Answer fractional powers of exact numbers past a double's range (closes #16)

A fractional power turned its base into a double first and refused a
base outside the normal range of a double, so (2^1200)^0.5, (2^1024)^0.5,
(2^-1200)^0.5 and 1e400^0.5 were refused although each answer is an
ordinary double. Such a base is now brought into that range by square
roots taken from its exact value in a big.Float, at most three under the
4096-bit limit, with the exponent doubled for each, and only a result
that is not a normal double is refused. The tests give these powers and
the edges of the range their values, and the bounded-work test covers a
base that needs three roots. The README's refusal sentence and example
are updated.

Model: opus-5-5

README.md

@@ -438,8 +438,11 @@ container.
   `4.9406564584124654e-324`. 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
-  (`2^4095`, `1e-1300 + 1`); and a power computed as a double whose base
+  (`2^4095`, `1e-1300 + 1`); and a power computed as a double whose
-  or result is outside the normal range of a double (`1e-400^0.5`).
+  result is outside the normal range of a double (`2^1500.5`). The base
+  of such a power may be outside that range: square roots taken from its
+  exact value bring it inside first, so `(2^1200)^0.5` is
+  `4.149515568880993e+180` and `1e-400^0.5` is `1e-200`.
 - **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

@@ -28,7 +28,8 @@ with no deprecation warning.
 # Completed Steps
 
 - 2026-09-29 Exact results past the range of a double, such as `2^1200`,
-  are written to 17 significant digits instead of being refused
+  are written to 17 significant digits instead of being refused, and a
+  fractional power of such a number, such as `(2^1200)^0.5`, is answered
 - 2026-09-29 `POST`, `GET` and `DELETE` on a chat's webhooks, under
   `/api/v1/chats/{id}/webhooks`, kept in `$DATA_DIR/webhooks.json`
 - 2026-09-29 `GET` and `POST /api/v1/chats/{id}/messages`: a chat's

internal/calc/calc.go

@@ -52,7 +52,9 @@ const (
 // significantDigits significant digits, the most the shortest form of a
 // double takes. It is rounded to them from a float of floatPrecision
 // bits, the bits a numerator or denominator can hold and 64 more for the
-// digits, so that the float rounds as the exact result would.
+// digits, so that the float rounds as the exact result would. The square
+// roots of a power's base past that range are taken in such a float too:
+// see nonNegativePower.
 const (
 	significantDigits = 17
 	floatPrecision    = bitLimit + 64
@@ -404,11 +406,25 @@ func nonNegativePower(x, y, n constant.Value) (constant.Value, error) {
 
 	xf, _ := constant.Float64Val(x)
 	yf, _ := constant.Float64Val(y)
-	f := math.Pow(xf, yf)
 
-	// Neither x nor x^y is zero. If either is not a normal double, it
+	// x^y is (√x)^(2y). An x outside the normal range of a double, such
-	// has lost digits, or all of them.
+	// as 2^1200, would lose digits as a double, or all of them, so square
-	if !normal(xf) || !normal(f) {
+	// roots taken from its exact value bring it into that range first. As
+	// x is between 2^-4096 and 2^4096 (see exact), three at most are
+	// needed.
+	r, _ := constant.Val(x).(*big.Rat)
+	root := new(big.Float).SetPrec(floatPrecision).SetRat(r)
+
+	for !normal(xf) {
+		root.Sqrt(root)
+		xf, _ = root.Float64()
+		yf *= 2
+	}
+
+	// x^y is not zero. If it is not a normal double, it has lost digits,
+	// or all of them.
+	f := math.Pow(xf, yf)
+	if !normal(f) {
 		return nil, ErrOutOfRange
 	}
 

internal/calc/calc_test.go

@@ -130,6 +130,21 @@ func TestEvaluatePowers(t *testing.T) {
 		// A power computed in float64 carries its rounding into the exact
 		// arithmetic after it, past the range of a double as within it.
 		"2^0.5 * 1e400": "1.4142135623730951e+400",
+		// A fractional power of an exact number outside the range of a
+		// double, taken from its exact value, up to the edges of that
+		// range.
+		"(2^1200)^0.5":         "4.149515568880993e+180",
+		"(2^1024)^0.5":         "1.3407807929942597e+154",
+		"(2^-1200)^0.5":        "2.409919865102884e-181",
+		"1e400^0.5":            "1e+200",
+		"1e-400^0.5":           "1e-200",
+		"1e-310^0.5":           "1e-155",
+		"(2^1200)^-0.5":        "2.409919865102884e-181",
+		"(2^1200)^0.5 / 2^600": "1",
+		"1e400^-0.001":         "0.39810717055349726",
+		"1e-400^0.001":         "0.39810717055349726",
+		"(2^2047)^0.5":         "1.2711610061536464e+308",
+		"(2^-2044)^0.5":        "2.2250738585072014e-308",
 	})
 }
 
@@ -252,15 +267,17 @@ func TestEvaluateOutOfRange(t *testing.T) {
 		"2^5000":                    calc.ErrOutOfRange,
 		"(-2)^5001":                 calc.ErrOutOfRange,
 		"0.5^-5000":                 calc.ErrOutOfRange,
-		// Powers computed in float64 whose base or result is not a
+		// Powers computed in float64 whose result is not a normal double,
-		// normal double, and so has lost digits, or all of them.
+		// and so has lost digits, or all of them, whatever the size of
+		// the base.
 		"2^-1073.5 * 2^1073":          calc.ErrOutOfRange,
-		"1e400^-0.001":                calc.ErrOutOfRange,
+		"2^1500.5":                    calc.ErrOutOfRange,
-		"1e-400^0.001":                calc.ErrOutOfRange,
+		"(2^1200)^0.9":                calc.ErrOutOfRange,
-		"1e-310^0.5":                  calc.ErrOutOfRange,
+		"1e-400^0.9":                  calc.ErrOutOfRange,
+		"(2^2048)^0.5":                calc.ErrOutOfRange,
+		"(2^-2046)^0.5":               calc.ErrOutOfRange,
 		"(0.5^1100)^4 / (0.5^1100)^4": calc.ErrOutOfRange,
 		"(1/3)^1e400":                 calc.ErrOutOfRange,
-		"(2^1200)^0.5":                calc.ErrOutOfRange,
 		// go/constant holds numbers of this size rounded. A sum of them
 		// can lose the answer (this one would be 0), and so can a
 		// remainder or the sign of -1 to such a power.
@@ -343,6 +360,10 @@ func TestEvaluateBoundsWork(t *testing.T) {
 		{in: "3^2583", want: "2.5363018640659988e+1232"},
 		{in: "2^-4094", want: "3.8299909843808741e-1233"},
 		{in: "-1/3^2583", want: "-3.9427483540814775e-1233"},
+		// Fractional powers of numbers just below the limit, whose bases
+		// take the most square roots to bring into the range of a double.
+		{in: "(2^-4094)^0.125", want: "8.869511863657883e-155"},
+		{in: "(1/3^2583)^0.5", err: calc.ErrOutOfRange},
 	}
 
 	for _, c := range cases {