2isqrt (example 3.6)

Percentage Accurate: 38.9% → 99.0%

Time: 8.7s

Alternatives: 9

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: 38.9% 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: 99.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{\left(0.5 - \frac{0.375}{x}\right) + \frac{0.3125}{x \cdot x}}{x}}{\sqrt{x}} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/ (/ (+ (- 0.5 (/ 0.375 x)) (/ 0.3125 (* x x))) x) (sqrt x)))

C

double code(double x) {
	return (((0.5 - (0.375 / x)) + (0.3125 / (x * x))) / x) / sqrt(x);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (((0.5d0 - (0.375d0 / x)) + (0.3125d0 / (x * x))) / x) / sqrt(x)
end function

Java

public static double code(double x) {
	return (((0.5 - (0.375 / x)) + (0.3125 / (x * x))) / x) / Math.sqrt(x);
}

Python

def code(x):
	return (((0.5 - (0.375 / x)) + (0.3125 / (x * x))) / x) / math.sqrt(x)

Julia

function code(x)
	return Float64(Float64(Float64(Float64(0.5 - Float64(0.375 / x)) + Float64(0.3125 / Float64(x * x))) / x) / sqrt(x))
end

MATLAB

function tmp = code(x)
	tmp = (((0.5 - (0.375 / x)) + (0.3125 / (x * x))) / x) / sqrt(x);
end

Wolfram

code[x_] := N[(N[(N[(N[(0.5 - N[(0.375 / x), $MachinePrecision]), $MachinePrecision] + N[(0.3125 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] / N[Sqrt[x], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 43.2%

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

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

   10. associate-/r* N/A

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

   11. lower-/.f64 N/A

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

4. Applied rewrites 43.2%

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

   1. lift-/.f64 N/A

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

   2. lift--.f64 N/A

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

   3. sub-div N/A

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

   4. frac-sub N/A

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

   5. *-commutative N/A

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

   6. lift-sqrt.f64 N/A

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

   7. lift-sqrt.f64 N/A

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

   8. rem-square-sqrt N/A

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

   9. lower-/.f64 N/A

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

6. Applied rewrites 7.5%

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

7. Taylor expanded in x around inf

   \[\leadsto \frac{\color{blue}{\frac{\left(\frac{1}{2} + \frac{\frac{5}{16}}{{x}^{2}}\right) - \frac{3}{8} \cdot \frac{1}{x}}{x}}}{\sqrt{x}} \]
8. Step-by-step derivation

   1. lower-/.f64 N/A

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

   2. lower--.f64 N/A

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

   3. +-commutative N/A

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

   4. lower-+.f64 N/A

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

   5. lower-/.f64 N/A

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

   6. unpow2 N/A

      \[\leadsto \frac{\frac{\left(\frac{\frac{5}{16}}{\color{blue}{x \cdot x}} + \frac{1}{2}\right) - \frac{3}{8} \cdot \frac{1}{x}}{x}}{\sqrt{x}} \]

   7. lower-*.f64 N/A

      \[\leadsto \frac{\frac{\left(\frac{\frac{5}{16}}{\color{blue}{x \cdot x}} + \frac{1}{2}\right) - \frac{3}{8} \cdot \frac{1}{x}}{x}}{\sqrt{x}} \]

   8. associate-*r/ N/A

      \[\leadsto \frac{\frac{\left(\frac{\frac{5}{16}}{x \cdot x} + \frac{1}{2}\right) - \color{blue}{\frac{\frac{3}{8} \cdot 1}{x}}}{x}}{\sqrt{x}} \]

   9. metadata-eval N/A

      \[\leadsto \frac{\frac{\left(\frac{\frac{5}{16}}{x \cdot x} + \frac{1}{2}\right) - \frac{\color{blue}{\frac{3}{8}}}{x}}{x}}{\sqrt{x}} \]

   10. lower-/.f64 98.7

       \[\leadsto \frac{\frac{\left(\frac{0.3125}{x \cdot x} + 0.5\right) - \color{blue}{\frac{0.375}{x}}}{x}}{\sqrt{x}} \]

9. Applied rewrites 98.7%

   \[\leadsto \frac{\color{blue}{\frac{\left(\frac{0.3125}{x \cdot x} + 0.5\right) - \frac{0.375}{x}}{x}}}{\sqrt{x}} \]
10. Step-by-step derivation

11. Applied rewrites 98.7%

    \[\leadsto \frac{\frac{\frac{0.3125}{x \cdot x} + \left(0.5 - \frac{0.375}{x}\right)}{x}}{\sqrt{x}} \]

12. Final simplification 98.7%

    \[\leadsto \frac{\frac{\left(0.5 - \frac{0.375}{x}\right) + \frac{0.3125}{x \cdot x}}{x}}{\sqrt{x}} \]
13. Add Preprocessing

Alternative 2: 99.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{\frac{\frac{0.3125}{x} - 0.375}{x} + 0.5}{x}}{\sqrt{x}} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/ (/ (+ (/ (- (/ 0.3125 x) 0.375) x) 0.5) x) (sqrt x)))

C

double code(double x) {
	return (((((0.3125 / x) - 0.375) / x) + 0.5) / x) / sqrt(x);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (((((0.3125d0 / x) - 0.375d0) / x) + 0.5d0) / x) / sqrt(x)
end function

Java

public static double code(double x) {
	return (((((0.3125 / x) - 0.375) / x) + 0.5) / x) / Math.sqrt(x);
}

Python

def code(x):
	return (((((0.3125 / x) - 0.375) / x) + 0.5) / x) / math.sqrt(x)

Julia

function code(x)
	return Float64(Float64(Float64(Float64(Float64(Float64(0.3125 / x) - 0.375) / x) + 0.5) / x) / sqrt(x))
end

MATLAB

function tmp = code(x)
	tmp = (((((0.3125 / x) - 0.375) / x) + 0.5) / x) / sqrt(x);
end

Wolfram

code[x_] := N[(N[(N[(N[(N[(N[(0.3125 / x), $MachinePrecision] - 0.375), $MachinePrecision] / x), $MachinePrecision] + 0.5), $MachinePrecision] / x), $MachinePrecision] / N[Sqrt[x], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 43.2%

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

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

   10. associate-/r* N/A

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

   11. lower-/.f64 N/A

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

4. Applied rewrites 43.2%

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

   1. lift-/.f64 N/A

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

   2. lift--.f64 N/A

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

   3. sub-div N/A

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

   4. frac-sub N/A

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

   5. *-commutative N/A

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

   6. lift-sqrt.f64 N/A

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

   7. lift-sqrt.f64 N/A

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

   8. rem-square-sqrt N/A

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

   9. lower-/.f64 N/A

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

6. Applied rewrites 7.5%

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

7. Taylor expanded in x around inf

   \[\leadsto \frac{\color{blue}{\frac{\left(\frac{1}{2} + \frac{\frac{5}{16}}{{x}^{2}}\right) - \frac{3}{8} \cdot \frac{1}{x}}{x}}}{\sqrt{x}} \]
8. Step-by-step derivation

   1. lower-/.f64 N/A

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

   2. lower--.f64 N/A

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

   3. +-commutative N/A

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

   4. lower-+.f64 N/A

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

   5. lower-/.f64 N/A

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

   6. unpow2 N/A

      \[\leadsto \frac{\frac{\left(\frac{\frac{5}{16}}{\color{blue}{x \cdot x}} + \frac{1}{2}\right) - \frac{3}{8} \cdot \frac{1}{x}}{x}}{\sqrt{x}} \]

   7. lower-*.f64 N/A

      \[\leadsto \frac{\frac{\left(\frac{\frac{5}{16}}{\color{blue}{x \cdot x}} + \frac{1}{2}\right) - \frac{3}{8} \cdot \frac{1}{x}}{x}}{\sqrt{x}} \]

   8. associate-*r/ N/A

      \[\leadsto \frac{\frac{\left(\frac{\frac{5}{16}}{x \cdot x} + \frac{1}{2}\right) - \color{blue}{\frac{\frac{3}{8} \cdot 1}{x}}}{x}}{\sqrt{x}} \]

   9. metadata-eval N/A

      \[\leadsto \frac{\frac{\left(\frac{\frac{5}{16}}{x \cdot x} + \frac{1}{2}\right) - \frac{\color{blue}{\frac{3}{8}}}{x}}{x}}{\sqrt{x}} \]

   10. lower-/.f64 98.7

       \[\leadsto \frac{\frac{\left(\frac{0.3125}{x \cdot x} + 0.5\right) - \color{blue}{\frac{0.375}{x}}}{x}}{\sqrt{x}} \]

9. Applied rewrites 98.7%

   \[\leadsto \frac{\color{blue}{\frac{\left(\frac{0.3125}{x \cdot x} + 0.5\right) - \frac{0.375}{x}}{x}}}{\sqrt{x}} \]

10. Taylor expanded in x around inf

    \[\leadsto \frac{\color{blue}{\frac{\left(\frac{1}{2} + \frac{\frac{5}{16}}{{x}^{2}}\right) - \frac{3}{8} \cdot \frac{1}{x}}{x}}}{\sqrt{x}} \]
11. Step-by-step derivation

    1. lower-/.f64 N/A

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

    2. associate--l+ N/A

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

    3. +-commutative N/A

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

    4. unpow2 N/A

       \[\leadsto \frac{\frac{\left(\frac{\frac{5}{16}}{\color{blue}{x \cdot x}} - \frac{3}{8} \cdot \frac{1}{x}\right) + \frac{1}{2}}{x}}{\sqrt{x}} \]

    5. associate-/r* N/A

       \[\leadsto \frac{\frac{\left(\color{blue}{\frac{\frac{\frac{5}{16}}{x}}{x}} - \frac{3}{8} \cdot \frac{1}{x}\right) + \frac{1}{2}}{x}}{\sqrt{x}} \]

    6. metadata-eval N/A

       \[\leadsto \frac{\frac{\left(\frac{\frac{\color{blue}{\frac{5}{16} \cdot 1}}{x}}{x} - \frac{3}{8} \cdot \frac{1}{x}\right) + \frac{1}{2}}{x}}{\sqrt{x}} \]

    7. associate-*r/ N/A

       \[\leadsto \frac{\frac{\left(\frac{\color{blue}{\frac{5}{16} \cdot \frac{1}{x}}}{x} - \frac{3}{8} \cdot \frac{1}{x}\right) + \frac{1}{2}}{x}}{\sqrt{x}} \]

    8. associate-*r/ N/A

       \[\leadsto \frac{\frac{\left(\frac{\frac{5}{16} \cdot \frac{1}{x}}{x} - \color{blue}{\frac{\frac{3}{8} \cdot 1}{x}}\right) + \frac{1}{2}}{x}}{\sqrt{x}} \]

    9. metadata-eval N/A

       \[\leadsto \frac{\frac{\left(\frac{\frac{5}{16} \cdot \frac{1}{x}}{x} - \frac{\color{blue}{\frac{3}{8}}}{x}\right) + \frac{1}{2}}{x}}{\sqrt{x}} \]

    10. div-sub N/A

        \[\leadsto \frac{\frac{\color{blue}{\frac{\frac{5}{16} \cdot \frac{1}{x} - \frac{3}{8}}{x}} + \frac{1}{2}}{x}}{\sqrt{x}} \]

    11. lower-+.f64 N/A

        \[\leadsto \frac{\frac{\color{blue}{\frac{\frac{5}{16} \cdot \frac{1}{x} - \frac{3}{8}}{x} + \frac{1}{2}}}{x}}{\sqrt{x}} \]

    12. lower-/.f64 N/A

        \[\leadsto \frac{\frac{\color{blue}{\frac{\frac{5}{16} \cdot \frac{1}{x} - \frac{3}{8}}{x}} + \frac{1}{2}}{x}}{\sqrt{x}} \]

    13. lower--.f64 N/A

        \[\leadsto \frac{\frac{\frac{\color{blue}{\frac{5}{16} \cdot \frac{1}{x} - \frac{3}{8}}}{x} + \frac{1}{2}}{x}}{\sqrt{x}} \]

    14. associate-*r/ N/A

        \[\leadsto \frac{\frac{\frac{\color{blue}{\frac{\frac{5}{16} \cdot 1}{x}} - \frac{3}{8}}{x} + \frac{1}{2}}{x}}{\sqrt{x}} \]

    15. metadata-eval N/A

        \[\leadsto \frac{\frac{\frac{\frac{\color{blue}{\frac{5}{16}}}{x} - \frac{3}{8}}{x} + \frac{1}{2}}{x}}{\sqrt{x}} \]

    16. lower-/.f64 98.7

        \[\leadsto \frac{\frac{\frac{\color{blue}{\frac{0.3125}{x}} - 0.375}{x} + 0.5}{x}}{\sqrt{x}} \]

12. Applied rewrites 98.7%

    \[\leadsto \frac{\color{blue}{\frac{\frac{\frac{0.3125}{x} - 0.375}{x} + 0.5}{x}}}{\sqrt{x}} \]
13. Add Preprocessing

Alternative 3: 98.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 43.2%

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

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

   10. associate-/r* N/A

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

   11. lower-/.f64 N/A

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

4. Applied rewrites 43.2%

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

   1. lift-/.f64 N/A

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

   2. lift--.f64 N/A

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

   3. sub-div N/A

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

   4. frac-sub N/A

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

   5. *-commutative N/A

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

   6. lift-sqrt.f64 N/A

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

   7. lift-sqrt.f64 N/A

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

   8. rem-square-sqrt N/A

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

   9. lower-/.f64 N/A

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

6. Applied rewrites 7.5%

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

7. Taylor expanded in x around inf

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

   1. +-commutative N/A

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

   2. lower-+.f64 N/A

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

   3. associate-*r/ N/A

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

   4. metadata-eval N/A

      \[\leadsto \frac{\frac{\frac{\color{blue}{\frac{1}{8}}}{x} + \frac{1}{2}}{x + 1}}{\sqrt{x}} \]

   5. lower-/.f64 98.5

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

9. Applied rewrites 98.5%

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

10. Final simplification 98.5%

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

Alternative 4: 98.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 43.2%

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

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

   10. associate-/r* N/A

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

   11. lower-/.f64 N/A

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

4. Applied rewrites 43.2%

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

   1. lift-/.f64 N/A

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

   2. lift--.f64 N/A

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

   3. sub-div N/A

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

   4. frac-sub N/A

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

   5. *-commutative N/A

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

   6. lift-sqrt.f64 N/A

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

   7. lift-sqrt.f64 N/A

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

   8. rem-square-sqrt N/A

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

   9. lower-/.f64 N/A

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

6. Applied rewrites 7.5%

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

7. Taylor expanded in x around inf

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

   1. +-commutative N/A

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

   2. lower-+.f64 N/A

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

   3. associate-*r/ N/A

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

   4. metadata-eval N/A

      \[\leadsto \frac{\frac{\frac{\color{blue}{\frac{1}{8}}}{x} + \frac{1}{2}}{x + 1}}{\sqrt{x}} \]

   5. lower-/.f64 98.5

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

9. Applied rewrites 98.5%

   \[\leadsto \frac{\frac{\color{blue}{\frac{0.125}{x} + 0.5}}{x + 1}}{\sqrt{x}} \]
10. Step-by-step derivation

    1. lift-/.f64 N/A

       \[\leadsto \color{blue}{\frac{\frac{\frac{\frac{1}{8}}{x} + \frac{1}{2}}{x + 1}}{\sqrt{x}}} \]

    2. lift-+.f64 N/A

       \[\leadsto \frac{\frac{\frac{\frac{1}{8}}{x} + \frac{1}{2}}{\color{blue}{x + 1}}}{\sqrt{x}} \]

    3. lift-/.f64 N/A

       \[\leadsto \frac{\color{blue}{\frac{\frac{\frac{1}{8}}{x} + \frac{1}{2}}{x + 1}}}{\sqrt{x}} \]

    4. associate-/l/ N/A

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

    5. associate-/r* N/A

       \[\leadsto \color{blue}{\frac{\frac{\frac{\frac{1}{8}}{x} + \frac{1}{2}}{\sqrt{x}}}{x + 1}} \]

    6. lower-/.f64 N/A

       \[\leadsto \color{blue}{\frac{\frac{\frac{\frac{1}{8}}{x} + \frac{1}{2}}{\sqrt{x}}}{x + 1}} \]

    7. lower-/.f64 N/A

       \[\leadsto \frac{\color{blue}{\frac{\frac{\frac{1}{8}}{x} + \frac{1}{2}}{\sqrt{x}}}}{x + 1} \]

    8. +-commutative N/A

       \[\leadsto \frac{\frac{\frac{\frac{1}{8}}{x} + \frac{1}{2}}{\sqrt{x}}}{\color{blue}{1 + x}} \]

    9. lower-+.f64 98.4

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

11. Applied rewrites 98.4%

    \[\leadsto \color{blue}{\frac{\frac{\frac{0.125}{x} + 0.5}{\sqrt{x}}}{1 + x}} \]
12. Add Preprocessing

Alternative 5: 98.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double x) {
	return ((0.5 - (0.375 / x)) / x) / sqrt(x);
}

Fortran

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

Java

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

Python

def code(x):
	return ((0.5 - (0.375 / x)) / x) / math.sqrt(x)

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 43.2%

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

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

   10. associate-/r* N/A

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

   11. lower-/.f64 N/A

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

4. Applied rewrites 43.2%

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

5. Taylor expanded in x around inf

   \[\leadsto \frac{\color{blue}{\frac{\frac{1}{2} + -1 \cdot \frac{\frac{1}{8} \cdot x + \frac{1}{4} \cdot x}{{x}^{2}}}{x}}}{\sqrt{x}} \]
6. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \frac{\color{blue}{\frac{\frac{1}{2} + -1 \cdot \frac{\frac{1}{8} \cdot x + \frac{1}{4} \cdot x}{{x}^{2}}}{x}}}{\sqrt{x}} \]

   2. mul-1-neg N/A

      \[\leadsto \frac{\frac{\frac{1}{2} + \color{blue}{\left(\mathsf{neg}\left(\frac{\frac{1}{8} \cdot x + \frac{1}{4} \cdot x}{{x}^{2}}\right)\right)}}{x}}{\sqrt{x}} \]

   3. unsub-neg N/A

      \[\leadsto \frac{\frac{\color{blue}{\frac{1}{2} - \frac{\frac{1}{8} \cdot x + \frac{1}{4} \cdot x}{{x}^{2}}}}{x}}{\sqrt{x}} \]

   4. distribute-rgt-out N/A

      \[\leadsto \frac{\frac{\frac{1}{2} - \frac{\color{blue}{x \cdot \left(\frac{1}{8} + \frac{1}{4}\right)}}{{x}^{2}}}{x}}{\sqrt{x}} \]

   5. metadata-eval N/A

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

   6. *-commutative N/A

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

   7. unpow2 N/A

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

   8. associate-/r* N/A

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

   9. *-lft-identity N/A

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

   10. times-frac N/A

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

   11. metadata-eval N/A

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

   12. *-inverses N/A

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

   13. metadata-eval N/A

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

   14. lower--.f64 N/A

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

   15. lower-/.f64 98.3

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

7. Applied rewrites 98.3%

   \[\leadsto \frac{\color{blue}{\frac{0.5 - \frac{0.375}{x}}{x}}}{\sqrt{x}} \]
8. Add Preprocessing

Alternative 6: 97.8% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 43.2%

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

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

   10. associate-/r* N/A

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

   11. lower-/.f64 N/A

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

4. Applied rewrites 43.2%

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

   1. lift-/.f64 N/A

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

   2. lift--.f64 N/A

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

   3. sub-div N/A

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

   4. frac-sub N/A

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

   5. *-commutative N/A

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

   6. lift-sqrt.f64 N/A

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

   7. lift-sqrt.f64 N/A

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

   8. rem-square-sqrt N/A

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

   9. lower-/.f64 N/A

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

6. Applied rewrites 7.5%

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

7. Taylor expanded in x around inf

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

9. Applied rewrites 97.5%

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

10. Final simplification 97.5%

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

Alternative 7: 97.6% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 43.2%

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

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

   10. associate-/r* N/A

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

   11. lower-/.f64 N/A

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

4. Applied rewrites 43.2%

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

5. Taylor expanded in x around inf

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

   1. lower-/.f64 97.3

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

7. Applied rewrites 97.3%

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

Alternative 8: 37.3% 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 43.2%

   \[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}} \]
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 41.2%

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

Alternative 9: 36.0% 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 43.2%

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

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

   10. associate-/r* N/A

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

   11. lower-/.f64 N/A

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

4. Applied rewrites 43.2%

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

   1. lift-/.f64 N/A

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

   2. lift--.f64 N/A

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

   3. sub-div N/A

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

   4. frac-sub N/A

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

   5. *-commutative N/A

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

   6. lift-sqrt.f64 N/A

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

   7. lift-sqrt.f64 N/A

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

   8. rem-square-sqrt N/A

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

   9. lower-/.f64 N/A

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

6. Applied rewrites 7.5%

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

7. Taylor expanded in x around -inf

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

   1. distribute-rgt-in N/A

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

   2. unpow2 N/A

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

   3. rem-square-sqrt N/A

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

   4. distribute-rgt-in N/A

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

   5. metadata-eval N/A

      \[\leadsto \sqrt{\frac{1}{x}} \cdot \color{blue}{0} \]

   6. mul0-rgt 39.9

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

9. Applied rewrites 39.9%

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

Developer Target 1: 38.9% 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 2024283 
(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)))))