2isqrt (example 3.6)

Average Accuracy: 68.6% → 99.7%

Time: 16.2s

Precision: binary64

Cost: 47428

Math

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

↓

\[\begin{array}{l} t_0 := \frac{x}{1 + x}\\ t_1 := {t_0}^{0.75}\\ \mathbf{if}\;\frac{1}{\sqrt{x}} + \frac{-1}{\sqrt{1 + x}} \leq 2 \cdot 10^{-7}:\\ \;\;\;\;\left(\frac{0.5}{x} + \left(\frac{0.3125}{{x}^{3}} - \left(\frac{0.375}{x \cdot x} + \frac{0.2734375}{{x}^{4}}\right)\right)\right) \cdot {x}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;{x}^{-0.5} \cdot \frac{\mathsf{fma}\left(t_1, -t_1, 1\right)}{1 + \left(t_0 + \sqrt{t_0}\right)}\\ \end{array} \]

FPCore

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

↓

(FPCore (x)
 :precision binary64
 (let* ((t_0 (/ x (+ 1.0 x))) (t_1 (pow t_0 0.75)))
   (if (<= (+ (/ 1.0 (sqrt x)) (/ -1.0 (sqrt (+ 1.0 x)))) 2e-7)
     (*
      (+
       (/ 0.5 x)
       (-
        (/ 0.3125 (pow x 3.0))
        (+ (/ 0.375 (* x x)) (/ 0.2734375 (pow x 4.0)))))
      (pow x -0.5))
     (* (pow x -0.5) (/ (fma t_1 (- t_1) 1.0) (+ 1.0 (+ t_0 (sqrt t_0))))))))

C

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

↓

double code(double x) {
	double t_0 = x / (1.0 + x);
	double t_1 = pow(t_0, 0.75);
	double tmp;
	if (((1.0 / sqrt(x)) + (-1.0 / sqrt((1.0 + x)))) <= 2e-7) {
		tmp = ((0.5 / x) + ((0.3125 / pow(x, 3.0)) - ((0.375 / (x * x)) + (0.2734375 / pow(x, 4.0))))) * pow(x, -0.5);
	} else {
		tmp = pow(x, -0.5) * (fma(t_1, -t_1, 1.0) / (1.0 + (t_0 + sqrt(t_0))));
	}
	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(x / Float64(1.0 + x))
	t_1 = t_0 ^ 0.75
	tmp = 0.0
	if (Float64(Float64(1.0 / sqrt(x)) + Float64(-1.0 / sqrt(Float64(1.0 + x)))) <= 2e-7)
		tmp = Float64(Float64(Float64(0.5 / x) + Float64(Float64(0.3125 / (x ^ 3.0)) - Float64(Float64(0.375 / Float64(x * x)) + Float64(0.2734375 / (x ^ 4.0))))) * (x ^ -0.5));
	else
		tmp = Float64((x ^ -0.5) * Float64(fma(t_1, Float64(-t_1), 1.0) / Float64(1.0 + Float64(t_0 + sqrt(t_0)))));
	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[(x / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Power[t$95$0, 0.75], $MachinePrecision]}, If[LessEqual[N[(N[(1.0 / N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + N[(-1.0 / N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2e-7], N[(N[(N[(0.5 / x), $MachinePrecision] + N[(N[(0.3125 / N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision] - N[(N[(0.375 / N[(x * x), $MachinePrecision]), $MachinePrecision] + N[(0.2734375 / N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Power[x, -0.5], $MachinePrecision]), $MachinePrecision], N[(N[Power[x, -0.5], $MachinePrecision] * N[(N[(t$95$1 * (-t$95$1) + 1.0), $MachinePrecision] / N[(1.0 + N[(t$95$0 + N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Target

Original	68.6%
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)))) < 1.9999999999999999e-7

   1. Initial program 39.6%

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

   2. Applied egg-rr 39.5%

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

      [Start] 39.6	\[ \frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}} \]
      frac-sub [=>] 39.5	\[ \color{blue}{\frac{1 \cdot \sqrt{x + 1} - \sqrt{x} \cdot 1}{\sqrt{x} \cdot \sqrt{x + 1}}} \]
      div-inv [=>] 39.5	\[ \color{blue}{\left(1 \cdot \sqrt{x + 1} - \sqrt{x} \cdot 1\right) \cdot \frac{1}{\sqrt{x} \cdot \sqrt{x + 1}}} \]
      *-un-lft-identity [<=] 39.5	\[ \left(\color{blue}{\sqrt{x + 1}} - \sqrt{x} \cdot 1\right) \cdot \frac{1}{\sqrt{x} \cdot \sqrt{x + 1}} \]
      +-commutative [=>] 39.5	\[ \left(\sqrt{\color{blue}{1 + x}} - \sqrt{x} \cdot 1\right) \cdot \frac{1}{\sqrt{x} \cdot \sqrt{x + 1}} \]
      *-rgt-identity [=>] 39.5	\[ \left(\sqrt{1 + x} - \color{blue}{\sqrt{x}}\right) \cdot \frac{1}{\sqrt{x} \cdot \sqrt{x + 1}} \]
      metadata-eval [<=] 39.5	\[ \left(\sqrt{1 + x} - \sqrt{x}\right) \cdot \frac{\color{blue}{1 \cdot 1}}{\sqrt{x} \cdot \sqrt{x + 1}} \]
      frac-times [<=] 39.5	\[ \left(\sqrt{1 + x} - \sqrt{x}\right) \cdot \color{blue}{\left(\frac{1}{\sqrt{x}} \cdot \frac{1}{\sqrt{x + 1}}\right)} \]
      un-div-inv [=>] 39.5	\[ \left(\sqrt{1 + x} - \sqrt{x}\right) \cdot \color{blue}{\frac{\frac{1}{\sqrt{x}}}{\sqrt{x + 1}}} \]
      pow1/2 [=>] 39.5	\[ \left(\sqrt{1 + x} - \sqrt{x}\right) \cdot \frac{\frac{1}{\color{blue}{{x}^{0.5}}}}{\sqrt{x + 1}} \]
      pow-flip [=>] 39.5	\[ \left(\sqrt{1 + x} - \sqrt{x}\right) \cdot \frac{\color{blue}{{x}^{\left(-0.5\right)}}}{\sqrt{x + 1}} \]
      metadata-eval [=>] 39.5	\[ \left(\sqrt{1 + x} - \sqrt{x}\right) \cdot \frac{{x}^{\color{blue}{-0.5}}}{\sqrt{x + 1}} \]
      +-commutative [=>] 39.5	\[ \left(\sqrt{1 + x} - \sqrt{x}\right) \cdot \frac{{x}^{-0.5}}{\sqrt{\color{blue}{1 + x}}} \]

   3. Simplified 39.6%

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

      [Start] 39.5	\[ \left(\sqrt{1 + x} - \sqrt{x}\right) \cdot \frac{{x}^{-0.5}}{\sqrt{1 + x}} \]
      associate-*r/ [=>] 39.5	\[ \color{blue}{\frac{\left(\sqrt{1 + x} - \sqrt{x}\right) \cdot {x}^{-0.5}}{\sqrt{1 + x}}} \]
      *-rgt-identity [<=] 39.5	\[ \frac{\left(\sqrt{1 + x} - \sqrt{x}\right) \cdot {x}^{-0.5}}{\color{blue}{\sqrt{1 + x} \cdot 1}} \]
      times-frac [=>] 39.5	\[ \color{blue}{\frac{\sqrt{1 + x} - \sqrt{x}}{\sqrt{1 + x}} \cdot \frac{{x}^{-0.5}}{1}} \]
      div-sub [=>] 39.6	\[ \color{blue}{\left(\frac{\sqrt{1 + x}}{\sqrt{1 + x}} - \frac{\sqrt{x}}{\sqrt{1 + x}}\right)} \cdot \frac{{x}^{-0.5}}{1} \]
      *-inverses [=>] 39.6	\[ \left(\color{blue}{1} - \frac{\sqrt{x}}{\sqrt{1 + x}}\right) \cdot \frac{{x}^{-0.5}}{1} \]
      /-rgt-identity [=>] 39.6	\[ \left(1 - \frac{\sqrt{x}}{\sqrt{1 + x}}\right) \cdot \color{blue}{{x}^{-0.5}} \]

   4. Applied egg-rr 39.7%

      \[\leadsto \color{blue}{\left(\left(1 - {\left(\frac{x}{1 + x}\right)}^{1.5}\right) \cdot \frac{1}{1 + \left(\sqrt{\frac{x}{1 + x}} + \frac{x}{1 + x}\right)}\right)} \cdot {x}^{-0.5} \]
      Proof

      [Start] 39.6	\[ \left(1 - \frac{\sqrt{x}}{\sqrt{1 + x}}\right) \cdot {x}^{-0.5} \]
      flip3-- [=>] 39.6	\[ \color{blue}{\frac{{1}^{3} - {\left(\frac{\sqrt{x}}{\sqrt{1 + x}}\right)}^{3}}{1 \cdot 1 + \left(\frac{\sqrt{x}}{\sqrt{1 + x}} \cdot \frac{\sqrt{x}}{\sqrt{1 + x}} + 1 \cdot \frac{\sqrt{x}}{\sqrt{1 + x}}\right)}} \cdot {x}^{-0.5} \]
      div-inv [=>] 39.6	\[ \color{blue}{\left(\left({1}^{3} - {\left(\frac{\sqrt{x}}{\sqrt{1 + x}}\right)}^{3}\right) \cdot \frac{1}{1 \cdot 1 + \left(\frac{\sqrt{x}}{\sqrt{1 + x}} \cdot \frac{\sqrt{x}}{\sqrt{1 + x}} + 1 \cdot \frac{\sqrt{x}}{\sqrt{1 + x}}\right)}\right)} \cdot {x}^{-0.5} \]
      metadata-eval [=>] 39.6	\[ \left(\left(\color{blue}{1} - {\left(\frac{\sqrt{x}}{\sqrt{1 + x}}\right)}^{3}\right) \cdot \frac{1}{1 \cdot 1 + \left(\frac{\sqrt{x}}{\sqrt{1 + x}} \cdot \frac{\sqrt{x}}{\sqrt{1 + x}} + 1 \cdot \frac{\sqrt{x}}{\sqrt{1 + x}}\right)}\right) \cdot {x}^{-0.5} \]
      add-sqr-sqrt [=>] 39.6	\[ \left(\left(1 - {\color{blue}{\left(\sqrt{\frac{\sqrt{x}}{\sqrt{1 + x}}} \cdot \sqrt{\frac{\sqrt{x}}{\sqrt{1 + x}}}\right)}}^{3}\right) \cdot \frac{1}{1 \cdot 1 + \left(\frac{\sqrt{x}}{\sqrt{1 + x}} \cdot \frac{\sqrt{x}}{\sqrt{1 + x}} + 1 \cdot \frac{\sqrt{x}}{\sqrt{1 + x}}\right)}\right) \cdot {x}^{-0.5} \]
      sqrt-unprod [=>] 39.6	\[ \left(\left(1 - {\color{blue}{\left(\sqrt{\frac{\sqrt{x}}{\sqrt{1 + x}} \cdot \frac{\sqrt{x}}{\sqrt{1 + x}}}\right)}}^{3}\right) \cdot \frac{1}{1 \cdot 1 + \left(\frac{\sqrt{x}}{\sqrt{1 + x}} \cdot \frac{\sqrt{x}}{\sqrt{1 + x}} + 1 \cdot \frac{\sqrt{x}}{\sqrt{1 + x}}\right)}\right) \cdot {x}^{-0.5} \]
      sqrt-pow2 [=>] 39.6	\[ \left(\left(1 - \color{blue}{{\left(\frac{\sqrt{x}}{\sqrt{1 + x}} \cdot \frac{\sqrt{x}}{\sqrt{1 + x}}\right)}^{\left(\frac{3}{2}\right)}}\right) \cdot \frac{1}{1 \cdot 1 + \left(\frac{\sqrt{x}}{\sqrt{1 + x}} \cdot \frac{\sqrt{x}}{\sqrt{1 + x}} + 1 \cdot \frac{\sqrt{x}}{\sqrt{1 + x}}\right)}\right) \cdot {x}^{-0.5} \]
      sqrt-undiv [=>] 39.6	\[ \left(\left(1 - {\left(\color{blue}{\sqrt{\frac{x}{1 + x}}} \cdot \frac{\sqrt{x}}{\sqrt{1 + x}}\right)}^{\left(\frac{3}{2}\right)}\right) \cdot \frac{1}{1 \cdot 1 + \left(\frac{\sqrt{x}}{\sqrt{1 + x}} \cdot \frac{\sqrt{x}}{\sqrt{1 + x}} + 1 \cdot \frac{\sqrt{x}}{\sqrt{1 + x}}\right)}\right) \cdot {x}^{-0.5} \]
      sqrt-undiv [=>] 39.7	\[ \left(\left(1 - {\left(\sqrt{\frac{x}{1 + x}} \cdot \color{blue}{\sqrt{\frac{x}{1 + x}}}\right)}^{\left(\frac{3}{2}\right)}\right) \cdot \frac{1}{1 \cdot 1 + \left(\frac{\sqrt{x}}{\sqrt{1 + x}} \cdot \frac{\sqrt{x}}{\sqrt{1 + x}} + 1 \cdot \frac{\sqrt{x}}{\sqrt{1 + x}}\right)}\right) \cdot {x}^{-0.5} \]
      add-sqr-sqrt [<=] 39.7	\[ \left(\left(1 - {\color{blue}{\left(\frac{x}{1 + x}\right)}}^{\left(\frac{3}{2}\right)}\right) \cdot \frac{1}{1 \cdot 1 + \left(\frac{\sqrt{x}}{\sqrt{1 + x}} \cdot \frac{\sqrt{x}}{\sqrt{1 + x}} + 1 \cdot \frac{\sqrt{x}}{\sqrt{1 + x}}\right)}\right) \cdot {x}^{-0.5} \]
      metadata-eval [=>] 39.7	\[ \left(\left(1 - {\left(\frac{x}{1 + x}\right)}^{\color{blue}{1.5}}\right) \cdot \frac{1}{1 \cdot 1 + \left(\frac{\sqrt{x}}{\sqrt{1 + x}} \cdot \frac{\sqrt{x}}{\sqrt{1 + x}} + 1 \cdot \frac{\sqrt{x}}{\sqrt{1 + x}}\right)}\right) \cdot {x}^{-0.5} \]

   5. Simplified 39.7%

      \[\leadsto \color{blue}{\frac{1 - {\left(\frac{x}{1 + x}\right)}^{1.5}}{1 + \left(\frac{x}{1 + x} + \sqrt{\frac{x}{1 + x}}\right)}} \cdot {x}^{-0.5} \]
      Proof

      [Start] 39.7	\[ \left(\left(1 - {\left(\frac{x}{1 + x}\right)}^{1.5}\right) \cdot \frac{1}{1 + \left(\sqrt{\frac{x}{1 + x}} + \frac{x}{1 + x}\right)}\right) \cdot {x}^{-0.5} \]
      associate-*r/ [=>] 39.7	\[ \color{blue}{\frac{\left(1 - {\left(\frac{x}{1 + x}\right)}^{1.5}\right) \cdot 1}{1 + \left(\sqrt{\frac{x}{1 + x}} + \frac{x}{1 + x}\right)}} \cdot {x}^{-0.5} \]
      *-rgt-identity [=>] 39.7	\[ \frac{\color{blue}{1 - {\left(\frac{x}{1 + x}\right)}^{1.5}}}{1 + \left(\sqrt{\frac{x}{1 + x}} + \frac{x}{1 + x}\right)} \cdot {x}^{-0.5} \]
      +-commutative [=>] 39.7	\[ \frac{1 - {\left(\frac{x}{1 + x}\right)}^{1.5}}{1 + \color{blue}{\left(\frac{x}{1 + x} + \sqrt{\frac{x}{1 + x}}\right)}} \cdot {x}^{-0.5} \]

   6. Taylor expanded in x around inf 99.6%

      \[\leadsto \color{blue}{\left(\left(0.5 \cdot \frac{1}{x} + 0.3125 \cdot \frac{1}{{x}^{3}}\right) - \left(0.2734375 \cdot \frac{1}{{x}^{4}} + 0.375 \cdot \frac{1}{{x}^{2}}\right)\right)} \cdot {x}^{-0.5} \]

   7. Simplified 99.6%

      \[\leadsto \color{blue}{\left(\frac{0.5}{x} + \left(\frac{0.3125}{{x}^{3}} - \left(\frac{0.375}{x \cdot x} + \frac{0.2734375}{{x}^{4}}\right)\right)\right)} \cdot {x}^{-0.5} \]
      Proof

      [Start] 99.6	\[ \left(\left(0.5 \cdot \frac{1}{x} + 0.3125 \cdot \frac{1}{{x}^{3}}\right) - \left(0.2734375 \cdot \frac{1}{{x}^{4}} + 0.375 \cdot \frac{1}{{x}^{2}}\right)\right) \cdot {x}^{-0.5} \]
      associate--l+ [=>] 99.6	\[ \color{blue}{\left(0.5 \cdot \frac{1}{x} + \left(0.3125 \cdot \frac{1}{{x}^{3}} - \left(0.2734375 \cdot \frac{1}{{x}^{4}} + 0.375 \cdot \frac{1}{{x}^{2}}\right)\right)\right)} \cdot {x}^{-0.5} \]
      associate-*r/ [=>] 99.6	\[ \left(\color{blue}{\frac{0.5 \cdot 1}{x}} + \left(0.3125 \cdot \frac{1}{{x}^{3}} - \left(0.2734375 \cdot \frac{1}{{x}^{4}} + 0.375 \cdot \frac{1}{{x}^{2}}\right)\right)\right) \cdot {x}^{-0.5} \]
      metadata-eval [=>] 99.6	\[ \left(\frac{\color{blue}{0.5}}{x} + \left(0.3125 \cdot \frac{1}{{x}^{3}} - \left(0.2734375 \cdot \frac{1}{{x}^{4}} + 0.375 \cdot \frac{1}{{x}^{2}}\right)\right)\right) \cdot {x}^{-0.5} \]
      associate-*r/ [=>] 99.6	\[ \left(\frac{0.5}{x} + \left(\color{blue}{\frac{0.3125 \cdot 1}{{x}^{3}}} - \left(0.2734375 \cdot \frac{1}{{x}^{4}} + 0.375 \cdot \frac{1}{{x}^{2}}\right)\right)\right) \cdot {x}^{-0.5} \]
      metadata-eval [=>] 99.6	\[ \left(\frac{0.5}{x} + \left(\frac{\color{blue}{0.3125}}{{x}^{3}} - \left(0.2734375 \cdot \frac{1}{{x}^{4}} + 0.375 \cdot \frac{1}{{x}^{2}}\right)\right)\right) \cdot {x}^{-0.5} \]
      +-commutative [=>] 99.6	\[ \left(\frac{0.5}{x} + \left(\frac{0.3125}{{x}^{3}} - \color{blue}{\left(0.375 \cdot \frac{1}{{x}^{2}} + 0.2734375 \cdot \frac{1}{{x}^{4}}\right)}\right)\right) \cdot {x}^{-0.5} \]
      associate-*r/ [=>] 99.6	\[ \left(\frac{0.5}{x} + \left(\frac{0.3125}{{x}^{3}} - \left(\color{blue}{\frac{0.375 \cdot 1}{{x}^{2}}} + 0.2734375 \cdot \frac{1}{{x}^{4}}\right)\right)\right) \cdot {x}^{-0.5} \]
      metadata-eval [=>] 99.6	\[ \left(\frac{0.5}{x} + \left(\frac{0.3125}{{x}^{3}} - \left(\frac{\color{blue}{0.375}}{{x}^{2}} + 0.2734375 \cdot \frac{1}{{x}^{4}}\right)\right)\right) \cdot {x}^{-0.5} \]
      unpow2 [=>] 99.6	\[ \left(\frac{0.5}{x} + \left(\frac{0.3125}{{x}^{3}} - \left(\frac{0.375}{\color{blue}{x \cdot x}} + 0.2734375 \cdot \frac{1}{{x}^{4}}\right)\right)\right) \cdot {x}^{-0.5} \]
      associate-*r/ [=>] 99.6	\[ \left(\frac{0.5}{x} + \left(\frac{0.3125}{{x}^{3}} - \left(\frac{0.375}{x \cdot x} + \color{blue}{\frac{0.2734375 \cdot 1}{{x}^{4}}}\right)\right)\right) \cdot {x}^{-0.5} \]
      metadata-eval [=>] 99.6	\[ \left(\frac{0.5}{x} + \left(\frac{0.3125}{{x}^{3}} - \left(\frac{0.375}{x \cdot x} + \frac{\color{blue}{0.2734375}}{{x}^{4}}\right)\right)\right) \cdot {x}^{-0.5} \]

   if 1.9999999999999999e-7 < (-.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%

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

      [Start] 99.4	\[ \frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}} \]
      frac-sub [=>] 99.4	\[ \color{blue}{\frac{1 \cdot \sqrt{x + 1} - \sqrt{x} \cdot 1}{\sqrt{x} \cdot \sqrt{x + 1}}} \]
      div-inv [=>] 99.4	\[ \color{blue}{\left(1 \cdot \sqrt{x + 1} - \sqrt{x} \cdot 1\right) \cdot \frac{1}{\sqrt{x} \cdot \sqrt{x + 1}}} \]
      *-un-lft-identity [<=] 99.4	\[ \left(\color{blue}{\sqrt{x + 1}} - \sqrt{x} \cdot 1\right) \cdot \frac{1}{\sqrt{x} \cdot \sqrt{x + 1}} \]
      +-commutative [=>] 99.4	\[ \left(\sqrt{\color{blue}{1 + x}} - \sqrt{x} \cdot 1\right) \cdot \frac{1}{\sqrt{x} \cdot \sqrt{x + 1}} \]
      *-rgt-identity [=>] 99.4	\[ \left(\sqrt{1 + x} - \color{blue}{\sqrt{x}}\right) \cdot \frac{1}{\sqrt{x} \cdot \sqrt{x + 1}} \]
      metadata-eval [<=] 99.4	\[ \left(\sqrt{1 + x} - \sqrt{x}\right) \cdot \frac{\color{blue}{1 \cdot 1}}{\sqrt{x} \cdot \sqrt{x + 1}} \]
      frac-times [<=] 99.4	\[ \left(\sqrt{1 + x} - \sqrt{x}\right) \cdot \color{blue}{\left(\frac{1}{\sqrt{x}} \cdot \frac{1}{\sqrt{x + 1}}\right)} \]
      un-div-inv [=>] 99.4	\[ \left(\sqrt{1 + x} - \sqrt{x}\right) \cdot \color{blue}{\frac{\frac{1}{\sqrt{x}}}{\sqrt{x + 1}}} \]
      pow1/2 [=>] 99.4	\[ \left(\sqrt{1 + x} - \sqrt{x}\right) \cdot \frac{\frac{1}{\color{blue}{{x}^{0.5}}}}{\sqrt{x + 1}} \]
      pow-flip [=>] 99.8	\[ \left(\sqrt{1 + x} - \sqrt{x}\right) \cdot \frac{\color{blue}{{x}^{\left(-0.5\right)}}}{\sqrt{x + 1}} \]
      metadata-eval [=>] 99.8	\[ \left(\sqrt{1 + x} - \sqrt{x}\right) \cdot \frac{{x}^{\color{blue}{-0.5}}}{\sqrt{x + 1}} \]
      +-commutative [=>] 99.8	\[ \left(\sqrt{1 + x} - \sqrt{x}\right) \cdot \frac{{x}^{-0.5}}{\sqrt{\color{blue}{1 + x}}} \]

   3. Simplified 99.8%

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

      [Start] 99.8	\[ \left(\sqrt{1 + x} - \sqrt{x}\right) \cdot \frac{{x}^{-0.5}}{\sqrt{1 + x}} \]
      associate-*r/ [=>] 99.8	\[ \color{blue}{\frac{\left(\sqrt{1 + x} - \sqrt{x}\right) \cdot {x}^{-0.5}}{\sqrt{1 + x}}} \]
      *-rgt-identity [<=] 99.8	\[ \frac{\left(\sqrt{1 + x} - \sqrt{x}\right) \cdot {x}^{-0.5}}{\color{blue}{\sqrt{1 + x} \cdot 1}} \]
      times-frac [=>] 99.8	\[ \color{blue}{\frac{\sqrt{1 + x} - \sqrt{x}}{\sqrt{1 + x}} \cdot \frac{{x}^{-0.5}}{1}} \]
      div-sub [=>] 99.8	\[ \color{blue}{\left(\frac{\sqrt{1 + x}}{\sqrt{1 + x}} - \frac{\sqrt{x}}{\sqrt{1 + x}}\right)} \cdot \frac{{x}^{-0.5}}{1} \]
      *-inverses [=>] 99.8	\[ \left(\color{blue}{1} - \frac{\sqrt{x}}{\sqrt{1 + x}}\right) \cdot \frac{{x}^{-0.5}}{1} \]
      /-rgt-identity [=>] 99.8	\[ \left(1 - \frac{\sqrt{x}}{\sqrt{1 + x}}\right) \cdot \color{blue}{{x}^{-0.5}} \]

   4. Applied egg-rr 99.9%

      \[\leadsto \color{blue}{\left(\left(1 - {\left(\frac{x}{1 + x}\right)}^{1.5}\right) \cdot \frac{1}{1 + \left(\sqrt{\frac{x}{1 + x}} + \frac{x}{1 + x}\right)}\right)} \cdot {x}^{-0.5} \]
      Proof

      [Start] 99.8	\[ \left(1 - \frac{\sqrt{x}}{\sqrt{1 + x}}\right) \cdot {x}^{-0.5} \]
      flip3-- [=>] 99.8	\[ \color{blue}{\frac{{1}^{3} - {\left(\frac{\sqrt{x}}{\sqrt{1 + x}}\right)}^{3}}{1 \cdot 1 + \left(\frac{\sqrt{x}}{\sqrt{1 + x}} \cdot \frac{\sqrt{x}}{\sqrt{1 + x}} + 1 \cdot \frac{\sqrt{x}}{\sqrt{1 + x}}\right)}} \cdot {x}^{-0.5} \]
      div-inv [=>] 99.8	\[ \color{blue}{\left(\left({1}^{3} - {\left(\frac{\sqrt{x}}{\sqrt{1 + x}}\right)}^{3}\right) \cdot \frac{1}{1 \cdot 1 + \left(\frac{\sqrt{x}}{\sqrt{1 + x}} \cdot \frac{\sqrt{x}}{\sqrt{1 + x}} + 1 \cdot \frac{\sqrt{x}}{\sqrt{1 + x}}\right)}\right)} \cdot {x}^{-0.5} \]
      metadata-eval [=>] 99.8	\[ \left(\left(\color{blue}{1} - {\left(\frac{\sqrt{x}}{\sqrt{1 + x}}\right)}^{3}\right) \cdot \frac{1}{1 \cdot 1 + \left(\frac{\sqrt{x}}{\sqrt{1 + x}} \cdot \frac{\sqrt{x}}{\sqrt{1 + x}} + 1 \cdot \frac{\sqrt{x}}{\sqrt{1 + x}}\right)}\right) \cdot {x}^{-0.5} \]
      add-sqr-sqrt [=>] 99.9	\[ \left(\left(1 - {\color{blue}{\left(\sqrt{\frac{\sqrt{x}}{\sqrt{1 + x}}} \cdot \sqrt{\frac{\sqrt{x}}{\sqrt{1 + x}}}\right)}}^{3}\right) \cdot \frac{1}{1 \cdot 1 + \left(\frac{\sqrt{x}}{\sqrt{1 + x}} \cdot \frac{\sqrt{x}}{\sqrt{1 + x}} + 1 \cdot \frac{\sqrt{x}}{\sqrt{1 + x}}\right)}\right) \cdot {x}^{-0.5} \]
      sqrt-unprod [=>] 99.8	\[ \left(\left(1 - {\color{blue}{\left(\sqrt{\frac{\sqrt{x}}{\sqrt{1 + x}} \cdot \frac{\sqrt{x}}{\sqrt{1 + x}}}\right)}}^{3}\right) \cdot \frac{1}{1 \cdot 1 + \left(\frac{\sqrt{x}}{\sqrt{1 + x}} \cdot \frac{\sqrt{x}}{\sqrt{1 + x}} + 1 \cdot \frac{\sqrt{x}}{\sqrt{1 + x}}\right)}\right) \cdot {x}^{-0.5} \]
      sqrt-pow2 [=>] 99.8	\[ \left(\left(1 - \color{blue}{{\left(\frac{\sqrt{x}}{\sqrt{1 + x}} \cdot \frac{\sqrt{x}}{\sqrt{1 + x}}\right)}^{\left(\frac{3}{2}\right)}}\right) \cdot \frac{1}{1 \cdot 1 + \left(\frac{\sqrt{x}}{\sqrt{1 + x}} \cdot \frac{\sqrt{x}}{\sqrt{1 + x}} + 1 \cdot \frac{\sqrt{x}}{\sqrt{1 + x}}\right)}\right) \cdot {x}^{-0.5} \]
      sqrt-undiv [=>] 99.8	\[ \left(\left(1 - {\left(\color{blue}{\sqrt{\frac{x}{1 + x}}} \cdot \frac{\sqrt{x}}{\sqrt{1 + x}}\right)}^{\left(\frac{3}{2}\right)}\right) \cdot \frac{1}{1 \cdot 1 + \left(\frac{\sqrt{x}}{\sqrt{1 + x}} \cdot \frac{\sqrt{x}}{\sqrt{1 + x}} + 1 \cdot \frac{\sqrt{x}}{\sqrt{1 + x}}\right)}\right) \cdot {x}^{-0.5} \]
      sqrt-undiv [=>] 99.9	\[ \left(\left(1 - {\left(\sqrt{\frac{x}{1 + x}} \cdot \color{blue}{\sqrt{\frac{x}{1 + x}}}\right)}^{\left(\frac{3}{2}\right)}\right) \cdot \frac{1}{1 \cdot 1 + \left(\frac{\sqrt{x}}{\sqrt{1 + x}} \cdot \frac{\sqrt{x}}{\sqrt{1 + x}} + 1 \cdot \frac{\sqrt{x}}{\sqrt{1 + x}}\right)}\right) \cdot {x}^{-0.5} \]
      add-sqr-sqrt [<=] 99.9	\[ \left(\left(1 - {\color{blue}{\left(\frac{x}{1 + x}\right)}}^{\left(\frac{3}{2}\right)}\right) \cdot \frac{1}{1 \cdot 1 + \left(\frac{\sqrt{x}}{\sqrt{1 + x}} \cdot \frac{\sqrt{x}}{\sqrt{1 + x}} + 1 \cdot \frac{\sqrt{x}}{\sqrt{1 + x}}\right)}\right) \cdot {x}^{-0.5} \]
      metadata-eval [=>] 99.9	\[ \left(\left(1 - {\left(\frac{x}{1 + x}\right)}^{\color{blue}{1.5}}\right) \cdot \frac{1}{1 \cdot 1 + \left(\frac{\sqrt{x}}{\sqrt{1 + x}} \cdot \frac{\sqrt{x}}{\sqrt{1 + x}} + 1 \cdot \frac{\sqrt{x}}{\sqrt{1 + x}}\right)}\right) \cdot {x}^{-0.5} \]

   5. Simplified 99.9%

      \[\leadsto \color{blue}{\frac{1 - {\left(\frac{x}{1 + x}\right)}^{1.5}}{1 + \left(\frac{x}{1 + x} + \sqrt{\frac{x}{1 + x}}\right)}} \cdot {x}^{-0.5} \]
      Proof

      [Start] 99.9	\[ \left(\left(1 - {\left(\frac{x}{1 + x}\right)}^{1.5}\right) \cdot \frac{1}{1 + \left(\sqrt{\frac{x}{1 + x}} + \frac{x}{1 + x}\right)}\right) \cdot {x}^{-0.5} \]
      associate-*r/ [=>] 99.9	\[ \color{blue}{\frac{\left(1 - {\left(\frac{x}{1 + x}\right)}^{1.5}\right) \cdot 1}{1 + \left(\sqrt{\frac{x}{1 + x}} + \frac{x}{1 + x}\right)}} \cdot {x}^{-0.5} \]
      *-rgt-identity [=>] 99.9	\[ \frac{\color{blue}{1 - {\left(\frac{x}{1 + x}\right)}^{1.5}}}{1 + \left(\sqrt{\frac{x}{1 + x}} + \frac{x}{1 + x}\right)} \cdot {x}^{-0.5} \]
      +-commutative [=>] 99.9	\[ \frac{1 - {\left(\frac{x}{1 + x}\right)}^{1.5}}{1 + \color{blue}{\left(\frac{x}{1 + x} + \sqrt{\frac{x}{1 + x}}\right)}} \cdot {x}^{-0.5} \]

   6. Applied egg-rr 99.9%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({\left(\frac{x}{1 + x}\right)}^{0.75}, -{\left(\frac{x}{1 + x}\right)}^{0.75}, 1\right)}}{1 + \left(\frac{x}{1 + x} + \sqrt{\frac{x}{1 + x}}\right)} \cdot {x}^{-0.5} \]
      Proof

      [Start] 99.9	\[ \frac{1 - {\left(\frac{x}{1 + x}\right)}^{1.5}}{1 + \left(\frac{x}{1 + x} + \sqrt{\frac{x}{1 + x}}\right)} \cdot {x}^{-0.5} \]
      sub-neg [=>] 99.9	\[ \frac{\color{blue}{1 + \left(-{\left(\frac{x}{1 + x}\right)}^{1.5}\right)}}{1 + \left(\frac{x}{1 + x} + \sqrt{\frac{x}{1 + x}}\right)} \cdot {x}^{-0.5} \]
      +-commutative [=>] 99.9	\[ \frac{\color{blue}{\left(-{\left(\frac{x}{1 + x}\right)}^{1.5}\right) + 1}}{1 + \left(\frac{x}{1 + x} + \sqrt{\frac{x}{1 + x}}\right)} \cdot {x}^{-0.5} \]
      sqr-pow [=>] 99.9	\[ \frac{\left(-\color{blue}{{\left(\frac{x}{1 + x}\right)}^{\left(\frac{1.5}{2}\right)} \cdot {\left(\frac{x}{1 + x}\right)}^{\left(\frac{1.5}{2}\right)}}\right) + 1}{1 + \left(\frac{x}{1 + x} + \sqrt{\frac{x}{1 + x}}\right)} \cdot {x}^{-0.5} \]
      distribute-rgt-neg-in [=>] 99.9	\[ \frac{\color{blue}{{\left(\frac{x}{1 + x}\right)}^{\left(\frac{1.5}{2}\right)} \cdot \left(-{\left(\frac{x}{1 + x}\right)}^{\left(\frac{1.5}{2}\right)}\right)} + 1}{1 + \left(\frac{x}{1 + x} + \sqrt{\frac{x}{1 + x}}\right)} \cdot {x}^{-0.5} \]
      fma-def [=>] 99.9	\[ \frac{\color{blue}{\mathsf{fma}\left({\left(\frac{x}{1 + x}\right)}^{\left(\frac{1.5}{2}\right)}, -{\left(\frac{x}{1 + x}\right)}^{\left(\frac{1.5}{2}\right)}, 1\right)}}{1 + \left(\frac{x}{1 + x} + \sqrt{\frac{x}{1 + x}}\right)} \cdot {x}^{-0.5} \]
      metadata-eval [=>] 99.9	\[ \frac{\mathsf{fma}\left({\left(\frac{x}{1 + x}\right)}^{\color{blue}{0.75}}, -{\left(\frac{x}{1 + x}\right)}^{\left(\frac{1.5}{2}\right)}, 1\right)}{1 + \left(\frac{x}{1 + x} + \sqrt{\frac{x}{1 + x}}\right)} \cdot {x}^{-0.5} \]
      metadata-eval [=>] 99.9	\[ \frac{\mathsf{fma}\left({\left(\frac{x}{1 + x}\right)}^{0.75}, -{\left(\frac{x}{1 + x}\right)}^{\color{blue}{0.75}}, 1\right)}{1 + \left(\frac{x}{1 + x} + \sqrt{\frac{x}{1 + x}}\right)} \cdot {x}^{-0.5} \]

3. Recombined 2 regimes into one program.

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{\sqrt{x}} + \frac{-1}{\sqrt{1 + x}} \leq 2 \cdot 10^{-7}:\\ \;\;\;\;\left(\frac{0.5}{x} + \left(\frac{0.3125}{{x}^{3}} - \left(\frac{0.375}{x \cdot x} + \frac{0.2734375}{{x}^{4}}\right)\right)\right) \cdot {x}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;{x}^{-0.5} \cdot \frac{\mathsf{fma}\left({\left(\frac{x}{1 + x}\right)}^{0.75}, -{\left(\frac{x}{1 + x}\right)}^{0.75}, 1\right)}{1 + \left(\frac{x}{1 + x} + \sqrt{\frac{x}{1 + x}}\right)}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	99.7%
Cost	34244

\[\begin{array}{l} t_0 := \frac{x}{1 + x}\\ \mathbf{if}\;\frac{1}{\sqrt{x}} + \frac{-1}{\sqrt{1 + x}} \leq 2 \cdot 10^{-7}:\\ \;\;\;\;\left(\frac{0.5}{x} + \left(\frac{0.3125}{{x}^{3}} - \left(\frac{0.375}{x \cdot x} + \frac{0.2734375}{{x}^{4}}\right)\right)\right) \cdot {x}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;{x}^{-0.5} \cdot \frac{1 - {t_0}^{1.5}}{1 + \left(t_0 + \sqrt{t_0}\right)}\\ \end{array} \]

Alternative 2
Accuracy	99.8%
Cost	34052

\[\begin{array}{l} \mathbf{if}\;\frac{1}{\sqrt{x}} + \frac{-1}{\sqrt{1 + x}} \leq 10^{-5}:\\ \;\;\;\;\left(\frac{0.5}{x} + \left(\frac{0.3125}{{x}^{3}} - \left(\frac{0.375}{x \cdot x} + \frac{0.2734375}{{x}^{4}}\right)\right)\right) \cdot {x}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;{x}^{-0.5} \cdot \left(1 - \sqrt{\frac{x}{1 + x}}\right)\\ \end{array} \]

Alternative 3
Accuracy	99.7%
Cost	27332

\[\begin{array}{l} \mathbf{if}\;\frac{1}{\sqrt{x}} + \frac{-1}{\sqrt{1 + x}} \leq 2 \cdot 10^{-7}:\\ \;\;\;\;{x}^{-0.5} \cdot \left(\frac{0.5}{x} + \left(\frac{0.3125}{{x}^{3}} - \frac{0.375}{x \cdot x}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{x}^{-0.5} \cdot \left(1 - \sqrt{\frac{x}{1 + x}}\right)\\ \end{array} \]

Alternative 4
Accuracy	99.6%
Cost	26884

\[\begin{array}{l} \mathbf{if}\;\frac{1}{\sqrt{x}} + \frac{-1}{\sqrt{1 + x}} \leq 5 \cdot 10^{-12}:\\ \;\;\;\;{x}^{-0.5} \cdot \left(\frac{0.5}{x} - \frac{0.375}{x \cdot x}\right)\\ \mathbf{else}:\\ \;\;\;\;{x}^{-0.5} \cdot \left(1 - \sqrt{\frac{x}{1 + x}}\right)\\ \end{array} \]

Alternative 5
Accuracy	99.6%
Cost	26692

\[\begin{array}{l} \mathbf{if}\;\frac{1}{\sqrt{x}} + \frac{-1}{\sqrt{1 + x}} \leq 5 \cdot 10^{-12}:\\ \;\;\;\;{x}^{-0.5} \cdot \left(\frac{0.5}{x} - \frac{0.375}{x \cdot x}\right)\\ \mathbf{else}:\\ \;\;\;\;{x}^{-0.5} - {\left(1 + x\right)}^{-0.5}\\ \end{array} \]

Alternative 6
Accuracy	98.8%
Cost	7300

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

Alternative 7
Accuracy	98.4%
Cost	7044

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

Alternative 8
Accuracy	98.1%
Cost	6916

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

Alternative 9
Accuracy	98.1%
Cost	6916

\[\begin{array}{l} \mathbf{if}\;x \leq 0.68:\\ \;\;\;\;{x}^{-0.5} + -1\\ \mathbf{else}:\\ \;\;\;\;\frac{{x}^{-0.5}}{x \cdot 2}\\ \end{array} \]

Alternative 10
Accuracy	49.9%
Cost	6528

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

Alternative 11
Accuracy	3.9%
Cost	192

\[x \cdot 0.5 \]

Alternative 12
Accuracy	1.9%
Cost	64

\[-1 \]

Reproduce

herbie shell --seed 2023160 
(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)))))