2sqrt (example 3.1)

Percentage Accurate: 6.5% → 99.6%

Time: 7.0s

Alternatives: 6

Speedup: 1.2×

Specification

\[x > 1 \land x < 10^{+308}\]

Math

\[\begin{array}{l} \\ \sqrt{x + 1} - \sqrt{x} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (- (sqrt (+ x 1.0)) (sqrt x)))

C

double code(double x) {
	return sqrt((x + 1.0)) - sqrt(x);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = sqrt((x + 1.0d0)) - sqrt(x)
end function

Java

public static double code(double x) {
	return Math.sqrt((x + 1.0)) - Math.sqrt(x);
}

Python

def code(x):
	return math.sqrt((x + 1.0)) - math.sqrt(x)

Julia

function code(x)
	return Float64(sqrt(Float64(x + 1.0)) - sqrt(x))
end

MATLAB

function tmp = code(x)
	tmp = sqrt((x + 1.0)) - sqrt(x);
end

Wolfram

code[x_] := N[(N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]

Initial Program: 6.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \sqrt{x + 1} - \sqrt{x} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (- (sqrt (+ x 1.0)) (sqrt x)))

C

double code(double x) {
	return sqrt((x + 1.0)) - sqrt(x);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = sqrt((x + 1.0d0)) - sqrt(x)
end function

Java

public static double code(double x) {
	return Math.sqrt((x + 1.0)) - Math.sqrt(x);
}

Python

def code(x):
	return math.sqrt((x + 1.0)) - math.sqrt(x)

Julia

function code(x)
	return Float64(sqrt(Float64(x + 1.0)) - sqrt(x))
end

MATLAB

function tmp = code(x)
	tmp = sqrt((x + 1.0)) - sqrt(x);
end

Wolfram

code[x_] := N[(N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]

Alternative 1: 99.6% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ {\left(\sqrt{x} + \sqrt{1 + x}\right)}^{-1} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (pow (+ (sqrt x) (sqrt (+ 1.0 x))) -1.0))

C

double code(double x) {
	return pow((sqrt(x) + sqrt((1.0 + x))), -1.0);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (sqrt(x) + sqrt((1.0d0 + x))) ** (-1.0d0)
end function

Java

public static double code(double x) {
	return Math.pow((Math.sqrt(x) + Math.sqrt((1.0 + x))), -1.0);
}

Python

def code(x):
	return math.pow((math.sqrt(x) + math.sqrt((1.0 + x))), -1.0)

Julia

function code(x)
	return Float64(sqrt(x) + sqrt(Float64(1.0 + x))) ^ -1.0
end

MATLAB

function tmp = code(x)
	tmp = (sqrt(x) + sqrt((1.0 + x))) ^ -1.0;
end

Wolfram

code[x_] := N[Power[N[(N[Sqrt[x], $MachinePrecision] + N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision]

Derivation

1. Initial program 7.6%

   \[\sqrt{x + 1} - \sqrt{x} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-sqrt.f64 N/A

      \[\leadsto \color{blue}{\sqrt{x + 1}} - \sqrt{x} \]

   2. pow1/2 N/A

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

   3. metadata-eval N/A

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

   4. pow-div N/A

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

   5. unpow1 N/A

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

   6. pow1/2 N/A

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

   7. lift-sqrt.f64 N/A

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

   8. lower-/.f64 7.6

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

   9. lift-+.f64 N/A

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

   10. +-commutative N/A

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

   11. lower-+.f64 7.6

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

   12. lift-+.f64 N/A

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

   13. +-commutative N/A

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

   14. lower-+.f64 7.6

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

4. Applied rewrites 7.6%

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

   1. lift--.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. div-inv N/A

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

   4. *-commutative N/A

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

   5. lift-sqrt.f64 N/A

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

   6. pow1/2 N/A

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

   7. pow-flip N/A

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

   8. metadata-eval N/A

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

   9. metadata-eval N/A

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

   10. pow-plus N/A

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

   11. metadata-eval N/A

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

   12. metadata-eval N/A

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

   13. lift-+.f64 N/A

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

   14. +-commutative N/A

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

   15. pow1/2 N/A

       \[\leadsto \color{blue}{\sqrt{x + 1}} - \sqrt{x} \]

   16. flip-- N/A

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

6. Applied rewrites 99.6%

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

7. Final simplification 99.6%

   \[\leadsto {\left(\sqrt{x} + \sqrt{1 + x}\right)}^{-1} \]
8. Add Preprocessing

Alternative 2: 98.9% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{x + 1} - \sqrt{x}\\ \mathbf{if}\;t\_0 \leq 5 \cdot 10^{-5}:\\ \;\;\;\;\sqrt{{x}^{-1}} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (- (sqrt (+ x 1.0)) (sqrt x))))
   (if (<= t_0 5e-5) (* (sqrt (pow x -1.0)) 0.5) t_0)))

C

double code(double x) {
	double t_0 = sqrt((x + 1.0)) - sqrt(x);
	double tmp;
	if (t_0 <= 5e-5) {
		tmp = sqrt(pow(x, -1.0)) * 0.5;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sqrt((x + 1.0d0)) - sqrt(x)
    if (t_0 <= 5d-5) then
        tmp = sqrt((x ** (-1.0d0))) * 0.5d0
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double t_0 = Math.sqrt((x + 1.0)) - Math.sqrt(x);
	double tmp;
	if (t_0 <= 5e-5) {
		tmp = Math.sqrt(Math.pow(x, -1.0)) * 0.5;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x):
	t_0 = math.sqrt((x + 1.0)) - math.sqrt(x)
	tmp = 0
	if t_0 <= 5e-5:
		tmp = math.sqrt(math.pow(x, -1.0)) * 0.5
	else:
		tmp = t_0
	return tmp

Julia

function code(x)
	t_0 = Float64(sqrt(Float64(x + 1.0)) - sqrt(x))
	tmp = 0.0
	if (t_0 <= 5e-5)
		tmp = Float64(sqrt((x ^ -1.0)) * 0.5);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = sqrt((x + 1.0)) - sqrt(x);
	tmp = 0.0;
	if (t_0 <= 5e-5)
		tmp = sqrt((x ^ -1.0)) * 0.5;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 5e-5], N[(N[Sqrt[N[Power[x, -1.0], $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision], t$95$0]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) < 5.00000000000000024e-5

   1. Initial program 5.0%

      \[\sqrt{x + 1} - \sqrt{x} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 99.2

         \[\leadsto \sqrt{\color{blue}{\frac{1}{x}}} \cdot 0.5 \]

   5. Applied rewrites 99.2%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{x}} \cdot 0.5} \]

   if 5.00000000000000024e-5 < (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x))

   1. Initial program 87.7%

      \[\sqrt{x + 1} - \sqrt{x} \]
   2. Add Preprocessing
3. Recombined 2 regimes into one program.

4. Final simplification 98.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{x + 1} - \sqrt{x} \leq 5 \cdot 10^{-5}:\\ \;\;\;\;\sqrt{{x}^{-1}} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x + 1} - \sqrt{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.1% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \sqrt{{x}^{-1}} \cdot 0.5 \end{array} \]

FPCore

(FPCore (x) :precision binary64 (* (sqrt (pow x -1.0)) 0.5))

C

double code(double x) {
	return sqrt(pow(x, -1.0)) * 0.5;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = sqrt((x ** (-1.0d0))) * 0.5d0
end function

Java

public static double code(double x) {
	return Math.sqrt(Math.pow(x, -1.0)) * 0.5;
}

Python

def code(x):
	return math.sqrt(math.pow(x, -1.0)) * 0.5

Julia

function code(x)
	return Float64(sqrt((x ^ -1.0)) * 0.5)
end

MATLAB

function tmp = code(x)
	tmp = sqrt((x ^ -1.0)) * 0.5;
end

Wolfram

code[x_] := N[(N[Sqrt[N[Power[x, -1.0], $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]

Derivation

1. Initial program 7.6%

   \[\sqrt{x + 1} - \sqrt{x} \]
2. Add Preprocessing

3. Taylor expanded in x around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower-sqrt.f64 N/A

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

   4. lower-/.f64 97.2

      \[\leadsto \sqrt{\color{blue}{\frac{1}{x}}} \cdot 0.5 \]

5. Applied rewrites 97.2%

   \[\leadsto \color{blue}{\sqrt{\frac{1}{x}} \cdot 0.5} \]

6. Final simplification 97.2%

   \[\leadsto \sqrt{{x}^{-1}} \cdot 0.5 \]
7. Add Preprocessing

Alternative 4: 97.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \frac{0.5}{\sqrt{x}} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ 0.5 (sqrt x)))

C

double code(double x) {
	return 0.5 / sqrt(x);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 0.5d0 / sqrt(x)
end function

Java

public static double code(double x) {
	return 0.5 / Math.sqrt(x);
}

Python

def code(x):
	return 0.5 / math.sqrt(x)

Julia

function code(x)
	return Float64(0.5 / sqrt(x))
end

MATLAB

function tmp = code(x)
	tmp = 0.5 / sqrt(x);
end

Wolfram

code[x_] := N[(0.5 / N[Sqrt[x], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 7.6%

   \[\sqrt{x + 1} - \sqrt{x} \]
2. Add Preprocessing

3. Taylor expanded in x around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower-sqrt.f64 N/A

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

   4. lower-/.f64 97.2

      \[\leadsto \sqrt{\color{blue}{\frac{1}{x}}} \cdot 0.5 \]

5. Applied rewrites 97.2%

   \[\leadsto \color{blue}{\sqrt{\frac{1}{x}} \cdot 0.5} \]
6. Step-by-step derivation

7. Applied rewrites 97.0%

   \[\leadsto \frac{0.5}{\color{blue}{\sqrt{x}}} \]
8. Add Preprocessing

Alternative 5: 4.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ 0.5 \cdot x - \sqrt{x} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (- (* 0.5 x) (sqrt x)))

C

double code(double x) {
	return (0.5 * x) - sqrt(x);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (0.5d0 * x) - sqrt(x)
end function

Java

public static double code(double x) {
	return (0.5 * x) - Math.sqrt(x);
}

Python

def code(x):
	return (0.5 * x) - math.sqrt(x)

Julia

function code(x)
	return Float64(Float64(0.5 * x) - sqrt(x))
end

MATLAB

function tmp = code(x)
	tmp = (0.5 * x) - sqrt(x);
end

Wolfram

code[x_] := N[(N[(0.5 * x), $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 7.6%

   \[\sqrt{x + 1} - \sqrt{x} \]
2. Add Preprocessing

3. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. lower-fma.f64 4.5

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.5, x, 1\right)} - \sqrt{x} \]

5. Applied rewrites 4.5%

   \[\leadsto \color{blue}{\mathsf{fma}\left(0.5, x, 1\right)} - \sqrt{x} \]

6. Taylor expanded in x around inf

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

8. Applied rewrites 4.5%

   \[\leadsto 0.5 \cdot \color{blue}{x} - \sqrt{x} \]
9. Add Preprocessing

Alternative 6: 3.8% accurate, 27.0× speedup

Math

\[\begin{array}{l} \\ 0 \end{array} \]

FPCore

(FPCore (x) :precision binary64 0.0)

C

double code(double x) {
	return 0.0;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 0.0d0
end function

Java

public static double code(double x) {
	return 0.0;
}

Python

def code(x):
	return 0.0

Julia

function code(x)
	return 0.0
end

MATLAB

function tmp = code(x)
	tmp = 0.0;
end

Wolfram

code[x_] := 0.0

Derivation

1. Initial program 7.6%

   \[\sqrt{x + 1} - \sqrt{x} \]
2. Add Preprocessing

4. Applied rewrites 9.5%

   \[\leadsto \color{blue}{\frac{\left(1 + x\right) - x}{\mathsf{fma}\left(\sqrt{x}, x, {\left(1 + x\right)}^{1.5}\right)} \cdot \left(\left(\left(1 + x\right) + x\right) - \mathsf{hypot}\left(\sqrt{x}, x\right)\right)} \]

5. Taylor expanded in x around -inf

   \[\leadsto \color{blue}{\frac{3}{x \cdot \left(-1 \cdot \left(\sqrt{\frac{1}{x}} \cdot {\left(\sqrt{-1}\right)}^{2}\right) + \sqrt{\frac{1}{x}} \cdot {\left(\sqrt{-1}\right)}^{2}\right)}} \]
6. Step-by-step derivation

   1. associate-/r* N/A

      \[\leadsto \color{blue}{\frac{\frac{3}{x}}{-1 \cdot \left(\sqrt{\frac{1}{x}} \cdot {\left(\sqrt{-1}\right)}^{2}\right) + \sqrt{\frac{1}{x}} \cdot {\left(\sqrt{-1}\right)}^{2}}} \]

   2. distribute-lft1-in N/A

      \[\leadsto \frac{\frac{3}{x}}{\color{blue}{\left(-1 + 1\right) \cdot \left(\sqrt{\frac{1}{x}} \cdot {\left(\sqrt{-1}\right)}^{2}\right)}} \]

   3. metadata-eval N/A

      \[\leadsto \frac{\frac{3}{x}}{\color{blue}{0} \cdot \left(\sqrt{\frac{1}{x}} \cdot {\left(\sqrt{-1}\right)}^{2}\right)} \]

   4. mul0-lft N/A

      \[\leadsto \frac{\frac{3}{x}}{\color{blue}{0}} \]

   5. associate-/l/ N/A

      \[\leadsto \color{blue}{\frac{3}{0 \cdot x}} \]

   6. mul0-lft N/A

      \[\leadsto \frac{3}{\color{blue}{0}} \]

   7. metadata-eval N/A

      \[\leadsto \frac{3}{\color{blue}{\mathsf{neg}\left(0\right)}} \]

   8. distribute-neg-frac2 N/A

      \[\leadsto \color{blue}{\mathsf{neg}\left(\frac{3}{0}\right)} \]

   9. distribute-neg-frac N/A

      \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(3\right)}{0}} \]

   10. lower-/.f64 N/A

       \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(3\right)}{0}} \]

   11. metadata-eval 0.9

       \[\leadsto \frac{\color{blue}{-3}}{0} \]

7. Applied rewrites 0.9%

   \[\leadsto \color{blue}{\frac{-3}{0}} \]
8. Step-by-step derivation

9. Applied rewrites 3.8%

   \[\leadsto \color{blue}{0} \]
10. Add Preprocessing

Developer Target 1: 98.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ 0.5 \cdot {x}^{-0.5} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (* 0.5 (pow x -0.5)))

C

double code(double x) {
	return 0.5 * pow(x, -0.5);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 0.5d0 * (x ** (-0.5d0))
end function

Java

public static double code(double x) {
	return 0.5 * Math.pow(x, -0.5);
}

Python

def code(x):
	return 0.5 * math.pow(x, -0.5)

Julia

function code(x)
	return Float64(0.5 * (x ^ -0.5))
end

MATLAB

function tmp = code(x)
	tmp = 0.5 * (x ^ -0.5);
end

Wolfram

code[x_] := N[(0.5 * N[Power[x, -0.5], $MachinePrecision]), $MachinePrecision]

Reproduce

herbie shell --seed 2024314 
(FPCore (x)
  :name "2sqrt (example 3.1)"
  :precision binary64
  :pre (and (> x 1.0) (< x 1e+308))

  :alt
  (! :herbie-platform default (* 1/2 (pow x -1/2)))

  (- (sqrt (+ x 1.0)) (sqrt x)))