2isqrt (example 3.6)

Percentage Accurate: 37.7% → 98.7%

Time: 7.7s

Alternatives: 5

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: 37.7% 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, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

function code(x)
	return Float64(Float64(Float64(-1.0) / sqrt(Float64(1.0 + x))) / fma(-2.0, x, -0.5))
end

Wolfram

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

Derivation

1. Initial program 38.0%

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

   1. lift--.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. frac-sub N/A

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

   5. div-inv N/A

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

   6. metadata-eval N/A

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

   7. div-inv N/A

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

   8. *-lft-identity N/A

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

   9. flip-- N/A

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

   10. metadata-eval N/A

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

   11. frac-times N/A

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

   12. frac-2neg N/A

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

   13. metadata-eval N/A

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

   14. lift-/.f64 N/A

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

   15. associate-*l/ N/A

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

   16. neg-mul-1 N/A

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

4. Applied rewrites 40.1%

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

5. Taylor expanded in x around inf

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 N/A

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

   3. distribute-rgt-in N/A

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

   4. associate-*l* N/A

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

   5. lft-mult-inverse N/A

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

   6. metadata-eval N/A

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

   7. lower-fma.f64 N/A

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

   8. associate-*r/ N/A

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

   9. metadata-eval N/A

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

   10. lower-/.f64 N/A

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

   11. unpow2 N/A

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

   12. lower-*.f64 39.3

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

7. Applied rewrites 39.3%

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

8. Taylor expanded in x around 0

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

10. Applied rewrites 84.0%

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

    1. lift-*.f64 N/A

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

    2. lift-/.f64 N/A

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

    3. frac-2neg N/A

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

    4. metadata-eval N/A

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

    5. un-div-inv N/A

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

    6. lower-/.f64 N/A

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

    7. lower-neg.f64 84.0

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

    8. lift-+.f64 N/A

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

    9. +-commutative N/A

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

    10. lower-+.f64 84.0

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

12. Applied rewrites 98.8%

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

13. Final simplification 98.8%

    \[\leadsto \frac{\frac{-1}{\sqrt{1 + x}}}{\mathsf{fma}\left(-2, x, -0.5\right)} \]
14. Add Preprocessing

Alternative 2: 5.7% accurate, 0.4× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 (sqrt (pow x -1.0)))

C

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

Fortran

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

Java

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

Python

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

Julia

function code(x)
	return sqrt((x ^ -1.0))
end

MATLAB

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

Wolfram

code[x_] := N[Sqrt[N[Power[x, -1.0], $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 38.0%

   \[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}} \]
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 5.7

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

5. Applied rewrites 5.7%

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

6. Final simplification 5.7%

   \[\leadsto \sqrt{{x}^{-1}} \]
7. Add Preprocessing

Alternative 3: 97.5% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 38.0%

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

3. Taylor expanded in x around inf

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

   1. div-sub N/A

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

   2. *-commutative N/A

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

   3. associate-/l* N/A

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

   4. *-commutative N/A

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

   5. associate-/l* N/A

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

   6. distribute-rgt-out-- N/A

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

   7. lower-*.f64 N/A

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

   8. unpow2 N/A

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

   9. associate-/r* N/A

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

   10. lower-/.f64 N/A

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

   11. lower-/.f64 N/A

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

   12. lower--.f64 N/A

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

   13. lower-sqrt.f64 N/A

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

   14. lower-/.f64 N/A

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

   15. lower-sqrt.f64 84.9

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

5. Applied rewrites 84.9%

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

7. Applied rewrites 97.8%

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

8. Taylor expanded in x around inf

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

10. Applied rewrites 97.7%

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

12. Applied rewrites 97.8%

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

Alternative 4: 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 38.0%

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

   1. lift--.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. clear-num N/A

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

   4. lift-/.f64 N/A

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

   5. frac-sub N/A

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

   6. div-inv N/A

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

   7. metadata-eval N/A

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

   8. *-rgt-identity N/A

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

   9. *-commutative N/A

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

   10. associate-/r* N/A

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

   11. lower-/.f64 N/A

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

4. Applied rewrites 38.0%

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

5. Taylor expanded in x around inf

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

   1. lower-/.f64 97.8

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

7. Applied rewrites 97.8%

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

Alternative 5: 36.0% 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 38.0%

   \[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}} \]
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 5.7

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

5. Applied rewrites 5.7%

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

7. Applied rewrites 5.7%

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

9. Applied rewrites 36.5%

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

Developer Target 1: 37.7% 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 2024302 
(FPCore (x)
  :name "2isqrt (example 3.6)"
  :precision binary64
  :pre (and (> x 1.0) (< x 1e+308))

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

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