2isqrt (example 3.6)

Percentage Accurate: 38.1% → 99.0%

Time: 7.5s

Alternatives: 10

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: 38.1% 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, 0.6× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{\frac{\frac{0.375}{x} - 0.5}{x} + 1}{x}}{\sqrt{1 + x} + \sqrt{x}} \end{array} \]

FPCore

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

C

double code(double x) {
	return (((((0.375 / x) - 0.5) / x) + 1.0) / x) / (sqrt((1.0 + x)) + sqrt(x));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (((((0.375d0 / x) - 0.5d0) / x) + 1.0d0) / x) / (sqrt((1.0d0 + x)) + sqrt(x))
end function

Java

public static double code(double x) {
	return (((((0.375 / x) - 0.5) / x) + 1.0) / x) / (Math.sqrt((1.0 + x)) + Math.sqrt(x));
}

Python

def code(x):
	return (((((0.375 / x) - 0.5) / x) + 1.0) / x) / (math.sqrt((1.0 + x)) + math.sqrt(x))

Julia

function code(x)
	return Float64(Float64(Float64(Float64(Float64(Float64(0.375 / x) - 0.5) / x) + 1.0) / x) / Float64(sqrt(Float64(1.0 + x)) + sqrt(x)))
end

MATLAB

function tmp = code(x)
	tmp = (((((0.375 / x) - 0.5) / x) + 1.0) / x) / (sqrt((1.0 + x)) + sqrt(x));
end

Wolfram

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

Derivation

1. Initial program 40.9%

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

4. Applied rewrites 43.1%

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

5. Taylor expanded in x around inf

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

   1. lower-/.f64 N/A

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

   2. lower--.f64 N/A

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

   3. +-commutative N/A

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

   4. lower-+.f64 N/A

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

   5. lower-/.f64 N/A

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

   6. unpow2 N/A

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

   7. lower-*.f64 N/A

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

   8. associate-*r/ N/A

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

   9. metadata-eval N/A

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

   10. lower-/.f64 99.4

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

7. Applied rewrites 99.4%

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

8. Taylor expanded in x around inf

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

   1. lower-/.f64 N/A

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

   2. associate--l+ N/A

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

   3. +-commutative N/A

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

   4. unpow2 N/A

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

   5. associate-/r* N/A

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

   6. metadata-eval N/A

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

   7. associate-*r/ N/A

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

   8. associate-*r/ N/A

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

   9. metadata-eval N/A

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

   10. div-sub N/A

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

   11. lower-+.f64 N/A

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

   12. lower-/.f64 N/A

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

   13. lower--.f64 N/A

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

   14. associate-*r/ N/A

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

   15. metadata-eval N/A

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

   16. lower-/.f64 99.4

       \[\leadsto \frac{\frac{\frac{\color{blue}{\frac{0.375}{x}} - 0.5}{x} + 1}{x}}{\sqrt{x} + \sqrt{x + 1}} \]

10. Applied rewrites 99.4%

    \[\leadsto \frac{\color{blue}{\frac{\frac{\frac{0.375}{x} - 0.5}{x} + 1}{x}}}{\sqrt{x} + \sqrt{x + 1}} \]

11. Final simplification 99.4%

    \[\leadsto \frac{\frac{\frac{\frac{0.375}{x} - 0.5}{x} + 1}{x}}{\sqrt{1 + x} + \sqrt{x}} \]
12. Add Preprocessing

Alternative 2: 98.9% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 40.9%

   \[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}} \]
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 40.9%

   \[\leadsto \color{blue}{\frac{\frac{\sqrt{x + 1} - \sqrt{x}}{\sqrt{x + 1}}}{\sqrt{x}}} \]
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. *-inverses N/A

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

   5. lower--.f64 N/A

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

   6. lift-sqrt.f64 N/A

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

   7. lift-sqrt.f64 N/A

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

   8. sqrt-undiv N/A

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

   9. lower-sqrt.f64 N/A

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

   10. lower-/.f64 41.0

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

6. Applied rewrites 41.0%

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

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 99.4

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

9. Applied rewrites 99.4%

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

Alternative 3: 98.6% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 40.9%

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

4. Applied rewrites 43.1%

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

5. Taylor expanded in x around inf

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

   1. lower-/.f64 N/A

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

   2. lower--.f64 N/A

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

   3. associate-*r/ N/A

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

   4. metadata-eval N/A

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

   5. lower-/.f64 99.1

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

7. Applied rewrites 99.1%

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

8. Final simplification 99.1%

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

Alternative 4: 98.6% 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 40.9%

   \[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}} \]
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 40.9%

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

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. mul-1-neg N/A

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

   2. unsub-neg N/A

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

   3. div-sub N/A

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

   4. distribute-rgt-out N/A

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

   5. metadata-eval N/A

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

   6. unpow2 N/A

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

   7. times-frac N/A

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

   8. *-rgt-identity N/A

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

   9. associate-*r/ N/A

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

   10. rgt-mult-inverse N/A

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

   11. metadata-eval N/A

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

   12. associate-*r/ N/A

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

   13. *-lft-identity N/A

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

   14. div-sub N/A

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

   15. lower-/.f64 N/A

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

7. Applied rewrites 99.1%

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

Alternative 5: 97.7% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 40.9%

   \[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}} \]
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 40.9%

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

5. Taylor expanded in x around inf

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

   1. lower-*.f64 N/A

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

   2. lower-sqrt.f64 N/A

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

   3. lower-/.f64 98.0

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

7. Applied rewrites 98.0%

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

   1. lift-/.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. frac-2neg N/A

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

   4. associate-/l/ N/A

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

   5. lower-/.f64 N/A

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

9. Applied rewrites 97.9%

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

10. Taylor expanded in x around inf

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

    1. distribute-lft-in N/A

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

    2. *-rgt-identity N/A

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

    3. distribute-lft-in N/A

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

    4. *-commutative N/A

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

    5. associate-*l* N/A

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

    6. lft-mult-inverse N/A

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

    7. metadata-eval N/A

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

    8. metadata-eval N/A

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

    9. lower-fma.f64 98.3

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

12. Applied rewrites 98.3%

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

Alternative 6: 97.6% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 40.9%

   \[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}} \]
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 40.9%

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

5. Taylor expanded in x around inf

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

   1. lower-*.f64 N/A

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

   2. lower-sqrt.f64 N/A

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

   3. lower-/.f64 98.0

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

7. Applied rewrites 98.0%

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

   1. lift-/.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. frac-2neg N/A

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

   4. associate-/l/ N/A

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

   5. lower-/.f64 N/A

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

9. Applied rewrites 97.9%

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

10. Taylor expanded in x around inf

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

    1. mul-1-neg N/A

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

    2. lower-neg.f64 98.2

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

12. Applied rewrites 98.2%

    \[\leadsto \frac{-\frac{0.5}{\sqrt{x}}}{\color{blue}{-x}} \]
13. 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 40.9%

   \[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}} \]
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 40.9%

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

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 98.1

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

7. Applied rewrites 98.1%

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

Alternative 8: 81.1% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 40.9%

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

3. Taylor expanded in x around inf

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

5. Applied rewrites 83.7%

   \[\leadsto \color{blue}{\frac{-0.5 \cdot \left(\sqrt{\frac{1}{x}} - \mathsf{fma}\left(\mathsf{fma}\left(0.25, x, 1\right), \sqrt{\frac{1}{{x}^{3}}}, \sqrt{x}\right)\right)}{x \cdot x}} \]

6. Taylor expanded in x around inf

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

8. Applied rewrites 82.8%

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

9. Final simplification 82.8%

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

Alternative 9: 36.5% 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 40.9%

   \[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}} \]
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 39.8%

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

Alternative 10: 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 40.9%

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

3. Taylor expanded in x around 0

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

   1. lower-sqrt.f64 N/A

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

   2. lower-/.f64 5.6

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

5. Applied rewrites 5.6%

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

Developer Target 1: 38.2% 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 2024304 
(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)))))