2sqrt (example 3.1)

Percentage Accurate: 6.6% → 100.0%

Time: 10.6s

Alternatives: 8

Speedup: 2.0×

Specification

\[x > 1 \land x < 10^{+308}\]

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 6.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sqrt{x + 1} - \sqrt{x} \leq 5 \cdot 10^{-5}:\\ \;\;\;\;\frac{-0.125}{{x}^{1.5}} - -0.5 \cdot {x}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + \left(1 - x\right)}{{x}^{1.5} + {\left(x + 1\right)}^{1.5}} \cdot \left(\left(x + \left(x + 1\right)\right) - \sqrt{x \cdot \left(x + 1\right)}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (- (sqrt (+ x 1.0)) (sqrt x)) 5e-5)
   (- (/ -0.125 (pow x 1.5)) (* -0.5 (pow x -0.5)))
   (*
    (/ (+ x (- 1.0 x)) (+ (pow x 1.5) (pow (+ x 1.0) 1.5)))
    (- (+ x (+ x 1.0)) (sqrt (* x (+ x 1.0)))))))

C

double code(double x) {
	double tmp;
	if ((sqrt((x + 1.0)) - sqrt(x)) <= 5e-5) {
		tmp = (-0.125 / pow(x, 1.5)) - (-0.5 * pow(x, -0.5));
	} else {
		tmp = ((x + (1.0 - x)) / (pow(x, 1.5) + pow((x + 1.0), 1.5))) * ((x + (x + 1.0)) - sqrt((x * (x + 1.0))));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((sqrt((x + 1.0d0)) - sqrt(x)) <= 5d-5) then
        tmp = ((-0.125d0) / (x ** 1.5d0)) - ((-0.5d0) * (x ** (-0.5d0)))
    else
        tmp = ((x + (1.0d0 - x)) / ((x ** 1.5d0) + ((x + 1.0d0) ** 1.5d0))) * ((x + (x + 1.0d0)) - sqrt((x * (x + 1.0d0))))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if ((Math.sqrt((x + 1.0)) - Math.sqrt(x)) <= 5e-5) {
		tmp = (-0.125 / Math.pow(x, 1.5)) - (-0.5 * Math.pow(x, -0.5));
	} else {
		tmp = ((x + (1.0 - x)) / (Math.pow(x, 1.5) + Math.pow((x + 1.0), 1.5))) * ((x + (x + 1.0)) - Math.sqrt((x * (x + 1.0))));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if (math.sqrt((x + 1.0)) - math.sqrt(x)) <= 5e-5:
		tmp = (-0.125 / math.pow(x, 1.5)) - (-0.5 * math.pow(x, -0.5))
	else:
		tmp = ((x + (1.0 - x)) / (math.pow(x, 1.5) + math.pow((x + 1.0), 1.5))) * ((x + (x + 1.0)) - math.sqrt((x * (x + 1.0))))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (Float64(sqrt(Float64(x + 1.0)) - sqrt(x)) <= 5e-5)
		tmp = Float64(Float64(-0.125 / (x ^ 1.5)) - Float64(-0.5 * (x ^ -0.5)));
	else
		tmp = Float64(Float64(Float64(x + Float64(1.0 - x)) / Float64((x ^ 1.5) + (Float64(x + 1.0) ^ 1.5))) * Float64(Float64(x + Float64(x + 1.0)) - sqrt(Float64(x * Float64(x + 1.0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if ((sqrt((x + 1.0)) - sqrt(x)) <= 5e-5)
		tmp = (-0.125 / (x ^ 1.5)) - (-0.5 * (x ^ -0.5));
	else
		tmp = ((x + (1.0 - x)) / ((x ^ 1.5) + ((x + 1.0) ^ 1.5))) * ((x + (x + 1.0)) - sqrt((x * (x + 1.0))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[N[(N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision], 5e-5], N[(N[(-0.125 / N[Power[x, 1.5], $MachinePrecision]), $MachinePrecision] - N[(-0.5 * N[Power[x, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(x + N[(1.0 - x), $MachinePrecision]), $MachinePrecision] / N[(N[Power[x, 1.5], $MachinePrecision] + N[Power[N[(x + 1.0), $MachinePrecision], 1.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(x + N[(x + 1.0), $MachinePrecision]), $MachinePrecision] - N[Sqrt[N[(x * N[(x + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) < 5.00000000000000024e-5

   1. Initial program 4.7%

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

   3. Taylor expanded in x around inf

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

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

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{-1}{8} \cdot \sqrt{\frac{1}{x}} + \frac{1}{2} \cdot \sqrt{x}\right), \color{blue}{x}\right) \]

      2. remove-double-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{-1}{8} \cdot \sqrt{\frac{1}{x}} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\frac{1}{2} \cdot \sqrt{x}\right)\right)\right)\right)\right), x\right) \]

      3. mul-1-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{-1}{8} \cdot \sqrt{\frac{1}{x}} + \left(\mathsf{neg}\left(-1 \cdot \left(\frac{1}{2} \cdot \sqrt{x}\right)\right)\right)\right), x\right) \]

      4. unsub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{-1}{8} \cdot \sqrt{\frac{1}{x}} - -1 \cdot \left(\frac{1}{2} \cdot \sqrt{x}\right)\right), x\right) \]

      5. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(\frac{-1}{8} \cdot \sqrt{\frac{1}{x}}\right), \left(-1 \cdot \left(\frac{1}{2} \cdot \sqrt{x}\right)\right)\right), x\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \left(\sqrt{\frac{1}{x}}\right)\right), \left(-1 \cdot \left(\frac{1}{2} \cdot \sqrt{x}\right)\right)\right), x\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{sqrt.f64}\left(\left(\frac{1}{x}\right)\right)\right), \left(-1 \cdot \left(\frac{1}{2} \cdot \sqrt{x}\right)\right)\right), x\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, x\right)\right)\right), \left(-1 \cdot \left(\frac{1}{2} \cdot \sqrt{x}\right)\right)\right), x\right) \]

      9. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, x\right)\right)\right), \left(\left(-1 \cdot \frac{1}{2}\right) \cdot \sqrt{x}\right)\right), x\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, x\right)\right)\right), \left(\frac{-1}{2} \cdot \sqrt{x}\right)\right), x\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, x\right)\right)\right), \left(\sqrt{x} \cdot \frac{-1}{2}\right)\right), x\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, x\right)\right)\right), \mathsf{*.f64}\left(\left(\sqrt{x}\right), \frac{-1}{2}\right)\right), x\right) \]

      13. sqrt-lowering-sqrt.f64 99.6%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, x\right)\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \frac{-1}{2}\right)\right), x\right) \]

   5. Simplified 99.6%

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

      1. div-sub N/A

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

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

         \[\leadsto \mathsf{\_.f64}\left(\left(\frac{\frac{-1}{8} \cdot \sqrt{\frac{1}{x}}}{x}\right), \color{blue}{\left(\frac{\sqrt{x} \cdot \frac{-1}{2}}{x}\right)}\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\frac{\sqrt{\frac{1}{x}} \cdot \frac{-1}{8}}{x}\right), \left(\frac{\color{blue}{\sqrt{x}} \cdot \frac{-1}{2}}{x}\right)\right) \]

      4. associate-/l* N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\sqrt{\frac{1}{x}} \cdot \frac{\frac{-1}{8}}{x}\right), \left(\frac{\color{blue}{\sqrt{x} \cdot \frac{-1}{2}}}{x}\right)\right) \]

      5. sqrt-div N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\frac{\sqrt{1}}{\sqrt{x}} \cdot \frac{\frac{-1}{8}}{x}\right), \left(\frac{\color{blue}{\sqrt{x}} \cdot \frac{-1}{2}}{x}\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\frac{1}{\sqrt{x}} \cdot \frac{\frac{-1}{8}}{x}\right), \left(\frac{\sqrt{\color{blue}{x}} \cdot \frac{-1}{2}}{x}\right)\right) \]

      7. frac-times N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\frac{1 \cdot \frac{-1}{8}}{\sqrt{x} \cdot x}\right), \left(\frac{\color{blue}{\sqrt{x} \cdot \frac{-1}{2}}}{x}\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\frac{\frac{-1}{8}}{\sqrt{x} \cdot x}\right), \left(\frac{\color{blue}{\sqrt{x}} \cdot \frac{-1}{2}}{x}\right)\right) \]

      9. rem-square-sqrt N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\frac{\frac{-1}{8}}{\sqrt{x} \cdot \left(\sqrt{x} \cdot \sqrt{x}\right)}\right), \left(\frac{\sqrt{x} \cdot \frac{-1}{2}}{x}\right)\right) \]

      10. cube-mult N/A

          \[\leadsto \mathsf{\_.f64}\left(\left(\frac{\frac{-1}{8}}{{\left(\sqrt{x}\right)}^{3}}\right), \left(\frac{\sqrt{x} \cdot \color{blue}{\frac{-1}{2}}}{x}\right)\right) \]

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

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\frac{-1}{8}, \left({\left(\sqrt{x}\right)}^{3}\right)\right), \left(\frac{\color{blue}{\sqrt{x} \cdot \frac{-1}{2}}}{x}\right)\right) \]

      12. pow1/2 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\frac{-1}{8}, \left({\left({x}^{\frac{1}{2}}\right)}^{3}\right)\right), \left(\frac{\sqrt{x} \cdot \frac{-1}{2}}{x}\right)\right) \]

      13. pow-pow N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\frac{-1}{8}, \left({x}^{\left(\frac{1}{2} \cdot 3\right)}\right)\right), \left(\frac{\sqrt{x} \cdot \color{blue}{\frac{-1}{2}}}{x}\right)\right) \]

      14. pow-lowering-pow.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\frac{-1}{8}, \mathsf{pow.f64}\left(x, \left(\frac{1}{2} \cdot 3\right)\right)\right), \left(\frac{\sqrt{x} \cdot \color{blue}{\frac{-1}{2}}}{x}\right)\right) \]

      15. metadata-eval N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\frac{-1}{8}, \mathsf{pow.f64}\left(x, \frac{3}{2}\right)\right), \left(\frac{\sqrt{x} \cdot \frac{-1}{2}}{x}\right)\right) \]

      16. *-rgt-identity N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\frac{-1}{8}, \mathsf{pow.f64}\left(x, \frac{3}{2}\right)\right), \left(\frac{\sqrt{x} \cdot \frac{-1}{2}}{x \cdot \color{blue}{1}}\right)\right) \]

      17. times-frac N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\frac{-1}{8}, \mathsf{pow.f64}\left(x, \frac{3}{2}\right)\right), \left(\frac{\sqrt{x}}{x} \cdot \color{blue}{\frac{\frac{-1}{2}}{1}}\right)\right) \]

   7. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{\frac{-0.125}{{x}^{1.5}} - {x}^{-0.5} \cdot -0.5} \]

   if 5.00000000000000024e-5 < (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x))

   1. Initial program 88.4%

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

   4. Applied egg-rr 99.5%

      \[\leadsto \color{blue}{\frac{x + \left(1 - x\right)}{{\left(x + 1\right)}^{1.5} + {x}^{1.5}} \cdot \left(\left(x + \left(x + 1\right)\right) - \sqrt{x \cdot \left(x + 1\right)}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{x + 1} - \sqrt{x} \leq 5 \cdot 10^{-5}:\\ \;\;\;\;\frac{-0.125}{{x}^{1.5}} - -0.5 \cdot {x}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + \left(1 - x\right)}{{x}^{1.5} + {\left(x + 1\right)}^{1.5}} \cdot \left(\left(x + \left(x + 1\right)\right) - \sqrt{x \cdot \left(x + 1\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{x + 1}\\ \mathbf{if}\;t\_0 - \sqrt{x} \leq 0:\\ \;\;\;\;{x}^{-0.5} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{x + \left(1 - x\right)}{t\_0 + \sqrt{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (sqrt (+ x 1.0))))
   (if (<= (- t_0 (sqrt x)) 0.0)
     (* (pow x -0.5) 0.5)
     (/ (+ x (- 1.0 x)) (+ t_0 (sqrt x))))))

C

double code(double x) {
	double t_0 = sqrt((x + 1.0));
	double tmp;
	if ((t_0 - sqrt(x)) <= 0.0) {
		tmp = pow(x, -0.5) * 0.5;
	} else {
		tmp = (x + (1.0 - x)) / (t_0 + sqrt(x));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sqrt((x + 1.0d0))
    if ((t_0 - sqrt(x)) <= 0.0d0) then
        tmp = (x ** (-0.5d0)) * 0.5d0
    else
        tmp = (x + (1.0d0 - x)) / (t_0 + sqrt(x))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double t_0 = Math.sqrt((x + 1.0));
	double tmp;
	if ((t_0 - Math.sqrt(x)) <= 0.0) {
		tmp = Math.pow(x, -0.5) * 0.5;
	} else {
		tmp = (x + (1.0 - x)) / (t_0 + Math.sqrt(x));
	}
	return tmp;
}

Python

def code(x):
	t_0 = math.sqrt((x + 1.0))
	tmp = 0
	if (t_0 - math.sqrt(x)) <= 0.0:
		tmp = math.pow(x, -0.5) * 0.5
	else:
		tmp = (x + (1.0 - x)) / (t_0 + math.sqrt(x))
	return tmp

Julia

function code(x)
	t_0 = sqrt(Float64(x + 1.0))
	tmp = 0.0
	if (Float64(t_0 - sqrt(x)) <= 0.0)
		tmp = Float64((x ^ -0.5) * 0.5);
	else
		tmp = Float64(Float64(x + Float64(1.0 - x)) / Float64(t_0 + sqrt(x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = sqrt((x + 1.0));
	tmp = 0.0;
	if ((t_0 - sqrt(x)) <= 0.0)
		tmp = (x ^ -0.5) * 0.5;
	else
		tmp = (x + (1.0 - x)) / (t_0 + sqrt(x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(t$95$0 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision], 0.0], N[(N[Power[x, -0.5], $MachinePrecision] * 0.5), $MachinePrecision], N[(N[(x + N[(1.0 - x), $MachinePrecision]), $MachinePrecision] / N[(t$95$0 + N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) < 0.0

   1. Initial program 3.9%

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

   3. Taylor expanded in x around inf

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

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

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \color{blue}{\left(\sqrt{\frac{1}{x}}\right)}\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\left(\frac{1}{x}\right)\right)\right) \]

      3. /-lowering-/.f64 99.8%

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, x\right)\right)\right) \]

   5. Simplified 99.8%

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

      1. *-commutative N/A

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

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

         \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{\frac{1}{x}}\right), \color{blue}{\frac{1}{2}}\right) \]

      3. inv-pow N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{{x}^{-1}}\right), \frac{1}{2}\right) \]

      4. sqrt-pow1 N/A

         \[\leadsto \mathsf{*.f64}\left(\left({x}^{\left(\frac{-1}{2}\right)}\right), \frac{1}{2}\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\left({x}^{\frac{-1}{2}}\right), \frac{1}{2}\right) \]

      6. pow-lowering-pow.f64 100.0%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(x, \frac{-1}{2}\right), \frac{1}{2}\right) \]

   7. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{{x}^{-0.5} \cdot 0.5} \]

   if 0.0 < (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x))

   1. Initial program 78.4%

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

      1. flip-- N/A

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

      2. clear-num N/A

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

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

         \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{\sqrt{x + 1} + \sqrt{x}}{\sqrt{x + 1} \cdot \sqrt{x + 1} - \sqrt{x} \cdot \sqrt{x}}\right)}\right) \]

      4. clear-num N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left(\frac{1}{\color{blue}{\frac{\sqrt{x + 1} \cdot \sqrt{x + 1} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}}}}\right)\right) \]

      5. flip-- N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left(\frac{1}{\sqrt{x + 1} - \color{blue}{\sqrt{x}}}\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(1, \color{blue}{\left(\sqrt{x + 1} - \sqrt{x}\right)}\right)\right) \]

      7. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\left(\sqrt{x + 1}\right), \color{blue}{\left(\sqrt{x}\right)}\right)\right)\right) \]

      8. pow1/2 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\left({\left(x + 1\right)}^{\frac{1}{2}}\right), \left(\sqrt{\color{blue}{x}}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\mathsf{pow.f64}\left(\left(x + 1\right), \frac{1}{2}\right), \left(\sqrt{\color{blue}{x}}\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(x, 1\right), \frac{1}{2}\right), \left(\sqrt{x}\right)\right)\right)\right) \]

      11. sqrt-lowering-sqrt.f64 78.3%

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(x, 1\right), \frac{1}{2}\right), \mathsf{sqrt.f64}\left(x\right)\right)\right)\right) \]

   4. Applied egg-rr 78.3%

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

      1. unpow1/2 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\left(\sqrt{x + 1}\right), \mathsf{sqrt.f64}\left(\color{blue}{x}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\left(x + 1\right)\right), \mathsf{sqrt.f64}\left(\color{blue}{x}\right)\right)\right)\right) \]

      3. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\left(1 + x\right)\right), \mathsf{sqrt.f64}\left(x\right)\right)\right)\right) \]

      4. +-lowering-+.f64 78.3%

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, x\right)\right), \mathsf{sqrt.f64}\left(x\right)\right)\right)\right) \]

   6. Applied egg-rr 78.3%

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

      1. remove-double-div N/A

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

      2. flip-- N/A

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

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

         \[\leadsto \mathsf{/.f64}\left(\left(\sqrt{1 + x} \cdot \sqrt{1 + x} - \sqrt{x} \cdot \sqrt{x}\right), \color{blue}{\left(\sqrt{1 + x} + \sqrt{x}\right)}\right) \]

      4. rem-square-sqrt N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(1 + x\right) - \sqrt{x} \cdot \sqrt{x}\right), \left(\sqrt{\color{blue}{1 + x}} + \sqrt{x}\right)\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(x + 1\right) - \sqrt{x} \cdot \sqrt{x}\right), \left(\sqrt{\color{blue}{1 + x}} + \sqrt{x}\right)\right) \]

      6. rem-square-sqrt N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(x + 1\right) - x\right), \left(\sqrt{1 + x} + \sqrt{x}\right)\right) \]

      7. associate--l+ N/A

         \[\leadsto \mathsf{/.f64}\left(\left(x + \left(1 - x\right)\right), \left(\color{blue}{\sqrt{1 + x}} + \sqrt{x}\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \left(1 - x\right)\right), \left(\color{blue}{\sqrt{1 + x}} + \sqrt{x}\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(1, x\right)\right), \left(\sqrt{1 + x} + \sqrt{x}\right)\right) \]

      10. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(1, x\right)\right), \left(\sqrt{x} + \color{blue}{\sqrt{1 + x}}\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(1, x\right)\right), \mathsf{+.f64}\left(\left(\sqrt{x}\right), \color{blue}{\left(\sqrt{1 + x}\right)}\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(1, x\right)\right), \mathsf{+.f64}\left(\mathsf{sqrt.f64}\left(x\right), \left(\sqrt{\color{blue}{1 + x}}\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(1, x\right)\right), \mathsf{+.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{sqrt.f64}\left(\left(1 + x\right)\right)\right)\right) \]

      14. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(1, x\right)\right), \mathsf{+.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{sqrt.f64}\left(\left(x + 1\right)\right)\right)\right) \]

      15. +-lowering-+.f64 99.5%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(1, x\right)\right), \mathsf{+.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(x, 1\right)\right)\right)\right) \]

   8. Applied egg-rr 99.5%

      \[\leadsto \color{blue}{\frac{x + \left(1 - x\right)}{\sqrt{x} + \sqrt{x + 1}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{x + 1} - \sqrt{x} \leq 0:\\ \;\;\;\;{x}^{-0.5} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{x + \left(1 - x\right)}{\sqrt{x + 1} + \sqrt{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 60000000:\\ \;\;\;\;\sqrt{x + 1} - \sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;{x}^{-0.5} \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 60000000.0) (- (sqrt (+ x 1.0)) (sqrt x)) (* (pow x -0.5) 0.5)))

C

double code(double x) {
	double tmp;
	if (x <= 60000000.0) {
		tmp = sqrt((x + 1.0)) - sqrt(x);
	} else {
		tmp = pow(x, -0.5) * 0.5;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 60000000.0d0) then
        tmp = sqrt((x + 1.0d0)) - sqrt(x)
    else
        tmp = (x ** (-0.5d0)) * 0.5d0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 60000000.0) {
		tmp = Math.sqrt((x + 1.0)) - Math.sqrt(x);
	} else {
		tmp = Math.pow(x, -0.5) * 0.5;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 60000000.0:
		tmp = math.sqrt((x + 1.0)) - math.sqrt(x)
	else:
		tmp = math.pow(x, -0.5) * 0.5
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 60000000.0)
		tmp = Float64(sqrt(Float64(x + 1.0)) - sqrt(x));
	else
		tmp = Float64((x ^ -0.5) * 0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 60000000.0)
		tmp = sqrt((x + 1.0)) - sqrt(x);
	else
		tmp = (x ^ -0.5) * 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 60000000.0], N[(N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision], N[(N[Power[x, -0.5], $MachinePrecision] * 0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 6e7

   1. Initial program 88.4%

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

   if 6e7 < x

   1. Initial program 4.7%

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

   3. Taylor expanded in x around inf

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

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

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \color{blue}{\left(\sqrt{\frac{1}{x}}\right)}\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\left(\frac{1}{x}\right)\right)\right) \]

      3. /-lowering-/.f64 99.3%

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, x\right)\right)\right) \]

   5. Simplified 99.3%

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

      1. *-commutative N/A

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

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

         \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{\frac{1}{x}}\right), \color{blue}{\frac{1}{2}}\right) \]

      3. inv-pow N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{{x}^{-1}}\right), \frac{1}{2}\right) \]

      4. sqrt-pow1 N/A

         \[\leadsto \mathsf{*.f64}\left(\left({x}^{\left(\frac{-1}{2}\right)}\right), \frac{1}{2}\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\left({x}^{\frac{-1}{2}}\right), \frac{1}{2}\right) \]

      6. pow-lowering-pow.f64 99.5%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(x, \frac{-1}{2}\right), \frac{1}{2}\right) \]

   7. Applied egg-rr 99.5%

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

Alternative 4: 99.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{-0.125}{{x}^{1.5}} - -0.5 \cdot {x}^{-0.5} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (- (/ -0.125 (pow x 1.5)) (* -0.5 (pow x -0.5))))

C

double code(double x) {
	return (-0.125 / pow(x, 1.5)) - (-0.5 * pow(x, -0.5));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = ((-0.125d0) / (x ** 1.5d0)) - ((-0.5d0) * (x ** (-0.5d0)))
end function

Java

public static double code(double x) {
	return (-0.125 / Math.pow(x, 1.5)) - (-0.5 * Math.pow(x, -0.5));
}

Python

def code(x):
	return (-0.125 / math.pow(x, 1.5)) - (-0.5 * math.pow(x, -0.5))

Julia

function code(x)
	return Float64(Float64(-0.125 / (x ^ 1.5)) - Float64(-0.5 * (x ^ -0.5)))
end

MATLAB

function tmp = code(x)
	tmp = (-0.125 / (x ^ 1.5)) - (-0.5 * (x ^ -0.5));
end

Wolfram

code[x_] := N[(N[(-0.125 / N[Power[x, 1.5], $MachinePrecision]), $MachinePrecision] - N[(-0.5 * N[Power[x, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 7.7%

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

3. Taylor expanded in x around inf

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

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

      \[\leadsto \mathsf{/.f64}\left(\left(\frac{-1}{8} \cdot \sqrt{\frac{1}{x}} + \frac{1}{2} \cdot \sqrt{x}\right), \color{blue}{x}\right) \]

   2. remove-double-neg N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\frac{-1}{8} \cdot \sqrt{\frac{1}{x}} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\frac{1}{2} \cdot \sqrt{x}\right)\right)\right)\right)\right), x\right) \]

   3. mul-1-neg N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\frac{-1}{8} \cdot \sqrt{\frac{1}{x}} + \left(\mathsf{neg}\left(-1 \cdot \left(\frac{1}{2} \cdot \sqrt{x}\right)\right)\right)\right), x\right) \]

   4. unsub-neg N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\frac{-1}{8} \cdot \sqrt{\frac{1}{x}} - -1 \cdot \left(\frac{1}{2} \cdot \sqrt{x}\right)\right), x\right) \]

   5. --lowering--.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(\frac{-1}{8} \cdot \sqrt{\frac{1}{x}}\right), \left(-1 \cdot \left(\frac{1}{2} \cdot \sqrt{x}\right)\right)\right), x\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \left(\sqrt{\frac{1}{x}}\right)\right), \left(-1 \cdot \left(\frac{1}{2} \cdot \sqrt{x}\right)\right)\right), x\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{sqrt.f64}\left(\left(\frac{1}{x}\right)\right)\right), \left(-1 \cdot \left(\frac{1}{2} \cdot \sqrt{x}\right)\right)\right), x\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, x\right)\right)\right), \left(-1 \cdot \left(\frac{1}{2} \cdot \sqrt{x}\right)\right)\right), x\right) \]

   9. associate-*r* N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, x\right)\right)\right), \left(\left(-1 \cdot \frac{1}{2}\right) \cdot \sqrt{x}\right)\right), x\right) \]

   10. metadata-eval N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, x\right)\right)\right), \left(\frac{-1}{2} \cdot \sqrt{x}\right)\right), x\right) \]

   11. *-commutative N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, x\right)\right)\right), \left(\sqrt{x} \cdot \frac{-1}{2}\right)\right), x\right) \]

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

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, x\right)\right)\right), \mathsf{*.f64}\left(\left(\sqrt{x}\right), \frac{-1}{2}\right)\right), x\right) \]

   13. sqrt-lowering-sqrt.f64 97.8%

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, x\right)\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \frac{-1}{2}\right)\right), x\right) \]

5. Simplified 97.8%

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

   1. div-sub N/A

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

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

      \[\leadsto \mathsf{\_.f64}\left(\left(\frac{\frac{-1}{8} \cdot \sqrt{\frac{1}{x}}}{x}\right), \color{blue}{\left(\frac{\sqrt{x} \cdot \frac{-1}{2}}{x}\right)}\right) \]

   3. *-commutative N/A

      \[\leadsto \mathsf{\_.f64}\left(\left(\frac{\sqrt{\frac{1}{x}} \cdot \frac{-1}{8}}{x}\right), \left(\frac{\color{blue}{\sqrt{x}} \cdot \frac{-1}{2}}{x}\right)\right) \]

   4. associate-/l* N/A

      \[\leadsto \mathsf{\_.f64}\left(\left(\sqrt{\frac{1}{x}} \cdot \frac{\frac{-1}{8}}{x}\right), \left(\frac{\color{blue}{\sqrt{x} \cdot \frac{-1}{2}}}{x}\right)\right) \]

   5. sqrt-div N/A

      \[\leadsto \mathsf{\_.f64}\left(\left(\frac{\sqrt{1}}{\sqrt{x}} \cdot \frac{\frac{-1}{8}}{x}\right), \left(\frac{\color{blue}{\sqrt{x}} \cdot \frac{-1}{2}}{x}\right)\right) \]

   6. metadata-eval N/A

      \[\leadsto \mathsf{\_.f64}\left(\left(\frac{1}{\sqrt{x}} \cdot \frac{\frac{-1}{8}}{x}\right), \left(\frac{\sqrt{\color{blue}{x}} \cdot \frac{-1}{2}}{x}\right)\right) \]

   7. frac-times N/A

      \[\leadsto \mathsf{\_.f64}\left(\left(\frac{1 \cdot \frac{-1}{8}}{\sqrt{x} \cdot x}\right), \left(\frac{\color{blue}{\sqrt{x} \cdot \frac{-1}{2}}}{x}\right)\right) \]

   8. metadata-eval N/A

      \[\leadsto \mathsf{\_.f64}\left(\left(\frac{\frac{-1}{8}}{\sqrt{x} \cdot x}\right), \left(\frac{\color{blue}{\sqrt{x}} \cdot \frac{-1}{2}}{x}\right)\right) \]

   9. rem-square-sqrt N/A

      \[\leadsto \mathsf{\_.f64}\left(\left(\frac{\frac{-1}{8}}{\sqrt{x} \cdot \left(\sqrt{x} \cdot \sqrt{x}\right)}\right), \left(\frac{\sqrt{x} \cdot \frac{-1}{2}}{x}\right)\right) \]

   10. cube-mult N/A

       \[\leadsto \mathsf{\_.f64}\left(\left(\frac{\frac{-1}{8}}{{\left(\sqrt{x}\right)}^{3}}\right), \left(\frac{\sqrt{x} \cdot \color{blue}{\frac{-1}{2}}}{x}\right)\right) \]

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

       \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\frac{-1}{8}, \left({\left(\sqrt{x}\right)}^{3}\right)\right), \left(\frac{\color{blue}{\sqrt{x} \cdot \frac{-1}{2}}}{x}\right)\right) \]

   12. pow1/2 N/A

       \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\frac{-1}{8}, \left({\left({x}^{\frac{1}{2}}\right)}^{3}\right)\right), \left(\frac{\sqrt{x} \cdot \frac{-1}{2}}{x}\right)\right) \]

   13. pow-pow N/A

       \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\frac{-1}{8}, \left({x}^{\left(\frac{1}{2} \cdot 3\right)}\right)\right), \left(\frac{\sqrt{x} \cdot \color{blue}{\frac{-1}{2}}}{x}\right)\right) \]

   14. pow-lowering-pow.f64 N/A

       \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\frac{-1}{8}, \mathsf{pow.f64}\left(x, \left(\frac{1}{2} \cdot 3\right)\right)\right), \left(\frac{\sqrt{x} \cdot \color{blue}{\frac{-1}{2}}}{x}\right)\right) \]

   15. metadata-eval N/A

       \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\frac{-1}{8}, \mathsf{pow.f64}\left(x, \frac{3}{2}\right)\right), \left(\frac{\sqrt{x} \cdot \frac{-1}{2}}{x}\right)\right) \]

   16. *-rgt-identity N/A

       \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\frac{-1}{8}, \mathsf{pow.f64}\left(x, \frac{3}{2}\right)\right), \left(\frac{\sqrt{x} \cdot \frac{-1}{2}}{x \cdot \color{blue}{1}}\right)\right) \]

   17. times-frac N/A

       \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\frac{-1}{8}, \mathsf{pow.f64}\left(x, \frac{3}{2}\right)\right), \left(\frac{\sqrt{x}}{x} \cdot \color{blue}{\frac{\frac{-1}{2}}{1}}\right)\right) \]

7. Applied egg-rr 98.2%

   \[\leadsto \color{blue}{\frac{-0.125}{{x}^{1.5}} - {x}^{-0.5} \cdot -0.5} \]

8. Final simplification 98.2%

   \[\leadsto \frac{-0.125}{{x}^{1.5}} - -0.5 \cdot {x}^{-0.5} \]
9. Add Preprocessing

Alternative 5: 98.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{-0.125}{{x}^{1.5}} + \frac{0.5}{\sqrt{x}} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (+ (/ -0.125 (pow x 1.5)) (/ 0.5 (sqrt x))))

C

double code(double x) {
	return (-0.125 / pow(x, 1.5)) + (0.5 / sqrt(x));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = ((-0.125d0) / (x ** 1.5d0)) + (0.5d0 / sqrt(x))
end function

Java

public static double code(double x) {
	return (-0.125 / Math.pow(x, 1.5)) + (0.5 / Math.sqrt(x));
}

Python

def code(x):
	return (-0.125 / math.pow(x, 1.5)) + (0.5 / math.sqrt(x))

Julia

function code(x)
	return Float64(Float64(-0.125 / (x ^ 1.5)) + Float64(0.5 / sqrt(x)))
end

MATLAB

function tmp = code(x)
	tmp = (-0.125 / (x ^ 1.5)) + (0.5 / sqrt(x));
end

Wolfram

code[x_] := N[(N[(-0.125 / N[Power[x, 1.5], $MachinePrecision]), $MachinePrecision] + N[(0.5 / N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 7.7%

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

3. Taylor expanded in x around inf

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

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

      \[\leadsto \mathsf{/.f64}\left(\left(\frac{-1}{8} \cdot \sqrt{\frac{1}{x}} + \frac{1}{2} \cdot \sqrt{x}\right), \color{blue}{x}\right) \]

   2. remove-double-neg N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\frac{-1}{8} \cdot \sqrt{\frac{1}{x}} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\frac{1}{2} \cdot \sqrt{x}\right)\right)\right)\right)\right), x\right) \]

   3. mul-1-neg N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\frac{-1}{8} \cdot \sqrt{\frac{1}{x}} + \left(\mathsf{neg}\left(-1 \cdot \left(\frac{1}{2} \cdot \sqrt{x}\right)\right)\right)\right), x\right) \]

   4. unsub-neg N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\frac{-1}{8} \cdot \sqrt{\frac{1}{x}} - -1 \cdot \left(\frac{1}{2} \cdot \sqrt{x}\right)\right), x\right) \]

   5. --lowering--.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(\frac{-1}{8} \cdot \sqrt{\frac{1}{x}}\right), \left(-1 \cdot \left(\frac{1}{2} \cdot \sqrt{x}\right)\right)\right), x\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \left(\sqrt{\frac{1}{x}}\right)\right), \left(-1 \cdot \left(\frac{1}{2} \cdot \sqrt{x}\right)\right)\right), x\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{sqrt.f64}\left(\left(\frac{1}{x}\right)\right)\right), \left(-1 \cdot \left(\frac{1}{2} \cdot \sqrt{x}\right)\right)\right), x\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, x\right)\right)\right), \left(-1 \cdot \left(\frac{1}{2} \cdot \sqrt{x}\right)\right)\right), x\right) \]

   9. associate-*r* N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, x\right)\right)\right), \left(\left(-1 \cdot \frac{1}{2}\right) \cdot \sqrt{x}\right)\right), x\right) \]

   10. metadata-eval N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, x\right)\right)\right), \left(\frac{-1}{2} \cdot \sqrt{x}\right)\right), x\right) \]

   11. *-commutative N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, x\right)\right)\right), \left(\sqrt{x} \cdot \frac{-1}{2}\right)\right), x\right) \]

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

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, x\right)\right)\right), \mathsf{*.f64}\left(\left(\sqrt{x}\right), \frac{-1}{2}\right)\right), x\right) \]

   13. sqrt-lowering-sqrt.f64 97.8%

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, x\right)\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \frac{-1}{2}\right)\right), x\right) \]

5. Simplified 97.8%

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

   1. clear-num N/A

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

   2. associate-/r/ N/A

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

   3. sub-neg N/A

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

   4. +-commutative N/A

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

   5. distribute-rgt-in N/A

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

   6. distribute-lft-neg-in N/A

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

   7. div-inv N/A

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

   8. div-inv N/A

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

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

      \[\leadsto \mathsf{+.f64}\left(\left(\mathsf{neg}\left(\frac{\sqrt{x} \cdot \frac{-1}{2}}{x}\right)\right), \color{blue}{\left(\frac{\frac{-1}{8} \cdot \sqrt{\frac{1}{x}}}{x}\right)}\right) \]

7. Applied egg-rr 97.8%

   \[\leadsto \color{blue}{\frac{0.5}{\sqrt{x}} + \frac{-0.125}{{x}^{1.5}}} \]

8. Final simplification 97.8%

   \[\leadsto \frac{-0.125}{{x}^{1.5}} + \frac{0.5}{\sqrt{x}} \]
9. Add Preprocessing

Alternative 6: 98.1% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ {x}^{-0.5} \cdot 0.5 \end{array} \]

FPCore

(FPCore (x) :precision binary64 (* (pow x -0.5) 0.5))

C

double code(double x) {
	return pow(x, -0.5) * 0.5;
}

Fortran

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

Java

public static double code(double x) {
	return Math.pow(x, -0.5) * 0.5;
}

Python

def code(x):
	return math.pow(x, -0.5) * 0.5

Julia

function code(x)
	return Float64((x ^ -0.5) * 0.5)
end

MATLAB

function tmp = code(x)
	tmp = (x ^ -0.5) * 0.5;
end

Wolfram

code[x_] := N[(N[Power[x, -0.5], $MachinePrecision] * 0.5), $MachinePrecision]

Derivation

1. Initial program 7.7%

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

3. Taylor expanded in x around inf

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

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

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \color{blue}{\left(\sqrt{\frac{1}{x}}\right)}\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\left(\frac{1}{x}\right)\right)\right) \]

   3. /-lowering-/.f64 97.0%

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, x\right)\right)\right) \]

5. Simplified 97.0%

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

   1. *-commutative N/A

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

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

      \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{\frac{1}{x}}\right), \color{blue}{\frac{1}{2}}\right) \]

   3. inv-pow N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{{x}^{-1}}\right), \frac{1}{2}\right) \]

   4. sqrt-pow1 N/A

      \[\leadsto \mathsf{*.f64}\left(\left({x}^{\left(\frac{-1}{2}\right)}\right), \frac{1}{2}\right) \]

   5. metadata-eval N/A

      \[\leadsto \mathsf{*.f64}\left(\left({x}^{\frac{-1}{2}}\right), \frac{1}{2}\right) \]

   6. pow-lowering-pow.f64 97.2%

      \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(x, \frac{-1}{2}\right), \frac{1}{2}\right) \]

7. Applied egg-rr 97.2%

   \[\leadsto \color{blue}{{x}^{-0.5} \cdot 0.5} \]
8. Add Preprocessing

Alternative 7: 97.8% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 7.7%

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

3. Taylor expanded in x around inf

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

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

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \color{blue}{\left(\sqrt{\frac{1}{x}}\right)}\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\left(\frac{1}{x}\right)\right)\right) \]

   3. /-lowering-/.f64 97.0%

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, x\right)\right)\right) \]

5. Simplified 97.0%

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

   1. sqrt-div N/A

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

   2. metadata-eval N/A

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

   3. un-div-inv N/A

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

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

      \[\leadsto \mathsf{/.f64}\left(\frac{1}{2}, \color{blue}{\left(\sqrt{x}\right)}\right) \]

   5. sqrt-lowering-sqrt.f64 96.8%

      \[\leadsto \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(x\right)\right) \]

7. Applied egg-rr 96.8%

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

Alternative 8: 1.6% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 7.7%

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

3. Taylor expanded in x around 0

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

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

      \[\leadsto \mathsf{\_.f64}\left(1, \color{blue}{\left(\sqrt{x}\right)}\right) \]

   2. sqrt-lowering-sqrt.f64 1.6%

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{sqrt.f64}\left(x\right)\right) \]

5. Simplified 1.6%

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

Developer Target 1: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Developer Target 2: 98.1% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ 0.5 \cdot {x}^{-0.5} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (* 0.5 (pow x -0.5)))

C

double code(double x) {
	return 0.5 * pow(x, -0.5);
}

Fortran

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

Java

public static double code(double x) {
	return 0.5 * Math.pow(x, -0.5);
}

Python

def code(x):
	return 0.5 * math.pow(x, -0.5)

Julia

function code(x)
	return Float64(0.5 * (x ^ -0.5))
end

MATLAB

function tmp = code(x)
	tmp = 0.5 * (x ^ -0.5);
end

Wolfram

code[x_] := N[(0.5 * N[Power[x, -0.5], $MachinePrecision]), $MachinePrecision]

Reproduce

herbie shell --seed 2024191 
(FPCore (x)
  :name "2sqrt (example 3.1)"
  :precision binary64
  :pre (and (> x 1.0) (< x 1e+308))

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

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

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