2isqrt (example 3.6)

Percentage Accurate: 37.8% → 99.6%

Time: 12.5s

Alternatives: 9

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: 37.8% 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: 99.6% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := N[(N[(1.0 / N[Sqrt[N[Sqrt[x], $MachinePrecision] ^ 2 + x ^ 2], $MachinePrecision]), $MachinePrecision] / N[(N[Sqrt[x], $MachinePrecision] + N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 39.6%

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

   1. frac-sub 39.7%

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

   2. *-un-lft-identity 39.7%

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

   3. +-commutative 39.7%

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

   4. *-rgt-identity 39.7%

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

   5. sqrt-unprod 39.7%

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

   6. +-commutative 39.7%

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

4. Applied egg-rr 39.7%

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

   1. flip-- 40.6%

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

   2. add-sqr-sqrt 41.3%

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

   3. add-sqr-sqrt 43.1%

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

6. Applied egg-rr 43.1%

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

   1. associate--l+ 84.7%

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

   2. +-inverses 84.7%

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

   3. metadata-eval 84.7%

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

   4. +-commutative 84.7%

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

8. Simplified 84.7%

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

   1. *-un-lft-identity 84.7%

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

   2. *-commutative 84.7%

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

   3. associate-/l/ 84.6%

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

   4. associate-/r* 84.7%

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

   5. distribute-rgt-in 84.7%

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

   6. *-un-lft-identity 84.7%

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

   7. add-sqr-sqrt 84.7%

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

   8. hypot-define 99.7%

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

10. Applied egg-rr 99.7%

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

11. Final simplification 99.7%

    \[\leadsto \frac{\frac{1}{\mathsf{hypot}\left(\sqrt{x}, x\right)}}{\sqrt{x} + \sqrt{1 + x}} \]
12. Add Preprocessing

Alternative 2: 98.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 39.6%

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

   1. frac-sub 39.7%

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

   2. *-un-lft-identity 39.7%

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

   3. +-commutative 39.7%

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

   4. *-rgt-identity 39.7%

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

   5. sqrt-unprod 39.7%

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

   6. +-commutative 39.7%

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

4. Applied egg-rr 39.7%

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

   1. flip-- 40.6%

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

   2. add-sqr-sqrt 41.3%

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

   3. add-sqr-sqrt 43.1%

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

6. Applied egg-rr 43.1%

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

   1. associate--l+ 84.7%

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

   2. +-inverses 84.7%

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

   3. metadata-eval 84.7%

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

   4. +-commutative 84.7%

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

8. Simplified 84.7%

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

9. Taylor expanded in x around inf 99.2%

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

    1. associate-*r/ 99.2%

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

    2. metadata-eval 99.2%

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

11. Simplified 99.2%

    \[\leadsto \frac{\frac{1}{\sqrt{x} + \sqrt{1 + x}}}{\color{blue}{x \cdot \left(1 + \frac{0.5}{x}\right)}} \]
12. Add Preprocessing

Alternative 3: 98.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \frac{0.5 \cdot {x}^{-1.5}}{1 + \frac{0.5}{x}} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ (* 0.5 (pow x -1.5)) (+ 1.0 (/ 0.5 x))))

C

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

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (0.5d0 * (x ** (-1.5d0))) / (1.0d0 + (0.5d0 / x))
end function

Java

public static double code(double x) {
	return (0.5 * Math.pow(x, -1.5)) / (1.0 + (0.5 / x));
}

Python

def code(x):
	return (0.5 * math.pow(x, -1.5)) / (1.0 + (0.5 / x))

Julia

function code(x)
	return Float64(Float64(0.5 * (x ^ -1.5)) / Float64(1.0 + Float64(0.5 / x)))
end

MATLAB

function tmp = code(x)
	tmp = (0.5 * (x ^ -1.5)) / (1.0 + (0.5 / x));
end

Wolfram

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

Derivation

1. Initial program 39.6%

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

   1. frac-sub 39.7%

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

   2. *-un-lft-identity 39.7%

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

   3. +-commutative 39.7%

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

   4. *-rgt-identity 39.7%

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

   5. sqrt-unprod 39.7%

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

   6. +-commutative 39.7%

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

4. Applied egg-rr 39.7%

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

5. Taylor expanded in x around inf 82.5%

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

   1. *-commutative 82.5%

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

7. Simplified 82.5%

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

8. Taylor expanded in x around inf 97.6%

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

   1. associate-*r/ 99.2%

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

   2. metadata-eval 99.2%

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

10. Simplified 97.6%

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

    1. *-un-lft-identity 97.6%

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

    2. *-commutative 97.6%

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

    3. inv-pow 97.6%

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

    4. sqrt-pow1 97.6%

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

    5. metadata-eval 97.6%

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

    6. *-commutative 97.6%

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

    7. times-frac 97.6%

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

    8. *-un-lft-identity 97.6%

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

    9. add-sqr-sqrt 97.3%

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

    10. times-frac 97.2%

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

    11. pow1/2 97.2%

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

    12. pow-flip 97.3%

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

    13. metadata-eval 97.3%

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

    14. un-div-inv 97.3%

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

    15. pow1/2 97.3%

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

    16. pow-flip 97.2%

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

    17. metadata-eval 97.2%

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

    18. cube-mult 97.2%

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

    19. pow-pow 97.8%

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

    20. metadata-eval 97.8%

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

12. Applied egg-rr 97.8%

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

    1. *-lft-identity 97.8%

       \[\leadsto \color{blue}{\frac{0.5}{1 + \frac{0.5}{x}} \cdot {x}^{-1.5}} \]

    2. associate-*l/ 97.8%

       \[\leadsto \color{blue}{\frac{0.5 \cdot {x}^{-1.5}}{1 + \frac{0.5}{x}}} \]

14. Simplified 97.8%

    \[\leadsto \color{blue}{\frac{0.5 \cdot {x}^{-1.5}}{1 + \frac{0.5}{x}}} \]
15. Add Preprocessing

Alternative 4: 97.8% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 39.6%

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

   1. frac-sub 39.7%

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

   2. *-un-lft-identity 39.7%

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

   3. +-commutative 39.7%

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

   4. *-rgt-identity 39.7%

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

   5. sqrt-unprod 39.7%

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

   6. +-commutative 39.7%

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

4. Applied egg-rr 39.7%

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

5. Taylor expanded in x around inf 82.5%

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

   1. *-commutative 82.5%

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

7. Simplified 82.5%

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

8. Taylor expanded in x around inf 97.6%

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

   1. associate-*r/ 99.2%

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

   2. metadata-eval 99.2%

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

10. Simplified 97.6%

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

    1. *-commutative 97.6%

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

    2. inv-pow 97.6%

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

    3. sqrt-pow1 97.6%

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

    4. metadata-eval 97.6%

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

    5. *-un-lft-identity 97.6%

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

    6. times-frac 97.6%

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

    7. metadata-eval 97.6%

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

12. Applied egg-rr 97.6%

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

    1. associate-*r/ 97.6%

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

    2. distribute-lft-in 97.6%

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

    3. *-rgt-identity 97.6%

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

    4. associate-*r/ 97.6%

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

    5. *-commutative 97.6%

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

    6. associate-/l* 97.6%

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

    7. *-inverses 97.6%

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

    8. metadata-eval 97.6%

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

14. Simplified 97.6%

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

Alternative 5: 97.7% accurate, 2.0× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 (/ (/ 0.5 (sqrt x)) (+ x 0.5)))

C

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

Fortran

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

Java

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

Python

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

Julia

function code(x)
	return Float64(Float64(0.5 / sqrt(x)) / Float64(x + 0.5))
end

MATLAB

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

Wolfram

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

Derivation

1. Initial program 39.6%

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

   1. frac-sub 39.7%

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

   2. *-un-lft-identity 39.7%

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

   3. +-commutative 39.7%

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

   4. *-rgt-identity 39.7%

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

   5. sqrt-unprod 39.7%

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

   6. +-commutative 39.7%

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

4. Applied egg-rr 39.7%

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

5. Taylor expanded in x around inf 82.5%

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

   1. *-commutative 82.5%

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

7. Simplified 82.5%

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

8. Taylor expanded in x around inf 97.6%

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

   1. associate-*r/ 99.2%

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

   2. metadata-eval 99.2%

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

10. Simplified 97.6%

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

    1. *-un-lft-identity 97.6%

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

    2. times-frac 97.4%

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

    3. *-commutative 97.4%

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

    4. sqrt-div 97.4%

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

    5. metadata-eval 97.4%

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

    6. un-div-inv 97.4%

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

12. Applied egg-rr 97.4%

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

    1. *-commutative 97.4%

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

    2. times-frac 97.5%

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

    3. *-commutative 97.5%

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

    4. *-rgt-identity 97.5%

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

    5. distribute-lft-in 97.5%

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

    6. *-rgt-identity 97.5%

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

    7. associate-*r/ 97.5%

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

    8. *-commutative 97.5%

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

    9. associate-/l* 97.5%

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

    10. *-inverses 97.5%

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

    11. metadata-eval 97.5%

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

14. Simplified 97.5%

    \[\leadsto \color{blue}{\frac{\frac{0.5}{\sqrt{x}}}{x + 0.5}} \]
15. Add Preprocessing

Alternative 6: 97.7% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 39.6%

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

   1. frac-sub 39.7%

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

   2. *-un-lft-identity 39.7%

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

   3. +-commutative 39.7%

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

   4. *-rgt-identity 39.7%

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

   5. sqrt-unprod 39.7%

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

   6. +-commutative 39.7%

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

4. Applied egg-rr 39.7%

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

5. Taylor expanded in x around inf 82.5%

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

   1. *-commutative 82.5%

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

7. Simplified 82.5%

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

   1. associate-/l* 82.4%

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

   2. inv-pow 82.4%

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

   3. sqrt-pow1 82.5%

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

   4. metadata-eval 82.5%

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

   5. *-un-lft-identity 82.5%

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

   6. associate-*r/ 82.5%

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

   7. sqrt-prod 97.3%

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

   8. times-frac 97.2%

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

   9. metadata-eval 97.2%

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

   10. pow-flip 97.1%

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

   11. pow1/2 97.1%

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

   12. associate-/r* 97.1%

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

   13. add-sqr-sqrt 97.4%

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

9. Applied egg-rr 97.4%

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

    1. *-lft-identity 97.4%

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

    2. associate-*r/ 97.5%

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

    3. associate-*l/ 97.5%

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

    4. metadata-eval 97.5%

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

11. Simplified 97.5%

    \[\leadsto \color{blue}{\frac{\frac{0.5}{x}}{\sqrt{1 + x}}} \]
12. Add Preprocessing

Alternative 7: 97.7% 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 39.6%

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

   1. frac-sub 39.7%

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

   2. *-un-lft-identity 39.7%

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

   3. +-commutative 39.7%

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

   4. *-rgt-identity 39.7%

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

   5. sqrt-unprod 39.7%

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

   6. +-commutative 39.7%

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

4. Applied egg-rr 39.7%

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

5. Taylor expanded in x around inf 82.5%

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

   1. *-commutative 82.5%

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

7. Simplified 82.5%

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

8. Taylor expanded in x around inf 97.4%

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

9. Final simplification 97.4%

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

Alternative 8: 37.0% accurate, 26.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 6.4 \cdot 10^{+153}:\\ \;\;\;\;\frac{0.5}{x}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (if (<= x 6.4e+153) (/ 0.5 x) 0.0))

C

double code(double x) {
	double tmp;
	if (x <= 6.4e+153) {
		tmp = 0.5 / x;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 6.4d+153) then
        tmp = 0.5d0 / x
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 6.4e+153) {
		tmp = 0.5 / x;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 6.4e+153:
		tmp = 0.5 / x
	else:
		tmp = 0.0
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 6.4e+153)
		tmp = Float64(0.5 / x);
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 6.4e+153)
		tmp = 0.5 / x;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 6.4e+153], N[(0.5 / x), $MachinePrecision], 0.0]

Derivation

1. Split input into 2 regimes

2. if x < 6.4000000000000003e153

   1. Initial program 11.4%

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

      1. frac-sub 11.6%

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

      2. *-un-lft-identity 11.6%

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

      3. +-commutative 11.6%

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

      4. *-rgt-identity 11.6%

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

      5. sqrt-unprod 11.6%

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

      6. +-commutative 11.6%

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

   4. Applied egg-rr 11.6%

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

   5. Taylor expanded in x around inf 95.4%

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

      1. *-commutative 95.4%

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

   7. Simplified 95.4%

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

   8. Taylor expanded in x around 0 8.5%

      \[\leadsto \color{blue}{\frac{0.5}{x}} \]

   if 6.4000000000000003e153 < x

   1. Initial program 69.1%

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

      1. sub-neg 69.1%

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

      2. +-commutative 69.1%

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

      3. add-sqr-sqrt 38.5%

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

      4. distribute-rgt-neg-in 38.5%

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

      5. fma-define 4.5%

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

      6. pow1/2 4.5%

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

      7. pow-flip 4.5%

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

      8. +-commutative 4.5%

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

      9. metadata-eval 4.5%

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

      10. pow1/2 4.5%

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

      11. pow-flip 4.5%

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

      12. +-commutative 4.5%

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

      13. metadata-eval 4.5%

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

      14. inv-pow 4.5%

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

      15. sqrt-pow2 4.5%

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

      16. metadata-eval 4.5%

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

   4. Applied egg-rr 4.5%

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

   5. Taylor expanded in x around inf 69.1%

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

      1. distribute-rgt1-in 69.1%

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

      2. metadata-eval 69.1%

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

      3. mul0-lft 69.1%

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

   7. Simplified 69.1%

      \[\leadsto \color{blue}{0} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 34.9% 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 39.6%

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

   1. sub-neg 39.6%

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

   2. +-commutative 39.6%

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

   3. add-sqr-sqrt 24.7%

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

   4. distribute-rgt-neg-in 24.7%

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

   5. fma-define 8.2%

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

   6. pow1/2 8.2%

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

   7. pow-flip 8.3%

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

   8. +-commutative 8.3%

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

   9. metadata-eval 8.3%

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

   10. pow1/2 8.3%

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

   11. pow-flip 8.2%

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

   12. +-commutative 8.2%

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

   13. metadata-eval 8.2%

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

   14. inv-pow 8.2%

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

   15. sqrt-pow2 8.3%

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

   16. metadata-eval 8.3%

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

4. Applied egg-rr 8.3%

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

5. Taylor expanded in x around inf 36.0%

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

   1. distribute-rgt1-in 36.0%

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

   2. metadata-eval 36.0%

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

   3. mul0-lft 36.0%

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

7. Simplified 36.0%

   \[\leadsto \color{blue}{0} \]
8. 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 2024089 
(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)))))