sqrt D (should all be same)

Percentage Accurate: 54.9% → 99.3%

Time: 11.9s

Alternatives: 5

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: 54.9% 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: 99.3% accurate, 3.5× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -5e-310) (* (- x) (sqrt 2.0)) (/ (* x 2.0) (sqrt 2.0))))

C

double code(double x) {
	double tmp;
	if (x <= -5e-310) {
		tmp = -x * sqrt(2.0);
	} else {
		tmp = (x * 2.0) / sqrt(2.0);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-5d-310)) then
        tmp = -x * sqrt(2.0d0)
    else
        tmp = (x * 2.0d0) / sqrt(2.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= -5e-310) {
		tmp = -x * Math.sqrt(2.0);
	} else {
		tmp = (x * 2.0) / Math.sqrt(2.0);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -5e-310:
		tmp = -x * math.sqrt(2.0)
	else:
		tmp = (x * 2.0) / math.sqrt(2.0)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -5e-310)
		tmp = Float64(Float64(-x) * sqrt(2.0));
	else
		tmp = Float64(Float64(x * 2.0) / sqrt(2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -5e-310)
		tmp = -x * sqrt(2.0);
	else
		tmp = (x * 2.0) / sqrt(2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -5e-310], N[((-x) * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision], N[(N[(x * 2.0), $MachinePrecision] / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -4.999999999999985e-310

   1. Initial program 56.1%

      \[\sqrt{2 \cdot {x}^{2}} \]
   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 -4.999999999999985e-310 < x

   1. Initial program 52.3%

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

   4. Applied rewrites 99.1%

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

      1. lift-*.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. unpow1 N/A

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

      4. metadata-eval N/A

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

      5. sqrt-pow1 N/A

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

      6. sqrt-prod N/A

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

      7. pow1/2 N/A

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

      8. rem-exp-log N/A

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

      9. metadata-eval N/A

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

      10. metadata-eval N/A

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

      11. pow-div N/A

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

      12. metadata-eval N/A

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

      13. unpow1 N/A

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

      14. rem-exp-log N/A

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

      15. pow1/2 N/A

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

      16. lower-/.f64 N/A

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

      17. rem-exp-log N/A

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

      18. pow2 N/A

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

      19. associate-*r* N/A

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

      20. lower-*.f64 N/A

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

      21. *-commutative N/A

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

      22. lower-*.f64 N/A

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

      23. sqrt-prod N/A

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

      24. lift-sqrt.f64 N/A

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

      25. sqrt-pow1 N/A

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

      26. metadata-eval N/A

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

      27. unpow1 N/A

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

   6. Applied rewrites 52.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. times-frac N/A

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

      5. *-inverses N/A

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

      6. *-rgt-identity N/A

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

      7. lower-/.f64 99.4

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 99.4

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

   8. Applied rewrites 99.4%

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

4. Final simplification 99.4%

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

Alternative 2: 99.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \frac{\sqrt{2}}{\left|{x}^{-1}\right|} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ (sqrt 2.0) (fabs (pow x -1.0))))

C

double code(double x) {
	return sqrt(2.0) / fabs(pow(x, -1.0));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = sqrt(2.0d0) / abs((x ** (-1.0d0)))
end function

Java

public static double code(double x) {
	return Math.sqrt(2.0) / Math.abs(Math.pow(x, -1.0));
}

Python

def code(x):
	return math.sqrt(2.0) / math.fabs(math.pow(x, -1.0))

Julia

function code(x)
	return Float64(sqrt(2.0) / abs((x ^ -1.0)))
end

MATLAB

function tmp = code(x)
	tmp = sqrt(2.0) / abs((x ^ -1.0));
end

Wolfram

code[x_] := N[(N[Sqrt[2.0], $MachinePrecision] / N[Abs[N[Power[x, -1.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 54.4%

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

4. Applied rewrites 46.3%

   \[\leadsto \color{blue}{\sqrt{2} \cdot x} \]
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. clear-num N/A

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

6. Applied rewrites 46.3%

   \[\leadsto \color{blue}{\frac{\sqrt{2}}{{x}^{-1}}} \]
7. Step-by-step derivation

   1. rem-square-sqrt N/A

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

   2. sqrt-unprod N/A

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

   3. rem-sqrt-square N/A

      \[\leadsto \frac{\sqrt{2}}{\color{blue}{\left|{x}^{-1}\right|}} \]

   4. lower-fabs.f64 99.3

      \[\leadsto \frac{\sqrt{2}}{\color{blue}{\left|{x}^{-1}\right|}} \]

8. Applied rewrites 99.3%

   \[\leadsto \frac{\sqrt{2}}{\color{blue}{\left|{x}^{-1}\right|}} \]
9. Add Preprocessing

Alternative 3: 99.3% accurate, 4.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\left(-x\right) \cdot \sqrt{2}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \sqrt{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -5e-310) (* (- x) (sqrt 2.0)) (* x (sqrt 2.0))))

C

double code(double x) {
	double tmp;
	if (x <= -5e-310) {
		tmp = -x * sqrt(2.0);
	} else {
		tmp = x * sqrt(2.0);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-5d-310)) then
        tmp = -x * sqrt(2.0d0)
    else
        tmp = x * sqrt(2.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= -5e-310) {
		tmp = -x * Math.sqrt(2.0);
	} else {
		tmp = x * Math.sqrt(2.0);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -5e-310:
		tmp = -x * math.sqrt(2.0)
	else:
		tmp = x * math.sqrt(2.0)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -5e-310)
		tmp = Float64(Float64(-x) * sqrt(2.0));
	else
		tmp = Float64(x * sqrt(2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -5e-310)
		tmp = -x * sqrt(2.0);
	else
		tmp = x * sqrt(2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -5e-310], N[((-x) * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision], N[(x * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -4.999999999999985e-310

   1. Initial program 56.1%

      \[\sqrt{2 \cdot {x}^{2}} \]
   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 -4.999999999999985e-310 < x

   1. Initial program 52.3%

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

   4. Applied rewrites 99.1%

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

4. Final simplification 99.2%

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

Alternative 4: 52.7% accurate, 5.3× speedup

Math

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

FPCore

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

C

double code(double x) {
	double tmp;
	if (x <= -4e-206) {
		tmp = sqrt(2.0);
	} else {
		tmp = x * sqrt(2.0);
	}
	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 = x * sqrt(2.0d0)
    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 = x * Math.sqrt(2.0);
	}
	return tmp;
}

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < -4.00000000000000011e-206

   1. Initial program 66.3%

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

   4. Applied rewrites 5.9%

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

   if -4.00000000000000011e-206 < x

   1. Initial program 44.7%

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

   4. Applied rewrites 82.9%

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

4. Final simplification 48.3%

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

Alternative 5: 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 54.4%

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

4. Applied rewrites 5.2%

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

Reproduce

herbie shell --seed 2024243 
(FPCore (x)
  :name "sqrt D (should all be same)"
  :precision binary64
  (sqrt (* 2.0 (pow x 2.0))))