Beckmann Distribution sample, tan2theta, alphax == alphay

Percentage Accurate: 55.6% → 98.9%

Time: 11.6s

Alternatives: 10

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: 55.6% 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: 98.9% 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 58.4%

   \[\left(\left(-\alpha\right) \cdot \alpha\right) \cdot \log \left(1 - u0\right) \]
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. lower-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. lower-neg.f32 98.9

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

4. Applied egg-rr 98.9%

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

   1. neg-sub0 N/A

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

   2. flip3-- N/A

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

   3. clear-num N/A

      \[\leadsto \left(\color{blue}{\frac{1}{\frac{0 \cdot 0 + \left(\alpha \cdot \alpha + 0 \cdot \alpha\right)}{{0}^{3} - {\alpha}^{3}}}} \cdot \alpha\right) \cdot \mathsf{log1p}\left(\mathsf{neg}\left(u0\right)\right) \]

   4. lower-/.f32 N/A

      \[\leadsto \left(\color{blue}{\frac{1}{\frac{0 \cdot 0 + \left(\alpha \cdot \alpha + 0 \cdot \alpha\right)}{{0}^{3} - {\alpha}^{3}}}} \cdot \alpha\right) \cdot \mathsf{log1p}\left(\mathsf{neg}\left(u0\right)\right) \]

   5. lower-/.f32 N/A

      \[\leadsto \left(\frac{1}{\color{blue}{\frac{0 \cdot 0 + \left(\alpha \cdot \alpha + 0 \cdot \alpha\right)}{{0}^{3} - {\alpha}^{3}}}} \cdot \alpha\right) \cdot \mathsf{log1p}\left(\mathsf{neg}\left(u0\right)\right) \]

   6. metadata-eval N/A

      \[\leadsto \left(\frac{1}{\frac{\color{blue}{0} + \left(\alpha \cdot \alpha + 0 \cdot \alpha\right)}{{0}^{3} - {\alpha}^{3}}} \cdot \alpha\right) \cdot \mathsf{log1p}\left(\mathsf{neg}\left(u0\right)\right) \]

   7. lower-+.f32 N/A

      \[\leadsto \left(\frac{1}{\frac{\color{blue}{0 + \left(\alpha \cdot \alpha + 0 \cdot \alpha\right)}}{{0}^{3} - {\alpha}^{3}}} \cdot \alpha\right) \cdot \mathsf{log1p}\left(\mathsf{neg}\left(u0\right)\right) \]

   8. lower-fma.f32 N/A

      \[\leadsto \left(\frac{1}{\frac{0 + \color{blue}{\mathsf{fma}\left(\alpha, \alpha, 0 \cdot \alpha\right)}}{{0}^{3} - {\alpha}^{3}}} \cdot \alpha\right) \cdot \mathsf{log1p}\left(\mathsf{neg}\left(u0\right)\right) \]

   9. lower-*.f32 N/A

      \[\leadsto \left(\frac{1}{\frac{0 + \mathsf{fma}\left(\alpha, \alpha, \color{blue}{0 \cdot \alpha}\right)}{{0}^{3} - {\alpha}^{3}}} \cdot \alpha\right) \cdot \mathsf{log1p}\left(\mathsf{neg}\left(u0\right)\right) \]

   10. metadata-eval N/A

       \[\leadsto \left(\frac{1}{\frac{0 + \mathsf{fma}\left(\alpha, \alpha, 0 \cdot \alpha\right)}{\color{blue}{0} - {\alpha}^{3}}} \cdot \alpha\right) \cdot \mathsf{log1p}\left(\mathsf{neg}\left(u0\right)\right) \]

   11. lower--.f32 N/A

       \[\leadsto \left(\frac{1}{\frac{0 + \mathsf{fma}\left(\alpha, \alpha, 0 \cdot \alpha\right)}{\color{blue}{0 - {\alpha}^{3}}}} \cdot \alpha\right) \cdot \mathsf{log1p}\left(\mathsf{neg}\left(u0\right)\right) \]

   12. cube-unmult N/A

       \[\leadsto \left(\frac{1}{\frac{0 + \mathsf{fma}\left(\alpha, \alpha, 0 \cdot \alpha\right)}{0 - \color{blue}{\alpha \cdot \left(\alpha \cdot \alpha\right)}}} \cdot \alpha\right) \cdot \mathsf{log1p}\left(\mathsf{neg}\left(u0\right)\right) \]

   13. lift-*.f32 N/A

       \[\leadsto \left(\frac{1}{\frac{0 + \mathsf{fma}\left(\alpha, \alpha, 0 \cdot \alpha\right)}{0 - \alpha \cdot \color{blue}{\left(\alpha \cdot \alpha\right)}}} \cdot \alpha\right) \cdot \mathsf{log1p}\left(\mathsf{neg}\left(u0\right)\right) \]

   14. lower-*.f32 98.3

       \[\leadsto \left(\frac{1}{\frac{0 + \mathsf{fma}\left(\alpha, \alpha, 0 \cdot \alpha\right)}{0 - \color{blue}{\alpha \cdot \left(\alpha \cdot \alpha\right)}}} \cdot \alpha\right) \cdot \mathsf{log1p}\left(-u0\right) \]

6. Applied egg-rr 98.3%

   \[\leadsto \left(\color{blue}{\frac{1}{\frac{0 + \mathsf{fma}\left(\alpha, \alpha, 0 \cdot \alpha\right)}{0 - \alpha \cdot \left(\alpha \cdot \alpha\right)}}} \cdot \alpha\right) \cdot \mathsf{log1p}\left(-u0\right) \]

7. Taylor expanded in alpha around 0

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

   1. lower-/.f32 98.6

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

9. Simplified 98.6%

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

    1. lift-/.f32 N/A

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

    2. lift-/.f32 N/A

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

    3. lift-neg.f32 N/A

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

    4. lift-log1p.f32 N/A

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

    5. associate-*l* N/A

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

    6. lift-/.f32 N/A

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

    7. lift-/.f32 N/A

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

    8. associate-/r/ N/A

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

    9. metadata-eval N/A

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

    10. neg-mul-1 N/A

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

    11. *-commutative N/A

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

    12. *-commutative N/A

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

    13. associate-*r* N/A

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

    14. *-commutative N/A

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

    15. associate-*r* N/A

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

    16. lower-*.f32 N/A

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

    17. lower-*.f32 N/A

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

    18. lower-neg.f32 98.9

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

11. Applied egg-rr 98.9%

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

12. Final simplification 98.9%

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

Alternative 2: 93.3% accurate, 3.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 58.4%

   \[\left(\left(-\alpha\right) \cdot \alpha\right) \cdot \log \left(1 - u0\right) \]
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. lower-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. lower-neg.f32 98.9

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

4. Applied egg-rr 98.9%

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

   1. neg-sub0 N/A

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

   2. flip3-- N/A

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

   3. clear-num N/A

      \[\leadsto \left(\color{blue}{\frac{1}{\frac{0 \cdot 0 + \left(\alpha \cdot \alpha + 0 \cdot \alpha\right)}{{0}^{3} - {\alpha}^{3}}}} \cdot \alpha\right) \cdot \mathsf{log1p}\left(\mathsf{neg}\left(u0\right)\right) \]

   4. lower-/.f32 N/A

      \[\leadsto \left(\color{blue}{\frac{1}{\frac{0 \cdot 0 + \left(\alpha \cdot \alpha + 0 \cdot \alpha\right)}{{0}^{3} - {\alpha}^{3}}}} \cdot \alpha\right) \cdot \mathsf{log1p}\left(\mathsf{neg}\left(u0\right)\right) \]

   5. lower-/.f32 N/A

      \[\leadsto \left(\frac{1}{\color{blue}{\frac{0 \cdot 0 + \left(\alpha \cdot \alpha + 0 \cdot \alpha\right)}{{0}^{3} - {\alpha}^{3}}}} \cdot \alpha\right) \cdot \mathsf{log1p}\left(\mathsf{neg}\left(u0\right)\right) \]

   6. metadata-eval N/A

      \[\leadsto \left(\frac{1}{\frac{\color{blue}{0} + \left(\alpha \cdot \alpha + 0 \cdot \alpha\right)}{{0}^{3} - {\alpha}^{3}}} \cdot \alpha\right) \cdot \mathsf{log1p}\left(\mathsf{neg}\left(u0\right)\right) \]

   7. lower-+.f32 N/A

      \[\leadsto \left(\frac{1}{\frac{\color{blue}{0 + \left(\alpha \cdot \alpha + 0 \cdot \alpha\right)}}{{0}^{3} - {\alpha}^{3}}} \cdot \alpha\right) \cdot \mathsf{log1p}\left(\mathsf{neg}\left(u0\right)\right) \]

   8. lower-fma.f32 N/A

      \[\leadsto \left(\frac{1}{\frac{0 + \color{blue}{\mathsf{fma}\left(\alpha, \alpha, 0 \cdot \alpha\right)}}{{0}^{3} - {\alpha}^{3}}} \cdot \alpha\right) \cdot \mathsf{log1p}\left(\mathsf{neg}\left(u0\right)\right) \]

   9. lower-*.f32 N/A

      \[\leadsto \left(\frac{1}{\frac{0 + \mathsf{fma}\left(\alpha, \alpha, \color{blue}{0 \cdot \alpha}\right)}{{0}^{3} - {\alpha}^{3}}} \cdot \alpha\right) \cdot \mathsf{log1p}\left(\mathsf{neg}\left(u0\right)\right) \]

   10. metadata-eval N/A

       \[\leadsto \left(\frac{1}{\frac{0 + \mathsf{fma}\left(\alpha, \alpha, 0 \cdot \alpha\right)}{\color{blue}{0} - {\alpha}^{3}}} \cdot \alpha\right) \cdot \mathsf{log1p}\left(\mathsf{neg}\left(u0\right)\right) \]

   11. lower--.f32 N/A

       \[\leadsto \left(\frac{1}{\frac{0 + \mathsf{fma}\left(\alpha, \alpha, 0 \cdot \alpha\right)}{\color{blue}{0 - {\alpha}^{3}}}} \cdot \alpha\right) \cdot \mathsf{log1p}\left(\mathsf{neg}\left(u0\right)\right) \]

   12. cube-unmult N/A

       \[\leadsto \left(\frac{1}{\frac{0 + \mathsf{fma}\left(\alpha, \alpha, 0 \cdot \alpha\right)}{0 - \color{blue}{\alpha \cdot \left(\alpha \cdot \alpha\right)}}} \cdot \alpha\right) \cdot \mathsf{log1p}\left(\mathsf{neg}\left(u0\right)\right) \]

   13. lift-*.f32 N/A

       \[\leadsto \left(\frac{1}{\frac{0 + \mathsf{fma}\left(\alpha, \alpha, 0 \cdot \alpha\right)}{0 - \alpha \cdot \color{blue}{\left(\alpha \cdot \alpha\right)}}} \cdot \alpha\right) \cdot \mathsf{log1p}\left(\mathsf{neg}\left(u0\right)\right) \]

   14. lower-*.f32 98.3

       \[\leadsto \left(\frac{1}{\frac{0 + \mathsf{fma}\left(\alpha, \alpha, 0 \cdot \alpha\right)}{0 - \color{blue}{\alpha \cdot \left(\alpha \cdot \alpha\right)}}} \cdot \alpha\right) \cdot \mathsf{log1p}\left(-u0\right) \]

6. Applied egg-rr 98.3%

   \[\leadsto \left(\color{blue}{\frac{1}{\frac{0 + \mathsf{fma}\left(\alpha, \alpha, 0 \cdot \alpha\right)}{0 - \alpha \cdot \left(\alpha \cdot \alpha\right)}}} \cdot \alpha\right) \cdot \mathsf{log1p}\left(-u0\right) \]

7. Taylor expanded in alpha around 0

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

   1. lower-/.f32 98.6

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

9. Simplified 98.6%

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

10. Taylor expanded in u0 around 0

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

12. Simplified 91.9%

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

Alternative 3: 91.5% accurate, 3.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 58.4%

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

3. Taylor expanded in u0 around 0

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

   1. lower-*.f32 N/A

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

   2. lower-fma.f32 N/A

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

   3. *-commutative N/A

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

   4. associate-*r* N/A

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

   5. distribute-rgt-out N/A

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

   6. unpow2 N/A

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

   7. +-commutative N/A

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

   8. associate-*l* N/A

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

   9. lower-*.f32 N/A

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

   10. lower-*.f32 N/A

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

   11. +-commutative N/A

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

   12. *-commutative N/A

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

   13. lower-fma.f32 N/A

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

   14. unpow2 N/A

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

   15. lower-*.f32 89.3

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

5. Simplified 89.3%

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

   1. lift-fma.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. lift-*.f32 N/A

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

   5. +-commutative N/A

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

   6. lift-*.f32 N/A

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

   7. lower-fma.f32 N/A

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

   8. lift-*.f32 N/A

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

   9. lift-*.f32 N/A

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

   10. associate-*r* N/A

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

   11. lift-*.f32 N/A

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

   12. associate-*r* N/A

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

   13. lift-*.f32 N/A

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

   14. lower-*.f32 89.6

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

7. Applied egg-rr 89.6%

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

Alternative 4: 91.3% accurate, 4.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 58.4%

   \[\left(\left(-\alpha\right) \cdot \alpha\right) \cdot \log \left(1 - u0\right) \]
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. lower-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. lower-neg.f32 98.9

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

4. Applied egg-rr 98.9%

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

   1. neg-sub0 N/A

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

   2. flip3-- N/A

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

   3. clear-num N/A

      \[\leadsto \left(\color{blue}{\frac{1}{\frac{0 \cdot 0 + \left(\alpha \cdot \alpha + 0 \cdot \alpha\right)}{{0}^{3} - {\alpha}^{3}}}} \cdot \alpha\right) \cdot \mathsf{log1p}\left(\mathsf{neg}\left(u0\right)\right) \]

   4. lower-/.f32 N/A

      \[\leadsto \left(\color{blue}{\frac{1}{\frac{0 \cdot 0 + \left(\alpha \cdot \alpha + 0 \cdot \alpha\right)}{{0}^{3} - {\alpha}^{3}}}} \cdot \alpha\right) \cdot \mathsf{log1p}\left(\mathsf{neg}\left(u0\right)\right) \]

   5. lower-/.f32 N/A

      \[\leadsto \left(\frac{1}{\color{blue}{\frac{0 \cdot 0 + \left(\alpha \cdot \alpha + 0 \cdot \alpha\right)}{{0}^{3} - {\alpha}^{3}}}} \cdot \alpha\right) \cdot \mathsf{log1p}\left(\mathsf{neg}\left(u0\right)\right) \]

   6. metadata-eval N/A

      \[\leadsto \left(\frac{1}{\frac{\color{blue}{0} + \left(\alpha \cdot \alpha + 0 \cdot \alpha\right)}{{0}^{3} - {\alpha}^{3}}} \cdot \alpha\right) \cdot \mathsf{log1p}\left(\mathsf{neg}\left(u0\right)\right) \]

   7. lower-+.f32 N/A

      \[\leadsto \left(\frac{1}{\frac{\color{blue}{0 + \left(\alpha \cdot \alpha + 0 \cdot \alpha\right)}}{{0}^{3} - {\alpha}^{3}}} \cdot \alpha\right) \cdot \mathsf{log1p}\left(\mathsf{neg}\left(u0\right)\right) \]

   8. lower-fma.f32 N/A

      \[\leadsto \left(\frac{1}{\frac{0 + \color{blue}{\mathsf{fma}\left(\alpha, \alpha, 0 \cdot \alpha\right)}}{{0}^{3} - {\alpha}^{3}}} \cdot \alpha\right) \cdot \mathsf{log1p}\left(\mathsf{neg}\left(u0\right)\right) \]

   9. lower-*.f32 N/A

      \[\leadsto \left(\frac{1}{\frac{0 + \mathsf{fma}\left(\alpha, \alpha, \color{blue}{0 \cdot \alpha}\right)}{{0}^{3} - {\alpha}^{3}}} \cdot \alpha\right) \cdot \mathsf{log1p}\left(\mathsf{neg}\left(u0\right)\right) \]

   10. metadata-eval N/A

       \[\leadsto \left(\frac{1}{\frac{0 + \mathsf{fma}\left(\alpha, \alpha, 0 \cdot \alpha\right)}{\color{blue}{0} - {\alpha}^{3}}} \cdot \alpha\right) \cdot \mathsf{log1p}\left(\mathsf{neg}\left(u0\right)\right) \]

   11. lower--.f32 N/A

       \[\leadsto \left(\frac{1}{\frac{0 + \mathsf{fma}\left(\alpha, \alpha, 0 \cdot \alpha\right)}{\color{blue}{0 - {\alpha}^{3}}}} \cdot \alpha\right) \cdot \mathsf{log1p}\left(\mathsf{neg}\left(u0\right)\right) \]

   12. cube-unmult N/A

       \[\leadsto \left(\frac{1}{\frac{0 + \mathsf{fma}\left(\alpha, \alpha, 0 \cdot \alpha\right)}{0 - \color{blue}{\alpha \cdot \left(\alpha \cdot \alpha\right)}}} \cdot \alpha\right) \cdot \mathsf{log1p}\left(\mathsf{neg}\left(u0\right)\right) \]

   13. lift-*.f32 N/A

       \[\leadsto \left(\frac{1}{\frac{0 + \mathsf{fma}\left(\alpha, \alpha, 0 \cdot \alpha\right)}{0 - \alpha \cdot \color{blue}{\left(\alpha \cdot \alpha\right)}}} \cdot \alpha\right) \cdot \mathsf{log1p}\left(\mathsf{neg}\left(u0\right)\right) \]

   14. lower-*.f32 98.3

       \[\leadsto \left(\frac{1}{\frac{0 + \mathsf{fma}\left(\alpha, \alpha, 0 \cdot \alpha\right)}{0 - \color{blue}{\alpha \cdot \left(\alpha \cdot \alpha\right)}}} \cdot \alpha\right) \cdot \mathsf{log1p}\left(-u0\right) \]

6. Applied egg-rr 98.3%

   \[\leadsto \left(\color{blue}{\frac{1}{\frac{0 + \mathsf{fma}\left(\alpha, \alpha, 0 \cdot \alpha\right)}{0 - \alpha \cdot \left(\alpha \cdot \alpha\right)}}} \cdot \alpha\right) \cdot \mathsf{log1p}\left(-u0\right) \]

7. Taylor expanded in alpha around 0

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

   1. lower-/.f32 98.6

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

9. Simplified 98.6%

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

10. Taylor expanded in u0 around 0

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

    1. *-commutative N/A

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

    2. +-commutative N/A

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

    3. *-rgt-identity N/A

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

    4. distribute-rgt-in N/A

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

    5. *-commutative N/A

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

    6. associate-*l* N/A

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

    7. associate-*r* N/A

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

    8. *-commutative N/A

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

    9. distribute-lft-in N/A

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

    10. +-commutative N/A

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

    11. associate-*r* N/A

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

    12. distribute-lft-in N/A

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

    13. associate-*r* N/A

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

    14. *-commutative N/A

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

    15. unpow2 N/A

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

12. Simplified 89.5%

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

Alternative 5: 87.3% accurate, 4.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 58.4%

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

3. Taylor expanded in u0 around 0

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

   1. lower-*.f32 N/A

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

   2. lower-fma.f32 N/A

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

   3. *-commutative N/A

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

   4. associate-*r* N/A

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

   5. distribute-rgt-out N/A

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

   6. unpow2 N/A

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

   7. +-commutative N/A

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

   8. associate-*l* N/A

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

   9. lower-*.f32 N/A

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

   10. lower-*.f32 N/A

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

   11. +-commutative N/A

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

   12. *-commutative N/A

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

   13. lower-fma.f32 N/A

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

   14. unpow2 N/A

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

   15. lower-*.f32 89.3

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

5. Simplified 89.3%

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

   1. lift-fma.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. lift-*.f32 N/A

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

   5. +-commutative N/A

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

   6. lift-*.f32 N/A

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

   7. lower-fma.f32 N/A

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

   8. lift-*.f32 N/A

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

   9. lift-*.f32 N/A

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

   10. associate-*r* N/A

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

   11. lift-*.f32 N/A

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

   12. associate-*r* N/A

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

   13. lift-*.f32 N/A

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

   14. lower-*.f32 89.6

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

7. Applied egg-rr 89.6%

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

8. Taylor expanded in u0 around 0

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

   1. associate-*r* N/A

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

   2. *-commutative N/A

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

   3. lower-*.f32 N/A

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

   4. *-commutative N/A

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

   5. lower-*.f32 N/A

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

   6. unpow2 N/A

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

   7. lower-*.f32 85.4

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

10. Simplified 85.4%

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

11. Final simplification 85.4%

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

Alternative 6: 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 58.4%

   \[\left(\left(-\alpha\right) \cdot \alpha\right) \cdot \log \left(1 - u0\right) \]
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. lower-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. lower-neg.f32 98.9

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

4. Applied egg-rr 98.9%

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

5. Taylor expanded in u0 around 0

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

   1. lower-*.f32 N/A

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

   2. *-commutative N/A

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

   3. associate-*l* N/A

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

   4. *-commutative N/A

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

   5. *-rgt-identity N/A

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

   6. distribute-lft-out N/A

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

   7. +-commutative N/A

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

   8. lower-*.f32 N/A

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

   9. unpow2 N/A

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

   10. lower-*.f32 N/A

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

   11. +-commutative N/A

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

   12. *-commutative N/A

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

   13. lower-fma.f32 85.2

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

7. Simplified 85.2%

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

9. Applied egg-rr 85.3%

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

10. Final simplification 85.3%

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

Alternative 7: 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 58.4%

   \[\left(\left(-\alpha\right) \cdot \alpha\right) \cdot \log \left(1 - u0\right) \]
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. lower-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. lower-neg.f32 98.9

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

4. Applied egg-rr 98.9%

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

5. Taylor expanded in u0 around 0

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

   1. lower-*.f32 N/A

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

   2. *-commutative N/A

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

   3. associate-*l* N/A

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

   4. *-commutative N/A

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

   5. *-rgt-identity N/A

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

   6. distribute-lft-out N/A

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

   7. +-commutative N/A

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

   8. lower-*.f32 N/A

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

   9. unpow2 N/A

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

   10. lower-*.f32 N/A

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

   11. +-commutative N/A

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

   12. *-commutative N/A

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

   13. lower-fma.f32 85.2

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

7. Simplified 85.2%

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

9. Applied egg-rr 85.3%

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

10. Final simplification 85.3%

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

Alternative 8: 87.0% accurate, 5.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 58.4%

   \[\left(\left(-\alpha\right) \cdot \alpha\right) \cdot \log \left(1 - u0\right) \]
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. lower-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. lower-neg.f32 98.9

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

4. Applied egg-rr 98.9%

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

5. Taylor expanded in u0 around 0

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

   1. lower-*.f32 N/A

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

   2. *-commutative N/A

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

   3. associate-*l* N/A

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

   4. *-commutative N/A

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

   5. *-rgt-identity N/A

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

   6. distribute-lft-out N/A

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

   7. +-commutative N/A

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

   8. lower-*.f32 N/A

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

   9. unpow2 N/A

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

   10. lower-*.f32 N/A

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

   11. +-commutative N/A

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

   12. *-commutative N/A

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

   13. lower-fma.f32 85.2

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

7. Simplified 85.2%

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

   1. unpow1 N/A

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

   2. metadata-eval N/A

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

   3. pow-div N/A

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

   4. cube-unmult N/A

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

   5. lift-*.f32 N/A

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

   6. lift-*.f32 N/A

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

   7. pow2 N/A

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

   8. lift-*.f32 N/A

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

   9. frac-2neg N/A

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

   10. lift-neg.f32 N/A

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

   11. associate-/l* N/A

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

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

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

   13. lift-*.f32 N/A

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

   14. lift-neg.f32 N/A

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

   15. clear-num N/A

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

   16. lift-/.f32 N/A

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

   17. lift-/.f32 N/A

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

   18. lift-fma.f32 N/A

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

9. Applied egg-rr 85.1%

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

10. Final simplification 85.1%

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

Alternative 9: 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 58.4%

   \[\left(\left(-\alpha\right) \cdot \alpha\right) \cdot \log \left(1 - u0\right) \]
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. lower-*.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. lower-*.f32 72.8

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

5. Simplified 72.8%

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

   1. associate-*r* N/A

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

   2. lower-*.f32 N/A

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

   3. *-commutative N/A

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

   4. lower-*.f32 72.8

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

7. Applied egg-rr 72.8%

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

8. Final simplification 72.8%

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

Alternative 10: 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 58.4%

   \[\left(\left(-\alpha\right) \cdot \alpha\right) \cdot \log \left(1 - u0\right) \]
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. lower-*.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. lower-*.f32 72.8

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

5. Simplified 72.8%

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

Reproduce

herbie shell --seed 2024208 
(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))))