2isqrt (example 3.6)

Percentage Accurate: 38.3% → 99.0%

Time: 9.3s

Alternatives: 9

Speedup: 1.5×

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: 38.3% 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: 99.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{1}{\left(2 - \frac{0.125}{x \cdot x}\right) \cdot x + 0.5}}{\sqrt{1 + x}} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/ (/ 1.0 (+ (* (- 2.0 (/ 0.125 (* x x))) x) 0.5)) (sqrt (+ 1.0 x))))

C

double code(double x) {
	return (1.0 / (((2.0 - (0.125 / (x * x))) * x) + 0.5)) / sqrt((1.0 + x));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (1.0d0 / (((2.0d0 - (0.125d0 / (x * x))) * x) + 0.5d0)) / sqrt((1.0d0 + x))
end function

Java

public static double code(double x) {
	return (1.0 / (((2.0 - (0.125 / (x * x))) * x) + 0.5)) / Math.sqrt((1.0 + x));
}

Python

def code(x):
	return (1.0 / (((2.0 - (0.125 / (x * x))) * x) + 0.5)) / math.sqrt((1.0 + x))

Julia

function code(x)
	return Float64(Float64(1.0 / Float64(Float64(Float64(2.0 - Float64(0.125 / Float64(x * x))) * x) + 0.5)) / sqrt(Float64(1.0 + x)))
end

MATLAB

function tmp = code(x)
	tmp = (1.0 / (((2.0 - (0.125 / (x * x))) * x) + 0.5)) / sqrt((1.0 + x));
end

Wolfram

code[x_] := N[(N[(1.0 / N[(N[(N[(2.0 - N[(0.125 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 38.9%

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

4. Applied rewrites 40.2%

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

5. Taylor expanded in x around 0

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

7. Applied rewrites 84.3%

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

8. Taylor expanded in x around inf

   \[\leadsto \frac{\frac{1}{\color{blue}{x \cdot \left(\left(2 + \frac{1}{2} \cdot \frac{1}{x}\right) - \frac{\frac{1}{8}}{{x}^{2}}\right)}}}{\sqrt{x + 1}} \]
9. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \frac{\frac{1}{x \cdot \left(\color{blue}{\left(\frac{1}{2} \cdot \frac{1}{x} + 2\right)} - \frac{\frac{1}{8}}{{x}^{2}}\right)}}{\sqrt{x + 1}} \]

   2. associate--l+ N/A

      \[\leadsto \frac{\frac{1}{x \cdot \color{blue}{\left(\frac{1}{2} \cdot \frac{1}{x} + \left(2 - \frac{\frac{1}{8}}{{x}^{2}}\right)\right)}}}{\sqrt{x + 1}} \]

   3. distribute-lft-in N/A

      \[\leadsto \frac{\frac{1}{\color{blue}{x \cdot \left(\frac{1}{2} \cdot \frac{1}{x}\right) + x \cdot \left(2 - \frac{\frac{1}{8}}{{x}^{2}}\right)}}}{\sqrt{x + 1}} \]

   4. *-commutative N/A

      \[\leadsto \frac{\frac{1}{\color{blue}{\left(\frac{1}{2} \cdot \frac{1}{x}\right) \cdot x} + x \cdot \left(2 - \frac{\frac{1}{8}}{{x}^{2}}\right)}}{\sqrt{x + 1}} \]

   5. associate-*l* N/A

      \[\leadsto \frac{\frac{1}{\color{blue}{\frac{1}{2} \cdot \left(\frac{1}{x} \cdot x\right)} + x \cdot \left(2 - \frac{\frac{1}{8}}{{x}^{2}}\right)}}{\sqrt{x + 1}} \]

   6. lft-mult-inverse N/A

      \[\leadsto \frac{\frac{1}{\frac{1}{2} \cdot \color{blue}{1} + x \cdot \left(2 - \frac{\frac{1}{8}}{{x}^{2}}\right)}}{\sqrt{x + 1}} \]

   7. metadata-eval N/A

      \[\leadsto \frac{\frac{1}{\color{blue}{\frac{1}{2}} + x \cdot \left(2 - \frac{\frac{1}{8}}{{x}^{2}}\right)}}{\sqrt{x + 1}} \]

   8. lower-+.f64 N/A

      \[\leadsto \frac{\frac{1}{\color{blue}{\frac{1}{2} + x \cdot \left(2 - \frac{\frac{1}{8}}{{x}^{2}}\right)}}}{\sqrt{x + 1}} \]

   9. lower-*.f64 N/A

      \[\leadsto \frac{\frac{1}{\frac{1}{2} + \color{blue}{x \cdot \left(2 - \frac{\frac{1}{8}}{{x}^{2}}\right)}}}{\sqrt{x + 1}} \]

   10. lower--.f64 N/A

       \[\leadsto \frac{\frac{1}{\frac{1}{2} + x \cdot \color{blue}{\left(2 - \frac{\frac{1}{8}}{{x}^{2}}\right)}}}{\sqrt{x + 1}} \]

   11. lower-/.f64 N/A

       \[\leadsto \frac{\frac{1}{\frac{1}{2} + x \cdot \left(2 - \color{blue}{\frac{\frac{1}{8}}{{x}^{2}}}\right)}}{\sqrt{x + 1}} \]

   12. unpow2 N/A

       \[\leadsto \frac{\frac{1}{\frac{1}{2} + x \cdot \left(2 - \frac{\frac{1}{8}}{\color{blue}{x \cdot x}}\right)}}{\sqrt{x + 1}} \]

   13. lower-*.f64 99.4

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

10. Applied rewrites 99.4%

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

11. Final simplification 99.4%

    \[\leadsto \frac{\frac{1}{\left(2 - \frac{0.125}{x \cdot x}\right) \cdot x + 0.5}}{\sqrt{1 + x}} \]
12. Add Preprocessing

Alternative 2: 98.7% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 38.9%

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

4. Applied rewrites 40.2%

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

5. Taylor expanded in x around 0

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

7. Applied rewrites 84.3%

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

   1. lift-/.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-/l/ N/A

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

   4. associate-/r* N/A

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

   5. lower-/.f64 N/A

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

   6. lower-/.f64 84.3

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

9. Applied rewrites 84.3%

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

10. Taylor expanded in x around inf

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

    1. +-commutative N/A

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

    2. distribute-rgt-in N/A

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

    3. associate-*l* N/A

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

    4. lft-mult-inverse N/A

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

    5. metadata-eval N/A

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

    6. *-lft-identity N/A

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

    7. lower-+.f64 99.0

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

12. Applied rewrites 99.0%

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

13. Final simplification 99.0%

    \[\leadsto \frac{\frac{1}{\sqrt{1 + x}}}{\left(0.5 + x\right) + x} \]
14. Add Preprocessing

Alternative 3: 97.7% accurate, 1.4× 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 38.9%

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

4. Applied rewrites 40.2%

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

5. Taylor expanded in x around inf

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

   1. lower-/.f64 98.2

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

7. Applied rewrites 98.2%

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

8. Final simplification 98.2%

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

Alternative 4: 97.5% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 38.9%

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

4. Applied rewrites 40.2%

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

5. Taylor expanded in x around inf

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

   1. lower-/.f64 98.2

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

7. Applied rewrites 98.2%

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

8. Taylor expanded in x around inf

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

   1. lower-sqrt.f64 98.1

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

10. Applied rewrites 98.1%

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

Alternative 5: 81.5% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 38.9%

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

3. Taylor expanded in x around inf

   \[\leadsto \color{blue}{\frac{\frac{1}{2} \cdot \left(\sqrt{\frac{1}{{x}^{3}}} \cdot \left(1 + \frac{1}{4} \cdot x\right)\right) - \left(\frac{-1}{2} \cdot \sqrt{x} + \frac{1}{2} \cdot \sqrt{\frac{1}{x}}\right)}{{x}^{2}}} \]

5. Applied rewrites 83.6%

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

6. Taylor expanded in x around inf

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

8. Applied rewrites 82.8%

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

9. Final simplification 82.8%

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

Alternative 6: 37.5% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{0.5}{x}}{\mathsf{fma}\left(0.5, x, 1\right)} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ (/ 0.5 x) (fma 0.5 x 1.0)))

C

double code(double x) {
	return (0.5 / x) / fma(0.5, x, 1.0);
}

Julia

function code(x)
	return Float64(Float64(0.5 / x) / fma(0.5, x, 1.0))
end

Wolfram

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

Derivation

1. Initial program 38.9%

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

4. Applied rewrites 40.2%

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

5. Taylor expanded in x around inf

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

   1. lower-/.f64 98.2

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

7. Applied rewrites 98.2%

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

8. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. lower-fma.f64 38.8

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

10. Applied rewrites 38.8%

    \[\leadsto \frac{\frac{0.5}{x}}{\color{blue}{\mathsf{fma}\left(0.5, x, 1\right)}} \]
11. Add Preprocessing

Alternative 7: 36.7% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 38.9%

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

3. Taylor expanded in x around 0

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

   1. lower-sqrt.f64 N/A

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

   2. lower-/.f64 5.5

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

5. Applied rewrites 5.5%

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

7. Applied rewrites 38.0%

   \[\leadsto \sqrt{\frac{x}{x \cdot x}} \]
8. Add Preprocessing

Alternative 8: 7.9% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

public static double code(double x) {
	return (0.5 / x) / 1.0;
}

Python

def code(x):
	return (0.5 / x) / 1.0

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 38.9%

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

4. Applied rewrites 40.2%

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

5. Taylor expanded in x around inf

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

   1. lower-/.f64 98.2

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

7. Applied rewrites 98.2%

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

8. Taylor expanded in x around 0

   \[\leadsto \frac{\frac{\frac{1}{2}}{x}}{\color{blue}{1}} \]
9. Step-by-step derivation

10. Applied rewrites 7.8%

    \[\leadsto \frac{\frac{0.5}{x}}{\color{blue}{1}} \]
11. Add Preprocessing

Alternative 9: 5.7% accurate, 2.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 38.9%

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

3. Taylor expanded in x around 0

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

   1. lower-sqrt.f64 N/A

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

   2. lower-/.f64 5.5

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

5. Applied rewrites 5.5%

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

Developer Target 1: 98.5% 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]

Developer Target 2: 38.4% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ {x}^{-0.5} - {\left(x + 1\right)}^{-0.5} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024240 
(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))))))

  :alt
  (! :herbie-platform default (- (pow x -1/2) (pow (+ x 1) -1/2)))

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