Beckmann Distribution sample, tan2theta, alphax == alphay

Percentage Accurate: 55.9% → 99.0%

Time: 11.0s

Alternatives: 13

Speedup: 21.6×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Initial Program: 55.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Alternative 1: 99.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 56.5%

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

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

      \[\leadsto \mathsf{*.f32}\left(\left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right), \color{blue}{\log \left(1 - u0\right)}\right) \]

   2. *-commutative N/A

      \[\leadsto \mathsf{*.f32}\left(\left(\alpha \cdot \left(\mathsf{neg}\left(\alpha\right)\right)\right), \log \color{blue}{\left(1 - u0\right)}\right) \]

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

      \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(\alpha, \left(\mathsf{neg}\left(\alpha\right)\right)\right), \log \color{blue}{\left(1 - u0\right)}\right) \]

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

      \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(\alpha, \mathsf{neg.f32}\left(\alpha\right)\right), \log \left(1 - \color{blue}{u0}\right)\right) \]

   5. sub-neg N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(\alpha, \mathsf{neg.f32}\left(\alpha\right)\right), \log \left(1 + \left(\mathsf{neg}\left(u0\right)\right)\right)\right) \]

   6. log1p-define N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(\alpha, \mathsf{neg.f32}\left(\alpha\right)\right), \left(\mathsf{log1p}\left(\mathsf{neg}\left(u0\right)\right)\right)\right) \]

   7. log1p-lowering-log1p.f32 N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(\alpha, \mathsf{neg.f32}\left(\alpha\right)\right), \mathsf{log1p.f32}\left(\left(\mathsf{neg}\left(u0\right)\right)\right)\right) \]

   8. neg-lowering-neg.f32 99.0%

      \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(\alpha, \mathsf{neg.f32}\left(\alpha\right)\right), \mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right)\right) \]

3. Simplified 99.0%

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

   1. distribute-rgt-neg-out N/A

      \[\leadsto \mathsf{*.f32}\left(\left(\mathsf{neg}\left(\alpha \cdot \alpha\right)\right), \mathsf{log1p.f32}\left(\color{blue}{\mathsf{neg.f32}\left(u0\right)}\right)\right) \]

   2. neg-sub0 N/A

      \[\leadsto \mathsf{*.f32}\left(\left(0 - \alpha \cdot \alpha\right), \mathsf{log1p.f32}\left(\color{blue}{\mathsf{neg.f32}\left(u0\right)}\right)\right) \]

   3. flip-- N/A

      \[\leadsto \mathsf{*.f32}\left(\left(\frac{0 \cdot 0 - \left(\alpha \cdot \alpha\right) \cdot \left(\alpha \cdot \alpha\right)}{0 + \alpha \cdot \alpha}\right), \mathsf{log1p.f32}\left(\color{blue}{\mathsf{neg.f32}\left(u0\right)}\right)\right) \]

   4. +-commutative N/A

      \[\leadsto \mathsf{*.f32}\left(\left(\frac{0 \cdot 0 - \left(\alpha \cdot \alpha\right) \cdot \left(\alpha \cdot \alpha\right)}{\alpha \cdot \alpha + 0}\right), \mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right)\right) \]

   5. mul0-lft N/A

      \[\leadsto \mathsf{*.f32}\left(\left(\frac{0 \cdot 0 - \left(\alpha \cdot \alpha\right) \cdot \left(\alpha \cdot \alpha\right)}{\alpha \cdot \alpha + 0 \cdot \alpha}\right), \mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right)\right) \]

   6. distribute-rgt-out N/A

      \[\leadsto \mathsf{*.f32}\left(\left(\frac{0 \cdot 0 - \left(\alpha \cdot \alpha\right) \cdot \left(\alpha \cdot \alpha\right)}{\alpha \cdot \left(\alpha + 0\right)}\right), \mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right)\right) \]

   7. +-commutative N/A

      \[\leadsto \mathsf{*.f32}\left(\left(\frac{0 \cdot 0 - \left(\alpha \cdot \alpha\right) \cdot \left(\alpha \cdot \alpha\right)}{\alpha \cdot \left(0 + \alpha\right)}\right), \mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right)\right) \]

   8. +-lft-identity N/A

      \[\leadsto \mathsf{*.f32}\left(\left(\frac{0 \cdot 0 - \left(\alpha \cdot \alpha\right) \cdot \left(\alpha \cdot \alpha\right)}{\alpha \cdot \alpha}\right), \mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right)\right) \]

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

      \[\leadsto \mathsf{*.f32}\left(\mathsf{/.f32}\left(\left(0 \cdot 0 - \left(\alpha \cdot \alpha\right) \cdot \left(\alpha \cdot \alpha\right)\right), \left(\alpha \cdot \alpha\right)\right), \mathsf{log1p.f32}\left(\color{blue}{\mathsf{neg.f32}\left(u0\right)}\right)\right) \]

   10. metadata-eval N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{/.f32}\left(\left(0 - \left(\alpha \cdot \alpha\right) \cdot \left(\alpha \cdot \alpha\right)\right), \left(\alpha \cdot \alpha\right)\right), \mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f32}\left(\mathsf{/.f32}\left(\mathsf{\_.f32}\left(0, \left(\left(\alpha \cdot \alpha\right) \cdot \left(\alpha \cdot \alpha\right)\right)\right), \left(\alpha \cdot \alpha\right)\right), \mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(\color{blue}{u0}\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f32}\left(\mathsf{/.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{*.f32}\left(\left(\alpha \cdot \alpha\right), \left(\alpha \cdot \alpha\right)\right)\right), \left(\alpha \cdot \alpha\right)\right), \mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f32}\left(\mathsf{/.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{*.f32}\left(\mathsf{*.f32}\left(\alpha, \alpha\right), \left(\alpha \cdot \alpha\right)\right)\right), \left(\alpha \cdot \alpha\right)\right), \mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f32}\left(\mathsf{/.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{*.f32}\left(\mathsf{*.f32}\left(\alpha, \alpha\right), \mathsf{*.f32}\left(\alpha, \alpha\right)\right)\right), \left(\alpha \cdot \alpha\right)\right), \mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right)\right) \]

   15. *-lowering-*.f32 98.9%

       \[\leadsto \mathsf{*.f32}\left(\mathsf{/.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{*.f32}\left(\mathsf{*.f32}\left(\alpha, \alpha\right), \mathsf{*.f32}\left(\alpha, \alpha\right)\right)\right), \mathsf{*.f32}\left(\alpha, \alpha\right)\right), \mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right)\right) \]

6. Applied egg-rr 98.9%

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

   1. sub0-neg N/A

      \[\leadsto \mathsf{*.f32}\left(\left(\frac{\mathsf{neg}\left(\left(\alpha \cdot \alpha\right) \cdot \left(\alpha \cdot \alpha\right)\right)}{\alpha \cdot \alpha}\right), \mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(\color{blue}{u0}\right)\right)\right) \]

   2. associate-*l* N/A

      \[\leadsto \mathsf{*.f32}\left(\left(\frac{\mathsf{neg}\left(\alpha \cdot \left(\alpha \cdot \left(\alpha \cdot \alpha\right)\right)\right)}{\alpha \cdot \alpha}\right), \mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right)\right) \]

   3. distribute-rgt-neg-in N/A

      \[\leadsto \mathsf{*.f32}\left(\left(\frac{\alpha \cdot \left(\mathsf{neg}\left(\alpha \cdot \left(\alpha \cdot \alpha\right)\right)\right)}{\alpha \cdot \alpha}\right), \mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(\color{blue}{u0}\right)\right)\right) \]

   4. sub0-neg N/A

      \[\leadsto \mathsf{*.f32}\left(\left(\frac{\alpha \cdot \left(0 - \alpha \cdot \left(\alpha \cdot \alpha\right)\right)}{\alpha \cdot \alpha}\right), \mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right)\right) \]

   5. associate-*r/ N/A

      \[\leadsto \mathsf{*.f32}\left(\left(\alpha \cdot \frac{0 - \alpha \cdot \left(\alpha \cdot \alpha\right)}{\alpha \cdot \alpha}\right), \mathsf{log1p.f32}\left(\color{blue}{\mathsf{neg.f32}\left(u0\right)}\right)\right) \]

   6. mul0-lft N/A

      \[\leadsto \mathsf{*.f32}\left(\left(\alpha \cdot \frac{0 \cdot \left(\left(\alpha \cdot \alpha\right) \cdot \left(\alpha \cdot \alpha\right)\right) - \alpha \cdot \left(\alpha \cdot \alpha\right)}{\alpha \cdot \alpha}\right), \mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right)\right) \]

   7. associate-*r* N/A

      \[\leadsto \mathsf{*.f32}\left(\left(\alpha \cdot \frac{0 \cdot \left(\left(\left(\alpha \cdot \alpha\right) \cdot \alpha\right) \cdot \alpha\right) - \alpha \cdot \left(\alpha \cdot \alpha\right)}{\alpha \cdot \alpha}\right), \mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right)\right) \]

   8. associate-*r* N/A

      \[\leadsto \mathsf{*.f32}\left(\left(\alpha \cdot \frac{0 \cdot \left(\left(\alpha \cdot \left(\alpha \cdot \alpha\right)\right) \cdot \alpha\right) - \alpha \cdot \left(\alpha \cdot \alpha\right)}{\alpha \cdot \alpha}\right), \mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right)\right) \]

   9. associate-*r* N/A

      \[\leadsto \mathsf{*.f32}\left(\left(\alpha \cdot \frac{\left(0 \cdot \left(\alpha \cdot \left(\alpha \cdot \alpha\right)\right)\right) \cdot \alpha - \alpha \cdot \left(\alpha \cdot \alpha\right)}{\alpha \cdot \alpha}\right), \mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right)\right) \]

   10. mul0-lft N/A

       \[\leadsto \mathsf{*.f32}\left(\left(\alpha \cdot \frac{0 \cdot \alpha - \alpha \cdot \left(\alpha \cdot \alpha\right)}{\alpha \cdot \alpha}\right), \mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right)\right) \]

   11. frac-sub N/A

       \[\leadsto \mathsf{*.f32}\left(\left(\alpha \cdot \left(\frac{0}{\alpha} - \frac{\alpha \cdot \alpha}{\alpha}\right)\right), \mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right)\right) \]

   12. div-sub N/A

       \[\leadsto \mathsf{*.f32}\left(\left(\alpha \cdot \frac{0 - \alpha \cdot \alpha}{\alpha}\right), \mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right)\right) \]

   13. neg-sub0 N/A

       \[\leadsto \mathsf{*.f32}\left(\left(\alpha \cdot \frac{\mathsf{neg}\left(\alpha \cdot \alpha\right)}{\alpha}\right), \mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right)\right) \]

   14. clear-num N/A

       \[\leadsto \mathsf{*.f32}\left(\left(\alpha \cdot \frac{1}{\frac{\alpha}{\mathsf{neg}\left(\alpha \cdot \alpha\right)}}\right), \mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right)\right) \]

   15. un-div-inv N/A

       \[\leadsto \mathsf{*.f32}\left(\left(\frac{\alpha}{\frac{\alpha}{\mathsf{neg}\left(\alpha \cdot \alpha\right)}}\right), \mathsf{log1p.f32}\left(\color{blue}{\mathsf{neg.f32}\left(u0\right)}\right)\right) \]

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

       \[\leadsto \mathsf{*.f32}\left(\mathsf{/.f32}\left(\alpha, \left(\frac{\alpha}{\mathsf{neg}\left(\alpha \cdot \alpha\right)}\right)\right), \mathsf{log1p.f32}\left(\color{blue}{\mathsf{neg.f32}\left(u0\right)}\right)\right) \]

   17. distribute-frac-neg2 N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{/.f32}\left(\alpha, \left(\mathsf{neg}\left(\frac{\alpha}{\alpha \cdot \alpha}\right)\right)\right), \mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right)\right) \]

   18. neg-mul-1 N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{/.f32}\left(\alpha, \left(-1 \cdot \frac{\alpha}{\alpha \cdot \alpha}\right)\right), \mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right)\right) \]

   19. div-inv N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{/.f32}\left(\alpha, \left(-1 \cdot \left(\alpha \cdot \frac{1}{\alpha \cdot \alpha}\right)\right)\right), \mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right)\right) \]

   20. *-commutative N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{/.f32}\left(\alpha, \left(-1 \cdot \left(\frac{1}{\alpha \cdot \alpha} \cdot \alpha\right)\right)\right), \mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right)\right) \]

8. Applied egg-rr 98.8%

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

   1. div-inv N/A

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

   2. associate-/r* N/A

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

   3. clear-num N/A

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

   4. frac-2neg N/A

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

   5. metadata-eval N/A

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

   6. remove-double-div N/A

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

   7. associate-*l/ N/A

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

   8. div-inv N/A

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

   9. remove-double-div N/A

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

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

       \[\leadsto \mathsf{*.f32}\left(\left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \log \left(1 + \left(\mathsf{neg}\left(u0\right)\right)\right)\right), \color{blue}{\alpha}\right) \]

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

       \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(\left(\mathsf{neg}\left(\alpha\right)\right), \log \left(1 + \left(\mathsf{neg}\left(u0\right)\right)\right)\right), \alpha\right) \]

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

       \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{neg.f32}\left(\alpha\right), \log \left(1 + \left(\mathsf{neg}\left(u0\right)\right)\right)\right), \alpha\right) \]

   13. log1p-define N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{neg.f32}\left(\alpha\right), \left(\mathsf{log1p}\left(\mathsf{neg}\left(u0\right)\right)\right)\right), \alpha\right) \]

   14. log1p-lowering-log1p.f32 N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{neg.f32}\left(\alpha\right), \mathsf{log1p.f32}\left(\left(\mathsf{neg}\left(u0\right)\right)\right)\right), \alpha\right) \]

   15. neg-lowering-neg.f32 99.0%

       \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{neg.f32}\left(\alpha\right), \mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right)\right), \alpha\right) \]

10. Applied egg-rr 99.0%

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

11. Final simplification 99.0%

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

Alternative 2: 99.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 56.5%

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

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

      \[\leadsto \mathsf{*.f32}\left(\left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right), \color{blue}{\log \left(1 - u0\right)}\right) \]

   2. *-commutative N/A

      \[\leadsto \mathsf{*.f32}\left(\left(\alpha \cdot \left(\mathsf{neg}\left(\alpha\right)\right)\right), \log \color{blue}{\left(1 - u0\right)}\right) \]

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

      \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(\alpha, \left(\mathsf{neg}\left(\alpha\right)\right)\right), \log \color{blue}{\left(1 - u0\right)}\right) \]

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

      \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(\alpha, \mathsf{neg.f32}\left(\alpha\right)\right), \log \left(1 - \color{blue}{u0}\right)\right) \]

   5. sub-neg N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(\alpha, \mathsf{neg.f32}\left(\alpha\right)\right), \log \left(1 + \left(\mathsf{neg}\left(u0\right)\right)\right)\right) \]

   6. log1p-define N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(\alpha, \mathsf{neg.f32}\left(\alpha\right)\right), \left(\mathsf{log1p}\left(\mathsf{neg}\left(u0\right)\right)\right)\right) \]

   7. log1p-lowering-log1p.f32 N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(\alpha, \mathsf{neg.f32}\left(\alpha\right)\right), \mathsf{log1p.f32}\left(\left(\mathsf{neg}\left(u0\right)\right)\right)\right) \]

   8. neg-lowering-neg.f32 99.0%

      \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(\alpha, \mathsf{neg.f32}\left(\alpha\right)\right), \mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right)\right) \]

3. Simplified 99.0%

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

5. Final simplification 99.0%

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

Alternative 3: 93.5% accurate, 5.7× speedup

Math

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

FPCore

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

C

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

Fortran

real(4) function code(alpha, u0)
    real(4), intent (in) :: alpha
    real(4), intent (in) :: u0
    code = (alpha * alpha) * (u0 - (((-0.5e0) + (u0 * ((-0.3333333333333333e0) + (u0 * (-0.25e0))))) * (u0 * u0)))
end function

Julia

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

MATLAB

function tmp = code(alpha, u0)
	tmp = (alpha * alpha) * (u0 - ((single(-0.5) + (u0 * (single(-0.3333333333333333) + (u0 * single(-0.25))))) * (u0 * u0)));
end

Derivation

1. Initial program 56.5%

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

3. Taylor expanded in u0 around 0

   \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{neg.f32}\left(\alpha\right), \alpha\right), \color{blue}{\left(u0 \cdot \left(u0 \cdot \left(u0 \cdot \left(\frac{-1}{4} \cdot u0 - \frac{1}{3}\right) - \frac{1}{2}\right) - 1\right)\right)}\right) \]
4. Step-by-step derivation

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

      \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{neg.f32}\left(\alpha\right), \alpha\right), \mathsf{*.f32}\left(u0, \color{blue}{\left(u0 \cdot \left(u0 \cdot \left(\frac{-1}{4} \cdot u0 - \frac{1}{3}\right) - \frac{1}{2}\right) - 1\right)}\right)\right) \]

   2. sub-neg N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{neg.f32}\left(\alpha\right), \alpha\right), \mathsf{*.f32}\left(u0, \left(u0 \cdot \left(u0 \cdot \left(\frac{-1}{4} \cdot u0 - \frac{1}{3}\right) - \frac{1}{2}\right) + \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}\right)\right)\right) \]

   3. metadata-eval N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{neg.f32}\left(\alpha\right), \alpha\right), \mathsf{*.f32}\left(u0, \left(u0 \cdot \left(u0 \cdot \left(\frac{-1}{4} \cdot u0 - \frac{1}{3}\right) - \frac{1}{2}\right) + -1\right)\right)\right) \]

   4. +-commutative N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{neg.f32}\left(\alpha\right), \alpha\right), \mathsf{*.f32}\left(u0, \left(-1 + \color{blue}{u0 \cdot \left(u0 \cdot \left(\frac{-1}{4} \cdot u0 - \frac{1}{3}\right) - \frac{1}{2}\right)}\right)\right)\right) \]

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

      \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{neg.f32}\left(\alpha\right), \alpha\right), \mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(-1, \color{blue}{\left(u0 \cdot \left(u0 \cdot \left(\frac{-1}{4} \cdot u0 - \frac{1}{3}\right) - \frac{1}{2}\right)\right)}\right)\right)\right) \]

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

      \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{neg.f32}\left(\alpha\right), \alpha\right), \mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(-1, \mathsf{*.f32}\left(u0, \color{blue}{\left(u0 \cdot \left(\frac{-1}{4} \cdot u0 - \frac{1}{3}\right) - \frac{1}{2}\right)}\right)\right)\right)\right) \]

   7. sub-neg N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{neg.f32}\left(\alpha\right), \alpha\right), \mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(-1, \mathsf{*.f32}\left(u0, \left(u0 \cdot \left(\frac{-1}{4} \cdot u0 - \frac{1}{3}\right) + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right)\right)\right)\right)\right) \]

   8. metadata-eval N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{neg.f32}\left(\alpha\right), \alpha\right), \mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(-1, \mathsf{*.f32}\left(u0, \left(u0 \cdot \left(\frac{-1}{4} \cdot u0 - \frac{1}{3}\right) + \frac{-1}{2}\right)\right)\right)\right)\right) \]

   9. +-commutative N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{neg.f32}\left(\alpha\right), \alpha\right), \mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(-1, \mathsf{*.f32}\left(u0, \left(\frac{-1}{2} + \color{blue}{u0 \cdot \left(\frac{-1}{4} \cdot u0 - \frac{1}{3}\right)}\right)\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{neg.f32}\left(\alpha\right), \alpha\right), \mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(-1, \mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\frac{-1}{2}, \color{blue}{\left(u0 \cdot \left(\frac{-1}{4} \cdot u0 - \frac{1}{3}\right)\right)}\right)\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{neg.f32}\left(\alpha\right), \alpha\right), \mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(-1, \mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\frac{-1}{2}, \mathsf{*.f32}\left(u0, \color{blue}{\left(\frac{-1}{4} \cdot u0 - \frac{1}{3}\right)}\right)\right)\right)\right)\right)\right) \]

   12. sub-neg N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{neg.f32}\left(\alpha\right), \alpha\right), \mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(-1, \mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\frac{-1}{2}, \mathsf{*.f32}\left(u0, \left(\frac{-1}{4} \cdot u0 + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{3}\right)\right)}\right)\right)\right)\right)\right)\right)\right) \]

   13. metadata-eval N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{neg.f32}\left(\alpha\right), \alpha\right), \mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(-1, \mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\frac{-1}{2}, \mathsf{*.f32}\left(u0, \left(\frac{-1}{4} \cdot u0 + \frac{-1}{3}\right)\right)\right)\right)\right)\right)\right) \]

   14. +-commutative N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{neg.f32}\left(\alpha\right), \alpha\right), \mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(-1, \mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\frac{-1}{2}, \mathsf{*.f32}\left(u0, \left(\frac{-1}{3} + \color{blue}{\frac{-1}{4} \cdot u0}\right)\right)\right)\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{neg.f32}\left(\alpha\right), \alpha\right), \mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(-1, \mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\frac{-1}{2}, \mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\frac{-1}{3}, \color{blue}{\left(\frac{-1}{4} \cdot u0\right)}\right)\right)\right)\right)\right)\right)\right) \]

   16. *-commutative N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{neg.f32}\left(\alpha\right), \alpha\right), \mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(-1, \mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\frac{-1}{2}, \mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\frac{-1}{3}, \left(u0 \cdot \color{blue}{\frac{-1}{4}}\right)\right)\right)\right)\right)\right)\right)\right) \]

   17. *-lowering-*.f32 92.9%

       \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{neg.f32}\left(\alpha\right), \alpha\right), \mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(-1, \mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\frac{-1}{2}, \mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\frac{-1}{3}, \mathsf{*.f32}\left(u0, \color{blue}{\frac{-1}{4}}\right)\right)\right)\right)\right)\right)\right)\right) \]

5. Simplified 92.9%

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

   1. +-commutative N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{neg.f32}\left(\alpha\right), \alpha\right), \left(u0 \cdot \left(u0 \cdot \left(\frac{-1}{2} + u0 \cdot \left(\frac{-1}{3} + u0 \cdot \frac{-1}{4}\right)\right) + \color{blue}{-1}\right)\right)\right) \]

   2. distribute-rgt-in N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{neg.f32}\left(\alpha\right), \alpha\right), \left(\left(u0 \cdot \left(\frac{-1}{2} + u0 \cdot \left(\frac{-1}{3} + u0 \cdot \frac{-1}{4}\right)\right)\right) \cdot u0 + \color{blue}{-1 \cdot u0}\right)\right) \]

   3. neg-mul-1 N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{neg.f32}\left(\alpha\right), \alpha\right), \left(\left(u0 \cdot \left(\frac{-1}{2} + u0 \cdot \left(\frac{-1}{3} + u0 \cdot \frac{-1}{4}\right)\right)\right) \cdot u0 + \left(\mathsf{neg}\left(u0\right)\right)\right)\right) \]

   4. unsub-neg N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{neg.f32}\left(\alpha\right), \alpha\right), \left(\left(u0 \cdot \left(\frac{-1}{2} + u0 \cdot \left(\frac{-1}{3} + u0 \cdot \frac{-1}{4}\right)\right)\right) \cdot u0 - \color{blue}{u0}\right)\right) \]

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

      \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{neg.f32}\left(\alpha\right), \alpha\right), \mathsf{\_.f32}\left(\left(\left(u0 \cdot \left(\frac{-1}{2} + u0 \cdot \left(\frac{-1}{3} + u0 \cdot \frac{-1}{4}\right)\right)\right) \cdot u0\right), \color{blue}{u0}\right)\right) \]

   6. *-commutative N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{neg.f32}\left(\alpha\right), \alpha\right), \mathsf{\_.f32}\left(\left(\left(\left(\frac{-1}{2} + u0 \cdot \left(\frac{-1}{3} + u0 \cdot \frac{-1}{4}\right)\right) \cdot u0\right) \cdot u0\right), u0\right)\right) \]

   7. associate-*l* N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{neg.f32}\left(\alpha\right), \alpha\right), \mathsf{\_.f32}\left(\left(\left(\frac{-1}{2} + u0 \cdot \left(\frac{-1}{3} + u0 \cdot \frac{-1}{4}\right)\right) \cdot \left(u0 \cdot u0\right)\right), u0\right)\right) \]

   8. sqr-neg N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{neg.f32}\left(\alpha\right), \alpha\right), \mathsf{\_.f32}\left(\left(\left(\frac{-1}{2} + u0 \cdot \left(\frac{-1}{3} + u0 \cdot \frac{-1}{4}\right)\right) \cdot \left(\left(\mathsf{neg}\left(u0\right)\right) \cdot \left(\mathsf{neg}\left(u0\right)\right)\right)\right), u0\right)\right) \]

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

      \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{neg.f32}\left(\alpha\right), \alpha\right), \mathsf{\_.f32}\left(\mathsf{*.f32}\left(\left(\frac{-1}{2} + u0 \cdot \left(\frac{-1}{3} + u0 \cdot \frac{-1}{4}\right)\right), \left(\left(\mathsf{neg}\left(u0\right)\right) \cdot \left(\mathsf{neg}\left(u0\right)\right)\right)\right), u0\right)\right) \]

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

       \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{neg.f32}\left(\alpha\right), \alpha\right), \mathsf{\_.f32}\left(\mathsf{*.f32}\left(\mathsf{+.f32}\left(\frac{-1}{2}, \left(u0 \cdot \left(\frac{-1}{3} + u0 \cdot \frac{-1}{4}\right)\right)\right), \left(\left(\mathsf{neg}\left(u0\right)\right) \cdot \left(\mathsf{neg}\left(u0\right)\right)\right)\right), u0\right)\right) \]

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

       \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{neg.f32}\left(\alpha\right), \alpha\right), \mathsf{\_.f32}\left(\mathsf{*.f32}\left(\mathsf{+.f32}\left(\frac{-1}{2}, \mathsf{*.f32}\left(u0, \left(\frac{-1}{3} + u0 \cdot \frac{-1}{4}\right)\right)\right), \left(\left(\mathsf{neg}\left(u0\right)\right) \cdot \left(\mathsf{neg}\left(u0\right)\right)\right)\right), u0\right)\right) \]

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

       \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{neg.f32}\left(\alpha\right), \alpha\right), \mathsf{\_.f32}\left(\mathsf{*.f32}\left(\mathsf{+.f32}\left(\frac{-1}{2}, \mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\frac{-1}{3}, \left(u0 \cdot \frac{-1}{4}\right)\right)\right)\right), \left(\left(\mathsf{neg}\left(u0\right)\right) \cdot \left(\mathsf{neg}\left(u0\right)\right)\right)\right), u0\right)\right) \]

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

       \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{neg.f32}\left(\alpha\right), \alpha\right), \mathsf{\_.f32}\left(\mathsf{*.f32}\left(\mathsf{+.f32}\left(\frac{-1}{2}, \mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\frac{-1}{3}, \mathsf{*.f32}\left(u0, \frac{-1}{4}\right)\right)\right)\right), \left(\left(\mathsf{neg}\left(u0\right)\right) \cdot \left(\mathsf{neg}\left(u0\right)\right)\right)\right), u0\right)\right) \]

   14. sqr-neg N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{neg.f32}\left(\alpha\right), \alpha\right), \mathsf{\_.f32}\left(\mathsf{*.f32}\left(\mathsf{+.f32}\left(\frac{-1}{2}, \mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\frac{-1}{3}, \mathsf{*.f32}\left(u0, \frac{-1}{4}\right)\right)\right)\right), \left(u0 \cdot u0\right)\right), u0\right)\right) \]

   15. *-lowering-*.f32 93.3%

       \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{neg.f32}\left(\alpha\right), \alpha\right), \mathsf{\_.f32}\left(\mathsf{*.f32}\left(\mathsf{+.f32}\left(\frac{-1}{2}, \mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\frac{-1}{3}, \mathsf{*.f32}\left(u0, \frac{-1}{4}\right)\right)\right)\right), \mathsf{*.f32}\left(u0, u0\right)\right), u0\right)\right) \]

7. Applied egg-rr 93.3%

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

8. Final simplification 93.3%

   \[\leadsto \left(\alpha \cdot \alpha\right) \cdot \left(u0 - \left(-0.5 + u0 \cdot \left(-0.3333333333333333 + u0 \cdot -0.25\right)\right) \cdot \left(u0 \cdot u0\right)\right) \]
9. Add Preprocessing

Alternative 4: 93.2% accurate, 5.7× speedup

Math

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

FPCore

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

C

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

Fortran

real(4) function code(alpha, u0)
    real(4), intent (in) :: alpha
    real(4), intent (in) :: u0
    code = alpha * (alpha * (u0 * (1.0e0 - (u0 * ((-0.5e0) + (u0 * ((-0.3333333333333333e0) + (u0 * (-0.25e0)))))))))
end function

Julia

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

MATLAB

function tmp = code(alpha, u0)
	tmp = alpha * (alpha * (u0 * (single(1.0) - (u0 * (single(-0.5) + (u0 * (single(-0.3333333333333333) + (u0 * single(-0.25)))))))));
end

Derivation

1. Initial program 56.5%

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

3. Taylor expanded in u0 around 0

   \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{neg.f32}\left(\alpha\right), \alpha\right), \color{blue}{\left(u0 \cdot \left(u0 \cdot \left(u0 \cdot \left(\frac{-1}{4} \cdot u0 - \frac{1}{3}\right) - \frac{1}{2}\right) - 1\right)\right)}\right) \]
4. Step-by-step derivation

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

      \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{neg.f32}\left(\alpha\right), \alpha\right), \mathsf{*.f32}\left(u0, \color{blue}{\left(u0 \cdot \left(u0 \cdot \left(\frac{-1}{4} \cdot u0 - \frac{1}{3}\right) - \frac{1}{2}\right) - 1\right)}\right)\right) \]

   2. sub-neg N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{neg.f32}\left(\alpha\right), \alpha\right), \mathsf{*.f32}\left(u0, \left(u0 \cdot \left(u0 \cdot \left(\frac{-1}{4} \cdot u0 - \frac{1}{3}\right) - \frac{1}{2}\right) + \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}\right)\right)\right) \]

   3. metadata-eval N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{neg.f32}\left(\alpha\right), \alpha\right), \mathsf{*.f32}\left(u0, \left(u0 \cdot \left(u0 \cdot \left(\frac{-1}{4} \cdot u0 - \frac{1}{3}\right) - \frac{1}{2}\right) + -1\right)\right)\right) \]

   4. +-commutative N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{neg.f32}\left(\alpha\right), \alpha\right), \mathsf{*.f32}\left(u0, \left(-1 + \color{blue}{u0 \cdot \left(u0 \cdot \left(\frac{-1}{4} \cdot u0 - \frac{1}{3}\right) - \frac{1}{2}\right)}\right)\right)\right) \]

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

      \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{neg.f32}\left(\alpha\right), \alpha\right), \mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(-1, \color{blue}{\left(u0 \cdot \left(u0 \cdot \left(\frac{-1}{4} \cdot u0 - \frac{1}{3}\right) - \frac{1}{2}\right)\right)}\right)\right)\right) \]

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

      \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{neg.f32}\left(\alpha\right), \alpha\right), \mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(-1, \mathsf{*.f32}\left(u0, \color{blue}{\left(u0 \cdot \left(\frac{-1}{4} \cdot u0 - \frac{1}{3}\right) - \frac{1}{2}\right)}\right)\right)\right)\right) \]

   7. sub-neg N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{neg.f32}\left(\alpha\right), \alpha\right), \mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(-1, \mathsf{*.f32}\left(u0, \left(u0 \cdot \left(\frac{-1}{4} \cdot u0 - \frac{1}{3}\right) + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right)\right)\right)\right)\right) \]

   8. metadata-eval N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{neg.f32}\left(\alpha\right), \alpha\right), \mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(-1, \mathsf{*.f32}\left(u0, \left(u0 \cdot \left(\frac{-1}{4} \cdot u0 - \frac{1}{3}\right) + \frac{-1}{2}\right)\right)\right)\right)\right) \]

   9. +-commutative N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{neg.f32}\left(\alpha\right), \alpha\right), \mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(-1, \mathsf{*.f32}\left(u0, \left(\frac{-1}{2} + \color{blue}{u0 \cdot \left(\frac{-1}{4} \cdot u0 - \frac{1}{3}\right)}\right)\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{neg.f32}\left(\alpha\right), \alpha\right), \mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(-1, \mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\frac{-1}{2}, \color{blue}{\left(u0 \cdot \left(\frac{-1}{4} \cdot u0 - \frac{1}{3}\right)\right)}\right)\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{neg.f32}\left(\alpha\right), \alpha\right), \mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(-1, \mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\frac{-1}{2}, \mathsf{*.f32}\left(u0, \color{blue}{\left(\frac{-1}{4} \cdot u0 - \frac{1}{3}\right)}\right)\right)\right)\right)\right)\right) \]

   12. sub-neg N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{neg.f32}\left(\alpha\right), \alpha\right), \mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(-1, \mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\frac{-1}{2}, \mathsf{*.f32}\left(u0, \left(\frac{-1}{4} \cdot u0 + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{3}\right)\right)}\right)\right)\right)\right)\right)\right)\right) \]

   13. metadata-eval N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{neg.f32}\left(\alpha\right), \alpha\right), \mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(-1, \mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\frac{-1}{2}, \mathsf{*.f32}\left(u0, \left(\frac{-1}{4} \cdot u0 + \frac{-1}{3}\right)\right)\right)\right)\right)\right)\right) \]

   14. +-commutative N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{neg.f32}\left(\alpha\right), \alpha\right), \mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(-1, \mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\frac{-1}{2}, \mathsf{*.f32}\left(u0, \left(\frac{-1}{3} + \color{blue}{\frac{-1}{4} \cdot u0}\right)\right)\right)\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{neg.f32}\left(\alpha\right), \alpha\right), \mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(-1, \mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\frac{-1}{2}, \mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\frac{-1}{3}, \color{blue}{\left(\frac{-1}{4} \cdot u0\right)}\right)\right)\right)\right)\right)\right)\right) \]

   16. *-commutative N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{neg.f32}\left(\alpha\right), \alpha\right), \mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(-1, \mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\frac{-1}{2}, \mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\frac{-1}{3}, \left(u0 \cdot \color{blue}{\frac{-1}{4}}\right)\right)\right)\right)\right)\right)\right)\right) \]

   17. *-lowering-*.f32 92.9%

       \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{neg.f32}\left(\alpha\right), \alpha\right), \mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(-1, \mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\frac{-1}{2}, \mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\frac{-1}{3}, \mathsf{*.f32}\left(u0, \color{blue}{\frac{-1}{4}}\right)\right)\right)\right)\right)\right)\right)\right) \]

5. Simplified 92.9%

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

   1. associate-*l* N/A

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

   2. *-commutative N/A

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

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

      \[\leadsto \mathsf{*.f32}\left(\left(\alpha \cdot \left(u0 \cdot \left(-1 + u0 \cdot \left(\frac{-1}{2} + u0 \cdot \left(\frac{-1}{3} + u0 \cdot \frac{-1}{4}\right)\right)\right)\right)\right), \color{blue}{\left(\mathsf{neg}\left(\alpha\right)\right)}\right) \]

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

      \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(\alpha, \left(u0 \cdot \left(-1 + u0 \cdot \left(\frac{-1}{2} + u0 \cdot \left(\frac{-1}{3} + u0 \cdot \frac{-1}{4}\right)\right)\right)\right)\right), \left(\mathsf{neg}\left(\color{blue}{\alpha}\right)\right)\right) \]

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

      \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(\alpha, \mathsf{*.f32}\left(u0, \left(-1 + u0 \cdot \left(\frac{-1}{2} + u0 \cdot \left(\frac{-1}{3} + u0 \cdot \frac{-1}{4}\right)\right)\right)\right)\right), \left(\mathsf{neg}\left(\alpha\right)\right)\right) \]

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

      \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(\alpha, \mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(-1, \left(u0 \cdot \left(\frac{-1}{2} + u0 \cdot \left(\frac{-1}{3} + u0 \cdot \frac{-1}{4}\right)\right)\right)\right)\right)\right), \left(\mathsf{neg}\left(\alpha\right)\right)\right) \]

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

      \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(\alpha, \mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(-1, \mathsf{*.f32}\left(u0, \left(\frac{-1}{2} + u0 \cdot \left(\frac{-1}{3} + u0 \cdot \frac{-1}{4}\right)\right)\right)\right)\right)\right), \left(\mathsf{neg}\left(\alpha\right)\right)\right) \]

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

      \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(\alpha, \mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(-1, \mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\frac{-1}{2}, \left(u0 \cdot \left(\frac{-1}{3} + u0 \cdot \frac{-1}{4}\right)\right)\right)\right)\right)\right)\right), \left(\mathsf{neg}\left(\alpha\right)\right)\right) \]

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

      \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(\alpha, \mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(-1, \mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\frac{-1}{2}, \mathsf{*.f32}\left(u0, \left(\frac{-1}{3} + u0 \cdot \frac{-1}{4}\right)\right)\right)\right)\right)\right)\right), \left(\mathsf{neg}\left(\alpha\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(\alpha, \mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(-1, \mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\frac{-1}{2}, \mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\frac{-1}{3}, \left(u0 \cdot \frac{-1}{4}\right)\right)\right)\right)\right)\right)\right)\right), \left(\mathsf{neg}\left(\alpha\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(\alpha, \mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(-1, \mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\frac{-1}{2}, \mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\frac{-1}{3}, \mathsf{*.f32}\left(u0, \frac{-1}{4}\right)\right)\right)\right)\right)\right)\right)\right), \left(\mathsf{neg}\left(\alpha\right)\right)\right) \]

   12. neg-lowering-neg.f32 93.0%

       \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(\alpha, \mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(-1, \mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\frac{-1}{2}, \mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\frac{-1}{3}, \mathsf{*.f32}\left(u0, \frac{-1}{4}\right)\right)\right)\right)\right)\right)\right)\right), \mathsf{neg.f32}\left(\alpha\right)\right) \]

7. Applied egg-rr 93.0%

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

8. Final simplification 93.0%

   \[\leadsto \alpha \cdot \left(\alpha \cdot \left(u0 \cdot \left(1 - u0 \cdot \left(-0.5 + u0 \cdot \left(-0.3333333333333333 + u0 \cdot -0.25\right)\right)\right)\right)\right) \]
9. Add Preprocessing

Alternative 5: 92.9% accurate, 7.2× speedup

Math

\[\begin{array}{l} \\ \frac{u0 \cdot \left(\alpha \cdot \alpha\right)}{u0 \cdot \left(-0.5 + u0 \cdot -0.08333333333333333\right) + 1} \end{array} \]

FPCore

(FPCore (alpha u0)
 :precision binary32
 (/
  (* u0 (* alpha alpha))
  (+ (* u0 (+ -0.5 (* u0 -0.08333333333333333))) 1.0)))

C

float code(float alpha, float u0) {
	return (u0 * (alpha * alpha)) / ((u0 * (-0.5f + (u0 * -0.08333333333333333f))) + 1.0f);
}

Fortran

real(4) function code(alpha, u0)
    real(4), intent (in) :: alpha
    real(4), intent (in) :: u0
    code = (u0 * (alpha * alpha)) / ((u0 * ((-0.5e0) + (u0 * (-0.08333333333333333e0)))) + 1.0e0)
end function

Julia

function code(alpha, u0)
	return Float32(Float32(u0 * Float32(alpha * alpha)) / Float32(Float32(u0 * Float32(Float32(-0.5) + Float32(u0 * Float32(-0.08333333333333333)))) + Float32(1.0)))
end

MATLAB

function tmp = code(alpha, u0)
	tmp = (u0 * (alpha * alpha)) / ((u0 * (single(-0.5) + (u0 * single(-0.08333333333333333)))) + single(1.0));
end

Derivation

1. Initial program 56.5%

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

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

      \[\leadsto \mathsf{*.f32}\left(\left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right), \color{blue}{\log \left(1 - u0\right)}\right) \]

   2. *-commutative N/A

      \[\leadsto \mathsf{*.f32}\left(\left(\alpha \cdot \left(\mathsf{neg}\left(\alpha\right)\right)\right), \log \color{blue}{\left(1 - u0\right)}\right) \]

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

      \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(\alpha, \left(\mathsf{neg}\left(\alpha\right)\right)\right), \log \color{blue}{\left(1 - u0\right)}\right) \]

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

      \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(\alpha, \mathsf{neg.f32}\left(\alpha\right)\right), \log \left(1 - \color{blue}{u0}\right)\right) \]

   5. sub-neg N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(\alpha, \mathsf{neg.f32}\left(\alpha\right)\right), \log \left(1 + \left(\mathsf{neg}\left(u0\right)\right)\right)\right) \]

   6. log1p-define N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(\alpha, \mathsf{neg.f32}\left(\alpha\right)\right), \left(\mathsf{log1p}\left(\mathsf{neg}\left(u0\right)\right)\right)\right) \]

   7. log1p-lowering-log1p.f32 N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(\alpha, \mathsf{neg.f32}\left(\alpha\right)\right), \mathsf{log1p.f32}\left(\left(\mathsf{neg}\left(u0\right)\right)\right)\right) \]

   8. neg-lowering-neg.f32 99.0%

      \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(\alpha, \mathsf{neg.f32}\left(\alpha\right)\right), \mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right)\right) \]

3. Simplified 99.0%

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

5. Taylor expanded in u0 around 0

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

   1. distribute-rgt-in N/A

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

   2. associate-*r* N/A

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

   3. associate-+l+ N/A

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

   4. *-commutative N/A

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

   5. distribute-lft-in N/A

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

   6. associate-*r* N/A

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

   7. associate-*r* N/A

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

   8. distribute-rgt-in N/A

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

   9. *-commutative N/A

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

   10. associate-*r* N/A

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

   11. distribute-lft1-in N/A

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

   12. distribute-rgt-out N/A

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

7. Simplified 90.9%

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

   1. flip3-+ N/A

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

   2. clear-num N/A

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

   3. un-div-inv N/A

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

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

      \[\leadsto \mathsf{/.f32}\left(\left(\left(\alpha \cdot \alpha\right) \cdot u0\right), \color{blue}{\left(\frac{\left(\frac{1}{3} \cdot \left(u0 \cdot u0\right)\right) \cdot \left(\frac{1}{3} \cdot \left(u0 \cdot u0\right)\right) + \left(\left(u0 \cdot \frac{1}{2} + 1\right) \cdot \left(u0 \cdot \frac{1}{2} + 1\right) - \left(\frac{1}{3} \cdot \left(u0 \cdot u0\right)\right) \cdot \left(u0 \cdot \frac{1}{2} + 1\right)\right)}{{\left(\frac{1}{3} \cdot \left(u0 \cdot u0\right)\right)}^{3} + {\left(u0 \cdot \frac{1}{2} + 1\right)}^{3}}\right)}\right) \]

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

      \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(\left(\alpha \cdot \alpha\right), u0\right), \left(\frac{\color{blue}{\left(\frac{1}{3} \cdot \left(u0 \cdot u0\right)\right) \cdot \left(\frac{1}{3} \cdot \left(u0 \cdot u0\right)\right) + \left(\left(u0 \cdot \frac{1}{2} + 1\right) \cdot \left(u0 \cdot \frac{1}{2} + 1\right) - \left(\frac{1}{3} \cdot \left(u0 \cdot u0\right)\right) \cdot \left(u0 \cdot \frac{1}{2} + 1\right)\right)}}{{\left(\frac{1}{3} \cdot \left(u0 \cdot u0\right)\right)}^{3} + {\left(u0 \cdot \frac{1}{2} + 1\right)}^{3}}\right)\right) \]

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

      \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(\alpha, \alpha\right), u0\right), \left(\frac{\color{blue}{\left(\frac{1}{3} \cdot \left(u0 \cdot u0\right)\right) \cdot \left(\frac{1}{3} \cdot \left(u0 \cdot u0\right)\right)} + \left(\left(u0 \cdot \frac{1}{2} + 1\right) \cdot \left(u0 \cdot \frac{1}{2} + 1\right) - \left(\frac{1}{3} \cdot \left(u0 \cdot u0\right)\right) \cdot \left(u0 \cdot \frac{1}{2} + 1\right)\right)}{{\left(\frac{1}{3} \cdot \left(u0 \cdot u0\right)\right)}^{3} + {\left(u0 \cdot \frac{1}{2} + 1\right)}^{3}}\right)\right) \]

   7. clear-num N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(\alpha, \alpha\right), u0\right), \left(\frac{1}{\color{blue}{\frac{{\left(\frac{1}{3} \cdot \left(u0 \cdot u0\right)\right)}^{3} + {\left(u0 \cdot \frac{1}{2} + 1\right)}^{3}}{\left(\frac{1}{3} \cdot \left(u0 \cdot u0\right)\right) \cdot \left(\frac{1}{3} \cdot \left(u0 \cdot u0\right)\right) + \left(\left(u0 \cdot \frac{1}{2} + 1\right) \cdot \left(u0 \cdot \frac{1}{2} + 1\right) - \left(\frac{1}{3} \cdot \left(u0 \cdot u0\right)\right) \cdot \left(u0 \cdot \frac{1}{2} + 1\right)\right)}}}\right)\right) \]

9. Applied egg-rr 90.9%

   \[\leadsto \color{blue}{\frac{\left(\alpha \cdot \alpha\right) \cdot u0}{\frac{1}{1 + u0 \cdot \left(u0 \cdot 0.3333333333333333 + 0.5\right)}}} \]

10. Taylor expanded in u0 around 0

    \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(\alpha, \alpha\right), u0\right), \color{blue}{\left(1 + u0 \cdot \left(\frac{-1}{12} \cdot u0 - \frac{1}{2}\right)\right)}\right) \]
11. Step-by-step derivation

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

       \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(\alpha, \alpha\right), u0\right), \mathsf{+.f32}\left(1, \color{blue}{\left(u0 \cdot \left(\frac{-1}{12} \cdot u0 - \frac{1}{2}\right)\right)}\right)\right) \]

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

       \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(\alpha, \alpha\right), u0\right), \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(u0, \color{blue}{\left(\frac{-1}{12} \cdot u0 - \frac{1}{2}\right)}\right)\right)\right) \]

    3. sub-neg N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(\alpha, \alpha\right), u0\right), \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(u0, \left(\frac{-1}{12} \cdot u0 + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right)\right)\right)\right) \]

    4. metadata-eval N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(\alpha, \alpha\right), u0\right), \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(u0, \left(\frac{-1}{12} \cdot u0 + \frac{-1}{2}\right)\right)\right)\right) \]

    5. +-commutative N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(\alpha, \alpha\right), u0\right), \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(u0, \left(\frac{-1}{2} + \color{blue}{\frac{-1}{12} \cdot u0}\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(\alpha, \alpha\right), u0\right), \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\frac{-1}{2}, \color{blue}{\left(\frac{-1}{12} \cdot u0\right)}\right)\right)\right)\right) \]

    7. *-commutative N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(\alpha, \alpha\right), u0\right), \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\frac{-1}{2}, \left(u0 \cdot \color{blue}{\frac{-1}{12}}\right)\right)\right)\right)\right) \]

    8. *-lowering-*.f32 92.4%

       \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(\alpha, \alpha\right), u0\right), \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\frac{-1}{2}, \mathsf{*.f32}\left(u0, \color{blue}{\frac{-1}{12}}\right)\right)\right)\right)\right) \]

12. Simplified 92.4%

    \[\leadsto \frac{\left(\alpha \cdot \alpha\right) \cdot u0}{\color{blue}{1 + u0 \cdot \left(-0.5 + u0 \cdot -0.08333333333333333\right)}} \]

13. Final simplification 92.4%

    \[\leadsto \frac{u0 \cdot \left(\alpha \cdot \alpha\right)}{u0 \cdot \left(-0.5 + u0 \cdot -0.08333333333333333\right) + 1} \]
14. Add Preprocessing

Alternative 6: 91.3% accurate, 7.2× speedup

Math

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

FPCore

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

C

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

Fortran

real(4) function code(alpha, u0)
    real(4), intent (in) :: alpha
    real(4), intent (in) :: u0
    code = alpha * (u0 * (alpha * ((u0 * ((u0 * 0.3333333333333333e0) + 0.5e0)) + 1.0e0)))
end function

Julia

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

MATLAB

function tmp = code(alpha, u0)
	tmp = alpha * (u0 * (alpha * ((u0 * ((u0 * single(0.3333333333333333)) + single(0.5))) + single(1.0))));
end

Derivation

1. Initial program 56.5%

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

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

      \[\leadsto \mathsf{*.f32}\left(\left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right), \color{blue}{\log \left(1 - u0\right)}\right) \]

   2. *-commutative N/A

      \[\leadsto \mathsf{*.f32}\left(\left(\alpha \cdot \left(\mathsf{neg}\left(\alpha\right)\right)\right), \log \color{blue}{\left(1 - u0\right)}\right) \]

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

      \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(\alpha, \left(\mathsf{neg}\left(\alpha\right)\right)\right), \log \color{blue}{\left(1 - u0\right)}\right) \]

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

      \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(\alpha, \mathsf{neg.f32}\left(\alpha\right)\right), \log \left(1 - \color{blue}{u0}\right)\right) \]

   5. sub-neg N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(\alpha, \mathsf{neg.f32}\left(\alpha\right)\right), \log \left(1 + \left(\mathsf{neg}\left(u0\right)\right)\right)\right) \]

   6. log1p-define N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(\alpha, \mathsf{neg.f32}\left(\alpha\right)\right), \left(\mathsf{log1p}\left(\mathsf{neg}\left(u0\right)\right)\right)\right) \]

   7. log1p-lowering-log1p.f32 N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(\alpha, \mathsf{neg.f32}\left(\alpha\right)\right), \mathsf{log1p.f32}\left(\left(\mathsf{neg}\left(u0\right)\right)\right)\right) \]

   8. neg-lowering-neg.f32 99.0%

      \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(\alpha, \mathsf{neg.f32}\left(\alpha\right)\right), \mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right)\right) \]

3. Simplified 99.0%

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

5. Taylor expanded in u0 around 0

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

   1. distribute-rgt-in N/A

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

   2. associate-*r* N/A

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

   3. associate-+l+ N/A

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

   4. *-commutative N/A

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

   5. distribute-lft-in N/A

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

   6. associate-*r* N/A

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

   7. associate-*r* N/A

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

   8. distribute-rgt-in N/A

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

   9. *-commutative N/A

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

   10. associate-*r* N/A

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

   11. distribute-lft1-in N/A

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

   12. distribute-rgt-out N/A

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

7. Simplified 90.9%

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

   1. *-commutative N/A

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

   2. associate-*r* N/A

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

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

      \[\leadsto \mathsf{*.f32}\left(\left(\left(\frac{1}{3} \cdot \left(u0 \cdot u0\right) + \left(u0 \cdot \frac{1}{2} + 1\right)\right) \cdot \left(\alpha \cdot \alpha\right)\right), \color{blue}{u0}\right) \]

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

      \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(\left(\frac{1}{3} \cdot \left(u0 \cdot u0\right) + \left(u0 \cdot \frac{1}{2} + 1\right)\right), \left(\alpha \cdot \alpha\right)\right), u0\right) \]

   5. associate-+r+ N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(\left(\left(\frac{1}{3} \cdot \left(u0 \cdot u0\right) + u0 \cdot \frac{1}{2}\right) + 1\right), \left(\alpha \cdot \alpha\right)\right), u0\right) \]

   6. +-commutative N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(\left(1 + \left(\frac{1}{3} \cdot \left(u0 \cdot u0\right) + u0 \cdot \frac{1}{2}\right)\right), \left(\alpha \cdot \alpha\right)\right), u0\right) \]

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

      \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{+.f32}\left(1, \left(\frac{1}{3} \cdot \left(u0 \cdot u0\right) + u0 \cdot \frac{1}{2}\right)\right), \left(\alpha \cdot \alpha\right)\right), u0\right) \]

   8. associate-*r* N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{+.f32}\left(1, \left(\left(\frac{1}{3} \cdot u0\right) \cdot u0 + u0 \cdot \frac{1}{2}\right)\right), \left(\alpha \cdot \alpha\right)\right), u0\right) \]

   9. *-commutative N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{+.f32}\left(1, \left(\left(\frac{1}{3} \cdot u0\right) \cdot u0 + \frac{1}{2} \cdot u0\right)\right), \left(\alpha \cdot \alpha\right)\right), u0\right) \]

   10. distribute-rgt-out N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{+.f32}\left(1, \left(u0 \cdot \left(\frac{1}{3} \cdot u0 + \frac{1}{2}\right)\right)\right), \left(\alpha \cdot \alpha\right)\right), u0\right) \]

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

       \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{+.f32}\left(1, \mathsf{*.f32}\left(u0, \left(\frac{1}{3} \cdot u0 + \frac{1}{2}\right)\right)\right), \left(\alpha \cdot \alpha\right)\right), u0\right) \]

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

       \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{+.f32}\left(1, \mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\left(\frac{1}{3} \cdot u0\right), \frac{1}{2}\right)\right)\right), \left(\alpha \cdot \alpha\right)\right), u0\right) \]

   13. *-commutative N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{+.f32}\left(1, \mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\left(u0 \cdot \frac{1}{3}\right), \frac{1}{2}\right)\right)\right), \left(\alpha \cdot \alpha\right)\right), u0\right) \]

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

       \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{+.f32}\left(1, \mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{*.f32}\left(u0, \frac{1}{3}\right), \frac{1}{2}\right)\right)\right), \left(\alpha \cdot \alpha\right)\right), u0\right) \]

   15. *-lowering-*.f32 90.9%

       \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{+.f32}\left(1, \mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{*.f32}\left(u0, \frac{1}{3}\right), \frac{1}{2}\right)\right)\right), \mathsf{*.f32}\left(\alpha, \alpha\right)\right), u0\right) \]

9. Applied egg-rr 90.9%

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

    1. *-commutative N/A

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

    2. associate-*r* N/A

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

    3. associate-*r* N/A

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

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

       \[\leadsto \mathsf{*.f32}\left(\left(u0 \cdot \left(\left(1 + u0 \cdot \left(u0 \cdot \frac{1}{3} + \frac{1}{2}\right)\right) \cdot \alpha\right)\right), \color{blue}{\alpha}\right) \]

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

       \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(u0, \left(\left(1 + u0 \cdot \left(u0 \cdot \frac{1}{3} + \frac{1}{2}\right)\right) \cdot \alpha\right)\right), \alpha\right) \]

    6. *-commutative N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(u0, \left(\alpha \cdot \left(1 + u0 \cdot \left(u0 \cdot \frac{1}{3} + \frac{1}{2}\right)\right)\right)\right), \alpha\right) \]

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

       \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(u0, \mathsf{*.f32}\left(\alpha, \left(1 + u0 \cdot \left(u0 \cdot \frac{1}{3} + \frac{1}{2}\right)\right)\right)\right), \alpha\right) \]

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

       \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(u0, \mathsf{*.f32}\left(\alpha, \mathsf{+.f32}\left(1, \left(u0 \cdot \left(u0 \cdot \frac{1}{3} + \frac{1}{2}\right)\right)\right)\right)\right), \alpha\right) \]

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

       \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(u0, \mathsf{*.f32}\left(\alpha, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(u0, \left(u0 \cdot \frac{1}{3} + \frac{1}{2}\right)\right)\right)\right)\right), \alpha\right) \]

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

        \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(u0, \mathsf{*.f32}\left(\alpha, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\left(u0 \cdot \frac{1}{3}\right), \frac{1}{2}\right)\right)\right)\right)\right), \alpha\right) \]

    11. *-lowering-*.f32 91.1%

        \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(u0, \mathsf{*.f32}\left(\alpha, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{*.f32}\left(u0, \frac{1}{3}\right), \frac{1}{2}\right)\right)\right)\right)\right), \alpha\right) \]

11. Applied egg-rr 91.1%

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

12. Final simplification 91.1%

    \[\leadsto \alpha \cdot \left(u0 \cdot \left(\alpha \cdot \left(u0 \cdot \left(u0 \cdot 0.3333333333333333 + 0.5\right) + 1\right)\right)\right) \]
13. Add Preprocessing

Alternative 7: 91.3% accurate, 7.2× speedup

Math

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

FPCore

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

C

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

Fortran

real(4) function code(alpha, u0)
    real(4), intent (in) :: alpha
    real(4), intent (in) :: u0
    code = alpha * (((u0 * ((u0 * 0.3333333333333333e0) + 0.5e0)) + 1.0e0) * (alpha * u0))
end function

Julia

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

MATLAB

function tmp = code(alpha, u0)
	tmp = alpha * (((u0 * ((u0 * single(0.3333333333333333)) + single(0.5))) + single(1.0)) * (alpha * u0));
end

Derivation

1. Initial program 56.5%

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

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

      \[\leadsto \mathsf{*.f32}\left(\left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right), \color{blue}{\log \left(1 - u0\right)}\right) \]

   2. *-commutative N/A

      \[\leadsto \mathsf{*.f32}\left(\left(\alpha \cdot \left(\mathsf{neg}\left(\alpha\right)\right)\right), \log \color{blue}{\left(1 - u0\right)}\right) \]

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

      \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(\alpha, \left(\mathsf{neg}\left(\alpha\right)\right)\right), \log \color{blue}{\left(1 - u0\right)}\right) \]

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

      \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(\alpha, \mathsf{neg.f32}\left(\alpha\right)\right), \log \left(1 - \color{blue}{u0}\right)\right) \]

   5. sub-neg N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(\alpha, \mathsf{neg.f32}\left(\alpha\right)\right), \log \left(1 + \left(\mathsf{neg}\left(u0\right)\right)\right)\right) \]

   6. log1p-define N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(\alpha, \mathsf{neg.f32}\left(\alpha\right)\right), \left(\mathsf{log1p}\left(\mathsf{neg}\left(u0\right)\right)\right)\right) \]

   7. log1p-lowering-log1p.f32 N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(\alpha, \mathsf{neg.f32}\left(\alpha\right)\right), \mathsf{log1p.f32}\left(\left(\mathsf{neg}\left(u0\right)\right)\right)\right) \]

   8. neg-lowering-neg.f32 99.0%

      \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(\alpha, \mathsf{neg.f32}\left(\alpha\right)\right), \mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right)\right) \]

3. Simplified 99.0%

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

5. Taylor expanded in u0 around 0

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

   1. distribute-rgt-in N/A

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

   2. associate-*r* N/A

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

   3. associate-+l+ N/A

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

   4. *-commutative N/A

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

   5. distribute-lft-in N/A

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

   6. associate-*r* N/A

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

   7. associate-*r* N/A

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

   8. distribute-rgt-in N/A

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

   9. *-commutative N/A

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

   10. associate-*r* N/A

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

   11. distribute-lft1-in N/A

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

   12. distribute-rgt-out N/A

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

7. Simplified 90.9%

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

   1. associate-*l* N/A

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

   2. associate-*l* N/A

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

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

      \[\leadsto \mathsf{*.f32}\left(\alpha, \color{blue}{\left(\left(\alpha \cdot u0\right) \cdot \left(\frac{1}{3} \cdot \left(u0 \cdot u0\right) + \left(u0 \cdot \frac{1}{2} + 1\right)\right)\right)}\right) \]

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

      \[\leadsto \mathsf{*.f32}\left(\alpha, \mathsf{*.f32}\left(\left(\alpha \cdot u0\right), \color{blue}{\left(\frac{1}{3} \cdot \left(u0 \cdot u0\right) + \left(u0 \cdot \frac{1}{2} + 1\right)\right)}\right)\right) \]

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

      \[\leadsto \mathsf{*.f32}\left(\alpha, \mathsf{*.f32}\left(\mathsf{*.f32}\left(\alpha, u0\right), \left(\color{blue}{\frac{1}{3} \cdot \left(u0 \cdot u0\right)} + \left(u0 \cdot \frac{1}{2} + 1\right)\right)\right)\right) \]

   6. associate-+r+ N/A

      \[\leadsto \mathsf{*.f32}\left(\alpha, \mathsf{*.f32}\left(\mathsf{*.f32}\left(\alpha, u0\right), \left(\left(\frac{1}{3} \cdot \left(u0 \cdot u0\right) + u0 \cdot \frac{1}{2}\right) + \color{blue}{1}\right)\right)\right) \]

   7. +-commutative N/A

      \[\leadsto \mathsf{*.f32}\left(\alpha, \mathsf{*.f32}\left(\mathsf{*.f32}\left(\alpha, u0\right), \left(1 + \color{blue}{\left(\frac{1}{3} \cdot \left(u0 \cdot u0\right) + u0 \cdot \frac{1}{2}\right)}\right)\right)\right) \]

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

      \[\leadsto \mathsf{*.f32}\left(\alpha, \mathsf{*.f32}\left(\mathsf{*.f32}\left(\alpha, u0\right), \mathsf{+.f32}\left(1, \color{blue}{\left(\frac{1}{3} \cdot \left(u0 \cdot u0\right) + u0 \cdot \frac{1}{2}\right)}\right)\right)\right) \]

   9. associate-*r* N/A

      \[\leadsto \mathsf{*.f32}\left(\alpha, \mathsf{*.f32}\left(\mathsf{*.f32}\left(\alpha, u0\right), \mathsf{+.f32}\left(1, \left(\left(\frac{1}{3} \cdot u0\right) \cdot u0 + \color{blue}{u0} \cdot \frac{1}{2}\right)\right)\right)\right) \]

   10. *-commutative N/A

       \[\leadsto \mathsf{*.f32}\left(\alpha, \mathsf{*.f32}\left(\mathsf{*.f32}\left(\alpha, u0\right), \mathsf{+.f32}\left(1, \left(\left(\frac{1}{3} \cdot u0\right) \cdot u0 + \frac{1}{2} \cdot \color{blue}{u0}\right)\right)\right)\right) \]

   11. distribute-rgt-out N/A

       \[\leadsto \mathsf{*.f32}\left(\alpha, \mathsf{*.f32}\left(\mathsf{*.f32}\left(\alpha, u0\right), \mathsf{+.f32}\left(1, \left(u0 \cdot \color{blue}{\left(\frac{1}{3} \cdot u0 + \frac{1}{2}\right)}\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f32}\left(\alpha, \mathsf{*.f32}\left(\mathsf{*.f32}\left(\alpha, u0\right), \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(u0, \color{blue}{\left(\frac{1}{3} \cdot u0 + \frac{1}{2}\right)}\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f32}\left(\alpha, \mathsf{*.f32}\left(\mathsf{*.f32}\left(\alpha, u0\right), \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\left(\frac{1}{3} \cdot u0\right), \color{blue}{\frac{1}{2}}\right)\right)\right)\right)\right) \]

   14. *-commutative N/A

       \[\leadsto \mathsf{*.f32}\left(\alpha, \mathsf{*.f32}\left(\mathsf{*.f32}\left(\alpha, u0\right), \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\left(u0 \cdot \frac{1}{3}\right), \frac{1}{2}\right)\right)\right)\right)\right) \]

   15. *-lowering-*.f32 91.0%

       \[\leadsto \mathsf{*.f32}\left(\alpha, \mathsf{*.f32}\left(\mathsf{*.f32}\left(\alpha, u0\right), \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{*.f32}\left(u0, \frac{1}{3}\right), \frac{1}{2}\right)\right)\right)\right)\right) \]

9. Applied egg-rr 91.0%

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

10. Final simplification 91.0%

    \[\leadsto \alpha \cdot \left(\left(u0 \cdot \left(u0 \cdot 0.3333333333333333 + 0.5\right) + 1\right) \cdot \left(\alpha \cdot u0\right)\right) \]
11. Add Preprocessing

Alternative 8: 91.3% accurate, 7.2× speedup

Math

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

FPCore

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

C

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

Fortran

real(4) function code(alpha, u0)
    real(4), intent (in) :: alpha
    real(4), intent (in) :: u0
    code = alpha * (alpha * (u0 * ((u0 * ((u0 * 0.3333333333333333e0) + 0.5e0)) + 1.0e0)))
end function

Julia

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

MATLAB

function tmp = code(alpha, u0)
	tmp = alpha * (alpha * (u0 * ((u0 * ((u0 * single(0.3333333333333333)) + single(0.5))) + single(1.0))));
end

Derivation

1. Initial program 56.5%

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

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

      \[\leadsto \mathsf{*.f32}\left(\left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right), \color{blue}{\log \left(1 - u0\right)}\right) \]

   2. *-commutative N/A

      \[\leadsto \mathsf{*.f32}\left(\left(\alpha \cdot \left(\mathsf{neg}\left(\alpha\right)\right)\right), \log \color{blue}{\left(1 - u0\right)}\right) \]

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

      \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(\alpha, \left(\mathsf{neg}\left(\alpha\right)\right)\right), \log \color{blue}{\left(1 - u0\right)}\right) \]

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

      \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(\alpha, \mathsf{neg.f32}\left(\alpha\right)\right), \log \left(1 - \color{blue}{u0}\right)\right) \]

   5. sub-neg N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(\alpha, \mathsf{neg.f32}\left(\alpha\right)\right), \log \left(1 + \left(\mathsf{neg}\left(u0\right)\right)\right)\right) \]

   6. log1p-define N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(\alpha, \mathsf{neg.f32}\left(\alpha\right)\right), \left(\mathsf{log1p}\left(\mathsf{neg}\left(u0\right)\right)\right)\right) \]

   7. log1p-lowering-log1p.f32 N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(\alpha, \mathsf{neg.f32}\left(\alpha\right)\right), \mathsf{log1p.f32}\left(\left(\mathsf{neg}\left(u0\right)\right)\right)\right) \]

   8. neg-lowering-neg.f32 99.0%

      \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(\alpha, \mathsf{neg.f32}\left(\alpha\right)\right), \mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right)\right) \]

3. Simplified 99.0%

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

5. Taylor expanded in u0 around 0

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

   1. distribute-rgt-in N/A

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

   2. associate-*r* N/A

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

   3. associate-+l+ N/A

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

   4. *-commutative N/A

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

   5. distribute-lft-in N/A

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

   6. associate-*r* N/A

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

   7. associate-*r* N/A

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

   8. distribute-rgt-in N/A

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

   9. *-commutative N/A

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

   10. associate-*r* N/A

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

   11. distribute-lft1-in N/A

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

   12. distribute-rgt-out N/A

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

7. Simplified 90.9%

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

   1. associate-*l* N/A

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

   2. associate-*l* N/A

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

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

      \[\leadsto \mathsf{*.f32}\left(\alpha, \color{blue}{\left(\alpha \cdot \left(u0 \cdot \left(\frac{1}{3} \cdot \left(u0 \cdot u0\right) + \left(u0 \cdot \frac{1}{2} + 1\right)\right)\right)\right)}\right) \]

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

      \[\leadsto \mathsf{*.f32}\left(\alpha, \mathsf{*.f32}\left(\alpha, \color{blue}{\left(u0 \cdot \left(\frac{1}{3} \cdot \left(u0 \cdot u0\right) + \left(u0 \cdot \frac{1}{2} + 1\right)\right)\right)}\right)\right) \]

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

      \[\leadsto \mathsf{*.f32}\left(\alpha, \mathsf{*.f32}\left(\alpha, \mathsf{*.f32}\left(u0, \color{blue}{\left(\frac{1}{3} \cdot \left(u0 \cdot u0\right) + \left(u0 \cdot \frac{1}{2} + 1\right)\right)}\right)\right)\right) \]

   6. associate-+r+ N/A

      \[\leadsto \mathsf{*.f32}\left(\alpha, \mathsf{*.f32}\left(\alpha, \mathsf{*.f32}\left(u0, \left(\left(\frac{1}{3} \cdot \left(u0 \cdot u0\right) + u0 \cdot \frac{1}{2}\right) + \color{blue}{1}\right)\right)\right)\right) \]

   7. +-commutative N/A

      \[\leadsto \mathsf{*.f32}\left(\alpha, \mathsf{*.f32}\left(\alpha, \mathsf{*.f32}\left(u0, \left(1 + \color{blue}{\left(\frac{1}{3} \cdot \left(u0 \cdot u0\right) + u0 \cdot \frac{1}{2}\right)}\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{*.f32}\left(\alpha, \mathsf{*.f32}\left(\alpha, \mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(1, \color{blue}{\left(\frac{1}{3} \cdot \left(u0 \cdot u0\right) + u0 \cdot \frac{1}{2}\right)}\right)\right)\right)\right) \]

   9. associate-*r* N/A

      \[\leadsto \mathsf{*.f32}\left(\alpha, \mathsf{*.f32}\left(\alpha, \mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(1, \left(\left(\frac{1}{3} \cdot u0\right) \cdot u0 + \color{blue}{u0} \cdot \frac{1}{2}\right)\right)\right)\right)\right) \]

   10. *-commutative N/A

       \[\leadsto \mathsf{*.f32}\left(\alpha, \mathsf{*.f32}\left(\alpha, \mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(1, \left(\left(\frac{1}{3} \cdot u0\right) \cdot u0 + \frac{1}{2} \cdot \color{blue}{u0}\right)\right)\right)\right)\right) \]

   11. distribute-rgt-out N/A

       \[\leadsto \mathsf{*.f32}\left(\alpha, \mathsf{*.f32}\left(\alpha, \mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(1, \left(u0 \cdot \color{blue}{\left(\frac{1}{3} \cdot u0 + \frac{1}{2}\right)}\right)\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f32}\left(\alpha, \mathsf{*.f32}\left(\alpha, \mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(u0, \color{blue}{\left(\frac{1}{3} \cdot u0 + \frac{1}{2}\right)}\right)\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f32}\left(\alpha, \mathsf{*.f32}\left(\alpha, \mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\left(\frac{1}{3} \cdot u0\right), \color{blue}{\frac{1}{2}}\right)\right)\right)\right)\right)\right) \]

   14. *-commutative N/A

       \[\leadsto \mathsf{*.f32}\left(\alpha, \mathsf{*.f32}\left(\alpha, \mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\left(u0 \cdot \frac{1}{3}\right), \frac{1}{2}\right)\right)\right)\right)\right)\right) \]

   15. *-lowering-*.f32 91.0%

       \[\leadsto \mathsf{*.f32}\left(\alpha, \mathsf{*.f32}\left(\alpha, \mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{*.f32}\left(u0, \frac{1}{3}\right), \frac{1}{2}\right)\right)\right)\right)\right)\right) \]

9. Applied egg-rr 91.0%

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

10. Final simplification 91.0%

    \[\leadsto \alpha \cdot \left(\alpha \cdot \left(u0 \cdot \left(u0 \cdot \left(u0 \cdot 0.3333333333333333 + 0.5\right) + 1\right)\right)\right) \]
11. Add Preprocessing

Alternative 9: 89.2% accurate, 9.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 56.5%

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

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

      \[\leadsto \mathsf{*.f32}\left(\left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right), \color{blue}{\log \left(1 - u0\right)}\right) \]

   2. *-commutative N/A

      \[\leadsto \mathsf{*.f32}\left(\left(\alpha \cdot \left(\mathsf{neg}\left(\alpha\right)\right)\right), \log \color{blue}{\left(1 - u0\right)}\right) \]

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

      \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(\alpha, \left(\mathsf{neg}\left(\alpha\right)\right)\right), \log \color{blue}{\left(1 - u0\right)}\right) \]

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

      \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(\alpha, \mathsf{neg.f32}\left(\alpha\right)\right), \log \left(1 - \color{blue}{u0}\right)\right) \]

   5. sub-neg N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(\alpha, \mathsf{neg.f32}\left(\alpha\right)\right), \log \left(1 + \left(\mathsf{neg}\left(u0\right)\right)\right)\right) \]

   6. log1p-define N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(\alpha, \mathsf{neg.f32}\left(\alpha\right)\right), \left(\mathsf{log1p}\left(\mathsf{neg}\left(u0\right)\right)\right)\right) \]

   7. log1p-lowering-log1p.f32 N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(\alpha, \mathsf{neg.f32}\left(\alpha\right)\right), \mathsf{log1p.f32}\left(\left(\mathsf{neg}\left(u0\right)\right)\right)\right) \]

   8. neg-lowering-neg.f32 99.0%

      \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(\alpha, \mathsf{neg.f32}\left(\alpha\right)\right), \mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right)\right) \]

3. Simplified 99.0%

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

5. Taylor expanded in u0 around 0

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

   1. distribute-rgt-in N/A

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

   2. associate-*r* N/A

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

   3. associate-+l+ N/A

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

   4. *-commutative N/A

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

   5. distribute-lft-in N/A

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

   6. associate-*r* N/A

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

   7. associate-*r* N/A

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

   8. distribute-rgt-in N/A

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

   9. *-commutative N/A

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

   10. associate-*r* N/A

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

   11. distribute-lft1-in N/A

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

   12. distribute-rgt-out N/A

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

7. Simplified 90.9%

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

   1. flip3-+ N/A

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

   2. clear-num N/A

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

   3. un-div-inv N/A

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

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

      \[\leadsto \mathsf{/.f32}\left(\left(\left(\alpha \cdot \alpha\right) \cdot u0\right), \color{blue}{\left(\frac{\left(\frac{1}{3} \cdot \left(u0 \cdot u0\right)\right) \cdot \left(\frac{1}{3} \cdot \left(u0 \cdot u0\right)\right) + \left(\left(u0 \cdot \frac{1}{2} + 1\right) \cdot \left(u0 \cdot \frac{1}{2} + 1\right) - \left(\frac{1}{3} \cdot \left(u0 \cdot u0\right)\right) \cdot \left(u0 \cdot \frac{1}{2} + 1\right)\right)}{{\left(\frac{1}{3} \cdot \left(u0 \cdot u0\right)\right)}^{3} + {\left(u0 \cdot \frac{1}{2} + 1\right)}^{3}}\right)}\right) \]

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

      \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(\left(\alpha \cdot \alpha\right), u0\right), \left(\frac{\color{blue}{\left(\frac{1}{3} \cdot \left(u0 \cdot u0\right)\right) \cdot \left(\frac{1}{3} \cdot \left(u0 \cdot u0\right)\right) + \left(\left(u0 \cdot \frac{1}{2} + 1\right) \cdot \left(u0 \cdot \frac{1}{2} + 1\right) - \left(\frac{1}{3} \cdot \left(u0 \cdot u0\right)\right) \cdot \left(u0 \cdot \frac{1}{2} + 1\right)\right)}}{{\left(\frac{1}{3} \cdot \left(u0 \cdot u0\right)\right)}^{3} + {\left(u0 \cdot \frac{1}{2} + 1\right)}^{3}}\right)\right) \]

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

      \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(\alpha, \alpha\right), u0\right), \left(\frac{\color{blue}{\left(\frac{1}{3} \cdot \left(u0 \cdot u0\right)\right) \cdot \left(\frac{1}{3} \cdot \left(u0 \cdot u0\right)\right)} + \left(\left(u0 \cdot \frac{1}{2} + 1\right) \cdot \left(u0 \cdot \frac{1}{2} + 1\right) - \left(\frac{1}{3} \cdot \left(u0 \cdot u0\right)\right) \cdot \left(u0 \cdot \frac{1}{2} + 1\right)\right)}{{\left(\frac{1}{3} \cdot \left(u0 \cdot u0\right)\right)}^{3} + {\left(u0 \cdot \frac{1}{2} + 1\right)}^{3}}\right)\right) \]

   7. clear-num N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(\alpha, \alpha\right), u0\right), \left(\frac{1}{\color{blue}{\frac{{\left(\frac{1}{3} \cdot \left(u0 \cdot u0\right)\right)}^{3} + {\left(u0 \cdot \frac{1}{2} + 1\right)}^{3}}{\left(\frac{1}{3} \cdot \left(u0 \cdot u0\right)\right) \cdot \left(\frac{1}{3} \cdot \left(u0 \cdot u0\right)\right) + \left(\left(u0 \cdot \frac{1}{2} + 1\right) \cdot \left(u0 \cdot \frac{1}{2} + 1\right) - \left(\frac{1}{3} \cdot \left(u0 \cdot u0\right)\right) \cdot \left(u0 \cdot \frac{1}{2} + 1\right)\right)}}}\right)\right) \]

9. Applied egg-rr 90.9%

   \[\leadsto \color{blue}{\frac{\left(\alpha \cdot \alpha\right) \cdot u0}{\frac{1}{1 + u0 \cdot \left(u0 \cdot 0.3333333333333333 + 0.5\right)}}} \]

10. Taylor expanded in u0 around 0

    \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(\alpha, \alpha\right), u0\right), \color{blue}{\left(1 + \frac{-1}{2} \cdot u0\right)}\right) \]
11. Step-by-step derivation

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

       \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(\alpha, \alpha\right), u0\right), \mathsf{+.f32}\left(1, \color{blue}{\left(\frac{-1}{2} \cdot u0\right)}\right)\right) \]

    2. *-commutative N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(\alpha, \alpha\right), u0\right), \mathsf{+.f32}\left(1, \left(u0 \cdot \color{blue}{\frac{-1}{2}}\right)\right)\right) \]

    3. *-lowering-*.f32 88.3%

       \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(\alpha, \alpha\right), u0\right), \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(u0, \color{blue}{\frac{-1}{2}}\right)\right)\right) \]

12. Simplified 88.3%

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

13. Final simplification 88.3%

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

Alternative 10: 87.2% accurate, 9.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 56.5%

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

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

      \[\leadsto \mathsf{*.f32}\left(\left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right), \color{blue}{\log \left(1 - u0\right)}\right) \]

   2. *-commutative N/A

      \[\leadsto \mathsf{*.f32}\left(\left(\alpha \cdot \left(\mathsf{neg}\left(\alpha\right)\right)\right), \log \color{blue}{\left(1 - u0\right)}\right) \]

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

      \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(\alpha, \left(\mathsf{neg}\left(\alpha\right)\right)\right), \log \color{blue}{\left(1 - u0\right)}\right) \]

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

      \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(\alpha, \mathsf{neg.f32}\left(\alpha\right)\right), \log \left(1 - \color{blue}{u0}\right)\right) \]

   5. sub-neg N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(\alpha, \mathsf{neg.f32}\left(\alpha\right)\right), \log \left(1 + \left(\mathsf{neg}\left(u0\right)\right)\right)\right) \]

   6. log1p-define N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(\alpha, \mathsf{neg.f32}\left(\alpha\right)\right), \left(\mathsf{log1p}\left(\mathsf{neg}\left(u0\right)\right)\right)\right) \]

   7. log1p-lowering-log1p.f32 N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(\alpha, \mathsf{neg.f32}\left(\alpha\right)\right), \mathsf{log1p.f32}\left(\left(\mathsf{neg}\left(u0\right)\right)\right)\right) \]

   8. neg-lowering-neg.f32 99.0%

      \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(\alpha, \mathsf{neg.f32}\left(\alpha\right)\right), \mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right)\right) \]

3. Simplified 99.0%

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

5. Taylor expanded in u0 around 0

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

   1. distribute-rgt-in N/A

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

   2. associate-*r* N/A

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

   3. associate-+l+ N/A

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

   4. *-commutative N/A

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

   5. distribute-lft-in N/A

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

   6. associate-*r* N/A

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

   7. associate-*r* N/A

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

   8. distribute-rgt-in N/A

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

   9. *-commutative N/A

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

   10. associate-*r* N/A

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

   11. distribute-lft1-in N/A

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

   12. distribute-rgt-out N/A

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

7. Simplified 90.9%

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

8. Taylor expanded in u0 around 0

   \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(\alpha, \alpha\right), u0\right), \color{blue}{\left(1 + \frac{1}{2} \cdot u0\right)}\right) \]
9. Step-by-step derivation

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

      \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(\alpha, \alpha\right), u0\right), \mathsf{+.f32}\left(1, \color{blue}{\left(\frac{1}{2} \cdot u0\right)}\right)\right) \]

   2. *-commutative N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(\alpha, \alpha\right), u0\right), \mathsf{+.f32}\left(1, \left(u0 \cdot \color{blue}{\frac{1}{2}}\right)\right)\right) \]

   3. *-lowering-*.f32 86.5%

      \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(\alpha, \alpha\right), u0\right), \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(u0, \color{blue}{\frac{1}{2}}\right)\right)\right) \]

10. Simplified 86.5%

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

11. Final simplification 86.5%

    \[\leadsto \left(u0 \cdot \left(\alpha \cdot \alpha\right)\right) \cdot \left(u0 \cdot 0.5 + 1\right) \]
12. Add Preprocessing

Alternative 11: 87.2% accurate, 9.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 56.5%

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

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

      \[\leadsto \mathsf{*.f32}\left(\left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right), \color{blue}{\log \left(1 - u0\right)}\right) \]

   2. *-commutative N/A

      \[\leadsto \mathsf{*.f32}\left(\left(\alpha \cdot \left(\mathsf{neg}\left(\alpha\right)\right)\right), \log \color{blue}{\left(1 - u0\right)}\right) \]

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

      \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(\alpha, \left(\mathsf{neg}\left(\alpha\right)\right)\right), \log \color{blue}{\left(1 - u0\right)}\right) \]

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

      \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(\alpha, \mathsf{neg.f32}\left(\alpha\right)\right), \log \left(1 - \color{blue}{u0}\right)\right) \]

   5. sub-neg N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(\alpha, \mathsf{neg.f32}\left(\alpha\right)\right), \log \left(1 + \left(\mathsf{neg}\left(u0\right)\right)\right)\right) \]

   6. log1p-define N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(\alpha, \mathsf{neg.f32}\left(\alpha\right)\right), \left(\mathsf{log1p}\left(\mathsf{neg}\left(u0\right)\right)\right)\right) \]

   7. log1p-lowering-log1p.f32 N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(\alpha, \mathsf{neg.f32}\left(\alpha\right)\right), \mathsf{log1p.f32}\left(\left(\mathsf{neg}\left(u0\right)\right)\right)\right) \]

   8. neg-lowering-neg.f32 99.0%

      \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(\alpha, \mathsf{neg.f32}\left(\alpha\right)\right), \mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right)\right) \]

3. Simplified 99.0%

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

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

      \[\leadsto \mathsf{*.f32}\left(u0, \color{blue}{\left(\frac{1}{2} \cdot \left({\alpha}^{2} \cdot u0\right) + {\alpha}^{2}\right)}\right) \]

   2. *-commutative N/A

      \[\leadsto \mathsf{*.f32}\left(u0, \left(\frac{1}{2} \cdot \left(u0 \cdot {\alpha}^{2}\right) + {\alpha}^{2}\right)\right) \]

   3. associate-*r* N/A

      \[\leadsto \mathsf{*.f32}\left(u0, \left(\left(\frac{1}{2} \cdot u0\right) \cdot {\alpha}^{2} + {\color{blue}{\alpha}}^{2}\right)\right) \]

   4. distribute-lft1-in N/A

      \[\leadsto \mathsf{*.f32}\left(u0, \left(\left(\frac{1}{2} \cdot u0 + 1\right) \cdot \color{blue}{{\alpha}^{2}}\right)\right) \]

   5. *-commutative N/A

      \[\leadsto \mathsf{*.f32}\left(u0, \left(\left(u0 \cdot \frac{1}{2} + 1\right) \cdot {\alpha}^{2}\right)\right) \]

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

      \[\leadsto \mathsf{*.f32}\left(u0, \mathsf{*.f32}\left(\left(u0 \cdot \frac{1}{2} + 1\right), \color{blue}{\left({\alpha}^{2}\right)}\right)\right) \]

   7. *-commutative N/A

      \[\leadsto \mathsf{*.f32}\left(u0, \mathsf{*.f32}\left(\left(\frac{1}{2} \cdot u0 + 1\right), \left({\alpha}^{2}\right)\right)\right) \]

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

      \[\leadsto \mathsf{*.f32}\left(u0, \mathsf{*.f32}\left(\mathsf{+.f32}\left(\left(\frac{1}{2} \cdot u0\right), 1\right), \left({\color{blue}{\alpha}}^{2}\right)\right)\right) \]

   9. *-commutative N/A

      \[\leadsto \mathsf{*.f32}\left(u0, \mathsf{*.f32}\left(\mathsf{+.f32}\left(\left(u0 \cdot \frac{1}{2}\right), 1\right), \left({\alpha}^{2}\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f32}\left(u0, \mathsf{*.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(u0, \frac{1}{2}\right), 1\right), \left({\alpha}^{2}\right)\right)\right) \]

   11. unpow2 N/A

       \[\leadsto \mathsf{*.f32}\left(u0, \mathsf{*.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(u0, \frac{1}{2}\right), 1\right), \left(\alpha \cdot \color{blue}{\alpha}\right)\right)\right) \]

   12. *-lowering-*.f32 86.4%

       \[\leadsto \mathsf{*.f32}\left(u0, \mathsf{*.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(u0, \frac{1}{2}\right), 1\right), \mathsf{*.f32}\left(\alpha, \color{blue}{\alpha}\right)\right)\right) \]

7. Simplified 86.4%

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

8. Final simplification 86.4%

   \[\leadsto u0 \cdot \left(\left(\alpha \cdot \alpha\right) \cdot \left(u0 \cdot 0.5 + 1\right)\right) \]
9. Add Preprocessing

Alternative 12: 74.6% accurate, 21.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 56.5%

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

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

      \[\leadsto \mathsf{*.f32}\left(\left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right), \color{blue}{\log \left(1 - u0\right)}\right) \]

   2. *-commutative N/A

      \[\leadsto \mathsf{*.f32}\left(\left(\alpha \cdot \left(\mathsf{neg}\left(\alpha\right)\right)\right), \log \color{blue}{\left(1 - u0\right)}\right) \]

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

      \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(\alpha, \left(\mathsf{neg}\left(\alpha\right)\right)\right), \log \color{blue}{\left(1 - u0\right)}\right) \]

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

      \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(\alpha, \mathsf{neg.f32}\left(\alpha\right)\right), \log \left(1 - \color{blue}{u0}\right)\right) \]

   5. sub-neg N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(\alpha, \mathsf{neg.f32}\left(\alpha\right)\right), \log \left(1 + \left(\mathsf{neg}\left(u0\right)\right)\right)\right) \]

   6. log1p-define N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(\alpha, \mathsf{neg.f32}\left(\alpha\right)\right), \left(\mathsf{log1p}\left(\mathsf{neg}\left(u0\right)\right)\right)\right) \]

   7. log1p-lowering-log1p.f32 N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(\alpha, \mathsf{neg.f32}\left(\alpha\right)\right), \mathsf{log1p.f32}\left(\left(\mathsf{neg}\left(u0\right)\right)\right)\right) \]

   8. neg-lowering-neg.f32 99.0%

      \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(\alpha, \mathsf{neg.f32}\left(\alpha\right)\right), \mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right)\right) \]

3. Simplified 99.0%

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

5. Taylor expanded in u0 around 0

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

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

      \[\leadsto \mathsf{*.f32}\left(\left({\alpha}^{2}\right), \color{blue}{u0}\right) \]

   2. unpow2 N/A

      \[\leadsto \mathsf{*.f32}\left(\left(\alpha \cdot \alpha\right), u0\right) \]

   3. *-lowering-*.f32 73.5%

      \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(\alpha, \alpha\right), u0\right) \]

7. Simplified 73.5%

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

   1. associate-*l* N/A

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

   2. *-commutative N/A

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

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

      \[\leadsto \mathsf{*.f32}\left(\left(\alpha \cdot u0\right), \color{blue}{\alpha}\right) \]

   4. *-lowering-*.f32 73.5%

      \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(\alpha, u0\right), \alpha\right) \]

9. Applied egg-rr 73.5%

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

10. Final simplification 73.5%

    \[\leadsto \alpha \cdot \left(\alpha \cdot u0\right) \]
11. Add Preprocessing

Alternative 13: 74.6% accurate, 21.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 56.5%

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

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

      \[\leadsto \mathsf{*.f32}\left(\left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right), \color{blue}{\log \left(1 - u0\right)}\right) \]

   2. *-commutative N/A

      \[\leadsto \mathsf{*.f32}\left(\left(\alpha \cdot \left(\mathsf{neg}\left(\alpha\right)\right)\right), \log \color{blue}{\left(1 - u0\right)}\right) \]

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

      \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(\alpha, \left(\mathsf{neg}\left(\alpha\right)\right)\right), \log \color{blue}{\left(1 - u0\right)}\right) \]

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

      \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(\alpha, \mathsf{neg.f32}\left(\alpha\right)\right), \log \left(1 - \color{blue}{u0}\right)\right) \]

   5. sub-neg N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(\alpha, \mathsf{neg.f32}\left(\alpha\right)\right), \log \left(1 + \left(\mathsf{neg}\left(u0\right)\right)\right)\right) \]

   6. log1p-define N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(\alpha, \mathsf{neg.f32}\left(\alpha\right)\right), \left(\mathsf{log1p}\left(\mathsf{neg}\left(u0\right)\right)\right)\right) \]

   7. log1p-lowering-log1p.f32 N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(\alpha, \mathsf{neg.f32}\left(\alpha\right)\right), \mathsf{log1p.f32}\left(\left(\mathsf{neg}\left(u0\right)\right)\right)\right) \]

   8. neg-lowering-neg.f32 99.0%

      \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(\alpha, \mathsf{neg.f32}\left(\alpha\right)\right), \mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right)\right) \]

3. Simplified 99.0%

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

5. Taylor expanded in u0 around 0

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

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

      \[\leadsto \mathsf{*.f32}\left(\left({\alpha}^{2}\right), \color{blue}{u0}\right) \]

   2. unpow2 N/A

      \[\leadsto \mathsf{*.f32}\left(\left(\alpha \cdot \alpha\right), u0\right) \]

   3. *-lowering-*.f32 73.5%

      \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(\alpha, \alpha\right), u0\right) \]

7. Simplified 73.5%

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

8. Final simplification 73.5%

   \[\leadsto u0 \cdot \left(\alpha \cdot \alpha\right) \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024158 
(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))))