2isqrt (example 3.6)

Percentage Accurate: 38.2% → 98.7%

Time: 13.7s

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: 38.2% 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: 98.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(-0.75, \frac{0.5 \cdot {x}^{-1.5}}{x}, \frac{1}{x} \cdot \left(0.5 \cdot \sqrt{\frac{1}{x}}\right)\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (fma -0.75 (/ (* 0.5 (pow x -1.5)) x) (* (/ 1.0 x) (* 0.5 (sqrt (/ 1.0 x))))))

C

double code(double x) {
	return fma(-0.75, ((0.5 * pow(x, -1.5)) / x), ((1.0 / x) * (0.5 * sqrt((1.0 / x)))));
}

Julia

function code(x)
	return fma(-0.75, Float64(Float64(0.5 * (x ^ -1.5)) / x), Float64(Float64(1.0 / x) * Float64(0.5 * sqrt(Float64(1.0 / x)))))
end

Wolfram

code[x_] := N[(-0.75 * N[(N[(0.5 * N[Power[x, -1.5], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] + N[(N[(1.0 / x), $MachinePrecision] * N[(0.5 * N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 33.4%

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

   1. add-exp-log 33.4%

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

   2. inv-pow 33.4%

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

   3. sqrt-pow2 24.2%

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

   4. metadata-eval 24.2%

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

   5. pow1/2 24.2%

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

   6. pow-flip 33.4%

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

   7. +-commutative 33.4%

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

   8. metadata-eval 33.4%

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

4. Applied egg-rr 33.4%

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

5. Taylor expanded in x around inf 93.0%

   \[\leadsto e^{\color{blue}{\left(\log \left(--0.5 \cdot \sqrt{x}\right) + \left(0.5 \cdot \frac{2 \cdot \left(1 + 0.25 \cdot x\right) - 1}{{x}^{2}} + 2 \cdot \log \left(\frac{1}{x}\right)\right)\right) - \frac{1}{x}}} \]

6. Taylor expanded in x around inf 93.4%

   \[\leadsto \color{blue}{e^{\log \left(--0.5 \cdot \sqrt{\frac{1}{x}}\right) + \left(-1 \cdot \log \left(\frac{1}{x}\right) + 2 \cdot \log \left(\frac{1}{x}\right)\right)} + -0.75 \cdot \frac{e^{\log \left(--0.5 \cdot \sqrt{\frac{1}{x}}\right) + \left(-1 \cdot \log \left(\frac{1}{x}\right) + 2 \cdot \log \left(\frac{1}{x}\right)\right)}}{x}} \]
7. Step-by-step derivation

   1. +-commutative 93.4%

      \[\leadsto \color{blue}{-0.75 \cdot \frac{e^{\log \left(--0.5 \cdot \sqrt{\frac{1}{x}}\right) + \left(-1 \cdot \log \left(\frac{1}{x}\right) + 2 \cdot \log \left(\frac{1}{x}\right)\right)}}{x} + e^{\log \left(--0.5 \cdot \sqrt{\frac{1}{x}}\right) + \left(-1 \cdot \log \left(\frac{1}{x}\right) + 2 \cdot \log \left(\frac{1}{x}\right)\right)}} \]

   2. fma-define 93.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-0.75, \frac{e^{\log \left(--0.5 \cdot \sqrt{\frac{1}{x}}\right) + \left(-1 \cdot \log \left(\frac{1}{x}\right) + 2 \cdot \log \left(\frac{1}{x}\right)\right)}}{x}, e^{\log \left(--0.5 \cdot \sqrt{\frac{1}{x}}\right) + \left(-1 \cdot \log \left(\frac{1}{x}\right) + 2 \cdot \log \left(\frac{1}{x}\right)\right)}\right)} \]

8. Simplified 99.5%

   \[\leadsto \color{blue}{\mathsf{fma}\left(-0.75, \left(0.5 \cdot \sqrt{\frac{1}{x}}\right) \cdot \frac{1}{{x}^{2}}, \frac{1}{x} \cdot \left(0.5 \cdot \sqrt{\frac{1}{x}}\right)\right)} \]
9. Step-by-step derivation

   1. un-div-inv 99.5%

      \[\leadsto \mathsf{fma}\left(-0.75, \color{blue}{\frac{0.5 \cdot \sqrt{\frac{1}{x}}}{{x}^{2}}}, \frac{1}{x} \cdot \left(0.5 \cdot \sqrt{\frac{1}{x}}\right)\right) \]

   2. unpow2 99.5%

      \[\leadsto \mathsf{fma}\left(-0.75, \frac{0.5 \cdot \sqrt{\frac{1}{x}}}{\color{blue}{x \cdot x}}, \frac{1}{x} \cdot \left(0.5 \cdot \sqrt{\frac{1}{x}}\right)\right) \]

   3. associate-/r* 99.5%

      \[\leadsto \mathsf{fma}\left(-0.75, \color{blue}{\frac{\frac{0.5 \cdot \sqrt{\frac{1}{x}}}{x}}{x}}, \frac{1}{x} \cdot \left(0.5 \cdot \sqrt{\frac{1}{x}}\right)\right) \]

   4. div-inv 99.5%

      \[\leadsto \mathsf{fma}\left(-0.75, \frac{\color{blue}{\left(0.5 \cdot \sqrt{\frac{1}{x}}\right) \cdot \frac{1}{x}}}{x}, \frac{1}{x} \cdot \left(0.5 \cdot \sqrt{\frac{1}{x}}\right)\right) \]

   5. *-commutative 99.5%

      \[\leadsto \mathsf{fma}\left(-0.75, \frac{\color{blue}{\frac{1}{x} \cdot \left(0.5 \cdot \sqrt{\frac{1}{x}}\right)}}{x}, \frac{1}{x} \cdot \left(0.5 \cdot \sqrt{\frac{1}{x}}\right)\right) \]

   6. associate-*r* 99.5%

      \[\leadsto \mathsf{fma}\left(-0.75, \frac{\color{blue}{\left(\frac{1}{x} \cdot 0.5\right) \cdot \sqrt{\frac{1}{x}}}}{x}, \frac{1}{x} \cdot \left(0.5 \cdot \sqrt{\frac{1}{x}}\right)\right) \]

   7. *-commutative 99.5%

      \[\leadsto \mathsf{fma}\left(-0.75, \frac{\color{blue}{\left(0.5 \cdot \frac{1}{x}\right)} \cdot \sqrt{\frac{1}{x}}}{x}, \frac{1}{x} \cdot \left(0.5 \cdot \sqrt{\frac{1}{x}}\right)\right) \]

   8. inv-pow 99.5%

      \[\leadsto \mathsf{fma}\left(-0.75, \frac{\left(0.5 \cdot \frac{1}{x}\right) \cdot \sqrt{\color{blue}{{x}^{-1}}}}{x}, \frac{1}{x} \cdot \left(0.5 \cdot \sqrt{\frac{1}{x}}\right)\right) \]

   9. sqrt-pow1 99.5%

      \[\leadsto \mathsf{fma}\left(-0.75, \frac{\left(0.5 \cdot \frac{1}{x}\right) \cdot \color{blue}{{x}^{\left(\frac{-1}{2}\right)}}}{x}, \frac{1}{x} \cdot \left(0.5 \cdot \sqrt{\frac{1}{x}}\right)\right) \]

   10. metadata-eval 99.5%

       \[\leadsto \mathsf{fma}\left(-0.75, \frac{\left(0.5 \cdot \frac{1}{x}\right) \cdot {x}^{\color{blue}{-0.5}}}{x}, \frac{1}{x} \cdot \left(0.5 \cdot \sqrt{\frac{1}{x}}\right)\right) \]

   11. associate-*r* 99.5%

       \[\leadsto \mathsf{fma}\left(-0.75, \frac{\color{blue}{0.5 \cdot \left(\frac{1}{x} \cdot {x}^{-0.5}\right)}}{x}, \frac{1}{x} \cdot \left(0.5 \cdot \sqrt{\frac{1}{x}}\right)\right) \]

   12. inv-pow 99.5%

       \[\leadsto \mathsf{fma}\left(-0.75, \frac{0.5 \cdot \left(\color{blue}{{x}^{-1}} \cdot {x}^{-0.5}\right)}{x}, \frac{1}{x} \cdot \left(0.5 \cdot \sqrt{\frac{1}{x}}\right)\right) \]

   13. metadata-eval 99.5%

       \[\leadsto \mathsf{fma}\left(-0.75, \frac{0.5 \cdot \left({x}^{\color{blue}{\left(-0.5 + -0.5\right)}} \cdot {x}^{-0.5}\right)}{x}, \frac{1}{x} \cdot \left(0.5 \cdot \sqrt{\frac{1}{x}}\right)\right) \]

   14. pow-prod-up 99.5%

       \[\leadsto \mathsf{fma}\left(-0.75, \frac{0.5 \cdot \left(\color{blue}{\left({x}^{-0.5} \cdot {x}^{-0.5}\right)} \cdot {x}^{-0.5}\right)}{x}, \frac{1}{x} \cdot \left(0.5 \cdot \sqrt{\frac{1}{x}}\right)\right) \]

   15. pow3 99.5%

       \[\leadsto \mathsf{fma}\left(-0.75, \frac{0.5 \cdot \color{blue}{{\left({x}^{-0.5}\right)}^{3}}}{x}, \frac{1}{x} \cdot \left(0.5 \cdot \sqrt{\frac{1}{x}}\right)\right) \]

   16. pow-pow 99.5%

       \[\leadsto \mathsf{fma}\left(-0.75, \frac{0.5 \cdot \color{blue}{{x}^{\left(-0.5 \cdot 3\right)}}}{x}, \frac{1}{x} \cdot \left(0.5 \cdot \sqrt{\frac{1}{x}}\right)\right) \]

   17. metadata-eval 99.5%

       \[\leadsto \mathsf{fma}\left(-0.75, \frac{0.5 \cdot {x}^{\color{blue}{-1.5}}}{x}, \frac{1}{x} \cdot \left(0.5 \cdot \sqrt{\frac{1}{x}}\right)\right) \]

10. Applied egg-rr 99.5%

    \[\leadsto \mathsf{fma}\left(-0.75, \color{blue}{\frac{0.5 \cdot {x}^{-1.5}}{x}}, \frac{1}{x} \cdot \left(0.5 \cdot \sqrt{\frac{1}{x}}\right)\right) \]

11. Final simplification 99.5%

    \[\leadsto \mathsf{fma}\left(-0.75, \frac{0.5 \cdot {x}^{-1.5}}{x}, \frac{1}{x} \cdot \left(0.5 \cdot \sqrt{\frac{1}{x}}\right)\right) \]
12. Add Preprocessing

Alternative 2: 98.1% 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 33.4%

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

3. Taylor expanded in x around inf 62.5%

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

   1. *-un-lft-identity 62.5%

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

   2. pow-flip 64.4%

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

   3. sqrt-pow1 99.0%

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

   4. metadata-eval 99.0%

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

   5. metadata-eval 99.0%

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

5. Applied egg-rr 99.0%

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

   1. *-lft-identity 99.0%

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

7. Simplified 99.0%

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

8. Final simplification 99.0%

   \[\leadsto 0.5 \cdot {x}^{-1.5} \]
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 33.4%

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

   1. expm1-log1p-u 33.4%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{\sqrt{x}}\right)\right)} - \frac{1}{\sqrt{x + 1}} \]

   2. expm1-undefine 4.2%

      \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{1}{\sqrt{x}}\right)} - 1\right)} - \frac{1}{\sqrt{x + 1}} \]

   3. inv-pow 4.2%

      \[\leadsto \left(e^{\mathsf{log1p}\left(\color{blue}{{\left(\sqrt{x}\right)}^{-1}}\right)} - 1\right) - \frac{1}{\sqrt{x + 1}} \]

   4. sqrt-pow2 4.2%

      \[\leadsto \left(e^{\mathsf{log1p}\left(\color{blue}{{x}^{\left(\frac{-1}{2}\right)}}\right)} - 1\right) - \frac{1}{\sqrt{x + 1}} \]

   5. metadata-eval 4.2%

      \[\leadsto \left(e^{\mathsf{log1p}\left({x}^{\color{blue}{-0.5}}\right)} - 1\right) - \frac{1}{\sqrt{x + 1}} \]

4. Applied egg-rr 4.2%

   \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left({x}^{-0.5}\right)} - 1\right)} - \frac{1}{\sqrt{x + 1}} \]
5. Step-by-step derivation

   1. sub-neg 4.2%

      \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left({x}^{-0.5}\right)} + \left(-1\right)\right)} - \frac{1}{\sqrt{x + 1}} \]

   2. log1p-undefine 4.2%

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

   3. rem-exp-log 4.2%

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

   4. +-commutative 4.2%

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

   5. metadata-eval 4.2%

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

   6. associate-+l+ 24.6%

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

   7. metadata-eval 24.6%

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

   8. +-rgt-identity 24.6%

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

6. Simplified 24.6%

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

7. Taylor expanded in x around 0 5.6%

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

   1. unpow1/2 5.6%

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

   2. exp-to-pow 5.6%

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

   3. log-rec 5.6%

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

   4. distribute-lft-neg-out 5.6%

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

   5. distribute-rgt-neg-in 5.6%

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

   6. metadata-eval 5.6%

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

   7. exp-to-pow 5.6%

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

9. Simplified 5.6%

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

10. Final simplification 5.6%

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

Developer target: 98.3% 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 2024067 
(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)))))