2sqrt (example 3.1)

Percentage Accurate: 6.8% → 99.6%

Time: 7.4s

Alternatives: 3

Speedup: 2.0×

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: 6.8% 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, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 7.6%

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

   1. flip-- 8.4%

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

   2. div-inv 8.4%

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

   3. add-sqr-sqrt 9.0%

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

   4. add-sqr-sqrt 11.0%

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

   5. associate--l+ 11.0%

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

4. Applied egg-rr 11.0%

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

   1. associate-*r/ 11.0%

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

   2. *-rgt-identity 11.0%

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

   3. +-commutative 11.0%

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

   4. associate-+l- 99.6%

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

   5. div-sub 99.6%

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

   6. +-inverses 99.6%

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

   7. div0 99.6%

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

   8. --rgt-identity 99.6%

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

   9. +-commutative 99.6%

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

6. Simplified 99.6%

   \[\leadsto \color{blue}{\frac{1}{\sqrt{1 + x} + \sqrt{x}}} \]
7. Add Preprocessing

Alternative 2: 98.1% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 7.6%

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

   1. flip-- 8.4%

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

   2. div-inv 8.4%

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

   3. add-sqr-sqrt 9.0%

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

   4. add-sqr-sqrt 11.0%

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

   5. associate--l+ 11.0%

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

4. Applied egg-rr 11.0%

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

   1. associate-*r/ 11.0%

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

   2. *-rgt-identity 11.0%

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

   3. +-commutative 11.0%

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

   4. associate-+l- 99.6%

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

   5. div-sub 99.6%

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

   6. +-inverses 99.6%

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

   7. div0 99.6%

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

   8. --rgt-identity 99.6%

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

   9. +-commutative 99.6%

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

6. Simplified 99.6%

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

7. Taylor expanded in x around inf 97.6%

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

   1. *-commutative 97.6%

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

   2. unpow1/2 97.6%

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

   3. rem-exp-log 90.5%

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

   4. exp-neg 90.5%

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

   5. exp-prod 90.5%

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

   6. distribute-lft-neg-out 90.5%

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

   7. distribute-rgt-neg-in 90.5%

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

   8. metadata-eval 90.5%

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

   9. exp-to-pow 97.8%

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

9. Simplified 97.8%

   \[\leadsto \color{blue}{{x}^{-0.5} \cdot 0.5} \]
10. Add Preprocessing

Alternative 3: 5.4% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \sqrt{x \cdot 0.140625} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (sqrt (* x 0.140625)))

C

double code(double x) {
	return sqrt((x * 0.140625));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = sqrt((x * 0.140625d0))
end function

Java

public static double code(double x) {
	return Math.sqrt((x * 0.140625));
}

Python

def code(x):
	return math.sqrt((x * 0.140625))

Julia

function code(x)
	return sqrt(Float64(x * 0.140625))
end

MATLAB

function tmp = code(x)
	tmp = sqrt((x * 0.140625));
end

Wolfram

code[x_] := N[Sqrt[N[(x * 0.140625), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 7.6%

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

3. Taylor expanded in x around inf 99.1%

   \[\leadsto \color{blue}{\frac{-0.125 \cdot \sqrt{\frac{1}{x}} + 0.5 \cdot \sqrt{x}}{x}} \]
4. Step-by-step derivation

   1. *-un-lft-identity 99.1%

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

   2. inv-pow 99.1%

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

   3. sqrt-pow1 99.1%

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

   4. metadata-eval 99.1%

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

5. Applied egg-rr 99.1%

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

   1. *-lft-identity 99.1%

      \[\leadsto \frac{-0.125 \cdot \color{blue}{{x}^{-0.5}} + 0.5 \cdot \sqrt{x}}{x} \]

7. Simplified 99.1%

   \[\leadsto \frac{-0.125 \cdot \color{blue}{{x}^{-0.5}} + 0.5 \cdot \sqrt{x}}{x} \]
8. Step-by-step derivation

   1. *-un-lft-identity 99.1%

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

   2. add-sqr-sqrt 99.0%

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

   3. times-frac 99.1%

      \[\leadsto \color{blue}{\frac{1}{\sqrt{x}} \cdot \frac{-0.125 \cdot {x}^{-0.5} + 0.5 \cdot \sqrt{x}}{\sqrt{x}}} \]

   4. pow1/2 99.1%

      \[\leadsto \frac{1}{\color{blue}{{x}^{0.5}}} \cdot \frac{-0.125 \cdot {x}^{-0.5} + 0.5 \cdot \sqrt{x}}{\sqrt{x}} \]

   5. pow-to-exp 92.1%

      \[\leadsto \frac{1}{\color{blue}{e^{\log x \cdot 0.5}}} \cdot \frac{-0.125 \cdot {x}^{-0.5} + 0.5 \cdot \sqrt{x}}{\sqrt{x}} \]

   6. exp-neg 92.1%

      \[\leadsto \color{blue}{e^{-\log x \cdot 0.5}} \cdot \frac{-0.125 \cdot {x}^{-0.5} + 0.5 \cdot \sqrt{x}}{\sqrt{x}} \]

   7. distribute-lft-neg-out 92.1%

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

   8. exp-prod 92.1%

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

   9. unpow1/2 92.1%

      \[\leadsto \color{blue}{\sqrt{e^{-\log x}}} \cdot \frac{-0.125 \cdot {x}^{-0.5} + 0.5 \cdot \sqrt{x}}{\sqrt{x}} \]

   10. add-sqr-sqrt 0.0%

       \[\leadsto \sqrt{e^{\color{blue}{\sqrt{-\log x} \cdot \sqrt{-\log x}}}} \cdot \frac{-0.125 \cdot {x}^{-0.5} + 0.5 \cdot \sqrt{x}}{\sqrt{x}} \]

   11. sqrt-unprod 5.4%

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

   12. sqr-neg 5.4%

       \[\leadsto \sqrt{e^{\sqrt{\color{blue}{\log x \cdot \log x}}}} \cdot \frac{-0.125 \cdot {x}^{-0.5} + 0.5 \cdot \sqrt{x}}{\sqrt{x}} \]

   13. sqrt-unprod 5.4%

       \[\leadsto \sqrt{e^{\color{blue}{\sqrt{\log x} \cdot \sqrt{\log x}}}} \cdot \frac{-0.125 \cdot {x}^{-0.5} + 0.5 \cdot \sqrt{x}}{\sqrt{x}} \]

   14. add-sqr-sqrt 5.4%

       \[\leadsto \sqrt{e^{\color{blue}{\log x}}} \cdot \frac{-0.125 \cdot {x}^{-0.5} + 0.5 \cdot \sqrt{x}}{\sqrt{x}} \]

   15. add-exp-log 5.4%

       \[\leadsto \sqrt{\color{blue}{x}} \cdot \frac{-0.125 \cdot {x}^{-0.5} + 0.5 \cdot \sqrt{x}}{\sqrt{x}} \]

9. Applied egg-rr 5.4%

   \[\leadsto \color{blue}{\sqrt{x} \cdot \frac{\mathsf{fma}\left(0.5, \sqrt{x}, -0.125 \cdot \sqrt{x}\right)}{\sqrt{x}}} \]
10. Step-by-step derivation

    1. associate-*r/ 5.4%

       \[\leadsto \color{blue}{\frac{\sqrt{x} \cdot \mathsf{fma}\left(0.5, \sqrt{x}, -0.125 \cdot \sqrt{x}\right)}{\sqrt{x}}} \]

    2. *-rgt-identity 5.4%

       \[\leadsto \frac{\sqrt{x} \cdot \mathsf{fma}\left(0.5, \sqrt{x}, -0.125 \cdot \sqrt{x}\right)}{\color{blue}{\sqrt{x} \cdot 1}} \]

    3. times-frac 5.4%

       \[\leadsto \color{blue}{\frac{\sqrt{x}}{\sqrt{x}} \cdot \frac{\mathsf{fma}\left(0.5, \sqrt{x}, -0.125 \cdot \sqrt{x}\right)}{1}} \]

    4. *-inverses 5.4%

       \[\leadsto \color{blue}{1} \cdot \frac{\mathsf{fma}\left(0.5, \sqrt{x}, -0.125 \cdot \sqrt{x}\right)}{1} \]

    5. associate-*r/ 5.4%

       \[\leadsto \color{blue}{\frac{1 \cdot \mathsf{fma}\left(0.5, \sqrt{x}, -0.125 \cdot \sqrt{x}\right)}{1}} \]

    6. *-lft-identity 5.4%

       \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(0.5, \sqrt{x}, -0.125 \cdot \sqrt{x}\right)}}{1} \]

    7. /-rgt-identity 5.4%

       \[\leadsto \color{blue}{\mathsf{fma}\left(0.5, \sqrt{x}, -0.125 \cdot \sqrt{x}\right)} \]

    8. fma-undefine 5.4%

       \[\leadsto \color{blue}{0.5 \cdot \sqrt{x} + -0.125 \cdot \sqrt{x}} \]

    9. distribute-rgt-out 5.4%

       \[\leadsto \color{blue}{\sqrt{x} \cdot \left(0.5 + -0.125\right)} \]

    10. metadata-eval 5.4%

        \[\leadsto \sqrt{x} \cdot \color{blue}{0.375} \]

11. Simplified 5.4%

    \[\leadsto \color{blue}{\sqrt{x} \cdot 0.375} \]
12. Step-by-step derivation

    1. add-sqr-sqrt 5.4%

       \[\leadsto \color{blue}{\sqrt{\sqrt{x} \cdot 0.375} \cdot \sqrt{\sqrt{x} \cdot 0.375}} \]

    2. sqrt-unprod 5.4%

       \[\leadsto \color{blue}{\sqrt{\left(\sqrt{x} \cdot 0.375\right) \cdot \left(\sqrt{x} \cdot 0.375\right)}} \]

    3. swap-sqr 5.4%

       \[\leadsto \sqrt{\color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right) \cdot \left(0.375 \cdot 0.375\right)}} \]

    4. add-sqr-sqrt 5.4%

       \[\leadsto \sqrt{\color{blue}{x} \cdot \left(0.375 \cdot 0.375\right)} \]

    5. metadata-eval 5.4%

       \[\leadsto \sqrt{x \cdot \color{blue}{0.140625}} \]

13. Applied egg-rr 5.4%

    \[\leadsto \color{blue}{\sqrt{x \cdot 0.140625}} \]
14. Add Preprocessing

Developer target: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024089 
(FPCore (x)
  :name "2sqrt (example 3.1)"
  :precision binary64
  :pre (and (> x 1.0) (< x 1e+308))

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

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