sqrt D (should all be same)

Percentage Accurate: 53.6% → 99.4%

Time: 12.8s

Alternatives: 7

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: 53.6% 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: 99.4% accurate, 3.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < -1.999999999999994e-310

   1. Initial program 58.7%

      \[\sqrt{2 \cdot {x}^{2}} \]
   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. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sqrt.f64 99.3

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

   5. Applied rewrites 99.3%

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

      1. lift-sqrt.f64 N/A

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

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

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

      3. neg-sub0 N/A

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

      4. flip-- N/A

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

      5. +-lft-identity N/A

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

      6. associate-*r/ N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. metadata-eval N/A

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

      10. lift-sqrt.f64 N/A

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

      11. lift-sqrt.f64 N/A

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

      12. rem-square-sqrt N/A

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

      13. metadata-eval 99.3

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

   7. Applied rewrites 99.3%

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

   9. Applied rewrites 99.5%

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

   if -1.999999999999994e-310 < x

   1. Initial program 53.5%

      \[\sqrt{2 \cdot {x}^{2}} \]
   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. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sqrt.f64 2.4

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

   5. Applied rewrites 2.4%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. +-lft-identity N/A

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

      4. flip3-+ N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\frac{{0}^{3} + {\left(x \cdot \sqrt{2}\right)}^{3}}{0 \cdot 0 + \left(\left(x \cdot \sqrt{2}\right) \cdot \left(x \cdot \sqrt{2}\right) - 0 \cdot \left(x \cdot \sqrt{2}\right)\right)}}\right) \]

      5. distribute-neg-frac N/A

         \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(\left({0}^{3} + {\left(x \cdot \sqrt{2}\right)}^{3}\right)\right)}{0 \cdot 0 + \left(\left(x \cdot \sqrt{2}\right) \cdot \left(x \cdot \sqrt{2}\right) - 0 \cdot \left(x \cdot \sqrt{2}\right)\right)}} \]

   7. Applied rewrites 99.5%

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

Alternative 2: 99.4% accurate, 3.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < -1.999999999999994e-310

   1. Initial program 58.7%

      \[\sqrt{2 \cdot {x}^{2}} \]
   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. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sqrt.f64 99.3

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

   5. Applied rewrites 99.3%

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

      1. lift-sqrt.f64 N/A

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

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

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

      3. neg-sub0 N/A

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

      4. flip-- N/A

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

      5. +-lft-identity N/A

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

      6. associate-*r/ N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. metadata-eval N/A

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

      10. lift-sqrt.f64 N/A

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

      11. lift-sqrt.f64 N/A

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

      12. rem-square-sqrt N/A

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

      13. metadata-eval 99.3

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

   7. Applied rewrites 99.3%

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

   if -1.999999999999994e-310 < x

   1. Initial program 53.5%

      \[\sqrt{2 \cdot {x}^{2}} \]
   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. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sqrt.f64 2.4

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

   5. Applied rewrites 2.4%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. +-lft-identity N/A

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

      4. flip3-+ N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\frac{{0}^{3} + {\left(x \cdot \sqrt{2}\right)}^{3}}{0 \cdot 0 + \left(\left(x \cdot \sqrt{2}\right) \cdot \left(x \cdot \sqrt{2}\right) - 0 \cdot \left(x \cdot \sqrt{2}\right)\right)}}\right) \]

      5. distribute-neg-frac N/A

         \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(\left({0}^{3} + {\left(x \cdot \sqrt{2}\right)}^{3}\right)\right)}{0 \cdot 0 + \left(\left(x \cdot \sqrt{2}\right) \cdot \left(x \cdot \sqrt{2}\right) - 0 \cdot \left(x \cdot \sqrt{2}\right)\right)}} \]

   7. Applied rewrites 99.5%

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

Alternative 3: 99.3% accurate, 3.5× speedup

Math

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

FPCore

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

C

double code(double x) {
	double tmp;
	if (x <= -2e-310) {
		tmp = (x * -2.0) / 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 <= (-2d-310)) then
        tmp = (x * (-2.0d0)) / 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 <= -2e-310) {
		tmp = (x * -2.0) / Math.sqrt(2.0);
	} else {
		tmp = (x * 2.0) / Math.sqrt(2.0);
	}
	return tmp;
}

Python

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

Julia

function code(x)
	tmp = 0.0
	if (x <= -2e-310)
		tmp = Float64(Float64(x * -2.0) / 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 <= -2e-310)
		tmp = (x * -2.0) / sqrt(2.0);
	else
		tmp = (x * 2.0) / sqrt(2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -2e-310], N[(N[(x * -2.0), $MachinePrecision] / 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 < -1.999999999999994e-310

   1. Initial program 58.7%

      \[\sqrt{2 \cdot {x}^{2}} \]
   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. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sqrt.f64 99.3

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

   5. Applied rewrites 99.3%

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

      1. lift-sqrt.f64 N/A

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

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

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

      3. neg-sub0 N/A

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

      4. flip-- N/A

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

      5. +-lft-identity N/A

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

      6. associate-*r/ N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. metadata-eval N/A

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

      10. lift-sqrt.f64 N/A

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

      11. lift-sqrt.f64 N/A

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

      12. rem-square-sqrt N/A

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

      13. metadata-eval 99.3

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

   7. Applied rewrites 99.3%

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

   if -1.999999999999994e-310 < x

   1. Initial program 53.5%

      \[\sqrt{2 \cdot {x}^{2}} \]
   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. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sqrt.f64 2.4

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

   5. Applied rewrites 2.4%

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

      1. lift-sqrt.f64 N/A

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

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

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

      3. neg-sub0 N/A

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

      4. flip-- N/A

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

      5. +-lft-identity N/A

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

      6. associate-*r/ N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. metadata-eval N/A

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

      10. lift-sqrt.f64 N/A

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

      11. lift-sqrt.f64 N/A

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

      12. rem-square-sqrt N/A

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

      13. metadata-eval 2.4

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

   7. Applied rewrites 2.4%

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

      1. lift-*.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-/.f64 2.4

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

      4. unpow1 N/A

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

      5. metadata-eval N/A

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

      6. pow-sqr N/A

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

      7. pow-prod-down N/A

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

   9. Applied rewrites 99.3%

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

Alternative 4: 99.3% accurate, 3.5× speedup

Math

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

FPCore

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

C

double code(double x) {
	double tmp;
	if (x <= -2e-310) {
		tmp = (x * -2.0) / 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 <= (-2d-310)) then
        tmp = (x * (-2.0d0)) / 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 <= -2e-310) {
		tmp = (x * -2.0) / Math.sqrt(2.0);
	} else {
		tmp = x * Math.sqrt(2.0);
	}
	return tmp;
}

Python

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

Julia

function code(x)
	tmp = 0.0
	if (x <= -2e-310)
		tmp = Float64(Float64(x * -2.0) / 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 <= -2e-310)
		tmp = (x * -2.0) / sqrt(2.0);
	else
		tmp = x * sqrt(2.0);
	end
	tmp_2 = tmp;
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < -1.999999999999994e-310

   1. Initial program 58.7%

      \[\sqrt{2 \cdot {x}^{2}} \]
   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. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sqrt.f64 99.3

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

   5. Applied rewrites 99.3%

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

      1. lift-sqrt.f64 N/A

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

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

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

      3. neg-sub0 N/A

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

      4. flip-- N/A

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

      5. +-lft-identity N/A

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

      6. associate-*r/ N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. metadata-eval N/A

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

      10. lift-sqrt.f64 N/A

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

      11. lift-sqrt.f64 N/A

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

      12. rem-square-sqrt N/A

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

      13. metadata-eval 99.3

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

   7. Applied rewrites 99.3%

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

   if -1.999999999999994e-310 < x

   1. Initial program 53.5%

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

   3. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 99.3

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

   5. Applied rewrites 99.3%

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

Alternative 5: 99.3% accurate, 4.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \sqrt{2}\\ \mathbf{if}\;x \leq -2 \cdot 10^{-310}:\\ \;\;\;\;-t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* x (sqrt 2.0)))) (if (<= x -2e-310) (- t_0) t_0)))

C

double code(double x) {
	double t_0 = x * sqrt(2.0);
	double tmp;
	if (x <= -2e-310) {
		tmp = -t_0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x * sqrt(2.0d0)
    if (x <= (-2d-310)) then
        tmp = -t_0
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double t_0 = x * Math.sqrt(2.0);
	double tmp;
	if (x <= -2e-310) {
		tmp = -t_0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x):
	t_0 = x * math.sqrt(2.0)
	tmp = 0
	if x <= -2e-310:
		tmp = -t_0
	else:
		tmp = t_0
	return tmp

Julia

function code(x)
	t_0 = Float64(x * sqrt(2.0))
	tmp = 0.0
	if (x <= -2e-310)
		tmp = Float64(-t_0);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = x * sqrt(2.0);
	tmp = 0.0;
	if (x <= -2e-310)
		tmp = -t_0;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[(x * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -2e-310], (-t$95$0), t$95$0]]

Derivation

1. Split input into 2 regimes

2. if x < -1.999999999999994e-310

   1. Initial program 58.7%

      \[\sqrt{2 \cdot {x}^{2}} \]
   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. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sqrt.f64 99.3

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

   5. Applied rewrites 99.3%

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

   if -1.999999999999994e-310 < x

   1. Initial program 53.5%

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

   3. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 99.3

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

   5. Applied rewrites 99.3%

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

Alternative 6: 51.1% 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 70.4%

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

   4. Applied rewrites 6.0%

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

   if -4.00000000000000011e-206 < x

   1. Initial program 46.2%

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

   3. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 84.5

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

   5. Applied rewrites 84.5%

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

Alternative 7: 5.3% 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 56.1%

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

4. Applied rewrites 5.4%

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

Reproduce

herbie shell --seed 2024219 
(FPCore (x)
  :name "sqrt D (should all be same)"
  :precision binary64
  (sqrt (* 2.0 (pow x 2.0))))