sqrt D (should all be same)

Percentage Accurate: 53.9% → 99.5%

Time: 14.4s

Alternatives: 8

Speedup: 4.9×

Specification

Math

\[\begin{array}{l} \\ \sqrt{2 \cdot {x}^{2}} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (sqrt (* 2.0 (pow x 2.0))))

C

double code(double x) {
	return sqrt((2.0 * pow(x, 2.0)));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = sqrt((2.0d0 * (x ** 2.0d0)))
end function

Java

public static double code(double x) {
	return Math.sqrt((2.0 * Math.pow(x, 2.0)));
}

Python

def code(x):
	return math.sqrt((2.0 * math.pow(x, 2.0)))

Julia

function code(x)
	return sqrt(Float64(2.0 * (x ^ 2.0)))
end

MATLAB

function tmp = code(x)
	tmp = sqrt((2.0 * (x ^ 2.0)));
end

Wolfram

code[x_] := N[Sqrt[N[(2.0 * N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Initial Program: 53.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \sqrt{2 \cdot {x}^{2}} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (sqrt (* 2.0 (pow x 2.0))))

C

double code(double x) {
	return sqrt((2.0 * pow(x, 2.0)));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = sqrt((2.0d0 * (x ** 2.0d0)))
end function

Java

public static double code(double x) {
	return Math.sqrt((2.0 * Math.pow(x, 2.0)));
}

Python

def code(x):
	return math.sqrt((2.0 * math.pow(x, 2.0)))

Julia

function code(x)
	return sqrt(Float64(2.0 * (x ^ 2.0)))
end

MATLAB

function tmp = code(x)
	tmp = sqrt((2.0 * (x ^ 2.0)));
end

Wolfram

code[x_] := N[Sqrt[N[(2.0 * N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Alternative 1: 99.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\left(-{16}^{0.03125}\right) \cdot \left({64}^{0.0625} \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;{x}^{0.75} \cdot {\left(4 \cdot x\right)}^{0.25}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -2e-310)
   (* (- (pow 16.0 0.03125)) (* (pow 64.0 0.0625) x))
   (* (pow x 0.75) (pow (* 4.0 x) 0.25))))

C

double code(double x) {
	double tmp;
	if (x <= -2e-310) {
		tmp = -pow(16.0, 0.03125) * (pow(64.0, 0.0625) * x);
	} else {
		tmp = pow(x, 0.75) * pow((4.0 * x), 0.25);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-2d-310)) then
        tmp = -(16.0d0 ** 0.03125d0) * ((64.0d0 ** 0.0625d0) * x)
    else
        tmp = (x ** 0.75d0) * ((4.0d0 * x) ** 0.25d0)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= -2e-310) {
		tmp = -Math.pow(16.0, 0.03125) * (Math.pow(64.0, 0.0625) * x);
	} else {
		tmp = Math.pow(x, 0.75) * Math.pow((4.0 * x), 0.25);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -2e-310:
		tmp = -math.pow(16.0, 0.03125) * (math.pow(64.0, 0.0625) * x)
	else:
		tmp = math.pow(x, 0.75) * math.pow((4.0 * x), 0.25)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -2e-310)
		tmp = Float64(Float64(-(16.0 ^ 0.03125)) * Float64((64.0 ^ 0.0625) * x));
	else
		tmp = Float64((x ^ 0.75) * (Float64(4.0 * x) ^ 0.25));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -2e-310)
		tmp = -(16.0 ^ 0.03125) * ((64.0 ^ 0.0625) * x);
	else
		tmp = (x ^ 0.75) * ((4.0 * x) ^ 0.25);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -2e-310], N[((-N[Power[16.0, 0.03125], $MachinePrecision]) * N[(N[Power[64.0, 0.0625], $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision], N[(N[Power[x, 0.75], $MachinePrecision] * N[Power[N[(4.0 * x), $MachinePrecision], 0.25], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.999999999999994e-310

   1. Initial program 56.2%

      \[\sqrt{2 \cdot {x}^{2}} \]
   2. Add Preprocessing

   4. Applied rewrites 2.1%

      \[\leadsto \color{blue}{\sqrt{2} \cdot x} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\sqrt{2} \cdot x} \]

      2. +-lft-identity N/A

         \[\leadsto \sqrt{2} \cdot \color{blue}{\left(0 + x\right)} \]

      3. flip-+ N/A

         \[\leadsto \sqrt{2} \cdot \color{blue}{\frac{0 \cdot 0 - x \cdot x}{0 - x}} \]

      4. neg-sub0 N/A

         \[\leadsto \sqrt{2} \cdot \frac{0 \cdot 0 - x \cdot x}{\color{blue}{\mathsf{neg}\left(x\right)}} \]

      5. lift-neg.f64 N/A

         \[\leadsto \sqrt{2} \cdot \frac{0 \cdot 0 - x \cdot x}{\color{blue}{-x}} \]

      6. div-inv N/A

         \[\leadsto \sqrt{2} \cdot \color{blue}{\left(\left(0 \cdot 0 - x \cdot x\right) \cdot \frac{1}{-x}\right)} \]

      7. inv-pow N/A

         \[\leadsto \sqrt{2} \cdot \left(\left(0 \cdot 0 - x \cdot x\right) \cdot \color{blue}{{\left(-x\right)}^{-1}}\right) \]

      8. metadata-eval N/A

         \[\leadsto \sqrt{2} \cdot \left(\left(0 \cdot 0 - x \cdot x\right) \cdot {\left(-x\right)}^{\color{blue}{\left(2 \cdot \frac{-1}{2}\right)}}\right) \]

      9. metadata-eval N/A

         \[\leadsto \sqrt{2} \cdot \left(\left(0 \cdot 0 - x \cdot x\right) \cdot {\left(-x\right)}^{\left(2 \cdot \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right)}\right) \]

      10. pow-pow N/A

          \[\leadsto \sqrt{2} \cdot \left(\left(0 \cdot 0 - x \cdot x\right) \cdot \color{blue}{{\left({\left(-x\right)}^{2}\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}\right) \]

      11. pow2 N/A

          \[\leadsto \sqrt{2} \cdot \left(\left(0 \cdot 0 - x \cdot x\right) \cdot {\color{blue}{\left(\left(-x\right) \cdot \left(-x\right)\right)}}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right) \]

      12. lift-neg.f64 N/A

          \[\leadsto \sqrt{2} \cdot \left(\left(0 \cdot 0 - x \cdot x\right) \cdot {\left(\color{blue}{\left(\mathsf{neg}\left(x\right)\right)} \cdot \left(-x\right)\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right) \]

      13. lift-neg.f64 N/A

          \[\leadsto \sqrt{2} \cdot \left(\left(0 \cdot 0 - x \cdot x\right) \cdot {\left(\left(\mathsf{neg}\left(x\right)\right) \cdot \color{blue}{\left(\mathsf{neg}\left(x\right)\right)}\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right) \]

      14. sqr-neg N/A

          \[\leadsto \sqrt{2} \cdot \left(\left(0 \cdot 0 - x \cdot x\right) \cdot {\color{blue}{\left(x \cdot x\right)}}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right) \]

      15. pow2 N/A

          \[\leadsto \sqrt{2} \cdot \left(\left(0 \cdot 0 - x \cdot x\right) \cdot {\color{blue}{\left({x}^{2}\right)}}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right) \]

      16. pow-pow N/A

          \[\leadsto \sqrt{2} \cdot \left(\left(0 \cdot 0 - x \cdot x\right) \cdot \color{blue}{{x}^{\left(2 \cdot \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto \sqrt{2} \cdot \left(\left(0 \cdot 0 - x \cdot x\right) \cdot {x}^{\left(2 \cdot \color{blue}{\frac{-1}{2}}\right)}\right) \]

      18. metadata-eval N/A

          \[\leadsto \sqrt{2} \cdot \left(\left(0 \cdot 0 - x \cdot x\right) \cdot {x}^{\color{blue}{-1}}\right) \]

      19. inv-pow N/A

          \[\leadsto \sqrt{2} \cdot \left(\left(0 \cdot 0 - x \cdot x\right) \cdot \color{blue}{\frac{1}{x}}\right) \]

      20. +-lft-identity N/A

          \[\leadsto \sqrt{2} \cdot \left(\left(0 \cdot 0 - x \cdot x\right) \cdot \frac{1}{\color{blue}{0 + x}}\right) \]

      21. div-inv N/A

          \[\leadsto \sqrt{2} \cdot \color{blue}{\frac{0 \cdot 0 - x \cdot x}{0 + x}} \]

      22. flip-- N/A

          \[\leadsto \sqrt{2} \cdot \color{blue}{\left(0 - x\right)} \]

      23. neg-sub0 N/A

          \[\leadsto \sqrt{2} \cdot \color{blue}{\left(\mathsf{neg}\left(x\right)\right)} \]

   6. Applied rewrites 99.3%

      \[\leadsto \color{blue}{\left({4}^{0.125} \cdot x\right) \cdot \left(-{4}^{0.125}\right)} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left({4}^{\frac{1}{8}} \cdot x\right) \cdot \left(-{4}^{\frac{1}{8}}\right)} \]

      2. lift-neg.f64 N/A

         \[\leadsto \left({4}^{\frac{1}{8}} \cdot x\right) \cdot \color{blue}{\left(\mathsf{neg}\left({4}^{\frac{1}{8}}\right)\right)} \]

      3. lift-pow.f64 N/A

         \[\leadsto \left({4}^{\frac{1}{8}} \cdot x\right) \cdot \left(\mathsf{neg}\left(\color{blue}{{4}^{\frac{1}{8}}}\right)\right) \]

      4. sqr-pow N/A

         \[\leadsto \left({4}^{\frac{1}{8}} \cdot x\right) \cdot \left(\mathsf{neg}\left(\color{blue}{{4}^{\left(\frac{\frac{1}{8}}{2}\right)} \cdot {4}^{\left(\frac{\frac{1}{8}}{2}\right)}}\right)\right) \]

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

         \[\leadsto \left({4}^{\frac{1}{8}} \cdot x\right) \cdot \color{blue}{\left({4}^{\left(\frac{\frac{1}{8}}{2}\right)} \cdot \left(\mathsf{neg}\left({4}^{\left(\frac{\frac{1}{8}}{2}\right)}\right)\right)\right)} \]

      6. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\left({4}^{\frac{1}{8}} \cdot x\right) \cdot {4}^{\left(\frac{\frac{1}{8}}{2}\right)}\right) \cdot \left(\mathsf{neg}\left({4}^{\left(\frac{\frac{1}{8}}{2}\right)}\right)\right)} \]

      7. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left({4}^{\frac{1}{8}} \cdot x\right) \cdot {4}^{\left(\frac{\frac{1}{8}}{2}\right)}\right) \cdot \left(\mathsf{neg}\left({4}^{\left(\frac{\frac{1}{8}}{2}\right)}\right)\right)} \]

      8. lift-*.f64 N/A

         \[\leadsto \left(\color{blue}{\left({4}^{\frac{1}{8}} \cdot x\right)} \cdot {4}^{\left(\frac{\frac{1}{8}}{2}\right)}\right) \cdot \left(\mathsf{neg}\left({4}^{\left(\frac{\frac{1}{8}}{2}\right)}\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \left(\color{blue}{\left(x \cdot {4}^{\frac{1}{8}}\right)} \cdot {4}^{\left(\frac{\frac{1}{8}}{2}\right)}\right) \cdot \left(\mathsf{neg}\left({4}^{\left(\frac{\frac{1}{8}}{2}\right)}\right)\right) \]

      10. associate-*l* N/A

          \[\leadsto \color{blue}{\left(x \cdot \left({4}^{\frac{1}{8}} \cdot {4}^{\left(\frac{\frac{1}{8}}{2}\right)}\right)\right)} \cdot \left(\mathsf{neg}\left({4}^{\left(\frac{\frac{1}{8}}{2}\right)}\right)\right) \]

      11. lower-*.f64 N/A

          \[\leadsto \color{blue}{\left(x \cdot \left({4}^{\frac{1}{8}} \cdot {4}^{\left(\frac{\frac{1}{8}}{2}\right)}\right)\right)} \cdot \left(\mathsf{neg}\left({4}^{\left(\frac{\frac{1}{8}}{2}\right)}\right)\right) \]

      12. lift-pow.f64 N/A

          \[\leadsto \left(x \cdot \left(\color{blue}{{4}^{\frac{1}{8}}} \cdot {4}^{\left(\frac{\frac{1}{8}}{2}\right)}\right)\right) \cdot \left(\mathsf{neg}\left({4}^{\left(\frac{\frac{1}{8}}{2}\right)}\right)\right) \]

      13. sqr-pow N/A

          \[\leadsto \left(x \cdot \left(\color{blue}{\left({4}^{\left(\frac{\frac{1}{8}}{2}\right)} \cdot {4}^{\left(\frac{\frac{1}{8}}{2}\right)}\right)} \cdot {4}^{\left(\frac{\frac{1}{8}}{2}\right)}\right)\right) \cdot \left(\mathsf{neg}\left({4}^{\left(\frac{\frac{1}{8}}{2}\right)}\right)\right) \]

      14. pow-prod-down N/A

          \[\leadsto \left(x \cdot \left(\color{blue}{{\left(4 \cdot 4\right)}^{\left(\frac{\frac{1}{8}}{2}\right)}} \cdot {4}^{\left(\frac{\frac{1}{8}}{2}\right)}\right)\right) \cdot \left(\mathsf{neg}\left({4}^{\left(\frac{\frac{1}{8}}{2}\right)}\right)\right) \]

      15. pow-prod-down N/A

          \[\leadsto \left(x \cdot \color{blue}{{\left(\left(4 \cdot 4\right) \cdot 4\right)}^{\left(\frac{\frac{1}{8}}{2}\right)}}\right) \cdot \left(\mathsf{neg}\left({4}^{\left(\frac{\frac{1}{8}}{2}\right)}\right)\right) \]

      16. lower-pow.f64 N/A

          \[\leadsto \left(x \cdot \color{blue}{{\left(\left(4 \cdot 4\right) \cdot 4\right)}^{\left(\frac{\frac{1}{8}}{2}\right)}}\right) \cdot \left(\mathsf{neg}\left({4}^{\left(\frac{\frac{1}{8}}{2}\right)}\right)\right) \]

      17. metadata-eval N/A

          \[\leadsto \left(x \cdot {\left(\color{blue}{16} \cdot 4\right)}^{\left(\frac{\frac{1}{8}}{2}\right)}\right) \cdot \left(\mathsf{neg}\left({4}^{\left(\frac{\frac{1}{8}}{2}\right)}\right)\right) \]

      18. metadata-eval N/A

          \[\leadsto \left(x \cdot {\color{blue}{64}}^{\left(\frac{\frac{1}{8}}{2}\right)}\right) \cdot \left(\mathsf{neg}\left({4}^{\left(\frac{\frac{1}{8}}{2}\right)}\right)\right) \]

      19. metadata-eval N/A

          \[\leadsto \left(x \cdot {64}^{\color{blue}{\frac{1}{16}}}\right) \cdot \left(\mathsf{neg}\left({4}^{\left(\frac{\frac{1}{8}}{2}\right)}\right)\right) \]

      20. lower-neg.f64 N/A

          \[\leadsto \left(x \cdot {64}^{\frac{1}{16}}\right) \cdot \color{blue}{\left(-{4}^{\left(\frac{\frac{1}{8}}{2}\right)}\right)} \]

      21. sqr-pow N/A

          \[\leadsto \left(x \cdot {64}^{\frac{1}{16}}\right) \cdot \left(-\color{blue}{{4}^{\left(\frac{\frac{\frac{1}{8}}{2}}{2}\right)} \cdot {4}^{\left(\frac{\frac{\frac{1}{8}}{2}}{2}\right)}}\right) \]

      22. pow-prod-down N/A

          \[\leadsto \left(x \cdot {64}^{\frac{1}{16}}\right) \cdot \left(-\color{blue}{{\left(4 \cdot 4\right)}^{\left(\frac{\frac{\frac{1}{8}}{2}}{2}\right)}}\right) \]

   8. Applied rewrites 99.6%

      \[\leadsto \color{blue}{\left(x \cdot {64}^{0.0625}\right) \cdot \left(-{16}^{0.03125}\right)} \]

   if -1.999999999999994e-310 < x

   1. Initial program 52.2%

      \[\sqrt{2 \cdot {x}^{2}} \]
   2. Add Preprocessing

   4. Applied rewrites 99.3%

      \[\leadsto \color{blue}{\sqrt{2} \cdot x} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\sqrt{2} \cdot x} \]

      2. +-lft-identity N/A

         \[\leadsto \sqrt{2} \cdot \color{blue}{\left(0 + x\right)} \]

      3. flip-+ N/A

         \[\leadsto \sqrt{2} \cdot \color{blue}{\frac{0 \cdot 0 - x \cdot x}{0 - x}} \]

      4. neg-sub0 N/A

         \[\leadsto \sqrt{2} \cdot \frac{0 \cdot 0 - x \cdot x}{\color{blue}{\mathsf{neg}\left(x\right)}} \]

      5. lift-neg.f64 N/A

         \[\leadsto \sqrt{2} \cdot \frac{0 \cdot 0 - x \cdot x}{\color{blue}{-x}} \]

      6. div-inv N/A

         \[\leadsto \sqrt{2} \cdot \color{blue}{\left(\left(0 \cdot 0 - x \cdot x\right) \cdot \frac{1}{-x}\right)} \]

      7. inv-pow N/A

         \[\leadsto \sqrt{2} \cdot \left(\left(0 \cdot 0 - x \cdot x\right) \cdot \color{blue}{{\left(-x\right)}^{-1}}\right) \]

      8. metadata-eval N/A

         \[\leadsto \sqrt{2} \cdot \left(\left(0 \cdot 0 - x \cdot x\right) \cdot {\left(-x\right)}^{\color{blue}{\left(2 \cdot \frac{-1}{2}\right)}}\right) \]

      9. metadata-eval N/A

         \[\leadsto \sqrt{2} \cdot \left(\left(0 \cdot 0 - x \cdot x\right) \cdot {\left(-x\right)}^{\left(2 \cdot \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right)}\right) \]

      10. pow-pow N/A

          \[\leadsto \sqrt{2} \cdot \left(\left(0 \cdot 0 - x \cdot x\right) \cdot \color{blue}{{\left({\left(-x\right)}^{2}\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}\right) \]

      11. pow2 N/A

          \[\leadsto \sqrt{2} \cdot \left(\left(0 \cdot 0 - x \cdot x\right) \cdot {\color{blue}{\left(\left(-x\right) \cdot \left(-x\right)\right)}}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right) \]

      12. lift-neg.f64 N/A

          \[\leadsto \sqrt{2} \cdot \left(\left(0 \cdot 0 - x \cdot x\right) \cdot {\left(\color{blue}{\left(\mathsf{neg}\left(x\right)\right)} \cdot \left(-x\right)\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right) \]

      13. lift-neg.f64 N/A

          \[\leadsto \sqrt{2} \cdot \left(\left(0 \cdot 0 - x \cdot x\right) \cdot {\left(\left(\mathsf{neg}\left(x\right)\right) \cdot \color{blue}{\left(\mathsf{neg}\left(x\right)\right)}\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right) \]

      14. sqr-neg N/A

          \[\leadsto \sqrt{2} \cdot \left(\left(0 \cdot 0 - x \cdot x\right) \cdot {\color{blue}{\left(x \cdot x\right)}}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right) \]

      15. pow2 N/A

          \[\leadsto \sqrt{2} \cdot \left(\left(0 \cdot 0 - x \cdot x\right) \cdot {\color{blue}{\left({x}^{2}\right)}}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right) \]

      16. pow-pow N/A

          \[\leadsto \sqrt{2} \cdot \left(\left(0 \cdot 0 - x \cdot x\right) \cdot \color{blue}{{x}^{\left(2 \cdot \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto \sqrt{2} \cdot \left(\left(0 \cdot 0 - x \cdot x\right) \cdot {x}^{\left(2 \cdot \color{blue}{\frac{-1}{2}}\right)}\right) \]

      18. metadata-eval N/A

          \[\leadsto \sqrt{2} \cdot \left(\left(0 \cdot 0 - x \cdot x\right) \cdot {x}^{\color{blue}{-1}}\right) \]

      19. inv-pow N/A

          \[\leadsto \sqrt{2} \cdot \left(\left(0 \cdot 0 - x \cdot x\right) \cdot \color{blue}{\frac{1}{x}}\right) \]

      20. +-lft-identity N/A

          \[\leadsto \sqrt{2} \cdot \left(\left(0 \cdot 0 - x \cdot x\right) \cdot \frac{1}{\color{blue}{0 + x}}\right) \]

      21. div-inv N/A

          \[\leadsto \sqrt{2} \cdot \color{blue}{\frac{0 \cdot 0 - x \cdot x}{0 + x}} \]

      22. flip-- N/A

          \[\leadsto \sqrt{2} \cdot \color{blue}{\left(0 - x\right)} \]

      23. neg-sub0 N/A

          \[\leadsto \sqrt{2} \cdot \color{blue}{\left(\mathsf{neg}\left(x\right)\right)} \]

   6. Applied rewrites 2.4%

      \[\leadsto \color{blue}{\left({4}^{0.125} \cdot x\right) \cdot \left(-{4}^{0.125}\right)} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left({4}^{\frac{1}{8}} \cdot x\right) \cdot \left(-{4}^{\frac{1}{8}}\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left({4}^{\frac{1}{8}} \cdot x\right)} \cdot \left(-{4}^{\frac{1}{8}}\right) \]

      3. associate-*l* N/A

         \[\leadsto \color{blue}{{4}^{\frac{1}{8}} \cdot \left(x \cdot \left(-{4}^{\frac{1}{8}}\right)\right)} \]

      4. *-commutative N/A

         \[\leadsto \color{blue}{\left(x \cdot \left(-{4}^{\frac{1}{8}}\right)\right) \cdot {4}^{\frac{1}{8}}} \]

      5. lift-neg.f64 N/A

         \[\leadsto \left(x \cdot \color{blue}{\left(\mathsf{neg}\left({4}^{\frac{1}{8}}\right)\right)}\right) \cdot {4}^{\frac{1}{8}} \]

      6. neg-mul-1 N/A

         \[\leadsto \left(x \cdot \color{blue}{\left(-1 \cdot {4}^{\frac{1}{8}}\right)}\right) \cdot {4}^{\frac{1}{8}} \]

      7. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\left(x \cdot -1\right) \cdot {4}^{\frac{1}{8}}\right)} \cdot {4}^{\frac{1}{8}} \]

      8. *-commutative N/A

         \[\leadsto \left(\color{blue}{\left(-1 \cdot x\right)} \cdot {4}^{\frac{1}{8}}\right) \cdot {4}^{\frac{1}{8}} \]

      9. metadata-eval N/A

         \[\leadsto \left(\left(\color{blue}{\frac{-1}{1}} \cdot x\right) \cdot {4}^{\frac{1}{8}}\right) \cdot {4}^{\frac{1}{8}} \]

      10. associate-/r/ N/A

          \[\leadsto \left(\color{blue}{\frac{-1}{\frac{1}{x}}} \cdot {4}^{\frac{1}{8}}\right) \cdot {4}^{\frac{1}{8}} \]

      11. unpow-1 N/A

          \[\leadsto \left(\frac{-1}{\color{blue}{{x}^{-1}}} \cdot {4}^{\frac{1}{8}}\right) \cdot {4}^{\frac{1}{8}} \]

      12. lift-pow.f64 N/A

          \[\leadsto \left(\frac{-1}{\color{blue}{{x}^{-1}}} \cdot {4}^{\frac{1}{8}}\right) \cdot {4}^{\frac{1}{8}} \]

      13. lift-/.f64 N/A

          \[\leadsto \left(\color{blue}{\frac{-1}{{x}^{-1}}} \cdot {4}^{\frac{1}{8}}\right) \cdot {4}^{\frac{1}{8}} \]

      14. associate-*l* N/A

          \[\leadsto \color{blue}{\frac{-1}{{x}^{-1}} \cdot \left({4}^{\frac{1}{8}} \cdot {4}^{\frac{1}{8}}\right)} \]

   8. Applied rewrites 99.5%

      \[\leadsto \color{blue}{{\left(x \cdot 4\right)}^{0.25} \cdot {x}^{0.75}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\left(-{16}^{0.03125}\right) \cdot \left({64}^{0.0625} \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;{x}^{0.75} \cdot {\left(4 \cdot x\right)}^{0.25}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\frac{-2 \cdot x}{\sqrt{2}}\\ \mathbf{else}:\\ \;\;\;\;{x}^{0.75} \cdot {\left(4 \cdot x\right)}^{0.25}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -2e-310)
   (/ (* -2.0 x) (sqrt 2.0))
   (* (pow x 0.75) (pow (* 4.0 x) 0.25))))

C

double code(double x) {
	double tmp;
	if (x <= -2e-310) {
		tmp = (-2.0 * x) / sqrt(2.0);
	} else {
		tmp = pow(x, 0.75) * pow((4.0 * x), 0.25);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-2d-310)) then
        tmp = ((-2.0d0) * x) / sqrt(2.0d0)
    else
        tmp = (x ** 0.75d0) * ((4.0d0 * x) ** 0.25d0)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= -2e-310) {
		tmp = (-2.0 * x) / Math.sqrt(2.0);
	} else {
		tmp = Math.pow(x, 0.75) * Math.pow((4.0 * x), 0.25);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -2e-310:
		tmp = (-2.0 * x) / math.sqrt(2.0)
	else:
		tmp = math.pow(x, 0.75) * math.pow((4.0 * x), 0.25)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -2e-310)
		tmp = Float64(Float64(-2.0 * x) / sqrt(2.0));
	else
		tmp = Float64((x ^ 0.75) * (Float64(4.0 * x) ^ 0.25));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -2e-310)
		tmp = (-2.0 * x) / sqrt(2.0);
	else
		tmp = (x ^ 0.75) * ((4.0 * x) ^ 0.25);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -2e-310], N[(N[(-2.0 * x), $MachinePrecision] / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision], N[(N[Power[x, 0.75], $MachinePrecision] * N[Power[N[(4.0 * x), $MachinePrecision], 0.25], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.999999999999994e-310

   1. Initial program 56.2%

      \[\sqrt{2 \cdot {x}^{2}} \]
   2. Add Preprocessing

   4. Applied rewrites 2.1%

      \[\leadsto \color{blue}{\sqrt{2} \cdot x} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\sqrt{2} \cdot x} \]

      2. +-lft-identity N/A

         \[\leadsto \sqrt{2} \cdot \color{blue}{\left(0 + x\right)} \]

      3. flip-+ N/A

         \[\leadsto \sqrt{2} \cdot \color{blue}{\frac{0 \cdot 0 - x \cdot x}{0 - x}} \]

      4. neg-sub0 N/A

         \[\leadsto \sqrt{2} \cdot \frac{0 \cdot 0 - x \cdot x}{\color{blue}{\mathsf{neg}\left(x\right)}} \]

      5. lift-neg.f64 N/A

         \[\leadsto \sqrt{2} \cdot \frac{0 \cdot 0 - x \cdot x}{\color{blue}{-x}} \]

      6. div-inv N/A

         \[\leadsto \sqrt{2} \cdot \color{blue}{\left(\left(0 \cdot 0 - x \cdot x\right) \cdot \frac{1}{-x}\right)} \]

      7. inv-pow N/A

         \[\leadsto \sqrt{2} \cdot \left(\left(0 \cdot 0 - x \cdot x\right) \cdot \color{blue}{{\left(-x\right)}^{-1}}\right) \]

      8. metadata-eval N/A

         \[\leadsto \sqrt{2} \cdot \left(\left(0 \cdot 0 - x \cdot x\right) \cdot {\left(-x\right)}^{\color{blue}{\left(2 \cdot \frac{-1}{2}\right)}}\right) \]

      9. metadata-eval N/A

         \[\leadsto \sqrt{2} \cdot \left(\left(0 \cdot 0 - x \cdot x\right) \cdot {\left(-x\right)}^{\left(2 \cdot \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right)}\right) \]

      10. pow-pow N/A

          \[\leadsto \sqrt{2} \cdot \left(\left(0 \cdot 0 - x \cdot x\right) \cdot \color{blue}{{\left({\left(-x\right)}^{2}\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}\right) \]

      11. pow2 N/A

          \[\leadsto \sqrt{2} \cdot \left(\left(0 \cdot 0 - x \cdot x\right) \cdot {\color{blue}{\left(\left(-x\right) \cdot \left(-x\right)\right)}}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right) \]

      12. lift-neg.f64 N/A

          \[\leadsto \sqrt{2} \cdot \left(\left(0 \cdot 0 - x \cdot x\right) \cdot {\left(\color{blue}{\left(\mathsf{neg}\left(x\right)\right)} \cdot \left(-x\right)\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right) \]

      13. lift-neg.f64 N/A

          \[\leadsto \sqrt{2} \cdot \left(\left(0 \cdot 0 - x \cdot x\right) \cdot {\left(\left(\mathsf{neg}\left(x\right)\right) \cdot \color{blue}{\left(\mathsf{neg}\left(x\right)\right)}\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right) \]

      14. sqr-neg N/A

          \[\leadsto \sqrt{2} \cdot \left(\left(0 \cdot 0 - x \cdot x\right) \cdot {\color{blue}{\left(x \cdot x\right)}}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right) \]

      15. pow2 N/A

          \[\leadsto \sqrt{2} \cdot \left(\left(0 \cdot 0 - x \cdot x\right) \cdot {\color{blue}{\left({x}^{2}\right)}}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right) \]

      16. pow-pow N/A

          \[\leadsto \sqrt{2} \cdot \left(\left(0 \cdot 0 - x \cdot x\right) \cdot \color{blue}{{x}^{\left(2 \cdot \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto \sqrt{2} \cdot \left(\left(0 \cdot 0 - x \cdot x\right) \cdot {x}^{\left(2 \cdot \color{blue}{\frac{-1}{2}}\right)}\right) \]

      18. metadata-eval N/A

          \[\leadsto \sqrt{2} \cdot \left(\left(0 \cdot 0 - x \cdot x\right) \cdot {x}^{\color{blue}{-1}}\right) \]

      19. inv-pow N/A

          \[\leadsto \sqrt{2} \cdot \left(\left(0 \cdot 0 - x \cdot x\right) \cdot \color{blue}{\frac{1}{x}}\right) \]

      20. +-lft-identity N/A

          \[\leadsto \sqrt{2} \cdot \left(\left(0 \cdot 0 - x \cdot x\right) \cdot \frac{1}{\color{blue}{0 + x}}\right) \]

      21. div-inv N/A

          \[\leadsto \sqrt{2} \cdot \color{blue}{\frac{0 \cdot 0 - x \cdot x}{0 + x}} \]

      22. flip-- N/A

          \[\leadsto \sqrt{2} \cdot \color{blue}{\left(0 - x\right)} \]

      23. neg-sub0 N/A

          \[\leadsto \sqrt{2} \cdot \color{blue}{\left(\mathsf{neg}\left(x\right)\right)} \]

   6. Applied rewrites 99.3%

      \[\leadsto \color{blue}{\left({4}^{0.125} \cdot x\right) \cdot \left(-{4}^{0.125}\right)} \]

   8. Applied rewrites 55.9%

      \[\leadsto \color{blue}{\frac{0 - \left(x \cdot x\right) \cdot 2}{\sqrt{2} \cdot x}} \]
   9. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{0 - \left(x \cdot x\right) \cdot 2}{\sqrt{2} \cdot x}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{0 - \left(x \cdot x\right) \cdot 2}{\color{blue}{\sqrt{2} \cdot x}} \]

      3. associate-/l/ N/A

         \[\leadsto \color{blue}{\frac{\frac{0 - \left(x \cdot x\right) \cdot 2}{x}}{\sqrt{2}}} \]

      4. lift--.f64 N/A

         \[\leadsto \frac{\frac{\color{blue}{0 - \left(x \cdot x\right) \cdot 2}}{x}}{\sqrt{2}} \]

      5. sub0-neg N/A

         \[\leadsto \frac{\frac{\color{blue}{\mathsf{neg}\left(\left(x \cdot x\right) \cdot 2\right)}}{x}}{\sqrt{2}} \]

      6. distribute-frac-neg N/A

         \[\leadsto \frac{\color{blue}{\mathsf{neg}\left(\frac{\left(x \cdot x\right) \cdot 2}{x}\right)}}{\sqrt{2}} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{\mathsf{neg}\left(\frac{\color{blue}{\left(x \cdot x\right) \cdot 2}}{x}\right)}{\sqrt{2}} \]

      8. lift-*.f64 N/A

         \[\leadsto \frac{\mathsf{neg}\left(\frac{\color{blue}{\left(x \cdot x\right)} \cdot 2}{x}\right)}{\sqrt{2}} \]

      9. associate-*l* N/A

         \[\leadsto \frac{\mathsf{neg}\left(\frac{\color{blue}{x \cdot \left(x \cdot 2\right)}}{x}\right)}{\sqrt{2}} \]

      10. associate-/l* N/A

          \[\leadsto \frac{\mathsf{neg}\left(\color{blue}{x \cdot \frac{x \cdot 2}{x}}\right)}{\sqrt{2}} \]

      11. *-commutative N/A

          \[\leadsto \frac{\mathsf{neg}\left(x \cdot \frac{\color{blue}{2 \cdot x}}{x}\right)}{\sqrt{2}} \]

      12. associate-/l* N/A

          \[\leadsto \frac{\mathsf{neg}\left(x \cdot \color{blue}{\left(2 \cdot \frac{x}{x}\right)}\right)}{\sqrt{2}} \]

      13. *-inverses N/A

          \[\leadsto \frac{\mathsf{neg}\left(x \cdot \left(2 \cdot \color{blue}{1}\right)\right)}{\sqrt{2}} \]

      14. metadata-eval N/A

          \[\leadsto \frac{\mathsf{neg}\left(x \cdot \color{blue}{2}\right)}{\sqrt{2}} \]

      15. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(x \cdot 2\right)}{\sqrt{2}}} \]

      16. *-commutative N/A

          \[\leadsto \frac{\mathsf{neg}\left(\color{blue}{2 \cdot x}\right)}{\sqrt{2}} \]

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

          \[\leadsto \frac{\color{blue}{\left(\mathsf{neg}\left(2\right)\right) \cdot x}}{\sqrt{2}} \]

      18. lower-*.f64 N/A

          \[\leadsto \frac{\color{blue}{\left(\mathsf{neg}\left(2\right)\right) \cdot x}}{\sqrt{2}} \]

      19. metadata-eval 99.4

          \[\leadsto \frac{\color{blue}{-2} \cdot x}{\sqrt{2}} \]

   10. Applied rewrites 99.4%

       \[\leadsto \color{blue}{\frac{-2 \cdot x}{\sqrt{2}}} \]

   if -1.999999999999994e-310 < x

   1. Initial program 52.2%

      \[\sqrt{2 \cdot {x}^{2}} \]
   2. Add Preprocessing

   4. Applied rewrites 99.3%

      \[\leadsto \color{blue}{\sqrt{2} \cdot x} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\sqrt{2} \cdot x} \]

      2. +-lft-identity N/A

         \[\leadsto \sqrt{2} \cdot \color{blue}{\left(0 + x\right)} \]

      3. flip-+ N/A

         \[\leadsto \sqrt{2} \cdot \color{blue}{\frac{0 \cdot 0 - x \cdot x}{0 - x}} \]

      4. neg-sub0 N/A

         \[\leadsto \sqrt{2} \cdot \frac{0 \cdot 0 - x \cdot x}{\color{blue}{\mathsf{neg}\left(x\right)}} \]

      5. lift-neg.f64 N/A

         \[\leadsto \sqrt{2} \cdot \frac{0 \cdot 0 - x \cdot x}{\color{blue}{-x}} \]

      6. div-inv N/A

         \[\leadsto \sqrt{2} \cdot \color{blue}{\left(\left(0 \cdot 0 - x \cdot x\right) \cdot \frac{1}{-x}\right)} \]

      7. inv-pow N/A

         \[\leadsto \sqrt{2} \cdot \left(\left(0 \cdot 0 - x \cdot x\right) \cdot \color{blue}{{\left(-x\right)}^{-1}}\right) \]

      8. metadata-eval N/A

         \[\leadsto \sqrt{2} \cdot \left(\left(0 \cdot 0 - x \cdot x\right) \cdot {\left(-x\right)}^{\color{blue}{\left(2 \cdot \frac{-1}{2}\right)}}\right) \]

      9. metadata-eval N/A

         \[\leadsto \sqrt{2} \cdot \left(\left(0 \cdot 0 - x \cdot x\right) \cdot {\left(-x\right)}^{\left(2 \cdot \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right)}\right) \]

      10. pow-pow N/A

          \[\leadsto \sqrt{2} \cdot \left(\left(0 \cdot 0 - x \cdot x\right) \cdot \color{blue}{{\left({\left(-x\right)}^{2}\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}\right) \]

      11. pow2 N/A

          \[\leadsto \sqrt{2} \cdot \left(\left(0 \cdot 0 - x \cdot x\right) \cdot {\color{blue}{\left(\left(-x\right) \cdot \left(-x\right)\right)}}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right) \]

      12. lift-neg.f64 N/A

          \[\leadsto \sqrt{2} \cdot \left(\left(0 \cdot 0 - x \cdot x\right) \cdot {\left(\color{blue}{\left(\mathsf{neg}\left(x\right)\right)} \cdot \left(-x\right)\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right) \]

      13. lift-neg.f64 N/A

          \[\leadsto \sqrt{2} \cdot \left(\left(0 \cdot 0 - x \cdot x\right) \cdot {\left(\left(\mathsf{neg}\left(x\right)\right) \cdot \color{blue}{\left(\mathsf{neg}\left(x\right)\right)}\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right) \]

      14. sqr-neg N/A

          \[\leadsto \sqrt{2} \cdot \left(\left(0 \cdot 0 - x \cdot x\right) \cdot {\color{blue}{\left(x \cdot x\right)}}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right) \]

      15. pow2 N/A

          \[\leadsto \sqrt{2} \cdot \left(\left(0 \cdot 0 - x \cdot x\right) \cdot {\color{blue}{\left({x}^{2}\right)}}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right) \]

      16. pow-pow N/A

          \[\leadsto \sqrt{2} \cdot \left(\left(0 \cdot 0 - x \cdot x\right) \cdot \color{blue}{{x}^{\left(2 \cdot \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto \sqrt{2} \cdot \left(\left(0 \cdot 0 - x \cdot x\right) \cdot {x}^{\left(2 \cdot \color{blue}{\frac{-1}{2}}\right)}\right) \]

      18. metadata-eval N/A

          \[\leadsto \sqrt{2} \cdot \left(\left(0 \cdot 0 - x \cdot x\right) \cdot {x}^{\color{blue}{-1}}\right) \]

      19. inv-pow N/A

          \[\leadsto \sqrt{2} \cdot \left(\left(0 \cdot 0 - x \cdot x\right) \cdot \color{blue}{\frac{1}{x}}\right) \]

      20. +-lft-identity N/A

          \[\leadsto \sqrt{2} \cdot \left(\left(0 \cdot 0 - x \cdot x\right) \cdot \frac{1}{\color{blue}{0 + x}}\right) \]

      21. div-inv N/A

          \[\leadsto \sqrt{2} \cdot \color{blue}{\frac{0 \cdot 0 - x \cdot x}{0 + x}} \]

      22. flip-- N/A

          \[\leadsto \sqrt{2} \cdot \color{blue}{\left(0 - x\right)} \]

      23. neg-sub0 N/A

          \[\leadsto \sqrt{2} \cdot \color{blue}{\left(\mathsf{neg}\left(x\right)\right)} \]

   6. Applied rewrites 2.4%

      \[\leadsto \color{blue}{\left({4}^{0.125} \cdot x\right) \cdot \left(-{4}^{0.125}\right)} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left({4}^{\frac{1}{8}} \cdot x\right) \cdot \left(-{4}^{\frac{1}{8}}\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left({4}^{\frac{1}{8}} \cdot x\right)} \cdot \left(-{4}^{\frac{1}{8}}\right) \]

      3. associate-*l* N/A

         \[\leadsto \color{blue}{{4}^{\frac{1}{8}} \cdot \left(x \cdot \left(-{4}^{\frac{1}{8}}\right)\right)} \]

      4. *-commutative N/A

         \[\leadsto \color{blue}{\left(x \cdot \left(-{4}^{\frac{1}{8}}\right)\right) \cdot {4}^{\frac{1}{8}}} \]

      5. lift-neg.f64 N/A

         \[\leadsto \left(x \cdot \color{blue}{\left(\mathsf{neg}\left({4}^{\frac{1}{8}}\right)\right)}\right) \cdot {4}^{\frac{1}{8}} \]

      6. neg-mul-1 N/A

         \[\leadsto \left(x \cdot \color{blue}{\left(-1 \cdot {4}^{\frac{1}{8}}\right)}\right) \cdot {4}^{\frac{1}{8}} \]

      7. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\left(x \cdot -1\right) \cdot {4}^{\frac{1}{8}}\right)} \cdot {4}^{\frac{1}{8}} \]

      8. *-commutative N/A

         \[\leadsto \left(\color{blue}{\left(-1 \cdot x\right)} \cdot {4}^{\frac{1}{8}}\right) \cdot {4}^{\frac{1}{8}} \]

      9. metadata-eval N/A

         \[\leadsto \left(\left(\color{blue}{\frac{-1}{1}} \cdot x\right) \cdot {4}^{\frac{1}{8}}\right) \cdot {4}^{\frac{1}{8}} \]

      10. associate-/r/ N/A

          \[\leadsto \left(\color{blue}{\frac{-1}{\frac{1}{x}}} \cdot {4}^{\frac{1}{8}}\right) \cdot {4}^{\frac{1}{8}} \]

      11. unpow-1 N/A

          \[\leadsto \left(\frac{-1}{\color{blue}{{x}^{-1}}} \cdot {4}^{\frac{1}{8}}\right) \cdot {4}^{\frac{1}{8}} \]

      12. lift-pow.f64 N/A

          \[\leadsto \left(\frac{-1}{\color{blue}{{x}^{-1}}} \cdot {4}^{\frac{1}{8}}\right) \cdot {4}^{\frac{1}{8}} \]

      13. lift-/.f64 N/A

          \[\leadsto \left(\color{blue}{\frac{-1}{{x}^{-1}}} \cdot {4}^{\frac{1}{8}}\right) \cdot {4}^{\frac{1}{8}} \]

      14. associate-*l* N/A

          \[\leadsto \color{blue}{\frac{-1}{{x}^{-1}} \cdot \left({4}^{\frac{1}{8}} \cdot {4}^{\frac{1}{8}}\right)} \]

   8. Applied rewrites 99.5%

      \[\leadsto \color{blue}{{\left(x \cdot 4\right)}^{0.25} \cdot {x}^{0.75}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\frac{-2 \cdot x}{\sqrt{2}}\\ \mathbf{else}:\\ \;\;\;\;{x}^{0.75} \cdot {\left(4 \cdot x\right)}^{0.25}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\frac{-2 \cdot x}{\sqrt{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2 \cdot x}}{{x}^{-0.5}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -2e-310)
   (/ (* -2.0 x) (sqrt 2.0))
   (/ (sqrt (* 2.0 x)) (pow x -0.5))))

C

double code(double x) {
	double tmp;
	if (x <= -2e-310) {
		tmp = (-2.0 * x) / sqrt(2.0);
	} else {
		tmp = sqrt((2.0 * x)) / pow(x, -0.5);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-2d-310)) then
        tmp = ((-2.0d0) * x) / sqrt(2.0d0)
    else
        tmp = sqrt((2.0d0 * x)) / (x ** (-0.5d0))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= -2e-310) {
		tmp = (-2.0 * x) / Math.sqrt(2.0);
	} else {
		tmp = Math.sqrt((2.0 * x)) / Math.pow(x, -0.5);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -2e-310:
		tmp = (-2.0 * x) / math.sqrt(2.0)
	else:
		tmp = math.sqrt((2.0 * x)) / math.pow(x, -0.5)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -2e-310)
		tmp = Float64(Float64(-2.0 * x) / sqrt(2.0));
	else
		tmp = Float64(sqrt(Float64(2.0 * x)) / (x ^ -0.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -2e-310)
		tmp = (-2.0 * x) / sqrt(2.0);
	else
		tmp = sqrt((2.0 * x)) / (x ^ -0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -2e-310], N[(N[(-2.0 * x), $MachinePrecision] / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(2.0 * x), $MachinePrecision]], $MachinePrecision] / N[Power[x, -0.5], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.999999999999994e-310

   1. Initial program 56.2%

      \[\sqrt{2 \cdot {x}^{2}} \]
   2. Add Preprocessing

   4. Applied rewrites 2.1%

      \[\leadsto \color{blue}{\sqrt{2} \cdot x} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\sqrt{2} \cdot x} \]

      2. +-lft-identity N/A

         \[\leadsto \sqrt{2} \cdot \color{blue}{\left(0 + x\right)} \]

      3. flip-+ N/A

         \[\leadsto \sqrt{2} \cdot \color{blue}{\frac{0 \cdot 0 - x \cdot x}{0 - x}} \]

      4. neg-sub0 N/A

         \[\leadsto \sqrt{2} \cdot \frac{0 \cdot 0 - x \cdot x}{\color{blue}{\mathsf{neg}\left(x\right)}} \]

      5. lift-neg.f64 N/A

         \[\leadsto \sqrt{2} \cdot \frac{0 \cdot 0 - x \cdot x}{\color{blue}{-x}} \]

      6. div-inv N/A

         \[\leadsto \sqrt{2} \cdot \color{blue}{\left(\left(0 \cdot 0 - x \cdot x\right) \cdot \frac{1}{-x}\right)} \]

      7. inv-pow N/A

         \[\leadsto \sqrt{2} \cdot \left(\left(0 \cdot 0 - x \cdot x\right) \cdot \color{blue}{{\left(-x\right)}^{-1}}\right) \]

      8. metadata-eval N/A

         \[\leadsto \sqrt{2} \cdot \left(\left(0 \cdot 0 - x \cdot x\right) \cdot {\left(-x\right)}^{\color{blue}{\left(2 \cdot \frac{-1}{2}\right)}}\right) \]

      9. metadata-eval N/A

         \[\leadsto \sqrt{2} \cdot \left(\left(0 \cdot 0 - x \cdot x\right) \cdot {\left(-x\right)}^{\left(2 \cdot \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right)}\right) \]

      10. pow-pow N/A

          \[\leadsto \sqrt{2} \cdot \left(\left(0 \cdot 0 - x \cdot x\right) \cdot \color{blue}{{\left({\left(-x\right)}^{2}\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}\right) \]

      11. pow2 N/A

          \[\leadsto \sqrt{2} \cdot \left(\left(0 \cdot 0 - x \cdot x\right) \cdot {\color{blue}{\left(\left(-x\right) \cdot \left(-x\right)\right)}}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right) \]

      12. lift-neg.f64 N/A

          \[\leadsto \sqrt{2} \cdot \left(\left(0 \cdot 0 - x \cdot x\right) \cdot {\left(\color{blue}{\left(\mathsf{neg}\left(x\right)\right)} \cdot \left(-x\right)\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right) \]

      13. lift-neg.f64 N/A

          \[\leadsto \sqrt{2} \cdot \left(\left(0 \cdot 0 - x \cdot x\right) \cdot {\left(\left(\mathsf{neg}\left(x\right)\right) \cdot \color{blue}{\left(\mathsf{neg}\left(x\right)\right)}\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right) \]

      14. sqr-neg N/A

          \[\leadsto \sqrt{2} \cdot \left(\left(0 \cdot 0 - x \cdot x\right) \cdot {\color{blue}{\left(x \cdot x\right)}}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right) \]

      15. pow2 N/A

          \[\leadsto \sqrt{2} \cdot \left(\left(0 \cdot 0 - x \cdot x\right) \cdot {\color{blue}{\left({x}^{2}\right)}}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right) \]

      16. pow-pow N/A

          \[\leadsto \sqrt{2} \cdot \left(\left(0 \cdot 0 - x \cdot x\right) \cdot \color{blue}{{x}^{\left(2 \cdot \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto \sqrt{2} \cdot \left(\left(0 \cdot 0 - x \cdot x\right) \cdot {x}^{\left(2 \cdot \color{blue}{\frac{-1}{2}}\right)}\right) \]

      18. metadata-eval N/A

          \[\leadsto \sqrt{2} \cdot \left(\left(0 \cdot 0 - x \cdot x\right) \cdot {x}^{\color{blue}{-1}}\right) \]

      19. inv-pow N/A

          \[\leadsto \sqrt{2} \cdot \left(\left(0 \cdot 0 - x \cdot x\right) \cdot \color{blue}{\frac{1}{x}}\right) \]

      20. +-lft-identity N/A

          \[\leadsto \sqrt{2} \cdot \left(\left(0 \cdot 0 - x \cdot x\right) \cdot \frac{1}{\color{blue}{0 + x}}\right) \]

      21. div-inv N/A

          \[\leadsto \sqrt{2} \cdot \color{blue}{\frac{0 \cdot 0 - x \cdot x}{0 + x}} \]

      22. flip-- N/A

          \[\leadsto \sqrt{2} \cdot \color{blue}{\left(0 - x\right)} \]

      23. neg-sub0 N/A

          \[\leadsto \sqrt{2} \cdot \color{blue}{\left(\mathsf{neg}\left(x\right)\right)} \]

   6. Applied rewrites 99.3%

      \[\leadsto \color{blue}{\left({4}^{0.125} \cdot x\right) \cdot \left(-{4}^{0.125}\right)} \]

   8. Applied rewrites 55.9%

      \[\leadsto \color{blue}{\frac{0 - \left(x \cdot x\right) \cdot 2}{\sqrt{2} \cdot x}} \]
   9. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{0 - \left(x \cdot x\right) \cdot 2}{\sqrt{2} \cdot x}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{0 - \left(x \cdot x\right) \cdot 2}{\color{blue}{\sqrt{2} \cdot x}} \]

      3. associate-/l/ N/A

         \[\leadsto \color{blue}{\frac{\frac{0 - \left(x \cdot x\right) \cdot 2}{x}}{\sqrt{2}}} \]

      4. lift--.f64 N/A

         \[\leadsto \frac{\frac{\color{blue}{0 - \left(x \cdot x\right) \cdot 2}}{x}}{\sqrt{2}} \]

      5. sub0-neg N/A

         \[\leadsto \frac{\frac{\color{blue}{\mathsf{neg}\left(\left(x \cdot x\right) \cdot 2\right)}}{x}}{\sqrt{2}} \]

      6. distribute-frac-neg N/A

         \[\leadsto \frac{\color{blue}{\mathsf{neg}\left(\frac{\left(x \cdot x\right) \cdot 2}{x}\right)}}{\sqrt{2}} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{\mathsf{neg}\left(\frac{\color{blue}{\left(x \cdot x\right) \cdot 2}}{x}\right)}{\sqrt{2}} \]

      8. lift-*.f64 N/A

         \[\leadsto \frac{\mathsf{neg}\left(\frac{\color{blue}{\left(x \cdot x\right)} \cdot 2}{x}\right)}{\sqrt{2}} \]

      9. associate-*l* N/A

         \[\leadsto \frac{\mathsf{neg}\left(\frac{\color{blue}{x \cdot \left(x \cdot 2\right)}}{x}\right)}{\sqrt{2}} \]

      10. associate-/l* N/A

          \[\leadsto \frac{\mathsf{neg}\left(\color{blue}{x \cdot \frac{x \cdot 2}{x}}\right)}{\sqrt{2}} \]

      11. *-commutative N/A

          \[\leadsto \frac{\mathsf{neg}\left(x \cdot \frac{\color{blue}{2 \cdot x}}{x}\right)}{\sqrt{2}} \]

      12. associate-/l* N/A

          \[\leadsto \frac{\mathsf{neg}\left(x \cdot \color{blue}{\left(2 \cdot \frac{x}{x}\right)}\right)}{\sqrt{2}} \]

      13. *-inverses N/A

          \[\leadsto \frac{\mathsf{neg}\left(x \cdot \left(2 \cdot \color{blue}{1}\right)\right)}{\sqrt{2}} \]

      14. metadata-eval N/A

          \[\leadsto \frac{\mathsf{neg}\left(x \cdot \color{blue}{2}\right)}{\sqrt{2}} \]

      15. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(x \cdot 2\right)}{\sqrt{2}}} \]

      16. *-commutative N/A

          \[\leadsto \frac{\mathsf{neg}\left(\color{blue}{2 \cdot x}\right)}{\sqrt{2}} \]

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

          \[\leadsto \frac{\color{blue}{\left(\mathsf{neg}\left(2\right)\right) \cdot x}}{\sqrt{2}} \]

      18. lower-*.f64 N/A

          \[\leadsto \frac{\color{blue}{\left(\mathsf{neg}\left(2\right)\right) \cdot x}}{\sqrt{2}} \]

      19. metadata-eval 99.4

          \[\leadsto \frac{\color{blue}{-2} \cdot x}{\sqrt{2}} \]

   10. Applied rewrites 99.4%

       \[\leadsto \color{blue}{\frac{-2 \cdot x}{\sqrt{2}}} \]

   if -1.999999999999994e-310 < x

   1. Initial program 52.2%

      \[\sqrt{2 \cdot {x}^{2}} \]
   2. Add Preprocessing

   4. Applied rewrites 99.3%

      \[\leadsto \color{blue}{\sqrt{2} \cdot x} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\sqrt{2} \cdot x} \]

      2. +-lft-identity N/A

         \[\leadsto \sqrt{2} \cdot \color{blue}{\left(0 + x\right)} \]

      3. flip-+ N/A

         \[\leadsto \sqrt{2} \cdot \color{blue}{\frac{0 \cdot 0 - x \cdot x}{0 - x}} \]

      4. neg-sub0 N/A

         \[\leadsto \sqrt{2} \cdot \frac{0 \cdot 0 - x \cdot x}{\color{blue}{\mathsf{neg}\left(x\right)}} \]

      5. lift-neg.f64 N/A

         \[\leadsto \sqrt{2} \cdot \frac{0 \cdot 0 - x \cdot x}{\color{blue}{-x}} \]

      6. div-inv N/A

         \[\leadsto \sqrt{2} \cdot \color{blue}{\left(\left(0 \cdot 0 - x \cdot x\right) \cdot \frac{1}{-x}\right)} \]

      7. inv-pow N/A

         \[\leadsto \sqrt{2} \cdot \left(\left(0 \cdot 0 - x \cdot x\right) \cdot \color{blue}{{\left(-x\right)}^{-1}}\right) \]

      8. metadata-eval N/A

         \[\leadsto \sqrt{2} \cdot \left(\left(0 \cdot 0 - x \cdot x\right) \cdot {\left(-x\right)}^{\color{blue}{\left(2 \cdot \frac{-1}{2}\right)}}\right) \]

      9. metadata-eval N/A

         \[\leadsto \sqrt{2} \cdot \left(\left(0 \cdot 0 - x \cdot x\right) \cdot {\left(-x\right)}^{\left(2 \cdot \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right)}\right) \]

      10. pow-pow N/A

          \[\leadsto \sqrt{2} \cdot \left(\left(0 \cdot 0 - x \cdot x\right) \cdot \color{blue}{{\left({\left(-x\right)}^{2}\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}\right) \]

      11. pow2 N/A

          \[\leadsto \sqrt{2} \cdot \left(\left(0 \cdot 0 - x \cdot x\right) \cdot {\color{blue}{\left(\left(-x\right) \cdot \left(-x\right)\right)}}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right) \]

      12. lift-neg.f64 N/A

          \[\leadsto \sqrt{2} \cdot \left(\left(0 \cdot 0 - x \cdot x\right) \cdot {\left(\color{blue}{\left(\mathsf{neg}\left(x\right)\right)} \cdot \left(-x\right)\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right) \]

      13. lift-neg.f64 N/A

          \[\leadsto \sqrt{2} \cdot \left(\left(0 \cdot 0 - x \cdot x\right) \cdot {\left(\left(\mathsf{neg}\left(x\right)\right) \cdot \color{blue}{\left(\mathsf{neg}\left(x\right)\right)}\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right) \]

      14. sqr-neg N/A

          \[\leadsto \sqrt{2} \cdot \left(\left(0 \cdot 0 - x \cdot x\right) \cdot {\color{blue}{\left(x \cdot x\right)}}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right) \]

      15. pow2 N/A

          \[\leadsto \sqrt{2} \cdot \left(\left(0 \cdot 0 - x \cdot x\right) \cdot {\color{blue}{\left({x}^{2}\right)}}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right) \]

      16. pow-pow N/A

          \[\leadsto \sqrt{2} \cdot \left(\left(0 \cdot 0 - x \cdot x\right) \cdot \color{blue}{{x}^{\left(2 \cdot \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto \sqrt{2} \cdot \left(\left(0 \cdot 0 - x \cdot x\right) \cdot {x}^{\left(2 \cdot \color{blue}{\frac{-1}{2}}\right)}\right) \]

      18. metadata-eval N/A

          \[\leadsto \sqrt{2} \cdot \left(\left(0 \cdot 0 - x \cdot x\right) \cdot {x}^{\color{blue}{-1}}\right) \]

      19. inv-pow N/A

          \[\leadsto \sqrt{2} \cdot \left(\left(0 \cdot 0 - x \cdot x\right) \cdot \color{blue}{\frac{1}{x}}\right) \]

      20. +-lft-identity N/A

          \[\leadsto \sqrt{2} \cdot \left(\left(0 \cdot 0 - x \cdot x\right) \cdot \frac{1}{\color{blue}{0 + x}}\right) \]

      21. div-inv N/A

          \[\leadsto \sqrt{2} \cdot \color{blue}{\frac{0 \cdot 0 - x \cdot x}{0 + x}} \]

      22. flip-- N/A

          \[\leadsto \sqrt{2} \cdot \color{blue}{\left(0 - x\right)} \]

      23. neg-sub0 N/A

          \[\leadsto \sqrt{2} \cdot \color{blue}{\left(\mathsf{neg}\left(x\right)\right)} \]

   6. Applied rewrites 2.4%

      \[\leadsto \color{blue}{\left({4}^{0.125} \cdot x\right) \cdot \left(-{4}^{0.125}\right)} \]

   8. Applied rewrites 99.4%

      \[\leadsto \color{blue}{\frac{\sqrt{2 \cdot x}}{{x}^{-0.5}}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 99.4% accurate, 3.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\frac{-2 \cdot x}{\sqrt{2}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x} \cdot \sqrt{2 \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -2e-310) (/ (* -2.0 x) (sqrt 2.0)) (* (sqrt x) (sqrt (* 2.0 x)))))

C

double code(double x) {
	double tmp;
	if (x <= -2e-310) {
		tmp = (-2.0 * x) / sqrt(2.0);
	} else {
		tmp = sqrt(x) * sqrt((2.0 * x));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-2d-310)) then
        tmp = ((-2.0d0) * x) / sqrt(2.0d0)
    else
        tmp = sqrt(x) * sqrt((2.0d0 * x))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= -2e-310) {
		tmp = (-2.0 * x) / Math.sqrt(2.0);
	} else {
		tmp = Math.sqrt(x) * Math.sqrt((2.0 * x));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -2e-310:
		tmp = (-2.0 * x) / math.sqrt(2.0)
	else:
		tmp = math.sqrt(x) * math.sqrt((2.0 * x))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -2e-310)
		tmp = Float64(Float64(-2.0 * x) / sqrt(2.0));
	else
		tmp = Float64(sqrt(x) * sqrt(Float64(2.0 * x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -2e-310)
		tmp = (-2.0 * x) / sqrt(2.0);
	else
		tmp = sqrt(x) * sqrt((2.0 * x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -2e-310], N[(N[(-2.0 * x), $MachinePrecision] / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[x], $MachinePrecision] * N[Sqrt[N[(2.0 * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.999999999999994e-310

   1. Initial program 56.2%

      \[\sqrt{2 \cdot {x}^{2}} \]
   2. Add Preprocessing

   4. Applied rewrites 2.1%

      \[\leadsto \color{blue}{\sqrt{2} \cdot x} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\sqrt{2} \cdot x} \]

      2. +-lft-identity N/A

         \[\leadsto \sqrt{2} \cdot \color{blue}{\left(0 + x\right)} \]

      3. flip-+ N/A

         \[\leadsto \sqrt{2} \cdot \color{blue}{\frac{0 \cdot 0 - x \cdot x}{0 - x}} \]

      4. neg-sub0 N/A

         \[\leadsto \sqrt{2} \cdot \frac{0 \cdot 0 - x \cdot x}{\color{blue}{\mathsf{neg}\left(x\right)}} \]

      5. lift-neg.f64 N/A

         \[\leadsto \sqrt{2} \cdot \frac{0 \cdot 0 - x \cdot x}{\color{blue}{-x}} \]

      6. div-inv N/A

         \[\leadsto \sqrt{2} \cdot \color{blue}{\left(\left(0 \cdot 0 - x \cdot x\right) \cdot \frac{1}{-x}\right)} \]

      7. inv-pow N/A

         \[\leadsto \sqrt{2} \cdot \left(\left(0 \cdot 0 - x \cdot x\right) \cdot \color{blue}{{\left(-x\right)}^{-1}}\right) \]

      8. metadata-eval N/A

         \[\leadsto \sqrt{2} \cdot \left(\left(0 \cdot 0 - x \cdot x\right) \cdot {\left(-x\right)}^{\color{blue}{\left(2 \cdot \frac{-1}{2}\right)}}\right) \]

      9. metadata-eval N/A

         \[\leadsto \sqrt{2} \cdot \left(\left(0 \cdot 0 - x \cdot x\right) \cdot {\left(-x\right)}^{\left(2 \cdot \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right)}\right) \]

      10. pow-pow N/A

          \[\leadsto \sqrt{2} \cdot \left(\left(0 \cdot 0 - x \cdot x\right) \cdot \color{blue}{{\left({\left(-x\right)}^{2}\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}\right) \]

      11. pow2 N/A

          \[\leadsto \sqrt{2} \cdot \left(\left(0 \cdot 0 - x \cdot x\right) \cdot {\color{blue}{\left(\left(-x\right) \cdot \left(-x\right)\right)}}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right) \]

      12. lift-neg.f64 N/A

          \[\leadsto \sqrt{2} \cdot \left(\left(0 \cdot 0 - x \cdot x\right) \cdot {\left(\color{blue}{\left(\mathsf{neg}\left(x\right)\right)} \cdot \left(-x\right)\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right) \]

      13. lift-neg.f64 N/A

          \[\leadsto \sqrt{2} \cdot \left(\left(0 \cdot 0 - x \cdot x\right) \cdot {\left(\left(\mathsf{neg}\left(x\right)\right) \cdot \color{blue}{\left(\mathsf{neg}\left(x\right)\right)}\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right) \]

      14. sqr-neg N/A

          \[\leadsto \sqrt{2} \cdot \left(\left(0 \cdot 0 - x \cdot x\right) \cdot {\color{blue}{\left(x \cdot x\right)}}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right) \]

      15. pow2 N/A

          \[\leadsto \sqrt{2} \cdot \left(\left(0 \cdot 0 - x \cdot x\right) \cdot {\color{blue}{\left({x}^{2}\right)}}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right) \]

      16. pow-pow N/A

          \[\leadsto \sqrt{2} \cdot \left(\left(0 \cdot 0 - x \cdot x\right) \cdot \color{blue}{{x}^{\left(2 \cdot \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto \sqrt{2} \cdot \left(\left(0 \cdot 0 - x \cdot x\right) \cdot {x}^{\left(2 \cdot \color{blue}{\frac{-1}{2}}\right)}\right) \]

      18. metadata-eval N/A

          \[\leadsto \sqrt{2} \cdot \left(\left(0 \cdot 0 - x \cdot x\right) \cdot {x}^{\color{blue}{-1}}\right) \]

      19. inv-pow N/A

          \[\leadsto \sqrt{2} \cdot \left(\left(0 \cdot 0 - x \cdot x\right) \cdot \color{blue}{\frac{1}{x}}\right) \]

      20. +-lft-identity N/A

          \[\leadsto \sqrt{2} \cdot \left(\left(0 \cdot 0 - x \cdot x\right) \cdot \frac{1}{\color{blue}{0 + x}}\right) \]

      21. div-inv N/A

          \[\leadsto \sqrt{2} \cdot \color{blue}{\frac{0 \cdot 0 - x \cdot x}{0 + x}} \]

      22. flip-- N/A

          \[\leadsto \sqrt{2} \cdot \color{blue}{\left(0 - x\right)} \]

      23. neg-sub0 N/A

          \[\leadsto \sqrt{2} \cdot \color{blue}{\left(\mathsf{neg}\left(x\right)\right)} \]

   6. Applied rewrites 99.3%

      \[\leadsto \color{blue}{\left({4}^{0.125} \cdot x\right) \cdot \left(-{4}^{0.125}\right)} \]

   8. Applied rewrites 55.9%

      \[\leadsto \color{blue}{\frac{0 - \left(x \cdot x\right) \cdot 2}{\sqrt{2} \cdot x}} \]
   9. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{0 - \left(x \cdot x\right) \cdot 2}{\sqrt{2} \cdot x}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{0 - \left(x \cdot x\right) \cdot 2}{\color{blue}{\sqrt{2} \cdot x}} \]

      3. associate-/l/ N/A

         \[\leadsto \color{blue}{\frac{\frac{0 - \left(x \cdot x\right) \cdot 2}{x}}{\sqrt{2}}} \]

      4. lift--.f64 N/A

         \[\leadsto \frac{\frac{\color{blue}{0 - \left(x \cdot x\right) \cdot 2}}{x}}{\sqrt{2}} \]

      5. sub0-neg N/A

         \[\leadsto \frac{\frac{\color{blue}{\mathsf{neg}\left(\left(x \cdot x\right) \cdot 2\right)}}{x}}{\sqrt{2}} \]

      6. distribute-frac-neg N/A

         \[\leadsto \frac{\color{blue}{\mathsf{neg}\left(\frac{\left(x \cdot x\right) \cdot 2}{x}\right)}}{\sqrt{2}} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{\mathsf{neg}\left(\frac{\color{blue}{\left(x \cdot x\right) \cdot 2}}{x}\right)}{\sqrt{2}} \]

      8. lift-*.f64 N/A

         \[\leadsto \frac{\mathsf{neg}\left(\frac{\color{blue}{\left(x \cdot x\right)} \cdot 2}{x}\right)}{\sqrt{2}} \]

      9. associate-*l* N/A

         \[\leadsto \frac{\mathsf{neg}\left(\frac{\color{blue}{x \cdot \left(x \cdot 2\right)}}{x}\right)}{\sqrt{2}} \]

      10. associate-/l* N/A

          \[\leadsto \frac{\mathsf{neg}\left(\color{blue}{x \cdot \frac{x \cdot 2}{x}}\right)}{\sqrt{2}} \]

      11. *-commutative N/A

          \[\leadsto \frac{\mathsf{neg}\left(x \cdot \frac{\color{blue}{2 \cdot x}}{x}\right)}{\sqrt{2}} \]

      12. associate-/l* N/A

          \[\leadsto \frac{\mathsf{neg}\left(x \cdot \color{blue}{\left(2 \cdot \frac{x}{x}\right)}\right)}{\sqrt{2}} \]

      13. *-inverses N/A

          \[\leadsto \frac{\mathsf{neg}\left(x \cdot \left(2 \cdot \color{blue}{1}\right)\right)}{\sqrt{2}} \]

      14. metadata-eval N/A

          \[\leadsto \frac{\mathsf{neg}\left(x \cdot \color{blue}{2}\right)}{\sqrt{2}} \]

      15. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(x \cdot 2\right)}{\sqrt{2}}} \]

      16. *-commutative N/A

          \[\leadsto \frac{\mathsf{neg}\left(\color{blue}{2 \cdot x}\right)}{\sqrt{2}} \]

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

          \[\leadsto \frac{\color{blue}{\left(\mathsf{neg}\left(2\right)\right) \cdot x}}{\sqrt{2}} \]

      18. lower-*.f64 N/A

          \[\leadsto \frac{\color{blue}{\left(\mathsf{neg}\left(2\right)\right) \cdot x}}{\sqrt{2}} \]

      19. metadata-eval 99.4

          \[\leadsto \frac{\color{blue}{-2} \cdot x}{\sqrt{2}} \]

   10. Applied rewrites 99.4%

       \[\leadsto \color{blue}{\frac{-2 \cdot x}{\sqrt{2}}} \]

   if -1.999999999999994e-310 < x

   1. Initial program 52.2%

      \[\sqrt{2 \cdot {x}^{2}} \]
   2. Add Preprocessing

   4. Applied rewrites 99.3%

      \[\leadsto \color{blue}{\sqrt{2} \cdot x} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\sqrt{2} \cdot x} \]

      2. unpow1 N/A

         \[\leadsto \sqrt{2} \cdot \color{blue}{{x}^{1}} \]

      3. metadata-eval N/A

         \[\leadsto \sqrt{2} \cdot {x}^{\color{blue}{\left(\frac{2}{2}\right)}} \]

      4. sqr-pow N/A

         \[\leadsto \sqrt{2} \cdot \color{blue}{\left({x}^{\left(\frac{\frac{2}{2}}{2}\right)} \cdot {x}^{\left(\frac{\frac{2}{2}}{2}\right)}\right)} \]

      5. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\sqrt{2} \cdot {x}^{\left(\frac{\frac{2}{2}}{2}\right)}\right) \cdot {x}^{\left(\frac{\frac{2}{2}}{2}\right)}} \]

      6. lift-sqrt.f64 N/A

         \[\leadsto \left(\color{blue}{\sqrt{2}} \cdot {x}^{\left(\frac{\frac{2}{2}}{2}\right)}\right) \cdot {x}^{\left(\frac{\frac{2}{2}}{2}\right)} \]

      7. pow1/2 N/A

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

      8. metadata-eval N/A

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

      9. metadata-eval N/A

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

      10. unpow-prod-down N/A

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

      11. lower-*.f64 N/A

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

      12. pow1/2 N/A

          \[\leadsto \color{blue}{\sqrt{2 \cdot x}} \cdot {x}^{\left(\frac{\frac{2}{2}}{2}\right)} \]

      13. lower-sqrt.f64 N/A

          \[\leadsto \color{blue}{\sqrt{2 \cdot x}} \cdot {x}^{\left(\frac{\frac{2}{2}}{2}\right)} \]

      14. *-commutative N/A

          \[\leadsto \sqrt{\color{blue}{x \cdot 2}} \cdot {x}^{\left(\frac{\frac{2}{2}}{2}\right)} \]

      15. lower-*.f64 N/A

          \[\leadsto \sqrt{\color{blue}{x \cdot 2}} \cdot {x}^{\left(\frac{\frac{2}{2}}{2}\right)} \]

      16. metadata-eval N/A

          \[\leadsto \sqrt{x \cdot 2} \cdot {x}^{\left(\frac{\color{blue}{1}}{2}\right)} \]

      17. metadata-eval N/A

          \[\leadsto \sqrt{x \cdot 2} \cdot {x}^{\color{blue}{\frac{1}{2}}} \]

      18. pow1/2 N/A

          \[\leadsto \sqrt{x \cdot 2} \cdot \color{blue}{\sqrt{x}} \]

      19. lower-sqrt.f64 99.4

          \[\leadsto \sqrt{x \cdot 2} \cdot \color{blue}{\sqrt{x}} \]

   6. Applied rewrites 99.4%

      \[\leadsto \color{blue}{\sqrt{x \cdot 2} \cdot \sqrt{x}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\frac{-2 \cdot x}{\sqrt{2}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x} \cdot \sqrt{2 \cdot x}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.3% accurate, 3.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\frac{-2 \cdot x}{\sqrt{2}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -2e-310) (/ (* -2.0 x) (sqrt 2.0)) (* (sqrt 2.0) x)))

C

double code(double x) {
	double tmp;
	if (x <= -2e-310) {
		tmp = (-2.0 * x) / sqrt(2.0);
	} else {
		tmp = sqrt(2.0) * x;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-2d-310)) then
        tmp = ((-2.0d0) * x) / sqrt(2.0d0)
    else
        tmp = sqrt(2.0d0) * x
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= -2e-310) {
		tmp = (-2.0 * x) / Math.sqrt(2.0);
	} else {
		tmp = Math.sqrt(2.0) * x;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -2e-310:
		tmp = (-2.0 * x) / math.sqrt(2.0)
	else:
		tmp = math.sqrt(2.0) * x
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -2e-310)
		tmp = Float64(Float64(-2.0 * x) / sqrt(2.0));
	else
		tmp = Float64(sqrt(2.0) * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -2e-310)
		tmp = (-2.0 * x) / sqrt(2.0);
	else
		tmp = sqrt(2.0) * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -2e-310], N[(N[(-2.0 * x), $MachinePrecision] / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[2.0], $MachinePrecision] * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.999999999999994e-310

   1. Initial program 56.2%

      \[\sqrt{2 \cdot {x}^{2}} \]
   2. Add Preprocessing

   4. Applied rewrites 2.1%

      \[\leadsto \color{blue}{\sqrt{2} \cdot x} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\sqrt{2} \cdot x} \]

      2. +-lft-identity N/A

         \[\leadsto \sqrt{2} \cdot \color{blue}{\left(0 + x\right)} \]

      3. flip-+ N/A

         \[\leadsto \sqrt{2} \cdot \color{blue}{\frac{0 \cdot 0 - x \cdot x}{0 - x}} \]

      4. neg-sub0 N/A

         \[\leadsto \sqrt{2} \cdot \frac{0 \cdot 0 - x \cdot x}{\color{blue}{\mathsf{neg}\left(x\right)}} \]

      5. lift-neg.f64 N/A

         \[\leadsto \sqrt{2} \cdot \frac{0 \cdot 0 - x \cdot x}{\color{blue}{-x}} \]

      6. div-inv N/A

         \[\leadsto \sqrt{2} \cdot \color{blue}{\left(\left(0 \cdot 0 - x \cdot x\right) \cdot \frac{1}{-x}\right)} \]

      7. inv-pow N/A

         \[\leadsto \sqrt{2} \cdot \left(\left(0 \cdot 0 - x \cdot x\right) \cdot \color{blue}{{\left(-x\right)}^{-1}}\right) \]

      8. metadata-eval N/A

         \[\leadsto \sqrt{2} \cdot \left(\left(0 \cdot 0 - x \cdot x\right) \cdot {\left(-x\right)}^{\color{blue}{\left(2 \cdot \frac{-1}{2}\right)}}\right) \]

      9. metadata-eval N/A

         \[\leadsto \sqrt{2} \cdot \left(\left(0 \cdot 0 - x \cdot x\right) \cdot {\left(-x\right)}^{\left(2 \cdot \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right)}\right) \]

      10. pow-pow N/A

          \[\leadsto \sqrt{2} \cdot \left(\left(0 \cdot 0 - x \cdot x\right) \cdot \color{blue}{{\left({\left(-x\right)}^{2}\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}\right) \]

      11. pow2 N/A

          \[\leadsto \sqrt{2} \cdot \left(\left(0 \cdot 0 - x \cdot x\right) \cdot {\color{blue}{\left(\left(-x\right) \cdot \left(-x\right)\right)}}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right) \]

      12. lift-neg.f64 N/A

          \[\leadsto \sqrt{2} \cdot \left(\left(0 \cdot 0 - x \cdot x\right) \cdot {\left(\color{blue}{\left(\mathsf{neg}\left(x\right)\right)} \cdot \left(-x\right)\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right) \]

      13. lift-neg.f64 N/A

          \[\leadsto \sqrt{2} \cdot \left(\left(0 \cdot 0 - x \cdot x\right) \cdot {\left(\left(\mathsf{neg}\left(x\right)\right) \cdot \color{blue}{\left(\mathsf{neg}\left(x\right)\right)}\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right) \]

      14. sqr-neg N/A

          \[\leadsto \sqrt{2} \cdot \left(\left(0 \cdot 0 - x \cdot x\right) \cdot {\color{blue}{\left(x \cdot x\right)}}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right) \]

      15. pow2 N/A

          \[\leadsto \sqrt{2} \cdot \left(\left(0 \cdot 0 - x \cdot x\right) \cdot {\color{blue}{\left({x}^{2}\right)}}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right) \]

      16. pow-pow N/A

          \[\leadsto \sqrt{2} \cdot \left(\left(0 \cdot 0 - x \cdot x\right) \cdot \color{blue}{{x}^{\left(2 \cdot \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto \sqrt{2} \cdot \left(\left(0 \cdot 0 - x \cdot x\right) \cdot {x}^{\left(2 \cdot \color{blue}{\frac{-1}{2}}\right)}\right) \]

      18. metadata-eval N/A

          \[\leadsto \sqrt{2} \cdot \left(\left(0 \cdot 0 - x \cdot x\right) \cdot {x}^{\color{blue}{-1}}\right) \]

      19. inv-pow N/A

          \[\leadsto \sqrt{2} \cdot \left(\left(0 \cdot 0 - x \cdot x\right) \cdot \color{blue}{\frac{1}{x}}\right) \]

      20. +-lft-identity N/A

          \[\leadsto \sqrt{2} \cdot \left(\left(0 \cdot 0 - x \cdot x\right) \cdot \frac{1}{\color{blue}{0 + x}}\right) \]

      21. div-inv N/A

          \[\leadsto \sqrt{2} \cdot \color{blue}{\frac{0 \cdot 0 - x \cdot x}{0 + x}} \]

      22. flip-- N/A

          \[\leadsto \sqrt{2} \cdot \color{blue}{\left(0 - x\right)} \]

      23. neg-sub0 N/A

          \[\leadsto \sqrt{2} \cdot \color{blue}{\left(\mathsf{neg}\left(x\right)\right)} \]

   6. Applied rewrites 99.3%

      \[\leadsto \color{blue}{\left({4}^{0.125} \cdot x\right) \cdot \left(-{4}^{0.125}\right)} \]

   8. Applied rewrites 55.9%

      \[\leadsto \color{blue}{\frac{0 - \left(x \cdot x\right) \cdot 2}{\sqrt{2} \cdot x}} \]
   9. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{0 - \left(x \cdot x\right) \cdot 2}{\sqrt{2} \cdot x}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{0 - \left(x \cdot x\right) \cdot 2}{\color{blue}{\sqrt{2} \cdot x}} \]

      3. associate-/l/ N/A

         \[\leadsto \color{blue}{\frac{\frac{0 - \left(x \cdot x\right) \cdot 2}{x}}{\sqrt{2}}} \]

      4. lift--.f64 N/A

         \[\leadsto \frac{\frac{\color{blue}{0 - \left(x \cdot x\right) \cdot 2}}{x}}{\sqrt{2}} \]

      5. sub0-neg N/A

         \[\leadsto \frac{\frac{\color{blue}{\mathsf{neg}\left(\left(x \cdot x\right) \cdot 2\right)}}{x}}{\sqrt{2}} \]

      6. distribute-frac-neg N/A

         \[\leadsto \frac{\color{blue}{\mathsf{neg}\left(\frac{\left(x \cdot x\right) \cdot 2}{x}\right)}}{\sqrt{2}} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{\mathsf{neg}\left(\frac{\color{blue}{\left(x \cdot x\right) \cdot 2}}{x}\right)}{\sqrt{2}} \]

      8. lift-*.f64 N/A

         \[\leadsto \frac{\mathsf{neg}\left(\frac{\color{blue}{\left(x \cdot x\right)} \cdot 2}{x}\right)}{\sqrt{2}} \]

      9. associate-*l* N/A

         \[\leadsto \frac{\mathsf{neg}\left(\frac{\color{blue}{x \cdot \left(x \cdot 2\right)}}{x}\right)}{\sqrt{2}} \]

      10. associate-/l* N/A

          \[\leadsto \frac{\mathsf{neg}\left(\color{blue}{x \cdot \frac{x \cdot 2}{x}}\right)}{\sqrt{2}} \]

      11. *-commutative N/A

          \[\leadsto \frac{\mathsf{neg}\left(x \cdot \frac{\color{blue}{2 \cdot x}}{x}\right)}{\sqrt{2}} \]

      12. associate-/l* N/A

          \[\leadsto \frac{\mathsf{neg}\left(x \cdot \color{blue}{\left(2 \cdot \frac{x}{x}\right)}\right)}{\sqrt{2}} \]

      13. *-inverses N/A

          \[\leadsto \frac{\mathsf{neg}\left(x \cdot \left(2 \cdot \color{blue}{1}\right)\right)}{\sqrt{2}} \]

      14. metadata-eval N/A

          \[\leadsto \frac{\mathsf{neg}\left(x \cdot \color{blue}{2}\right)}{\sqrt{2}} \]

      15. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(x \cdot 2\right)}{\sqrt{2}}} \]

      16. *-commutative N/A

          \[\leadsto \frac{\mathsf{neg}\left(\color{blue}{2 \cdot x}\right)}{\sqrt{2}} \]

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

          \[\leadsto \frac{\color{blue}{\left(\mathsf{neg}\left(2\right)\right) \cdot x}}{\sqrt{2}} \]

      18. lower-*.f64 N/A

          \[\leadsto \frac{\color{blue}{\left(\mathsf{neg}\left(2\right)\right) \cdot x}}{\sqrt{2}} \]

      19. metadata-eval 99.4

          \[\leadsto \frac{\color{blue}{-2} \cdot x}{\sqrt{2}} \]

   10. Applied rewrites 99.4%

       \[\leadsto \color{blue}{\frac{-2 \cdot x}{\sqrt{2}}} \]

   if -1.999999999999994e-310 < x

   1. Initial program 52.2%

      \[\sqrt{2 \cdot {x}^{2}} \]
   2. Add Preprocessing

   4. Applied rewrites 99.3%

      \[\leadsto \color{blue}{\sqrt{2} \cdot x} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 99.3% accurate, 4.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\left(-x\right) \cdot \sqrt{2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -2e-310) (* (- x) (sqrt 2.0)) (* (sqrt 2.0) x)))

C

double code(double x) {
	double tmp;
	if (x <= -2e-310) {
		tmp = -x * sqrt(2.0);
	} else {
		tmp = sqrt(2.0) * x;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-2d-310)) then
        tmp = -x * sqrt(2.0d0)
    else
        tmp = sqrt(2.0d0) * x
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= -2e-310) {
		tmp = -x * Math.sqrt(2.0);
	} else {
		tmp = Math.sqrt(2.0) * x;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -2e-310:
		tmp = -x * math.sqrt(2.0)
	else:
		tmp = math.sqrt(2.0) * x
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -2e-310)
		tmp = Float64(Float64(-x) * sqrt(2.0));
	else
		tmp = Float64(sqrt(2.0) * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -2e-310)
		tmp = -x * sqrt(2.0);
	else
		tmp = sqrt(2.0) * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -2e-310], N[((-x) * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[2.0], $MachinePrecision] * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.999999999999994e-310

   1. Initial program 56.2%

      \[\sqrt{2 \cdot {x}^{2}} \]
   2. Add Preprocessing

   3. Taylor expanded in x around -inf

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

      1. associate-*r* N/A

         \[\leadsto \color{blue}{\left(-1 \cdot x\right) \cdot \sqrt{2}} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(-1 \cdot x\right) \cdot \sqrt{2}} \]

      3. mul-1-neg N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(x\right)\right)} \cdot \sqrt{2} \]

      4. lower-neg.f64 N/A

         \[\leadsto \color{blue}{\left(-x\right)} \cdot \sqrt{2} \]

      5. lower-sqrt.f64 99.3

         \[\leadsto \left(-x\right) \cdot \color{blue}{\sqrt{2}} \]

   5. Applied rewrites 99.3%

      \[\leadsto \color{blue}{\left(-x\right) \cdot \sqrt{2}} \]

   if -1.999999999999994e-310 < x

   1. Initial program 52.2%

      \[\sqrt{2 \cdot {x}^{2}} \]
   2. Add Preprocessing

   4. Applied rewrites 99.3%

      \[\leadsto \color{blue}{\sqrt{2} \cdot x} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 52.5% accurate, 5.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4 \cdot 10^{-206}:\\ \;\;\;\;\sqrt{2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -4e-206) (sqrt 2.0) (* (sqrt 2.0) x)))

C

double code(double x) {
	double tmp;
	if (x <= -4e-206) {
		tmp = sqrt(2.0);
	} else {
		tmp = sqrt(2.0) * x;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-4d-206)) then
        tmp = sqrt(2.0d0)
    else
        tmp = sqrt(2.0d0) * x
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= -4e-206) {
		tmp = Math.sqrt(2.0);
	} else {
		tmp = Math.sqrt(2.0) * x;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -4e-206:
		tmp = math.sqrt(2.0)
	else:
		tmp = math.sqrt(2.0) * x
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -4e-206)
		tmp = sqrt(2.0);
	else
		tmp = Float64(sqrt(2.0) * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -4e-206)
		tmp = sqrt(2.0);
	else
		tmp = sqrt(2.0) * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -4e-206], N[Sqrt[2.0], $MachinePrecision], N[(N[Sqrt[2.0], $MachinePrecision] * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -4.00000000000000011e-206

   1. Initial program 65.8%

      \[\sqrt{2 \cdot {x}^{2}} \]
   2. Add Preprocessing

   4. Applied rewrites 5.9%

      \[\leadsto \color{blue}{\sqrt{2}} \]

   if -4.00000000000000011e-206 < x

   1. Initial program 45.3%

      \[\sqrt{2 \cdot {x}^{2}} \]
   2. Add Preprocessing

   4. Applied rewrites 84.9%

      \[\leadsto \color{blue}{\sqrt{2} \cdot x} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 5.4% accurate, 10.6× speedup

Math

\[\begin{array}{l} \\ \sqrt{2} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (sqrt 2.0))

C

double code(double x) {
	return sqrt(2.0);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = sqrt(2.0d0)
end function

Java

public static double code(double x) {
	return Math.sqrt(2.0);
}

Python

def code(x):
	return math.sqrt(2.0)

Julia

function code(x)
	return sqrt(2.0)
end

MATLAB

function tmp = code(x)
	tmp = sqrt(2.0);
end

Wolfram

code[x_] := N[Sqrt[2.0], $MachinePrecision]

Derivation

1. Initial program 54.3%

   \[\sqrt{2 \cdot {x}^{2}} \]
2. Add Preprocessing

4. Applied rewrites 5.4%

   \[\leadsto \color{blue}{\sqrt{2}} \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2024241 
(FPCore (x)
  :name "sqrt D (should all be same)"
  :precision binary64
  (sqrt (* 2.0 (pow x 2.0))))