Powers and remainders in the calculator (closes #3)

internal/calc gets its own tokenizer and precedence-climbing parser in
place of go/parser, which cannot express a power operator; go/constant
still computes. ^ (also **) binds tighter than * / % and a sign on its
left, and groups to the right. % takes the sign of the divisor and is
exact on decimals. A whole exponent is computed exactly while the
result's numerator and denominator stay within 4096 bits; otherwise,
and for a fractional exponent, in float64. A remainder, or the sign of
a negative base's power, that depends on a value go/constant holds
rounded is refused as too large. The welcome text and the replies
mention the new operators; a negative base with a fractional exponent
gets its own reply.

Model: opus-5-5

AGENTS.md

@@ -120,7 +120,7 @@ Do not weaken them.
 ```
 cmd/simplexcalc/    main(), a single call into internal/cli
 internal/bot/       startup, address setup, and the reply to a message
-internal/calc/      the arithmetic: go/parser and go/constant
+internal/calc/      the arithmetic: its own parser, and go/constant
 internal/cli/       cobra command tree: run and version
 internal/config/    viper-backed configuration; the abort-on-garbage rule
 internal/logger/    log/slog, JSON always

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
+`12.5`. It understands decimal numbers, `+ - * /`, powers written `2^10`
-parentheses, and computes exactly, so `0.1 + 0.2` is `0.3`. Anything
+or `2**10`, remainders written `7 % 3`, signs and parentheses, and
-else gets a short explanation instead of a result.
+computes exactly, so `0.1 + 0.2` is `0.3`. Anything else gets a short
+explanation instead of a result.
 
 ## Getting Started
 
@@ -157,16 +158,25 @@ 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
+- **Arithmetic** (`internal/calc`): a small parser of its own reads
-  expression with `go/parser`, and only numbers, `+ - * /`, unary signs
+  numbers, `+ - * / % ^`, signs and parentheses, and refuses anything
-  and parentheses are evaluated; anything else in the syntax tree is
+  else. `go/constant` computes with exact rationals. `^`, also written
-  refused. `go/constant` computes with exact rationals. Numbers are read
+  `**`, is a power: it binds tighter than `*`, `/`, `%` and a sign on
-  as decimal, so `010` is ten. Input over 256 bytes is refused, so a
+  its left, and groups to the right, so `2^3^2` is `512`, `-2^2` is
-  message cannot make the bot do unbounded work. Whole numbers below
+  `-4`, `(-2)^2` is `4` and `2^-1` is `0.5`. `%` is the remainder and
-  10<sup>21</sup> are written exactly; other results in the shortest
+  ranks with `*` and `/`; its result takes the sign of the divisor, as
-  form that reads back as the same double, in exponent notation from
+  in Python, so `7 % 3` is `1`, `-7 % 3` is `2` and `7.5 % 2` is `1.5`.
-  10<sup>21</sup> up and below 10<sup>-6</sup>. A result beyond the
+  A power with a whole exponent is exact, so `0.1^2` is `0.01`, unless
-  range of a double is refused as too large.
+  its numerator and denominator together could pass 4096 bits; 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 and exact
+  powers are capped, 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.
 - **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

docs/TODO.md

@@ -27,6 +27,9 @@ with no deprecation warning.
 
 # Completed Steps
 
+- 2026-09-28 Powers (`^`, also written `**`) and remainders (`%`) in
+  `internal/calc`, which now reads expressions with a parser of its own
+  in place of `go/parser`
 - 2026-09-28 Moved the command tree and the `run` and `version` commands
   from `cmd/simplexcalc/` into `internal/cli`; `cmd/simplexcalc/main.go`
   is now a single call to `cli.Main`

internal/bot/bot.go

@@ -22,7 +22,7 @@ import (
 const DisplayName = "calc"
 
 // Welcome is sent to everyone whose contact request the bot accepts.
-const Welcome = "Send me arithmetic, such as 2 + 2 or 5 * 5/2, " +
+const Welcome = "Send me arithmetic, such as 2 + 2, 5 * 5/2, 2^10 or 7 % 3, " +
 	"and I will reply with the result."
 
 const (
@@ -223,8 +223,10 @@ func Reply(text string) string {
 		return "I cannot divide by zero."
 	case errors.Is(err, calc.ErrTooLarge):
 		return "The result is too large for me."
+	case errors.Is(err, calc.ErrNoRealResult):
+		return "A negative number to a fractional power has no real result."
 	default:
-		return "I only understand arithmetic: numbers, + - * / and " +
+		return "I only understand arithmetic: numbers, + - * /, ^ for a power, " +
-			"parentheses, such as 5 * 5/2."
+			"% for a remainder, and parentheses, such as 5 * 5/2 or 2^10."
 	}
 }

internal/bot/bot_test.go

@@ -16,6 +16,8 @@ func TestReply(t *testing.T) {
 	for in, want := range map[string]string{
 		"2 + 2":   "4",
 		"5 * 5/2": "12.5",
+		"2^10":    "1024",
+		"7 % 3":   "1",
 	} {
 		if got := bot.Reply(in); got != want {
 			t.Errorf("Reply(%q) = %q, want %q", in, got, want)
@@ -26,6 +28,7 @@ func TestReply(t *testing.T) {
 		"hello":    "I only understand arithmetic",
 		"1 / 0":    "I cannot divide by zero.",
 		"1e400":    "The result is too large for me.",
+		"(-8)^0.5": "A negative number to a fractional power has no real",
 		strings.Repeat("1+", calc.MaxInputLength) + "1": "That is too long for me",
 	} {
 		if got := bot.Reply(in); !strings.HasPrefix(got, want) {

internal/calc/calc.go

@@ -1,29 +1,37 @@
 // Package calc evaluates the arithmetic people send the bot: decimal
-// numbers, + - * /, unary minus and parentheses.
+// numbers, + - * / % ^, signs and parentheses.
 //
-// The expression is parsed by go/parser and computed by go/constant,
+// The expression is read by a small parser of its own, because Go's
-// which does exact rational arithmetic: 5 * 5/2 is exactly 12.5, and
+// grammar has no power operator (^ is XOR there), and computed by
-// 0.1 + 0.2 is exactly 0.3, so a result carries no binary floating
+// go/constant, which does exact rational arithmetic: 5 * 5/2 is exactly
-// point noise until the moment it is formatted.
+// 12.5, and 0.1 + 0.2 is exactly 0.3, so a result carries no binary
+// floating point noise until the moment it is formatted. A power is the
+// exception: one with a fractional exponent, or too large to compute
+// exactly, is computed in float64.
 package calc
 
 import (
 	"errors"
-	"go/ast"
 	"go/constant"
-	"go/parser"
 	"go/token"
 	"math"
+	"math/big"
 	"regexp"
 	"strconv"
 	"strings"
 )
 
-// MaxInputLength caps an expression, in bytes, so a message cannot make
+// MaxInputLength caps an expression, in bytes, and maxExactPowerBits
-// the bot do unbounded work. Every operation's cost grows with the size
+// caps a power, so that a message cannot make the bot do unbounded work.
-// of its operands, and the operands can only grow with the input.
 const MaxInputLength = 256
 
+// maxExactPowerBits caps a power computed exactly: its numerator and
+// denominator together have at most this many bits, estimated before
+// multiplying as the exponent times the bits in the base's numerator and
+// denominator. A larger power is computed in float64, whose cost does not
+// grow with it.
+const maxExactPowerBits = 4096
+
 // Results of magnitude plainUpper or more are written in exponent form
 // (1e+21 rather than twenty-two digits), and so are fractions smaller
 // than plainLower (1e-07 rather than 0.0000001).
@@ -32,20 +40,37 @@ const (
 	plainLower = 1e-6
 )
 
+// The precedence of the binary operators: the higher, the tighter the
+// operator binds.
+const (
+	sumPrecedence = iota + 1
+	productPrecedence
+	powerPrecedence
+)
+
 // Errors returned by Evaluate. The bot turns each into a reply.
 var (
 	ErrTooLong        = errors.New("expression too long")
 	ErrNotArithmetic  = errors.New("not an arithmetic expression")
 	ErrDivisionByZero = errors.New("division by zero")
 	ErrTooLarge       = errors.New("result too large")
+	ErrNoRealResult   = errors.New("no real result")
 )
 
-// decimalLiteral is the only number syntax accepted. Go's own literal
+// decimal is the only number syntax accepted. Go's own literal syntax is
-// syntax is wider, and parts of it are traps for someone typing
+// wider, and parts of it are traps for someone typing arithmetic: 010 is
-// arithmetic: 010 is octal 8, and 0x10, 1_000 and 1i are not what a
+// octal 8, and 0x10, 1_000 and 1i are not what a calculator user means
-// calculator user means by a number.
+// by a number. Here the x, _ or i matches no token and is refused.
-var decimalLiteral = regexp.MustCompile(
+const decimal = `([0-9]+\.?[0-9]*|\.[0-9]+)([eE][+-]?[0-9]+)?`
-	`^([0-9]+\.?[0-9]*|\.[0-9]+)([eE][+-]?[0-9]+)?$`,
+
+var (
+	// nextToken matches the token at the start of the input, after any
+	// whitespace: an operator, a parenthesis or a number. ** comes
+	// before * so that it is read as one token.
+	nextToken = regexp.MustCompile(`^\s*(\*\*|[-+*/%^()]|` + decimal + `)`)
+
+	// decimalLiteral matches a token that is a number.
+	decimalLiteral = regexp.MustCompile(`^` + decimal + `$`)
 )
 
 // Evaluate computes an arithmetic expression and returns its result as
@@ -57,75 +82,206 @@ func Evaluate(input string) (string, error) {
 		return "", ErrTooLong
 	}
 
-	if s == "" {
+	tokens, err := tokenize(s)
-		return "", ErrNotArithmetic
-	}
-
-	expr, err := parser.ParseExpr(s)
-	if err != nil {
-		return "", ErrNotArithmetic
-	}
-
-	v, err := eval(expr)
 	if err != nil {
 		return "", err
 	}
 
+	p := parser{tokens: tokens}
+
+	v, err := p.expr(sumPrecedence)
+	if err != nil {
+		return "", err
+	}
+
+	if p.next() != "" {
+		return "", ErrNotArithmetic
+	}
+
 	return format(v)
 }
 
-// eval walks the syntax tree, allowing only the node types and
+// tokenize splits an expression into operators, parentheses and
-// operators of arithmetic. Anything else — identifiers, calls, strings,
+// numbers, and refuses anything else. ** is returned as ^.
-// shifts, comparisons — is refused, not evaluated.
+func tokenize(s string) ([]string, error) {
-func eval(e ast.Expr) (constant.Value, error) {
+	var tokens []string
-	switch n := e.(type) {
+
-	case *ast.BasicLit:
+	for strings.TrimSpace(s) != "" {
-		return literal(n)
+		m := nextToken.FindStringSubmatch(s)
-	case *ast.ParenExpr:
+		if m == nil {
-		return eval(n.X)
-	case *ast.UnaryExpr:
-		if n.Op != token.ADD && n.Op != token.SUB {
 			return nil, ErrNotArithmetic
 		}
 
-		x, err := eval(n.X)
+		tok := m[1]
+		if tok == "**" {
+			tok = "^"
+		}
+
+		tokens = append(tokens, tok)
+		s = s[len(m[0]):]
+	}
+
+	return tokens, nil
+}
+
+// parser computes an expression as it reads it, by precedence climbing:
+// expr reads operands joined by operators of at least a given
+// precedence, and hands the right operand of each to a deeper call that
+// takes only the operators that bind tighter, so those are applied
+// first.
+type parser struct {
+	tokens []string
+}
+
+// next removes and returns the next token, or "" at the end.
+func (p *parser) next() string {
+	tok := p.peek()
+	if tok != "" {
+		p.tokens = p.tokens[1:]
+	}
+
+	return tok
+}
+
+// peek returns the next token, or "" at the end, and leaves it unread.
+func (p *parser) peek() string {
+	if len(p.tokens) == 0 {
+		return ""
+	}
+
+	return p.tokens[0]
+}
+
+// expr reads and computes an expression whose binary operators all have
+// at least minPrecedence. Operators of equal precedence group to the
+// left, 8/2/2 is (8/2)/2, except ^, which groups to the right: 2^3^2 is
+// 2^(3^2).
+func (p *parser) expr(minPrecedence int) (constant.Value, error) {
+	x, err := p.operand()
 	if err != nil {
 		return nil, err
 	}
 
-		return constant.UnaryOp(n.Op, x, 0), nil
+	for {
-	case *ast.BinaryExpr:
+		op := p.peek()
-		return binary(n)
+
-	default:
+		prec := precedence(op)
-		return nil, ErrNotArithmetic
+		if prec < minPrecedence {
+			return x, nil
+		}
+
+		p.next()
+
+		rightPrecedence := prec + 1
+		if op == "^" {
+			rightPrecedence = prec
+		}
+
+		y, err := p.expr(rightPrecedence)
+		if err != nil {
+			return nil, err
+		}
+
+		x, err = apply(x, op, y)
+		if err != nil {
+			return nil, err
+		}
 	}
 }
 
-func binary(n *ast.BinaryExpr) (constant.Value, error) {
+// operand reads a number, an expression in parentheses, or a sign and
-	switch n.Op { //nolint:exhaustive // every other operator is refused.
+// its operand. A sign binds more loosely than a power that follows it,
-	case token.ADD, token.SUB, token.MUL, token.QUO:
+// so -2^2 is -(2^2), and 2^-1 is 2^(-1).
-	default:
+func (p *parser) operand() (constant.Value, error) {
+	switch tok := p.next(); tok {
+	case "+", "-":
+		x, err := p.expr(powerPrecedence)
+		if err != nil {
+			return nil, err
+		}
+
+		if tok == "-" {
+			x = constant.UnaryOp(token.SUB, x, 0)
+		}
+
+		return x, nil
+	case "(":
+		x, err := p.expr(sumPrecedence)
+		if err != nil {
+			return nil, err
+		}
+
+		if p.next() != ")" {
 			return nil, ErrNotArithmetic
 		}
 
-	x, err := eval(n.X)
+		return x, nil
+	default:
+		return number(tok)
+	}
+}
+
+// precedence returns the precedence of a binary operator, and 0 for any
+// other token, which ends an expression.
+func precedence(op string) int {
+	switch op {
+	case "+", "-":
+		return sumPrecedence
+	case "*", "/", "%":
+		return productPrecedence
+	case "^":
+		return powerPrecedence
+	default:
+		return 0
+	}
+}
+
+func number(tok string) (constant.Value, error) {
+	if !decimalLiteral.MatchString(tok) {
+		return nil, ErrNotArithmetic
+	}
+
+	// Read as FLOAT, which makes every literal decimal: as INT, a
+	// leading zero would make it octal.
+	v := constant.MakeFromLiteral(tok, token.FLOAT, 0)
+
+	// The syntax was checked above, so Unknown here means the exponent
+	// overflowed.
+	if v.Kind() == constant.Unknown {
+		return nil, ErrTooLarge
+	}
+
+	return v, nil
+}
+
+// apply computes x op y.
+func apply(x constant.Value, op string, y constant.Value) (constant.Value, error) {
+	var (
+		v   constant.Value
+		err error
+	)
+
+	switch op {
+	case "+":
+		v = constant.BinaryOp(x, token.ADD, y)
+	case "-":
+		v = constant.BinaryOp(x, token.SUB, y)
+	case "*":
+		v = constant.BinaryOp(x, token.MUL, y)
+	case "/":
+		v, err = divide(x, y)
+	case "%":
+		v, err = modulo(x, y)
+	case "^":
+		v, err = power(x, y)
+	default:
+		err = ErrNotArithmetic
+	}
+
 	if err != nil {
 		return nil, err
 	}
 
-	y, err := eval(n.Y)
-	if err != nil {
-		return nil, err
-	}
-
-	// constant.BinaryOp panics on a zero divisor.
-	if n.Op == token.QUO && constant.Sign(y) == 0 {
-		return nil, ErrDivisionByZero
-	}
-
-	// token.QUO divides exactly, integers included: 25/2 is 12.5.
-	v := constant.BinaryOp(x, n.Op, y)
-
 	// go/constant represents an overflow to infinity as Unknown.
 	if v.Kind() == constant.Unknown {
 		return nil, ErrTooLarge
@@ -134,28 +290,139 @@ func binary(n *ast.BinaryExpr) (constant.Value, error) {
 	return v, nil
 }
 
-func literal(n *ast.BasicLit) (constant.Value, error) {
+func divide(x, y constant.Value) (constant.Value, error) {
-	if n.Kind != token.INT && n.Kind != token.FLOAT {
+	// constant.BinaryOp panics on a zero divisor.
-		return nil, ErrNotArithmetic
+	if constant.Sign(y) == 0 {
+		return nil, ErrDivisionByZero
 	}
 
-	if !decimalLiteral.MatchString(n.Value) {
+	// token.QUO divides exactly, integers included: 25/2 is 12.5.
-		return nil, ErrNotArithmetic
+	return constant.BinaryOp(x, token.QUO, y), nil
+}
+
+// modulo computes x % y, whose result takes the sign of y, as in Python:
+// -7 % 3 is 2 and 7 % -3 is -2. It is exact for decimals too: 7.5 % 2
+// is 1.5.
+func modulo(x, y constant.Value) (constant.Value, error) {
+	q, err := divide(x, y)
+	if err != nil {
+		return nil, err
 	}
 
-	// Read as FLOAT whatever the token says, which makes every literal
+	// The whole part of a rounded quotient, and so the remainder, would
-	// decimal: as INT, a leading zero would make it octal.
+	// be wrong.
-	v := constant.MakeFromLiteral(n.Value, token.FLOAT, 0)
+	if rounded(q) {
-
-	// The syntax was checked above, so Unknown here means the exponent
-	// overflowed.
-	if v.Kind() == constant.Unknown {
 		return nil, ErrTooLarge
 	}
 
+	// token.QUO_ASSIGN is go/constant's integer division, which
+	// truncates, so r has the sign of x and is less than y in size.
+	whole := constant.BinaryOp(constant.Num(q), token.QUO_ASSIGN, constant.Denom(q))
+	r := constant.BinaryOp(x, token.SUB, constant.BinaryOp(y, token.MUL, whole))
+
+	if constant.Sign(r) != 0 && constant.Sign(r) != constant.Sign(y) {
+		r = constant.BinaryOp(r, token.ADD, y)
+	}
+
+	return r, nil
+}
+
+// power computes x^y. A negative x needs a whole y, and its sign is
+// applied here: math.Pow would take it from the parity of y's float64
+// value, and every float64 from 2^53 up is even.
+func power(x, y constant.Value) (constant.Value, error) {
+	// n is y if y is a whole number, and Unknown otherwise.
+	n := constant.ToInt(y)
+
+	switch {
+	case constant.Sign(x) == 0 && constant.Sign(y) < 0:
+		return nil, ErrDivisionByZero
+	case constant.Sign(x) >= 0:
+		return nonNegativePower(x, y, n), nil
+	case rounded(y):
+		// Whether a rounded y is whole, or odd, is unknown.
+		return nil, ErrTooLarge
+	case n.Kind() != constant.Int:
+		return nil, ErrNoRealResult
+	}
+
+	// x is negative and n whole: x^n is (-x)^n, negated if n is odd.
+	v := nonNegativePower(constant.UnaryOp(token.SUB, x, 0), y, n)
+
+	odd := constant.BinaryOp(n, token.AND, constant.MakeInt64(1))
+	if constant.Sign(odd) != 0 {
+		v = constant.UnaryOp(token.SUB, v, 0)
+	}
+
 	return v, nil
 }
 
+// nonNegativePower computes x^y for x of at least zero, and y not below
+// zero if x is zero: exactly if y is a whole number n and the result
+// fits in maxExactPowerBits, otherwise in float64.
+func nonNegativePower(x, y, n constant.Value) constant.Value {
+	e, ok := constant.Int64Val(n)
+	if ok && exactPowerFits(x, e) {
+		return exactPower(x, e)
+	}
+
+	xf, _ := constant.Float64Val(x)
+	yf, _ := constant.Float64Val(y)
+
+	// An infinite result becomes Unknown, which apply refuses as too
+	// large.
+	return constant.MakeFloat64(math.Pow(xf, yf))
+}
+
+// exactPowerFits reports whether x^e fits in maxExactPowerBits.
+func exactPowerFits(x constant.Value, e int64) bool {
+	// Checked first so that the product below cannot overflow.
+	if e < -maxExactPowerBits || e > maxExactPowerBits {
+		return false
+	}
+
+	// Num and Denom are Unknown for a value too large or too small to
+	// be held as a fraction.
+	num, den := constant.Num(x), constant.Denom(x)
+	if num.Kind() != constant.Int {
+		return false
+	}
+
+	bits := int64(constant.BitLen(num) + constant.BitLen(den))
+
+	return bits*max(e, -e) <= maxExactPowerBits
+}
+
+// exactPower computes x^e by repeated squaring. x is not zero if e is
+// negative.
+func exactPower(x constant.Value, e int64) constant.Value {
+	result := constant.MakeInt64(1)
+
+	for n := max(e, -e); n > 0; n >>= 1 {
+		if n&1 == 1 {
+			result = constant.BinaryOp(result, token.MUL, x)
+		}
+
+		x = constant.BinaryOp(x, token.MUL, x)
+	}
+
+	if e < 0 {
+		result = constant.BinaryOp(constant.MakeInt64(1), token.QUO, result)
+	}
+
+	return result
+}
+
+// rounded reports whether go/constant holds v rounded. It holds a
+// number exactly, as a fraction, only while the numerator and the
+// denominator each stay under 4096 bits; past that, and for a literal of
+// that size, it holds a 512-bit float.
+func rounded(v constant.Value) bool {
+	_, isFloat := constant.Val(v).(*big.Float)
+
+	return isFloat
+}
+
 // format writes a result for a person to read. A whole number of
 // ordinary size is written exactly, digit for digit; anything else goes
 // through float64, whose shortest round-trip form is free of the noise

internal/calc/calc_test.go

@@ -4,6 +4,7 @@ import (
 	"errors"
 	"strings"
 	"testing"
+	"time"
 
 	"sneak.berlin/go/simplexcalc/internal/calc"
 )
@@ -13,7 +14,7 @@ import (
 func TestEvaluate(t *testing.T) {
 	t.Parallel()
 
-	cases := map[string]string{
+	expectResults(t, map[string]string{
 		// The specification's own examples.
 		"2 + 2":   "4",
 		"5 * 5/2": "12.5",
@@ -51,7 +52,81 @@ func TestEvaluate(t *testing.T) {
 		"1234567.5":               "1234567.5",
 		"-1 / 4":                  "-0.25",
 		"1e300 * 1e8":             "1e+308",
-	}
+	})
+}
+
+// TestEvaluatePowers: ^ and ** are one operator, binding tighter than
+// * / % and a sign on its left, and grouping to the right.
+func TestEvaluatePowers(t *testing.T) {
+	t.Parallel()
+
+	expectResults(t, map[string]string{
+		"2^3":               "8",
+		"2**3":              "8",
+		"2 ** 3 ^ 2":        "512",
+		"2^3^2":             "512",
+		"(2^3)^2":           "64",
+		"-2^2":              "-4",
+		"(-2)^2":            "4",
+		"(-2)^3":            "-8",
+		"(-2)^-3":           "-0.125",
+		"2^-1":              "0.5",
+		"2**-1":             "0.5",
+		"-2^-2":             "-0.25",
+		"2^-3^2":            "0.001953125",
+		"2 * 3^2":           "18",
+		"3^2 * 2":           "18",
+		"2^3 / 2^2":         "2",
+		"1 + 2^3 - 3^2":     "0",
+		"010^2":             "100",
+		"0.1^2":             "0.01",
+		"2^100 - 2^100 + 1": "1",
+		"2^64":              "18446744073709551616",
+		"0^0":               "1",
+		"0^3":               "0",
+		"1.5^2":             "2.25",
+		"2^0.5":             "1.4142135623730951",
+		"-2^0.5":            "-1.4142135623730951",
+		"4^0.5":             "2",
+		"0^0.5":             "0",
+		"2^1023":            "8.98846567431158e+307",
+		// Past 2^53 a float64 cannot tell odd from even.
+		"(-1)^(2^53 + 1)": "-1",
+		"(-1)^(10^30)":    "1",
+	})
+}
+
+// TestEvaluateModulo: % sits with * and /, left to right, and its result
+// takes the sign of the divisor.
+func TestEvaluateModulo(t *testing.T) {
+	t.Parallel()
+
+	expectResults(t, map[string]string{
+		"7 % 3":       "1",
+		"-7 % 3":      "2",
+		"7 % -3":      "-2",
+		"-7 % -3":     "-1",
+		"6 % 3":       "0",
+		"-6 % 3":      "0",
+		"7.5 % 2":     "1.5",
+		"0.3 % 0.1":   "0",
+		"-0.3 % 0.2":  "0.1",
+		"10 % 4 * 3":  "6",
+		"2 * 7 % 4":   "2",
+		"1 + 7 % 3":   "2",
+		"2^10 % 7":    "2",
+		"10^400 % 7":  "4",
+		"1e-30 % 1":   "1e-30",
+		"-1e-30 % 1":  "1",
+		"10 / 8 % 1":  "0.25",
+		"(7 % 3)^2":   "1",
+		"7 % (3 ^ 2)": "7",
+	})
+}
+
+// expectResults checks that each expression evaluates to its result.
+func expectResults(t *testing.T, cases map[string]string) {
+	t.Helper()
 
 	for in, want := range cases {
 		t.Run(in, func(t *testing.T) {
@@ -89,19 +164,45 @@ func TestEvaluateRefuses(t *testing.T) {
 		"2i * 2i":                   calc.ErrNotArithmetic,
 		"0x10 + 1":                  calc.ErrNotArithmetic,
 		"1_000 + 1":                 calc.ErrNotArithmetic,
-		"7 % 2":                     calc.ErrNotArithmetic,
-		"2 ^ 3":                     calc.ErrNotArithmetic,
 		"1 << 10":                   calc.ErrNotArithmetic,
 		"1 == 1":                    calc.ErrNotArithmetic,
 		"!1":                        calc.ErrNotArithmetic,
 		"func() int { return 1 }()": calc.ErrNotArithmetic,
+		"(1 + 2":                    calc.ErrNotArithmetic,
+		"1 + 2)":                    calc.ErrNotArithmetic,
+		"()":                        calc.ErrNotArithmetic,
+		"(2)(3)":                    calc.ErrNotArithmetic,
+		"2 ^":                       calc.ErrNotArithmetic,
+		"^ 2":                       calc.ErrNotArithmetic,
+		"2 ^^ 3":                    calc.ErrNotArithmetic,
+		"2 *** 3":                   calc.ErrNotArithmetic,
+		"2 * * 3":                   calc.ErrNotArithmetic,
+		"% 3":                       calc.ErrNotArithmetic,
+		"50%":                       calc.ErrNotArithmetic,
+		"2 × 3":                     calc.ErrNotArithmetic,
 		"1 / 0":                     calc.ErrDivisionByZero,
 		"1 / (2 - 2)":               calc.ErrDivisionByZero,
 		"5 / 0.0":                   calc.ErrDivisionByZero,
+		"7 % 0":                     calc.ErrDivisionByZero,
+		"7.5 % (1 - 1)":             calc.ErrDivisionByZero,
+		"0^-1":                      calc.ErrDivisionByZero,
+		"0^-0.5":                    calc.ErrDivisionByZero,
+		"(-2)^0.5":                  calc.ErrNoRealResult,
+		"(-8)^(1/3)":                calc.ErrNoRealResult,
+		"(-1)^-0.5":                 calc.ErrNoRealResult,
 		"1e400":                     calc.ErrTooLarge,
 		"1e300 * 1e300":             calc.ErrTooLarge,
 		"1e999999999 * 1e999999999": calc.ErrTooLarge,
 		"1 / 1e-400":                calc.ErrTooLarge,
+		"2^1024":                    calc.ErrTooLarge,
+		"2^5000":                    calc.ErrTooLarge,
+		"(-2)^5001":                 calc.ErrTooLarge,
+		"0.5^-5000":                 calc.ErrTooLarge,
+		// go/constant holds a product of this size rounded, so the
+		// remainder, or the sign of -1 to its power, cannot be known.
+		"7^1000 * 7^1000 / 7^1000 % 10":   calc.ErrTooLarge,
+		"(-1)^(3^1365 * 3^1365 / 3^1365)": calc.ErrTooLarge,
+		"(-1)^1e1300":                     calc.ErrTooLarge,
 	}
 
 	for in, want := range cases {
@@ -116,6 +217,46 @@ func TestEvaluateRefuses(t *testing.T) {
 	}
 }
 
+// TestEvaluateBoundsWork: computed exactly, each of these powers would
+// need more time and memory than any machine has. They must be answered
+// at once.
+func TestEvaluateBoundsWork(t *testing.T) {
+	t.Parallel()
+
+	cases := []struct {
+		in   string
+		want string
+		err  error
+	}{
+		{in: "9^9^9^9^9", err: calc.ErrTooLarge},
+		{in: "((9^999)^999)^999", err: calc.ErrTooLarge},
+		{in: "1.0000001^99999", want: "1.01005006557947"},
+		{in: "0.5^99999999999999999999", want: "0"},
+		{in: "(-1)^99999999999999999999", want: "-1"},
+		// The longest tower that fits.
+		{in: strings.Repeat("9^", 127) + "9", err: calc.ErrTooLarge},
+		// The largest power computed exactly, as often as fits.
+		{in: strings.Repeat("3^1365*", 36) + "0", want: "0"},
+	}
+
+	for _, c := range cases {
+		t.Run(c.in, func(t *testing.T) {
+			t.Parallel()
+
+			start := time.Now()
+			got, err := calc.Evaluate(c.in)
+
+			if elapsed := time.Since(start); elapsed > time.Second {
+				t.Errorf("Evaluate(%q) took %v", c.in, elapsed)
+			}
+
+			if !errors.Is(err, c.err) || got != c.want {
+				t.Errorf("Evaluate(%q) = %q, %v; want %q, %v", c.in, got, err, c.want, c.err)
+			}
+		})
+	}
+}
+
 // TestEvaluateCapsInput: the length cap is what bounds the work a
 // message can cause, so it must hold exactly at the boundary.
 func TestEvaluateCapsInput(t *testing.T) {