Merge next into issue-4-api-chats (closes #4)

Brings in the calculator's powers and remainders. docs/TODO.md keeps
both Completed Steps lines, the API one first.

Model: opus-5-5

README.md

@@ -5,9 +5,10 @@ SimpleX Chat network: it accepts every contact request and answers
 arithmetic such as `2 + 2` with the result.
 
 Send it `2 + 2` and it replies `4`; send `5 * 5/2` and it replies
-`12.5`. It understands decimal numbers, `+ - * /`, unary minus and
-parentheses, and computes exactly, so `0.1 + 0.2` is `0.3`. Anything
-else gets a short explanation instead of a result.
+`12.5`. It understands decimal numbers, `+ - * /`, powers written `2^10`
+or `2**10`, remainders written `7 % 3`, signs and parentheses, and
+computes exactly, so `0.1 + 0.2` is `0.3`. Anything else gets a short
+explanation instead of a result.
 
 ## Getting Started
 
@@ -232,16 +233,33 @@ container.
 - **Replies**: for each text message a contact sends in a direct chat,
   the bot sends back the result, as a reply quoting the message. Group
   messages, files and the bot's own messages are ignored.
-- **Arithmetic** (`internal/calc`): the text is parsed as a Go
-  expression with `go/parser`, and only numbers, `+ - * /`, unary signs
-  and parentheses are evaluated; anything else in the syntax tree is
-  refused. `go/constant` computes with exact rationals. Numbers are read
-  as decimal, so `010` is ten. Input over 256 bytes is refused, so a
+- **Arithmetic** (`internal/calc`): a small parser of its own reads
+  numbers, `+ - * / % ^`, signs and parentheses, and refuses anything
+  else. `go/constant` computes with exact rationals. `^`, also written
+  `**`, is a power: it binds tighter than `*`, `/`, `%` and a sign on
+  its left, and groups to the right, so `2^3^2` is `512`, `-2^2` is
+  `-4`, `(-2)^2` is `4` and `2^-1` is `0.5`. `%` is the remainder and
+  ranks with `*` and `/`; its result takes the sign of the divisor, as
+  in Python, so `7 % 3` is `1`, `-7 % 3` is `2` and `7.5 % 2` is `1.5`.
+  A power with a whole exponent is exact, so `0.1^2` is `0.01` and
+  `2^-1400 * 2^1400` is `1`, unless `go/constant` could hold the result
+  only rounded; that power, and one with a fractional exponent, is
+  computed as a double, so `2^0.5` is `1.4142135623730951`. A negative
+  number to a fractional power is refused, as having no real result.
+  Numbers are read as decimal, so `010` is ten. Input over 256 bytes is
+  refused, exact powers are capped, and every number is held as a
+  fraction, whole numbers too, under the 4096-bit limit below, so a
   message cannot make the bot do unbounded work. Whole numbers below
   10<sup>21</sup> are written exactly; other results in the shortest
   form that reads back as the same double, in exponent notation from
-  10<sup>21</sup> up and below 10<sup>-6</sup>. A result beyond the
-  range of a double is refused as too large.
+  10<sup>21</sup> up and below 10<sup>-6</sup>. Refused as too large or
+  too small: any number whose numerator or denominator reaches 4096
+  bits, wherever it appears, as `go/constant` rounds a fraction that
+  grows that large (`1e-1300 + 1`); a power computed as a double whose
+  base or result is outside the normal range of a double, about 2.2e-308
+  to 1.8e308 in magnitude, where a double keeps all its digits
+  (`1e-400^0.5`); and a result other than zero outside that range, as it
+  is written through a double (`1e400`, `2^-1400`).
 - **Failure is an exit.** If the chat client exits or the connection to
   it drops, the bot exits with an error and the container's restart
   policy starts both again. `SIGTERM` stops the bot, which stops the