Beckmann Distribution sample, tan2theta, alphax == alphay

Percentage Accurate: 56.2% → 99.0%

Time: 11.3s

Alternatives: 17

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

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

FPCore

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

C

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

Julia

function code(alpha, u0)
	return Float32(Float32(-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%

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

Alternative 2: 99.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 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%

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

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

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

   2. *-commutative N/A

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

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

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

   4. associate-*r* N/A

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

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

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

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

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

   7. diff-log N/A

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

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

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

   9. sub-neg N/A

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

   10. metadata-eval N/A

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

   11. flip-- N/A

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

   12. sub-neg N/A

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

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

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

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

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

   15. neg-lowering-neg.f32 98.9

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

8. Applied egg-rr 98.9%

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

9. Final simplification 98.9%

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

Alternative 3: 93.3% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Julia

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

   \[\leadsto \color{blue}{u0 \cdot \left(u0 \cdot \left(\frac{1}{2} \cdot {\alpha}^{2} + u0 \cdot \left(\frac{1}{4} \cdot \left({\alpha}^{2} \cdot u0\right) + \frac{1}{3} \cdot {\alpha}^{2}\right)\right) + {\alpha}^{2}\right)} \]
6. 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

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

7. Simplified 93.6%

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

8. Final simplification 93.6%

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

Alternative 4: 93.3% accurate, 2.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 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%

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

Alternative 5: 93.3% accurate, 3.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 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

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

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

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

   2. sub-neg N/A

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

   3. metadata-eval N/A

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

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

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

   5. sub-neg N/A

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

   6. metadata-eval N/A

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

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

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

   8. sub-neg N/A

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

   9. *-commutative N/A

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

   10. metadata-eval N/A

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

   11. accelerator-lowering-fma.f32 93.2

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

5. Simplified 93.2%

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

   1. distribute-rgt-in N/A

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

   2. neg-mul-1 N/A

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

   3. *-commutative N/A

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

   4. associate-*l* N/A

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

   5. sqr-neg N/A

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

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

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

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

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

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

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

   9. sqr-neg N/A

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

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

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

   11. neg-lowering-neg.f32 93.6

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

7. Applied egg-rr 93.6%

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

8. Final simplification 93.6%

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

Alternative 6: 93.1% accurate, 3.2× speedup

Math

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

FPCore

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

C

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

Julia

function code(alpha, u0)
	return Float32(u0 * Float32(Float32(-Float32(alpha * alpha)) * 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. Taylor expanded in u0 around 0

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

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

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

   2. sub-neg N/A

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

   3. metadata-eval N/A

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

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

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

   5. sub-neg N/A

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

   6. metadata-eval N/A

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

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

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

   8. sub-neg N/A

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

   9. *-commutative N/A

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

   10. metadata-eval N/A

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

   11. accelerator-lowering-fma.f32 93.2

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

5. Simplified 93.2%

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

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

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

   2. *-commutative N/A

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

   3. associate-*r* N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

   11. accelerator-lowering-fma.f32 93.2

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

7. Applied egg-rr 93.2%

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

8. Final simplification 93.2%

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

Alternative 7: 93.1% accurate, 3.2× 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(-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. Taylor expanded in u0 around 0

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

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

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

   2. sub-neg N/A

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

   3. metadata-eval N/A

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

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

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

   5. sub-neg N/A

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

   6. metadata-eval N/A

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

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

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

   8. sub-neg N/A

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

   9. *-commutative N/A

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

   10. metadata-eval N/A

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

   11. accelerator-lowering-fma.f32 93.2

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

5. Simplified 93.2%

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

6. Final simplification 93.2%

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

Alternative 8: 91.5% accurate, 3.5× speedup

Math

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

FPCore

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

C

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

Julia

function code(alpha, u0)
	return Float32(u0 * fma(alpha, alpha, Float32(Float32(alpha * alpha) * Float32(u0 * fma(u0, Float32(0.3333333333333333), Float32(0.5))))))
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. *-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. accelerator-lowering-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. *-lowering-*.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. *-lowering-*.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. accelerator-lowering-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. *-lowering-*.f32 91.9

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

5. Simplified 91.9%

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

6. Taylor expanded in u0 around 0

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

   1. +-commutative N/A

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

   2. *-rgt-identity N/A

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

   3. *-commutative N/A

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

   4. *-commutative N/A

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

   5. associate-*r* N/A

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

   6. distribute-rgt-in N/A

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

   7. +-commutative N/A

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

   8. associate-*r* N/A

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

   9. *-commutative N/A

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

   10. distribute-lft-in N/A

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

   11. unpow2 N/A

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

   12. associate-*l* N/A

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

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

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

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

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

   15. +-commutative N/A

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

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

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

8. Simplified 91.5%

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

   1. associate-*r* N/A

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

   2. +-commutative N/A

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

   3. distribute-rgt-in N/A

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

   4. *-lft-identity N/A

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

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

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

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

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

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

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

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

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

   9. *-lowering-*.f32 92.0

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

10. Applied egg-rr 92.0%

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

11. Final simplification 92.0%

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

Alternative 9: 91.5% accurate, 3.5× speedup

Math

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

FPCore

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

C

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

Julia

function code(alpha, u0)
	return Float32(u0 * fma(alpha, alpha, Float32(alpha * Float32(u0 * Float32(alpha * fma(u0, Float32(0.3333333333333333), Float32(0.5)))))))
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. *-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. accelerator-lowering-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. *-lowering-*.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. *-lowering-*.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. accelerator-lowering-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. *-lowering-*.f32 91.9

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

5. Simplified 91.9%

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

   1. +-commutative N/A

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

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

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

   3. *-commutative N/A

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

   4. associate-*l* N/A

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

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

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

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

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

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

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

   8. accelerator-lowering-fma.f32 92.0

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

7. Applied egg-rr 92.0%

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

8. Final simplification 92.0%

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

Alternative 10: 91.3% accurate, 4.1× speedup

Math

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

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(Float32(alpha * alpha) * fma(Float32(u0 * u0), fma(u0, Float32(0.3333333333333333), Float32(0.5)), 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. 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. *-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. *-commutative N/A

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

   5. *-commutative N/A

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

   6. associate-*r* N/A

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

   7. distribute-rgt-in N/A

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

   8. +-commutative N/A

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

   9. associate-*r* N/A

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

   10. *-commutative N/A

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

   11. distribute-lft-in N/A

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

   12. associate-*r* N/A

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

   13. *-commutative N/A

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

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

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

7. Simplified 91.8%

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

Alternative 11: 91.3% accurate, 4.1× speedup

Math

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

FPCore

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

C

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

Julia

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

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

   1. *-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. accelerator-lowering-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. *-lowering-*.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. *-lowering-*.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. accelerator-lowering-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. *-lowering-*.f32 91.9

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

5. Simplified 91.9%

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

6. Taylor expanded in u0 around 0

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

   1. +-commutative N/A

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

   2. *-rgt-identity N/A

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

   3. *-commutative N/A

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

   4. *-commutative N/A

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

   5. associate-*r* N/A

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

   6. distribute-rgt-in N/A

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

   7. +-commutative N/A

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

   8. associate-*r* N/A

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

   9. *-commutative N/A

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

   10. distribute-lft-in N/A

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

   11. unpow2 N/A

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

   12. associate-*l* N/A

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

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

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

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

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

   15. +-commutative N/A

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

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

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

8. Simplified 91.5%

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

   1. distribute-rgt-in N/A

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

   2. *-commutative N/A

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

   3. associate-*l* N/A

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

   4. *-lft-identity N/A

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

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

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

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

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

   7. *-lowering-*.f32 91.8

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

10. Applied egg-rr 91.8%

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

11. Final simplification 91.8%

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

Alternative 12: 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. *-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. accelerator-lowering-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. *-lowering-*.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. *-lowering-*.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. accelerator-lowering-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. *-lowering-*.f32 91.9

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

5. Simplified 91.9%

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

6. Taylor expanded in u0 around 0

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

   1. +-commutative N/A

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

   2. *-rgt-identity N/A

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

   3. *-commutative N/A

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

   4. *-commutative N/A

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

   5. associate-*r* N/A

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

   6. distribute-rgt-in N/A

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

   7. +-commutative N/A

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

   8. associate-*r* N/A

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

   9. *-commutative N/A

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

   10. distribute-lft-in N/A

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

   11. unpow2 N/A

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

   12. associate-*l* N/A

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

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

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

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

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

   15. +-commutative N/A

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

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

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

8. Simplified 91.5%

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

Alternative 13: 87.3% accurate, 4.3× speedup

Math

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

FPCore

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

C

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

Julia

function code(alpha, u0)
	return Float32(u0 * fma(alpha, alpha, Float32(u0 * Float32(Float32(alpha * alpha) * Float32(0.5)))))
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. distribute-lft-in N/A

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

   2. associate-*r* N/A

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

   3. associate-*r* N/A

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

   4. metadata-eval N/A

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

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

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

   6. *-commutative N/A

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

   7. metadata-eval N/A

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

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

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

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

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

   10. mul-1-neg N/A

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

   11. *-commutative N/A

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

   12. distribute-lft-in N/A

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

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

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

   14. +-commutative N/A

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

   15. unpow2 N/A

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

7. Simplified 88.3%

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

Alternative 14: 87.2% accurate, 5.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 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

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

   1. distribute-lft-in N/A

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

   2. associate-*r* N/A

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

   3. *-commutative N/A

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

   4. associate-*r* N/A

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

   5. distribute-rgt-out N/A

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

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

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

   7. unpow2 N/A

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

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

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

   9. associate-*r* N/A

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

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

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

   11. *-commutative N/A

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

   12. *-lowering-*.f32 88.2

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

5. Simplified 88.2%

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

Alternative 15: 87.2% accurate, 5.3× speedup

Math

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

FPCore

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

C

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

Julia

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

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

6. Applied egg-rr 98.8%

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

7. Taylor expanded in u0 around 0

   \[\leadsto \color{blue}{u0 \cdot \left(\frac{1}{2} \cdot \left({\alpha}^{2} \cdot u0\right) + {\alpha}^{2}\right)} \]
8. 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

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

   3. *-commutative N/A

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

   4. associate-*r* N/A

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

   5. *-commutative N/A

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

   6. associate-*r* N/A

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

   7. *-commutative N/A

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

   8. distribute-lft1-in N/A

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

   9. +-commutative N/A

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

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

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

   11. +-commutative N/A

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

   12. *-commutative N/A

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

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

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

   14. unpow2 N/A

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

   15. *-lowering-*.f32 88.0

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

9. Simplified 88.0%

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

10. Taylor expanded in u0 around 0

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

    1. *-commutative N/A

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

    2. associate-*r* N/A

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

    3. *-commutative N/A

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

    4. unpow2 N/A

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

    5. associate-*l* N/A

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

    6. *-commutative N/A

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

    7. associate-*l* N/A

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

    8. *-commutative N/A

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

    9. unpow2 N/A

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

    10. distribute-lft-out N/A

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

    11. +-commutative N/A

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

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

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

    13. +-commutative N/A

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

    14. associate-*r* N/A

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

    15. *-commutative N/A

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

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

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

    17. *-commutative N/A

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

    18. *-lowering-*.f32 88.2

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

12. Simplified 88.2%

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

Alternative 16: 87.2% accurate, 5.3× speedup

Math

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

FPCore

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

C

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

Julia

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

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

6. Applied egg-rr 98.8%

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

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

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

   2. *-commutative N/A

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

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

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

   4. associate-*r* N/A

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

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

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

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

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

   7. diff-log N/A

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

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

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

   9. sub-neg N/A

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

   10. metadata-eval N/A

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

   11. flip-- N/A

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

   12. sub-neg N/A

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

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

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

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

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

   15. neg-lowering-neg.f32 98.9

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

8. Applied egg-rr 98.9%

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

9. Taylor expanded in u0 around 0

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

    1. distribute-lft-in N/A

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

    2. associate-*r* N/A

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

    3. *-commutative N/A

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

    4. *-commutative N/A

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

    5. distribute-lft1-in N/A

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

    6. +-commutative N/A

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

    7. *-commutative N/A

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

    8. associate-*r* N/A

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

    9. unpow2 N/A

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

    10. associate-*l* N/A

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

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

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

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

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

    13. *-commutative N/A

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

    14. +-commutative N/A

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

    15. distribute-lft1-in N/A

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

    16. associate-*l* N/A

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

    17. unpow2 N/A

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

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

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

    19. unpow2 N/A

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

    20. *-lowering-*.f32 88.1

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

11. Simplified 88.1%

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

Alternative 17: 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. unpow2 N/A

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

   4. *-lowering-*.f32 75.2

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

5. Simplified 75.2%

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