2isqrt (example 3.6)

Percentage Accurate: 38.1% → 98.9%

Time: 9.6s

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: 38.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ 0.5 \cdot {x}^{-1.5} - \frac{0.375}{{x}^{2.5}} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (- (* 0.5 (pow x -1.5)) (/ 0.375 (pow x 2.5))))

C

double code(double x) {
	return (0.5 * pow(x, -1.5)) - (0.375 / pow(x, 2.5));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (0.5d0 * (x ** (-1.5d0))) - (0.375d0 / (x ** 2.5d0))
end function

Java

public static double code(double x) {
	return (0.5 * Math.pow(x, -1.5)) - (0.375 / Math.pow(x, 2.5));
}

Python

def code(x):
	return (0.5 * math.pow(x, -1.5)) - (0.375 / math.pow(x, 2.5))

Julia

function code(x)
	return Float64(Float64(0.5 * (x ^ -1.5)) - Float64(0.375 / (x ^ 2.5)))
end

MATLAB

function tmp = code(x)
	tmp = (0.5 * (x ^ -1.5)) - (0.375 / (x ^ 2.5));
end

Wolfram

code[x_] := N[(N[(0.5 * N[Power[x, -1.5], $MachinePrecision]), $MachinePrecision] - N[(0.375 / N[Power[x, 2.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 38.8%

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

3. Taylor expanded in x around inf 81.0%

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

4. Taylor expanded in x around inf 98.5%

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

   1. +-commutative 98.5%

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

   2. mul-1-neg 98.5%

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

   3. unsub-neg 98.5%

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

   4. *-commutative 98.5%

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

   5. distribute-rgt-out 98.5%

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

   6. metadata-eval 98.5%

      \[\leadsto \frac{\sqrt{\frac{1}{x}} \cdot 0.5 - \frac{\sqrt{\frac{1}{x}} \cdot \color{blue}{0.375}}{x}}{x} \]

   7. *-rgt-identity 98.5%

      \[\leadsto \frac{\sqrt{\frac{1}{x}} \cdot 0.5 - \frac{\sqrt{\frac{1}{x}} \cdot 0.375}{\color{blue}{x \cdot 1}}}{x} \]

   8. times-frac 98.5%

      \[\leadsto \frac{\sqrt{\frac{1}{x}} \cdot 0.5 - \color{blue}{\frac{\sqrt{\frac{1}{x}}}{x} \cdot \frac{0.375}{1}}}{x} \]

   9. metadata-eval 98.5%

      \[\leadsto \frac{\sqrt{\frac{1}{x}} \cdot 0.5 - \frac{\sqrt{\frac{1}{x}}}{x} \cdot \color{blue}{0.375}}{x} \]

6. Simplified 98.5%

   \[\leadsto \color{blue}{\frac{\sqrt{\frac{1}{x}} \cdot 0.5 - \frac{\sqrt{\frac{1}{x}}}{x} \cdot 0.375}{x}} \]
7. Step-by-step derivation

   1. div-sub 98.5%

      \[\leadsto \color{blue}{\frac{\sqrt{\frac{1}{x}} \cdot 0.5}{x} - \frac{\frac{\sqrt{\frac{1}{x}}}{x} \cdot 0.375}{x}} \]

8. Applied egg-rr 98.9%

   \[\leadsto \color{blue}{0.5 \cdot {x}^{-1.5} - 0.375 \cdot \frac{{x}^{-1.5}}{x}} \]
9. Step-by-step derivation

   1. clear-num 98.9%

      \[\leadsto 0.5 \cdot {x}^{-1.5} - 0.375 \cdot \color{blue}{\frac{1}{\frac{x}{{x}^{-1.5}}}} \]

   2. un-div-inv 98.9%

      \[\leadsto 0.5 \cdot {x}^{-1.5} - \color{blue}{\frac{0.375}{\frac{x}{{x}^{-1.5}}}} \]

   3. pow1 98.9%

      \[\leadsto 0.5 \cdot {x}^{-1.5} - \frac{0.375}{\frac{\color{blue}{{x}^{1}}}{{x}^{-1.5}}} \]

   4. pow-div 98.9%

      \[\leadsto 0.5 \cdot {x}^{-1.5} - \frac{0.375}{\color{blue}{{x}^{\left(1 - -1.5\right)}}} \]

   5. metadata-eval 98.9%

      \[\leadsto 0.5 \cdot {x}^{-1.5} - \frac{0.375}{{x}^{\color{blue}{2.5}}} \]

10. Applied egg-rr 98.9%

    \[\leadsto 0.5 \cdot {x}^{-1.5} - \color{blue}{\frac{0.375}{{x}^{2.5}}} \]
11. Add Preprocessing

Alternative 2: 97.7% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 38.8%

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

3. Taylor expanded in x around inf 81.0%

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

4. Taylor expanded in x around inf 98.5%

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

   1. +-commutative 98.5%

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

   2. mul-1-neg 98.5%

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

   3. unsub-neg 98.5%

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

   4. *-commutative 98.5%

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

   5. distribute-rgt-out 98.5%

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

   6. metadata-eval 98.5%

      \[\leadsto \frac{\sqrt{\frac{1}{x}} \cdot 0.5 - \frac{\sqrt{\frac{1}{x}} \cdot \color{blue}{0.375}}{x}}{x} \]

   7. *-rgt-identity 98.5%

      \[\leadsto \frac{\sqrt{\frac{1}{x}} \cdot 0.5 - \frac{\sqrt{\frac{1}{x}} \cdot 0.375}{\color{blue}{x \cdot 1}}}{x} \]

   8. times-frac 98.5%

      \[\leadsto \frac{\sqrt{\frac{1}{x}} \cdot 0.5 - \color{blue}{\frac{\sqrt{\frac{1}{x}}}{x} \cdot \frac{0.375}{1}}}{x} \]

   9. metadata-eval 98.5%

      \[\leadsto \frac{\sqrt{\frac{1}{x}} \cdot 0.5 - \frac{\sqrt{\frac{1}{x}}}{x} \cdot \color{blue}{0.375}}{x} \]

6. Simplified 98.5%

   \[\leadsto \color{blue}{\frac{\sqrt{\frac{1}{x}} \cdot 0.5 - \frac{\sqrt{\frac{1}{x}}}{x} \cdot 0.375}{x}} \]

7. Taylor expanded in x around inf 97.8%

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

   1. unpow1/2 97.8%

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

   2. unpow-1 97.8%

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

   3. exp-to-pow 93.8%

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

   4. *-commutative 93.8%

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

   5. neg-mul-1 93.8%

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

   6. exp-prod 93.8%

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

   7. distribute-lft-neg-out 93.8%

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

   8. distribute-rgt-neg-in 93.8%

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

   9. metadata-eval 93.8%

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

   10. exp-to-pow 97.8%

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

9. Simplified 97.8%

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

Alternative 3: 97.6% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 38.8%

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

3. Taylor expanded in x around inf 80.3%

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

   1. distribute-lft-out-- 80.3%

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

5. Simplified 80.3%

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

6. Taylor expanded in x around inf 80.2%

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

   1. *-commutative 80.2%

      \[\leadsto \frac{\color{blue}{\sqrt{x} \cdot 0.5}}{{x}^{2}} \]

8. Simplified 80.2%

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

   1. unpow2 80.2%

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

   2. times-frac 97.7%

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

10. Applied egg-rr 97.7%

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

    1. pow1/2 97.7%

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

    2. pow1 97.7%

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

    3. pow-div 97.8%

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

    4. metadata-eval 97.8%

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

    5. *-un-lft-identity 97.8%

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

12. Applied egg-rr 97.8%

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

    1. *-lft-identity 97.8%

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

14. Simplified 97.8%

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

Alternative 4: 5.7% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 38.8%

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

   1. flip-- 38.8%

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

   2. frac-times 22.9%

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

   3. metadata-eval 22.9%

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

   4. add-sqr-sqrt 20.9%

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

   5. frac-times 25.1%

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

   6. metadata-eval 25.1%

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

   7. add-sqr-sqrt 39.0%

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

   8. +-commutative 39.0%

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

   9. inv-pow 39.0%

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

   10. sqrt-pow2 39.0%

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

   11. metadata-eval 39.0%

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

   12. inv-pow 39.0%

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

   13. sqrt-pow2 39.0%

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

   14. +-commutative 39.0%

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

   15. metadata-eval 39.0%

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

4. Applied egg-rr 39.0%

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

5. Taylor expanded in x around inf 37.8%

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

   1. *-commutative 37.8%

      \[\leadsto \frac{\frac{1}{x} - \frac{1}{1 + x}}{\color{blue}{\sqrt{\frac{1}{x}} \cdot 2}} \]

7. Simplified 37.8%

   \[\leadsto \frac{\frac{1}{x} - \frac{1}{1 + x}}{\color{blue}{\sqrt{\frac{1}{x}} \cdot 2}} \]

8. Taylor expanded in x around 0 5.7%

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

Alternative 5: 5.7% 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 38.8%

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

3. Taylor expanded in x around 0 5.6%

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

   1. pow1/2 5.6%

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

   2. inv-pow 5.6%

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

   3. pow-pow 5.6%

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

   4. metadata-eval 5.6%

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

   5. *-un-lft-identity 5.6%

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

5. Applied egg-rr 5.6%

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

   1. *-lft-identity 5.6%

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

7. Simplified 5.6%

   \[\leadsto \color{blue}{{x}^{-0.5}} \]
8. 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 2024165 
(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)))))