Beckmann Distribution sample, tan2theta, alphax == alphay

Percentage Accurate: 56.2% → 99.0%

Time: 11.3s

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: 56.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Alternative 1: 99.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(0 - \alpha \cdot \alpha\right) \cdot \mathsf{log1p}\left(-u0\right) \end{array} \]

FPCore

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

C

float code(float alpha, float u0) {
	return (0.0f - (alpha * alpha)) * log1pf(-u0);
}

Julia

function code(alpha, u0)
	return Float32(Float32(Float32(0.0) - Float32(alpha * alpha)) * log1p(Float32(-u0)))
end

Derivation

1. Initial program 55.8%

   \[\left(\left(-\alpha\right) \cdot \alpha\right) \cdot \log \left(1 - u0\right) \]
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. Final simplification 99.0%

   \[\leadsto \left(0 - \alpha \cdot \alpha\right) \cdot \mathsf{log1p}\left(-u0\right) \]
6. Add Preprocessing

Alternative 2: 93.3% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ u0 \cdot \mathsf{fma}\left(u0, \mathsf{fma}\left(\alpha, \alpha, 0\right) \cdot \mathsf{fma}\left(u0, u0 \cdot 0.25, \mathsf{fma}\left(u0, 0.3333333333333333, 0.5\right)\right), \mathsf{fma}\left(\alpha, \alpha, 0\right)\right) \end{array} \]

FPCore

(FPCore (alpha u0)
 :precision binary32
 (*
  u0
  (fma
   u0
   (*
    (fma alpha alpha 0.0)
    (fma u0 (* u0 0.25) (fma u0 0.3333333333333333 0.5)))
   (fma alpha alpha 0.0))))

C

float code(float alpha, float u0) {
	return u0 * fmaf(u0, (fmaf(alpha, alpha, 0.0f) * fmaf(u0, (u0 * 0.25f), fmaf(u0, 0.3333333333333333f, 0.5f))), fmaf(alpha, alpha, 0.0f));
}

Julia

function code(alpha, u0)
	return Float32(u0 * fma(u0, Float32(fma(alpha, alpha, Float32(0.0)) * fma(u0, Float32(u0 * Float32(0.25)), fma(u0, Float32(0.3333333333333333), Float32(0.5)))), fma(alpha, alpha, Float32(0.0))))
end

Derivation

1. Initial program 55.8%

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

3. Taylor expanded in u0 around 0

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

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

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

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

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

5. Simplified 93.6%

   \[\leadsto \color{blue}{u0 \cdot \mathsf{fma}\left(u0, \mathsf{fma}\left(\alpha, \alpha, 0\right) \cdot \mathsf{fma}\left(u0, u0 \cdot 0.25, \mathsf{fma}\left(u0, 0.3333333333333333, 0.5\right)\right), \mathsf{fma}\left(\alpha, \alpha, 0\right)\right)} \]
6. Add Preprocessing

Alternative 3: 93.1% accurate, 3.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 55.8%

   \[\left(\left(-\alpha\right) \cdot \alpha\right) \cdot \log \left(1 - u0\right) \]
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%

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

6. Applied egg-rr 98.8%

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

7. Taylor expanded in alpha around 0

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

   1. mul-1-neg N/A

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

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

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

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

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

   4. unpow2 N/A

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

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

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

   6. sub-neg N/A

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

   7. +-commutative N/A

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

   8. distribute-neg-in N/A

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

   9. unsub-neg N/A

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

   10. remove-double-neg N/A

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

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

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

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

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

   13. sub-neg N/A

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

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

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

   15. neg-sub0 N/A

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

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

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

   17. unpow2 N/A

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

   18. *-lowering-*.f32 98.8

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

9. Simplified 98.8%

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

10. Taylor expanded in u0 around 0

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

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

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

    2. +-commutative N/A

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

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

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

    4. +-commutative N/A

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

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

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

    6. +-commutative N/A

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

    7. *-commutative N/A

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

    8. accelerator-lowering-fma.f32 93.2

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

12. Simplified 93.2%

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

Alternative 4: 91.5% accurate, 3.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 55.8%

   \[\left(\left(-\alpha\right) \cdot \alpha\right) \cdot \log \left(1 - u0\right) \]
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. Taylor expanded in u0 around 0

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

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

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

   2. +-commutative N/A

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

   3. unpow2 N/A

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

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

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

   5. distribute-rgt-in N/A

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

   6. *-commutative N/A

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

   7. associate-*r* N/A

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

   8. *-commutative N/A

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

   9. associate-*r* N/A

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

   10. distribute-rgt-out N/A

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

   11. +-commutative N/A

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

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

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

   13. *-commutative N/A

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

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

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

   15. unpow2 N/A

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

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

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

   17. +-commutative N/A

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

   18. *-commutative N/A

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

   19. accelerator-lowering-fma.f32 92.0

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

7. Simplified 92.0%

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

8. Final simplification 92.0%

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

Alternative 5: 91.1% accurate, 4.0× speedup

Math

\[\begin{array}{l} \\ u0 \cdot \left(\mathsf{fma}\left(\alpha, \alpha, 0\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, u0, 0.5\right), u0, 1\right)\right) \end{array} \]

FPCore

(FPCore (alpha u0)
 :precision binary32
 (* u0 (* (fma alpha alpha 0.0) (fma (fma 0.3333333333333333 u0 0.5) u0 1.0))))

C

float code(float alpha, float u0) {
	return u0 * (fmaf(alpha, alpha, 0.0f) * fmaf(fmaf(0.3333333333333333f, u0, 0.5f), u0, 1.0f));
}

Julia

function code(alpha, u0)
	return Float32(u0 * Float32(fma(alpha, alpha, Float32(0.0)) * fma(fma(Float32(0.3333333333333333), u0, Float32(0.5)), u0, Float32(1.0))))
end

Derivation

1. Initial program 55.8%

   \[\left(\left(-\alpha\right) \cdot \alpha\right) \cdot \log \left(1 - u0\right) \]
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. distribute-rgt-in N/A

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

   2. associate-*r* N/A

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

   3. associate-+l+ N/A

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

   4. distribute-lft-in N/A

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

   5. *-commutative N/A

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

   6. associate-*r* N/A

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

   7. associate-*r* N/A

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

   8. *-commutative N/A

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

   9. distribute-lft-in N/A

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

   10. associate-*r* N/A

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

   11. *-commutative N/A

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

   12. distribute-lft1-in N/A

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

5. Simplified 91.5%

   \[\leadsto \color{blue}{\left(u0 \cdot \mathsf{fma}\left(\alpha, \alpha, 0\right)\right) \cdot \mathsf{fma}\left(u0, u0 \cdot 0.3333333333333333, \mathsf{fma}\left(u0, 0.5, 1\right)\right)} \]
6. Step-by-step derivation

   1. associate-+r+ N/A

      \[\leadsto \left(u0 \cdot \mathsf{fma}\left(\alpha, \alpha, 0\right)\right) \cdot \color{blue}{\left(\left(u0 \cdot \left(u0 \cdot \frac{1}{3}\right) + u0 \cdot \frac{1}{2}\right) + 1\right)} \]

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

      \[\leadsto \left(u0 \cdot \mathsf{fma}\left(\alpha, \alpha, 0\right)\right) \cdot \color{blue}{\left(\left(u0 \cdot \left(u0 \cdot \frac{1}{3}\right) + u0 \cdot \frac{1}{2}\right) + 1\right)} \]

   3. distribute-lft-out N/A

      \[\leadsto \left(u0 \cdot \mathsf{fma}\left(\alpha, \alpha, 0\right)\right) \cdot \left(\color{blue}{u0 \cdot \left(u0 \cdot \frac{1}{3} + \frac{1}{2}\right)} + 1\right) \]

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

      \[\leadsto \left(u0 \cdot \mathsf{fma}\left(\alpha, \alpha, 0\right)\right) \cdot \left(\color{blue}{u0 \cdot \left(u0 \cdot \frac{1}{3} + \frac{1}{2}\right)} + 1\right) \]

   5. accelerator-lowering-fma.f32 91.5

      \[\leadsto \left(u0 \cdot \mathsf{fma}\left(\alpha, \alpha, 0\right)\right) \cdot \left(u0 \cdot \color{blue}{\mathsf{fma}\left(u0, 0.3333333333333333, 0.5\right)} + 1\right) \]

7. Applied egg-rr 91.5%

   \[\leadsto \left(u0 \cdot \mathsf{fma}\left(\alpha, \alpha, 0\right)\right) \cdot \color{blue}{\left(u0 \cdot \mathsf{fma}\left(u0, 0.3333333333333333, 0.5\right) + 1\right)} \]

9. Applied egg-rr 91.6%

   \[\leadsto \color{blue}{\left(\mathsf{fma}\left(\alpha, \alpha, 0\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, u0, 0.5\right), u0, 1\right)\right) \cdot u0} \]

10. Final simplification 91.6%

    \[\leadsto u0 \cdot \left(\mathsf{fma}\left(\alpha, \alpha, 0\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, u0, 0.5\right), u0, 1\right)\right) \]
11. Add Preprocessing

Alternative 6: 91.1% accurate, 4.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 55.8%

   \[\left(\left(-\alpha\right) \cdot \alpha\right) \cdot \log \left(1 - u0\right) \]
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. distribute-rgt-in N/A

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

   2. associate-*r* N/A

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

   3. associate-+l+ N/A

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

   4. distribute-lft-in N/A

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

   5. *-commutative N/A

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

   6. associate-*r* N/A

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

   7. associate-*r* N/A

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

   8. *-commutative N/A

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

   9. distribute-lft-in N/A

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

   10. associate-*r* N/A

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

   11. *-commutative N/A

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

   12. distribute-lft1-in N/A

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

5. Simplified 91.5%

   \[\leadsto \color{blue}{\left(u0 \cdot \mathsf{fma}\left(\alpha, \alpha, 0\right)\right) \cdot \mathsf{fma}\left(u0, u0 \cdot 0.3333333333333333, \mathsf{fma}\left(u0, 0.5, 1\right)\right)} \]
6. Step-by-step derivation

   1. +-rgt-identity N/A

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

   2. associate-*l* N/A

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

   3. *-commutative N/A

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

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

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

   5. associate-*l* N/A

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

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

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

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

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

   8. associate-+r+ N/A

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

   9. distribute-lft-out N/A

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

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

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

   11. accelerator-lowering-fma.f32 91.5

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

7. Applied egg-rr 91.5%

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

8. Final simplification 91.5%

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

Alternative 7: 91.1% accurate, 4.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 55.8%

   \[\left(\left(-\alpha\right) \cdot \alpha\right) \cdot \log \left(1 - u0\right) \]
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. distribute-rgt-in N/A

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

   2. associate-*r* N/A

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

   3. associate-+l+ N/A

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

   4. distribute-lft-in N/A

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

   5. *-commutative N/A

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

   6. associate-*r* N/A

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

   7. associate-*r* N/A

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

   8. *-commutative N/A

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

   9. distribute-lft-in N/A

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

   10. associate-*r* N/A

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

   11. *-commutative N/A

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

   12. distribute-lft1-in N/A

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

5. Simplified 91.5%

   \[\leadsto \color{blue}{\left(u0 \cdot \mathsf{fma}\left(\alpha, \alpha, 0\right)\right) \cdot \mathsf{fma}\left(u0, u0 \cdot 0.3333333333333333, \mathsf{fma}\left(u0, 0.5, 1\right)\right)} \]

6. Taylor expanded in alpha around 0

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

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

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

   2. unpow2 N/A

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

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

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

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

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

   5. +-commutative N/A

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

   6. associate-+r+ N/A

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

   7. unpow2 N/A

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

   8. associate-*r* N/A

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

   9. associate-+r+ N/A

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

   10. distribute-rgt-in N/A

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

   11. +-commutative N/A

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

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

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

   13. +-commutative N/A

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

   14. *-commutative N/A

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

   15. accelerator-lowering-fma.f32 91.5

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

8. Simplified 91.5%

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

Alternative 8: 91.1% accurate, 4.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 55.8%

   \[\left(\left(-\alpha\right) \cdot \alpha\right) \cdot \log \left(1 - u0\right) \]
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%

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

6. Applied egg-rr 98.8%

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

7. Taylor expanded in u0 around 0

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

   1. distribute-lft-in N/A

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

   2. associate-*r* N/A

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

   3. unpow2 N/A

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

   4. +-commutative N/A

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

   5. *-commutative N/A

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

   6. associate-*r* N/A

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

   7. distribute-rgt-in N/A

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

   8. *-commutative N/A

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

   9. associate-*l* N/A

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

   10. distribute-rgt-in N/A

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

   11. unpow2 N/A

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

   12. associate-*r* N/A

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

   13. *-rgt-identity N/A

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

   14. distribute-lft-in N/A

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

   15. +-commutative N/A

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

   16. associate-*r* N/A

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

9. Simplified 91.5%

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

    1. associate-*l* N/A

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

    2. associate-*l* N/A

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

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

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

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

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

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

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

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

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

    7. accelerator-lowering-fma.f32 91.4

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

11. Applied egg-rr 91.4%

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

Alternative 9: 87.0% accurate, 5.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 55.8%

   \[\left(\left(-\alpha\right) \cdot \alpha\right) \cdot \log \left(1 - u0\right) \]
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. Taylor expanded in u0 around 0

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

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

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

   2. *-commutative N/A

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

   3. associate-*r* N/A

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

   4. distribute-lft1-in N/A

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

   5. lft-mult-inverse N/A

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

   6. distribute-rgt-in N/A

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

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

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

   8. distribute-lft-in N/A

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

   9. rgt-mult-inverse N/A

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

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

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

   11. unpow2 N/A

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

   12. *-lowering-*.f32 88.0

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

7. Simplified 88.0%

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

8. Final simplification 88.0%

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

Alternative 10: 74.3% 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 55.8%

   \[\left(\left(-\alpha\right) \cdot \alpha\right) \cdot \log \left(1 - u0\right) \]
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. --rgt-identity N/A

      \[\leadsto u0 \cdot \color{blue}{\left({\alpha}^{2} - 0\right)} \]

   4. sub-neg N/A

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

   5. unpow2 N/A

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

   6. metadata-eval N/A

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

   7. accelerator-lowering-fma.f32 75.2

      \[\leadsto u0 \cdot \color{blue}{\mathsf{fma}\left(\alpha, \alpha, 0\right)} \]

5. Simplified 75.2%

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

   1. +-rgt-identity N/A

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

   2. *-lowering-*.f32 75.2

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

7. Applied egg-rr 75.2%

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

Reproduce

herbie shell --seed 2024198 
(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))))