Beckmann Distribution sample, tan2theta, alphax == alphay

Percentage Accurate: 56.2% → 89.5%

Time: 7.3s

Alternatives: 6

Speedup: 10.5×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Initial Program: 56.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Alternative 1: 89.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;1 - u0 \leq 0.999875009059906:\\ \;\;\;\;\log \left(1 - u0\right) \cdot \frac{-1}{{\alpha}^{-2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\alpha} \cdot \left(\left(\left(\alpha \cdot \alpha\right) \cdot \left(-u0\right)\right) \cdot \alpha\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha u0)
 :precision binary32
 (if (<= (- 1.0 u0) 0.999875009059906)
   (* (log (- 1.0 u0)) (/ -1.0 (pow alpha -2.0)))
   (* (/ -1.0 alpha) (* (* (* alpha alpha) (- u0)) alpha))))

C

float code(float alpha, float u0) {
	float tmp;
	if ((1.0f - u0) <= 0.999875009059906f) {
		tmp = logf((1.0f - u0)) * (-1.0f / powf(alpha, -2.0f));
	} else {
		tmp = (-1.0f / alpha) * (((alpha * alpha) * -u0) * alpha);
	}
	return tmp;
}

Fortran

real(4) function code(alpha, u0)
    real(4), intent (in) :: alpha
    real(4), intent (in) :: u0
    real(4) :: tmp
    if ((1.0e0 - u0) <= 0.999875009059906e0) then
        tmp = log((1.0e0 - u0)) * ((-1.0e0) / (alpha ** (-2.0e0)))
    else
        tmp = ((-1.0e0) / alpha) * (((alpha * alpha) * -u0) * alpha)
    end if
    code = tmp
end function

Julia

function code(alpha, u0)
	tmp = Float32(0.0)
	if (Float32(Float32(1.0) - u0) <= Float32(0.999875009059906))
		tmp = Float32(log(Float32(Float32(1.0) - u0)) * Float32(Float32(-1.0) / (alpha ^ Float32(-2.0))));
	else
		tmp = Float32(Float32(Float32(-1.0) / alpha) * Float32(Float32(Float32(alpha * alpha) * Float32(-u0)) * alpha));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, u0)
	tmp = single(0.0);
	if ((single(1.0) - u0) <= single(0.999875009059906))
		tmp = log((single(1.0) - u0)) * (single(-1.0) / (alpha ^ single(-2.0)));
	else
		tmp = (single(-1.0) / alpha) * (((alpha * alpha) * -u0) * alpha);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (-.f32 #s(literal 1 binary32) u0) < 0.999875009

   1. Initial program 87.4%

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

      1. +-lft-identity N/A

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

      2. flip-+ N/A

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

      3. neg-sub0 N/A

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

      4. lift-*.f32 N/A

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

      5. lift-neg.f32 N/A

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

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

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

      7. remove-double-neg N/A

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

      8. lower-/.f32 N/A

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

      9. metadata-eval N/A

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

      10. lower--.f32 N/A

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

      11. lower-*.f32 N/A

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

      12. lower-*.f32 87.4

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

   4. Applied rewrites 87.4%

      \[\leadsto \color{blue}{\frac{0 - \left(\left(-\alpha\right) \cdot \alpha\right) \cdot \left(\left(-\alpha\right) \cdot \alpha\right)}{\alpha \cdot \alpha}} \cdot \log \left(1 - u0\right) \]
   5. Step-by-step derivation

      1. lift-/.f32 N/A

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

      2. clear-num N/A

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

      3. frac-2neg N/A

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

      4. metadata-eval N/A

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

      5. lift--.f32 N/A

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

      6. sub0-neg N/A

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

      7. distribute-frac-neg2 N/A

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

      8. lift-*.f32 N/A

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

      9. pow2 N/A

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

      10. lift-*.f32 N/A

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

      11. lift-*.f32 N/A

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

      12. lift-*.f32 N/A

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

      13. swap-sqr N/A

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

      14. lift-neg.f32 N/A

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

      15. lift-neg.f32 N/A

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

      16. sqr-neg N/A

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

      17. pow2 N/A

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

      18. pow2 N/A

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

      19. pow-sqr N/A

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

      20. metadata-eval N/A

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

   6. Applied rewrites 87.5%

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

   if 0.999875009 < (-.f32 #s(literal 1 binary32) u0)

   1. Initial program 35.1%

      \[\left(\left(-\alpha\right) \cdot \alpha\right) \cdot \log \left(1 - u0\right) \]
   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 35.1

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

   4. Applied rewrites 35.1%

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

   5. Taylor expanded in u0 around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f32 91.2

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

   7. Applied rewrites 91.2%

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

      1. lift-*.f32 N/A

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

      2. lift-/.f32 N/A

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

      3. associate-*l/ N/A

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

      4. lift-*.f32 N/A

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

      5. lift-*.f32 N/A

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

      6. associate-*l* N/A

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

      7. lift-*.f32 N/A

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

      8. associate-*l* N/A

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

      9. associate-/l* N/A

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

      10. lower-*.f32 N/A

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

      11. lower-/.f32 N/A

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

      12. lower-*.f32 91.2

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

   9. Applied rewrites 91.2%

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

       1. lift-*.f32 N/A

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

       2. lift-/.f32 N/A

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

       3. associate-*r/ N/A

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

       4. div-inv N/A

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

       5. lower-*.f32 N/A

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

       6. lower-*.f32 N/A

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

       7. lift-*.f32 N/A

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

       8. *-commutative N/A

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

       9. lower-*.f32 N/A

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

       10. lower-/.f32 91.3

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

   11. Applied rewrites 91.3%

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

4. Final simplification 89.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 - u0 \leq 0.999875009059906:\\ \;\;\;\;\log \left(1 - u0\right) \cdot \frac{-1}{{\alpha}^{-2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\alpha} \cdot \left(\left(\left(\alpha \cdot \alpha\right) \cdot \left(-u0\right)\right) \cdot \alpha\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 89.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;1 - u0 \leq 0.999875009059906:\\ \;\;\;\;\frac{1}{\frac{-1}{\alpha \cdot \alpha}} \cdot \log \left(1 - u0\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\alpha} \cdot \left(\left(\left(\alpha \cdot \alpha\right) \cdot \left(-u0\right)\right) \cdot \alpha\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha u0)
 :precision binary32
 (if (<= (- 1.0 u0) 0.999875009059906)
   (* (/ 1.0 (/ -1.0 (* alpha alpha))) (log (- 1.0 u0)))
   (* (/ -1.0 alpha) (* (* (* alpha alpha) (- u0)) alpha))))

C

float code(float alpha, float u0) {
	float tmp;
	if ((1.0f - u0) <= 0.999875009059906f) {
		tmp = (1.0f / (-1.0f / (alpha * alpha))) * logf((1.0f - u0));
	} else {
		tmp = (-1.0f / alpha) * (((alpha * alpha) * -u0) * alpha);
	}
	return tmp;
}

Fortran

real(4) function code(alpha, u0)
    real(4), intent (in) :: alpha
    real(4), intent (in) :: u0
    real(4) :: tmp
    if ((1.0e0 - u0) <= 0.999875009059906e0) then
        tmp = (1.0e0 / ((-1.0e0) / (alpha * alpha))) * log((1.0e0 - u0))
    else
        tmp = ((-1.0e0) / alpha) * (((alpha * alpha) * -u0) * alpha)
    end if
    code = tmp
end function

Julia

function code(alpha, u0)
	tmp = Float32(0.0)
	if (Float32(Float32(1.0) - u0) <= Float32(0.999875009059906))
		tmp = Float32(Float32(Float32(1.0) / Float32(Float32(-1.0) / Float32(alpha * alpha))) * log(Float32(Float32(1.0) - u0)));
	else
		tmp = Float32(Float32(Float32(-1.0) / alpha) * Float32(Float32(Float32(alpha * alpha) * Float32(-u0)) * alpha));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, u0)
	tmp = single(0.0);
	if ((single(1.0) - u0) <= single(0.999875009059906))
		tmp = (single(1.0) / (single(-1.0) / (alpha * alpha))) * log((single(1.0) - u0));
	else
		tmp = (single(-1.0) / alpha) * (((alpha * alpha) * -u0) * alpha);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (-.f32 #s(literal 1 binary32) u0) < 0.999875009

   1. Initial program 87.4%

      \[\left(\left(-\alpha\right) \cdot \alpha\right) \cdot \log \left(1 - u0\right) \]
   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. 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. flip3-- N/A

         \[\leadsto \color{blue}{\frac{{0}^{3} - {\left(\alpha \cdot \alpha\right)}^{3}}{0 \cdot 0 + \left(\left(\alpha \cdot \alpha\right) \cdot \left(\alpha \cdot \alpha\right) + 0 \cdot \left(\alpha \cdot \alpha\right)\right)}} \cdot \log \left(1 - u0\right) \]

      6. clear-num N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{0 \cdot 0 + \left(\left(\alpha \cdot \alpha\right) \cdot \left(\alpha \cdot \alpha\right) + 0 \cdot \left(\alpha \cdot \alpha\right)\right)}{{0}^{3} - {\left(\alpha \cdot \alpha\right)}^{3}}}} \cdot \log \left(1 - u0\right) \]

      7. lower-/.f32 N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{0 \cdot 0 + \left(\left(\alpha \cdot \alpha\right) \cdot \left(\alpha \cdot \alpha\right) + 0 \cdot \left(\alpha \cdot \alpha\right)\right)}{{0}^{3} - {\left(\alpha \cdot \alpha\right)}^{3}}}} \cdot \log \left(1 - u0\right) \]

      8. metadata-eval N/A

         \[\leadsto \frac{1}{\frac{\color{blue}{0} + \left(\left(\alpha \cdot \alpha\right) \cdot \left(\alpha \cdot \alpha\right) + 0 \cdot \left(\alpha \cdot \alpha\right)\right)}{{0}^{3} - {\left(\alpha \cdot \alpha\right)}^{3}}} \cdot \log \left(1 - u0\right) \]

      9. +-lft-identity N/A

         \[\leadsto \frac{1}{\frac{\color{blue}{\left(\alpha \cdot \alpha\right) \cdot \left(\alpha \cdot \alpha\right) + 0 \cdot \left(\alpha \cdot \alpha\right)}}{{0}^{3} - {\left(\alpha \cdot \alpha\right)}^{3}}} \cdot \log \left(1 - u0\right) \]

      10. mul0-lft N/A

          \[\leadsto \frac{1}{\frac{\left(\alpha \cdot \alpha\right) \cdot \left(\alpha \cdot \alpha\right) + \color{blue}{0}}{{0}^{3} - {\left(\alpha \cdot \alpha\right)}^{3}}} \cdot \log \left(1 - u0\right) \]

      11. metadata-eval N/A

          \[\leadsto \frac{1}{\frac{\left(\alpha \cdot \alpha\right) \cdot \left(\alpha \cdot \alpha\right) + \color{blue}{\left(\mathsf{neg}\left(0\right)\right)}}{{0}^{3} - {\left(\alpha \cdot \alpha\right)}^{3}}} \cdot \log \left(1 - u0\right) \]

      12. sub-neg N/A

          \[\leadsto \frac{1}{\frac{\color{blue}{\left(\alpha \cdot \alpha\right) \cdot \left(\alpha \cdot \alpha\right) - 0}}{{0}^{3} - {\left(\alpha \cdot \alpha\right)}^{3}}} \cdot \log \left(1 - u0\right) \]

      13. --rgt-identity N/A

          \[\leadsto \frac{1}{\frac{\color{blue}{\left(\alpha \cdot \alpha\right) \cdot \left(\alpha \cdot \alpha\right)}}{{0}^{3} - {\left(\alpha \cdot \alpha\right)}^{3}}} \cdot \log \left(1 - u0\right) \]

      14. lower-/.f32 N/A

          \[\leadsto \frac{1}{\color{blue}{\frac{\left(\alpha \cdot \alpha\right) \cdot \left(\alpha \cdot \alpha\right)}{{0}^{3} - {\left(\alpha \cdot \alpha\right)}^{3}}}} \cdot \log \left(1 - u0\right) \]

      15. pow2 N/A

          \[\leadsto \frac{1}{\frac{\color{blue}{{\alpha}^{2}} \cdot \left(\alpha \cdot \alpha\right)}{{0}^{3} - {\left(\alpha \cdot \alpha\right)}^{3}}} \cdot \log \left(1 - u0\right) \]

      16. pow2 N/A

          \[\leadsto \frac{1}{\frac{{\alpha}^{2} \cdot \color{blue}{{\alpha}^{2}}}{{0}^{3} - {\left(\alpha \cdot \alpha\right)}^{3}}} \cdot \log \left(1 - u0\right) \]

      17. pow-prod-up N/A

          \[\leadsto \frac{1}{\frac{\color{blue}{{\alpha}^{\left(2 + 2\right)}}}{{0}^{3} - {\left(\alpha \cdot \alpha\right)}^{3}}} \cdot \log \left(1 - u0\right) \]

      18. lower-pow.f32 N/A

          \[\leadsto \frac{1}{\frac{\color{blue}{{\alpha}^{\left(2 + 2\right)}}}{{0}^{3} - {\left(\alpha \cdot \alpha\right)}^{3}}} \cdot \log \left(1 - u0\right) \]

      19. metadata-eval N/A

          \[\leadsto \frac{1}{\frac{{\alpha}^{\color{blue}{4}}}{{0}^{3} - {\left(\alpha \cdot \alpha\right)}^{3}}} \cdot \log \left(1 - u0\right) \]

      20. metadata-eval N/A

          \[\leadsto \frac{1}{\frac{{\alpha}^{4}}{\color{blue}{0} - {\left(\alpha \cdot \alpha\right)}^{3}}} \cdot \log \left(1 - u0\right) \]

      21. sub0-neg N/A

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

   4. Applied rewrites 87.4%

      \[\leadsto \color{blue}{\frac{1}{\frac{{\alpha}^{4}}{-{\alpha}^{6}}}} \cdot \log \left(1 - u0\right) \]

   5. Taylor expanded in alpha around 0

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

      1. lower-/.f32 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f32 87.4

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

   7. Applied rewrites 87.4%

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

   if 0.999875009 < (-.f32 #s(literal 1 binary32) u0)

   1. Initial program 35.1%

      \[\left(\left(-\alpha\right) \cdot \alpha\right) \cdot \log \left(1 - u0\right) \]
   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 35.1

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

   4. Applied rewrites 35.1%

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

   5. Taylor expanded in u0 around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f32 91.2

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

   7. Applied rewrites 91.2%

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

      1. lift-*.f32 N/A

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

      2. lift-/.f32 N/A

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

      3. associate-*l/ N/A

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

      4. lift-*.f32 N/A

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

      5. lift-*.f32 N/A

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

      6. associate-*l* N/A

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

      7. lift-*.f32 N/A

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

      8. associate-*l* N/A

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

      9. associate-/l* N/A

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

      10. lower-*.f32 N/A

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

      11. lower-/.f32 N/A

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

      12. lower-*.f32 91.2

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

   9. Applied rewrites 91.2%

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

       1. lift-*.f32 N/A

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

       2. lift-/.f32 N/A

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

       3. associate-*r/ N/A

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

       4. div-inv N/A

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

       5. lower-*.f32 N/A

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

       6. lower-*.f32 N/A

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

       7. lift-*.f32 N/A

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

       8. *-commutative N/A

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

       9. lower-*.f32 N/A

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

       10. lower-/.f32 91.3

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

   11. Applied rewrites 91.3%

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

4. Final simplification 89.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 - u0 \leq 0.999875009059906:\\ \;\;\;\;\frac{1}{\frac{-1}{\alpha \cdot \alpha}} \cdot \log \left(1 - u0\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\alpha} \cdot \left(\left(\left(\alpha \cdot \alpha\right) \cdot \left(-u0\right)\right) \cdot \alpha\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 89.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;1 - u0 \leq 0.999875009059906:\\ \;\;\;\;\left(\left(-\alpha\right) \cdot \alpha\right) \cdot \log \left(1 - u0\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\alpha} \cdot \left(\left(\left(\alpha \cdot \alpha\right) \cdot \left(-u0\right)\right) \cdot \alpha\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha u0)
 :precision binary32
 (if (<= (- 1.0 u0) 0.999875009059906)
   (* (* (- alpha) alpha) (log (- 1.0 u0)))
   (* (/ -1.0 alpha) (* (* (* alpha alpha) (- u0)) alpha))))

C

float code(float alpha, float u0) {
	float tmp;
	if ((1.0f - u0) <= 0.999875009059906f) {
		tmp = (-alpha * alpha) * logf((1.0f - u0));
	} else {
		tmp = (-1.0f / alpha) * (((alpha * alpha) * -u0) * alpha);
	}
	return tmp;
}

Fortran

real(4) function code(alpha, u0)
    real(4), intent (in) :: alpha
    real(4), intent (in) :: u0
    real(4) :: tmp
    if ((1.0e0 - u0) <= 0.999875009059906e0) then
        tmp = (-alpha * alpha) * log((1.0e0 - u0))
    else
        tmp = ((-1.0e0) / alpha) * (((alpha * alpha) * -u0) * alpha)
    end if
    code = tmp
end function

Julia

function code(alpha, u0)
	tmp = Float32(0.0)
	if (Float32(Float32(1.0) - u0) <= Float32(0.999875009059906))
		tmp = Float32(Float32(Float32(-alpha) * alpha) * log(Float32(Float32(1.0) - u0)));
	else
		tmp = Float32(Float32(Float32(-1.0) / alpha) * Float32(Float32(Float32(alpha * alpha) * Float32(-u0)) * alpha));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, u0)
	tmp = single(0.0);
	if ((single(1.0) - u0) <= single(0.999875009059906))
		tmp = (-alpha * alpha) * log((single(1.0) - u0));
	else
		tmp = (single(-1.0) / alpha) * (((alpha * alpha) * -u0) * alpha);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (-.f32 #s(literal 1 binary32) u0) < 0.999875009

   1. Initial program 87.4%

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

   if 0.999875009 < (-.f32 #s(literal 1 binary32) u0)

   1. Initial program 35.1%

      \[\left(\left(-\alpha\right) \cdot \alpha\right) \cdot \log \left(1 - u0\right) \]
   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 35.1

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

   4. Applied rewrites 35.1%

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

   5. Taylor expanded in u0 around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f32 91.2

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

   7. Applied rewrites 91.2%

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

      1. lift-*.f32 N/A

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

      2. lift-/.f32 N/A

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

      3. associate-*l/ N/A

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

      4. lift-*.f32 N/A

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

      5. lift-*.f32 N/A

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

      6. associate-*l* N/A

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

      7. lift-*.f32 N/A

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

      8. associate-*l* N/A

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

      9. associate-/l* N/A

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

      10. lower-*.f32 N/A

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

      11. lower-/.f32 N/A

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

      12. lower-*.f32 91.2

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

   9. Applied rewrites 91.2%

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

       1. lift-*.f32 N/A

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

       2. lift-/.f32 N/A

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

       3. associate-*r/ N/A

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

       4. div-inv N/A

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

       5. lower-*.f32 N/A

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

       6. lower-*.f32 N/A

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

       7. lift-*.f32 N/A

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

       8. *-commutative N/A

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

       9. lower-*.f32 N/A

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

       10. lower-/.f32 91.3

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

   11. Applied rewrites 91.3%

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

4. Final simplification 89.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 - u0 \leq 0.999875009059906:\\ \;\;\;\;\left(\left(-\alpha\right) \cdot \alpha\right) \cdot \log \left(1 - u0\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\alpha} \cdot \left(\left(\left(\alpha \cdot \alpha\right) \cdot \left(-u0\right)\right) \cdot \alpha\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 74.3% accurate, 3.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 57.4%

   \[\left(\left(-\alpha\right) \cdot \alpha\right) \cdot \log \left(1 - u0\right) \]
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. 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 57.3

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

4. Applied rewrites 57.3%

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

5. Taylor expanded in u0 around 0

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

   1. mul-1-neg N/A

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

   2. lower-neg.f32 72.9

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

7. Applied rewrites 72.9%

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

   1. lift-*.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. associate-*l/ N/A

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

   4. lift-*.f32 N/A

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

   5. lift-*.f32 N/A

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

   6. associate-*l* N/A

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

   7. lift-*.f32 N/A

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

   8. associate-*l* N/A

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

   9. associate-/l* N/A

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

   10. lower-*.f32 N/A

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

   11. lower-/.f32 N/A

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

   12. lower-*.f32 72.9

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

9. Applied rewrites 72.9%

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

    1. lift-*.f32 N/A

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

    2. lift-/.f32 N/A

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

    3. associate-*r/ N/A

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

    4. div-inv N/A

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

    5. lower-*.f32 N/A

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

    6. lower-*.f32 N/A

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

    7. lift-*.f32 N/A

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

    8. *-commutative N/A

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

    9. lower-*.f32 N/A

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

    10. lower-/.f32 73.0

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

11. Applied rewrites 73.0%

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

12. Final simplification 73.0%

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

Alternative 5: 74.4% 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 57.4%

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

3. Taylor expanded in u0 around 0

   \[\leadsto \color{blue}{{\alpha}^{2} \cdot u0} \]
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 73.0

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

5. Applied rewrites 73.0%

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

7. Applied rewrites 73.0%

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

8. Final simplification 73.0%

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

Alternative 6: 74.4% 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 57.4%

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

3. Taylor expanded in u0 around 0

   \[\leadsto \color{blue}{{\alpha}^{2} \cdot u0} \]
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 73.0

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

5. Applied rewrites 73.0%

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

6. Final simplification 73.0%

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

Reproduce

herbie shell --seed 2024249 
(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))))