2isqrt (example 3.6)

Average Accuracy: 68.6% → 99.7%

Time: 13.1s

Precision: binary64

Cost: 28484

Math

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

↓

\[\begin{array}{l} t_0 := {\left(\frac{-1}{\frac{-1}{x}}\right)}^{-0.5}\\ \mathbf{if}\;\frac{1}{\sqrt{x}} + \frac{-1}{\sqrt{1 + x}} \leq 5 \cdot 10^{-8}:\\ \;\;\;\;\frac{t_0}{x} \cdot \left(0.5 + \frac{-0.375}{x}\right) + \frac{t_0}{x \cdot x} \cdot \left(\frac{0.3125}{x} + \frac{-0.2734375}{x \cdot x}\right)\\ \mathbf{else}:\\ \;\;\;\;{x}^{-0.5} - {\left(1 + x\right)}^{-0.5}\\ \end{array} \]

FPCore

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

↓

(FPCore (x)
 :precision binary64
 (let* ((t_0 (pow (/ -1.0 (/ -1.0 x)) -0.5)))
   (if (<= (+ (/ 1.0 (sqrt x)) (/ -1.0 (sqrt (+ 1.0 x)))) 5e-8)
     (+
      (* (/ t_0 x) (+ 0.5 (/ -0.375 x)))
      (* (/ t_0 (* x x)) (+ (/ 0.3125 x) (/ -0.2734375 (* x x)))))
     (- (pow x -0.5) (pow (+ 1.0 x) -0.5)))))

C

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

↓

double code(double x) {
	double t_0 = pow((-1.0 / (-1.0 / x)), -0.5);
	double tmp;
	if (((1.0 / sqrt(x)) + (-1.0 / sqrt((1.0 + x)))) <= 5e-8) {
		tmp = ((t_0 / x) * (0.5 + (-0.375 / x))) + ((t_0 / (x * x)) * ((0.3125 / x) + (-0.2734375 / (x * x))));
	} else {
		tmp = pow(x, -0.5) - pow((1.0 + x), -0.5);
	}
	return tmp;
}

Fortran

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

↓

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = ((-1.0d0) / ((-1.0d0) / x)) ** (-0.5d0)
    if (((1.0d0 / sqrt(x)) + ((-1.0d0) / sqrt((1.0d0 + x)))) <= 5d-8) then
        tmp = ((t_0 / x) * (0.5d0 + ((-0.375d0) / x))) + ((t_0 / (x * x)) * ((0.3125d0 / x) + ((-0.2734375d0) / (x * x))))
    else
        tmp = (x ** (-0.5d0)) - ((1.0d0 + x) ** (-0.5d0))
    end if
    code = tmp
end function

Java

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

↓

public static double code(double x) {
	double t_0 = Math.pow((-1.0 / (-1.0 / x)), -0.5);
	double tmp;
	if (((1.0 / Math.sqrt(x)) + (-1.0 / Math.sqrt((1.0 + x)))) <= 5e-8) {
		tmp = ((t_0 / x) * (0.5 + (-0.375 / x))) + ((t_0 / (x * x)) * ((0.3125 / x) + (-0.2734375 / (x * x))));
	} else {
		tmp = Math.pow(x, -0.5) - Math.pow((1.0 + x), -0.5);
	}
	return tmp;
}

Python

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

↓

def code(x):
	t_0 = math.pow((-1.0 / (-1.0 / x)), -0.5)
	tmp = 0
	if ((1.0 / math.sqrt(x)) + (-1.0 / math.sqrt((1.0 + x)))) <= 5e-8:
		tmp = ((t_0 / x) * (0.5 + (-0.375 / x))) + ((t_0 / (x * x)) * ((0.3125 / x) + (-0.2734375 / (x * x))))
	else:
		tmp = math.pow(x, -0.5) - math.pow((1.0 + x), -0.5)
	return tmp

Julia

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

↓

function code(x)
	t_0 = Float64(-1.0 / Float64(-1.0 / x)) ^ -0.5
	tmp = 0.0
	if (Float64(Float64(1.0 / sqrt(x)) + Float64(-1.0 / sqrt(Float64(1.0 + x)))) <= 5e-8)
		tmp = Float64(Float64(Float64(t_0 / x) * Float64(0.5 + Float64(-0.375 / x))) + Float64(Float64(t_0 / Float64(x * x)) * Float64(Float64(0.3125 / x) + Float64(-0.2734375 / Float64(x * x)))));
	else
		tmp = Float64((x ^ -0.5) - (Float64(1.0 + x) ^ -0.5));
	end
	return tmp
end

MATLAB

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

↓

function tmp_2 = code(x)
	t_0 = (-1.0 / (-1.0 / x)) ^ -0.5;
	tmp = 0.0;
	if (((1.0 / sqrt(x)) + (-1.0 / sqrt((1.0 + x)))) <= 5e-8)
		tmp = ((t_0 / x) * (0.5 + (-0.375 / x))) + ((t_0 / (x * x)) * ((0.3125 / x) + (-0.2734375 / (x * x))));
	else
		tmp = (x ^ -0.5) - ((1.0 + x) ^ -0.5);
	end
	tmp_2 = tmp;
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]

↓

code[x_] := Block[{t$95$0 = N[Power[N[(-1.0 / N[(-1.0 / x), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision]}, If[LessEqual[N[(N[(1.0 / N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + N[(-1.0 / N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 5e-8], N[(N[(N[(t$95$0 / x), $MachinePrecision] * N[(0.5 + N[(-0.375 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(t$95$0 / N[(x * x), $MachinePrecision]), $MachinePrecision] * N[(N[(0.3125 / x), $MachinePrecision] + N[(-0.2734375 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Power[x, -0.5], $MachinePrecision] - N[Power[N[(1.0 + x), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]]]

Target

Original	68.6%
Target	98.9%
Herbie	99.7%

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

Derivation

1. Split input into 2 regimes

2. if (-.f64 (/.f64 1 (sqrt.f64 x)) (/.f64 1 (sqrt.f64 (+.f64 x 1)))) < 4.9999999999999998e-8

   1. Initial program 37.9%

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

   2. Applied egg-rr 6.1%

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

      [Start] 37.9	\[ \frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}} \]
      inv-pow [=>] 37.9	\[ \color{blue}{{\left(\sqrt{x}\right)}^{-1}} - \frac{1}{\sqrt{x + 1}} \]
      pow1/2 [=>] 37.9	\[ {\color{blue}{\left({x}^{0.5}\right)}}^{-1} - \frac{1}{\sqrt{x + 1}} \]
      pow-pow [=>] 29.3	\[ \color{blue}{{x}^{\left(0.5 \cdot -1\right)}} - \frac{1}{\sqrt{x + 1}} \]
      add-exp-log [=>] 6.2	\[ {\color{blue}{\left(e^{\log x}\right)}}^{\left(0.5 \cdot -1\right)} - \frac{1}{\sqrt{x + 1}} \]
      pow-exp [=>] 6.1	\[ \color{blue}{e^{\log x \cdot \left(0.5 \cdot -1\right)}} - \frac{1}{\sqrt{x + 1}} \]
      metadata-eval [=>] 6.1	\[ e^{\log x \cdot \color{blue}{-0.5}} - \frac{1}{\sqrt{x + 1}} \]

   3. Applied egg-rr 37.9%

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

      [Start] 6.1	\[ e^{\log x \cdot -0.5} - \frac{1}{\sqrt{x + 1}} \]
      add-sqr-sqrt [=>] 6.2	\[ \color{blue}{\sqrt{e^{\log x \cdot -0.5}} \cdot \sqrt{e^{\log x \cdot -0.5}}} - \frac{1}{\sqrt{x + 1}} \]
      clear-num [=>] 6.2	\[ \sqrt{e^{\log x \cdot -0.5}} \cdot \sqrt{e^{\log x \cdot -0.5}} - \color{blue}{\frac{1}{\frac{\sqrt{x + 1}}{1}}} \]
      associate-/r/ [=>] 6.2	\[ \sqrt{e^{\log x \cdot -0.5}} \cdot \sqrt{e^{\log x \cdot -0.5}} - \color{blue}{\frac{1}{\sqrt{x + 1}} \cdot 1} \]
      prod-diff [=>] 6.1	\[ \color{blue}{\mathsf{fma}\left(\sqrt{e^{\log x \cdot -0.5}}, \sqrt{e^{\log x \cdot -0.5}}, -1 \cdot \frac{1}{\sqrt{x + 1}}\right) + \mathsf{fma}\left(-1, \frac{1}{\sqrt{x + 1}}, 1 \cdot \frac{1}{\sqrt{x + 1}}\right)} \]
      *-un-lft-identity [<=] 6.1	\[ \mathsf{fma}\left(\sqrt{e^{\log x \cdot -0.5}}, \sqrt{e^{\log x \cdot -0.5}}, -\color{blue}{\frac{1}{\sqrt{x + 1}}}\right) + \mathsf{fma}\left(-1, \frac{1}{\sqrt{x + 1}}, 1 \cdot \frac{1}{\sqrt{x + 1}}\right) \]
      fma-neg [<=] 6.2	\[ \color{blue}{\left(\sqrt{e^{\log x \cdot -0.5}} \cdot \sqrt{e^{\log x \cdot -0.5}} - \frac{1}{\sqrt{x + 1}}\right)} + \mathsf{fma}\left(-1, \frac{1}{\sqrt{x + 1}}, 1 \cdot \frac{1}{\sqrt{x + 1}}\right) \]
      add-sqr-sqrt [<=] 6.1	\[ \left(\color{blue}{e^{\log x \cdot -0.5}} - \frac{1}{\sqrt{x + 1}}\right) + \mathsf{fma}\left(-1, \frac{1}{\sqrt{x + 1}}, 1 \cdot \frac{1}{\sqrt{x + 1}}\right) \]
      exp-to-pow [=>] 29.3	\[ \left(\color{blue}{{x}^{-0.5}} - \frac{1}{\sqrt{x + 1}}\right) + \mathsf{fma}\left(-1, \frac{1}{\sqrt{x + 1}}, 1 \cdot \frac{1}{\sqrt{x + 1}}\right) \]
      pow1/2 [=>] 29.3	\[ \left({x}^{-0.5} - \frac{1}{\color{blue}{{\left(x + 1\right)}^{0.5}}}\right) + \mathsf{fma}\left(-1, \frac{1}{\sqrt{x + 1}}, 1 \cdot \frac{1}{\sqrt{x + 1}}\right) \]
      pow-flip [=>] 37.9	\[ \left({x}^{-0.5} - \color{blue}{{\left(x + 1\right)}^{\left(-0.5\right)}}\right) + \mathsf{fma}\left(-1, \frac{1}{\sqrt{x + 1}}, 1 \cdot \frac{1}{\sqrt{x + 1}}\right) \]
      metadata-eval [=>] 37.9	\[ \left({x}^{-0.5} - {\left(x + 1\right)}^{\color{blue}{-0.5}}\right) + \mathsf{fma}\left(-1, \frac{1}{\sqrt{x + 1}}, 1 \cdot \frac{1}{\sqrt{x + 1}}\right) \]

   4. Simplified 37.9%

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

      [Start] 37.9	\[ \left({x}^{-0.5} - {\left(x + 1\right)}^{-0.5}\right) + \mathsf{fma}\left(-1, {\left(x + 1\right)}^{-0.5}, {\left(x + 1\right)}^{-0.5}\right) \]
      fma-udef [=>] 37.9	\[ \left({x}^{-0.5} - {\left(x + 1\right)}^{-0.5}\right) + \color{blue}{\left(-1 \cdot {\left(x + 1\right)}^{-0.5} + {\left(x + 1\right)}^{-0.5}\right)} \]
      neg-mul-1 [<=] 37.9	\[ \left({x}^{-0.5} - {\left(x + 1\right)}^{-0.5}\right) + \left(\color{blue}{\left(-{\left(x + 1\right)}^{-0.5}\right)} + {\left(x + 1\right)}^{-0.5}\right) \]
      rem-log-exp [<=] 6.0	\[ \left({x}^{-0.5} - {\left(x + 1\right)}^{-0.5}\right) + \left(\color{blue}{\log \left(e^{-{\left(x + 1\right)}^{-0.5}}\right)} + {\left(x + 1\right)}^{-0.5}\right) \]
      rec-exp [<=] 6.0	\[ \left({x}^{-0.5} - {\left(x + 1\right)}^{-0.5}\right) + \left(\log \color{blue}{\left(\frac{1}{e^{{\left(x + 1\right)}^{-0.5}}}\right)} + {\left(x + 1\right)}^{-0.5}\right) \]
      +-commutative [=>] 6.0	\[ \left({x}^{-0.5} - {\left(x + 1\right)}^{-0.5}\right) + \color{blue}{\left({\left(x + 1\right)}^{-0.5} + \log \left(\frac{1}{e^{{\left(x + 1\right)}^{-0.5}}}\right)\right)} \]
      rec-exp [=>] 6.0	\[ \left({x}^{-0.5} - {\left(x + 1\right)}^{-0.5}\right) + \left({\left(x + 1\right)}^{-0.5} + \log \color{blue}{\left(e^{-{\left(x + 1\right)}^{-0.5}}\right)}\right) \]
      rem-log-exp [=>] 37.9	\[ \left({x}^{-0.5} - {\left(x + 1\right)}^{-0.5}\right) + \left({\left(x + 1\right)}^{-0.5} + \color{blue}{\left(-{\left(x + 1\right)}^{-0.5}\right)}\right) \]
      sub-neg [<=] 37.9	\[ \left({x}^{-0.5} - {\left(x + 1\right)}^{-0.5}\right) + \color{blue}{\left({\left(x + 1\right)}^{-0.5} - {\left(x + 1\right)}^{-0.5}\right)} \]
      +-inverses [=>] 37.9	\[ \left({x}^{-0.5} - {\left(x + 1\right)}^{-0.5}\right) + \color{blue}{0} \]
      +-rgt-identity [=>] 37.9	\[ \color{blue}{{x}^{-0.5} - {\left(x + 1\right)}^{-0.5}} \]

   5. Taylor expanded in x around -inf 0.0%

      \[\leadsto \color{blue}{-0.2734375 \cdot \frac{e^{-0.5 \cdot \left(-1 \cdot \log \left(\frac{-1}{x}\right) + \log -1\right)}}{{x}^{4}} + \left(0.3125 \cdot \frac{e^{-0.5 \cdot \left(-1 \cdot \log \left(\frac{-1}{x}\right) + \log -1\right)}}{{x}^{3}} + \left(0.5 \cdot \frac{e^{-0.5 \cdot \left(-1 \cdot \log \left(\frac{-1}{x}\right) + \log -1\right)}}{x} + -0.375 \cdot \frac{e^{-0.5 \cdot \left(-1 \cdot \log \left(\frac{-1}{x}\right) + \log -1\right)}}{{x}^{2}}\right)\right)} \]

   6. Simplified 99.7%

      \[\leadsto \color{blue}{\frac{{\left(\frac{-1}{\frac{-1}{x}}\right)}^{-0.5}}{x} \cdot \left(0.5 + \frac{-0.375}{x}\right) + \frac{{\left(\frac{-1}{\frac{-1}{x}}\right)}^{-0.5}}{x \cdot x} \cdot \left(\frac{0.3125}{x} + \frac{-0.2734375}{x \cdot x}\right)} \]
      Proof

      [Start] 0.0	\[ -0.2734375 \cdot \frac{e^{-0.5 \cdot \left(-1 \cdot \log \left(\frac{-1}{x}\right) + \log -1\right)}}{{x}^{4}} + \left(0.3125 \cdot \frac{e^{-0.5 \cdot \left(-1 \cdot \log \left(\frac{-1}{x}\right) + \log -1\right)}}{{x}^{3}} + \left(0.5 \cdot \frac{e^{-0.5 \cdot \left(-1 \cdot \log \left(\frac{-1}{x}\right) + \log -1\right)}}{x} + -0.375 \cdot \frac{e^{-0.5 \cdot \left(-1 \cdot \log \left(\frac{-1}{x}\right) + \log -1\right)}}{{x}^{2}}\right)\right) \]
      +-commutative [=>] 0.0	\[ \color{blue}{\left(0.3125 \cdot \frac{e^{-0.5 \cdot \left(-1 \cdot \log \left(\frac{-1}{x}\right) + \log -1\right)}}{{x}^{3}} + \left(0.5 \cdot \frac{e^{-0.5 \cdot \left(-1 \cdot \log \left(\frac{-1}{x}\right) + \log -1\right)}}{x} + -0.375 \cdot \frac{e^{-0.5 \cdot \left(-1 \cdot \log \left(\frac{-1}{x}\right) + \log -1\right)}}{{x}^{2}}\right)\right) + -0.2734375 \cdot \frac{e^{-0.5 \cdot \left(-1 \cdot \log \left(\frac{-1}{x}\right) + \log -1\right)}}{{x}^{4}}} \]
      +-commutative [=>] 0.0	\[ \color{blue}{\left(\left(0.5 \cdot \frac{e^{-0.5 \cdot \left(-1 \cdot \log \left(\frac{-1}{x}\right) + \log -1\right)}}{x} + -0.375 \cdot \frac{e^{-0.5 \cdot \left(-1 \cdot \log \left(\frac{-1}{x}\right) + \log -1\right)}}{{x}^{2}}\right) + 0.3125 \cdot \frac{e^{-0.5 \cdot \left(-1 \cdot \log \left(\frac{-1}{x}\right) + \log -1\right)}}{{x}^{3}}\right)} + -0.2734375 \cdot \frac{e^{-0.5 \cdot \left(-1 \cdot \log \left(\frac{-1}{x}\right) + \log -1\right)}}{{x}^{4}} \]
      associate-+l+ [=>] 0.0	\[ \color{blue}{\left(0.5 \cdot \frac{e^{-0.5 \cdot \left(-1 \cdot \log \left(\frac{-1}{x}\right) + \log -1\right)}}{x} + -0.375 \cdot \frac{e^{-0.5 \cdot \left(-1 \cdot \log \left(\frac{-1}{x}\right) + \log -1\right)}}{{x}^{2}}\right) + \left(0.3125 \cdot \frac{e^{-0.5 \cdot \left(-1 \cdot \log \left(\frac{-1}{x}\right) + \log -1\right)}}{{x}^{3}} + -0.2734375 \cdot \frac{e^{-0.5 \cdot \left(-1 \cdot \log \left(\frac{-1}{x}\right) + \log -1\right)}}{{x}^{4}}\right)} \]

   if 4.9999999999999998e-8 < (-.f64 (/.f64 1 (sqrt.f64 x)) (/.f64 1 (sqrt.f64 (+.f64 x 1))))

   1. Initial program 99.4%

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

   2. Applied egg-rr 99.8%

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

      [Start] 99.4	\[ \frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}} \]
      *-un-lft-identity [=>] 99.4	\[ \color{blue}{1 \cdot \frac{1}{\sqrt{x}}} - \frac{1}{\sqrt{x + 1}} \]
      clear-num [=>] 99.4	\[ 1 \cdot \frac{1}{\sqrt{x}} - \color{blue}{\frac{1}{\frac{\sqrt{x + 1}}{1}}} \]
      associate-/r/ [=>] 99.4	\[ 1 \cdot \frac{1}{\sqrt{x}} - \color{blue}{\frac{1}{\sqrt{x + 1}} \cdot 1} \]
      prod-diff [=>] 99.4	\[ \color{blue}{\mathsf{fma}\left(1, \frac{1}{\sqrt{x}}, -1 \cdot \frac{1}{\sqrt{x + 1}}\right) + \mathsf{fma}\left(-1, \frac{1}{\sqrt{x + 1}}, 1 \cdot \frac{1}{\sqrt{x + 1}}\right)} \]
      *-un-lft-identity [<=] 99.4	\[ \mathsf{fma}\left(1, \frac{1}{\sqrt{x}}, -\color{blue}{\frac{1}{\sqrt{x + 1}}}\right) + \mathsf{fma}\left(-1, \frac{1}{\sqrt{x + 1}}, 1 \cdot \frac{1}{\sqrt{x + 1}}\right) \]
      fma-neg [<=] 99.4	\[ \color{blue}{\left(1 \cdot \frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}}\right)} + \mathsf{fma}\left(-1, \frac{1}{\sqrt{x + 1}}, 1 \cdot \frac{1}{\sqrt{x + 1}}\right) \]
      *-un-lft-identity [<=] 99.4	\[ \left(\color{blue}{\frac{1}{\sqrt{x}}} - \frac{1}{\sqrt{x + 1}}\right) + \mathsf{fma}\left(-1, \frac{1}{\sqrt{x + 1}}, 1 \cdot \frac{1}{\sqrt{x + 1}}\right) \]
      inv-pow [=>] 99.4	\[ \left(\color{blue}{{\left(\sqrt{x}\right)}^{-1}} - \frac{1}{\sqrt{x + 1}}\right) + \mathsf{fma}\left(-1, \frac{1}{\sqrt{x + 1}}, 1 \cdot \frac{1}{\sqrt{x + 1}}\right) \]
      sqrt-pow2 [=>] 99.8	\[ \left(\color{blue}{{x}^{\left(\frac{-1}{2}\right)}} - \frac{1}{\sqrt{x + 1}}\right) + \mathsf{fma}\left(-1, \frac{1}{\sqrt{x + 1}}, 1 \cdot \frac{1}{\sqrt{x + 1}}\right) \]
      metadata-eval [=>] 99.8	\[ \left({x}^{\color{blue}{-0.5}} - \frac{1}{\sqrt{x + 1}}\right) + \mathsf{fma}\left(-1, \frac{1}{\sqrt{x + 1}}, 1 \cdot \frac{1}{\sqrt{x + 1}}\right) \]
      pow1/2 [=>] 99.8	\[ \left({x}^{-0.5} - \frac{1}{\color{blue}{{\left(x + 1\right)}^{0.5}}}\right) + \mathsf{fma}\left(-1, \frac{1}{\sqrt{x + 1}}, 1 \cdot \frac{1}{\sqrt{x + 1}}\right) \]
      pow-flip [=>] 99.8	\[ \left({x}^{-0.5} - \color{blue}{{\left(x + 1\right)}^{\left(-0.5\right)}}\right) + \mathsf{fma}\left(-1, \frac{1}{\sqrt{x + 1}}, 1 \cdot \frac{1}{\sqrt{x + 1}}\right) \]
      +-commutative [=>] 99.8	\[ \left({x}^{-0.5} - {\color{blue}{\left(1 + x\right)}}^{\left(-0.5\right)}\right) + \mathsf{fma}\left(-1, \frac{1}{\sqrt{x + 1}}, 1 \cdot \frac{1}{\sqrt{x + 1}}\right) \]
      metadata-eval [=>] 99.8	\[ \left({x}^{-0.5} - {\left(1 + x\right)}^{\color{blue}{-0.5}}\right) + \mathsf{fma}\left(-1, \frac{1}{\sqrt{x + 1}}, 1 \cdot \frac{1}{\sqrt{x + 1}}\right) \]

   3. Simplified 99.8%

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

      [Start] 99.8	\[ \left({x}^{-0.5} - {\left(1 + x\right)}^{-0.5}\right) + \mathsf{fma}\left(-1, {\left(1 + x\right)}^{-0.5}, {\left(1 + x\right)}^{-0.5}\right) \]
      fma-udef [=>] 99.8	\[ \left({x}^{-0.5} - {\left(1 + x\right)}^{-0.5}\right) + \color{blue}{\left(-1 \cdot {\left(1 + x\right)}^{-0.5} + {\left(1 + x\right)}^{-0.5}\right)} \]
      distribute-lft1-in [=>] 99.8	\[ \left({x}^{-0.5} - {\left(1 + x\right)}^{-0.5}\right) + \color{blue}{\left(-1 + 1\right) \cdot {\left(1 + x\right)}^{-0.5}} \]
      metadata-eval [=>] 99.8	\[ \left({x}^{-0.5} - {\left(1 + x\right)}^{-0.5}\right) + \color{blue}{0} \cdot {\left(1 + x\right)}^{-0.5} \]
      mul0-lft [=>] 99.8	\[ \left({x}^{-0.5} - {\left(1 + x\right)}^{-0.5}\right) + \color{blue}{0} \]
      +-rgt-identity [=>] 99.8	\[ \color{blue}{{x}^{-0.5} - {\left(1 + x\right)}^{-0.5}} \]

3. Recombined 2 regimes into one program.

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{\sqrt{x}} + \frac{-1}{\sqrt{1 + x}} \leq 5 \cdot 10^{-8}:\\ \;\;\;\;\frac{{\left(\frac{-1}{\frac{-1}{x}}\right)}^{-0.5}}{x} \cdot \left(0.5 + \frac{-0.375}{x}\right) + \frac{{\left(\frac{-1}{\frac{-1}{x}}\right)}^{-0.5}}{x \cdot x} \cdot \left(\frac{0.3125}{x} + \frac{-0.2734375}{x \cdot x}\right)\\ \mathbf{else}:\\ \;\;\;\;{x}^{-0.5} - {\left(1 + x\right)}^{-0.5}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	99.7%
Cost	27332

\[\begin{array}{l} \mathbf{if}\;\frac{1}{\sqrt{x}} + \frac{-1}{\sqrt{1 + x}} \leq 5 \cdot 10^{-8}:\\ \;\;\;\;{x}^{-0.5} \cdot \left(\frac{0.5}{x} + \left(\frac{0.3125}{{x}^{3}} - \frac{0.375}{x \cdot x}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{x}^{-0.5} - {\left(1 + x\right)}^{-0.5}\\ \end{array} \]

Alternative 2
Accuracy	99.6%
Cost	27268

\[\begin{array}{l} \mathbf{if}\;\frac{1}{\sqrt{x}} + \frac{-1}{\sqrt{1 + x}} \leq 10^{-11}:\\ \;\;\;\;\frac{{\left(\frac{-1}{\frac{-1}{x}}\right)}^{-0.5}}{x} \cdot \left(0.5 + \frac{-0.375}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;{x}^{-0.5} \cdot \left(1 - \sqrt{\frac{x}{1 - x \cdot x} \cdot \left(1 - x\right)}\right)\\ \end{array} \]

Alternative 3
Accuracy	99.6%
Cost	26884

\[\begin{array}{l} \mathbf{if}\;\frac{1}{\sqrt{x}} + \frac{-1}{\sqrt{1 + x}} \leq 10^{-11}:\\ \;\;\;\;\frac{{\left(\frac{-1}{\frac{-1}{x}}\right)}^{-0.5}}{x} \cdot \left(0.5 + \frac{-0.375}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;{x}^{-0.5} \cdot \left(1 - \sqrt{\frac{x}{1 + x}}\right)\\ \end{array} \]

Alternative 4
Accuracy	99.6%
Cost	26692

\[\begin{array}{l} \mathbf{if}\;\frac{1}{\sqrt{x}} + \frac{-1}{\sqrt{1 + x}} \leq 10^{-11}:\\ \;\;\;\;\frac{{\left(\frac{-1}{\frac{-1}{x}}\right)}^{-0.5}}{x} \cdot \left(0.5 + \frac{-0.375}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;{x}^{-0.5} - {\left(1 + x\right)}^{-0.5}\\ \end{array} \]

Alternative 5
Accuracy	98.9%
Cost	7428

\[\begin{array}{l} \mathbf{if}\;x \leq 1.1:\\ \;\;\;\;-1 + \left({x}^{-0.5} + x \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(\frac{-1}{\frac{-1}{x}}\right)}^{-0.5}}{x} \cdot \left(0.5 + \frac{-0.375}{x}\right)\\ \end{array} \]

Alternative 6
Accuracy	98.9%
Cost	7300

\[\begin{array}{l} \mathbf{if}\;x \leq 1.1:\\ \;\;\;\;-1 + \left({x}^{-0.5} + x \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;{x}^{-0.5} \cdot \left(\frac{0.5}{x} - \frac{0.375}{x \cdot x}\right)\\ \end{array} \]

Alternative 7
Accuracy	98.9%
Cost	7172

\[\begin{array}{l} \mathbf{if}\;x \leq 1.1:\\ \;\;\;\;-1 + \left({x}^{-0.5} + x \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;{x}^{-0.5} \cdot \frac{0.5 - \frac{0.375}{x}}{x}\\ \end{array} \]

Alternative 8
Accuracy	98.5%
Cost	7044

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

Alternative 9
Accuracy	97.1%
Cost	6788

\[\begin{array}{l} \mathbf{if}\;x \leq 0.5:\\ \;\;\;\;{x}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot {x}^{-1.5}\\ \end{array} \]

Alternative 10
Accuracy	98.3%
Cost	6788

\[\begin{array}{l} \mathbf{if}\;x \leq 0.66:\\ \;\;\;\;-1 + {x}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot {x}^{-1.5}\\ \end{array} \]

Alternative 11
Accuracy	50.2%
Cost	6528

\[{x}^{-0.5} \]

Alternative 12
Accuracy	2.4%
Cost	320

\[-1 + x \cdot 0.5 \]

Alternative 13
Accuracy	1.9%
Cost	64

\[-1 \]

Reproduce

herbie shell --seed 2023152 
(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)))))