2isqrt (example 3.6)

Average Accuracy: 69.2% → 99.8%

Time: 12.4s

Precision: binary64

Cost: 39684

Math

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

↓

\[\begin{array}{l} t_0 := \sqrt{x + 1}\\ t_1 := \sqrt{x} + t_0\\ \mathbf{if}\;\frac{1}{\sqrt{x}} + \frac{-1}{t_0} \leq 0:\\ \;\;\;\;\frac{\frac{1}{t_1}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(\mathsf{fma}\left(x, x, x\right)\right)}^{-0.5}}{t_1}\\ \end{array} \]

FPCore

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

↓

(FPCore (x)
 :precision binary64
 (let* ((t_0 (sqrt (+ x 1.0))) (t_1 (+ (sqrt x) t_0)))
   (if (<= (+ (/ 1.0 (sqrt x)) (/ -1.0 t_0)) 0.0)
     (/ (/ 1.0 t_1) x)
     (/ (pow (fma x x x) -0.5) t_1))))

C

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

↓

double code(double x) {
	double t_0 = sqrt((x + 1.0));
	double t_1 = sqrt(x) + t_0;
	double tmp;
	if (((1.0 / sqrt(x)) + (-1.0 / t_0)) <= 0.0) {
		tmp = (1.0 / t_1) / x;
	} else {
		tmp = pow(fma(x, x, x), -0.5) / t_1;
	}
	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 = sqrt(Float64(x + 1.0))
	t_1 = Float64(sqrt(x) + t_0)
	tmp = 0.0
	if (Float64(Float64(1.0 / sqrt(x)) + Float64(-1.0 / t_0)) <= 0.0)
		tmp = Float64(Float64(1.0 / t_1) / x);
	else
		tmp = Float64((fma(x, x, x) ^ -0.5) / t_1);
	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_] := Block[{t$95$0 = N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[Sqrt[x], $MachinePrecision] + t$95$0), $MachinePrecision]}, If[LessEqual[N[(N[(1.0 / N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + N[(-1.0 / t$95$0), $MachinePrecision]), $MachinePrecision], 0.0], N[(N[(1.0 / t$95$1), $MachinePrecision] / x), $MachinePrecision], N[(N[Power[N[(x * x + x), $MachinePrecision], -0.5], $MachinePrecision] / t$95$1), $MachinePrecision]]]]

Target

Original	69.2%
Target	99.0%
Herbie	99.8%

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

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 36.4%

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

   2. Applied egg-rr 36.4%

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

      [Start] 36.4	\[ \frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}} \]
      frac-sub [=>] 36.4	\[ \color{blue}{\frac{1 \cdot \sqrt{x + 1} - \sqrt{x} \cdot 1}{\sqrt{x} \cdot \sqrt{x + 1}}} \]
      clear-num [=>] 36.4	\[ \color{blue}{\frac{1}{\frac{\sqrt{x} \cdot \sqrt{x + 1}}{1 \cdot \sqrt{x + 1} - \sqrt{x} \cdot 1}}} \]
      sqrt-unprod [=>] 36.4	\[ \frac{1}{\frac{\color{blue}{\sqrt{x \cdot \left(x + 1\right)}}}{1 \cdot \sqrt{x + 1} - \sqrt{x} \cdot 1}} \]
      +-commutative [=>] 36.4	\[ \frac{1}{\frac{\sqrt{x \cdot \color{blue}{\left(1 + x\right)}}}{1 \cdot \sqrt{x + 1} - \sqrt{x} \cdot 1}} \]
      *-un-lft-identity [<=] 36.4	\[ \frac{1}{\frac{\sqrt{x \cdot \left(1 + x\right)}}{\color{blue}{\sqrt{x + 1}} - \sqrt{x} \cdot 1}} \]
      *-rgt-identity [=>] 36.4	\[ \frac{1}{\frac{\sqrt{x \cdot \left(1 + x\right)}}{\sqrt{x + 1} - \color{blue}{\sqrt{x}}}} \]
      +-commutative [=>] 36.4	\[ \frac{1}{\frac{\sqrt{x \cdot \left(1 + x\right)}}{\sqrt{\color{blue}{1 + x}} - \sqrt{x}}} \]

   3. Simplified 36.4%

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

      [Start] 36.4	\[ \frac{1}{\frac{\sqrt{x \cdot \left(1 + x\right)}}{\sqrt{1 + x} - \sqrt{x}}} \]
      associate-/r/ [=>] 36.4	\[ \color{blue}{\frac{1}{\sqrt{x \cdot \left(1 + x\right)}} \cdot \left(\sqrt{1 + x} - \sqrt{x}\right)} \]
      associate-*l/ [=>] 36.4	\[ \color{blue}{\frac{1 \cdot \left(\sqrt{1 + x} - \sqrt{x}\right)}{\sqrt{x \cdot \left(1 + x\right)}}} \]
      *-lft-identity [=>] 36.4	\[ \frac{\color{blue}{\sqrt{1 + x} - \sqrt{x}}}{\sqrt{x \cdot \left(1 + x\right)}} \]
      distribute-rgt-in [=>] 36.4	\[ \frac{\sqrt{1 + x} - \sqrt{x}}{\sqrt{\color{blue}{1 \cdot x + x \cdot x}}} \]
      *-lft-identity [=>] 36.4	\[ \frac{\sqrt{1 + x} - \sqrt{x}}{\sqrt{\color{blue}{x} + x \cdot x}} \]

   4. Applied egg-rr 36.5%

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

      [Start] 36.4	\[ \frac{\sqrt{1 + x} - \sqrt{x}}{\sqrt{x + x \cdot x}} \]
      flip-- [=>] 36.4	\[ \frac{\color{blue}{\frac{\sqrt{1 + x} \cdot \sqrt{1 + x} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{1 + x} + \sqrt{x}}}}{\sqrt{x + x \cdot x}} \]
      div-inv [=>] 36.4	\[ \frac{\color{blue}{\left(\sqrt{1 + x} \cdot \sqrt{1 + x} - \sqrt{x} \cdot \sqrt{x}\right) \cdot \frac{1}{\sqrt{1 + x} + \sqrt{x}}}}{\sqrt{x + x \cdot x}} \]
      add-sqr-sqrt [<=] 36.5	\[ \frac{\left(\color{blue}{\left(1 + x\right)} - \sqrt{x} \cdot \sqrt{x}\right) \cdot \frac{1}{\sqrt{1 + x} + \sqrt{x}}}{\sqrt{x + x \cdot x}} \]
      +-commutative [=>] 36.5	\[ \frac{\left(\color{blue}{\left(x + 1\right)} - \sqrt{x} \cdot \sqrt{x}\right) \cdot \frac{1}{\sqrt{1 + x} + \sqrt{x}}}{\sqrt{x + x \cdot x}} \]
      add-sqr-sqrt [<=] 36.5	\[ \frac{\left(\left(x + 1\right) - \color{blue}{x}\right) \cdot \frac{1}{\sqrt{1 + x} + \sqrt{x}}}{\sqrt{x + x \cdot x}} \]
      associate--l+ [=>] 36.5	\[ \frac{\color{blue}{\left(x + \left(1 - x\right)\right)} \cdot \frac{1}{\sqrt{1 + x} + \sqrt{x}}}{\sqrt{x + x \cdot x}} \]
      +-commutative [=>] 36.5	\[ \frac{\left(x + \left(1 - x\right)\right) \cdot \frac{1}{\sqrt{\color{blue}{x + 1}} + \sqrt{x}}}{\sqrt{x + x \cdot x}} \]

   5. Simplified 82.4%

      \[\leadsto \frac{\color{blue}{\frac{1}{\sqrt{x} + \sqrt{x + 1}}}}{\sqrt{x + x \cdot x}} \]
      Proof

      [Start] 36.5	\[ \frac{\left(x + \left(1 - x\right)\right) \cdot \frac{1}{\sqrt{x + 1} + \sqrt{x}}}{\sqrt{x + x \cdot x}} \]
      associate-*r/ [=>] 36.5	\[ \frac{\color{blue}{\frac{\left(x + \left(1 - x\right)\right) \cdot 1}{\sqrt{x + 1} + \sqrt{x}}}}{\sqrt{x + x \cdot x}} \]
      *-rgt-identity [=>] 36.5	\[ \frac{\frac{\color{blue}{x + \left(1 - x\right)}}{\sqrt{x + 1} + \sqrt{x}}}{\sqrt{x + x \cdot x}} \]
      +-commutative [=>] 36.5	\[ \frac{\frac{\color{blue}{\left(1 - x\right) + x}}{\sqrt{x + 1} + \sqrt{x}}}{\sqrt{x + x \cdot x}} \]
      associate-+l- [=>] 82.4	\[ \frac{\frac{\color{blue}{1 - \left(x - x\right)}}{\sqrt{x + 1} + \sqrt{x}}}{\sqrt{x + x \cdot x}} \]
      +-inverses [=>] 82.4	\[ \frac{\frac{1 - \color{blue}{0}}{\sqrt{x + 1} + \sqrt{x}}}{\sqrt{x + x \cdot x}} \]
      metadata-eval [=>] 82.4	\[ \frac{\frac{\color{blue}{1}}{\sqrt{x + 1} + \sqrt{x}}}{\sqrt{x + x \cdot x}} \]
      +-commutative [=>] 82.4	\[ \frac{\frac{1}{\color{blue}{\sqrt{x} + \sqrt{x + 1}}}}{\sqrt{x + x \cdot x}} \]

   6. Taylor expanded in x around inf 99.6%

      \[\leadsto \frac{\frac{1}{\sqrt{x} + \sqrt{x + 1}}}{\color{blue}{x}} \]

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

   1. Initial program 97.5%

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

   2. Applied egg-rr 97.5%

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

      [Start] 97.5	\[ \frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}} \]
      frac-sub [=>] 97.5	\[ \color{blue}{\frac{1 \cdot \sqrt{x + 1} - \sqrt{x} \cdot 1}{\sqrt{x} \cdot \sqrt{x + 1}}} \]
      clear-num [=>] 97.5	\[ \color{blue}{\frac{1}{\frac{\sqrt{x} \cdot \sqrt{x + 1}}{1 \cdot \sqrt{x + 1} - \sqrt{x} \cdot 1}}} \]
      sqrt-unprod [=>] 97.5	\[ \frac{1}{\frac{\color{blue}{\sqrt{x \cdot \left(x + 1\right)}}}{1 \cdot \sqrt{x + 1} - \sqrt{x} \cdot 1}} \]
      +-commutative [=>] 97.5	\[ \frac{1}{\frac{\sqrt{x \cdot \color{blue}{\left(1 + x\right)}}}{1 \cdot \sqrt{x + 1} - \sqrt{x} \cdot 1}} \]
      *-un-lft-identity [<=] 97.5	\[ \frac{1}{\frac{\sqrt{x \cdot \left(1 + x\right)}}{\color{blue}{\sqrt{x + 1}} - \sqrt{x} \cdot 1}} \]
      *-rgt-identity [=>] 97.5	\[ \frac{1}{\frac{\sqrt{x \cdot \left(1 + x\right)}}{\sqrt{x + 1} - \color{blue}{\sqrt{x}}}} \]
      +-commutative [=>] 97.5	\[ \frac{1}{\frac{\sqrt{x \cdot \left(1 + x\right)}}{\sqrt{\color{blue}{1 + x}} - \sqrt{x}}} \]

   3. Simplified 97.5%

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

      [Start] 97.5	\[ \frac{1}{\frac{\sqrt{x \cdot \left(1 + x\right)}}{\sqrt{1 + x} - \sqrt{x}}} \]
      associate-/r/ [=>] 97.5	\[ \color{blue}{\frac{1}{\sqrt{x \cdot \left(1 + x\right)}} \cdot \left(\sqrt{1 + x} - \sqrt{x}\right)} \]
      associate-*l/ [=>] 97.5	\[ \color{blue}{\frac{1 \cdot \left(\sqrt{1 + x} - \sqrt{x}\right)}{\sqrt{x \cdot \left(1 + x\right)}}} \]
      *-lft-identity [=>] 97.5	\[ \frac{\color{blue}{\sqrt{1 + x} - \sqrt{x}}}{\sqrt{x \cdot \left(1 + x\right)}} \]
      distribute-rgt-in [=>] 97.5	\[ \frac{\sqrt{1 + x} - \sqrt{x}}{\sqrt{\color{blue}{1 \cdot x + x \cdot x}}} \]
      *-lft-identity [=>] 97.5	\[ \frac{\sqrt{1 + x} - \sqrt{x}}{\sqrt{\color{blue}{x} + x \cdot x}} \]

   4. Applied egg-rr 99.5%

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

      [Start] 97.5	\[ \frac{\sqrt{1 + x} - \sqrt{x}}{\sqrt{x + x \cdot x}} \]
      flip-- [=>] 98.2	\[ \frac{\color{blue}{\frac{\sqrt{1 + x} \cdot \sqrt{1 + x} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{1 + x} + \sqrt{x}}}}{\sqrt{x + x \cdot x}} \]
      div-inv [=>] 98.2	\[ \frac{\color{blue}{\left(\sqrt{1 + x} \cdot \sqrt{1 + x} - \sqrt{x} \cdot \sqrt{x}\right) \cdot \frac{1}{\sqrt{1 + x} + \sqrt{x}}}}{\sqrt{x + x \cdot x}} \]
      add-sqr-sqrt [<=] 98.6	\[ \frac{\left(\color{blue}{\left(1 + x\right)} - \sqrt{x} \cdot \sqrt{x}\right) \cdot \frac{1}{\sqrt{1 + x} + \sqrt{x}}}{\sqrt{x + x \cdot x}} \]
      +-commutative [=>] 98.6	\[ \frac{\left(\color{blue}{\left(x + 1\right)} - \sqrt{x} \cdot \sqrt{x}\right) \cdot \frac{1}{\sqrt{1 + x} + \sqrt{x}}}{\sqrt{x + x \cdot x}} \]
      add-sqr-sqrt [<=] 99.5	\[ \frac{\left(\left(x + 1\right) - \color{blue}{x}\right) \cdot \frac{1}{\sqrt{1 + x} + \sqrt{x}}}{\sqrt{x + x \cdot x}} \]
      associate--l+ [=>] 99.5	\[ \frac{\color{blue}{\left(x + \left(1 - x\right)\right)} \cdot \frac{1}{\sqrt{1 + x} + \sqrt{x}}}{\sqrt{x + x \cdot x}} \]
      +-commutative [=>] 99.5	\[ \frac{\left(x + \left(1 - x\right)\right) \cdot \frac{1}{\sqrt{\color{blue}{x + 1}} + \sqrt{x}}}{\sqrt{x + x \cdot x}} \]

   5. Simplified 99.5%

      \[\leadsto \frac{\color{blue}{\frac{1}{\sqrt{x} + \sqrt{x + 1}}}}{\sqrt{x + x \cdot x}} \]
      Proof

      [Start] 99.5	\[ \frac{\left(x + \left(1 - x\right)\right) \cdot \frac{1}{\sqrt{x + 1} + \sqrt{x}}}{\sqrt{x + x \cdot x}} \]
      associate-*r/ [=>] 99.5	\[ \frac{\color{blue}{\frac{\left(x + \left(1 - x\right)\right) \cdot 1}{\sqrt{x + 1} + \sqrt{x}}}}{\sqrt{x + x \cdot x}} \]
      *-rgt-identity [=>] 99.5	\[ \frac{\frac{\color{blue}{x + \left(1 - x\right)}}{\sqrt{x + 1} + \sqrt{x}}}{\sqrt{x + x \cdot x}} \]
      +-commutative [=>] 99.5	\[ \frac{\frac{\color{blue}{\left(1 - x\right) + x}}{\sqrt{x + 1} + \sqrt{x}}}{\sqrt{x + x \cdot x}} \]
      associate-+l- [=>] 99.5	\[ \frac{\frac{\color{blue}{1 - \left(x - x\right)}}{\sqrt{x + 1} + \sqrt{x}}}{\sqrt{x + x \cdot x}} \]
      +-inverses [=>] 99.5	\[ \frac{\frac{1 - \color{blue}{0}}{\sqrt{x + 1} + \sqrt{x}}}{\sqrt{x + x \cdot x}} \]
      metadata-eval [=>] 99.5	\[ \frac{\frac{\color{blue}{1}}{\sqrt{x + 1} + \sqrt{x}}}{\sqrt{x + x \cdot x}} \]
      +-commutative [=>] 99.5	\[ \frac{\frac{1}{\color{blue}{\sqrt{x} + \sqrt{x + 1}}}}{\sqrt{x + x \cdot x}} \]

   6. Applied egg-rr 99.9%

      \[\leadsto \color{blue}{0 + \frac{{\left(\mathsf{fma}\left(x, x, x\right)\right)}^{-0.5}}{\sqrt{x} + \sqrt{1 + x}}} \]
      Proof

      [Start] 99.5	\[ \frac{\frac{1}{\sqrt{x} + \sqrt{x + 1}}}{\sqrt{x + x \cdot x}} \]
      add-log-exp [=>] 3.8	\[ \color{blue}{\log \left(e^{\frac{\frac{1}{\sqrt{x} + \sqrt{x + 1}}}{\sqrt{x + x \cdot x}}}\right)} \]
      *-un-lft-identity [=>] 3.8	\[ \log \color{blue}{\left(1 \cdot e^{\frac{\frac{1}{\sqrt{x} + \sqrt{x + 1}}}{\sqrt{x + x \cdot x}}}\right)} \]
      log-prod [=>] 3.8	\[ \color{blue}{\log 1 + \log \left(e^{\frac{\frac{1}{\sqrt{x} + \sqrt{x + 1}}}{\sqrt{x + x \cdot x}}}\right)} \]
      metadata-eval [=>] 3.8	\[ \color{blue}{0} + \log \left(e^{\frac{\frac{1}{\sqrt{x} + \sqrt{x + 1}}}{\sqrt{x + x \cdot x}}}\right) \]
      add-log-exp [<=] 99.5	\[ 0 + \color{blue}{\frac{\frac{1}{\sqrt{x} + \sqrt{x + 1}}}{\sqrt{x + x \cdot x}}} \]
      associate-/l/ [=>] 99.5	\[ 0 + \color{blue}{\frac{1}{\sqrt{x + x \cdot x} \cdot \left(\sqrt{x} + \sqrt{x + 1}\right)}} \]
      associate-/r* [=>] 99.5	\[ 0 + \color{blue}{\frac{\frac{1}{\sqrt{x + x \cdot x}}}{\sqrt{x} + \sqrt{x + 1}}} \]
      pow1/2 [=>] 99.5	\[ 0 + \frac{\frac{1}{\color{blue}{{\left(x + x \cdot x\right)}^{0.5}}}}{\sqrt{x} + \sqrt{x + 1}} \]
      pow-flip [=>] 99.9	\[ 0 + \frac{\color{blue}{{\left(x + x \cdot x\right)}^{\left(-0.5\right)}}}{\sqrt{x} + \sqrt{x + 1}} \]
      +-commutative [=>] 99.9	\[ 0 + \frac{{\color{blue}{\left(x \cdot x + x\right)}}^{\left(-0.5\right)}}{\sqrt{x} + \sqrt{x + 1}} \]
      fma-def [=>] 99.9	\[ 0 + \frac{{\color{blue}{\left(\mathsf{fma}\left(x, x, x\right)\right)}}^{\left(-0.5\right)}}{\sqrt{x} + \sqrt{x + 1}} \]
      metadata-eval [=>] 99.9	\[ 0 + \frac{{\left(\mathsf{fma}\left(x, x, x\right)\right)}^{\color{blue}{-0.5}}}{\sqrt{x} + \sqrt{x + 1}} \]
      +-commutative [=>] 99.9	\[ 0 + \frac{{\left(\mathsf{fma}\left(x, x, x\right)\right)}^{-0.5}}{\sqrt{x} + \sqrt{\color{blue}{1 + x}}} \]

   7. Simplified 99.9%

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

      [Start] 99.9	\[ 0 + \frac{{\left(\mathsf{fma}\left(x, x, x\right)\right)}^{-0.5}}{\sqrt{x} + \sqrt{1 + x}} \]
      +-lft-identity [=>] 99.9	\[ \color{blue}{\frac{{\left(\mathsf{fma}\left(x, x, x\right)\right)}^{-0.5}}{\sqrt{x} + \sqrt{1 + x}}} \]

3. Recombined 2 regimes into one program.

4. Final simplification 99.8%

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

Alternatives

Alternative 1
Accuracy	99.5%
Cost	52224

\[\begin{array}{l} t_0 := {\left(\sqrt{x} + \sqrt{x + 1}\right)}^{-0.5}\\ t_0 \cdot \frac{t_0}{\mathsf{hypot}\left(x, \sqrt{x}\right)} \end{array} \]

Alternative 2
Accuracy	99.2%
Cost	33088

\[\begin{array}{l} t_0 := \sqrt{x + 1}\\ \frac{\frac{1}{\sqrt{x} + t_0}}{{x}^{0.25} \cdot \left(t_0 \cdot {x}^{0.25}\right)} \end{array} \]

Alternative 3
Accuracy	98.9%
Cost	32576

\[{\left(\left(\sqrt{x} + \sqrt{x + 1}\right) \cdot \mathsf{hypot}\left(x, \sqrt{x}\right)\right)}^{-1} \]

Alternative 4
Accuracy	99.6%
Cost	26948

\[\begin{array}{l} t_0 := \sqrt{x + 1}\\ \mathbf{if}\;\frac{1}{\sqrt{x}} + \frac{-1}{t_0} \leq 10^{-11}:\\ \;\;\;\;\frac{\frac{1}{\sqrt{x} + t_0}}{x + 0.5}\\ \mathbf{else}:\\ \;\;\;\;{x}^{-0.5} - {\left(x + 1\right)}^{-0.5}\\ \end{array} \]

Alternative 5
Accuracy	99.3%
Cost	26820

\[\begin{array}{l} t_0 := \sqrt{x + 1}\\ \mathbf{if}\;\frac{1}{\sqrt{x}} + \frac{-1}{t_0} \leq 5 \cdot 10^{-13}:\\ \;\;\;\;\frac{\frac{1}{\sqrt{x} + t_0}}{x}\\ \mathbf{else}:\\ \;\;\;\;{x}^{-0.5} - {\left(x + 1\right)}^{-0.5}\\ \end{array} \]

Alternative 6
Accuracy	83.4%
Cost	13380

\[\begin{array}{l} \mathbf{if}\;x \leq 52000000:\\ \;\;\;\;{x}^{-0.5} - {\left(x + 1\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{\frac{1}{{x}^{3}}}\\ \end{array} \]

Alternative 7
Accuracy	82.5%
Cost	13316

\[\begin{array}{l} \mathbf{if}\;x \leq 1.7:\\ \;\;\;\;{x}^{-0.5} + \frac{-1}{1 + x \cdot 0.5}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{\frac{1}{{x}^{3}}}\\ \end{array} \]

Alternative 8
Accuracy	68.5%
Cost	7492

\[\begin{array}{l} \mathbf{if}\;x \leq 1.22:\\ \;\;\;\;{x}^{-0.5} + \frac{-1}{1 + x \cdot 0.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x} + \frac{-1}{x + 1}}{2 \cdot \sqrt{\frac{1}{x}}}\\ \end{array} \]

Alternative 9
Accuracy	68.5%
Cost	7172

\[\begin{array}{l} \mathbf{if}\;x \leq 5.9:\\ \;\;\;\;{x}^{-0.5} + \frac{-1}{1 + x \cdot 0.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{x + x \cdot x}}\\ \end{array} \]

Alternative 10
Accuracy	68.4%
Cost	7044

\[\begin{array}{l} \mathbf{if}\;x \leq 1.4:\\ \;\;\;\;{x}^{-0.5} + \left(-1 - x \cdot -0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{x + x \cdot x}}\\ \end{array} \]

Alternative 11
Accuracy	68.1%
Cost	6980

\[\begin{array}{l} \mathbf{if}\;x \leq 0.7:\\ \;\;\;\;{x}^{-0.5} + -1\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{x + x \cdot x}}\\ \end{array} \]

Alternative 12
Accuracy	67.1%
Cost	6788

\[\begin{array}{l} \mathbf{if}\;x \leq 1:\\ \;\;\;\;{x}^{-0.5} + -1\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 13
Accuracy	66.2%
Cost	6724

\[\begin{array}{l} \mathbf{if}\;x \leq 8.5 \cdot 10^{+122}:\\ \;\;\;\;\frac{1}{\sqrt{x}}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 14
Accuracy	21.6%
Cost	452

\[\begin{array}{l} \mathbf{if}\;x \leq 4.6 \cdot 10^{+153}:\\ \;\;\;\;\frac{1}{x + 0.5}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 15
Accuracy	21.6%
Cost	324

\[\begin{array}{l} \mathbf{if}\;x \leq 4.6 \cdot 10^{+153}:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 16
Accuracy	18.3%
Cost	64

\[0 \]

Reproduce

herbie shell --seed 2023151 
(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)))))