2isqrt (example 3.6)

Percentage Accurate: 38.6% → 99.2%

Time: 10.5s

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: 38.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.2% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 37.3%

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

4. Applied egg-rr 39.2%

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

5. Taylor expanded in x around inf

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

7. Simplified 99.2%

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

   1. clear-num N/A

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

   2. associate-/r/ N/A

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

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

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

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

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

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

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

   6. associate-*r* N/A

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

   7. associate-/r* N/A

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

   8. sub-div N/A

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

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

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

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

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

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

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

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

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

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

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

   14. *-lowering-*.f64 99.2%

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

9. Applied egg-rr 99.2%

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

Alternative 2: 99.2% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 37.3%

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

4. Applied egg-rr 39.2%

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

5. Taylor expanded in x around inf

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

7. Simplified 99.2%

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

   1. clear-num N/A

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

   2. associate-/r/ N/A

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

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

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

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

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

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

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

   6. associate-*r* N/A

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

   7. associate-/r* N/A

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

   8. sub-div N/A

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

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

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

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

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

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

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

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

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

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

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

   14. *-lowering-*.f64 99.2%

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

9. Applied egg-rr 99.2%

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

    1. +-commutative N/A

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

    2. distribute-lft-in N/A

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

    3. associate-*l/ N/A

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

    4. metadata-eval N/A

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

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

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

11. Applied egg-rr 99.2%

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

Alternative 3: 99.2% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Python

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

MATLAB

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

Derivation

1. Initial program 37.3%

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

4. Applied egg-rr 39.2%

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

5. Taylor expanded in x around inf

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

7. Simplified 99.2%

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

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 99.2%

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

Alternative 4: 99.1% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 37.3%

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

4. Applied egg-rr 39.2%

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

5. Taylor expanded in x around inf

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

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

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

   2. associate--l+ N/A

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

   3. unpow2 N/A

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

   4. associate-/r* N/A

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

   5. metadata-eval N/A

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

   6. associate-*r/ N/A

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

   7. associate-*r/ N/A

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

   8. metadata-eval N/A

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

   9. div-sub N/A

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

   10. *-commutative N/A

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

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

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

   12. distribute-neg-frac2 N/A

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

   13. mul-1-neg N/A

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

   14. rem-square-sqrt N/A

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

   15. unpow2 N/A

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

   16. *-commutative N/A

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

   17. *-commutative N/A

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

7. Simplified 38.4%

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

   1. div-inv N/A

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

   2. associate-/r* N/A

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

   3. associate-*l/ N/A

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

   4. associate--l+ N/A

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

   5. +-inverses N/A

      \[\leadsto \frac{\frac{1 + 0}{x} \cdot \frac{1}{{\left(1 + x\right)}^{\frac{1}{2}}}}{2 + \frac{\frac{1}{2} + \frac{\frac{-1}{8}}{x}}{x}} \]

   6. metadata-eval N/A

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

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

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

9. Applied egg-rr 97.0%

   \[\leadsto \color{blue}{\frac{\frac{1}{x \cdot \sqrt{x + 1}}}{2 + \frac{0.5 + \frac{-0.125}{x}}{x}}} \]
10. Step-by-step derivation

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

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

    2. *-commutative N/A

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

    3. pow1/2 N/A

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

    4. metadata-eval N/A

       \[\leadsto \mathsf{/.f64}\left(\left(\frac{1}{{\left(x + 1\right)}^{\left(2 \cdot \frac{1}{4}\right)} \cdot x}\right), \left(2 + \frac{\frac{1}{2} + \frac{\frac{-1}{8}}{x}}{x}\right)\right) \]

    5. pow-pow N/A

       \[\leadsto \mathsf{/.f64}\left(\left(\frac{1}{{\left({\left(x + 1\right)}^{2}\right)}^{\frac{1}{4}} \cdot x}\right), \left(2 + \frac{\frac{1}{2} + \frac{\frac{-1}{8}}{x}}{x}\right)\right) \]

    6. pow2 N/A

       \[\leadsto \mathsf{/.f64}\left(\left(\frac{1}{{\left(\left(x + 1\right) \cdot \left(x + 1\right)\right)}^{\frac{1}{4}} \cdot x}\right), \left(2 + \frac{\frac{1}{2} + \frac{\frac{-1}{8}}{x}}{x}\right)\right) \]

    7. associate-/r* N/A

       \[\leadsto \mathsf{/.f64}\left(\left(\frac{\frac{1}{{\left(\left(x + 1\right) \cdot \left(x + 1\right)\right)}^{\frac{1}{4}}}}{x}\right), \left(\color{blue}{2} + \frac{\frac{1}{2} + \frac{\frac{-1}{8}}{x}}{x}\right)\right) \]

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

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{1}{{\left(\left(x + 1\right) \cdot \left(x + 1\right)\right)}^{\frac{1}{4}}}\right), x\right), \left(\color{blue}{2} + \frac{\frac{1}{2} + \frac{\frac{-1}{8}}{x}}{x}\right)\right) \]

    9. pow-flip N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left({\left(\left(x + 1\right) \cdot \left(x + 1\right)\right)}^{\left(\mathsf{neg}\left(\frac{1}{4}\right)\right)}\right), x\right), \left(2 + \frac{\frac{1}{2} + \frac{\frac{-1}{8}}{x}}{x}\right)\right) \]

    10. metadata-eval N/A

        \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left({\left(\left(x + 1\right) \cdot \left(x + 1\right)\right)}^{\frac{-1}{4}}\right), x\right), \left(2 + \frac{\frac{1}{2} + \frac{\frac{-1}{8}}{x}}{x}\right)\right) \]

    11. metadata-eval N/A

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

    12. pow-prod-down N/A

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

    13. sqr-pow N/A

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

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

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

    15. +-commutative N/A

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

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

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

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

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

    18. /-lowering-/.f64 N/A

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

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

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

    20. /-lowering-/.f64 98.8%

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

11. Applied egg-rr 98.8%

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

Alternative 5: 99.0% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 37.3%

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

4. Applied egg-rr 39.2%

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

5. Taylor expanded in x around inf

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

7. Simplified 98.8%

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

Alternative 6: 99.1% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 37.3%

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

4. Applied egg-rr 39.2%

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

5. Taylor expanded in x around inf

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

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

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

   2. associate--l+ N/A

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

   3. unpow2 N/A

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

   4. associate-/r* N/A

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

   5. metadata-eval N/A

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

   6. associate-*r/ N/A

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

   7. associate-*r/ N/A

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

   8. metadata-eval N/A

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

   9. div-sub N/A

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

   10. *-commutative N/A

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

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

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

   12. distribute-neg-frac2 N/A

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

   13. mul-1-neg N/A

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

   14. rem-square-sqrt N/A

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

   15. unpow2 N/A

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

   16. *-commutative N/A

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

   17. *-commutative N/A

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

7. Simplified 38.4%

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

   1. associate-/l/ N/A

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

   2. associate--l+ N/A

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

   3. +-inverses N/A

      \[\leadsto \frac{1 + 0}{{\left(1 + x\right)}^{\color{blue}{\frac{1}{2}}} \cdot \left(x \cdot \left(2 + \frac{\frac{1}{2} + \frac{\frac{-1}{8}}{x}}{x}\right)\right)} \]

   4. metadata-eval N/A

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

   5. associate-/r* N/A

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

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

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

   7. pow-flip N/A

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

   8. metadata-eval N/A

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

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

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

   10. +-commutative N/A

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

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

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

   12. distribute-lft-in N/A

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

   13. metadata-eval N/A

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

   14. div-inv N/A

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

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

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

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

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

   17. *-commutative N/A

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

   18. div-inv N/A

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

   19. associate-*l* N/A

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

   20. inv-pow N/A

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

   21. pow-plus N/A

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

   22. metadata-eval N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(x, 1\right), \frac{-1}{2}\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(x, \frac{1}{2}\right), \left(\left(\frac{1}{2} + \frac{\frac{-1}{8}}{x}\right) \cdot {x}^{0}\right)\right)\right) \]

   23. metadata-eval N/A

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

9. Applied egg-rr 98.8%

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

10. Final simplification 98.8%

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

Alternative 7: 98.7% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 37.3%

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

4. Applied egg-rr 39.2%

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

5. Taylor expanded in x around inf

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

   1. *-commutative N/A

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

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

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

   3. distribute-neg-frac2 N/A

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

   4. mul-1-neg N/A

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

   5. rem-square-sqrt N/A

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

   6. unpow2 N/A

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

   7. *-commutative N/A

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

   8. *-commutative N/A

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

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

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

7. Simplified 98.1%

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

Alternative 8: 97.8% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 37.3%

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

4. Applied egg-rr 39.2%

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

5. Taylor expanded in x around inf

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

   1. /-lowering-/.f64 96.8%

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

7. Simplified 96.8%

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

   1. div-inv N/A

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

   2. clear-num N/A

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

   3. associate-*l/ N/A

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

   4. div-inv N/A

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

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

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

   6. pow-flip N/A

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

   7. metadata-eval N/A

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

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

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

   9. +-commutative N/A

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

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

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

   11. /-lowering-/.f64 96.8%

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

9. Applied egg-rr 96.8%

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

10. Final simplification 96.8%

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

Alternative 9: 97.7% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 37.3%

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

4. Applied egg-rr 39.2%

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

5. Taylor expanded in x around inf

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

   1. /-lowering-/.f64 96.8%

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

7. Simplified 96.8%

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

Alternative 10: 97.7% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 37.3%

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

4. Applied egg-rr 39.2%

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

5. Taylor expanded in x around inf

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

   1. /-lowering-/.f64 96.8%

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

7. Simplified 96.8%

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

   1. associate-/l/ N/A

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

   2. associate-/r* N/A

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

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

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

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

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

   5. unpow1/2 N/A

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

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

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

   7. +-commutative N/A

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

   8. +-lowering-+.f64 96.7%

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

9. Applied egg-rr 96.7%

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

10. Final simplification 96.7%

    \[\leadsto \frac{\frac{0.5}{\sqrt{1 + x}}}{x} \]
11. Add Preprocessing

Alternative 11: 97.6% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 37.3%

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

4. Applied egg-rr 39.2%

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

5. Taylor expanded in x around inf

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

   1. /-lowering-/.f64 96.8%

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

7. Simplified 96.8%

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

8. Taylor expanded in x around inf

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

   1. sqrt-lowering-sqrt.f64 96.6%

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

10. Simplified 96.6%

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

Alternative 12: 37.9% accurate, 23.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 37.3%

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

4. Applied egg-rr 39.2%

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

5. Taylor expanded in x around inf

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

   1. /-lowering-/.f64 96.8%

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

7. Simplified 96.8%

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

8. Taylor expanded in x around 0

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

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

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

   2. *-commutative N/A

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

   3. *-lowering-*.f64 35.3%

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

10. Simplified 35.3%

    \[\leadsto \frac{\frac{0.5}{x}}{\color{blue}{1 + x \cdot 0.5}} \]
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 37.3%

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

4. Applied egg-rr 39.2%

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

5. Taylor expanded in x around inf

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

   1. /-lowering-/.f64 96.8%

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

7. Simplified 96.8%

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

8. Taylor expanded in x around 0

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

   1. /-lowering-/.f64 7.9%

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

10. Simplified 7.9%

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

Developer Target 1: 98.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024158 
(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)))))