2isqrt (example 3.6)

Percentage Accurate: 69.0% → 83.7%

Time: 20.1s

Precision: binary64

Cost: 66372

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{1 + x}} \leq 5 \cdot 10^{-8}:\\ \;\;\;\;0.3125 \cdot \sqrt{\frac{1}{{x}^{7}}} + \left(0.5 \cdot \sqrt{\frac{1}{{x}^{3}}} + \left(-0.2734375 \cdot \sqrt{\frac{1}{{x}^{9}}} + -0.375 \cdot \sqrt{\frac{1}{{x}^{5}}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{x}^{-0.5} - \frac{{\left(1 + x\right)}^{-0.16666666666666666}}{\sqrt[3]{1 + x}}\\ \end{array} \]

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-8)
   (+
    (* 0.3125 (sqrt (/ 1.0 (pow x 7.0))))
    (+
     (* 0.5 (sqrt (/ 1.0 (pow x 3.0))))
     (+
      (* -0.2734375 (sqrt (/ 1.0 (pow x 9.0))))
      (* -0.375 (sqrt (/ 1.0 (pow x 5.0)))))))
   (- (pow x -0.5) (/ (pow (+ 1.0 x) -0.16666666666666666) (cbrt (+ 1.0 x))))))

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-8) {
		tmp = (0.3125 * sqrt((1.0 / pow(x, 7.0)))) + ((0.5 * sqrt((1.0 / pow(x, 3.0)))) + ((-0.2734375 * sqrt((1.0 / pow(x, 9.0)))) + (-0.375 * sqrt((1.0 / pow(x, 5.0))))));
	} else {
		tmp = pow(x, -0.5) - (pow((1.0 + x), -0.16666666666666666) / cbrt((1.0 + x)));
	}
	return tmp;
}

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)))) <= 5e-8) {
		tmp = (0.3125 * Math.sqrt((1.0 / Math.pow(x, 7.0)))) + ((0.5 * Math.sqrt((1.0 / Math.pow(x, 3.0)))) + ((-0.2734375 * Math.sqrt((1.0 / Math.pow(x, 9.0)))) + (-0.375 * Math.sqrt((1.0 / Math.pow(x, 5.0))))));
	} else {
		tmp = Math.pow(x, -0.5) - (Math.pow((1.0 + x), -0.16666666666666666) / Math.cbrt((1.0 + x)));
	}
	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-8)
		tmp = Float64(Float64(0.3125 * sqrt(Float64(1.0 / (x ^ 7.0)))) + Float64(Float64(0.5 * sqrt(Float64(1.0 / (x ^ 3.0)))) + Float64(Float64(-0.2734375 * sqrt(Float64(1.0 / (x ^ 9.0)))) + Float64(-0.375 * sqrt(Float64(1.0 / (x ^ 5.0)))))));
	else
		tmp = Float64((x ^ -0.5) - Float64((Float64(1.0 + x) ^ -0.16666666666666666) / cbrt(Float64(1.0 + x))));
	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-8], N[(N[(0.3125 * N[Sqrt[N[(1.0 / N[Power[x, 7.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + N[(N[(0.5 * N[Sqrt[N[(1.0 / N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + N[(N[(-0.2734375 * N[Sqrt[N[(1.0 / N[Power[x, 9.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + N[(-0.375 * N[Sqrt[N[(1.0 / N[Power[x, 5.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Power[x, -0.5], $MachinePrecision] - N[(N[Power[N[(1.0 + x), $MachinePrecision], -0.16666666666666666], $MachinePrecision] / N[Power[N[(1.0 + x), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Target

Original	69.0%
Target	99.0%
Herbie	83.7%

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

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 40.1%

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

   2. Applied egg-rr 40.1%

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

      [Start] 40.1%	\[ \frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}} \]
      *-un-lft-identity [=>] 40.1%	\[ \color{blue}{1 \cdot \frac{1}{\sqrt{x}}} - \frac{1}{\sqrt{x + 1}} \]
      clear-num [=>] 40.1%	\[ 1 \cdot \frac{1}{\sqrt{x}} - \color{blue}{\frac{1}{\frac{\sqrt{x + 1}}{1}}} \]
      associate-/r/ [=>] 40.1%	\[ 1 \cdot \frac{1}{\sqrt{x}} - \color{blue}{\frac{1}{\sqrt{x + 1}} \cdot 1} \]
      prod-diff [=>] 40.1%	\[ \color{blue}{\mathsf{fma}\left(1, \frac{1}{\sqrt{x}}, -1 \cdot \frac{1}{\sqrt{x + 1}}\right) + \mathsf{fma}\left(-1, \frac{1}{\sqrt{x + 1}}, 1 \cdot \frac{1}{\sqrt{x + 1}}\right)} \]
      *-un-lft-identity [<=] 40.1%	\[ \mathsf{fma}\left(1, \frac{1}{\sqrt{x}}, -\color{blue}{\frac{1}{\sqrt{x + 1}}}\right) + \mathsf{fma}\left(-1, \frac{1}{\sqrt{x + 1}}, 1 \cdot \frac{1}{\sqrt{x + 1}}\right) \]
      fma-neg [<=] 40.1%	\[ \color{blue}{\left(1 \cdot \frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}}\right)} + \mathsf{fma}\left(-1, \frac{1}{\sqrt{x + 1}}, 1 \cdot \frac{1}{\sqrt{x + 1}}\right) \]
      *-un-lft-identity [<=] 40.1%	\[ \left(\color{blue}{\frac{1}{\sqrt{x}}} - \frac{1}{\sqrt{x + 1}}\right) + \mathsf{fma}\left(-1, \frac{1}{\sqrt{x + 1}}, 1 \cdot \frac{1}{\sqrt{x + 1}}\right) \]
      inv-pow [=>] 40.1%	\[ \left(\color{blue}{{\left(\sqrt{x}\right)}^{-1}} - \frac{1}{\sqrt{x + 1}}\right) + \mathsf{fma}\left(-1, \frac{1}{\sqrt{x + 1}}, 1 \cdot \frac{1}{\sqrt{x + 1}}\right) \]
      sqrt-pow2 [=>] 32.2%	\[ \left(\color{blue}{{x}^{\left(\frac{-1}{2}\right)}} - \frac{1}{\sqrt{x + 1}}\right) + \mathsf{fma}\left(-1, \frac{1}{\sqrt{x + 1}}, 1 \cdot \frac{1}{\sqrt{x + 1}}\right) \]
      metadata-eval [=>] 32.2%	\[ \left({x}^{\color{blue}{-0.5}} - \frac{1}{\sqrt{x + 1}}\right) + \mathsf{fma}\left(-1, \frac{1}{\sqrt{x + 1}}, 1 \cdot \frac{1}{\sqrt{x + 1}}\right) \]
      pow1/2 [=>] 32.2%	\[ \left({x}^{-0.5} - \frac{1}{\color{blue}{{\left(x + 1\right)}^{0.5}}}\right) + \mathsf{fma}\left(-1, \frac{1}{\sqrt{x + 1}}, 1 \cdot \frac{1}{\sqrt{x + 1}}\right) \]
      pow-flip [=>] 40.1%	\[ \left({x}^{-0.5} - \color{blue}{{\left(x + 1\right)}^{\left(-0.5\right)}}\right) + \mathsf{fma}\left(-1, \frac{1}{\sqrt{x + 1}}, 1 \cdot \frac{1}{\sqrt{x + 1}}\right) \]
      +-commutative [=>] 40.1%	\[ \left({x}^{-0.5} - {\color{blue}{\left(1 + x\right)}}^{\left(-0.5\right)}\right) + \mathsf{fma}\left(-1, \frac{1}{\sqrt{x + 1}}, 1 \cdot \frac{1}{\sqrt{x + 1}}\right) \]
      metadata-eval [=>] 40.1%	\[ \left({x}^{-0.5} - {\left(1 + x\right)}^{\color{blue}{-0.5}}\right) + \mathsf{fma}\left(-1, \frac{1}{\sqrt{x + 1}}, 1 \cdot \frac{1}{\sqrt{x + 1}}\right) \]

   3. Simplified 40.1%

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

      [Start] 40.1%	\[ \left({x}^{-0.5} - {\left(1 + x\right)}^{-0.5}\right) + \mathsf{fma}\left(-1, {\left(1 + x\right)}^{-0.5}, {\left(1 + x\right)}^{-0.5}\right) \]
      fma-udef [=>] 40.1%	\[ \left({x}^{-0.5} - {\left(1 + x\right)}^{-0.5}\right) + \color{blue}{\left(-1 \cdot {\left(1 + x\right)}^{-0.5} + {\left(1 + x\right)}^{-0.5}\right)} \]
      distribute-lft1-in [=>] 40.1%	\[ \left({x}^{-0.5} - {\left(1 + x\right)}^{-0.5}\right) + \color{blue}{\left(-1 + 1\right) \cdot {\left(1 + x\right)}^{-0.5}} \]
      metadata-eval [=>] 40.1%	\[ \left({x}^{-0.5} - {\left(1 + x\right)}^{-0.5}\right) + \color{blue}{0} \cdot {\left(1 + x\right)}^{-0.5} \]
      mul0-lft [=>] 40.1%	\[ \left({x}^{-0.5} - {\left(1 + x\right)}^{-0.5}\right) + \color{blue}{0} \]
      +-rgt-identity [=>] 40.1%	\[ \color{blue}{{x}^{-0.5} - {\left(1 + x\right)}^{-0.5}} \]

   4. Taylor expanded in x around inf 64.8%

      \[\leadsto \color{blue}{0.3125 \cdot \sqrt{\frac{1}{{x}^{7}}} + \left(0.5 \cdot \sqrt{\frac{1}{{x}^{3}}} + \left(-0.2734375 \cdot \sqrt{\frac{1}{{x}^{9}}} + -0.375 \cdot \sqrt{\frac{1}{{x}^{5}}}\right)\right)} \]

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

   1. Initial program 99.5%

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

   2. Applied egg-rr 100.0%

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

      [Start] 99.5%	\[ \frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}} \]
      *-un-lft-identity [=>] 99.5%	\[ \color{blue}{1 \cdot \frac{1}{\sqrt{x}}} - \frac{1}{\sqrt{x + 1}} \]
      clear-num [=>] 99.5%	\[ 1 \cdot \frac{1}{\sqrt{x}} - \color{blue}{\frac{1}{\frac{\sqrt{x + 1}}{1}}} \]
      associate-/r/ [=>] 99.5%	\[ 1 \cdot \frac{1}{\sqrt{x}} - \color{blue}{\frac{1}{\sqrt{x + 1}} \cdot 1} \]
      prod-diff [=>] 99.5%	\[ \color{blue}{\mathsf{fma}\left(1, \frac{1}{\sqrt{x}}, -1 \cdot \frac{1}{\sqrt{x + 1}}\right) + \mathsf{fma}\left(-1, \frac{1}{\sqrt{x + 1}}, 1 \cdot \frac{1}{\sqrt{x + 1}}\right)} \]
      *-un-lft-identity [<=] 99.5%	\[ \mathsf{fma}\left(1, \frac{1}{\sqrt{x}}, -\color{blue}{\frac{1}{\sqrt{x + 1}}}\right) + \mathsf{fma}\left(-1, \frac{1}{\sqrt{x + 1}}, 1 \cdot \frac{1}{\sqrt{x + 1}}\right) \]
      fma-neg [<=] 99.5%	\[ \color{blue}{\left(1 \cdot \frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}}\right)} + \mathsf{fma}\left(-1, \frac{1}{\sqrt{x + 1}}, 1 \cdot \frac{1}{\sqrt{x + 1}}\right) \]
      *-un-lft-identity [<=] 99.5%	\[ \left(\color{blue}{\frac{1}{\sqrt{x}}} - \frac{1}{\sqrt{x + 1}}\right) + \mathsf{fma}\left(-1, \frac{1}{\sqrt{x + 1}}, 1 \cdot \frac{1}{\sqrt{x + 1}}\right) \]
      inv-pow [=>] 99.5%	\[ \left(\color{blue}{{\left(\sqrt{x}\right)}^{-1}} - \frac{1}{\sqrt{x + 1}}\right) + \mathsf{fma}\left(-1, \frac{1}{\sqrt{x + 1}}, 1 \cdot \frac{1}{\sqrt{x + 1}}\right) \]
      sqrt-pow2 [=>] 99.9%	\[ \left(\color{blue}{{x}^{\left(\frac{-1}{2}\right)}} - \frac{1}{\sqrt{x + 1}}\right) + \mathsf{fma}\left(-1, \frac{1}{\sqrt{x + 1}}, 1 \cdot \frac{1}{\sqrt{x + 1}}\right) \]
      metadata-eval [=>] 99.9%	\[ \left({x}^{\color{blue}{-0.5}} - \frac{1}{\sqrt{x + 1}}\right) + \mathsf{fma}\left(-1, \frac{1}{\sqrt{x + 1}}, 1 \cdot \frac{1}{\sqrt{x + 1}}\right) \]
      pow1/2 [=>] 99.9%	\[ \left({x}^{-0.5} - \frac{1}{\color{blue}{{\left(x + 1\right)}^{0.5}}}\right) + \mathsf{fma}\left(-1, \frac{1}{\sqrt{x + 1}}, 1 \cdot \frac{1}{\sqrt{x + 1}}\right) \]
      pow-flip [=>] 100.0%	\[ \left({x}^{-0.5} - \color{blue}{{\left(x + 1\right)}^{\left(-0.5\right)}}\right) + \mathsf{fma}\left(-1, \frac{1}{\sqrt{x + 1}}, 1 \cdot \frac{1}{\sqrt{x + 1}}\right) \]
      +-commutative [=>] 100.0%	\[ \left({x}^{-0.5} - {\color{blue}{\left(1 + x\right)}}^{\left(-0.5\right)}\right) + \mathsf{fma}\left(-1, \frac{1}{\sqrt{x + 1}}, 1 \cdot \frac{1}{\sqrt{x + 1}}\right) \]
      metadata-eval [=>] 100.0%	\[ \left({x}^{-0.5} - {\left(1 + x\right)}^{\color{blue}{-0.5}}\right) + \mathsf{fma}\left(-1, \frac{1}{\sqrt{x + 1}}, 1 \cdot \frac{1}{\sqrt{x + 1}}\right) \]

   3. Simplified 100.0%

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

      [Start] 100.0%	\[ \left({x}^{-0.5} - {\left(1 + x\right)}^{-0.5}\right) + \mathsf{fma}\left(-1, {\left(1 + x\right)}^{-0.5}, {\left(1 + x\right)}^{-0.5}\right) \]
      fma-udef [=>] 100.0%	\[ \left({x}^{-0.5} - {\left(1 + x\right)}^{-0.5}\right) + \color{blue}{\left(-1 \cdot {\left(1 + x\right)}^{-0.5} + {\left(1 + x\right)}^{-0.5}\right)} \]
      distribute-lft1-in [=>] 100.0%	\[ \left({x}^{-0.5} - {\left(1 + x\right)}^{-0.5}\right) + \color{blue}{\left(-1 + 1\right) \cdot {\left(1 + x\right)}^{-0.5}} \]
      metadata-eval [=>] 100.0%	\[ \left({x}^{-0.5} - {\left(1 + x\right)}^{-0.5}\right) + \color{blue}{0} \cdot {\left(1 + x\right)}^{-0.5} \]
      mul0-lft [=>] 100.0%	\[ \left({x}^{-0.5} - {\left(1 + x\right)}^{-0.5}\right) + \color{blue}{0} \]
      +-rgt-identity [=>] 100.0%	\[ \color{blue}{{x}^{-0.5} - {\left(1 + x\right)}^{-0.5}} \]

   4. Applied egg-rr 100.0%

      \[\leadsto {x}^{-0.5} - \color{blue}{{\left(x + 1\right)}^{-0.16666666666666666} \cdot \frac{1}{\sqrt[3]{x + 1}}} \]
      Step-by-step derivation

      [Start] 100.0%	\[ {x}^{-0.5} - {\left(1 + x\right)}^{-0.5} \]
      add-cube-cbrt [=>] 99.9%	\[ {x}^{-0.5} - \color{blue}{\left(\sqrt[3]{{\left(1 + x\right)}^{-0.5}} \cdot \sqrt[3]{{\left(1 + x\right)}^{-0.5}}\right) \cdot \sqrt[3]{{\left(1 + x\right)}^{-0.5}}} \]
      associate-*l* [=>] 99.9%	\[ {x}^{-0.5} - \color{blue}{\sqrt[3]{{\left(1 + x\right)}^{-0.5}} \cdot \left(\sqrt[3]{{\left(1 + x\right)}^{-0.5}} \cdot \sqrt[3]{{\left(1 + x\right)}^{-0.5}}\right)} \]
      pow1/3 [=>] 99.9%	\[ {x}^{-0.5} - \color{blue}{{\left({\left(1 + x\right)}^{-0.5}\right)}^{0.3333333333333333}} \cdot \left(\sqrt[3]{{\left(1 + x\right)}^{-0.5}} \cdot \sqrt[3]{{\left(1 + x\right)}^{-0.5}}\right) \]
      pow-pow [=>] 99.9%	\[ {x}^{-0.5} - \color{blue}{{\left(1 + x\right)}^{\left(-0.5 \cdot 0.3333333333333333\right)}} \cdot \left(\sqrt[3]{{\left(1 + x\right)}^{-0.5}} \cdot \sqrt[3]{{\left(1 + x\right)}^{-0.5}}\right) \]
      +-commutative [=>] 99.9%	\[ {x}^{-0.5} - {\color{blue}{\left(x + 1\right)}}^{\left(-0.5 \cdot 0.3333333333333333\right)} \cdot \left(\sqrt[3]{{\left(1 + x\right)}^{-0.5}} \cdot \sqrt[3]{{\left(1 + x\right)}^{-0.5}}\right) \]
      metadata-eval [=>] 99.9%	\[ {x}^{-0.5} - {\left(x + 1\right)}^{\color{blue}{-0.16666666666666666}} \cdot \left(\sqrt[3]{{\left(1 + x\right)}^{-0.5}} \cdot \sqrt[3]{{\left(1 + x\right)}^{-0.5}}\right) \]
      cbrt-unprod [=>] 99.9%	\[ {x}^{-0.5} - {\left(x + 1\right)}^{-0.16666666666666666} \cdot \color{blue}{\sqrt[3]{{\left(1 + x\right)}^{-0.5} \cdot {\left(1 + x\right)}^{-0.5}}} \]
      pow-prod-up [=>] 100.0%	\[ {x}^{-0.5} - {\left(x + 1\right)}^{-0.16666666666666666} \cdot \sqrt[3]{\color{blue}{{\left(1 + x\right)}^{\left(-0.5 + -0.5\right)}}} \]
      metadata-eval [=>] 100.0%	\[ {x}^{-0.5} - {\left(x + 1\right)}^{-0.16666666666666666} \cdot \sqrt[3]{{\left(1 + x\right)}^{\color{blue}{-1}}} \]
      inv-pow [<=] 100.0%	\[ {x}^{-0.5} - {\left(x + 1\right)}^{-0.16666666666666666} \cdot \sqrt[3]{\color{blue}{\frac{1}{1 + x}}} \]
      cbrt-div [=>] 100.0%	\[ {x}^{-0.5} - {\left(x + 1\right)}^{-0.16666666666666666} \cdot \color{blue}{\frac{\sqrt[3]{1}}{\sqrt[3]{1 + x}}} \]
      metadata-eval [=>] 100.0%	\[ {x}^{-0.5} - {\left(x + 1\right)}^{-0.16666666666666666} \cdot \frac{\color{blue}{1}}{\sqrt[3]{1 + x}} \]
      +-commutative [=>] 100.0%	\[ {x}^{-0.5} - {\left(x + 1\right)}^{-0.16666666666666666} \cdot \frac{1}{\sqrt[3]{\color{blue}{x + 1}}} \]

   5. Simplified 100.0%

      \[\leadsto {x}^{-0.5} - \color{blue}{\frac{{\left(x + 1\right)}^{-0.16666666666666666}}{\sqrt[3]{x + 1}}} \]
      Step-by-step derivation

      [Start] 100.0%	\[ {x}^{-0.5} - {\left(x + 1\right)}^{-0.16666666666666666} \cdot \frac{1}{\sqrt[3]{x + 1}} \]
      associate-*r/ [=>] 100.0%	\[ {x}^{-0.5} - \color{blue}{\frac{{\left(x + 1\right)}^{-0.16666666666666666} \cdot 1}{\sqrt[3]{x + 1}}} \]
      *-rgt-identity [=>] 100.0%	\[ {x}^{-0.5} - \frac{\color{blue}{{\left(x + 1\right)}^{-0.16666666666666666}}}{\sqrt[3]{x + 1}} \]

3. Recombined 2 regimes into one program.

4. Final simplification 83.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{1 + x}} \leq 5 \cdot 10^{-8}:\\ \;\;\;\;0.3125 \cdot \sqrt{\frac{1}{{x}^{7}}} + \left(0.5 \cdot \sqrt{\frac{1}{{x}^{3}}} + \left(-0.2734375 \cdot \sqrt{\frac{1}{{x}^{9}}} + -0.375 \cdot \sqrt{\frac{1}{{x}^{5}}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{x}^{-0.5} - \frac{{\left(1 + x\right)}^{-0.16666666666666666}}{\sqrt[3]{1 + x}}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	83.7%
Cost	66372

\[\begin{array}{l} \mathbf{if}\;\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{1 + x}} \leq 5 \cdot 10^{-8}:\\ \;\;\;\;0.3125 \cdot \sqrt{\frac{1}{{x}^{7}}} + \left(0.5 \cdot \sqrt{\frac{1}{{x}^{3}}} + \left(-0.2734375 \cdot \sqrt{\frac{1}{{x}^{9}}} + -0.375 \cdot \sqrt{\frac{1}{{x}^{5}}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{x}^{-0.5} - \frac{{\left(1 + x\right)}^{-0.16666666666666666}}{\sqrt[3]{1 + x}}\\ \end{array} \]

Alternative 2
Accuracy	83.7%
Cost	53124

\[\begin{array}{l} \mathbf{if}\;\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{1 + x}} \leq 5 \cdot 10^{-8}:\\ \;\;\;\;0.3125 \cdot \sqrt{\frac{1}{{x}^{7}}} + \left(0.5 \cdot \sqrt{\frac{1}{{x}^{3}}} + -0.375 \cdot \sqrt{\frac{1}{{x}^{5}}}\right)\\ \mathbf{else}:\\ \;\;\;\;{x}^{-0.5} - \frac{{\left(1 + x\right)}^{-0.16666666666666666}}{\sqrt[3]{1 + x}}\\ \end{array} \]

Alternative 3
Accuracy	83.3%
Cost	39876

\[\begin{array}{l} \mathbf{if}\;\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{1 + x}} \leq 2 \cdot 10^{-16}:\\ \;\;\;\;0.5 \cdot \sqrt{\frac{1}{{x}^{3}}} + -0.375 \cdot \sqrt{\frac{1}{{x}^{5}}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-\sqrt[3]{\frac{1}{1 + x}}, {\left(1 + x\right)}^{-0.16666666666666666}, {x}^{-0.5}\right)\\ \end{array} \]

Alternative 4
Accuracy	83.2%
Cost	39812

\[\begin{array}{l} \mathbf{if}\;\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{1 + x}} \leq 2 \cdot 10^{-16}:\\ \;\;\;\;0.5 \cdot \sqrt{\frac{1}{{x}^{3}}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-\sqrt[3]{\frac{1}{1 + x}}, {\left(1 + x\right)}^{-0.16666666666666666}, {x}^{-0.5}\right)\\ \end{array} \]

Alternative 5
Accuracy	83.3%
Cost	26756

\[\begin{array}{l} \mathbf{if}\;\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{1 + x}} \leq 2 \cdot 10^{-16}:\\ \;\;\;\;0.5 \cdot \sqrt{\frac{1}{{x}^{3}}}\\ \mathbf{else}:\\ \;\;\;\;{x}^{-0.5} - \sqrt{\frac{1}{1 + x}}\\ \end{array} \]

Alternative 6
Accuracy	83.3%
Cost	13380

\[\begin{array}{l} \mathbf{if}\;x \leq 126000000:\\ \;\;\;\;{x}^{-0.5} - {\left(1 + x\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{\frac{1}{{x}^{3}}}\\ \end{array} \]

Alternative 7
Accuracy	82.3%
Cost	13316

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

Alternative 8
Accuracy	51.7%
Cost	13056

\[\left|{x}^{-0.5} + -1\right| \]

Alternative 9
Accuracy	51.4%
Cost	6912

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

Alternative 10
Accuracy	50.6%
Cost	6656

\[{x}^{-0.5} + -1 \]

Alternative 11
Accuracy	50.8%
Cost	6528

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

Alternative 12
Accuracy	1.9%
Cost	64

\[-1 \]

Reproduce

herbie shell --seed 2023277 
(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)))))