2isqrt (example 3.6)

Average Accuracy: 68.7% → 99.8%

Time: 14.9s

Precision: binary64

Cost: 26372

Math

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

↓

\[\begin{array}{l} t_0 := \sqrt{x} + \sqrt{1 + x}\\ \mathbf{if}\;x \leq 1.5 \cdot 10^{+38}:\\ \;\;\;\;\frac{{\left(\mathsf{fma}\left(x, x, x\right)\right)}^{-0.5}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-1}{t_0}}{-0.5 - x}\\ \end{array} \]

FPCore

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

↓

(FPCore (x)
 :precision binary64
 (let* ((t_0 (+ (sqrt x) (sqrt (+ 1.0 x)))))
   (if (<= x 1.5e+38)
     (/ (pow (fma x x x) -0.5) t_0)
     (/ (/ -1.0 t_0) (- -0.5 x)))))

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) + sqrt((1.0 + x));
	double tmp;
	if (x <= 1.5e+38) {
		tmp = pow(fma(x, x, x), -0.5) / t_0;
	} else {
		tmp = (-1.0 / t_0) / (-0.5 - 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)
	t_0 = Float64(sqrt(x) + sqrt(Float64(1.0 + x)))
	tmp = 0.0
	if (x <= 1.5e+38)
		tmp = Float64((fma(x, x, x) ^ -0.5) / t_0);
	else
		tmp = Float64(Float64(-1.0 / t_0) / Float64(-0.5 - 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_] := Block[{t$95$0 = N[(N[Sqrt[x], $MachinePrecision] + N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, 1.5e+38], N[(N[Power[N[(x * x + x), $MachinePrecision], -0.5], $MachinePrecision] / t$95$0), $MachinePrecision], N[(N[(-1.0 / t$95$0), $MachinePrecision] / N[(-0.5 - x), $MachinePrecision]), $MachinePrecision]]]

Target

Original	68.7%
Target	98.9%
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 x < 1.5000000000000001e38

   1. Initial program 91.5%

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

   2. Applied egg-rr 99.5%

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

      [Start] 91.5	\[ \frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}} \]
      frac-sub [=>] 91.6	\[ \color{blue}{\frac{1 \cdot \sqrt{x + 1} - \sqrt{x} \cdot 1}{\sqrt{x} \cdot \sqrt{x + 1}}} \]
      *-un-lft-identity [<=] 91.6	\[ \frac{\color{blue}{\sqrt{x + 1}} - \sqrt{x} \cdot 1}{\sqrt{x} \cdot \sqrt{x + 1}} \]
      *-rgt-identity [=>] 91.6	\[ \frac{\sqrt{x + 1} - \color{blue}{\sqrt{x}}}{\sqrt{x} \cdot \sqrt{x + 1}} \]
      flip-- [=>] 92.2	\[ \frac{\color{blue}{\frac{\sqrt{x + 1} \cdot \sqrt{x + 1} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}}}}{\sqrt{x} \cdot \sqrt{x + 1}} \]
      associate-/l/ [=>] 92.2	\[ \color{blue}{\frac{\sqrt{x + 1} \cdot \sqrt{x + 1} - \sqrt{x} \cdot \sqrt{x}}{\left(\sqrt{x} \cdot \sqrt{x + 1}\right) \cdot \left(\sqrt{x + 1} + \sqrt{x}\right)}} \]
      add-sqr-sqrt [<=] 92.4	\[ \frac{\color{blue}{\left(x + 1\right)} - \sqrt{x} \cdot \sqrt{x}}{\left(\sqrt{x} \cdot \sqrt{x + 1}\right) \cdot \left(\sqrt{x + 1} + \sqrt{x}\right)} \]
      +-commutative [=>] 92.4	\[ \frac{\color{blue}{\left(1 + x\right)} - \sqrt{x} \cdot \sqrt{x}}{\left(\sqrt{x} \cdot \sqrt{x + 1}\right) \cdot \left(\sqrt{x + 1} + \sqrt{x}\right)} \]
      add-sqr-sqrt [<=] 93.4	\[ \frac{\left(1 + x\right) - \color{blue}{x}}{\left(\sqrt{x} \cdot \sqrt{x + 1}\right) \cdot \left(\sqrt{x + 1} + \sqrt{x}\right)} \]
      associate--l+ [=>] 99.5	\[ \frac{\color{blue}{1 + \left(x - x\right)}}{\left(\sqrt{x} \cdot \sqrt{x + 1}\right) \cdot \left(\sqrt{x + 1} + \sqrt{x}\right)} \]
      sqrt-unprod [=>] 99.5	\[ \frac{1 + \left(x - x\right)}{\color{blue}{\sqrt{x \cdot \left(x + 1\right)}} \cdot \left(\sqrt{x + 1} + \sqrt{x}\right)} \]
      +-commutative [=>] 99.5	\[ \frac{1 + \left(x - x\right)}{\sqrt{x \cdot \color{blue}{\left(1 + x\right)}} \cdot \left(\sqrt{x + 1} + \sqrt{x}\right)} \]
      distribute-rgt-in [=>] 99.5	\[ \frac{1 + \left(x - x\right)}{\sqrt{\color{blue}{1 \cdot x + x \cdot x}} \cdot \left(\sqrt{x + 1} + \sqrt{x}\right)} \]
      *-un-lft-identity [<=] 99.5	\[ \frac{1 + \left(x - x\right)}{\sqrt{\color{blue}{x} + x \cdot x} \cdot \left(\sqrt{x + 1} + \sqrt{x}\right)} \]
      +-commutative [=>] 99.5	\[ \frac{1 + \left(x - x\right)}{\sqrt{x + x \cdot x} \cdot \color{blue}{\left(\sqrt{x} + \sqrt{x + 1}\right)}} \]

   3. Simplified 99.5%

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

      [Start] 99.5	\[ \frac{1 + \left(x - x\right)}{\sqrt{x + x \cdot x} \cdot \left(\sqrt{x} + \sqrt{1 + x}\right)} \]
      +-commutative [=>] 99.5	\[ \frac{\color{blue}{\left(x - x\right) + 1}}{\sqrt{x + x \cdot x} \cdot \left(\sqrt{x} + \sqrt{1 + x}\right)} \]
      +-inverses [=>] 99.5	\[ \frac{\color{blue}{0} + 1}{\sqrt{x + x \cdot x} \cdot \left(\sqrt{x} + \sqrt{1 + x}\right)} \]
      metadata-eval [=>] 99.5	\[ \frac{\color{blue}{1}}{\sqrt{x + x \cdot x} \cdot \left(\sqrt{x} + \sqrt{1 + x}\right)} \]
      +-commutative [=>] 99.5	\[ \frac{1}{\sqrt{x + x \cdot x} \cdot \color{blue}{\left(\sqrt{1 + x} + \sqrt{x}\right)}} \]

   4. Applied egg-rr 84.4%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{{\left(\mathsf{fma}\left(x, x, x\right)\right)}^{-0.5}}{\sqrt{x + 1} + \sqrt{x}}\right)} - 1} \]
      Proof

      [Start] 99.5	\[ \frac{1}{\sqrt{x + x \cdot x} \cdot \left(\sqrt{1 + x} + \sqrt{x}\right)} \]
      expm1-log1p-u [=>] 93.4	\[ \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{\sqrt{x + x \cdot x} \cdot \left(\sqrt{1 + x} + \sqrt{x}\right)}\right)\right)} \]
      expm1-udef [=>] 84.4	\[ \color{blue}{e^{\mathsf{log1p}\left(\frac{1}{\sqrt{x + x \cdot x} \cdot \left(\sqrt{1 + x} + \sqrt{x}\right)}\right)} - 1} \]
      associate-/r* [=>] 84.4	\[ e^{\mathsf{log1p}\left(\color{blue}{\frac{\frac{1}{\sqrt{x + x \cdot x}}}{\sqrt{1 + x} + \sqrt{x}}}\right)} - 1 \]
      pow1/2 [=>] 84.4	\[ e^{\mathsf{log1p}\left(\frac{\frac{1}{\color{blue}{{\left(x + x \cdot x\right)}^{0.5}}}}{\sqrt{1 + x} + \sqrt{x}}\right)} - 1 \]
      pow-flip [=>] 84.4	\[ e^{\mathsf{log1p}\left(\frac{\color{blue}{{\left(x + x \cdot x\right)}^{\left(-0.5\right)}}}{\sqrt{1 + x} + \sqrt{x}}\right)} - 1 \]
      +-commutative [=>] 84.4	\[ e^{\mathsf{log1p}\left(\frac{{\color{blue}{\left(x \cdot x + x\right)}}^{\left(-0.5\right)}}{\sqrt{1 + x} + \sqrt{x}}\right)} - 1 \]
      fma-def [=>] 84.4	\[ e^{\mathsf{log1p}\left(\frac{{\color{blue}{\left(\mathsf{fma}\left(x, x, x\right)\right)}}^{\left(-0.5\right)}}{\sqrt{1 + x} + \sqrt{x}}\right)} - 1 \]
      metadata-eval [=>] 84.4	\[ e^{\mathsf{log1p}\left(\frac{{\left(\mathsf{fma}\left(x, x, x\right)\right)}^{\color{blue}{-0.5}}}{\sqrt{1 + x} + \sqrt{x}}\right)} - 1 \]
      +-commutative [=>] 84.4	\[ e^{\mathsf{log1p}\left(\frac{{\left(\mathsf{fma}\left(x, x, x\right)\right)}^{-0.5}}{\sqrt{\color{blue}{x + 1}} + \sqrt{x}}\right)} - 1 \]

   5. Simplified 99.9%

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

      [Start] 84.4	\[ e^{\mathsf{log1p}\left(\frac{{\left(\mathsf{fma}\left(x, x, x\right)\right)}^{-0.5}}{\sqrt{x + 1} + \sqrt{x}}\right)} - 1 \]
      expm1-def [=>] 93.4	\[ \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{{\left(\mathsf{fma}\left(x, x, x\right)\right)}^{-0.5}}{\sqrt{x + 1} + \sqrt{x}}\right)\right)} \]
      expm1-log1p [=>] 99.9	\[ \color{blue}{\frac{{\left(\mathsf{fma}\left(x, x, x\right)\right)}^{-0.5}}{\sqrt{x + 1} + \sqrt{x}}} \]
      +-commutative [=>] 99.9	\[ \frac{{\left(\mathsf{fma}\left(x, x, x\right)\right)}^{-0.5}}{\color{blue}{\sqrt{x} + \sqrt{x + 1}}} \]

   if 1.5000000000000001e38 < x

   1. Initial program 39.9%

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

   2. Applied egg-rr 79.9%

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

      [Start] 39.9	\[ \frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}} \]
      frac-sub [=>] 39.9	\[ \color{blue}{\frac{1 \cdot \sqrt{x + 1} - \sqrt{x} \cdot 1}{\sqrt{x} \cdot \sqrt{x + 1}}} \]
      *-un-lft-identity [<=] 39.9	\[ \frac{\color{blue}{\sqrt{x + 1}} - \sqrt{x} \cdot 1}{\sqrt{x} \cdot \sqrt{x + 1}} \]
      *-rgt-identity [=>] 39.9	\[ \frac{\sqrt{x + 1} - \color{blue}{\sqrt{x}}}{\sqrt{x} \cdot \sqrt{x + 1}} \]
      flip-- [=>] 39.9	\[ \frac{\color{blue}{\frac{\sqrt{x + 1} \cdot \sqrt{x + 1} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}}}}{\sqrt{x} \cdot \sqrt{x + 1}} \]
      associate-/l/ [=>] 39.9	\[ \color{blue}{\frac{\sqrt{x + 1} \cdot \sqrt{x + 1} - \sqrt{x} \cdot \sqrt{x}}{\left(\sqrt{x} \cdot \sqrt{x + 1}\right) \cdot \left(\sqrt{x + 1} + \sqrt{x}\right)}} \]
      add-sqr-sqrt [<=] 39.4	\[ \frac{\color{blue}{\left(x + 1\right)} - \sqrt{x} \cdot \sqrt{x}}{\left(\sqrt{x} \cdot \sqrt{x + 1}\right) \cdot \left(\sqrt{x + 1} + \sqrt{x}\right)} \]
      +-commutative [=>] 39.4	\[ \frac{\color{blue}{\left(1 + x\right)} - \sqrt{x} \cdot \sqrt{x}}{\left(\sqrt{x} \cdot \sqrt{x + 1}\right) \cdot \left(\sqrt{x + 1} + \sqrt{x}\right)} \]
      add-sqr-sqrt [<=] 39.9	\[ \frac{\left(1 + x\right) - \color{blue}{x}}{\left(\sqrt{x} \cdot \sqrt{x + 1}\right) \cdot \left(\sqrt{x + 1} + \sqrt{x}\right)} \]
      associate--l+ [=>] 97.8	\[ \frac{\color{blue}{1 + \left(x - x\right)}}{\left(\sqrt{x} \cdot \sqrt{x + 1}\right) \cdot \left(\sqrt{x + 1} + \sqrt{x}\right)} \]
      sqrt-unprod [=>] 79.9	\[ \frac{1 + \left(x - x\right)}{\color{blue}{\sqrt{x \cdot \left(x + 1\right)}} \cdot \left(\sqrt{x + 1} + \sqrt{x}\right)} \]
      +-commutative [=>] 79.9	\[ \frac{1 + \left(x - x\right)}{\sqrt{x \cdot \color{blue}{\left(1 + x\right)}} \cdot \left(\sqrt{x + 1} + \sqrt{x}\right)} \]
      distribute-rgt-in [=>] 79.9	\[ \frac{1 + \left(x - x\right)}{\sqrt{\color{blue}{1 \cdot x + x \cdot x}} \cdot \left(\sqrt{x + 1} + \sqrt{x}\right)} \]
      *-un-lft-identity [<=] 79.9	\[ \frac{1 + \left(x - x\right)}{\sqrt{\color{blue}{x} + x \cdot x} \cdot \left(\sqrt{x + 1} + \sqrt{x}\right)} \]
      +-commutative [=>] 79.9	\[ \frac{1 + \left(x - x\right)}{\sqrt{x + x \cdot x} \cdot \color{blue}{\left(\sqrt{x} + \sqrt{x + 1}\right)}} \]

   3. Simplified 79.9%

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

      [Start] 79.9	\[ \frac{1 + \left(x - x\right)}{\sqrt{x + x \cdot x} \cdot \left(\sqrt{x} + \sqrt{1 + x}\right)} \]
      +-commutative [=>] 79.9	\[ \frac{\color{blue}{\left(x - x\right) + 1}}{\sqrt{x + x \cdot x} \cdot \left(\sqrt{x} + \sqrt{1 + x}\right)} \]
      +-inverses [=>] 79.9	\[ \frac{\color{blue}{0} + 1}{\sqrt{x + x \cdot x} \cdot \left(\sqrt{x} + \sqrt{1 + x}\right)} \]
      metadata-eval [=>] 79.9	\[ \frac{\color{blue}{1}}{\sqrt{x + x \cdot x} \cdot \left(\sqrt{x} + \sqrt{1 + x}\right)} \]
      +-commutative [=>] 79.9	\[ \frac{1}{\sqrt{x + x \cdot x} \cdot \color{blue}{\left(\sqrt{1 + x} + \sqrt{x}\right)}} \]

   4. Taylor expanded in x around inf 98.2%

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

   5. Simplified 98.2%

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

      [Start] 98.2	\[ \frac{1}{\left(0.5 + x\right) \cdot \left(\sqrt{1 + x} + \sqrt{x}\right)} \]
      +-commutative [=>] 98.2	\[ \frac{1}{\color{blue}{\left(x + 0.5\right)} \cdot \left(\sqrt{1 + x} + \sqrt{x}\right)} \]

   6. Applied egg-rr 98.1%

      \[\leadsto \frac{1}{\color{blue}{\frac{x + 0.5}{\frac{1}{\sqrt{1 + x} + \sqrt{x}}}}} \]
      Proof

      [Start] 98.2	\[ \frac{1}{\left(x + 0.5\right) \cdot \left(\sqrt{1 + x} + \sqrt{x}\right)} \]
      /-rgt-identity [<=] 98.2	\[ \frac{1}{\color{blue}{\frac{\left(x + 0.5\right) \cdot \left(\sqrt{1 + x} + \sqrt{x}\right)}{1}}} \]
      associate-/l* [=>] 98.1	\[ \frac{1}{\color{blue}{\frac{x + 0.5}{\frac{1}{\sqrt{1 + x} + \sqrt{x}}}}} \]

   7. Applied egg-rr 39.9%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{1}{\left(x + 0.5\right) \cdot \left(\sqrt{1 + x} + \sqrt{x}\right)}\right)} - 1} \]
      Proof

      [Start] 98.1	\[ \frac{1}{\frac{x + 0.5}{\frac{1}{\sqrt{1 + x} + \sqrt{x}}}} \]
      expm1-log1p-u [=>] 98.1	\[ \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{\frac{x + 0.5}{\frac{1}{\sqrt{1 + x} + \sqrt{x}}}}\right)\right)} \]
      expm1-udef [=>] 39.9	\[ \color{blue}{e^{\mathsf{log1p}\left(\frac{1}{\frac{x + 0.5}{\frac{1}{\sqrt{1 + x} + \sqrt{x}}}}\right)} - 1} \]
      div-inv [=>] 39.9	\[ e^{\mathsf{log1p}\left(\frac{1}{\color{blue}{\left(x + 0.5\right) \cdot \frac{1}{\frac{1}{\sqrt{1 + x} + \sqrt{x}}}}}\right)} - 1 \]
      associate-/r/ [=>] 39.9	\[ e^{\mathsf{log1p}\left(\frac{1}{\left(x + 0.5\right) \cdot \color{blue}{\left(\frac{1}{1} \cdot \left(\sqrt{1 + x} + \sqrt{x}\right)\right)}}\right)} - 1 \]
      metadata-eval [=>] 39.9	\[ e^{\mathsf{log1p}\left(\frac{1}{\left(x + 0.5\right) \cdot \left(\color{blue}{1} \cdot \left(\sqrt{1 + x} + \sqrt{x}\right)\right)}\right)} - 1 \]
      *-un-lft-identity [<=] 39.9	\[ e^{\mathsf{log1p}\left(\frac{1}{\left(x + 0.5\right) \cdot \color{blue}{\left(\sqrt{1 + x} + \sqrt{x}\right)}}\right)} - 1 \]

   8. Simplified 99.7%

      \[\leadsto \color{blue}{\frac{\frac{-1}{\sqrt{1 + x} + \sqrt{x}}}{-0.5 - x}} \]
      Proof

      [Start] 39.9	\[ e^{\mathsf{log1p}\left(\frac{1}{\left(x + 0.5\right) \cdot \left(\sqrt{1 + x} + \sqrt{x}\right)}\right)} - 1 \]
      expm1-def [=>] 98.2	\[ \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{\left(x + 0.5\right) \cdot \left(\sqrt{1 + x} + \sqrt{x}\right)}\right)\right)} \]
      expm1-log1p [=>] 98.2	\[ \color{blue}{\frac{1}{\left(x + 0.5\right) \cdot \left(\sqrt{1 + x} + \sqrt{x}\right)}} \]
      *-lft-identity [<=] 98.2	\[ \color{blue}{1 \cdot \frac{1}{\left(x + 0.5\right) \cdot \left(\sqrt{1 + x} + \sqrt{x}\right)}} \]
      associate-/r* [=>] 99.7	\[ 1 \cdot \color{blue}{\frac{\frac{1}{x + 0.5}}{\sqrt{1 + x} + \sqrt{x}}} \]
      metadata-eval [<=] 99.7	\[ \color{blue}{\frac{-1}{-1}} \cdot \frac{\frac{1}{x + 0.5}}{\sqrt{1 + x} + \sqrt{x}} \]
      times-frac [<=] 99.7	\[ \color{blue}{\frac{-1 \cdot \frac{1}{x + 0.5}}{-1 \cdot \left(\sqrt{1 + x} + \sqrt{x}\right)}} \]
      *-commutative [<=] 99.7	\[ \frac{\color{blue}{\frac{1}{x + 0.5} \cdot -1}}{-1 \cdot \left(\sqrt{1 + x} + \sqrt{x}\right)} \]
      times-frac [=>] 99.6	\[ \color{blue}{\frac{\frac{1}{x + 0.5}}{-1} \cdot \frac{-1}{\sqrt{1 + x} + \sqrt{x}}} \]
      associate-/l/ [=>] 99.6	\[ \color{blue}{\frac{1}{-1 \cdot \left(x + 0.5\right)}} \cdot \frac{-1}{\sqrt{1 + x} + \sqrt{x}} \]
      associate-*l/ [=>] 99.7	\[ \color{blue}{\frac{1 \cdot \frac{-1}{\sqrt{1 + x} + \sqrt{x}}}{-1 \cdot \left(x + 0.5\right)}} \]
      *-lft-identity [=>] 99.7	\[ \frac{\color{blue}{\frac{-1}{\sqrt{1 + x} + \sqrt{x}}}}{-1 \cdot \left(x + 0.5\right)} \]
      distribute-lft-in [=>] 99.7	\[ \frac{\frac{-1}{\sqrt{1 + x} + \sqrt{x}}}{\color{blue}{-1 \cdot x + -1 \cdot 0.5}} \]
      neg-mul-1 [<=] 99.7	\[ \frac{\frac{-1}{\sqrt{1 + x} + \sqrt{x}}}{\color{blue}{\left(-x\right)} + -1 \cdot 0.5} \]
      metadata-eval [=>] 99.7	\[ \frac{\frac{-1}{\sqrt{1 + x} + \sqrt{x}}}{\left(-x\right) + \color{blue}{-0.5}} \]
      +-commutative [<=] 99.7	\[ \frac{\frac{-1}{\sqrt{1 + x} + \sqrt{x}}}{\color{blue}{-0.5 + \left(-x\right)}} \]
      sub-neg [<=] 99.7	\[ \frac{\frac{-1}{\sqrt{1 + x} + \sqrt{x}}}{\color{blue}{-0.5 - x}} \]

3. Recombined 2 regimes into one program.

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.5 \cdot 10^{+38}:\\ \;\;\;\;\frac{{\left(\mathsf{fma}\left(x, x, x\right)\right)}^{-0.5}}{\sqrt{x} + \sqrt{1 + x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-1}{\sqrt{x} + \sqrt{1 + x}}}{-0.5 - x}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	99.7%
Cost	27588

\[\begin{array}{l} t_0 := \sqrt{1 + x}\\ \mathbf{if}\;\frac{1}{\sqrt{x}} + \frac{-1}{t_0} \leq 2 \cdot 10^{-8}:\\ \;\;\;\;\frac{\left(1 + \left(x - x\right)\right) \cdot \frac{1}{0.5 + \left(x + \frac{-0.125}{x}\right)}}{\sqrt{x} + t_0}\\ \mathbf{else}:\\ \;\;\;\;{x}^{-0.5} - {\left(1 + x\right)}^{-0.5}\\ \end{array} \]

Alternative 2
Accuracy	98.7%
Cost	26948

\[\begin{array}{l} t_0 := \sqrt{1 + x}\\ \mathbf{if}\;\frac{1}{\sqrt{x}} + \frac{-1}{t_0} \leq 2 \cdot 10^{-17}:\\ \;\;\;\;\frac{1}{\left(\sqrt{x} + t_0\right) \cdot \left(x + 0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;{x}^{-0.5} - {\left(1 + x\right)}^{-0.5}\\ \end{array} \]

Alternative 3
Accuracy	99.3%
Cost	26948

\[\begin{array}{l} t_0 := \sqrt{1 + x}\\ \mathbf{if}\;\frac{1}{\sqrt{x}} + \frac{-1}{t_0} \leq 2 \cdot 10^{-17}:\\ \;\;\;\;\frac{\frac{-1}{\sqrt{x} + t_0}}{-0.5 - x}\\ \mathbf{else}:\\ \;\;\;\;{x}^{-0.5} - {\left(1 + x\right)}^{-0.5}\\ \end{array} \]

Alternative 4
Accuracy	98.9%
Cost	26368

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

Alternative 5
Accuracy	98.7%
Cost	13508

\[\begin{array}{l} \mathbf{if}\;x \leq 132000000:\\ \;\;\;\;{x}^{-0.5} - {\left(1 + x\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x \cdot \left(\sqrt{x} + \sqrt{1 + x}\right)}\\ \end{array} \]

Alternative 6
Accuracy	83.5%
Cost	13448

\[\begin{array}{l} t_0 := -1 + \left(x \cdot x\right) \cdot 0.25\\ \mathbf{if}\;x \leq 1.7:\\ \;\;\;\;{x}^{-0.5} + \left(\frac{1}{t_0} + \left(x \cdot 0.5\right) \cdot \frac{-1}{t_0}\right)\\ \mathbf{elif}\;x \leq 5.5 \cdot 10^{+102}:\\ \;\;\;\;0.5 \cdot \sqrt{\frac{1}{{x}^{3}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(x + 0.5\right) \cdot \left(1 + \sqrt{x}\right)}\\ \end{array} \]

Alternative 7
Accuracy	83.9%
Cost	13448

\[\begin{array}{l} t_0 := -1 + \left(x \cdot x\right) \cdot 0.25\\ \mathbf{if}\;x \leq 1.7:\\ \;\;\;\;{x}^{-0.5} + \left(\frac{1}{t_0} + \left(x \cdot 0.5\right) \cdot \frac{-1}{t_0}\right)\\ \mathbf{elif}\;x \leq 7.2 \cdot 10^{+107}:\\ \;\;\;\;0.5 \cdot \sqrt{{\left(\frac{1}{x}\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(x + 0.5\right) \cdot \left(1 + \sqrt{x}\right)}\\ \end{array} \]

Alternative 8
Accuracy	84.9%
Cost	13448

\[\begin{array}{l} \mathbf{if}\;x \leq 132000000:\\ \;\;\;\;{x}^{-0.5} - {\left(1 + x\right)}^{-0.5}\\ \mathbf{elif}\;x \leq 7.2 \cdot 10^{+107}:\\ \;\;\;\;0.5 \cdot \sqrt{{\left(\frac{1}{x}\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(x + 0.5\right) \cdot \left(1 + \sqrt{x}\right)}\\ \end{array} \]

Alternative 9
Accuracy	71.5%
Cost	8196

\[\begin{array}{l} t_0 := -1 + \left(x \cdot x\right) \cdot 0.25\\ \mathbf{if}\;x \leq 1:\\ \;\;\;\;{x}^{-0.5} + \left(\frac{1}{t_0} + \left(x \cdot 0.5\right) \cdot \frac{-1}{t_0}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(x + 0.5\right) \cdot \left(1 + \sqrt{x}\right)}\\ \end{array} \]

Alternative 10
Accuracy	71.5%
Cost	7684

\[\begin{array}{l} \mathbf{if}\;x \leq 1:\\ \;\;\;\;{x}^{-0.5} + \frac{1}{1 - \left(x \cdot x\right) \cdot 0.25} \cdot \left(-1 + x \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(x + 0.5\right) \cdot \left(1 + \sqrt{x}\right)}\\ \end{array} \]

Alternative 11
Accuracy	71.5%
Cost	7172

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

Alternative 12
Accuracy	71.5%
Cost	7108

\[\begin{array}{l} \mathbf{if}\;x \leq 0.41:\\ \;\;\;\;{x}^{-0.5} + \left(-1 + x \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(x + 0.5\right) \cdot \left(1 + \sqrt{x}\right)}\\ \end{array} \]

Alternative 13
Accuracy	68.0%
Cost	7044

\[\begin{array}{l} \mathbf{if}\;x \leq 1.35:\\ \;\;\;\;{x}^{-0.5} + \left(-1 + x \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x + -0.5}{\mathsf{fma}\left(x, x, -0.25\right)}\\ \end{array} \]

Alternative 14
Accuracy	67.7%
Cost	6980

\[\begin{array}{l} \mathbf{if}\;x \leq 0.6:\\ \;\;\;\;-1 + {x}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + -0.5}{\mathsf{fma}\left(x, x, -0.25\right)}\\ \end{array} \]

Alternative 15
Accuracy	67.1%
Cost	6852

\[\begin{array}{l} \mathbf{if}\;x \leq 4.6 \cdot 10^{+153}:\\ \;\;\;\;\frac{1}{x + \sqrt{x}}\\ \mathbf{else}:\\ \;\;\;\;-1 + \left(1 + \frac{1}{x + 0.5}\right)\\ \end{array} \]

Alternative 16
Accuracy	67.0%
Cost	6788

\[\begin{array}{l} \mathbf{if}\;x \leq 0.6:\\ \;\;\;\;-1 + {x}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;-1 + \left(1 + \frac{1}{x + 0.5}\right)\\ \end{array} \]

Alternative 17
Accuracy	65.9%
Cost	6660

\[\begin{array}{l} \mathbf{if}\;x \leq 8.2 \cdot 10^{+122}:\\ \;\;\;\;{x}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;-1 + \left(1 + \frac{1}{x + 0.5}\right)\\ \end{array} \]

Alternative 18
Accuracy	21.6%
Cost	576

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

Alternative 19
Accuracy	7.4%
Cost	320

\[\frac{1}{x + 0.5} \]

Alternative 20
Accuracy	7.4%
Cost	192

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

Alternative 21
Accuracy	5.8%
Cost	64

\[2 \]

Reproduce

herbie shell --seed 2023143 
(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)))))