2isqrt (example 3.6)

Percentage Accurate: 37.8% → 99.3%

Time: 10.2s

Alternatives: 7

Speedup: 2.0×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 37.8% 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

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 41.0%

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

4. Applied egg-rr 43.7%

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

5. Taylor expanded in x around inf

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

7. Simplified 99.6%

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

   1. div-inv N/A

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

   2. clear-num N/A

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

   3. associate-*l/ N/A

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

   4. div-inv N/A

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

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

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

   6. pow1/2 N/A

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

   7. +-commutative N/A

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

   8. pow-flip N/A

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

   9. metadata-eval N/A

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

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

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

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

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

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

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

   13. associate--l+ N/A

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

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

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

9. Applied egg-rr 99.6%

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

Alternative 2: 99.2% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

function tmp = code(x)
	tmp = ((x + 1.0) ^ -0.5) / (x * (2.0 + ((0.5 + (-0.125 / 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[(-0.125 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 41.0%

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

4. Applied egg-rr 43.7%

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

5. Taylor expanded in x around inf

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

7. Simplified 99.6%

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

   1. div-inv N/A

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

   2. clear-num N/A

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

   3. associate-*l/ N/A

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

   4. div-inv N/A

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

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

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

   6. pow1/2 N/A

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

   7. +-commutative N/A

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

   8. pow-flip N/A

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

   9. metadata-eval N/A

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

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

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

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

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

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

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

   13. associate--l+ N/A

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

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

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

9. Applied egg-rr 99.6%

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

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. *-lowering-*.f64 N/A

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

    2. associate--l+ N/A

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

    3. associate-*r/ N/A

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

    4. metadata-eval N/A

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

    5. unpow2 N/A

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

    6. associate-/r* N/A

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

    7. metadata-eval N/A

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

    8. associate-*r/ N/A

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

    9. div-sub N/A

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

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

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

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

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

    12. sub-neg N/A

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

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

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

    14. associate-*r/ N/A

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

    15. metadata-eval N/A

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

    16. distribute-neg-frac N/A

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

    17. metadata-eval N/A

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

    18. /-lowering-/.f64 99.6%

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

12. Simplified 99.6%

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

Alternative 3: 99.1% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 41.0%

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

4. Applied egg-rr 43.7%

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

5. Taylor expanded in x around inf

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

7. Simplified 99.6%

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

8. Final simplification 99.6%

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

Alternative 4: 98.9% 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 41.0%

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

4. Applied egg-rr 43.7%

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

5. Taylor expanded in x around inf

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

7. Simplified 99.6%

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

   1. div-inv N/A

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

   2. clear-num N/A

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

   3. associate-*l/ N/A

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

   4. div-inv N/A

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

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

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

   6. pow1/2 N/A

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

   7. +-commutative N/A

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

   8. pow-flip N/A

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

   9. metadata-eval N/A

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

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

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

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

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

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

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

   13. associate--l+ N/A

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

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

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

9. Applied egg-rr 99.6%

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

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. +-commutative N/A

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

    2. distribute-rgt-in N/A

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

    3. associate-*l* N/A

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

    4. lft-mult-inverse N/A

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

    5. metadata-eval N/A

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

    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 99.3%

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

12. Simplified 99.3%

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

Alternative 5: 97.9% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 41.0%

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

3. Taylor expanded in x around inf

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

5. Simplified 85.3%

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

6. Taylor expanded in x around inf

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

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

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

   2. sqrt-lowering-sqrt.f64 84.0%

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

8. Simplified 84.0%

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

   1. associate-/l* N/A

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

   2. *-commutative N/A

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

   3. pow1/2 N/A

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

   4. pow2 N/A

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

   5. pow-div N/A

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

   6. metadata-eval N/A

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

   7. metadata-eval N/A

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

   8. metadata-eval N/A

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

   9. pow-pow N/A

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

   10. inv-pow N/A

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

   11. metadata-eval N/A

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

   12. metadata-eval N/A

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

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

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

   14. inv-pow N/A

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

   15. metadata-eval N/A

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

   16. metadata-eval N/A

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

   17. pow-pow N/A

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

   18. metadata-eval N/A

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

   19. metadata-eval N/A

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

   20. metadata-eval N/A

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

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

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

   22. metadata-eval 98.3%

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

10. Applied egg-rr 98.3%

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

11. Final simplification 98.3%

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

Alternative 6: 36.9% accurate, 8.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 6.5 \cdot 10^{+153}:\\ \;\;\;\;\frac{1}{\frac{x}{0.5 + \frac{\left(-0.125 + \frac{0.0625}{x}\right) - \frac{0.0390625}{x \cdot x}}{x}}}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 6.5e+153)
   (/
    1.0
    (/ x (+ 0.5 (/ (- (+ -0.125 (/ 0.0625 x)) (/ 0.0390625 (* x x))) x))))
   0.0))

C

double code(double x) {
	double tmp;
	if (x <= 6.5e+153) {
		tmp = 1.0 / (x / (0.5 + (((-0.125 + (0.0625 / x)) - (0.0390625 / (x * x))) / x)));
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 6.5d+153) then
        tmp = 1.0d0 / (x / (0.5d0 + ((((-0.125d0) + (0.0625d0 / x)) - (0.0390625d0 / (x * x))) / x)))
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 6.5e+153) {
		tmp = 1.0 / (x / (0.5 + (((-0.125 + (0.0625 / x)) - (0.0390625 / (x * x))) / x)));
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 6.5e+153:
		tmp = 1.0 / (x / (0.5 + (((-0.125 + (0.0625 / x)) - (0.0390625 / (x * x))) / x)))
	else:
		tmp = 0.0
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 6.5e+153)
		tmp = Float64(1.0 / Float64(x / Float64(0.5 + Float64(Float64(Float64(-0.125 + Float64(0.0625 / x)) - Float64(0.0390625 / Float64(x * x))) / x))));
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 6.5e+153)
		tmp = 1.0 / (x / (0.5 + (((-0.125 + (0.0625 / x)) - (0.0390625 / (x * x))) / x)));
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 6.5e+153], N[(1.0 / N[(x / 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]), $MachinePrecision], 0.0]

Derivation

1. Split input into 2 regimes

2. if x < 6.49999999999999972e153

   1. Initial program 9.5%

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

   4. Applied egg-rr 14.9%

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

   5. Taylor expanded in x around inf

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

   7. Simplified 99.4%

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

      1. div-inv N/A

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

      2. clear-num N/A

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

      3. associate-*l/ N/A

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

      4. div-inv N/A

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

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

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

      6. pow1/2 N/A

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

      7. +-commutative N/A

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

      8. pow-flip N/A

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

      9. metadata-eval N/A

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

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

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

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

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

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

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

      13. associate--l+ N/A

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

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

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

   9. Applied egg-rr 99.4%

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

   10. Taylor expanded in x around 0

       \[\leadsto \mathsf{/.f64}\left(\color{blue}{1}, \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(\frac{-1}{8}, \mathsf{/.f64}\left(\frac{1}{16}, x\right)\right), \mathsf{/.f64}\left(\frac{5}{128}, \mathsf{*.f64}\left(x, x\right)\right)\right), x\right)\right)\right)\right) \]
   11. Step-by-step derivation

   12. Simplified 8.5%

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

   if 6.49999999999999972e153 < x

   1. Initial program 72.1%

      \[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}} \]
   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 48.4%

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

   5. Simplified 48.4%

      \[\leadsto \frac{1}{\sqrt{x}} - \color{blue}{\sqrt{\frac{1}{x}}} \]
   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 72.1%

         \[\leadsto 0 \]

   7. Applied egg-rr 72.1%

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

Alternative 7: 34.8% 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 41.0%

   \[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}} \]
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 26.8%

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

5. Simplified 26.8%

   \[\leadsto \frac{1}{\sqrt{x}} - \color{blue}{\sqrt{\frac{1}{x}}} \]
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 38.4%

      \[\leadsto 0 \]

7. Applied egg-rr 38.4%

   \[\leadsto \color{blue}{0} \]
8. 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 2024170 
(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)))))