2isqrt (example 3.6)

Percentage Accurate: 39.1% → 98.9%

Time: 8.0s

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: 39.1% 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: 98.9% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 35.6%

   \[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}} \]
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. clear-num N/A

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

   11. lower-/.f64 N/A

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

   12. lower-/.f64 N/A

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

   13. lower-/.f64 N/A

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

4. Applied rewrites 35.6%

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

5. Taylor expanded in x around inf

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

   1. lower-/.f64 97.2

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

7. Applied rewrites 97.2%

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

   1. lift-/.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. frac-2neg N/A

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

   4. associate-/r/ N/A

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

   5. distribute-neg-frac2 N/A

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

   6. lift-sqrt.f64 N/A

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

   7. lift-+.f64 N/A

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

   8. lower-*.f64 N/A

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

9. Applied rewrites 98.0%

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

10. Taylor expanded in x around inf

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

    1. lower-/.f64 N/A

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

    2. lower--.f64 N/A

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

    3. +-commutative N/A

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

    4. lower-+.f64 N/A

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

    5. associate-*r/ N/A

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

    6. metadata-eval N/A

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

    7. lower-/.f64 N/A

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

    8. unpow2 N/A

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

    9. lower-*.f64 N/A

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

    10. lower-/.f64 99.1

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

12. Applied rewrites 99.1%

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

13. Final simplification 99.1%

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

Alternative 2: 98.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \frac{\mathsf{fma}\left(\frac{--0.75}{\sqrt{x} \cdot x}, -0.5, 0.5 \cdot \sqrt{{x}^{-1}}\right)}{x} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/ (fma (/ (- -0.75) (* (sqrt x) x)) -0.5 (* 0.5 (sqrt (pow x -1.0)))) x))

C

double code(double x) {
	return fma((-(-0.75) / (sqrt(x) * x)), -0.5, (0.5 * sqrt(pow(x, -1.0)))) / x;
}

Julia

function code(x)
	return Float64(fma(Float64(Float64(-(-0.75)) / Float64(sqrt(x) * x)), -0.5, Float64(0.5 * sqrt((x ^ -1.0)))) / x)
end

Wolfram

code[x_] := N[(N[(N[((--0.75) / N[(N[Sqrt[x], $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision] * -0.5 + N[(0.5 * N[Sqrt[N[Power[x, -1.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]

Derivation

1. Initial program 35.6%

   \[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}} \]
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 80.1%

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

6. Taylor expanded in x around inf

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

8. Applied rewrites 99.0%

   \[\leadsto \frac{\mathsf{fma}\left(\frac{0.75 \cdot \sqrt{\frac{1}{x}}}{x}, -0.5, 0.5 \cdot \sqrt{\frac{1}{x}}\right)}{\color{blue}{x}} \]
9. Step-by-step derivation

10. Applied rewrites 99.0%

    \[\leadsto \frac{\mathsf{fma}\left(\frac{\frac{0.75}{\sqrt{x}}}{x}, -0.5, 0.5 \cdot \sqrt{\frac{1}{x}}\right)}{x} \]
11. Step-by-step derivation

12. Applied rewrites 99.0%

    \[\leadsto \frac{\mathsf{fma}\left(\frac{-0.75}{\sqrt{x} \cdot \left(-x\right)}, -0.5, 0.5 \cdot \sqrt{\frac{1}{x}}\right)}{x} \]

13. Final simplification 99.0%

    \[\leadsto \frac{\mathsf{fma}\left(\frac{--0.75}{\sqrt{x} \cdot x}, -0.5, 0.5 \cdot \sqrt{{x}^{-1}}\right)}{x} \]
14. Add Preprocessing

Alternative 3: 98.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \frac{\mathsf{fma}\left(\frac{0.75}{\sqrt{x}}, \frac{-0.5}{x}, \frac{0.5}{\sqrt{x}}\right)}{x} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/ (fma (/ 0.75 (sqrt x)) (/ -0.5 x) (/ 0.5 (sqrt x))) x))

C

double code(double x) {
	return fma((0.75 / sqrt(x)), (-0.5 / x), (0.5 / sqrt(x))) / x;
}

Julia

function code(x)
	return Float64(fma(Float64(0.75 / sqrt(x)), Float64(-0.5 / x), Float64(0.5 / sqrt(x))) / x)
end

Wolfram

code[x_] := N[(N[(N[(0.75 / N[Sqrt[x], $MachinePrecision]), $MachinePrecision] * N[(-0.5 / x), $MachinePrecision] + N[(0.5 / N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]

Derivation

1. Initial program 35.6%

   \[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}} \]
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 80.1%

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

6. Taylor expanded in x around inf

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

8. Applied rewrites 99.0%

   \[\leadsto \frac{\mathsf{fma}\left(\frac{0.75 \cdot \sqrt{\frac{1}{x}}}{x}, -0.5, 0.5 \cdot \sqrt{\frac{1}{x}}\right)}{\color{blue}{x}} \]
9. Step-by-step derivation

10. Applied rewrites 99.0%

    \[\leadsto \frac{\mathsf{fma}\left(\frac{0.75}{\sqrt{x}}, \frac{-0.5}{x}, \frac{0.5}{\sqrt{x}}\right)}{x} \]
11. Add Preprocessing

Alternative 4: 97.6% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 35.6%

   \[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}} \]
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. clear-num N/A

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

   11. lower-/.f64 N/A

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

   12. lower-/.f64 N/A

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

   13. lower-/.f64 N/A

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

4. Applied rewrites 35.6%

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

5. Taylor expanded in x around inf

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

   1. lower-/.f64 97.2

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

7. Applied rewrites 97.2%

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

   1. lift-/.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. clear-num N/A

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

   4. lower-/.f64 98.2

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

9. Applied rewrites 98.2%

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

Alternative 5: 81.1% 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 35.6%

   \[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}} \]
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 80.1%

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

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 79.2%

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

Alternative 6: 37.4% 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 35.6%

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

7. Applied rewrites 34.6%

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

Alternative 7: 7.9% accurate, 2.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 35.6%

   \[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}} \]
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. clear-num N/A

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

   11. lower-/.f64 N/A

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

   12. lower-/.f64 N/A

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

   13. lower-/.f64 N/A

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

4. Applied rewrites 35.6%

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

5. Taylor expanded in x around inf

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

   1. lower-/.f64 97.2

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

7. Applied rewrites 97.2%

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

   1. lift-/.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. frac-2neg N/A

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

   4. associate-/r/ N/A

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

   5. distribute-neg-frac2 N/A

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

   6. lift-sqrt.f64 N/A

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

   7. lift-+.f64 N/A

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

   8. lower-*.f64 N/A

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

9. Applied rewrites 98.0%

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

10. Taylor expanded in x around 0

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

12. Applied rewrites 7.9%

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

13. Final simplification 7.9%

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

Developer Target 1: 39.2% 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 2024323 
(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)))))