2isqrt (example 3.6)

Percentage Accurate: 38.8% → 99.2%

Time: 8.5s

Alternatives: 7

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: 38.8% 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: 99.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{\left(0.5 + \frac{0.0625}{x \cdot x}\right) - \frac{0.125}{x}}{x}}{\sqrt{1 + x}} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/ (/ (- (+ 0.5 (/ 0.0625 (* x x))) (/ 0.125 x)) x) (sqrt (+ 1.0 x))))

C

double code(double x) {
	return (((0.5 + (0.0625 / (x * x))) - (0.125 / x)) / x) / sqrt((1.0 + x));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (((0.5d0 + (0.0625d0 / (x * x))) - (0.125d0 / x)) / x) / sqrt((1.0d0 + x))
end function

Java

public static double code(double x) {
	return (((0.5 + (0.0625 / (x * x))) - (0.125 / x)) / x) / Math.sqrt((1.0 + x));
}

Python

def code(x):
	return (((0.5 + (0.0625 / (x * x))) - (0.125 / x)) / x) / math.sqrt((1.0 + x))

Julia

function code(x)
	return Float64(Float64(Float64(Float64(0.5 + Float64(0.0625 / Float64(x * x))) - Float64(0.125 / x)) / x) / sqrt(Float64(1.0 + x)))
end

MATLAB

function tmp = code(x)
	tmp = (((0.5 + (0.0625 / (x * x))) - (0.125 / x)) / x) / sqrt((1.0 + x));
end

Wolfram

code[x_] := N[(N[(N[(N[(0.5 + N[(0.0625 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(0.125 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] / N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 37.6%

   \[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}} \]
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. associate-/r* N/A

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

   10. lower-/.f64 N/A

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

4. Applied rewrites 37.6%

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

5. Taylor expanded in x around inf

   \[\leadsto \frac{\color{blue}{\frac{\left(\frac{1}{2} + \frac{1}{16} \cdot \frac{1}{{x}^{2}}\right) - \frac{\frac{1}{8}}{x}}{x}}}{\sqrt{x + 1}} \]
6. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \frac{\color{blue}{\frac{\left(\frac{1}{2} + \frac{1}{16} \cdot \frac{1}{{x}^{2}}\right) - \frac{\frac{1}{8}}{x}}{x}}}{\sqrt{x + 1}} \]

   2. lower--.f64 N/A

      \[\leadsto \frac{\frac{\color{blue}{\left(\frac{1}{2} + \frac{1}{16} \cdot \frac{1}{{x}^{2}}\right) - \frac{\frac{1}{8}}{x}}}{x}}{\sqrt{x + 1}} \]

   3. +-commutative N/A

      \[\leadsto \frac{\frac{\color{blue}{\left(\frac{1}{16} \cdot \frac{1}{{x}^{2}} + \frac{1}{2}\right)} - \frac{\frac{1}{8}}{x}}{x}}{\sqrt{x + 1}} \]

   4. lower-+.f64 N/A

      \[\leadsto \frac{\frac{\color{blue}{\left(\frac{1}{16} \cdot \frac{1}{{x}^{2}} + \frac{1}{2}\right)} - \frac{\frac{1}{8}}{x}}{x}}{\sqrt{x + 1}} \]

   5. associate-*r/ N/A

      \[\leadsto \frac{\frac{\left(\color{blue}{\frac{\frac{1}{16} \cdot 1}{{x}^{2}}} + \frac{1}{2}\right) - \frac{\frac{1}{8}}{x}}{x}}{\sqrt{x + 1}} \]

   6. metadata-eval N/A

      \[\leadsto \frac{\frac{\left(\frac{\color{blue}{\frac{1}{16}}}{{x}^{2}} + \frac{1}{2}\right) - \frac{\frac{1}{8}}{x}}{x}}{\sqrt{x + 1}} \]

   7. lower-/.f64 N/A

      \[\leadsto \frac{\frac{\left(\color{blue}{\frac{\frac{1}{16}}{{x}^{2}}} + \frac{1}{2}\right) - \frac{\frac{1}{8}}{x}}{x}}{\sqrt{x + 1}} \]

   8. unpow2 N/A

      \[\leadsto \frac{\frac{\left(\frac{\frac{1}{16}}{\color{blue}{x \cdot x}} + \frac{1}{2}\right) - \frac{\frac{1}{8}}{x}}{x}}{\sqrt{x + 1}} \]

   9. lower-*.f64 N/A

      \[\leadsto \frac{\frac{\left(\frac{\frac{1}{16}}{\color{blue}{x \cdot x}} + \frac{1}{2}\right) - \frac{\frac{1}{8}}{x}}{x}}{\sqrt{x + 1}} \]

   10. lower-/.f64 99.4

       \[\leadsto \frac{\frac{\left(\frac{0.0625}{x \cdot x} + 0.5\right) - \color{blue}{\frac{0.125}{x}}}{x}}{\sqrt{x + 1}} \]

7. Applied rewrites 99.4%

   \[\leadsto \frac{\color{blue}{\frac{\left(\frac{0.0625}{x \cdot x} + 0.5\right) - \frac{0.125}{x}}{x}}}{\sqrt{x + 1}} \]

8. Final simplification 99.4%

   \[\leadsto \frac{\frac{\left(0.5 + \frac{0.0625}{x \cdot x}\right) - \frac{0.125}{x}}{x}}{\sqrt{1 + x}} \]
9. Add Preprocessing

Alternative 2: 99.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{\frac{\frac{0.0625}{x} - 0.125}{x} + 0.5}{x}}{\sqrt{1 + x}} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/ (/ (+ (/ (- (/ 0.0625 x) 0.125) x) 0.5) x) (sqrt (+ 1.0 x))))

C

double code(double x) {
	return (((((0.0625 / x) - 0.125) / x) + 0.5) / x) / sqrt((1.0 + x));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (((((0.0625d0 / x) - 0.125d0) / x) + 0.5d0) / x) / sqrt((1.0d0 + x))
end function

Java

public static double code(double x) {
	return (((((0.0625 / x) - 0.125) / x) + 0.5) / x) / Math.sqrt((1.0 + x));
}

Python

def code(x):
	return (((((0.0625 / x) - 0.125) / x) + 0.5) / x) / math.sqrt((1.0 + x))

Julia

function code(x)
	return Float64(Float64(Float64(Float64(Float64(Float64(0.0625 / x) - 0.125) / x) + 0.5) / x) / sqrt(Float64(1.0 + x)))
end

MATLAB

function tmp = code(x)
	tmp = (((((0.0625 / x) - 0.125) / x) + 0.5) / x) / sqrt((1.0 + x));
end

Wolfram

code[x_] := N[(N[(N[(N[(N[(N[(0.0625 / x), $MachinePrecision] - 0.125), $MachinePrecision] / x), $MachinePrecision] + 0.5), $MachinePrecision] / x), $MachinePrecision] / N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 37.6%

   \[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}} \]
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. associate-/r* N/A

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

   10. lower-/.f64 N/A

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

4. Applied rewrites 37.6%

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

5. Taylor expanded in x around inf

   \[\leadsto \frac{\color{blue}{\frac{\left(\frac{1}{2} + \frac{1}{16} \cdot \frac{1}{{x}^{2}}\right) - \frac{\frac{1}{8}}{x}}{x}}}{\sqrt{x + 1}} \]
6. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \frac{\color{blue}{\frac{\left(\frac{1}{2} + \frac{1}{16} \cdot \frac{1}{{x}^{2}}\right) - \frac{\frac{1}{8}}{x}}{x}}}{\sqrt{x + 1}} \]

   2. lower--.f64 N/A

      \[\leadsto \frac{\frac{\color{blue}{\left(\frac{1}{2} + \frac{1}{16} \cdot \frac{1}{{x}^{2}}\right) - \frac{\frac{1}{8}}{x}}}{x}}{\sqrt{x + 1}} \]

   3. +-commutative N/A

      \[\leadsto \frac{\frac{\color{blue}{\left(\frac{1}{16} \cdot \frac{1}{{x}^{2}} + \frac{1}{2}\right)} - \frac{\frac{1}{8}}{x}}{x}}{\sqrt{x + 1}} \]

   4. lower-+.f64 N/A

      \[\leadsto \frac{\frac{\color{blue}{\left(\frac{1}{16} \cdot \frac{1}{{x}^{2}} + \frac{1}{2}\right)} - \frac{\frac{1}{8}}{x}}{x}}{\sqrt{x + 1}} \]

   5. associate-*r/ N/A

      \[\leadsto \frac{\frac{\left(\color{blue}{\frac{\frac{1}{16} \cdot 1}{{x}^{2}}} + \frac{1}{2}\right) - \frac{\frac{1}{8}}{x}}{x}}{\sqrt{x + 1}} \]

   6. metadata-eval N/A

      \[\leadsto \frac{\frac{\left(\frac{\color{blue}{\frac{1}{16}}}{{x}^{2}} + \frac{1}{2}\right) - \frac{\frac{1}{8}}{x}}{x}}{\sqrt{x + 1}} \]

   7. lower-/.f64 N/A

      \[\leadsto \frac{\frac{\left(\color{blue}{\frac{\frac{1}{16}}{{x}^{2}}} + \frac{1}{2}\right) - \frac{\frac{1}{8}}{x}}{x}}{\sqrt{x + 1}} \]

   8. unpow2 N/A

      \[\leadsto \frac{\frac{\left(\frac{\frac{1}{16}}{\color{blue}{x \cdot x}} + \frac{1}{2}\right) - \frac{\frac{1}{8}}{x}}{x}}{\sqrt{x + 1}} \]

   9. lower-*.f64 N/A

      \[\leadsto \frac{\frac{\left(\frac{\frac{1}{16}}{\color{blue}{x \cdot x}} + \frac{1}{2}\right) - \frac{\frac{1}{8}}{x}}{x}}{\sqrt{x + 1}} \]

   10. lower-/.f64 99.4

       \[\leadsto \frac{\frac{\left(\frac{0.0625}{x \cdot x} + 0.5\right) - \color{blue}{\frac{0.125}{x}}}{x}}{\sqrt{x + 1}} \]

7. Applied rewrites 99.4%

   \[\leadsto \frac{\color{blue}{\frac{\left(\frac{0.0625}{x \cdot x} + 0.5\right) - \frac{0.125}{x}}{x}}}{\sqrt{x + 1}} \]

8. Taylor expanded in x around inf

   \[\leadsto \frac{\color{blue}{\frac{\left(\frac{1}{2} + \frac{1}{16} \cdot \frac{1}{{x}^{2}}\right) - \frac{\frac{1}{8}}{x}}{x}}}{\sqrt{x + 1}} \]
9. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \frac{\color{blue}{\frac{\left(\frac{1}{2} + \frac{1}{16} \cdot \frac{1}{{x}^{2}}\right) - \frac{\frac{1}{8}}{x}}{x}}}{\sqrt{x + 1}} \]

10. Applied rewrites 99.4%

    \[\leadsto \frac{\color{blue}{\frac{\frac{\frac{0.0625}{x} - 0.125}{x} + 0.5}{x}}}{\sqrt{x + 1}} \]

11. Final simplification 99.4%

    \[\leadsto \frac{\frac{\frac{\frac{0.0625}{x} - 0.125}{x} + 0.5}{x}}{\sqrt{1 + x}} \]
12. Add Preprocessing

Alternative 3: 99.0% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \frac{-\sqrt{\frac{1}{x}}}{\mathsf{fma}\left(-2, x, -1.5\right)} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ (- (sqrt (/ 1.0 x))) (fma -2.0 x -1.5)))

C

double code(double x) {
	return -sqrt((1.0 / x)) / fma(-2.0, x, -1.5);
}

Julia

function code(x)
	return Float64(Float64(-sqrt(Float64(1.0 / x))) / fma(-2.0, x, -1.5))
end

Wolfram

code[x_] := N[((-N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision]) / N[(-2.0 * x + -1.5), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 37.6%

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

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

   13. metadata-eval N/A

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

   14. lift-/.f64 N/A

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

   15. associate-*r/ N/A

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

4. Applied rewrites 40.3%

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

5. Taylor expanded in x around 0

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 N/A

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

   3. lower-sqrt.f64 N/A

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

   4. lower-/.f64 99.4

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

7. Applied rewrites 99.4%

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

8. Taylor expanded in x around inf

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

   1. mul-1-neg N/A

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

   2. distribute-lft-in N/A

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

   3. distribute-neg-in N/A

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

   4. *-commutative N/A

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

   5. associate-*l* N/A

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

   6. distribute-lft-neg-in N/A

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

   7. metadata-eval N/A

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

   8. unpow2 N/A

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

   9. associate-*r* N/A

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

   10. lft-mult-inverse N/A

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

   11. *-lft-identity N/A

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

   12. *-commutative N/A

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

   13. lower-fma.f64 N/A

       \[\leadsto \frac{-\sqrt{\frac{1}{x}}}{\color{blue}{\mathsf{fma}\left(-2, x, \mathsf{neg}\left(\left(\frac{3}{2} \cdot \frac{1}{{x}^{2}}\right) \cdot {x}^{2}\right)\right)}} \]

   14. associate-*l* N/A

       \[\leadsto \frac{-\sqrt{\frac{1}{x}}}{\mathsf{fma}\left(-2, x, \mathsf{neg}\left(\color{blue}{\frac{3}{2} \cdot \left(\frac{1}{{x}^{2}} \cdot {x}^{2}\right)}\right)\right)} \]

   15. lft-mult-inverse N/A

       \[\leadsto \frac{-\sqrt{\frac{1}{x}}}{\mathsf{fma}\left(-2, x, \mathsf{neg}\left(\frac{3}{2} \cdot \color{blue}{1}\right)\right)} \]

10. Applied rewrites 99.2%

    \[\leadsto \frac{-\sqrt{\frac{1}{x}}}{\color{blue}{\mathsf{fma}\left(-2, x, -1.5\right)}} \]
11. Add Preprocessing

Alternative 4: 97.9% 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 37.6%

   \[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}} \]
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. associate-/r* N/A

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

   10. lower-/.f64 N/A

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

4. Applied rewrites 37.6%

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

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 97.6

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

7. Applied rewrites 97.6%

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

8. Final simplification 97.6%

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

Alternative 5: 82.2% 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 37.6%

   \[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}} \]
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 85.0%

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

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 83.4%

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

9. Final simplification 83.4%

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

Alternative 6: 36.6% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 37.6%

   \[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}} \]
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. associate-/r* N/A

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

   10. lower-/.f64 N/A

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

4. Applied rewrites 37.6%

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

   1. lift-/.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-/l/ N/A

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

   4. lift--.f64 N/A

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

   5. flip-- N/A

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

   6. +-commutative N/A

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

   7. lift-+.f64 N/A

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

   8. associate-/l/ N/A

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

6. Applied rewrites 40.3%

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

7. Taylor expanded in x around 0

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

   1. distribute-lft-in N/A

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

   2. *-rgt-identity N/A

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

   3. rem-square-sqrt N/A

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

   4. lower-+.f64 N/A

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

   5. lower-sqrt.f64 34.9

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

9. Applied rewrites 34.9%

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

10. Final simplification 34.9%

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

Alternative 7: 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 37.6%

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

Developer Target 1: 38.8% 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 2024248 
(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)))))