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Official technical manual: IEEE Arithmetic.

Relevant locators: IEEE Arithmetic Model, significand precision and overall bit count; Double Format, field layout and Table 2-5; and Ranges and Precisions in Decimal Representation. The source describes the traditional IEEE double format, now called binary64. It supports the precision and storage calculations for the double-precision floating-point format; no claim about a current processor's execution speed is inferred.

JSON, short for JavaScript Object Notation, is a textual data interchange format used for serialization. Its values are objects, arrays, strings, numbers, Boolean values and null. Object member names are strings; array elements are ordered. A complete JSON text may represent a single value, including a number or string.^[ref]

The format's syntax and an implementation's numerical precision are separate. For an original example, consider {"n":9007199254740993}. A JavaScript call to JSON.parse produces a Number whose decimal value is 9007199254740992; the requested integer $2^{53}+1$ has rounded to $2^{53}$ in the double-precision floating-point format. The same input decoded by Python's default json.loads retains the integer 9007199254740993 exactly.

Encoding the digits as {"n":"9007199254740993"} preserves them as a string in both examples, but changes the represented type. Any later numerical interpretation needs an agreed rule. RFC8259 identifies the integer interval $[-(2^{53})+1,(2^{53})-1]$ for exact agreement with commonly used numerical implementations; it does not restrict valid JSON numerals to that interval.^[ref]

For exchange between systems outside a closed ecosystem, RFC8259 requires UTF-8.^[ref] Defining a textual format therefore addresses portable syntax, while faithful deserialization also requires agreement about the values an application expects.

Choose the reciprocal-factorial summation eb for asymptotically high accuracy, counting multiplications and divisions as arithmetic work. For tolerance $ϵ$, linear powering in part (a)(i) takes $O({ϵ}^{-1})$ multiplications; binary exponentiation in part (a)(ii) reduces this to $O(\log (1/ϵ))$. Part (c) also has $O(\log (1/ϵ))$ arithmetic work, since its required $n$ grows like ${ϵ}^{-1/2}$, though its constants and cancellation error differ.

By the reciprocal-factorial series remainder bound, the omitted positive tail satisfiesThus the factorial denominator grows much faster than an exponential in the number of iterations. For $d$ decimal places, setting $ϵ$ of order $10^{-d}$ gives $n=\Theta (d/\log d)$ by Stirling's approximation, rather than $\Theta (d)$ binary-power arithmetic steps. This is a comparison in an exact-arithmetic operation-count model; the stated Big O notation does not determine the exact winner for every small tolerance. Taken literally, eb uses zero multiplications and only divisions. Fixed double-precision floating-point format cannot deliver arbitrarily many correct decimal places, and the precision-dependent cost of arbitrary-precision arithmetic is a separate issue.

About five steps are certified by the uniform worst-case estimate; the actual iteration reaches double-precision accuracy in about four. Put $K=15/14$. The initial enclosure gives $e_0=|x_0-r|<1/5$, and repeated squaring yieldsFor $k=4$ this is about $1.84\times 10^{-11}$, while for $k=5$ it is below $4\times 10^{-22}$. Thus five ideal-arithmetic steps push this bound well below the approximately $10^{-16}$ relative resolution of double-precision floating-point format.

The bound is conservative. Factoring $f(x)=(x^2+1)(x^4-6x^2+6)$ identifiesHigh-precision evaluation of the actual successive errors gives approximately $7.40\times 10^{-2}$, $3.56\times 10^{-3}$, $9.91\times 10^{-6}$, $7.73\times 10^{-11}$ and $4.70\times 10^{-21}$. Thus the fourth step would already exceed double-precision accuracy in exact arithmetic. In floating-point evaluation, cancellation in the residual and other roundoff errors prevent the subsequent theoretical digits from being realized.

With the gap-at-one convention, machine epsilon is the difference between $1$ and the next larger representable floating-point number. The double-precision floating-point format has 53 significant binary digits, including the implicit leading digit, soA related convention uses unit roundoff, the worst relative rounding scale for round-to-nearest: $u=2^{-53}\approx 1.11\times 10^{-16}$. Stating the convention distinguishes these two common values.^[ref]