sqrt D (should all be same)

Percentage Accurate: 53.6% → 99.4%

Time: 12.3s

Alternatives: 6

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, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\left({2}^{0.125} \cdot \left(x \cdot {2}^{0.25}\right)\right) \cdot \left(-{2}^{0.125}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x \cdot 2} \cdot \sqrt{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -5e-310)
   (* (* (pow 2.0 0.125) (* x (pow 2.0 0.25))) (- (pow 2.0 0.125)))
   (* (sqrt (* x 2.0)) (sqrt x))))

C

double code(double x) {
	double tmp;
	if (x <= -5e-310) {
		tmp = (pow(2.0, 0.125) * (x * pow(2.0, 0.25))) * -pow(2.0, 0.125);
	} 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 <= (-5d-310)) then
        tmp = ((2.0d0 ** 0.125d0) * (x * (2.0d0 ** 0.25d0))) * -(2.0d0 ** 0.125d0)
    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 <= -5e-310) {
		tmp = (Math.pow(2.0, 0.125) * (x * Math.pow(2.0, 0.25))) * -Math.pow(2.0, 0.125);
	} else {
		tmp = Math.sqrt((x * 2.0)) * Math.sqrt(x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -5e-310:
		tmp = (math.pow(2.0, 0.125) * (x * math.pow(2.0, 0.25))) * -math.pow(2.0, 0.125)
	else:
		tmp = math.sqrt((x * 2.0)) * math.sqrt(x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -5e-310)
		tmp = Float64(Float64((2.0 ^ 0.125) * Float64(x * (2.0 ^ 0.25))) * Float64(-(2.0 ^ 0.125)));
	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 <= -5e-310)
		tmp = ((2.0 ^ 0.125) * (x * (2.0 ^ 0.25))) * -(2.0 ^ 0.125);
	else
		tmp = sqrt((x * 2.0)) * sqrt(x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -5e-310], N[(N[(N[Power[2.0, 0.125], $MachinePrecision] * N[(x * N[Power[2.0, 0.25], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * (-N[Power[2.0, 0.125], $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 < -4.999999999999985e-310

   1. Initial program 51.9%

      \[\sqrt{2 \cdot {x}^{2}} \]
   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. pow1/2 N/A

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

      2. sqr-pow N/A

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

      3. associate-*r* N/A

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

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

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

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

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

      6. *-commutative N/A

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

      7. sqr-pow N/A

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

      8. *-commutative N/A

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

      9. associate-*l* N/A

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

      10. lower-*.f64 N/A

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

      11. lower-pow.f64 N/A

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

      12. metadata-eval N/A

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

      13. metadata-eval N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

   7. Applied rewrites 99.4%

      \[\leadsto \color{blue}{{2}^{0.125} \cdot \left({2}^{0.125} \cdot \left(-x \cdot {2}^{0.25}\right)\right)} \]

   if -4.999999999999985e-310 < x

   1. Initial program 44.9%

      \[\sqrt{2 \cdot {x}^{2}} \]
   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. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\left({2}^{0.125} \cdot \left(x \cdot {2}^{0.25}\right)\right) \cdot \left(-{2}^{0.125}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x \cdot 2} \cdot \sqrt{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.4% accurate, 3.2× speedup

Math

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

FPCore

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

C

double code(double x) {
	double tmp;
	if (x <= -5e-310) {
		tmp = sqrt(2.0) * -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 <= (-5d-310)) then
        tmp = sqrt(2.0d0) * -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 <= -5e-310) {
		tmp = Math.sqrt(2.0) * -x;
	} else {
		tmp = Math.sqrt((x * 2.0)) * Math.sqrt(x);
	}
	return tmp;
}

Python

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

Julia

function code(x)
	tmp = 0.0
	if (x <= -5e-310)
		tmp = Float64(sqrt(2.0) * 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 <= -5e-310)
		tmp = sqrt(2.0) * -x;
	else
		tmp = sqrt((x * 2.0)) * sqrt(x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -5e-310], N[(N[Sqrt[2.0], $MachinePrecision] * (-x)), $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 < -4.999999999999985e-310

   1. Initial program 51.9%

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

   1. Initial program 44.9%

      \[\sqrt{2 \cdot {x}^{2}} \]
   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. Final simplification 99.4%

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

Alternative 3: 99.3% accurate, 3.5× speedup

Math

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

FPCore

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

C

double code(double x) {
	double tmp;
	if (x <= -5e-310) {
		tmp = sqrt(2.0) * -x;
	} 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 = sqrt(2.0d0) * -x
    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 = Math.sqrt(2.0) * -x;
	} else {
		tmp = (x * 2.0) / Math.sqrt(2.0);
	}
	return tmp;
}

Python

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

Julia

function code(x)
	tmp = 0.0
	if (x <= -5e-310)
		tmp = Float64(sqrt(2.0) * Float64(-x));
	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 = sqrt(2.0) * -x;
	else
		tmp = (x * 2.0) / sqrt(2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -5e-310], N[(N[Sqrt[2.0], $MachinePrecision] * (-x)), $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 51.9%

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

   1. Initial program 44.9%

      \[\sqrt{2 \cdot {x}^{2}} \]
   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 2.4

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

      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. metadata-eval N/A

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

      6. +-lft-identity N/A

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

      7. sqr-pow N/A

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

      8. pow-prod-down N/A

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

      9. sqr-neg N/A

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

      10. lift-neg.f64 N/A

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

      11. lift-neg.f64 N/A

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

      12. pow-prod-down N/A

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

      13. sqr-pow N/A

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

      14. lift-neg.f64 N/A

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

      15. cube-neg N/A

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

      16. +-lft-identity N/A

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

      17. metadata-eval N/A

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

   7. Applied rewrites 99.3%

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

      1. lift-sqrt.f64 N/A

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

      2. associate-/l* N/A

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

      3. frac-2neg N/A

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

      4. metadata-eval N/A

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

      5. associate-*r/ N/A

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

      6. lift-*.f64 N/A

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

      7. distribute-neg-frac2 N/A

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

      8. remove-double-neg N/A

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

      9. lower-/.f64 99.3

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

   9. Applied rewrites 99.3%

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

4. Final simplification 99.3%

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

Alternative 4: 99.3% accurate, 4.9× speedup

Math

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

FPCore

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

C

double code(double x) {
	double tmp;
	if (x <= -5e-310) {
		tmp = sqrt(2.0) * -x;
	} 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 = sqrt(2.0d0) * -x
    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 = Math.sqrt(2.0) * -x;
	} else {
		tmp = x * Math.sqrt(2.0);
	}
	return tmp;
}

Python

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

Julia

function code(x)
	tmp = 0.0
	if (x <= -5e-310)
		tmp = Float64(sqrt(2.0) * Float64(-x));
	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 = sqrt(2.0) * -x;
	else
		tmp = x * sqrt(2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -5e-310], N[(N[Sqrt[2.0], $MachinePrecision] * (-x)), $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 51.9%

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

   1. Initial program 44.9%

      \[\sqrt{2 \cdot {x}^{2}} \]
   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. Final simplification 99.3%

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

Alternative 5: 52.7% accurate, 5.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4.1 \cdot 10^{-206}:\\ \;\;\;\;\sqrt{2}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \sqrt{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -4.1e-206) (sqrt 2.0) (* x (sqrt 2.0))))

C

double code(double x) {
	double tmp;
	if (x <= -4.1e-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 <= (-4.1d-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 <= -4.1e-206) {
		tmp = Math.sqrt(2.0);
	} else {
		tmp = x * Math.sqrt(2.0);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -4.1e-206:
		tmp = math.sqrt(2.0)
	else:
		tmp = x * math.sqrt(2.0)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -4.1e-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 <= -4.1e-206)
		tmp = sqrt(2.0);
	else
		tmp = x * sqrt(2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -4.1e-206], N[Sqrt[2.0], $MachinePrecision], N[(x * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -4.10000000000000016e-206

   1. Initial program 60.1%

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

   4. Applied rewrites 5.6%

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

   if -4.10000000000000016e-206 < x

   1. Initial program 40.3%

      \[\sqrt{2 \cdot {x}^{2}} \]
   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 87.7

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

   5. Applied rewrites 87.7%

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

Alternative 6: 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 48.3%

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

4. Applied rewrites 5.3%

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

Reproduce

herbie shell --seed 2024214 
(FPCore (x)
  :name "sqrt D (should all be same)"
  :precision binary64
  (sqrt (* 2.0 (pow x 2.0))))