2isqrt (example 3.6)

Average Error: 19.6 → 0.2

Time: 9.5s

Precision: binary64

Cost: 26692

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{1 + x}} \leq 4 \cdot 10^{-8}:\\ \;\;\;\;\frac{{x}^{-0.5}}{x} \cdot \left(0.5 + \left(\frac{0.3125}{x \cdot x} + \frac{-0.375}{x}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{x}^{-0.5} - {\left(1 + x\right)}^{-0.5}\\ \end{array} \]

FPCore

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

↓

(FPCore (x)
 :precision binary64
 (if (<= (- (/ 1.0 (sqrt x)) (/ 1.0 (sqrt (+ 1.0 x)))) 4e-8)
   (* (/ (pow x -0.5) x) (+ 0.5 (+ (/ 0.3125 (* x x)) (/ -0.375 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 tmp;
	if (((1.0 / sqrt(x)) - (1.0 / sqrt((1.0 + x)))) <= 4e-8) {
		tmp = (pow(x, -0.5) / x) * (0.5 + ((0.3125 / (x * x)) + (-0.375 / 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) :: tmp
    if (((1.0d0 / sqrt(x)) - (1.0d0 / sqrt((1.0d0 + x)))) <= 4d-8) then
        tmp = ((x ** (-0.5d0)) / x) * (0.5d0 + ((0.3125d0 / (x * x)) + ((-0.375d0) / 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 tmp;
	if (((1.0 / Math.sqrt(x)) - (1.0 / Math.sqrt((1.0 + x)))) <= 4e-8) {
		tmp = (Math.pow(x, -0.5) / x) * (0.5 + ((0.3125 / (x * x)) + (-0.375 / 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):
	tmp = 0
	if ((1.0 / math.sqrt(x)) - (1.0 / math.sqrt((1.0 + x)))) <= 4e-8:
		tmp = (math.pow(x, -0.5) / x) * (0.5 + ((0.3125 / (x * x)) + (-0.375 / 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)
	tmp = 0.0
	if (Float64(Float64(1.0 / sqrt(x)) - Float64(1.0 / sqrt(Float64(1.0 + x)))) <= 4e-8)
		tmp = Float64(Float64((x ^ -0.5) / x) * Float64(0.5 + Float64(Float64(0.3125 / Float64(x * x)) + Float64(-0.375 / 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)
	tmp = 0.0;
	if (((1.0 / sqrt(x)) - (1.0 / sqrt((1.0 + x)))) <= 4e-8)
		tmp = ((x ^ -0.5) / x) * (0.5 + ((0.3125 / (x * x)) + (-0.375 / 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_] := If[LessEqual[N[(N[(1.0 / N[Sqrt[x], $MachinePrecision]), $MachinePrecision] - N[(1.0 / N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 4e-8], N[(N[(N[Power[x, -0.5], $MachinePrecision] / x), $MachinePrecision] * N[(0.5 + N[(N[(0.3125 / N[(x * x), $MachinePrecision]), $MachinePrecision] + N[(-0.375 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Power[x, -0.5], $MachinePrecision] - N[Power[N[(1.0 + x), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]]

Target

Original	19.6
Target	0.6
Herbie	0.2

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

Derivation

1. Split input into 2 regimes

2. if (-.f64 (/.f64 1 (sqrt.f64 x)) (/.f64 1 (sqrt.f64 (+.f64 x 1)))) < 4.0000000000000001e-8

   1. Initial program 39.6

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

   2. Applied egg-rr 39.5

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

   3. Applied egg-rr 39.6

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

   4. Taylor expanded in x around -inf 64.0

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

   5. Simplified 0.2

      \[\leadsto \color{blue}{\frac{{x}^{-0.5}}{x} \cdot \left(0.5 + \left(\frac{0.3125}{x \cdot x} + \frac{-0.375}{x}\right)\right)} \]
      Proof
      (*.f64 (/.f64 (pow.f64 x -1/2) x) (+.f64 1/2 (+.f64 (/.f64 5/16 (*.f64 x x)) (/.f64 -3/8 x)))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 (Rewrite<= exp-to-pow_binary64 (exp.f64 (*.f64 (log.f64 x) -1/2))) x) (+.f64 1/2 (+.f64 (/.f64 5/16 (*.f64 x x)) (/.f64 -3/8 x)))): 99 points increase in error, 18 points decrease in error
      (*.f64 (/.f64 (exp.f64 (Rewrite<= *-commutative_binary64 (*.f64 -1/2 (log.f64 x)))) x) (+.f64 1/2 (+.f64 (/.f64 5/16 (*.f64 x x)) (/.f64 -3/8 x)))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 (Rewrite=> exp-prod_binary64 (pow.f64 (exp.f64 -1/2) (log.f64 x))) x) (+.f64 1/2 (+.f64 (/.f64 5/16 (*.f64 x x)) (/.f64 -3/8 x)))): 42 points increase in error, 51 points decrease in error
      (*.f64 (/.f64 (pow.f64 (exp.f64 -1/2) (Rewrite<= +-lft-identity_binary64 (+.f64 0 (log.f64 x)))) x) (+.f64 1/2 (+.f64 (/.f64 5/16 (*.f64 x x)) (/.f64 -3/8 x)))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 (pow.f64 (exp.f64 -1/2) (+.f64 (Rewrite<= +-inverses_binary64 (-.f64 (log.f64 -1) (log.f64 -1))) (log.f64 x))) x) (+.f64 1/2 (+.f64 (/.f64 5/16 (*.f64 x x)) (/.f64 -3/8 x)))): 169 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 (pow.f64 (exp.f64 -1/2) (Rewrite<= associate--r-_binary64 (-.f64 (log.f64 -1) (-.f64 (log.f64 -1) (log.f64 x))))) x) (+.f64 1/2 (+.f64 (/.f64 5/16 (*.f64 x x)) (/.f64 -3/8 x)))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 (pow.f64 (exp.f64 -1/2) (-.f64 (log.f64 -1) (Rewrite<= log-div_binary64 (log.f64 (/.f64 -1 x))))) x) (+.f64 1/2 (+.f64 (/.f64 5/16 (*.f64 x x)) (/.f64 -3/8 x)))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 (pow.f64 (exp.f64 -1/2) (Rewrite<= unsub-neg_binary64 (+.f64 (log.f64 -1) (neg.f64 (log.f64 (/.f64 -1 x)))))) x) (+.f64 1/2 (+.f64 (/.f64 5/16 (*.f64 x x)) (/.f64 -3/8 x)))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 (pow.f64 (exp.f64 -1/2) (+.f64 (log.f64 -1) (Rewrite<= mul-1-neg_binary64 (*.f64 -1 (log.f64 (/.f64 -1 x)))))) x) (+.f64 1/2 (+.f64 (/.f64 5/16 (*.f64 x x)) (/.f64 -3/8 x)))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 (pow.f64 (exp.f64 -1/2) (Rewrite<= +-commutative_binary64 (+.f64 (*.f64 -1 (log.f64 (/.f64 -1 x))) (log.f64 -1)))) x) (+.f64 1/2 (+.f64 (/.f64 5/16 (*.f64 x x)) (/.f64 -3/8 x)))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 (Rewrite<= exp-prod_binary64 (exp.f64 (*.f64 -1/2 (+.f64 (*.f64 -1 (log.f64 (/.f64 -1 x))) (log.f64 -1))))) x) (+.f64 1/2 (+.f64 (/.f64 5/16 (*.f64 x x)) (/.f64 -3/8 x)))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 (exp.f64 (*.f64 -1/2 (+.f64 (*.f64 -1 (log.f64 (/.f64 -1 x))) (log.f64 -1)))) x) (+.f64 1/2 (+.f64 (/.f64 5/16 (Rewrite<= unpow2_binary64 (pow.f64 x 2))) (/.f64 -3/8 x)))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= distribute-lft-out_binary64 (+.f64 (*.f64 (/.f64 (exp.f64 (*.f64 -1/2 (+.f64 (*.f64 -1 (log.f64 (/.f64 -1 x))) (log.f64 -1)))) x) 1/2) (*.f64 (/.f64 (exp.f64 (*.f64 -1/2 (+.f64 (*.f64 -1 (log.f64 (/.f64 -1 x))) (log.f64 -1)))) x) (+.f64 (/.f64 5/16 (pow.f64 x 2)) (/.f64 -3/8 x))))): 0 points increase in error, 0 points decrease in error
      (+.f64 (Rewrite<= *-commutative_binary64 (*.f64 1/2 (/.f64 (exp.f64 (*.f64 -1/2 (+.f64 (*.f64 -1 (log.f64 (/.f64 -1 x))) (log.f64 -1)))) x))) (*.f64 (/.f64 (exp.f64 (*.f64 -1/2 (+.f64 (*.f64 -1 (log.f64 (/.f64 -1 x))) (log.f64 -1)))) x) (+.f64 (/.f64 5/16 (pow.f64 x 2)) (/.f64 -3/8 x)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (*.f64 1/2 (/.f64 (exp.f64 (*.f64 -1/2 (+.f64 (*.f64 -1 (log.f64 (/.f64 -1 x))) (log.f64 -1)))) x)) (Rewrite<= distribute-rgt-out_binary64 (+.f64 (*.f64 (/.f64 5/16 (pow.f64 x 2)) (/.f64 (exp.f64 (*.f64 -1/2 (+.f64 (*.f64 -1 (log.f64 (/.f64 -1 x))) (log.f64 -1)))) x)) (*.f64 (/.f64 -3/8 x) (/.f64 (exp.f64 (*.f64 -1/2 (+.f64 (*.f64 -1 (log.f64 (/.f64 -1 x))) (log.f64 -1)))) x))))): 0 points increase in error, 0 points decrease in error
      (+.f64 (*.f64 1/2 (/.f64 (exp.f64 (*.f64 -1/2 (+.f64 (*.f64 -1 (log.f64 (/.f64 -1 x))) (log.f64 -1)))) x)) (+.f64 (Rewrite<= times-frac_binary64 (/.f64 (*.f64 5/16 (exp.f64 (*.f64 -1/2 (+.f64 (*.f64 -1 (log.f64 (/.f64 -1 x))) (log.f64 -1))))) (*.f64 (pow.f64 x 2) x))) (*.f64 (/.f64 -3/8 x) (/.f64 (exp.f64 (*.f64 -1/2 (+.f64 (*.f64 -1 (log.f64 (/.f64 -1 x))) (log.f64 -1)))) x)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (*.f64 1/2 (/.f64 (exp.f64 (*.f64 -1/2 (+.f64 (*.f64 -1 (log.f64 (/.f64 -1 x))) (log.f64 -1)))) x)) (+.f64 (/.f64 (*.f64 5/16 (exp.f64 (*.f64 -1/2 (+.f64 (*.f64 -1 (log.f64 (/.f64 -1 x))) (log.f64 -1))))) (*.f64 (Rewrite=> unpow2_binary64 (*.f64 x x)) x)) (*.f64 (/.f64 -3/8 x) (/.f64 (exp.f64 (*.f64 -1/2 (+.f64 (*.f64 -1 (log.f64 (/.f64 -1 x))) (log.f64 -1)))) x)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (*.f64 1/2 (/.f64 (exp.f64 (*.f64 -1/2 (+.f64 (*.f64 -1 (log.f64 (/.f64 -1 x))) (log.f64 -1)))) x)) (+.f64 (/.f64 (*.f64 5/16 (exp.f64 (*.f64 -1/2 (+.f64 (*.f64 -1 (log.f64 (/.f64 -1 x))) (log.f64 -1))))) (Rewrite<= unpow3_binary64 (pow.f64 x 3))) (*.f64 (/.f64 -3/8 x) (/.f64 (exp.f64 (*.f64 -1/2 (+.f64 (*.f64 -1 (log.f64 (/.f64 -1 x))) (log.f64 -1)))) x)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (*.f64 1/2 (/.f64 (exp.f64 (*.f64 -1/2 (+.f64 (*.f64 -1 (log.f64 (/.f64 -1 x))) (log.f64 -1)))) x)) (+.f64 (Rewrite<= associate-*r/_binary64 (*.f64 5/16 (/.f64 (exp.f64 (*.f64 -1/2 (+.f64 (*.f64 -1 (log.f64 (/.f64 -1 x))) (log.f64 -1)))) (pow.f64 x 3)))) (*.f64 (/.f64 -3/8 x) (/.f64 (exp.f64 (*.f64 -1/2 (+.f64 (*.f64 -1 (log.f64 (/.f64 -1 x))) (log.f64 -1)))) x)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (*.f64 1/2 (/.f64 (exp.f64 (*.f64 -1/2 (+.f64 (*.f64 -1 (log.f64 (/.f64 -1 x))) (log.f64 -1)))) x)) (+.f64 (*.f64 5/16 (/.f64 (exp.f64 (*.f64 -1/2 (+.f64 (*.f64 -1 (log.f64 (/.f64 -1 x))) (log.f64 -1)))) (pow.f64 x 3))) (Rewrite<= times-frac_binary64 (/.f64 (*.f64 -3/8 (exp.f64 (*.f64 -1/2 (+.f64 (*.f64 -1 (log.f64 (/.f64 -1 x))) (log.f64 -1))))) (*.f64 x x))))): 0 points increase in error, 0 points decrease in error
      (+.f64 (*.f64 1/2 (/.f64 (exp.f64 (*.f64 -1/2 (+.f64 (*.f64 -1 (log.f64 (/.f64 -1 x))) (log.f64 -1)))) x)) (+.f64 (*.f64 5/16 (/.f64 (exp.f64 (*.f64 -1/2 (+.f64 (*.f64 -1 (log.f64 (/.f64 -1 x))) (log.f64 -1)))) (pow.f64 x 3))) (/.f64 (*.f64 -3/8 (exp.f64 (*.f64 -1/2 (+.f64 (*.f64 -1 (log.f64 (/.f64 -1 x))) (log.f64 -1))))) (Rewrite<= unpow2_binary64 (pow.f64 x 2))))): 0 points increase in error, 0 points decrease in error
      (+.f64 (*.f64 1/2 (/.f64 (exp.f64 (*.f64 -1/2 (+.f64 (*.f64 -1 (log.f64 (/.f64 -1 x))) (log.f64 -1)))) x)) (+.f64 (*.f64 5/16 (/.f64 (exp.f64 (*.f64 -1/2 (+.f64 (*.f64 -1 (log.f64 (/.f64 -1 x))) (log.f64 -1)))) (pow.f64 x 3))) (Rewrite<= associate-*r/_binary64 (*.f64 -3/8 (/.f64 (exp.f64 (*.f64 -1/2 (+.f64 (*.f64 -1 (log.f64 (/.f64 -1 x))) (log.f64 -1)))) (pow.f64 x 2)))))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= associate-+l+_binary64 (+.f64 (+.f64 (*.f64 1/2 (/.f64 (exp.f64 (*.f64 -1/2 (+.f64 (*.f64 -1 (log.f64 (/.f64 -1 x))) (log.f64 -1)))) x)) (*.f64 5/16 (/.f64 (exp.f64 (*.f64 -1/2 (+.f64 (*.f64 -1 (log.f64 (/.f64 -1 x))) (log.f64 -1)))) (pow.f64 x 3)))) (*.f64 -3/8 (/.f64 (exp.f64 (*.f64 -1/2 (+.f64 (*.f64 -1 (log.f64 (/.f64 -1 x))) (log.f64 -1)))) (pow.f64 x 2))))): 0 points increase in error, 0 points decrease in error
      (+.f64 (Rewrite<= +-commutative_binary64 (+.f64 (*.f64 5/16 (/.f64 (exp.f64 (*.f64 -1/2 (+.f64 (*.f64 -1 (log.f64 (/.f64 -1 x))) (log.f64 -1)))) (pow.f64 x 3))) (*.f64 1/2 (/.f64 (exp.f64 (*.f64 -1/2 (+.f64 (*.f64 -1 (log.f64 (/.f64 -1 x))) (log.f64 -1)))) x)))) (*.f64 -3/8 (/.f64 (exp.f64 (*.f64 -1/2 (+.f64 (*.f64 -1 (log.f64 (/.f64 -1 x))) (log.f64 -1)))) (pow.f64 x 2)))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= associate-+r+_binary64 (+.f64 (*.f64 5/16 (/.f64 (exp.f64 (*.f64 -1/2 (+.f64 (*.f64 -1 (log.f64 (/.f64 -1 x))) (log.f64 -1)))) (pow.f64 x 3))) (+.f64 (*.f64 1/2 (/.f64 (exp.f64 (*.f64 -1/2 (+.f64 (*.f64 -1 (log.f64 (/.f64 -1 x))) (log.f64 -1)))) x)) (*.f64 -3/8 (/.f64 (exp.f64 (*.f64 -1/2 (+.f64 (*.f64 -1 (log.f64 (/.f64 -1 x))) (log.f64 -1)))) (pow.f64 x 2)))))): 0 points increase in error, 0 points decrease in error

   if 4.0000000000000001e-8 < (-.f64 (/.f64 1 (sqrt.f64 x)) (/.f64 1 (sqrt.f64 (+.f64 x 1))))

   1. Initial program 0.4

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

   2. Applied egg-rr 0.6

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

   3. Applied egg-rr 0.1

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

4. Final simplification 0.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{1 + x}} \leq 4 \cdot 10^{-8}:\\ \;\;\;\;\frac{{x}^{-0.5}}{x} \cdot \left(0.5 + \left(\frac{0.3125}{x \cdot x} + \frac{-0.375}{x}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{x}^{-0.5} - {\left(1 + x\right)}^{-0.5}\\ \end{array} \]

Alternatives

Alternative 1
Error	0.4
Cost	13760

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

Alternative 2
Error	0.6
Cost	7556

\[\begin{array}{l} \mathbf{if}\;x \leq 0.6722496367662038:\\ \;\;\;\;{x}^{-0.5} + \left(-1 + x \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{{x}^{-0.5}}{x} \cdot \left(0.5 + \left(\frac{0.3125}{x \cdot x} + \frac{-0.375}{x}\right)\right)\\ \end{array} \]

Alternative 3
Error	0.7
Cost	7172

\[\begin{array}{l} \mathbf{if}\;x \leq 0.6722496367662038:\\ \;\;\;\;{x}^{-0.5} + \left(-1 + x \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{{x}^{-0.5}}{x} \cdot \left(0.5 + \frac{-0.375}{x}\right)\\ \end{array} \]

Alternative 4
Error	1.0
Cost	7044

\[\begin{array}{l} \mathbf{if}\;x \leq 0.6722496367662038:\\ \;\;\;\;{x}^{-0.5} + \left(-1 + x \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot {x}^{-1.5}\\ \end{array} \]

Alternative 5
Error	2.2
Cost	6788

\[\begin{array}{l} \mathbf{if}\;x \leq 0.11024660055317465:\\ \;\;\;\;{x}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot {x}^{-1.5}\\ \end{array} \]

Alternative 6
Error	1.2
Cost	6788

\[\begin{array}{l} \mathbf{if}\;x \leq 0.6722496367662038:\\ \;\;\;\;{x}^{-0.5} + -1\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot {x}^{-1.5}\\ \end{array} \]

Alternative 7
Error	31.7
Cost	6528

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

Alternative 8
Error	59.2
Cost	192

\[\frac{1}{x} \]

Alternative 9
Error	62.8
Cost	64

\[-1 \]

Reproduce

herbie shell --seed 2022290 
(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)))))