2isqrt (example 3.6)

Average Error: 19.8 → 0.2

Time: 13.1s

Precision: binary64

Cost: 26500

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \leq 5 \cdot 10^{+56}:\\ \;\;\;\;\frac{1}{\sqrt{x} + \sqrt{1 + x}} \cdot {\left(\mathsf{fma}\left(x, x, x\right)\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;{x}^{-0.5} \cdot \frac{0.5 - \frac{0.375}{x}}{x}\\ \end{array} \]

FPCore

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

↓

(FPCore (x)
 :precision binary64
 (if (<= x 5e+56)
   (* (/ 1.0 (+ (sqrt x) (sqrt (+ 1.0 x)))) (pow (fma x x x) -0.5))
   (* (pow x -0.5) (/ (- 0.5 (/ 0.375 x)) x))))

C

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

↓

double code(double x) {
	double tmp;
	if (x <= 5e+56) {
		tmp = (1.0 / (sqrt(x) + sqrt((1.0 + x)))) * pow(fma(x, x, x), -0.5);
	} else {
		tmp = pow(x, -0.5) * ((0.5 - (0.375 / x)) / 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 (x <= 5e+56)
		tmp = Float64(Float64(1.0 / Float64(sqrt(x) + sqrt(Float64(1.0 + x)))) * (fma(x, x, x) ^ -0.5));
	else
		tmp = Float64((x ^ -0.5) * Float64(Float64(0.5 - Float64(0.375 / x)) / 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[x, 5e+56], N[(N[(1.0 / N[(N[Sqrt[x], $MachinePrecision] + N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Power[N[(x * x + x), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision], N[(N[Power[x, -0.5], $MachinePrecision] * N[(N[(0.5 - N[(0.375 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]]

Target

Original	19.8
Target	0.7
Herbie	0.2

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

Derivation

1. Split input into 2 regimes

2. if x < 5.00000000000000024e56

   1. Initial program 8.4

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

   2. Applied egg-rr 0.3

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

   3. Simplified 0.3

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

      [Start] 0.3	\[ \frac{1 + \left(x - x\right)}{\sqrt{x + x \cdot x} \cdot \left(\sqrt{x} + \sqrt{1 + x}\right)} \]
      +-commutative [=>] 0.3	\[ \frac{\color{blue}{\left(x - x\right) + 1}}{\sqrt{x + x \cdot x} \cdot \left(\sqrt{x} + \sqrt{1 + x}\right)} \]
      +-inverses [=>] 0.3	\[ \frac{\color{blue}{0} + 1}{\sqrt{x + x \cdot x} \cdot \left(\sqrt{x} + \sqrt{1 + x}\right)} \]
      metadata-eval [=>] 0.3	\[ \frac{\color{blue}{1}}{\sqrt{x + x \cdot x} \cdot \left(\sqrt{x} + \sqrt{1 + x}\right)} \]
      +-commutative [=>] 0.3	\[ \frac{1}{\sqrt{x + x \cdot x} \cdot \color{blue}{\left(\sqrt{1 + x} + \sqrt{x}\right)}} \]

   4. Applied egg-rr 0.1

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

   if 5.00000000000000024e56 < x

   1. Initial program 36.7

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

   2. Applied egg-rr 36.7

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

   3. Simplified 36.7

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

      [Start] 36.7	\[ \frac{\frac{\sqrt{1 + x} - \sqrt{x}}{\sqrt{x}}}{-\sqrt{1 + x}} \cdot -1 \]
      associate-*l/ [=>] 36.7	\[ \color{blue}{\frac{\frac{\sqrt{1 + x} - \sqrt{x}}{\sqrt{x}} \cdot -1}{-\sqrt{1 + x}}} \]
      associate-/l* [=>] 36.7	\[ \color{blue}{\frac{\frac{\sqrt{1 + x} - \sqrt{x}}{\sqrt{x}}}{\frac{-\sqrt{1 + x}}{-1}}} \]
      metadata-eval [<=] 36.7	\[ \frac{\frac{\sqrt{1 + x} - \sqrt{x}}{\sqrt{x}}}{\frac{-\sqrt{1 + x}}{\color{blue}{\frac{1}{-1}}}} \]
      associate-/l* [<=] 36.7	\[ \frac{\frac{\sqrt{1 + x} - \sqrt{x}}{\sqrt{x}}}{\color{blue}{\frac{\left(-\sqrt{1 + x}\right) \cdot -1}{1}}} \]
      *-commutative [<=] 36.7	\[ \frac{\frac{\sqrt{1 + x} - \sqrt{x}}{\sqrt{x}}}{\frac{\color{blue}{-1 \cdot \left(-\sqrt{1 + x}\right)}}{1}} \]
      mul-1-neg [=>] 36.7	\[ \frac{\frac{\sqrt{1 + x} - \sqrt{x}}{\sqrt{x}}}{\frac{\color{blue}{-\left(-\sqrt{1 + x}\right)}}{1}} \]
      remove-double-neg [=>] 36.7	\[ \frac{\frac{\sqrt{1 + x} - \sqrt{x}}{\sqrt{x}}}{\frac{\color{blue}{\sqrt{1 + x}}}{1}} \]
      /-rgt-identity [=>] 36.7	\[ \frac{\frac{\sqrt{1 + x} - \sqrt{x}}{\sqrt{x}}}{\color{blue}{\sqrt{1 + x}}} \]
      associate-/l/ [=>] 36.7	\[ \color{blue}{\frac{\sqrt{1 + x} - \sqrt{x}}{\sqrt{1 + x} \cdot \sqrt{x}}} \]
      associate-/r* [=>] 36.7	\[ \color{blue}{\frac{\frac{\sqrt{1 + x} - \sqrt{x}}{\sqrt{1 + x}}}{\sqrt{x}}} \]
      div-sub [=>] 36.7	\[ \frac{\color{blue}{\frac{\sqrt{1 + x}}{\sqrt{1 + x}} - \frac{\sqrt{x}}{\sqrt{1 + x}}}}{\sqrt{x}} \]
      *-inverses [=>] 36.7	\[ \frac{\color{blue}{1} - \frac{\sqrt{x}}{\sqrt{1 + x}}}{\sqrt{x}} \]

   4. Taylor expanded in x around inf 0.2

      \[\leadsto \frac{\color{blue}{0.5 \cdot \frac{1}{x} - 0.375 \cdot \frac{1}{{x}^{2}}}}{\sqrt{x}} \]

   5. Simplified 0.2

      \[\leadsto \frac{\color{blue}{\frac{0.5}{x} - \frac{0.375}{x \cdot x}}}{\sqrt{x}} \]
      Proof

      [Start] 0.2	\[ \frac{0.5 \cdot \frac{1}{x} - 0.375 \cdot \frac{1}{{x}^{2}}}{\sqrt{x}} \]
      associate-*r/ [=>] 0.2	\[ \frac{\color{blue}{\frac{0.5 \cdot 1}{x}} - 0.375 \cdot \frac{1}{{x}^{2}}}{\sqrt{x}} \]
      metadata-eval [=>] 0.2	\[ \frac{\frac{\color{blue}{0.5}}{x} - 0.375 \cdot \frac{1}{{x}^{2}}}{\sqrt{x}} \]
      associate-*r/ [=>] 0.2	\[ \frac{\frac{0.5}{x} - \color{blue}{\frac{0.375 \cdot 1}{{x}^{2}}}}{\sqrt{x}} \]
      metadata-eval [=>] 0.2	\[ \frac{\frac{0.5}{x} - \frac{\color{blue}{0.375}}{{x}^{2}}}{\sqrt{x}} \]
      unpow2 [=>] 0.2	\[ \frac{\frac{0.5}{x} - \frac{0.375}{\color{blue}{x \cdot x}}}{\sqrt{x}} \]

   6. Applied egg-rr 0.2

      \[\leadsto \color{blue}{\frac{0.5 - \frac{0.375}{x}}{x} \cdot {x}^{-0.5}} \]

   7. Simplified 0.2

      \[\leadsto \color{blue}{{x}^{-0.5} \cdot \frac{0.5 - \frac{0.375}{x}}{x}} \]
      Proof

      [Start] 0.2	\[ \frac{0.5 - \frac{0.375}{x}}{x} \cdot {x}^{-0.5} \]
      *-commutative [=>] 0.2	\[ \color{blue}{{x}^{-0.5} \cdot \frac{0.5 - \frac{0.375}{x}}{x}} \]

3. Recombined 2 regimes into one program.

4. Final simplification 0.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 5 \cdot 10^{+56}:\\ \;\;\;\;\frac{1}{\sqrt{x} + \sqrt{1 + x}} \cdot {\left(\mathsf{fma}\left(x, x, x\right)\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;{x}^{-0.5} \cdot \frac{0.5 - \frac{0.375}{x}}{x}\\ \end{array} \]

Alternatives

Alternative 1
Error	0.2
Cost	33988

\[\begin{array}{l} t_0 := \frac{-1}{\sqrt{1 + x}}\\ \mathbf{if}\;\frac{1}{\sqrt{x}} + t_0 \leq 5 \cdot 10^{-9}:\\ \;\;\;\;\frac{\left(0.5 \cdot \frac{1}{x} + 0.3125 \cdot \frac{1}{{x}^{3}}\right) + 0.375 \cdot \frac{-1}{{x}^{2}}}{\sqrt{x}}\\ \mathbf{else}:\\ \;\;\;\;{x}^{-0.5} + t_0\\ \end{array} \]

Alternative 2
Error	0.2
Cost	27268

\[\begin{array}{l} t_0 := \frac{-1}{\sqrt{1 + x}}\\ \mathbf{if}\;\frac{1}{\sqrt{x}} + t_0 \leq 5 \cdot 10^{-9}:\\ \;\;\;\;\frac{\frac{0.5}{x} + \left(\frac{0.3125}{{x}^{3}} + \frac{-0.375}{x \cdot x}\right)}{\sqrt{x}}\\ \mathbf{else}:\\ \;\;\;\;{x}^{-0.5} + t_0\\ \end{array} \]

Alternative 3
Error	0.2
Cost	26756

\[\begin{array}{l} t_0 := \frac{-1}{\sqrt{1 + x}}\\ \mathbf{if}\;\frac{1}{\sqrt{x}} + t_0 \leq 5 \cdot 10^{-9}:\\ \;\;\;\;{x}^{-0.5} \cdot \frac{0.5 - \frac{0.375}{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;{x}^{-0.5} + t_0\\ \end{array} \]

Alternative 4
Error	0.7
Cost	26368

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

Alternative 5
Error	0.2
Cost	13636

\[\begin{array}{l} t_0 := \sqrt{1 + x}\\ \mathbf{if}\;x \leq 112000:\\ \;\;\;\;{x}^{-0.5} + \frac{-1}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x + 0.5}}{\sqrt{x} + t_0}\\ \end{array} \]

Alternative 6
Error	0.2
Cost	13636

\[\begin{array}{l} t_0 := \sqrt{1 + x}\\ \mathbf{if}\;x \leq 112000:\\ \;\;\;\;{x}^{-0.5} + \frac{-1}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\sqrt{x} + t_0}}{x + 0.5}\\ \end{array} \]

Alternative 7
Error	0.2
Cost	13380

\[\begin{array}{l} \mathbf{if}\;x \leq 160000:\\ \;\;\;\;{x}^{-0.5} - {\left(1 + x\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;{x}^{-0.5} \cdot \frac{0.5 - \frac{0.375}{x}}{x}\\ \end{array} \]

Alternative 8
Error	0.8
Cost	7172

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

Alternative 9
Error	0.7
Cost	7172

\[\begin{array}{l} \mathbf{if}\;x \leq 1.4:\\ \;\;\;\;{x}^{-0.5} + \frac{-1}{1 - x \cdot -0.5}\\ \mathbf{else}:\\ \;\;\;\;{x}^{-0.5} \cdot \frac{0.5 - \frac{0.375}{x}}{x}\\ \end{array} \]

Alternative 10
Error	0.8
Cost	7108

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

Alternative 11
Error	1.1
Cost	7044

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

Alternative 12
Error	1.1
Cost	7044

\[\begin{array}{l} \mathbf{if}\;x \leq 1.1:\\ \;\;\;\;{x}^{-0.5} + \left(-1 - x \cdot -0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 - \frac{0.375}{x}}{{x}^{1.5}}\\ \end{array} \]

Alternative 13
Error	1.4
Cost	6852

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

Alternative 14
Error	21.0
Cost	6788

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

Alternative 15
Error	21.7
Cost	6660

\[\begin{array}{l} \mathbf{if}\;x \leq 8.5 \cdot 10^{+122}:\\ \;\;\;\;{x}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\frac{1}{x + 0.5} + -1\right)\\ \end{array} \]

Alternative 16
Error	50.4
Cost	576

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

Alternative 17
Error	50.4
Cost	576

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

Alternative 18
Error	59.2
Cost	320

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

Alternative 19
Error	59.2
Cost	192

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

Alternative 20
Error	60.3
Cost	64

\[2 \]

Reproduce

herbie shell --seed 2023187 
(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)))))