Beckmann Distribution sample, tan2theta, alphax == alphay

Percentage Accurate: 55.7% → 99.0%

Time: 10.9s

Alternatives: 10

Speedup: 10.5×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Initial Program: 55.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Alternative 1: 99.0% accurate, 1.0× speedup

Math

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

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(alpha * Float32(-alpha)) * log1p(Float32(-u0)))
end

Derivation

1. Initial program 56.5%

   \[\left(\left(-\alpha\right) \cdot \alpha\right) \cdot \log \left(1 - u0\right) \]
2. 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 99.1

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

4. Applied egg-rr 99.1%

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

5. Final simplification 99.1%

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

Alternative 2: 93.2% accurate, 3.2× speedup

Math

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

FPCore

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

C

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

Julia

function code(alpha, u0)
	return Float32(Float32(alpha * Float32(-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 56.5%

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

3. Taylor expanded in u0 around 0

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

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

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

   2. sub-neg N/A

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

   3. metadata-eval N/A

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

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

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

   5. sub-neg N/A

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

   6. metadata-eval N/A

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

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

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

   8. sub-neg N/A

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

   9. *-commutative N/A

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

   10. metadata-eval N/A

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

   11. accelerator-lowering-fma.f32 92.3

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

5. Simplified 92.3%

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

6. Final simplification 92.3%

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

Alternative 3: 91.5% accurate, 4.1× speedup

Math

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

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(u0, Float32(u0 * fma(u0, Float32(0.3333333333333333), Float32(0.5))), u0))
end

Derivation

1. Initial program 56.5%

   \[\left(\left(-\alpha\right) \cdot \alpha\right) \cdot \log \left(1 - u0\right) \]
2. Add Preprocessing
3. 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 99.1

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

4. Applied egg-rr 99.1%

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

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

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

   5. distribute-rgt-in N/A

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

   6. associate-*r* N/A

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

   7. *-commutative N/A

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

   8. associate-*r* N/A

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

   9. *-commutative N/A

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

   10. distribute-rgt-in N/A

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

   11. associate-*r* N/A

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

   12. distribute-lft-in N/A

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

   13. associate-*r* N/A

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

   14. *-commutative N/A

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

   15. *-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 90.5%

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

Alternative 4: 91.5% accurate, 4.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 56.5%

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

3. Taylor expanded in u0 around 0

   \[\leadsto \color{blue}{u0 \cdot \left(u0 \cdot \left(\frac{1}{3} \cdot \left({\alpha}^{2} \cdot u0\right) + \frac{1}{2} \cdot {\alpha}^{2}\right) + {\alpha}^{2}\right)} \]
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 90.3

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

5. Simplified 90.3%

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

6. 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

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

   4. distribute-rgt-in N/A

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

   5. associate-*r* N/A

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

   6. *-commutative N/A

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

   7. associate-*r* N/A

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

   8. *-commutative N/A

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

   9. distribute-rgt-in N/A

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

   10. associate-*r* N/A

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

   11. distribute-lft-in N/A

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

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

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

   13. unpow2 N/A

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

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

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

   15. +-commutative N/A

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

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

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

   17. +-commutative N/A

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

   18. *-commutative N/A

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

   19. accelerator-lowering-fma.f32 90.2

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

8. Simplified 90.2%

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

9. Taylor expanded in alpha around 0

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

    1. unpow2 N/A

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

    2. associate-*l* N/A

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

    3. *-commutative N/A

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

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

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

    5. +-commutative N/A

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

    6. distribute-lft1-in N/A

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

    7. *-commutative N/A

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

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

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

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

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

    10. +-commutative N/A

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

    11. *-commutative N/A

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

    12. accelerator-lowering-fma.f32 90.3

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

11. Simplified 90.3%

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

Alternative 5: 91.3% accurate, 4.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 56.5%

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

3. Taylor expanded in u0 around 0

   \[\leadsto \color{blue}{u0 \cdot \left(u0 \cdot \left(\frac{1}{3} \cdot \left({\alpha}^{2} \cdot u0\right) + \frac{1}{2} \cdot {\alpha}^{2}\right) + {\alpha}^{2}\right)} \]
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 90.3

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

5. Simplified 90.3%

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

6. 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

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

   4. distribute-rgt-in N/A

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

   5. associate-*r* N/A

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

   6. *-commutative N/A

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

   7. associate-*r* N/A

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

   8. *-commutative N/A

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

   9. distribute-rgt-in N/A

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

   10. associate-*r* N/A

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

   11. distribute-lft-in N/A

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

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

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

   13. unpow2 N/A

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

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

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

   15. +-commutative N/A

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

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

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

   17. +-commutative N/A

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

   18. *-commutative N/A

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

   19. accelerator-lowering-fma.f32 90.2

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

8. Simplified 90.2%

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

   1. *-commutative N/A

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

   2. associate-*l* N/A

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

   3. associate-*l* N/A

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

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

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

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

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

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

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

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

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

   8. accelerator-lowering-fma.f32 90.2

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

10. Applied egg-rr 90.2%

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

11. Final simplification 90.2%

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

Alternative 6: 87.4% 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 56.5%

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

3. Taylor expanded in u0 around 0

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

   1. distribute-lft-in N/A

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

   2. associate-*r* N/A

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

   3. *-commutative N/A

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

   4. associate-*r* N/A

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

   5. distribute-rgt-out N/A

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

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

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

   7. unpow2 N/A

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

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

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

   9. associate-*r* N/A

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

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

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

   11. *-commutative N/A

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

   12. *-lowering-*.f32 86.7

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

5. Simplified 86.7%

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

Alternative 7: 87.4% accurate, 5.3× speedup

Math

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

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(Float32(alpha * alpha) * fma(Float32(0.5), Float32(u0 * u0), u0))
end

Derivation

1. Initial program 56.5%

   \[\left(\left(-\alpha\right) \cdot \alpha\right) \cdot \log \left(1 - u0\right) \]
2. Add Preprocessing
3. 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 99.1

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

4. Applied egg-rr 99.1%

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

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

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

   5. distribute-rgt-in N/A

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

   6. associate-*r* N/A

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

   7. *-commutative N/A

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

   8. associate-*r* N/A

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

   9. *-commutative N/A

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

   10. distribute-rgt-in N/A

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

   11. associate-*r* N/A

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

   12. distribute-lft-in N/A

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

   13. associate-*r* N/A

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

   14. *-commutative N/A

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

   15. *-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 90.5%

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

8. Taylor expanded in u0 around 0

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

   1. *-commutative N/A

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

   2. +-commutative N/A

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

   3. distribute-lft1-in N/A

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

   4. associate-*r* N/A

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

   5. unpow2 N/A

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

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

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

   7. unpow2 N/A

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

   8. *-lowering-*.f32 86.7

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

10. Simplified 86.7%

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

Alternative 8: 87.4% accurate, 5.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 56.5%

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

3. Taylor expanded in u0 around 0

   \[\leadsto \color{blue}{u0 \cdot \left(u0 \cdot \left(\frac{1}{3} \cdot \left({\alpha}^{2} \cdot u0\right) + \frac{1}{2} \cdot {\alpha}^{2}\right) + {\alpha}^{2}\right)} \]
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 90.3

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

5. Simplified 90.3%

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

6. Taylor expanded in u0 around 0

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

   1. *-commutative N/A

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

   2. associate-*r* N/A

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

   3. *-lft-identity N/A

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

   4. distribute-rgt-out N/A

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

   5. +-commutative N/A

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

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

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

   7. unpow2 N/A

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

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

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

   9. +-commutative N/A

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

   10. *-commutative N/A

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

   11. accelerator-lowering-fma.f32 86.5

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

8. Simplified 86.5%

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

   1. *-commutative N/A

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

   2. associate-*l* N/A

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

   3. associate-*l* N/A

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

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

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

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

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

   6. distribute-lft-in N/A

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

   7. metadata-eval N/A

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

   8. div-inv N/A

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

   9. /-rgt-identity N/A

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

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

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

   11. *-lowering-*.f32 86.6

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

10. Applied egg-rr 86.6%

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

11. Final simplification 86.6%

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

Alternative 9: 87.4% 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 56.5%

   \[\left(\left(-\alpha\right) \cdot \alpha\right) \cdot \log \left(1 - u0\right) \]
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 99.1

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

4. Applied egg-rr 99.1%

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

   1. /-rgt-identity N/A

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

   2. div-inv N/A

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

   3. metadata-eval N/A

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

   4. associate-*l* N/A

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

   5. rgt-mult-inverse N/A

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

   6. un-div-inv N/A

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

   7. associate-/l* N/A

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

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

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

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

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

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

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

   11. *-commutative N/A

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

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

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

   13. *-commutative N/A

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

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

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

   15. neg-lowering-neg.f32 98.9

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

6. Applied egg-rr 98.9%

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

   1. *-commutative N/A

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

   2. associate-/l* N/A

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

   3. *-inverses N/A

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

   4. metadata-eval N/A

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

   5. div-inv N/A

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

   6. associate-*l/ N/A

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

   7. associate-/r/ N/A

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

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

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

   9. metadata-eval N/A

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

   10. frac-2neg N/A

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

   11. /-lowering-/.f32 98.8

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

8. Applied egg-rr 98.8%

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

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. *-commutative N/A

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

    3. associate-*r* N/A

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

    4. *-commutative N/A

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

    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-*r* 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 86.5

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

11. Simplified 86.5%

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

Alternative 10: 74.7% 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 56.5%

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

3. Taylor expanded in u0 around 0

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

   1. *-commutative N/A

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

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

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

   3. unpow2 N/A

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

   4. *-lowering-*.f32 73.6

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

5. Simplified 73.6%

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

Reproduce

herbie shell --seed 2024204 
(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))))