Beckmann Distribution sample, tan2theta, alphax == alphay

Percentage Accurate: 55.9% → 99.0%

Time: 12.2s

Alternatives: 14

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: 55.9% 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 54.0%

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

   1. sub-neg N/A

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

   2. lower-log1p.f32 N/A

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

   3. lower-neg.f32 98.9

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

4. Applied 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 54.0%

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

   1. sub-neg N/A

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

   2. lower-log1p.f32 N/A

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

   3. lower-neg.f32 98.9

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

4. Applied rewrites 98.9%

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

6. Applied rewrites 98.8%

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

   1. lift-neg.f32 N/A

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

   2. lift-neg.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. lift-log1p.f32 N/A

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

   5. lift-log1p.f32 N/A

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

   6. lift--.f32 N/A

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

   7. associate-*l* N/A

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

   8. *-commutative N/A

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

   9. *-commutative N/A

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

   10. associate-*r* N/A

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

   11. *-commutative N/A

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

   12. associate-*r* N/A

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

   13. lower-*.f32 N/A

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

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: 93.3% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 54.0%

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

   1. sub-neg N/A

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

   2. lower-log1p.f32 N/A

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

   3. lower-neg.f32 98.9

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

4. Applied rewrites 98.9%

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

5. Taylor expanded in u0 around 0

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

   1. 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. +-commutative N/A

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

   3. distribute-rgt-in N/A

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

   4. associate-*r* N/A

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

   5. associate-+r+ N/A

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

   6. +-commutative N/A

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

   7. *-commutative N/A

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

   8. +-commutative N/A

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

   9. lower-fma.f32 N/A

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

7. Applied rewrites 92.8%

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

   1. lift-*.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. lift-fma.f32 N/A

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

   4. lift-*.f32 N/A

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

   5. lift-*.f32 N/A

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

   6. lift-fma.f32 N/A

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

   7. lift-*.f32 N/A

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

   8. +-commutative N/A

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

   9. distribute-lft-in N/A

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

9. Applied rewrites 93.0%

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

Alternative 4: 93.3% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 54.0%

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

   1. sub-neg N/A

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

   2. lower-log1p.f32 N/A

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

   3. lower-neg.f32 98.9

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

4. Applied rewrites 98.9%

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

5. Taylor expanded in u0 around 0

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

   1. 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. +-commutative N/A

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

   3. distribute-rgt-in N/A

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

   4. associate-*r* N/A

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

   5. associate-+r+ N/A

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

   6. +-commutative N/A

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

   7. *-commutative N/A

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

   8. +-commutative N/A

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

   9. lower-fma.f32 N/A

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

7. Applied rewrites 92.8%

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

   1. lift-*.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. lift-fma.f32 N/A

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

   4. lift-*.f32 N/A

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

   5. lift-*.f32 N/A

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

   6. lift-fma.f32 N/A

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

   7. lift-*.f32 N/A

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

   8. +-commutative N/A

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

   9. lift-*.f32 N/A

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

   10. lift-fma.f32 N/A

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

   11. distribute-lft-in N/A

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

   12. *-rgt-identity N/A

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

   13. associate-+l+ N/A

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

   14. *-commutative N/A

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

   15. lift-*.f32 N/A

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

   16. associate-*r* N/A

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

9. Applied rewrites 93.0%

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

10. Final simplification 93.0%

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

Alternative 5: 93.3% accurate, 2.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 54.0%

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

   1. sub-neg N/A

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

   2. lower-log1p.f32 N/A

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

   3. lower-neg.f32 98.9

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

4. Applied rewrites 98.9%

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

6. Applied rewrites 98.8%

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

7. Taylor expanded in u0 around 0

   \[\leadsto \color{blue}{u0 \cdot \left(u0 \cdot \left(\frac{1}{2} \cdot {\alpha}^{2} + u0 \cdot \left(\frac{1}{4} \cdot \left({\alpha}^{2} \cdot u0\right) + \frac{1}{3} \cdot {\alpha}^{2}\right)\right) + {\alpha}^{2}\right)} \]
8. 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

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

9. Applied rewrites 93.0%

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

Alternative 6: 93.3% accurate, 3.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 54.0%

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

   1. sub-neg N/A

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

   2. lower-log1p.f32 N/A

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

   3. lower-neg.f32 98.9

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

4. Applied rewrites 98.9%

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

6. Applied rewrites 98.8%

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

   1. lift-neg.f32 N/A

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

   2. *-commutative N/A

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

   3. lift-*.f32 N/A

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

   4. lift-neg.f32 N/A

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

   5. lift-*.f32 N/A

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

   6. lift-log1p.f32 N/A

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

   7. lift-log1p.f32 N/A

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

   8. sub-neg N/A

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

   9. distribute-lft-in N/A

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

   10. lower-fma.f32 N/A

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

   11. lift-*.f32 N/A

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

   12. lift-neg.f32 N/A

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

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

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

   14. lift-*.f32 N/A

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

   15. lower-neg.f32 N/A

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

   16. lower-*.f32 N/A

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

8. Applied rewrites 98.9%

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

9. Taylor expanded in u0 around 0

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

11. Applied rewrites 93.0%

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

Alternative 7: 93.0% accurate, 3.4× speedup

Math

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

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 * u0)) * fma(u0, fma(u0, fma(u0, Float32(0.25), Float32(0.3333333333333333)), Float32(0.5)), Float32(1.0)))
end

Derivation

1. Initial program 54.0%

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

   1. sub-neg N/A

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

   2. lower-log1p.f32 N/A

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

   3. lower-neg.f32 98.9

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

4. Applied rewrites 98.9%

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

5. Taylor expanded in u0 around 0

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

   1. 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. +-commutative N/A

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

   3. distribute-rgt-in N/A

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

   4. associate-*r* N/A

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

   5. associate-+r+ N/A

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

   6. +-commutative N/A

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

   7. *-commutative N/A

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

   8. +-commutative N/A

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

   9. lower-fma.f32 N/A

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

7. Applied rewrites 92.8%

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

8. Taylor expanded in alpha around 0

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

   1. associate-*r* N/A

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

   2. lower-*.f32 N/A

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

   3. unpow2 N/A

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

   4. associate-*l* N/A

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

   5. lower-*.f32 N/A

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

   6. lower-*.f32 N/A

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

   7. +-commutative N/A

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

   8. +-commutative N/A

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

   9. unpow2 N/A

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

   10. associate-*l* N/A

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

   11. *-commutative N/A

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

   12. distribute-lft-out N/A

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

   13. lower-fma.f32 N/A

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

   14. lower-fma.f32 N/A

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

   15. +-commutative N/A

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

   16. *-commutative N/A

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

   17. lower-fma.f32 92.8

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

10. Applied rewrites 92.8%

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

Alternative 8: 93.1% accurate, 3.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 54.0%

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

   1. sub-neg N/A

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

   2. lower-log1p.f32 N/A

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

   3. lower-neg.f32 98.9

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

4. Applied rewrites 98.9%

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

5. Taylor expanded in u0 around 0

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

   1. 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. +-commutative N/A

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

   3. distribute-rgt-in N/A

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

   4. associate-*r* N/A

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

   5. associate-+r+ N/A

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

   6. +-commutative N/A

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

   7. *-commutative N/A

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

   8. +-commutative N/A

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

   9. lower-fma.f32 N/A

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

7. Applied rewrites 92.8%

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

8. Taylor expanded in alpha around 0

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

   1. lower-*.f32 N/A

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

   2. unpow2 N/A

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

   3. lower-*.f32 N/A

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

   4. +-commutative N/A

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

   5. +-commutative N/A

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

   6. unpow2 N/A

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

   7. associate-*l* N/A

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

   8. *-commutative N/A

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

   9. distribute-lft-out N/A

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

   10. lower-fma.f32 N/A

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

   11. lower-fma.f32 N/A

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

   12. +-commutative N/A

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

   13. *-commutative N/A

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

   14. lower-fma.f32 92.8

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

10. Applied rewrites 92.8%

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

Alternative 9: 91.5% accurate, 3.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 54.0%

   \[\left(\left(-\alpha\right) \cdot \alpha\right) \cdot \log \left(1 - u0\right) \]
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 91.1

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

5. Applied rewrites 91.1%

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

   1. lift-fma.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. lift-*.f32 N/A

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

   5. +-commutative N/A

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

   6. lift-*.f32 N/A

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

   7. lower-fma.f32 N/A

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

   8. *-commutative N/A

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

   9. lift-*.f32 N/A

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

   10. lift-*.f32 N/A

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

   11. associate-*r* N/A

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

   12. lift-*.f32 N/A

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

   13. associate-*l* N/A

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

   14. lower-*.f32 N/A

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

   15. lower-*.f32 91.4

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

7. Applied rewrites 91.4%

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

8. Final simplification 91.4%

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

Alternative 10: 91.3% accurate, 4.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 54.0%

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

   1. sub-neg N/A

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

   2. lower-log1p.f32 N/A

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

   3. lower-neg.f32 98.9

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

4. Applied rewrites 98.9%

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

5. Taylor expanded in u0 around 0

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

   1. *-commutative N/A

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

7. Applied rewrites 91.2%

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

Alternative 11: 91.1% accurate, 4.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 54.0%

   \[\left(\left(-\alpha\right) \cdot \alpha\right) \cdot \log \left(1 - u0\right) \]
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 91.1

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

5. Applied rewrites 91.1%

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

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. distribute-lft-in N/A

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

   3. associate-+r+ N/A

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

   4. +-commutative N/A

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

   5. associate-*r* N/A

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

   6. *-commutative N/A

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

   7. *-commutative N/A

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

   8. associate-*r* N/A

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

   9. associate-*r* N/A

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

   10. unpow2 N/A

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

   11. *-commutative N/A

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

   12. associate-*r* N/A

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

   13. unpow2 N/A

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

   14. associate-*r* N/A

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

   15. *-commutative N/A

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

   16. distribute-rgt1-in N/A

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

8. Applied rewrites 90.9%

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

Alternative 12: 87.3% accurate, 4.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 54.0%

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

   1. sub-neg N/A

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

   2. lower-log1p.f32 N/A

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

   3. lower-neg.f32 98.9

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

4. Applied rewrites 98.9%

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

5. Taylor expanded in u0 around 0

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

   1. lower-*.f32 N/A

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

   2. *-commutative N/A

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

   3. associate-*l* N/A

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

   4. *-commutative N/A

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

   5. *-rgt-identity N/A

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

   6. distribute-lft-out N/A

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

   7. +-commutative N/A

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

   8. lower-*.f32 N/A

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

   9. unpow2 N/A

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

   10. lower-*.f32 N/A

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

   11. +-commutative N/A

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

   12. *-commutative N/A

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

   13. lower-fma.f32 87.3

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

7. Applied rewrites 87.3%

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

   1. lift-*.f32 N/A

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

   2. +-commutative N/A

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

   3. distribute-lft-in N/A

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

   4. *-rgt-identity N/A

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

   5. lift-*.f32 N/A

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

   6. lower-fma.f32 N/A

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

   7. lift-*.f32 N/A

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

   8. associate-*l* N/A

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

   9. lower-*.f32 N/A

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

   10. lower-*.f32 N/A

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

   11. lower-*.f32 87.5

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

9. Applied rewrites 87.5%

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

Alternative 13: 87.2% accurate, 5.3× speedup

Math

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

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

Derivation

1. Initial program 54.0%

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

   1. sub-neg N/A

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

   2. lower-log1p.f32 N/A

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

   3. lower-neg.f32 98.9

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

4. Applied rewrites 98.9%

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

6. Applied rewrites 98.8%

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

   1. lift-neg.f32 N/A

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

   2. *-commutative N/A

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

   3. lift-*.f32 N/A

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

   4. lift-neg.f32 N/A

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

   5. lift-*.f32 N/A

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

   6. lift-log1p.f32 N/A

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

   7. lift-log1p.f32 N/A

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

   8. sub-neg N/A

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

   9. distribute-lft-in N/A

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

   10. lower-fma.f32 N/A

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

   11. lift-*.f32 N/A

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

   12. lift-neg.f32 N/A

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

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

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

   14. lift-*.f32 N/A

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

   15. lower-neg.f32 N/A

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

   16. lower-*.f32 N/A

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

8. Applied rewrites 98.9%

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

9. Taylor expanded in u0 around 0

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

    1. distribute-lft1-in N/A

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

    2. metadata-eval N/A

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

    3. associate-*r* N/A

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

    4. *-commutative N/A

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

    5. *-lft-identity N/A

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

    6. distribute-rgt-in N/A

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

    7. +-commutative N/A

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

    8. associate-*l* N/A

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

    9. *-commutative N/A

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

    10. associate-*r* N/A

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

    11. unpow2 N/A

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

    12. associate-*l* N/A

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

    13. lower-*.f32 N/A

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

    14. lower-*.f32 N/A

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

    15. +-commutative N/A

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

    16. distribute-lft-in N/A

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

    17. *-rgt-identity N/A

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

    18. lower-fma.f32 N/A

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

    19. *-commutative N/A

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

    20. lower-*.f32 87.4

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

11. Applied rewrites 87.4%

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

Alternative 14: 74.5% 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 54.0%

   \[\left(\left(-\alpha\right) \cdot \alpha\right) \cdot \log \left(1 - u0\right) \]
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 75.3

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

5. Applied rewrites 75.3%

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

Reproduce

herbie shell --seed 2024214 
(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))))