2isqrt (example 3.6)

Percentage Accurate: 38.5% → 99.7%

Time: 10.3s

Alternatives: 7

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: 38.5% 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: 99.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 10^{+57}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{fma}\left(x, x, x\right)} \cdot \left(\sqrt{x + 1} + \sqrt{x}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 \cdot \sqrt{\frac{1}{x}}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1e+57)
   (/ 1.0 (* (sqrt (fma x x x)) (+ (sqrt (+ x 1.0)) (sqrt x))))
   (/ (* 0.5 (sqrt (/ 1.0 x))) x)))

C

double code(double x) {
	double tmp;
	if (x <= 1e+57) {
		tmp = 1.0 / (sqrt(fma(x, x, x)) * (sqrt((x + 1.0)) + sqrt(x)));
	} else {
		tmp = (0.5 * sqrt((1.0 / x))) / x;
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 1e+57)
		tmp = Float64(1.0 / Float64(sqrt(fma(x, x, x)) * Float64(sqrt(Float64(x + 1.0)) + sqrt(x))));
	else
		tmp = Float64(Float64(0.5 * sqrt(Float64(1.0 / x))) / x);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 1e+57], N[(1.0 / N[(N[Sqrt[N[(x * x + x), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision] + N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 * N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.00000000000000005e57

   1. Initial program 20.2%

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

   4. Applied rewrites 31.6%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 99.5%

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

   if 1.00000000000000005e57 < x

   1. Initial program 38.7%

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

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 79.6

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

   5. Applied rewrites 79.6%

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

   7. Applied rewrites 99.6%

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

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 99.8%

       \[\leadsto \frac{0.5 \cdot \sqrt{\frac{1}{x}}}{x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 10^{+57}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{fma}\left(x, x, x\right)} \cdot \left(\sqrt{x + 1} + \sqrt{x}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 \cdot \sqrt{\frac{1}{x}}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 97.6% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 35.7%

   \[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift--.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. clear-num N/A

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

   4. lift-/.f64 N/A

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

   5. frac-sub N/A

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

   6. div-inv N/A

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

   7. metadata-eval N/A

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

   8. *-rgt-identity N/A

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

   9. lower-/.f64 N/A

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

4. Applied rewrites 35.7%

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

5. Taylor expanded in x around inf

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

   1. distribute-rgt-in N/A

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

   2. *-lft-identity N/A

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

   3. associate-*l* N/A

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

   4. lft-mult-inverse N/A

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

   5. metadata-eval N/A

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

   6. lower-+.f64 35.2

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

7. Applied rewrites 35.2%

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

8. Taylor expanded in x around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower-sqrt.f64 N/A

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

   4. lower-/.f64 98.0

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

10. Applied rewrites 98.0%

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

11. Final simplification 98.0%

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

Alternative 3: 97.5% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 35.7%

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

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

   2. *-commutative N/A

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

   3. associate-/l* N/A

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

   4. lower-*.f64 N/A

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

   5. lower--.f64 N/A

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

   6. lower-sqrt.f64 N/A

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

   7. lower-/.f64 N/A

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

   8. lower-sqrt.f64 N/A

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

   9. lower-/.f64 N/A

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

   10. unpow2 N/A

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

   11. lower-*.f64 81.0

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

5. Applied rewrites 81.0%

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

7. Applied rewrites 97.8%

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

8. Taylor expanded in x around inf

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

10. Applied rewrites 97.9%

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

Alternative 4: 97.4% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 35.7%

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

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

   2. *-commutative N/A

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

   3. associate-/l* N/A

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

   4. lower-*.f64 N/A

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

   5. lower--.f64 N/A

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

   6. lower-sqrt.f64 N/A

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

   7. lower-/.f64 N/A

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

   8. lower-sqrt.f64 N/A

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

   9. lower-/.f64 N/A

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

   10. unpow2 N/A

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

   11. lower-*.f64 81.0

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

5. Applied rewrites 81.0%

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

7. Applied rewrites 97.8%

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

8. Taylor expanded in x around inf

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

10. Applied rewrites 97.9%

    \[\leadsto \frac{0.5 \cdot \sqrt{\frac{1}{x}}}{x} \]
11. Step-by-step derivation

12. Applied rewrites 97.8%

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

Alternative 5: 96.3% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \frac{1}{x \cdot \left(\sqrt{x} \cdot 2\right)} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ 1.0 (* x (* (sqrt x) 2.0))))

C

double code(double x) {
	return 1.0 / (x * (sqrt(x) * 2.0));
}

Fortran

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

Java

public static double code(double x) {
	return 1.0 / (x * (Math.sqrt(x) * 2.0));
}

Python

def code(x):
	return 1.0 / (x * (math.sqrt(x) * 2.0))

Julia

function code(x)
	return Float64(1.0 / Float64(x * Float64(sqrt(x) * 2.0)))
end

MATLAB

function tmp = code(x)
	tmp = 1.0 / (x * (sqrt(x) * 2.0));
end

Wolfram

code[x_] := N[(1.0 / N[(x * N[(N[Sqrt[x], $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 35.7%

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

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

   2. *-commutative N/A

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

   3. associate-/l* N/A

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

   4. lower-*.f64 N/A

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

   5. lower--.f64 N/A

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

   6. lower-sqrt.f64 N/A

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

   7. lower-/.f64 N/A

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

   8. lower-sqrt.f64 N/A

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

   9. lower-/.f64 N/A

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

   10. unpow2 N/A

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

   11. lower-*.f64 81.0

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

5. Applied rewrites 81.0%

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

7. Applied rewrites 97.8%

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

9. Applied rewrites 97.4%

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

10. Taylor expanded in x around inf

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

12. Applied rewrites 97.3%

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

Alternative 6: 36.7% accurate, 1.8× 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 35.7%

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

3. Taylor expanded in x around 0

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

   1. lower-sqrt.f64 N/A

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

   2. lower-/.f64 5.6

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

5. Applied rewrites 5.6%

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

7. Applied rewrites 5.6%

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

9. Applied rewrites 34.3%

   \[\leadsto \sqrt{\frac{x}{x \cdot x}} \]
10. Add Preprocessing

Alternative 7: 5.7% accurate, 2.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 35.7%

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

3. Taylor expanded in x around 0

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

   1. lower-sqrt.f64 N/A

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

   2. lower-/.f64 5.6

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

5. Applied rewrites 5.6%

   \[\leadsto \color{blue}{\sqrt{\frac{1}{x}}} \]
6. 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]

Developer Target 2: 38.5% 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 2024235 
(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)))))