2isqrt (example 3.6)

Percentage Accurate: 38.6% → 98.6%

Time: 10.3s

Alternatives: 6

Speedup: 1.8×

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: 38.6% 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: 98.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 10^{+67}:\\ \;\;\;\;\frac{\frac{-0.375}{x} + 0.5}{x \cdot \sqrt{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.5}{x}}{\sqrt{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1e+67)
   (/ (+ (/ -0.375 x) 0.5) (* x (sqrt x)))
   (/ (/ 0.5 x) (sqrt x))))

C

double code(double x) {
	double tmp;
	if (x <= 1e+67) {
		tmp = ((-0.375 / x) + 0.5) / (x * sqrt(x));
	} else {
		tmp = (0.5 / x) / sqrt(x);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 1d+67) then
        tmp = (((-0.375d0) / x) + 0.5d0) / (x * sqrt(x))
    else
        tmp = (0.5d0 / x) / sqrt(x)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 1e+67) {
		tmp = ((-0.375 / x) + 0.5) / (x * Math.sqrt(x));
	} else {
		tmp = (0.5 / x) / Math.sqrt(x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 1e+67:
		tmp = ((-0.375 / x) + 0.5) / (x * math.sqrt(x))
	else:
		tmp = (0.5 / x) / math.sqrt(x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 1e+67)
		tmp = Float64(Float64(Float64(-0.375 / x) + 0.5) / Float64(x * sqrt(x)));
	else
		tmp = Float64(Float64(0.5 / x) / sqrt(x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 1e+67)
		tmp = ((-0.375 / x) + 0.5) / (x * sqrt(x));
	else
		tmp = (0.5 / x) / sqrt(x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 1e+67], N[(N[(N[(-0.375 / x), $MachinePrecision] + 0.5), $MachinePrecision] / N[(x * N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 / x), $MachinePrecision] / N[Sqrt[x], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 9.99999999999999983e66

   1. Initial program 22.5%

      \[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}} \]
   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. Simplified 94.1%

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

      1. associate-/r* N/A

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

      2. /-lowering-/.f64 N/A

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

   7. Applied egg-rr 94.1%

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

   8. Taylor expanded in x around -inf

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

      1. associate-*r/ N/A

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

      2. /-lowering-/.f64 N/A

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

   10. Simplified 94.0%

       \[\leadsto \color{blue}{\frac{\sqrt{\frac{1}{x}} \cdot \left(\frac{-0.375}{x} + 0.5\right)}{x}} \]
   11. Step-by-step derivation

       1. div-inv N/A

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

       2. *-commutative N/A

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

       3. associate-*l* N/A

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

       4. div-inv N/A

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

       5. sqrt-div N/A

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

       6. metadata-eval N/A

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

       7. associate-/r* N/A

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

       8. div-inv N/A

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

       9. /-lowering-/.f64 N/A

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

       10. +-lowering-+.f64 N/A

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

       11. /-lowering-/.f64 N/A

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

       12. *-commutative N/A

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

       13. *-lowering-*.f64 N/A

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

       14. sqrt-lowering-sqrt.f64 94.2

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

   12. Applied egg-rr 94.2%

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

   if 9.99999999999999983e66 < x

   1. Initial program 44.3%

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

   3. Taylor expanded in x around inf

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

      1. *-lowering-*.f64 N/A

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

      2. sqrt-lowering-sqrt.f64 N/A

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

      3. /-lowering-/.f64 N/A

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

      4. cube-mult N/A

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

      5. unpow2 N/A

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

      6. *-lowering-*.f64 N/A

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

      7. unpow2 N/A

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

      8. *-lowering-*.f64 55.5

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

   5. Simplified 55.5%

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

      1. associate-/r* N/A

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

      2. sqrt-div N/A

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

      3. sqrt-div N/A

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

      4. metadata-eval N/A

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

      5. sqrt-unprod N/A

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

      6. rem-square-sqrt N/A

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

      7. associate-/r* N/A

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

      8. *-commutative N/A

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

      9. associate-/r* N/A

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

      10. /-lowering-/.f64 N/A

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

      11. /-lowering-/.f64 N/A

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

      12. sqrt-lowering-sqrt.f64 99.7

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

   7. Applied egg-rr 99.7%

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

      1. associate-*r/ N/A

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

      2. /-lowering-/.f64 N/A

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

      3. un-div-inv N/A

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

      4. /-lowering-/.f64 N/A

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

      5. sqrt-lowering-sqrt.f64 99.8

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

   9. Applied egg-rr 99.8%

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

Alternative 2: 98.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \frac{\sqrt{\frac{1}{x}} \cdot \left(\frac{-0.375}{x} + 0.5\right)}{x} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

5. Simplified 80.3%

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

   1. associate-/r* N/A

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

   2. /-lowering-/.f64 N/A

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

7. Applied egg-rr 98.4%

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

8. Taylor expanded in x around -inf

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

   1. associate-*r/ N/A

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

   2. /-lowering-/.f64 N/A

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

10. Simplified 98.4%

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

Alternative 3: 98.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := N[(N[(N[(N[(-0.375 / x), $MachinePrecision] + 0.5), $MachinePrecision] / x), $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. Simplified 80.3%

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

   1. associate-/r* N/A

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

   2. /-lowering-/.f64 N/A

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

7. Applied egg-rr 98.4%

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

8. Taylor expanded in x around -inf

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

   1. associate-*r/ N/A

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

   2. /-lowering-/.f64 N/A

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

10. Simplified 98.4%

    \[\leadsto \color{blue}{\frac{\sqrt{\frac{1}{x}} \cdot \left(\frac{-0.375}{x} + 0.5\right)}{x}} \]
11. Step-by-step derivation

    1. associate-/l* N/A

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

    2. sqrt-div N/A

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

    3. metadata-eval N/A

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

    4. associate-*l/ N/A

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

    5. clear-num N/A

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

    6. div-inv N/A

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

    7. clear-num N/A

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

    8. /-lowering-/.f64 N/A

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

    9. /-lowering-/.f64 N/A

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

    10. +-lowering-+.f64 N/A

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

    11. /-lowering-/.f64 N/A

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

    12. sqrt-lowering-sqrt.f64 98.4

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

12. Applied egg-rr 98.4%

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

Alternative 4: 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 39.2%

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

3. Taylor expanded in x around inf

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

   1. *-lowering-*.f64 N/A

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

   2. sqrt-lowering-sqrt.f64 N/A

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

   3. /-lowering-/.f64 N/A

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

   4. cube-mult N/A

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

   5. unpow2 N/A

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

   6. *-lowering-*.f64 N/A

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

   7. unpow2 N/A

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

   8. *-lowering-*.f64 62.7

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

5. Simplified 62.7%

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

   1. associate-/r* N/A

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

   2. sqrt-div N/A

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

   3. sqrt-div N/A

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

   4. metadata-eval N/A

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

   5. sqrt-unprod N/A

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

   6. rem-square-sqrt N/A

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

   7. associate-/r* N/A

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

   8. *-commutative N/A

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

   9. associate-/r* N/A

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

   10. /-lowering-/.f64 N/A

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

   11. /-lowering-/.f64 N/A

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

   12. sqrt-lowering-sqrt.f64 96.5

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

7. Applied egg-rr 96.5%

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

   1. associate-*r/ N/A

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

   2. /-lowering-/.f64 N/A

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

   3. un-div-inv N/A

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

   4. /-lowering-/.f64 N/A

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

   5. sqrt-lowering-sqrt.f64 96.5

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

9. Applied egg-rr 96.5%

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

Alternative 5: 96.4% accurate, 1.8× speedup

Math

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

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(0.5 / Float64(x * sqrt(x)))
end

MATLAB

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

Wolfram

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

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

   1. *-lowering-*.f64 N/A

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

   2. sqrt-lowering-sqrt.f64 N/A

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

   3. /-lowering-/.f64 N/A

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

   4. cube-mult N/A

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

   5. unpow2 N/A

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

   6. *-lowering-*.f64 N/A

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

   7. unpow2 N/A

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

   8. *-lowering-*.f64 62.7

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

5. Simplified 62.7%

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

   1. associate-/r* N/A

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

   2. sqrt-div N/A

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

   3. sqrt-unprod N/A

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

   4. rem-square-sqrt N/A

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

   5. sqrt-div N/A

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

   6. metadata-eval N/A

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

   7. associate-/r* N/A

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

   8. *-commutative N/A

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

   9. un-div-inv N/A

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

   10. /-lowering-/.f64 N/A

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

   11. *-lowering-*.f64 N/A

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

   12. sqrt-lowering-sqrt.f64 94.9

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

7. Applied egg-rr 94.9%

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

Alternative 6: 35.7% 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 39.2%

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

3. Taylor expanded in x around inf

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

   1. sqrt-lowering-sqrt.f64 N/A

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

   2. /-lowering-/.f64 28.8

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

5. Simplified 28.8%

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

   1. metadata-eval N/A

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

   2. sqrt-div N/A

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

   3. +-inverses 34.7

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

7. Applied egg-rr 34.7%

   \[\leadsto \color{blue}{0} \]
8. 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: 38.7% 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 2024205 
(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)))))