Beckmann Distribution sample, tan2theta, alphax == alphay

Percentage Accurate: 55.6% → 89.8%

Time: 7.8s

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: 55.6% 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: 89.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;1 - u0 \leq 0.9998400211334229:\\ \;\;\;\;\log \left(1 - u0\right) \cdot \left(\left(\left(\alpha \cdot \alpha\right) \cdot \frac{\alpha}{\frac{-1}{\frac{1}{\alpha}} \cdot \alpha}\right) \cdot \alpha\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\alpha \cdot \alpha\right) \cdot u0\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha u0)
 :precision binary32
 (if (<= (- 1.0 u0) 0.9998400211334229)
   (*
    (log (- 1.0 u0))
    (* (* (* alpha alpha) (/ alpha (* (/ -1.0 (/ 1.0 alpha)) alpha))) alpha))
   (* (* alpha alpha) u0)))

C

float code(float alpha, float u0) {
	float tmp;
	if ((1.0f - u0) <= 0.9998400211334229f) {
		tmp = logf((1.0f - u0)) * (((alpha * alpha) * (alpha / ((-1.0f / (1.0f / alpha)) * alpha))) * alpha);
	} else {
		tmp = (alpha * alpha) * 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) <= 0.9998400211334229e0) then
        tmp = log((1.0e0 - u0)) * (((alpha * alpha) * (alpha / (((-1.0e0) / (1.0e0 / alpha)) * alpha))) * alpha)
    else
        tmp = (alpha * alpha) * u0
    end if
    code = tmp
end function

Julia

function code(alpha, u0)
	tmp = Float32(0.0)
	if (Float32(Float32(1.0) - u0) <= Float32(0.9998400211334229))
		tmp = Float32(log(Float32(Float32(1.0) - u0)) * Float32(Float32(Float32(alpha * alpha) * Float32(alpha / Float32(Float32(Float32(-1.0) / Float32(Float32(1.0) / alpha)) * alpha))) * alpha));
	else
		tmp = Float32(Float32(alpha * alpha) * u0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, u0)
	tmp = single(0.0);
	if ((single(1.0) - u0) <= single(0.9998400211334229))
		tmp = log((single(1.0) - u0)) * (((alpha * alpha) * (alpha / ((single(-1.0) / (single(1.0) / alpha)) * alpha))) * alpha);
	else
		tmp = (alpha * alpha) * u0;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (-.f32 #s(literal 1 binary32) u0) < 0.999840021

   1. Initial program 86.0%

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

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

      4. frac-2neg N/A

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

      5. metadata-eval N/A

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

      6. +-lft-identity N/A

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

      7. mul0-lft N/A

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

      8. +-rgt-identity N/A

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

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

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

      10. lift-neg.f32 N/A

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

      11. lift-*.f32 N/A

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

      12. div-inv N/A

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

      13. metadata-eval N/A

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

      14. sub0-neg N/A

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

      15. remove-double-neg N/A

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

      16. lower-*.f32 N/A

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

      17. lower-pow.f32 N/A

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

      18. lower-/.f32 85.9

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

   4. Applied rewrites 85.9%

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

      1. lift-*.f32 N/A

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

      2. *-commutative N/A

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

      3. lift-pow.f32 N/A

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

      4. cube-mult N/A

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

      5. lift-*.f32 N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f32 N/A

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

      8. lift-/.f32 N/A

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

      9. associate-*l/ N/A

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

      10. *-lft-identity N/A

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

      11. lower-/.f32 86.0

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

   6. Applied rewrites 86.0%

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

      1. remove-double-div N/A

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

      2. lift-/.f32 N/A

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

      3. lift-/.f32 86.1

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

      4. lift-/.f32 N/A

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

      5. metadata-eval N/A

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

      6. lift-neg.f32 N/A

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

      7. frac-2neg N/A

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

      8. lower-/.f32 86.1

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

   8. Applied rewrites 86.1%

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

   if 0.999840021 < (-.f32 #s(literal 1 binary32) u0)

   1. Initial program 33.9%

      \[\left(\left(-\alpha\right) \cdot \alpha\right) \cdot \log \left(1 - u0\right) \]
   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 91.9

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

   5. Applied rewrites 91.9%

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

4. Final simplification 90.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 - u0 \leq 0.9998400211334229:\\ \;\;\;\;\log \left(1 - u0\right) \cdot \left(\left(\left(\alpha \cdot \alpha\right) \cdot \frac{\alpha}{\frac{-1}{\frac{1}{\alpha}} \cdot \alpha}\right) \cdot \alpha\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\alpha \cdot \alpha\right) \cdot u0\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 89.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;1 - u0 \leq 0.9998400211334229:\\ \;\;\;\;\frac{1}{\frac{-1}{\alpha \cdot \alpha}} \cdot \log \left(1 - u0\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\alpha \cdot \alpha\right) \cdot u0\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha u0)
 :precision binary32
 (if (<= (- 1.0 u0) 0.9998400211334229)
   (* (/ 1.0 (/ -1.0 (* alpha alpha))) (log (- 1.0 u0)))
   (* (* alpha alpha) u0)))

C

float code(float alpha, float u0) {
	float tmp;
	if ((1.0f - u0) <= 0.9998400211334229f) {
		tmp = (1.0f / (-1.0f / (alpha * alpha))) * logf((1.0f - u0));
	} else {
		tmp = (alpha * alpha) * 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) <= 0.9998400211334229e0) then
        tmp = (1.0e0 / ((-1.0e0) / (alpha * alpha))) * log((1.0e0 - u0))
    else
        tmp = (alpha * alpha) * u0
    end if
    code = tmp
end function

Julia

function code(alpha, u0)
	tmp = Float32(0.0)
	if (Float32(Float32(1.0) - u0) <= Float32(0.9998400211334229))
		tmp = Float32(Float32(Float32(1.0) / Float32(Float32(-1.0) / Float32(alpha * alpha))) * log(Float32(Float32(1.0) - u0)));
	else
		tmp = Float32(Float32(alpha * alpha) * u0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, u0)
	tmp = single(0.0);
	if ((single(1.0) - u0) <= single(0.9998400211334229))
		tmp = (single(1.0) / (single(-1.0) / (alpha * alpha))) * log((single(1.0) - u0));
	else
		tmp = (alpha * alpha) * u0;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (-.f32 #s(literal 1 binary32) u0) < 0.999840021

   1. Initial program 86.0%

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

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

      7. metadata-eval N/A

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

      8. +-lft-identity N/A

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

      9. mul0-lft N/A

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

      12. mul0-rgt N/A

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

      13. mul0-lft N/A

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

      14. mul0-lft N/A

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

      15. mul0-lft N/A

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

      16. lower-*.f32 N/A

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

   4. Applied rewrites 86.0%

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

      1. lift-*.f32 N/A

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

      2. lift-neg.f32 N/A

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

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

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

      4. lift-pow.f32 N/A

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

      5. lift-/.f32 N/A

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

      6. lift-pow.f32 N/A

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

      7. pow-flip N/A

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

      8. pow-prod-up N/A

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

      9. metadata-eval N/A

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

      10. metadata-eval N/A

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

      11. pow2 N/A

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

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

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

      13. lift-neg.f32 N/A

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

      14. lift-*.f32 86.0

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

      15. /-rgt-identity N/A

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

      16. clear-num N/A

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

      17. lift-/.f32 N/A

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

      18. lower-/.f32 86.0

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

      19. lift-/.f32 N/A

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

      20. frac-2neg N/A

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

      21. metadata-eval N/A

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

      22. lift-*.f32 N/A

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

      23. lift-neg.f32 N/A

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

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

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

      25. lift-*.f32 N/A

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

      26. remove-double-neg N/A

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

      27. lower-/.f32 86.0

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

   6. Applied rewrites 86.0%

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

   if 0.999840021 < (-.f32 #s(literal 1 binary32) u0)

   1. Initial program 33.9%

      \[\left(\left(-\alpha\right) \cdot \alpha\right) \cdot \log \left(1 - u0\right) \]
   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 91.9

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

   5. Applied rewrites 91.9%

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

4. Final simplification 90.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 - u0 \leq 0.9998400211334229:\\ \;\;\;\;\frac{1}{\frac{-1}{\alpha \cdot \alpha}} \cdot \log \left(1 - u0\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\alpha \cdot \alpha\right) \cdot u0\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 89.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;1 - u0 \leq 0.9998400211334229:\\ \;\;\;\;\left(\frac{-1}{\alpha} \cdot \left(\left(\alpha \cdot \alpha\right) \cdot \alpha\right)\right) \cdot \log \left(1 - u0\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\alpha \cdot \alpha\right) \cdot u0\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha u0)
 :precision binary32
 (if (<= (- 1.0 u0) 0.9998400211334229)
   (* (* (/ -1.0 alpha) (* (* alpha alpha) alpha)) (log (- 1.0 u0)))
   (* (* alpha alpha) u0)))

C

float code(float alpha, float u0) {
	float tmp;
	if ((1.0f - u0) <= 0.9998400211334229f) {
		tmp = ((-1.0f / alpha) * ((alpha * alpha) * alpha)) * logf((1.0f - u0));
	} else {
		tmp = (alpha * alpha) * 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) <= 0.9998400211334229e0) then
        tmp = (((-1.0e0) / alpha) * ((alpha * alpha) * alpha)) * log((1.0e0 - u0))
    else
        tmp = (alpha * alpha) * u0
    end if
    code = tmp
end function

Julia

function code(alpha, u0)
	tmp = Float32(0.0)
	if (Float32(Float32(1.0) - u0) <= Float32(0.9998400211334229))
		tmp = Float32(Float32(Float32(Float32(-1.0) / alpha) * Float32(Float32(alpha * alpha) * alpha)) * log(Float32(Float32(1.0) - u0)));
	else
		tmp = Float32(Float32(alpha * alpha) * u0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, u0)
	tmp = single(0.0);
	if ((single(1.0) - u0) <= single(0.9998400211334229))
		tmp = ((single(-1.0) / alpha) * ((alpha * alpha) * alpha)) * log((single(1.0) - u0));
	else
		tmp = (alpha * alpha) * u0;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (-.f32 #s(literal 1 binary32) u0) < 0.999840021

   1. Initial program 86.0%

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

      1. lift-*.f32 N/A

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

      2. lift-neg.f32 N/A

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

      3. neg-sub0 N/A

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

      4. flip-- N/A

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

      5. metadata-eval N/A

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

      6. neg-sub0 N/A

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

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

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

      8. lift-neg.f32 N/A

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

      9. lift-*.f32 N/A

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

      10. +-lft-identity N/A

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

      11. associate-*l/ N/A

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

      12. div-inv N/A

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

      13. +-lft-identity N/A

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

      14. lower-*.f32 N/A

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

      15. lower-*.f32 N/A

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

      16. +-lft-identity N/A

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

      17. lower-/.f32 86.0

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

   4. Applied rewrites 86.0%

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

   if 0.999840021 < (-.f32 #s(literal 1 binary32) u0)

   1. Initial program 33.9%

      \[\left(\left(-\alpha\right) \cdot \alpha\right) \cdot \log \left(1 - u0\right) \]
   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 91.9

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

   5. Applied rewrites 91.9%

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

4. Final simplification 90.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 - u0 \leq 0.9998400211334229:\\ \;\;\;\;\left(\frac{-1}{\alpha} \cdot \left(\left(\alpha \cdot \alpha\right) \cdot \alpha\right)\right) \cdot \log \left(1 - u0\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\alpha \cdot \alpha\right) \cdot u0\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 89.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;1 - u0 \leq 0.9998400211334229:\\ \;\;\;\;\left(\left(-\alpha\right) \cdot \alpha\right) \cdot \log \left(1 - u0\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\alpha \cdot \alpha\right) \cdot u0\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha u0)
 :precision binary32
 (if (<= (- 1.0 u0) 0.9998400211334229)
   (* (* (- alpha) alpha) (log (- 1.0 u0)))
   (* (* alpha alpha) u0)))

C

float code(float alpha, float u0) {
	float tmp;
	if ((1.0f - u0) <= 0.9998400211334229f) {
		tmp = (-alpha * alpha) * logf((1.0f - u0));
	} else {
		tmp = (alpha * alpha) * 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) <= 0.9998400211334229e0) then
        tmp = (-alpha * alpha) * log((1.0e0 - u0))
    else
        tmp = (alpha * alpha) * u0
    end if
    code = tmp
end function

Julia

function code(alpha, u0)
	tmp = Float32(0.0)
	if (Float32(Float32(1.0) - u0) <= Float32(0.9998400211334229))
		tmp = Float32(Float32(Float32(-alpha) * alpha) * log(Float32(Float32(1.0) - u0)));
	else
		tmp = Float32(Float32(alpha * alpha) * u0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, u0)
	tmp = single(0.0);
	if ((single(1.0) - u0) <= single(0.9998400211334229))
		tmp = (-alpha * alpha) * log((single(1.0) - u0));
	else
		tmp = (alpha * alpha) * u0;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (-.f32 #s(literal 1 binary32) u0) < 0.999840021

   1. Initial program 86.0%

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

   if 0.999840021 < (-.f32 #s(literal 1 binary32) u0)

   1. Initial program 33.9%

      \[\left(\left(-\alpha\right) \cdot \alpha\right) \cdot \log \left(1 - u0\right) \]
   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 91.9

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

   5. Applied rewrites 91.9%

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

4. Final simplification 90.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 - u0 \leq 0.9998400211334229:\\ \;\;\;\;\left(\left(-\alpha\right) \cdot \alpha\right) \cdot \log \left(1 - u0\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\alpha \cdot \alpha\right) \cdot u0\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 89.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;1 - u0 \leq 0.9998400211334229:\\ \;\;\;\;\left(\left(-\alpha\right) \cdot \log \left(1 - u0\right)\right) \cdot \alpha\\ \mathbf{else}:\\ \;\;\;\;\left(\alpha \cdot \alpha\right) \cdot u0\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha u0)
 :precision binary32
 (if (<= (- 1.0 u0) 0.9998400211334229)
   (* (* (- alpha) (log (- 1.0 u0))) alpha)
   (* (* alpha alpha) u0)))

C

float code(float alpha, float u0) {
	float tmp;
	if ((1.0f - u0) <= 0.9998400211334229f) {
		tmp = (-alpha * logf((1.0f - u0))) * alpha;
	} else {
		tmp = (alpha * alpha) * 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) <= 0.9998400211334229e0) then
        tmp = (-alpha * log((1.0e0 - u0))) * alpha
    else
        tmp = (alpha * alpha) * u0
    end if
    code = tmp
end function

Julia

function code(alpha, u0)
	tmp = Float32(0.0)
	if (Float32(Float32(1.0) - u0) <= Float32(0.9998400211334229))
		tmp = Float32(Float32(Float32(-alpha) * log(Float32(Float32(1.0) - u0))) * alpha);
	else
		tmp = Float32(Float32(alpha * alpha) * u0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, u0)
	tmp = single(0.0);
	if ((single(1.0) - u0) <= single(0.9998400211334229))
		tmp = (-alpha * log((single(1.0) - u0))) * alpha;
	else
		tmp = (alpha * alpha) * u0;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (-.f32 #s(literal 1 binary32) u0) < 0.999840021

   1. Initial program 86.0%

      \[\left(\left(-\alpha\right) \cdot \alpha\right) \cdot \log \left(1 - u0\right) \]
   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 51.0

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

   5. Applied rewrites 51.0%

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

   7. Applied rewrites 85.9%

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

   if 0.999840021 < (-.f32 #s(literal 1 binary32) u0)

   1. Initial program 33.9%

      \[\left(\left(-\alpha\right) \cdot \alpha\right) \cdot \log \left(1 - u0\right) \]
   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 91.9

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

   5. Applied rewrites 91.9%

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

4. Final simplification 90.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 - u0 \leq 0.9998400211334229:\\ \;\;\;\;\left(\left(-\alpha\right) \cdot \log \left(1 - u0\right)\right) \cdot \alpha\\ \mathbf{else}:\\ \;\;\;\;\left(\alpha \cdot \alpha\right) \cdot u0\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 74.5% accurate, 10.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 50.6%

   \[\left(\left(-\alpha\right) \cdot \alpha\right) \cdot \log \left(1 - u0\right) \]
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 78.8

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

5. Applied rewrites 78.8%

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

6. Final simplification 78.8%

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

Reproduce

herbie shell --seed 2024266 
(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))))