2isqrt (example 3.6)

Average Accuracy: 69.4% → 99.7%

Time: 13.1s

Precision: binary64

Cost: 39812

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{1 + x}} \leq 5 \cdot 10^{-9}:\\ \;\;\;\;\left(\frac{0.5}{x} + \left(\frac{0.3125}{{x}^{3}} - \frac{0.375}{x \cdot x}\right)\right) \cdot {x}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-\sqrt[3]{\frac{1}{1 + x}}, {\left(1 + x\right)}^{-0.16666666666666666}, {x}^{-0.5}\right)\\ \end{array} \]

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)))) 5e-9)
   (* (+ (/ 0.5 x) (- (/ 0.3125 (pow x 3.0)) (/ 0.375 (* x x)))) (pow x -0.5))
   (fma
    (- (cbrt (/ 1.0 (+ 1.0 x))))
    (pow (+ 1.0 x) -0.16666666666666666)
    (pow 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)))) <= 5e-9) {
		tmp = ((0.5 / x) + ((0.3125 / pow(x, 3.0)) - (0.375 / (x * x)))) * pow(x, -0.5);
	} else {
		tmp = fma(-cbrt((1.0 / (1.0 + x))), pow((1.0 + x), -0.16666666666666666), pow(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)))) <= 5e-9)
		tmp = Float64(Float64(Float64(0.5 / x) + Float64(Float64(0.3125 / (x ^ 3.0)) - Float64(0.375 / Float64(x * x)))) * (x ^ -0.5));
	else
		tmp = fma(Float64(-cbrt(Float64(1.0 / Float64(1.0 + x)))), (Float64(1.0 + x) ^ -0.16666666666666666), (x ^ -0.5));
	end
	return 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], 5e-9], N[(N[(N[(0.5 / x), $MachinePrecision] + N[(N[(0.3125 / N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision] - N[(0.375 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Power[x, -0.5], $MachinePrecision]), $MachinePrecision], N[((-N[Power[N[(1.0 / N[(1.0 + x), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]) * N[Power[N[(1.0 + x), $MachinePrecision], -0.16666666666666666], $MachinePrecision] + N[Power[x, -0.5], $MachinePrecision]), $MachinePrecision]]

Target

Original	69.4%
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)))) < 5.0000000000000001e-9

   1. Initial program 32.4%

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

   2. Applied egg-rr 32.4%

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

      [Start] 32.4	\[ \frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}} \]
      frac-sub [=>] 32.4	\[ \color{blue}{\frac{1 \cdot \sqrt{x + 1} - \sqrt{x} \cdot 1}{\sqrt{x} \cdot \sqrt{x + 1}}} \]
      div-inv [=>] 32.4	\[ \color{blue}{\left(1 \cdot \sqrt{x + 1} - \sqrt{x} \cdot 1\right) \cdot \frac{1}{\sqrt{x} \cdot \sqrt{x + 1}}} \]
      *-un-lft-identity [<=] 32.4	\[ \left(\color{blue}{\sqrt{x + 1}} - \sqrt{x} \cdot 1\right) \cdot \frac{1}{\sqrt{x} \cdot \sqrt{x + 1}} \]
      +-commutative [=>] 32.4	\[ \left(\sqrt{\color{blue}{1 + x}} - \sqrt{x} \cdot 1\right) \cdot \frac{1}{\sqrt{x} \cdot \sqrt{x + 1}} \]
      *-rgt-identity [=>] 32.4	\[ \left(\sqrt{1 + x} - \color{blue}{\sqrt{x}}\right) \cdot \frac{1}{\sqrt{x} \cdot \sqrt{x + 1}} \]
      metadata-eval [<=] 32.4	\[ \left(\sqrt{1 + x} - \sqrt{x}\right) \cdot \frac{\color{blue}{1 \cdot 1}}{\sqrt{x} \cdot \sqrt{x + 1}} \]
      frac-times [<=] 32.4	\[ \left(\sqrt{1 + x} - \sqrt{x}\right) \cdot \color{blue}{\left(\frac{1}{\sqrt{x}} \cdot \frac{1}{\sqrt{x + 1}}\right)} \]
      un-div-inv [=>] 32.4	\[ \left(\sqrt{1 + x} - \sqrt{x}\right) \cdot \color{blue}{\frac{\frac{1}{\sqrt{x}}}{\sqrt{x + 1}}} \]
      pow1/2 [=>] 32.4	\[ \left(\sqrt{1 + x} - \sqrt{x}\right) \cdot \frac{\frac{1}{\color{blue}{{x}^{0.5}}}}{\sqrt{x + 1}} \]
      pow-flip [=>] 32.4	\[ \left(\sqrt{1 + x} - \sqrt{x}\right) \cdot \frac{\color{blue}{{x}^{\left(-0.5\right)}}}{\sqrt{x + 1}} \]
      metadata-eval [=>] 32.4	\[ \left(\sqrt{1 + x} - \sqrt{x}\right) \cdot \frac{{x}^{\color{blue}{-0.5}}}{\sqrt{x + 1}} \]
      +-commutative [=>] 32.4	\[ \left(\sqrt{1 + x} - \sqrt{x}\right) \cdot \frac{{x}^{-0.5}}{\sqrt{\color{blue}{1 + x}}} \]

   3. Simplified 32.4%

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

      [Start] 32.4	\[ \left(\sqrt{1 + x} - \sqrt{x}\right) \cdot \frac{{x}^{-0.5}}{\sqrt{1 + x}} \]
      associate-*r/ [=>] 32.4	\[ \color{blue}{\frac{\left(\sqrt{1 + x} - \sqrt{x}\right) \cdot {x}^{-0.5}}{\sqrt{1 + x}}} \]
      *-rgt-identity [<=] 32.4	\[ \frac{\left(\sqrt{1 + x} - \sqrt{x}\right) \cdot {x}^{-0.5}}{\color{blue}{\sqrt{1 + x} \cdot 1}} \]
      times-frac [=>] 32.4	\[ \color{blue}{\frac{\sqrt{1 + x} - \sqrt{x}}{\sqrt{1 + x}} \cdot \frac{{x}^{-0.5}}{1}} \]
      div-sub [=>] 32.4	\[ \color{blue}{\left(\frac{\sqrt{1 + x}}{\sqrt{1 + x}} - \frac{\sqrt{x}}{\sqrt{1 + x}}\right)} \cdot \frac{{x}^{-0.5}}{1} \]
      *-inverses [=>] 32.4	\[ \left(\color{blue}{1} - \frac{\sqrt{x}}{\sqrt{1 + x}}\right) \cdot \frac{{x}^{-0.5}}{1} \]
      /-rgt-identity [=>] 32.4	\[ \left(1 - \frac{\sqrt{x}}{\sqrt{1 + x}}\right) \cdot \color{blue}{{x}^{-0.5}} \]

   4. Taylor expanded in x around inf 99.6%

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

   5. Simplified 99.6%

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

      [Start] 99.6	\[ \left(\left(0.5 \cdot \frac{1}{x} + 0.3125 \cdot \frac{1}{{x}^{3}}\right) - 0.375 \cdot \frac{1}{{x}^{2}}\right) \cdot {x}^{-0.5} \]
      associate--l+ [=>] 99.6	\[ \color{blue}{\left(0.5 \cdot \frac{1}{x} + \left(0.3125 \cdot \frac{1}{{x}^{3}} - 0.375 \cdot \frac{1}{{x}^{2}}\right)\right)} \cdot {x}^{-0.5} \]
      associate-*r/ [=>] 99.6	\[ \left(\color{blue}{\frac{0.5 \cdot 1}{x}} + \left(0.3125 \cdot \frac{1}{{x}^{3}} - 0.375 \cdot \frac{1}{{x}^{2}}\right)\right) \cdot {x}^{-0.5} \]
      metadata-eval [=>] 99.6	\[ \left(\frac{\color{blue}{0.5}}{x} + \left(0.3125 \cdot \frac{1}{{x}^{3}} - 0.375 \cdot \frac{1}{{x}^{2}}\right)\right) \cdot {x}^{-0.5} \]
      associate-*r/ [=>] 99.6	\[ \left(\frac{0.5}{x} + \left(\color{blue}{\frac{0.3125 \cdot 1}{{x}^{3}}} - 0.375 \cdot \frac{1}{{x}^{2}}\right)\right) \cdot {x}^{-0.5} \]
      metadata-eval [=>] 99.6	\[ \left(\frac{0.5}{x} + \left(\frac{\color{blue}{0.3125}}{{x}^{3}} - 0.375 \cdot \frac{1}{{x}^{2}}\right)\right) \cdot {x}^{-0.5} \]
      associate-*r/ [=>] 99.6	\[ \left(\frac{0.5}{x} + \left(\frac{0.3125}{{x}^{3}} - \color{blue}{\frac{0.375 \cdot 1}{{x}^{2}}}\right)\right) \cdot {x}^{-0.5} \]
      metadata-eval [=>] 99.6	\[ \left(\frac{0.5}{x} + \left(\frac{0.3125}{{x}^{3}} - \frac{\color{blue}{0.375}}{{x}^{2}}\right)\right) \cdot {x}^{-0.5} \]
      unpow2 [=>] 99.6	\[ \left(\frac{0.5}{x} + \left(\frac{0.3125}{{x}^{3}} - \frac{0.375}{\color{blue}{x \cdot x}}\right)\right) \cdot {x}^{-0.5} \]

   if 5.0000000000000001e-9 < (-.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	\[ \color{blue}{{x}^{-0.5} - {\left(1 + x\right)}^{-0.5}} \]

   4. Applied egg-rr 99.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-\sqrt[3]{\frac{1}{x + 1}}, {\left(x + 1\right)}^{-0.16666666666666666}, {x}^{-0.5}\right)} \]
      Proof

      [Start] 99.8	\[ {x}^{-0.5} - {\left(1 + x\right)}^{-0.5} \]
      sub-neg [=>] 99.8	\[ \color{blue}{{x}^{-0.5} + \left(-{\left(1 + x\right)}^{-0.5}\right)} \]
      +-commutative [=>] 99.8	\[ \color{blue}{\left(-{\left(1 + x\right)}^{-0.5}\right) + {x}^{-0.5}} \]
      add-cube-cbrt [=>] 99.8	\[ \left(-\color{blue}{\left(\sqrt[3]{{\left(1 + x\right)}^{-0.5}} \cdot \sqrt[3]{{\left(1 + x\right)}^{-0.5}}\right) \cdot \sqrt[3]{{\left(1 + x\right)}^{-0.5}}}\right) + {x}^{-0.5} \]
      distribute-lft-neg-in [=>] 99.8	\[ \color{blue}{\left(-\sqrt[3]{{\left(1 + x\right)}^{-0.5}} \cdot \sqrt[3]{{\left(1 + x\right)}^{-0.5}}\right) \cdot \sqrt[3]{{\left(1 + x\right)}^{-0.5}}} + {x}^{-0.5} \]
      fma-def [=>] 99.9	\[ \color{blue}{\mathsf{fma}\left(-\sqrt[3]{{\left(1 + x\right)}^{-0.5}} \cdot \sqrt[3]{{\left(1 + x\right)}^{-0.5}}, \sqrt[3]{{\left(1 + x\right)}^{-0.5}}, {x}^{-0.5}\right)} \]
      cbrt-unprod [=>] 99.9	\[ \mathsf{fma}\left(-\color{blue}{\sqrt[3]{{\left(1 + x\right)}^{-0.5} \cdot {\left(1 + x\right)}^{-0.5}}}, \sqrt[3]{{\left(1 + x\right)}^{-0.5}}, {x}^{-0.5}\right) \]
      pow-prod-up [=>] 99.9	\[ \mathsf{fma}\left(-\sqrt[3]{\color{blue}{{\left(1 + x\right)}^{\left(-0.5 + -0.5\right)}}}, \sqrt[3]{{\left(1 + x\right)}^{-0.5}}, {x}^{-0.5}\right) \]
      metadata-eval [=>] 99.9	\[ \mathsf{fma}\left(-\sqrt[3]{{\left(1 + x\right)}^{\color{blue}{-1}}}, \sqrt[3]{{\left(1 + x\right)}^{-0.5}}, {x}^{-0.5}\right) \]
      unpow-1 [=>] 99.9	\[ \mathsf{fma}\left(-\sqrt[3]{\color{blue}{\frac{1}{1 + x}}}, \sqrt[3]{{\left(1 + x\right)}^{-0.5}}, {x}^{-0.5}\right) \]
      +-commutative [=>] 99.9	\[ \mathsf{fma}\left(-\sqrt[3]{\frac{1}{\color{blue}{x + 1}}}, \sqrt[3]{{\left(1 + x\right)}^{-0.5}}, {x}^{-0.5}\right) \]
      pow1/3 [=>] 99.9	\[ \mathsf{fma}\left(-\sqrt[3]{\frac{1}{x + 1}}, \color{blue}{{\left({\left(1 + x\right)}^{-0.5}\right)}^{0.3333333333333333}}, {x}^{-0.5}\right) \]
      pow-pow [=>] 99.9	\[ \mathsf{fma}\left(-\sqrt[3]{\frac{1}{x + 1}}, \color{blue}{{\left(1 + x\right)}^{\left(-0.5 \cdot 0.3333333333333333\right)}}, {x}^{-0.5}\right) \]
      +-commutative [=>] 99.9	\[ \mathsf{fma}\left(-\sqrt[3]{\frac{1}{x + 1}}, {\color{blue}{\left(x + 1\right)}}^{\left(-0.5 \cdot 0.3333333333333333\right)}, {x}^{-0.5}\right) \]
      metadata-eval [=>] 99.9	\[ \mathsf{fma}\left(-\sqrt[3]{\frac{1}{x + 1}}, {\left(x + 1\right)}^{\color{blue}{-0.16666666666666666}}, {x}^{-0.5}\right) \]

3. Recombined 2 regimes into one program.

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{1 + x}} \leq 5 \cdot 10^{-9}:\\ \;\;\;\;\left(\frac{0.5}{x} + \left(\frac{0.3125}{{x}^{3}} - \frac{0.375}{x \cdot x}\right)\right) \cdot {x}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-\sqrt[3]{\frac{1}{1 + x}}, {\left(1 + x\right)}^{-0.16666666666666666}, {x}^{-0.5}\right)\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	99.7%
Cost	27332

\[\begin{array}{l} \mathbf{if}\;\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{1 + x}} \leq 5 \cdot 10^{-9}:\\ \;\;\;\;\left(\frac{0.5}{x} + \left(\frac{0.3125}{{x}^{3}} - \frac{0.375}{x \cdot x}\right)\right) \cdot {x}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;{x}^{-0.5} - {\left(1 + x\right)}^{-0.5}\\ \end{array} \]

Alternative 2
Accuracy	99.7%
Cost	27204

\[\begin{array}{l} t_0 := \sqrt{1 + x}\\ \mathbf{if}\;\frac{1}{\sqrt{x}} - \frac{1}{t_0} \leq 5 \cdot 10^{-9}:\\ \;\;\;\;\frac{\frac{1}{\sqrt{x} + t_0}}{\left(x + 0.5\right) - \frac{0.125}{x}}\\ \mathbf{else}:\\ \;\;\;\;{x}^{-0.5} - {\left(1 + x\right)}^{-0.5}\\ \end{array} \]

Alternative 3
Accuracy	99.7%
Cost	26948

\[\begin{array}{l} t_0 := \sqrt{1 + x}\\ \mathbf{if}\;\frac{1}{\sqrt{x}} - \frac{1}{t_0} \leq 5 \cdot 10^{-9}:\\ \;\;\;\;\frac{\frac{1}{\sqrt{x} + t_0}}{x + 0.5}\\ \mathbf{else}:\\ \;\;\;\;{x}^{-0.5} - {\left(1 + x\right)}^{-0.5}\\ \end{array} \]

Alternative 4
Accuracy	99.7%
Cost	26692

\[\begin{array}{l} \mathbf{if}\;\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{1 + x}} \leq 5 \cdot 10^{-9}:\\ \;\;\;\;{x}^{-0.5} \cdot \frac{0.5 - \frac{0.375}{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;{x}^{-0.5} - {\left(1 + x\right)}^{-0.5}\\ \end{array} \]

Alternative 5
Accuracy	99.0%
Cost	7172

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

Alternative 6
Accuracy	98.7%
Cost	7044

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

Alternative 7
Accuracy	97.0%
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 8
Accuracy	98.4%
Cost	6788

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

Alternative 9
Accuracy	51.6%
Cost	6528

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

Alternative 10
Accuracy	4.8%
Cost	576

\[\frac{1}{\left(x + 0.5\right) - \frac{0.125}{x}} \]

Alternative 11
Accuracy	1.9%
Cost	64

\[-1 \]

Reproduce

herbie shell --seed 2023159 
(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)))))