Beckmann Distribution sample, tan2theta, alphax == alphay

Percentage Accurate: 55.3% → 96.7%

Time: 6.2s

Alternatives: 9

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: 55.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Alternative 1: 96.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;1 - u0 \leq 0.9965999722480774:\\ \;\;\;\;\left(\left(-{\alpha}^{4}\right) \cdot {\alpha}^{-2}\right) \cdot \log \left(1 - u0\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-\alpha\right) \cdot \alpha\right) \cdot \left(\left(u0 \cdot -0.5\right) \cdot u0 - u0\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha u0)
 :precision binary32
 (if (<= (- 1.0 u0) 0.9965999722480774)
   (* (* (- (pow alpha 4.0)) (pow alpha -2.0)) (log (- 1.0 u0)))
   (* (* (- alpha) alpha) (- (* (* u0 -0.5) u0) u0))))

C

float code(float alpha, float u0) {
	float tmp;
	if ((1.0f - u0) <= 0.9965999722480774f) {
		tmp = (-powf(alpha, 4.0f) * powf(alpha, -2.0f)) * logf((1.0f - u0));
	} else {
		tmp = (-alpha * alpha) * (((u0 * -0.5f) * u0) - u0);
	}
	return tmp;
}

Fortran

real(4) function code(alpha, u0)
    real(4), intent (in) :: alpha
    real(4), intent (in) :: u0
    real(4) :: tmp
    if ((1.0e0 - u0) <= 0.9965999722480774e0) then
        tmp = (-(alpha ** 4.0e0) * (alpha ** (-2.0e0))) * log((1.0e0 - u0))
    else
        tmp = (-alpha * alpha) * (((u0 * (-0.5e0)) * u0) - u0)
    end if
    code = tmp
end function

Julia

function code(alpha, u0)
	tmp = Float32(0.0)
	if (Float32(Float32(1.0) - u0) <= Float32(0.9965999722480774))
		tmp = Float32(Float32(Float32(-(alpha ^ Float32(4.0))) * (alpha ^ Float32(-2.0))) * log(Float32(Float32(1.0) - u0)));
	else
		tmp = Float32(Float32(Float32(-alpha) * alpha) * Float32(Float32(Float32(u0 * Float32(-0.5)) * u0) - u0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, u0)
	tmp = single(0.0);
	if ((single(1.0) - u0) <= single(0.9965999722480774))
		tmp = (-(alpha ^ single(4.0)) * (alpha ^ single(-2.0))) * log((single(1.0) - u0));
	else
		tmp = (-alpha * alpha) * (((u0 * single(-0.5)) * u0) - u0);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (-.f32 #s(literal 1 binary32) u0) < 0.996599972

   1. Initial program 93.1%

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

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

      8. +-rgt-identity N/A

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

      9. mul0-lft N/A

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

      10. +-lft-identity N/A

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

      11. metadata-eval N/A

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

      12. lower-*.f32 N/A

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

      13. metadata-eval N/A

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

      14. sub0-neg N/A

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

      15. lower-neg.f32 N/A

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

      16. pow2 N/A

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

      17. pow2 N/A

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

      18. pow-prod-up N/A

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

      19. lower-pow.f32 N/A

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

      20. metadata-eval N/A

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

      21. metadata-eval N/A

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

   4. Applied rewrites 93.3%

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

   if 0.996599972 < (-.f32 #s(literal 1 binary32) u0)

   1. Initial program 42.0%

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

   3. Taylor expanded in u0 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

      5. lower-fma.f32 86.6

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

   5. Applied rewrites 86.1%

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

   7. Applied rewrites 98.4%

      \[\leadsto \left(\left(-\alpha\right) \cdot \alpha\right) \cdot \left(\left(-0.5 \cdot u0\right) \cdot u0 + \color{blue}{\left(-u0\right)}\right) \]
   8. Step-by-step derivation

   9. Applied rewrites 98.4%

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

Alternative 2: 96.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;1 - u0 \leq 0.9965999722480774:\\ \;\;\;\;\frac{\alpha}{\frac{-1}{\left(\left(\alpha \cdot \alpha\right) \cdot \alpha\right) \cdot \frac{\frac{1}{\alpha}}{\alpha}}} \cdot \log \left(1 - u0\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-\alpha\right) \cdot \alpha\right) \cdot \left(\left(u0 \cdot -0.5\right) \cdot u0 - u0\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha u0)
 :precision binary32
 (if (<= (- 1.0 u0) 0.9965999722480774)
   (*
    (/ alpha (/ -1.0 (* (* (* alpha alpha) alpha) (/ (/ 1.0 alpha) alpha))))
    (log (- 1.0 u0)))
   (* (* (- alpha) alpha) (- (* (* u0 -0.5) u0) u0))))

C

float code(float alpha, float u0) {
	float tmp;
	if ((1.0f - u0) <= 0.9965999722480774f) {
		tmp = (alpha / (-1.0f / (((alpha * alpha) * alpha) * ((1.0f / alpha) / alpha)))) * logf((1.0f - u0));
	} else {
		tmp = (-alpha * alpha) * (((u0 * -0.5f) * u0) - u0);
	}
	return tmp;
}

Fortran

real(4) function code(alpha, u0)
    real(4), intent (in) :: alpha
    real(4), intent (in) :: u0
    real(4) :: tmp
    if ((1.0e0 - u0) <= 0.9965999722480774e0) then
        tmp = (alpha / ((-1.0e0) / (((alpha * alpha) * alpha) * ((1.0e0 / alpha) / alpha)))) * log((1.0e0 - u0))
    else
        tmp = (-alpha * alpha) * (((u0 * (-0.5e0)) * u0) - u0)
    end if
    code = tmp
end function

Julia

function code(alpha, u0)
	tmp = Float32(0.0)
	if (Float32(Float32(1.0) - u0) <= Float32(0.9965999722480774))
		tmp = Float32(Float32(alpha / Float32(Float32(-1.0) / Float32(Float32(Float32(alpha * alpha) * alpha) * Float32(Float32(Float32(1.0) / alpha) / alpha)))) * log(Float32(Float32(1.0) - u0)));
	else
		tmp = Float32(Float32(Float32(-alpha) * alpha) * Float32(Float32(Float32(u0 * Float32(-0.5)) * u0) - u0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, u0)
	tmp = single(0.0);
	if ((single(1.0) - u0) <= single(0.9965999722480774))
		tmp = (alpha / (single(-1.0) / (((alpha * alpha) * alpha) * ((single(1.0) / alpha) / alpha)))) * log((single(1.0) - u0));
	else
		tmp = (-alpha * alpha) * (((u0 * single(-0.5)) * u0) - u0);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (-.f32 #s(literal 1 binary32) u0) < 0.996599972

   1. Initial program 93.1%

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

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

      3. lift-neg.f32 N/A

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

      4. neg-sub0 N/A

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

      5. flip3-- N/A

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

      6. clear-num N/A

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

      7. un-div-inv N/A

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

      8. lower-/.f32 N/A

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

      9. metadata-eval N/A

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

      10. +-lft-identity N/A

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

      11. mul0-lft N/A

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

      12. +-rgt-identity N/A

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

      13. clear-num N/A

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

      14. +-rgt-identity N/A

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

      15. mul0-lft N/A

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

      16. +-lft-identity N/A

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

      17. metadata-eval N/A

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

      18. flip3-- N/A

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

      19. neg-sub0 N/A

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

      20. lift-neg.f32 N/A

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

      21. lower-/.f32 93.0

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

   4. Applied rewrites 93.0%

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

      1. /-rgt-identity N/A

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

      2. div-inv N/A

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

      3. remove-double-div N/A

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

      4. associate-/r/ N/A

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

      5. metadata-eval N/A

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

      6. lft-mult-inverse N/A

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

      7. lift-/.f32 N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. lift-*.f32 N/A

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

      11. metadata-eval N/A

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

      12. lft-mult-inverse N/A

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

      13. lift-/.f32 N/A

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

      14. *-commutative N/A

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

      15. associate-*l* N/A

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

      16. associate-*r* N/A

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

      17. lift-*.f32 N/A

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

      18. lift-/.f32 N/A

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

      19. associate-*l/ N/A

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

      20. associate-*l/ N/A

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

      21. associate-/l* N/A

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

      22. lower-*.f32 N/A

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

   6. Applied rewrites 93.2%

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

   if 0.996599972 < (-.f32 #s(literal 1 binary32) u0)

   1. Initial program 42.0%

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

   3. Taylor expanded in u0 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

      5. lower-fma.f32 86.6

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

   5. Applied rewrites 86.6%

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

   7. Applied rewrites 98.4%

      \[\leadsto \left(\left(-\alpha\right) \cdot \alpha\right) \cdot \left(\left(-0.5 \cdot u0\right) \cdot u0 + \color{blue}{\left(-u0\right)}\right) \]
   8. Step-by-step derivation

   9. Applied rewrites 98.4%

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

4. Final simplification 97.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 - u0 \leq 0.9965999722480774:\\ \;\;\;\;\frac{\alpha}{\frac{-1}{\left(\left(\alpha \cdot \alpha\right) \cdot \alpha\right) \cdot \frac{\frac{1}{\alpha}}{\alpha}}} \cdot \log \left(1 - u0\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-\alpha\right) \cdot \alpha\right) \cdot \left(\left(u0 \cdot -0.5\right) \cdot u0 - u0\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 96.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;1 - u0 \leq 0.9965999722480774:\\ \;\;\;\;\left(\left(\frac{1}{\frac{1}{\alpha \cdot \alpha}} \cdot \frac{-1}{\alpha}\right) \cdot \alpha\right) \cdot \log \left(1 - u0\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-\alpha\right) \cdot \alpha\right) \cdot \left(\left(u0 \cdot -0.5\right) \cdot u0 - u0\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha u0)
 :precision binary32
 (if (<= (- 1.0 u0) 0.9965999722480774)
   (*
    (* (* (/ 1.0 (/ 1.0 (* alpha alpha))) (/ -1.0 alpha)) alpha)
    (log (- 1.0 u0)))
   (* (* (- alpha) alpha) (- (* (* u0 -0.5) u0) u0))))

C

float code(float alpha, float u0) {
	float tmp;
	if ((1.0f - u0) <= 0.9965999722480774f) {
		tmp = (((1.0f / (1.0f / (alpha * alpha))) * (-1.0f / alpha)) * alpha) * logf((1.0f - u0));
	} else {
		tmp = (-alpha * alpha) * (((u0 * -0.5f) * u0) - u0);
	}
	return tmp;
}

Fortran

real(4) function code(alpha, u0)
    real(4), intent (in) :: alpha
    real(4), intent (in) :: u0
    real(4) :: tmp
    if ((1.0e0 - u0) <= 0.9965999722480774e0) then
        tmp = (((1.0e0 / (1.0e0 / (alpha * alpha))) * ((-1.0e0) / alpha)) * alpha) * log((1.0e0 - u0))
    else
        tmp = (-alpha * alpha) * (((u0 * (-0.5e0)) * u0) - u0)
    end if
    code = tmp
end function

Julia

function code(alpha, u0)
	tmp = Float32(0.0)
	if (Float32(Float32(1.0) - u0) <= Float32(0.9965999722480774))
		tmp = Float32(Float32(Float32(Float32(Float32(1.0) / Float32(Float32(1.0) / Float32(alpha * alpha))) * Float32(Float32(-1.0) / alpha)) * alpha) * log(Float32(Float32(1.0) - u0)));
	else
		tmp = Float32(Float32(Float32(-alpha) * alpha) * Float32(Float32(Float32(u0 * Float32(-0.5)) * u0) - u0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, u0)
	tmp = single(0.0);
	if ((single(1.0) - u0) <= single(0.9965999722480774))
		tmp = (((single(1.0) / (single(1.0) / (alpha * alpha))) * (single(-1.0) / alpha)) * alpha) * log((single(1.0) - u0));
	else
		tmp = (-alpha * alpha) * (((u0 * single(-0.5)) * u0) - u0);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (-.f32 #s(literal 1 binary32) u0) < 0.996599972

   1. Initial program 93.1%

      \[\left(\left(-\alpha\right) \cdot \alpha\right) \cdot \log \left(1 - u0\right) \]
   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 93.0

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

   4. Applied rewrites 93.0%

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

   6. Applied rewrites 93.2%

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

   if 0.996599972 < (-.f32 #s(literal 1 binary32) u0)

   1. Initial program 42.0%

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

   3. Taylor expanded in u0 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

      5. lower-fma.f32 86.6

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

   5. Applied rewrites 86.1%

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

   7. Applied rewrites 98.4%

      \[\leadsto \left(\left(-\alpha\right) \cdot \alpha\right) \cdot \left(\left(-0.5 \cdot u0\right) \cdot u0 + \color{blue}{\left(-u0\right)}\right) \]
   8. Step-by-step derivation

   9. Applied rewrites 98.4%

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

4. Final simplification 97.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 - u0 \leq 0.9965999722480774:\\ \;\;\;\;\left(\left(\frac{1}{\frac{1}{\alpha \cdot \alpha}} \cdot \frac{-1}{\alpha}\right) \cdot \alpha\right) \cdot \log \left(1 - u0\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-\alpha\right) \cdot \alpha\right) \cdot \left(\left(u0 \cdot -0.5\right) \cdot u0 - u0\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 96.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;1 - u0 \leq 0.9965999722480774:\\ \;\;\;\;\frac{\alpha}{\frac{-1}{\frac{\alpha \cdot \alpha}{\alpha}}} \cdot \log \left(1 - u0\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-\alpha\right) \cdot \alpha\right) \cdot \left(\left(u0 \cdot -0.5\right) \cdot u0 - u0\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha u0)
 :precision binary32
 (if (<= (- 1.0 u0) 0.9965999722480774)
   (* (/ alpha (/ -1.0 (/ (* alpha alpha) alpha))) (log (- 1.0 u0)))
   (* (* (- alpha) alpha) (- (* (* u0 -0.5) u0) u0))))

C

float code(float alpha, float u0) {
	float tmp;
	if ((1.0f - u0) <= 0.9965999722480774f) {
		tmp = (alpha / (-1.0f / ((alpha * alpha) / alpha))) * logf((1.0f - u0));
	} else {
		tmp = (-alpha * alpha) * (((u0 * -0.5f) * u0) - u0);
	}
	return tmp;
}

Fortran

real(4) function code(alpha, u0)
    real(4), intent (in) :: alpha
    real(4), intent (in) :: u0
    real(4) :: tmp
    if ((1.0e0 - u0) <= 0.9965999722480774e0) then
        tmp = (alpha / ((-1.0e0) / ((alpha * alpha) / alpha))) * log((1.0e0 - u0))
    else
        tmp = (-alpha * alpha) * (((u0 * (-0.5e0)) * u0) - u0)
    end if
    code = tmp
end function

Julia

function code(alpha, u0)
	tmp = Float32(0.0)
	if (Float32(Float32(1.0) - u0) <= Float32(0.9965999722480774))
		tmp = Float32(Float32(alpha / Float32(Float32(-1.0) / Float32(Float32(alpha * alpha) / alpha))) * log(Float32(Float32(1.0) - u0)));
	else
		tmp = Float32(Float32(Float32(-alpha) * alpha) * Float32(Float32(Float32(u0 * Float32(-0.5)) * u0) - u0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, u0)
	tmp = single(0.0);
	if ((single(1.0) - u0) <= single(0.9965999722480774))
		tmp = (alpha / (single(-1.0) / ((alpha * alpha) / alpha))) * log((single(1.0) - u0));
	else
		tmp = (-alpha * alpha) * (((u0 * single(-0.5)) * u0) - u0);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (-.f32 #s(literal 1 binary32) u0) < 0.996599972

   1. Initial program 93.1%

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

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

      3. lift-neg.f32 N/A

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

      4. neg-sub0 N/A

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

      5. flip3-- N/A

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

      6. clear-num N/A

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

      7. un-div-inv N/A

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

      8. lower-/.f32 N/A

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

      9. metadata-eval N/A

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

      10. +-lft-identity N/A

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

      11. mul0-lft N/A

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

      12. +-rgt-identity N/A

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

      13. clear-num N/A

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

      14. +-rgt-identity N/A

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

      15. mul0-lft N/A

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

      16. +-lft-identity N/A

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

      17. metadata-eval N/A

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

      18. flip3-- N/A

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

      19. neg-sub0 N/A

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

      20. lift-neg.f32 N/A

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

      21. lower-/.f32 93.0

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

   4. Applied rewrites 93.0%

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

      1. /-rgt-identity N/A

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

      2. div-inv N/A

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

      3. metadata-eval N/A

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

      4. lft-mult-inverse N/A

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

      5. associate-*l/ N/A

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

      6. metadata-eval N/A

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

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

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

      8. neg-mul-1 N/A

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

      9. remove-double-neg N/A

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

      10. associate-/l* N/A

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

      11. lift-*.f32 N/A

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

      12. lower-/.f32 93.1

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

   6. Applied rewrites 93.1%

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

   if 0.996599972 < (-.f32 #s(literal 1 binary32) u0)

   1. Initial program 42.0%

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

   3. Taylor expanded in u0 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

      5. lower-fma.f32 86.6

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

   5. Applied rewrites 86.6%

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

   7. Applied rewrites 98.4%

      \[\leadsto \left(\left(-\alpha\right) \cdot \alpha\right) \cdot \left(\left(-0.5 \cdot u0\right) \cdot u0 + \color{blue}{\left(-u0\right)}\right) \]
   8. Step-by-step derivation

   9. Applied rewrites 98.4%

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

4. Final simplification 97.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 - u0 \leq 0.9965999722480774:\\ \;\;\;\;\frac{\alpha}{\frac{-1}{\frac{\alpha \cdot \alpha}{\alpha}}} \cdot \log \left(1 - u0\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-\alpha\right) \cdot \alpha\right) \cdot \left(\left(u0 \cdot -0.5\right) \cdot u0 - u0\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 96.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;1 - u0 \leq 0.9965999722480774:\\ \;\;\;\;\left(\frac{-1}{\frac{1}{\alpha}} \cdot \alpha\right) \cdot \log \left(1 - u0\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-\alpha\right) \cdot \alpha\right) \cdot \left(\left(u0 \cdot -0.5\right) \cdot u0 - u0\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha u0)
 :precision binary32
 (if (<= (- 1.0 u0) 0.9965999722480774)
   (* (* (/ -1.0 (/ 1.0 alpha)) alpha) (log (- 1.0 u0)))
   (* (* (- alpha) alpha) (- (* (* u0 -0.5) u0) u0))))

C

float code(float alpha, float u0) {
	float tmp;
	if ((1.0f - u0) <= 0.9965999722480774f) {
		tmp = ((-1.0f / (1.0f / alpha)) * alpha) * logf((1.0f - u0));
	} else {
		tmp = (-alpha * alpha) * (((u0 * -0.5f) * u0) - u0);
	}
	return tmp;
}

Fortran

real(4) function code(alpha, u0)
    real(4), intent (in) :: alpha
    real(4), intent (in) :: u0
    real(4) :: tmp
    if ((1.0e0 - u0) <= 0.9965999722480774e0) then
        tmp = (((-1.0e0) / (1.0e0 / alpha)) * alpha) * log((1.0e0 - u0))
    else
        tmp = (-alpha * alpha) * (((u0 * (-0.5e0)) * u0) - u0)
    end if
    code = tmp
end function

Julia

function code(alpha, u0)
	tmp = Float32(0.0)
	if (Float32(Float32(1.0) - u0) <= Float32(0.9965999722480774))
		tmp = Float32(Float32(Float32(Float32(-1.0) / Float32(Float32(1.0) / alpha)) * alpha) * log(Float32(Float32(1.0) - u0)));
	else
		tmp = Float32(Float32(Float32(-alpha) * alpha) * Float32(Float32(Float32(u0 * Float32(-0.5)) * u0) - u0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, u0)
	tmp = single(0.0);
	if ((single(1.0) - u0) <= single(0.9965999722480774))
		tmp = ((single(-1.0) / (single(1.0) / alpha)) * alpha) * log((single(1.0) - u0));
	else
		tmp = (-alpha * alpha) * (((u0 * single(-0.5)) * u0) - u0);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (-.f32 #s(literal 1 binary32) u0) < 0.996599972

   1. Initial program 93.1%

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

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

      5. lower-/.f32 N/A

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

      6. metadata-eval N/A

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

      7. +-lft-identity N/A

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

      8. mul0-lft N/A

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

      9. +-rgt-identity N/A

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

      10. clear-num N/A

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

      11. +-rgt-identity N/A

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

      12. mul0-lft N/A

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

      13. +-lft-identity N/A

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

      14. metadata-eval N/A

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

      15. flip3-- N/A

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

      16. neg-sub0 N/A

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

      17. lift-neg.f32 N/A

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

      18. lower-/.f32 93.1

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

   4. Applied rewrites 93.1%

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

   if 0.996599972 < (-.f32 #s(literal 1 binary32) u0)

   1. Initial program 42.0%

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

   3. Taylor expanded in u0 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

      5. lower-fma.f32 86.6

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

   5. Applied rewrites 86.1%

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

   7. Applied rewrites 98.4%

      \[\leadsto \left(\left(-\alpha\right) \cdot \alpha\right) \cdot \left(\left(-0.5 \cdot u0\right) \cdot u0 + \color{blue}{\left(-u0\right)}\right) \]
   8. Step-by-step derivation

   9. Applied rewrites 98.4%

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

4. Final simplification 97.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 - u0 \leq 0.9965999722480774:\\ \;\;\;\;\left(\frac{-1}{\frac{1}{\alpha}} \cdot \alpha\right) \cdot \log \left(1 - u0\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-\alpha\right) \cdot \alpha\right) \cdot \left(\left(u0 \cdot -0.5\right) \cdot u0 - u0\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 96.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(-\alpha\right) \cdot \alpha\\ \mathbf{if}\;1 - u0 \leq 0.9965999722480774:\\ \;\;\;\;t\_0 \cdot \log \left(1 - u0\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \left(\left(u0 \cdot -0.5\right) \cdot u0 - u0\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha u0)
 :precision binary32
 (let* ((t_0 (* (- alpha) alpha)))
   (if (<= (- 1.0 u0) 0.9965999722480774)
     (* t_0 (log (- 1.0 u0)))
     (* t_0 (- (* (* u0 -0.5) u0) u0)))))

C

float code(float alpha, float u0) {
	float t_0 = -alpha * alpha;
	float tmp;
	if ((1.0f - u0) <= 0.9965999722480774f) {
		tmp = t_0 * logf((1.0f - u0));
	} else {
		tmp = t_0 * (((u0 * -0.5f) * u0) - u0);
	}
	return tmp;
}

Fortran

real(4) function code(alpha, u0)
    real(4), intent (in) :: alpha
    real(4), intent (in) :: u0
    real(4) :: t_0
    real(4) :: tmp
    t_0 = -alpha * alpha
    if ((1.0e0 - u0) <= 0.9965999722480774e0) then
        tmp = t_0 * log((1.0e0 - u0))
    else
        tmp = t_0 * (((u0 * (-0.5e0)) * u0) - u0)
    end if
    code = tmp
end function

Julia

function code(alpha, u0)
	t_0 = Float32(Float32(-alpha) * alpha)
	tmp = Float32(0.0)
	if (Float32(Float32(1.0) - u0) <= Float32(0.9965999722480774))
		tmp = Float32(t_0 * log(Float32(Float32(1.0) - u0)));
	else
		tmp = Float32(t_0 * Float32(Float32(Float32(u0 * Float32(-0.5)) * u0) - u0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, u0)
	t_0 = -alpha * alpha;
	tmp = single(0.0);
	if ((single(1.0) - u0) <= single(0.9965999722480774))
		tmp = t_0 * log((single(1.0) - u0));
	else
		tmp = t_0 * (((u0 * single(-0.5)) * u0) - u0);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (-.f32 #s(literal 1 binary32) u0) < 0.996599972

   1. Initial program 93.1%

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

   if 0.996599972 < (-.f32 #s(literal 1 binary32) u0)

   1. Initial program 42.0%

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

   3. Taylor expanded in u0 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

      5. lower-fma.f32 86.6

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

   5. Applied rewrites 86.1%

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

   7. Applied rewrites 98.4%

      \[\leadsto \left(\left(-\alpha\right) \cdot \alpha\right) \cdot \left(\left(-0.5 \cdot u0\right) \cdot u0 + \color{blue}{\left(-u0\right)}\right) \]
   8. Step-by-step derivation

   9. Applied rewrites 98.4%

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

Alternative 7: 87.7% accurate, 4.5× speedup

Math

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

FPCore

(FPCore (alpha u0)
 :precision binary32
 (* (* (- alpha) alpha) (- (* (* u0 -0.5) u0) u0)))

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 55.0%

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

3. Taylor expanded in u0 around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. sub-neg N/A

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

   4. metadata-eval N/A

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

   5. lower-fma.f32 75.6

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

5. Applied rewrites 75.2%

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

7. Applied rewrites 88.4%

   \[\leadsto \left(\left(-\alpha\right) \cdot \alpha\right) \cdot \left(\left(-0.5 \cdot u0\right) \cdot u0 + \color{blue}{\left(-u0\right)}\right) \]
8. Step-by-step derivation

9. Applied rewrites 88.4%

   \[\leadsto \left(\left(-\alpha\right) \cdot \alpha\right) \cdot \left(\left(u0 \cdot -0.5\right) \cdot u0 - \color{blue}{u0}\right) \]
10. Add Preprocessing

Alternative 8: 87.5% accurate, 4.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 55.0%

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

3. Taylor expanded in u0 around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. sub-neg N/A

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

   4. metadata-eval N/A

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

   5. lower-fma.f32 75.6

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

5. Applied rewrites 75.2%

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

7. Applied rewrites 88.2%

   \[\leadsto \left(\left(-\alpha\right) \cdot \alpha\right) \cdot \left(\left(-0.5 \cdot u0 + -1\right) \cdot u0\right) \]
8. Add Preprocessing

Alternative 9: 75.0% 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 55.0%

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

3. Taylor expanded in u0 around 0

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

   1. lower-*.f32 N/A

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

   2. unpow2 N/A

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

   3. lower-*.f32 75.6

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

5. Applied rewrites 75.6%

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

Reproduce

herbie shell --seed 2024298 
(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))))