2isqrt (example 3.6)

Percentage Accurate: 37.2% → 98.7%

Time: 16.8s

Alternatives: 4

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: 37.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{{x}^{-1.5} \cdot -0.375 + 0.5 \cdot \sqrt{\frac{1}{x}}}{x} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/ (+ (* (pow x -1.5) -0.375) (* 0.5 (sqrt (/ 1.0 x)))) x))

C

double code(double x) {
	return ((pow(x, -1.5) * -0.375) + (0.5 * sqrt((1.0 / x)))) / x;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (((x ** (-1.5d0)) * (-0.375d0)) + (0.5d0 * sqrt((1.0d0 / x)))) / x
end function

Java

public static double code(double x) {
	return ((Math.pow(x, -1.5) * -0.375) + (0.5 * Math.sqrt((1.0 / x)))) / x;
}

Python

def code(x):
	return ((math.pow(x, -1.5) * -0.375) + (0.5 * math.sqrt((1.0 / x)))) / x

Julia

function code(x)
	return Float64(Float64(Float64((x ^ -1.5) * -0.375) + Float64(0.5 * sqrt(Float64(1.0 / x)))) / x)
end

MATLAB

function tmp = code(x)
	tmp = (((x ^ -1.5) * -0.375) + (0.5 * sqrt((1.0 / x)))) / x;
end

Wolfram

code[x_] := N[(N[(N[(N[Power[x, -1.5], $MachinePrecision] * -0.375), $MachinePrecision] + N[(0.5 * N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]

Derivation

1. Initial program 34.1%

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

3. Taylor expanded in x around inf 80.9%

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

4. Taylor expanded in x around inf 98.9%

   \[\leadsto \color{blue}{\frac{\left(0.125 \cdot \sqrt{\frac{1}{{x}^{3}}} + 0.5 \cdot \sqrt{\frac{1}{x}}\right) - 0.5 \cdot \sqrt{\frac{1}{{x}^{3}}}}{x}} \]
5. Step-by-step derivation

6. Simplified 98.9%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left({x}^{-1.5}, 0.125, 0.5 \cdot \left(\sqrt{\frac{1}{x}} - {x}^{-1.5}\right)\right)}{x}} \]

7. Taylor expanded in x around inf 98.9%

   \[\leadsto \color{blue}{\frac{-0.5 \cdot \sqrt{\frac{1}{{x}^{3}}} + \left(0.125 \cdot \sqrt{\frac{1}{{x}^{3}}} + 0.5 \cdot \sqrt{\frac{1}{x}}\right)}{x}} \]
8. Step-by-step derivation

   1. associate-+r+ 98.9%

      \[\leadsto \frac{\color{blue}{\left(-0.5 \cdot \sqrt{\frac{1}{{x}^{3}}} + 0.125 \cdot \sqrt{\frac{1}{{x}^{3}}}\right) + 0.5 \cdot \sqrt{\frac{1}{x}}}}{x} \]

   2. distribute-rgt-out 98.9%

      \[\leadsto \frac{\color{blue}{\sqrt{\frac{1}{{x}^{3}}} \cdot \left(-0.5 + 0.125\right)} + 0.5 \cdot \sqrt{\frac{1}{x}}}{x} \]

   3. metadata-eval 98.9%

      \[\leadsto \frac{\sqrt{\frac{1}{{x}^{3}}} \cdot \color{blue}{-0.375} + 0.5 \cdot \sqrt{\frac{1}{x}}}{x} \]

9. Simplified 98.9%

   \[\leadsto \color{blue}{\frac{\sqrt{\frac{1}{{x}^{3}}} \cdot -0.375 + 0.5 \cdot \sqrt{\frac{1}{x}}}{x}} \]

10. Taylor expanded in x around -inf 0.0%

    \[\leadsto \frac{\color{blue}{\left(-1 \cdot \left(\sqrt{\frac{1}{{x}^{3}}} \cdot {\left(\sqrt{-1}\right)}^{2}\right)\right)} \cdot -0.375 + 0.5 \cdot \sqrt{\frac{1}{x}}}{x} \]
11. Step-by-step derivation

    1. *-commutative 0.0%

       \[\leadsto \frac{\left(-1 \cdot \color{blue}{\left({\left(\sqrt{-1}\right)}^{2} \cdot \sqrt{\frac{1}{{x}^{3}}}\right)}\right) \cdot -0.375 + 0.5 \cdot \sqrt{\frac{1}{x}}}{x} \]

    2. unpow2 0.0%

       \[\leadsto \frac{\left(-1 \cdot \left(\color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)} \cdot \sqrt{\frac{1}{{x}^{3}}}\right)\right) \cdot -0.375 + 0.5 \cdot \sqrt{\frac{1}{x}}}{x} \]

    3. rem-square-sqrt 98.9%

       \[\leadsto \frac{\left(-1 \cdot \left(\color{blue}{-1} \cdot \sqrt{\frac{1}{{x}^{3}}}\right)\right) \cdot -0.375 + 0.5 \cdot \sqrt{\frac{1}{x}}}{x} \]

    4. associate-*r* 98.9%

       \[\leadsto \frac{\color{blue}{\left(\left(-1 \cdot -1\right) \cdot \sqrt{\frac{1}{{x}^{3}}}\right)} \cdot -0.375 + 0.5 \cdot \sqrt{\frac{1}{x}}}{x} \]

    5. metadata-eval 98.9%

       \[\leadsto \frac{\left(\color{blue}{1} \cdot \sqrt{\frac{1}{{x}^{3}}}\right) \cdot -0.375 + 0.5 \cdot \sqrt{\frac{1}{x}}}{x} \]

    6. *-lft-identity 98.9%

       \[\leadsto \frac{\color{blue}{\sqrt{\frac{1}{{x}^{3}}}} \cdot -0.375 + 0.5 \cdot \sqrt{\frac{1}{x}}}{x} \]

    7. unpow-1 98.9%

       \[\leadsto \frac{\sqrt{\color{blue}{{\left({x}^{3}\right)}^{-1}}} \cdot -0.375 + 0.5 \cdot \sqrt{\frac{1}{x}}}{x} \]

    8. exp-to-pow 98.9%

       \[\leadsto \frac{\sqrt{{\color{blue}{\left(e^{\log x \cdot 3}\right)}}^{-1}} \cdot -0.375 + 0.5 \cdot \sqrt{\frac{1}{x}}}{x} \]

    9. *-commutative 98.9%

       \[\leadsto \frac{\sqrt{{\left(e^{\color{blue}{3 \cdot \log x}}\right)}^{-1}} \cdot -0.375 + 0.5 \cdot \sqrt{\frac{1}{x}}}{x} \]

    10. exp-prod 98.9%

        \[\leadsto \frac{\sqrt{\color{blue}{e^{\left(3 \cdot \log x\right) \cdot -1}}} \cdot -0.375 + 0.5 \cdot \sqrt{\frac{1}{x}}}{x} \]

    11. *-commutative 98.9%

        \[\leadsto \frac{\sqrt{e^{\color{blue}{-1 \cdot \left(3 \cdot \log x\right)}}} \cdot -0.375 + 0.5 \cdot \sqrt{\frac{1}{x}}}{x} \]

    12. associate-*r* 98.9%

        \[\leadsto \frac{\sqrt{e^{\color{blue}{\left(-1 \cdot 3\right) \cdot \log x}}} \cdot -0.375 + 0.5 \cdot \sqrt{\frac{1}{x}}}{x} \]

    13. metadata-eval 98.9%

        \[\leadsto \frac{\sqrt{e^{\color{blue}{-3} \cdot \log x}} \cdot -0.375 + 0.5 \cdot \sqrt{\frac{1}{x}}}{x} \]

    14. *-commutative 98.9%

        \[\leadsto \frac{\sqrt{e^{\color{blue}{\log x \cdot -3}}} \cdot -0.375 + 0.5 \cdot \sqrt{\frac{1}{x}}}{x} \]

    15. exp-to-pow 98.9%

        \[\leadsto \frac{\sqrt{\color{blue}{{x}^{-3}}} \cdot -0.375 + 0.5 \cdot \sqrt{\frac{1}{x}}}{x} \]

    16. metadata-eval 98.9%

        \[\leadsto \frac{\sqrt{{x}^{\color{blue}{\left(2 \cdot -1.5\right)}}} \cdot -0.375 + 0.5 \cdot \sqrt{\frac{1}{x}}}{x} \]

    17. pow-sqr 98.9%

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

    18. rem-sqrt-square 98.9%

        \[\leadsto \frac{\color{blue}{\left|{x}^{-1.5}\right|} \cdot -0.375 + 0.5 \cdot \sqrt{\frac{1}{x}}}{x} \]

    19. rem-square-sqrt 98.9%

        \[\leadsto \frac{\left|\color{blue}{\sqrt{{x}^{-1.5}} \cdot \sqrt{{x}^{-1.5}}}\right| \cdot -0.375 + 0.5 \cdot \sqrt{\frac{1}{x}}}{x} \]

    20. fabs-sqr 98.9%

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

    21. rem-square-sqrt 98.9%

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

12. Simplified 98.9%

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

13. Final simplification 98.9%

    \[\leadsto \frac{{x}^{-1.5} \cdot -0.375 + 0.5 \cdot \sqrt{\frac{1}{x}}}{x} \]
14. Add Preprocessing

Alternative 2: 97.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 34.1%

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

3. Taylor expanded in x around inf 79.6%

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

   1. distribute-lft-out-- 79.6%

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

5. Simplified 79.6%

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

6. Taylor expanded in x around inf 97.6%

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

8. Simplified 97.6%

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

9. Final simplification 97.6%

   \[\leadsto \frac{0.5 \cdot \left(\sqrt{\frac{1}{x}} - {x}^{-1.5}\right)}{x} \]
10. Add Preprocessing

Alternative 3: 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 34.1%

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

3. Taylor expanded in x around inf 80.9%

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

4. Taylor expanded in x around inf 98.9%

   \[\leadsto \color{blue}{\frac{\left(0.125 \cdot \sqrt{\frac{1}{{x}^{3}}} + 0.5 \cdot \sqrt{\frac{1}{x}}\right) - 0.5 \cdot \sqrt{\frac{1}{{x}^{3}}}}{x}} \]
5. Step-by-step derivation

6. Simplified 98.9%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left({x}^{-1.5}, 0.125, 0.5 \cdot \left(\sqrt{\frac{1}{x}} - {x}^{-1.5}\right)\right)}{x}} \]

7. Taylor expanded in x around inf 97.5%

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

8. Final simplification 97.5%

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

Alternative 4: 34.3% 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 34.1%

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

   1. inv-pow 34.1%

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

   2. add-sqr-sqrt 21.7%

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

   3. unpow-prod-down 18.8%

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

   4. fma-neg 7.8%

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

   5. pow1/2 7.8%

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

   6. sqrt-pow1 7.7%

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

   7. metadata-eval 7.7%

      \[\leadsto \mathsf{fma}\left({\left({x}^{\color{blue}{0.25}}\right)}^{-1}, {\left(\sqrt{\sqrt{x}}\right)}^{-1}, -\frac{1}{\sqrt{x + 1}}\right) \]

   8. pow1/2 7.7%

      \[\leadsto \mathsf{fma}\left({\left({x}^{0.25}\right)}^{-1}, {\left(\sqrt{\color{blue}{{x}^{0.5}}}\right)}^{-1}, -\frac{1}{\sqrt{x + 1}}\right) \]

   9. sqrt-pow1 7.8%

      \[\leadsto \mathsf{fma}\left({\left({x}^{0.25}\right)}^{-1}, {\color{blue}{\left({x}^{\left(\frac{0.5}{2}\right)}\right)}}^{-1}, -\frac{1}{\sqrt{x + 1}}\right) \]

   10. metadata-eval 7.8%

       \[\leadsto \mathsf{fma}\left({\left({x}^{0.25}\right)}^{-1}, {\left({x}^{\color{blue}{0.25}}\right)}^{-1}, -\frac{1}{\sqrt{x + 1}}\right) \]

   11. distribute-neg-frac 7.8%

       \[\leadsto \mathsf{fma}\left({\left({x}^{0.25}\right)}^{-1}, {\left({x}^{0.25}\right)}^{-1}, \color{blue}{\frac{-1}{\sqrt{x + 1}}}\right) \]

   12. metadata-eval 7.8%

       \[\leadsto \mathsf{fma}\left({\left({x}^{0.25}\right)}^{-1}, {\left({x}^{0.25}\right)}^{-1}, \frac{\color{blue}{-1}}{\sqrt{x + 1}}\right) \]

   13. +-commutative 7.8%

       \[\leadsto \mathsf{fma}\left({\left({x}^{0.25}\right)}^{-1}, {\left({x}^{0.25}\right)}^{-1}, \frac{-1}{\sqrt{\color{blue}{1 + x}}}\right) \]

4. Applied egg-rr 7.8%

   \[\leadsto \color{blue}{\mathsf{fma}\left({\left({x}^{0.25}\right)}^{-1}, {\left({x}^{0.25}\right)}^{-1}, \frac{-1}{\sqrt{1 + x}}\right)} \]

5. Taylor expanded in x around inf 30.7%

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

   1. distribute-rgt1-in 30.7%

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

   2. metadata-eval 30.7%

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

   3. mul0-lft 30.7%

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

7. Simplified 30.7%

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

8. Final simplification 30.7%

   \[\leadsto 0 \]
9. 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 2024053 
(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)))))