2isqrt (example 3.6)

Percentage Accurate: 38.4% → 98.4%

Time: 10.8s

Alternatives: 3

Speedup: 2.0×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 38.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\left(1 + x\right) \cdot \left(\sqrt{x} + x \cdot {\left(1 + x\right)}^{-0.5}\right)} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 40.0%

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

   1. flip-- 40.1%

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

   2. clear-num 40.1%

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

   3. inv-pow 40.1%

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

   4. sqrt-pow2 40.1%

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

   5. metadata-eval 40.1%

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

   6. pow1/2 40.1%

      \[\leadsto \frac{1}{\frac{{x}^{-0.5} + \frac{1}{\color{blue}{{\left(x + 1\right)}^{0.5}}}}{\frac{1}{\sqrt{x}} \cdot \frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}} \cdot \frac{1}{\sqrt{x + 1}}}} \]

   7. pow-flip 40.1%

      \[\leadsto \frac{1}{\frac{{x}^{-0.5} + \color{blue}{{\left(x + 1\right)}^{\left(-0.5\right)}}}{\frac{1}{\sqrt{x}} \cdot \frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}} \cdot \frac{1}{\sqrt{x + 1}}}} \]

   8. +-commutative 40.1%

      \[\leadsto \frac{1}{\frac{{x}^{-0.5} + {\color{blue}{\left(1 + x\right)}}^{\left(-0.5\right)}}{\frac{1}{\sqrt{x}} \cdot \frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}} \cdot \frac{1}{\sqrt{x + 1}}}} \]

   9. metadata-eval 40.1%

      \[\leadsto \frac{1}{\frac{{x}^{-0.5} + {\left(1 + x\right)}^{\color{blue}{-0.5}}}{\frac{1}{\sqrt{x}} \cdot \frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}} \cdot \frac{1}{\sqrt{x + 1}}}} \]

   10. frac-times 22.4%

       \[\leadsto \frac{1}{\frac{{x}^{-0.5} + {\left(1 + x\right)}^{-0.5}}{\color{blue}{\frac{1 \cdot 1}{\sqrt{x} \cdot \sqrt{x}}} - \frac{1}{\sqrt{x + 1}} \cdot \frac{1}{\sqrt{x + 1}}}} \]

   11. metadata-eval 22.4%

       \[\leadsto \frac{1}{\frac{{x}^{-0.5} + {\left(1 + x\right)}^{-0.5}}{\frac{\color{blue}{1}}{\sqrt{x} \cdot \sqrt{x}} - \frac{1}{\sqrt{x + 1}} \cdot \frac{1}{\sqrt{x + 1}}}} \]

   12. add-sqr-sqrt 18.0%

       \[\leadsto \frac{1}{\frac{{x}^{-0.5} + {\left(1 + x\right)}^{-0.5}}{\frac{1}{\color{blue}{x}} - \frac{1}{\sqrt{x + 1}} \cdot \frac{1}{\sqrt{x + 1}}}} \]

   13. frac-times 25.8%

       \[\leadsto \frac{1}{\frac{{x}^{-0.5} + {\left(1 + x\right)}^{-0.5}}{\frac{1}{x} - \color{blue}{\frac{1 \cdot 1}{\sqrt{x + 1} \cdot \sqrt{x + 1}}}}} \]

   14. metadata-eval 25.8%

       \[\leadsto \frac{1}{\frac{{x}^{-0.5} + {\left(1 + x\right)}^{-0.5}}{\frac{1}{x} - \frac{\color{blue}{1}}{\sqrt{x + 1} \cdot \sqrt{x + 1}}}} \]

   15. add-sqr-sqrt 40.1%

       \[\leadsto \frac{1}{\frac{{x}^{-0.5} + {\left(1 + x\right)}^{-0.5}}{\frac{1}{x} - \frac{1}{\color{blue}{x + 1}}}} \]

4. Applied egg-rr 40.1%

   \[\leadsto \color{blue}{\frac{1}{\frac{{x}^{-0.5} + {\left(1 + x\right)}^{-0.5}}{\frac{1}{x} - \frac{1}{1 + x}}}} \]
5. Step-by-step derivation

   1. frac-sub 40.8%

      \[\leadsto \frac{1}{\frac{{x}^{-0.5} + {\left(1 + x\right)}^{-0.5}}{\color{blue}{\frac{1 \cdot \left(1 + x\right) - x \cdot 1}{x \cdot \left(1 + x\right)}}}} \]

   2. *-un-lft-identity 40.8%

      \[\leadsto \frac{1}{\frac{{x}^{-0.5} + {\left(1 + x\right)}^{-0.5}}{\frac{\color{blue}{\left(1 + x\right)} - x \cdot 1}{x \cdot \left(1 + x\right)}}} \]

6. Applied egg-rr 40.8%

   \[\leadsto \frac{1}{\frac{{x}^{-0.5} + {\left(1 + x\right)}^{-0.5}}{\color{blue}{\frac{\left(1 + x\right) - x \cdot 1}{x \cdot \left(1 + x\right)}}}} \]
7. Step-by-step derivation

   1. *-rgt-identity 40.8%

      \[\leadsto \frac{1}{\frac{{x}^{-0.5} + {\left(1 + x\right)}^{-0.5}}{\frac{\left(1 + x\right) - \color{blue}{x}}{x \cdot \left(1 + x\right)}}} \]

   2. associate--l+ 85.3%

      \[\leadsto \frac{1}{\frac{{x}^{-0.5} + {\left(1 + x\right)}^{-0.5}}{\frac{\color{blue}{1 + \left(x - x\right)}}{x \cdot \left(1 + x\right)}}} \]

   3. +-inverses 85.3%

      \[\leadsto \frac{1}{\frac{{x}^{-0.5} + {\left(1 + x\right)}^{-0.5}}{\frac{1 + \color{blue}{0}}{x \cdot \left(1 + x\right)}}} \]

   4. metadata-eval 85.3%

      \[\leadsto \frac{1}{\frac{{x}^{-0.5} + {\left(1 + x\right)}^{-0.5}}{\frac{\color{blue}{1}}{x \cdot \left(1 + x\right)}}} \]

8. Simplified 85.3%

   \[\leadsto \frac{1}{\frac{{x}^{-0.5} + {\left(1 + x\right)}^{-0.5}}{\color{blue}{\frac{1}{x \cdot \left(1 + x\right)}}}} \]
9. Step-by-step derivation

   1. associate-/r/ 85.3%

      \[\leadsto \frac{1}{\color{blue}{\frac{{x}^{-0.5} + {\left(1 + x\right)}^{-0.5}}{1} \cdot \left(x \cdot \left(1 + x\right)\right)}} \]

   2. /-rgt-identity 85.3%

      \[\leadsto \frac{1}{\color{blue}{\left({x}^{-0.5} + {\left(1 + x\right)}^{-0.5}\right)} \cdot \left(x \cdot \left(1 + x\right)\right)} \]

   3. distribute-rgt-in 85.3%

      \[\leadsto \frac{1}{\left({x}^{-0.5} + {\left(1 + x\right)}^{-0.5}\right) \cdot \color{blue}{\left(1 \cdot x + x \cdot x\right)}} \]

   4. *-un-lft-identity 85.3%

      \[\leadsto \frac{1}{\left({x}^{-0.5} + {\left(1 + x\right)}^{-0.5}\right) \cdot \left(\color{blue}{x} + x \cdot x\right)} \]

   5. distribute-lft-in 85.3%

      \[\leadsto \frac{1}{\color{blue}{\left({x}^{-0.5} + {\left(1 + x\right)}^{-0.5}\right) \cdot x + \left({x}^{-0.5} + {\left(1 + x\right)}^{-0.5}\right) \cdot \left(x \cdot x\right)}} \]

   6. +-commutative 85.3%

      \[\leadsto \frac{1}{\left({x}^{-0.5} + {\color{blue}{\left(x + 1\right)}}^{-0.5}\right) \cdot x + \left({x}^{-0.5} + {\left(1 + x\right)}^{-0.5}\right) \cdot \left(x \cdot x\right)} \]

   7. +-commutative 85.3%

      \[\leadsto \frac{1}{\left({x}^{-0.5} + {\left(x + 1\right)}^{-0.5}\right) \cdot x + \left({x}^{-0.5} + {\color{blue}{\left(x + 1\right)}}^{-0.5}\right) \cdot \left(x \cdot x\right)} \]

   8. pow2 85.3%

      \[\leadsto \frac{1}{\left({x}^{-0.5} + {\left(x + 1\right)}^{-0.5}\right) \cdot x + \left({x}^{-0.5} + {\left(x + 1\right)}^{-0.5}\right) \cdot \color{blue}{{x}^{2}}} \]

10. Applied egg-rr 85.3%

    \[\leadsto \frac{1}{\color{blue}{\left({x}^{-0.5} + {\left(x + 1\right)}^{-0.5}\right) \cdot x + \left({x}^{-0.5} + {\left(x + 1\right)}^{-0.5}\right) \cdot {x}^{2}}} \]
11. Step-by-step derivation

    1. distribute-lft-in 85.3%

       \[\leadsto \frac{1}{\color{blue}{\left({x}^{-0.5} + {\left(x + 1\right)}^{-0.5}\right) \cdot \left(x + {x}^{2}\right)}} \]

    2. *-commutative 85.3%

       \[\leadsto \frac{1}{\color{blue}{\left(x + {x}^{2}\right) \cdot \left({x}^{-0.5} + {\left(x + 1\right)}^{-0.5}\right)}} \]

    3. unpow2 85.3%

       \[\leadsto \frac{1}{\left(x + \color{blue}{x \cdot x}\right) \cdot \left({x}^{-0.5} + {\left(x + 1\right)}^{-0.5}\right)} \]

    4. distribute-rgt1-in 85.3%

       \[\leadsto \frac{1}{\color{blue}{\left(\left(x + 1\right) \cdot x\right)} \cdot \left({x}^{-0.5} + {\left(x + 1\right)}^{-0.5}\right)} \]

    5. associate-*r* 98.8%

       \[\leadsto \frac{1}{\color{blue}{\left(x + 1\right) \cdot \left(x \cdot \left({x}^{-0.5} + {\left(x + 1\right)}^{-0.5}\right)\right)}} \]

    6. distribute-rgt-in 98.8%

       \[\leadsto \frac{1}{\left(x + 1\right) \cdot \color{blue}{\left({x}^{-0.5} \cdot x + {\left(x + 1\right)}^{-0.5} \cdot x\right)}} \]

    7. +-commutative 98.8%

       \[\leadsto \frac{1}{\color{blue}{\left(1 + x\right)} \cdot \left({x}^{-0.5} \cdot x + {\left(x + 1\right)}^{-0.5} \cdot x\right)} \]

    8. pow-plus 98.9%

       \[\leadsto \frac{1}{\left(1 + x\right) \cdot \left(\color{blue}{{x}^{\left(-0.5 + 1\right)}} + {\left(x + 1\right)}^{-0.5} \cdot x\right)} \]

    9. metadata-eval 98.9%

       \[\leadsto \frac{1}{\left(1 + x\right) \cdot \left({x}^{\color{blue}{0.5}} + {\left(x + 1\right)}^{-0.5} \cdot x\right)} \]

    10. unpow1/2 98.9%

        \[\leadsto \frac{1}{\left(1 + x\right) \cdot \left(\color{blue}{\sqrt{x}} + {\left(x + 1\right)}^{-0.5} \cdot x\right)} \]

    11. *-commutative 98.9%

        \[\leadsto \frac{1}{\left(1 + x\right) \cdot \left(\sqrt{x} + \color{blue}{x \cdot {\left(x + 1\right)}^{-0.5}}\right)} \]

    12. +-commutative 98.9%

        \[\leadsto \frac{1}{\left(1 + x\right) \cdot \left(\sqrt{x} + x \cdot {\color{blue}{\left(1 + x\right)}}^{-0.5}\right)} \]

12. Simplified 98.9%

    \[\leadsto \frac{1}{\color{blue}{\left(1 + x\right) \cdot \left(\sqrt{x} + x \cdot {\left(1 + x\right)}^{-0.5}\right)}} \]

13. Final simplification 98.9%

    \[\leadsto \frac{1}{\left(1 + x\right) \cdot \left(\sqrt{x} + x \cdot {\left(1 + x\right)}^{-0.5}\right)} \]
14. Add Preprocessing

Alternative 2: 97.8% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ 0.5 \cdot {x}^{-1.5} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (* 0.5 (pow x -1.5)))

C

double code(double x) {
	return 0.5 * pow(x, -1.5);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 0.5d0 * (x ** (-1.5d0))
end function

Java

public static double code(double x) {
	return 0.5 * Math.pow(x, -1.5);
}

Python

def code(x):
	return 0.5 * math.pow(x, -1.5)

Julia

function code(x)
	return Float64(0.5 * (x ^ -1.5))
end

MATLAB

function tmp = code(x)
	tmp = 0.5 * (x ^ -1.5);
end

Wolfram

code[x_] := N[(0.5 * N[Power[x, -1.5], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 40.0%

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

3. Taylor expanded in x around inf 67.3%

   \[\leadsto \color{blue}{0.5 \cdot \sqrt{\frac{1}{{x}^{3}}}} \]
4. Step-by-step derivation

   1. pow1 67.3%

      \[\leadsto \color{blue}{{\left(0.5 \cdot \sqrt{\frac{1}{{x}^{3}}}\right)}^{1}} \]

   2. *-commutative 67.3%

      \[\leadsto {\color{blue}{\left(\sqrt{\frac{1}{{x}^{3}}} \cdot 0.5\right)}}^{1} \]

   3. pow-flip 68.8%

      \[\leadsto {\left(\sqrt{\color{blue}{{x}^{\left(-3\right)}}} \cdot 0.5\right)}^{1} \]

   4. sqrt-pow1 98.0%

      \[\leadsto {\left(\color{blue}{{x}^{\left(\frac{-3}{2}\right)}} \cdot 0.5\right)}^{1} \]

   5. metadata-eval 98.0%

      \[\leadsto {\left({x}^{\left(\frac{\color{blue}{-3}}{2}\right)} \cdot 0.5\right)}^{1} \]

   6. metadata-eval 98.0%

      \[\leadsto {\left({x}^{\color{blue}{-1.5}} \cdot 0.5\right)}^{1} \]

5. Applied egg-rr 98.0%

   \[\leadsto \color{blue}{{\left({x}^{-1.5} \cdot 0.5\right)}^{1}} \]
6. Step-by-step derivation

   1. unpow1 98.0%

      \[\leadsto \color{blue}{{x}^{-1.5} \cdot 0.5} \]

   2. *-commutative 98.0%

      \[\leadsto \color{blue}{0.5 \cdot {x}^{-1.5}} \]

7. Simplified 98.0%

   \[\leadsto \color{blue}{0.5 \cdot {x}^{-1.5}} \]

8. Final simplification 98.0%

   \[\leadsto 0.5 \cdot {x}^{-1.5} \]
9. Add Preprocessing

Alternative 3: 5.6% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

function code(x)
	return x ^ -0.5
end

MATLAB

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

Wolfram

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

Derivation

1. Initial program 40.0%

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

3. Taylor expanded in x around 0 5.6%

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

   1. inv-pow 5.6%

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

   2. sqrt-pow1 5.6%

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

   3. metadata-eval 5.6%

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

   4. *-un-lft-identity 5.6%

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

5. Applied egg-rr 5.6%

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

   1. *-lft-identity 5.6%

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

7. Simplified 5.6%

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

8. Final simplification 5.6%

   \[\leadsto {x}^{-0.5} \]
9. Add Preprocessing

Developer target: 98.4% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (/ 1.0 (+ (* (+ x 1.0) (sqrt x)) (* x (sqrt (+ x 1.0))))))

C

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

Fortran

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

Java

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

Python

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

Julia

function code(x)
	return Float64(1.0 / Float64(Float64(Float64(x + 1.0) * sqrt(x)) + Float64(x * sqrt(Float64(x + 1.0)))))
end

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024095 
(FPCore (x)
  :name "2isqrt (example 3.6)"
  :precision binary64
  :pre (and (> x 1.0) (< x 1e+308))

  :alt
  (/ 1.0 (+ (* (+ x 1.0) (sqrt x)) (* x (sqrt (+ x 1.0)))))

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