Beckmann Distribution sample, tan2theta, alphax == alphay

Percentage Accurate: 56.2% → 99.0%

Time: 10.3s

Alternatives: 11

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

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

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 57.1%

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

   1. lift-log.f32 N/A

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

   2. lift--.f32 N/A

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

   3. sub-neg N/A

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

   4. lower-log1p.f32 N/A

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

   5. lower-neg.f32 98.9

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

4. Applied rewrites 98.9%

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

5. Final simplification 98.9%

   \[\leadsto \left(\alpha \cdot \left(-\alpha\right)\right) \cdot \mathsf{log1p}\left(-u0\right) \]
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 57.1%

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

   1. lift-log.f32 N/A

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

   2. lift--.f32 N/A

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

   3. sub-neg N/A

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

   4. lower-log1p.f32 N/A

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

   5. lower-neg.f32 98.9

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

4. Applied rewrites 98.9%

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

   1. lift-neg.f32 N/A

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

   2. neg-sub0 N/A

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

   3. flip-- N/A

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

   4. lower-/.f32 N/A

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

   5. metadata-eval N/A

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

   6. lift-*.f32 N/A

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

   7. lower--.f32 N/A

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

   8. lower-+.f32 98.9

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

6. Applied rewrites 98.9%

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

   1. lift-*.f32 N/A

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

   2. *-commutative N/A

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

   3. lift-*.f32 N/A

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

   4. lift-/.f32 N/A

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

   5. lift--.f32 N/A

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

   6. metadata-eval N/A

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

   7. lift-*.f32 N/A

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

   8. lift-+.f32 N/A

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

   9. flip-- N/A

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

   10. neg-sub0 N/A

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

   11. lift-neg.f32 N/A

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

   12. lift-log1p.f32 N/A

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

   13. lift-neg.f32 N/A

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

   14. sub-neg N/A

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

   15. associate-*r* N/A

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

   16. lower-*.f32 N/A

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

   17. lower-*.f32 N/A

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

   18. sub-neg N/A

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

   19. lift-neg.f32 N/A

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

   20. lift-log1p.f32 98.9

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

8. Applied rewrites 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: 93.1% 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 57.1%

   \[\left(\left(-\alpha\right) \cdot \alpha\right) \cdot \log \left(1 - u0\right) \]
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. lower-*.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. lower-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. Applied rewrites 93.5%

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

Alternative 4: 92.9% 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 57.1%

   \[\left(\left(-\alpha\right) \cdot \alpha\right) \cdot \log \left(1 - u0\right) \]
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. lower-*.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. lower-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. lower-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. lower-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. Applied rewrites 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%

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

Alternative 5: 91.2% 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 57.1%

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

3. Taylor expanded in u0 around 0

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

   1. lower-*.f32 N/A

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

   2. lower-fma.f32 N/A

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

   3. *-commutative N/A

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

   4. associate-*r* N/A

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

   5. distribute-rgt-out N/A

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

   6. unpow2 N/A

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

   7. +-commutative N/A

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

   8. associate-*l* N/A

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

   9. lower-*.f32 N/A

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

   10. lower-*.f32 N/A

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

   11. +-commutative N/A

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

   12. *-commutative N/A

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

   13. lower-fma.f32 N/A

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

   14. unpow2 N/A

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

   15. lower-*.f32 91.7

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

5. Applied rewrites 91.7%

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

7. Applied rewrites 91.8%

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

8. Final simplification 91.8%

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

Alternative 6: 91.1% accurate, 4.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 57.1%

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

   1. lift-log.f32 N/A

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

   2. lift--.f32 N/A

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

   3. sub-neg N/A

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

   4. lower-log1p.f32 N/A

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

   5. lower-neg.f32 98.9

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

4. Applied rewrites 98.9%

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

   1. lift-*.f32 N/A

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

   2. lift-neg.f32 N/A

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

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

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

   4. pow2 N/A

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

   5. metadata-eval N/A

      \[\leadsto \left(\mathsf{neg}\left({\alpha}^{\color{blue}{\left(4 - 2\right)}}\right)\right) \cdot \mathsf{log1p}\left(\mathsf{neg}\left(u0\right)\right) \]

   6. metadata-eval N/A

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

   7. pow-div N/A

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

   8. pow-prod-up N/A

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

   9. pow-prod-down N/A

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

   10. lift-*.f32 N/A

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

   11. pow2 N/A

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

   12. pow2 N/A

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

   13. lift-*.f32 N/A

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

   14. distribute-frac-neg N/A

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

   15. lift-*.f32 N/A

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

   16. associate-*l* N/A

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

   17. lift-*.f32 N/A

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

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

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

   19. lift-neg.f32 N/A

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

   20. *-commutative N/A

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

   21. associate-/l* N/A

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

   22. lift-*.f32 N/A

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

   23. associate-/r* N/A

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

   24. *-inverses N/A

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

6. Applied rewrites 98.9%

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

7. Taylor expanded in u0 around 0

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

   1. lower-*.f32 N/A

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

   2. +-commutative N/A

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

   3. unpow2 N/A

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

   4. *-commutative N/A

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

   5. associate-*r* N/A

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

   6. distribute-rgt-in N/A

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

   7. +-commutative N/A

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

   8. *-commutative N/A

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

   9. associate-*l* N/A

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

   10. unpow2 N/A

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

   11. associate-*r* N/A

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

   12. distribute-rgt-out N/A

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

   13. lower-*.f32 N/A

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

   14. distribute-rgt1-in N/A

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

   15. distribute-lft1-in N/A

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

   16. associate-*l* N/A

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

9. Applied rewrites 91.7%

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

10. Final simplification 91.7%

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

Alternative 7: 90.9% 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 57.1%

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

3. Taylor expanded in u0 around 0

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

   1. lower-*.f32 N/A

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

   2. lower-fma.f32 N/A

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

   3. *-commutative N/A

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

   4. associate-*r* N/A

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

   5. distribute-rgt-out N/A

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

   6. unpow2 N/A

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

   7. +-commutative N/A

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

   8. associate-*l* N/A

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

   9. lower-*.f32 N/A

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

   10. lower-*.f32 N/A

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

   11. +-commutative N/A

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

   12. *-commutative N/A

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

   13. lower-fma.f32 N/A

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

   14. unpow2 N/A

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

   15. lower-*.f32 91.7

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

5. Applied rewrites 91.7%

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

6. Taylor expanded in u0 around 0

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

8. Applied rewrites 91.4%

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

Alternative 8: 86.9% accurate, 4.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 57.1%

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

3. Taylor expanded in u0 around 0

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

   1. lower-*.f32 N/A

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

   2. lower-fma.f32 N/A

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

   3. *-commutative N/A

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

   4. associate-*r* N/A

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

   5. distribute-rgt-out N/A

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

   6. unpow2 N/A

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

   7. +-commutative N/A

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

   8. associate-*l* N/A

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

   9. lower-*.f32 N/A

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

   10. lower-*.f32 N/A

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

   11. +-commutative N/A

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

   12. *-commutative N/A

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

   13. lower-fma.f32 N/A

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

   14. unpow2 N/A

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

   15. lower-*.f32 91.7

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

5. Applied rewrites 91.7%

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

6. Taylor expanded in u0 around 0

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

8. Applied rewrites 87.9%

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

Alternative 9: 86.9% accurate, 5.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 57.1%

   \[\left(\left(-\alpha\right) \cdot \alpha\right) \cdot \log \left(1 - u0\right) \]
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. lower-*.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. lower-*.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. lower-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. lower-*.f32 87.8

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

5. Applied rewrites 87.8%

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

Alternative 10: 86.8% accurate, 5.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 57.1%

   \[\left(\left(-\alpha\right) \cdot \alpha\right) \cdot \log \left(1 - u0\right) \]
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. lower-*.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. lower-*.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. lower-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. lower-*.f32 87.8

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

5. Applied rewrites 87.8%

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

7. Applied rewrites 87.6%

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

8. Final simplification 87.6%

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

Alternative 11: 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 57.1%

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

3. Taylor expanded in u0 around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. unpow2 N/A

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

   4. lower-*.f32 74.8

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

5. Applied rewrites 74.8%

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

Reproduce

herbie shell --seed 2024216 
(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))))