2isqrt (example 3.6)

Percentage Accurate: 68.7% → 99.6%

Time: 13.2s

Precision: binary64

Cost: 33860

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{1}{\sqrt{x}} + \frac{-1}{\sqrt{1 + x}} \leq 0:\\ \;\;\;\;0.5 \cdot {x}^{-1.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x + \left(1 - x\right)}{x + x \cdot x}}{{x}^{-0.5} + \frac{1}{\mathsf{hypot}\left(1, \sqrt{x}\right)}}\\ \end{array} \]

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)))) 0.0)
   (* 0.5 (pow x -1.5))
   (/
    (/ (+ x (- 1.0 x)) (+ x (* x x)))
    (+ (pow x -0.5) (/ 1.0 (hypot 1.0 (sqrt 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)))) <= 0.0) {
		tmp = 0.5 * pow(x, -1.5);
	} else {
		tmp = ((x + (1.0 - x)) / (x + (x * x))) / (pow(x, -0.5) + (1.0 / hypot(1.0, sqrt(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)))) <= 0.0) {
		tmp = 0.5 * Math.pow(x, -1.5);
	} else {
		tmp = ((x + (1.0 - x)) / (x + (x * x))) / (Math.pow(x, -0.5) + (1.0 / Math.hypot(1.0, Math.sqrt(x))));
	}
	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)))) <= 0.0:
		tmp = 0.5 * math.pow(x, -1.5)
	else:
		tmp = ((x + (1.0 - x)) / (x + (x * x))) / (math.pow(x, -0.5) + (1.0 / math.hypot(1.0, math.sqrt(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)))) <= 0.0)
		tmp = Float64(0.5 * (x ^ -1.5));
	else
		tmp = Float64(Float64(Float64(x + Float64(1.0 - x)) / Float64(x + Float64(x * x))) / Float64((x ^ -0.5) + Float64(1.0 / hypot(1.0, sqrt(x)))));
	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)))) <= 0.0)
		tmp = 0.5 * (x ^ -1.5);
	else
		tmp = ((x + (1.0 - x)) / (x + (x * x))) / ((x ^ -0.5) + (1.0 / hypot(1.0, sqrt(x))));
	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], 0.0], N[(0.5 * N[Power[x, -1.5], $MachinePrecision]), $MachinePrecision], N[(N[(N[(x + N[(1.0 - x), $MachinePrecision]), $MachinePrecision] / N[(x + N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[Power[x, -0.5], $MachinePrecision] + N[(1.0 / N[Sqrt[1.0 ^ 2 + N[Sqrt[x], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Target

Original	68.7%
Target	98.9%
Herbie	99.6%

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

Derivation

1. Split input into 2 regimes

2. if (-.f64 (/.f64 1 (sqrt.f64 x)) (/.f64 1 (sqrt.f64 (+.f64 x 1)))) < 0.0

   1. Initial program 38.2%

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

   2. Applied egg-rr 38.2%

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

      [Start] 38.2%	\[ \frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}} \]
      *-un-lft-identity [=>] 38.2%	\[ \color{blue}{1 \cdot \frac{1}{\sqrt{x}}} - \frac{1}{\sqrt{x + 1}} \]
      clear-num [=>] 38.2%	\[ 1 \cdot \frac{1}{\sqrt{x}} - \color{blue}{\frac{1}{\frac{\sqrt{x + 1}}{1}}} \]
      associate-/r/ [=>] 38.2%	\[ 1 \cdot \frac{1}{\sqrt{x}} - \color{blue}{\frac{1}{\sqrt{x + 1}} \cdot 1} \]
      prod-diff [=>] 38.2%	\[ \color{blue}{\mathsf{fma}\left(1, \frac{1}{\sqrt{x}}, -1 \cdot \frac{1}{\sqrt{x + 1}}\right) + \mathsf{fma}\left(-1, \frac{1}{\sqrt{x + 1}}, 1 \cdot \frac{1}{\sqrt{x + 1}}\right)} \]
      *-un-lft-identity [<=] 38.2%	\[ \mathsf{fma}\left(1, \frac{1}{\sqrt{x}}, -\color{blue}{\frac{1}{\sqrt{x + 1}}}\right) + \mathsf{fma}\left(-1, \frac{1}{\sqrt{x + 1}}, 1 \cdot \frac{1}{\sqrt{x + 1}}\right) \]
      fma-neg [<=] 38.2%	\[ \color{blue}{\left(1 \cdot \frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}}\right)} + \mathsf{fma}\left(-1, \frac{1}{\sqrt{x + 1}}, 1 \cdot \frac{1}{\sqrt{x + 1}}\right) \]
      *-un-lft-identity [<=] 38.2%	\[ \left(\color{blue}{\frac{1}{\sqrt{x}}} - \frac{1}{\sqrt{x + 1}}\right) + \mathsf{fma}\left(-1, \frac{1}{\sqrt{x + 1}}, 1 \cdot \frac{1}{\sqrt{x + 1}}\right) \]
      inv-pow [=>] 38.2%	\[ \left(\color{blue}{{\left(\sqrt{x}\right)}^{-1}} - \frac{1}{\sqrt{x + 1}}\right) + \mathsf{fma}\left(-1, \frac{1}{\sqrt{x + 1}}, 1 \cdot \frac{1}{\sqrt{x + 1}}\right) \]
      sqrt-pow2 [=>] 26.9%	\[ \left(\color{blue}{{x}^{\left(\frac{-1}{2}\right)}} - \frac{1}{\sqrt{x + 1}}\right) + \mathsf{fma}\left(-1, \frac{1}{\sqrt{x + 1}}, 1 \cdot \frac{1}{\sqrt{x + 1}}\right) \]
      metadata-eval [=>] 26.9%	\[ \left({x}^{\color{blue}{-0.5}} - \frac{1}{\sqrt{x + 1}}\right) + \mathsf{fma}\left(-1, \frac{1}{\sqrt{x + 1}}, 1 \cdot \frac{1}{\sqrt{x + 1}}\right) \]
      pow1/2 [=>] 26.9%	\[ \left({x}^{-0.5} - \frac{1}{\color{blue}{{\left(x + 1\right)}^{0.5}}}\right) + \mathsf{fma}\left(-1, \frac{1}{\sqrt{x + 1}}, 1 \cdot \frac{1}{\sqrt{x + 1}}\right) \]
      pow-flip [=>] 38.2%	\[ \left({x}^{-0.5} - \color{blue}{{\left(x + 1\right)}^{\left(-0.5\right)}}\right) + \mathsf{fma}\left(-1, \frac{1}{\sqrt{x + 1}}, 1 \cdot \frac{1}{\sqrt{x + 1}}\right) \]
      +-commutative [=>] 38.2%	\[ \left({x}^{-0.5} - {\color{blue}{\left(1 + x\right)}}^{\left(-0.5\right)}\right) + \mathsf{fma}\left(-1, \frac{1}{\sqrt{x + 1}}, 1 \cdot \frac{1}{\sqrt{x + 1}}\right) \]
      metadata-eval [=>] 38.2%	\[ \left({x}^{-0.5} - {\left(1 + x\right)}^{\color{blue}{-0.5}}\right) + \mathsf{fma}\left(-1, \frac{1}{\sqrt{x + 1}}, 1 \cdot \frac{1}{\sqrt{x + 1}}\right) \]

   3. Simplified 38.2%

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

      [Start] 38.2%	\[ \left({x}^{-0.5} - {\left(1 + x\right)}^{-0.5}\right) + \mathsf{fma}\left(-1, {\left(1 + x\right)}^{-0.5}, {\left(1 + x\right)}^{-0.5}\right) \]
      fma-udef [=>] 38.2%	\[ \left({x}^{-0.5} - {\left(1 + x\right)}^{-0.5}\right) + \color{blue}{\left(-1 \cdot {\left(1 + x\right)}^{-0.5} + {\left(1 + x\right)}^{-0.5}\right)} \]
      distribute-lft1-in [=>] 38.2%	\[ \left({x}^{-0.5} - {\left(1 + x\right)}^{-0.5}\right) + \color{blue}{\left(-1 + 1\right) \cdot {\left(1 + x\right)}^{-0.5}} \]
      metadata-eval [=>] 38.2%	\[ \left({x}^{-0.5} - {\left(1 + x\right)}^{-0.5}\right) + \color{blue}{0} \cdot {\left(1 + x\right)}^{-0.5} \]
      mul0-lft [=>] 38.2%	\[ \left({x}^{-0.5} - {\left(1 + x\right)}^{-0.5}\right) + \color{blue}{0} \]
      +-rgt-identity [=>] 38.2%	\[ \color{blue}{{x}^{-0.5} - {\left(1 + x\right)}^{-0.5}} \]

   4. Applied egg-rr 38.2%

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

      [Start] 38.2%	\[ {x}^{-0.5} - {\left(1 + x\right)}^{-0.5} \]
      flip-- [=>] 38.2%	\[ \color{blue}{\frac{{x}^{-0.5} \cdot {x}^{-0.5} - {\left(1 + x\right)}^{-0.5} \cdot {\left(1 + x\right)}^{-0.5}}{{x}^{-0.5} + {\left(1 + x\right)}^{-0.5}}} \]
      pow-prod-up [=>] 25.5%	\[ \frac{\color{blue}{{x}^{\left(-0.5 + -0.5\right)}} - {\left(1 + x\right)}^{-0.5} \cdot {\left(1 + x\right)}^{-0.5}}{{x}^{-0.5} + {\left(1 + x\right)}^{-0.5}} \]
      metadata-eval [=>] 25.5%	\[ \frac{{x}^{\color{blue}{-1}} - {\left(1 + x\right)}^{-0.5} \cdot {\left(1 + x\right)}^{-0.5}}{{x}^{-0.5} + {\left(1 + x\right)}^{-0.5}} \]
      inv-pow [<=] 25.5%	\[ \frac{\color{blue}{\frac{1}{x}} - {\left(1 + x\right)}^{-0.5} \cdot {\left(1 + x\right)}^{-0.5}}{{x}^{-0.5} + {\left(1 + x\right)}^{-0.5}} \]
      pow-prod-up [=>] 38.2%	\[ \frac{\frac{1}{x} - \color{blue}{{\left(1 + x\right)}^{\left(-0.5 + -0.5\right)}}}{{x}^{-0.5} + {\left(1 + x\right)}^{-0.5}} \]
      metadata-eval [=>] 38.2%	\[ \frac{\frac{1}{x} - {\left(1 + x\right)}^{\color{blue}{-1}}}{{x}^{-0.5} + {\left(1 + x\right)}^{-0.5}} \]
      inv-pow [<=] 38.2%	\[ \frac{\frac{1}{x} - \color{blue}{\frac{1}{1 + x}}}{{x}^{-0.5} + {\left(1 + x\right)}^{-0.5}} \]
      +-commutative [<=] 38.2%	\[ \frac{\frac{1}{x} - \frac{1}{\color{blue}{x + 1}}}{{x}^{-0.5} + {\left(1 + x\right)}^{-0.5}} \]
      +-commutative [<=] 38.2%	\[ \frac{\frac{1}{x} - \frac{1}{x + 1}}{{x}^{-0.5} + {\color{blue}{\left(x + 1\right)}}^{-0.5}} \]

   5. Taylor expanded in x around inf 66.1%

      \[\leadsto \color{blue}{0.5 \cdot \sqrt{\frac{1}{{x}^{3}}}} \]

   6. Simplified 100.0%

      \[\leadsto \color{blue}{0.5 \cdot {x}^{-1.5}} \]
      Step-by-step derivation

      [Start] 66.1%	\[ 0.5 \cdot \sqrt{\frac{1}{{x}^{3}}} \]
      rem-exp-log [<=] 63.9%	\[ 0.5 \cdot \sqrt{\frac{1}{\color{blue}{e^{\log \left({x}^{3}\right)}}}} \]
      log-pow [=>] 63.7%	\[ 0.5 \cdot \sqrt{\frac{1}{e^{\color{blue}{3 \cdot \log x}}}} \]
      *-commutative [<=] 63.7%	\[ 0.5 \cdot \sqrt{\frac{1}{e^{\color{blue}{\log x \cdot 3}}}} \]
      exp-neg [<=] 65.2%	\[ 0.5 \cdot \sqrt{\color{blue}{e^{-\log x \cdot 3}}} \]
      distribute-rgt-neg-in [=>] 65.2%	\[ 0.5 \cdot \sqrt{e^{\color{blue}{\log x \cdot \left(-3\right)}}} \]
      metadata-eval [=>] 65.2%	\[ 0.5 \cdot \sqrt{e^{\log x \cdot \color{blue}{-3}}} \]
      exp-to-pow [=>] 67.7%	\[ 0.5 \cdot \sqrt{\color{blue}{{x}^{-3}}} \]
      unpow1/2 [<=] 67.7%	\[ 0.5 \cdot \color{blue}{{\left({x}^{-3}\right)}^{0.5}} \]
      exp-to-pow [<=] 65.2%	\[ 0.5 \cdot {\color{blue}{\left(e^{\log x \cdot -3}\right)}}^{0.5} \]
      *-commutative [=>] 65.2%	\[ 0.5 \cdot {\left(e^{\color{blue}{-3 \cdot \log x}}\right)}^{0.5} \]
      exp-prod [<=] 93.6%	\[ 0.5 \cdot \color{blue}{e^{\left(-3 \cdot \log x\right) \cdot 0.5}} \]
      *-commutative [=>] 93.6%	\[ 0.5 \cdot e^{\color{blue}{0.5 \cdot \left(-3 \cdot \log x\right)}} \]
      associate-*r* [=>] 93.6%	\[ 0.5 \cdot e^{\color{blue}{\left(0.5 \cdot -3\right) \cdot \log x}} \]
      metadata-eval [=>] 93.6%	\[ 0.5 \cdot e^{\color{blue}{-1.5} \cdot \log x} \]
      log-pow [<=] 94.0%	\[ 0.5 \cdot e^{\color{blue}{\log \left({x}^{-1.5}\right)}} \]
      rem-exp-log [=>] 100.0%	\[ 0.5 \cdot \color{blue}{{x}^{-1.5}} \]

   if 0.0 < (-.f64 (/.f64 1 (sqrt.f64 x)) (/.f64 1 (sqrt.f64 (+.f64 x 1))))

   1. Initial program 97.3%

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

   2. Applied egg-rr 97.0%

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

      [Start] 97.3%	\[ \frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}} \]
      sub-neg [=>] 97.3%	\[ \color{blue}{\frac{1}{\sqrt{x}} + \left(-\frac{1}{\sqrt{x + 1}}\right)} \]
      flip-+ [=>] 97.2%	\[ \color{blue}{\frac{\frac{1}{\sqrt{x}} \cdot \frac{1}{\sqrt{x}} - \left(-\frac{1}{\sqrt{x + 1}}\right) \cdot \left(-\frac{1}{\sqrt{x + 1}}\right)}{\frac{1}{\sqrt{x}} - \left(-\frac{1}{\sqrt{x + 1}}\right)}} \]
      frac-times [=>] 96.8%	\[ \frac{\color{blue}{\frac{1 \cdot 1}{\sqrt{x} \cdot \sqrt{x}}} - \left(-\frac{1}{\sqrt{x + 1}}\right) \cdot \left(-\frac{1}{\sqrt{x + 1}}\right)}{\frac{1}{\sqrt{x}} - \left(-\frac{1}{\sqrt{x + 1}}\right)} \]
      metadata-eval [=>] 96.8%	\[ \frac{\frac{\color{blue}{1}}{\sqrt{x} \cdot \sqrt{x}} - \left(-\frac{1}{\sqrt{x + 1}}\right) \cdot \left(-\frac{1}{\sqrt{x + 1}}\right)}{\frac{1}{\sqrt{x}} - \left(-\frac{1}{\sqrt{x + 1}}\right)} \]
      add-sqr-sqrt [<=] 97.1%	\[ \frac{\frac{1}{\color{blue}{x}} - \left(-\frac{1}{\sqrt{x + 1}}\right) \cdot \left(-\frac{1}{\sqrt{x + 1}}\right)}{\frac{1}{\sqrt{x}} - \left(-\frac{1}{\sqrt{x + 1}}\right)} \]
      distribute-neg-frac [=>] 97.1%	\[ \frac{\frac{1}{x} - \color{blue}{\frac{-1}{\sqrt{x + 1}}} \cdot \left(-\frac{1}{\sqrt{x + 1}}\right)}{\frac{1}{\sqrt{x}} - \left(-\frac{1}{\sqrt{x + 1}}\right)} \]
      metadata-eval [=>] 97.1%	\[ \frac{\frac{1}{x} - \frac{\color{blue}{-1}}{\sqrt{x + 1}} \cdot \left(-\frac{1}{\sqrt{x + 1}}\right)}{\frac{1}{\sqrt{x}} - \left(-\frac{1}{\sqrt{x + 1}}\right)} \]
      +-commutative [=>] 97.1%	\[ \frac{\frac{1}{x} - \frac{-1}{\sqrt{\color{blue}{1 + x}}} \cdot \left(-\frac{1}{\sqrt{x + 1}}\right)}{\frac{1}{\sqrt{x}} - \left(-\frac{1}{\sqrt{x + 1}}\right)} \]
      distribute-neg-frac [=>] 97.1%	\[ \frac{\frac{1}{x} - \frac{-1}{\sqrt{1 + x}} \cdot \color{blue}{\frac{-1}{\sqrt{x + 1}}}}{\frac{1}{\sqrt{x}} - \left(-\frac{1}{\sqrt{x + 1}}\right)} \]
      metadata-eval [=>] 97.1%	\[ \frac{\frac{1}{x} - \frac{-1}{\sqrt{1 + x}} \cdot \frac{\color{blue}{-1}}{\sqrt{x + 1}}}{\frac{1}{\sqrt{x}} - \left(-\frac{1}{\sqrt{x + 1}}\right)} \]
      +-commutative [=>] 97.1%	\[ \frac{\frac{1}{x} - \frac{-1}{\sqrt{1 + x}} \cdot \frac{-1}{\sqrt{\color{blue}{1 + x}}}}{\frac{1}{\sqrt{x}} - \left(-\frac{1}{\sqrt{x + 1}}\right)} \]
      pow1/2 [=>] 97.1%	\[ \frac{\frac{1}{x} - \frac{-1}{\sqrt{1 + x}} \cdot \frac{-1}{\sqrt{1 + x}}}{\frac{1}{\color{blue}{{x}^{0.5}}} - \left(-\frac{1}{\sqrt{x + 1}}\right)} \]
      pow-flip [=>] 97.0%	\[ \frac{\frac{1}{x} - \frac{-1}{\sqrt{1 + x}} \cdot \frac{-1}{\sqrt{1 + x}}}{\color{blue}{{x}^{\left(-0.5\right)}} - \left(-\frac{1}{\sqrt{x + 1}}\right)} \]
      metadata-eval [=>] 97.0%	\[ \frac{\frac{1}{x} - \frac{-1}{\sqrt{1 + x}} \cdot \frac{-1}{\sqrt{1 + x}}}{{x}^{\color{blue}{-0.5}} - \left(-\frac{1}{\sqrt{x + 1}}\right)} \]

   3. Simplified 97.1%

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

      [Start] 97.0%	\[ \frac{\frac{1}{x} - \frac{-1}{\sqrt{1 + x}} \cdot \frac{-1}{\sqrt{1 + x}}}{{x}^{-0.5} - \frac{-1}{\sqrt{1 + x}}} \]
      associate-*r/ [=>] 97.0%	\[ \frac{\frac{1}{x} - \color{blue}{\frac{\frac{-1}{\sqrt{1 + x}} \cdot -1}{\sqrt{1 + x}}}}{{x}^{-0.5} - \frac{-1}{\sqrt{1 + x}}} \]
      associate-*l/ [=>] 97.0%	\[ \frac{\frac{1}{x} - \frac{\color{blue}{\frac{-1 \cdot -1}{\sqrt{1 + x}}}}{\sqrt{1 + x}}}{{x}^{-0.5} - \frac{-1}{\sqrt{1 + x}}} \]
      metadata-eval [=>] 97.0%	\[ \frac{\frac{1}{x} - \frac{\frac{\color{blue}{1}}{\sqrt{1 + x}}}{\sqrt{1 + x}}}{{x}^{-0.5} - \frac{-1}{\sqrt{1 + x}}} \]
      associate-/l/ [=>] 97.0%	\[ \frac{\frac{1}{x} - \color{blue}{\frac{1}{\sqrt{1 + x} \cdot \sqrt{1 + x}}}}{{x}^{-0.5} - \frac{-1}{\sqrt{1 + x}}} \]
      rem-square-sqrt [=>] 97.1%	\[ \frac{\frac{1}{x} - \frac{1}{\color{blue}{1 + x}}}{{x}^{-0.5} - \frac{-1}{\sqrt{1 + x}}} \]
      sub-neg [=>] 97.1%	\[ \frac{\color{blue}{\frac{1}{x} + \left(-\frac{1}{1 + x}\right)}}{{x}^{-0.5} - \frac{-1}{\sqrt{1 + x}}} \]
      distribute-neg-frac [=>] 97.1%	\[ \frac{\frac{1}{x} + \color{blue}{\frac{-1}{1 + x}}}{{x}^{-0.5} - \frac{-1}{\sqrt{1 + x}}} \]
      metadata-eval [=>] 97.1%	\[ \frac{\frac{1}{x} + \frac{\color{blue}{-1}}{1 + x}}{{x}^{-0.5} - \frac{-1}{\sqrt{1 + x}}} \]
      *-rgt-identity [<=] 97.1%	\[ \frac{\frac{1}{x} + \frac{-1}{1 + x}}{{x}^{-0.5} - \color{blue}{\frac{-1}{\sqrt{1 + x}} \cdot 1}} \]
      metadata-eval [<=] 97.1%	\[ \frac{\frac{1}{x} + \frac{-1}{1 + x}}{{x}^{-0.5} - \frac{-1}{\sqrt{1 + x}} \cdot \color{blue}{\left(--1\right)}} \]
      distribute-rgt-neg-in [<=] 97.1%	\[ \frac{\frac{1}{x} + \frac{-1}{1 + x}}{{x}^{-0.5} - \color{blue}{\left(-\frac{-1}{\sqrt{1 + x}} \cdot -1\right)}} \]
      neg-mul-1 [=>] 97.1%	\[ \frac{\frac{1}{x} + \frac{-1}{1 + x}}{{x}^{-0.5} - \color{blue}{-1 \cdot \left(\frac{-1}{\sqrt{1 + x}} \cdot -1\right)}} \]
      cancel-sign-sub-inv [=>] 97.1%	\[ \frac{\frac{1}{x} + \frac{-1}{1 + x}}{\color{blue}{{x}^{-0.5} + \left(--1\right) \cdot \left(\frac{-1}{\sqrt{1 + x}} \cdot -1\right)}} \]
      metadata-eval [=>] 97.1%	\[ \frac{\frac{1}{x} + \frac{-1}{1 + x}}{{x}^{-0.5} + \color{blue}{1} \cdot \left(\frac{-1}{\sqrt{1 + x}} \cdot -1\right)} \]
      *-lft-identity [=>] 97.1%	\[ \frac{\frac{1}{x} + \frac{-1}{1 + x}}{{x}^{-0.5} + \color{blue}{\frac{-1}{\sqrt{1 + x}} \cdot -1}} \]

   4. Applied egg-rr 99.2%

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

      [Start] 97.1%	\[ \frac{\frac{1}{x} + \frac{-1}{1 + x}}{{x}^{-0.5} + \frac{1}{\mathsf{hypot}\left(1, \sqrt{x}\right)}} \]
      frac-add [=>] 99.2%	\[ \frac{\color{blue}{\frac{1 \cdot \left(1 + x\right) + x \cdot -1}{x \cdot \left(1 + x\right)}}}{{x}^{-0.5} + \frac{1}{\mathsf{hypot}\left(1, \sqrt{x}\right)}} \]
      *-un-lft-identity [<=] 99.2%	\[ \frac{\frac{\color{blue}{\left(1 + x\right)} + x \cdot -1}{x \cdot \left(1 + x\right)}}{{x}^{-0.5} + \frac{1}{\mathsf{hypot}\left(1, \sqrt{x}\right)}} \]
      +-commutative [<=] 99.2%	\[ \frac{\frac{\color{blue}{\left(x + 1\right)} + x \cdot -1}{x \cdot \left(1 + x\right)}}{{x}^{-0.5} + \frac{1}{\mathsf{hypot}\left(1, \sqrt{x}\right)}} \]
      +-commutative [<=] 99.2%	\[ \frac{\frac{\left(x + 1\right) + x \cdot -1}{x \cdot \color{blue}{\left(x + 1\right)}}}{{x}^{-0.5} + \frac{1}{\mathsf{hypot}\left(1, \sqrt{x}\right)}} \]

   5. Simplified 99.2%

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

      [Start] 99.2%	\[ \frac{\frac{\left(x + 1\right) + x \cdot -1}{x \cdot \left(x + 1\right)}}{{x}^{-0.5} + \frac{1}{\mathsf{hypot}\left(1, \sqrt{x}\right)}} \]
      /-rgt-identity [<=] 99.2%	\[ \frac{\frac{\left(x + 1\right) + \color{blue}{\frac{x}{1}} \cdot -1}{x \cdot \left(x + 1\right)}}{{x}^{-0.5} + \frac{1}{\mathsf{hypot}\left(1, \sqrt{x}\right)}} \]
      /-rgt-identity [=>] 99.2%	\[ \frac{\frac{\left(x + 1\right) + \color{blue}{x} \cdot -1}{x \cdot \left(x + 1\right)}}{{x}^{-0.5} + \frac{1}{\mathsf{hypot}\left(1, \sqrt{x}\right)}} \]
      *-commutative [<=] 99.2%	\[ \frac{\frac{\left(x + 1\right) + \color{blue}{-1 \cdot x}}{x \cdot \left(x + 1\right)}}{{x}^{-0.5} + \frac{1}{\mathsf{hypot}\left(1, \sqrt{x}\right)}} \]
      neg-mul-1 [<=] 99.2%	\[ \frac{\frac{\left(x + 1\right) + \color{blue}{\left(-x\right)}}{x \cdot \left(x + 1\right)}}{{x}^{-0.5} + \frac{1}{\mathsf{hypot}\left(1, \sqrt{x}\right)}} \]
      associate-+r+ [<=] 99.2%	\[ \frac{\frac{\color{blue}{x + \left(1 + \left(-x\right)\right)}}{x \cdot \left(x + 1\right)}}{{x}^{-0.5} + \frac{1}{\mathsf{hypot}\left(1, \sqrt{x}\right)}} \]
      sub-neg [<=] 99.2%	\[ \frac{\frac{x + \color{blue}{\left(1 - x\right)}}{x \cdot \left(x + 1\right)}}{{x}^{-0.5} + \frac{1}{\mathsf{hypot}\left(1, \sqrt{x}\right)}} \]
      distribute-lft-in [=>] 99.2%	\[ \frac{\frac{x + \left(1 - x\right)}{\color{blue}{x \cdot x + x \cdot 1}}}{{x}^{-0.5} + \frac{1}{\mathsf{hypot}\left(1, \sqrt{x}\right)}} \]
      *-rgt-identity [=>] 99.2%	\[ \frac{\frac{x + \left(1 - x\right)}{x \cdot x + \color{blue}{x}}}{{x}^{-0.5} + \frac{1}{\mathsf{hypot}\left(1, \sqrt{x}\right)}} \]

3. Recombined 2 regimes into one program.

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{\sqrt{x}} + \frac{-1}{\sqrt{1 + x}} \leq 0:\\ \;\;\;\;0.5 \cdot {x}^{-1.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x + \left(1 - x\right)}{x + x \cdot x}}{{x}^{-0.5} + \frac{1}{\mathsf{hypot}\left(1, \sqrt{x}\right)}}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	99.6%
Cost	33860

\[\begin{array}{l} \mathbf{if}\;\frac{1}{\sqrt{x}} + \frac{-1}{\sqrt{1 + x}} \leq 0:\\ \;\;\;\;0.5 \cdot {x}^{-1.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x + \left(1 - x\right)}{x + x \cdot x}}{{x}^{-0.5} + \frac{1}{\mathsf{hypot}\left(1, \sqrt{x}\right)}}\\ \end{array} \]

Alternative 2
Accuracy	99.6%
Cost	27524

\[\begin{array}{l} t_0 := \sqrt{1 + x}\\ \mathbf{if}\;\frac{1}{\sqrt{x}} + \frac{-1}{t_0} \leq 0:\\ \;\;\;\;0.5 \cdot {x}^{-1.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x + \left(1 - x\right)}{x + x \cdot x}}{\frac{1}{t_0} + {x}^{-0.5}}\\ \end{array} \]

Alternative 3
Accuracy	99.5%
Cost	26692

\[\begin{array}{l} \mathbf{if}\;\frac{1}{\sqrt{x}} + \frac{-1}{\sqrt{1 + x}} \leq 5 \cdot 10^{-15}:\\ \;\;\;\;0.5 \cdot {x}^{-1.5}\\ \mathbf{else}:\\ \;\;\;\;{x}^{-0.5} - {\left(1 + x\right)}^{-0.5}\\ \end{array} \]

Alternative 4
Accuracy	98.6%
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 5
Accuracy	98.3%
Cost	6788

\[\begin{array}{l} \mathbf{if}\;x \leq 0.68:\\ \;\;\;\;{x}^{-0.5} + -1\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot {x}^{-1.5}\\ \end{array} \]

Alternative 6
Accuracy	52.6%
Cost	6656

\[0.5 \cdot {x}^{-1.5} \]

Alternative 7
Accuracy	1.9%
Cost	64

\[-1 \]

Reproduce

herbie shell --seed 2023271 
(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)))))