2isqrt (example 3.6)

Percentage Accurate: 38.7% → 98.8%

Time: 10.3s

Alternatives: 5

Speedup: 1.5×

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: 98.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\frac{1}{x}}\\ \frac{-0.5 \cdot \left(\mathsf{fma}\left(0.75, \frac{t\_0}{x}, t\_0 \cdot \frac{-0.5}{x \cdot x}\right) - t\_0\right)}{x} \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (sqrt (/ 1.0 x))))
   (/ (* -0.5 (- (fma 0.75 (/ t_0 x) (* t_0 (/ -0.5 (* x x)))) t_0)) x)))

C

double code(double x) {
	double t_0 = sqrt((1.0 / x));
	return (-0.5 * (fma(0.75, (t_0 / x), (t_0 * (-0.5 / (x * x)))) - t_0)) / x;
}

Julia

function code(x)
	t_0 = sqrt(Float64(1.0 / x))
	return Float64(Float64(-0.5 * Float64(fma(0.75, Float64(t_0 / x), Float64(t_0 * Float64(-0.5 / Float64(x * x)))) - t_0)) / x)
end

Wolfram

code[x_] := Block[{t$95$0 = N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision]}, N[(N[(-0.5 * N[(N[(0.75 * N[(t$95$0 / x), $MachinePrecision] + N[(t$95$0 * N[(-0.5 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]]

Derivation

1. Initial program 39.1%

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

3. Taylor expanded in x around inf

   \[\leadsto \color{blue}{\frac{\frac{-1}{2} \cdot \left(\sqrt{\frac{1}{{x}^{5}}} \cdot \left(1 + \frac{1}{2} \cdot x\right)\right) - \left(\frac{-1}{2} \cdot \sqrt{x} + \left(\frac{-1}{2} \cdot \left(\sqrt{\frac{1}{{x}^{3}}} \cdot \left(1 + \frac{1}{4} \cdot x\right)\right) + \frac{1}{2} \cdot \sqrt{\frac{1}{x}}\right)\right)}{{x}^{2}}} \]

5. Simplified 83.7%

   \[\leadsto \color{blue}{\frac{-0.5 \cdot \left(\mathsf{fma}\left(\sqrt{\frac{1}{{x}^{5}}}, \mathsf{fma}\left(x, 0.5, 1\right), -\mathsf{fma}\left(\sqrt{\frac{1}{x \cdot \left(x \cdot x\right)}}, \mathsf{fma}\left(x, 0.25, 1\right), \sqrt{x}\right)\right) + \sqrt{\frac{1}{x}}\right)}{x \cdot x}} \]

6. Taylor expanded in x around inf

   \[\leadsto \color{blue}{\frac{\frac{-1}{2} \cdot \frac{\sqrt{\frac{1}{x}} - \frac{1}{4} \cdot \sqrt{\frac{1}{x}}}{x} + \left(\frac{-1}{2} \cdot \frac{\frac{1}{2} \cdot \sqrt{\frac{1}{x}} - \sqrt{\frac{1}{x}}}{{x}^{2}} + \frac{1}{2} \cdot \sqrt{\frac{1}{x}}\right)}{x}} \]
7. Step-by-step derivation

   1. lower-/.f64 N/A

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

8. Simplified 98.6%

   \[\leadsto \color{blue}{\frac{-0.5 \cdot \left(\left(-\sqrt{\frac{1}{x}}\right) + \mathsf{fma}\left(0.75, \frac{\sqrt{\frac{1}{x}}}{x}, \sqrt{\frac{1}{x}} \cdot \frac{-0.5}{x \cdot x}\right)\right)}{x}} \]

9. Final simplification 98.6%

   \[\leadsto \frac{-0.5 \cdot \left(\mathsf{fma}\left(0.75, \frac{\sqrt{\frac{1}{x}}}{x}, \sqrt{\frac{1}{x}} \cdot \frac{-0.5}{x \cdot x}\right) - \sqrt{\frac{1}{x}}\right)}{x} \]
10. Add Preprocessing

Alternative 2: 98.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \frac{\mathsf{fma}\left(\sqrt{\frac{1}{x \cdot \left(x \cdot x\right)}}, -0.375, \sqrt{\frac{1}{x}} \cdot 0.5\right)}{x} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/ (fma (sqrt (/ 1.0 (* x (* x x)))) -0.375 (* (sqrt (/ 1.0 x)) 0.5)) x))

C

double code(double x) {
	return fma(sqrt((1.0 / (x * (x * x)))), -0.375, (sqrt((1.0 / x)) * 0.5)) / x;
}

Julia

function code(x)
	return Float64(fma(sqrt(Float64(1.0 / Float64(x * Float64(x * x)))), -0.375, Float64(sqrt(Float64(1.0 / x)) * 0.5)) / x)
end

Wolfram

code[x_] := N[(N[(N[Sqrt[N[(1.0 / N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * -0.375 + N[(N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]

Derivation

1. Initial program 39.1%

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

3. Taylor expanded in x around inf

   \[\leadsto \color{blue}{\frac{\frac{-1}{2} \cdot \left(\sqrt{\frac{1}{{x}^{5}}} \cdot \left(1 + \frac{1}{2} \cdot x\right)\right) - \left(\frac{-1}{2} \cdot \sqrt{x} + \left(\frac{-1}{2} \cdot \left(\sqrt{\frac{1}{{x}^{3}}} \cdot \left(1 + \frac{1}{4} \cdot x\right)\right) + \frac{1}{2} \cdot \sqrt{\frac{1}{x}}\right)\right)}{{x}^{2}}} \]

5. Simplified 83.7%

   \[\leadsto \color{blue}{\frac{-0.5 \cdot \left(\mathsf{fma}\left(\sqrt{\frac{1}{{x}^{5}}}, \mathsf{fma}\left(x, 0.5, 1\right), -\mathsf{fma}\left(\sqrt{\frac{1}{x \cdot \left(x \cdot x\right)}}, \mathsf{fma}\left(x, 0.25, 1\right), \sqrt{x}\right)\right) + \sqrt{\frac{1}{x}}\right)}{x \cdot x}} \]

6. Taylor expanded in x around inf

   \[\leadsto \color{blue}{\frac{\frac{-1}{2} \cdot \frac{\sqrt{\frac{1}{x}} - \frac{1}{4} \cdot \sqrt{\frac{1}{x}}}{x} + \left(\frac{-1}{2} \cdot \frac{\frac{1}{2} \cdot \sqrt{\frac{1}{x}} - \sqrt{\frac{1}{x}}}{{x}^{2}} + \frac{1}{2} \cdot \sqrt{\frac{1}{x}}\right)}{x}} \]
7. Step-by-step derivation

   1. lower-/.f64 N/A

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

8. Simplified 98.6%

   \[\leadsto \color{blue}{\frac{-0.5 \cdot \left(\left(-\sqrt{\frac{1}{x}}\right) + \mathsf{fma}\left(0.75, \frac{\sqrt{\frac{1}{x}}}{x}, \sqrt{\frac{1}{x}} \cdot \frac{-0.5}{x \cdot x}\right)\right)}{x}} \]

9. Taylor expanded in x around inf

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

    1. +-commutative N/A

       \[\leadsto \frac{\color{blue}{\frac{1}{2} \cdot \sqrt{\frac{1}{x}} + \frac{-3}{8} \cdot \sqrt{\frac{1}{{x}^{3}}}}}{x} \]

    2. metadata-eval N/A

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

    3. associate-*r* N/A

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

    4. rem-square-sqrt N/A

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

    5. unpow2 N/A

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

    6. *-commutative N/A

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

    7. metadata-eval N/A

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

    8. associate-*r* N/A

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

    9. rem-square-sqrt N/A

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

    10. unpow2 N/A

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

    11. *-commutative N/A

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

    12. lower-/.f64 N/A

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

11. Simplified 98.5%

    \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\sqrt{\frac{1}{x \cdot \left(x \cdot x\right)}}, -0.375, \sqrt{\frac{1}{x}} \cdot 0.5\right)}{x}} \]
12. Add Preprocessing

Alternative 3: 97.7% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 39.1%

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

3. Taylor expanded in x around inf

   \[\leadsto \color{blue}{\frac{\frac{-1}{2} \cdot \left(\sqrt{\frac{1}{{x}^{5}}} \cdot \left(1 + \frac{1}{2} \cdot x\right)\right) - \left(\frac{-1}{2} \cdot \sqrt{x} + \left(\frac{-1}{2} \cdot \left(\sqrt{\frac{1}{{x}^{3}}} \cdot \left(1 + \frac{1}{4} \cdot x\right)\right) + \frac{1}{2} \cdot \sqrt{\frac{1}{x}}\right)\right)}{{x}^{2}}} \]

5. Simplified 83.7%

   \[\leadsto \color{blue}{\frac{-0.5 \cdot \left(\mathsf{fma}\left(\sqrt{\frac{1}{{x}^{5}}}, \mathsf{fma}\left(x, 0.5, 1\right), -\mathsf{fma}\left(\sqrt{\frac{1}{x \cdot \left(x \cdot x\right)}}, \mathsf{fma}\left(x, 0.25, 1\right), \sqrt{x}\right)\right) + \sqrt{\frac{1}{x}}\right)}{x \cdot x}} \]

6. Taylor expanded in x around inf

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

   1. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\frac{-1}{2} \cdot \frac{\sqrt{\frac{1}{x}} - \frac{1}{4} \cdot \sqrt{\frac{1}{x}}}{x} + \frac{1}{2} \cdot \sqrt{\frac{1}{x}}}{x}} \]

8. Simplified 98.5%

   \[\leadsto \color{blue}{\frac{-0.5 \cdot \mathsf{fma}\left(0.75, \frac{\sqrt{\frac{1}{x}}}{x}, -\sqrt{\frac{1}{x}}\right)}{x}} \]

9. Taylor expanded in x around inf

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

    1. mul-1-neg N/A

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

    2. lower-neg.f64 N/A

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

    3. lower-sqrt.f64 N/A

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

    4. lower-/.f64 97.3

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

11. Simplified 97.3%

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

12. Final simplification 97.3%

    \[\leadsto \frac{\sqrt{\frac{1}{x}} \cdot \left(--0.5\right)}{x} \]
13. Add Preprocessing

Alternative 4: 81.0% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 39.1%

   \[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}} \]
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. metadata-eval N/A

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

   2. distribute-lft-neg-in N/A

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

   3. unsub-neg N/A

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

   4. distribute-neg-in N/A

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

   5. +-commutative N/A

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

   6. lower-/.f64 N/A

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

5. Simplified 82.6%

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

6. Taylor expanded in x around inf

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

   1. lower-sqrt.f64 82.4

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

8. Simplified 82.4%

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

Alternative 5: 35.8% accurate, 49.0× 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 39.1%

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

3. Taylor expanded in x around inf

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

   1. lower-sqrt.f64 N/A

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

   2. lower-/.f64 26.7

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

5. Simplified 26.7%

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

6. Taylor expanded in x around 0

   \[\leadsto \color{blue}{0} \]
7. Step-by-step derivation

8. Simplified 35.7%

   \[\leadsto \color{blue}{0} \]
9. 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]

Developer Target 2: 38.8% accurate, 0.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024215 
(FPCore (x)
  :name "2isqrt (example 3.6)"
  :precision binary64
  :pre (and (> x 1.0) (< x 1e+308))

  :alt
  (! :herbie-platform default (/ 1 (+ (* (+ x 1) (sqrt x)) (* x (sqrt (+ x 1))))))

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

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