2isqrt (example 3.6)

Percentage Accurate: 38.9% → 99.3%

Time: 10.3s

Alternatives: 13

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: 38.9% 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

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 42.3%

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

4. Applied egg-rr 44.2%

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

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 99.4%

   \[\leadsto \color{blue}{\frac{0.5 + \left(\frac{-0.125 + \frac{0.0625}{x}}{x} - \frac{0.0390625}{x \cdot \left(x \cdot x\right)}\right)}{x}} \cdot {\left(1 + x\right)}^{-0.5} \]
8. Step-by-step derivation

   1. associate-*l/ N/A

      \[\leadsto \frac{\left(\frac{1}{2} + \left(\frac{\frac{-1}{8} + \frac{\frac{1}{16}}{x}}{x} - \frac{\frac{5}{128}}{x \cdot \left(x \cdot x\right)}\right)\right) \cdot {\left(1 + x\right)}^{\frac{-1}{2}}}{\color{blue}{x}} \]

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

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

9. Applied egg-rr 99.5%

   \[\leadsto \color{blue}{\frac{\left(0.5 + \frac{\left(-0.125 + \frac{0.0625}{x}\right) - \frac{0.0390625}{x \cdot x}}{x}\right) \cdot {\left(x + 1\right)}^{-0.5}}{x}} \]
10. Step-by-step derivation

    1. *-commutative N/A

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

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

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

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

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

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

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

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

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

    6. /-lowering-/.f64 N/A

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

    7. associate--l+ N/A

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

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

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

    9. associate-/r* N/A

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

    10. sub-div N/A

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

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

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

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

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

    13. /-lowering-/.f64 99.5%

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

11. Applied egg-rr 99.5%

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

12. Final simplification 99.5%

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

Alternative 2: 99.2% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 42.3%

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

4. Applied egg-rr 44.2%

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

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 99.4%

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

8. Taylor expanded in x around inf

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

10. Simplified 99.4%

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

11. Final simplification 99.4%

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

Alternative 3: 99.1% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

Wolfram

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

Derivation

1. Initial program 42.3%

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

4. Applied egg-rr 44.2%

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

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 99.0%

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

   1. associate-*l/ N/A

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

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

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

9. Applied egg-rr 99.1%

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

Alternative 4: 99.1% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 42.3%

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

4. Applied egg-rr 44.2%

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

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 99.0%

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

   1. associate-*l/ N/A

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

   2. associate-/l* N/A

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

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

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

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

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

   5. /-lowering-/.f64 N/A

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

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

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

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

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

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

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

9. Applied egg-rr 99.1%

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

Alternative 5: 98.8% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 42.3%

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

4. Applied egg-rr 44.2%

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

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 99.0%

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

   1. clear-num N/A

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

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

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

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

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

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

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

   5. /-lowering-/.f64 N/A

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

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

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

   7. /-lowering-/.f64 99.0%

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

9. Applied egg-rr 99.0%

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

10. Taylor expanded in x around inf

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

    1. +-commutative N/A

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

    2. distribute-rgt-in N/A

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

    3. associate-*l* N/A

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

    4. lft-mult-inverse N/A

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

    5. metadata-eval N/A

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

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

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

    7. *-commutative N/A

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

    8. *-lowering-*.f64 98.5%

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

12. Simplified 98.5%

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

13. Final simplification 98.5%

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

Alternative 6: 97.9% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 42.3%

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

4. Applied egg-rr 44.2%

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

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 97.5%

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

7. Simplified 97.5%

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

   1. *-commutative N/A

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

   2. clear-num N/A

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

   3. un-div-inv N/A

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

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

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

   5. +-commutative N/A

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

   6. rem-square-sqrt N/A

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

   7. metadata-eval N/A

      \[\leadsto \mathsf{/.f64}\left(\left({\left(\sqrt{x} \cdot \sqrt{x} + {x}^{0}\right)}^{\frac{-1}{2}}\right), \left(\frac{x}{\frac{1}{2}}\right)\right) \]

   8. metadata-eval N/A

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

   9. pow-prod-up N/A

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

   10. pow1/2 N/A

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

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

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

   12. rem-square-sqrt N/A

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

   13. pow1/2 N/A

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

   14. pow-prod-up N/A

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

   15. metadata-eval N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\left(x + {x}^{0}\right), \frac{-1}{2}\right), \left(\frac{x}{\frac{1}{2}}\right)\right) \]

   16. metadata-eval N/A

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

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

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

   18. /-lowering-/.f64 97.6%

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

9. Applied egg-rr 97.6%

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

Alternative 7: 97.8% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 42.3%

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

4. Applied egg-rr 44.2%

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

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 97.5%

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

7. Simplified 97.5%

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

8. Final simplification 97.5%

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

Alternative 8: 97.7% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 42.3%

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

4. Applied egg-rr 44.2%

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

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 97.5%

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

7. Simplified 97.5%

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

   1. *-commutative N/A

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

   2. clear-num N/A

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

   3. un-div-inv N/A

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

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

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

   5. +-commutative N/A

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

   6. rem-square-sqrt N/A

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

   7. metadata-eval N/A

      \[\leadsto \mathsf{/.f64}\left(\left({\left(\sqrt{x} \cdot \sqrt{x} + {x}^{0}\right)}^{\frac{-1}{2}}\right), \left(\frac{x}{\frac{1}{2}}\right)\right) \]

   8. metadata-eval N/A

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

   9. pow-prod-up N/A

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

   10. pow1/2 N/A

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

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

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

   12. rem-square-sqrt N/A

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

   13. pow1/2 N/A

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

   14. pow-prod-up N/A

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

   15. metadata-eval N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\left(x + {x}^{0}\right), \frac{-1}{2}\right), \left(\frac{x}{\frac{1}{2}}\right)\right) \]

   16. metadata-eval N/A

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

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

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

   18. /-lowering-/.f64 97.6%

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

9. Applied egg-rr 97.6%

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

10. Taylor expanded in x around inf

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

12. Simplified 97.5%

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

Alternative 9: 97.6% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 42.3%

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

4. Applied egg-rr 44.2%

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

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 97.5%

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

7. Simplified 97.5%

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

8. Taylor expanded in x around inf

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

10. Simplified 97.4%

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

Alternative 10: 37.3% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ {\left(x \cdot x\right)}^{-0.25} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (pow (* x x) -0.25))

C

double code(double x) {
	return pow((x * x), -0.25);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (x * x) ** (-0.25d0)
end function

Java

public static double code(double x) {
	return Math.pow((x * x), -0.25);
}

Python

def code(x):
	return math.pow((x * x), -0.25)

Julia

function code(x)
	return Float64(x * x) ^ -0.25
end

MATLAB

function tmp = code(x)
	tmp = (x * x) ^ -0.25;
end

Wolfram

code[x_] := N[Power[N[(x * x), $MachinePrecision], -0.25], $MachinePrecision]

Derivation

1. Initial program 42.3%

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

3. Taylor expanded in x around 0

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

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

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

   2. /-lowering-/.f64 5.5%

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

5. Simplified 5.5%

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

   1. inv-pow N/A

      \[\leadsto \sqrt{{x}^{-1}} \]

   2. sqrt-pow1 N/A

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

   3. metadata-eval N/A

      \[\leadsto {x}^{\frac{-1}{2}} \]

   4. sqr-pow N/A

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

   5. pow-prod-down N/A

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

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

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

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

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

   8. metadata-eval 40.5%

      \[\leadsto \mathsf{pow.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{-1}{4}\right) \]

7. Applied egg-rr 40.5%

   \[\leadsto \color{blue}{{\left(x \cdot x\right)}^{-0.25}} \]
8. Add Preprocessing

Alternative 11: 37.1% accurate, 29.9× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{0.0625}{x \cdot x}}{x} \end{array} \]

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(Float64(0.0625 / Float64(x * x)) / x)
end

MATLAB

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

Wolfram

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

Derivation

1. Initial program 42.3%

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

4. Applied egg-rr 44.2%

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

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 99.0%

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

   1. clear-num N/A

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

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

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

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

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

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

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

   5. /-lowering-/.f64 N/A

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

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

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

   7. /-lowering-/.f64 99.0%

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

9. Applied egg-rr 99.0%

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

10. Taylor expanded in x around 0

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

    1. unpow3 N/A

       \[\leadsto \frac{\frac{1}{16}}{\left(x \cdot x\right) \cdot \color{blue}{x}} \]

    2. unpow2 N/A

       \[\leadsto \frac{\frac{1}{16}}{{x}^{2} \cdot x} \]

    3. associate-/r* N/A

       \[\leadsto \frac{\frac{\frac{1}{16}}{{x}^{2}}}{\color{blue}{x}} \]

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

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

    5. /-lowering-/.f64 N/A

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

    6. unpow2 N/A

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

    7. *-lowering-*.f64 40.2%

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

12. Simplified 40.2%

    \[\leadsto \color{blue}{\frac{\frac{0.0625}{x \cdot x}}{x}} \]
13. Add Preprocessing

Alternative 12: 37.1% 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 42.3%

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

4. Applied egg-rr 44.2%

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

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 99.0%

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

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 40.2%

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

10. Simplified 40.2%

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

Alternative 13: 7.9% 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 42.3%

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

4. Applied egg-rr 44.2%

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

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 97.5%

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

7. Simplified 97.5%

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

8. Taylor expanded in x around 0

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

   1. /-lowering-/.f64 7.9%

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

10. Simplified 7.9%

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

Developer Target 1: 98.2% 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 2024145 
(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)))))