Beckmann Distribution sample, tan2theta, alphax == alphay

Percentage Accurate: 55.7% → 99.0%

Time: 11.1s

Alternatives: 13

Speedup: 10.5×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Initial Program: 55.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Alternative 1: 99.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 57.6%

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

   1. sub-neg N/A

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

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

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

   3. neg-lowering-neg.f32 99.0

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

4. Applied egg-rr 99.0%

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

   1. neg-sub0 N/A

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

   2. flip3-- N/A

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

   3. metadata-eval N/A

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

   4. cube-unmult N/A

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

   5. neg-sub0 N/A

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

   6. cube-unmult N/A

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

   7. cube-neg N/A

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

   8. sqr-pow N/A

      \[\leadsto \left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right) \cdot \mathsf{log1p}\left(\frac{\color{blue}{{\left(\mathsf{neg}\left(u0\right)\right)}^{\left(\frac{3}{2}\right)} \cdot {\left(\mathsf{neg}\left(u0\right)\right)}^{\left(\frac{3}{2}\right)}}}{0 \cdot 0 + \left(u0 \cdot u0 + 0 \cdot u0\right)}\right) \]

   9. unpow-prod-down N/A

      \[\leadsto \left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right) \cdot \mathsf{log1p}\left(\frac{\color{blue}{{\left(\left(\mathsf{neg}\left(u0\right)\right) \cdot \left(\mathsf{neg}\left(u0\right)\right)\right)}^{\left(\frac{3}{2}\right)}}}{0 \cdot 0 + \left(u0 \cdot u0 + 0 \cdot u0\right)}\right) \]

   10. sqr-neg N/A

       \[\leadsto \left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right) \cdot \mathsf{log1p}\left(\frac{{\color{blue}{\left(u0 \cdot u0\right)}}^{\left(\frac{3}{2}\right)}}{0 \cdot 0 + \left(u0 \cdot u0 + 0 \cdot u0\right)}\right) \]

   11. unpow-prod-down N/A

       \[\leadsto \left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right) \cdot \mathsf{log1p}\left(\frac{\color{blue}{{u0}^{\left(\frac{3}{2}\right)} \cdot {u0}^{\left(\frac{3}{2}\right)}}}{0 \cdot 0 + \left(u0 \cdot u0 + 0 \cdot u0\right)}\right) \]

   12. sqr-pow N/A

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

   13. cube-unmult N/A

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

   14. frac-2neg N/A

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

   15. cube-unmult N/A

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

   16. cube-neg N/A

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

   17. sqr-pow N/A

       \[\leadsto \left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right) \cdot \mathsf{log1p}\left(\frac{\color{blue}{{\left(\mathsf{neg}\left(u0\right)\right)}^{\left(\frac{3}{2}\right)} \cdot {\left(\mathsf{neg}\left(u0\right)\right)}^{\left(\frac{3}{2}\right)}}}{\mathsf{neg}\left(\left(0 \cdot 0 + \left(u0 \cdot u0 + 0 \cdot u0\right)\right)\right)}\right) \]

   18. unpow-prod-down N/A

       \[\leadsto \left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right) \cdot \mathsf{log1p}\left(\frac{\color{blue}{{\left(\left(\mathsf{neg}\left(u0\right)\right) \cdot \left(\mathsf{neg}\left(u0\right)\right)\right)}^{\left(\frac{3}{2}\right)}}}{\mathsf{neg}\left(\left(0 \cdot 0 + \left(u0 \cdot u0 + 0 \cdot u0\right)\right)\right)}\right) \]

   19. sqr-neg N/A

       \[\leadsto \left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right) \cdot \mathsf{log1p}\left(\frac{{\color{blue}{\left(u0 \cdot u0\right)}}^{\left(\frac{3}{2}\right)}}{\mathsf{neg}\left(\left(0 \cdot 0 + \left(u0 \cdot u0 + 0 \cdot u0\right)\right)\right)}\right) \]

   20. unpow-prod-down N/A

       \[\leadsto \left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right) \cdot \mathsf{log1p}\left(\frac{\color{blue}{{u0}^{\left(\frac{3}{2}\right)} \cdot {u0}^{\left(\frac{3}{2}\right)}}}{\mathsf{neg}\left(\left(0 \cdot 0 + \left(u0 \cdot u0 + 0 \cdot u0\right)\right)\right)}\right) \]

   21. sqr-pow N/A

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

   22. cube-unmult N/A

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

   23. /-lowering-/.f32 N/A

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

6. Applied egg-rr 98.8%

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

   1. *-commutative N/A

      \[\leadsto \color{blue}{\log \left(1 + \frac{u0 \cdot \left(u0 \cdot u0\right)}{\mathsf{neg}\left(\left(0 + \left(u0 \cdot u0 + 0 \cdot u0\right)\right)\right)}\right) \cdot \left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right)} \]

   2. associate-*r* N/A

      \[\leadsto \color{blue}{\left(\log \left(1 + \frac{u0 \cdot \left(u0 \cdot u0\right)}{\mathsf{neg}\left(\left(0 + \left(u0 \cdot u0 + 0 \cdot u0\right)\right)\right)}\right) \cdot \left(\mathsf{neg}\left(\alpha\right)\right)\right) \cdot \alpha} \]

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

      \[\leadsto \color{blue}{\left(\log \left(1 + \frac{u0 \cdot \left(u0 \cdot u0\right)}{\mathsf{neg}\left(\left(0 + \left(u0 \cdot u0 + 0 \cdot u0\right)\right)\right)}\right) \cdot \left(\mathsf{neg}\left(\alpha\right)\right)\right) \cdot \alpha} \]

8. Applied egg-rr 99.1%

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

9. Final simplification 99.1%

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

Alternative 2: 93.8% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 57.6%

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

3. Taylor expanded in u0 around 0

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

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

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

   2. sub-neg N/A

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

   3. metadata-eval N/A

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

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

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

   5. sub-neg N/A

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

   6. metadata-eval N/A

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

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

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

   8. sub-neg N/A

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

   9. *-commutative N/A

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

   10. metadata-eval N/A

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

   11. accelerator-lowering-fma.f32 92.9

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

5. Simplified 92.9%

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

   1. flip-+ N/A

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

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

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

7. Applied egg-rr 92.8%

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

8. Taylor expanded in u0 around 0

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

10. Simplified 93.6%

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

11. Final simplification 93.6%

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

Alternative 3: 93.5% accurate, 2.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 57.6%

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

3. Taylor expanded in u0 around 0

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

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

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

   2. sub-neg N/A

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

   3. metadata-eval N/A

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

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

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

   5. sub-neg N/A

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

   6. metadata-eval N/A

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

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

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

   8. sub-neg N/A

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

   9. *-commutative N/A

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

   10. metadata-eval N/A

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

   11. accelerator-lowering-fma.f32 92.9

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

5. Simplified 92.9%

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

   1. associate-*l* N/A

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

   2. *-commutative N/A

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

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

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

   4. associate-*r* N/A

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

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

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

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

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

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

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

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

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

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

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

   10. neg-lowering-neg.f32 93.0

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

7. Applied egg-rr 93.0%

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

   1. +-commutative N/A

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

   2. distribute-rgt-in N/A

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

   3. neg-mul-1 N/A

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

   4. *-commutative N/A

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

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

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

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

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

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

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

   8. *-commutative N/A

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

   9. associate-*l* N/A

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

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

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

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

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

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

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

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

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

   14. *-commutative N/A

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

   15. *-lowering-*.f32 93.4

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

9. Applied egg-rr 93.4%

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

10. Final simplification 93.4%

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

Alternative 4: 93.4% accurate, 3.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 57.6%

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

3. Taylor expanded in u0 around 0

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

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

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

   2. sub-neg N/A

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

   3. metadata-eval N/A

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

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

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

   5. sub-neg N/A

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

   6. metadata-eval N/A

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

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

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

   8. sub-neg N/A

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

   9. *-commutative N/A

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

   10. metadata-eval N/A

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

   11. accelerator-lowering-fma.f32 92.9

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

5. Simplified 92.9%

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

   1. associate-*l* N/A

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

   2. *-commutative N/A

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

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

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

   4. associate-*r* N/A

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

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

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

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

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

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

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

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

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

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

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

   10. neg-lowering-neg.f32 93.0

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

7. Applied egg-rr 93.0%

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

   1. associate-*l* N/A

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

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

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

   3. *-commutative N/A

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

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

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

   5. metadata-eval N/A

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

   6. sub-neg N/A

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

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

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

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

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

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

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

   10. sub-neg N/A

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

   11. metadata-eval N/A

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

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

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

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

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

   14. accelerator-lowering-fma.f32 93.0

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

9. Applied egg-rr 93.0%

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

10. Taylor expanded in alpha around 0

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

12. Simplified 93.2%

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

Alternative 5: 93.1% accurate, 3.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 57.6%

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

3. Taylor expanded in u0 around 0

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

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

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

   2. sub-neg N/A

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

   3. metadata-eval N/A

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

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

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

   5. sub-neg N/A

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

   6. metadata-eval N/A

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

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

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

   8. sub-neg N/A

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

   9. *-commutative N/A

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

   10. metadata-eval N/A

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

   11. accelerator-lowering-fma.f32 92.9

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

5. Simplified 92.9%

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

   1. associate-*l* N/A

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

   2. *-commutative N/A

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

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

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

   4. associate-*r* N/A

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

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

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

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

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

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

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

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

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

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

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

   10. neg-lowering-neg.f32 93.0

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

7. Applied egg-rr 93.0%

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

   1. associate-*l* N/A

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

   2. *-commutative N/A

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

   3. associate-*l* N/A

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

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

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

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

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

   6. metadata-eval N/A

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

   7. sub-neg N/A

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

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

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

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

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

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

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

9. Applied egg-rr 92.9%

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

10. Final simplification 92.9%

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

Alternative 6: 91.3% accurate, 4.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 57.6%

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

   1. sub-neg N/A

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

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

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

   3. neg-lowering-neg.f32 99.0

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

4. Applied egg-rr 99.0%

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

   1. neg-sub0 N/A

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

   2. flip3-- N/A

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

   3. metadata-eval N/A

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

   4. cube-unmult N/A

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

   5. neg-sub0 N/A

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

   6. cube-unmult N/A

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

   7. cube-neg N/A

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

   8. sqr-pow N/A

      \[\leadsto \left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right) \cdot \mathsf{log1p}\left(\frac{\color{blue}{{\left(\mathsf{neg}\left(u0\right)\right)}^{\left(\frac{3}{2}\right)} \cdot {\left(\mathsf{neg}\left(u0\right)\right)}^{\left(\frac{3}{2}\right)}}}{0 \cdot 0 + \left(u0 \cdot u0 + 0 \cdot u0\right)}\right) \]

   9. unpow-prod-down N/A

      \[\leadsto \left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right) \cdot \mathsf{log1p}\left(\frac{\color{blue}{{\left(\left(\mathsf{neg}\left(u0\right)\right) \cdot \left(\mathsf{neg}\left(u0\right)\right)\right)}^{\left(\frac{3}{2}\right)}}}{0 \cdot 0 + \left(u0 \cdot u0 + 0 \cdot u0\right)}\right) \]

   10. sqr-neg N/A

       \[\leadsto \left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right) \cdot \mathsf{log1p}\left(\frac{{\color{blue}{\left(u0 \cdot u0\right)}}^{\left(\frac{3}{2}\right)}}{0 \cdot 0 + \left(u0 \cdot u0 + 0 \cdot u0\right)}\right) \]

   11. unpow-prod-down N/A

       \[\leadsto \left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right) \cdot \mathsf{log1p}\left(\frac{\color{blue}{{u0}^{\left(\frac{3}{2}\right)} \cdot {u0}^{\left(\frac{3}{2}\right)}}}{0 \cdot 0 + \left(u0 \cdot u0 + 0 \cdot u0\right)}\right) \]

   12. sqr-pow N/A

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

   13. cube-unmult N/A

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

   14. frac-2neg N/A

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

   15. cube-unmult N/A

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

   16. cube-neg N/A

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

   17. sqr-pow N/A

       \[\leadsto \left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right) \cdot \mathsf{log1p}\left(\frac{\color{blue}{{\left(\mathsf{neg}\left(u0\right)\right)}^{\left(\frac{3}{2}\right)} \cdot {\left(\mathsf{neg}\left(u0\right)\right)}^{\left(\frac{3}{2}\right)}}}{\mathsf{neg}\left(\left(0 \cdot 0 + \left(u0 \cdot u0 + 0 \cdot u0\right)\right)\right)}\right) \]

   18. unpow-prod-down N/A

       \[\leadsto \left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right) \cdot \mathsf{log1p}\left(\frac{\color{blue}{{\left(\left(\mathsf{neg}\left(u0\right)\right) \cdot \left(\mathsf{neg}\left(u0\right)\right)\right)}^{\left(\frac{3}{2}\right)}}}{\mathsf{neg}\left(\left(0 \cdot 0 + \left(u0 \cdot u0 + 0 \cdot u0\right)\right)\right)}\right) \]

   19. sqr-neg N/A

       \[\leadsto \left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right) \cdot \mathsf{log1p}\left(\frac{{\color{blue}{\left(u0 \cdot u0\right)}}^{\left(\frac{3}{2}\right)}}{\mathsf{neg}\left(\left(0 \cdot 0 + \left(u0 \cdot u0 + 0 \cdot u0\right)\right)\right)}\right) \]

   20. unpow-prod-down N/A

       \[\leadsto \left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right) \cdot \mathsf{log1p}\left(\frac{\color{blue}{{u0}^{\left(\frac{3}{2}\right)} \cdot {u0}^{\left(\frac{3}{2}\right)}}}{\mathsf{neg}\left(\left(0 \cdot 0 + \left(u0 \cdot u0 + 0 \cdot u0\right)\right)\right)}\right) \]

   21. sqr-pow N/A

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

   22. cube-unmult N/A

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

   23. /-lowering-/.f32 N/A

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

6. Applied egg-rr 98.8%

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

   1. *-commutative N/A

      \[\leadsto \color{blue}{\log \left(1 + \frac{u0 \cdot \left(u0 \cdot u0\right)}{\mathsf{neg}\left(\left(0 + \left(u0 \cdot u0 + 0 \cdot u0\right)\right)\right)}\right) \cdot \left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right)} \]

   2. associate-*r* N/A

      \[\leadsto \color{blue}{\left(\log \left(1 + \frac{u0 \cdot \left(u0 \cdot u0\right)}{\mathsf{neg}\left(\left(0 + \left(u0 \cdot u0 + 0 \cdot u0\right)\right)\right)}\right) \cdot \left(\mathsf{neg}\left(\alpha\right)\right)\right) \cdot \alpha} \]

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

      \[\leadsto \color{blue}{\left(\log \left(1 + \frac{u0 \cdot \left(u0 \cdot u0\right)}{\mathsf{neg}\left(\left(0 + \left(u0 \cdot u0 + 0 \cdot u0\right)\right)\right)}\right) \cdot \left(\mathsf{neg}\left(\alpha\right)\right)\right) \cdot \alpha} \]

8. Applied egg-rr 99.1%

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

9. Taylor expanded in u0 around 0

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

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

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

    2. +-commutative N/A

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

    3. distribute-lft-in N/A

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

    4. *-commutative N/A

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

    5. *-commutative N/A

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

    6. associate-*l* N/A

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

    7. *-commutative N/A

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

    8. associate-*r* N/A

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

    9. *-commutative N/A

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

    10. distribute-lft-out N/A

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

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

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

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

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

    13. accelerator-lowering-fma.f32 91.1

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

11. Simplified 91.1%

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

12. Final simplification 91.1%

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

Alternative 7: 91.3% accurate, 4.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 57.6%

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

3. Taylor expanded in u0 around 0

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

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

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

   2. sub-neg N/A

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

   3. metadata-eval N/A

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

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

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

   5. sub-neg N/A

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

   6. metadata-eval N/A

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

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

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

   8. sub-neg N/A

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

   9. *-commutative N/A

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

   10. metadata-eval N/A

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

   11. accelerator-lowering-fma.f32 92.9

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

5. Simplified 92.9%

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

   1. associate-*l* N/A

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

   2. *-commutative N/A

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

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

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

   4. associate-*r* N/A

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

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

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

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

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

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

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

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

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

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

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

   10. neg-lowering-neg.f32 93.0

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

7. Applied egg-rr 93.0%

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

   1. associate-*l* N/A

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

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

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

   3. *-commutative N/A

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

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

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

   5. metadata-eval N/A

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

   6. sub-neg N/A

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

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

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

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

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

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

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

   10. sub-neg N/A

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

   11. metadata-eval N/A

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

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

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

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

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

   14. accelerator-lowering-fma.f32 93.0

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

9. Applied egg-rr 93.0%

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

10. Taylor expanded in u0 around 0

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

    1. sub-neg N/A

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

    2. cancel-sign-sub-inv N/A

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

    3. metadata-eval N/A

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

    4. distribute-rgt-in N/A

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

    5. *-commutative N/A

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

    6. associate-*l* N/A

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

    7. associate-*r* N/A

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

    8. *-commutative N/A

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

    9. distribute-lft-out N/A

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

    10. mul-1-neg N/A

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

    11. remove-double-neg N/A

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

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

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

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

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

    14. accelerator-lowering-fma.f32 91.0

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

12. Simplified 91.0%

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

13. Final simplification 91.0%

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

Alternative 8: 87.2% accurate, 5.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 57.6%

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

   1. sub-neg N/A

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

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

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

   3. neg-lowering-neg.f32 99.0

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

4. Applied egg-rr 99.0%

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

   1. neg-sub0 N/A

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

   2. flip3-- N/A

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

   3. metadata-eval N/A

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

   4. cube-unmult N/A

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

   5. neg-sub0 N/A

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

   6. cube-unmult N/A

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

   7. cube-neg N/A

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

   8. sqr-pow N/A

      \[\leadsto \left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right) \cdot \mathsf{log1p}\left(\frac{\color{blue}{{\left(\mathsf{neg}\left(u0\right)\right)}^{\left(\frac{3}{2}\right)} \cdot {\left(\mathsf{neg}\left(u0\right)\right)}^{\left(\frac{3}{2}\right)}}}{0 \cdot 0 + \left(u0 \cdot u0 + 0 \cdot u0\right)}\right) \]

   9. unpow-prod-down N/A

      \[\leadsto \left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right) \cdot \mathsf{log1p}\left(\frac{\color{blue}{{\left(\left(\mathsf{neg}\left(u0\right)\right) \cdot \left(\mathsf{neg}\left(u0\right)\right)\right)}^{\left(\frac{3}{2}\right)}}}{0 \cdot 0 + \left(u0 \cdot u0 + 0 \cdot u0\right)}\right) \]

   10. sqr-neg N/A

       \[\leadsto \left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right) \cdot \mathsf{log1p}\left(\frac{{\color{blue}{\left(u0 \cdot u0\right)}}^{\left(\frac{3}{2}\right)}}{0 \cdot 0 + \left(u0 \cdot u0 + 0 \cdot u0\right)}\right) \]

   11. unpow-prod-down N/A

       \[\leadsto \left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right) \cdot \mathsf{log1p}\left(\frac{\color{blue}{{u0}^{\left(\frac{3}{2}\right)} \cdot {u0}^{\left(\frac{3}{2}\right)}}}{0 \cdot 0 + \left(u0 \cdot u0 + 0 \cdot u0\right)}\right) \]

   12. sqr-pow N/A

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

   13. cube-unmult N/A

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

   14. frac-2neg N/A

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

   15. cube-unmult N/A

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

   16. cube-neg N/A

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

   17. sqr-pow N/A

       \[\leadsto \left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right) \cdot \mathsf{log1p}\left(\frac{\color{blue}{{\left(\mathsf{neg}\left(u0\right)\right)}^{\left(\frac{3}{2}\right)} \cdot {\left(\mathsf{neg}\left(u0\right)\right)}^{\left(\frac{3}{2}\right)}}}{\mathsf{neg}\left(\left(0 \cdot 0 + \left(u0 \cdot u0 + 0 \cdot u0\right)\right)\right)}\right) \]

   18. unpow-prod-down N/A

       \[\leadsto \left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right) \cdot \mathsf{log1p}\left(\frac{\color{blue}{{\left(\left(\mathsf{neg}\left(u0\right)\right) \cdot \left(\mathsf{neg}\left(u0\right)\right)\right)}^{\left(\frac{3}{2}\right)}}}{\mathsf{neg}\left(\left(0 \cdot 0 + \left(u0 \cdot u0 + 0 \cdot u0\right)\right)\right)}\right) \]

   19. sqr-neg N/A

       \[\leadsto \left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right) \cdot \mathsf{log1p}\left(\frac{{\color{blue}{\left(u0 \cdot u0\right)}}^{\left(\frac{3}{2}\right)}}{\mathsf{neg}\left(\left(0 \cdot 0 + \left(u0 \cdot u0 + 0 \cdot u0\right)\right)\right)}\right) \]

   20. unpow-prod-down N/A

       \[\leadsto \left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right) \cdot \mathsf{log1p}\left(\frac{\color{blue}{{u0}^{\left(\frac{3}{2}\right)} \cdot {u0}^{\left(\frac{3}{2}\right)}}}{\mathsf{neg}\left(\left(0 \cdot 0 + \left(u0 \cdot u0 + 0 \cdot u0\right)\right)\right)}\right) \]

   21. sqr-pow N/A

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

   22. cube-unmult N/A

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

   23. /-lowering-/.f32 N/A

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

6. Applied egg-rr 98.8%

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

   1. *-commutative N/A

      \[\leadsto \color{blue}{\log \left(1 + \frac{u0 \cdot \left(u0 \cdot u0\right)}{\mathsf{neg}\left(\left(0 + \left(u0 \cdot u0 + 0 \cdot u0\right)\right)\right)}\right) \cdot \left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right)} \]

   2. associate-*r* N/A

      \[\leadsto \color{blue}{\left(\log \left(1 + \frac{u0 \cdot \left(u0 \cdot u0\right)}{\mathsf{neg}\left(\left(0 + \left(u0 \cdot u0 + 0 \cdot u0\right)\right)\right)}\right) \cdot \left(\mathsf{neg}\left(\alpha\right)\right)\right) \cdot \alpha} \]

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

      \[\leadsto \color{blue}{\left(\log \left(1 + \frac{u0 \cdot \left(u0 \cdot u0\right)}{\mathsf{neg}\left(\left(0 + \left(u0 \cdot u0 + 0 \cdot u0\right)\right)\right)}\right) \cdot \left(\mathsf{neg}\left(\alpha\right)\right)\right) \cdot \alpha} \]

8. Applied egg-rr 99.1%

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

9. Taylor expanded in u0 around 0

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

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

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

    2. +-commutative N/A

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

    3. *-commutative N/A

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

    4. associate-*l* N/A

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

    5. *-commutative N/A

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

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

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

    7. *-commutative N/A

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

    8. *-lowering-*.f32 86.2

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

11. Simplified 86.2%

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

12. Final simplification 86.2%

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

Alternative 9: 87.2% accurate, 5.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 57.6%

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

3. Taylor expanded in u0 around 0

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

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

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

   2. sub-neg N/A

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

   3. metadata-eval N/A

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

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

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

   5. sub-neg N/A

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

   6. metadata-eval N/A

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

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

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

   8. sub-neg N/A

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

   9. *-commutative N/A

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

   10. metadata-eval N/A

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

   11. accelerator-lowering-fma.f32 92.9

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

5. Simplified 92.9%

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

   1. associate-*l* N/A

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

   2. *-commutative N/A

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

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

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

   4. associate-*r* N/A

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

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

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

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

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

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

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

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

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

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

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

   10. neg-lowering-neg.f32 93.0

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

7. Applied egg-rr 93.0%

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

   1. associate-*l* N/A

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

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

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

   3. *-commutative N/A

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

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

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

   5. metadata-eval N/A

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

   6. sub-neg N/A

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

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

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

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

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

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

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

   10. sub-neg N/A

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

   11. metadata-eval N/A

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

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

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

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

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

   14. accelerator-lowering-fma.f32 93.0

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

9. Applied egg-rr 93.0%

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

10. Taylor expanded in u0 around 0

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

    1. sub-neg N/A

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

    2. *-commutative N/A

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

    3. associate-*l* N/A

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

    4. *-commutative N/A

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

    5. mul-1-neg N/A

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

    6. remove-double-neg N/A

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

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

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

    8. *-commutative N/A

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

    9. *-lowering-*.f32 86.1

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

12. Simplified 86.1%

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

Alternative 10: 87.2% accurate, 5.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 57.6%

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

3. Taylor expanded in u0 around 0

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

   1. distribute-lft-in N/A

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

   2. associate-*r* N/A

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

   3. *-commutative N/A

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

   4. associate-*r* N/A

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

   5. distribute-rgt-out N/A

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

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

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

   7. unpow2 N/A

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

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

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

   9. associate-*r* N/A

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

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

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

   11. *-commutative N/A

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

   12. *-lowering-*.f32 86.0

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

5. Simplified 86.0%

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

Alternative 11: 87.0% accurate, 5.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 57.6%

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

3. Taylor expanded in u0 around 0

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

   1. distribute-lft-in N/A

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

   2. associate-*r* N/A

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

   3. *-commutative N/A

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

   4. associate-*r* N/A

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

   5. distribute-rgt-out N/A

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

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

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

   7. unpow2 N/A

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

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

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

   9. associate-*r* N/A

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

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

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

   11. *-commutative N/A

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

   12. *-lowering-*.f32 86.0

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

5. Simplified 86.0%

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

   1. *-commutative N/A

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

   2. distribute-lft1-in N/A

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

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

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

   4. accelerator-lowering-fma.f32 85.8

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

7. Applied egg-rr 85.8%

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

8. Final simplification 85.8%

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

Alternative 12: 74.6% accurate, 10.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 57.6%

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

3. Taylor expanded in u0 around 0

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

   1. *-commutative N/A

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

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

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

   3. unpow2 N/A

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

   4. *-lowering-*.f32 73.0

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

5. Simplified 73.0%

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

   1. associate-*r* N/A

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

   2. *-commutative N/A

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

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

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

   4. *-lowering-*.f32 73.1

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

7. Applied egg-rr 73.1%

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

8. Final simplification 73.1%

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

Alternative 13: 74.6% accurate, 10.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 57.6%

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

3. Taylor expanded in u0 around 0

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

   1. *-commutative N/A

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

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

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

   3. unpow2 N/A

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

   4. *-lowering-*.f32 73.0

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

5. Simplified 73.0%

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

Reproduce

herbie shell --seed 2024199 
(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))))