Beckmann Distribution sample, tan2theta, alphax == alphay

Percentage Accurate: 55.9% → 96.6%

Time: 7.3s

Alternatives: 10

Speedup: 10.5×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Initial Program: 55.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Alternative 1: 96.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\alpha \cdot \alpha\right) \cdot u0\\ \mathbf{if}\;1 - u0 \leq 0.9972000122070313:\\ \;\;\;\;\log \left(1 - u0\right) \cdot \left({\alpha}^{-2} \cdot \left(-{\alpha}^{4}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot u0\right) \cdot t\_0 + t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha u0)
 :precision binary32
 (let* ((t_0 (* (* alpha alpha) u0)))
   (if (<= (- 1.0 u0) 0.9972000122070313)
     (* (log (- 1.0 u0)) (* (pow alpha -2.0) (- (pow alpha 4.0))))
     (+ (* (* 0.5 u0) t_0) t_0))))

C

float code(float alpha, float u0) {
	float t_0 = (alpha * alpha) * u0;
	float tmp;
	if ((1.0f - u0) <= 0.9972000122070313f) {
		tmp = logf((1.0f - u0)) * (powf(alpha, -2.0f) * -powf(alpha, 4.0f));
	} else {
		tmp = ((0.5f * u0) * t_0) + t_0;
	}
	return tmp;
}

Fortran

real(4) function code(alpha, u0)
    real(4), intent (in) :: alpha
    real(4), intent (in) :: u0
    real(4) :: t_0
    real(4) :: tmp
    t_0 = (alpha * alpha) * u0
    if ((1.0e0 - u0) <= 0.9972000122070313e0) then
        tmp = log((1.0e0 - u0)) * ((alpha ** (-2.0e0)) * -(alpha ** 4.0e0))
    else
        tmp = ((0.5e0 * u0) * t_0) + t_0
    end if
    code = tmp
end function

Julia

function code(alpha, u0)
	t_0 = Float32(Float32(alpha * alpha) * u0)
	tmp = Float32(0.0)
	if (Float32(Float32(1.0) - u0) <= Float32(0.9972000122070313))
		tmp = Float32(log(Float32(Float32(1.0) - u0)) * Float32((alpha ^ Float32(-2.0)) * Float32(-(alpha ^ Float32(4.0)))));
	else
		tmp = Float32(Float32(Float32(Float32(0.5) * u0) * t_0) + t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, u0)
	t_0 = (alpha * alpha) * u0;
	tmp = single(0.0);
	if ((single(1.0) - u0) <= single(0.9972000122070313))
		tmp = log((single(1.0) - u0)) * ((alpha ^ single(-2.0)) * -(alpha ^ single(4.0)));
	else
		tmp = ((single(0.5) * u0) * t_0) + t_0;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (-.f32 #s(literal 1 binary32) u0) < 0.99720001

   1. Initial program 93.5%

      \[\left(\left(-\alpha\right) \cdot \alpha\right) \cdot \log \left(1 - u0\right) \]
   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. distribute-lft-neg-out N/A

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

      4. neg-sub0 N/A

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

      5. flip-- N/A

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

      6. +-lft-identity N/A

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

      7. div-inv N/A

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

      8. +-rgt-identity N/A

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

      9. mul0-lft N/A

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

      10. +-lft-identity N/A

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

      11. metadata-eval N/A

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

      12. lower-*.f32 N/A

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

      13. metadata-eval N/A

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

      14. sub0-neg N/A

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

      15. lower-neg.f32 N/A

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

      16. pow2 N/A

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

      17. pow2 N/A

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

      18. pow-prod-up N/A

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

      19. lower-pow.f32 N/A

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

      20. metadata-eval N/A

          \[\leadsto \left(\left(-{\alpha}^{\color{blue}{4}}\right) \cdot \frac{1}{0 \cdot 0 + \left(\alpha \cdot \alpha + 0 \cdot \alpha\right)}\right) \cdot \log \left(1 - u0\right) \]

      21. metadata-eval N/A

          \[\leadsto \left(\left(-{\alpha}^{4}\right) \cdot \frac{1}{\color{blue}{0} + \left(\alpha \cdot \alpha + 0 \cdot \alpha\right)}\right) \cdot \log \left(1 - u0\right) \]

   4. Applied rewrites 93.8%

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

   if 0.99720001 < (-.f32 #s(literal 1 binary32) u0)

   1. Initial program 39.8%

      \[\left(\left(-\alpha\right) \cdot \alpha\right) \cdot \log \left(1 - u0\right) \]
   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 39.8

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

   4. Applied rewrites 39.8%

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

      1. lift-*.f32 N/A

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

      2. /-rgt-identity N/A

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

      3. div-inv N/A

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

      4. metadata-eval N/A

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

      5. associate-*l* N/A

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

      6. lift-*.f32 N/A

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

      7. rgt-mult-inverse N/A

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

      8. lift-/.f32 N/A

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

      9. associate-*l* N/A

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

      10. lift-*.f32 N/A

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

      11. *-commutative N/A

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

      12. lift-*.f32 N/A

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

      13. lift-*.f32 N/A

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

      14. associate-*l* N/A

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

      15. lift-*.f32 N/A

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

      16. associate-*r* N/A

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

      17. lower-*.f32 N/A

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

      18. lower-*.f32 39.8

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

   6. Applied rewrites 39.8%

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

   7. Taylor expanded in u0 around 0

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

      1. distribute-lft-in N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. distribute-lft1-in N/A

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

      5. lower-*.f32 N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f32 N/A

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

      8. lower-*.f32 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f32 82.3

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

   9. Applied rewrites 86.2%

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

   11. Applied rewrites 98.1%

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

4. Final simplification 97.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 - u0 \leq 0.9972000122070313:\\ \;\;\;\;\log \left(1 - u0\right) \cdot \left({\alpha}^{-2} \cdot \left(-{\alpha}^{4}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot u0\right) \cdot \left(\left(\alpha \cdot \alpha\right) \cdot u0\right) + \left(\alpha \cdot \alpha\right) \cdot u0\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 96.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\alpha \cdot \alpha\right) \cdot u0\\ \mathbf{if}\;1 - u0 \leq 0.9972000122070313:\\ \;\;\;\;\frac{{\left(-\alpha\right)}^{3}}{\alpha} \cdot \log \left(1 - u0\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot u0\right) \cdot t\_0 + t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha u0)
 :precision binary32
 (let* ((t_0 (* (* alpha alpha) u0)))
   (if (<= (- 1.0 u0) 0.9972000122070313)
     (* (/ (pow (- alpha) 3.0) alpha) (log (- 1.0 u0)))
     (+ (* (* 0.5 u0) t_0) t_0))))

C

float code(float alpha, float u0) {
	float t_0 = (alpha * alpha) * u0;
	float tmp;
	if ((1.0f - u0) <= 0.9972000122070313f) {
		tmp = (powf(-alpha, 3.0f) / alpha) * logf((1.0f - u0));
	} else {
		tmp = ((0.5f * u0) * t_0) + t_0;
	}
	return tmp;
}

Fortran

real(4) function code(alpha, u0)
    real(4), intent (in) :: alpha
    real(4), intent (in) :: u0
    real(4) :: t_0
    real(4) :: tmp
    t_0 = (alpha * alpha) * u0
    if ((1.0e0 - u0) <= 0.9972000122070313e0) then
        tmp = ((-alpha ** 3.0e0) / alpha) * log((1.0e0 - u0))
    else
        tmp = ((0.5e0 * u0) * t_0) + t_0
    end if
    code = tmp
end function

Julia

function code(alpha, u0)
	t_0 = Float32(Float32(alpha * alpha) * u0)
	tmp = Float32(0.0)
	if (Float32(Float32(1.0) - u0) <= Float32(0.9972000122070313))
		tmp = Float32(Float32((Float32(-alpha) ^ Float32(3.0)) / alpha) * log(Float32(Float32(1.0) - u0)));
	else
		tmp = Float32(Float32(Float32(Float32(0.5) * u0) * t_0) + t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, u0)
	t_0 = (alpha * alpha) * u0;
	tmp = single(0.0);
	if ((single(1.0) - u0) <= single(0.9972000122070313))
		tmp = ((-alpha ^ single(3.0)) / alpha) * log((single(1.0) - u0));
	else
		tmp = ((single(0.5) * u0) * t_0) + t_0;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (-.f32 #s(literal 1 binary32) u0) < 0.99720001

   1. Initial program 93.5%

      \[\left(\left(-\alpha\right) \cdot \alpha\right) \cdot \log \left(1 - u0\right) \]
   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 93.5

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

   4. Applied rewrites 93.5%

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

      1. lift-*.f32 N/A

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

      2. lift-/.f32 N/A

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

      3. un-div-inv N/A

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

      4. lift-*.f32 N/A

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

      5. lift-*.f32 N/A

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

      6. associate-*l* N/A

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

      7. sqr-neg N/A

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

      8. lift-neg.f32 N/A

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

      9. lift-neg.f32 N/A

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

      10. cube-mult N/A

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

      11. lift-pow.f32 N/A

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

      12. lift-/.f32 93.7

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

   6. Applied rewrites 93.7%

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

   if 0.99720001 < (-.f32 #s(literal 1 binary32) u0)

   1. Initial program 39.8%

      \[\left(\left(-\alpha\right) \cdot \alpha\right) \cdot \log \left(1 - u0\right) \]
   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 39.8

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

   4. Applied rewrites 39.8%

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

      1. lift-*.f32 N/A

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

      2. /-rgt-identity N/A

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

      3. div-inv N/A

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

      4. metadata-eval N/A

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

      5. associate-*l* N/A

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

      6. lift-*.f32 N/A

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

      7. rgt-mult-inverse N/A

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

      8. lift-/.f32 N/A

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

      9. associate-*l* N/A

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

      10. lift-*.f32 N/A

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

      11. *-commutative N/A

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

      12. lift-*.f32 N/A

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

      13. lift-*.f32 N/A

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

      14. associate-*l* N/A

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

      15. lift-*.f32 N/A

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

      16. associate-*r* N/A

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

      17. lower-*.f32 N/A

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

      18. lower-*.f32 39.8

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

   6. Applied rewrites 39.8%

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

   7. Taylor expanded in u0 around 0

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

      1. distribute-lft-in N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. distribute-lft1-in N/A

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

      5. lower-*.f32 N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f32 N/A

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

      8. lower-*.f32 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f32 86.5

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

   9. Applied rewrites 86.5%

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

   11. Applied rewrites 98.1%

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

4. Final simplification 97.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 - u0 \leq 0.9972000122070313:\\ \;\;\;\;\frac{{\left(-\alpha\right)}^{3}}{\alpha} \cdot \log \left(1 - u0\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot u0\right) \cdot \left(\left(\alpha \cdot \alpha\right) \cdot u0\right) + \left(\alpha \cdot \alpha\right) \cdot u0\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 96.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\alpha \cdot \alpha\right) \cdot u0\\ \mathbf{if}\;1 - u0 \leq 0.9972000122070313:\\ \;\;\;\;\left(\frac{1}{\frac{-1}{\alpha}} \cdot \alpha\right) \cdot \log \left(1 - u0\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot u0\right) \cdot t\_0 + t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha u0)
 :precision binary32
 (let* ((t_0 (* (* alpha alpha) u0)))
   (if (<= (- 1.0 u0) 0.9972000122070313)
     (* (* (/ 1.0 (/ -1.0 alpha)) alpha) (log (- 1.0 u0)))
     (+ (* (* 0.5 u0) t_0) t_0))))

C

float code(float alpha, float u0) {
	float t_0 = (alpha * alpha) * u0;
	float tmp;
	if ((1.0f - u0) <= 0.9972000122070313f) {
		tmp = ((1.0f / (-1.0f / alpha)) * alpha) * logf((1.0f - u0));
	} else {
		tmp = ((0.5f * u0) * t_0) + t_0;
	}
	return tmp;
}

Fortran

real(4) function code(alpha, u0)
    real(4), intent (in) :: alpha
    real(4), intent (in) :: u0
    real(4) :: t_0
    real(4) :: tmp
    t_0 = (alpha * alpha) * u0
    if ((1.0e0 - u0) <= 0.9972000122070313e0) then
        tmp = ((1.0e0 / ((-1.0e0) / alpha)) * alpha) * log((1.0e0 - u0))
    else
        tmp = ((0.5e0 * u0) * t_0) + t_0
    end if
    code = tmp
end function

Julia

function code(alpha, u0)
	t_0 = Float32(Float32(alpha * alpha) * u0)
	tmp = Float32(0.0)
	if (Float32(Float32(1.0) - u0) <= Float32(0.9972000122070313))
		tmp = Float32(Float32(Float32(Float32(1.0) / Float32(Float32(-1.0) / alpha)) * alpha) * log(Float32(Float32(1.0) - u0)));
	else
		tmp = Float32(Float32(Float32(Float32(0.5) * u0) * t_0) + t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, u0)
	t_0 = (alpha * alpha) * u0;
	tmp = single(0.0);
	if ((single(1.0) - u0) <= single(0.9972000122070313))
		tmp = ((single(1.0) / (single(-1.0) / alpha)) * alpha) * log((single(1.0) - u0));
	else
		tmp = ((single(0.5) * u0) * t_0) + t_0;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (-.f32 #s(literal 1 binary32) u0) < 0.99720001

   1. Initial program 93.5%

      \[\left(\left(-\alpha\right) \cdot \alpha\right) \cdot \log \left(1 - u0\right) \]
   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 93.5

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

   4. Applied rewrites 93.5%

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

      1. lift-*.f32 N/A

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

      2. /-rgt-identity N/A

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

      3. div-inv N/A

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

      4. metadata-eval N/A

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

      5. associate-*l* N/A

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

      6. lift-*.f32 N/A

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

      7. rgt-mult-inverse N/A

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

      8. lift-/.f32 N/A

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

      9. associate-*l* N/A

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

      10. lift-*.f32 N/A

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

      11. *-commutative N/A

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

      12. lift-*.f32 N/A

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

      13. lift-*.f32 N/A

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

      14. associate-*l* N/A

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

      15. lift-*.f32 N/A

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

      16. associate-*r* N/A

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

      17. lower-*.f32 N/A

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

      18. lower-*.f32 93.4

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

   6. Applied rewrites 93.4%

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

      1. lift-*.f32 N/A

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

      2. lift-*.f32 N/A

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

      3. associate-*l* N/A

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

      4. lift-/.f32 N/A

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

      5. rgt-mult-inverse N/A

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

      6. lift-*.f32 N/A

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

      7. lift-*.f32 N/A

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

      8. associate-*r* N/A

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

      9. associate-*l* N/A

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

      10. lift-*.f32 N/A

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

      11. *-commutative N/A

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

      12. lift-neg.f32 N/A

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

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

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

      14. lift-/.f32 N/A

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

      15. rgt-mult-inverse N/A

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

      16. metadata-eval N/A

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

      17. neg-mul-1 N/A

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

      18. lift-neg.f32 N/A

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

      19. metadata-eval N/A

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

      20. div-inv N/A

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

      21. /-rgt-identity N/A

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

   8. Applied rewrites 93.6%

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

   if 0.99720001 < (-.f32 #s(literal 1 binary32) u0)

   1. Initial program 39.8%

      \[\left(\left(-\alpha\right) \cdot \alpha\right) \cdot \log \left(1 - u0\right) \]
   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 39.8

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

   4. Applied rewrites 39.8%

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

      1. lift-*.f32 N/A

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

      2. /-rgt-identity N/A

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

      3. div-inv N/A

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

      4. metadata-eval N/A

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

      5. associate-*l* N/A

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

      6. lift-*.f32 N/A

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

      7. rgt-mult-inverse N/A

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

      8. lift-/.f32 N/A

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

      9. associate-*l* N/A

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

      10. lift-*.f32 N/A

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

      11. *-commutative N/A

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

      12. lift-*.f32 N/A

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

      13. lift-*.f32 N/A

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

      14. associate-*l* N/A

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

      15. lift-*.f32 N/A

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

      16. associate-*r* N/A

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

      17. lower-*.f32 N/A

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

      18. lower-*.f32 39.8

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

   6. Applied rewrites 39.8%

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

   7. Taylor expanded in u0 around 0

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

      1. distribute-lft-in N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. distribute-lft1-in N/A

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

      5. lower-*.f32 N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f32 N/A

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

      8. lower-*.f32 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f32 51.6

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

   9. Applied rewrites 86.2%

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

   11. Applied rewrites 98.1%

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

4. Final simplification 96.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 - u0 \leq 0.9972000122070313:\\ \;\;\;\;\left(\frac{1}{\frac{-1}{\alpha}} \cdot \alpha\right) \cdot \log \left(1 - u0\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot u0\right) \cdot \left(\left(\alpha \cdot \alpha\right) \cdot u0\right) + \left(\alpha \cdot \alpha\right) \cdot u0\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 96.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\alpha \cdot \alpha\right) \cdot u0\\ \mathbf{if}\;1 - u0 \leq 0.9972000122070313:\\ \;\;\;\;\frac{\left(\left(-\alpha\right) \cdot \alpha\right) \cdot \alpha}{\alpha} \cdot \log \left(1 - u0\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot u0\right) \cdot t\_0 + t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha u0)
 :precision binary32
 (let* ((t_0 (* (* alpha alpha) u0)))
   (if (<= (- 1.0 u0) 0.9972000122070313)
     (* (/ (* (* (- alpha) alpha) alpha) alpha) (log (- 1.0 u0)))
     (+ (* (* 0.5 u0) t_0) t_0))))

C

float code(float alpha, float u0) {
	float t_0 = (alpha * alpha) * u0;
	float tmp;
	if ((1.0f - u0) <= 0.9972000122070313f) {
		tmp = (((-alpha * alpha) * alpha) / alpha) * logf((1.0f - u0));
	} else {
		tmp = ((0.5f * u0) * t_0) + t_0;
	}
	return tmp;
}

Fortran

real(4) function code(alpha, u0)
    real(4), intent (in) :: alpha
    real(4), intent (in) :: u0
    real(4) :: t_0
    real(4) :: tmp
    t_0 = (alpha * alpha) * u0
    if ((1.0e0 - u0) <= 0.9972000122070313e0) then
        tmp = (((-alpha * alpha) * alpha) / alpha) * log((1.0e0 - u0))
    else
        tmp = ((0.5e0 * u0) * t_0) + t_0
    end if
    code = tmp
end function

Julia

function code(alpha, u0)
	t_0 = Float32(Float32(alpha * alpha) * u0)
	tmp = Float32(0.0)
	if (Float32(Float32(1.0) - u0) <= Float32(0.9972000122070313))
		tmp = Float32(Float32(Float32(Float32(Float32(-alpha) * alpha) * alpha) / alpha) * log(Float32(Float32(1.0) - u0)));
	else
		tmp = Float32(Float32(Float32(Float32(0.5) * u0) * t_0) + t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, u0)
	t_0 = (alpha * alpha) * u0;
	tmp = single(0.0);
	if ((single(1.0) - u0) <= single(0.9972000122070313))
		tmp = (((-alpha * alpha) * alpha) / alpha) * log((single(1.0) - u0));
	else
		tmp = ((single(0.5) * u0) * t_0) + t_0;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (-.f32 #s(literal 1 binary32) u0) < 0.99720001

   1. Initial program 93.5%

      \[\left(\left(-\alpha\right) \cdot \alpha\right) \cdot \log \left(1 - u0\right) \]
   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. lower-/.f32 N/A

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

      13. lower-*.f32 93.6

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

   4. Applied rewrites 93.6%

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

   if 0.99720001 < (-.f32 #s(literal 1 binary32) u0)

   1. Initial program 39.8%

      \[\left(\left(-\alpha\right) \cdot \alpha\right) \cdot \log \left(1 - u0\right) \]
   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 39.8

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

   4. Applied rewrites 39.8%

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

      1. lift-*.f32 N/A

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

      2. /-rgt-identity N/A

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

      3. div-inv N/A

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

      4. metadata-eval N/A

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

      5. associate-*l* N/A

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

      6. lift-*.f32 N/A

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

      7. rgt-mult-inverse N/A

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

      8. lift-/.f32 N/A

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

      9. associate-*l* N/A

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

      10. lift-*.f32 N/A

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

      11. *-commutative N/A

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

      12. lift-*.f32 N/A

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

      13. lift-*.f32 N/A

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

      14. associate-*l* N/A

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

      15. lift-*.f32 N/A

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

      16. associate-*r* N/A

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

      17. lower-*.f32 N/A

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

      18. lower-*.f32 39.8

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

   6. Applied rewrites 39.8%

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

   7. Taylor expanded in u0 around 0

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

      1. distribute-lft-in N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. distribute-lft1-in N/A

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

      5. lower-*.f32 N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f32 N/A

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

      8. lower-*.f32 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f32 56.2

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

   9. Applied rewrites 86.2%

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

   11. Applied rewrites 98.1%

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

4. Final simplification 96.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 - u0 \leq 0.9972000122070313:\\ \;\;\;\;\frac{\left(\left(-\alpha\right) \cdot \alpha\right) \cdot \alpha}{\alpha} \cdot \log \left(1 - u0\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot u0\right) \cdot \left(\left(\alpha \cdot \alpha\right) \cdot u0\right) + \left(\alpha \cdot \alpha\right) \cdot u0\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 96.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\alpha \cdot \alpha\right) \cdot u0\\ \mathbf{if}\;1 - u0 \leq 0.9972000122070313:\\ \;\;\;\;\left(\left(-\alpha\right) \cdot \alpha\right) \cdot \log \left(1 - u0\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot u0\right) \cdot t\_0 + t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha u0)
 :precision binary32
 (let* ((t_0 (* (* alpha alpha) u0)))
   (if (<= (- 1.0 u0) 0.9972000122070313)
     (* (* (- alpha) alpha) (log (- 1.0 u0)))
     (+ (* (* 0.5 u0) t_0) t_0))))

C

float code(float alpha, float u0) {
	float t_0 = (alpha * alpha) * u0;
	float tmp;
	if ((1.0f - u0) <= 0.9972000122070313f) {
		tmp = (-alpha * alpha) * logf((1.0f - u0));
	} else {
		tmp = ((0.5f * u0) * t_0) + t_0;
	}
	return tmp;
}

Fortran

real(4) function code(alpha, u0)
    real(4), intent (in) :: alpha
    real(4), intent (in) :: u0
    real(4) :: t_0
    real(4) :: tmp
    t_0 = (alpha * alpha) * u0
    if ((1.0e0 - u0) <= 0.9972000122070313e0) then
        tmp = (-alpha * alpha) * log((1.0e0 - u0))
    else
        tmp = ((0.5e0 * u0) * t_0) + t_0
    end if
    code = tmp
end function

Julia

function code(alpha, u0)
	t_0 = Float32(Float32(alpha * alpha) * u0)
	tmp = Float32(0.0)
	if (Float32(Float32(1.0) - u0) <= Float32(0.9972000122070313))
		tmp = Float32(Float32(Float32(-alpha) * alpha) * log(Float32(Float32(1.0) - u0)));
	else
		tmp = Float32(Float32(Float32(Float32(0.5) * u0) * t_0) + t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, u0)
	t_0 = (alpha * alpha) * u0;
	tmp = single(0.0);
	if ((single(1.0) - u0) <= single(0.9972000122070313))
		tmp = (-alpha * alpha) * log((single(1.0) - u0));
	else
		tmp = ((single(0.5) * u0) * t_0) + t_0;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (-.f32 #s(literal 1 binary32) u0) < 0.99720001

   1. Initial program 93.5%

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

   if 0.99720001 < (-.f32 #s(literal 1 binary32) u0)

   1. Initial program 39.8%

      \[\left(\left(-\alpha\right) \cdot \alpha\right) \cdot \log \left(1 - u0\right) \]
   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 39.8

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

   4. Applied rewrites 39.8%

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

      1. lift-*.f32 N/A

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

      2. /-rgt-identity N/A

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

      3. div-inv N/A

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

      4. metadata-eval N/A

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

      5. associate-*l* N/A

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

      6. lift-*.f32 N/A

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

      7. rgt-mult-inverse N/A

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

      8. lift-/.f32 N/A

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

      9. associate-*l* N/A

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

      10. lift-*.f32 N/A

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

      11. *-commutative N/A

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

      12. lift-*.f32 N/A

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

      13. lift-*.f32 N/A

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

      14. associate-*l* N/A

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

      15. lift-*.f32 N/A

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

      16. associate-*r* N/A

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

      17. lower-*.f32 N/A

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

      18. lower-*.f32 39.8

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

   6. Applied rewrites 39.8%

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

   7. Taylor expanded in u0 around 0

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

      1. distribute-lft-in N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. distribute-lft1-in N/A

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

      5. lower-*.f32 N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f32 N/A

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

      8. lower-*.f32 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f32 86.5

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

   9. Applied rewrites 86.2%

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

   11. Applied rewrites 98.1%

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

4. Final simplification 96.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 - u0 \leq 0.9972000122070313:\\ \;\;\;\;\left(\left(-\alpha\right) \cdot \alpha\right) \cdot \log \left(1 - u0\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot u0\right) \cdot \left(\left(\alpha \cdot \alpha\right) \cdot u0\right) + \left(\alpha \cdot \alpha\right) \cdot u0\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 96.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\alpha \cdot \alpha\right) \cdot u0\\ \mathbf{if}\;1 - u0 \leq 0.9972000122070313:\\ \;\;\;\;\left(\left(-\alpha\right) \cdot \log \left(1 - u0\right)\right) \cdot \alpha\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot u0\right) \cdot t\_0 + t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha u0)
 :precision binary32
 (let* ((t_0 (* (* alpha alpha) u0)))
   (if (<= (- 1.0 u0) 0.9972000122070313)
     (* (* (- alpha) (log (- 1.0 u0))) alpha)
     (+ (* (* 0.5 u0) t_0) t_0))))

C

float code(float alpha, float u0) {
	float t_0 = (alpha * alpha) * u0;
	float tmp;
	if ((1.0f - u0) <= 0.9972000122070313f) {
		tmp = (-alpha * logf((1.0f - u0))) * alpha;
	} else {
		tmp = ((0.5f * u0) * t_0) + t_0;
	}
	return tmp;
}

Fortran

real(4) function code(alpha, u0)
    real(4), intent (in) :: alpha
    real(4), intent (in) :: u0
    real(4) :: t_0
    real(4) :: tmp
    t_0 = (alpha * alpha) * u0
    if ((1.0e0 - u0) <= 0.9972000122070313e0) then
        tmp = (-alpha * log((1.0e0 - u0))) * alpha
    else
        tmp = ((0.5e0 * u0) * t_0) + t_0
    end if
    code = tmp
end function

Julia

function code(alpha, u0)
	t_0 = Float32(Float32(alpha * alpha) * u0)
	tmp = Float32(0.0)
	if (Float32(Float32(1.0) - u0) <= Float32(0.9972000122070313))
		tmp = Float32(Float32(Float32(-alpha) * log(Float32(Float32(1.0) - u0))) * alpha);
	else
		tmp = Float32(Float32(Float32(Float32(0.5) * u0) * t_0) + t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, u0)
	t_0 = (alpha * alpha) * u0;
	tmp = single(0.0);
	if ((single(1.0) - u0) <= single(0.9972000122070313))
		tmp = (-alpha * log((single(1.0) - u0))) * alpha;
	else
		tmp = ((single(0.5) * u0) * t_0) + t_0;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (-.f32 #s(literal 1 binary32) u0) < 0.99720001

   1. Initial program 93.5%

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

   3. Taylor expanded in alpha around 0

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

      1. mul-1-neg N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

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

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

      5. *-commutative N/A

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

      6. lower-*.f32 N/A

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

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

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

      8. mul-1-neg N/A

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

      9. lower-*.f32 N/A

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

      10. mul-1-neg N/A

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

      11. lower-neg.f32 N/A

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

      12. sub-neg N/A

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

      13. lower-log1p.f32 N/A

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

      14. lower-neg.f32 41.2

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

   5. Applied rewrites 41.2%

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

   7. Applied rewrites 93.3%

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

   if 0.99720001 < (-.f32 #s(literal 1 binary32) u0)

   1. Initial program 39.8%

      \[\left(\left(-\alpha\right) \cdot \alpha\right) \cdot \log \left(1 - u0\right) \]
   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 39.8

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

   4. Applied rewrites 39.8%

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

      1. lift-*.f32 N/A

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

      2. /-rgt-identity N/A

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

      3. div-inv N/A

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

      4. metadata-eval N/A

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

      5. associate-*l* N/A

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

      6. lift-*.f32 N/A

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

      7. rgt-mult-inverse N/A

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

      8. lift-/.f32 N/A

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

      9. associate-*l* N/A

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

      10. lift-*.f32 N/A

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

      11. *-commutative N/A

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

      12. lift-*.f32 N/A

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

      13. lift-*.f32 N/A

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

      14. associate-*l* N/A

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

      15. lift-*.f32 N/A

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

      16. associate-*r* N/A

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

      17. lower-*.f32 N/A

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

      18. lower-*.f32 39.8

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

   6. Applied rewrites 39.8%

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

   7. Taylor expanded in u0 around 0

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

      1. distribute-lft-in N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. distribute-lft1-in N/A

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

      5. lower-*.f32 N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f32 N/A

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

      8. lower-*.f32 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f32 84.0

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

   9. Applied rewrites 86.2%

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

   11. Applied rewrites 98.1%

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

4. Final simplification 96.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 - u0 \leq 0.9972000122070313:\\ \;\;\;\;\left(\left(-\alpha\right) \cdot \log \left(1 - u0\right)\right) \cdot \alpha\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot u0\right) \cdot \left(\left(\alpha \cdot \alpha\right) \cdot u0\right) + \left(\alpha \cdot \alpha\right) \cdot u0\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 87.3% accurate, 3.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\alpha \cdot \alpha\right) \cdot u0\\ \left(0.5 \cdot u0\right) \cdot t\_0 + t\_0 \end{array} \end{array} \]

FPCore

(FPCore (alpha u0)
 :precision binary32
 (let* ((t_0 (* (* alpha alpha) u0))) (+ (* (* 0.5 u0) t_0) t_0)))

C

float code(float alpha, float u0) {
	float t_0 = (alpha * alpha) * u0;
	return ((0.5f * u0) * t_0) + t_0;
}

Fortran

real(4) function code(alpha, u0)
    real(4), intent (in) :: alpha
    real(4), intent (in) :: u0
    real(4) :: t_0
    t_0 = (alpha * alpha) * u0
    code = ((0.5e0 * u0) * t_0) + t_0
end function

Julia

function code(alpha, u0)
	t_0 = Float32(Float32(alpha * alpha) * u0)
	return Float32(Float32(Float32(Float32(0.5) * u0) * t_0) + t_0)
end

MATLAB

function tmp = code(alpha, u0)
	t_0 = (alpha * alpha) * u0;
	tmp = ((single(0.5) * u0) * t_0) + t_0;
end

Derivation

1. Initial program 53.4%

   \[\left(\left(-\alpha\right) \cdot \alpha\right) \cdot \log \left(1 - u0\right) \]
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 53.4

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

4. Applied rewrites 53.4%

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

   1. lift-*.f32 N/A

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

   2. /-rgt-identity N/A

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

   3. div-inv N/A

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

   4. metadata-eval N/A

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

   5. associate-*l* N/A

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

   6. lift-*.f32 N/A

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

   7. rgt-mult-inverse N/A

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

   8. lift-/.f32 N/A

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

   9. associate-*l* N/A

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

   10. lift-*.f32 N/A

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

   11. *-commutative N/A

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

   12. lift-*.f32 N/A

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

   13. lift-*.f32 N/A

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

   14. associate-*l* N/A

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

   15. lift-*.f32 N/A

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

   16. associate-*r* N/A

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

   17. lower-*.f32 N/A

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

   18. lower-*.f32 53.4

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

6. Applied rewrites 53.4%

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

7. Taylor expanded in u0 around 0

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

   1. distribute-lft-in N/A

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

   2. associate-*r* N/A

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

   3. *-commutative N/A

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

   4. distribute-lft1-in N/A

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

   5. lower-*.f32 N/A

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

   6. *-commutative N/A

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

   7. lower-fma.f32 N/A

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

   8. lower-*.f32 N/A

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

   9. unpow2 N/A

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

   10. lower-*.f32 43.4

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

9. Applied rewrites 74.8%

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

11. Applied rewrites 87.2%

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

12. Final simplification 87.2%

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

Alternative 8: 87.1% accurate, 4.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 53.4%

   \[\left(\left(-\alpha\right) \cdot \alpha\right) \cdot \log \left(1 - u0\right) \]
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 53.4

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

4. Applied rewrites 53.4%

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

   1. lift-*.f32 N/A

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

   2. /-rgt-identity N/A

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

   3. div-inv N/A

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

   4. metadata-eval N/A

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

   5. associate-*l* N/A

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

   6. lift-*.f32 N/A

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

   7. rgt-mult-inverse N/A

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

   8. lift-/.f32 N/A

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

   9. associate-*l* N/A

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

   10. lift-*.f32 N/A

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

   11. *-commutative N/A

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

   12. lift-*.f32 N/A

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

   13. lift-*.f32 N/A

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

   14. associate-*l* N/A

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

   15. lift-*.f32 N/A

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

   16. associate-*r* N/A

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

   17. lower-*.f32 N/A

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

   18. lower-*.f32 53.4

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

6. Applied rewrites 53.4%

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

7. Taylor expanded in u0 around 0

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

   1. distribute-lft-in N/A

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

   2. associate-*r* N/A

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

   3. *-commutative N/A

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

   4. distribute-lft1-in N/A

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

   5. lower-*.f32 N/A

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

   6. *-commutative N/A

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

   7. lower-fma.f32 N/A

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

   8. lower-*.f32 N/A

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

   9. unpow2 N/A

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

   10. lower-*.f32 75.0

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

9. Applied rewrites 74.8%

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

11. Applied rewrites 87.0%

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

Alternative 9: 74.5% accurate, 10.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 53.4%

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

3. Taylor expanded in u0 around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. unpow2 N/A

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

   4. lower-*.f32 75.0

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

5. Applied rewrites 75.0%

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

7. Applied rewrites 75.0%

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

8. Final simplification 75.0%

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

Alternative 10: 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 53.4%

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

3. Taylor expanded in u0 around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. unpow2 N/A

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

   4. lower-*.f32 75.0

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

5. Applied rewrites 75.0%

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

6. Final simplification 75.0%

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

Reproduce

herbie shell --seed 2024264 
(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))))