Every number a fraction under the 4096-bit limit (closes #3)

go/constant never rounds an integer, and x^0 returned one, so whole
numbers built from it escaped the 4096-bit limit:
(((2^0+2^0)^4096)^4096)^4096 asked for about 69 billion bits and
stalled the bot. A power now starts from the fraction 1, and a number
is exact only as a fraction whose numerator and denominator are below
4096 bits, which also refuses a literal that go/constant reads exactly
past the limit, such as 1e-1233. The bounded-work test holds the tower
and the other short inputs tried against the change.

Model: opus-5-5

internal/calc/calc.go

@@ -21,11 +21,15 @@ import (
 	"strings"
 )
 
-// MaxInputLength caps an expression, in bytes, and maxExactExponent caps
-// a power computed exactly, so that a message cannot make the bot do
-// unbounded work.
+// MaxInputLength caps an expression, in bytes. With maxExactExponent,
+// which caps a power computed exactly, and bitLimit, which caps every
+// number, it keeps a message from making the bot do unbounded work.
 const MaxInputLength = 256
 
+// bitLimit caps the numerator and denominator of every number: see
+// exact.
+const bitLimit = 4096
+
 // maxExactExponent is the largest exponent, either way, of a power
 // computed exactly. Past it, x^n has a numerator or denominator of more
 // than 4096 bits, which go/constant holds only rounded, unless x is 0 or
@@ -245,11 +249,11 @@ func number(tok string) (constant.Value, error) {
 		return nil, ErrNotArithmetic
 	}
 
-	// Read as FLOAT, which makes every literal decimal: as INT, a
-	// leading zero would make it octal.
+	// Read as FLOAT, which makes every literal decimal and a fraction
+	// (see exact): as INT, a leading zero would make it octal.
 	v := constant.MakeFromLiteral(tok, token.FLOAT, 0)
 
-	// A literal such as 1e1300 or 1e-1300 is held rounded: see exact.
+	// A literal such as 1e1300 or 1e-1233 is past bitLimit: see exact.
 	if !exact(v) {
 		return nil, ErrOutOfRange
 	}
@@ -403,10 +407,13 @@ func nonNegativePower(x, y, n constant.Value) (constant.Value, error) {
 }
 
 // exactPower computes x^e by repeated squaring. x is not zero if e is
-// negative. Each step's numbers stay small: go/constant holds one whose
-// numerator or denominator reaches 4096 bits as a 512-bit float.
+// negative. It starts from 1 as a fraction, a Float to go/constant, so
+// that x^0 is a fraction like every other number (see exact). Each
+// step's numbers stay small: go/constant holds one whose numerator or
+// denominator reaches 4096 bits as a 512-bit float.
 func exactPower(x constant.Value, e int64) constant.Value {
-	result := constant.MakeInt64(1)
+	one := constant.MakeFloat64(1)
+	result := one
 
 	for n := max(e, -e); n > 0; n >>= 1 {
 		if n&1 == 1 {
@@ -417,26 +424,29 @@ func exactPower(x constant.Value, e int64) constant.Value {
 	}
 
 	if e < 0 {
-		result = constant.BinaryOp(constant.MakeInt64(1), token.QUO, result)
+		result = constant.BinaryOp(one, token.QUO, result)
 	}
 
 	return result
 }
 
-// exact reports whether go/constant holds v exactly. It holds a number
-// as a fraction until its numerator or denominator reaches 4096 bits,
-// then as a 512-bit float, and past that float's range as Unknown. A
-// number not held exactly is refused wherever it appears: a sum can lose
-// the answer entirely (7^1000*7^1000 + 5 - 7^1000*7^1000 would be 0), and
-// a remainder, or whether an exponent is whole or odd, cannot be read
-// from one.
+// exact reports whether v is a fraction whose numerator and denominator
+// are both below bitLimit bits, as every number here must be, so that
+// each step of arithmetic stays small. go/constant never rounds an
+// integer, however large, so every number is made a fraction: literals
+// are read as FLOAT, and a power starts from the fraction 1. It rounds
+// a fraction that grows past the limit, to a 512-bit float and past
+// that float's range to Unknown, but not one it reads from a literal,
+// such as 1e-1233, so the limit is checked here.
+//
+// A number that is not exact is refused wherever it appears: a sum can
+// lose the answer entirely (7^1000*7^1000 + 5 - 7^1000*7^1000 would be
+// 0), and a remainder, or whether an exponent is whole or odd, cannot be
+// read from one.
 func exact(v constant.Value) bool {
-	switch constant.Val(v).(type) {
-	case int64, *big.Int, *big.Rat:
-		return true
-	default:
-		return false
-	}
+	r, ok := constant.Val(v).(*big.Rat)
+
+	return ok && r.Num().BitLen() < bitLimit && r.Denom().BitLen() < bitLimit
 }
 
 // normal reports whether f is a normal double, finite and at least

internal/calc/calc_test.go

@@ -130,6 +130,8 @@ func TestEvaluateModulo(t *testing.T) {
 		// Both operands and their quotient are held exactly, but y times
 		// the whole part of x/y is too large to be.
 		"(5^860*3^630/7) % (5^860/2^998/2^998)": "0.5179219763783696",
+		// A whole number made from x^0, just below the 4096-bit limit.
+		"(3^0 + 3^0 + 3^0)^2583 % 10": "7",
 	})
 }
 
@@ -243,6 +245,11 @@ func TestEvaluateOutOfRange(t *testing.T) {
 		"0.1^800 * 0.1^800":                     calc.ErrOutOfRange,
 		// Both operands are held exactly, but their quotient is not.
 		"3^1365 % 7^-1000": calc.ErrOutOfRange,
+		// The same limit for a whole number made from x^0, which
+		// go/constant would hold as an integer and never round, and for
+		// a literal it reads exactly as a fraction past the limit.
+		"(2^0 + 2^0)^4095 % 10": calc.ErrOutOfRange,
+		"1e-1233 * 0":           calc.ErrOutOfRange,
 		// go/constant reads this literal as 0.
 		"1e-999999999":     calc.ErrOutOfRange,
 		"1 / 1e-999999999": calc.ErrOutOfRange,
@@ -288,6 +295,20 @@ func TestEvaluateBoundsWork(t *testing.T) {
 		{in: strings.Repeat("9^", 127) + "9", err: calc.ErrOutOfRange},
 		// The largest power of 3 computed exactly, as often as fits.
 		{in: "0" + strings.Repeat("*3^2583", 36), want: "0"},
+		// Whole numbers made from x^0, through each operation. Held as
+		// integers, which go/constant never rounds, they would escape
+		// the 4096-bit limit: the first needs about 69 billion bits.
+		{in: "(((2^0+2^0)^4096)^4096)^4096", err: calc.ErrOutOfRange},
+		{in: "(((0^0+0^0)^4096)^4096)^4096", err: calc.ErrOutOfRange},
+		{in: "(((-2^0-2^0)^4096)^4096)^4096", err: calc.ErrOutOfRange},
+		{in: "((2^0+2^0)^4000*(2^0+2^0)^4000)^4096", err: calc.ErrOutOfRange},
+		{in: "((((2^0+2^0)/2^0)^4096)^4096)^4096", err: calc.ErrOutOfRange},
+		{in: "((((2^0+2^0) % 3)^4096)^4096)^4096", err: calc.ErrOutOfRange},
+		{in: "(((2^0+2^0)^4096)^4096)^4096 * 0", err: calc.ErrOutOfRange},
+		// A fraction whose numerator and denominator are both just below
+		// the limit, and a literal whose exponent is too large to read.
+		{in: "(3^2583/5^1760)^4096", err: calc.ErrOutOfRange},
+		{in: "1e99999999999999999999", err: calc.ErrOutOfRange},
 	}
 
 	for _, c := range cases {