2isqrt (example 3.6)

Percentage Accurate: 38.7% → 98.7%

Time: 10.2s

Alternatives: 11

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: 38.7% 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, 1.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 40.4%

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

4. Applied rewrites 41.7%

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

   1. lift-/.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-/l/ N/A

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

   4. lift--.f64 N/A

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

   5. lift-+.f64 N/A

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

   6. associate--l+ N/A

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

   7. +-inverses N/A

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

   8. metadata-eval N/A

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

   9. lower-/.f64 N/A

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

   10. lower-*.f64 81.0

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

6. Applied rewrites 81.0%

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

7. Taylor expanded in x around inf

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

   1. +-commutative N/A

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

   2. distribute-rgt-in N/A

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

   3. associate-*l* N/A

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

   4. lft-mult-inverse N/A

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

   5. metadata-eval N/A

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

   6. *-lft-identity N/A

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

   7. lower-+.f64 97.5

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

9. Applied rewrites 97.5%

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

    1. lift-/.f64 N/A

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

    2. lift-*.f64 N/A

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

    3. lift-+.f64 N/A

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

    4. lift-sqrt.f64 N/A

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

    5. *-commutative N/A

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

    6. associate-/r* N/A

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

    7. lower-/.f64 N/A

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

    8. lower-/.f64 N/A

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

    9. lift-sqrt.f64 N/A

       \[\leadsto \frac{\frac{1}{x + \mathsf{Rewrite=>}\left(lower-+.f64, \left(x + \frac{1}{2}\right)\right)}}{\color{blue}{\sqrt{1 + x}}} \]

    10. lift-+.f64 N/A

        \[\leadsto \frac{\frac{1}{x + \mathsf{Rewrite=>}\left(lower-+.f64, \left(x + \frac{1}{2}\right)\right)}}{\sqrt{\color{blue}{1 + x}}} \]

11. Applied rewrites 99.4%

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

Alternative 2: 97.8% 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 40.4%

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

4. Applied rewrites 41.7%

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

5. Taylor expanded in x around inf

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

   1. lower-/.f64 98.8

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

7. Applied rewrites 98.8%

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

Alternative 3: 97.6% 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 40.4%

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

4. Applied rewrites 41.7%

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

5. Taylor expanded in x around inf

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

   1. lower-/.f64 98.8

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

7. Applied rewrites 98.8%

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

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 98.7

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

10. Applied rewrites 98.7%

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

Alternative 4: 96.4% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 40.4%

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

4. Applied rewrites 41.7%

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

   1. lift-/.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-/l/ N/A

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

   4. lift--.f64 N/A

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

   5. lift-+.f64 N/A

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

   6. associate--l+ N/A

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

   7. +-inverses N/A

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

   8. metadata-eval N/A

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

   9. lower-/.f64 N/A

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

   10. lower-*.f64 81.0

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

6. Applied rewrites 81.0%

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

7. Taylor expanded in x around inf

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

   1. +-commutative N/A

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

   2. distribute-rgt-in N/A

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

   3. associate-*l* N/A

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

   4. lft-mult-inverse N/A

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

   5. metadata-eval N/A

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

   6. *-lft-identity N/A

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

   7. lower-+.f64 97.5

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

9. Applied rewrites 97.5%

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

10. Taylor expanded in x around inf

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

    1. lower-sqrt.f64 96.8

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

12. Applied rewrites 96.8%

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

13. Final simplification 96.8%

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

Alternative 5: 80.7% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 40.4%

   \[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}} \]
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 80.6%

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

6. Taylor expanded in x around inf

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

8. Applied rewrites 80.0%

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

Alternative 6: 80.7% accurate, 1.5× speedup

Math

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

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(sqrt(x) * Float64(0.5 / Float64(x * x)))
end

MATLAB

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

Wolfram

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

Derivation

1. Initial program 40.4%

   \[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}} \]
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 80.6%

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

6. Taylor expanded in x around inf

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

8. Applied rewrites 80.0%

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

10. Applied rewrites 80.0%

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

Alternative 7: 38.0% 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 40.4%

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

4. Applied rewrites 41.7%

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

5. Taylor expanded in x around inf

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

   1. lower-/.f64 98.8

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

7. Applied rewrites 98.8%

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

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 40.7

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

10. Applied rewrites 40.7%

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

Alternative 8: 38.0% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 40.4%

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

4. Applied rewrites 41.7%

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

   1. lift-/.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-/l/ N/A

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

   4. lift--.f64 N/A

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

   5. lift-+.f64 N/A

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

   6. associate--l+ N/A

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

   7. +-inverses N/A

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

   8. metadata-eval N/A

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

   9. lower-/.f64 N/A

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

   10. lower-*.f64 81.0

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

6. Applied rewrites 81.0%

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

7. Taylor expanded in x around inf

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

   1. +-commutative N/A

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

   2. distribute-rgt-in N/A

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

   3. associate-*l* N/A

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

   4. lft-mult-inverse N/A

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

   5. metadata-eval N/A

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

   6. *-lft-identity N/A

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

   7. lower-+.f64 97.5

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

9. Applied rewrites 97.5%

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

10. Taylor expanded in x around 0

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

    1. +-commutative N/A

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

    2. *-commutative N/A

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

    3. lower-fma.f64 40.7

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

12. Applied rewrites 40.7%

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

13. Final simplification 40.7%

    \[\leadsto \frac{1}{\left(x + \left(x + 0.5\right)\right) \cdot \mathsf{fma}\left(x, 0.5, 1\right)} \]
14. Add Preprocessing

Alternative 9: 7.9% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 40.4%

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

4. Applied rewrites 41.7%

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

   1. lift-/.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-/l/ N/A

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

   4. lift--.f64 N/A

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

   5. lift-+.f64 N/A

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

   6. associate--l+ N/A

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

   7. +-inverses N/A

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

   8. metadata-eval N/A

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

   9. lower-/.f64 N/A

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

   10. lower-*.f64 81.0

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

6. Applied rewrites 81.0%

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

7. Taylor expanded in x around inf

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

   1. +-commutative N/A

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

   2. distribute-rgt-in N/A

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

   3. associate-*l* N/A

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

   4. lft-mult-inverse N/A

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

   5. metadata-eval N/A

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

   6. *-lft-identity N/A

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

   7. lower-+.f64 97.5

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

9. Applied rewrites 97.5%

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

10. Taylor expanded in x around 0

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

12. Applied rewrites 7.6%

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

13. Final simplification 7.6%

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

Alternative 10: 5.6% 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 40.4%

   \[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}} \]
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 5.4

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

5. Applied rewrites 5.4%

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

Alternative 11: 2.6% accurate, 4.1× speedup

Math

\[\begin{array}{l} \\ \frac{-2}{1} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ -2.0 1.0))

C

double code(double x) {
	return -2.0 / 1.0;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (-2.0d0) / 1.0d0
end function

Java

public static double code(double x) {
	return -2.0 / 1.0;
}

Python

def code(x):
	return -2.0 / 1.0

Julia

function code(x)
	return Float64(-2.0 / 1.0)
end

MATLAB

function tmp = code(x)
	tmp = -2.0 / 1.0;
end

Wolfram

code[x_] := N[(-2.0 / 1.0), $MachinePrecision]

Derivation

1. Initial program 40.4%

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

4. Applied rewrites 41.7%

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

5. Taylor expanded in x around inf

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

   1. lower-/.f64 98.8

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

7. Applied rewrites 98.8%

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

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 7.6%

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

11. Taylor expanded in x around -inf

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

13. Applied rewrites 2.6%

    \[\leadsto \frac{\color{blue}{-2}}{1} \]
14. Add Preprocessing

Developer Target 1: 98.4% 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: 38.7% 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 2024238 
(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)))))