Beckmann Distribution sample, tan2theta, alphax == alphay

Percentage Accurate: 55.2% → 99.0%

Time: 12.3s

Alternatives: 14

Speedup: 21.6×

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: 55.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Alternative 1: 99.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 57.4%

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

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

      \[\leadsto \mathsf{*.f32}\left(\left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right), \color{blue}{\log \left(1 - u0\right)}\right) \]

   2. *-commutative N/A

      \[\leadsto \mathsf{*.f32}\left(\left(\alpha \cdot \left(\mathsf{neg}\left(\alpha\right)\right)\right), \log \color{blue}{\left(1 - u0\right)}\right) \]

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

      \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(\alpha, \left(\mathsf{neg}\left(\alpha\right)\right)\right), \log \color{blue}{\left(1 - u0\right)}\right) \]

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

      \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(\alpha, \mathsf{neg.f32}\left(\alpha\right)\right), \log \left(1 - \color{blue}{u0}\right)\right) \]

   5. sub-neg N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(\alpha, \mathsf{neg.f32}\left(\alpha\right)\right), \log \left(1 + \left(\mathsf{neg}\left(u0\right)\right)\right)\right) \]

   6. log1p-define N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(\alpha, \mathsf{neg.f32}\left(\alpha\right)\right), \left(\mathsf{log1p}\left(\mathsf{neg}\left(u0\right)\right)\right)\right) \]

   7. log1p-lowering-log1p.f32 N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(\alpha, \mathsf{neg.f32}\left(\alpha\right)\right), \mathsf{log1p.f32}\left(\left(\mathsf{neg}\left(u0\right)\right)\right)\right) \]

   8. neg-lowering-neg.f32 99.1%

      \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(\alpha, \mathsf{neg.f32}\left(\alpha\right)\right), \mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right)\right) \]

3. Simplified 99.1%

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

5. Final simplification 99.1%

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

Alternative 2: 93.6% accurate, 4.0× speedup

Math

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

FPCore

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

C

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

Fortran

real(4) function code(alpha, u0)
    real(4), intent (in) :: alpha
    real(4), intent (in) :: u0
    code = u0 * (((alpha * (alpha * u0)) * (0.5e0 + (u0 * 0.3333333333333333e0))) + (alpha * (alpha * ((u0 * (u0 * (u0 * 0.25e0))) + 1.0e0))))
end function

Julia

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

MATLAB

function tmp = code(alpha, u0)
	tmp = u0 * (((alpha * (alpha * u0)) * (single(0.5) + (u0 * single(0.3333333333333333)))) + (alpha * (alpha * ((u0 * (u0 * (u0 * single(0.25)))) + single(1.0)))));
end

Derivation

1. Initial program 57.4%

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

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

      \[\leadsto \mathsf{*.f32}\left(\left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right), \color{blue}{\log \left(1 - u0\right)}\right) \]

   2. *-commutative N/A

      \[\leadsto \mathsf{*.f32}\left(\left(\alpha \cdot \left(\mathsf{neg}\left(\alpha\right)\right)\right), \log \color{blue}{\left(1 - u0\right)}\right) \]

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

      \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(\alpha, \left(\mathsf{neg}\left(\alpha\right)\right)\right), \log \color{blue}{\left(1 - u0\right)}\right) \]

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

      \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(\alpha, \mathsf{neg.f32}\left(\alpha\right)\right), \log \left(1 - \color{blue}{u0}\right)\right) \]

   5. sub-neg N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(\alpha, \mathsf{neg.f32}\left(\alpha\right)\right), \log \left(1 + \left(\mathsf{neg}\left(u0\right)\right)\right)\right) \]

   6. log1p-define N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(\alpha, \mathsf{neg.f32}\left(\alpha\right)\right), \left(\mathsf{log1p}\left(\mathsf{neg}\left(u0\right)\right)\right)\right) \]

   7. log1p-lowering-log1p.f32 N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(\alpha, \mathsf{neg.f32}\left(\alpha\right)\right), \mathsf{log1p.f32}\left(\left(\mathsf{neg}\left(u0\right)\right)\right)\right) \]

   8. neg-lowering-neg.f32 99.1%

      \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(\alpha, \mathsf{neg.f32}\left(\alpha\right)\right), \mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right)\right) \]

3. Simplified 99.1%

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

5. Taylor expanded in u0 around 0

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

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

      \[\leadsto \mathsf{*.f32}\left(u0, \color{blue}{\left(u0 \cdot \left(\frac{1}{2} \cdot {\alpha}^{2} + u0 \cdot \left(\frac{1}{4} \cdot \left({\alpha}^{2} \cdot u0\right) + \frac{1}{3} \cdot {\alpha}^{2}\right)\right) + {\alpha}^{2}\right)}\right) \]

   2. +-commutative N/A

      \[\leadsto \mathsf{*.f32}\left(u0, \left({\alpha}^{2} + \color{blue}{u0 \cdot \left(\frac{1}{2} \cdot {\alpha}^{2} + u0 \cdot \left(\frac{1}{4} \cdot \left({\alpha}^{2} \cdot u0\right) + \frac{1}{3} \cdot {\alpha}^{2}\right)\right)}\right)\right) \]

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

      \[\leadsto \mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\left({\alpha}^{2}\right), \color{blue}{\left(u0 \cdot \left(\frac{1}{2} \cdot {\alpha}^{2} + u0 \cdot \left(\frac{1}{4} \cdot \left({\alpha}^{2} \cdot u0\right) + \frac{1}{3} \cdot {\alpha}^{2}\right)\right)\right)}\right)\right) \]

   4. unpow2 N/A

      \[\leadsto \mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\left(\alpha \cdot \alpha\right), \left(\color{blue}{u0} \cdot \left(\frac{1}{2} \cdot {\alpha}^{2} + u0 \cdot \left(\frac{1}{4} \cdot \left({\alpha}^{2} \cdot u0\right) + \frac{1}{3} \cdot {\alpha}^{2}\right)\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{*.f32}\left(\alpha, \alpha\right), \left(\color{blue}{u0} \cdot \left(\frac{1}{2} \cdot {\alpha}^{2} + u0 \cdot \left(\frac{1}{4} \cdot \left({\alpha}^{2} \cdot u0\right) + \frac{1}{3} \cdot {\alpha}^{2}\right)\right)\right)\right)\right) \]

   6. +-commutative N/A

      \[\leadsto \mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{*.f32}\left(\alpha, \alpha\right), \left(u0 \cdot \left(u0 \cdot \left(\frac{1}{4} \cdot \left({\alpha}^{2} \cdot u0\right) + \frac{1}{3} \cdot {\alpha}^{2}\right) + \color{blue}{\frac{1}{2} \cdot {\alpha}^{2}}\right)\right)\right)\right) \]

   7. distribute-rgt-in N/A

      \[\leadsto \mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{*.f32}\left(\alpha, \alpha\right), \left(u0 \cdot \left(\left(\left(\frac{1}{4} \cdot \left({\alpha}^{2} \cdot u0\right)\right) \cdot u0 + \left(\frac{1}{3} \cdot {\alpha}^{2}\right) \cdot u0\right) + \color{blue}{\frac{1}{2}} \cdot {\alpha}^{2}\right)\right)\right)\right) \]

   8. associate-*r* N/A

      \[\leadsto \mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{*.f32}\left(\alpha, \alpha\right), \left(u0 \cdot \left(\left(\left(\frac{1}{4} \cdot \left({\alpha}^{2} \cdot u0\right)\right) \cdot u0 + \frac{1}{3} \cdot \left({\alpha}^{2} \cdot u0\right)\right) + \frac{1}{2} \cdot {\alpha}^{2}\right)\right)\right)\right) \]

   9. associate-+l+ N/A

      \[\leadsto \mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{*.f32}\left(\alpha, \alpha\right), \left(u0 \cdot \left(\left(\frac{1}{4} \cdot \left({\alpha}^{2} \cdot u0\right)\right) \cdot u0 + \color{blue}{\left(\frac{1}{3} \cdot \left({\alpha}^{2} \cdot u0\right) + \frac{1}{2} \cdot {\alpha}^{2}\right)}\right)\right)\right)\right) \]

   10. *-commutative N/A

       \[\leadsto \mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{*.f32}\left(\alpha, \alpha\right), \left(u0 \cdot \left(u0 \cdot \left(\frac{1}{4} \cdot \left({\alpha}^{2} \cdot u0\right)\right) + \left(\color{blue}{\frac{1}{3} \cdot \left({\alpha}^{2} \cdot u0\right)} + \frac{1}{2} \cdot {\alpha}^{2}\right)\right)\right)\right)\right) \]

7. Simplified 92.4%

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

   1. distribute-rgt-in N/A

      \[\leadsto \mathsf{*.f32}\left(u0, \left(\alpha \cdot \alpha + \left(\left(u0 \cdot \left(u0 \cdot \frac{1}{4}\right)\right) \cdot \left(\left(\alpha \cdot \alpha\right) \cdot u0\right) + \color{blue}{\left(u0 \cdot \frac{1}{3} + \frac{1}{2}\right) \cdot \left(\left(\alpha \cdot \alpha\right) \cdot u0\right)}\right)\right)\right) \]

   2. associate-+r+ N/A

      \[\leadsto \mathsf{*.f32}\left(u0, \left(\left(\alpha \cdot \alpha + \left(u0 \cdot \left(u0 \cdot \frac{1}{4}\right)\right) \cdot \left(\left(\alpha \cdot \alpha\right) \cdot u0\right)\right) + \color{blue}{\left(u0 \cdot \frac{1}{3} + \frac{1}{2}\right) \cdot \left(\left(\alpha \cdot \alpha\right) \cdot u0\right)}\right)\right) \]

   3. +-commutative N/A

      \[\leadsto \mathsf{*.f32}\left(u0, \left(\left(u0 \cdot \frac{1}{3} + \frac{1}{2}\right) \cdot \left(\left(\alpha \cdot \alpha\right) \cdot u0\right) + \color{blue}{\left(\alpha \cdot \alpha + \left(u0 \cdot \left(u0 \cdot \frac{1}{4}\right)\right) \cdot \left(\left(\alpha \cdot \alpha\right) \cdot u0\right)\right)}\right)\right) \]

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

      \[\leadsto \mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\left(\left(u0 \cdot \frac{1}{3} + \frac{1}{2}\right) \cdot \left(\left(\alpha \cdot \alpha\right) \cdot u0\right)\right), \color{blue}{\left(\alpha \cdot \alpha + \left(u0 \cdot \left(u0 \cdot \frac{1}{4}\right)\right) \cdot \left(\left(\alpha \cdot \alpha\right) \cdot u0\right)\right)}\right)\right) \]

   5. *-commutative N/A

      \[\leadsto \mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\left(\left(\left(\alpha \cdot \alpha\right) \cdot u0\right) \cdot \left(u0 \cdot \frac{1}{3} + \frac{1}{2}\right)\right), \left(\color{blue}{\alpha \cdot \alpha} + \left(u0 \cdot \left(u0 \cdot \frac{1}{4}\right)\right) \cdot \left(\left(\alpha \cdot \alpha\right) \cdot u0\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{*.f32}\left(\left(\left(\alpha \cdot \alpha\right) \cdot u0\right), \left(u0 \cdot \frac{1}{3} + \frac{1}{2}\right)\right), \left(\color{blue}{\alpha \cdot \alpha} + \left(u0 \cdot \left(u0 \cdot \frac{1}{4}\right)\right) \cdot \left(\left(\alpha \cdot \alpha\right) \cdot u0\right)\right)\right)\right) \]

   7. associate-*l* N/A

      \[\leadsto \mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{*.f32}\left(\left(\alpha \cdot \left(\alpha \cdot u0\right)\right), \left(u0 \cdot \frac{1}{3} + \frac{1}{2}\right)\right), \left(\color{blue}{\alpha} \cdot \alpha + \left(u0 \cdot \left(u0 \cdot \frac{1}{4}\right)\right) \cdot \left(\left(\alpha \cdot \alpha\right) \cdot u0\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(\alpha, \left(\alpha \cdot u0\right)\right), \left(u0 \cdot \frac{1}{3} + \frac{1}{2}\right)\right), \left(\color{blue}{\alpha} \cdot \alpha + \left(u0 \cdot \left(u0 \cdot \frac{1}{4}\right)\right) \cdot \left(\left(\alpha \cdot \alpha\right) \cdot u0\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(\alpha, \mathsf{*.f32}\left(\alpha, u0\right)\right), \left(u0 \cdot \frac{1}{3} + \frac{1}{2}\right)\right), \left(\alpha \cdot \alpha + \left(u0 \cdot \left(u0 \cdot \frac{1}{4}\right)\right) \cdot \left(\left(\alpha \cdot \alpha\right) \cdot u0\right)\right)\right)\right) \]

   10. +-commutative N/A

       \[\leadsto \mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(\alpha, \mathsf{*.f32}\left(\alpha, u0\right)\right), \left(\frac{1}{2} + u0 \cdot \frac{1}{3}\right)\right), \left(\alpha \cdot \color{blue}{\alpha} + \left(u0 \cdot \left(u0 \cdot \frac{1}{4}\right)\right) \cdot \left(\left(\alpha \cdot \alpha\right) \cdot u0\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(\alpha, \mathsf{*.f32}\left(\alpha, u0\right)\right), \mathsf{+.f32}\left(\frac{1}{2}, \left(u0 \cdot \frac{1}{3}\right)\right)\right), \left(\alpha \cdot \color{blue}{\alpha} + \left(u0 \cdot \left(u0 \cdot \frac{1}{4}\right)\right) \cdot \left(\left(\alpha \cdot \alpha\right) \cdot u0\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(\alpha, \mathsf{*.f32}\left(\alpha, u0\right)\right), \mathsf{+.f32}\left(\frac{1}{2}, \mathsf{*.f32}\left(u0, \frac{1}{3}\right)\right)\right), \left(\alpha \cdot \alpha + \left(u0 \cdot \left(u0 \cdot \frac{1}{4}\right)\right) \cdot \left(\left(\alpha \cdot \alpha\right) \cdot u0\right)\right)\right)\right) \]

   13. *-commutative N/A

       \[\leadsto \mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(\alpha, \mathsf{*.f32}\left(\alpha, u0\right)\right), \mathsf{+.f32}\left(\frac{1}{2}, \mathsf{*.f32}\left(u0, \frac{1}{3}\right)\right)\right), \left(\alpha \cdot \alpha + \left(\left(\alpha \cdot \alpha\right) \cdot u0\right) \cdot \color{blue}{\left(u0 \cdot \left(u0 \cdot \frac{1}{4}\right)\right)}\right)\right)\right) \]

   14. associate-*l* N/A

       \[\leadsto \mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(\alpha, \mathsf{*.f32}\left(\alpha, u0\right)\right), \mathsf{+.f32}\left(\frac{1}{2}, \mathsf{*.f32}\left(u0, \frac{1}{3}\right)\right)\right), \left(\alpha \cdot \alpha + \left(\alpha \cdot \left(\alpha \cdot u0\right)\right) \cdot \left(\color{blue}{u0} \cdot \left(u0 \cdot \frac{1}{4}\right)\right)\right)\right)\right) \]

   15. associate-*l* N/A

       \[\leadsto \mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(\alpha, \mathsf{*.f32}\left(\alpha, u0\right)\right), \mathsf{+.f32}\left(\frac{1}{2}, \mathsf{*.f32}\left(u0, \frac{1}{3}\right)\right)\right), \left(\alpha \cdot \alpha + \alpha \cdot \color{blue}{\left(\left(\alpha \cdot u0\right) \cdot \left(u0 \cdot \left(u0 \cdot \frac{1}{4}\right)\right)\right)}\right)\right)\right) \]

   16. distribute-lft-out N/A

       \[\leadsto \mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(\alpha, \mathsf{*.f32}\left(\alpha, u0\right)\right), \mathsf{+.f32}\left(\frac{1}{2}, \mathsf{*.f32}\left(u0, \frac{1}{3}\right)\right)\right), \left(\alpha \cdot \color{blue}{\left(\alpha + \left(\alpha \cdot u0\right) \cdot \left(u0 \cdot \left(u0 \cdot \frac{1}{4}\right)\right)\right)}\right)\right)\right) \]

9. Applied egg-rr 92.4%

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

10. Final simplification 92.4%

    \[\leadsto u0 \cdot \left(\left(\alpha \cdot \left(\alpha \cdot u0\right)\right) \cdot \left(0.5 + u0 \cdot 0.3333333333333333\right) + \alpha \cdot \left(\alpha \cdot \left(u0 \cdot \left(u0 \cdot \left(u0 \cdot 0.25\right)\right) + 1\right)\right)\right) \]
11. Add Preprocessing

Alternative 3: 93.7% accurate, 4.7× speedup

Math

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

FPCore

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

C

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

Fortran

real(4) function code(alpha, u0)
    real(4), intent (in) :: alpha
    real(4), intent (in) :: u0
    code = u0 * ((alpha * alpha) + ((u0 * (alpha * alpha)) * ((0.5e0 + (u0 * 0.3333333333333333e0)) + (u0 * (u0 * 0.25e0)))))
end function

Julia

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

MATLAB

function tmp = code(alpha, u0)
	tmp = u0 * ((alpha * alpha) + ((u0 * (alpha * alpha)) * ((single(0.5) + (u0 * single(0.3333333333333333))) + (u0 * (u0 * single(0.25))))));
end

Derivation

1. Initial program 57.4%

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

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

      \[\leadsto \mathsf{*.f32}\left(\left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right), \color{blue}{\log \left(1 - u0\right)}\right) \]

   2. *-commutative N/A

      \[\leadsto \mathsf{*.f32}\left(\left(\alpha \cdot \left(\mathsf{neg}\left(\alpha\right)\right)\right), \log \color{blue}{\left(1 - u0\right)}\right) \]

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

      \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(\alpha, \left(\mathsf{neg}\left(\alpha\right)\right)\right), \log \color{blue}{\left(1 - u0\right)}\right) \]

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

      \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(\alpha, \mathsf{neg.f32}\left(\alpha\right)\right), \log \left(1 - \color{blue}{u0}\right)\right) \]

   5. sub-neg N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(\alpha, \mathsf{neg.f32}\left(\alpha\right)\right), \log \left(1 + \left(\mathsf{neg}\left(u0\right)\right)\right)\right) \]

   6. log1p-define N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(\alpha, \mathsf{neg.f32}\left(\alpha\right)\right), \left(\mathsf{log1p}\left(\mathsf{neg}\left(u0\right)\right)\right)\right) \]

   7. log1p-lowering-log1p.f32 N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(\alpha, \mathsf{neg.f32}\left(\alpha\right)\right), \mathsf{log1p.f32}\left(\left(\mathsf{neg}\left(u0\right)\right)\right)\right) \]

   8. neg-lowering-neg.f32 99.1%

      \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(\alpha, \mathsf{neg.f32}\left(\alpha\right)\right), \mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right)\right) \]

3. Simplified 99.1%

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

5. Taylor expanded in u0 around 0

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

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

      \[\leadsto \mathsf{*.f32}\left(u0, \color{blue}{\left(u0 \cdot \left(\frac{1}{2} \cdot {\alpha}^{2} + u0 \cdot \left(\frac{1}{4} \cdot \left({\alpha}^{2} \cdot u0\right) + \frac{1}{3} \cdot {\alpha}^{2}\right)\right) + {\alpha}^{2}\right)}\right) \]

   2. +-commutative N/A

      \[\leadsto \mathsf{*.f32}\left(u0, \left({\alpha}^{2} + \color{blue}{u0 \cdot \left(\frac{1}{2} \cdot {\alpha}^{2} + u0 \cdot \left(\frac{1}{4} \cdot \left({\alpha}^{2} \cdot u0\right) + \frac{1}{3} \cdot {\alpha}^{2}\right)\right)}\right)\right) \]

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

      \[\leadsto \mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\left({\alpha}^{2}\right), \color{blue}{\left(u0 \cdot \left(\frac{1}{2} \cdot {\alpha}^{2} + u0 \cdot \left(\frac{1}{4} \cdot \left({\alpha}^{2} \cdot u0\right) + \frac{1}{3} \cdot {\alpha}^{2}\right)\right)\right)}\right)\right) \]

   4. unpow2 N/A

      \[\leadsto \mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\left(\alpha \cdot \alpha\right), \left(\color{blue}{u0} \cdot \left(\frac{1}{2} \cdot {\alpha}^{2} + u0 \cdot \left(\frac{1}{4} \cdot \left({\alpha}^{2} \cdot u0\right) + \frac{1}{3} \cdot {\alpha}^{2}\right)\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{*.f32}\left(\alpha, \alpha\right), \left(\color{blue}{u0} \cdot \left(\frac{1}{2} \cdot {\alpha}^{2} + u0 \cdot \left(\frac{1}{4} \cdot \left({\alpha}^{2} \cdot u0\right) + \frac{1}{3} \cdot {\alpha}^{2}\right)\right)\right)\right)\right) \]

   6. +-commutative N/A

      \[\leadsto \mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{*.f32}\left(\alpha, \alpha\right), \left(u0 \cdot \left(u0 \cdot \left(\frac{1}{4} \cdot \left({\alpha}^{2} \cdot u0\right) + \frac{1}{3} \cdot {\alpha}^{2}\right) + \color{blue}{\frac{1}{2} \cdot {\alpha}^{2}}\right)\right)\right)\right) \]

   7. distribute-rgt-in N/A

      \[\leadsto \mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{*.f32}\left(\alpha, \alpha\right), \left(u0 \cdot \left(\left(\left(\frac{1}{4} \cdot \left({\alpha}^{2} \cdot u0\right)\right) \cdot u0 + \left(\frac{1}{3} \cdot {\alpha}^{2}\right) \cdot u0\right) + \color{blue}{\frac{1}{2}} \cdot {\alpha}^{2}\right)\right)\right)\right) \]

   8. associate-*r* N/A

      \[\leadsto \mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{*.f32}\left(\alpha, \alpha\right), \left(u0 \cdot \left(\left(\left(\frac{1}{4} \cdot \left({\alpha}^{2} \cdot u0\right)\right) \cdot u0 + \frac{1}{3} \cdot \left({\alpha}^{2} \cdot u0\right)\right) + \frac{1}{2} \cdot {\alpha}^{2}\right)\right)\right)\right) \]

   9. associate-+l+ N/A

      \[\leadsto \mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{*.f32}\left(\alpha, \alpha\right), \left(u0 \cdot \left(\left(\frac{1}{4} \cdot \left({\alpha}^{2} \cdot u0\right)\right) \cdot u0 + \color{blue}{\left(\frac{1}{3} \cdot \left({\alpha}^{2} \cdot u0\right) + \frac{1}{2} \cdot {\alpha}^{2}\right)}\right)\right)\right)\right) \]

   10. *-commutative N/A

       \[\leadsto \mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{*.f32}\left(\alpha, \alpha\right), \left(u0 \cdot \left(u0 \cdot \left(\frac{1}{4} \cdot \left({\alpha}^{2} \cdot u0\right)\right) + \left(\color{blue}{\frac{1}{3} \cdot \left({\alpha}^{2} \cdot u0\right)} + \frac{1}{2} \cdot {\alpha}^{2}\right)\right)\right)\right)\right) \]

7. Simplified 92.4%

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

8. Final simplification 92.4%

   \[\leadsto u0 \cdot \left(\alpha \cdot \alpha + \left(u0 \cdot \left(\alpha \cdot \alpha\right)\right) \cdot \left(\left(0.5 + u0 \cdot 0.3333333333333333\right) + u0 \cdot \left(u0 \cdot 0.25\right)\right)\right) \]
9. Add Preprocessing

Alternative 4: 93.7% accurate, 5.1× speedup

Math

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

FPCore

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

C

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

Fortran

real(4) function code(alpha, u0)
    real(4), intent (in) :: alpha
    real(4), intent (in) :: u0
    code = u0 * ((alpha * alpha) + ((alpha * (alpha * u0)) * (0.5e0 + (u0 * (0.3333333333333333e0 + (u0 * 0.25e0))))))
end function

Julia

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

MATLAB

function tmp = code(alpha, u0)
	tmp = u0 * ((alpha * alpha) + ((alpha * (alpha * u0)) * (single(0.5) + (u0 * (single(0.3333333333333333) + (u0 * single(0.25)))))));
end

Derivation

1. Initial program 57.4%

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

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

      \[\leadsto \mathsf{*.f32}\left(\left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right), \color{blue}{\log \left(1 - u0\right)}\right) \]

   2. *-commutative N/A

      \[\leadsto \mathsf{*.f32}\left(\left(\alpha \cdot \left(\mathsf{neg}\left(\alpha\right)\right)\right), \log \color{blue}{\left(1 - u0\right)}\right) \]

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

      \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(\alpha, \left(\mathsf{neg}\left(\alpha\right)\right)\right), \log \color{blue}{\left(1 - u0\right)}\right) \]

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

      \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(\alpha, \mathsf{neg.f32}\left(\alpha\right)\right), \log \left(1 - \color{blue}{u0}\right)\right) \]

   5. sub-neg N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(\alpha, \mathsf{neg.f32}\left(\alpha\right)\right), \log \left(1 + \left(\mathsf{neg}\left(u0\right)\right)\right)\right) \]

   6. log1p-define N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(\alpha, \mathsf{neg.f32}\left(\alpha\right)\right), \left(\mathsf{log1p}\left(\mathsf{neg}\left(u0\right)\right)\right)\right) \]

   7. log1p-lowering-log1p.f32 N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(\alpha, \mathsf{neg.f32}\left(\alpha\right)\right), \mathsf{log1p.f32}\left(\left(\mathsf{neg}\left(u0\right)\right)\right)\right) \]

   8. neg-lowering-neg.f32 99.1%

      \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(\alpha, \mathsf{neg.f32}\left(\alpha\right)\right), \mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right)\right) \]

3. Simplified 99.1%

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

5. Taylor expanded in u0 around 0

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

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

      \[\leadsto \mathsf{*.f32}\left(u0, \color{blue}{\left(u0 \cdot \left(\frac{1}{2} \cdot {\alpha}^{2} + u0 \cdot \left(\frac{1}{4} \cdot \left({\alpha}^{2} \cdot u0\right) + \frac{1}{3} \cdot {\alpha}^{2}\right)\right) + {\alpha}^{2}\right)}\right) \]

   2. +-commutative N/A

      \[\leadsto \mathsf{*.f32}\left(u0, \left({\alpha}^{2} + \color{blue}{u0 \cdot \left(\frac{1}{2} \cdot {\alpha}^{2} + u0 \cdot \left(\frac{1}{4} \cdot \left({\alpha}^{2} \cdot u0\right) + \frac{1}{3} \cdot {\alpha}^{2}\right)\right)}\right)\right) \]

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

      \[\leadsto \mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\left({\alpha}^{2}\right), \color{blue}{\left(u0 \cdot \left(\frac{1}{2} \cdot {\alpha}^{2} + u0 \cdot \left(\frac{1}{4} \cdot \left({\alpha}^{2} \cdot u0\right) + \frac{1}{3} \cdot {\alpha}^{2}\right)\right)\right)}\right)\right) \]

   4. unpow2 N/A

      \[\leadsto \mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\left(\alpha \cdot \alpha\right), \left(\color{blue}{u0} \cdot \left(\frac{1}{2} \cdot {\alpha}^{2} + u0 \cdot \left(\frac{1}{4} \cdot \left({\alpha}^{2} \cdot u0\right) + \frac{1}{3} \cdot {\alpha}^{2}\right)\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{*.f32}\left(\alpha, \alpha\right), \left(\color{blue}{u0} \cdot \left(\frac{1}{2} \cdot {\alpha}^{2} + u0 \cdot \left(\frac{1}{4} \cdot \left({\alpha}^{2} \cdot u0\right) + \frac{1}{3} \cdot {\alpha}^{2}\right)\right)\right)\right)\right) \]

   6. +-commutative N/A

      \[\leadsto \mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{*.f32}\left(\alpha, \alpha\right), \left(u0 \cdot \left(u0 \cdot \left(\frac{1}{4} \cdot \left({\alpha}^{2} \cdot u0\right) + \frac{1}{3} \cdot {\alpha}^{2}\right) + \color{blue}{\frac{1}{2} \cdot {\alpha}^{2}}\right)\right)\right)\right) \]

   7. distribute-rgt-in N/A

      \[\leadsto \mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{*.f32}\left(\alpha, \alpha\right), \left(u0 \cdot \left(\left(\left(\frac{1}{4} \cdot \left({\alpha}^{2} \cdot u0\right)\right) \cdot u0 + \left(\frac{1}{3} \cdot {\alpha}^{2}\right) \cdot u0\right) + \color{blue}{\frac{1}{2}} \cdot {\alpha}^{2}\right)\right)\right)\right) \]

   8. associate-*r* N/A

      \[\leadsto \mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{*.f32}\left(\alpha, \alpha\right), \left(u0 \cdot \left(\left(\left(\frac{1}{4} \cdot \left({\alpha}^{2} \cdot u0\right)\right) \cdot u0 + \frac{1}{3} \cdot \left({\alpha}^{2} \cdot u0\right)\right) + \frac{1}{2} \cdot {\alpha}^{2}\right)\right)\right)\right) \]

   9. associate-+l+ N/A

      \[\leadsto \mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{*.f32}\left(\alpha, \alpha\right), \left(u0 \cdot \left(\left(\frac{1}{4} \cdot \left({\alpha}^{2} \cdot u0\right)\right) \cdot u0 + \color{blue}{\left(\frac{1}{3} \cdot \left({\alpha}^{2} \cdot u0\right) + \frac{1}{2} \cdot {\alpha}^{2}\right)}\right)\right)\right)\right) \]

   10. *-commutative N/A

       \[\leadsto \mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{*.f32}\left(\alpha, \alpha\right), \left(u0 \cdot \left(u0 \cdot \left(\frac{1}{4} \cdot \left({\alpha}^{2} \cdot u0\right)\right) + \left(\color{blue}{\frac{1}{3} \cdot \left({\alpha}^{2} \cdot u0\right)} + \frac{1}{2} \cdot {\alpha}^{2}\right)\right)\right)\right)\right) \]

7. Simplified 92.4%

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

   1. +-commutative N/A

      \[\leadsto \mathsf{*.f32}\left(u0, \left(\left(\left(\alpha \cdot \alpha\right) \cdot u0\right) \cdot \left(u0 \cdot \left(u0 \cdot \frac{1}{4}\right) + \left(u0 \cdot \frac{1}{3} + \frac{1}{2}\right)\right) + \color{blue}{\alpha \cdot \alpha}\right)\right) \]

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

      \[\leadsto \mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\left(\left(\left(\alpha \cdot \alpha\right) \cdot u0\right) \cdot \left(u0 \cdot \left(u0 \cdot \frac{1}{4}\right) + \left(u0 \cdot \frac{1}{3} + \frac{1}{2}\right)\right)\right), \color{blue}{\left(\alpha \cdot \alpha\right)}\right)\right) \]

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

      \[\leadsto \mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{*.f32}\left(\left(\left(\alpha \cdot \alpha\right) \cdot u0\right), \left(u0 \cdot \left(u0 \cdot \frac{1}{4}\right) + \left(u0 \cdot \frac{1}{3} + \frac{1}{2}\right)\right)\right), \left(\color{blue}{\alpha} \cdot \alpha\right)\right)\right) \]

   4. associate-*l* N/A

      \[\leadsto \mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{*.f32}\left(\left(\alpha \cdot \left(\alpha \cdot u0\right)\right), \left(u0 \cdot \left(u0 \cdot \frac{1}{4}\right) + \left(u0 \cdot \frac{1}{3} + \frac{1}{2}\right)\right)\right), \left(\alpha \cdot \alpha\right)\right)\right) \]

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

      \[\leadsto \mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(\alpha, \left(\alpha \cdot u0\right)\right), \left(u0 \cdot \left(u0 \cdot \frac{1}{4}\right) + \left(u0 \cdot \frac{1}{3} + \frac{1}{2}\right)\right)\right), \left(\alpha \cdot \alpha\right)\right)\right) \]

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

      \[\leadsto \mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(\alpha, \mathsf{*.f32}\left(\alpha, u0\right)\right), \left(u0 \cdot \left(u0 \cdot \frac{1}{4}\right) + \left(u0 \cdot \frac{1}{3} + \frac{1}{2}\right)\right)\right), \left(\alpha \cdot \alpha\right)\right)\right) \]

   7. associate-+r+ N/A

      \[\leadsto \mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(\alpha, \mathsf{*.f32}\left(\alpha, u0\right)\right), \left(\left(u0 \cdot \left(u0 \cdot \frac{1}{4}\right) + u0 \cdot \frac{1}{3}\right) + \frac{1}{2}\right)\right), \left(\alpha \cdot \alpha\right)\right)\right) \]

   8. +-commutative N/A

      \[\leadsto \mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(\alpha, \mathsf{*.f32}\left(\alpha, u0\right)\right), \left(\frac{1}{2} + \left(u0 \cdot \left(u0 \cdot \frac{1}{4}\right) + u0 \cdot \frac{1}{3}\right)\right)\right), \left(\alpha \cdot \alpha\right)\right)\right) \]

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

      \[\leadsto \mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(\alpha, \mathsf{*.f32}\left(\alpha, u0\right)\right), \mathsf{+.f32}\left(\frac{1}{2}, \left(u0 \cdot \left(u0 \cdot \frac{1}{4}\right) + u0 \cdot \frac{1}{3}\right)\right)\right), \left(\alpha \cdot \alpha\right)\right)\right) \]

   10. distribute-lft-out N/A

       \[\leadsto \mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(\alpha, \mathsf{*.f32}\left(\alpha, u0\right)\right), \mathsf{+.f32}\left(\frac{1}{2}, \left(u0 \cdot \left(u0 \cdot \frac{1}{4} + \frac{1}{3}\right)\right)\right)\right), \left(\alpha \cdot \alpha\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(\alpha, \mathsf{*.f32}\left(\alpha, u0\right)\right), \mathsf{+.f32}\left(\frac{1}{2}, \mathsf{*.f32}\left(u0, \left(u0 \cdot \frac{1}{4} + \frac{1}{3}\right)\right)\right)\right), \left(\alpha \cdot \alpha\right)\right)\right) \]

   12. +-commutative N/A

       \[\leadsto \mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(\alpha, \mathsf{*.f32}\left(\alpha, u0\right)\right), \mathsf{+.f32}\left(\frac{1}{2}, \mathsf{*.f32}\left(u0, \left(\frac{1}{3} + u0 \cdot \frac{1}{4}\right)\right)\right)\right), \left(\alpha \cdot \alpha\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(\alpha, \mathsf{*.f32}\left(\alpha, u0\right)\right), \mathsf{+.f32}\left(\frac{1}{2}, \mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\frac{1}{3}, \left(u0 \cdot \frac{1}{4}\right)\right)\right)\right)\right), \left(\alpha \cdot \alpha\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(\alpha, \mathsf{*.f32}\left(\alpha, u0\right)\right), \mathsf{+.f32}\left(\frac{1}{2}, \mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\frac{1}{3}, \mathsf{*.f32}\left(u0, \frac{1}{4}\right)\right)\right)\right)\right), \left(\alpha \cdot \alpha\right)\right)\right) \]

   15. *-lowering-*.f32 92.4%

       \[\leadsto \mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(\alpha, \mathsf{*.f32}\left(\alpha, u0\right)\right), \mathsf{+.f32}\left(\frac{1}{2}, \mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\frac{1}{3}, \mathsf{*.f32}\left(u0, \frac{1}{4}\right)\right)\right)\right)\right), \mathsf{*.f32}\left(\alpha, \color{blue}{\alpha}\right)\right)\right) \]

9. Applied egg-rr 92.4%

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

10. Final simplification 92.4%

    \[\leadsto u0 \cdot \left(\alpha \cdot \alpha + \left(\alpha \cdot \left(\alpha \cdot u0\right)\right) \cdot \left(0.5 + u0 \cdot \left(0.3333333333333333 + u0 \cdot 0.25\right)\right)\right) \]
11. Add Preprocessing

Alternative 5: 93.4% accurate, 5.7× speedup

Math

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

FPCore

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

C

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

Fortran

real(4) function code(alpha, u0)
    real(4), intent (in) :: alpha
    real(4), intent (in) :: u0
    code = (alpha * alpha) * (u0 * ((u0 * (0.5e0 + (u0 * (0.3333333333333333e0 + (u0 * 0.25e0))))) + 1.0e0))
end function

Julia

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

MATLAB

function tmp = code(alpha, u0)
	tmp = (alpha * alpha) * (u0 * ((u0 * (single(0.5) + (u0 * (single(0.3333333333333333) + (u0 * single(0.25)))))) + single(1.0)));
end

Derivation

1. Initial program 57.4%

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

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

      \[\leadsto \mathsf{*.f32}\left(\left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right), \color{blue}{\log \left(1 - u0\right)}\right) \]

   2. *-commutative N/A

      \[\leadsto \mathsf{*.f32}\left(\left(\alpha \cdot \left(\mathsf{neg}\left(\alpha\right)\right)\right), \log \color{blue}{\left(1 - u0\right)}\right) \]

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

      \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(\alpha, \left(\mathsf{neg}\left(\alpha\right)\right)\right), \log \color{blue}{\left(1 - u0\right)}\right) \]

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

      \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(\alpha, \mathsf{neg.f32}\left(\alpha\right)\right), \log \left(1 - \color{blue}{u0}\right)\right) \]

   5. sub-neg N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(\alpha, \mathsf{neg.f32}\left(\alpha\right)\right), \log \left(1 + \left(\mathsf{neg}\left(u0\right)\right)\right)\right) \]

   6. log1p-define N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(\alpha, \mathsf{neg.f32}\left(\alpha\right)\right), \left(\mathsf{log1p}\left(\mathsf{neg}\left(u0\right)\right)\right)\right) \]

   7. log1p-lowering-log1p.f32 N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(\alpha, \mathsf{neg.f32}\left(\alpha\right)\right), \mathsf{log1p.f32}\left(\left(\mathsf{neg}\left(u0\right)\right)\right)\right) \]

   8. neg-lowering-neg.f32 99.1%

      \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(\alpha, \mathsf{neg.f32}\left(\alpha\right)\right), \mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right)\right) \]

3. Simplified 99.1%

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

5. Taylor expanded in u0 around 0

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

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

      \[\leadsto \mathsf{*.f32}\left(u0, \color{blue}{\left(u0 \cdot \left(\frac{1}{2} \cdot {\alpha}^{2} + u0 \cdot \left(\frac{1}{4} \cdot \left({\alpha}^{2} \cdot u0\right) + \frac{1}{3} \cdot {\alpha}^{2}\right)\right) + {\alpha}^{2}\right)}\right) \]

   2. +-commutative N/A

      \[\leadsto \mathsf{*.f32}\left(u0, \left({\alpha}^{2} + \color{blue}{u0 \cdot \left(\frac{1}{2} \cdot {\alpha}^{2} + u0 \cdot \left(\frac{1}{4} \cdot \left({\alpha}^{2} \cdot u0\right) + \frac{1}{3} \cdot {\alpha}^{2}\right)\right)}\right)\right) \]

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

      \[\leadsto \mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\left({\alpha}^{2}\right), \color{blue}{\left(u0 \cdot \left(\frac{1}{2} \cdot {\alpha}^{2} + u0 \cdot \left(\frac{1}{4} \cdot \left({\alpha}^{2} \cdot u0\right) + \frac{1}{3} \cdot {\alpha}^{2}\right)\right)\right)}\right)\right) \]

   4. unpow2 N/A

      \[\leadsto \mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\left(\alpha \cdot \alpha\right), \left(\color{blue}{u0} \cdot \left(\frac{1}{2} \cdot {\alpha}^{2} + u0 \cdot \left(\frac{1}{4} \cdot \left({\alpha}^{2} \cdot u0\right) + \frac{1}{3} \cdot {\alpha}^{2}\right)\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{*.f32}\left(\alpha, \alpha\right), \left(\color{blue}{u0} \cdot \left(\frac{1}{2} \cdot {\alpha}^{2} + u0 \cdot \left(\frac{1}{4} \cdot \left({\alpha}^{2} \cdot u0\right) + \frac{1}{3} \cdot {\alpha}^{2}\right)\right)\right)\right)\right) \]

   6. +-commutative N/A

      \[\leadsto \mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{*.f32}\left(\alpha, \alpha\right), \left(u0 \cdot \left(u0 \cdot \left(\frac{1}{4} \cdot \left({\alpha}^{2} \cdot u0\right) + \frac{1}{3} \cdot {\alpha}^{2}\right) + \color{blue}{\frac{1}{2} \cdot {\alpha}^{2}}\right)\right)\right)\right) \]

   7. distribute-rgt-in N/A

      \[\leadsto \mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{*.f32}\left(\alpha, \alpha\right), \left(u0 \cdot \left(\left(\left(\frac{1}{4} \cdot \left({\alpha}^{2} \cdot u0\right)\right) \cdot u0 + \left(\frac{1}{3} \cdot {\alpha}^{2}\right) \cdot u0\right) + \color{blue}{\frac{1}{2}} \cdot {\alpha}^{2}\right)\right)\right)\right) \]

   8. associate-*r* N/A

      \[\leadsto \mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{*.f32}\left(\alpha, \alpha\right), \left(u0 \cdot \left(\left(\left(\frac{1}{4} \cdot \left({\alpha}^{2} \cdot u0\right)\right) \cdot u0 + \frac{1}{3} \cdot \left({\alpha}^{2} \cdot u0\right)\right) + \frac{1}{2} \cdot {\alpha}^{2}\right)\right)\right)\right) \]

   9. associate-+l+ N/A

      \[\leadsto \mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{*.f32}\left(\alpha, \alpha\right), \left(u0 \cdot \left(\left(\frac{1}{4} \cdot \left({\alpha}^{2} \cdot u0\right)\right) \cdot u0 + \color{blue}{\left(\frac{1}{3} \cdot \left({\alpha}^{2} \cdot u0\right) + \frac{1}{2} \cdot {\alpha}^{2}\right)}\right)\right)\right)\right) \]

   10. *-commutative N/A

       \[\leadsto \mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{*.f32}\left(\alpha, \alpha\right), \left(u0 \cdot \left(u0 \cdot \left(\frac{1}{4} \cdot \left({\alpha}^{2} \cdot u0\right)\right) + \left(\color{blue}{\frac{1}{3} \cdot \left({\alpha}^{2} \cdot u0\right)} + \frac{1}{2} \cdot {\alpha}^{2}\right)\right)\right)\right)\right) \]

7. Simplified 92.4%

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

   1. distribute-rgt-in N/A

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

   2. associate-*l* N/A

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

   3. associate-*l* N/A

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

   4. distribute-lft-out N/A

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

   5. *-commutative N/A

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

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

      \[\leadsto \mathsf{*.f32}\left(\left(u0 + \left(u0 \cdot \left(u0 \cdot \left(u0 \cdot \frac{1}{4}\right) + \left(u0 \cdot \frac{1}{3} + \frac{1}{2}\right)\right)\right) \cdot u0\right), \color{blue}{\left(\alpha \cdot \alpha\right)}\right) \]

9. Applied egg-rr 92.4%

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

10. Final simplification 92.4%

    \[\leadsto \left(\alpha \cdot \alpha\right) \cdot \left(u0 \cdot \left(u0 \cdot \left(0.5 + u0 \cdot \left(0.3333333333333333 + u0 \cdot 0.25\right)\right) + 1\right)\right) \]
11. Add Preprocessing

Alternative 6: 93.4% accurate, 5.7× speedup

Math

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

FPCore

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

C

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

Fortran

real(4) function code(alpha, u0)
    real(4), intent (in) :: alpha
    real(4), intent (in) :: u0
    code = u0 * (alpha * (alpha * ((u0 * (0.5e0 + (u0 * (0.3333333333333333e0 + (u0 * 0.25e0))))) + 1.0e0)))
end function

Julia

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

MATLAB

function tmp = code(alpha, u0)
	tmp = u0 * (alpha * (alpha * ((u0 * (single(0.5) + (u0 * (single(0.3333333333333333) + (u0 * single(0.25)))))) + single(1.0))));
end

Derivation

1. Initial program 57.4%

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

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

      \[\leadsto \mathsf{*.f32}\left(\left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right), \color{blue}{\log \left(1 - u0\right)}\right) \]

   2. *-commutative N/A

      \[\leadsto \mathsf{*.f32}\left(\left(\alpha \cdot \left(\mathsf{neg}\left(\alpha\right)\right)\right), \log \color{blue}{\left(1 - u0\right)}\right) \]

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

      \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(\alpha, \left(\mathsf{neg}\left(\alpha\right)\right)\right), \log \color{blue}{\left(1 - u0\right)}\right) \]

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

      \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(\alpha, \mathsf{neg.f32}\left(\alpha\right)\right), \log \left(1 - \color{blue}{u0}\right)\right) \]

   5. sub-neg N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(\alpha, \mathsf{neg.f32}\left(\alpha\right)\right), \log \left(1 + \left(\mathsf{neg}\left(u0\right)\right)\right)\right) \]

   6. log1p-define N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(\alpha, \mathsf{neg.f32}\left(\alpha\right)\right), \left(\mathsf{log1p}\left(\mathsf{neg}\left(u0\right)\right)\right)\right) \]

   7. log1p-lowering-log1p.f32 N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(\alpha, \mathsf{neg.f32}\left(\alpha\right)\right), \mathsf{log1p.f32}\left(\left(\mathsf{neg}\left(u0\right)\right)\right)\right) \]

   8. neg-lowering-neg.f32 99.1%

      \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(\alpha, \mathsf{neg.f32}\left(\alpha\right)\right), \mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right)\right) \]

3. Simplified 99.1%

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

5. Taylor expanded in u0 around 0

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

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

      \[\leadsto \mathsf{*.f32}\left(u0, \color{blue}{\left(u0 \cdot \left(\frac{1}{2} \cdot {\alpha}^{2} + u0 \cdot \left(\frac{1}{4} \cdot \left({\alpha}^{2} \cdot u0\right) + \frac{1}{3} \cdot {\alpha}^{2}\right)\right) + {\alpha}^{2}\right)}\right) \]

   2. +-commutative N/A

      \[\leadsto \mathsf{*.f32}\left(u0, \left({\alpha}^{2} + \color{blue}{u0 \cdot \left(\frac{1}{2} \cdot {\alpha}^{2} + u0 \cdot \left(\frac{1}{4} \cdot \left({\alpha}^{2} \cdot u0\right) + \frac{1}{3} \cdot {\alpha}^{2}\right)\right)}\right)\right) \]

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

      \[\leadsto \mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\left({\alpha}^{2}\right), \color{blue}{\left(u0 \cdot \left(\frac{1}{2} \cdot {\alpha}^{2} + u0 \cdot \left(\frac{1}{4} \cdot \left({\alpha}^{2} \cdot u0\right) + \frac{1}{3} \cdot {\alpha}^{2}\right)\right)\right)}\right)\right) \]

   4. unpow2 N/A

      \[\leadsto \mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\left(\alpha \cdot \alpha\right), \left(\color{blue}{u0} \cdot \left(\frac{1}{2} \cdot {\alpha}^{2} + u0 \cdot \left(\frac{1}{4} \cdot \left({\alpha}^{2} \cdot u0\right) + \frac{1}{3} \cdot {\alpha}^{2}\right)\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{*.f32}\left(\alpha, \alpha\right), \left(\color{blue}{u0} \cdot \left(\frac{1}{2} \cdot {\alpha}^{2} + u0 \cdot \left(\frac{1}{4} \cdot \left({\alpha}^{2} \cdot u0\right) + \frac{1}{3} \cdot {\alpha}^{2}\right)\right)\right)\right)\right) \]

   6. +-commutative N/A

      \[\leadsto \mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{*.f32}\left(\alpha, \alpha\right), \left(u0 \cdot \left(u0 \cdot \left(\frac{1}{4} \cdot \left({\alpha}^{2} \cdot u0\right) + \frac{1}{3} \cdot {\alpha}^{2}\right) + \color{blue}{\frac{1}{2} \cdot {\alpha}^{2}}\right)\right)\right)\right) \]

   7. distribute-rgt-in N/A

      \[\leadsto \mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{*.f32}\left(\alpha, \alpha\right), \left(u0 \cdot \left(\left(\left(\frac{1}{4} \cdot \left({\alpha}^{2} \cdot u0\right)\right) \cdot u0 + \left(\frac{1}{3} \cdot {\alpha}^{2}\right) \cdot u0\right) + \color{blue}{\frac{1}{2}} \cdot {\alpha}^{2}\right)\right)\right)\right) \]

   8. associate-*r* N/A

      \[\leadsto \mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{*.f32}\left(\alpha, \alpha\right), \left(u0 \cdot \left(\left(\left(\frac{1}{4} \cdot \left({\alpha}^{2} \cdot u0\right)\right) \cdot u0 + \frac{1}{3} \cdot \left({\alpha}^{2} \cdot u0\right)\right) + \frac{1}{2} \cdot {\alpha}^{2}\right)\right)\right)\right) \]

   9. associate-+l+ N/A

      \[\leadsto \mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{*.f32}\left(\alpha, \alpha\right), \left(u0 \cdot \left(\left(\frac{1}{4} \cdot \left({\alpha}^{2} \cdot u0\right)\right) \cdot u0 + \color{blue}{\left(\frac{1}{3} \cdot \left({\alpha}^{2} \cdot u0\right) + \frac{1}{2} \cdot {\alpha}^{2}\right)}\right)\right)\right)\right) \]

   10. *-commutative N/A

       \[\leadsto \mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\mathsf{*.f32}\left(\alpha, \alpha\right), \left(u0 \cdot \left(u0 \cdot \left(\frac{1}{4} \cdot \left({\alpha}^{2} \cdot u0\right)\right) + \left(\color{blue}{\frac{1}{3} \cdot \left({\alpha}^{2} \cdot u0\right)} + \frac{1}{2} \cdot {\alpha}^{2}\right)\right)\right)\right)\right) \]

7. Simplified 92.4%

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

   1. associate-*l* N/A

      \[\leadsto \mathsf{*.f32}\left(u0, \left(\alpha \cdot \alpha + \left(\alpha \cdot \alpha\right) \cdot \color{blue}{\left(u0 \cdot \left(u0 \cdot \left(u0 \cdot \frac{1}{4}\right) + \left(u0 \cdot \frac{1}{3} + \frac{1}{2}\right)\right)\right)}\right)\right) \]

   2. associate-*l* N/A

      \[\leadsto \mathsf{*.f32}\left(u0, \left(\alpha \cdot \alpha + \alpha \cdot \color{blue}{\left(\alpha \cdot \left(u0 \cdot \left(u0 \cdot \left(u0 \cdot \frac{1}{4}\right) + \left(u0 \cdot \frac{1}{3} + \frac{1}{2}\right)\right)\right)\right)}\right)\right) \]

   3. distribute-lft-out N/A

      \[\leadsto \mathsf{*.f32}\left(u0, \left(\alpha \cdot \color{blue}{\left(\alpha + \alpha \cdot \left(u0 \cdot \left(u0 \cdot \left(u0 \cdot \frac{1}{4}\right) + \left(u0 \cdot \frac{1}{3} + \frac{1}{2}\right)\right)\right)\right)}\right)\right) \]

   4. *-commutative N/A

      \[\leadsto \mathsf{*.f32}\left(u0, \left(\left(\alpha + \alpha \cdot \left(u0 \cdot \left(u0 \cdot \left(u0 \cdot \frac{1}{4}\right) + \left(u0 \cdot \frac{1}{3} + \frac{1}{2}\right)\right)\right)\right) \cdot \color{blue}{\alpha}\right)\right) \]

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

      \[\leadsto \mathsf{*.f32}\left(u0, \mathsf{*.f32}\left(\left(\alpha + \alpha \cdot \left(u0 \cdot \left(u0 \cdot \left(u0 \cdot \frac{1}{4}\right) + \left(u0 \cdot \frac{1}{3} + \frac{1}{2}\right)\right)\right)\right), \color{blue}{\alpha}\right)\right) \]

9. Applied egg-rr 92.4%

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

10. Final simplification 92.4%

    \[\leadsto u0 \cdot \left(\alpha \cdot \left(\alpha \cdot \left(u0 \cdot \left(0.5 + u0 \cdot \left(0.3333333333333333 + u0 \cdot 0.25\right)\right) + 1\right)\right)\right) \]
11. Add Preprocessing

Alternative 7: 91.7% accurate, 7.2× speedup

Math

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

FPCore

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

C

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

Fortran

real(4) function code(alpha, u0)
    real(4), intent (in) :: alpha
    real(4), intent (in) :: u0
    code = u0 * (alpha * (alpha + (u0 * (alpha * (0.5e0 + (u0 * 0.3333333333333333e0))))))
end function

Julia

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

MATLAB

function tmp = code(alpha, u0)
	tmp = u0 * (alpha * (alpha + (u0 * (alpha * (single(0.5) + (u0 * single(0.3333333333333333)))))));
end

Derivation

1. Initial program 57.4%

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

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

      \[\leadsto \mathsf{*.f32}\left(\left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right), \color{blue}{\log \left(1 - u0\right)}\right) \]

   2. *-commutative N/A

      \[\leadsto \mathsf{*.f32}\left(\left(\alpha \cdot \left(\mathsf{neg}\left(\alpha\right)\right)\right), \log \color{blue}{\left(1 - u0\right)}\right) \]

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

      \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(\alpha, \left(\mathsf{neg}\left(\alpha\right)\right)\right), \log \color{blue}{\left(1 - u0\right)}\right) \]

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

      \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(\alpha, \mathsf{neg.f32}\left(\alpha\right)\right), \log \left(1 - \color{blue}{u0}\right)\right) \]

   5. sub-neg N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(\alpha, \mathsf{neg.f32}\left(\alpha\right)\right), \log \left(1 + \left(\mathsf{neg}\left(u0\right)\right)\right)\right) \]

   6. log1p-define N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(\alpha, \mathsf{neg.f32}\left(\alpha\right)\right), \left(\mathsf{log1p}\left(\mathsf{neg}\left(u0\right)\right)\right)\right) \]

   7. log1p-lowering-log1p.f32 N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(\alpha, \mathsf{neg.f32}\left(\alpha\right)\right), \mathsf{log1p.f32}\left(\left(\mathsf{neg}\left(u0\right)\right)\right)\right) \]

   8. neg-lowering-neg.f32 99.1%

      \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(\alpha, \mathsf{neg.f32}\left(\alpha\right)\right), \mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right)\right) \]

3. Simplified 99.1%

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

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. distribute-rgt-in N/A

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

   2. associate-*r* N/A

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

   3. associate-+l+ N/A

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

   4. *-commutative N/A

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

   5. distribute-lft-in N/A

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

   6. associate-*r* N/A

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

   7. associate-*r* N/A

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

   8. distribute-rgt-in N/A

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

   9. *-commutative N/A

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

   10. associate-*r* N/A

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

   11. distribute-lft1-in N/A

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

   12. distribute-rgt-out N/A

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

7. Simplified 90.1%

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

   1. associate-+r+ N/A

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

   2. distribute-rgt-in N/A

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

   3. *-commutative N/A

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

   4. associate-*l* N/A

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

   5. distribute-lft-in N/A

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

   6. associate-*r* N/A

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

   7. *-commutative N/A

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

   8. *-lft-identity N/A

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

   9. distribute-rgt-in N/A

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

   10. *-commutative N/A

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

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

       \[\leadsto \mathsf{*.f32}\left(\left(\left(u0 \cdot \frac{1}{3} + \frac{1}{2}\right) \cdot \left(\left(\alpha \cdot \alpha\right) \cdot u0\right) + \alpha \cdot \alpha\right), \color{blue}{u0}\right) \]

9. Applied egg-rr 90.2%

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

    1. +-commutative N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(\alpha, \left(\alpha \cdot \left(u0 \cdot \left(\frac{1}{2} + u0 \cdot \frac{1}{3}\right) + 1\right)\right)\right), u0\right) \]

    2. distribute-lft-in N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(\alpha, \left(\alpha \cdot \left(u0 \cdot \left(\frac{1}{2} + u0 \cdot \frac{1}{3}\right)\right) + \alpha \cdot 1\right)\right), u0\right) \]

    3. *-rgt-identity N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(\alpha, \left(\alpha \cdot \left(u0 \cdot \left(\frac{1}{2} + u0 \cdot \frac{1}{3}\right)\right) + \alpha\right)\right), u0\right) \]

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

       \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(\alpha, \mathsf{+.f32}\left(\left(\alpha \cdot \left(u0 \cdot \left(\frac{1}{2} + u0 \cdot \frac{1}{3}\right)\right)\right), \alpha\right)\right), u0\right) \]

    5. associate-*r* N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(\alpha, \mathsf{+.f32}\left(\left(\left(\alpha \cdot u0\right) \cdot \left(\frac{1}{2} + u0 \cdot \frac{1}{3}\right)\right), \alpha\right)\right), u0\right) \]

    6. *-commutative N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(\alpha, \mathsf{+.f32}\left(\left(\left(u0 \cdot \alpha\right) \cdot \left(\frac{1}{2} + u0 \cdot \frac{1}{3}\right)\right), \alpha\right)\right), u0\right) \]

    7. associate-*l* N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(\alpha, \mathsf{+.f32}\left(\left(u0 \cdot \left(\alpha \cdot \left(\frac{1}{2} + u0 \cdot \frac{1}{3}\right)\right)\right), \alpha\right)\right), u0\right) \]

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

       \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(\alpha, \mathsf{+.f32}\left(\mathsf{*.f32}\left(u0, \left(\alpha \cdot \left(\frac{1}{2} + u0 \cdot \frac{1}{3}\right)\right)\right), \alpha\right)\right), u0\right) \]

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

       \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(\alpha, \mathsf{+.f32}\left(\mathsf{*.f32}\left(u0, \mathsf{*.f32}\left(\alpha, \left(\frac{1}{2} + u0 \cdot \frac{1}{3}\right)\right)\right), \alpha\right)\right), u0\right) \]

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

        \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(\alpha, \mathsf{+.f32}\left(\mathsf{*.f32}\left(u0, \mathsf{*.f32}\left(\alpha, \mathsf{+.f32}\left(\frac{1}{2}, \left(u0 \cdot \frac{1}{3}\right)\right)\right)\right), \alpha\right)\right), u0\right) \]

    11. *-lowering-*.f32 90.2%

        \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(\alpha, \mathsf{+.f32}\left(\mathsf{*.f32}\left(u0, \mathsf{*.f32}\left(\alpha, \mathsf{+.f32}\left(\frac{1}{2}, \mathsf{*.f32}\left(u0, \frac{1}{3}\right)\right)\right)\right), \alpha\right)\right), u0\right) \]

11. Applied egg-rr 90.2%

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

12. Final simplification 90.2%

    \[\leadsto u0 \cdot \left(\alpha \cdot \left(\alpha + u0 \cdot \left(\alpha \cdot \left(0.5 + u0 \cdot 0.3333333333333333\right)\right)\right)\right) \]
13. Add Preprocessing

Alternative 8: 91.5% accurate, 7.2× speedup

Math

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

FPCore

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

C

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

Fortran

real(4) function code(alpha, u0)
    real(4), intent (in) :: alpha
    real(4), intent (in) :: u0
    code = u0 * (alpha * (alpha * ((u0 * (0.5e0 + (u0 * 0.3333333333333333e0))) + 1.0e0)))
end function

Julia

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

MATLAB

function tmp = code(alpha, u0)
	tmp = u0 * (alpha * (alpha * ((u0 * (single(0.5) + (u0 * single(0.3333333333333333)))) + single(1.0))));
end

Derivation

1. Initial program 57.4%

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

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

      \[\leadsto \mathsf{*.f32}\left(\left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right), \color{blue}{\log \left(1 - u0\right)}\right) \]

   2. *-commutative N/A

      \[\leadsto \mathsf{*.f32}\left(\left(\alpha \cdot \left(\mathsf{neg}\left(\alpha\right)\right)\right), \log \color{blue}{\left(1 - u0\right)}\right) \]

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

      \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(\alpha, \left(\mathsf{neg}\left(\alpha\right)\right)\right), \log \color{blue}{\left(1 - u0\right)}\right) \]

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

      \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(\alpha, \mathsf{neg.f32}\left(\alpha\right)\right), \log \left(1 - \color{blue}{u0}\right)\right) \]

   5. sub-neg N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(\alpha, \mathsf{neg.f32}\left(\alpha\right)\right), \log \left(1 + \left(\mathsf{neg}\left(u0\right)\right)\right)\right) \]

   6. log1p-define N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(\alpha, \mathsf{neg.f32}\left(\alpha\right)\right), \left(\mathsf{log1p}\left(\mathsf{neg}\left(u0\right)\right)\right)\right) \]

   7. log1p-lowering-log1p.f32 N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(\alpha, \mathsf{neg.f32}\left(\alpha\right)\right), \mathsf{log1p.f32}\left(\left(\mathsf{neg}\left(u0\right)\right)\right)\right) \]

   8. neg-lowering-neg.f32 99.1%

      \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(\alpha, \mathsf{neg.f32}\left(\alpha\right)\right), \mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right)\right) \]

3. Simplified 99.1%

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

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. distribute-rgt-in N/A

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

   2. associate-*r* N/A

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

   3. associate-+l+ N/A

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

   4. *-commutative N/A

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

   5. distribute-lft-in N/A

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

   6. associate-*r* N/A

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

   7. associate-*r* N/A

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

   8. distribute-rgt-in N/A

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

   9. *-commutative N/A

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

   10. associate-*r* N/A

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

   11. distribute-lft1-in N/A

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

   12. distribute-rgt-out N/A

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

7. Simplified 90.1%

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

   1. associate-+r+ N/A

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

   2. distribute-rgt-in N/A

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

   3. *-commutative N/A

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

   4. associate-*l* N/A

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

   5. distribute-lft-in N/A

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

   6. associate-*r* N/A

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

   7. *-commutative N/A

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

   8. *-lft-identity N/A

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

   9. distribute-rgt-in N/A

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

   10. *-commutative N/A

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

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

       \[\leadsto \mathsf{*.f32}\left(\left(\left(u0 \cdot \frac{1}{3} + \frac{1}{2}\right) \cdot \left(\left(\alpha \cdot \alpha\right) \cdot u0\right) + \alpha \cdot \alpha\right), \color{blue}{u0}\right) \]

9. Applied egg-rr 90.2%

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

10. Final simplification 90.2%

    \[\leadsto u0 \cdot \left(\alpha \cdot \left(\alpha \cdot \left(u0 \cdot \left(0.5 + u0 \cdot 0.3333333333333333\right) + 1\right)\right)\right) \]
11. Add Preprocessing

Alternative 9: 91.5% accurate, 7.2× speedup

Math

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

FPCore

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

C

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

Fortran

real(4) function code(alpha, u0)
    real(4), intent (in) :: alpha
    real(4), intent (in) :: u0
    code = alpha * (alpha * (u0 * ((u0 * (0.5e0 + (u0 * 0.3333333333333333e0))) + 1.0e0)))
end function

Julia

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

MATLAB

function tmp = code(alpha, u0)
	tmp = alpha * (alpha * (u0 * ((u0 * (single(0.5) + (u0 * single(0.3333333333333333)))) + single(1.0))));
end

Derivation

1. Initial program 57.4%

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

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

      \[\leadsto \mathsf{*.f32}\left(\left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right), \color{blue}{\log \left(1 - u0\right)}\right) \]

   2. *-commutative N/A

      \[\leadsto \mathsf{*.f32}\left(\left(\alpha \cdot \left(\mathsf{neg}\left(\alpha\right)\right)\right), \log \color{blue}{\left(1 - u0\right)}\right) \]

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

      \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(\alpha, \left(\mathsf{neg}\left(\alpha\right)\right)\right), \log \color{blue}{\left(1 - u0\right)}\right) \]

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

      \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(\alpha, \mathsf{neg.f32}\left(\alpha\right)\right), \log \left(1 - \color{blue}{u0}\right)\right) \]

   5. sub-neg N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(\alpha, \mathsf{neg.f32}\left(\alpha\right)\right), \log \left(1 + \left(\mathsf{neg}\left(u0\right)\right)\right)\right) \]

   6. log1p-define N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(\alpha, \mathsf{neg.f32}\left(\alpha\right)\right), \left(\mathsf{log1p}\left(\mathsf{neg}\left(u0\right)\right)\right)\right) \]

   7. log1p-lowering-log1p.f32 N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(\alpha, \mathsf{neg.f32}\left(\alpha\right)\right), \mathsf{log1p.f32}\left(\left(\mathsf{neg}\left(u0\right)\right)\right)\right) \]

   8. neg-lowering-neg.f32 99.1%

      \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(\alpha, \mathsf{neg.f32}\left(\alpha\right)\right), \mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right)\right) \]

3. Simplified 99.1%

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

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. distribute-rgt-in N/A

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

   2. associate-*r* N/A

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

   3. associate-+l+ N/A

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

   4. *-commutative N/A

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

   5. distribute-lft-in N/A

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

   6. associate-*r* N/A

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

   7. associate-*r* N/A

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

   8. distribute-rgt-in N/A

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

   9. *-commutative N/A

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

   10. associate-*r* N/A

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

   11. distribute-lft1-in N/A

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

   12. distribute-rgt-out N/A

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

7. Simplified 90.1%

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

8. Taylor expanded in alpha around 0

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

   1. unpow2 N/A

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

   2. associate-*l* N/A

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

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

      \[\leadsto \mathsf{*.f32}\left(\alpha, \color{blue}{\left(\alpha \cdot \left(u0 \cdot \left(1 + \left(\frac{1}{3} \cdot {u0}^{2} + \frac{1}{2} \cdot u0\right)\right)\right)\right)}\right) \]

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

      \[\leadsto \mathsf{*.f32}\left(\alpha, \mathsf{*.f32}\left(\alpha, \color{blue}{\left(u0 \cdot \left(1 + \left(\frac{1}{3} \cdot {u0}^{2} + \frac{1}{2} \cdot u0\right)\right)\right)}\right)\right) \]

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

      \[\leadsto \mathsf{*.f32}\left(\alpha, \mathsf{*.f32}\left(\alpha, \mathsf{*.f32}\left(u0, \color{blue}{\left(1 + \left(\frac{1}{3} \cdot {u0}^{2} + \frac{1}{2} \cdot u0\right)\right)}\right)\right)\right) \]

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

      \[\leadsto \mathsf{*.f32}\left(\alpha, \mathsf{*.f32}\left(\alpha, \mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(1, \color{blue}{\left(\frac{1}{3} \cdot {u0}^{2} + \frac{1}{2} \cdot u0\right)}\right)\right)\right)\right) \]

   7. +-commutative N/A

      \[\leadsto \mathsf{*.f32}\left(\alpha, \mathsf{*.f32}\left(\alpha, \mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(1, \left(\frac{1}{2} \cdot u0 + \color{blue}{\frac{1}{3} \cdot {u0}^{2}}\right)\right)\right)\right)\right) \]

   8. unpow2 N/A

      \[\leadsto \mathsf{*.f32}\left(\alpha, \mathsf{*.f32}\left(\alpha, \mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(1, \left(\frac{1}{2} \cdot u0 + \frac{1}{3} \cdot \left(u0 \cdot \color{blue}{u0}\right)\right)\right)\right)\right)\right) \]

   9. associate-*r* N/A

      \[\leadsto \mathsf{*.f32}\left(\alpha, \mathsf{*.f32}\left(\alpha, \mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(1, \left(\frac{1}{2} \cdot u0 + \left(\frac{1}{3} \cdot u0\right) \cdot \color{blue}{u0}\right)\right)\right)\right)\right) \]

   10. distribute-rgt-in N/A

       \[\leadsto \mathsf{*.f32}\left(\alpha, \mathsf{*.f32}\left(\alpha, \mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(1, \left(u0 \cdot \color{blue}{\left(\frac{1}{2} + \frac{1}{3} \cdot u0\right)}\right)\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f32}\left(\alpha, \mathsf{*.f32}\left(\alpha, \mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(u0, \color{blue}{\left(\frac{1}{2} + \frac{1}{3} \cdot u0\right)}\right)\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f32}\left(\alpha, \mathsf{*.f32}\left(\alpha, \mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\frac{1}{2}, \color{blue}{\left(\frac{1}{3} \cdot u0\right)}\right)\right)\right)\right)\right)\right) \]

   13. *-commutative N/A

       \[\leadsto \mathsf{*.f32}\left(\alpha, \mathsf{*.f32}\left(\alpha, \mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\frac{1}{2}, \left(u0 \cdot \color{blue}{\frac{1}{3}}\right)\right)\right)\right)\right)\right)\right) \]

   14. *-lowering-*.f32 90.0%

       \[\leadsto \mathsf{*.f32}\left(\alpha, \mathsf{*.f32}\left(\alpha, \mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(\frac{1}{2}, \mathsf{*.f32}\left(u0, \color{blue}{\frac{1}{3}}\right)\right)\right)\right)\right)\right)\right) \]

10. Simplified 90.0%

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

11. Final simplification 90.0%

    \[\leadsto \alpha \cdot \left(\alpha \cdot \left(u0 \cdot \left(u0 \cdot \left(0.5 + u0 \cdot 0.3333333333333333\right) + 1\right)\right)\right) \]
12. Add Preprocessing

Alternative 10: 87.6% accurate, 9.8× speedup

Math

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

FPCore

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

C

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

Fortran

real(4) function code(alpha, u0)
    real(4), intent (in) :: alpha
    real(4), intent (in) :: u0
    code = (alpha * alpha) * (u0 - (u0 * (u0 * (-0.5e0))))
end function

Julia

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

MATLAB

function tmp = code(alpha, u0)
	tmp = (alpha * alpha) * (u0 - (u0 * (u0 * single(-0.5))));
end

Derivation

1. Initial program 57.4%

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

3. Taylor expanded in u0 around 0

   \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{neg.f32}\left(\alpha\right), \alpha\right), \color{blue}{\left(u0 \cdot \left(\frac{-1}{2} \cdot u0 - 1\right)\right)}\right) \]
4. Step-by-step derivation

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

      \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{neg.f32}\left(\alpha\right), \alpha\right), \mathsf{*.f32}\left(u0, \color{blue}{\left(\frac{-1}{2} \cdot u0 - 1\right)}\right)\right) \]

   2. sub-neg N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{neg.f32}\left(\alpha\right), \alpha\right), \mathsf{*.f32}\left(u0, \left(\frac{-1}{2} \cdot u0 + \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}\right)\right)\right) \]

   3. metadata-eval N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{neg.f32}\left(\alpha\right), \alpha\right), \mathsf{*.f32}\left(u0, \left(\frac{-1}{2} \cdot u0 + -1\right)\right)\right) \]

   4. +-commutative N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{neg.f32}\left(\alpha\right), \alpha\right), \mathsf{*.f32}\left(u0, \left(-1 + \color{blue}{\frac{-1}{2} \cdot u0}\right)\right)\right) \]

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

      \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{neg.f32}\left(\alpha\right), \alpha\right), \mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(-1, \color{blue}{\left(\frac{-1}{2} \cdot u0\right)}\right)\right)\right) \]

   6. *-commutative N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{neg.f32}\left(\alpha\right), \alpha\right), \mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(-1, \left(u0 \cdot \color{blue}{\frac{-1}{2}}\right)\right)\right)\right) \]

   7. *-lowering-*.f32 85.6%

      \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{neg.f32}\left(\alpha\right), \alpha\right), \mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(-1, \mathsf{*.f32}\left(u0, \color{blue}{\frac{-1}{2}}\right)\right)\right)\right) \]

5. Simplified 85.6%

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

   1. +-commutative N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{neg.f32}\left(\alpha\right), \alpha\right), \left(u0 \cdot \left(u0 \cdot \frac{-1}{2} + \color{blue}{-1}\right)\right)\right) \]

   2. distribute-rgt-in N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{neg.f32}\left(\alpha\right), \alpha\right), \left(\left(u0 \cdot \frac{-1}{2}\right) \cdot u0 + \color{blue}{-1 \cdot u0}\right)\right) \]

   3. neg-mul-1 N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{neg.f32}\left(\alpha\right), \alpha\right), \left(\left(u0 \cdot \frac{-1}{2}\right) \cdot u0 + \left(\mathsf{neg}\left(u0\right)\right)\right)\right) \]

   4. unsub-neg N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{neg.f32}\left(\alpha\right), \alpha\right), \left(\left(u0 \cdot \frac{-1}{2}\right) \cdot u0 - \color{blue}{u0}\right)\right) \]

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

      \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{neg.f32}\left(\alpha\right), \alpha\right), \mathsf{\_.f32}\left(\left(\left(u0 \cdot \frac{-1}{2}\right) \cdot u0\right), \color{blue}{u0}\right)\right) \]

   6. *-commutative N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{neg.f32}\left(\alpha\right), \alpha\right), \mathsf{\_.f32}\left(\left(u0 \cdot \left(u0 \cdot \frac{-1}{2}\right)\right), u0\right)\right) \]

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

      \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{neg.f32}\left(\alpha\right), \alpha\right), \mathsf{\_.f32}\left(\mathsf{*.f32}\left(u0, \left(u0 \cdot \frac{-1}{2}\right)\right), u0\right)\right) \]

   8. *-lowering-*.f32 85.7%

      \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{neg.f32}\left(\alpha\right), \alpha\right), \mathsf{\_.f32}\left(\mathsf{*.f32}\left(u0, \mathsf{*.f32}\left(u0, \frac{-1}{2}\right)\right), u0\right)\right) \]

7. Applied egg-rr 85.7%

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

8. Final simplification 85.7%

   \[\leadsto \left(\alpha \cdot \alpha\right) \cdot \left(u0 - u0 \cdot \left(u0 \cdot -0.5\right)\right) \]
9. Add Preprocessing

Alternative 11: 87.4% accurate, 9.8× speedup

Math

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

FPCore

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

C

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

Fortran

real(4) function code(alpha, u0)
    real(4), intent (in) :: alpha
    real(4), intent (in) :: u0
    code = (alpha * alpha) * (u0 * ((u0 * 0.5e0) + 1.0e0))
end function

Julia

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

MATLAB

function tmp = code(alpha, u0)
	tmp = (alpha * alpha) * (u0 * ((u0 * single(0.5)) + single(1.0)));
end

Derivation

1. Initial program 57.4%

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

3. Taylor expanded in u0 around 0

   \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{neg.f32}\left(\alpha\right), \alpha\right), \color{blue}{\left(u0 \cdot \left(\frac{-1}{2} \cdot u0 - 1\right)\right)}\right) \]
4. Step-by-step derivation

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

      \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{neg.f32}\left(\alpha\right), \alpha\right), \mathsf{*.f32}\left(u0, \color{blue}{\left(\frac{-1}{2} \cdot u0 - 1\right)}\right)\right) \]

   2. sub-neg N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{neg.f32}\left(\alpha\right), \alpha\right), \mathsf{*.f32}\left(u0, \left(\frac{-1}{2} \cdot u0 + \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}\right)\right)\right) \]

   3. metadata-eval N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{neg.f32}\left(\alpha\right), \alpha\right), \mathsf{*.f32}\left(u0, \left(\frac{-1}{2} \cdot u0 + -1\right)\right)\right) \]

   4. +-commutative N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{neg.f32}\left(\alpha\right), \alpha\right), \mathsf{*.f32}\left(u0, \left(-1 + \color{blue}{\frac{-1}{2} \cdot u0}\right)\right)\right) \]

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

      \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{neg.f32}\left(\alpha\right), \alpha\right), \mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(-1, \color{blue}{\left(\frac{-1}{2} \cdot u0\right)}\right)\right)\right) \]

   6. *-commutative N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{neg.f32}\left(\alpha\right), \alpha\right), \mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(-1, \left(u0 \cdot \color{blue}{\frac{-1}{2}}\right)\right)\right)\right) \]

   7. *-lowering-*.f32 85.6%

      \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{neg.f32}\left(\alpha\right), \alpha\right), \mathsf{*.f32}\left(u0, \mathsf{+.f32}\left(-1, \mathsf{*.f32}\left(u0, \color{blue}{\frac{-1}{2}}\right)\right)\right)\right) \]

5. Simplified 85.6%

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

   1. *-commutative N/A

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

   2. distribute-rgt-in N/A

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

   3. neg-mul-1 N/A

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

   4. neg-sub0 N/A

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

   5. associate-+l- N/A

      \[\leadsto \left(0 - \left(u0 - \left(u0 \cdot \frac{-1}{2}\right) \cdot u0\right)\right) \cdot \left(\color{blue}{\left(\mathsf{neg}\left(\alpha\right)\right)} \cdot \alpha\right) \]

   6. sub0-neg N/A

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

   7. *-lft-identity N/A

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

   8. fmm-def N/A

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

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

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

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

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

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

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

   12. remove-double-neg N/A

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

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

       \[\leadsto \mathsf{*.f32}\left(\left(\mathsf{fma}\left(1, u0, \mathsf{neg}\left(\left(u0 \cdot \frac{-1}{2}\right) \cdot u0\right)\right)\right), \color{blue}{\left(\alpha \cdot \alpha\right)}\right) \]

7. Applied egg-rr 85.6%

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

8. Final simplification 85.6%

   \[\leadsto \left(\alpha \cdot \alpha\right) \cdot \left(u0 \cdot \left(u0 \cdot 0.5 + 1\right)\right) \]
9. Add Preprocessing

Alternative 12: 87.6% accurate, 9.8× speedup

Math

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

FPCore

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

C

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

Fortran

real(4) function code(alpha, u0)
    real(4), intent (in) :: alpha
    real(4), intent (in) :: u0
    code = u0 * (alpha * (alpha + (u0 * (alpha * 0.5e0))))
end function

Julia

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

MATLAB

function tmp = code(alpha, u0)
	tmp = u0 * (alpha * (alpha + (u0 * (alpha * single(0.5)))));
end

Derivation

1. Initial program 57.4%

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

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

      \[\leadsto \mathsf{*.f32}\left(\left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right), \color{blue}{\log \left(1 - u0\right)}\right) \]

   2. *-commutative N/A

      \[\leadsto \mathsf{*.f32}\left(\left(\alpha \cdot \left(\mathsf{neg}\left(\alpha\right)\right)\right), \log \color{blue}{\left(1 - u0\right)}\right) \]

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

      \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(\alpha, \left(\mathsf{neg}\left(\alpha\right)\right)\right), \log \color{blue}{\left(1 - u0\right)}\right) \]

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

      \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(\alpha, \mathsf{neg.f32}\left(\alpha\right)\right), \log \left(1 - \color{blue}{u0}\right)\right) \]

   5. sub-neg N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(\alpha, \mathsf{neg.f32}\left(\alpha\right)\right), \log \left(1 + \left(\mathsf{neg}\left(u0\right)\right)\right)\right) \]

   6. log1p-define N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(\alpha, \mathsf{neg.f32}\left(\alpha\right)\right), \left(\mathsf{log1p}\left(\mathsf{neg}\left(u0\right)\right)\right)\right) \]

   7. log1p-lowering-log1p.f32 N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(\alpha, \mathsf{neg.f32}\left(\alpha\right)\right), \mathsf{log1p.f32}\left(\left(\mathsf{neg}\left(u0\right)\right)\right)\right) \]

   8. neg-lowering-neg.f32 99.1%

      \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(\alpha, \mathsf{neg.f32}\left(\alpha\right)\right), \mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right)\right) \]

3. Simplified 99.1%

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

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. distribute-rgt-in N/A

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

   2. associate-*r* N/A

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

   3. associate-+l+ N/A

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

   4. *-commutative N/A

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

   5. distribute-lft-in N/A

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

   6. associate-*r* N/A

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

   7. associate-*r* N/A

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

   8. distribute-rgt-in N/A

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

   9. *-commutative N/A

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

   10. associate-*r* N/A

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

   11. distribute-lft1-in N/A

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

   12. distribute-rgt-out N/A

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

7. Simplified 90.1%

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

   1. associate-+r+ N/A

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

   2. distribute-rgt-in N/A

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

   3. *-commutative N/A

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

   4. associate-*l* N/A

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

   5. distribute-lft-in N/A

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

   6. associate-*r* N/A

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

   7. *-commutative N/A

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

   8. *-lft-identity N/A

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

   9. distribute-rgt-in N/A

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

   10. *-commutative N/A

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

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

       \[\leadsto \mathsf{*.f32}\left(\left(\left(u0 \cdot \frac{1}{3} + \frac{1}{2}\right) \cdot \left(\left(\alpha \cdot \alpha\right) \cdot u0\right) + \alpha \cdot \alpha\right), \color{blue}{u0}\right) \]

9. Applied egg-rr 90.2%

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

    1. +-commutative N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(\alpha, \left(\alpha \cdot \left(u0 \cdot \left(\frac{1}{2} + u0 \cdot \frac{1}{3}\right) + 1\right)\right)\right), u0\right) \]

    2. distribute-lft-in N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(\alpha, \left(\alpha \cdot \left(u0 \cdot \left(\frac{1}{2} + u0 \cdot \frac{1}{3}\right)\right) + \alpha \cdot 1\right)\right), u0\right) \]

    3. *-rgt-identity N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(\alpha, \left(\alpha \cdot \left(u0 \cdot \left(\frac{1}{2} + u0 \cdot \frac{1}{3}\right)\right) + \alpha\right)\right), u0\right) \]

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

       \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(\alpha, \mathsf{+.f32}\left(\left(\alpha \cdot \left(u0 \cdot \left(\frac{1}{2} + u0 \cdot \frac{1}{3}\right)\right)\right), \alpha\right)\right), u0\right) \]

    5. associate-*r* N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(\alpha, \mathsf{+.f32}\left(\left(\left(\alpha \cdot u0\right) \cdot \left(\frac{1}{2} + u0 \cdot \frac{1}{3}\right)\right), \alpha\right)\right), u0\right) \]

    6. *-commutative N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(\alpha, \mathsf{+.f32}\left(\left(\left(u0 \cdot \alpha\right) \cdot \left(\frac{1}{2} + u0 \cdot \frac{1}{3}\right)\right), \alpha\right)\right), u0\right) \]

    7. associate-*l* N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(\alpha, \mathsf{+.f32}\left(\left(u0 \cdot \left(\alpha \cdot \left(\frac{1}{2} + u0 \cdot \frac{1}{3}\right)\right)\right), \alpha\right)\right), u0\right) \]

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

       \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(\alpha, \mathsf{+.f32}\left(\mathsf{*.f32}\left(u0, \left(\alpha \cdot \left(\frac{1}{2} + u0 \cdot \frac{1}{3}\right)\right)\right), \alpha\right)\right), u0\right) \]

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

       \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(\alpha, \mathsf{+.f32}\left(\mathsf{*.f32}\left(u0, \mathsf{*.f32}\left(\alpha, \left(\frac{1}{2} + u0 \cdot \frac{1}{3}\right)\right)\right), \alpha\right)\right), u0\right) \]

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

        \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(\alpha, \mathsf{+.f32}\left(\mathsf{*.f32}\left(u0, \mathsf{*.f32}\left(\alpha, \mathsf{+.f32}\left(\frac{1}{2}, \left(u0 \cdot \frac{1}{3}\right)\right)\right)\right), \alpha\right)\right), u0\right) \]

    11. *-lowering-*.f32 90.2%

        \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(\alpha, \mathsf{+.f32}\left(\mathsf{*.f32}\left(u0, \mathsf{*.f32}\left(\alpha, \mathsf{+.f32}\left(\frac{1}{2}, \mathsf{*.f32}\left(u0, \frac{1}{3}\right)\right)\right)\right), \alpha\right)\right), u0\right) \]

11. Applied egg-rr 90.2%

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

12. Taylor expanded in u0 around 0

    \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(\alpha, \mathsf{+.f32}\left(\mathsf{*.f32}\left(u0, \color{blue}{\left(\frac{1}{2} \cdot \alpha\right)}\right), \alpha\right)\right), u0\right) \]
13. Step-by-step derivation

    1. *-commutative N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(\alpha, \mathsf{+.f32}\left(\mathsf{*.f32}\left(u0, \left(\alpha \cdot \frac{1}{2}\right)\right), \alpha\right)\right), u0\right) \]

    2. *-lowering-*.f32 85.6%

       \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(\alpha, \mathsf{+.f32}\left(\mathsf{*.f32}\left(u0, \mathsf{*.f32}\left(\alpha, \frac{1}{2}\right)\right), \alpha\right)\right), u0\right) \]

14. Simplified 85.6%

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

15. Final simplification 85.6%

    \[\leadsto u0 \cdot \left(\alpha \cdot \left(\alpha + u0 \cdot \left(\alpha \cdot 0.5\right)\right)\right) \]
16. Add Preprocessing

Alternative 13: 87.4% accurate, 9.8× speedup

Math

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

FPCore

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

C

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

Fortran

real(4) function code(alpha, u0)
    real(4), intent (in) :: alpha
    real(4), intent (in) :: u0
    code = u0 * ((alpha * alpha) * ((u0 * 0.5e0) + 1.0e0))
end function

Julia

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

MATLAB

function tmp = code(alpha, u0)
	tmp = u0 * ((alpha * alpha) * ((u0 * single(0.5)) + single(1.0)));
end

Derivation

1. Initial program 57.4%

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

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

      \[\leadsto \mathsf{*.f32}\left(\left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right), \color{blue}{\log \left(1 - u0\right)}\right) \]

   2. *-commutative N/A

      \[\leadsto \mathsf{*.f32}\left(\left(\alpha \cdot \left(\mathsf{neg}\left(\alpha\right)\right)\right), \log \color{blue}{\left(1 - u0\right)}\right) \]

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

      \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(\alpha, \left(\mathsf{neg}\left(\alpha\right)\right)\right), \log \color{blue}{\left(1 - u0\right)}\right) \]

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

      \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(\alpha, \mathsf{neg.f32}\left(\alpha\right)\right), \log \left(1 - \color{blue}{u0}\right)\right) \]

   5. sub-neg N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(\alpha, \mathsf{neg.f32}\left(\alpha\right)\right), \log \left(1 + \left(\mathsf{neg}\left(u0\right)\right)\right)\right) \]

   6. log1p-define N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(\alpha, \mathsf{neg.f32}\left(\alpha\right)\right), \left(\mathsf{log1p}\left(\mathsf{neg}\left(u0\right)\right)\right)\right) \]

   7. log1p-lowering-log1p.f32 N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(\alpha, \mathsf{neg.f32}\left(\alpha\right)\right), \mathsf{log1p.f32}\left(\left(\mathsf{neg}\left(u0\right)\right)\right)\right) \]

   8. neg-lowering-neg.f32 99.1%

      \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(\alpha, \mathsf{neg.f32}\left(\alpha\right)\right), \mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right)\right) \]

3. Simplified 99.1%

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

5. Taylor expanded in u0 around 0

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

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

      \[\leadsto \mathsf{*.f32}\left(u0, \color{blue}{\left(\frac{1}{2} \cdot \left({\alpha}^{2} \cdot u0\right) + {\alpha}^{2}\right)}\right) \]

   2. *-commutative N/A

      \[\leadsto \mathsf{*.f32}\left(u0, \left(\frac{1}{2} \cdot \left(u0 \cdot {\alpha}^{2}\right) + {\alpha}^{2}\right)\right) \]

   3. associate-*r* N/A

      \[\leadsto \mathsf{*.f32}\left(u0, \left(\left(\frac{1}{2} \cdot u0\right) \cdot {\alpha}^{2} + {\color{blue}{\alpha}}^{2}\right)\right) \]

   4. distribute-lft1-in N/A

      \[\leadsto \mathsf{*.f32}\left(u0, \left(\left(\frac{1}{2} \cdot u0 + 1\right) \cdot \color{blue}{{\alpha}^{2}}\right)\right) \]

   5. *-commutative N/A

      \[\leadsto \mathsf{*.f32}\left(u0, \left(\left(u0 \cdot \frac{1}{2} + 1\right) \cdot {\alpha}^{2}\right)\right) \]

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

      \[\leadsto \mathsf{*.f32}\left(u0, \mathsf{*.f32}\left(\left(u0 \cdot \frac{1}{2} + 1\right), \color{blue}{\left({\alpha}^{2}\right)}\right)\right) \]

   7. *-commutative N/A

      \[\leadsto \mathsf{*.f32}\left(u0, \mathsf{*.f32}\left(\left(\frac{1}{2} \cdot u0 + 1\right), \left({\alpha}^{2}\right)\right)\right) \]

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

      \[\leadsto \mathsf{*.f32}\left(u0, \mathsf{*.f32}\left(\mathsf{+.f32}\left(\left(\frac{1}{2} \cdot u0\right), 1\right), \left({\color{blue}{\alpha}}^{2}\right)\right)\right) \]

   9. *-commutative N/A

      \[\leadsto \mathsf{*.f32}\left(u0, \mathsf{*.f32}\left(\mathsf{+.f32}\left(\left(u0 \cdot \frac{1}{2}\right), 1\right), \left({\alpha}^{2}\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f32}\left(u0, \mathsf{*.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(u0, \frac{1}{2}\right), 1\right), \left({\alpha}^{2}\right)\right)\right) \]

   11. unpow2 N/A

       \[\leadsto \mathsf{*.f32}\left(u0, \mathsf{*.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(u0, \frac{1}{2}\right), 1\right), \left(\alpha \cdot \color{blue}{\alpha}\right)\right)\right) \]

   12. *-lowering-*.f32 85.6%

       \[\leadsto \mathsf{*.f32}\left(u0, \mathsf{*.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(u0, \frac{1}{2}\right), 1\right), \mathsf{*.f32}\left(\alpha, \color{blue}{\alpha}\right)\right)\right) \]

7. Simplified 85.6%

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

8. Final simplification 85.6%

   \[\leadsto u0 \cdot \left(\left(\alpha \cdot \alpha\right) \cdot \left(u0 \cdot 0.5 + 1\right)\right) \]
9. Add Preprocessing

Alternative 14: 75.0% accurate, 21.6× 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 57.4%

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

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

      \[\leadsto \mathsf{*.f32}\left(\left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right), \color{blue}{\log \left(1 - u0\right)}\right) \]

   2. *-commutative N/A

      \[\leadsto \mathsf{*.f32}\left(\left(\alpha \cdot \left(\mathsf{neg}\left(\alpha\right)\right)\right), \log \color{blue}{\left(1 - u0\right)}\right) \]

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

      \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(\alpha, \left(\mathsf{neg}\left(\alpha\right)\right)\right), \log \color{blue}{\left(1 - u0\right)}\right) \]

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

      \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(\alpha, \mathsf{neg.f32}\left(\alpha\right)\right), \log \left(1 - \color{blue}{u0}\right)\right) \]

   5. sub-neg N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(\alpha, \mathsf{neg.f32}\left(\alpha\right)\right), \log \left(1 + \left(\mathsf{neg}\left(u0\right)\right)\right)\right) \]

   6. log1p-define N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(\alpha, \mathsf{neg.f32}\left(\alpha\right)\right), \left(\mathsf{log1p}\left(\mathsf{neg}\left(u0\right)\right)\right)\right) \]

   7. log1p-lowering-log1p.f32 N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(\alpha, \mathsf{neg.f32}\left(\alpha\right)\right), \mathsf{log1p.f32}\left(\left(\mathsf{neg}\left(u0\right)\right)\right)\right) \]

   8. neg-lowering-neg.f32 99.1%

      \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(\alpha, \mathsf{neg.f32}\left(\alpha\right)\right), \mathsf{log1p.f32}\left(\mathsf{neg.f32}\left(u0\right)\right)\right) \]

3. Simplified 99.1%

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

5. Taylor expanded in u0 around 0

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

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

      \[\leadsto \mathsf{*.f32}\left(\left({\alpha}^{2}\right), \color{blue}{u0}\right) \]

   2. unpow2 N/A

      \[\leadsto \mathsf{*.f32}\left(\left(\alpha \cdot \alpha\right), u0\right) \]

   3. *-lowering-*.f32 73.4%

      \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(\alpha, \alpha\right), u0\right) \]

7. Simplified 73.4%

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

8. Final simplification 73.4%

   \[\leadsto u0 \cdot \left(\alpha \cdot \alpha\right) \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024164 
(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))))