2isqrt (example 3.6)

Percentage Accurate: 38.9% → 99.2%

Time: 10.1s

Alternatives: 8

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: 99.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{0.5 + \left(\frac{-0.125 + \frac{0.0625}{x}}{x} + \frac{-0.0390625}{x \cdot \left(x \cdot x\right)}\right)}{x}}{\sqrt{1 + x}} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/
  (/ (+ 0.5 (+ (/ (+ -0.125 (/ 0.0625 x)) x) (/ -0.0390625 (* x (* x x))))) x)
  (sqrt (+ 1.0 x))))

C

double code(double x) {
	return ((0.5 + (((-0.125 + (0.0625 / x)) / x) + (-0.0390625 / (x * (x * x))))) / x) / sqrt((1.0 + x));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = ((0.5d0 + ((((-0.125d0) + (0.0625d0 / x)) / x) + ((-0.0390625d0) / (x * (x * x))))) / x) / sqrt((1.0d0 + x))
end function

Java

public static double code(double x) {
	return ((0.5 + (((-0.125 + (0.0625 / x)) / x) + (-0.0390625 / (x * (x * x))))) / x) / Math.sqrt((1.0 + x));
}

Python

def code(x):
	return ((0.5 + (((-0.125 + (0.0625 / x)) / x) + (-0.0390625 / (x * (x * x))))) / x) / math.sqrt((1.0 + x))

Julia

function code(x)
	return Float64(Float64(Float64(0.5 + Float64(Float64(Float64(-0.125 + Float64(0.0625 / x)) / x) + Float64(-0.0390625 / Float64(x * Float64(x * x))))) / x) / sqrt(Float64(1.0 + x)))
end

MATLAB

function tmp = code(x)
	tmp = ((0.5 + (((-0.125 + (0.0625 / x)) / x) + (-0.0390625 / (x * (x * x))))) / x) / sqrt((1.0 + x));
end

Wolfram

code[x_] := N[(N[(N[(0.5 + N[(N[(N[(-0.125 + N[(0.0625 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] + N[(-0.0390625 / N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] / N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 39.2%

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

4. Applied rewrites 40.9%

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

   1. lift-/.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-/r/ N/A

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

   4. lift-/.f64 N/A

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

   5. frac-times N/A

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

   6. lift--.f64 N/A

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

   7. lift-+.f64 N/A

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

   8. associate--l+ N/A

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

   9. +-inverses N/A

      \[\leadsto \frac{1 \cdot \left(1 + \color{blue}{0}\right)}{\sqrt{1 + x} \cdot \left(x + \sqrt{\mathsf{fma}\left(x, x, x\right)}\right)} \]

   10. metadata-eval N/A

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

   11. metadata-eval N/A

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

   12. associate-/l/ N/A

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

   13. lower-/.f64 N/A

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

   14. lower-/.f64 84.3

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

6. Applied rewrites 84.3%

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

7. Taylor expanded in x around inf

   \[\leadsto \frac{\color{blue}{\frac{\left(\frac{1}{2} + \frac{\frac{1}{16}}{{x}^{2}}\right) - \left(\frac{1}{8} \cdot \frac{1}{x} + \frac{5}{128} \cdot \frac{1}{{x}^{3}}\right)}{x}}}{\sqrt{1 + x}} \]
8. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \frac{\color{blue}{\frac{\left(\frac{1}{2} + \frac{\frac{1}{16}}{{x}^{2}}\right) - \left(\frac{1}{8} \cdot \frac{1}{x} + \frac{5}{128} \cdot \frac{1}{{x}^{3}}\right)}{x}}}{\sqrt{1 + x}} \]

9. Applied rewrites 99.7%

   \[\leadsto \frac{\color{blue}{\frac{0.5 + \left(\frac{-0.125 + \frac{0.0625}{x}}{x} + \frac{-0.0390625}{x \cdot \left(x \cdot x\right)}\right)}{x}}}{\sqrt{1 + x}} \]
10. Add Preprocessing

Alternative 2: 98.8% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 39.2%

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

4. Applied rewrites 40.9%

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

   1. lift-/.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-/r/ N/A

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

   4. lift-/.f64 N/A

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

   5. frac-times N/A

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

   6. lift--.f64 N/A

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

   7. lift-+.f64 N/A

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

   8. associate--l+ N/A

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

   9. +-inverses N/A

      \[\leadsto \frac{1 \cdot \left(1 + \color{blue}{0}\right)}{\sqrt{1 + x} \cdot \left(x + \sqrt{\mathsf{fma}\left(x, x, x\right)}\right)} \]

   10. metadata-eval N/A

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

   11. metadata-eval N/A

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

   12. associate-/l/ N/A

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

   13. lower-/.f64 N/A

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

   14. lower-/.f64 84.3

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

6. Applied rewrites 84.3%

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

7. Taylor expanded in x around inf

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

   1. distribute-rgt-in N/A

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

   2. *-lft-identity N/A

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

   3. associate-*l* N/A

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

   4. lft-mult-inverse N/A

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

   5. metadata-eval N/A

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

   6. lower-+.f64 99.4

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

9. Applied rewrites 99.4%

   \[\leadsto \frac{\frac{1}{x + \color{blue}{\left(x + 0.5\right)}}}{\sqrt{1 + x}} \]
10. Add Preprocessing

Alternative 3: 97.8% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 39.2%

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

4. Applied rewrites 40.9%

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

   1. lift-/.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-/r/ N/A

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

   4. lift-/.f64 N/A

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

   5. frac-times N/A

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

   6. lift--.f64 N/A

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

   7. lift-+.f64 N/A

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

   8. associate--l+ N/A

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

   9. +-inverses N/A

      \[\leadsto \frac{1 \cdot \left(1 + \color{blue}{0}\right)}{\sqrt{1 + x} \cdot \left(x + \sqrt{\mathsf{fma}\left(x, x, x\right)}\right)} \]

   10. metadata-eval N/A

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

   11. metadata-eval N/A

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

   12. associate-/l/ N/A

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

   13. lower-/.f64 N/A

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

   14. lower-/.f64 84.3

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

6. Applied rewrites 84.3%

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

7. Taylor expanded in x around inf

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

   1. lower-/.f64 98.6

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

9. Applied rewrites 98.6%

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

Alternative 4: 97.6% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 39.2%

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

3. Taylor expanded in x around inf

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

   1. distribute-lft-out-- N/A

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

   2. *-commutative N/A

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

   3. associate-/l* N/A

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

   4. lower-*.f64 N/A

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

   5. lower--.f64 N/A

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

   6. lower-sqrt.f64 N/A

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

   7. lower-/.f64 N/A

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

   8. lower-sqrt.f64 N/A

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

   9. lower-/.f64 N/A

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

   10. unpow2 N/A

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

   11. lower-*.f64 83.2

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

5. Applied rewrites 83.2%

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

7. Applied rewrites 98.5%

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

8. Taylor expanded in x around inf

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

10. Applied rewrites 98.5%

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

12. Applied rewrites 98.5%

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

Alternative 5: 96.4% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 39.2%

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

4. Applied rewrites 40.9%

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

5. Taylor expanded in x around inf

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

   1. lower-*.f64 N/A

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

   2. lower-sqrt.f64 N/A

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

   3. cube-mult N/A

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

   4. unpow2 N/A

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

   5. lower-*.f64 N/A

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

   6. unpow2 N/A

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

   7. lower-*.f64 69.2

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

7. Applied rewrites 69.2%

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

9. Applied rewrites 97.4%

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

Alternative 6: 81.1% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 39.2%

   \[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}} \]
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 84.1%

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

6. Taylor expanded in x around inf

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

8. Applied rewrites 83.2%

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

Alternative 7: 81.0% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 39.2%

   \[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}} \]
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 84.1%

   \[\leadsto \color{blue}{\frac{0.5 \cdot \left(\mathsf{fma}\left(\sqrt{\frac{1}{x \cdot \left(x \cdot x\right)}}, \mathsf{fma}\left(x, 0.25, 1\right), \sqrt{x}\right) - \sqrt{\frac{1}{x}}\right)}{x \cdot x}} \]
6. Step-by-step derivation

7. Applied rewrites 84.0%

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

8. Taylor expanded in x around inf

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

10. Applied rewrites 83.2%

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

12. Applied rewrites 83.1%

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

Alternative 8: 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 39.2%

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

Developer Target 1: 98.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Developer Target 2: 38.9% 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 2024223 
(FPCore (x)
  :name "2isqrt (example 3.6)"
  :precision binary64
  :pre (and (> x 1.0) (< x 1e+308))

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

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

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