2isqrt (example 3.6)

Percentage Accurate: 38.5% → 98.9%

Time: 8.8s

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: 38.5% 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: 98.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{x \cdot \left(0.125 \cdot {x}^{-2.5} - {x}^{-1.5} \cdot 0.5\right)}{-0.5 - x} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/ (* x (- (* 0.125 (pow x -2.5)) (* (pow x -1.5) 0.5))) (- -0.5 x)))

C

double code(double x) {
	return (x * ((0.125 * pow(x, -2.5)) - (pow(x, -1.5) * 0.5))) / (-0.5 - x);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (x * ((0.125d0 * (x ** (-2.5d0))) - ((x ** (-1.5d0)) * 0.5d0))) / ((-0.5d0) - x)
end function

Java

public static double code(double x) {
	return (x * ((0.125 * Math.pow(x, -2.5)) - (Math.pow(x, -1.5) * 0.5))) / (-0.5 - x);
}

Python

def code(x):
	return (x * ((0.125 * math.pow(x, -2.5)) - (math.pow(x, -1.5) * 0.5))) / (-0.5 - x)

Julia

function code(x)
	return Float64(Float64(x * Float64(Float64(0.125 * (x ^ -2.5)) - Float64((x ^ -1.5) * 0.5))) / Float64(-0.5 - x))
end

MATLAB

function tmp = code(x)
	tmp = (x * ((0.125 * (x ^ -2.5)) - ((x ^ -1.5) * 0.5))) / (-0.5 - x);
end

Wolfram

code[x_] := N[(N[(x * N[(N[(0.125 * N[Power[x, -2.5], $MachinePrecision]), $MachinePrecision] - N[(N[Power[x, -1.5], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(-0.5 - x), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 37.6%

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

   1. frac-2neg 37.6%

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

   2. metadata-eval 37.6%

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

   3. frac-sub 37.6%

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

   4. *-un-lft-identity 37.6%

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

   5. +-commutative 37.6%

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

   6. +-commutative 37.6%

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

4. Applied egg-rr 37.6%

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

5. Taylor expanded in x around inf 36.9%

   \[\leadsto \frac{\color{blue}{x \cdot \left(0.125 \cdot \sqrt{\frac{1}{{x}^{5}}} - \left(\sqrt{\frac{1}{x}} + \left(-1 \cdot \sqrt{\frac{1}{x}} + 0.5 \cdot \sqrt{\frac{1}{{x}^{3}}}\right)\right)\right)}}{\sqrt{x} \cdot \left(-\sqrt{1 + x}\right)} \]
6. Step-by-step derivation

   1. associate-+r+ 67.9%

      \[\leadsto \frac{x \cdot \left(0.125 \cdot \sqrt{\frac{1}{{x}^{5}}} - \color{blue}{\left(\left(\sqrt{\frac{1}{x}} + -1 \cdot \sqrt{\frac{1}{x}}\right) + 0.5 \cdot \sqrt{\frac{1}{{x}^{3}}}\right)}\right)}{\sqrt{x} \cdot \left(-\sqrt{1 + x}\right)} \]

   2. distribute-rgt1-in 67.9%

      \[\leadsto \frac{x \cdot \left(0.125 \cdot \sqrt{\frac{1}{{x}^{5}}} - \left(\color{blue}{\left(-1 + 1\right) \cdot \sqrt{\frac{1}{x}}} + 0.5 \cdot \sqrt{\frac{1}{{x}^{3}}}\right)\right)}{\sqrt{x} \cdot \left(-\sqrt{1 + x}\right)} \]

   3. metadata-eval 67.9%

      \[\leadsto \frac{x \cdot \left(0.125 \cdot \sqrt{\frac{1}{{x}^{5}}} - \left(\color{blue}{0} \cdot \sqrt{\frac{1}{x}} + 0.5 \cdot \sqrt{\frac{1}{{x}^{3}}}\right)\right)}{\sqrt{x} \cdot \left(-\sqrt{1 + x}\right)} \]

   4. mul0-lft 67.9%

      \[\leadsto \frac{x \cdot \left(0.125 \cdot \sqrt{\frac{1}{{x}^{5}}} - \left(\color{blue}{0} + 0.5 \cdot \sqrt{\frac{1}{{x}^{3}}}\right)\right)}{\sqrt{x} \cdot \left(-\sqrt{1 + x}\right)} \]

   5. associate--r+ 67.9%

      \[\leadsto \frac{x \cdot \color{blue}{\left(\left(0.125 \cdot \sqrt{\frac{1}{{x}^{5}}} - 0\right) - 0.5 \cdot \sqrt{\frac{1}{{x}^{3}}}\right)}}{\sqrt{x} \cdot \left(-\sqrt{1 + x}\right)} \]

7. Simplified 67.9%

   \[\leadsto \frac{\color{blue}{x \cdot \left(\left(0.125 \cdot {x}^{-2.5} - 0\right) - \sqrt{\frac{1}{{x}^{3}}} \cdot 0.5\right)}}{\sqrt{x} \cdot \left(-\sqrt{1 + x}\right)} \]
8. Step-by-step derivation

   1. *-un-lft-identity 67.9%

      \[\leadsto \frac{x \cdot \left(\left(0.125 \cdot {x}^{-2.5} - 0\right) - \color{blue}{\left(1 \cdot \sqrt{\frac{1}{{x}^{3}}}\right)} \cdot 0.5\right)}{\sqrt{x} \cdot \left(-\sqrt{1 + x}\right)} \]

   2. pow-flip 69.2%

      \[\leadsto \frac{x \cdot \left(\left(0.125 \cdot {x}^{-2.5} - 0\right) - \left(1 \cdot \sqrt{\color{blue}{{x}^{\left(-3\right)}}}\right) \cdot 0.5\right)}{\sqrt{x} \cdot \left(-\sqrt{1 + x}\right)} \]

   3. sqrt-pow1 98.7%

      \[\leadsto \frac{x \cdot \left(\left(0.125 \cdot {x}^{-2.5} - 0\right) - \left(1 \cdot \color{blue}{{x}^{\left(\frac{-3}{2}\right)}}\right) \cdot 0.5\right)}{\sqrt{x} \cdot \left(-\sqrt{1 + x}\right)} \]

   4. metadata-eval 98.7%

      \[\leadsto \frac{x \cdot \left(\left(0.125 \cdot {x}^{-2.5} - 0\right) - \left(1 \cdot {x}^{\left(\frac{\color{blue}{-3}}{2}\right)}\right) \cdot 0.5\right)}{\sqrt{x} \cdot \left(-\sqrt{1 + x}\right)} \]

   5. metadata-eval 98.7%

      \[\leadsto \frac{x \cdot \left(\left(0.125 \cdot {x}^{-2.5} - 0\right) - \left(1 \cdot {x}^{\color{blue}{-1.5}}\right) \cdot 0.5\right)}{\sqrt{x} \cdot \left(-\sqrt{1 + x}\right)} \]

9. Applied egg-rr 98.7%

   \[\leadsto \frac{x \cdot \left(\left(0.125 \cdot {x}^{-2.5} - 0\right) - \color{blue}{\left(1 \cdot {x}^{-1.5}\right)} \cdot 0.5\right)}{\sqrt{x} \cdot \left(-\sqrt{1 + x}\right)} \]
10. Step-by-step derivation

    1. *-lft-identity 98.7%

       \[\leadsto \frac{x \cdot \left(\left(0.125 \cdot {x}^{-2.5} - 0\right) - \color{blue}{{x}^{-1.5}} \cdot 0.5\right)}{\sqrt{x} \cdot \left(-\sqrt{1 + x}\right)} \]

11. Simplified 98.7%

    \[\leadsto \frac{x \cdot \left(\left(0.125 \cdot {x}^{-2.5} - 0\right) - \color{blue}{{x}^{-1.5}} \cdot 0.5\right)}{\sqrt{x} \cdot \left(-\sqrt{1 + x}\right)} \]

12. Taylor expanded in x around inf 99.1%

    \[\leadsto \frac{x \cdot \left(\left(0.125 \cdot {x}^{-2.5} - 0\right) - {x}^{-1.5} \cdot 0.5\right)}{\color{blue}{-1 \cdot \left(x \cdot \left(1 + 0.5 \cdot \frac{1}{x}\right)\right)}} \]
13. Step-by-step derivation

    1. mul-1-neg 99.1%

       \[\leadsto \frac{x \cdot \left(\left(0.125 \cdot {x}^{-2.5} - 0\right) - {x}^{-1.5} \cdot 0.5\right)}{\color{blue}{-x \cdot \left(1 + 0.5 \cdot \frac{1}{x}\right)}} \]

    2. distribute-lft-in 99.1%

       \[\leadsto \frac{x \cdot \left(\left(0.125 \cdot {x}^{-2.5} - 0\right) - {x}^{-1.5} \cdot 0.5\right)}{-\color{blue}{\left(x \cdot 1 + x \cdot \left(0.5 \cdot \frac{1}{x}\right)\right)}} \]

    3. *-rgt-identity 99.1%

       \[\leadsto \frac{x \cdot \left(\left(0.125 \cdot {x}^{-2.5} - 0\right) - {x}^{-1.5} \cdot 0.5\right)}{-\left(\color{blue}{x} + x \cdot \left(0.5 \cdot \frac{1}{x}\right)\right)} \]

    4. distribute-neg-in 99.1%

       \[\leadsto \frac{x \cdot \left(\left(0.125 \cdot {x}^{-2.5} - 0\right) - {x}^{-1.5} \cdot 0.5\right)}{\color{blue}{\left(-x\right) + \left(-x \cdot \left(0.5 \cdot \frac{1}{x}\right)\right)}} \]

    5. *-commutative 99.1%

       \[\leadsto \frac{x \cdot \left(\left(0.125 \cdot {x}^{-2.5} - 0\right) - {x}^{-1.5} \cdot 0.5\right)}{\left(-x\right) + \left(-x \cdot \color{blue}{\left(\frac{1}{x} \cdot 0.5\right)}\right)} \]

    6. associate-*r* 99.1%

       \[\leadsto \frac{x \cdot \left(\left(0.125 \cdot {x}^{-2.5} - 0\right) - {x}^{-1.5} \cdot 0.5\right)}{\left(-x\right) + \left(-\color{blue}{\left(x \cdot \frac{1}{x}\right) \cdot 0.5}\right)} \]

    7. rgt-mult-inverse 99.1%

       \[\leadsto \frac{x \cdot \left(\left(0.125 \cdot {x}^{-2.5} - 0\right) - {x}^{-1.5} \cdot 0.5\right)}{\left(-x\right) + \left(-\color{blue}{1} \cdot 0.5\right)} \]

    8. metadata-eval 99.1%

       \[\leadsto \frac{x \cdot \left(\left(0.125 \cdot {x}^{-2.5} - 0\right) - {x}^{-1.5} \cdot 0.5\right)}{\left(-x\right) + \left(-\color{blue}{0.5}\right)} \]

    9. metadata-eval 99.1%

       \[\leadsto \frac{x \cdot \left(\left(0.125 \cdot {x}^{-2.5} - 0\right) - {x}^{-1.5} \cdot 0.5\right)}{\left(-x\right) + \color{blue}{-0.5}} \]

14. Simplified 99.1%

    \[\leadsto \frac{x \cdot \left(\left(0.125 \cdot {x}^{-2.5} - 0\right) - {x}^{-1.5} \cdot 0.5\right)}{\color{blue}{\left(-x\right) + -0.5}} \]

15. Final simplification 99.1%

    \[\leadsto \frac{x \cdot \left(0.125 \cdot {x}^{-2.5} - {x}^{-1.5} \cdot 0.5\right)}{-0.5 - x} \]
16. Add Preprocessing

Alternative 2: 97.8% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 37.6%

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

3. Taylor expanded in x around inf 67.2%

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

   1. *-un-lft-identity 67.9%

      \[\leadsto \frac{x \cdot \left(\left(0.125 \cdot {x}^{-2.5} - 0\right) - \color{blue}{\left(1 \cdot \sqrt{\frac{1}{{x}^{3}}}\right)} \cdot 0.5\right)}{\sqrt{x} \cdot \left(-\sqrt{1 + x}\right)} \]

   2. pow-flip 69.2%

      \[\leadsto \frac{x \cdot \left(\left(0.125 \cdot {x}^{-2.5} - 0\right) - \left(1 \cdot \sqrt{\color{blue}{{x}^{\left(-3\right)}}}\right) \cdot 0.5\right)}{\sqrt{x} \cdot \left(-\sqrt{1 + x}\right)} \]

   3. sqrt-pow1 98.7%

      \[\leadsto \frac{x \cdot \left(\left(0.125 \cdot {x}^{-2.5} - 0\right) - \left(1 \cdot \color{blue}{{x}^{\left(\frac{-3}{2}\right)}}\right) \cdot 0.5\right)}{\sqrt{x} \cdot \left(-\sqrt{1 + x}\right)} \]

   4. metadata-eval 98.7%

      \[\leadsto \frac{x \cdot \left(\left(0.125 \cdot {x}^{-2.5} - 0\right) - \left(1 \cdot {x}^{\left(\frac{\color{blue}{-3}}{2}\right)}\right) \cdot 0.5\right)}{\sqrt{x} \cdot \left(-\sqrt{1 + x}\right)} \]

   5. metadata-eval 98.7%

      \[\leadsto \frac{x \cdot \left(\left(0.125 \cdot {x}^{-2.5} - 0\right) - \left(1 \cdot {x}^{\color{blue}{-1.5}}\right) \cdot 0.5\right)}{\sqrt{x} \cdot \left(-\sqrt{1 + x}\right)} \]

5. Applied egg-rr 98.3%

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

   1. *-lft-identity 98.7%

      \[\leadsto \frac{x \cdot \left(\left(0.125 \cdot {x}^{-2.5} - 0\right) - \color{blue}{{x}^{-1.5}} \cdot 0.5\right)}{\sqrt{x} \cdot \left(-\sqrt{1 + x}\right)} \]

7. Simplified 98.3%

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

8. Final simplification 98.3%

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

Alternative 3: 35.4% accurate, 209.0× 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 37.6%

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

   1. frac-2neg 37.6%

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

   2. metadata-eval 37.6%

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

   3. frac-sub 37.6%

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

   4. *-un-lft-identity 37.6%

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

   5. +-commutative 37.6%

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

   6. +-commutative 37.6%

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

4. Applied egg-rr 37.6%

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

5. Taylor expanded in x around inf 35.2%

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

   1. distribute-rgt1-in 35.2%

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

   2. metadata-eval 35.2%

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

   3. mul0-lft 35.2%

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

7. Simplified 35.2%

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

Developer Target 1: 98.5% 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 2024123 
(FPCore (x)
  :name "2isqrt (example 3.6)"
  :precision binary64
  :pre (and (> x 1.0) (< x 1e+308))

  :alt
  (! :herbie-platform default (/ 1 (+ (* (+ x 1) (sqrt x)) (* x (sqrt (+ x 1))))))

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