2isqrt (example 3.6)

Percentage Accurate: 37.7% → 99.2%

Time: 10.8s

Alternatives: 10

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: 37.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.2% accurate, 1.7× speedup

Math

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

FPCore

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

C

double code(double x) {
	return pow((1.0 + x), -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 = ((1.0d0 + x) ** (-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((1.0 + x), -0.5) / (x * (2.0 + ((0.5 + ((-0.125 + (0.0625 / x)) / x)) / x)));
}

Python

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

Julia

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

Wolfram

code[x_] := N[(N[Power[N[(1.0 + x), $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 37.0%

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

4. Applied egg-rr 39.2%

   \[\leadsto \color{blue}{\frac{\frac{\left(1 + x\right) - x}{x + \sqrt{x \cdot \left(1 + x\right)}}}{{\left(1 + x\right)}^{0.5}}} \]
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 81.6%

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

6. Applied egg-rr 81.6%

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

7. Taylor expanded in x around inf

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

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

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

   2. associate-+r+ N/A

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

   3. associate--l+ N/A

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

   4. associate-*r/ N/A

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

   5. metadata-eval N/A

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

   6. unpow3 N/A

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

   7. unpow2 N/A

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

   8. associate-/r* N/A

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

   9. unpow2 N/A

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

   10. associate-/r* N/A

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

   11. metadata-eval N/A

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

   12. associate-*r/ N/A

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

   13. div-sub N/A

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

9. Simplified 99.0%

   \[\leadsto \frac{{\left(1 + x\right)}^{-0.5}}{\color{blue}{x \cdot \left(\left(2 + \frac{0.5}{x}\right) + \frac{\frac{-0.125 + \frac{0.0625}{x}}{x}}{x}\right)}} \]
10. Step-by-step derivation

    1. *-commutative N/A

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

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

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

    3. associate-+l+ N/A

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

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

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

    5. div-inv N/A

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

    6. div-inv N/A

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

    7. distribute-rgt-out N/A

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

    8. *-commutative N/A

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

    9. un-div-inv N/A

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

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

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

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

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

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

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

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

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

    14. /-lowering-/.f64 99.0%

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

11. Applied egg-rr 99.0%

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

12. Final simplification 99.0%

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

Alternative 2: 99.1% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 37.0%

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

4. Applied egg-rr 39.2%

   \[\leadsto \color{blue}{\frac{\frac{\left(1 + x\right) - x}{x + \sqrt{x \cdot \left(1 + x\right)}}}{{\left(1 + x\right)}^{0.5}}} \]
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 81.6%

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

6. Applied egg-rr 81.6%

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

7. Taylor expanded in x around inf

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

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

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

   2. associate--l+ N/A

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

   3. unpow2 N/A

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

   4. associate-/r* N/A

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

   5. metadata-eval N/A

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

   6. associate-*r/ N/A

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

   7. associate-*r/ N/A

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

   8. metadata-eval N/A

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

   9. div-sub N/A

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

   10. *-commutative N/A

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

   11. cancel-sign-sub-inv N/A

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

   12. distribute-neg-frac2 N/A

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

   13. mul-1-neg N/A

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

   14. rem-square-sqrt N/A

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

   15. unpow2 N/A

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

   16. *-commutative N/A

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

   17. *-commutative N/A

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

9. Simplified 98.9%

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

Alternative 3: 98.9% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 37.0%

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

4. Applied egg-rr 39.2%

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

5. Taylor expanded in x around inf

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

7. Simplified 98.8%

   \[\leadsto \frac{\color{blue}{\frac{0.5 + \frac{-0.125 - \frac{-0.0625}{x}}{x}}{x}}}{{\left(1 + x\right)}^{0.5}} \]
8. Step-by-step derivation

   1. unpow1/2 N/A

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

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

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

   3. +-lowering-+.f64 98.8%

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

9. Applied egg-rr 98.8%

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

Alternative 4: 98.8% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 37.0%

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

4. Applied egg-rr 39.2%

   \[\leadsto \color{blue}{\frac{\frac{\left(1 + x\right) - x}{x + \sqrt{x \cdot \left(1 + x\right)}}}{{\left(1 + x\right)}^{0.5}}} \]
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 81.6%

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

6. Applied egg-rr 81.6%

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

7. Taylor expanded in x around inf

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

   1. +-commutative N/A

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

   2. distribute-rgt-in N/A

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

   3. associate-*l* N/A

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

   4. lft-mult-inverse N/A

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

   5. metadata-eval N/A

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

   6. *-lft-identity N/A

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

   7. +-lowering-+.f64 98.5%

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

9. Simplified 98.5%

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

10. Final simplification 98.5%

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

Alternative 5: 97.9% 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 37.0%

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

3. Taylor expanded in x around inf

   \[\leadsto \color{blue}{\frac{\frac{1}{2} \cdot \left(\sqrt{\frac{1}{{x}^{3}}} \cdot \left(1 + \frac{1}{4} \cdot x\right)\right) - \left(\frac{-1}{2} \cdot \sqrt{x} + \frac{1}{2} \cdot \sqrt{\frac{1}{x}}\right)}{{x}^{2}}} \]

5. Simplified 80.2%

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

6. Taylor expanded in x around inf

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

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

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

   2. sqrt-lowering-sqrt.f64 78.9%

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

8. Simplified 78.9%

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

   1. associate-/l* N/A

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

   2. *-commutative N/A

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

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

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

   4. pow1/2 N/A

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

   5. pow2 N/A

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

   6. pow-div N/A

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

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

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

   8. metadata-eval 97.4%

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

10. Applied egg-rr 97.4%

    \[\leadsto \color{blue}{{x}^{-1.5} \cdot 0.5} \]

11. Final simplification 97.4%

    \[\leadsto 0.5 \cdot {x}^{-1.5} \]
12. Add Preprocessing

Alternative 6: 36.8% accurate, 12.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 37.0%

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

4. Applied egg-rr 39.2%

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

5. Taylor expanded in x around inf

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

7. Simplified 98.8%

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

8. Taylor expanded in x around 0

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

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

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

   2. *-lowering-*.f64 35.9%

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

10. Simplified 35.9%

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

11. Final simplification 35.9%

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

Alternative 7: 36.8% accurate, 16.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(x):
	return ((0.5 + (-0.125 / x)) / x) / (1.0 + (x * 0.5))

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 37.0%

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

4. Applied egg-rr 39.2%

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

5. Taylor expanded in x around inf

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

   1. *-commutative N/A

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

   2. cancel-sign-sub-inv N/A

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

   3. distribute-neg-frac2 N/A

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

   4. mul-1-neg N/A

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

   5. rem-square-sqrt N/A

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

   6. unpow2 N/A

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

   7. *-commutative N/A

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

   8. *-commutative N/A

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

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

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

7. Simplified 98.4%

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

8. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

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

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

   3. *-commutative N/A

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

   4. *-lowering-*.f64 35.9%

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

10. Simplified 35.9%

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

11. Final simplification 35.9%

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

Alternative 8: 35.7% 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 37.0%

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

4. Applied egg-rr 39.2%

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

5. Taylor expanded in x around inf

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

7. Simplified 98.8%

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

8. Taylor expanded in x around 0

   \[\leadsto \color{blue}{\frac{\frac{1}{16}}{{x}^{3}}} \]
9. 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 34.7%

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

10. Simplified 34.7%

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

Alternative 9: 34.1% accurate, 41.8× speedup

Math

\[\begin{array}{l} \\ \frac{-0.125}{x \cdot x} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ -0.125 (* x x)))

C

double code(double x) {
	return -0.125 / (x * x);
}

Fortran

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

Java

public static double code(double x) {
	return -0.125 / (x * x);
}

Python

def code(x):
	return -0.125 / (x * x)

Julia

function code(x)
	return Float64(-0.125 / Float64(x * x))
end

MATLAB

function tmp = code(x)
	tmp = -0.125 / (x * x);
end

Wolfram

code[x_] := N[(-0.125 / N[(x * x), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 37.0%

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

4. Applied egg-rr 39.2%

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

5. Taylor expanded in x around inf

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

   1. *-commutative N/A

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

   2. cancel-sign-sub-inv N/A

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

   3. distribute-neg-frac2 N/A

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

   4. mul-1-neg N/A

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

   5. rem-square-sqrt N/A

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

   6. unpow2 N/A

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

   7. *-commutative N/A

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

   8. *-commutative N/A

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

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

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

7. Simplified 98.4%

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

8. Taylor expanded in x around 0

   \[\leadsto \color{blue}{\frac{\frac{-1}{8}}{{x}^{2}}} \]
9. Step-by-step derivation

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

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

   2. unpow2 N/A

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

   3. *-lowering-*.f64 32.9%

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

10. Simplified 32.9%

    \[\leadsto \color{blue}{\frac{-0.125}{x \cdot x}} \]
11. Add Preprocessing

Alternative 10: 2.8% accurate, 41.8× speedup

Math

\[\begin{array}{l} \\ x \cdot \left(x \cdot 16\right) \end{array} \]

FPCore

(FPCore (x) :precision binary64 (* x (* x 16.0)))

C

double code(double x) {
	return x * (x * 16.0);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = x * (x * 16.0d0)
end function

Java

public static double code(double x) {
	return x * (x * 16.0);
}

Python

def code(x):
	return x * (x * 16.0)

Julia

function code(x)
	return Float64(x * Float64(x * 16.0))
end

MATLAB

function tmp = code(x)
	tmp = x * (x * 16.0);
end

Wolfram

code[x_] := N[(x * N[(x * 16.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 37.0%

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

4. Applied egg-rr 39.2%

   \[\leadsto \color{blue}{\frac{\frac{\left(1 + x\right) - x}{x + \sqrt{x \cdot \left(1 + x\right)}}}{{\left(1 + x\right)}^{0.5}}} \]
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 81.6%

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

6. Applied egg-rr 81.6%

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

7. Taylor expanded in x around inf

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

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

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

   2. associate-+r+ N/A

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

   3. associate--l+ N/A

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

   4. associate-*r/ N/A

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

   5. metadata-eval N/A

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

   6. unpow3 N/A

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

   7. unpow2 N/A

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

   8. associate-/r* N/A

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

   9. unpow2 N/A

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

   10. associate-/r* N/A

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

   11. metadata-eval N/A

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

   12. associate-*r/ N/A

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

   13. div-sub N/A

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

9. Simplified 99.0%

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

10. Taylor expanded in x around 0

    \[\leadsto \color{blue}{16 \cdot {x}^{2}} \]
11. Step-by-step derivation

    1. *-commutative N/A

       \[\leadsto {x}^{2} \cdot \color{blue}{16} \]

    2. unpow2 N/A

       \[\leadsto \left(x \cdot x\right) \cdot 16 \]

    3. associate-*l* N/A

       \[\leadsto x \cdot \color{blue}{\left(x \cdot 16\right)} \]

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

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

    5. *-lowering-*.f64 3.0%

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

12. Simplified 3.0%

    \[\leadsto \color{blue}{x \cdot \left(x \cdot 16\right)} \]
13. 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 2024163 
(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)))))