2isqrt (example 3.6)

Percentage Accurate: 39.0% → 98.6%

Time: 9.4s

Alternatives: 5

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: 39.0% 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: 98.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left|{x}^{-0.5}\right|\\ \frac{t\_0 \cdot 0.5 - t\_0 \cdot \frac{0.375}{x}}{x} \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (fabs (pow x -0.5)))) (/ (- (* t_0 0.5) (* t_0 (/ 0.375 x))) x)))

C

double code(double x) {
	double t_0 = fabs(pow(x, -0.5));
	return ((t_0 * 0.5) - (t_0 * (0.375 / x))) / x;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    t_0 = abs((x ** (-0.5d0)))
    code = ((t_0 * 0.5d0) - (t_0 * (0.375d0 / x))) / x
end function

Java

public static double code(double x) {
	double t_0 = Math.abs(Math.pow(x, -0.5));
	return ((t_0 * 0.5) - (t_0 * (0.375 / x))) / x;
}

Python

def code(x):
	t_0 = math.fabs(math.pow(x, -0.5))
	return ((t_0 * 0.5) - (t_0 * (0.375 / x))) / x

Julia

function code(x)
	t_0 = abs((x ^ -0.5))
	return Float64(Float64(Float64(t_0 * 0.5) - Float64(t_0 * Float64(0.375 / x))) / x)
end

MATLAB

function tmp = code(x)
	t_0 = abs((x ^ -0.5));
	tmp = ((t_0 * 0.5) - (t_0 * (0.375 / x))) / x;
end

Wolfram

code[x_] := Block[{t$95$0 = N[Abs[N[Power[x, -0.5], $MachinePrecision]], $MachinePrecision]}, N[(N[(N[(t$95$0 * 0.5), $MachinePrecision] - N[(t$95$0 * N[(0.375 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]]

Derivation

1. Initial program 42.1%

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

3. Taylor expanded in x around inf 80.7%

   \[\leadsto \color{blue}{\frac{-0.5 \cdot \left(\sqrt{\frac{1}{{x}^{5}}} \cdot \left(1 + 0.5 \cdot x\right)\right) - \left(-0.5 \cdot \sqrt{x} + \left(-0.5 \cdot \left(\sqrt{\frac{1}{{x}^{3}}} \cdot \left(1 + 0.25 \cdot x\right)\right) + 0.5 \cdot \sqrt{\frac{1}{x}}\right)\right)}{{x}^{2}}} \]

4. Taylor expanded in x around inf 98.8%

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

6. Simplified 98.8%

   \[\leadsto \color{blue}{\frac{\left|{x}^{-0.5}\right| \cdot 0.5 - \left|{x}^{-0.5}\right| \cdot \frac{0.375}{x}}{x}} \]
7. Add Preprocessing

Alternative 2: 98.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \frac{{x}^{-0.5} \cdot \left(0.5 - \frac{0.375}{x}\right)}{x} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ (* (pow x -0.5) (- 0.5 (/ 0.375 x))) x))

C

double code(double x) {
	return (pow(x, -0.5) * (0.5 - (0.375 / x))) / x;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = ((x ** (-0.5d0)) * (0.5d0 - (0.375d0 / x))) / x
end function

Java

public static double code(double x) {
	return (Math.pow(x, -0.5) * (0.5 - (0.375 / x))) / x;
}

Python

def code(x):
	return (math.pow(x, -0.5) * (0.5 - (0.375 / x))) / x

Julia

function code(x)
	return Float64(Float64((x ^ -0.5) * Float64(0.5 - Float64(0.375 / x))) / x)
end

MATLAB

function tmp = code(x)
	tmp = ((x ^ -0.5) * (0.5 - (0.375 / x))) / x;
end

Wolfram

code[x_] := N[(N[(N[Power[x, -0.5], $MachinePrecision] * N[(0.5 - N[(0.375 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]

Derivation

1. Initial program 42.1%

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

3. Taylor expanded in x around inf 80.7%

   \[\leadsto \color{blue}{\frac{-0.5 \cdot \left(\sqrt{\frac{1}{{x}^{5}}} \cdot \left(1 + 0.5 \cdot x\right)\right) - \left(-0.5 \cdot \sqrt{x} + \left(-0.5 \cdot \left(\sqrt{\frac{1}{{x}^{3}}} \cdot \left(1 + 0.25 \cdot x\right)\right) + 0.5 \cdot \sqrt{\frac{1}{x}}\right)\right)}{{x}^{2}}} \]

4. Taylor expanded in x around inf 98.8%

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

6. Simplified 98.8%

   \[\leadsto \color{blue}{\frac{\left|{x}^{-0.5}\right| \cdot 0.5 - \left|{x}^{-0.5}\right| \cdot \frac{0.375}{x}}{x}} \]
7. Step-by-step derivation

   1. distribute-lft-out-- 98.8%

      \[\leadsto \frac{\color{blue}{\left|{x}^{-0.5}\right| \cdot \left(0.5 - \frac{0.375}{x}\right)}}{x} \]

8. Applied egg-rr 98.8%

   \[\leadsto \frac{\color{blue}{\left|{x}^{-0.5}\right| \cdot \left(0.5 - \frac{0.375}{x}\right)}}{x} \]
9. Step-by-step derivation

   1. pow1 98.8%

      \[\leadsto \frac{\color{blue}{{\left(\left|{x}^{-0.5}\right| \cdot \left(0.5 - \frac{0.375}{x}\right)\right)}^{1}}}{x} \]

   2. add-sqr-sqrt 98.5%

      \[\leadsto \frac{{\left(\left|\color{blue}{\sqrt{{x}^{-0.5}} \cdot \sqrt{{x}^{-0.5}}}\right| \cdot \left(0.5 - \frac{0.375}{x}\right)\right)}^{1}}{x} \]

   3. fabs-sqr 98.5%

      \[\leadsto \frac{{\left(\color{blue}{\left(\sqrt{{x}^{-0.5}} \cdot \sqrt{{x}^{-0.5}}\right)} \cdot \left(0.5 - \frac{0.375}{x}\right)\right)}^{1}}{x} \]

   4. add-sqr-sqrt 98.8%

      \[\leadsto \frac{{\left(\color{blue}{{x}^{-0.5}} \cdot \left(0.5 - \frac{0.375}{x}\right)\right)}^{1}}{x} \]

10. Applied egg-rr 98.8%

    \[\leadsto \frac{\color{blue}{{\left({x}^{-0.5} \cdot \left(0.5 - \frac{0.375}{x}\right)\right)}^{1}}}{x} \]

12. Simplified 98.8%

    \[\leadsto \frac{\color{blue}{{x}^{-0.5} \cdot \left(0.5 - \frac{0.375}{x}\right)}}{x} \]
13. Add Preprocessing

Alternative 3: 98.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ {x}^{-0.5} \cdot \frac{0.5 - \frac{0.375}{x}}{x} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (* (pow x -0.5) (/ (- 0.5 (/ 0.375 x)) x)))

C

double code(double x) {
	return pow(x, -0.5) * ((0.5 - (0.375 / x)) / x);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (x ** (-0.5d0)) * ((0.5d0 - (0.375d0 / x)) / x)
end function

Java

public static double code(double x) {
	return Math.pow(x, -0.5) * ((0.5 - (0.375 / x)) / x);
}

Python

def code(x):
	return math.pow(x, -0.5) * ((0.5 - (0.375 / x)) / x)

Julia

function code(x)
	return Float64((x ^ -0.5) * Float64(Float64(0.5 - Float64(0.375 / x)) / x))
end

MATLAB

function tmp = code(x)
	tmp = (x ^ -0.5) * ((0.5 - (0.375 / x)) / x);
end

Wolfram

code[x_] := N[(N[Power[x, -0.5], $MachinePrecision] * N[(N[(0.5 - N[(0.375 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 42.1%

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

3. Taylor expanded in x around inf 80.7%

   \[\leadsto \color{blue}{\frac{-0.5 \cdot \left(\sqrt{\frac{1}{{x}^{5}}} \cdot \left(1 + 0.5 \cdot x\right)\right) - \left(-0.5 \cdot \sqrt{x} + \left(-0.5 \cdot \left(\sqrt{\frac{1}{{x}^{3}}} \cdot \left(1 + 0.25 \cdot x\right)\right) + 0.5 \cdot \sqrt{\frac{1}{x}}\right)\right)}{{x}^{2}}} \]

4. Taylor expanded in x around inf 98.8%

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

6. Simplified 98.8%

   \[\leadsto \color{blue}{\frac{\left|{x}^{-0.5}\right| \cdot 0.5 - \left|{x}^{-0.5}\right| \cdot \frac{0.375}{x}}{x}} \]
7. Step-by-step derivation

   1. distribute-lft-out-- 98.8%

      \[\leadsto \frac{\color{blue}{\left|{x}^{-0.5}\right| \cdot \left(0.5 - \frac{0.375}{x}\right)}}{x} \]

8. Applied egg-rr 98.8%

   \[\leadsto \frac{\color{blue}{\left|{x}^{-0.5}\right| \cdot \left(0.5 - \frac{0.375}{x}\right)}}{x} \]
9. Step-by-step derivation

   1. associate-/l* 98.8%

      \[\leadsto \color{blue}{\left|{x}^{-0.5}\right| \cdot \frac{0.5 - \frac{0.375}{x}}{x}} \]

   2. add-sqr-sqrt 98.5%

      \[\leadsto \left|\color{blue}{\sqrt{{x}^{-0.5}} \cdot \sqrt{{x}^{-0.5}}}\right| \cdot \frac{0.5 - \frac{0.375}{x}}{x} \]

   3. fabs-sqr 98.5%

      \[\leadsto \color{blue}{\left(\sqrt{{x}^{-0.5}} \cdot \sqrt{{x}^{-0.5}}\right)} \cdot \frac{0.5 - \frac{0.375}{x}}{x} \]

   4. add-sqr-sqrt 98.8%

      \[\leadsto \color{blue}{{x}^{-0.5}} \cdot \frac{0.5 - \frac{0.375}{x}}{x} \]

10. Applied egg-rr 98.8%

    \[\leadsto \color{blue}{{x}^{-0.5} \cdot \frac{0.5 - \frac{0.375}{x}}{x}} \]
11. Add Preprocessing

Alternative 4: 97.5% accurate, 2.0× speedup

Math

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

FPCore

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

C

double code(double x) {
	return (0.5 * sqrt((1.0 / x))) / x;
}

Fortran

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

Java

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

Python

def code(x):
	return (0.5 * math.sqrt((1.0 / x))) / x

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 42.1%

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

3. Taylor expanded in x around inf 79.5%

   \[\leadsto \color{blue}{\frac{-0.5 \cdot \sqrt{\frac{1}{x}} - -0.5 \cdot \sqrt{x}}{{x}^{2}}} \]
4. Step-by-step derivation

   1. distribute-lft-out-- 79.5%

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

5. Simplified 79.5%

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

6. Taylor expanded in x around inf 97.7%

   \[\leadsto \color{blue}{\frac{-0.5 \cdot \sqrt{\frac{1}{{x}^{3}}} + 0.5 \cdot \sqrt{\frac{1}{x}}}{x}} \]

7. Taylor expanded in x around inf 97.6%

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

9. Simplified 97.6%

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

10. Final simplification 97.6%

    \[\leadsto \frac{0.5 \cdot \sqrt{\frac{1}{x}}}{x} \]
11. Add Preprocessing

Alternative 5: 36.0% 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 42.1%

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

   1. add-cube-cbrt 13.4%

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

   2. associate-*l* 13.4%

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

   3. frac-2neg 13.4%

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

   4. metadata-eval 13.4%

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

   5. div-inv 13.4%

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

   6. distribute-neg-frac2 13.4%

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

   7. prod-diff 6.9%

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

4. Applied egg-rr 6.9%

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

6. Simplified 7.0%

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

7. Taylor expanded in x around inf 39.2%

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

   1. distribute-rgt1-in 39.2%

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

   2. metadata-eval 39.2%

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

   3. mul0-lft 39.2%

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

9. Simplified 39.2%

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

Developer Target 1: 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 2024130 
(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)))))