2isqrt (example 3.6)

Percentage Accurate: 39.0% → 99.3%

Time: 11.5s

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: 39.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.3% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 41.4%

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

4. Applied egg-rr 42.5%

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

5. Taylor expanded in x around inf

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

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

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

7. Simplified 98.8%

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

   1. *-commutative N/A

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

   2. clear-num N/A

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

   3. associate-*r/ N/A

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

   4. div-inv N/A

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

   5. times-frac N/A

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

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

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

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

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

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

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

   9. +-commutative N/A

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

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

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

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

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

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

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

9. Applied egg-rr 98.9%

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

10. Taylor expanded in x around inf

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

12. Simplified 98.9%

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

13. Final simplification 98.9%

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

Alternative 2: 99.3% accurate, 1.7× speedup

Math

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

FPCore

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

C

double code(double x) {
	return (pow((x + 1.0), -0.5) * (0.5 + ((-0.125 - (((0.0390625 / x) - 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) - (((0.0390625d0 / x) - 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 - (((0.0390625 / x) - 0.0625) / x)) / x))) / x;
}

Python

def code(x):
	return (math.pow((x + 1.0), -0.5) * (0.5 + ((-0.125 - (((0.0390625 / x) - 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(Float64(Float64(0.0390625 / x) - 0.0625) / x)) / x))) / x)
end

MATLAB

function tmp = code(x)
	tmp = (((x + 1.0) ^ -0.5) * (0.5 + ((-0.125 - (((0.0390625 / x) - 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[(N[(N[(0.0390625 / x), $MachinePrecision] - 0.0625), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]

Derivation

1. Initial program 41.4%

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

4. Applied egg-rr 42.5%

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

5. Taylor expanded in x around inf

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

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

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

7. Simplified 98.8%

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

   1. associate-*l/ N/A

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

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

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

9. Applied egg-rr 98.9%

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

10. Final simplification 98.9%

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

Alternative 3: 99.3% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 41.4%

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

4. Applied egg-rr 42.5%

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

5. Taylor expanded in x around inf

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

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

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

7. Simplified 98.8%

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

   1. associate-*l/ N/A

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

   2. associate-/l* N/A

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

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

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

   4. +-commutative N/A

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

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

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

   6. sub-div N/A

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

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

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

   8. associate--l+ N/A

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

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

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

   10. associate-/r* N/A

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

   11. sub-div N/A

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

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

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

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

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

   14. /-lowering-/.f64 N/A

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

9. Applied egg-rr 98.9%

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

10. Final simplification 98.9%

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

Alternative 4: 99.2% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 41.4%

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

4. Applied egg-rr 42.5%

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

5. Taylor expanded in x around inf

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

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

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

7. Simplified 98.8%

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

   1. *-commutative N/A

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

   2. clear-num N/A

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

   3. associate-*r/ N/A

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

   4. div-inv N/A

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

   5. times-frac N/A

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

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

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

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

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

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

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

   9. +-commutative N/A

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

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

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

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

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

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

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

9. Applied egg-rr 98.9%

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

10. Taylor expanded in x around inf

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

    1. +-commutative N/A

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

    2. associate--l+ N/A

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

    3. +-commutative N/A

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

    4. associate--r- N/A

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

    5. unpow2 N/A

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

    6. associate-/r* N/A

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

    7. metadata-eval N/A

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

    8. associate-*r/ N/A

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

    9. associate-*r/ N/A

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

    10. metadata-eval N/A

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

    11. div-sub N/A

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

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

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

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

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

    14. sub-neg N/A

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

    15. metadata-eval N/A

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

    16. +-commutative N/A

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

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

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

    18. associate-*r/ N/A

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

    19. metadata-eval N/A

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

    20. /-lowering-/.f64 98.5%

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

12. Simplified 98.5%

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

13. Final simplification 98.5%

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

Alternative 5: 99.2% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 41.4%

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

4. Applied egg-rr 42.5%

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

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 98.4%

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

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

   3. *-commutative N/A

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

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

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

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

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

   6. +-commutative N/A

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

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

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

   8. +-commutative N/A

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

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

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

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

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

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

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

   12. /-lowering-/.f64 98.5%

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

9. Applied egg-rr 98.5%

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

10. Final simplification 98.5%

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

Alternative 6: 99.1% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 41.4%

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

4. Applied egg-rr 42.5%

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

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 98.4%

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

8. Final simplification 98.4%

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

Alternative 7: 98.9% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 41.4%

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

4. Applied egg-rr 42.5%

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

5. Taylor expanded in x around inf

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

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

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

7. Simplified 98.8%

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

   1. *-commutative N/A

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

   2. clear-num N/A

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

   3. associate-*r/ N/A

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

   4. div-inv N/A

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

   5. times-frac N/A

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

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

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

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

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

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

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

   9. +-commutative N/A

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

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

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

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

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

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

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

9. Applied egg-rr 98.9%

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

10. Taylor expanded in x around inf

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

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

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

    2. associate-*r/ N/A

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

    3. metadata-eval N/A

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

    4. /-lowering-/.f64 98.0%

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

12. Simplified 98.0%

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

Alternative 8: 98.9% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 41.4%

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

4. Applied egg-rr 42.5%

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

5. Taylor expanded in x around inf

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

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

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

7. Simplified 98.8%

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

   1. *-commutative N/A

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

   2. clear-num N/A

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

   3. associate-*r/ N/A

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

   4. div-inv N/A

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

   5. times-frac N/A

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

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

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

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

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

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

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

   9. +-commutative N/A

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

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

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

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

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

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

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

9. Applied egg-rr 98.9%

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

10. Taylor expanded in x around inf

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

    1. sub-neg N/A

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

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

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

    3. associate-*r/ N/A

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

    4. metadata-eval N/A

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

    5. distribute-neg-frac N/A

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

    6. metadata-eval N/A

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

    7. /-lowering-/.f64 98.0%

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

12. Simplified 98.0%

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

Alternative 9: 98.8% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 41.4%

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

4. Applied egg-rr 42.5%

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

5. Taylor expanded in x around inf

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

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

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

   2. sub-neg N/A

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

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

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

   4. associate-*r/ N/A

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

   5. metadata-eval N/A

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

   6. distribute-neg-frac N/A

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

   7. metadata-eval N/A

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

   8. /-lowering-/.f64 97.9%

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

7. Simplified 97.9%

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

8. Final simplification 97.9%

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

Alternative 10: 98.0% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 41.4%

   \[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}} \]
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 79.2%

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

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 78.1%

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

8. Simplified 78.1%

   \[\leadsto \frac{\color{blue}{0.5 \cdot \sqrt{x}}}{x \cdot x} \]
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. pow1/2 N/A

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

   4. pow2 N/A

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

   5. pow-div N/A

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

   6. metadata-eval N/A

      \[\leadsto {x}^{\frac{-3}{2}} \cdot \frac{1}{2} \]

   7. metadata-eval N/A

      \[\leadsto {x}^{\left(-1 \cdot \frac{3}{2}\right)} \cdot \frac{1}{2} \]

   8. metadata-eval N/A

      \[\leadsto {x}^{\left(-1 \cdot \left(\frac{1}{2} \cdot 3\right)\right)} \cdot \frac{1}{2} \]

   9. pow-pow N/A

      \[\leadsto {\left({x}^{-1}\right)}^{\left(\frac{1}{2} \cdot 3\right)} \cdot \frac{1}{2} \]

   10. inv-pow N/A

       \[\leadsto {\left(\frac{1}{x}\right)}^{\left(\frac{1}{2} \cdot 3\right)} \cdot \frac{1}{2} \]

   11. pow-pow N/A

       \[\leadsto {\left({\left(\frac{1}{x}\right)}^{\frac{1}{2}}\right)}^{3} \cdot \frac{1}{2} \]

   12. pow1/2 N/A

       \[\leadsto {\left(\sqrt{\frac{1}{x}}\right)}^{3} \cdot \frac{1}{2} \]

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

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

   14. pow1/2 N/A

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

   15. pow-pow N/A

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

   16. inv-pow N/A

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

   17. pow-pow N/A

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

   18. metadata-eval N/A

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

   19. metadata-eval N/A

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

   20. metadata-eval N/A

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

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

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

   22. metadata-eval 97.0%

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

10. Applied egg-rr 97.0%

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

11. Final simplification 97.0%

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

Alternative 11: 37.2% accurate, 19.0× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{0.08203125 + \frac{-0.0390625}{x}}{x \cdot x}}{x} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/ (/ (+ 0.08203125 (/ -0.0390625 x)) (* x x)) x))

C

double code(double x) {
	return ((0.08203125 + (-0.0390625 / x)) / (x * x)) / x;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = ((0.08203125d0 + ((-0.0390625d0) / x)) / (x * x)) / x
end function

Java

public static double code(double x) {
	return ((0.08203125 + (-0.0390625 / x)) / (x * x)) / x;
}

Python

def code(x):
	return ((0.08203125 + (-0.0390625 / x)) / (x * x)) / x

Julia

function code(x)
	return Float64(Float64(Float64(0.08203125 + Float64(-0.0390625 / x)) / Float64(x * x)) / x)
end

MATLAB

function tmp = code(x)
	tmp = ((0.08203125 + (-0.0390625 / x)) / (x * x)) / x;
end

Wolfram

code[x_] := N[(N[(N[(0.08203125 + N[(-0.0390625 / x), $MachinePrecision]), $MachinePrecision] / N[(x * x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]

Derivation

1. Initial program 41.4%

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

4. Applied egg-rr 42.5%

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

5. Taylor expanded in x around inf

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

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

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

7. Simplified 98.8%

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

   1. associate-*l/ N/A

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

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

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

9. Applied egg-rr 98.9%

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

10. Taylor expanded in x around 0

    \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(\frac{\frac{21}{256} \cdot x - \frac{5}{128}}{{x}^{3}}\right)}, x\right) \]
11. Step-by-step derivation

    1. cube-mult N/A

       \[\leadsto \mathsf{/.f64}\left(\left(\frac{\frac{21}{256} \cdot x - \frac{5}{128}}{x \cdot \left(x \cdot x\right)}\right), x\right) \]

    2. unpow2 N/A

       \[\leadsto \mathsf{/.f64}\left(\left(\frac{\frac{21}{256} \cdot x - \frac{5}{128}}{x \cdot {x}^{2}}\right), x\right) \]

    3. associate-/r* N/A

       \[\leadsto \mathsf{/.f64}\left(\left(\frac{\frac{\frac{21}{256} \cdot x - \frac{5}{128}}{x}}{{x}^{2}}\right), x\right) \]

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

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{\frac{21}{256} \cdot x - \frac{5}{128}}{x}\right), \left({x}^{2}\right)\right), x\right) \]

    5. div-sub N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{\frac{21}{256} \cdot x}{x} - \frac{\frac{5}{128}}{x}\right), \left({x}^{2}\right)\right), x\right) \]

    6. associate-/l* N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{21}{256} \cdot \frac{x}{x} - \frac{\frac{5}{128}}{x}\right), \left({x}^{2}\right)\right), x\right) \]

    7. *-rgt-identity N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{21}{256} \cdot \frac{x \cdot 1}{x} - \frac{\frac{5}{128}}{x}\right), \left({x}^{2}\right)\right), x\right) \]

    8. associate-*r/ N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{21}{256} \cdot \left(x \cdot \frac{1}{x}\right) - \frac{\frac{5}{128}}{x}\right), \left({x}^{2}\right)\right), x\right) \]

    9. rgt-mult-inverse N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{21}{256} \cdot 1 - \frac{\frac{5}{128}}{x}\right), \left({x}^{2}\right)\right), x\right) \]

    10. metadata-eval N/A

        \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{21}{256} - \frac{\frac{5}{128}}{x}\right), \left({x}^{2}\right)\right), x\right) \]

    11. metadata-eval N/A

        \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{21}{256} - \frac{\frac{5}{128} \cdot 1}{x}\right), \left({x}^{2}\right)\right), x\right) \]

    12. associate-*r/ N/A

        \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{21}{256} - \frac{5}{128} \cdot \frac{1}{x}\right), \left({x}^{2}\right)\right), x\right) \]

    13. sub-neg N/A

        \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{21}{256} + \left(\mathsf{neg}\left(\frac{5}{128} \cdot \frac{1}{x}\right)\right)\right), \left({x}^{2}\right)\right), x\right) \]

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

        \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{21}{256}, \left(\mathsf{neg}\left(\frac{5}{128} \cdot \frac{1}{x}\right)\right)\right), \left({x}^{2}\right)\right), x\right) \]

    15. associate-*r/ N/A

        \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{21}{256}, \left(\mathsf{neg}\left(\frac{\frac{5}{128} \cdot 1}{x}\right)\right)\right), \left({x}^{2}\right)\right), x\right) \]

    16. metadata-eval N/A

        \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{21}{256}, \left(\mathsf{neg}\left(\frac{\frac{5}{128}}{x}\right)\right)\right), \left({x}^{2}\right)\right), x\right) \]

    17. distribute-neg-frac N/A

        \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{21}{256}, \left(\frac{\mathsf{neg}\left(\frac{5}{128}\right)}{x}\right)\right), \left({x}^{2}\right)\right), x\right) \]

    18. metadata-eval N/A

        \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{21}{256}, \left(\frac{\frac{-5}{128}}{x}\right)\right), \left({x}^{2}\right)\right), x\right) \]

    19. /-lowering-/.f64 N/A

        \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{21}{256}, \mathsf{/.f64}\left(\frac{-5}{128}, x\right)\right), \left({x}^{2}\right)\right), x\right) \]

    20. unpow2 N/A

        \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{21}{256}, \mathsf{/.f64}\left(\frac{-5}{128}, x\right)\right), \left(x \cdot x\right)\right), x\right) \]

    21. *-lowering-*.f64 38.5%

        \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{21}{256}, \mathsf{/.f64}\left(\frac{-5}{128}, x\right)\right), \mathsf{*.f64}\left(x, x\right)\right), x\right) \]

12. Simplified 38.5%

    \[\leadsto \frac{\color{blue}{\frac{0.08203125 + \frac{-0.0390625}{x}}{x \cdot x}}}{x} \]
13. Add Preprocessing

Alternative 12: 37.2% accurate, 29.9× speedup

Math

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

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 41.4%

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

4. Applied egg-rr 42.5%

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

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 98.4%

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

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 38.5%

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

10. Simplified 38.5%

    \[\leadsto \color{blue}{\frac{0.0625}{x \cdot \left(x \cdot x\right)}} \]
11. Step-by-step derivation

    1. associate-/r* N/A

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

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

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

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

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

    4. *-lowering-*.f64 38.5%

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

12. Applied egg-rr 38.5%

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

Alternative 13: 37.2% 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 41.4%

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

4. Applied egg-rr 42.5%

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

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 98.4%

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

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 38.5%

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

10. Simplified 38.5%

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

Developer Target 1: 98.4% 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]

Developer Target 2: 39.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024191 
(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))))))

  :alt
  (! :herbie-platform default (- (pow x -1/2) (pow (+ x 1) -1/2)))

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