2isqrt (example 3.6)

Percentage Accurate: 68.5% → 99.6%

Time: 12.0s

Alternatives: 18

Speedup: 1.9×

Specification

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: 68.5% 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: 99.6% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 70.9%

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

   1. frac-sub 70.9%

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

   2. div-inv 70.9%

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

   3. *-un-lft-identity 70.9%

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

   4. +-commutative 70.9%

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

   5. *-rgt-identity 70.9%

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

   6. metadata-eval 70.9%

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

   7. frac-times 70.9%

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

   8. un-div-inv 70.9%

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

   9. pow1/2 71.0%

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

   10. pow-flip 71.2%

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

   11. metadata-eval 71.2%

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

   12. +-commutative 71.2%

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

3. Applied egg-rr 71.2%

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

   1. associate-*r/ 71.2%

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

5. Simplified 71.2%

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

   1. flip-- 71.4%

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

   2. div-inv 71.4%

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

   3. add-sqr-sqrt 62.8%

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

   4. add-sqr-sqrt 72.0%

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

   5. associate--l+ 99.6%

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

7. Applied egg-rr 99.6%

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

   1. +-inverses 99.6%

      \[\leadsto \frac{\left(\left(1 + \color{blue}{0}\right) \cdot \frac{1}{\sqrt{1 + x} + \sqrt{x}}\right) \cdot {x}^{-0.5}}{\sqrt{1 + x}} \]

   2. metadata-eval 99.6%

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

   3. *-lft-identity 99.6%

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

9. Simplified 99.6%

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

    1. div-inv 99.6%

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

    2. pow1/2 99.6%

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

    3. pow-flip 99.7%

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

    4. metadata-eval 99.7%

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

    5. metadata-eval 99.7%

       \[\leadsto \left(\frac{1}{\sqrt{1 + x} + \sqrt{x}} \cdot {x}^{-0.5}\right) \cdot {\left(1 + x\right)}^{\color{blue}{\left(-1.5 \cdot 0.3333333333333333\right)}} \]

    6. pow-pow 97.5%

       \[\leadsto \left(\frac{1}{\sqrt{1 + x} + \sqrt{x}} \cdot {x}^{-0.5}\right) \cdot \color{blue}{{\left({\left(1 + x\right)}^{-1.5}\right)}^{0.3333333333333333}} \]

    7. pow1/3 99.5%

       \[\leadsto \left(\frac{1}{\sqrt{1 + x} + \sqrt{x}} \cdot {x}^{-0.5}\right) \cdot \color{blue}{\sqrt[3]{{\left(1 + x\right)}^{-1.5}}} \]

    8. metadata-eval 99.5%

       \[\leadsto \left(\frac{1}{\sqrt{1 + x} + \sqrt{x}} \cdot {x}^{\color{blue}{\left(-0.5\right)}}\right) \cdot \sqrt[3]{{\left(1 + x\right)}^{-1.5}} \]

    9. pow-flip 99.3%

       \[\leadsto \left(\frac{1}{\sqrt{1 + x} + \sqrt{x}} \cdot \color{blue}{\frac{1}{{x}^{0.5}}}\right) \cdot \sqrt[3]{{\left(1 + x\right)}^{-1.5}} \]

    10. pow1/2 99.3%

        \[\leadsto \left(\frac{1}{\sqrt{1 + x} + \sqrt{x}} \cdot \frac{1}{\color{blue}{\sqrt{x}}}\right) \cdot \sqrt[3]{{\left(1 + x\right)}^{-1.5}} \]

    11. *-commutative 99.3%

        \[\leadsto \color{blue}{\left(\frac{1}{\sqrt{x}} \cdot \frac{1}{\sqrt{1 + x} + \sqrt{x}}\right)} \cdot \sqrt[3]{{\left(1 + x\right)}^{-1.5}} \]

    12. associate-*l* 99.3%

        \[\leadsto \color{blue}{\frac{1}{\sqrt{x}} \cdot \left(\frac{1}{\sqrt{1 + x} + \sqrt{x}} \cdot \sqrt[3]{{\left(1 + x\right)}^{-1.5}}\right)} \]

    13. pow1/2 99.3%

        \[\leadsto \frac{1}{\color{blue}{{x}^{0.5}}} \cdot \left(\frac{1}{\sqrt{1 + x} + \sqrt{x}} \cdot \sqrt[3]{{\left(1 + x\right)}^{-1.5}}\right) \]

    14. pow-flip 99.5%

        \[\leadsto \color{blue}{{x}^{\left(-0.5\right)}} \cdot \left(\frac{1}{\sqrt{1 + x} + \sqrt{x}} \cdot \sqrt[3]{{\left(1 + x\right)}^{-1.5}}\right) \]

    15. metadata-eval 99.5%

        \[\leadsto {x}^{\color{blue}{-0.5}} \cdot \left(\frac{1}{\sqrt{1 + x} + \sqrt{x}} \cdot \sqrt[3]{{\left(1 + x\right)}^{-1.5}}\right) \]

    16. pow1/3 97.5%

        \[\leadsto {x}^{-0.5} \cdot \left(\frac{1}{\sqrt{1 + x} + \sqrt{x}} \cdot \color{blue}{{\left({\left(1 + x\right)}^{-1.5}\right)}^{0.3333333333333333}}\right) \]

    17. pow-pow 99.7%

        \[\leadsto {x}^{-0.5} \cdot \left(\frac{1}{\sqrt{1 + x} + \sqrt{x}} \cdot \color{blue}{{\left(1 + x\right)}^{\left(-1.5 \cdot 0.3333333333333333\right)}}\right) \]

    18. metadata-eval 99.7%

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

11. Applied egg-rr 99.7%

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

12. Final simplification 99.7%

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

Alternative 2: 99.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{x + 1}\\ \mathbf{if}\;\frac{1}{\sqrt{x}} + \frac{-1}{t_0} \leq 5 \cdot 10^{-17}:\\ \;\;\;\;\frac{\frac{0.5}{x}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;{x}^{-0.5} - {\left(x + 1\right)}^{-0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (sqrt (+ x 1.0))))
   (if (<= (+ (/ 1.0 (sqrt x)) (/ -1.0 t_0)) 5e-17)
     (/ (/ 0.5 x) t_0)
     (- (pow x -0.5) (pow (+ x 1.0) -0.5)))))

C

double code(double x) {
	double t_0 = sqrt((x + 1.0));
	double tmp;
	if (((1.0 / sqrt(x)) + (-1.0 / t_0)) <= 5e-17) {
		tmp = (0.5 / x) / t_0;
	} else {
		tmp = pow(x, -0.5) - pow((x + 1.0), -0.5);
	}
	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 (((1.0d0 / sqrt(x)) + ((-1.0d0) / t_0)) <= 5d-17) then
        tmp = (0.5d0 / x) / t_0
    else
        tmp = (x ** (-0.5d0)) - ((x + 1.0d0) ** (-0.5d0))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double t_0 = Math.sqrt((x + 1.0));
	double tmp;
	if (((1.0 / Math.sqrt(x)) + (-1.0 / t_0)) <= 5e-17) {
		tmp = (0.5 / x) / t_0;
	} else {
		tmp = Math.pow(x, -0.5) - Math.pow((x + 1.0), -0.5);
	}
	return tmp;
}

Python

def code(x):
	t_0 = math.sqrt((x + 1.0))
	tmp = 0
	if ((1.0 / math.sqrt(x)) + (-1.0 / t_0)) <= 5e-17:
		tmp = (0.5 / x) / t_0
	else:
		tmp = math.pow(x, -0.5) - math.pow((x + 1.0), -0.5)
	return tmp

Julia

function code(x)
	t_0 = sqrt(Float64(x + 1.0))
	tmp = 0.0
	if (Float64(Float64(1.0 / sqrt(x)) + Float64(-1.0 / t_0)) <= 5e-17)
		tmp = Float64(Float64(0.5 / x) / t_0);
	else
		tmp = Float64((x ^ -0.5) - (Float64(x + 1.0) ^ -0.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = sqrt((x + 1.0));
	tmp = 0.0;
	if (((1.0 / sqrt(x)) + (-1.0 / t_0)) <= 5e-17)
		tmp = (0.5 / x) / t_0;
	else
		tmp = (x ^ -0.5) - ((x + 1.0) ^ -0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(N[(1.0 / N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + N[(-1.0 / t$95$0), $MachinePrecision]), $MachinePrecision], 5e-17], N[(N[(0.5 / x), $MachinePrecision] / t$95$0), $MachinePrecision], N[(N[Power[x, -0.5], $MachinePrecision] - N[Power[N[(x + 1.0), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (/.f64 1 (sqrt.f64 x)) (/.f64 1 (sqrt.f64 (+.f64 x 1)))) < 4.9999999999999999e-17

   1. Initial program 42.9%

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

      1. frac-sub 42.9%

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

      2. div-inv 42.9%

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

      3. *-un-lft-identity 42.9%

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

      4. +-commutative 42.9%

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

      5. *-rgt-identity 42.9%

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

      6. metadata-eval 42.9%

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

      7. frac-times 42.9%

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

      8. un-div-inv 42.9%

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

      9. pow1/2 42.9%

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

      10. pow-flip 42.9%

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

      11. metadata-eval 42.9%

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

      12. +-commutative 42.9%

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

   3. Applied egg-rr 42.9%

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

      1. associate-*r/ 42.9%

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

   5. Simplified 42.9%

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

      1. flip-- 43.3%

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

      2. div-inv 43.3%

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

      3. add-sqr-sqrt 26.1%

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

      4. add-sqr-sqrt 44.2%

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

      5. associate--l+ 99.4%

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

   7. Applied egg-rr 99.4%

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

      1. +-inverses 99.4%

         \[\leadsto \frac{\left(\left(1 + \color{blue}{0}\right) \cdot \frac{1}{\sqrt{1 + x} + \sqrt{x}}\right) \cdot {x}^{-0.5}}{\sqrt{1 + x}} \]

      2. metadata-eval 99.4%

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

      3. *-lft-identity 99.4%

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

   9. Simplified 99.4%

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

   10. Taylor expanded in x around inf 99.2%

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

   if 4.9999999999999999e-17 < (-.f64 (/.f64 1 (sqrt.f64 x)) (/.f64 1 (sqrt.f64 (+.f64 x 1))))

   1. Initial program 98.9%

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

      1. *-un-lft-identity 98.9%

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

      2. clear-num 98.9%

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

      3. associate-/r/ 98.9%

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

      4. prod-diff 98.9%

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

      5. *-un-lft-identity 98.9%

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

      6. fma-neg 98.9%

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

      7. *-un-lft-identity 98.9%

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

      8. inv-pow 98.9%

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

      9. sqrt-pow2 99.4%

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

      10. metadata-eval 99.4%

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

      11. pow1/2 99.4%

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

      12. pow-flip 99.4%

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

      13. +-commutative 99.4%

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

      14. metadata-eval 99.4%

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

   3. Applied egg-rr 99.4%

      \[\leadsto \color{blue}{\left({x}^{-0.5} - {\left(1 + x\right)}^{-0.5}\right) + \mathsf{fma}\left(-1, {\left(1 + x\right)}^{-0.5}, {\left(1 + x\right)}^{-0.5}\right)} \]
   4. Step-by-step derivation

      1. fma-udef 99.4%

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

      2. distribute-lft1-in 99.4%

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

      3. metadata-eval 99.4%

         \[\leadsto \left({x}^{-0.5} - {\left(1 + x\right)}^{-0.5}\right) + \color{blue}{0} \cdot {\left(1 + x\right)}^{-0.5} \]

      4. mul0-lft 99.4%

         \[\leadsto \left({x}^{-0.5} - {\left(1 + x\right)}^{-0.5}\right) + \color{blue}{0} \]

      5. +-rgt-identity 99.4%

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

   5. Simplified 99.4%

      \[\leadsto \color{blue}{{x}^{-0.5} - {\left(1 + x\right)}^{-0.5}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{\sqrt{x}} + \frac{-1}{\sqrt{x + 1}} \leq 5 \cdot 10^{-17}:\\ \;\;\;\;\frac{\frac{0.5}{x}}{\sqrt{x + 1}}\\ \mathbf{else}:\\ \;\;\;\;{x}^{-0.5} - {\left(x + 1\right)}^{-0.5}\\ \end{array} \]

Alternative 3: 99.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 4 \cdot 10^{+80}:\\ \;\;\;\;\sqrt{\frac{{\left(\sqrt{x} + \sqrt{x + 1}\right)}^{-2}}{x \cdot \left(x + 1\right)}}\\ \mathbf{else}:\\ \;\;\;\;{x}^{-0.5} \cdot \frac{0.5}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 4e+80)
   (sqrt (/ (pow (+ (sqrt x) (sqrt (+ x 1.0))) -2.0) (* x (+ x 1.0))))
   (* (pow x -0.5) (/ 0.5 x))))

C

double code(double x) {
	double tmp;
	if (x <= 4e+80) {
		tmp = sqrt((pow((sqrt(x) + sqrt((x + 1.0))), -2.0) / (x * (x + 1.0))));
	} else {
		tmp = pow(x, -0.5) * (0.5 / x);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 4d+80) then
        tmp = sqrt((((sqrt(x) + sqrt((x + 1.0d0))) ** (-2.0d0)) / (x * (x + 1.0d0))))
    else
        tmp = (x ** (-0.5d0)) * (0.5d0 / x)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 4e+80) {
		tmp = Math.sqrt((Math.pow((Math.sqrt(x) + Math.sqrt((x + 1.0))), -2.0) / (x * (x + 1.0))));
	} else {
		tmp = Math.pow(x, -0.5) * (0.5 / x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 4e+80:
		tmp = math.sqrt((math.pow((math.sqrt(x) + math.sqrt((x + 1.0))), -2.0) / (x * (x + 1.0))))
	else:
		tmp = math.pow(x, -0.5) * (0.5 / x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 4e+80)
		tmp = sqrt(Float64((Float64(sqrt(x) + sqrt(Float64(x + 1.0))) ^ -2.0) / Float64(x * Float64(x + 1.0))));
	else
		tmp = Float64((x ^ -0.5) * Float64(0.5 / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 4e+80)
		tmp = sqrt((((sqrt(x) + sqrt((x + 1.0))) ^ -2.0) / (x * (x + 1.0))));
	else
		tmp = (x ^ -0.5) * (0.5 / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 4e+80], N[Sqrt[N[(N[Power[N[(N[Sqrt[x], $MachinePrecision] + N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], -2.0], $MachinePrecision] / N[(x * N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[Power[x, -0.5], $MachinePrecision] * N[(0.5 / x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 4e80

   1. Initial program 81.1%

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

      1. frac-sub 81.1%

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

      2. div-inv 81.1%

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

      3. *-un-lft-identity 81.1%

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

      4. +-commutative 81.1%

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

      5. *-rgt-identity 81.1%

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

      6. metadata-eval 81.1%

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

      7. frac-times 81.1%

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

      8. un-div-inv 81.1%

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

      9. pow1/2 81.1%

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

      10. pow-flip 81.5%

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

      11. metadata-eval 81.5%

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

      12. +-commutative 81.5%

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

   3. Applied egg-rr 81.5%

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

      1. associate-*r/ 81.5%

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

   5. Simplified 81.5%

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

      1. flip-- 81.8%

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

      2. div-inv 81.8%

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

      3. add-sqr-sqrt 82.4%

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

      4. add-sqr-sqrt 82.9%

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

      5. associate--l+ 99.7%

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

   7. Applied egg-rr 99.7%

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

      1. +-inverses 99.7%

         \[\leadsto \frac{\left(\left(1 + \color{blue}{0}\right) \cdot \frac{1}{\sqrt{1 + x} + \sqrt{x}}\right) \cdot {x}^{-0.5}}{\sqrt{1 + x}} \]

      2. metadata-eval 99.7%

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

      3. *-lft-identity 99.7%

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

   9. Simplified 99.7%

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

       1. add-sqr-sqrt 99.1%

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

       2. sqrt-unprod 99.7%

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

       3. frac-times 99.7%

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

   11. Applied egg-rr 99.7%

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

       1. associate-*l/ 99.7%

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

       2. *-lft-identity 99.7%

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

       3. associate-/l/ 99.7%

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

       4. +-commutative 99.7%

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

       5. *-commutative 99.7%

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

       6. +-commutative 99.7%

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

   13. Simplified 99.7%

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

   if 4e80 < x

   1. Initial program 54.2%

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

      1. frac-sub 54.2%

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

      2. div-inv 54.2%

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

      3. *-un-lft-identity 54.2%

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

      4. +-commutative 54.2%

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

      5. *-rgt-identity 54.2%

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

      6. metadata-eval 54.2%

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

      7. frac-times 54.2%

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

      8. un-div-inv 54.2%

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

      9. pow1/2 54.2%

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

      10. pow-flip 54.2%

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

      11. metadata-eval 54.2%

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

      12. +-commutative 54.2%

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

   3. Applied egg-rr 54.2%

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

      1. associate-*r/ 54.2%

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

   5. Simplified 54.2%

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

      1. flip-- 54.2%

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

      2. div-inv 54.2%

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

      3. add-sqr-sqrt 30.6%

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

      4. add-sqr-sqrt 54.2%

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

      5. associate--l+ 99.5%

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

   7. Applied egg-rr 99.5%

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

      1. +-inverses 99.5%

         \[\leadsto \frac{\left(\left(1 + \color{blue}{0}\right) \cdot \frac{1}{\sqrt{1 + x} + \sqrt{x}}\right) \cdot {x}^{-0.5}}{\sqrt{1 + x}} \]

      2. metadata-eval 99.5%

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

      3. *-lft-identity 99.5%

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

   9. Simplified 99.5%

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

       1. div-inv 99.5%

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

       2. pow1/2 99.5%

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

       3. pow-flip 99.6%

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

       4. metadata-eval 99.6%

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

       5. metadata-eval 99.6%

          \[\leadsto \left(\frac{1}{\sqrt{1 + x} + \sqrt{x}} \cdot {x}^{-0.5}\right) \cdot {\left(1 + x\right)}^{\color{blue}{\left(-1.5 \cdot 0.3333333333333333\right)}} \]

       6. pow-pow 95.6%

          \[\leadsto \left(\frac{1}{\sqrt{1 + x} + \sqrt{x}} \cdot {x}^{-0.5}\right) \cdot \color{blue}{{\left({\left(1 + x\right)}^{-1.5}\right)}^{0.3333333333333333}} \]

       7. pow1/3 99.3%

          \[\leadsto \left(\frac{1}{\sqrt{1 + x} + \sqrt{x}} \cdot {x}^{-0.5}\right) \cdot \color{blue}{\sqrt[3]{{\left(1 + x\right)}^{-1.5}}} \]

       8. metadata-eval 99.3%

          \[\leadsto \left(\frac{1}{\sqrt{1 + x} + \sqrt{x}} \cdot {x}^{\color{blue}{\left(-0.5\right)}}\right) \cdot \sqrt[3]{{\left(1 + x\right)}^{-1.5}} \]

       9. pow-flip 99.2%

          \[\leadsto \left(\frac{1}{\sqrt{1 + x} + \sqrt{x}} \cdot \color{blue}{\frac{1}{{x}^{0.5}}}\right) \cdot \sqrt[3]{{\left(1 + x\right)}^{-1.5}} \]

       10. pow1/2 99.2%

           \[\leadsto \left(\frac{1}{\sqrt{1 + x} + \sqrt{x}} \cdot \frac{1}{\color{blue}{\sqrt{x}}}\right) \cdot \sqrt[3]{{\left(1 + x\right)}^{-1.5}} \]

       11. *-commutative 99.2%

           \[\leadsto \color{blue}{\left(\frac{1}{\sqrt{x}} \cdot \frac{1}{\sqrt{1 + x} + \sqrt{x}}\right)} \cdot \sqrt[3]{{\left(1 + x\right)}^{-1.5}} \]

       12. associate-*l* 99.2%

           \[\leadsto \color{blue}{\frac{1}{\sqrt{x}} \cdot \left(\frac{1}{\sqrt{1 + x} + \sqrt{x}} \cdot \sqrt[3]{{\left(1 + x\right)}^{-1.5}}\right)} \]

       13. pow1/2 99.2%

           \[\leadsto \frac{1}{\color{blue}{{x}^{0.5}}} \cdot \left(\frac{1}{\sqrt{1 + x} + \sqrt{x}} \cdot \sqrt[3]{{\left(1 + x\right)}^{-1.5}}\right) \]

       14. pow-flip 99.3%

           \[\leadsto \color{blue}{{x}^{\left(-0.5\right)}} \cdot \left(\frac{1}{\sqrt{1 + x} + \sqrt{x}} \cdot \sqrt[3]{{\left(1 + x\right)}^{-1.5}}\right) \]

       15. metadata-eval 99.3%

           \[\leadsto {x}^{\color{blue}{-0.5}} \cdot \left(\frac{1}{\sqrt{1 + x} + \sqrt{x}} \cdot \sqrt[3]{{\left(1 + x\right)}^{-1.5}}\right) \]

       16. pow1/3 95.6%

           \[\leadsto {x}^{-0.5} \cdot \left(\frac{1}{\sqrt{1 + x} + \sqrt{x}} \cdot \color{blue}{{\left({\left(1 + x\right)}^{-1.5}\right)}^{0.3333333333333333}}\right) \]

       17. pow-pow 99.6%

           \[\leadsto {x}^{-0.5} \cdot \left(\frac{1}{\sqrt{1 + x} + \sqrt{x}} \cdot \color{blue}{{\left(1 + x\right)}^{\left(-1.5 \cdot 0.3333333333333333\right)}}\right) \]

       18. metadata-eval 99.6%

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

   11. Applied egg-rr 99.6%

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

   12. Taylor expanded in x around inf 99.7%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 4 \cdot 10^{+80}:\\ \;\;\;\;\sqrt{\frac{{\left(\sqrt{x} + \sqrt{x + 1}\right)}^{-2}}{x \cdot \left(x + 1\right)}}\\ \mathbf{else}:\\ \;\;\;\;{x}^{-0.5} \cdot \frac{0.5}{x}\\ \end{array} \]

Alternative 4: 99.7% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 70.9%

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

   1. frac-sub 70.9%

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

   2. div-inv 70.9%

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

   3. *-un-lft-identity 70.9%

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

   4. +-commutative 70.9%

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

   5. *-rgt-identity 70.9%

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

   6. metadata-eval 70.9%

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

   7. frac-times 70.9%

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

   8. un-div-inv 70.9%

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

   9. pow1/2 71.0%

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

   10. pow-flip 71.2%

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

   11. metadata-eval 71.2%

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

   12. +-commutative 71.2%

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

3. Applied egg-rr 71.2%

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

   1. associate-*r/ 71.2%

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

5. Simplified 71.2%

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

   1. flip-- 71.4%

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

   2. div-inv 71.4%

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

   3. add-sqr-sqrt 62.8%

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

   4. add-sqr-sqrt 72.0%

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

   5. associate--l+ 99.6%

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

7. Applied egg-rr 99.6%

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

   1. +-inverses 99.6%

      \[\leadsto \frac{\left(\left(1 + \color{blue}{0}\right) \cdot \frac{1}{\sqrt{1 + x} + \sqrt{x}}\right) \cdot {x}^{-0.5}}{\sqrt{1 + x}} \]

   2. metadata-eval 99.6%

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

   3. *-lft-identity 99.6%

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

9. Simplified 99.6%

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

    1. clear-num 98.7%

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

    2. associate-/r/ 99.6%

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

    3. pow1/2 99.6%

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

    4. pow-flip 99.7%

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

    5. metadata-eval 99.7%

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

    6. associate-*l/ 99.6%

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

    7. *-un-lft-identity 99.6%

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

11. Applied egg-rr 99.6%

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

12. Final simplification 99.6%

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

Alternative 5: 99.6% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 70.9%

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

   1. frac-sub 70.9%

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

   2. div-inv 70.9%

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

   3. *-un-lft-identity 70.9%

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

   4. +-commutative 70.9%

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

   5. *-rgt-identity 70.9%

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

   6. metadata-eval 70.9%

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

   7. frac-times 70.9%

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

   8. un-div-inv 70.9%

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

   9. pow1/2 71.0%

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

   10. pow-flip 71.2%

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

   11. metadata-eval 71.2%

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

   12. +-commutative 71.2%

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

3. Applied egg-rr 71.2%

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

   1. associate-*r/ 71.2%

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

5. Simplified 71.2%

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

   1. flip-- 71.4%

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

   2. div-inv 71.4%

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

   3. add-sqr-sqrt 62.8%

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

   4. add-sqr-sqrt 72.0%

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

   5. associate--l+ 99.6%

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

7. Applied egg-rr 99.6%

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

   1. +-inverses 99.6%

      \[\leadsto \frac{\left(\left(1 + \color{blue}{0}\right) \cdot \frac{1}{\sqrt{1 + x} + \sqrt{x}}\right) \cdot {x}^{-0.5}}{\sqrt{1 + x}} \]

   2. metadata-eval 99.6%

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

   3. *-lft-identity 99.6%

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

9. Simplified 99.6%

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

    1. div-inv 99.6%

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

    2. pow1/2 99.6%

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

    3. pow-flip 99.7%

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

    4. metadata-eval 99.7%

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

    5. metadata-eval 99.7%

       \[\leadsto \left(\frac{1}{\sqrt{1 + x} + \sqrt{x}} \cdot {x}^{-0.5}\right) \cdot {\left(1 + x\right)}^{\color{blue}{\left(-1.5 \cdot 0.3333333333333333\right)}} \]

    6. pow-pow 97.5%

       \[\leadsto \left(\frac{1}{\sqrt{1 + x} + \sqrt{x}} \cdot {x}^{-0.5}\right) \cdot \color{blue}{{\left({\left(1 + x\right)}^{-1.5}\right)}^{0.3333333333333333}} \]

    7. pow1/3 99.5%

       \[\leadsto \left(\frac{1}{\sqrt{1 + x} + \sqrt{x}} \cdot {x}^{-0.5}\right) \cdot \color{blue}{\sqrt[3]{{\left(1 + x\right)}^{-1.5}}} \]

    8. metadata-eval 99.5%

       \[\leadsto \left(\frac{1}{\sqrt{1 + x} + \sqrt{x}} \cdot {x}^{\color{blue}{\left(-0.5\right)}}\right) \cdot \sqrt[3]{{\left(1 + x\right)}^{-1.5}} \]

    9. pow-flip 99.3%

       \[\leadsto \left(\frac{1}{\sqrt{1 + x} + \sqrt{x}} \cdot \color{blue}{\frac{1}{{x}^{0.5}}}\right) \cdot \sqrt[3]{{\left(1 + x\right)}^{-1.5}} \]

    10. pow1/2 99.3%

        \[\leadsto \left(\frac{1}{\sqrt{1 + x} + \sqrt{x}} \cdot \frac{1}{\color{blue}{\sqrt{x}}}\right) \cdot \sqrt[3]{{\left(1 + x\right)}^{-1.5}} \]

    11. *-commutative 99.3%

        \[\leadsto \color{blue}{\left(\frac{1}{\sqrt{x}} \cdot \frac{1}{\sqrt{1 + x} + \sqrt{x}}\right)} \cdot \sqrt[3]{{\left(1 + x\right)}^{-1.5}} \]

    12. associate-*l* 99.3%

        \[\leadsto \color{blue}{\frac{1}{\sqrt{x}} \cdot \left(\frac{1}{\sqrt{1 + x} + \sqrt{x}} \cdot \sqrt[3]{{\left(1 + x\right)}^{-1.5}}\right)} \]

    13. pow1/2 99.3%

        \[\leadsto \frac{1}{\color{blue}{{x}^{0.5}}} \cdot \left(\frac{1}{\sqrt{1 + x} + \sqrt{x}} \cdot \sqrt[3]{{\left(1 + x\right)}^{-1.5}}\right) \]

    14. pow-flip 99.5%

        \[\leadsto \color{blue}{{x}^{\left(-0.5\right)}} \cdot \left(\frac{1}{\sqrt{1 + x} + \sqrt{x}} \cdot \sqrt[3]{{\left(1 + x\right)}^{-1.5}}\right) \]

    15. metadata-eval 99.5%

        \[\leadsto {x}^{\color{blue}{-0.5}} \cdot \left(\frac{1}{\sqrt{1 + x} + \sqrt{x}} \cdot \sqrt[3]{{\left(1 + x\right)}^{-1.5}}\right) \]

    16. pow1/3 97.5%

        \[\leadsto {x}^{-0.5} \cdot \left(\frac{1}{\sqrt{1 + x} + \sqrt{x}} \cdot \color{blue}{{\left({\left(1 + x\right)}^{-1.5}\right)}^{0.3333333333333333}}\right) \]

    17. pow-pow 99.7%

        \[\leadsto {x}^{-0.5} \cdot \left(\frac{1}{\sqrt{1 + x} + \sqrt{x}} \cdot \color{blue}{{\left(1 + x\right)}^{\left(-1.5 \cdot 0.3333333333333333\right)}}\right) \]

    18. metadata-eval 99.7%

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

11. Applied egg-rr 99.7%

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

    1. associate-*r* 99.7%

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

    2. div-inv 99.6%

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

    3. *-commutative 99.6%

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

    4. clear-num 99.6%

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

    5. un-div-inv 99.6%

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

13. Applied egg-rr 99.6%

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

14. Final simplification 99.6%

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

Alternative 6: 99.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{x + 1}\\ \frac{{x}^{-0.5}}{t_0 \cdot \left(\sqrt{x} + t_0\right)} \end{array} \end{array} \]

FPCore

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

C

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

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    t_0 = sqrt((x + 1.0d0))
    code = (x ** (-0.5d0)) / (t_0 * (sqrt(x) + t_0))
end function

Java

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

Python

def code(x):
	t_0 = math.sqrt((x + 1.0))
	return math.pow(x, -0.5) / (t_0 * (math.sqrt(x) + t_0))

Julia

function code(x)
	t_0 = sqrt(Float64(x + 1.0))
	return Float64((x ^ -0.5) / Float64(t_0 * Float64(sqrt(x) + t_0)))
end

MATLAB

function tmp = code(x)
	t_0 = sqrt((x + 1.0));
	tmp = (x ^ -0.5) / (t_0 * (sqrt(x) + t_0));
end

Wolfram

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

Derivation

1. Initial program 70.9%

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

   1. frac-sub 70.9%

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

   2. div-inv 70.9%

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

   3. *-un-lft-identity 70.9%

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

   4. +-commutative 70.9%

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

   5. *-rgt-identity 70.9%

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

   6. metadata-eval 70.9%

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

   7. frac-times 70.9%

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

   8. un-div-inv 70.9%

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

   9. pow1/2 71.0%

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

   10. pow-flip 71.2%

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

   11. metadata-eval 71.2%

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

   12. +-commutative 71.2%

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

3. Applied egg-rr 71.2%

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

   1. associate-*r/ 71.2%

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

5. Simplified 71.2%

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

   1. flip-- 71.4%

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

   2. div-inv 71.4%

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

   3. add-sqr-sqrt 62.8%

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

   4. add-sqr-sqrt 72.0%

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

   5. associate--l+ 99.6%

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

7. Applied egg-rr 99.6%

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

   1. +-inverses 99.6%

      \[\leadsto \frac{\left(\left(1 + \color{blue}{0}\right) \cdot \frac{1}{\sqrt{1 + x} + \sqrt{x}}\right) \cdot {x}^{-0.5}}{\sqrt{1 + x}} \]

   2. metadata-eval 99.6%

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

   3. *-lft-identity 99.6%

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

9. Simplified 99.6%

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

    1. metadata-eval 99.6%

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

    2. pow-flip 99.3%

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

    3. pow1/2 99.3%

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

    4. *-commutative 99.3%

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

    5. *-un-lft-identity 99.3%

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

    6. times-frac 99.3%

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

    7. pow1/2 99.3%

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

    8. pow-flip 99.6%

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

    9. metadata-eval 99.6%

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

11. Applied egg-rr 99.6%

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

    1. /-rgt-identity 99.6%

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

    2. associate-/l/ 99.6%

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

    3. associate-*r/ 99.6%

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

    4. *-commutative 99.6%

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

    5. *-lft-identity 99.6%

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

    6. +-commutative 99.6%

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

    7. +-commutative 99.6%

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

13. Simplified 99.6%

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

14. Final simplification 99.6%

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

Alternative 7: 99.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 5 \cdot 10^{+116}:\\ \;\;\;\;\frac{1}{\left(\sqrt{x} + \sqrt{x + 1}\right) \cdot \sqrt{x \cdot \left(x + 1\right)}}\\ \mathbf{else}:\\ \;\;\;\;{x}^{-0.5} \cdot \frac{0.5}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 5e+116)
   (/ 1.0 (* (+ (sqrt x) (sqrt (+ x 1.0))) (sqrt (* x (+ x 1.0)))))
   (* (pow x -0.5) (/ 0.5 x))))

C

double code(double x) {
	double tmp;
	if (x <= 5e+116) {
		tmp = 1.0 / ((sqrt(x) + sqrt((x + 1.0))) * sqrt((x * (x + 1.0))));
	} else {
		tmp = pow(x, -0.5) * (0.5 / x);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 5d+116) then
        tmp = 1.0d0 / ((sqrt(x) + sqrt((x + 1.0d0))) * sqrt((x * (x + 1.0d0))))
    else
        tmp = (x ** (-0.5d0)) * (0.5d0 / x)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 5e+116) {
		tmp = 1.0 / ((Math.sqrt(x) + Math.sqrt((x + 1.0))) * Math.sqrt((x * (x + 1.0))));
	} else {
		tmp = Math.pow(x, -0.5) * (0.5 / x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 5e+116:
		tmp = 1.0 / ((math.sqrt(x) + math.sqrt((x + 1.0))) * math.sqrt((x * (x + 1.0))))
	else:
		tmp = math.pow(x, -0.5) * (0.5 / x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 5e+116)
		tmp = Float64(1.0 / Float64(Float64(sqrt(x) + sqrt(Float64(x + 1.0))) * sqrt(Float64(x * Float64(x + 1.0)))));
	else
		tmp = Float64((x ^ -0.5) * Float64(0.5 / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 5e+116)
		tmp = 1.0 / ((sqrt(x) + sqrt((x + 1.0))) * sqrt((x * (x + 1.0))));
	else
		tmp = (x ^ -0.5) * (0.5 / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 5e+116], N[(1.0 / N[(N[(N[Sqrt[x], $MachinePrecision] + N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(x * N[(x + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Power[x, -0.5], $MachinePrecision] * N[(0.5 / x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 5.00000000000000025e116

   1. Initial program 77.0%

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

      1. frac-sub 77.0%

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

      2. div-inv 77.0%

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

      3. *-un-lft-identity 77.0%

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

      4. +-commutative 77.0%

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

      5. *-rgt-identity 77.0%

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

      6. metadata-eval 77.0%

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

      7. frac-times 77.0%

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

      8. un-div-inv 77.0%

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

      9. pow1/2 77.0%

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

      10. pow-flip 77.4%

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

      11. metadata-eval 77.4%

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

      12. +-commutative 77.4%

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

   3. Applied egg-rr 77.4%

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

      1. associate-*r/ 77.4%

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

   5. Simplified 77.4%

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

      1. flip-- 77.7%

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

      2. div-inv 77.7%

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

      3. add-sqr-sqrt 78.2%

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

      4. add-sqr-sqrt 78.7%

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

      5. associate--l+ 99.7%

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

   7. Applied egg-rr 99.7%

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

      1. +-inverses 99.7%

         \[\leadsto \frac{\left(\left(1 + \color{blue}{0}\right) \cdot \frac{1}{\sqrt{1 + x} + \sqrt{x}}\right) \cdot {x}^{-0.5}}{\sqrt{1 + x}} \]

      2. metadata-eval 99.7%

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

      3. *-lft-identity 99.7%

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

   9. Simplified 99.7%

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

       1. expm1-log1p-u 94.4%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\frac{1}{\sqrt{1 + x} + \sqrt{x}} \cdot {x}^{-0.5}}{\sqrt{1 + x}}\right)\right)} \]

       2. expm1-udef 71.2%

          \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\frac{1}{\sqrt{1 + x} + \sqrt{x}} \cdot {x}^{-0.5}}{\sqrt{1 + x}}\right)} - 1} \]

   11. Applied egg-rr 71.2%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\frac{1}{\sqrt{\left(1 + x\right) \cdot x}}}{\sqrt{1 + x} + \sqrt{x}}\right)} - 1} \]
   12. Step-by-step derivation

       1. expm1-def 94.5%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\frac{1}{\sqrt{\left(1 + x\right) \cdot x}}}{\sqrt{1 + x} + \sqrt{x}}\right)\right)} \]

       2. expm1-log1p 99.4%

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

       3. associate-/l/ 99.4%

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

       4. rem-exp-log 92.6%

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

       5. rem-exp-log 99.4%

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

       6. +-commutative 99.4%

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

       7. *-commutative 99.4%

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

       8. +-commutative 99.4%

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

   13. Simplified 99.4%

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

   if 5.00000000000000025e116 < x

   1. Initial program 59.3%

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

      1. frac-sub 59.3%

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

      2. div-inv 59.3%

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

      3. *-un-lft-identity 59.3%

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

      4. +-commutative 59.3%

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

      5. *-rgt-identity 59.3%

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

      6. metadata-eval 59.3%

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

      7. frac-times 59.3%

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

      8. un-div-inv 59.3%

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

      9. pow1/2 59.3%

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

      10. pow-flip 59.3%

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

      11. metadata-eval 59.3%

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

      12. +-commutative 59.3%

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

   3. Applied egg-rr 59.3%

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

      1. associate-*r/ 59.3%

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

   5. Simplified 59.3%

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

      1. flip-- 59.3%

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

      2. div-inv 59.3%

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

      3. add-sqr-sqrt 33.3%

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

      4. add-sqr-sqrt 59.3%

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

      5. associate--l+ 99.6%

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

   7. Applied egg-rr 99.6%

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

      1. +-inverses 99.6%

         \[\leadsto \frac{\left(\left(1 + \color{blue}{0}\right) \cdot \frac{1}{\sqrt{1 + x} + \sqrt{x}}\right) \cdot {x}^{-0.5}}{\sqrt{1 + x}} \]

      2. metadata-eval 99.6%

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

      3. *-lft-identity 99.6%

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

   9. Simplified 99.6%

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

       1. div-inv 99.5%

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

       2. pow1/2 99.5%

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

       3. pow-flip 99.7%

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

       4. metadata-eval 99.7%

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

       5. metadata-eval 99.7%

          \[\leadsto \left(\frac{1}{\sqrt{1 + x} + \sqrt{x}} \cdot {x}^{-0.5}\right) \cdot {\left(1 + x\right)}^{\color{blue}{\left(-1.5 \cdot 0.3333333333333333\right)}} \]

       6. pow-pow 96.1%

          \[\leadsto \left(\frac{1}{\sqrt{1 + x} + \sqrt{x}} \cdot {x}^{-0.5}\right) \cdot \color{blue}{{\left({\left(1 + x\right)}^{-1.5}\right)}^{0.3333333333333333}} \]

       7. pow1/3 99.4%

          \[\leadsto \left(\frac{1}{\sqrt{1 + x} + \sqrt{x}} \cdot {x}^{-0.5}\right) \cdot \color{blue}{\sqrt[3]{{\left(1 + x\right)}^{-1.5}}} \]

       8. metadata-eval 99.4%

          \[\leadsto \left(\frac{1}{\sqrt{1 + x} + \sqrt{x}} \cdot {x}^{\color{blue}{\left(-0.5\right)}}\right) \cdot \sqrt[3]{{\left(1 + x\right)}^{-1.5}} \]

       9. pow-flip 99.3%

          \[\leadsto \left(\frac{1}{\sqrt{1 + x} + \sqrt{x}} \cdot \color{blue}{\frac{1}{{x}^{0.5}}}\right) \cdot \sqrt[3]{{\left(1 + x\right)}^{-1.5}} \]

       10. pow1/2 99.3%

           \[\leadsto \left(\frac{1}{\sqrt{1 + x} + \sqrt{x}} \cdot \frac{1}{\color{blue}{\sqrt{x}}}\right) \cdot \sqrt[3]{{\left(1 + x\right)}^{-1.5}} \]

       11. *-commutative 99.3%

           \[\leadsto \color{blue}{\left(\frac{1}{\sqrt{x}} \cdot \frac{1}{\sqrt{1 + x} + \sqrt{x}}\right)} \cdot \sqrt[3]{{\left(1 + x\right)}^{-1.5}} \]

       12. associate-*l* 99.3%

           \[\leadsto \color{blue}{\frac{1}{\sqrt{x}} \cdot \left(\frac{1}{\sqrt{1 + x} + \sqrt{x}} \cdot \sqrt[3]{{\left(1 + x\right)}^{-1.5}}\right)} \]

       13. pow1/2 99.3%

           \[\leadsto \frac{1}{\color{blue}{{x}^{0.5}}} \cdot \left(\frac{1}{\sqrt{1 + x} + \sqrt{x}} \cdot \sqrt[3]{{\left(1 + x\right)}^{-1.5}}\right) \]

       14. pow-flip 99.5%

           \[\leadsto \color{blue}{{x}^{\left(-0.5\right)}} \cdot \left(\frac{1}{\sqrt{1 + x} + \sqrt{x}} \cdot \sqrt[3]{{\left(1 + x\right)}^{-1.5}}\right) \]

       15. metadata-eval 99.5%

           \[\leadsto {x}^{\color{blue}{-0.5}} \cdot \left(\frac{1}{\sqrt{1 + x} + \sqrt{x}} \cdot \sqrt[3]{{\left(1 + x\right)}^{-1.5}}\right) \]

       16. pow1/3 96.0%

           \[\leadsto {x}^{-0.5} \cdot \left(\frac{1}{\sqrt{1 + x} + \sqrt{x}} \cdot \color{blue}{{\left({\left(1 + x\right)}^{-1.5}\right)}^{0.3333333333333333}}\right) \]

       17. pow-pow 99.7%

           \[\leadsto {x}^{-0.5} \cdot \left(\frac{1}{\sqrt{1 + x} + \sqrt{x}} \cdot \color{blue}{{\left(1 + x\right)}^{\left(-1.5 \cdot 0.3333333333333333\right)}}\right) \]

       18. metadata-eval 99.7%

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

   11. Applied egg-rr 99.7%

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

   12. Taylor expanded in x around inf 99.8%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 5 \cdot 10^{+116}:\\ \;\;\;\;\frac{1}{\left(\sqrt{x} + \sqrt{x + 1}\right) \cdot \sqrt{x \cdot \left(x + 1\right)}}\\ \mathbf{else}:\\ \;\;\;\;{x}^{-0.5} \cdot \frac{0.5}{x}\\ \end{array} \]

Alternative 8: 99.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 10^{+72}:\\ \;\;\;\;\frac{\frac{1}{\sqrt{x \cdot \left(x + 1\right)}}}{\sqrt{x} + \sqrt{x + 1}}\\ \mathbf{else}:\\ \;\;\;\;{x}^{-0.5} \cdot \frac{0.5}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1e+72)
   (/ (/ 1.0 (sqrt (* x (+ x 1.0)))) (+ (sqrt x) (sqrt (+ x 1.0))))
   (* (pow x -0.5) (/ 0.5 x))))

C

double code(double x) {
	double tmp;
	if (x <= 1e+72) {
		tmp = (1.0 / sqrt((x * (x + 1.0)))) / (sqrt(x) + sqrt((x + 1.0)));
	} else {
		tmp = pow(x, -0.5) * (0.5 / x);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 1d+72) then
        tmp = (1.0d0 / sqrt((x * (x + 1.0d0)))) / (sqrt(x) + sqrt((x + 1.0d0)))
    else
        tmp = (x ** (-0.5d0)) * (0.5d0 / x)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 1e+72) {
		tmp = (1.0 / Math.sqrt((x * (x + 1.0)))) / (Math.sqrt(x) + Math.sqrt((x + 1.0)));
	} else {
		tmp = Math.pow(x, -0.5) * (0.5 / x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 1e+72:
		tmp = (1.0 / math.sqrt((x * (x + 1.0)))) / (math.sqrt(x) + math.sqrt((x + 1.0)))
	else:
		tmp = math.pow(x, -0.5) * (0.5 / x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 1e+72)
		tmp = Float64(Float64(1.0 / sqrt(Float64(x * Float64(x + 1.0)))) / Float64(sqrt(x) + sqrt(Float64(x + 1.0))));
	else
		tmp = Float64((x ^ -0.5) * Float64(0.5 / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 1e+72)
		tmp = (1.0 / sqrt((x * (x + 1.0)))) / (sqrt(x) + sqrt((x + 1.0)));
	else
		tmp = (x ^ -0.5) * (0.5 / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 1e+72], N[(N[(1.0 / N[Sqrt[N[(x * N[(x + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(N[Sqrt[x], $MachinePrecision] + N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Power[x, -0.5], $MachinePrecision] * N[(0.5 / x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 9.99999999999999944e71

   1. Initial program 83.1%

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

      1. frac-sub 83.1%

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

      2. div-inv 83.1%

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

      3. *-un-lft-identity 83.1%

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

      4. +-commutative 83.1%

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

      5. *-rgt-identity 83.1%

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

      6. metadata-eval 83.1%

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

      7. frac-times 83.1%

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

      8. un-div-inv 83.1%

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

      9. pow1/2 83.1%

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

      10. pow-flip 83.5%

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

      11. metadata-eval 83.5%

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

      12. +-commutative 83.5%

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

   3. Applied egg-rr 83.5%

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

      1. associate-*r/ 83.5%

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

   5. Simplified 83.5%

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

      1. flip-- 83.8%

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

      2. div-inv 83.8%

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

      3. add-sqr-sqrt 84.4%

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

      4. add-sqr-sqrt 84.9%

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

      5. associate--l+ 99.7%

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

   7. Applied egg-rr 99.7%

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

      1. +-inverses 99.7%

         \[\leadsto \frac{\left(\left(1 + \color{blue}{0}\right) \cdot \frac{1}{\sqrt{1 + x} + \sqrt{x}}\right) \cdot {x}^{-0.5}}{\sqrt{1 + x}} \]

      2. metadata-eval 99.7%

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

      3. *-lft-identity 99.7%

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

   9. Simplified 99.7%

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

       1. expm1-log1p-u 94.0%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\frac{1}{\sqrt{1 + x} + \sqrt{x}} \cdot {x}^{-0.5}}{\sqrt{1 + x}}\right)\right)} \]

       2. expm1-udef 76.8%

          \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\frac{1}{\sqrt{1 + x} + \sqrt{x}} \cdot {x}^{-0.5}}{\sqrt{1 + x}}\right)} - 1} \]

   11. Applied egg-rr 76.8%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\frac{1}{\sqrt{\left(1 + x\right) \cdot x}}}{\sqrt{1 + x} + \sqrt{x}}\right)} - 1} \]
   12. Step-by-step derivation

       1. expm1-def 94.1%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\frac{1}{\sqrt{\left(1 + x\right) \cdot x}}}{\sqrt{1 + x} + \sqrt{x}}\right)\right)} \]

       2. expm1-log1p 99.4%

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

       3. *-commutative 99.4%

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

       4. +-commutative 99.4%

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

       5. +-commutative 99.4%

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

   13. Simplified 99.4%

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

   if 9.99999999999999944e71 < x

   1. Initial program 52.3%

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

      1. frac-sub 52.3%

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

      2. div-inv 52.3%

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

      3. *-un-lft-identity 52.3%

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

      4. +-commutative 52.3%

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

      5. *-rgt-identity 52.3%

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

      6. metadata-eval 52.3%

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

      7. frac-times 52.3%

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

      8. un-div-inv 52.3%

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

      9. pow1/2 52.3%

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

      10. pow-flip 52.3%

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

      11. metadata-eval 52.3%

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

      12. +-commutative 52.3%

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

   3. Applied egg-rr 52.3%

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

      1. associate-*r/ 52.3%

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

   5. Simplified 52.3%

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

      1. flip-- 52.3%

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

      2. div-inv 52.3%

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

      3. add-sqr-sqrt 29.5%

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

      4. add-sqr-sqrt 52.3%

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

      5. associate--l+ 99.5%

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

   7. Applied egg-rr 99.5%

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

      1. +-inverses 99.5%

         \[\leadsto \frac{\left(\left(1 + \color{blue}{0}\right) \cdot \frac{1}{\sqrt{1 + x} + \sqrt{x}}\right) \cdot {x}^{-0.5}}{\sqrt{1 + x}} \]

      2. metadata-eval 99.5%

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

      3. *-lft-identity 99.5%

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

   9. Simplified 99.5%

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

       1. div-inv 99.5%

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

       2. pow1/2 99.5%

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

       3. pow-flip 99.6%

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

       4. metadata-eval 99.6%

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

       5. metadata-eval 99.6%

          \[\leadsto \left(\frac{1}{\sqrt{1 + x} + \sqrt{x}} \cdot {x}^{-0.5}\right) \cdot {\left(1 + x\right)}^{\color{blue}{\left(-1.5 \cdot 0.3333333333333333\right)}} \]

       6. pow-pow 95.5%

          \[\leadsto \left(\frac{1}{\sqrt{1 + x} + \sqrt{x}} \cdot {x}^{-0.5}\right) \cdot \color{blue}{{\left({\left(1 + x\right)}^{-1.5}\right)}^{0.3333333333333333}} \]

       7. pow1/3 99.3%

          \[\leadsto \left(\frac{1}{\sqrt{1 + x} + \sqrt{x}} \cdot {x}^{-0.5}\right) \cdot \color{blue}{\sqrt[3]{{\left(1 + x\right)}^{-1.5}}} \]

       8. metadata-eval 99.3%

          \[\leadsto \left(\frac{1}{\sqrt{1 + x} + \sqrt{x}} \cdot {x}^{\color{blue}{\left(-0.5\right)}}\right) \cdot \sqrt[3]{{\left(1 + x\right)}^{-1.5}} \]

       9. pow-flip 99.2%

          \[\leadsto \left(\frac{1}{\sqrt{1 + x} + \sqrt{x}} \cdot \color{blue}{\frac{1}{{x}^{0.5}}}\right) \cdot \sqrt[3]{{\left(1 + x\right)}^{-1.5}} \]

       10. pow1/2 99.2%

           \[\leadsto \left(\frac{1}{\sqrt{1 + x} + \sqrt{x}} \cdot \frac{1}{\color{blue}{\sqrt{x}}}\right) \cdot \sqrt[3]{{\left(1 + x\right)}^{-1.5}} \]

       11. *-commutative 99.2%

           \[\leadsto \color{blue}{\left(\frac{1}{\sqrt{x}} \cdot \frac{1}{\sqrt{1 + x} + \sqrt{x}}\right)} \cdot \sqrt[3]{{\left(1 + x\right)}^{-1.5}} \]

       12. associate-*l* 99.2%

           \[\leadsto \color{blue}{\frac{1}{\sqrt{x}} \cdot \left(\frac{1}{\sqrt{1 + x} + \sqrt{x}} \cdot \sqrt[3]{{\left(1 + x\right)}^{-1.5}}\right)} \]

       13. pow1/2 99.2%

           \[\leadsto \frac{1}{\color{blue}{{x}^{0.5}}} \cdot \left(\frac{1}{\sqrt{1 + x} + \sqrt{x}} \cdot \sqrt[3]{{\left(1 + x\right)}^{-1.5}}\right) \]

       14. pow-flip 99.3%

           \[\leadsto \color{blue}{{x}^{\left(-0.5\right)}} \cdot \left(\frac{1}{\sqrt{1 + x} + \sqrt{x}} \cdot \sqrt[3]{{\left(1 + x\right)}^{-1.5}}\right) \]

       15. metadata-eval 99.3%

           \[\leadsto {x}^{\color{blue}{-0.5}} \cdot \left(\frac{1}{\sqrt{1 + x} + \sqrt{x}} \cdot \sqrt[3]{{\left(1 + x\right)}^{-1.5}}\right) \]

       16. pow1/3 95.5%

           \[\leadsto {x}^{-0.5} \cdot \left(\frac{1}{\sqrt{1 + x} + \sqrt{x}} \cdot \color{blue}{{\left({\left(1 + x\right)}^{-1.5}\right)}^{0.3333333333333333}}\right) \]

       17. pow-pow 99.6%

           \[\leadsto {x}^{-0.5} \cdot \left(\frac{1}{\sqrt{1 + x} + \sqrt{x}} \cdot \color{blue}{{\left(1 + x\right)}^{\left(-1.5 \cdot 0.3333333333333333\right)}}\right) \]

       18. metadata-eval 99.6%

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

   11. Applied egg-rr 99.6%

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

   12. Taylor expanded in x around inf 99.7%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 10^{+72}:\\ \;\;\;\;\frac{\frac{1}{\sqrt{x \cdot \left(x + 1\right)}}}{\sqrt{x} + \sqrt{x + 1}}\\ \mathbf{else}:\\ \;\;\;\;{x}^{-0.5} \cdot \frac{0.5}{x}\\ \end{array} \]

Alternative 9: 99.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 8 \cdot 10^{+153}:\\ \;\;\;\;\frac{\frac{1}{\sqrt{x} + \sqrt{x + 1}}}{\sqrt{x + x \cdot x}}\\ \mathbf{else}:\\ \;\;\;\;{x}^{-0.5} \cdot \frac{0.5}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 8e+153)
   (/ (/ 1.0 (+ (sqrt x) (sqrt (+ x 1.0)))) (sqrt (+ x (* x x))))
   (* (pow x -0.5) (/ 0.5 x))))

C

double code(double x) {
	double tmp;
	if (x <= 8e+153) {
		tmp = (1.0 / (sqrt(x) + sqrt((x + 1.0)))) / sqrt((x + (x * x)));
	} else {
		tmp = pow(x, -0.5) * (0.5 / x);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 8d+153) then
        tmp = (1.0d0 / (sqrt(x) + sqrt((x + 1.0d0)))) / sqrt((x + (x * x)))
    else
        tmp = (x ** (-0.5d0)) * (0.5d0 / x)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 8e+153) {
		tmp = (1.0 / (Math.sqrt(x) + Math.sqrt((x + 1.0)))) / Math.sqrt((x + (x * x)));
	} else {
		tmp = Math.pow(x, -0.5) * (0.5 / x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 8e+153:
		tmp = (1.0 / (math.sqrt(x) + math.sqrt((x + 1.0)))) / math.sqrt((x + (x * x)))
	else:
		tmp = math.pow(x, -0.5) * (0.5 / x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 8e+153)
		tmp = Float64(Float64(1.0 / Float64(sqrt(x) + sqrt(Float64(x + 1.0)))) / sqrt(Float64(x + Float64(x * x))));
	else
		tmp = Float64((x ^ -0.5) * Float64(0.5 / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 8e+153)
		tmp = (1.0 / (sqrt(x) + sqrt((x + 1.0)))) / sqrt((x + (x * x)));
	else
		tmp = (x ^ -0.5) * (0.5 / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 8e+153], N[(N[(1.0 / N[(N[Sqrt[x], $MachinePrecision] + N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[(x + N[(x * x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Power[x, -0.5], $MachinePrecision] * N[(0.5 / x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 8e153

   1. Initial program 70.1%

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

      1. frac-sub 70.1%

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

      2. clear-num 70.1%

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

      3. sqrt-unprod 70.1%

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

      4. +-commutative 70.1%

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

      5. *-un-lft-identity 70.1%

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

      6. *-rgt-identity 70.1%

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

      7. +-commutative 70.1%

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

   3. Applied egg-rr 70.1%

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

      1. associate-/r/ 70.1%

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

      2. associate-*l/ 70.1%

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

      3. *-lft-identity 70.1%

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

      4. distribute-rgt-in 70.1%

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

      5. *-lft-identity 70.1%

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

   5. Simplified 70.1%

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

      1. flip-- 70.7%

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

      2. div-inv 70.7%

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

      3. add-sqr-sqrt 71.1%

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

      4. add-sqr-sqrt 71.6%

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

      5. associate--l+ 99.6%

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

   7. Applied egg-rr 99.4%

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

      1. +-inverses 99.6%

         \[\leadsto \frac{\left(\left(1 + \color{blue}{0}\right) \cdot \frac{1}{\sqrt{1 + x} + \sqrt{x}}\right) \cdot {x}^{-0.5}}{\sqrt{1 + x}} \]

      2. metadata-eval 99.6%

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

      3. *-lft-identity 99.6%

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

   9. Simplified 99.4%

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

   if 8e153 < x

   1. Initial program 73.1%

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

      1. frac-sub 73.1%

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

      2. div-inv 73.1%

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

      3. *-un-lft-identity 73.1%

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

      4. +-commutative 73.1%

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

      5. *-rgt-identity 73.1%

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

      6. metadata-eval 73.1%

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

      7. frac-times 73.1%

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

      8. un-div-inv 73.1%

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

      9. pow1/2 73.1%

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

      10. pow-flip 73.1%

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

      11. metadata-eval 73.1%

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

      12. +-commutative 73.1%

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

   3. Applied egg-rr 73.1%

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

      1. associate-*r/ 73.1%

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

   5. Simplified 73.1%

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

      1. flip-- 73.1%

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

      2. div-inv 73.1%

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

      3. add-sqr-sqrt 40.6%

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

      4. add-sqr-sqrt 73.1%

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

      5. associate--l+ 99.7%

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

   7. Applied egg-rr 99.7%

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

      1. +-inverses 99.7%

         \[\leadsto \frac{\left(\left(1 + \color{blue}{0}\right) \cdot \frac{1}{\sqrt{1 + x} + \sqrt{x}}\right) \cdot {x}^{-0.5}}{\sqrt{1 + x}} \]

      2. metadata-eval 99.7%

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

      3. *-lft-identity 99.7%

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

   9. Simplified 99.7%

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

       1. div-inv 99.7%

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

       2. pow1/2 99.7%

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

       3. pow-flip 99.9%

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

       4. metadata-eval 99.9%

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

       5. metadata-eval 99.9%

          \[\leadsto \left(\frac{1}{\sqrt{1 + x} + \sqrt{x}} \cdot {x}^{-0.5}\right) \cdot {\left(1 + x\right)}^{\color{blue}{\left(-1.5 \cdot 0.3333333333333333\right)}} \]

       6. pow-pow 97.4%

          \[\leadsto \left(\frac{1}{\sqrt{1 + x} + \sqrt{x}} \cdot {x}^{-0.5}\right) \cdot \color{blue}{{\left({\left(1 + x\right)}^{-1.5}\right)}^{0.3333333333333333}} \]

       7. pow1/3 99.6%

          \[\leadsto \left(\frac{1}{\sqrt{1 + x} + \sqrt{x}} \cdot {x}^{-0.5}\right) \cdot \color{blue}{\sqrt[3]{{\left(1 + x\right)}^{-1.5}}} \]

       8. metadata-eval 99.6%

          \[\leadsto \left(\frac{1}{\sqrt{1 + x} + \sqrt{x}} \cdot {x}^{\color{blue}{\left(-0.5\right)}}\right) \cdot \sqrt[3]{{\left(1 + x\right)}^{-1.5}} \]

       9. pow-flip 99.5%

          \[\leadsto \left(\frac{1}{\sqrt{1 + x} + \sqrt{x}} \cdot \color{blue}{\frac{1}{{x}^{0.5}}}\right) \cdot \sqrt[3]{{\left(1 + x\right)}^{-1.5}} \]

       10. pow1/2 99.5%

           \[\leadsto \left(\frac{1}{\sqrt{1 + x} + \sqrt{x}} \cdot \frac{1}{\color{blue}{\sqrt{x}}}\right) \cdot \sqrt[3]{{\left(1 + x\right)}^{-1.5}} \]

       11. *-commutative 99.5%

           \[\leadsto \color{blue}{\left(\frac{1}{\sqrt{x}} \cdot \frac{1}{\sqrt{1 + x} + \sqrt{x}}\right)} \cdot \sqrt[3]{{\left(1 + x\right)}^{-1.5}} \]

       12. associate-*l* 99.5%

           \[\leadsto \color{blue}{\frac{1}{\sqrt{x}} \cdot \left(\frac{1}{\sqrt{1 + x} + \sqrt{x}} \cdot \sqrt[3]{{\left(1 + x\right)}^{-1.5}}\right)} \]

       13. pow1/2 99.5%

           \[\leadsto \frac{1}{\color{blue}{{x}^{0.5}}} \cdot \left(\frac{1}{\sqrt{1 + x} + \sqrt{x}} \cdot \sqrt[3]{{\left(1 + x\right)}^{-1.5}}\right) \]

       14. pow-flip 99.6%

           \[\leadsto \color{blue}{{x}^{\left(-0.5\right)}} \cdot \left(\frac{1}{\sqrt{1 + x} + \sqrt{x}} \cdot \sqrt[3]{{\left(1 + x\right)}^{-1.5}}\right) \]

       15. metadata-eval 99.6%

           \[\leadsto {x}^{\color{blue}{-0.5}} \cdot \left(\frac{1}{\sqrt{1 + x} + \sqrt{x}} \cdot \sqrt[3]{{\left(1 + x\right)}^{-1.5}}\right) \]

       16. pow1/3 97.4%

           \[\leadsto {x}^{-0.5} \cdot \left(\frac{1}{\sqrt{1 + x} + \sqrt{x}} \cdot \color{blue}{{\left({\left(1 + x\right)}^{-1.5}\right)}^{0.3333333333333333}}\right) \]

       17. pow-pow 99.9%

           \[\leadsto {x}^{-0.5} \cdot \left(\frac{1}{\sqrt{1 + x} + \sqrt{x}} \cdot \color{blue}{{\left(1 + x\right)}^{\left(-1.5 \cdot 0.3333333333333333\right)}}\right) \]

       18. metadata-eval 99.9%

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

   11. Applied egg-rr 99.9%

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

   12. Taylor expanded in x around inf 99.9%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 8 \cdot 10^{+153}:\\ \;\;\;\;\frac{\frac{1}{\sqrt{x} + \sqrt{x + 1}}}{\sqrt{x + x \cdot x}}\\ \mathbf{else}:\\ \;\;\;\;{x}^{-0.5} \cdot \frac{0.5}{x}\\ \end{array} \]

Alternative 10: 99.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{x + 1}\\ \mathbf{if}\;x \leq 44000000:\\ \;\;\;\;{x}^{-0.5} \cdot \left(1 - \frac{\sqrt{x}}{t_0}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.5}{x}}{t_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (sqrt (+ x 1.0))))
   (if (<= x 44000000.0)
     (* (pow x -0.5) (- 1.0 (/ (sqrt x) t_0)))
     (/ (/ 0.5 x) t_0))))

C

double code(double x) {
	double t_0 = sqrt((x + 1.0));
	double tmp;
	if (x <= 44000000.0) {
		tmp = pow(x, -0.5) * (1.0 - (sqrt(x) / t_0));
	} else {
		tmp = (0.5 / x) / t_0;
	}
	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 (x <= 44000000.0d0) then
        tmp = (x ** (-0.5d0)) * (1.0d0 - (sqrt(x) / t_0))
    else
        tmp = (0.5d0 / x) / t_0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double t_0 = Math.sqrt((x + 1.0));
	double tmp;
	if (x <= 44000000.0) {
		tmp = Math.pow(x, -0.5) * (1.0 - (Math.sqrt(x) / t_0));
	} else {
		tmp = (0.5 / x) / t_0;
	}
	return tmp;
}

Python

def code(x):
	t_0 = math.sqrt((x + 1.0))
	tmp = 0
	if x <= 44000000.0:
		tmp = math.pow(x, -0.5) * (1.0 - (math.sqrt(x) / t_0))
	else:
		tmp = (0.5 / x) / t_0
	return tmp

Julia

function code(x)
	t_0 = sqrt(Float64(x + 1.0))
	tmp = 0.0
	if (x <= 44000000.0)
		tmp = Float64((x ^ -0.5) * Float64(1.0 - Float64(sqrt(x) / t_0)));
	else
		tmp = Float64(Float64(0.5 / x) / t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = sqrt((x + 1.0));
	tmp = 0.0;
	if (x <= 44000000.0)
		tmp = (x ^ -0.5) * (1.0 - (sqrt(x) / t_0));
	else
		tmp = (0.5 / x) / t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x, 44000000.0], N[(N[Power[x, -0.5], $MachinePrecision] * N[(1.0 - N[(N[Sqrt[x], $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 / x), $MachinePrecision] / t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 4.4e7

   1. Initial program 98.9%

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

      1. frac-sub 99.0%

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

      2. div-inv 99.0%

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

      3. *-un-lft-identity 99.0%

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

      4. +-commutative 99.0%

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

      5. *-rgt-identity 99.0%

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

      6. metadata-eval 99.0%

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

      7. frac-times 99.0%

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

      8. un-div-inv 99.0%

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

      9. pow1/2 99.0%

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

      10. pow-flip 99.4%

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

      11. metadata-eval 99.4%

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

      12. +-commutative 99.4%

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

   3. Applied egg-rr 99.4%

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

      1. associate-*r/ 99.5%

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

      2. *-rgt-identity 99.5%

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

      3. times-frac 99.4%

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

      4. div-sub 99.4%

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

      5. *-inverses 99.4%

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

      6. /-rgt-identity 99.4%

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

   5. Simplified 99.4%

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

   if 4.4e7 < x

   1. Initial program 42.9%

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

      1. frac-sub 42.9%

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

      2. div-inv 42.9%

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

      3. *-un-lft-identity 42.9%

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

      4. +-commutative 42.9%

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

      5. *-rgt-identity 42.9%

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

      6. metadata-eval 42.9%

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

      7. frac-times 42.9%

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

      8. un-div-inv 42.9%

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

      9. pow1/2 42.9%

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

      10. pow-flip 42.9%

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

      11. metadata-eval 42.9%

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

      12. +-commutative 42.9%

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

   3. Applied egg-rr 42.9%

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

      1. associate-*r/ 42.9%

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

   5. Simplified 42.9%

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

      1. flip-- 43.3%

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

      2. div-inv 43.3%

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

      3. add-sqr-sqrt 26.1%

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

      4. add-sqr-sqrt 44.2%

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

      5. associate--l+ 99.4%

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

   7. Applied egg-rr 99.4%

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

      1. +-inverses 99.4%

         \[\leadsto \frac{\left(\left(1 + \color{blue}{0}\right) \cdot \frac{1}{\sqrt{1 + x} + \sqrt{x}}\right) \cdot {x}^{-0.5}}{\sqrt{1 + x}} \]

      2. metadata-eval 99.4%

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

      3. *-lft-identity 99.4%

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

   9. Simplified 99.4%

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

   10. Taylor expanded in x around inf 99.2%

       \[\leadsto \frac{\color{blue}{\frac{0.5}{x}}}{\sqrt{1 + x}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 44000000:\\ \;\;\;\;{x}^{-0.5} \cdot \left(1 - \frac{\sqrt{x}}{\sqrt{x + 1}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.5}{x}}{\sqrt{x + 1}}\\ \end{array} \]

Alternative 11: 98.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.72:\\ \;\;\;\;\left({x}^{-0.5} + x \cdot 0.5\right) + -1\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.5}{x}}{\sqrt{x + 1}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 0.72)
   (+ (+ (pow x -0.5) (* x 0.5)) -1.0)
   (/ (/ 0.5 x) (sqrt (+ x 1.0)))))

C

double code(double x) {
	double tmp;
	if (x <= 0.72) {
		tmp = (pow(x, -0.5) + (x * 0.5)) + -1.0;
	} else {
		tmp = (0.5 / x) / sqrt((x + 1.0));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 0.72d0) then
        tmp = ((x ** (-0.5d0)) + (x * 0.5d0)) + (-1.0d0)
    else
        tmp = (0.5d0 / x) / sqrt((x + 1.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 0.72) {
		tmp = (Math.pow(x, -0.5) + (x * 0.5)) + -1.0;
	} else {
		tmp = (0.5 / x) / Math.sqrt((x + 1.0));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 0.72:
		tmp = (math.pow(x, -0.5) + (x * 0.5)) + -1.0
	else:
		tmp = (0.5 / x) / math.sqrt((x + 1.0))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 0.72)
		tmp = Float64(Float64((x ^ -0.5) + Float64(x * 0.5)) + -1.0);
	else
		tmp = Float64(Float64(0.5 / x) / sqrt(Float64(x + 1.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 0.72)
		tmp = ((x ^ -0.5) + (x * 0.5)) + -1.0;
	else
		tmp = (0.5 / x) / sqrt((x + 1.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 0.72], N[(N[(N[Power[x, -0.5], $MachinePrecision] + N[(x * 0.5), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision], N[(N[(0.5 / x), $MachinePrecision] / N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.71999999999999997

   1. Initial program 99.5%

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

      1. inv-pow 99.5%

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

      2. pow1/2 99.5%

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

      3. pow-pow 100.0%

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

      4. add-exp-log 92.8%

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

      5. pow-exp 92.8%

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

      6. metadata-eval 92.8%

         \[\leadsto e^{\log x \cdot \color{blue}{-0.5}} - \frac{1}{\sqrt{x + 1}} \]

   3. Applied egg-rr 92.8%

      \[\leadsto \color{blue}{e^{\log x \cdot -0.5}} - \frac{1}{\sqrt{x + 1}} \]

   4. Taylor expanded in x around 0 99.5%

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

   if 0.71999999999999997 < x

   1. Initial program 44.0%

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

      1. frac-sub 44.1%

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

      2. div-inv 44.1%

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

      3. *-un-lft-identity 44.1%

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

      4. +-commutative 44.1%

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

      5. *-rgt-identity 44.1%

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

      6. metadata-eval 44.1%

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

      7. frac-times 44.1%

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

      8. un-div-inv 44.1%

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

      9. pow1/2 44.1%

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

      10. pow-flip 44.1%

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

      11. metadata-eval 44.1%

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

      12. +-commutative 44.1%

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

   3. Applied egg-rr 44.1%

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

      1. associate-*r/ 44.1%

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

   5. Simplified 44.1%

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

      1. flip-- 44.6%

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

      2. div-inv 44.6%

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

      3. add-sqr-sqrt 27.9%

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

      4. add-sqr-sqrt 45.9%

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

      5. associate--l+ 99.4%

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

   7. Applied egg-rr 99.4%

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

      1. +-inverses 99.4%

         \[\leadsto \frac{\left(\left(1 + \color{blue}{0}\right) \cdot \frac{1}{\sqrt{1 + x} + \sqrt{x}}\right) \cdot {x}^{-0.5}}{\sqrt{1 + x}} \]

      2. metadata-eval 99.4%

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

      3. *-lft-identity 99.4%

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

   9. Simplified 99.4%

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

   10. Taylor expanded in x around inf 97.4%

       \[\leadsto \frac{\color{blue}{\frac{0.5}{x}}}{\sqrt{1 + x}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.72:\\ \;\;\;\;\left({x}^{-0.5} + x \cdot 0.5\right) + -1\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.5}{x}}{\sqrt{x + 1}}\\ \end{array} \]

Alternative 12: 98.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.72:\\ \;\;\;\;{x}^{-0.5} + \left(-1 - x \cdot -0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.5}{x}}{\sqrt{x + 1}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 0.72)
   (+ (pow x -0.5) (- -1.0 (* x -0.5)))
   (/ (/ 0.5 x) (sqrt (+ x 1.0)))))

C

double code(double x) {
	double tmp;
	if (x <= 0.72) {
		tmp = pow(x, -0.5) + (-1.0 - (x * -0.5));
	} else {
		tmp = (0.5 / x) / sqrt((x + 1.0));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 0.72d0) then
        tmp = (x ** (-0.5d0)) + ((-1.0d0) - (x * (-0.5d0)))
    else
        tmp = (0.5d0 / x) / sqrt((x + 1.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 0.72) {
		tmp = Math.pow(x, -0.5) + (-1.0 - (x * -0.5));
	} else {
		tmp = (0.5 / x) / Math.sqrt((x + 1.0));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 0.72:
		tmp = math.pow(x, -0.5) + (-1.0 - (x * -0.5))
	else:
		tmp = (0.5 / x) / math.sqrt((x + 1.0))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 0.72)
		tmp = Float64((x ^ -0.5) + Float64(-1.0 - Float64(x * -0.5)));
	else
		tmp = Float64(Float64(0.5 / x) / sqrt(Float64(x + 1.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 0.72)
		tmp = (x ^ -0.5) + (-1.0 - (x * -0.5));
	else
		tmp = (0.5 / x) / sqrt((x + 1.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 0.72], N[(N[Power[x, -0.5], $MachinePrecision] + N[(-1.0 - N[(x * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 / x), $MachinePrecision] / N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.71999999999999997

   1. Initial program 99.5%

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

   2. Taylor expanded in x around 0 99.1%

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

      1. add-log-exp 4.8%

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

      2. *-un-lft-identity 4.8%

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

      3. log-prod 4.8%

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

      4. metadata-eval 4.8%

         \[\leadsto \left(\color{blue}{0} + \log \left(e^{\frac{1}{\sqrt{x}}}\right)\right) - \left(-0.5 \cdot x + 1\right) \]

      5. add-log-exp 99.1%

         \[\leadsto \left(0 + \color{blue}{\frac{1}{\sqrt{x}}}\right) - \left(-0.5 \cdot x + 1\right) \]

      6. pow1/2 99.1%

         \[\leadsto \left(0 + \frac{1}{\color{blue}{{x}^{0.5}}}\right) - \left(-0.5 \cdot x + 1\right) \]

      7. pow-flip 99.5%

         \[\leadsto \left(0 + \color{blue}{{x}^{\left(-0.5\right)}}\right) - \left(-0.5 \cdot x + 1\right) \]

      8. metadata-eval 99.5%

         \[\leadsto \left(0 + {x}^{\color{blue}{-0.5}}\right) - \left(-0.5 \cdot x + 1\right) \]

   4. Applied egg-rr 99.5%

      \[\leadsto \color{blue}{\left(0 + {x}^{-0.5}\right)} - \left(-0.5 \cdot x + 1\right) \]
   5. Step-by-step derivation

      1. +-lft-identity 99.5%

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

   6. Simplified 99.5%

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

   if 0.71999999999999997 < x

   1. Initial program 44.0%

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

      1. frac-sub 44.1%

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

      2. div-inv 44.1%

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

      3. *-un-lft-identity 44.1%

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

      4. +-commutative 44.1%

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

      5. *-rgt-identity 44.1%

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

      6. metadata-eval 44.1%

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

      7. frac-times 44.1%

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

      8. un-div-inv 44.1%

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

      9. pow1/2 44.1%

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

      10. pow-flip 44.1%

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

      11. metadata-eval 44.1%

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

      12. +-commutative 44.1%

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

   3. Applied egg-rr 44.1%

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

      1. associate-*r/ 44.1%

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

   5. Simplified 44.1%

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

      1. flip-- 44.6%

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

      2. div-inv 44.6%

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

      3. add-sqr-sqrt 27.9%

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

      4. add-sqr-sqrt 45.9%

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

      5. associate--l+ 99.4%

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

   7. Applied egg-rr 99.4%

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

      1. +-inverses 99.4%

         \[\leadsto \frac{\left(\left(1 + \color{blue}{0}\right) \cdot \frac{1}{\sqrt{1 + x} + \sqrt{x}}\right) \cdot {x}^{-0.5}}{\sqrt{1 + x}} \]

      2. metadata-eval 99.4%

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

      3. *-lft-identity 99.4%

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

   9. Simplified 99.4%

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

   10. Taylor expanded in x around inf 97.4%

       \[\leadsto \frac{\color{blue}{\frac{0.5}{x}}}{\sqrt{1 + x}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.72:\\ \;\;\;\;{x}^{-0.5} + \left(-1 - x \cdot -0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.5}{x}}{\sqrt{x + 1}}\\ \end{array} \]

Alternative 13: 98.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.48:\\ \;\;\;\;{x}^{-0.5} + -1\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.5}{x}}{\sqrt{x + 1}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 0.48) (+ (pow x -0.5) -1.0) (/ (/ 0.5 x) (sqrt (+ x 1.0)))))

C

double code(double x) {
	double tmp;
	if (x <= 0.48) {
		tmp = pow(x, -0.5) + -1.0;
	} else {
		tmp = (0.5 / x) / sqrt((x + 1.0));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 0.48d0) then
        tmp = (x ** (-0.5d0)) + (-1.0d0)
    else
        tmp = (0.5d0 / x) / sqrt((x + 1.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 0.48) {
		tmp = Math.pow(x, -0.5) + -1.0;
	} else {
		tmp = (0.5 / x) / Math.sqrt((x + 1.0));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 0.48:
		tmp = math.pow(x, -0.5) + -1.0
	else:
		tmp = (0.5 / x) / math.sqrt((x + 1.0))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 0.48)
		tmp = Float64((x ^ -0.5) + -1.0);
	else
		tmp = Float64(Float64(0.5 / x) / sqrt(Float64(x + 1.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 0.48)
		tmp = (x ^ -0.5) + -1.0;
	else
		tmp = (0.5 / x) / sqrt((x + 1.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 0.48], N[(N[Power[x, -0.5], $MachinePrecision] + -1.0), $MachinePrecision], N[(N[(0.5 / x), $MachinePrecision] / N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.47999999999999998

   1. Initial program 99.5%

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

      1. inv-pow 99.5%

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

      2. pow1/2 99.5%

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

      3. pow-pow 100.0%

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

      4. add-exp-log 92.8%

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

      5. pow-exp 92.8%

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

      6. metadata-eval 92.8%

         \[\leadsto e^{\log x \cdot \color{blue}{-0.5}} - \frac{1}{\sqrt{x + 1}} \]

   3. Applied egg-rr 92.8%

      \[\leadsto \color{blue}{e^{\log x \cdot -0.5}} - \frac{1}{\sqrt{x + 1}} \]

   4. Taylor expanded in x around 0 98.2%

      \[\leadsto \color{blue}{{x}^{-0.5} - 1} \]

   if 0.47999999999999998 < x

   1. Initial program 44.0%

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

      1. frac-sub 44.1%

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

      2. div-inv 44.1%

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

      3. *-un-lft-identity 44.1%

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

      4. +-commutative 44.1%

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

      5. *-rgt-identity 44.1%

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

      6. metadata-eval 44.1%

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

      7. frac-times 44.1%

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

      8. un-div-inv 44.1%

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

      9. pow1/2 44.1%

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

      10. pow-flip 44.1%

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

      11. metadata-eval 44.1%

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

      12. +-commutative 44.1%

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

   3. Applied egg-rr 44.1%

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

      1. associate-*r/ 44.1%

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

   5. Simplified 44.1%

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

      1. flip-- 44.6%

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

      2. div-inv 44.6%

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

      3. add-sqr-sqrt 27.9%

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

      4. add-sqr-sqrt 45.9%

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

      5. associate--l+ 99.4%

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

   7. Applied egg-rr 99.4%

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

      1. +-inverses 99.4%

         \[\leadsto \frac{\left(\left(1 + \color{blue}{0}\right) \cdot \frac{1}{\sqrt{1 + x} + \sqrt{x}}\right) \cdot {x}^{-0.5}}{\sqrt{1 + x}} \]

      2. metadata-eval 99.4%

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

      3. *-lft-identity 99.4%

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

   9. Simplified 99.4%

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

   10. Taylor expanded in x around inf 97.4%

       \[\leadsto \frac{\color{blue}{\frac{0.5}{x}}}{\sqrt{1 + x}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 97.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.48:\\ \;\;\;\;{x}^{-0.5} + -1\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.5}{x}}{\sqrt{x + 1}}\\ \end{array} \]

Alternative 14: 98.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.7:\\ \;\;\;\;{x}^{-0.5} + -1\\ \mathbf{else}:\\ \;\;\;\;{x}^{-0.5} \cdot \frac{0.5}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 0.7) (+ (pow x -0.5) -1.0) (* (pow x -0.5) (/ 0.5 x))))

C

double code(double x) {
	double tmp;
	if (x <= 0.7) {
		tmp = pow(x, -0.5) + -1.0;
	} else {
		tmp = pow(x, -0.5) * (0.5 / x);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 0.7d0) then
        tmp = (x ** (-0.5d0)) + (-1.0d0)
    else
        tmp = (x ** (-0.5d0)) * (0.5d0 / x)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 0.7) {
		tmp = Math.pow(x, -0.5) + -1.0;
	} else {
		tmp = Math.pow(x, -0.5) * (0.5 / x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 0.7:
		tmp = math.pow(x, -0.5) + -1.0
	else:
		tmp = math.pow(x, -0.5) * (0.5 / x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 0.7)
		tmp = Float64((x ^ -0.5) + -1.0);
	else
		tmp = Float64((x ^ -0.5) * Float64(0.5 / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 0.7)
		tmp = (x ^ -0.5) + -1.0;
	else
		tmp = (x ^ -0.5) * (0.5 / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 0.7], N[(N[Power[x, -0.5], $MachinePrecision] + -1.0), $MachinePrecision], N[(N[Power[x, -0.5], $MachinePrecision] * N[(0.5 / x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.69999999999999996

   1. Initial program 99.5%

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

      1. inv-pow 99.5%

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

      2. pow1/2 99.5%

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

      3. pow-pow 100.0%

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

      4. add-exp-log 92.8%

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

      5. pow-exp 92.8%

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

      6. metadata-eval 92.8%

         \[\leadsto e^{\log x \cdot \color{blue}{-0.5}} - \frac{1}{\sqrt{x + 1}} \]

   3. Applied egg-rr 92.8%

      \[\leadsto \color{blue}{e^{\log x \cdot -0.5}} - \frac{1}{\sqrt{x + 1}} \]

   4. Taylor expanded in x around 0 98.2%

      \[\leadsto \color{blue}{{x}^{-0.5} - 1} \]

   if 0.69999999999999996 < x

   1. Initial program 44.0%

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

      1. frac-sub 44.1%

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

      2. div-inv 44.1%

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

      3. *-un-lft-identity 44.1%

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

      4. +-commutative 44.1%

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

      5. *-rgt-identity 44.1%

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

      6. metadata-eval 44.1%

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

      7. frac-times 44.1%

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

      8. un-div-inv 44.1%

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

      9. pow1/2 44.1%

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

      10. pow-flip 44.1%

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

      11. metadata-eval 44.1%

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

      12. +-commutative 44.1%

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

   3. Applied egg-rr 44.1%

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

      1. associate-*r/ 44.1%

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

   5. Simplified 44.1%

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

      1. flip-- 44.6%

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

      2. div-inv 44.6%

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

      3. add-sqr-sqrt 27.9%

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

      4. add-sqr-sqrt 45.9%

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

      5. associate--l+ 99.4%

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

   7. Applied egg-rr 99.4%

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

      1. +-inverses 99.4%

         \[\leadsto \frac{\left(\left(1 + \color{blue}{0}\right) \cdot \frac{1}{\sqrt{1 + x} + \sqrt{x}}\right) \cdot {x}^{-0.5}}{\sqrt{1 + x}} \]

      2. metadata-eval 99.4%

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

      3. *-lft-identity 99.4%

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

   9. Simplified 99.4%

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

       1. div-inv 99.3%

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

       2. pow1/2 99.3%

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

       3. pow-flip 99.4%

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

       4. metadata-eval 99.4%

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

       5. metadata-eval 99.4%

          \[\leadsto \left(\frac{1}{\sqrt{1 + x} + \sqrt{x}} \cdot {x}^{-0.5}\right) \cdot {\left(1 + x\right)}^{\color{blue}{\left(-1.5 \cdot 0.3333333333333333\right)}} \]

       6. pow-pow 95.2%

          \[\leadsto \left(\frac{1}{\sqrt{1 + x} + \sqrt{x}} \cdot {x}^{-0.5}\right) \cdot \color{blue}{{\left({\left(1 + x\right)}^{-1.5}\right)}^{0.3333333333333333}} \]

       7. pow1/3 99.2%

          \[\leadsto \left(\frac{1}{\sqrt{1 + x} + \sqrt{x}} \cdot {x}^{-0.5}\right) \cdot \color{blue}{\sqrt[3]{{\left(1 + x\right)}^{-1.5}}} \]

       8. metadata-eval 99.2%

          \[\leadsto \left(\frac{1}{\sqrt{1 + x} + \sqrt{x}} \cdot {x}^{\color{blue}{\left(-0.5\right)}}\right) \cdot \sqrt[3]{{\left(1 + x\right)}^{-1.5}} \]

       9. pow-flip 99.1%

          \[\leadsto \left(\frac{1}{\sqrt{1 + x} + \sqrt{x}} \cdot \color{blue}{\frac{1}{{x}^{0.5}}}\right) \cdot \sqrt[3]{{\left(1 + x\right)}^{-1.5}} \]

       10. pow1/2 99.1%

           \[\leadsto \left(\frac{1}{\sqrt{1 + x} + \sqrt{x}} \cdot \frac{1}{\color{blue}{\sqrt{x}}}\right) \cdot \sqrt[3]{{\left(1 + x\right)}^{-1.5}} \]

       11. *-commutative 99.1%

           \[\leadsto \color{blue}{\left(\frac{1}{\sqrt{x}} \cdot \frac{1}{\sqrt{1 + x} + \sqrt{x}}\right)} \cdot \sqrt[3]{{\left(1 + x\right)}^{-1.5}} \]

       12. associate-*l* 99.0%

           \[\leadsto \color{blue}{\frac{1}{\sqrt{x}} \cdot \left(\frac{1}{\sqrt{1 + x} + \sqrt{x}} \cdot \sqrt[3]{{\left(1 + x\right)}^{-1.5}}\right)} \]

       13. pow1/2 99.0%

           \[\leadsto \frac{1}{\color{blue}{{x}^{0.5}}} \cdot \left(\frac{1}{\sqrt{1 + x} + \sqrt{x}} \cdot \sqrt[3]{{\left(1 + x\right)}^{-1.5}}\right) \]

       14. pow-flip 99.2%

           \[\leadsto \color{blue}{{x}^{\left(-0.5\right)}} \cdot \left(\frac{1}{\sqrt{1 + x} + \sqrt{x}} \cdot \sqrt[3]{{\left(1 + x\right)}^{-1.5}}\right) \]

       15. metadata-eval 99.2%

           \[\leadsto {x}^{\color{blue}{-0.5}} \cdot \left(\frac{1}{\sqrt{1 + x} + \sqrt{x}} \cdot \sqrt[3]{{\left(1 + x\right)}^{-1.5}}\right) \]

       16. pow1/3 95.2%

           \[\leadsto {x}^{-0.5} \cdot \left(\frac{1}{\sqrt{1 + x} + \sqrt{x}} \cdot \color{blue}{{\left({\left(1 + x\right)}^{-1.5}\right)}^{0.3333333333333333}}\right) \]

       17. pow-pow 99.4%

           \[\leadsto {x}^{-0.5} \cdot \left(\frac{1}{\sqrt{1 + x} + \sqrt{x}} \cdot \color{blue}{{\left(1 + x\right)}^{\left(-1.5 \cdot 0.3333333333333333\right)}}\right) \]

       18. metadata-eval 99.4%

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

   11. Applied egg-rr 99.4%

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

   12. Taylor expanded in x around inf 97.3%

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

4. Final simplification 97.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.7:\\ \;\;\;\;{x}^{-0.5} + -1\\ \mathbf{else}:\\ \;\;\;\;{x}^{-0.5} \cdot \frac{0.5}{x}\\ \end{array} \]

Alternative 15: 52.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.6:\\ \;\;\;\;{x}^{-0.5} + -1\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x} - \frac{0.5}{x \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 0.6) (+ (pow x -0.5) -1.0) (- (/ 1.0 x) (/ 0.5 (* x x)))))

C

double code(double x) {
	double tmp;
	if (x <= 0.6) {
		tmp = pow(x, -0.5) + -1.0;
	} else {
		tmp = (1.0 / x) - (0.5 / (x * x));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 0.6d0) then
        tmp = (x ** (-0.5d0)) + (-1.0d0)
    else
        tmp = (1.0d0 / x) - (0.5d0 / (x * x))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 0.6) {
		tmp = Math.pow(x, -0.5) + -1.0;
	} else {
		tmp = (1.0 / x) - (0.5 / (x * x));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 0.6:
		tmp = math.pow(x, -0.5) + -1.0
	else:
		tmp = (1.0 / x) - (0.5 / (x * x))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 0.6)
		tmp = Float64((x ^ -0.5) + -1.0);
	else
		tmp = Float64(Float64(1.0 / x) - Float64(0.5 / Float64(x * x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 0.6)
		tmp = (x ^ -0.5) + -1.0;
	else
		tmp = (1.0 / x) - (0.5 / (x * x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 0.6], N[(N[Power[x, -0.5], $MachinePrecision] + -1.0), $MachinePrecision], N[(N[(1.0 / x), $MachinePrecision] - N[(0.5 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.599999999999999978

   1. Initial program 99.5%

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

      1. inv-pow 99.5%

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

      2. pow1/2 99.5%

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

      3. pow-pow 100.0%

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

      4. add-exp-log 92.8%

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

      5. pow-exp 92.8%

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

      6. metadata-eval 92.8%

         \[\leadsto e^{\log x \cdot \color{blue}{-0.5}} - \frac{1}{\sqrt{x + 1}} \]

   3. Applied egg-rr 92.8%

      \[\leadsto \color{blue}{e^{\log x \cdot -0.5}} - \frac{1}{\sqrt{x + 1}} \]

   4. Taylor expanded in x around 0 98.2%

      \[\leadsto \color{blue}{{x}^{-0.5} - 1} \]

   if 0.599999999999999978 < x

   1. Initial program 44.0%

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

      1. frac-sub 44.1%

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

      2. div-inv 44.1%

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

      3. *-un-lft-identity 44.1%

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

      4. +-commutative 44.1%

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

      5. *-rgt-identity 44.1%

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

      6. metadata-eval 44.1%

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

      7. frac-times 44.1%

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

      8. un-div-inv 44.1%

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

      9. pow1/2 44.1%

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

      10. pow-flip 44.1%

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

      11. metadata-eval 44.1%

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

      12. +-commutative 44.1%

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

   3. Applied egg-rr 44.1%

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

      1. associate-*r/ 44.1%

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

   5. Simplified 44.1%

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

   6. Taylor expanded in x around 0 7.9%

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

   7. Taylor expanded in x around inf 7.9%

      \[\leadsto \color{blue}{\frac{1}{x} - 0.5 \cdot \frac{1}{{x}^{2}}} \]
   8. Step-by-step derivation

      1. associate-*r/ 7.9%

         \[\leadsto \frac{1}{x} - \color{blue}{\frac{0.5 \cdot 1}{{x}^{2}}} \]

      2. metadata-eval 7.9%

         \[\leadsto \frac{1}{x} - \frac{\color{blue}{0.5}}{{x}^{2}} \]

      3. unpow2 7.9%

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

   9. Simplified 7.9%

      \[\leadsto \color{blue}{\frac{1}{x} - \frac{0.5}{x \cdot x}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 51.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.6:\\ \;\;\;\;{x}^{-0.5} + -1\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x} - \frac{0.5}{x \cdot x}\\ \end{array} \]

Alternative 16: 50.4% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

function code(x)
	return x ^ -0.5
end

MATLAB

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

Wolfram

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

Derivation

1. Initial program 70.9%

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

   1. inv-pow 70.9%

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

   2. pow1/2 70.9%

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

   3. pow-pow 65.8%

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

   4. add-exp-log 48.8%

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

   5. pow-exp 48.8%

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

   6. metadata-eval 48.8%

      \[\leadsto e^{\log x \cdot \color{blue}{-0.5}} - \frac{1}{\sqrt{x + 1}} \]

3. Applied egg-rr 48.8%

   \[\leadsto \color{blue}{e^{\log x \cdot -0.5}} - \frac{1}{\sqrt{x + 1}} \]

4. Taylor expanded in x around inf 48.6%

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

   1. unpow-1 48.6%

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

   2. metadata-eval 48.6%

      \[\leadsto \sqrt{{x}^{\color{blue}{\left(2 \cdot -0.5\right)}}} \]

   3. pow-sqr 48.6%

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

   4. rem-sqrt-square 48.6%

      \[\leadsto \color{blue}{\left|{x}^{-0.5}\right|} \]

   5. unpow1 48.6%

      \[\leadsto \left|\color{blue}{{\left({x}^{-0.5}\right)}^{1}}\right| \]

   6. sqr-pow 48.3%

      \[\leadsto \left|\color{blue}{{\left({x}^{-0.5}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left({x}^{-0.5}\right)}^{\left(\frac{1}{2}\right)}}\right| \]

   7. fabs-sqr 48.3%

      \[\leadsto \color{blue}{{\left({x}^{-0.5}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left({x}^{-0.5}\right)}^{\left(\frac{1}{2}\right)}} \]

   8. sqr-pow 48.6%

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

   9. unpow1 48.6%

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

6. Simplified 48.6%

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

7. Final simplification 48.6%

   \[\leadsto {x}^{-0.5} \]

Alternative 17: 7.4% accurate, 69.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

public static double code(double x) {
	return 1.0 / x;
}

Python

def code(x):
	return 1.0 / x

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 70.9%

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

   1. frac-sub 70.9%

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

   2. div-inv 70.9%

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

   3. *-un-lft-identity 70.9%

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

   4. +-commutative 70.9%

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

   5. *-rgt-identity 70.9%

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

   6. metadata-eval 70.9%

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

   7. frac-times 70.9%

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

   8. un-div-inv 70.9%

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

   9. pow1/2 71.0%

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

   10. pow-flip 71.2%

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

   11. metadata-eval 71.2%

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

   12. +-commutative 71.2%

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

3. Applied egg-rr 71.2%

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

   1. associate-*r/ 71.2%

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

5. Simplified 71.2%

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

6. Taylor expanded in x around 0 49.8%

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

7. Taylor expanded in x around inf 7.6%

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

8. Final simplification 7.6%

   \[\leadsto \frac{1}{x} \]

Alternative 18: 1.9% accurate, 209.0× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 -1.0)

C

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

Fortran

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

Java

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

Python

def code(x):
	return -1.0

Julia

function code(x)
	return -1.0
end

MATLAB

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

Wolfram

code[x_] := -1.0

Derivation

1. Initial program 70.9%

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

   1. add-cube-cbrt 52.8%

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

   2. associate-*l* 52.8%

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

   3. frac-2neg 52.8%

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

   4. metadata-eval 52.8%

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

   5. div-inv 52.8%

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

   6. metadata-eval 52.8%

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

   7. distribute-neg-frac 52.8%

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

   8. metadata-eval 52.8%

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

   9. frac-2neg 52.8%

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

   10. prod-diff 51.2%

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

3. Applied egg-rr 51.6%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\sqrt[3]{{x}^{-0.5}}, \sqrt[3]{\frac{1}{x}}, -\frac{-1}{\sqrt{1 + x}} \cdot -1\right) + \mathsf{fma}\left(-\frac{-1}{\sqrt{1 + x}}, -1, \frac{-1}{\sqrt{1 + x}} \cdot -1\right)} \]
4. Step-by-step derivation

   1. fma-udef 58.1%

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

   2. distribute-rgt-neg-in 58.1%

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

   3. metadata-eval 58.1%

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

   4. *-rgt-identity 58.1%

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

   5. *-commutative 58.1%

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

   6. associate-+l+ 58.1%

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

   7. *-commutative 58.1%

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

   8. fma-udef 58.1%

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

5. Simplified 51.6%

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

6. Taylor expanded in x around 0 2.0%

   \[\leadsto \color{blue}{-1} \]

7. Final simplification 2.0%

   \[\leadsto -1 \]

Developer target: 99.0% 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]

Reproduce

herbie shell --seed 2023217 
(FPCore (x)
  :name "2isqrt (example 3.6)"
  :precision binary64

  :herbie-target
  (/ 1.0 (+ (* (+ x 1.0) (sqrt x)) (* x (sqrt (+ x 1.0)))))

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