2isqrt (example 3.6)

Percentage Accurate: 38.9% → 99.3%

Time: 10.9s

Alternatives: 12

Speedup: 2.0×

Specification

\[x > 1 \land x < 10^{+308}\]

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 38.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.3% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 38.5%

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

4. Applied egg-rr 39.6%

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

5. Taylor expanded in x around inf

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

7. Simplified 99.5%

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

   1. *-commutative N/A

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

   2. clear-num N/A

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

   3. un-div-inv N/A

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

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

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

   5. +-commutative N/A

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

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

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

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

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

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

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

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

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

   10. associate-*r* N/A

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

   11. associate-/r* N/A

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

   12. sub-div N/A

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

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

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

9. Applied egg-rr 99.5%

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

10. Taylor expanded in x around -inf

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

12. Simplified 99.6%

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

Alternative 2: 99.2% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 38.5%

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

4. Applied egg-rr 39.6%

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

5. Taylor expanded in x around inf

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

7. Simplified 99.5%

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

   1. *-commutative N/A

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

   2. clear-num N/A

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

   3. un-div-inv N/A

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

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

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

   5. +-commutative N/A

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

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

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

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

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

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

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

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

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

   10. associate-*r* N/A

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

   11. associate-/r* N/A

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

   12. sub-div N/A

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

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

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

9. Applied egg-rr 99.5%

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

10. Taylor expanded in x around inf

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

    1. associate-*r/ N/A

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

    2. metadata-eval N/A

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

    3. +-commutative N/A

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

    4. associate--l+ N/A

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

    5. distribute-lft-in N/A

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

    6. metadata-eval N/A

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

    7. associate-*r/ N/A

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

    8. *-commutative N/A

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

    9. associate-*r* N/A

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

    10. rgt-mult-inverse N/A

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

    11. metadata-eval N/A

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

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

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

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

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

    14. sub-neg N/A

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

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

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

    16. distribute-neg-frac N/A

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

    17. metadata-eval N/A

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

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

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

    19. unpow2 N/A

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

    20. *-lowering-*.f64 99.5%

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

12. Simplified 99.5%

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

Alternative 3: 99.0% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Derivation

1. Initial program 38.5%

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

4. Applied egg-rr 39.6%

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

5. Taylor expanded in x around inf

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

7. Simplified 99.5%

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

8. Final simplification 99.5%

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

Alternative 4: 98.8% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 38.5%

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

4. Applied egg-rr 39.6%

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

5. Taylor expanded in x around inf

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

7. Simplified 99.5%

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

   1. *-commutative N/A

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

   2. clear-num N/A

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

   3. un-div-inv N/A

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

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

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

   5. +-commutative N/A

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

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

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

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

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

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

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

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

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

   10. associate-*r* N/A

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

   11. associate-/r* N/A

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

   12. sub-div N/A

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

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

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

9. Applied egg-rr 99.5%

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

10. Taylor expanded in x around inf

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

    1. distribute-rgt-in N/A

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

    2. associate-*l* N/A

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

    3. lft-mult-inverse N/A

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

    4. metadata-eval N/A

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

    5. +-commutative N/A

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

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

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

    7. *-commutative N/A

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

    8. *-lowering-*.f64 99.4%

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

12. Simplified 99.4%

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

Alternative 5: 97.8% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 38.5%

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

4. Applied egg-rr 39.6%

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

5. Taylor expanded in x around inf

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

   1. /-lowering-/.f64 99.0%

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

7. Simplified 99.0%

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

   1. *-commutative N/A

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

   2. clear-num N/A

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

   3. un-div-inv N/A

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

   4. metadata-eval N/A

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

   5. distribute-neg-frac2 N/A

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

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

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

   7. +-commutative N/A

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

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

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

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

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

   10. distribute-neg-frac2 N/A

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

   11. metadata-eval N/A

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

   12. /-lowering-/.f64 99.1%

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

9. Applied egg-rr 99.1%

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

Alternative 6: 97.7% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 38.5%

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

4. Applied egg-rr 39.6%

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

5. Taylor expanded in x around inf

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

   1. /-lowering-/.f64 99.0%

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

7. Simplified 99.0%

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

8. Final simplification 99.0%

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

Alternative 7: 97.7% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 38.5%

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

3. Taylor expanded in x around inf

   \[\leadsto \color{blue}{\frac{\frac{-1}{2} \cdot \sqrt{\frac{1}{x}} - \frac{-1}{2} \cdot \sqrt{x}}{{x}^{2}}} \]
4. Step-by-step derivation

   1. distribute-lft-out-- N/A

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

   2. unpow2 N/A

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

   3. times-frac N/A

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

   4. associate-*r/ N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

   11. sqrt-lowering-sqrt.f64 98.9%

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

5. Simplified 98.9%

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

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

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

7. Applied egg-rr 99.0%

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

8. Taylor expanded in x around inf

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

   1. mul-1-neg N/A

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

   2. neg-sub0 N/A

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

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

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

   4. sqrt-lowering-sqrt.f64 98.9%

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

10. Simplified 98.9%

    \[\leadsto \frac{\frac{\color{blue}{0 - \sqrt{x}}}{\frac{x}{-0.5}}}{x} \]
11. Step-by-step derivation

    1. div-inv N/A

       \[\leadsto \frac{0 - \sqrt{x}}{\frac{x}{\frac{-1}{2}}} \cdot \color{blue}{\frac{1}{x}} \]

    2. frac-2neg N/A

       \[\leadsto \frac{\mathsf{neg}\left(\left(0 - \sqrt{x}\right)\right)}{\mathsf{neg}\left(\frac{x}{\frac{-1}{2}}\right)} \cdot \frac{\color{blue}{1}}{x} \]

    3. sub0-neg N/A

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

    4. remove-double-neg N/A

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

    5. associate-*l/ N/A

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

    6. pow1/2 N/A

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

    7. inv-pow N/A

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

    8. metadata-eval N/A

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

    9. pow-prod-up N/A

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

    10. metadata-eval N/A

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

    11. metadata-eval N/A

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

    12. metadata-eval N/A

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

    13. pow-flip N/A

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

    14. pow1/2 N/A

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

    15. +-lft-identity N/A

        \[\leadsto \frac{\frac{1}{0 + \sqrt{x}}}{\mathsf{neg}\left(\frac{x}{\color{blue}{\frac{-1}{2}}}\right)} \]

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

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

    17. +-lft-identity N/A

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

    18. pow1/2 N/A

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

    19. pow-flip N/A

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

    20. metadata-eval N/A

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

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

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

    22. distribute-neg-frac2 N/A

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

    23. metadata-eval N/A

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

    24. /-lowering-/.f64 99.0%

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

12. Applied egg-rr 99.0%

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

Alternative 8: 97.6% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 38.5%

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

4. Applied egg-rr 39.6%

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

5. Taylor expanded in x around inf

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

   1. /-lowering-/.f64 99.0%

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

7. Simplified 99.0%

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

8. Taylor expanded in x around inf

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

10. Simplified 98.9%

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

Alternative 9: 37.3% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 38.5%

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

3. Taylor expanded in x around 0

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

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

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

   2. /-lowering-/.f64 5.4%

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

5. Simplified 5.4%

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

   1. inv-pow N/A

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

   2. sqrt-pow1 N/A

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

   3. metadata-eval N/A

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

   4. metadata-eval N/A

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

   5. pow-div N/A

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

   6. pow1/2 N/A

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

   7. metadata-eval N/A

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

   8. sqrt-pow1 N/A

      \[\leadsto \frac{\sqrt{x}}{\sqrt{{x}^{2}}} \]

   9. pow2 N/A

      \[\leadsto \frac{\sqrt{x}}{\sqrt{x \cdot x}} \]

   10. sqrt-undiv N/A

       \[\leadsto \sqrt{\frac{x}{x \cdot x}} \]

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

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

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

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

   13. *-lowering-*.f64 38.4%

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

7. Applied egg-rr 38.4%

   \[\leadsto \color{blue}{\sqrt{\frac{x}{x \cdot x}}} \]
8. Add Preprocessing

Alternative 10: 36.6% accurate, 19.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 38.5%

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

4. Applied egg-rr 39.6%

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

5. Taylor expanded in x around inf

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

   1. distribute-rgt-in N/A

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

   2. associate-*l* N/A

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

   3. lft-mult-inverse N/A

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

   4. metadata-eval N/A

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

   5. *-lft-identity N/A

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

   6. +-lowering-+.f64 39.3%

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

7. Simplified 39.3%

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

8. Taylor expanded in x around 0

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

10. Simplified 37.6%

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

11. Final simplification 37.6%

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

Alternative 11: 38.1% accurate, 26.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 6.4 \cdot 10^{+153}:\\ \;\;\;\;\frac{0.5}{x}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (if (<= x 6.4e+153) (/ 0.5 x) 0.0))

C

double code(double x) {
	double tmp;
	if (x <= 6.4e+153) {
		tmp = 0.5 / x;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 6.4d+153) then
        tmp = 0.5d0 / x
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 6.4e+153) {
		tmp = 0.5 / x;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 6.4e+153:
		tmp = 0.5 / x
	else:
		tmp = 0.0
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 6.4e+153)
		tmp = Float64(0.5 / x);
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 6.4e+153)
		tmp = 0.5 / x;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 6.4e+153], N[(0.5 / x), $MachinePrecision], 0.0]

Derivation

1. Split input into 2 regimes

2. if x < 6.4000000000000003e153

   1. Initial program 6.9%

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

   4. Applied egg-rr 9.3%

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

   5. Taylor expanded in x around inf

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

      1. /-lowering-/.f64 98.0%

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

   7. Simplified 98.0%

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

   8. Taylor expanded in x around 0

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

      1. /-lowering-/.f64 8.2%

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

   10. Simplified 8.2%

       \[\leadsto \color{blue}{\frac{0.5}{x}} \]

   if 6.4000000000000003e153 < x

   1. Initial program 66.4%

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

   3. Taylor expanded in x around inf

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

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

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

      2. /-lowering-/.f64 52.7%

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

   5. Simplified 52.7%

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

      1. metadata-eval N/A

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

      2. sqrt-div N/A

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

      3. +-inverses 66.4%

         \[\leadsto 0 \]

   7. Applied egg-rr 66.4%

      \[\leadsto \color{blue}{0} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 12: 36.1% accurate, 209.0× speedup

Math

\[\begin{array}{l} \\ 0 \end{array} \]

FPCore

(FPCore (x) :precision binary64 0.0)

C

double code(double x) {
	return 0.0;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 0.0d0
end function

Java

public static double code(double x) {
	return 0.0;
}

Python

def code(x):
	return 0.0

Julia

function code(x)
	return 0.0
end

MATLAB

function tmp = code(x)
	tmp = 0.0;
end

Wolfram

code[x_] := 0.0

Derivation

1. Initial program 38.5%

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

3. Taylor expanded in x around inf

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

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

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

   2. /-lowering-/.f64 30.2%

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

5. Simplified 30.2%

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

   1. metadata-eval N/A

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

   2. sqrt-div N/A

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

   3. +-inverses 37.3%

      \[\leadsto 0 \]

7. Applied egg-rr 37.3%

   \[\leadsto \color{blue}{0} \]
8. Add Preprocessing

Developer Target 1: 98.5% 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 2024140 
(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)))))