2isqrt (example 3.6)

Percentage Accurate: 37.6% → 99.7%

Time: 10.7s

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: 37.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 5 \cdot 10^{+21}:\\ \;\;\;\;\frac{1}{\sqrt{1 + x} \cdot \left(x + \sqrt{\mathsf{fma}\left(x, x, x\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 \cdot \sqrt{\frac{1}{x}}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 5e+21)
   (/ 1.0 (* (sqrt (+ 1.0 x)) (+ x (sqrt (fma x x x)))))
   (/ (* 0.5 (sqrt (/ 1.0 x))) x)))

C

double code(double x) {
	double tmp;
	if (x <= 5e+21) {
		tmp = 1.0 / (sqrt((1.0 + x)) * (x + sqrt(fma(x, x, x))));
	} else {
		tmp = (0.5 * sqrt((1.0 / x))) / x;
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 5e+21)
		tmp = Float64(1.0 / Float64(sqrt(Float64(1.0 + x)) * Float64(x + sqrt(fma(x, x, x)))));
	else
		tmp = Float64(Float64(0.5 * sqrt(Float64(1.0 / x))) / x);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 5e+21], N[(1.0 / N[(N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision] * N[(x + N[Sqrt[N[(x * x + x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 * N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 5e21

   1. Initial program 35.5%

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

   4. Applied rewrites 62.9%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. lift--.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. associate--l+ N/A

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

      7. +-inverses N/A

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

      8. metadata-eval N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 99.3

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

   6. Applied rewrites 99.3%

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

   if 5e21 < x

   1. Initial program 37.7%

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

   3. Taylor expanded in x around inf

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

   5. Applied rewrites 82.2%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 99.7%

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

   9. Taylor expanded in x around inf

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

   11. Applied rewrites 99.7%

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

Alternative 2: 99.0% accurate, 0.8× speedup

Math

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

FPCore

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

C

double code(double x) {
	return ((0.5 + ((-0.125 + (0.0625 / 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)) / x) / sqrt((1.0d0 + x))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 37.5%

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

4. Applied rewrites 39.7%

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

5. Taylor expanded in x around inf

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

7. Applied rewrites 99.3%

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

Alternative 3: 98.6% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 37.5%

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

   1. lift--.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. frac-sub N/A

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

   5. div-inv N/A

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

   6. metadata-eval N/A

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

   7. frac-times N/A

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

   8. lift-/.f64 N/A

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

   9. lift-/.f64 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f64 N/A

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

4. Applied rewrites 37.5%

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

5. Taylor expanded in x around inf

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

   1. distribute-rgt-in N/A

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

   2. *-lft-identity N/A

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

   3. associate-*l* N/A

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

   4. lft-mult-inverse N/A

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

   5. metadata-eval N/A

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

   6. lower-+.f64 37.2

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

7. Applied rewrites 37.2%

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

   1. lift-*.f64 N/A

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

   2. lift--.f64 N/A

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

   3. flip-- N/A

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

   4. lift-+.f64 N/A

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

   5. associate-*r/ N/A

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

9. Applied rewrites 98.9%

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

10. Final simplification 98.9%

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

Alternative 4: 97.6% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 37.5%

   \[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}} \]
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 82.8%

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

6. Taylor expanded in x around inf

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

8. Applied rewrites 98.9%

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

9. Taylor expanded in x around inf

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

11. Applied rewrites 97.9%

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

Alternative 5: 97.7% 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 37.5%

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

4. Applied rewrites 39.7%

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

5. Taylor expanded in x around inf

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

   1. lower-/.f64 97.9

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

7. Applied rewrites 97.9%

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

Alternative 6: 80.9% 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 37.5%

   \[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}} \]
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 82.8%

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

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 81.8%

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

Alternative 7: 36.7% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 4.6 \cdot 10^{+153}:\\ \;\;\;\;\frac{1}{x + \sqrt{x}}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 4.6e+153) (/ 1.0 (+ x (sqrt x))) 0.0))

C

double code(double x) {
	double tmp;
	if (x <= 4.6e+153) {
		tmp = 1.0 / (x + sqrt(x));
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 4.6d+153) then
        tmp = 1.0d0 / (x + sqrt(x))
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 4.6e+153) {
		tmp = 1.0 / (x + Math.sqrt(x));
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 4.6e+153:
		tmp = 1.0 / (x + math.sqrt(x))
	else:
		tmp = 0.0
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 4.6e+153)
		tmp = Float64(1.0 / Float64(x + sqrt(x)));
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 4.6e+153)
		tmp = 1.0 / (x + sqrt(x));
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 4.6e+153], N[(1.0 / N[(x + N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0]

Derivation

1. Split input into 2 regimes

2. if x < 4.6000000000000003e153

   1. Initial program 9.5%

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

   4. Applied rewrites 13.9%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. rem-square-sqrt N/A

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

      4. *-rgt-identity N/A

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

      5. lower-+.f64 N/A

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

      6. lower-sqrt.f64 5.3

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

   7. Applied rewrites 5.3%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 8.4%

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

   if 4.6000000000000003e153 < x

   1. Initial program 66.8%

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

      1. lift--.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. clear-num N/A

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

      4. lift-/.f64 N/A

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

      5. frac-sub N/A

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

      6. div-inv N/A

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

      7. metadata-eval N/A

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

      8. *-rgt-identity N/A

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

      9. lower-/.f64 N/A

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

   4. Applied rewrites 66.8%

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

   6. Applied rewrites 4.3%

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

   7. Taylor expanded in x around inf

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

      1. distribute-rgt1-in N/A

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

      2. metadata-eval N/A

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

      3. mul0-lft 66.8

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

   9. Applied rewrites 66.8%

      \[\leadsto \color{blue}{0} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 35.2% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 37.5%

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

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

9. Applied rewrites 35.5%

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

Alternative 9: 34.7% 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 37.5%

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

   1. lift--.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. clear-num N/A

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

   4. lift-/.f64 N/A

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

   5. frac-sub N/A

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

   6. div-inv N/A

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

   7. metadata-eval N/A

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

   8. *-rgt-identity N/A

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

   9. lower-/.f64 N/A

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

4. Applied rewrites 37.5%

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

6. Applied rewrites 7.2%

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

7. Taylor expanded in x around inf

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

   1. distribute-rgt1-in N/A

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

   2. metadata-eval N/A

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

   3. mul0-lft 34.8

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

9. Applied rewrites 34.8%

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

Developer Target 1: 98.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Developer Target 2: 37.7% 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 2024226 
(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)))))