Beckmann Distribution sample, tan2theta, alphax == alphay

Percentage Accurate: 55.5% → 96.8%

Time: 6.6s

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: 55.5% 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: 96.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\alpha \cdot \alpha\right) \cdot u0\\ \mathbf{if}\;u0 \leq 0.003599999938160181:\\ \;\;\;\;\left(t\_0 \cdot 0.5\right) \cdot u0 + t\_0\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\alpha \cdot \alpha\right) \cdot \left(\alpha \cdot \frac{-1}{\alpha}\right)\right) \cdot \log \left(1 - u0\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha u0)
 :precision binary32
 (let* ((t_0 (* (* alpha alpha) u0)))
   (if (<= u0 0.003599999938160181)
     (+ (* (* t_0 0.5) u0) t_0)
     (* (* (* alpha alpha) (* alpha (/ -1.0 alpha))) (log (- 1.0 u0))))))

C

float code(float alpha, float u0) {
	float t_0 = (alpha * alpha) * u0;
	float tmp;
	if (u0 <= 0.003599999938160181f) {
		tmp = ((t_0 * 0.5f) * u0) + t_0;
	} else {
		tmp = ((alpha * alpha) * (alpha * (-1.0f / alpha))) * logf((1.0f - 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) * u0
    if (u0 <= 0.003599999938160181e0) then
        tmp = ((t_0 * 0.5e0) * u0) + t_0
    else
        tmp = ((alpha * alpha) * (alpha * ((-1.0e0) / alpha))) * log((1.0e0 - u0))
    end if
    code = tmp
end function

Julia

function code(alpha, u0)
	t_0 = Float32(Float32(alpha * alpha) * u0)
	tmp = Float32(0.0)
	if (u0 <= Float32(0.003599999938160181))
		tmp = Float32(Float32(Float32(t_0 * Float32(0.5)) * u0) + t_0);
	else
		tmp = Float32(Float32(Float32(alpha * alpha) * Float32(alpha * Float32(Float32(-1.0) / alpha))) * log(Float32(Float32(1.0) - u0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, u0)
	t_0 = (alpha * alpha) * u0;
	tmp = single(0.0);
	if (u0 <= single(0.003599999938160181))
		tmp = ((t_0 * single(0.5)) * u0) + t_0;
	else
		tmp = ((alpha * alpha) * (alpha * (single(-1.0) / alpha))) * log((single(1.0) - u0));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if u0 < 0.00359999994

   1. Initial program 39.2%

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

   3. Taylor expanded in u0 around 0

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

      1. distribute-lft-in N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. distribute-lft1-in N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f32 N/A

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

      8. lower-*.f32 N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f32 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f32 88.2

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

   5. Applied rewrites 87.8%

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

   7. Applied rewrites 98.2%

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

   9. Applied rewrites 98.4%

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

   if 0.00359999994 < u0

   1. Initial program 92.6%

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

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

      5. associate-*l/ N/A

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

      6. metadata-eval N/A

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

      7. +-lft-identity N/A

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

      8. mul0-lft N/A

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

      9. +-rgt-identity N/A

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

      10. lower-/.f32 N/A

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

      11. lower-*.f32 N/A

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

      12. metadata-eval N/A

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

      13. sub0-neg N/A

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

      14. cube-neg N/A

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

      15. lift-neg.f32 N/A

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

      16. lower-pow.f32 N/A

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

      17. lower-*.f32 92.4

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

   4. Applied rewrites 92.4%

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

      1. lift-/.f32 N/A

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

      2. lift-*.f32 N/A

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

      3. associate-/l* N/A

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

      4. lift-pow.f32 N/A

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

      5. unpow3 N/A

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

      6. lift-neg.f32 N/A

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

      7. lift-neg.f32 N/A

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

      8. sqr-neg N/A

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

      9. lift-*.f32 N/A

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

      10. associate-*l* N/A

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

      11. lower-*.f32 N/A

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

      12. lift-*.f32 N/A

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

      13. associate-/r* N/A

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

      14. *-inverses N/A

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

      15. remove-double-neg N/A

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

      16. distribute-frac-neg2 N/A

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

      17. lift-neg.f32 N/A

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

      18. lift-/.f32 N/A

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

      19. lower-*.f32 N/A

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

      20. lift-/.f32 N/A

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

      21. lift-neg.f32 N/A

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

      22. distribute-frac-neg2 N/A

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

      23. remove-double-neg N/A

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

      24. lower-/.f32 92.8

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

   6. Applied rewrites 92.8%

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

4. Final simplification 97.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u0 \leq 0.003599999938160181:\\ \;\;\;\;\left(\left(\left(\alpha \cdot \alpha\right) \cdot u0\right) \cdot 0.5\right) \cdot u0 + \left(\alpha \cdot \alpha\right) \cdot u0\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\alpha \cdot \alpha\right) \cdot \left(\alpha \cdot \frac{-1}{\alpha}\right)\right) \cdot \log \left(1 - u0\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 96.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\alpha \cdot \alpha\right) \cdot u0\\ \mathbf{if}\;u0 \leq 0.003599999938160181:\\ \;\;\;\;\left(t\_0 \cdot 0.5\right) \cdot u0 + t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{-1}{\alpha \cdot \alpha}} \cdot \log \left(1 - u0\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha u0)
 :precision binary32
 (let* ((t_0 (* (* alpha alpha) u0)))
   (if (<= u0 0.003599999938160181)
     (+ (* (* t_0 0.5) u0) t_0)
     (* (/ 1.0 (/ -1.0 (* alpha alpha))) (log (- 1.0 u0))))))

C

float code(float alpha, float u0) {
	float t_0 = (alpha * alpha) * u0;
	float tmp;
	if (u0 <= 0.003599999938160181f) {
		tmp = ((t_0 * 0.5f) * u0) + t_0;
	} else {
		tmp = (1.0f / (-1.0f / (alpha * alpha))) * logf((1.0f - 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) * u0
    if (u0 <= 0.003599999938160181e0) then
        tmp = ((t_0 * 0.5e0) * u0) + t_0
    else
        tmp = (1.0e0 / ((-1.0e0) / (alpha * alpha))) * log((1.0e0 - u0))
    end if
    code = tmp
end function

Julia

function code(alpha, u0)
	t_0 = Float32(Float32(alpha * alpha) * u0)
	tmp = Float32(0.0)
	if (u0 <= Float32(0.003599999938160181))
		tmp = Float32(Float32(Float32(t_0 * Float32(0.5)) * u0) + t_0);
	else
		tmp = Float32(Float32(Float32(1.0) / Float32(Float32(-1.0) / Float32(alpha * alpha))) * log(Float32(Float32(1.0) - u0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, u0)
	t_0 = (alpha * alpha) * u0;
	tmp = single(0.0);
	if (u0 <= single(0.003599999938160181))
		tmp = ((t_0 * single(0.5)) * u0) + t_0;
	else
		tmp = (single(1.0) / (single(-1.0) / (alpha * alpha))) * log((single(1.0) - u0));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if u0 < 0.00359999994

   1. Initial program 39.2%

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

   3. Taylor expanded in u0 around 0

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

      1. distribute-lft-in N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. distribute-lft1-in N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f32 N/A

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

      8. lower-*.f32 N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f32 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f32 88.2

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

   5. Applied rewrites 87.8%

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

   7. Applied rewrites 98.2%

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

   9. Applied rewrites 98.4%

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

   if 0.00359999994 < u0

   1. Initial program 92.6%

      \[\left(\left(-\alpha\right) \cdot \alpha\right) \cdot \log \left(1 - u0\right) \]
   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 92.7

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

   4. Applied rewrites 92.7%

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

      1. lift-*.f32 N/A

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

      2. lift-/.f32 N/A

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

      3. un-div-inv N/A

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

      4. lift-*.f32 N/A

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

      5. associate-*l/ N/A

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

      6. lift-*.f32 N/A

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

      7. lift-neg.f32 N/A

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

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

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

      9. lift-*.f32 N/A

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

      10. remove-double-neg N/A

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

      11. lift-neg.f32 N/A

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

      12. frac-2neg N/A

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

      13. associate-*l/ N/A

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

      14. lift-*.f32 N/A

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

      15. pow3 N/A

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

      16. metadata-eval N/A

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

      17. pow-prod-up N/A

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

      18. lift-pow.f32 N/A

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

      19. remove-double-neg N/A

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

      20. lift-neg.f32 N/A

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

      21. lift-neg.f32 N/A

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

      22. inv-pow N/A

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

      23. div-inv N/A

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

   6. Applied rewrites 92.7%

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

Alternative 3: 96.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\alpha \cdot \alpha\right) \cdot u0\\ \mathbf{if}\;u0 \leq 0.003599999938160181:\\ \;\;\;\;\left(t\_0 \cdot 0.5\right) \cdot u0 + t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left(-\alpha\right) \cdot \alpha\right) \cdot \alpha}{\alpha} \cdot \log \left(1 - u0\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha u0)
 :precision binary32
 (let* ((t_0 (* (* alpha alpha) u0)))
   (if (<= u0 0.003599999938160181)
     (+ (* (* t_0 0.5) u0) t_0)
     (* (/ (* (* (- alpha) alpha) alpha) alpha) (log (- 1.0 u0))))))

C

float code(float alpha, float u0) {
	float t_0 = (alpha * alpha) * u0;
	float tmp;
	if (u0 <= 0.003599999938160181f) {
		tmp = ((t_0 * 0.5f) * u0) + t_0;
	} else {
		tmp = (((-alpha * alpha) * alpha) / alpha) * logf((1.0f - 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) * u0
    if (u0 <= 0.003599999938160181e0) then
        tmp = ((t_0 * 0.5e0) * u0) + t_0
    else
        tmp = (((-alpha * alpha) * alpha) / alpha) * log((1.0e0 - u0))
    end if
    code = tmp
end function

Julia

function code(alpha, u0)
	t_0 = Float32(Float32(alpha * alpha) * u0)
	tmp = Float32(0.0)
	if (u0 <= Float32(0.003599999938160181))
		tmp = Float32(Float32(Float32(t_0 * Float32(0.5)) * u0) + t_0);
	else
		tmp = Float32(Float32(Float32(Float32(Float32(-alpha) * alpha) * alpha) / alpha) * log(Float32(Float32(1.0) - u0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, u0)
	t_0 = (alpha * alpha) * u0;
	tmp = single(0.0);
	if (u0 <= single(0.003599999938160181))
		tmp = ((t_0 * single(0.5)) * u0) + t_0;
	else
		tmp = (((-alpha * alpha) * alpha) / alpha) * log((single(1.0) - u0));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if u0 < 0.00359999994

   1. Initial program 39.2%

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

   3. Taylor expanded in u0 around 0

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

      1. distribute-lft-in N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. distribute-lft1-in N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f32 N/A

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

      8. lower-*.f32 N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f32 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f32 88.2

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

   5. Applied rewrites 87.8%

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

   7. Applied rewrites 98.2%

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

   9. Applied rewrites 98.4%

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

   if 0.00359999994 < u0

   1. Initial program 92.6%

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

      1. lift-*.f32 N/A

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

      2. lift-neg.f32 N/A

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

      3. neg-sub0 N/A

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

      4. flip-- N/A

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

      5. metadata-eval N/A

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

      6. neg-sub0 N/A

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

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

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

      8. lift-neg.f32 N/A

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

      9. lift-*.f32 N/A

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

      10. +-lft-identity N/A

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

      11. associate-*l/ N/A

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

      12. lower-/.f32 N/A

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

      13. lower-*.f32 92.7

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

   4. Applied rewrites 92.7%

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

Alternative 4: 96.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\alpha \cdot \alpha\right) \cdot u0\\ \mathbf{if}\;u0 \leq 0.003599999938160181:\\ \;\;\;\;\left(t\_0 \cdot 0.5\right) \cdot u0 + t\_0\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-\alpha\right) \cdot \alpha\right) \cdot \log \left(1 - u0\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha u0)
 :precision binary32
 (let* ((t_0 (* (* alpha alpha) u0)))
   (if (<= u0 0.003599999938160181)
     (+ (* (* t_0 0.5) u0) t_0)
     (* (* (- alpha) alpha) (log (- 1.0 u0))))))

C

float code(float alpha, float u0) {
	float t_0 = (alpha * alpha) * u0;
	float tmp;
	if (u0 <= 0.003599999938160181f) {
		tmp = ((t_0 * 0.5f) * u0) + t_0;
	} else {
		tmp = (-alpha * alpha) * logf((1.0f - 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) * u0
    if (u0 <= 0.003599999938160181e0) then
        tmp = ((t_0 * 0.5e0) * u0) + t_0
    else
        tmp = (-alpha * alpha) * log((1.0e0 - u0))
    end if
    code = tmp
end function

Julia

function code(alpha, u0)
	t_0 = Float32(Float32(alpha * alpha) * u0)
	tmp = Float32(0.0)
	if (u0 <= Float32(0.003599999938160181))
		tmp = Float32(Float32(Float32(t_0 * Float32(0.5)) * u0) + t_0);
	else
		tmp = Float32(Float32(Float32(-alpha) * alpha) * log(Float32(Float32(1.0) - u0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, u0)
	t_0 = (alpha * alpha) * u0;
	tmp = single(0.0);
	if (u0 <= single(0.003599999938160181))
		tmp = ((t_0 * single(0.5)) * u0) + t_0;
	else
		tmp = (-alpha * alpha) * log((single(1.0) - u0));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if u0 < 0.00359999994

   1. Initial program 39.2%

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

   3. Taylor expanded in u0 around 0

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

      1. distribute-lft-in N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. distribute-lft1-in N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f32 N/A

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

      8. lower-*.f32 N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f32 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f32 88.2

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

   5. Applied rewrites 87.8%

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

   7. Applied rewrites 98.2%

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

   9. Applied rewrites 98.4%

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

   if 0.00359999994 < u0

   1. Initial program 92.6%

      \[\left(\left(-\alpha\right) \cdot \alpha\right) \cdot \log \left(1 - u0\right) \]
   2. Add Preprocessing
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 87.4% accurate, 3.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 51.9%

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

3. Taylor expanded in u0 around 0

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

   1. distribute-lft-in N/A

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

   2. associate-*r* N/A

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

   3. *-commutative N/A

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

   4. distribute-lft1-in N/A

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

   5. *-commutative N/A

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

   6. associate-*r* N/A

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

   7. lower-*.f32 N/A

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

   8. lower-*.f32 N/A

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

   9. *-commutative N/A

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

   10. lower-fma.f32 N/A

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

   11. unpow2 N/A

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

   12. lower-*.f32 77.2

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

5. Applied rewrites 76.9%

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

7. Applied rewrites 88.4%

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

9. Applied rewrites 88.5%

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

Alternative 6: 87.4% accurate, 4.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 51.9%

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

3. Taylor expanded in u0 around 0

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

   1. distribute-lft-in N/A

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

   2. associate-*r* N/A

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

   3. *-commutative N/A

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

   4. distribute-lft1-in N/A

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

   5. *-commutative N/A

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

   6. associate-*r* N/A

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

   7. lower-*.f32 N/A

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

   8. lower-*.f32 N/A

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

   9. *-commutative N/A

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

   10. lower-fma.f32 N/A

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

   11. unpow2 N/A

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

   12. lower-*.f32 77.2

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

5. Applied rewrites 76.9%

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

7. Applied rewrites 88.4%

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

Alternative 7: 87.2% accurate, 4.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 51.9%

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

3. Taylor expanded in u0 around 0

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

   1. distribute-lft-in N/A

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

   2. associate-*r* N/A

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

   3. *-commutative N/A

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

   4. distribute-lft1-in N/A

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

   5. *-commutative N/A

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

   6. associate-*r* N/A

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

   7. lower-*.f32 N/A

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

   8. lower-*.f32 N/A

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

   9. *-commutative N/A

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

   10. lower-fma.f32 N/A

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

   11. unpow2 N/A

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

   12. lower-*.f32 77.2

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

5. Applied rewrites 76.9%

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

7. Applied rewrites 88.2%

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

Alternative 8: 74.8% accurate, 10.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 51.9%

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

3. Taylor expanded in u0 around 0

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

   1. 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 77.2

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

5. Applied rewrites 77.2%

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

7. Applied rewrites 77.2%

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

Alternative 9: 74.8% 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 51.9%

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

3. Taylor expanded in u0 around 0

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

   1. 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 77.2

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

5. Applied rewrites 77.2%

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

Reproduce

herbie shell --seed 2024318 
(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))))