2isqrt (example 3.6)

Percentage Accurate: 38.7% → 98.7%

Time: 11.2s

Alternatives: 4

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: 38.7% 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: 98.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{{x}^{-0.5}}{2} + \frac{-0.375}{{x}^{1.5}}}{x} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/ (+ (/ (pow x -0.5) 2.0) (/ -0.375 (pow x 1.5))) x))

C

double code(double x) {
	return ((pow(x, -0.5) / 2.0) + (-0.375 / pow(x, 1.5))) / x;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (((x ** (-0.5d0)) / 2.0d0) + ((-0.375d0) / (x ** 1.5d0))) / x
end function

Java

public static double code(double x) {
	return ((Math.pow(x, -0.5) / 2.0) + (-0.375 / Math.pow(x, 1.5))) / x;
}

Python

def code(x):
	return ((math.pow(x, -0.5) / 2.0) + (-0.375 / math.pow(x, 1.5))) / x

Julia

function code(x)
	return Float64(Float64(Float64((x ^ -0.5) / 2.0) + Float64(-0.375 / (x ^ 1.5))) / x)
end

MATLAB

function tmp = code(x)
	tmp = (((x ^ -0.5) / 2.0) + (-0.375 / (x ^ 1.5))) / x;
end

Wolfram

code[x_] := N[(N[(N[(N[Power[x, -0.5], $MachinePrecision] / 2.0), $MachinePrecision] + N[(-0.375 / N[Power[x, 1.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]

Derivation

1. Initial program 40.3%

   \[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}} \]
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 82.6%

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

6. Taylor expanded in x around inf

   \[\leadsto \color{blue}{\frac{\frac{-1}{2} \cdot \sqrt{\frac{1}{{x}^{3}}} + \left(\frac{1}{8} \cdot \sqrt{\frac{1}{{x}^{3}}} + \frac{1}{2} \cdot \sqrt{\frac{1}{x}}\right)}{x}} \]
7. Step-by-step derivation

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

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

8. Simplified 99.4%

   \[\leadsto \color{blue}{\frac{0.5 \cdot \sqrt{\frac{1}{x}} + \sqrt{\frac{1}{x \cdot \left(x \cdot x\right)}} \cdot -0.375}{x}} \]
9. Step-by-step derivation

   1. *-commutative N/A

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

   2. sqrt-div N/A

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

   3. metadata-eval N/A

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

   4. cube-unmult N/A

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

   5. sqrt-pow1 N/A

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

   6. metadata-eval N/A

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

   7. un-div-inv N/A

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

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

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

   9. pow-lowering-pow.f64 99.4%

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

10. Applied egg-rr 99.4%

    \[\leadsto \frac{0.5 \cdot \sqrt{\frac{1}{x}} + \color{blue}{\frac{-0.375}{{x}^{1.5}}}}{x} \]
11. Step-by-step derivation

    1. sqrt-div N/A

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

    2. metadata-eval N/A

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

    3. div-inv N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(\frac{\frac{1}{2}}{\sqrt{x}}\right), \mathsf{/.f64}\left(\frac{-3}{8}, \mathsf{pow.f64}\left(x, \frac{3}{2}\right)\right)\right), x\right) \]

    4. clear-num N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(\frac{1}{\frac{\sqrt{x}}{\frac{1}{2}}}\right), \mathsf{/.f64}\left(\frac{-3}{8}, \mathsf{pow.f64}\left(x, \frac{3}{2}\right)\right)\right), x\right) \]

    5. div-inv N/A

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

    6. metadata-eval N/A

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

    7. associate-/r* N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(\frac{\frac{1}{\sqrt{x}}}{2}\right), \mathsf{/.f64}\left(\frac{-3}{8}, \mathsf{pow.f64}\left(x, \frac{3}{2}\right)\right)\right), x\right) \]

    8. metadata-eval N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(\frac{\frac{\sqrt{1}}{\sqrt{x}}}{2}\right), \mathsf{/.f64}\left(\frac{-3}{8}, \mathsf{pow.f64}\left(x, \frac{3}{2}\right)\right)\right), x\right) \]

    9. sqrt-div N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(\frac{\sqrt{\frac{1}{x}}}{2}\right), \mathsf{/.f64}\left(\frac{-3}{8}, \mathsf{pow.f64}\left(x, \frac{3}{2}\right)\right)\right), x\right) \]

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

        \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\left(\sqrt{\frac{1}{x}}\right), 2\right), \mathsf{/.f64}\left(\frac{-3}{8}, \mathsf{pow.f64}\left(x, \frac{3}{2}\right)\right)\right), x\right) \]

    11. inv-pow N/A

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

    12. sqrt-pow1 N/A

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

    13. metadata-eval N/A

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

    14. pow-lowering-pow.f64 99.4%

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

12. Applied egg-rr 99.4%

    \[\leadsto \frac{\color{blue}{\frac{{x}^{-0.5}}{2}} + \frac{-0.375}{{x}^{1.5}}}{x} \]
13. Add Preprocessing

Alternative 2: 98.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{-0.375}{{x}^{1.5}} + \frac{0.5}{\sqrt{x}}}{x} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/ (+ (/ -0.375 (pow x 1.5)) (/ 0.5 (sqrt x))) x))

C

double code(double x) {
	return ((-0.375 / pow(x, 1.5)) + (0.5 / sqrt(x))) / x;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (((-0.375d0) / (x ** 1.5d0)) + (0.5d0 / sqrt(x))) / x
end function

Java

public static double code(double x) {
	return ((-0.375 / Math.pow(x, 1.5)) + (0.5 / Math.sqrt(x))) / x;
}

Python

def code(x):
	return ((-0.375 / math.pow(x, 1.5)) + (0.5 / math.sqrt(x))) / x

Julia

function code(x)
	return Float64(Float64(Float64(-0.375 / (x ^ 1.5)) + Float64(0.5 / sqrt(x))) / x)
end

MATLAB

function tmp = code(x)
	tmp = ((-0.375 / (x ^ 1.5)) + (0.5 / sqrt(x))) / x;
end

Wolfram

code[x_] := N[(N[(N[(-0.375 / N[Power[x, 1.5], $MachinePrecision]), $MachinePrecision] + N[(0.5 / N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]

Derivation

1. Initial program 40.3%

   \[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}} \]
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 82.6%

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

6. Taylor expanded in x around inf

   \[\leadsto \color{blue}{\frac{\frac{-1}{2} \cdot \sqrt{\frac{1}{{x}^{3}}} + \left(\frac{1}{8} \cdot \sqrt{\frac{1}{{x}^{3}}} + \frac{1}{2} \cdot \sqrt{\frac{1}{x}}\right)}{x}} \]
7. Step-by-step derivation

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

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

8. Simplified 99.4%

   \[\leadsto \color{blue}{\frac{0.5 \cdot \sqrt{\frac{1}{x}} + \sqrt{\frac{1}{x \cdot \left(x \cdot x\right)}} \cdot -0.375}{x}} \]

10. Applied egg-rr 99.3%

    \[\leadsto \color{blue}{\frac{\frac{0.5}{\sqrt{x}} + \frac{-0.375}{{x}^{1.5}}}{x}} \]

11. Final simplification 99.3%

    \[\leadsto \frac{\frac{-0.375}{{x}^{1.5}} + \frac{0.5}{\sqrt{x}}}{x} \]
12. Add Preprocessing

Alternative 3: 98.0% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 40.3%

   \[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}} \]
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 82.6%

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

6. Taylor expanded in x around inf

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

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

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

   2. sqrt-lowering-sqrt.f64 82.0%

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

8. Simplified 82.0%

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

   1. *-commutative N/A

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

   2. associate-*l/ N/A

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

   3. clear-num N/A

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

   4. pow2 N/A

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

   5. pow1/2 N/A

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

   6. pow-div N/A

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

   7. metadata-eval N/A

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

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

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

   9. pow-flip N/A

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

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

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

   11. metadata-eval 99.1%

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

10. Applied egg-rr 99.1%

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

11. Final simplification 99.1%

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

Alternative 4: 35.9% 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 40.3%

   \[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}} \]
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 26.9%

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

5. Simplified 26.9%

   \[\leadsto \frac{1}{\sqrt{x}} - \color{blue}{\sqrt{\frac{1}{x}}} \]
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 39.1%

      \[\leadsto 0 \]

7. Applied egg-rr 39.1%

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

Developer Target 1: 98.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024164 
(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)))))