2isqrt (example 3.6)

Percentage Accurate: 39.4% → 98.9%

Time: 7.8s

Alternatives: 9

Speedup: 1.4×

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: 39.4% 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: 98.9% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 38.3%

   \[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}} \]
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. associate-/r* N/A

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

   10. clear-num N/A

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

   11. lower-/.f64 N/A

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

   12. lower-/.f64 N/A

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

   13. lower-/.f64 N/A

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

4. Applied rewrites 38.3%

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

5. Taylor expanded in x around inf

   \[\leadsto \frac{1}{\frac{\sqrt{x + 1}}{\color{blue}{\frac{\left(\frac{1}{2} + \frac{1}{16} \cdot \frac{1}{{x}^{2}}\right) - \frac{\frac{1}{8}}{x}}{x}}}} \]
6. Step-by-step derivation

   1. lower-/.f64 N/A

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

   2. lower--.f64 N/A

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

   3. +-commutative N/A

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

   4. lower-+.f64 N/A

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

   5. associate-*r/ N/A

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

   6. metadata-eval N/A

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

   7. lower-/.f64 N/A

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

   8. unpow2 N/A

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

   9. lower-*.f64 N/A

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

   10. lower-/.f64 98.5

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

7. Applied rewrites 98.5%

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

   1. lift-/.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. div-inv N/A

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

   4. associate-/r* N/A

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

   5. lift-sqrt.f64 N/A

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

   6. lift-+.f64 N/A

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

   7. lower-/.f64 N/A

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

9. Applied rewrites 99.3%

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

    1. lift-pow.f64 N/A

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

    2. unpow-1 N/A

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

    3. lower-/.f64 99.3

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

11. Applied rewrites 99.3%

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

12. Final simplification 99.3%

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

Alternative 2: 98.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{1 + x}\\ \mathbf{if}\;\frac{1}{\sqrt{x}} - \frac{1}{t\_0} \leq 0:\\ \;\;\;\;\frac{\sqrt{\frac{1}{x}} \cdot 0.5}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + x\right) - x}{\left(\sqrt{x} + t\_0\right) \cdot \left(0.5 + x\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (sqrt (+ 1.0 x))))
   (if (<= (- (/ 1.0 (sqrt x)) (/ 1.0 t_0)) 0.0)
     (/ (* (sqrt (/ 1.0 x)) 0.5) x)
     (/ (- (+ 1.0 x) x) (* (+ (sqrt x) t_0) (+ 0.5 x))))))

C

double code(double x) {
	double t_0 = sqrt((1.0 + x));
	double tmp;
	if (((1.0 / sqrt(x)) - (1.0 / t_0)) <= 0.0) {
		tmp = (sqrt((1.0 / x)) * 0.5) / x;
	} else {
		tmp = ((1.0 + x) - x) / ((sqrt(x) + t_0) * (0.5 + x));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sqrt((1.0d0 + x))
    if (((1.0d0 / sqrt(x)) - (1.0d0 / t_0)) <= 0.0d0) then
        tmp = (sqrt((1.0d0 / x)) * 0.5d0) / x
    else
        tmp = ((1.0d0 + x) - x) / ((sqrt(x) + t_0) * (0.5d0 + x))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double t_0 = Math.sqrt((1.0 + x));
	double tmp;
	if (((1.0 / Math.sqrt(x)) - (1.0 / t_0)) <= 0.0) {
		tmp = (Math.sqrt((1.0 / x)) * 0.5) / x;
	} else {
		tmp = ((1.0 + x) - x) / ((Math.sqrt(x) + t_0) * (0.5 + x));
	}
	return tmp;
}

Python

def code(x):
	t_0 = math.sqrt((1.0 + x))
	tmp = 0
	if ((1.0 / math.sqrt(x)) - (1.0 / t_0)) <= 0.0:
		tmp = (math.sqrt((1.0 / x)) * 0.5) / x
	else:
		tmp = ((1.0 + x) - x) / ((math.sqrt(x) + t_0) * (0.5 + x))
	return tmp

Julia

function code(x)
	t_0 = sqrt(Float64(1.0 + x))
	tmp = 0.0
	if (Float64(Float64(1.0 / sqrt(x)) - Float64(1.0 / t_0)) <= 0.0)
		tmp = Float64(Float64(sqrt(Float64(1.0 / x)) * 0.5) / x);
	else
		tmp = Float64(Float64(Float64(1.0 + x) - x) / Float64(Float64(sqrt(x) + t_0) * Float64(0.5 + x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = sqrt((1.0 + x));
	tmp = 0.0;
	if (((1.0 / sqrt(x)) - (1.0 / t_0)) <= 0.0)
		tmp = (sqrt((1.0 / x)) * 0.5) / x;
	else
		tmp = ((1.0 + x) - x) / ((sqrt(x) + t_0) * (0.5 + x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(N[(1.0 / N[Sqrt[x], $MachinePrecision]), $MachinePrecision] - N[(1.0 / t$95$0), $MachinePrecision]), $MachinePrecision], 0.0], N[(N[(N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision] / x), $MachinePrecision], N[(N[(N[(1.0 + x), $MachinePrecision] - x), $MachinePrecision] / N[(N[(N[Sqrt[x], $MachinePrecision] + t$95$0), $MachinePrecision] * N[(0.5 + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (/.f64 #s(literal 1 binary64) (sqrt.f64 x)) (/.f64 #s(literal 1 binary64) (sqrt.f64 (+.f64 x #s(literal 1 binary64))))) < 0.0

   1. Initial program 36.8%

      \[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}} \]
   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. div-sub N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. associate-/l* N/A

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

      6. distribute-rgt-out-- N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. associate-/r* N/A

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

      10. lower-/.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower--.f64 N/A

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

      13. lower-sqrt.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower-sqrt.f64 85.2

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

   5. Applied rewrites 85.2%

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

   7. Applied rewrites 99.5%

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

   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%

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

   if 0.0 < (-.f64 (/.f64 #s(literal 1 binary64) (sqrt.f64 x)) (/.f64 #s(literal 1 binary64) (sqrt.f64 (+.f64 x #s(literal 1 binary64)))))

   1. Initial program 62.5%

      \[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}} \]
   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. lift-/.f64 N/A

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

      4. clear-num N/A

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

      5. frac-sub N/A

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

      6. /-rgt-identity N/A

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

      7. *-lft-identity N/A

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

      8. metadata-eval N/A

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

      9. div-inv N/A

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

      10. flip-- N/A

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

      11. *-rgt-identity N/A

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

      12. metadata-eval N/A

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

      13. div-inv N/A

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

      14. /-rgt-identity N/A

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

      15. associate-/l/ N/A

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

   4. Applied rewrites 99.6%

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

   5. Taylor expanded in x around inf

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. associate-*l* N/A

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

      4. lft-mult-inverse N/A

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

      5. metadata-eval N/A

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

      6. *-lft-identity N/A

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

      7. lower-+.f64 85.0

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

   7. Applied rewrites 85.0%

      \[\leadsto \frac{\left(x + 1\right) - x}{\color{blue}{\left(0.5 + x\right)} \cdot \left(\sqrt{x} + \sqrt{x + 1}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{1 + x}} \leq 0:\\ \;\;\;\;\frac{\sqrt{\frac{1}{x}} \cdot 0.5}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + x\right) - x}{\left(\sqrt{x} + \sqrt{1 + x}\right) \cdot \left(0.5 + x\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.9% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 38.3%

   \[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}} \]
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. associate-/r* N/A

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

   10. clear-num N/A

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

   11. lower-/.f64 N/A

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

   12. lower-/.f64 N/A

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

   13. lower-/.f64 N/A

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

4. Applied rewrites 38.3%

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

5. Taylor expanded in x around inf

   \[\leadsto \frac{1}{\frac{\sqrt{x + 1}}{\color{blue}{\frac{\left(\frac{1}{2} + \frac{1}{16} \cdot \frac{1}{{x}^{2}}\right) - \frac{\frac{1}{8}}{x}}{x}}}} \]
6. Step-by-step derivation

   1. lower-/.f64 N/A

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

   2. lower--.f64 N/A

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

   3. +-commutative N/A

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

   4. lower-+.f64 N/A

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

   5. associate-*r/ N/A

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

   6. metadata-eval N/A

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

   7. lower-/.f64 N/A

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

   8. unpow2 N/A

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

   9. lower-*.f64 N/A

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

   10. lower-/.f64 98.5

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

7. Applied rewrites 98.5%

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

   1. lift-/.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. clear-num N/A

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

   4. lower-/.f64 99.3

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

9. Applied rewrites 99.3%

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

10. Final simplification 99.3%

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

Alternative 4: 97.6% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 38.3%

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

3. Taylor expanded in x around inf

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

   1. div-sub N/A

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

   2. *-commutative N/A

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

   3. associate-/l* N/A

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

   4. *-commutative N/A

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

   5. associate-/l* N/A

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

   6. distribute-rgt-out-- N/A

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

   7. lower-*.f64 N/A

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

   8. unpow2 N/A

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

   9. associate-/r* N/A

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

   10. lower-/.f64 N/A

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

   11. lower-/.f64 N/A

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

   12. lower--.f64 N/A

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

   13. lower-sqrt.f64 N/A

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

   14. lower-/.f64 N/A

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

   15. lower-sqrt.f64 83.7

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

5. Applied rewrites 83.7%

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

6. Taylor expanded in x around inf

   \[\leadsto \color{blue}{\frac{\frac{-1}{2} \cdot \sqrt{\frac{1}{x}} - \frac{-1}{2} \cdot \sqrt{x}}{{x}^{2}}} \]
7. 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. associate-/l* N/A

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

   3. lower-*.f64 N/A

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

   4. lower-/.f64 N/A

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

   5. lower--.f64 N/A

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

   6. lower-sqrt.f64 N/A

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

   7. lower-/.f64 N/A

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

   8. lower-sqrt.f64 N/A

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

   9. unpow2 N/A

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

   10. lower-*.f64 82.1

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

8. Applied rewrites 82.1%

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

10. Applied rewrites 97.3%

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

Alternative 5: 97.5% accurate, 1.3× speedup

Math

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

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)) * 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 38.3%

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

3. Taylor expanded in x around inf

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

   1. div-sub N/A

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

   2. *-commutative N/A

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

   3. associate-/l* N/A

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

   4. *-commutative N/A

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

   5. associate-/l* N/A

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

   6. distribute-rgt-out-- N/A

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

   7. lower-*.f64 N/A

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

   8. unpow2 N/A

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

   9. associate-/r* N/A

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

   10. lower-/.f64 N/A

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

   11. lower-/.f64 N/A

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

   12. lower--.f64 N/A

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

   13. lower-sqrt.f64 N/A

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

   14. lower-/.f64 N/A

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

   15. lower-sqrt.f64 83.7

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

5. Applied rewrites 83.7%

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

7. Applied rewrites 97.2%

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

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 97.3%

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

11. Final simplification 97.3%

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

Alternative 6: 97.4% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 38.3%

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

3. Taylor expanded in x around inf

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

   1. div-sub N/A

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

   2. *-commutative N/A

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

   3. associate-/l* N/A

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

   4. *-commutative N/A

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

   5. associate-/l* N/A

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

   6. distribute-rgt-out-- N/A

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

   7. lower-*.f64 N/A

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

   8. unpow2 N/A

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

   9. associate-/r* N/A

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

   10. lower-/.f64 N/A

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

   11. lower-/.f64 N/A

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

   12. lower--.f64 N/A

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

   13. lower-sqrt.f64 N/A

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

   14. lower-/.f64 N/A

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

   15. lower-sqrt.f64 83.7

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

5. Applied rewrites 83.7%

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

7. Applied rewrites 97.2%

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

8. Taylor expanded in x around inf

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

10. Applied rewrites 97.1%

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

12. Applied rewrites 97.1%

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

13. Final simplification 97.1%

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

Alternative 7: 81.3% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 38.3%

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

3. Taylor expanded in x around inf

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

   1. div-sub N/A

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

   2. *-commutative N/A

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

   3. associate-/l* N/A

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

   4. *-commutative N/A

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

   5. associate-/l* N/A

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

   6. distribute-rgt-out-- N/A

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

   7. lower-*.f64 N/A

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

   8. unpow2 N/A

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

   9. associate-/r* N/A

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

   10. lower-/.f64 N/A

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

   11. lower-/.f64 N/A

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

   12. lower--.f64 N/A

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

   13. lower-sqrt.f64 N/A

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

   14. lower-/.f64 N/A

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

   15. lower-sqrt.f64 83.7

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

5. Applied rewrites 83.7%

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

6. Taylor expanded in x around inf

   \[\leadsto \color{blue}{\frac{\frac{-1}{2} \cdot \sqrt{\frac{1}{x}} - \frac{-1}{2} \cdot \sqrt{x}}{{x}^{2}}} \]
7. 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. associate-/l* N/A

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

   3. lower-*.f64 N/A

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

   4. lower-/.f64 N/A

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

   5. lower--.f64 N/A

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

   6. lower-sqrt.f64 N/A

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

   7. lower-/.f64 N/A

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

   8. lower-sqrt.f64 N/A

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

   9. unpow2 N/A

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

   10. lower-*.f64 82.1

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

8. Applied rewrites 82.1%

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

9. Taylor expanded in x around -inf

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

11. Applied rewrites 81.9%

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

12. Final simplification 81.9%

    \[\leadsto \frac{-\sqrt{x}}{x \cdot x} \cdot -0.5 \]
13. Add Preprocessing

Alternative 8: 37.6% 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 38.3%

   \[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}} \]
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 5.7

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

5. Applied rewrites 5.7%

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

7. Applied rewrites 5.7%

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

9. Applied rewrites 36.3%

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

Alternative 9: 5.6% 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 38.3%

   \[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}} \]
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 5.7

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

5. Applied rewrites 5.7%

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

Developer Target 1: 39.4% 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 2024327 
(FPCore (x)
  :name "2isqrt (example 3.6)"
  :precision binary64
  :pre (and (> x 1.0) (< x 1e+308))

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

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