Beckmann Distribution sample, tan2theta, alphax == alphay

Percentage Accurate: 55.8% → 97.6%

Time: 6.9s

Alternatives: 5

Speedup: 10.5×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Initial Program: 55.8% 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: 97.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;1 - u0 \leq 0.9869999885559082:\\ \;\;\;\;\left(\left(\left(\alpha \cdot \alpha\right) \cdot \alpha\right) \cdot \frac{-1}{\alpha}\right) \cdot \log \left(1 - u0\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(0.3333333333333333 \cdot \left(\alpha \cdot \alpha\right) - \frac{\left(\alpha \cdot \alpha\right) \cdot \left(-0.5 + \frac{-1}{u0}\right)}{u0}\right) \cdot u0\right) \cdot u0\right) \cdot u0\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha u0)
 :precision binary32
 (if (<= (- 1.0 u0) 0.9869999885559082)
   (* (* (* (* alpha alpha) alpha) (/ -1.0 alpha)) (log (- 1.0 u0)))
   (*
    (*
     (*
      (-
       (* 0.3333333333333333 (* alpha alpha))
       (/ (* (* alpha alpha) (+ -0.5 (/ -1.0 u0))) u0))
      u0)
     u0)
    u0)))

C

float code(float alpha, float u0) {
	float tmp;
	if ((1.0f - u0) <= 0.9869999885559082f) {
		tmp = (((alpha * alpha) * alpha) * (-1.0f / alpha)) * logf((1.0f - u0));
	} else {
		tmp = ((((0.3333333333333333f * (alpha * alpha)) - (((alpha * alpha) * (-0.5f + (-1.0f / u0))) / u0)) * u0) * u0) * u0;
	}
	return tmp;
}

Fortran

real(4) function code(alpha, u0)
    real(4), intent (in) :: alpha
    real(4), intent (in) :: u0
    real(4) :: tmp
    if ((1.0e0 - u0) <= 0.9869999885559082e0) then
        tmp = (((alpha * alpha) * alpha) * ((-1.0e0) / alpha)) * log((1.0e0 - u0))
    else
        tmp = ((((0.3333333333333333e0 * (alpha * alpha)) - (((alpha * alpha) * ((-0.5e0) + ((-1.0e0) / u0))) / u0)) * u0) * u0) * u0
    end if
    code = tmp
end function

Julia

function code(alpha, u0)
	tmp = Float32(0.0)
	if (Float32(Float32(1.0) - u0) <= Float32(0.9869999885559082))
		tmp = Float32(Float32(Float32(Float32(alpha * alpha) * alpha) * Float32(Float32(-1.0) / alpha)) * log(Float32(Float32(1.0) - u0)));
	else
		tmp = Float32(Float32(Float32(Float32(Float32(Float32(0.3333333333333333) * Float32(alpha * alpha)) - Float32(Float32(Float32(alpha * alpha) * Float32(Float32(-0.5) + Float32(Float32(-1.0) / u0))) / u0)) * u0) * u0) * u0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, u0)
	tmp = single(0.0);
	if ((single(1.0) - u0) <= single(0.9869999885559082))
		tmp = (((alpha * alpha) * alpha) * (single(-1.0) / alpha)) * log((single(1.0) - u0));
	else
		tmp = ((((single(0.3333333333333333) * (alpha * alpha)) - (((alpha * alpha) * (single(-0.5) + (single(-1.0) / u0))) / u0)) * u0) * u0) * u0;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (-.f32 #s(literal 1 binary32) u0) < 0.986999989

   1. Initial program 95.6%

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

      1. lift-*.f32 N/A

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

      2. lift-neg.f32 N/A

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

      3. neg-sub0 N/A

         \[\leadsto \left(\color{blue}{\left(0 - \alpha\right)} \cdot \alpha\right) \cdot \log \left(1 - u0\right) \]

      4. flip-- N/A

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

      5. metadata-eval N/A

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

      6. neg-sub0 N/A

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

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

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

      8. lift-neg.f32 N/A

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

      9. lift-*.f32 N/A

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

      10. +-lft-identity N/A

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

      11. associate-*l/ N/A

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

      12. div-inv N/A

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

      13. +-lft-identity N/A

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

      14. lower-*.f32 N/A

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

      15. lower-*.f32 N/A

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

      16. +-lft-identity N/A

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

      17. lower-/.f32 95.7

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

   4. Applied rewrites 95.7%

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

   if 0.986999989 < (-.f32 #s(literal 1 binary32) u0)

   1. Initial program 44.4%

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

      1. lift-neg.f32 N/A

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

      2. neg-sub0 N/A

         \[\leadsto \left(\color{blue}{\left(0 - \alpha\right)} \cdot \alpha\right) \cdot \log \left(1 - u0\right) \]

      3. flip-- N/A

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

      4. metadata-eval N/A

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

      5. neg-sub0 N/A

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

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

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

      7. lift-neg.f32 N/A

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

      8. lift-*.f32 N/A

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

      9. div-inv N/A

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

      10. lower-*.f32 N/A

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

      11. +-lft-identity N/A

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

      12. lower-/.f32 44.4

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

   4. Applied rewrites 44.4%

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

      1. lift-log.f32 N/A

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

      2. lift--.f32 N/A

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

      3. flip-- N/A

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

      4. log-div N/A

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

      5. sqr-neg N/A

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

      6. lift-neg.f32 N/A

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

      7. lift-neg.f32 N/A

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

      8. lower--.f32 N/A

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

      9. metadata-eval N/A

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

      10. lift-neg.f32 N/A

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

      11. lift-neg.f32 N/A

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

      12. sqr-neg N/A

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

      13. cancel-sign-sub-inv N/A

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

      14. lower-log1p.f32 N/A

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

      15. lift-neg.f32 N/A

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

      16. lower-*.f32 N/A

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

      17. lower-log1p.f32 81.7

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

   6. Applied rewrites 83.6%

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

   7. Taylor expanded in u0 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f32 N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. distribute-rgt-out N/A

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

      8. lower-*.f32 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f32 N/A

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

      11. lower-fma.f32 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f32 84.3

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

   9. Applied rewrites 83.8%

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

   10. Taylor expanded in u0 around -inf

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

   12. Applied rewrites 98.2%

       \[\leadsto \left(\left(\left(0.3333333333333333 \cdot \left(\alpha \cdot \alpha\right) - \frac{\left(\alpha \cdot \alpha\right) \cdot \left(-0.5 + \frac{-1}{u0}\right)}{u0}\right) \cdot u0\right) \cdot u0\right) \cdot u0 \]
3. Recombined 2 regimes into one program.

4. Final simplification 97.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 - u0 \leq 0.9869999885559082:\\ \;\;\;\;\left(\left(\left(\alpha \cdot \alpha\right) \cdot \alpha\right) \cdot \frac{-1}{\alpha}\right) \cdot \log \left(1 - u0\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(0.3333333333333333 \cdot \left(\alpha \cdot \alpha\right) - \frac{\left(\alpha \cdot \alpha\right) \cdot \left(-0.5 + \frac{-1}{u0}\right)}{u0}\right) \cdot u0\right) \cdot u0\right) \cdot u0\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 97.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;1 - u0 \leq 0.9869999885559082:\\ \;\;\;\;\left(\left(-\alpha\right) \cdot \alpha\right) \cdot \log \left(1 - u0\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(0.3333333333333333 \cdot \left(\alpha \cdot \alpha\right) - \frac{\left(\alpha \cdot \alpha\right) \cdot \left(-0.5 + \frac{-1}{u0}\right)}{u0}\right) \cdot u0\right) \cdot u0\right) \cdot u0\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha u0)
 :precision binary32
 (if (<= (- 1.0 u0) 0.9869999885559082)
   (* (* (- alpha) alpha) (log (- 1.0 u0)))
   (*
    (*
     (*
      (-
       (* 0.3333333333333333 (* alpha alpha))
       (/ (* (* alpha alpha) (+ -0.5 (/ -1.0 u0))) u0))
      u0)
     u0)
    u0)))

C

float code(float alpha, float u0) {
	float tmp;
	if ((1.0f - u0) <= 0.9869999885559082f) {
		tmp = (-alpha * alpha) * logf((1.0f - u0));
	} else {
		tmp = ((((0.3333333333333333f * (alpha * alpha)) - (((alpha * alpha) * (-0.5f + (-1.0f / u0))) / u0)) * u0) * u0) * u0;
	}
	return tmp;
}

Fortran

real(4) function code(alpha, u0)
    real(4), intent (in) :: alpha
    real(4), intent (in) :: u0
    real(4) :: tmp
    if ((1.0e0 - u0) <= 0.9869999885559082e0) then
        tmp = (-alpha * alpha) * log((1.0e0 - u0))
    else
        tmp = ((((0.3333333333333333e0 * (alpha * alpha)) - (((alpha * alpha) * ((-0.5e0) + ((-1.0e0) / u0))) / u0)) * u0) * u0) * u0
    end if
    code = tmp
end function

Julia

function code(alpha, u0)
	tmp = Float32(0.0)
	if (Float32(Float32(1.0) - u0) <= Float32(0.9869999885559082))
		tmp = Float32(Float32(Float32(-alpha) * alpha) * log(Float32(Float32(1.0) - u0)));
	else
		tmp = Float32(Float32(Float32(Float32(Float32(Float32(0.3333333333333333) * Float32(alpha * alpha)) - Float32(Float32(Float32(alpha * alpha) * Float32(Float32(-0.5) + Float32(Float32(-1.0) / u0))) / u0)) * u0) * u0) * u0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, u0)
	tmp = single(0.0);
	if ((single(1.0) - u0) <= single(0.9869999885559082))
		tmp = (-alpha * alpha) * log((single(1.0) - u0));
	else
		tmp = ((((single(0.3333333333333333) * (alpha * alpha)) - (((alpha * alpha) * (single(-0.5) + (single(-1.0) / u0))) / u0)) * u0) * u0) * u0;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (-.f32 #s(literal 1 binary32) u0) < 0.986999989

   1. Initial program 95.6%

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

   if 0.986999989 < (-.f32 #s(literal 1 binary32) u0)

   1. Initial program 44.4%

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

      1. lift-neg.f32 N/A

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

      2. neg-sub0 N/A

         \[\leadsto \left(\color{blue}{\left(0 - \alpha\right)} \cdot \alpha\right) \cdot \log \left(1 - u0\right) \]

      3. flip-- N/A

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

      4. metadata-eval N/A

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

      5. neg-sub0 N/A

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

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

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

      7. lift-neg.f32 N/A

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

      8. lift-*.f32 N/A

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

      9. div-inv N/A

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

      10. lower-*.f32 N/A

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

      11. +-lft-identity N/A

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

      12. lower-/.f32 44.4

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

   4. Applied rewrites 44.4%

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

      1. lift-log.f32 N/A

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

      2. lift--.f32 N/A

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

      3. flip-- N/A

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

      4. log-div N/A

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

      5. sqr-neg N/A

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

      6. lift-neg.f32 N/A

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

      7. lift-neg.f32 N/A

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

      8. lower--.f32 N/A

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

      9. metadata-eval N/A

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

      10. lift-neg.f32 N/A

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

      11. lift-neg.f32 N/A

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

      12. sqr-neg N/A

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

      13. cancel-sign-sub-inv N/A

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

      14. lower-log1p.f32 N/A

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

      15. lift-neg.f32 N/A

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

      16. lower-*.f32 N/A

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

      17. lower-log1p.f32 83.4

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

   6. Applied rewrites 76.9%

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

   7. Taylor expanded in u0 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f32 N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. distribute-rgt-out N/A

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

      8. lower-*.f32 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f32 N/A

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

      11. lower-fma.f32 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f32 84.3

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

   9. Applied rewrites 83.8%

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

   10. Taylor expanded in u0 around -inf

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

   12. Applied rewrites 98.2%

       \[\leadsto \left(\left(\left(0.3333333333333333 \cdot \left(\alpha \cdot \alpha\right) - \frac{\left(\alpha \cdot \alpha\right) \cdot \left(-0.5 + \frac{-1}{u0}\right)}{u0}\right) \cdot u0\right) \cdot u0\right) \cdot u0 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 90.6% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 56.8%

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

   1. lift-neg.f32 N/A

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

   2. neg-sub0 N/A

      \[\leadsto \left(\color{blue}{\left(0 - \alpha\right)} \cdot \alpha\right) \cdot \log \left(1 - u0\right) \]

   3. flip-- N/A

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

   4. metadata-eval N/A

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

   5. neg-sub0 N/A

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

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

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

   7. lift-neg.f32 N/A

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

   8. lift-*.f32 N/A

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

   9. div-inv N/A

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

   10. lower-*.f32 N/A

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

   11. +-lft-identity N/A

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

   12. lower-/.f32 56.8

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

4. Applied rewrites 56.8%

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

   1. lift-log.f32 N/A

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

   2. lift--.f32 N/A

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

   3. flip-- N/A

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

   4. log-div N/A

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

   5. sqr-neg N/A

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

   6. lift-neg.f32 N/A

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

   7. lift-neg.f32 N/A

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

   8. lower--.f32 N/A

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

   9. metadata-eval N/A

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

   10. lift-neg.f32 N/A

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

   11. lift-neg.f32 N/A

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

   12. sqr-neg N/A

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

   13. cancel-sign-sub-inv N/A

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

   14. lower-log1p.f32 N/A

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

   15. lift-neg.f32 N/A

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

   16. lower-*.f32 N/A

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

   17. lower-log1p.f32 73.1

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

6. Applied rewrites 72.4%

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

7. Taylor expanded in u0 around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. *-commutative N/A

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

   4. lower-fma.f32 N/A

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

   5. *-commutative N/A

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

   6. associate-*r* N/A

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

   7. distribute-rgt-out N/A

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

   8. lower-*.f32 N/A

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

   9. unpow2 N/A

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

   10. lower-*.f32 N/A

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

   11. lower-fma.f32 N/A

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

   12. unpow2 N/A

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

   13. lower-*.f32 72.9

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

9. Applied rewrites 72.6%

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

10. Taylor expanded in u0 around -inf

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

12. Applied rewrites 88.4%

    \[\leadsto \left(\left(\left(0.3333333333333333 \cdot \left(\alpha \cdot \alpha\right) - \frac{\left(\alpha \cdot \alpha\right) \cdot \left(-0.5 + \frac{-1}{u0}\right)}{u0}\right) \cdot u0\right) \cdot u0\right) \cdot u0 \]
13. Add Preprocessing

Alternative 4: 65.0% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;1 - u0 \leq 0.9999780058860779:\\ \;\;\;\;\left(\left(u0 \cdot \alpha\right) \cdot \left(\mathsf{fma}\left(0.3333333333333333, u0, 0.5\right) \cdot \alpha\right) + \alpha \cdot \alpha\right) \cdot u0\\ \mathbf{else}:\\ \;\;\;\;\left(\alpha \cdot \alpha\right) \cdot u0\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha u0)
 :precision binary32
 (if (<= (- 1.0 u0) 0.9999780058860779)
   (*
    (+
     (* (* u0 alpha) (* (fma 0.3333333333333333 u0 0.5) alpha))
     (* alpha alpha))
    u0)
   (* (* alpha alpha) u0)))

C

float code(float alpha, float u0) {
	float tmp;
	if ((1.0f - u0) <= 0.9999780058860779f) {
		tmp = (((u0 * alpha) * (fmaf(0.3333333333333333f, u0, 0.5f) * alpha)) + (alpha * alpha)) * u0;
	} else {
		tmp = (alpha * alpha) * u0;
	}
	return tmp;
}

Julia

function code(alpha, u0)
	tmp = Float32(0.0)
	if (Float32(Float32(1.0) - u0) <= Float32(0.9999780058860779))
		tmp = Float32(Float32(Float32(Float32(u0 * alpha) * Float32(fma(Float32(0.3333333333333333), u0, Float32(0.5)) * alpha)) + Float32(alpha * alpha)) * u0);
	else
		tmp = Float32(Float32(alpha * alpha) * u0);
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (-.f32 #s(literal 1 binary32) u0) < 0.999978006

   1. Initial program 84.5%

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

      1. lift-neg.f32 N/A

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

      2. neg-sub0 N/A

         \[\leadsto \left(\color{blue}{\left(0 - \alpha\right)} \cdot \alpha\right) \cdot \log \left(1 - u0\right) \]

      3. flip-- N/A

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

      4. metadata-eval N/A

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

      5. neg-sub0 N/A

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

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

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

      7. lift-neg.f32 N/A

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

      8. lift-*.f32 N/A

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

      9. div-inv N/A

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

      10. lower-*.f32 N/A

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

      11. +-lft-identity N/A

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

      12. lower-/.f32 84.5

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

   4. Applied rewrites 84.5%

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

      1. lift-log.f32 N/A

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

      2. lift--.f32 N/A

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

      3. flip-- N/A

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

      4. log-div N/A

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

      5. sqr-neg N/A

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

      6. lift-neg.f32 N/A

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

      7. lift-neg.f32 N/A

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

      8. lower--.f32 N/A

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

      9. metadata-eval N/A

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

      10. lift-neg.f32 N/A

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

      11. lift-neg.f32 N/A

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

      12. sqr-neg N/A

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

      13. cancel-sign-sub-inv N/A

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

      14. lower-log1p.f32 N/A

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

      15. lift-neg.f32 N/A

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

      16. lower-*.f32 N/A

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

      17. lower-log1p.f32 49.6

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

   6. Applied rewrites 52.0%

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

   7. Taylor expanded in u0 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f32 N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. distribute-rgt-out N/A

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

      8. lower-*.f32 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f32 N/A

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

      11. lower-fma.f32 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f32 51.4

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

   9. Applied rewrites 51.0%

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

   11. Applied rewrites 69.7%

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

   if 0.999978006 < (-.f32 #s(literal 1 binary32) u0)

   1. Initial program 28.6%

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

   3. Taylor expanded in u0 around 0

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

      1. lower-*.f32 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f32 94.9

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

   5. Applied rewrites 94.9%

      \[\leadsto \color{blue}{\left(\alpha \cdot \alpha\right) \cdot u0} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 74.5% accurate, 10.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 56.8%

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

3. Taylor expanded in u0 around 0

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

   1. lower-*.f32 N/A

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

   2. unpow2 N/A

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

   3. lower-*.f32 72.9

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

5. Applied rewrites 72.9%

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

Reproduce

herbie shell --seed 2024322 
(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))))