sqrt D (should all be same)

Percentage Accurate: 53.5% → 99.4%

Time: 13.4s

Alternatives: 7

Speedup: 4.9×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 53.5% 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, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\frac{1}{\frac{-1}{\sqrt{2} \cdot x}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x} \cdot \sqrt{2 \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -5e-310)
   (/ 1.0 (/ -1.0 (* (sqrt 2.0) x)))
   (* (sqrt x) (sqrt (* 2.0 x)))))

C

double code(double x) {
	double tmp;
	if (x <= -5e-310) {
		tmp = 1.0 / (-1.0 / (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 = 1.0d0 / ((-1.0d0) / (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 = 1.0 / (-1.0 / (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 = 1.0 / (-1.0 / (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(1.0 / Float64(-1.0 / Float64(sqrt(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 = 1.0 / (-1.0 / (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[(1.0 / N[(-1.0 / N[(N[Sqrt[2.0], $MachinePrecision] * 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 53.5%

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

   3. Taylor expanded in x around -inf

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

      1. 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 99.4

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

   5. Applied rewrites 99.4%

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

   7. Applied rewrites 99.2%

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

   9. Applied rewrites 99.4%

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

   if -4.999999999999985e-310 < x

   1. Initial program 59.8%

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

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\frac{1}{\frac{-1}{\sqrt{2} \cdot x}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x} \cdot \sqrt{2 \cdot x}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.4% accurate, 3.2× speedup

Math

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

FPCore

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

C

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

Python

def code(x):
	tmp = 0
	if x <= -5e-310:
		tmp = -x / math.sqrt(0.5)
	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(Float64(-x) / sqrt(0.5));
	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 = -x / sqrt(0.5);
	else
		tmp = sqrt(x) * sqrt((2.0 * x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -5e-310], N[((-x) / N[Sqrt[0.5], $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 53.5%

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

   3. Taylor expanded in x around -inf

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

      1. 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 99.4

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

   5. Applied rewrites 99.4%

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

   7. Applied rewrites 99.2%

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

   9. Applied rewrites 99.3%

      \[\leadsto \frac{-1}{\color{blue}{\frac{{2}^{-0.5}}{x}}} \]

   10. Taylor expanded in x around 0

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

   12. Applied rewrites 99.4%

       \[\leadsto \frac{-x}{\color{blue}{\sqrt{0.5}}} \]

   if -4.999999999999985e-310 < x

   1. Initial program 59.8%

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

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

Alternative 3: 99.3% accurate, 3.9× speedup

Math

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

FPCore

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

C

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

Python

def code(x):
	tmp = 0
	if x <= -5e-310:
		tmp = -x / math.sqrt(0.5)
	else:
		tmp = x / math.sqrt(0.5)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -5e-310)
		tmp = Float64(Float64(-x) / sqrt(0.5));
	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 = -x / sqrt(0.5);
	else
		tmp = x / sqrt(0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -5e-310], N[((-x) / N[Sqrt[0.5], $MachinePrecision]), $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 53.5%

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

   3. Taylor expanded in x around -inf

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

      1. 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 99.4

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

   5. Applied rewrites 99.4%

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

   7. Applied rewrites 99.2%

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

   9. Applied rewrites 99.3%

      \[\leadsto \frac{-1}{\color{blue}{\frac{{2}^{-0.5}}{x}}} \]

   10. Taylor expanded in x around 0

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

   12. Applied rewrites 99.4%

       \[\leadsto \frac{-x}{\color{blue}{\sqrt{0.5}}} \]

   if -4.999999999999985e-310 < x

   1. Initial program 59.8%

      \[\sqrt{2 \cdot {x}^{2}} \]
   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. +-lft-identity N/A

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

      3. flip-+ N/A

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

      4. neg-sub0 N/A

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

      5. lift-neg.f64 N/A

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

      6. div-inv N/A

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

      7. inv-pow N/A

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

      8. metadata-eval N/A

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

      9. metadata-eval N/A

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

      10. pow-pow N/A

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

      11. pow2 N/A

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

      12. lift-neg.f64 N/A

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

      13. lift-neg.f64 N/A

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

      14. sqr-neg N/A

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

      15. pow2 N/A

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

      16. pow-pow N/A

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

      17. metadata-eval N/A

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

      18. metadata-eval N/A

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

      19. inv-pow N/A

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

      20. +-lft-identity N/A

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

      21. div-inv N/A

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

      22. clear-num N/A

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

   6. Applied rewrites 99.3%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. div-inv N/A

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

      4. associate-/r* N/A

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

      5. lift-pow.f64 N/A

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

      6. unpow-1 N/A

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

      7. remove-double-div N/A

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

      8. lower-/.f64 N/A

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

      9. lift-sqrt.f64 N/A

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

      10. pow1/2 N/A

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

      11. pow-flip N/A

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

      12. metadata-eval N/A

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

      13. metadata-eval N/A

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

      14. lower-pow.f64 N/A

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

      15. metadata-eval 99.4

          \[\leadsto \frac{x}{{2}^{\color{blue}{-0.5}}} \]

   8. Applied rewrites 99.4%

      \[\leadsto \color{blue}{\frac{x}{{2}^{-0.5}}} \]

   9. Taylor expanded in x around 0

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

       1. lower-/.f64 N/A

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

       2. lower-sqrt.f64 99.4

          \[\leadsto \frac{x}{\color{blue}{\sqrt{0.5}}} \]

   11. Applied rewrites 99.4%

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

Alternative 4: 99.3% accurate, 4.2× speedup

Math

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

FPCore

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

C

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

Python

def code(x):
	tmp = 0
	if x <= -5e-310:
		tmp = -x * math.sqrt(2.0)
	else:
		tmp = x / math.sqrt(0.5)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -5e-310)
		tmp = Float64(Float64(-x) * sqrt(2.0));
	else
		tmp = Float64(x / sqrt(0.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -5e-310)
		tmp = -x * sqrt(2.0);
	else
		tmp = x / sqrt(0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -5e-310], N[((-x) * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision], N[(x / N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -4.999999999999985e-310

   1. Initial program 53.5%

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

   3. Taylor expanded in x around -inf

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

      1. 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 99.4

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

   5. Applied rewrites 99.4%

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

   if -4.999999999999985e-310 < x

   1. Initial program 59.8%

      \[\sqrt{2 \cdot {x}^{2}} \]
   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. +-lft-identity N/A

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

      3. flip-+ N/A

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

      4. neg-sub0 N/A

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

      5. lift-neg.f64 N/A

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

      6. div-inv N/A

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

      7. inv-pow N/A

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

      8. metadata-eval N/A

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

      9. metadata-eval N/A

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

      10. pow-pow N/A

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

      11. pow2 N/A

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

      12. lift-neg.f64 N/A

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

      13. lift-neg.f64 N/A

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

      14. sqr-neg N/A

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

      15. pow2 N/A

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

      16. pow-pow N/A

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

      17. metadata-eval N/A

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

      18. metadata-eval N/A

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

      19. inv-pow N/A

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

      20. +-lft-identity N/A

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

      21. div-inv N/A

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

      22. clear-num N/A

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

   6. Applied rewrites 99.3%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. div-inv N/A

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

      4. associate-/r* N/A

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

      5. lift-pow.f64 N/A

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

      6. unpow-1 N/A

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

      7. remove-double-div N/A

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

      8. lower-/.f64 N/A

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

      9. lift-sqrt.f64 N/A

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

      10. pow1/2 N/A

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

      11. pow-flip N/A

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

      12. metadata-eval N/A

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

      13. metadata-eval N/A

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

      14. lower-pow.f64 N/A

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

      15. metadata-eval 99.4

          \[\leadsto \frac{x}{{2}^{\color{blue}{-0.5}}} \]

   8. Applied rewrites 99.4%

      \[\leadsto \color{blue}{\frac{x}{{2}^{-0.5}}} \]

   9. Taylor expanded in x around 0

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

       1. lower-/.f64 N/A

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

       2. lower-sqrt.f64 99.4

          \[\leadsto \frac{x}{\color{blue}{\sqrt{0.5}}} \]

   11. Applied rewrites 99.4%

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

Alternative 5: 99.3% accurate, 4.9× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

Wolfram

code[x_] := If[LessEqual[x, -5e-310], N[((-x) * N[Sqrt[2.0], $MachinePrecision]), $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 53.5%

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

   3. Taylor expanded in x around -inf

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

      1. 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 99.4

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

   5. Applied rewrites 99.4%

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

   if -4.999999999999985e-310 < x

   1. Initial program 59.8%

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

   4. Applied rewrites 99.3%

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

Alternative 6: 52.6% 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 59.9%

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

   4. Applied rewrites 5.7%

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

   if -4.00000000000000011e-206 < x

   1. Initial program 54.5%

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

   4. Applied rewrites 90.0%

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

Alternative 7: 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 56.8%

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

4. Applied rewrites 5.5%

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

Reproduce

herbie shell --seed 2024257 
(FPCore (x)
  :name "sqrt D (should all be same)"
  :precision binary64
  (sqrt (* 2.0 (pow x 2.0))))