2isqrt (example 3.6)

Percentage Accurate: 38.5% → 99.3%

Time: 10.7s

Alternatives: 9

Speedup: 2.0×

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: 38.5% 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: 99.3% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \frac{{\left(x + 1\right)}^{-0.5}}{x \cdot \left(2 + \frac{0.5 + \frac{-0.125 + \frac{0.0625}{x}}{x}}{x}\right)} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/
  (pow (+ x 1.0) -0.5)
  (* x (+ 2.0 (/ (+ 0.5 (/ (+ -0.125 (/ 0.0625 x)) x)) x)))))

C

double code(double x) {
	return pow((x + 1.0), -0.5) / (x * (2.0 + ((0.5 + ((-0.125 + (0.0625 / x)) / x)) / x)));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = ((x + 1.0d0) ** (-0.5d0)) / (x * (2.0d0 + ((0.5d0 + (((-0.125d0) + (0.0625d0 / x)) / x)) / x)))
end function

Java

public static double code(double x) {
	return Math.pow((x + 1.0), -0.5) / (x * (2.0 + ((0.5 + ((-0.125 + (0.0625 / x)) / x)) / x)));
}

Python

def code(x):
	return math.pow((x + 1.0), -0.5) / (x * (2.0 + ((0.5 + ((-0.125 + (0.0625 / x)) / x)) / x)))

Julia

function code(x)
	return Float64((Float64(x + 1.0) ^ -0.5) / Float64(x * Float64(2.0 + Float64(Float64(0.5 + Float64(Float64(-0.125 + Float64(0.0625 / x)) / x)) / x))))
end

MATLAB

function tmp = code(x)
	tmp = ((x + 1.0) ^ -0.5) / (x * (2.0 + ((0.5 + ((-0.125 + (0.0625 / x)) / x)) / x)));
end

Wolfram

code[x_] := N[(N[Power[N[(x + 1.0), $MachinePrecision], -0.5], $MachinePrecision] / N[(x * N[(2.0 + N[(N[(0.5 + N[(N[(-0.125 + N[(0.0625 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 37.8%

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

4. Applied egg-rr 40.6%

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

   1. associate-/l/ N/A

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

   2. associate--l+ N/A

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

   3. +-inverses N/A

      \[\leadsto \frac{1 + 0}{{\left(1 + x\right)}^{\color{blue}{\frac{1}{2}}} \cdot \left(x + \sqrt{x \cdot \left(1 + x\right)}\right)} \]

   4. metadata-eval N/A

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

   5. associate-/r* N/A

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

   6. pow-flip N/A

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

   7. metadata-eval N/A

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

   8. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left({\left(1 + x\right)}^{\frac{-1}{2}}\right), \color{blue}{\left(x + \sqrt{x \cdot \left(1 + x\right)}\right)}\right) \]

   9. pow-lowering-pow.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\left(1 + x\right), \frac{-1}{2}\right), \left(\color{blue}{x} + \sqrt{x \cdot \left(1 + x\right)}\right)\right) \]

   10. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(1, x\right), \frac{-1}{2}\right), \left(x + \sqrt{x \cdot \left(1 + x\right)}\right)\right) \]

   11. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(1, x\right), \frac{-1}{2}\right), \mathsf{+.f64}\left(x, \color{blue}{\left(\sqrt{x \cdot \left(1 + x\right)}\right)}\right)\right) \]

   12. sqrt-lowering-sqrt.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(1, x\right), \frac{-1}{2}\right), \mathsf{+.f64}\left(x, \mathsf{sqrt.f64}\left(\left(x \cdot \left(1 + x\right)\right)\right)\right)\right) \]

   13. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(1, x\right), \frac{-1}{2}\right), \mathsf{+.f64}\left(x, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(x, \left(1 + x\right)\right)\right)\right)\right) \]

   14. +-lowering-+.f64 82.2%

       \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(1, x\right), \frac{-1}{2}\right), \mathsf{+.f64}\left(x, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, x\right)\right)\right)\right)\right) \]

6. Applied egg-rr 82.2%

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

7. Taylor expanded in x around inf

   \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(1, x\right), \frac{-1}{2}\right), \color{blue}{\left(x \cdot \left(\left(2 + \left(\frac{1}{2} \cdot \frac{1}{x} + \frac{1}{16} \cdot \frac{1}{{x}^{3}}\right)\right) - \frac{\frac{1}{8}}{{x}^{2}}\right)\right)}\right) \]

9. Simplified 99.1%

   \[\leadsto \frac{{\left(1 + x\right)}^{-0.5}}{\color{blue}{x \cdot \left(2 + \left(\frac{0.0625}{x \cdot \left(x \cdot x\right)} + \frac{0.5 + \frac{-0.125}{x}}{x}\right)\right)}} \]

10. Taylor expanded in x around inf

    \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(1, x\right), \frac{-1}{2}\right), \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(2, \color{blue}{\left(\frac{\left(\frac{1}{2} + \frac{\frac{1}{16}}{{x}^{2}}\right) - \frac{1}{8} \cdot \frac{1}{x}}{x}\right)}\right)\right)\right) \]
11. Step-by-step derivation

    1. sub-neg N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(1, x\right), \frac{-1}{2}\right), \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(2, \left(\frac{\left(\frac{1}{2} + \frac{\frac{1}{16}}{{x}^{2}}\right) + \left(\mathsf{neg}\left(\frac{1}{8} \cdot \frac{1}{x}\right)\right)}{x}\right)\right)\right)\right) \]

    2. +-commutative N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(1, x\right), \frac{-1}{2}\right), \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(2, \left(\frac{\left(\mathsf{neg}\left(\frac{1}{8} \cdot \frac{1}{x}\right)\right) + \left(\frac{1}{2} + \frac{\frac{1}{16}}{{x}^{2}}\right)}{x}\right)\right)\right)\right) \]

    3. associate-+r+ N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(1, x\right), \frac{-1}{2}\right), \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(2, \left(\frac{\left(\left(\mathsf{neg}\left(\frac{1}{8} \cdot \frac{1}{x}\right)\right) + \frac{1}{2}\right) + \frac{\frac{1}{16}}{{x}^{2}}}{x}\right)\right)\right)\right) \]

    4. +-commutative N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(1, x\right), \frac{-1}{2}\right), \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(2, \left(\frac{\left(\frac{1}{2} + \left(\mathsf{neg}\left(\frac{1}{8} \cdot \frac{1}{x}\right)\right)\right) + \frac{\frac{1}{16}}{{x}^{2}}}{x}\right)\right)\right)\right) \]

    5. sub-neg N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(1, x\right), \frac{-1}{2}\right), \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(2, \left(\frac{\left(\frac{1}{2} - \frac{1}{8} \cdot \frac{1}{x}\right) + \frac{\frac{1}{16}}{{x}^{2}}}{x}\right)\right)\right)\right) \]

    6. associate--r- N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(1, x\right), \frac{-1}{2}\right), \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(2, \left(\frac{\frac{1}{2} - \left(\frac{1}{8} \cdot \frac{1}{x} - \frac{\frac{1}{16}}{{x}^{2}}\right)}{x}\right)\right)\right)\right) \]

    7. associate-*r/ N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(1, x\right), \frac{-1}{2}\right), \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(2, \left(\frac{\frac{1}{2} - \left(\frac{\frac{1}{8} \cdot 1}{x} - \frac{\frac{1}{16}}{{x}^{2}}\right)}{x}\right)\right)\right)\right) \]

    8. metadata-eval N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(1, x\right), \frac{-1}{2}\right), \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(2, \left(\frac{\frac{1}{2} - \left(\frac{\frac{1}{8}}{x} - \frac{\frac{1}{16}}{{x}^{2}}\right)}{x}\right)\right)\right)\right) \]

    9. unpow2 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(1, x\right), \frac{-1}{2}\right), \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(2, \left(\frac{\frac{1}{2} - \left(\frac{\frac{1}{8}}{x} - \frac{\frac{1}{16}}{x \cdot x}\right)}{x}\right)\right)\right)\right) \]

    10. associate-/r* N/A

        \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(1, x\right), \frac{-1}{2}\right), \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(2, \left(\frac{\frac{1}{2} - \left(\frac{\frac{1}{8}}{x} - \frac{\frac{\frac{1}{16}}{x}}{x}\right)}{x}\right)\right)\right)\right) \]

    11. metadata-eval N/A

        \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(1, x\right), \frac{-1}{2}\right), \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(2, \left(\frac{\frac{1}{2} - \left(\frac{\frac{1}{8}}{x} - \frac{\frac{\frac{1}{16} \cdot 1}{x}}{x}\right)}{x}\right)\right)\right)\right) \]

    12. associate-*r/ N/A

        \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(1, x\right), \frac{-1}{2}\right), \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(2, \left(\frac{\frac{1}{2} - \left(\frac{\frac{1}{8}}{x} - \frac{\frac{1}{16} \cdot \frac{1}{x}}{x}\right)}{x}\right)\right)\right)\right) \]

    13. div-sub N/A

        \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(1, x\right), \frac{-1}{2}\right), \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(2, \left(\frac{\frac{1}{2} - \frac{\frac{1}{8} - \frac{1}{16} \cdot \frac{1}{x}}{x}}{x}\right)\right)\right)\right) \]

    14. unsub-neg N/A

        \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(1, x\right), \frac{-1}{2}\right), \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(2, \left(\frac{\frac{1}{2} + \left(\mathsf{neg}\left(\frac{\frac{1}{8} - \frac{1}{16} \cdot \frac{1}{x}}{x}\right)\right)}{x}\right)\right)\right)\right) \]

    15. mul-1-neg N/A

        \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(1, x\right), \frac{-1}{2}\right), \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(2, \left(\frac{\frac{1}{2} + -1 \cdot \frac{\frac{1}{8} - \frac{1}{16} \cdot \frac{1}{x}}{x}}{x}\right)\right)\right)\right) \]

    16. /-lowering-/.f64 N/A

        \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(1, x\right), \frac{-1}{2}\right), \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(2, \mathsf{/.f64}\left(\left(\frac{1}{2} + -1 \cdot \frac{\frac{1}{8} - \frac{1}{16} \cdot \frac{1}{x}}{x}\right), \color{blue}{x}\right)\right)\right)\right) \]

12. Simplified 99.1%

    \[\leadsto \frac{{\left(1 + x\right)}^{-0.5}}{x \cdot \left(2 + \color{blue}{\frac{0.5 + \frac{-0.125 + \frac{0.0625}{x}}{x}}{x}}\right)} \]

13. Final simplification 99.1%

    \[\leadsto \frac{{\left(x + 1\right)}^{-0.5}}{x \cdot \left(2 + \frac{0.5 + \frac{-0.125 + \frac{0.0625}{x}}{x}}{x}\right)} \]
14. Add Preprocessing

Alternative 2: 99.2% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{0.5 + \frac{\frac{0.0625 - \frac{0.0390625}{x}}{x} - 0.125}{x}}{x}}{{\left(x + 1\right)}^{0.5}} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/
  (/ (+ 0.5 (/ (- (/ (- 0.0625 (/ 0.0390625 x)) x) 0.125) x)) x)
  (pow (+ x 1.0) 0.5)))

C

double code(double x) {
	return ((0.5 + ((((0.0625 - (0.0390625 / x)) / x) - 0.125) / x)) / x) / pow((x + 1.0), 0.5);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = ((0.5d0 + ((((0.0625d0 - (0.0390625d0 / x)) / x) - 0.125d0) / x)) / x) / ((x + 1.0d0) ** 0.5d0)
end function

Java

public static double code(double x) {
	return ((0.5 + ((((0.0625 - (0.0390625 / x)) / x) - 0.125) / x)) / x) / Math.pow((x + 1.0), 0.5);
}

Python

def code(x):
	return ((0.5 + ((((0.0625 - (0.0390625 / x)) / x) - 0.125) / x)) / x) / math.pow((x + 1.0), 0.5)

Julia

function code(x)
	return Float64(Float64(Float64(0.5 + Float64(Float64(Float64(Float64(0.0625 - Float64(0.0390625 / x)) / x) - 0.125) / x)) / x) / (Float64(x + 1.0) ^ 0.5))
end

MATLAB

function tmp = code(x)
	tmp = ((0.5 + ((((0.0625 - (0.0390625 / x)) / x) - 0.125) / x)) / x) / ((x + 1.0) ^ 0.5);
end

Wolfram

code[x_] := N[(N[(N[(0.5 + N[(N[(N[(N[(0.0625 - N[(0.0390625 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] - 0.125), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] / N[Power[N[(x + 1.0), $MachinePrecision], 0.5], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 37.8%

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

4. Applied egg-rr 40.6%

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

5. Taylor expanded in x around inf

   \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(\frac{\left(\frac{1}{2} + \frac{\frac{1}{16}}{{x}^{2}}\right) - \left(\frac{1}{8} \cdot \frac{1}{x} + \frac{5}{128} \cdot \frac{1}{{x}^{3}}\right)}{x}\right)}, \mathsf{pow.f64}\left(\mathsf{+.f64}\left(1, x\right), \frac{1}{2}\right)\right) \]

7. Simplified 99.1%

   \[\leadsto \frac{\color{blue}{\frac{\left(0.5 + \frac{-0.125 - \frac{-0.0625}{x}}{x}\right) - \frac{0.0390625}{x \cdot \left(x \cdot x\right)}}{x}}}{{\left(1 + x\right)}^{0.5}} \]

8. Taylor expanded in x around -inf

   \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(-1 \cdot \frac{-1 \cdot \frac{-1 \cdot \frac{\frac{5}{128} \cdot \frac{1}{x} - \frac{1}{16}}{x} - \frac{1}{8}}{x} - \frac{1}{2}}{x}\right)}, \mathsf{pow.f64}\left(\mathsf{+.f64}\left(1, x\right), \frac{1}{2}\right)\right) \]

10. Simplified 99.1%

    \[\leadsto \frac{\color{blue}{\frac{0.5 - \frac{0.125 - \frac{0.0625 - \frac{0.0390625}{x}}{x}}{x}}{x}}}{{\left(1 + x\right)}^{0.5}} \]

11. Final simplification 99.1%

    \[\leadsto \frac{\frac{0.5 + \frac{\frac{0.0625 - \frac{0.0390625}{x}}{x} - 0.125}{x}}{x}}{{\left(x + 1\right)}^{0.5}} \]
12. Add Preprocessing

Alternative 3: 99.2% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 37.8%

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

4. Applied egg-rr 40.6%

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

   1. associate-/l/ N/A

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

   2. associate--l+ N/A

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

   3. +-inverses N/A

      \[\leadsto \frac{1 + 0}{{\left(1 + x\right)}^{\color{blue}{\frac{1}{2}}} \cdot \left(x + \sqrt{x \cdot \left(1 + x\right)}\right)} \]

   4. metadata-eval N/A

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

   5. associate-/r* N/A

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

   6. pow-flip N/A

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

   7. metadata-eval N/A

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

   8. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left({\left(1 + x\right)}^{\frac{-1}{2}}\right), \color{blue}{\left(x + \sqrt{x \cdot \left(1 + x\right)}\right)}\right) \]

   9. pow-lowering-pow.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\left(1 + x\right), \frac{-1}{2}\right), \left(\color{blue}{x} + \sqrt{x \cdot \left(1 + x\right)}\right)\right) \]

   10. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(1, x\right), \frac{-1}{2}\right), \left(x + \sqrt{x \cdot \left(1 + x\right)}\right)\right) \]

   11. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(1, x\right), \frac{-1}{2}\right), \mathsf{+.f64}\left(x, \color{blue}{\left(\sqrt{x \cdot \left(1 + x\right)}\right)}\right)\right) \]

   12. sqrt-lowering-sqrt.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(1, x\right), \frac{-1}{2}\right), \mathsf{+.f64}\left(x, \mathsf{sqrt.f64}\left(\left(x \cdot \left(1 + x\right)\right)\right)\right)\right) \]

   13. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(1, x\right), \frac{-1}{2}\right), \mathsf{+.f64}\left(x, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(x, \left(1 + x\right)\right)\right)\right)\right) \]

   14. +-lowering-+.f64 82.2%

       \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(1, x\right), \frac{-1}{2}\right), \mathsf{+.f64}\left(x, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, x\right)\right)\right)\right)\right) \]

6. Applied egg-rr 82.2%

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

7. Taylor expanded in x around inf

   \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(1, x\right), \frac{-1}{2}\right), \color{blue}{\left(x \cdot \left(\left(2 + \left(\frac{1}{2} \cdot \frac{1}{x} + \frac{1}{16} \cdot \frac{1}{{x}^{3}}\right)\right) - \frac{\frac{1}{8}}{{x}^{2}}\right)\right)}\right) \]

9. Simplified 99.1%

   \[\leadsto \frac{{\left(1 + x\right)}^{-0.5}}{\color{blue}{x \cdot \left(2 + \left(\frac{0.0625}{x \cdot \left(x \cdot x\right)} + \frac{0.5 + \frac{-0.125}{x}}{x}\right)\right)}} \]

10. Taylor expanded in x around inf

    \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(1, x\right), \frac{-1}{2}\right), \color{blue}{\left(x \cdot \left(\left(2 + \frac{1}{2} \cdot \frac{1}{x}\right) - \frac{\frac{1}{8}}{{x}^{2}}\right)\right)}\right) \]

12. Simplified 99.0%

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

13. Final simplification 99.0%

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

Alternative 4: 98.8% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 37.8%

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

4. Applied egg-rr 40.6%

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

5. Taylor expanded in x around inf

   \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, x\right), x\right), \mathsf{+.f64}\left(x, \color{blue}{\left(x \cdot \left(1 + \frac{1}{2} \cdot \frac{1}{x}\right)\right)}\right)\right), \mathsf{pow.f64}\left(\mathsf{+.f64}\left(1, x\right), \frac{1}{2}\right)\right) \]
6. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, x\right), x\right), \mathsf{+.f64}\left(x, \left(x \cdot \left(\frac{1}{2} \cdot \frac{1}{x} + 1\right)\right)\right)\right), \mathsf{pow.f64}\left(\mathsf{+.f64}\left(1, x\right), \frac{1}{2}\right)\right) \]

   2. distribute-rgt-in N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, x\right), x\right), \mathsf{+.f64}\left(x, \left(\left(\frac{1}{2} \cdot \frac{1}{x}\right) \cdot x + 1 \cdot x\right)\right)\right), \mathsf{pow.f64}\left(\mathsf{+.f64}\left(1, x\right), \frac{1}{2}\right)\right) \]

   3. associate-*l* N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, x\right), x\right), \mathsf{+.f64}\left(x, \left(\frac{1}{2} \cdot \left(\frac{1}{x} \cdot x\right) + 1 \cdot x\right)\right)\right), \mathsf{pow.f64}\left(\mathsf{+.f64}\left(1, x\right), \frac{1}{2}\right)\right) \]

   4. lft-mult-inverse N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, x\right), x\right), \mathsf{+.f64}\left(x, \left(\frac{1}{2} \cdot 1 + 1 \cdot x\right)\right)\right), \mathsf{pow.f64}\left(\mathsf{+.f64}\left(1, x\right), \frac{1}{2}\right)\right) \]

   5. metadata-eval N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, x\right), x\right), \mathsf{+.f64}\left(x, \left(\frac{1}{2} + 1 \cdot x\right)\right)\right), \mathsf{pow.f64}\left(\mathsf{+.f64}\left(1, x\right), \frac{1}{2}\right)\right) \]

   6. *-lft-identity N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, x\right), x\right), \mathsf{+.f64}\left(x, \left(\frac{1}{2} + x\right)\right)\right), \mathsf{pow.f64}\left(\mathsf{+.f64}\left(1, x\right), \frac{1}{2}\right)\right) \]

   7. +-lowering-+.f64 39.6%

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, x\right), x\right), \mathsf{+.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, x\right)\right)\right), \mathsf{pow.f64}\left(\mathsf{+.f64}\left(1, x\right), \frac{1}{2}\right)\right) \]

7. Simplified 39.6%

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

   1. associate-/l/ N/A

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

   2. associate--l+ N/A

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

   3. +-inverses N/A

      \[\leadsto \frac{1 + 0}{{\left(1 + x\right)}^{\color{blue}{\frac{1}{2}}} \cdot \left(x + \left(\frac{1}{2} + x\right)\right)} \]

   4. metadata-eval N/A

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

   5. associate-/r* N/A

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

   6. pow-flip N/A

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

   7. metadata-eval N/A

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

   8. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left({\left(1 + x\right)}^{\frac{-1}{2}}\right), \color{blue}{\left(x + \left(\frac{1}{2} + x\right)\right)}\right) \]

   9. pow-lowering-pow.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\left(1 + x\right), \frac{-1}{2}\right), \left(\color{blue}{x} + \left(\frac{1}{2} + x\right)\right)\right) \]

   10. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(1, x\right), \frac{-1}{2}\right), \left(x + \left(\frac{1}{2} + x\right)\right)\right) \]

   11. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(1, x\right), \frac{-1}{2}\right), \mathsf{+.f64}\left(x, \color{blue}{\left(\frac{1}{2} + x\right)}\right)\right) \]

   12. +-commutative N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(1, x\right), \frac{-1}{2}\right), \mathsf{+.f64}\left(x, \left(x + \color{blue}{\frac{1}{2}}\right)\right)\right) \]

   13. +-lowering-+.f64 98.7%

       \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(1, x\right), \frac{-1}{2}\right), \mathsf{+.f64}\left(x, \mathsf{+.f64}\left(x, \color{blue}{\frac{1}{2}}\right)\right)\right) \]

9. Applied egg-rr 98.7%

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

10. Final simplification 98.7%

    \[\leadsto \frac{{\left(x + 1\right)}^{-0.5}}{x + \left(x + 0.5\right)} \]
11. Add Preprocessing

Alternative 5: 98.0% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ 0.5 \cdot {x}^{-1.5} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (* 0.5 (pow x -1.5)))

C

double code(double x) {
	return 0.5 * pow(x, -1.5);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 0.5d0 * (x ** (-1.5d0))
end function

Java

public static double code(double x) {
	return 0.5 * Math.pow(x, -1.5);
}

Python

def code(x):
	return 0.5 * math.pow(x, -1.5)

Julia

function code(x)
	return Float64(0.5 * (x ^ -1.5))
end

MATLAB

function tmp = code(x)
	tmp = 0.5 * (x ^ -1.5);
end

Wolfram

code[x_] := N[(0.5 * N[Power[x, -1.5], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 37.8%

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

4. Applied egg-rr 40.6%

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

5. Taylor expanded in x around inf

   \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(\frac{\frac{1}{2}}{x}\right)}, \mathsf{pow.f64}\left(\mathsf{+.f64}\left(1, x\right), \frac{1}{2}\right)\right) \]
6. Step-by-step derivation

   1. /-lowering-/.f64 97.3%

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, x\right), \mathsf{pow.f64}\left(\color{blue}{\mathsf{+.f64}\left(1, x\right)}, \frac{1}{2}\right)\right) \]

7. Simplified 97.3%

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

8. Taylor expanded in x around inf

   \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, x\right), \color{blue}{\left(\sqrt{x}\right)}\right) \]
9. Step-by-step derivation

   1. sqrt-lowering-sqrt.f64 97.1%

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, x\right), \mathsf{sqrt.f64}\left(x\right)\right) \]

10. Simplified 97.1%

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

    1. associate-/l/ N/A

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

    2. div-inv N/A

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

    3. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \color{blue}{\left(\frac{1}{\sqrt{x} \cdot x}\right)}\right) \]

    4. pow1/2 N/A

       \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \left(\frac{1}{{x}^{\frac{1}{2}} \cdot x}\right)\right) \]

    5. pow-plus N/A

       \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \left(\frac{1}{{x}^{\color{blue}{\left(\frac{1}{2} + 1\right)}}}\right)\right) \]

    6. pow-flip N/A

       \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \left({x}^{\color{blue}{\left(\mathsf{neg}\left(\left(\frac{1}{2} + 1\right)\right)\right)}}\right)\right) \]

    7. metadata-eval N/A

       \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \left({x}^{\left(\mathsf{neg}\left(\frac{3}{2}\right)\right)}\right)\right) \]

    8. metadata-eval N/A

       \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \left({x}^{\frac{-3}{2}}\right)\right) \]

    9. metadata-eval N/A

       \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \left({x}^{\left(-1 \cdot \color{blue}{\frac{3}{2}}\right)}\right)\right) \]

    10. metadata-eval N/A

        \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \left({x}^{\left(-1 \cdot \left(\frac{1}{2} + \color{blue}{1}\right)\right)}\right)\right) \]

    11. pow-lowering-pow.f64 N/A

        \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{pow.f64}\left(x, \color{blue}{\left(-1 \cdot \left(\frac{1}{2} + 1\right)\right)}\right)\right) \]

    12. metadata-eval N/A

        \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{pow.f64}\left(x, \left(-1 \cdot \frac{3}{2}\right)\right)\right) \]

    13. metadata-eval 97.4%

        \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{pow.f64}\left(x, \frac{-3}{2}\right)\right) \]

12. Applied egg-rr 97.4%

    \[\leadsto \color{blue}{0.5 \cdot {x}^{-1.5}} \]
13. Add Preprocessing

Alternative 6: 37.8% accurate, 23.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 37.8%

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

4. Applied egg-rr 40.6%

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

5. Taylor expanded in x around inf

   \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(\frac{\frac{1}{2}}{x}\right)}, \mathsf{pow.f64}\left(\mathsf{+.f64}\left(1, x\right), \frac{1}{2}\right)\right) \]
6. Step-by-step derivation

   1. /-lowering-/.f64 97.3%

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, x\right), \mathsf{pow.f64}\left(\color{blue}{\mathsf{+.f64}\left(1, x\right)}, \frac{1}{2}\right)\right) \]

7. Simplified 97.3%

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

8. Taylor expanded in x around 0

   \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, x\right), \color{blue}{\left(1 + \frac{1}{2} \cdot x\right)}\right) \]
9. Step-by-step derivation

   1. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, x\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{1}{2} \cdot x\right)}\right)\right) \]

   2. *-lowering-*.f64 36.4%

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \color{blue}{x}\right)\right)\right) \]

10. Simplified 36.4%

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

11. Final simplification 36.4%

    \[\leadsto \frac{\frac{0.5}{x}}{x \cdot 0.5 + 1} \]
12. Add Preprocessing

Alternative 7: 36.7% accurate, 29.9× speedup

Math

\[\begin{array}{l} \\ \frac{0.0625}{x \cdot \left(x \cdot x\right)} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ 0.0625 (* x (* x x))))

C

double code(double x) {
	return 0.0625 / (x * (x * x));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 0.0625d0 / (x * (x * x))
end function

Java

public static double code(double x) {
	return 0.0625 / (x * (x * x));
}

Python

def code(x):
	return 0.0625 / (x * (x * x))

Julia

function code(x)
	return Float64(0.0625 / Float64(x * Float64(x * x)))
end

MATLAB

function tmp = code(x)
	tmp = 0.0625 / (x * (x * x));
end

Wolfram

code[x_] := N[(0.0625 / N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 37.8%

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

4. Applied egg-rr 40.6%

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

5. Taylor expanded in x around inf

   \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(\frac{\left(\frac{1}{2} + \frac{\frac{1}{16}}{{x}^{2}}\right) - \left(\frac{1}{8} \cdot \frac{1}{x} + \frac{5}{128} \cdot \frac{1}{{x}^{3}}\right)}{x}\right)}, \mathsf{pow.f64}\left(\mathsf{+.f64}\left(1, x\right), \frac{1}{2}\right)\right) \]

7. Simplified 99.1%

   \[\leadsto \frac{\color{blue}{\frac{\left(0.5 + \frac{-0.125 - \frac{-0.0625}{x}}{x}\right) - \frac{0.0390625}{x \cdot \left(x \cdot x\right)}}{x}}}{{\left(1 + x\right)}^{0.5}} \]

8. Taylor expanded in x around -inf

   \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(-1 \cdot \frac{-1 \cdot \frac{\frac{1}{16} \cdot \frac{1}{x} - \frac{1}{8}}{x} - \frac{1}{2}}{x}\right)}, \mathsf{pow.f64}\left(\mathsf{+.f64}\left(1, x\right), \frac{1}{2}\right)\right) \]
9. Step-by-step derivation

   1. associate-*r/ N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\frac{-1 \cdot \left(-1 \cdot \frac{\frac{1}{16} \cdot \frac{1}{x} - \frac{1}{8}}{x} - \frac{1}{2}\right)}{x}\right), \mathsf{pow.f64}\left(\color{blue}{\mathsf{+.f64}\left(1, x\right)}, \frac{1}{2}\right)\right) \]

   2. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(-1 \cdot \left(-1 \cdot \frac{\frac{1}{16} \cdot \frac{1}{x} - \frac{1}{8}}{x} - \frac{1}{2}\right)\right), x\right), \mathsf{pow.f64}\left(\color{blue}{\mathsf{+.f64}\left(1, x\right)}, \frac{1}{2}\right)\right) \]

10. Simplified 99.0%

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

11. Taylor expanded in x around 0

    \[\leadsto \color{blue}{\frac{\frac{1}{16}}{{x}^{3}}} \]
12. Step-by-step derivation

    1. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\frac{1}{16}, \color{blue}{\left({x}^{3}\right)}\right) \]

    2. cube-mult N/A

       \[\leadsto \mathsf{/.f64}\left(\frac{1}{16}, \left(x \cdot \color{blue}{\left(x \cdot x\right)}\right)\right) \]

    3. unpow2 N/A

       \[\leadsto \mathsf{/.f64}\left(\frac{1}{16}, \left(x \cdot {x}^{\color{blue}{2}}\right)\right) \]

    4. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\frac{1}{16}, \mathsf{*.f64}\left(x, \color{blue}{\left({x}^{2}\right)}\right)\right) \]

    5. unpow2 N/A

       \[\leadsto \mathsf{/.f64}\left(\frac{1}{16}, \mathsf{*.f64}\left(x, \left(x \cdot \color{blue}{x}\right)\right)\right) \]

    6. *-lowering-*.f64 35.3%

       \[\leadsto \mathsf{/.f64}\left(\frac{1}{16}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{x}\right)\right)\right) \]

13. Simplified 35.3%

    \[\leadsto \color{blue}{\frac{0.0625}{x \cdot \left(x \cdot x\right)}} \]
14. Add Preprocessing

Alternative 8: 7.8% accurate, 69.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(x):
	return 0.5 / x

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 37.8%

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

4. Applied egg-rr 40.6%

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

5. Taylor expanded in x around inf

   \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(\frac{\frac{1}{2}}{x}\right)}, \mathsf{pow.f64}\left(\mathsf{+.f64}\left(1, x\right), \frac{1}{2}\right)\right) \]
6. Step-by-step derivation

   1. /-lowering-/.f64 97.3%

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, x\right), \mathsf{pow.f64}\left(\color{blue}{\mathsf{+.f64}\left(1, x\right)}, \frac{1}{2}\right)\right) \]

7. Simplified 97.3%

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

8. Taylor expanded in x around 0

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

   1. /-lowering-/.f64 7.7%

      \[\leadsto \mathsf{/.f64}\left(\frac{1}{2}, \color{blue}{x}\right) \]

10. Simplified 7.7%

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

Alternative 9: 4.6% accurate, 209.0× speedup

Math

\[\begin{array}{l} \\ 2 \end{array} \]

FPCore

(FPCore (x) :precision binary64 2.0)

C

double code(double x) {
	return 2.0;
}

Fortran

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

Java

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

Python

def code(x):
	return 2.0

Julia

function code(x)
	return 2.0
end

MATLAB

function tmp = code(x)
	tmp = 2.0;
end

Wolfram

code[x_] := 2.0

Derivation

1. Initial program 37.8%

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

4. Applied egg-rr 40.6%

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

5. Taylor expanded in x around inf

   \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, x\right), x\right), \mathsf{+.f64}\left(x, \color{blue}{\left(x \cdot \left(1 + \frac{1}{2} \cdot \frac{1}{x}\right)\right)}\right)\right), \mathsf{pow.f64}\left(\mathsf{+.f64}\left(1, x\right), \frac{1}{2}\right)\right) \]
6. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, x\right), x\right), \mathsf{+.f64}\left(x, \left(x \cdot \left(\frac{1}{2} \cdot \frac{1}{x} + 1\right)\right)\right)\right), \mathsf{pow.f64}\left(\mathsf{+.f64}\left(1, x\right), \frac{1}{2}\right)\right) \]

   2. distribute-rgt-in N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, x\right), x\right), \mathsf{+.f64}\left(x, \left(\left(\frac{1}{2} \cdot \frac{1}{x}\right) \cdot x + 1 \cdot x\right)\right)\right), \mathsf{pow.f64}\left(\mathsf{+.f64}\left(1, x\right), \frac{1}{2}\right)\right) \]

   3. associate-*l* N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, x\right), x\right), \mathsf{+.f64}\left(x, \left(\frac{1}{2} \cdot \left(\frac{1}{x} \cdot x\right) + 1 \cdot x\right)\right)\right), \mathsf{pow.f64}\left(\mathsf{+.f64}\left(1, x\right), \frac{1}{2}\right)\right) \]

   4. lft-mult-inverse N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, x\right), x\right), \mathsf{+.f64}\left(x, \left(\frac{1}{2} \cdot 1 + 1 \cdot x\right)\right)\right), \mathsf{pow.f64}\left(\mathsf{+.f64}\left(1, x\right), \frac{1}{2}\right)\right) \]

   5. metadata-eval N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, x\right), x\right), \mathsf{+.f64}\left(x, \left(\frac{1}{2} + 1 \cdot x\right)\right)\right), \mathsf{pow.f64}\left(\mathsf{+.f64}\left(1, x\right), \frac{1}{2}\right)\right) \]

   6. *-lft-identity N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, x\right), x\right), \mathsf{+.f64}\left(x, \left(\frac{1}{2} + x\right)\right)\right), \mathsf{pow.f64}\left(\mathsf{+.f64}\left(1, x\right), \frac{1}{2}\right)\right) \]

   7. +-lowering-+.f64 39.6%

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, x\right), x\right), \mathsf{+.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, x\right)\right)\right), \mathsf{pow.f64}\left(\mathsf{+.f64}\left(1, x\right), \frac{1}{2}\right)\right) \]

7. Simplified 39.6%

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

8. Taylor expanded in x around 0

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

10. Simplified 4.6%

    \[\leadsto \color{blue}{2} \]
11. Add Preprocessing

Developer Target 1: 98.3% 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]

Reproduce

herbie shell --seed 2024162 
(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))))))

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