Beckmann Distribution sample, tan2theta, alphax == alphay

Percentage Accurate: 55.8% → 89.5%

Time: 8.8s

Alternatives: 11

Speedup: 10.5×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Initial Program: 55.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Alternative 1: 89.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;1 - u0 \leq 0.9998000264167786:\\ \;\;\;\;\log \left(1 - u0\right) \cdot \left(\frac{-\alpha}{{\alpha}^{4}} \cdot \left({\alpha}^{4} \cdot \alpha\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{{\alpha}^{-2}}{u0}}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha u0)
 :precision binary32
 (if (<= (- 1.0 u0) 0.9998000264167786)
   (*
    (log (- 1.0 u0))
    (* (/ (- alpha) (pow alpha 4.0)) (* (pow alpha 4.0) alpha)))
   (/ 1.0 (/ (pow alpha -2.0) u0))))

C

float code(float alpha, float u0) {
	float tmp;
	if ((1.0f - u0) <= 0.9998000264167786f) {
		tmp = logf((1.0f - u0)) * ((-alpha / powf(alpha, 4.0f)) * (powf(alpha, 4.0f) * alpha));
	} else {
		tmp = 1.0f / (powf(alpha, -2.0f) / u0);
	}
	return tmp;
}

Fortran

real(4) function code(alpha, u0)
    real(4), intent (in) :: alpha
    real(4), intent (in) :: u0
    real(4) :: tmp
    if ((1.0e0 - u0) <= 0.9998000264167786e0) then
        tmp = log((1.0e0 - u0)) * ((-alpha / (alpha ** 4.0e0)) * ((alpha ** 4.0e0) * alpha))
    else
        tmp = 1.0e0 / ((alpha ** (-2.0e0)) / u0)
    end if
    code = tmp
end function

Julia

function code(alpha, u0)
	tmp = Float32(0.0)
	if (Float32(Float32(1.0) - u0) <= Float32(0.9998000264167786))
		tmp = Float32(log(Float32(Float32(1.0) - u0)) * Float32(Float32(Float32(-alpha) / (alpha ^ Float32(4.0))) * Float32((alpha ^ Float32(4.0)) * alpha)));
	else
		tmp = Float32(Float32(1.0) / Float32((alpha ^ Float32(-2.0)) / u0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, u0)
	tmp = single(0.0);
	if ((single(1.0) - u0) <= single(0.9998000264167786))
		tmp = log((single(1.0) - u0)) * ((-alpha / (alpha ^ single(4.0))) * ((alpha ^ single(4.0)) * alpha));
	else
		tmp = single(1.0) / ((alpha ^ single(-2.0)) / u0);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (-.f32 #s(literal 1 binary32) u0) < 0.999800026

   1. Initial program 88.1%

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

      1. lift-neg.f32 N/A

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

      2. neg-sub0 N/A

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

      3. flip-- N/A

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

      4. metadata-eval N/A

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

      5. neg-sub0 N/A

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

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

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

      7. lift-neg.f32 N/A

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

      8. lift-*.f32 N/A

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

      9. div-inv N/A

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

      10. lower-*.f32 N/A

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

      11. +-lft-identity N/A

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

      12. lower-/.f32 87.9

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

   4. Applied rewrites 87.9%

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

      1. lift-*.f32 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f32 N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. lift-/.f32 N/A

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

      7. rgt-mult-inverse N/A

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

      8. *-lft-identity 88.1

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

      9. lift-*.f32 N/A

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

      10. lift-neg.f32 N/A

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

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

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

      12. pow2 N/A

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

      13. metadata-eval N/A

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

      14. pow-prod-up N/A

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

      15. lift-pow.f32 N/A

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

      16. lift-pow.f32 N/A

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

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

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

      18. neg-mul-1 N/A

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

      19. neg-mul-1 N/A

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

      20. lift-pow.f32 N/A

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

      21. metadata-eval N/A

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

      22. pow-div N/A

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

      23. pow2 N/A

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

      24. lift-*.f32 N/A

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

      25. lift-pow.f32 N/A

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

      26. distribute-frac-neg2 N/A

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

      27. lift-neg.f32 N/A

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

      28. lift-*.f32 N/A

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

   6. Applied rewrites 88.3%

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

   if 0.999800026 < (-.f32 #s(literal 1 binary32) u0)

   1. Initial program 32.8%

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

   3. Taylor expanded in u0 around 0

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

      1. *-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 91.1

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

   5. Applied rewrites 91.1%

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

   7. Applied rewrites 91.4%

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

   9. Applied rewrites 91.4%

      \[\leadsto \frac{1}{\color{blue}{\frac{{\alpha}^{-2}}{u0}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 90.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 - u0 \leq 0.9998000264167786:\\ \;\;\;\;\log \left(1 - u0\right) \cdot \left(\frac{-\alpha}{{\alpha}^{4}} \cdot \left({\alpha}^{4} \cdot \alpha\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{{\alpha}^{-2}}{u0}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 89.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;1 - u0 \leq 0.9998000264167786:\\ \;\;\;\;\left(\left(-{\alpha}^{-3}\right) \cdot \left({\alpha}^{4} \cdot \alpha\right)\right) \cdot \log \left(1 - u0\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{{\alpha}^{-2}}{u0}}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha u0)
 :precision binary32
 (if (<= (- 1.0 u0) 0.9998000264167786)
   (* (* (- (pow alpha -3.0)) (* (pow alpha 4.0) alpha)) (log (- 1.0 u0)))
   (/ 1.0 (/ (pow alpha -2.0) u0))))

C

float code(float alpha, float u0) {
	float tmp;
	if ((1.0f - u0) <= 0.9998000264167786f) {
		tmp = (-powf(alpha, -3.0f) * (powf(alpha, 4.0f) * alpha)) * logf((1.0f - u0));
	} else {
		tmp = 1.0f / (powf(alpha, -2.0f) / u0);
	}
	return tmp;
}

Fortran

real(4) function code(alpha, u0)
    real(4), intent (in) :: alpha
    real(4), intent (in) :: u0
    real(4) :: tmp
    if ((1.0e0 - u0) <= 0.9998000264167786e0) then
        tmp = (-(alpha ** (-3.0e0)) * ((alpha ** 4.0e0) * alpha)) * log((1.0e0 - u0))
    else
        tmp = 1.0e0 / ((alpha ** (-2.0e0)) / u0)
    end if
    code = tmp
end function

Julia

function code(alpha, u0)
	tmp = Float32(0.0)
	if (Float32(Float32(1.0) - u0) <= Float32(0.9998000264167786))
		tmp = Float32(Float32(Float32(-(alpha ^ Float32(-3.0))) * Float32((alpha ^ Float32(4.0)) * alpha)) * log(Float32(Float32(1.0) - u0)));
	else
		tmp = Float32(Float32(1.0) / Float32((alpha ^ Float32(-2.0)) / u0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, u0)
	tmp = single(0.0);
	if ((single(1.0) - u0) <= single(0.9998000264167786))
		tmp = (-(alpha ^ single(-3.0)) * ((alpha ^ single(4.0)) * alpha)) * log((single(1.0) - u0));
	else
		tmp = single(1.0) / ((alpha ^ single(-2.0)) / u0);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (-.f32 #s(literal 1 binary32) u0) < 0.999800026

   1. Initial program 88.1%

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

      1. lift-neg.f32 N/A

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

      2. neg-sub0 N/A

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

      3. flip-- N/A

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

      4. metadata-eval N/A

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

      5. neg-sub0 N/A

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

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

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

      7. lift-neg.f32 N/A

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

      8. lift-*.f32 N/A

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

      9. div-inv N/A

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

      10. lower-*.f32 N/A

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

      11. +-lft-identity N/A

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

      12. lower-/.f32 87.9

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

   4. Applied rewrites 87.9%

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

      1. lift-*.f32 N/A

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

      2. lift-*.f32 N/A

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

      3. associate-*l* N/A

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

      4. lift-/.f32 N/A

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

      5. rgt-mult-inverse N/A

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

      6. *-rgt-identity 88.1

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

      7. lift-neg.f32 N/A

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

      8. /-rgt-identity N/A

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

      9. distribute-neg-frac2 N/A

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

      10. metadata-eval N/A

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

      11. clear-num N/A

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

      12. lower-/.f32 N/A

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

      13. lower-/.f32 88.1

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

   6. Applied rewrites 88.1%

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

      1. lift-*.f32 N/A

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

      2. lift-/.f32 N/A

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

      3. lift-/.f32 N/A

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

      4. clear-num N/A

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

      5. *-rgt-identity N/A

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

      6. *-inverses N/A

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

      7. associate-/l* N/A

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

      8. times-frac N/A

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

      9. neg-mul-1 N/A

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

      10. lift-neg.f32 N/A

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

      11. associate-*r/ N/A

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

      12. div-inv N/A

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

      13. metadata-eval N/A

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

      14. lift-neg.f32 N/A

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

      15. frac-2neg N/A

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

      16. lift-/.f32 N/A

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

      17. pow2 N/A

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

      18. metadata-eval N/A

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

      19. pow-prod-up N/A

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

      20. lift-pow.f32 N/A

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

      21. lift-pow.f32 N/A

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

      22. associate-*r* N/A

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

      23. lift-/.f32 N/A

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

      24. div-inv N/A

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

      25. lift-/.f32 N/A

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

      26. mul-1-neg N/A

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

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

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

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

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

      29. lift-neg.f32 N/A

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

   8. Applied rewrites 88.2%

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

   if 0.999800026 < (-.f32 #s(literal 1 binary32) u0)

   1. Initial program 32.8%

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

   3. Taylor expanded in u0 around 0

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

      1. *-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 91.1

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

   5. Applied rewrites 91.1%

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

   7. Applied rewrites 91.4%

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

   9. Applied rewrites 91.4%

      \[\leadsto \frac{1}{\color{blue}{\frac{{\alpha}^{-2}}{u0}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 90.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 - u0 \leq 0.9998000264167786:\\ \;\;\;\;\left(\left(-{\alpha}^{-3}\right) \cdot \left({\alpha}^{4} \cdot \alpha\right)\right) \cdot \log \left(1 - u0\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{{\alpha}^{-2}}{u0}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 89.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;1 - u0 \leq 0.9998000264167786:\\ \;\;\;\;\frac{-1}{{\alpha}^{-2}} \cdot \log \left(1 - u0\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{{\alpha}^{-2}}{u0}}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha u0)
 :precision binary32
 (if (<= (- 1.0 u0) 0.9998000264167786)
   (* (/ -1.0 (pow alpha -2.0)) (log (- 1.0 u0)))
   (/ 1.0 (/ (pow alpha -2.0) u0))))

C

float code(float alpha, float u0) {
	float tmp;
	if ((1.0f - u0) <= 0.9998000264167786f) {
		tmp = (-1.0f / powf(alpha, -2.0f)) * logf((1.0f - u0));
	} else {
		tmp = 1.0f / (powf(alpha, -2.0f) / u0);
	}
	return tmp;
}

Fortran

real(4) function code(alpha, u0)
    real(4), intent (in) :: alpha
    real(4), intent (in) :: u0
    real(4) :: tmp
    if ((1.0e0 - u0) <= 0.9998000264167786e0) then
        tmp = ((-1.0e0) / (alpha ** (-2.0e0))) * log((1.0e0 - u0))
    else
        tmp = 1.0e0 / ((alpha ** (-2.0e0)) / u0)
    end if
    code = tmp
end function

Julia

function code(alpha, u0)
	tmp = Float32(0.0)
	if (Float32(Float32(1.0) - u0) <= Float32(0.9998000264167786))
		tmp = Float32(Float32(Float32(-1.0) / (alpha ^ Float32(-2.0))) * log(Float32(Float32(1.0) - u0)));
	else
		tmp = Float32(Float32(1.0) / Float32((alpha ^ Float32(-2.0)) / u0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, u0)
	tmp = single(0.0);
	if ((single(1.0) - u0) <= single(0.9998000264167786))
		tmp = (single(-1.0) / (alpha ^ single(-2.0))) * log((single(1.0) - u0));
	else
		tmp = single(1.0) / ((alpha ^ single(-2.0)) / u0);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (-.f32 #s(literal 1 binary32) u0) < 0.999800026

   1. Initial program 88.1%

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

      1. lift-neg.f32 N/A

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

      2. neg-sub0 N/A

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

      3. flip-- N/A

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

      4. metadata-eval N/A

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

      5. neg-sub0 N/A

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

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

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

      7. lift-neg.f32 N/A

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

      8. lift-*.f32 N/A

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

      9. div-inv N/A

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

      10. lower-*.f32 N/A

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

      11. +-lft-identity N/A

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

      12. lower-/.f32 87.9

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

   4. Applied rewrites 87.9%

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

      1. lift-*.f32 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f32 N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. lift-/.f32 N/A

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

      7. rgt-mult-inverse N/A

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

      8. *-lft-identity 88.1

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

      9. lift-*.f32 N/A

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

      10. /-rgt-identity N/A

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

      11. associate-*r/ N/A

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

      12. lift-neg.f32 N/A

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

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

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

      14. lift-*.f32 N/A

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

      15. metadata-eval N/A

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

      16. frac-2neg N/A

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

      17. clear-num N/A

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

      18. metadata-eval N/A

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

      19. distribute-neg-frac N/A

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

      20. lift-*.f32 N/A

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

      21. pow2 N/A

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

      22. pow-flip N/A

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

      23. metadata-eval N/A

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

      24. lift-pow.f32 N/A

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

      25. distribute-frac-neg2 N/A

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

      26. distribute-neg-frac N/A

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

      27. metadata-eval N/A

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

      28. lower-/.f32 88.2

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

   6. Applied rewrites 88.2%

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

   if 0.999800026 < (-.f32 #s(literal 1 binary32) u0)

   1. Initial program 32.8%

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

   3. Taylor expanded in u0 around 0

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

      1. *-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 91.1

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

   5. Applied rewrites 91.1%

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

   7. Applied rewrites 91.4%

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

   9. Applied rewrites 91.4%

      \[\leadsto \frac{1}{\color{blue}{\frac{{\alpha}^{-2}}{u0}}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 89.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;1 - u0 \leq 0.9998000264167786:\\ \;\;\;\;\frac{\left(\alpha \cdot \alpha\right) \cdot \left(\left(-\alpha\right) \cdot \alpha\right)}{\alpha \cdot \alpha} \cdot \log \left(1 - u0\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{{\alpha}^{-2}}{u0}}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha u0)
 :precision binary32
 (if (<= (- 1.0 u0) 0.9998000264167786)
   (*
    (/ (* (* alpha alpha) (* (- alpha) alpha)) (* alpha alpha))
    (log (- 1.0 u0)))
   (/ 1.0 (/ (pow alpha -2.0) u0))))

C

float code(float alpha, float u0) {
	float tmp;
	if ((1.0f - u0) <= 0.9998000264167786f) {
		tmp = (((alpha * alpha) * (-alpha * alpha)) / (alpha * alpha)) * logf((1.0f - u0));
	} else {
		tmp = 1.0f / (powf(alpha, -2.0f) / u0);
	}
	return tmp;
}

Fortran

real(4) function code(alpha, u0)
    real(4), intent (in) :: alpha
    real(4), intent (in) :: u0
    real(4) :: tmp
    if ((1.0e0 - u0) <= 0.9998000264167786e0) then
        tmp = (((alpha * alpha) * (-alpha * alpha)) / (alpha * alpha)) * log((1.0e0 - u0))
    else
        tmp = 1.0e0 / ((alpha ** (-2.0e0)) / u0)
    end if
    code = tmp
end function

Julia

function code(alpha, u0)
	tmp = Float32(0.0)
	if (Float32(Float32(1.0) - u0) <= Float32(0.9998000264167786))
		tmp = Float32(Float32(Float32(Float32(alpha * alpha) * Float32(Float32(-alpha) * alpha)) / Float32(alpha * alpha)) * log(Float32(Float32(1.0) - u0)));
	else
		tmp = Float32(Float32(1.0) / Float32((alpha ^ Float32(-2.0)) / u0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, u0)
	tmp = single(0.0);
	if ((single(1.0) - u0) <= single(0.9998000264167786))
		tmp = (((alpha * alpha) * (-alpha * alpha)) / (alpha * alpha)) * log((single(1.0) - u0));
	else
		tmp = single(1.0) / ((alpha ^ single(-2.0)) / u0);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (-.f32 #s(literal 1 binary32) u0) < 0.999800026

   1. Initial program 88.1%

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

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

      8. metadata-eval N/A

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

      9. sub0-neg N/A

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

      10. lower-neg.f32 N/A

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

      11. pow2 N/A

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

      12. pow2 N/A

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

      13. pow-prod-up N/A

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

      14. lower-pow.f32 N/A

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

      15. metadata-eval N/A

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

      16. lower-*.f32 88.2

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

   4. Applied rewrites 88.2%

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

      1. lift-neg.f32 N/A

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

      2. lift-pow.f32 N/A

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

      3. metadata-eval N/A

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

      4. pow-pow N/A

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

      5. pow2 N/A

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

      6. lift-*.f32 N/A

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

      7. pow2 N/A

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

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

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

      9. lift-*.f32 N/A

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

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

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

      11. lift-neg.f32 N/A

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

      12. lift-*.f32 N/A

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

      13. lower-*.f32 88.2

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

   6. Applied rewrites 88.2%

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

   if 0.999800026 < (-.f32 #s(literal 1 binary32) u0)

   1. Initial program 32.8%

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

   3. Taylor expanded in u0 around 0

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

      1. *-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 91.1

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

   5. Applied rewrites 91.1%

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

   7. Applied rewrites 91.4%

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

   9. Applied rewrites 91.4%

      \[\leadsto \frac{1}{\color{blue}{\frac{{\alpha}^{-2}}{u0}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 90.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 - u0 \leq 0.9998000264167786:\\ \;\;\;\;\frac{\left(\alpha \cdot \alpha\right) \cdot \left(\left(-\alpha\right) \cdot \alpha\right)}{\alpha \cdot \alpha} \cdot \log \left(1 - u0\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{{\alpha}^{-2}}{u0}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 89.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;1 - u0 \leq 0.9998000264167786:\\ \;\;\;\;\frac{\alpha}{\frac{-1}{\alpha}} \cdot \log \left(1 - u0\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{{\alpha}^{-2}}{u0}}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha u0)
 :precision binary32
 (if (<= (- 1.0 u0) 0.9998000264167786)
   (* (/ alpha (/ -1.0 alpha)) (log (- 1.0 u0)))
   (/ 1.0 (/ (pow alpha -2.0) u0))))

C

float code(float alpha, float u0) {
	float tmp;
	if ((1.0f - u0) <= 0.9998000264167786f) {
		tmp = (alpha / (-1.0f / alpha)) * logf((1.0f - u0));
	} else {
		tmp = 1.0f / (powf(alpha, -2.0f) / u0);
	}
	return tmp;
}

Fortran

real(4) function code(alpha, u0)
    real(4), intent (in) :: alpha
    real(4), intent (in) :: u0
    real(4) :: tmp
    if ((1.0e0 - u0) <= 0.9998000264167786e0) then
        tmp = (alpha / ((-1.0e0) / alpha)) * log((1.0e0 - u0))
    else
        tmp = 1.0e0 / ((alpha ** (-2.0e0)) / u0)
    end if
    code = tmp
end function

Julia

function code(alpha, u0)
	tmp = Float32(0.0)
	if (Float32(Float32(1.0) - u0) <= Float32(0.9998000264167786))
		tmp = Float32(Float32(alpha / Float32(Float32(-1.0) / alpha)) * log(Float32(Float32(1.0) - u0)));
	else
		tmp = Float32(Float32(1.0) / Float32((alpha ^ Float32(-2.0)) / u0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, u0)
	tmp = single(0.0);
	if ((single(1.0) - u0) <= single(0.9998000264167786))
		tmp = (alpha / (single(-1.0) / alpha)) * log((single(1.0) - u0));
	else
		tmp = single(1.0) / ((alpha ^ single(-2.0)) / u0);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (-.f32 #s(literal 1 binary32) u0) < 0.999800026

   1. Initial program 88.1%

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

      1. lift-neg.f32 N/A

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

      2. neg-sub0 N/A

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

      3. flip-- N/A

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

      4. metadata-eval N/A

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

      5. neg-sub0 N/A

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

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

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

      7. lift-neg.f32 N/A

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

      8. lift-*.f32 N/A

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

      9. div-inv N/A

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

      10. lower-*.f32 N/A

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

      11. +-lft-identity N/A

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

      12. lower-/.f32 87.9

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

   4. Applied rewrites 87.9%

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

      1. lift-*.f32 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f32 N/A

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

      4. lift-*.f32 N/A

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

      5. associate-*l* N/A

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

      6. lift-/.f32 N/A

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

      7. rgt-mult-inverse N/A

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

      8. *-rgt-identity N/A

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

      9. lift-neg.f32 N/A

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

      10. /-rgt-identity N/A

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

      11. distribute-neg-frac2 N/A

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

      12. metadata-eval N/A

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

      13. clear-num N/A

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

      14. un-div-inv N/A

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

      15. lower-/.f32 N/A

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

      16. lower-/.f32 88.2

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

   6. Applied rewrites 88.2%

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

   if 0.999800026 < (-.f32 #s(literal 1 binary32) u0)

   1. Initial program 32.8%

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

   3. Taylor expanded in u0 around 0

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

      1. *-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 91.1

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

   5. Applied rewrites 91.1%

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

   7. Applied rewrites 91.4%

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

   9. Applied rewrites 91.4%

      \[\leadsto \frac{1}{\color{blue}{\frac{{\alpha}^{-2}}{u0}}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 89.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;1 - u0 \leq 0.9998000264167786:\\ \;\;\;\;\left(\left(-\alpha\right) \cdot \alpha\right) \cdot \log \left(1 - u0\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{{\alpha}^{-2}}{u0}}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha u0)
 :precision binary32
 (if (<= (- 1.0 u0) 0.9998000264167786)
   (* (* (- alpha) alpha) (log (- 1.0 u0)))
   (/ 1.0 (/ (pow alpha -2.0) u0))))

C

float code(float alpha, float u0) {
	float tmp;
	if ((1.0f - u0) <= 0.9998000264167786f) {
		tmp = (-alpha * alpha) * logf((1.0f - u0));
	} else {
		tmp = 1.0f / (powf(alpha, -2.0f) / u0);
	}
	return tmp;
}

Fortran

real(4) function code(alpha, u0)
    real(4), intent (in) :: alpha
    real(4), intent (in) :: u0
    real(4) :: tmp
    if ((1.0e0 - u0) <= 0.9998000264167786e0) then
        tmp = (-alpha * alpha) * log((1.0e0 - u0))
    else
        tmp = 1.0e0 / ((alpha ** (-2.0e0)) / u0)
    end if
    code = tmp
end function

Julia

function code(alpha, u0)
	tmp = Float32(0.0)
	if (Float32(Float32(1.0) - u0) <= Float32(0.9998000264167786))
		tmp = Float32(Float32(Float32(-alpha) * alpha) * log(Float32(Float32(1.0) - u0)));
	else
		tmp = Float32(Float32(1.0) / Float32((alpha ^ Float32(-2.0)) / u0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, u0)
	tmp = single(0.0);
	if ((single(1.0) - u0) <= single(0.9998000264167786))
		tmp = (-alpha * alpha) * log((single(1.0) - u0));
	else
		tmp = single(1.0) / ((alpha ^ single(-2.0)) / u0);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (-.f32 #s(literal 1 binary32) u0) < 0.999800026

   1. Initial program 88.1%

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

   if 0.999800026 < (-.f32 #s(literal 1 binary32) u0)

   1. Initial program 32.8%

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

   3. Taylor expanded in u0 around 0

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

      1. *-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 91.1

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

   5. Applied rewrites 91.1%

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

   7. Applied rewrites 91.4%

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

   9. Applied rewrites 91.4%

      \[\leadsto \frac{1}{\color{blue}{\frac{{\alpha}^{-2}}{u0}}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 89.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;1 - u0 \leq 0.9998000264167786:\\ \;\;\;\;\left(\left(-\alpha\right) \cdot \alpha\right) \cdot \log \left(1 - u0\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{u0}{{\alpha}^{-2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha u0)
 :precision binary32
 (if (<= (- 1.0 u0) 0.9998000264167786)
   (* (* (- alpha) alpha) (log (- 1.0 u0)))
   (/ u0 (pow alpha -2.0))))

C

float code(float alpha, float u0) {
	float tmp;
	if ((1.0f - u0) <= 0.9998000264167786f) {
		tmp = (-alpha * alpha) * logf((1.0f - u0));
	} else {
		tmp = u0 / powf(alpha, -2.0f);
	}
	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) <= 0.9998000264167786e0) then
        tmp = (-alpha * alpha) * log((1.0e0 - u0))
    else
        tmp = u0 / (alpha ** (-2.0e0))
    end if
    code = tmp
end function

Julia

function code(alpha, u0)
	tmp = Float32(0.0)
	if (Float32(Float32(1.0) - u0) <= Float32(0.9998000264167786))
		tmp = Float32(Float32(Float32(-alpha) * alpha) * log(Float32(Float32(1.0) - u0)));
	else
		tmp = Float32(u0 / (alpha ^ Float32(-2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, u0)
	tmp = single(0.0);
	if ((single(1.0) - u0) <= single(0.9998000264167786))
		tmp = (-alpha * alpha) * log((single(1.0) - u0));
	else
		tmp = u0 / (alpha ^ single(-2.0));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (-.f32 #s(literal 1 binary32) u0) < 0.999800026

   1. Initial program 88.1%

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

   if 0.999800026 < (-.f32 #s(literal 1 binary32) u0)

   1. Initial program 32.8%

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

   3. Taylor expanded in u0 around 0

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

      1. *-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 91.1

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

   5. Applied rewrites 91.1%

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

   7. Applied rewrites 91.4%

      \[\leadsto \frac{u0}{\color{blue}{{\alpha}^{-2}}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 89.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;1 - u0 \leq 0.9998000264167786:\\ \;\;\;\;\left(\log \left(1 - u0\right) \cdot \left(-\alpha\right)\right) \cdot \alpha\\ \mathbf{else}:\\ \;\;\;\;\frac{u0}{{\alpha}^{-2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha u0)
 :precision binary32
 (if (<= (- 1.0 u0) 0.9998000264167786)
   (* (* (log (- 1.0 u0)) (- alpha)) alpha)
   (/ u0 (pow alpha -2.0))))

C

float code(float alpha, float u0) {
	float tmp;
	if ((1.0f - u0) <= 0.9998000264167786f) {
		tmp = (logf((1.0f - u0)) * -alpha) * alpha;
	} else {
		tmp = u0 / powf(alpha, -2.0f);
	}
	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) <= 0.9998000264167786e0) then
        tmp = (log((1.0e0 - u0)) * -alpha) * alpha
    else
        tmp = u0 / (alpha ** (-2.0e0))
    end if
    code = tmp
end function

Julia

function code(alpha, u0)
	tmp = Float32(0.0)
	if (Float32(Float32(1.0) - u0) <= Float32(0.9998000264167786))
		tmp = Float32(Float32(log(Float32(Float32(1.0) - u0)) * Float32(-alpha)) * alpha);
	else
		tmp = Float32(u0 / (alpha ^ Float32(-2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, u0)
	tmp = single(0.0);
	if ((single(1.0) - u0) <= single(0.9998000264167786))
		tmp = (log((single(1.0) - u0)) * -alpha) * alpha;
	else
		tmp = u0 / (alpha ^ single(-2.0));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (-.f32 #s(literal 1 binary32) u0) < 0.999800026

   1. Initial program 88.1%

      \[\left(\left(-\alpha\right) \cdot \alpha\right) \cdot \log \left(1 - u0\right) \]
   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 48.0

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

   5. Applied rewrites 48.0%

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

   7. Applied rewrites 88.1%

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

   if 0.999800026 < (-.f32 #s(literal 1 binary32) u0)

   1. Initial program 32.8%

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

   3. Taylor expanded in u0 around 0

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

      1. *-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 91.1

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

   5. Applied rewrites 91.1%

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

   7. Applied rewrites 91.4%

      \[\leadsto \frac{u0}{\color{blue}{{\alpha}^{-2}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 90.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 - u0 \leq 0.9998000264167786:\\ \;\;\;\;\left(\log \left(1 - u0\right) \cdot \left(-\alpha\right)\right) \cdot \alpha\\ \mathbf{else}:\\ \;\;\;\;\frac{u0}{{\alpha}^{-2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 74.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{u0}{{\alpha}^{-2}} \end{array} \]

FPCore

(FPCore (alpha u0) :precision binary32 (/ u0 (pow alpha -2.0)))

C

float code(float alpha, float u0) {
	return u0 / powf(alpha, -2.0f);
}

Fortran

real(4) function code(alpha, u0)
    real(4), intent (in) :: alpha
    real(4), intent (in) :: u0
    code = u0 / (alpha ** (-2.0e0))
end function

Julia

function code(alpha, u0)
	return Float32(u0 / (alpha ^ Float32(-2.0)))
end

MATLAB

function tmp = code(alpha, u0)
	tmp = u0 / (alpha ^ single(-2.0));
end

Derivation

1. Initial program 54.6%

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

3. Taylor expanded in u0 around 0

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

   1. *-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 74.1

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

5. Applied rewrites 74.1%

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

7. Applied rewrites 74.3%

   \[\leadsto \frac{u0}{\color{blue}{{\alpha}^{-2}}} \]
8. Add Preprocessing

Alternative 10: 74.7% 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 54.6%

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

3. Taylor expanded in u0 around 0

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

   1. *-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 74.1

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

5. Applied rewrites 74.1%

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

7. Applied rewrites 74.3%

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

8. Final simplification 74.3%

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

Alternative 11: 74.7% 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 54.6%

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

3. Taylor expanded in u0 around 0

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

   1. *-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 74.1

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

5. Applied rewrites 74.1%

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

6. Final simplification 74.1%

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

Reproduce

herbie shell --seed 2024268 
(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))))