) \cdot \color{blue}{\left(\left(u0 \cdot \left(u0 \cdot \left(u0 \cdot \frac{-1}{4} + \frac{-1}{3}\right) + \frac{-1}{2}\right)\right) \cdot u0 + -1 \cdot u0\right)} \]

   2. neg-mul-1 N/A

      \[\leadsto \left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right) \cdot \left(\left(u0 \cdot \left(u0 \cdot \left(u0 \cdot \frac{-1}{4} + \frac{-1}{3}\right) + \frac{-1}{2}\right)\right) \cdot u0 + \color{blue}{\left(\mathsf{neg}\left(u0\right)\right)}\right) \]

   3. accelerator-lowering-fma.f32 N/A

      \[\leadsto \left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right) \cdot \color{blue}{\mathsf{fma}\left(u0 \cdot \left(u0 \cdot \left(u0 \cdot \frac{-1}{4} + \frac{-1}{3}\right) + \frac{-1}{2}\right), u0, \mathsf{neg}\left(u0\right)\right)} \]

   4. *-lowering-*.f32 N/A

      \[\leadsto \left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right) \cdot \mathsf{fma}\left(\color{blue}{u0 \cdot \left(u0 \cdot \left(u0 \cdot \frac{-1}{4} + \frac{-1}{3}\right) + \frac{-1}{2}\right)}, u0, \mathsf{neg}\left(u0\right)\right) \]

   5. accelerator-lowering-fma.f32 N/A

      \[\leadsto \left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right) \cdot \mathsf{fma}\left(u0 \cdot \color{blue}{\mathsf{fma}\left(u0, u0 \cdot \frac{-1}{4} + \frac{-1}{3}, \frac{-1}{2}\right)}, u0, \mathsf{neg}\left(u0\right)\right) \]

   6. accelerator-lowering-fma.f32 N/A

      \[\leadsto \left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right) \cdot \mathsf{fma}\left(u0 \cdot \mathsf{fma}\left(u0, \color{blue}{\mathsf{fma}\left(u0, \frac{-1}{4}, \frac{-1}{3}\right)}, \frac{-1}{2}\right), u0, \mathsf{neg}\left(u0\right)\right) \]

   7. neg-lowering-neg.f32 93.7

      \[\leadsto \left(\left(-\alpha\right) \cdot \alpha\right) \cdot \mathsf{fma}\left(u0 \cdot \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, -0.25, -0.3333333333333333\right), -0.5\right), u0, \color{blue}{-u0}\right) \]

7. Applied egg-rr 93.7%

   \[\leadsto \left(\left(-\alpha\right) \cdot \alpha\right) \cdot \color{blue}{\mathsf{fma}\left(u0 \cdot \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, -0.25, -0.3333333333333333\right), -0.5\right), u0, -u0\right)} \]
8. Step-by-step derivation

   1. unsub-neg N/A

      \[\leadsto \left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right) \cdot \color{blue}{\left(\left(u0 \cdot \left(u0 \cdot \left(u0 \cdot \frac{-1}{4} + \frac{-1}{3}\right) + \frac{-1}{2}\right)\right) \cdot u0 - u0\right)} \]

   2. --lowering--.f32 N/A

      \[\leadsto \left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right) \cdot \color{blue}{\left(\left(u0 \cdot \left(u0 \cdot \left(u0 \cdot \frac{-1}{4} + \frac{-1}{3}\right) + \frac{-1}{2}\right)\right) \cdot u0 - u0\right)} \]

   3. *-commutative N/A

      \[\leadsto \left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right) \cdot \left(\color{blue}{u0 \cdot \left(u0 \cdot \left(u0 \cdot \left(u0 \cdot \frac{-1}{4} + \frac{-1}{3}\right) + \frac{-1}{2}\right)\right)} - u0\right) \]

   4. *-lowering-*.f32 N/A

      \[\leadsto \left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right) \cdot \left(\color{blue}{u0 \cdot \left(u0 \cdot \left(u0 \cdot \left(u0 \cdot \frac{-1}{4} + \frac{-1}{3}\right) + \frac{-1}{2}\right)\right)} - u0\right) \]

   5. *-lowering-*.f32 N/A

      \[\leadsto \left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right) \cdot \left(u0 \cdot \color{blue}{\left(u0 \cdot \left(u0 \cdot \left(u0 \cdot \frac{-1}{4} + \frac{-1}{3}\right) + \frac{-1}{2}\right)\right)} - u0\right) \]

   6. accelerator-lowering-fma.f32 N/A

      \[\leadsto \left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right) \cdot \left(u0 \cdot \left(u0 \cdot \color{blue}{\mathsf{fma}\left(u0, u0 \cdot \frac{-1}{4} + \frac{-1}{3}, \frac{-1}{2}\right)}\right) - u0\right) \]

   7. accelerator-lowering-fma.f32 93.7

      \[\leadsto \left(\left(-\alpha\right) \cdot \alpha\right) \cdot \left(u0 \cdot \left(u0 \cdot \mathsf{fma}\left(u0, \color{blue}{\mathsf{fma}\left(u0, -0.25, -0.3333333333333333\right)}, -0.5\right)\right) - u0\right) \]

9. Applied egg-rr 93.7%

   \[\leadsto \left(\left(-\alpha\right) \cdot \alpha\right) \cdot \color{blue}{\left(u0 \cdot \left(u0 \cdot \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, -0.25, -0.3333333333333333\right), -0.5\right)\right) - u0\right)} \]

10. Final simplification 93.7%

    \[\leadsto \left(\alpha \cdot \alpha\right) \cdot \left(u0 - u0 \cdot \left(u0 \cdot \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, -0.25, -0.3333333333333333\right), -0.5\right)\right)\right) \]
11. Add Preprocessing

Alternative 10: 93.3% accurate, 3.2× speedup

Math

\[\begin{array}{l} \\ \left(\left(\alpha \cdot \left(-\alpha\right)\right) \cdot u0\right) \cdot \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, -0.25, -0.3333333333333333\right), -0.5\right), -1\right) \end{array} \]

FPCore

(FPCore (alpha u0)
 :precision binary32
 (*
  (* (* alpha (- alpha)) u0)
  (fma u0 (fma u0 (fma u0 -0.25 -0.3333333333333333) -0.5) -1.0)))

C

float code(float alpha, float u0) {
	return ((alpha * -alpha) * u0) * fmaf(u0, fmaf(u0, fmaf(u0, -0.25f, -0.3333333333333333f), -0.5f), -1.0f);
}

Julia

function code(alpha, u0)
	return Float32(Float32(Float32(alpha * Float32(-alpha)) * u0) * fma(u0, fma(u0, fma(u0, Float32(-0.25), Float32(-0.3333333333333333)), Float32(-0.5)), Float32(-1.0)))
end

Derivation

1. Initial program 57.3%

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

3. Taylor expanded in u0 around 0

   \[\leadsto \left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right) \cdot \color{blue}{\left(u0 \cdot \left(u0 \cdot \left(u0 \cdot \left(\frac{-1}{4} \cdot u0 - \frac{1}{3}\right) - \frac{1}{2}\right) - 1\right)\right)} \]
4. Step-by-step derivation

   1. *-lowering-*.f32 N/A

      \[\leadsto \left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right) \cdot \color{blue}{\left(u0 \cdot \left(u0 \cdot \left(u0 \cdot \left(\frac{-1}{4} \cdot u0 - \frac{1}{3}\right) - \frac{1}{2}\right) - 1\right)\right)} \]

   2. sub-neg N/A

      \[\leadsto \left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right) \cdot \left(u0 \cdot \color{blue}{\left(u0 \cdot \left(u0 \cdot \left(\frac{-1}{4} \cdot u0 - \frac{1}{3}\right) - \frac{1}{2}\right) + \left(\mathsf{neg}\left(1\right)\right)\right)}\right) \]

   3. metadata-eval N/A

      \[\leadsto \left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right) \cdot \left(u0 \cdot \left(u0 \cdot \left(u0 \cdot \left(\frac{-1}{4} \cdot u0 - \frac{1}{3}\right) - \frac{1}{2}\right) + \color{blue}{-1}\right)\right) \]

   4. accelerator-lowering-fma.f32 N/A

      \[\leadsto \left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right) \cdot \left(u0 \cdot \color{blue}{\mathsf{fma}\left(u0, u0 \cdot \left(\frac{-1}{4} \cdot u0 - \frac{1}{3}\right) - \frac{1}{2}, -1\right)}\right) \]

   5. sub-neg N/A

      \[\leadsto \left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right) \cdot \left(u0 \cdot \mathsf{fma}\left(u0, \color{blue}{u0 \cdot \left(\frac{-1}{4} \cdot u0 - \frac{1}{3}\right) + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}, -1\right)\right) \]

   6. metadata-eval N/A

      \[\leadsto \left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right) \cdot \left(u0 \cdot \mathsf{fma}\left(u0, u0 \cdot \left(\frac{-1}{4} \cdot u0 - \frac{1}{3}\right) + \color{blue}{\frac{-1}{2}}, -1\right)\right) \]

   7. accelerator-lowering-fma.f32 N/A

      \[\leadsto \left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right) \cdot \left(u0 \cdot \mathsf{fma}\left(u0, \color{blue}{\mathsf{fma}\left(u0, \frac{-1}{4} \cdot u0 - \frac{1}{3}, \frac{-1}{2}\right)}, -1\right)\right) \]

   8. sub-neg N/A

      \[\leadsto \left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right) \cdot \left(u0 \cdot \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, \color{blue}{\frac{-1}{4} \cdot u0 + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)}, \frac{-1}{2}\right), -1\right)\right) \]

   9. *-commutative N/A

      \[\leadsto \left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right) \cdot \left(u0 \cdot \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, \color{blue}{u0 \cdot \frac{-1}{4}} + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right), \frac{-1}{2}\right), -1\right)\right) \]

   10. metadata-eval N/A

       \[\leadsto \left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right) \cdot \left(u0 \cdot \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, u0 \cdot \frac{-1}{4} + \color{blue}{\frac{-1}{3}}, \frac{-1}{2}\right), -1\right)\right) \]

   11. accelerator-lowering-fma.f32 93.5

       \[\leadsto \left(\left(-\alpha\right) \cdot \alpha\right) \cdot \left(u0 \cdot \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, \color{blue}{\mathsf{fma}\left(u0, -0.25, -0.3333333333333333\right)}, -0.5\right), -1\right)\right) \]

5. Simplified 93.5%

   \[\leadsto \left(\left(-\alpha\right) \cdot \alpha\right) \cdot \color{blue}{\left(u0 \cdot \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, -0.25, -0.3333333333333333\right), -0.5\right), -1\right)\right)} \]
6. Step-by-step derivation

   1. associate-*r* N/A

      \[\leadsto \color{blue}{\left(\left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right) \cdot u0\right) \cdot \left(u0 \cdot \left(u0 \cdot \left(u0 \cdot \frac{-1}{4} + \frac{-1}{3}\right) + \frac{-1}{2}\right) + -1\right)} \]

   2. *-commutative N/A

      \[\leadsto \color{blue}{\left(u0 \cdot \left(u0 \cdot \left(u0 \cdot \frac{-1}{4} + \frac{-1}{3}\right) + \frac{-1}{2}\right) + -1\right) \cdot \left(\left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right) \cdot u0\right)} \]

   3. *-lowering-*.f32 N/A

      \[\leadsto \color{blue}{\left(u0 \cdot \left(u0 \cdot \left(u0 \cdot \frac{-1}{4} + \frac{-1}{3}\right) + \frac{-1}{2}\right) + -1\right) \cdot \left(\left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right) \cdot u0\right)} \]

   4. accelerator-lowering-fma.f32 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(u0, u0 \cdot \left(u0 \cdot \frac{-1}{4} + \frac{-1}{3}\right) + \frac{-1}{2}, -1\right)} \cdot \left(\left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right) \cdot u0\right) \]

   5. accelerator-lowering-fma.f32 N/A

      \[\leadsto \mathsf{fma}\left(u0, \color{blue}{\mathsf{fma}\left(u0, u0 \cdot \frac{-1}{4} + \frac{-1}{3}, \frac{-1}{2}\right)}, -1\right) \cdot \left(\left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right) \cdot u0\right) \]

   6. accelerator-lowering-fma.f32 N/A

      \[\leadsto \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, \color{blue}{\mathsf{fma}\left(u0, \frac{-1}{4}, \frac{-1}{3}\right)}, \frac{-1}{2}\right), -1\right) \cdot \left(\left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right) \cdot u0\right) \]

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

      \[\leadsto \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, \frac{-1}{4}, \frac{-1}{3}\right), \frac{-1}{2}\right), -1\right) \cdot \left(\color{blue}{\left(\mathsf{neg}\left(\alpha \cdot \alpha\right)\right)} \cdot u0\right) \]

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

      \[\leadsto \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, \frac{-1}{4}, \frac{-1}{3}\right), \frac{-1}{2}\right), -1\right) \cdot \color{blue}{\left(\mathsf{neg}\left(\left(\alpha \cdot \alpha\right) \cdot u0\right)\right)} \]

   9. *-commutative N/A

      \[\leadsto \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, \frac{-1}{4}, \frac{-1}{3}\right), \frac{-1}{2}\right), -1\right) \cdot \left(\mathsf{neg}\left(\color{blue}{u0 \cdot \left(\alpha \cdot \alpha\right)}\right)\right) \]

   10. neg-lowering-neg.f32 N/A

       \[\leadsto \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, \frac{-1}{4}, \frac{-1}{3}\right), \frac{-1}{2}\right), -1\right) \cdot \color{blue}{\left(\mathsf{neg}\left(u0 \cdot \left(\alpha \cdot \alpha\right)\right)\right)} \]

   11. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, \frac{-1}{4}, \frac{-1}{3}\right), \frac{-1}{2}\right), -1\right) \cdot \left(\mathsf{neg}\left(\color{blue}{u0 \cdot \left(\alpha \cdot \alpha\right)}\right)\right) \]

   12. *-lowering-*.f32 93.6

       \[\leadsto \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, -0.25, -0.3333333333333333\right), -0.5\right), -1\right) \cdot \left(-u0 \cdot \color{blue}{\left(\alpha \cdot \alpha\right)}\right) \]

7. Applied egg-rr 93.6%

   \[\leadsto \color{blue}{\mathsf{fma}\left(u0, \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, -0.25, -0.3333333333333333\right), -0.5\right), -1\right) \cdot \left(-u0 \cdot \left(\alpha \cdot \alpha\right)\right)} \]

8. Final simplification 93.6%

   \[\leadsto \left(\left(\alpha \cdot \left(-\alpha\right)\right) \cdot u0\right) \cdot \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, -0.25, -0.3333333333333333\right), -0.5\right), -1\right) \]
9. Add Preprocessing

Alternative 11: 93.3% accurate, 3.2× speedup

Math

\[\begin{array}{l} \\ u0 \cdot \left(\left(\alpha \cdot \left(-\alpha\right)\right) \cdot \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, -0.25, -0.3333333333333333\right), -0.5\right), -1\right)\right) \end{array} \]

FPCore

(FPCore (alpha u0)
 :precision binary32
 (*
  u0
  (*
   (* alpha (- alpha))
   (fma u0 (fma u0 (fma u0 -0.25 -0.3333333333333333) -0.5) -1.0))))

C

float code(float alpha, float u0) {
	return u0 * ((alpha * -alpha) * fmaf(u0, fmaf(u0, fmaf(u0, -0.25f, -0.3333333333333333f), -0.5f), -1.0f));
}

Julia

function code(alpha, u0)
	return Float32(u0 * Float32(Float32(alpha * Float32(-alpha)) * fma(u0, fma(u0, fma(u0, Float32(-0.25), Float32(-0.3333333333333333)), Float32(-0.5)), Float32(-1.0))))
end

Derivation

1. Initial program 57.3%

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

3. Taylor expanded in u0 around 0

   \[\leadsto \left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right) \cdot \color{blue}{\left(u0 \cdot \left(u0 \cdot \left(u0 \cdot \left(\frac{-1}{4} \cdot u0 - \frac{1}{3}\right) - \frac{1}{2}\right) - 1\right)\right)} \]
4. Step-by-step derivation

   1. *-lowering-*.f32 N/A

      \[\leadsto \left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right) \cdot \color{blue}{\left(u0 \cdot \left(u0 \cdot \left(u0 \cdot \left(\frac{-1}{4} \cdot u0 - \frac{1}{3}\right) - \frac{1}{2}\right) - 1\right)\right)} \]

   2. sub-neg N/A

      \[\leadsto \left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right) \cdot \left(u0 \cdot \color{blue}{\left(u0 \cdot \left(u0 \cdot \left(\frac{-1}{4} \cdot u0 - \frac{1}{3}\right) - \frac{1}{2}\right) + \left(\mathsf{neg}\left(1\right)\right)\right)}\right) \]

   3. metadata-eval N/A

      \[\leadsto \left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right) \cdot \left(u0 \cdot \left(u0 \cdot \left(u0 \cdot \left(\frac{-1}{4} \cdot u0 - \frac{1}{3}\right) - \frac{1}{2}\right) + \color{blue}{-1}\right)\right) \]

   4. accelerator-lowering-fma.f32 N/A

      \[\leadsto \left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right) \cdot \left(u0 \cdot \color{blue}{\mathsf{fma}\left(u0, u0 \cdot \left(\frac{-1}{4} \cdot u0 - \frac{1}{3}\right) - \frac{1}{2}, -1\right)}\right) \]

   5. sub-neg N/A

      \[\leadsto \left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right) \cdot \left(u0 \cdot \mathsf{fma}\left(u0, \color{blue}{u0 \cdot \left(\frac{-1}{4} \cdot u0 - \frac{1}{3}\right) + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}, -1\right)\right) \]

   6. metadata-eval N/A

      \[\leadsto \left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right) \cdot \left(u0 \cdot \mathsf{fma}\left(u0, u0 \cdot \left(\frac{-1}{4} \cdot u0 - \frac{1}{3}\right) + \color{blue}{\frac{-1}{2}}, -1\right)\right) \]

   7. accelerator-lowering-fma.f32 N/A

      \[\leadsto \left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right) \cdot \left(u0 \cdot \mathsf{fma}\left(u0, \color{blue}{\mathsf{fma}\left(u0, \frac{-1}{4} \cdot u0 - \frac{1}{3}, \frac{-1}{2}\right)}, -1\right)\right) \]

   8. sub-neg N/A

      \[\leadsto \left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right) \cdot \left(u0 \cdot \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, \color{blue}{\frac{-1}{4} \cdot u0 + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)}, \frac{-1}{2}\right), -1\right)\right) \]

   9. *-commutative N/A

      \[\leadsto \left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right) \cdot \left(u0 \cdot \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, \color{blue}{u0 \cdot \frac{-1}{4}} + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right), \frac{-1}{2}\right), -1\right)\right) \]

   10. metadata-eval N/A

       \[\leadsto \left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right) \cdot \left(u0 \cdot \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, u0 \cdot \frac{-1}{4} + \color{blue}{\frac{-1}{3}}, \frac{-1}{2}\right), -1\right)\right) \]

   11. accelerator-lowering-fma.f32 93.5

       \[\leadsto \left(\left(-\alpha\right) \cdot \alpha\right) \cdot \left(u0 \cdot \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, \color{blue}{\mathsf{fma}\left(u0, -0.25, -0.3333333333333333\right)}, -0.5\right), -1\right)\right) \]

5. Simplified 93.5%

   \[\leadsto \left(\left(-\alpha\right) \cdot \alpha\right) \cdot \color{blue}{\left(u0 \cdot \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, -0.25, -0.3333333333333333\right), -0.5\right), -1\right)\right)} \]
6. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right) \cdot \color{blue}{\left(\left(u0 \cdot \left(u0 \cdot \left(u0 \cdot \frac{-1}{4} + \frac{-1}{3}\right) + \frac{-1}{2}\right) + -1\right) \cdot u0\right)} \]

   2. associate-*r* N/A

      \[\leadsto \color{blue}{\left(\left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right) \cdot \left(u0 \cdot \left(u0 \cdot \left(u0 \cdot \frac{-1}{4} + \frac{-1}{3}\right) + \frac{-1}{2}\right) + -1\right)\right) \cdot u0} \]

   3. *-lowering-*.f32 N/A

      \[\leadsto \color{blue}{\left(\left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right) \cdot \left(u0 \cdot \left(u0 \cdot \left(u0 \cdot \frac{-1}{4} + \frac{-1}{3}\right) + \frac{-1}{2}\right) + -1\right)\right) \cdot u0} \]

   4. *-lowering-*.f32 N/A

      \[\leadsto \color{blue}{\left(\left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right) \cdot \left(u0 \cdot \left(u0 \cdot \left(u0 \cdot \frac{-1}{4} + \frac{-1}{3}\right) + \frac{-1}{2}\right) + -1\right)\right)} \cdot u0 \]

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

      \[\leadsto \left(\color{blue}{\left(\mathsf{neg}\left(\alpha \cdot \alpha\right)\right)} \cdot \left(u0 \cdot \left(u0 \cdot \left(u0 \cdot \frac{-1}{4} + \frac{-1}{3}\right) + \frac{-1}{2}\right) + -1\right)\right) \cdot u0 \]

   6. neg-lowering-neg.f32 N/A

      \[\leadsto \left(\color{blue}{\left(\mathsf{neg}\left(\alpha \cdot \alpha\right)\right)} \cdot \left(u0 \cdot \left(u0 \cdot \left(u0 \cdot \frac{-1}{4} + \frac{-1}{3}\right) + \frac{-1}{2}\right) + -1\right)\right) \cdot u0 \]

   7. *-lowering-*.f32 N/A

      \[\leadsto \left(\left(\mathsf{neg}\left(\color{blue}{\alpha \cdot \alpha}\right)\right) \cdot \left(u0 \cdot \left(u0 \cdot \left(u0 \cdot \frac{-1}{4} + \frac{-1}{3}\right) + \frac{-1}{2}\right) + -1\right)\right) \cdot u0 \]

   8. accelerator-lowering-fma.f32 N/A

      \[\leadsto \left(\left(\mathsf{neg}\left(\alpha \cdot \alpha\right)\right) \cdot \color{blue}{\mathsf{fma}\left(u0, u0 \cdot \left(u0 \cdot \frac{-1}{4} + \frac{-1}{3}\right) + \frac{-1}{2}, -1\right)}\right) \cdot u0 \]

   9. accelerator-lowering-fma.f32 N/A

      \[\leadsto \left(\left(\mathsf{neg}\left(\alpha \cdot \alpha\right)\right) \cdot \mathsf{fma}\left(u0, \color{blue}{\mathsf{fma}\left(u0, u0 \cdot \frac{-1}{4} + \frac{-1}{3}, \frac{-1}{2}\right)}, -1\right)\right) \cdot u0 \]

   10. accelerator-lowering-fma.f32 93.5

       \[\leadsto \left(\left(-\alpha \cdot \alpha\right) \cdot \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, \color{blue}{\mathsf{fma}\left(u0, -0.25, -0.3333333333333333\right)}, -0.5\right), -1\right)\right) \cdot u0 \]

7. Applied egg-rr 93.5%

   \[\leadsto \color{blue}{\left(\left(-\alpha \cdot \alpha\right) \cdot \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, -0.25, -0.3333333333333333\right), -0.5\right), -1\right)\right) \cdot u0} \]

8. Final simplification 93.5%

   \[\leadsto u0 \cdot \left(\left(\alpha \cdot \left(-\alpha\right)\right) \cdot \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, -0.25, -0.3333333333333333\right), -0.5\right), -1\right)\right) \]
9. Add Preprocessing

Alternative 12: 93.3% accurate, 3.2× speedup

Math

\[\begin{array}{l} \\ \left(\alpha \cdot \left(-\alpha\right)\right) \cdot \left(u0 \cdot \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, -0.25, -0.3333333333333333\right), -0.5\right), -1\right)\right) \end{array} \]

FPCore

(FPCore (alpha u0)
 :precision binary32
 (*
  (* alpha (- alpha))
  (* u0 (fma u0 (fma u0 (fma u0 -0.25 -0.3333333333333333) -0.5) -1.0))))

C

float code(float alpha, float u0) {
	return (alpha * -alpha) * (u0 * fmaf(u0, fmaf(u0, fmaf(u0, -0.25f, -0.3333333333333333f), -0.5f), -1.0f));
}

Julia

function code(alpha, u0)
	return Float32(Float32(alpha * Float32(-alpha)) * Float32(u0 * fma(u0, fma(u0, fma(u0, Float32(-0.25), Float32(-0.3333333333333333)), Float32(-0.5)), Float32(-1.0))))
end

Derivation

1. Initial program 57.3%

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

3. Taylor expanded in u0 around 0

   \[\leadsto \left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right) \cdot \color{blue}{\left(u0 \cdot \left(u0 \cdot \left(u0 \cdot \left(\frac{-1}{4} \cdot u0 - \frac{1}{3}\right) - \frac{1}{2}\right) - 1\right)\right)} \]
4. Step-by-step derivation

   1. *-lowering-*.f32 N/A

      \[\leadsto \left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right) \cdot \color{blue}{\left(u0 \cdot \left(u0 \cdot \left(u0 \cdot \left(\frac{-1}{4} \cdot u0 - \frac{1}{3}\right) - \frac{1}{2}\right) - 1\right)\right)} \]

   2. sub-neg N/A

      \[\leadsto \left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right) \cdot \left(u0 \cdot \color{blue}{\left(u0 \cdot \left(u0 \cdot \left(\frac{-1}{4} \cdot u0 - \frac{1}{3}\right) - \frac{1}{2}\right) + \left(\mathsf{neg}\left(1\right)\right)\right)}\right) \]

   3. metadata-eval N/A

      \[\leadsto \left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right) \cdot \left(u0 \cdot \left(u0 \cdot \left(u0 \cdot \left(\frac{-1}{4} \cdot u0 - \frac{1}{3}\right) - \frac{1}{2}\right) + \color{blue}{-1}\right)\right) \]

   4. accelerator-lowering-fma.f32 N/A

      \[\leadsto \left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right) \cdot \left(u0 \cdot \color{blue}{\mathsf{fma}\left(u0, u0 \cdot \left(\frac{-1}{4} \cdot u0 - \frac{1}{3}\right) - \frac{1}{2}, -1\right)}\right) \]

   5. sub-neg N/A

      \[\leadsto \left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right) \cdot \left(u0 \cdot \mathsf{fma}\left(u0, \color{blue}{u0 \cdot \left(\frac{-1}{4} \cdot u0 - \frac{1}{3}\right) + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}, -1\right)\right) \]

   6. metadata-eval N/A

      \[\leadsto \left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right) \cdot \left(u0 \cdot \mathsf{fma}\left(u0, u0 \cdot \left(\frac{-1}{4} \cdot u0 - \frac{1}{3}\right) + \color{blue}{\frac{-1}{2}}, -1\right)\right) \]

   7. accelerator-lowering-fma.f32 N/A

      \[\leadsto \left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right) \cdot \left(u0 \cdot \mathsf{fma}\left(u0, \color{blue}{\mathsf{fma}\left(u0, \frac{-1}{4} \cdot u0 - \frac{1}{3}, \frac{-1}{2}\right)}, -1\right)\right) \]

   8. sub-neg N/A

      \[\leadsto \left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right) \cdot \left(u0 \cdot \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, \color{blue}{\frac{-1}{4} \cdot u0 + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)}, \frac{-1}{2}\right), -1\right)\right) \]

   9. *-commutative N/A

      \[\leadsto \left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right) \cdot \left(u0 \cdot \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, \color{blue}{u0 \cdot \frac{-1}{4}} + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right), \frac{-1}{2}\right), -1\right)\right) \]

   10. metadata-eval N/A

       \[\leadsto \left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right) \cdot \left(u0 \cdot \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, u0 \cdot \frac{-1}{4} + \color{blue}{\frac{-1}{3}}, \frac{-1}{2}\right), -1\right)\right) \]

   11. accelerator-lowering-fma.f32 93.5

       \[\leadsto \left(\left(-\alpha\right) \cdot \alpha\right) \cdot \left(u0 \cdot \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, \color{blue}{\mathsf{fma}\left(u0, -0.25, -0.3333333333333333\right)}, -0.5\right), -1\right)\right) \]

5. Simplified 93.5%

   \[\leadsto \left(\left(-\alpha\right) \cdot \alpha\right) \cdot \color{blue}{\left(u0 \cdot \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, -0.25, -0.3333333333333333\right), -0.5\right), -1\right)\right)} \]

6. Final simplification 93.5%

   \[\leadsto \left(\alpha \cdot \left(-\alpha\right)\right) \cdot \left(u0 \cdot \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, -0.25, -0.3333333333333333\right), -0.5\right), -1\right)\right) \]
7. Add Preprocessing

Alternative 13: 91.6% accurate, 3.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 57.3%

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

   1. sub-neg N/A

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

   2. accelerator-lowering-log1p.f32 N/A

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

   3. neg-lowering-neg.f32 98.9

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

4. Applied egg-rr 98.9%

   \[\leadsto \left(\left(-\alpha\right) \cdot \alpha\right) \cdot \color{blue}{\mathsf{log1p}\left(-u0\right)} \]
5. Step-by-step derivation

   1. neg-sub0 N/A

      \[\leadsto \left(\color{blue}{\left(0 - \alpha\right)} \cdot \alpha\right) \cdot \mathsf{log1p}\left(\mathsf{neg}\left(u0\right)\right) \]

   2. sub-neg N/A

      \[\leadsto \left(\color{blue}{\left(0 + \left(\mathsf{neg}\left(\alpha\right)\right)\right)} \cdot \alpha\right) \cdot \mathsf{log1p}\left(\mathsf{neg}\left(u0\right)\right) \]

   3. flip3-+ N/A

      \[\leadsto \left(\color{blue}{\frac{{0}^{3} + {\left(\mathsf{neg}\left(\alpha\right)\right)}^{3}}{0 \cdot 0 + \left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \left(\mathsf{neg}\left(\alpha\right)\right) - 0 \cdot \left(\mathsf{neg}\left(\alpha\right)\right)\right)}} \cdot \alpha\right) \cdot \mathsf{log1p}\left(\mathsf{neg}\left(u0\right)\right) \]

   4. /-lowering-/.f32 N/A

      \[\leadsto \left(\color{blue}{\frac{{0}^{3} + {\left(\mathsf{neg}\left(\alpha\right)\right)}^{3}}{0 \cdot 0 + \left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \left(\mathsf{neg}\left(\alpha\right)\right) - 0 \cdot \left(\mathsf{neg}\left(\alpha\right)\right)\right)}} \cdot \alpha\right) \cdot \mathsf{log1p}\left(\mathsf{neg}\left(u0\right)\right) \]

   5. metadata-eval N/A

      \[\leadsto \left(\frac{\color{blue}{0} + {\left(\mathsf{neg}\left(\alpha\right)\right)}^{3}}{0 \cdot 0 + \left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \left(\mathsf{neg}\left(\alpha\right)\right) - 0 \cdot \left(\mathsf{neg}\left(\alpha\right)\right)\right)} \cdot \alpha\right) \cdot \mathsf{log1p}\left(\mathsf{neg}\left(u0\right)\right) \]

   6. +-lowering-+.f32 N/A

      \[\leadsto \left(\frac{\color{blue}{0 + {\left(\mathsf{neg}\left(\alpha\right)\right)}^{3}}}{0 \cdot 0 + \left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \left(\mathsf{neg}\left(\alpha\right)\right) - 0 \cdot \left(\mathsf{neg}\left(\alpha\right)\right)\right)} \cdot \alpha\right) \cdot \mathsf{log1p}\left(\mathsf{neg}\left(u0\right)\right) \]

   7. pow-lowering-pow.f32 N/A

      \[\leadsto \left(\frac{0 + \color{blue}{{\left(\mathsf{neg}\left(\alpha\right)\right)}^{3}}}{0 \cdot 0 + \left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \left(\mathsf{neg}\left(\alpha\right)\right) - 0 \cdot \left(\mathsf{neg}\left(\alpha\right)\right)\right)} \cdot \alpha\right) \cdot \mathsf{log1p}\left(\mathsf{neg}\left(u0\right)\right) \]

   8. neg-lowering-neg.f32 N/A

      \[\leadsto \left(\frac{0 + {\color{blue}{\left(\mathsf{neg}\left(\alpha\right)\right)}}^{3}}{0 \cdot 0 + \left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \left(\mathsf{neg}\left(\alpha\right)\right) - 0 \cdot \left(\mathsf{neg}\left(\alpha\right)\right)\right)} \cdot \alpha\right) \cdot \mathsf{log1p}\left(\mathsf{neg}\left(u0\right)\right) \]

   9. metadata-eval N/A

      \[\leadsto \left(\frac{0 + {\left(\mathsf{neg}\left(\alpha\right)\right)}^{3}}{\color{blue}{0} + \left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \left(\mathsf{neg}\left(\alpha\right)\right) - 0 \cdot \left(\mathsf{neg}\left(\alpha\right)\right)\right)} \cdot \alpha\right) \cdot \mathsf{log1p}\left(\mathsf{neg}\left(u0\right)\right) \]

   10. +-lowering-+.f32 N/A

       \[\leadsto \left(\frac{0 + {\left(\mathsf{neg}\left(\alpha\right)\right)}^{3}}{\color{blue}{0 + \left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \left(\mathsf{neg}\left(\alpha\right)\right) - 0 \cdot \left(\mathsf{neg}\left(\alpha\right)\right)\right)}} \cdot \alpha\right) \cdot \mathsf{log1p}\left(\mathsf{neg}\left(u0\right)\right) \]

   11. --lowering--.f32 N/A

       \[\leadsto \left(\frac{0 + {\left(\mathsf{neg}\left(\alpha\right)\right)}^{3}}{0 + \color{blue}{\left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \left(\mathsf{neg}\left(\alpha\right)\right) - 0 \cdot \left(\mathsf{neg}\left(\alpha\right)\right)\right)}} \cdot \alpha\right) \cdot \mathsf{log1p}\left(\mathsf{neg}\left(u0\right)\right) \]

   12. *-lowering-*.f32 N/A

       \[\leadsto \left(\frac{0 + {\left(\mathsf{neg}\left(\alpha\right)\right)}^{3}}{0 + \left(\color{blue}{\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \left(\mathsf{neg}\left(\alpha\right)\right)} - 0 \cdot \left(\mathsf{neg}\left(\alpha\right)\right)\right)} \cdot \alpha\right) \cdot \mathsf{log1p}\left(\mathsf{neg}\left(u0\right)\right) \]

   13. neg-lowering-neg.f32 N/A

       \[\leadsto \left(\frac{0 + {\left(\mathsf{neg}\left(\alpha\right)\right)}^{3}}{0 + \left(\color{blue}{\left(\mathsf{neg}\left(\alpha\right)\right)} \cdot \left(\mathsf{neg}\left(\alpha\right)\right) - 0 \cdot \left(\mathsf{neg}\left(\alpha\right)\right)\right)} \cdot \alpha\right) \cdot \mathsf{log1p}\left(\mathsf{neg}\left(u0\right)\right) \]

   14. neg-lowering-neg.f32 N/A

       \[\leadsto \left(\frac{0 + {\left(\mathsf{neg}\left(\alpha\right)\right)}^{3}}{0 + \left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \color{blue}{\left(\mathsf{neg}\left(\alpha\right)\right)} - 0 \cdot \left(\mathsf{neg}\left(\alpha\right)\right)\right)} \cdot \alpha\right) \cdot \mathsf{log1p}\left(\mathsf{neg}\left(u0\right)\right) \]

   15. *-lowering-*.f32 N/A

       \[\leadsto \left(\frac{0 + {\left(\mathsf{neg}\left(\alpha\right)\right)}^{3}}{0 + \left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \left(\mathsf{neg}\left(\alpha\right)\right) - \color{blue}{0 \cdot \left(\mathsf{neg}\left(\alpha\right)\right)}\right)} \cdot \alpha\right) \cdot \mathsf{log1p}\left(\mathsf{neg}\left(u0\right)\right) \]

   16. neg-lowering-neg.f32 98.6

       \[\leadsto \left(\frac{0 + {\left(-\alpha\right)}^{3}}{0 + \left(\left(-\alpha\right) \cdot \left(-\alpha\right) - 0 \cdot \color{blue}{\left(-\alpha\right)}\right)} \cdot \alpha\right) \cdot \mathsf{log1p}\left(-u0\right) \]

6. Applied egg-rr 98.6%

   \[\leadsto \left(\color{blue}{\frac{0 + {\left(-\alpha\right)}^{3}}{0 + \left(\left(-\alpha\right) \cdot \left(-\alpha\right) - 0 \cdot \left(-\alpha\right)\right)}} \cdot \alpha\right) \cdot \mathsf{log1p}\left(-u0\right) \]
7. Step-by-step derivation

   1. +-lft-identity N/A

      \[\leadsto \left(\frac{\color{blue}{{\left(\mathsf{neg}\left(\alpha\right)\right)}^{3}}}{0 + \left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \left(\mathsf{neg}\left(\alpha\right)\right) - 0 \cdot \left(\mathsf{neg}\left(\alpha\right)\right)\right)} \cdot \alpha\right) \cdot \mathsf{log1p}\left(\mathsf{neg}\left(u0\right)\right) \]

   2. +-lft-identity N/A

      \[\leadsto \left(\frac{{\left(\mathsf{neg}\left(\alpha\right)\right)}^{3}}{\color{blue}{\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \left(\mathsf{neg}\left(\alpha\right)\right) - 0 \cdot \left(\mathsf{neg}\left(\alpha\right)\right)}} \cdot \alpha\right) \cdot \mathsf{log1p}\left(\mathsf{neg}\left(u0\right)\right) \]

   3. distribute-rgt-out-- N/A

      \[\leadsto \left(\frac{{\left(\mathsf{neg}\left(\alpha\right)\right)}^{3}}{\color{blue}{\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \left(\left(\mathsf{neg}\left(\alpha\right)\right) - 0\right)}} \cdot \alpha\right) \cdot \mathsf{log1p}\left(\mathsf{neg}\left(u0\right)\right) \]

   4. --rgt-identity N/A

      \[\leadsto \left(\frac{{\left(\mathsf{neg}\left(\alpha\right)\right)}^{3}}{\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \color{blue}{\left(\mathsf{neg}\left(\alpha\right)\right)}} \cdot \alpha\right) \cdot \mathsf{log1p}\left(\mathsf{neg}\left(u0\right)\right) \]

   5. pow2 N/A

      \[\leadsto \left(\frac{{\left(\mathsf{neg}\left(\alpha\right)\right)}^{3}}{\color{blue}{{\left(\mathsf{neg}\left(\alpha\right)\right)}^{2}}} \cdot \alpha\right) \cdot \mathsf{log1p}\left(\mathsf{neg}\left(u0\right)\right) \]

   6. pow-div N/A

      \[\leadsto \left(\color{blue}{{\left(\mathsf{neg}\left(\alpha\right)\right)}^{\left(3 - 2\right)}} \cdot \alpha\right) \cdot \mathsf{log1p}\left(\mathsf{neg}\left(u0\right)\right) \]

   7. metadata-eval N/A

      \[\leadsto \left({\left(\mathsf{neg}\left(\alpha\right)\right)}^{\color{blue}{1}} \cdot \alpha\right) \cdot \mathsf{log1p}\left(\mathsf{neg}\left(u0\right)\right) \]

   8. unpow1 N/A

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

   9. remove-double-div N/A

      \[\leadsto \left(\color{blue}{\frac{1}{\frac{1}{\mathsf{neg}\left(\alpha\right)}}} \cdot \alpha\right) \cdot \mathsf{log1p}\left(\mathsf{neg}\left(u0\right)\right) \]

   10. /-lowering-/.f32 N/A

       \[\leadsto \left(\color{blue}{\frac{1}{\frac{1}{\mathsf{neg}\left(\alpha\right)}}} \cdot \alpha\right) \cdot \mathsf{log1p}\left(\mathsf{neg}\left(u0\right)\right) \]

   11. inv-pow N/A

       \[\leadsto \left(\frac{1}{\color{blue}{{\left(\mathsf{neg}\left(\alpha\right)\right)}^{-1}}} \cdot \alpha\right) \cdot \mathsf{log1p}\left(\mathsf{neg}\left(u0\right)\right) \]

   12. metadata-eval N/A

       \[\leadsto \left(\frac{1}{{\left(\mathsf{neg}\left(\alpha\right)\right)}^{\color{blue}{\left(2 - 3\right)}}} \cdot \alpha\right) \cdot \mathsf{log1p}\left(\mathsf{neg}\left(u0\right)\right) \]

   13. pow-div N/A

       \[\leadsto \left(\frac{1}{\color{blue}{\frac{{\left(\mathsf{neg}\left(\alpha\right)\right)}^{2}}{{\left(\mathsf{neg}\left(\alpha\right)\right)}^{3}}}} \cdot \alpha\right) \cdot \mathsf{log1p}\left(\mathsf{neg}\left(u0\right)\right) \]

   14. pow2 N/A

       \[\leadsto \left(\frac{1}{\frac{\color{blue}{\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \left(\mathsf{neg}\left(\alpha\right)\right)}}{{\left(\mathsf{neg}\left(\alpha\right)\right)}^{3}}} \cdot \alpha\right) \cdot \mathsf{log1p}\left(\mathsf{neg}\left(u0\right)\right) \]

   15. sqr-neg N/A

       \[\leadsto \left(\frac{1}{\frac{\color{blue}{\alpha \cdot \alpha}}{{\left(\mathsf{neg}\left(\alpha\right)\right)}^{3}}} \cdot \alpha\right) \cdot \mathsf{log1p}\left(\mathsf{neg}\left(u0\right)\right) \]

   16. pow2 N/A

       \[\leadsto \left(\frac{1}{\frac{\color{blue}{{\alpha}^{2}}}{{\left(\mathsf{neg}\left(\alpha\right)\right)}^{3}}} \cdot \alpha\right) \cdot \mathsf{log1p}\left(\mathsf{neg}\left(u0\right)\right) \]

   17. sqr-pow N/A

       \[\leadsto \left(\frac{1}{\frac{{\alpha}^{2}}{\color{blue}{{\left(\mathsf{neg}\left(\alpha\right)\right)}^{\left(\frac{3}{2}\right)} \cdot {\left(\mathsf{neg}\left(\alpha\right)\right)}^{\left(\frac{3}{2}\right)}}}} \cdot \alpha\right) \cdot \mathsf{log1p}\left(\mathsf{neg}\left(u0\right)\right) \]

   18. pow-prod-down N/A

       \[\leadsto \left(\frac{1}{\frac{{\alpha}^{2}}{\color{blue}{{\left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \left(\mathsf{neg}\left(\alpha\right)\right)\right)}^{\left(\frac{3}{2}\right)}}}} \cdot \alpha\right) \cdot \mathsf{log1p}\left(\mathsf{neg}\left(u0\right)\right) \]

   19. sqr-neg N/A

       \[\leadsto \left(\frac{1}{\frac{{\alpha}^{2}}{{\color{blue}{\left(\alpha \cdot \alpha\right)}}^{\left(\frac{3}{2}\right)}}} \cdot \alpha\right) \cdot \mathsf{log1p}\left(\mathsf{neg}\left(u0\right)\right) \]

   20. unpow-prod-down N/A

       \[\leadsto \left(\frac{1}{\frac{{\alpha}^{2}}{\color{blue}{{\alpha}^{\left(\frac{3}{2}\right)} \cdot {\alpha}^{\left(\frac{3}{2}\right)}}}} \cdot \alpha\right) \cdot \mathsf{log1p}\left(\mathsf{neg}\left(u0\right)\right) \]

   21. sqr-pow N/A

       \[\leadsto \left(\frac{1}{\frac{{\alpha}^{2}}{\color{blue}{{\alpha}^{3}}}} \cdot \alpha\right) \cdot \mathsf{log1p}\left(\mathsf{neg}\left(u0\right)\right) \]

   22. pow-div N/A

       \[\leadsto \left(\frac{1}{\color{blue}{{\alpha}^{\left(2 - 3\right)}}} \cdot \alpha\right) \cdot \mathsf{log1p}\left(\mathsf{neg}\left(u0\right)\right) \]

   23. metadata-eval N/A

       \[\leadsto \left(\frac{1}{{\alpha}^{\color{blue}{-1}}} \cdot \alpha\right) \cdot \mathsf{log1p}\left(\mathsf{neg}\left(u0\right)\right) \]

   24. inv-pow N/A

       \[\leadsto \left(\frac{1}{\color{blue}{\frac{1}{\alpha}}} \cdot \alpha\right) \cdot \mathsf{log1p}\left(\mathsf{neg}\left(u0\right)\right) \]

   25. frac-2neg N/A

       \[\leadsto \left(\frac{1}{\color{blue}{\frac{\mathsf{neg}\left(1\right)}{\mathsf{neg}\left(\alpha\right)}}} \cdot \alpha\right) \cdot \mathsf{log1p}\left(\mathsf{neg}\left(u0\right)\right) \]

   26. metadata-eval N/A

       \[\leadsto \left(\frac{1}{\frac{\color{blue}{-1}}{\mathsf{neg}\left(\alpha\right)}} \cdot \alpha\right) \cdot \mathsf{log1p}\left(\mathsf{neg}\left(u0\right)\right) \]

   27. /-lowering-/.f32 N/A

       \[\leadsto \left(\frac{1}{\color{blue}{\frac{-1}{\mathsf{neg}\left(\alpha\right)}}} \cdot \alpha\right) \cdot \mathsf{log1p}\left(\mathsf{neg}\left(u0\right)\right) \]

8. Applied egg-rr 98.8%

   \[\leadsto \left(\color{blue}{\frac{1}{\frac{-1}{\alpha}}} \cdot \alpha\right) \cdot \mathsf{log1p}\left(-u0\right) \]

9. Taylor expanded in u0 around 0

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

    1. *-lowering-*.f32 N/A

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

    2. +-commutative N/A

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

    3. unpow2 N/A

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

    4. accelerator-lowering-fma.f32 N/A

       \[\leadsto u0 \cdot \color{blue}{\mathsf{fma}\left(\alpha, \alpha, u0 \cdot \left(\frac{1}{3} \cdot \left({\alpha}^{2} \cdot u0\right) + \frac{1}{2} \cdot {\alpha}^{2}\right)\right)} \]

    5. distribute-rgt-in N/A

       \[\leadsto u0 \cdot \mathsf{fma}\left(\alpha, \alpha, \color{blue}{\left(\frac{1}{3} \cdot \left({\alpha}^{2} \cdot u0\right)\right) \cdot u0 + \left(\frac{1}{2} \cdot {\alpha}^{2}\right) \cdot u0}\right) \]

    6. *-commutative N/A

       \[\leadsto u0 \cdot \mathsf{fma}\left(\alpha, \alpha, \color{blue}{\left(\left({\alpha}^{2} \cdot u0\right) \cdot \frac{1}{3}\right)} \cdot u0 + \left(\frac{1}{2} \cdot {\alpha}^{2}\right) \cdot u0\right) \]

    7. associate-*l* N/A

       \[\leadsto u0 \cdot \mathsf{fma}\left(\alpha, \alpha, \color{blue}{\left({\alpha}^{2} \cdot u0\right) \cdot \left(\frac{1}{3} \cdot u0\right)} + \left(\frac{1}{2} \cdot {\alpha}^{2}\right) \cdot u0\right) \]

    8. associate-*r* N/A

       \[\leadsto u0 \cdot \mathsf{fma}\left(\alpha, \alpha, \left({\alpha}^{2} \cdot u0\right) \cdot \left(\frac{1}{3} \cdot u0\right) + \color{blue}{\frac{1}{2} \cdot \left({\alpha}^{2} \cdot u0\right)}\right) \]

    9. *-commutative N/A

       \[\leadsto u0 \cdot \mathsf{fma}\left(\alpha, \alpha, \left({\alpha}^{2} \cdot u0\right) \cdot \left(\frac{1}{3} \cdot u0\right) + \color{blue}{\left({\alpha}^{2} \cdot u0\right) \cdot \frac{1}{2}}\right) \]

    10. distribute-lft-out N/A

        \[\leadsto u0 \cdot \mathsf{fma}\left(\alpha, \alpha, \color{blue}{\left({\alpha}^{2} \cdot u0\right) \cdot \left(\frac{1}{3} \cdot u0 + \frac{1}{2}\right)}\right) \]

    11. +-commutative N/A

        \[\leadsto u0 \cdot \mathsf{fma}\left(\alpha, \alpha, \left({\alpha}^{2} \cdot u0\right) \cdot \color{blue}{\left(\frac{1}{2} + \frac{1}{3} \cdot u0\right)}\right) \]

    12. *-lowering-*.f32 N/A

        \[\leadsto u0 \cdot \mathsf{fma}\left(\alpha, \alpha, \color{blue}{\left({\alpha}^{2} \cdot u0\right) \cdot \left(\frac{1}{2} + \frac{1}{3} \cdot u0\right)}\right) \]

    13. *-lowering-*.f32 N/A

        \[\leadsto u0 \cdot \mathsf{fma}\left(\alpha, \alpha, \color{blue}{\left({\alpha}^{2} \cdot u0\right)} \cdot \left(\frac{1}{2} + \frac{1}{3} \cdot u0\right)\right) \]

    14. unpow2 N/A

        \[\leadsto u0 \cdot \mathsf{fma}\left(\alpha, \alpha, \left(\color{blue}{\left(\alpha \cdot \alpha\right)} \cdot u0\right) \cdot \left(\frac{1}{2} + \frac{1}{3} \cdot u0\right)\right) \]

    15. *-lowering-*.f32 N/A

        \[\leadsto u0 \cdot \mathsf{fma}\left(\alpha, \alpha, \left(\color{blue}{\left(\alpha \cdot \alpha\right)} \cdot u0\right) \cdot \left(\frac{1}{2} + \frac{1}{3} \cdot u0\right)\right) \]

    16. +-commutative N/A

        \[\leadsto u0 \cdot \mathsf{fma}\left(\alpha, \alpha, \left(\left(\alpha \cdot \alpha\right) \cdot u0\right) \cdot \color{blue}{\left(\frac{1}{3} \cdot u0 + \frac{1}{2}\right)}\right) \]

    17. *-commutative N/A

        \[\leadsto u0 \cdot \mathsf{fma}\left(\alpha, \alpha, \left(\left(\alpha \cdot \alpha\right) \cdot u0\right) \cdot \left(\color{blue}{u0 \cdot \frac{1}{3}} + \frac{1}{2}\right)\right) \]

    18. accelerator-lowering-fma.f32 91.4

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

11. Simplified 91.4%

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

12. Final simplification 91.4%

    \[\leadsto u0 \cdot \mathsf{fma}\left(\alpha, \alpha, \left(u0 \cdot \left(\alpha \cdot \alpha\right)\right) \cdot \mathsf{fma}\left(u0, 0.3333333333333333, 0.5\right)\right) \]
13. Add Preprocessing

Alternative 14: 91.5% accurate, 3.6× speedup

Math

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

FPCore

(FPCore (alpha u0)
 :precision binary32
 (*
  (* alpha (- alpha))
  (fma (* u0 (fma u0 -0.3333333333333333 -0.5)) u0 (- u0))))

C

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

Julia

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

Derivation

1. Initial program 57.3%

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

3. Taylor expanded in u0 around 0

   \[\leadsto \left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right) \cdot \color{blue}{\left(u0 \cdot \left(u0 \cdot \left(u0 \cdot \left(\frac{-1}{4} \cdot u0 - \frac{1}{3}\right) - \frac{1}{2}\right) - 1\right)\right)} \]
4. Step-by-step derivation

   1. *-lowering-*.f32 N/A

      \[\leadsto \left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right) \cdot \color{blue}{\left(u0 \cdot \left(u0 \cdot \left(u0 \cdot \left(\frac{-1}{4} \cdot u0 - \frac{1}{3}\right) - \frac{1}{2}\right) - 1\right)\right)} \]

   2. sub-neg N/A

      \[\leadsto \left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right) \cdot \left(u0 \cdot \color{blue}{\left(u0 \cdot \left(u0 \cdot \left(\frac{-1}{4} \cdot u0 - \frac{1}{3}\right) - \frac{1}{2}\right) + \left(\mathsf{neg}\left(1\right)\right)\right)}\right) \]

   3. metadata-eval N/A

      \[\leadsto \left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right) \cdot \left(u0 \cdot \left(u0 \cdot \left(u0 \cdot \left(\frac{-1}{4} \cdot u0 - \frac{1}{3}\right) - \frac{1}{2}\right) + \color{blue}{-1}\right)\right) \]

   4. accelerator-lowering-fma.f32 N/A

      \[\leadsto \left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right) \cdot \left(u0 \cdot \color{blue}{\mathsf{fma}\left(u0, u0 \cdot \left(\frac{-1}{4} \cdot u0 - \frac{1}{3}\right) - \frac{1}{2}, -1\right)}\right) \]

   5. sub-neg N/A

      \[\leadsto \left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right) \cdot \left(u0 \cdot \mathsf{fma}\left(u0, \color{blue}{u0 \cdot \left(\frac{-1}{4} \cdot u0 - \frac{1}{3}\right) + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}, -1\right)\right) \]

   6. metadata-eval N/A

      \[\leadsto \left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right) \cdot \left(u0 \cdot \mathsf{fma}\left(u0, u0 \cdot \left(\frac{-1}{4} \cdot u0 - \frac{1}{3}\right) + \color{blue}{\frac{-1}{2}}, -1\right)\right) \]

   7. accelerator-lowering-fma.f32 N/A

      \[\leadsto \left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right) \cdot \left(u0 \cdot \mathsf{fma}\left(u0, \color{blue}{\mathsf{fma}\left(u0, \frac{-1}{4} \cdot u0 - \frac{1}{3}, \frac{-1}{2}\right)}, -1\right)\right) \]

   8. sub-neg N/A

      \[\leadsto \left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right) \cdot \left(u0 \cdot \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, \color{blue}{\frac{-1}{4} \cdot u0 + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)}, \frac{-1}{2}\right), -1\right)\right) \]

   9. *-commutative N/A

      \[\leadsto \left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right) \cdot \left(u0 \cdot \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, \color{blue}{u0 \cdot \frac{-1}{4}} + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right), \frac{-1}{2}\right), -1\right)\right) \]

   10. metadata-eval N/A

       \[\leadsto \left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right) \cdot \left(u0 \cdot \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, u0 \cdot \frac{-1}{4} + \color{blue}{\frac{-1}{3}}, \frac{-1}{2}\right), -1\right)\right) \]

   11. accelerator-lowering-fma.f32 93.5

       \[\leadsto \left(\left(-\alpha\right) \cdot \alpha\right) \cdot \left(u0 \cdot \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, \color{blue}{\mathsf{fma}\left(u0, -0.25, -0.3333333333333333\right)}, -0.5\right), -1\right)\right) \]

5. Simplified 93.5%

   \[\leadsto \left(\left(-\alpha\right) \cdot \alpha\right) \cdot \color{blue}{\left(u0 \cdot \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, -0.25, -0.3333333333333333\right), -0.5\right), -1\right)\right)} \]
6. Step-by-step derivation

   1. distribute-rgt-in N/A

      \[\leadsto \left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right) \cdot \color{blue}{\left(\left(u0 \cdot \left(u0 \cdot \left(u0 \cdot \frac{-1}{4} + \frac{-1}{3}\right) + \frac{-1}{2}\right)\right) \cdot u0 + -1 \cdot u0\right)} \]

   2. neg-mul-1 N/A

      \[\leadsto \left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right) \cdot \left(\left(u0 \cdot \left(u0 \cdot \left(u0 \cdot \frac{-1}{4} + \frac{-1}{3}\right) + \frac{-1}{2}\right)\right) \cdot u0 + \color{blue}{\left(\mathsf{neg}\left(u0\right)\right)}\right) \]

   3. accelerator-lowering-fma.f32 N/A

      \[\leadsto \left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right) \cdot \color{blue}{\mathsf{fma}\left(u0 \cdot \left(u0 \cdot \left(u0 \cdot \frac{-1}{4} + \frac{-1}{3}\right) + \frac{-1}{2}\right), u0, \mathsf{neg}\left(u0\right)\right)} \]

   4. *-lowering-*.f32 N/A

      \[\leadsto \left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right) \cdot \mathsf{fma}\left(\color{blue}{u0 \cdot \left(u0 \cdot \left(u0 \cdot \frac{-1}{4} + \frac{-1}{3}\right) + \frac{-1}{2}\right)}, u0, \mathsf{neg}\left(u0\right)\right) \]

   5. accelerator-lowering-fma.f32 N/A

      \[\leadsto \left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right) \cdot \mathsf{fma}\left(u0 \cdot \color{blue}{\mathsf{fma}\left(u0, u0 \cdot \frac{-1}{4} + \frac{-1}{3}, \frac{-1}{2}\right)}, u0, \mathsf{neg}\left(u0\right)\right) \]

   6. accelerator-lowering-fma.f32 N/A

      \[\leadsto \left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right) \cdot \mathsf{fma}\left(u0 \cdot \mathsf{fma}\left(u0, \color{blue}{\mathsf{fma}\left(u0, \frac{-1}{4}, \frac{-1}{3}\right)}, \frac{-1}{2}\right), u0, \mathsf{neg}\left(u0\right)\right) \]

   7. neg-lowering-neg.f32 93.7

      \[\leadsto \left(\left(-\alpha\right) \cdot \alpha\right) \cdot \mathsf{fma}\left(u0 \cdot \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, -0.25, -0.3333333333333333\right), -0.5\right), u0, \color{blue}{-u0}\right) \]

7. Applied egg-rr 93.7%

   \[\leadsto \left(\left(-\alpha\right) \cdot \alpha\right) \cdot \color{blue}{\mathsf{fma}\left(u0 \cdot \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, -0.25, -0.3333333333333333\right), -0.5\right), u0, -u0\right)} \]

8. Taylor expanded in u0 around 0

   \[\leadsto \left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right) \cdot \mathsf{fma}\left(\color{blue}{u0 \cdot \left(\frac{-1}{3} \cdot u0 - \frac{1}{2}\right)}, u0, \mathsf{neg}\left(u0\right)\right) \]
9. Step-by-step derivation

   1. *-lowering-*.f32 N/A

      \[\leadsto \left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right) \cdot \mathsf{fma}\left(\color{blue}{u0 \cdot \left(\frac{-1}{3} \cdot u0 - \frac{1}{2}\right)}, u0, \mathsf{neg}\left(u0\right)\right) \]

   2. sub-neg N/A

      \[\leadsto \left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right) \cdot \mathsf{fma}\left(u0 \cdot \color{blue}{\left(\frac{-1}{3} \cdot u0 + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right)}, u0, \mathsf{neg}\left(u0\right)\right) \]

   3. *-commutative N/A

      \[\leadsto \left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right) \cdot \mathsf{fma}\left(u0 \cdot \left(\color{blue}{u0 \cdot \frac{-1}{3}} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right), u0, \mathsf{neg}\left(u0\right)\right) \]

   4. metadata-eval N/A

      \[\leadsto \left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right) \cdot \mathsf{fma}\left(u0 \cdot \left(u0 \cdot \frac{-1}{3} + \color{blue}{\frac{-1}{2}}\right), u0, \mathsf{neg}\left(u0\right)\right) \]

   5. accelerator-lowering-fma.f32 91.1

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

10. Simplified 91.1%

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

11. Final simplification 91.1%

    \[\leadsto \left(\alpha \cdot \left(-\alpha\right)\right) \cdot \mathsf{fma}\left(u0 \cdot \mathsf{fma}\left(u0, -0.3333333333333333, -0.5\right), u0, -u0\right) \]
12. Add Preprocessing

Alternative 15: 91.3% accurate, 3.6× speedup

Math

\[\begin{array}{l} \\ \left(\alpha \cdot \left(-\alpha\right)\right) \cdot \left(u0 \cdot \left(-1 + u0 \cdot \mathsf{fma}\left(u0, -0.3333333333333333, -0.5\right)\right)\right) \end{array} \]

FPCore

(FPCore (alpha u0)
 :precision binary32
 (*
  (* alpha (- alpha))
  (* u0 (+ -1.0 (* u0 (fma u0 -0.3333333333333333 -0.5))))))

C

float code(float alpha, float u0) {
	return (alpha * -alpha) * (u0 * (-1.0f + (u0 * fmaf(u0, -0.3333333333333333f, -0.5f))));
}

Julia

function code(alpha, u0)
	return Float32(Float32(alpha * Float32(-alpha)) * Float32(u0 * Float32(Float32(-1.0) + Float32(u0 * fma(u0, Float32(-0.3333333333333333), Float32(-0.5))))))
end

Derivation

1. Initial program 57.3%

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

3. Taylor expanded in u0 around 0

   \[\leadsto \left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right) \cdot \color{blue}{\left(u0 \cdot \left(u0 \cdot \left(u0 \cdot \left(\frac{-1}{4} \cdot u0 - \frac{1}{3}\right) - \frac{1}{2}\right) - 1\right)\right)} \]
4. Step-by-step derivation

   1. *-lowering-*.f32 N/A

      \[\leadsto \left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right) \cdot \color{blue}{\left(u0 \cdot \left(u0 \cdot \left(u0 \cdot \left(\frac{-1}{4} \cdot u0 - \frac{1}{3}\right) - \frac{1}{2}\right) - 1\right)\right)} \]

   2. sub-neg N/A

      \[\leadsto \left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right) \cdot \left(u0 \cdot \color{blue}{\left(u0 \cdot \left(u0 \cdot \left(\frac{-1}{4} \cdot u0 - \frac{1}{3}\right) - \frac{1}{2}\right) + \left(\mathsf{neg}\left(1\right)\right)\right)}\right) \]

   3. metadata-eval N/A

      \[\leadsto \left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right) \cdot \left(u0 \cdot \left(u0 \cdot \left(u0 \cdot \left(\frac{-1}{4} \cdot u0 - \frac{1}{3}\right) - \frac{1}{2}\right) + \color{blue}{-1}\right)\right) \]

   4. accelerator-lowering-fma.f32 N/A

      \[\leadsto \left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right) \cdot \left(u0 \cdot \color{blue}{\mathsf{fma}\left(u0, u0 \cdot \left(\frac{-1}{4} \cdot u0 - \frac{1}{3}\right) - \frac{1}{2}, -1\right)}\right) \]

   5. sub-neg N/A

      \[\leadsto \left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right) \cdot \left(u0 \cdot \mathsf{fma}\left(u0, \color{blue}{u0 \cdot \left(\frac{-1}{4} \cdot u0 - \frac{1}{3}\right) + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}, -1\right)\right) \]

   6. metadata-eval N/A

      \[\leadsto \left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right) \cdot \left(u0 \cdot \mathsf{fma}\left(u0, u0 \cdot \left(\frac{-1}{4} \cdot u0 - \frac{1}{3}\right) + \color{blue}{\frac{-1}{2}}, -1\right)\right) \]

   7. accelerator-lowering-fma.f32 N/A

      \[\leadsto \left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right) \cdot \left(u0 \cdot \mathsf{fma}\left(u0, \color{blue}{\mathsf{fma}\left(u0, \frac{-1}{4} \cdot u0 - \frac{1}{3}, \frac{-1}{2}\right)}, -1\right)\right) \]

   8. sub-neg N/A

      \[\leadsto \left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right) \cdot \left(u0 \cdot \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, \color{blue}{\frac{-1}{4} \cdot u0 + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)}, \frac{-1}{2}\right), -1\right)\right) \]

   9. *-commutative N/A

      \[\leadsto \left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right) \cdot \left(u0 \cdot \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, \color{blue}{u0 \cdot \frac{-1}{4}} + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right), \frac{-1}{2}\right), -1\right)\right) \]

   10. metadata-eval N/A

       \[\leadsto \left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right) \cdot \left(u0 \cdot \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, u0 \cdot \frac{-1}{4} + \color{blue}{\frac{-1}{3}}, \frac{-1}{2}\right), -1\right)\right) \]

   11. accelerator-lowering-fma.f32 93.5

       \[\leadsto \left(\left(-\alpha\right) \cdot \alpha\right) \cdot \left(u0 \cdot \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, \color{blue}{\mathsf{fma}\left(u0, -0.25, -0.3333333333333333\right)}, -0.5\right), -1\right)\right) \]

5. Simplified 93.5%

   \[\leadsto \left(\left(-\alpha\right) \cdot \alpha\right) \cdot \color{blue}{\left(u0 \cdot \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, -0.25, -0.3333333333333333\right), -0.5\right), -1\right)\right)} \]
6. Step-by-step derivation

   1. distribute-rgt-in N/A

      \[\leadsto \left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right) \cdot \color{blue}{\left(\left(u0 \cdot \left(u0 \cdot \left(u0 \cdot \frac{-1}{4} + \frac{-1}{3}\right) + \frac{-1}{2}\right)\right) \cdot u0 + -1 \cdot u0\right)} \]

   2. neg-mul-1 N/A

      \[\leadsto \left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right) \cdot \left(\left(u0 \cdot \left(u0 \cdot \left(u0 \cdot \frac{-1}{4} + \frac{-1}{3}\right) + \frac{-1}{2}\right)\right) \cdot u0 + \color{blue}{\left(\mathsf{neg}\left(u0\right)\right)}\right) \]

   3. accelerator-lowering-fma.f32 N/A

      \[\leadsto \left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right) \cdot \color{blue}{\mathsf{fma}\left(u0 \cdot \left(u0 \cdot \left(u0 \cdot \frac{-1}{4} + \frac{-1}{3}\right) + \frac{-1}{2}\right), u0, \mathsf{neg}\left(u0\right)\right)} \]

   4. *-lowering-*.f32 N/A

      \[\leadsto \left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right) \cdot \mathsf{fma}\left(\color{blue}{u0 \cdot \left(u0 \cdot \left(u0 \cdot \frac{-1}{4} + \frac{-1}{3}\right) + \frac{-1}{2}\right)}, u0, \mathsf{neg}\left(u0\right)\right) \]

   5. accelerator-lowering-fma.f32 N/A

      \[\leadsto \left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right) \cdot \mathsf{fma}\left(u0 \cdot \color{blue}{\mathsf{fma}\left(u0, u0 \cdot \frac{-1}{4} + \frac{-1}{3}, \frac{-1}{2}\right)}, u0, \mathsf{neg}\left(u0\right)\right) \]

   6. accelerator-lowering-fma.f32 N/A

      \[\leadsto \left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right) \cdot \mathsf{fma}\left(u0 \cdot \mathsf{fma}\left(u0, \color{blue}{\mathsf{fma}\left(u0, \frac{-1}{4}, \frac{-1}{3}\right)}, \frac{-1}{2}\right), u0, \mathsf{neg}\left(u0\right)\right) \]

   7. neg-lowering-neg.f32 93.7

      \[\leadsto \left(\left(-\alpha\right) \cdot \alpha\right) \cdot \mathsf{fma}\left(u0 \cdot \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, -0.25, -0.3333333333333333\right), -0.5\right), u0, \color{blue}{-u0}\right) \]

7. Applied egg-rr 93.7%

   \[\leadsto \left(\left(-\alpha\right) \cdot \alpha\right) \cdot \color{blue}{\mathsf{fma}\left(u0 \cdot \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, -0.25, -0.3333333333333333\right), -0.5\right), u0, -u0\right)} \]
8. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right) \cdot \color{blue}{\left(\left(\mathsf{neg}\left(u0\right)\right) + \left(u0 \cdot \left(u0 \cdot \left(u0 \cdot \frac{-1}{4} + \frac{-1}{3}\right) + \frac{-1}{2}\right)\right) \cdot u0\right)} \]

   2. neg-mul-1 N/A

      \[\leadsto \left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right) \cdot \left(\color{blue}{-1 \cdot u0} + \left(u0 \cdot \left(u0 \cdot \left(u0 \cdot \frac{-1}{4} + \frac{-1}{3}\right) + \frac{-1}{2}\right)\right) \cdot u0\right) \]

   3. distribute-rgt-out N/A

      \[\leadsto \left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right) \cdot \color{blue}{\left(u0 \cdot \left(-1 + u0 \cdot \left(u0 \cdot \left(u0 \cdot \frac{-1}{4} + \frac{-1}{3}\right) + \frac{-1}{2}\right)\right)\right)} \]

   4. *-lowering-*.f32 N/A

      \[\leadsto \left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right) \cdot \color{blue}{\left(u0 \cdot \left(-1 + u0 \cdot \left(u0 \cdot \left(u0 \cdot \frac{-1}{4} + \frac{-1}{3}\right) + \frac{-1}{2}\right)\right)\right)} \]

   5. +-lowering-+.f32 N/A

      \[\leadsto \left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right) \cdot \left(u0 \cdot \color{blue}{\left(-1 + u0 \cdot \left(u0 \cdot \left(u0 \cdot \frac{-1}{4} + \frac{-1}{3}\right) + \frac{-1}{2}\right)\right)}\right) \]

   6. *-lowering-*.f32 N/A

      \[\leadsto \left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right) \cdot \left(u0 \cdot \left(-1 + \color{blue}{u0 \cdot \left(u0 \cdot \left(u0 \cdot \frac{-1}{4} + \frac{-1}{3}\right) + \frac{-1}{2}\right)}\right)\right) \]

   7. accelerator-lowering-fma.f32 N/A

      \[\leadsto \left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right) \cdot \left(u0 \cdot \left(-1 + u0 \cdot \color{blue}{\mathsf{fma}\left(u0, u0 \cdot \frac{-1}{4} + \frac{-1}{3}, \frac{-1}{2}\right)}\right)\right) \]

   8. accelerator-lowering-fma.f32 93.5

      \[\leadsto \left(\left(-\alpha\right) \cdot \alpha\right) \cdot \left(u0 \cdot \left(-1 + u0 \cdot \mathsf{fma}\left(u0, \color{blue}{\mathsf{fma}\left(u0, -0.25, -0.3333333333333333\right)}, -0.5\right)\right)\right) \]

9. Applied egg-rr 93.5%

   \[\leadsto \left(\left(-\alpha\right) \cdot \alpha\right) \cdot \color{blue}{\left(u0 \cdot \left(-1 + u0 \cdot \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, -0.25, -0.3333333333333333\right), -0.5\right)\right)\right)} \]

10. Taylor expanded in u0 around 0

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

    1. *-lowering-*.f32 N/A

       \[\leadsto \left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right) \cdot \left(u0 \cdot \left(-1 + \color{blue}{u0 \cdot \left(\frac{-1}{3} \cdot u0 - \frac{1}{2}\right)}\right)\right) \]

    2. sub-neg N/A

       \[\leadsto \left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right) \cdot \left(u0 \cdot \left(-1 + u0 \cdot \color{blue}{\left(\frac{-1}{3} \cdot u0 + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right)}\right)\right) \]

    3. *-commutative N/A

       \[\leadsto \left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right) \cdot \left(u0 \cdot \left(-1 + u0 \cdot \left(\color{blue}{u0 \cdot \frac{-1}{3}} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right)\right)\right) \]

    4. metadata-eval N/A

       \[\leadsto \left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right) \cdot \left(u0 \cdot \left(-1 + u0 \cdot \left(u0 \cdot \frac{-1}{3} + \color{blue}{\frac{-1}{2}}\right)\right)\right) \]

    5. accelerator-lowering-fma.f32 91.0

       \[\leadsto \left(\left(-\alpha\right) \cdot \alpha\right) \cdot \left(u0 \cdot \left(-1 + u0 \cdot \color{blue}{\mathsf{fma}\left(u0, -0.3333333333333333, -0.5\right)}\right)\right) \]

12. Simplified 91.0%

    \[\leadsto \left(\left(-\alpha\right) \cdot \alpha\right) \cdot \left(u0 \cdot \left(-1 + \color{blue}{u0 \cdot \mathsf{fma}\left(u0, -0.3333333333333333, -0.5\right)}\right)\right) \]

13. Final simplification 91.0%

    \[\leadsto \left(\alpha \cdot \left(-\alpha\right)\right) \cdot \left(u0 \cdot \left(-1 + u0 \cdot \mathsf{fma}\left(u0, -0.3333333333333333, -0.5\right)\right)\right) \]
14. Add Preprocessing

Alternative 16: 91.3% accurate, 3.9× speedup

Math

\[\begin{array}{l} \\ \left(\alpha \cdot \left(-\alpha\right)\right) \cdot \left(u0 \cdot \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, -0.3333333333333333, -0.5\right), -1\right)\right) \end{array} \]

FPCore

(FPCore (alpha u0)
 :precision binary32
 (*
  (* alpha (- alpha))
  (* u0 (fma u0 (fma u0 -0.3333333333333333 -0.5) -1.0))))

C

float code(float alpha, float u0) {
	return (alpha * -alpha) * (u0 * fmaf(u0, fmaf(u0, -0.3333333333333333f, -0.5f), -1.0f));
}

Julia

function code(alpha, u0)
	return Float32(Float32(alpha * Float32(-alpha)) * Float32(u0 * fma(u0, fma(u0, Float32(-0.3333333333333333), Float32(-0.5)), Float32(-1.0))))
end

Derivation

1. Initial program 57.3%

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

3. Taylor expanded in u0 around 0

   \[\leadsto \left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right) \cdot \color{blue}{\left(u0 \cdot \left(u0 \cdot \left(\frac{-1}{3} \cdot u0 - \frac{1}{2}\right) - 1\right)\right)} \]
4. Step-by-step derivation

   1. *-lowering-*.f32 N/A

      \[\leadsto \left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right) \cdot \color{blue}{\left(u0 \cdot \left(u0 \cdot \left(\frac{-1}{3} \cdot u0 - \frac{1}{2}\right) - 1\right)\right)} \]

   2. sub-neg N/A

      \[\leadsto \left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right) \cdot \left(u0 \cdot \color{blue}{\left(u0 \cdot \left(\frac{-1}{3} \cdot u0 - \frac{1}{2}\right) + \left(\mathsf{neg}\left(1\right)\right)\right)}\right) \]

   3. metadata-eval N/A

      \[\leadsto \left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right) \cdot \left(u0 \cdot \left(u0 \cdot \left(\frac{-1}{3} \cdot u0 - \frac{1}{2}\right) + \color{blue}{-1}\right)\right) \]

   4. accelerator-lowering-fma.f32 N/A

      \[\leadsto \left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right) \cdot \left(u0 \cdot \color{blue}{\mathsf{fma}\left(u0, \frac{-1}{3} \cdot u0 - \frac{1}{2}, -1\right)}\right) \]

   5. sub-neg N/A

      \[\leadsto \left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right) \cdot \left(u0 \cdot \mathsf{fma}\left(u0, \color{blue}{\frac{-1}{3} \cdot u0 + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}, -1\right)\right) \]

   6. *-commutative N/A

      \[\leadsto \left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right) \cdot \left(u0 \cdot \mathsf{fma}\left(u0, \color{blue}{u0 \cdot \frac{-1}{3}} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right), -1\right)\right) \]

   7. metadata-eval N/A

      \[\leadsto \left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right) \cdot \left(u0 \cdot \mathsf{fma}\left(u0, u0 \cdot \frac{-1}{3} + \color{blue}{\frac{-1}{2}}, -1\right)\right) \]

   8. accelerator-lowering-fma.f32 91.0

      \[\leadsto \left(\left(-\alpha\right) \cdot \alpha\right) \cdot \left(u0 \cdot \mathsf{fma}\left(u0, \color{blue}{\mathsf{fma}\left(u0, -0.3333333333333333, -0.5\right)}, -1\right)\right) \]

5. Simplified 91.0%

   \[\leadsto \left(\left(-\alpha\right) \cdot \alpha\right) \cdot \color{blue}{\left(u0 \cdot \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, -0.3333333333333333, -0.5\right), -1\right)\right)} \]

6. Final simplification 91.0%

   \[\leadsto \left(\alpha \cdot \left(-\alpha\right)\right) \cdot \left(u0 \cdot \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, -0.3333333333333333, -0.5\right), -1\right)\right) \]
7. Add Preprocessing

Alternative 17: 87.4% accurate, 4.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 57.3%

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

   1. sub-neg N/A

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

   2. accelerator-lowering-log1p.f32 N/A

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

   3. neg-lowering-neg.f32 98.9

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

4. Applied egg-rr 98.9%

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

5. Taylor expanded in u0 around 0

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

   1. *-lowering-*.f32 N/A

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

   2. +-commutative N/A

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

   3. *-commutative N/A

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

   4. *-commutative N/A

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

   5. associate-*r* N/A

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

   6. *-commutative N/A

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

   7. unpow2 N/A

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

   8. accelerator-lowering-fma.f32 N/A

      \[\leadsto u0 \cdot \color{blue}{\mathsf{fma}\left(\alpha, \alpha, u0 \cdot \left(\frac{1}{2} \cdot {\alpha}^{2}\right)\right)} \]

   9. *-commutative N/A

      \[\leadsto u0 \cdot \mathsf{fma}\left(\alpha, \alpha, u0 \cdot \color{blue}{\left({\alpha}^{2} \cdot \frac{1}{2}\right)}\right) \]

   10. associate-*r* N/A

       \[\leadsto u0 \cdot \mathsf{fma}\left(\alpha, \alpha, \color{blue}{\left(u0 \cdot {\alpha}^{2}\right) \cdot \frac{1}{2}}\right) \]

   11. *-commutative N/A

       \[\leadsto u0 \cdot \mathsf{fma}\left(\alpha, \alpha, \color{blue}{\left({\alpha}^{2} \cdot u0\right)} \cdot \frac{1}{2}\right) \]

   12. *-lowering-*.f32 N/A

       \[\leadsto u0 \cdot \mathsf{fma}\left(\alpha, \alpha, \color{blue}{\left({\alpha}^{2} \cdot u0\right) \cdot \frac{1}{2}}\right) \]

   13. *-lowering-*.f32 N/A

       \[\leadsto u0 \cdot \mathsf{fma}\left(\alpha, \alpha, \color{blue}{\left({\alpha}^{2} \cdot u0\right)} \cdot \frac{1}{2}\right) \]

   14. unpow2 N/A

       \[\leadsto u0 \cdot \mathsf{fma}\left(\alpha, \alpha, \left(\color{blue}{\left(\alpha \cdot \alpha\right)} \cdot u0\right) \cdot \frac{1}{2}\right) \]

   15. *-lowering-*.f32 86.7

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

7. Simplified 86.7%

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

8. Final simplification 86.7%

   \[\leadsto u0 \cdot \mathsf{fma}\left(\alpha, \alpha, \left(u0 \cdot \left(\alpha \cdot \alpha\right)\right) \cdot 0.5\right) \]
9. Add Preprocessing

Alternative 18: 87.3% accurate, 5.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 57.3%

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

3. Taylor expanded in u0 around 0

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

   1. distribute-lft-in N/A

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

   2. associate-*r* N/A

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

   3. *-commutative N/A

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

   4. associate-*r* N/A

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

   5. distribute-rgt-out N/A

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

   6. *-lowering-*.f32 N/A

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

   7. unpow2 N/A

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

   8. *-lowering-*.f32 N/A

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

   9. associate-*r* N/A

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

   10. accelerator-lowering-fma.f32 N/A

       \[\leadsto \left(\alpha \cdot \alpha\right) \cdot \color{blue}{\mathsf{fma}\left(u0, \frac{1}{2} \cdot u0, u0\right)} \]

   11. *-commutative N/A

       \[\leadsto \left(\alpha \cdot \alpha\right) \cdot \mathsf{fma}\left(u0, \color{blue}{u0 \cdot \frac{1}{2}}, u0\right) \]

   12. *-lowering-*.f32 86.5

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

5. Simplified 86.5%

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

   1. associate-*l* N/A

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

   2. *-commutative N/A

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

   3. *-lowering-*.f32 N/A

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

   4. *-lowering-*.f32 N/A

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

   5. accelerator-lowering-fma.f32 N/A

      \[\leadsto \left(\alpha \cdot \color{blue}{\mathsf{fma}\left(u0, u0 \cdot \frac{1}{2}, u0\right)}\right) \cdot \alpha \]

   6. *-lowering-*.f32 86.5

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

7. Applied egg-rr 86.5%

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

8. Final simplification 86.5%

   \[\leadsto \alpha \cdot \left(\alpha \cdot \mathsf{fma}\left(u0, u0 \cdot 0.5, u0\right)\right) \]
9. Add Preprocessing

Alternative 19: 87.3% accurate, 5.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 57.3%

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

3. Taylor expanded in u0 around 0

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

   1. distribute-lft-in N/A

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

   2. associate-*r* N/A

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

   3. *-commutative N/A

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

   4. associate-*r* N/A

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

   5. distribute-rgt-out N/A

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

   6. *-lowering-*.f32 N/A

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

   7. unpow2 N/A

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

   8. *-lowering-*.f32 N/A

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

   9. associate-*r* N/A

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

   10. accelerator-lowering-fma.f32 N/A

       \[\leadsto \left(\alpha \cdot \alpha\right) \cdot \color{blue}{\mathsf{fma}\left(u0, \frac{1}{2} \cdot u0, u0\right)} \]

   11. *-commutative N/A

       \[\leadsto \left(\alpha \cdot \alpha\right) \cdot \mathsf{fma}\left(u0, \color{blue}{u0 \cdot \frac{1}{2}}, u0\right) \]

   12. *-lowering-*.f32 86.5

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

5. Simplified 86.5%

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

Alternative 20: 87.1% accurate, 5.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 57.3%

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

3. Taylor expanded in u0 around 0

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

   1. distribute-lft-in N/A

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

   2. associate-*r* N/A

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

   3. *-commutative N/A

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

   4. associate-*r* N/A

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

   5. distribute-rgt-out N/A

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

   6. *-lowering-*.f32 N/A

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

   7. unpow2 N/A

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

   8. *-lowering-*.f32 N/A

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

   9. associate-*r* N/A

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

   10. accelerator-lowering-fma.f32 N/A

       \[\leadsto \left(\alpha \cdot \alpha\right) \cdot \color{blue}{\mathsf{fma}\left(u0, \frac{1}{2} \cdot u0, u0\right)} \]

   11. *-commutative N/A

       \[\leadsto \left(\alpha \cdot \alpha\right) \cdot \mathsf{fma}\left(u0, \color{blue}{u0 \cdot \frac{1}{2}}, u0\right) \]

   12. *-lowering-*.f32 86.5

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

5. Simplified 86.5%

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

   1. *-commutative N/A

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

   2. distribute-lft1-in N/A

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

   3. *-lowering-*.f32 N/A

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

   4. accelerator-lowering-fma.f32 86.4

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

7. Applied egg-rr 86.4%

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

8. Final simplification 86.4%

   \[\leadsto \left(\alpha \cdot \alpha\right) \cdot \left(u0 \cdot \mathsf{fma}\left(u0, 0.5, 1\right)\right) \]
9. Add Preprocessing

Alternative 21: 74.5% accurate, 10.5× 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 57.3%

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

3. Taylor expanded in u0 around 0

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

   1. *-commutative N/A

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

   2. *-lowering-*.f32 N/A

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

   3. unpow2 N/A

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

   4. *-lowering-*.f32 73.7

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

5. Simplified 73.7%

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

   1. associate-*r* N/A

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

   2. *-commutative N/A

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

   3. *-lowering-*.f32 N/A

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

   4. *-lowering-*.f32 73.7

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

7. Applied egg-rr 73.7%

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

8. Final simplification 73.7%

   \[\leadsto \alpha \cdot \left(\alpha \cdot u0\right) \]
9. Add Preprocessing

Alternative 22: 74.5% accurate, 10.5× 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 57.3%

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

3. Taylor expanded in u0 around 0

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

   1. *-commutative N/A

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

   2. *-lowering-*.f32 N/A

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

   3. unpow2 N/A

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

   4. *-lowering-*.f32 73.7

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

5. Simplified 73.7%

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

Reproduce

herbie shell --seed 2024201 
(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))))