sqrt D (should all be same)

Percentage Accurate: 54.1% → 99.4%

Time: 11.8s

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: 54.1% 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

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

FPCore

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

C

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

Python

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

Julia

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

Wolfram

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

   1. Initial program 50.6%

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

   4. Applied rewrites 2.3%

      \[\leadsto \color{blue}{\sqrt{2} \cdot x} \]
   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 0.0

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

   6. Applied rewrites 0.0%

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

      1. lift-*.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. sqrt-unprod N/A

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

      5. lift-*.f64 N/A

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

      6. remove-double-neg N/A

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

      7. lift-*.f64 N/A

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

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

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

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

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

      10. sqrt-prod N/A

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

      11. lower-*.f64 N/A

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

      12. lower-sqrt.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

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

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

      16. lower-*.f64 N/A

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

      17. metadata-eval N/A

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

      18. lower-sqrt.f64 N/A

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

      19. lower-neg.f64 99.5

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

   8. Applied rewrites 99.5%

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

   if -4.999999999999985e-310 < x

   1. Initial program 56.0%

      \[\sqrt{2 \cdot {x}^{2}} \]
   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 99.5

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

   6. 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%

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

Alternative 2: 99.4% accurate, 3.2× speedup

Math

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

FPCore

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

C

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

Python

def code(x):
	tmp = 0
	if x <= -5e-310:
		tmp = math.sqrt(2.0) * -x
	else:
		tmp = math.sqrt(x) * math.sqrt((2.0 * 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(x) * sqrt(Float64(2.0 * 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) * sqrt((2.0 * 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[x], $MachinePrecision] * N[Sqrt[N[(2.0 * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -4.999999999999985e-310

   1. Initial program 50.6%

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

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

      5. lower-sqrt.f64 99.4

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

   5. Applied rewrites 99.4%

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

   if -4.999999999999985e-310 < x

   1. Initial program 56.0%

      \[\sqrt{2 \cdot {x}^{2}} \]
   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 99.5

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

   6. 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%

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

Alternative 3: 99.3% accurate, 4.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\sqrt{2} \cdot \left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\sqrt{0.5}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -5e-310) (* (sqrt 2.0) (- x)) (/ x (sqrt 0.5))))

C

double code(double x) {
	double tmp;
	if (x <= -5e-310) {
		tmp = sqrt(2.0) * -x;
	} else {
		tmp = x / sqrt(0.5);
	}
	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(0.5d0)
    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(0.5);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -5e-310:
		tmp = math.sqrt(2.0) * -x
	else:
		tmp = x / math.sqrt(0.5)
	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(0.5));
	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(0.5);
	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[0.5], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -4.999999999999985e-310

   1. Initial program 50.6%

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

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

      5. lower-sqrt.f64 99.4

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

   5. Applied rewrites 99.4%

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

   if -4.999999999999985e-310 < x

   1. Initial program 56.0%

      \[\sqrt{2 \cdot {x}^{2}} \]
   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 99.5

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

   6. Applied rewrites 99.5%

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

   8. Applied rewrites 99.3%

      \[\leadsto \color{blue}{\frac{x}{\sqrt{0.5}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\sqrt{2} \cdot \left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\sqrt{0.5}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.3% accurate, 4.9× speedup

Math

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

FPCore

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

C

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

Python

def code(x):
	tmp = 0
	if x <= -5e-310:
		tmp = math.sqrt(2.0) * -x
	else:
		tmp = math.sqrt(2.0) * 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(2.0) * 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(2.0) * 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[2.0], $MachinePrecision] * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -4.999999999999985e-310

   1. Initial program 50.6%

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

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

      5. lower-sqrt.f64 99.4

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

   5. Applied rewrites 99.4%

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

   if -4.999999999999985e-310 < x

   1. Initial program 56.0%

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

   4. Applied rewrites 99.3%

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

4. Final simplification 99.3%

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

Alternative 5: 53.1% 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 61.1%

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

   4. Applied rewrites 5.4%

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

   if -4.00000000000000011e-206 < x

   1. Initial program 47.4%

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

   4. Applied rewrites 82.9%

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

Alternative 6: 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 53.2%

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

4. Applied rewrites 5.3%

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

Reproduce

herbie shell --seed 2024235 
(FPCore (x)
  :name "sqrt D (should all be same)"
  :precision binary64
  (sqrt (* 2.0 (pow x 2.0))))