sqrt D (should all be same)

Percentage Accurate: 53.3% → 99.5%

Time: 13.3s

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: 53.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.5% accurate, 0.5× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -2e-310)
   (* (- (pow 16.0 0.03125)) (* (pow 64.0 0.0625) x))
   (* (sqrt x) (sqrt (* 2.0 x)))))

C

double code(double x) {
	double tmp;
	if (x <= -2e-310) {
		tmp = -pow(16.0, 0.03125) * (pow(64.0, 0.0625) * x);
	} else {
		tmp = sqrt(x) * sqrt((2.0 * x));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-2d-310)) then
        tmp = -(16.0d0 ** 0.03125d0) * ((64.0d0 ** 0.0625d0) * 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 <= -2e-310) {
		tmp = -Math.pow(16.0, 0.03125) * (Math.pow(64.0, 0.0625) * x);
	} else {
		tmp = Math.sqrt(x) * Math.sqrt((2.0 * x));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -2e-310:
		tmp = -math.pow(16.0, 0.03125) * (math.pow(64.0, 0.0625) * x)
	else:
		tmp = math.sqrt(x) * math.sqrt((2.0 * x))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -2e-310)
		tmp = Float64(Float64(-(16.0 ^ 0.03125)) * Float64((64.0 ^ 0.0625) * x));
	else
		tmp = Float64(sqrt(x) * sqrt(Float64(2.0 * x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -2e-310)
		tmp = -(16.0 ^ 0.03125) * ((64.0 ^ 0.0625) * x);
	else
		tmp = sqrt(x) * sqrt((2.0 * x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -2e-310], N[((-N[Power[16.0, 0.03125], $MachinePrecision]) * N[(N[Power[64.0, 0.0625], $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[x], $MachinePrecision] * N[Sqrt[N[(2.0 * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.999999999999994e-310

   1. Initial program 50.0%

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

   4. Applied rewrites 2.4%

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

      1. lift-*.f64 N/A

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

      2. +-lft-identity N/A

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

      3. flip-+ N/A

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

      4. neg-sub0 N/A

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

      5. lift-neg.f64 N/A

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

      6. div-inv N/A

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

      7. inv-pow N/A

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

      8. metadata-eval N/A

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

      9. metadata-eval N/A

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

      10. pow-pow N/A

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

      11. pow2 N/A

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

      12. lift-neg.f64 N/A

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

      13. lift-neg.f64 N/A

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

      14. sqr-neg N/A

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

      15. pow2 N/A

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

      16. pow-pow N/A

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

      17. metadata-eval N/A

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

      18. metadata-eval N/A

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

      19. inv-pow N/A

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

      20. +-lft-identity N/A

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

      21. div-inv N/A

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

      22. flip-- N/A

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

      23. neg-sub0 N/A

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

   6. Applied rewrites 99.3%

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

      1. lift-*.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. sqr-pow N/A

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

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

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

   8. Applied rewrites 99.5%

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

   if -1.999999999999994e-310 < x

   1. Initial program 49.7%

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

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

Alternative 2: 99.3% accurate, 3.2× speedup

Math

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

FPCore

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

C

double code(double x) {
	double tmp;
	if (x <= -2e-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 <= (-2d-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 <= -2e-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 <= -2e-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 <= -2e-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 <= -2e-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, -2e-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 < -1.999999999999994e-310

   1. Initial program 50.0%

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

   3. Taylor expanded in x around -inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-sqrt.f64 99.3

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

   5. Applied rewrites 99.3%

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

   if -1.999999999999994e-310 < x

   1. Initial program 49.7%

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

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

Alternative 3: 99.3% accurate, 4.2× speedup

Math

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

FPCore

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

C

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

Python

def code(x):
	tmp = 0
	if x <= -2e-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 <= -2e-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 <= -2e-310)
		tmp = sqrt(2.0) * -x;
	else
		tmp = x / sqrt(0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -2e-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 < -1.999999999999994e-310

   1. Initial program 50.0%

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

   3. Taylor expanded in x around -inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-sqrt.f64 99.3

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

   5. Applied rewrites 99.3%

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

   if -1.999999999999994e-310 < x

   1. Initial program 49.7%

      \[\sqrt{2 \cdot {x}^{2}} \]
   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. lift-sqrt.f64 N/A

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

      3. unpow1 N/A

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

      4. metadata-eval N/A

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

      5. sqrt-pow1 N/A

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

      6. sqrt-prod N/A

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

      7. pow1/2 N/A

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

      8. rem-exp-log N/A

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

      9. metadata-eval N/A

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

      10. metadata-eval N/A

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

      11. pow-div N/A

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

      12. metadata-eval N/A

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

      13. unpow1 N/A

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

      14. rem-exp-log N/A

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

      15. pow1/2 N/A

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

      16. lower-/.f64 N/A

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

      17. rem-exp-log N/A

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

      18. pow2 N/A

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

      19. associate-*r* N/A

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

      20. lower-*.f64 N/A

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

      21. *-commutative N/A

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

      22. lower-*.f64 N/A

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

      23. sqrt-prod N/A

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

      24. lift-sqrt.f64 N/A

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

      25. sqrt-pow1 N/A

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

      26. metadata-eval N/A

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

      27. unpow1 N/A

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

   6. Applied rewrites 49.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. frac-times N/A

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

      5. *-inverses N/A

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

      6. associate-*l/ N/A

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

      7. *-rgt-identity N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*r/ N/A

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

      10. clear-num N/A

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

      11. associate-/r/ N/A

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

      12. lift-sqrt.f64 N/A

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

      13. pow1/2 N/A

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

      14. pow-flip N/A

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

      15. pow-plus N/A

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

      16. metadata-eval N/A

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

      17. metadata-eval N/A

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

      18. pow1/2 N/A

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

      19. lift-sqrt.f64 N/A

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

      20. *-rgt-identity N/A

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

      21. metadata-eval N/A

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

      22. div-inv N/A

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

      23. clear-num N/A

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

      24. div-inv N/A

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

      25. lower-/.f64 N/A

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

   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%

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

Alternative 4: 99.3% accurate, 4.9× speedup

Math

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

FPCore

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

C

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

Python

def code(x):
	tmp = 0
	if x <= -2e-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 <= -2e-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 <= -2e-310)
		tmp = sqrt(2.0) * -x;
	else
		tmp = sqrt(2.0) * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -2e-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 < -1.999999999999994e-310

   1. Initial program 50.0%

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

   3. Taylor expanded in x around -inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-sqrt.f64 99.3

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

   5. Applied rewrites 99.3%

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

   if -1.999999999999994e-310 < x

   1. Initial program 49.7%

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

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

Alternative 5: 53.2% 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 60.4%

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

   4. Applied rewrites 5.7%

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

   if -4.00000000000000011e-206 < x

   1. Initial program 42.1%

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

   4. Applied rewrites 82.2%

      \[\leadsto \color{blue}{\sqrt{2} \cdot x} \]
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 49.9%

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

4. Applied rewrites 5.2%

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

Reproduce

herbie shell --seed 2024242 
(FPCore (x)
  :name "sqrt D (should all be same)"
  :precision binary64
  (sqrt (* 2.0 (pow x 2.0))))