static class · System

NumberFormat

public static class NumberFormat

Numbers formatted as text (what the primitives' ToString(format) call).

Tipo: static class
Espacio de nombres: System
Ensamblado: System.Core
Acceso: public, static, abstract
Archivo fuente: Format.ant

Herencia

Clases base: Any NumberFormat

Métodos

FormatInteger(u64 bits, i32 width, bool signed, str format) : str

static

An integer (its bits, width of them, and whether its type is signed) by format.

FormatInteger(i64 value, str format) : str

static

ShortestDouble(f64 value) : str

static

A finite f64 as print writes it: the shortest digits that read back as the same number, in exponent form (1e+20, 1.5e-07) when the exponent is below -4 or at least 15.

FormatDouble(f64 value, str format) : str

static

An f64 by format (G or R: the shortest text that reads back as the same number; nan, inf, -inf).

FormatSingle(f32 value, str format) : str

static

An f32 by format (no format, G or R: the shortest text that reads back as the same f32).

FormatHalf(f16 value, str format) : str

static

An f16 by format (no format, G or R: the shortest text that reads back as the same f16).

FormatWide(u256 bits, i32 width, str format) : str

static

An f128 or f256 (its IEEE bits; width 128 or 256) by format: no format, G or R give the shortest text that reads back as the same number; the fixed formats show up to 33 (f128) or 71 (f256) digits.

Miembros privados e internos (25)

ExactDigits(f64 x) : DecimalDigits

private static

The exact digits of a finite f64 (every f64 is a finite decimal).

ExactOf(u64 mantissa, i32 e2, bool negative) : DecimalDigits

private static

The exact digits of mantissa × 2^e2.

Set(u32[] limbs, u64 value) : i32

private static

limbs = value (base 10^9); the count.

TimesPowerOfTwo(u32[] limbs, i32 used, i32 power) : i32

private static

TimesPowerOfFive(u32[] limbs, i32 used, i32 power) : i32

private static

DigitsOf(u32[] limbs, i32 used, i32 shift, bool negative, i32 limit) : DecimalDigits

private static

The decimal digits of a base-10^9 integer, over 10^shift: the first limit of them, then (when the rest is not all zeros) a 1 standing for the rest, which keeps comparisons and rounding to fewer digits exact.

ExactWide(u256 mantissa, i32 e2, bool negative) : DecimalDigits

private static

The exact digits of a wide mantissa × 2^e2 (an f128 or f256).

SetWide(u32[] limbs, u256 value) : i32

private static

limbs = value (base 10^9); the count.

TimesWide(u32[] a, i32 used, u256 factor, u32[] result) : i32

private static

a × factor (a wide integer: up to nine limbs), base 10^9, into result; the count.

ShortestWide(u256 mantissa, i32 e2, bool boundary, i32 most) : DecimalDigits

private static

The shortest digits that read back as the same wide float (see ShortestOf).

TimesLarge(u32[] a, i32 used, u64 factor, u32[] result) : i32

private static

a × factor (factor < 10^18: two limbs), base 10^9, into result; the count.

AddOrSubtract(u32[] a, i32 aUsed, u32[] b, i32 bUsed, bool add, u32[] result) : i32

private static

a + b or a - b (a ≥ b), base 10^9, into result; the count.

Multiply(u32[] limbs, i32 used, u64 factor) : i32

private static

limbs × factor (factor < 2^34), in place; the new count.

IntegerDigits(u64 magnitude, bool negative) : DecimalDigits

private static

The digits of an integer's magnitude.

Letter(str format) : u8

private static

A format's letter (0 for none: general) and precision (-1 for none).

Precision(str format) : i32

private static

Fixed(NumberText text, DecimalDigits d, i32 decimals, bool groups)

private static

Fixed point: the integer digits (grouped by three with commas when groups), then decimals decimals.

Scientific(NumberText text, DecimalDigits d, i32 decimals, u8 letter, i32 exponentDigits)

private static

Exponent form: one digit, decimals more after the point, then E (or e), the sign and exponentDigits or more.

General(NumberText text, DecimalDigits d, i32 precision, i32 scientificFrom, u8 letter)

private static

General: precision significant digits (all of them when 0 or less), in exponent form when the exponent is below -4 or at least scientificFrom; trailing zeros dropped.

Digits(NumberText text, DecimalDigits d, u8 letter, i32 precision, i32 generalPrecision, i32 scientificFrom) : bool

private static

The digits by a format letter (not D, X, B: those are integers' own).

Shortest(f64 value) : DecimalDigits

private static

The shortest digits that read back as the same f64 (the nearest of that length): those strictly between the midpoints to its neighbours (or on them, when its mantissa is even and reading rounds to it).

ShortestSingle(f32 value) : DecimalDigits

private static

ShortestOf(u64 mantissa, i32 e2, bool boundary, i32 most) : DecimalDigits

private static

RoundInto(DecimalDigits d, i32 keep, DecimalDigits target)

private static

d rounded to keep digits, into target (its array has room): only the digits that decide it are copied (what follows the next one, never all zeros, stands as a 1).

Compare(DecimalDigits a, DecimalDigits b) : i32

private static

Which of two positive numbers is larger (-1, 0, 1).

Miembros heredados (5)

GetType método de Any
Undescribed método de Any
ToString método de Any
Hash método de Any
destroy método de Any

Código fuente

NumberFormat en Antril, descompilado a partir de su Larva HIR.

System.NumberFormat.ant

// System.NumberFormat — System.Core (decompiled from Larva HIR)

into namespace System
{
	/// Numbers formatted as text (what the primitives' `ToString(format)` call).
	public static class NumberFormat
	{
		/// The exact digits of a finite f64 (every f64 is a finite decimal).
		ExactDigits(f64 x) : DecimalDigits
		{
			i64 bits = Float64.Bits(x);
			bool negative = bits < 0;
			i32 biased = (i32)(bits >> 52 & 2047);
			u64 mantissa = (u64)bits & 4503599627370495;
			i32 e2 = -1074;
			if (biased != 0)
			{
				mantissa |= 4503599627370496;
				e2 = biased - 1075;
			}
			return ExactOf(mantissa, e2, negative);
		}

		/// The exact digits of mantissa × 2^e2.
		ExactOf(u64 mantissa, i32 e2, bool negative) : DecimalDigits
		{
			if (mantissa == 0)
				return new DecimalDigits(new u8[1], 0, 0, negative);
			loop ((mantissa & 1) == 0)
			{
				mantissa = mantissa >> 1;
				e2++;
			}
			i32 k = e2 < 0 ? -e2 : 0;
			u32[] limbs = new u32[4 + (e2 >= 0 ? e2 / 29 : k * 7 / 29)];
			i32 used = Set(limbs, mantissa);
			used = e2 > 0 ? TimesPowerOfTwo(limbs, used, e2) : TimesPowerOfFive(limbs, used, k);
			return DigitsOf(limbs, used, k, negative, 2147483646);
		}

		/// limbs = value (base 10^9); the count.
		Set(u32[] limbs, u64 value) : i32
		{
			i32 used = 0;
			loop (value > 0)
			{
				limbs[used] = (u32)(value % 1000000000);
				value = value / 1000000000;
				used++;
			}
			return used;
		}

		TimesPowerOfTwo(u32[] limbs, i32 used, i32 power) : i32
		{
			loop (power > 0)
			{
				i32 step = power < 29 ? power : 29;
				used = Multiply(limbs, used, 1 << (u64)step);
				power -= step;
			}
			return used;
		}

		TimesPowerOfFive(u32[] limbs, i32 used, i32 power) : i32
		{
			loop (power > 0)
			{
				i32 step = power < 13 ? power : 13;
				u64 factor = 1;
				loop (i32 s = 0..step)
					factor *= 5;
				used = Multiply(limbs, used, factor);
				power -= step;
			}
			return used;
		}

		/// The decimal digits of a base-10^9 integer, over 10^shift: the first `limit` of them, then (when the rest is not
		/// all zeros) a 1 standing for the rest, which keeps comparisons and rounding to fewer digits exact.
		DigitsOf(u32[] limbs, i32 used, i32 shift, bool negative, i32 limit) : DecimalDigits
		{
			if (used == 0)
				return new DecimalDigits(new u8[1], 0, 0, negative);
			u32 top = limbs[used - 1];
			i32 topDigits = 1;
			loop (u32 t = top; t >= 10; t = t / 10)
				topDigits++;
			i32 total = topDigits + (used - 1) * 9;
			i32 keep = total < limit ? total : limit;
			u8[] digits = new u8[keep + 1];
			bool rest = false;
			i32 end = topDigits;
			loop (i32 l = used - 1; l >= 0; l--)
			{
				u32 limb = limbs[l];
				i32 width = l == used - 1 ? topDigits : 9;
				i32 start = end - width;
				if (start >= keep)
				{
					if (limb != 0)
						rest = true;
					if (rest)
						break;
				}
				else
				{
					loop (i32 at = end - 1; at >= start; at--)
					{
						u32 digit = limb % 10;
						limb = limb / 10;
						if (at < keep)
							digits[at] = (u8)digit;
						else if (digit != 0)
							rest = true;
					}
				}
				end += 9;
			}
			i32 count = keep;
			if (rest)
			{
				digits[count] = 1;
				count++;
			}
			DecimalDigits result = new DecimalDigits(digits, count, total - shift, negative);
			result.Trim();
			return result;
		}

		/// The exact digits of a wide mantissa × 2^e2 (an f128 or f256).
		ExactWide(u256 mantissa, i32 e2, bool negative) : DecimalDigits
		{
			if (mantissa == 0)
				return new DecimalDigits(new u8[1], 0, 0, negative);
			loop ((mantissa & 1) == 0)
			{
				mantissa = mantissa >> 1;
				e2++;
			}
			i32 k = e2 < 0 ? -e2 : 0;
			u32[] limbs = new u32[13 + (e2 >= 0 ? e2 / 29 : k * 7 / 29)];
			i32 used = SetWide(limbs, mantissa);
			used = e2 > 0 ? TimesPowerOfTwo(limbs, used, e2) : TimesPowerOfFive(limbs, used, k);
			return DigitsOf(limbs, used, k, negative, 2147483646);
		}

		/// limbs = value (base 10^9); the count.
		SetWide(u32[] limbs, u256 value) : i32
		{
			i32 used = 0;
			loop (value > 0)
			{
				limbs[used] = (u32)(value % 1000000000);
				value = value / 1000000000;
				used++;
			}
			return used;
		}

		/// a × factor (a wide integer: up to nine limbs), base 10^9, into `result`; the count.
		TimesWide(u32[] a, i32 used, u256 factor, u32[] result) : i32
		{
			u32[] f = new u32[9];
			i32 count = SetWide(f, factor);
			i32 end = used + count + 1;
			loop (i32 i = 0..end)
				result[i] = 0;
			loop (i32 i = 0..used)
			{
				u64 carry = 0;
				loop (i32 j = 0..count)
				{
					u64 product = (u64)a[i] * (u64)f[j] + (u64)result[i + j] + carry;
					result[i + j] = (u32)(product % 1000000000);
					carry = product / 1000000000;
				}
				i32 at = i + count;
				loop (carry > 0)
				{
					u64 sum = (u64)result[at] + carry;
					result[at] = (u32)(sum % 1000000000);
					carry = sum / 1000000000;
					at++;
				}
			}
			loop (end > 0 && result[end - 1] == 0)
				end--;
			return end;
		}

		/// The shortest digits that read back as the same wide float (see ShortestOf).
		ShortestWide(u256 mantissa, i32 e2, bool boundary, i32 most) : DecimalDigits
		{
			if (mantissa == 0)
				return new DecimalDigits(new u8[1], 0, 0, false);
			i32 scale = boundary ? 2 : 1;
			i32 unitExponent = e2 - scale;
			i32 k = unitExponent < 0 ? -unitExponent : 0;
			i32 room = 16 + (unitExponent >= 0 ? unitExponent / 29 : k * 7 / 29);
			u32[] unit = new u32[room];
			i32 unitUsed = Set(unit, 1);
			unitUsed = unitExponent > 0 ? TimesPowerOfTwo(unit, unitUsed, unitExponent) : TimesPowerOfFive(unit, unitUsed, k);
			u32[] center = new u32[room];
			i32 centerUsed = TimesWide(unit, unitUsed, mantissa << (u256)scale, center);
			u32[] step = new u32[room];
			loop (i32 i = 0..unitUsed)
				step[i] = unit[i];
			i32 stepUsed = boundary ? Multiply(step, unitUsed, 2) : unitUsed;
			u32[] side = new u32[room];
			i32 sideUsed = AddOrSubtract(center, centerUsed, step, stepUsed, true, side);
			DecimalDigits exact = DigitsOf(center, centerUsed, k, false, most + 2);
			DecimalDigits high = DigitsOf(side, sideUsed, k, false, most + 2);
			sideUsed = AddOrSubtract(center, centerUsed, unit, unitUsed, false, side);
			DecimalDigits low = DigitsOf(side, sideUsed, k, false, most + 2);
			bool even = (mantissa & 1) == 0;
			DecimalDigits scratch = new DecimalDigits(new u8[most + 2], 0, 0, false);
			i32 shared = 0;
			if (low.Point == high.Point)
			{
				loop (shared < most && low[shared] == high[shared])
					shared++;
			}
			i32 lowest = shared > 1 ? shared : 1;
			i32 last = most < exact.Count ? most : exact.Count;
			loop (lowest < last)
			{
				RoundInto(exact, lowest, scratch);
				i32 below = Compare(low, scratch);
				i32 above = Compare(scratch, high);
				if ((below < 0 || even && below == 0) && (above < 0 || even && above == 0))
					break;
				lowest++;
			}
			RoundInto(exact, lowest, scratch);
			u8[] digits = new u8[scratch.Count > 0 ? scratch.Count : 1];
			loop (i32 i = 0..scratch.Count)
				digits[i] = scratch.Digits[i];
			return new DecimalDigits(digits, scratch.Count, scratch.Point, false);
		}

		/// a × factor (factor < 10^18: two limbs), base 10^9, into `result`; the count.
		TimesLarge(u32[] a, i32 used, u64 factor, u32[] result) : i32
		{
			u64 f0 = factor % 1000000000;
			u64 f1 = factor / 1000000000;
			loop (i32 i = 0..used + 3)
				result[i] = 0;
			u64 carry = 0;
			loop (i32 i = 0..used)
			{
				u64 product = (u64)a[i] * f0 + (u64)result[i] + carry;
				result[i] = (u32)(product % 1000000000);
				carry = product / 1000000000;
			}
			i32 end = used;
			loop (carry > 0)
			{
				result[end] = (u32)(carry % 1000000000);
				carry = carry / 1000000000;
				end++;
			}
			if (f1 > 0)
			{
				carry = 0;
				loop (i32 i = 0..used)
				{
					u64 product = (u64)a[i] * f1 + (u64)result[i + 1] + carry;
					result[i + 1] = (u32)(product % 1000000000);
					carry = product / 1000000000;
				}
				i32 at = used + 1;
				loop (carry > 0)
				{
					u64 sum = (u64)result[at] + carry;
					result[at] = (u32)(sum % 1000000000);
					carry = sum / 1000000000;
					at++;
				}
				if (at > end)
					end = at;
			}
			loop (end > 0 && result[end - 1] == 0)
				end--;
			return end;
		}

		/// a + b or a - b (a ≥ b), base 10^9, into `result`; the count.
		AddOrSubtract(u32[] a, i32 aUsed, u32[] b, i32 bUsed, bool add, u32[] result) : i32
		{
			i64 carry = 0;
			i32 used = aUsed > bUsed ? aUsed : bUsed;
			loop (i32 i = 0..used)
			{
				i64 sum = (i < aUsed ? (i64)a[i] : 0) + carry + (add ? i < bUsed ? (i64)b[i] : 0 : -(i < bUsed ? (i64)b[i] : 0));
				carry = 0;
				if (sum >= 1000000000)
				{
					sum -= 1000000000;
					carry = 1;
				}
				else if (sum < 0)
				{
					sum += 1000000000;
					carry = -1;
				}
				result[i] = (u32)sum;
			}
			if (carry > 0)
			{
				result[used] = (u32)carry;
				used++;
			}
			loop (used > 0 && result[used - 1] == 0)
				used--;
			return used;
		}

		/// limbs × factor (factor < 2^34), in place; the new count.
		Multiply(u32[] limbs, i32 used, u64 factor) : i32
		{
			u64 carry = 0;
			loop (i32 i = 0..used)
			{
				u64 product = (u64)limbs[i] * factor + carry;
				limbs[i] = (u32)(product % 1000000000);
				carry = product / 1000000000;
			}
			loop (carry > 0)
			{
				limbs[used] = (u32)(carry % 1000000000);
				carry = carry / 1000000000;
				used++;
			}
			return used;
		}

		/// The digits of an integer's magnitude.
		IntegerDigits(u64 magnitude, bool negative) : DecimalDigits
		{
			u8[] digits = new u8[20];
			i32 count = 0;
			loop (u64 m = magnitude; m > 0; m = m / 10)
				count++;
			u64 rest = magnitude;
			loop (i32 d = count - 1; d >= 0; d--)
			{
				digits[d] = (u8)(rest % 10);
				rest = rest / 10;
			}
			DecimalDigits result = new DecimalDigits(digits, count, count, negative);
			result.Trim();
			return result;
		}

		/// A format's letter (0 for none: general) and precision (-1 for none).
		Letter(str format) : u8
		{
			if (format.Length == 0)
				return 71;
			u8 letter = format.Units()[0];
			return letter;
		}

		Precision(str format) : i32
		{
			u8[] bytes = format.Units();
			if (bytes.Length <= 1)
				return -1;
			i32 value = 0;
			loop (i32 i = 1..bytes.Length)
			{
				u8 c = bytes[i];
				if (c < 48 || c > 57 || value > 99999)
					throw new FormatError("bad number format: " + format);
				value = value * 10 + ((i32)c - 48);
			}
			return value;
		}

		/// Fixed point: the integer digits (grouped by three with commas when `groups`), then `decimals` decimals.
		Fixed(NumberText text, DecimalDigits d, i32 decimals, bool groups)
		{
			if (d.Point <= 0)
			{
				text.Add(48);
			}
			else
			{
				loop (i32 i = 0..d.Point)
				{
					if (groups && i > 0 && (d.Point - i) % 3 == 0)
						text.Add(44);
					text.Add(48 + d[i]);
				}
			}
			if (decimals > 0)
			{
				text.Add(46);
				loop (i32 i = d.Point..d.Point + decimals)
					text.Add(48 + d[i]);
			}
		}

		/// Exponent form: one digit, `decimals` more after the point, then E (or e), the sign and `exponentDigits` or more.
		Scientific(NumberText text, DecimalDigits d, i32 decimals, u8 letter, i32 exponentDigits)
		{
			text.Add(48 + d[0]);
			if (decimals > 0)
			{
				text.Add(46);
				loop (i32 i = 1..decimals + 1)
					text.Add(48 + d[i]);
			}
			i32 exponent = d.Count == 0 ? 0 : d.Point - 1;
			text.Add(letter);
			text.Add(exponent < 0 ? 45 : 43);
			i32 magnitude = exponent < 0 ? -exponent : exponent;
			i32 width = 1;
			loop (i32 m = magnitude; m >= 10; m = m / 10)
				width++;
			loop (i32 pad = width..exponentDigits)
				text.Add(48);
			text.Number(magnitude);
		}

		/// General: `precision` significant digits (all of them when 0 or less), in exponent form when the exponent is
		/// below -4 or at least `scientificFrom`; trailing zeros dropped.
		General(NumberText text, DecimalDigits d, i32 precision, i32 scientificFrom, u8 letter)
		{
			if (precision > 0)
				d.Round(precision);
			d.Trim();
			if (d.Count == 0)
			{
				text.Add(48);
				return;
			}
			i32 exponent = d.Point - 1;
			if (exponent < -4 || exponent >= scientificFrom)
			{
				Scientific(text, d, d.Count - 1, letter == 103 ? 101 : 69, 2);
				return;
			}
			i32 decimals = d.Count - d.Point;
			Fixed(text, d, decimals > 0 ? decimals : 0, false);
		}

		/// The digits by a format letter (not D, X, B: those are integers' own).
		Digits(NumberText text, DecimalDigits d, u8 letter, i32 precision, i32 generalPrecision, i32 scientificFrom) : bool
		{
			switch (letter)
			{
				70 or 102 =>
				{
					i32 decimals = precision < 0 ? 2 : precision;
					d.Round(d.Point + decimals);
					Fixed(text, d, decimals, false);
				}
				78 or 110 =>
				{
					i32 decimals = precision < 0 ? 2 : precision;
					d.Round(d.Point + decimals);
					Fixed(text, d, decimals, true);
				}
				69 or 101 =>
				{
					i32 decimals = precision < 0 ? 6 : precision;
					d.Round(decimals + 1);
					Scientific(text, d, decimals, letter, 3);
				}
				80 or 112 =>
				{
					i32 decimals = precision < 0 ? 2 : precision;
					d.Point += 2;
					d.Round(d.Point + decimals);
					Fixed(text, d, decimals, true);
					text.Add(32);
					text.Add(37);
				}
				71 or 103 =>
					General(text, d, precision > 0 ? precision : generalPrecision, precision > 0 ? precision : scientificFrom, letter);
				_ =>
					return false;
			}
			return true;
		}

		/// The shortest digits that read back as the same f64 (the nearest of that length): those strictly between
		/// the midpoints to its neighbours (or on them, when its mantissa is even and reading rounds to it).
		Shortest(f64 value) : DecimalDigits
		{
			i64 bits = Float64.Bits(value) & 9223372036854775807;
			i32 biased = (i32)(bits >> 52);
			u64 mantissa = (u64)bits & 4503599627370495;
			i32 e2 = -1074;
			if (biased != 0)
			{
				mantissa |= 4503599627370496;
				e2 = biased - 1075;
			}
			return ShortestOf(mantissa, e2, biased > 1 && mantissa == 4503599627370496, 17);
		}

		ShortestSingle(f32 value) : DecimalDigits
		{
			u32 bits = Float32.Bits(value) & 2147483647;
			i32 biased = (i32)(bits >> 23);
			u64 mantissa = (u64)bits & 8388607;
			i32 e2 = -149;
			if (biased != 0)
			{
				mantissa |= 8388608;
				e2 = biased - 150;
			}
			return ShortestOf(mantissa, e2, biased > 1 && mantissa == 8388608, 9);
		}

		ShortestOf(u64 mantissa, i32 e2, bool boundary, i32 most) : DecimalDigits
		{
			if (mantissa == 0)
				return new DecimalDigits(new u8[1], 0, 0, false);
			i32 scale = boundary ? 2 : 1;
			i32 unitExponent = e2 - scale;
			i32 k = unitExponent < 0 ? -unitExponent : 0;
			i32 room = 6 + (unitExponent >= 0 ? unitExponent / 29 : k * 7 / 29);
			u32[] unit = new u32[room];
			i32 unitUsed = Set(unit, 1);
			unitUsed = unitExponent > 0 ? TimesPowerOfTwo(unit, unitUsed, unitExponent) : TimesPowerOfFive(unit, unitUsed, k);
			u32[] center = new u32[room];
			i32 centerUsed = TimesLarge(unit, unitUsed, mantissa << (u64)scale, center);
			u32[] step = new u32[room];
			loop (i32 i = 0..unitUsed)
				step[i] = unit[i];
			i32 stepUsed = boundary ? Multiply(step, unitUsed, 2) : unitUsed;
			u32[] side = new u32[room];
			i32 sideUsed = AddOrSubtract(center, centerUsed, step, stepUsed, true, side);
			DecimalDigits exact = DigitsOf(center, centerUsed, k, false, most + 2);
			DecimalDigits high = DigitsOf(side, sideUsed, k, false, most + 2);
			sideUsed = AddOrSubtract(center, centerUsed, unit, unitUsed, false, side);
			DecimalDigits low = DigitsOf(side, sideUsed, k, false, most + 2);
			bool even = (mantissa & 1) == 0;
			DecimalDigits scratch = new DecimalDigits(new u8[most + 2], 0, 0, false);
			i32 shared = 0;
			if (low.Point == high.Point)
			{
				loop (shared < most && low[shared] == high[shared])
					shared++;
			}
			i32 lowest = shared > 1 ? shared : 1;
			i32 last = most < exact.Count ? most : exact.Count;
			loop (lowest < last)
			{
				RoundInto(exact, lowest, scratch);
				i32 below = Compare(low, scratch);
				i32 above = Compare(scratch, high);
				if ((below < 0 || even && below == 0) && (above < 0 || even && above == 0))
					break;
				lowest++;
			}
			RoundInto(exact, lowest, scratch);
			u8[] digits = new u8[scratch.Count > 0 ? scratch.Count : 1];
			loop (i32 i = 0..scratch.Count)
				digits[i] = scratch.Digits[i];
			return new DecimalDigits(digits, scratch.Count, scratch.Point, false);
		}

		/// `d` rounded to `keep` digits, into `target` (its array has room): only the digits that decide it are copied
		/// (what follows the next one, never all zeros, stands as a 1).
		RoundInto(DecimalDigits d, i32 keep, DecimalDigits target)
		{
			bool more = d.Count > keep + 1;
			i32 count = more ? keep + 2 : d.Count;
			u8[] digits = target.Digits;
			loop (i32 i = 0..count)
				digits[i] = i == keep + 1 ? 1 : d.Digits[i];
			target.Count = count;
			target.Point = d.Point;
			target.Round(keep);
		}

		/// Which of two positive numbers is larger (-1, 0, 1).
		Compare(DecimalDigits a, DecimalDigits b) : i32
		{
			if (a.Point != b.Point)
				return a.Point < b.Point ? -1 : 1;
			i32 n = a.Count > b.Count ? a.Count : b.Count;
			loop (i32 i = 0..n)
			{
				u8 x = a[i];
				u8 y = b[i];
				if (x != y)
					return x < y ? -1 : 1;
			}
			return 0;
		}

		public:
			/// An integer (its bits, `width` of them, and whether its type is signed) by `format`.
			FormatInteger(u64 bits, i32 width, bool signed, str format) : str
			{
				u8 letter = Letter(format);
				i32 precision = Precision(format);
				bool negative = signed && width < 64 ? (bits >> (u64)(width - 1) & 1) == 1 : signed && (i64)bits < 0;
				u64 masked = width < 64 ? bits & (1 << (u64)width) - 1 : bits;
				u64 magnitude = negative ? 0 - masked & (width < 64 ? (1 << (u64)width) - 1 : 18446744073709551615) : masked;
				if (letter == 68 || letter == 100)
				{
					i32 count = 1;
					loop (u64 m = magnitude / 10; m > 0; m = m / 10)
						count++;
					i32 size = count > precision ? count : precision;
					u8[] units = new u8[size + (negative ? 1 : 0)];
					i32 at = units.Length;
					u64 rest = magnitude;
					loop (i32 i = 0..size)
					{
						at--;
						units[at] = (u8)(48 + rest % 10);
						rest = rest / 10;
					}
					if (negative)
						units[0] = 45;
					return new str(units);
				}
				if (letter == 88 || letter == 120 || letter == 66 || letter == 98)
				{
					i32 shift = letter == 66 || letter == 98 ? 1 : 4;
					u64 digitMask = (1 << (u64)shift) - 1;
					i32 count = 1;
					loop (u64 m = masked >> (u64)shift; m > 0; m = m >> (u64)shift)
						count++;
					i32 size = count > precision ? count : precision;
					u8[] units = new u8[size];
					u8 letters = letter == 120 ? 87 : 55;
					u64 rest = masked;
					loop (i32 at = size - 1; at >= 0; at--)
					{
						u8 digit = (u8)(rest & digitMask);
						units[at] = digit < 10 ? 48 + digit : letters + digit;
						rest = rest >> (u64)shift;
					}
					return new str(units);
				}
				NumberText text = new NumberText();
				switch (letter)
				{
					_ =>
					{
						DecimalDigits d = IntegerDigits(magnitude, negative);
						if (negative)
							text.Add(45);
						if (!Digits(text, d, letter, precision, 0, 1000))
							throw new FormatError("bad number format: " + format);
					}
				}
				return text.Made();
			}

			FormatInteger(i64 value, str format) : str => FormatInteger((u64)value, 64, true, format);

			/// A finite f64 as `print` writes it: the shortest digits that read back as the same number, in exponent
			/// form (`1e+20`, `1.5e-07`) when the exponent is below -4 or at least 15.
			ShortestDouble(f64 value) : str
			{
				NumberText text = new NumberText();
				if (Float64.Bits(value) < 0)
					text.Add(45);
				General(text, Shortest(value), 0, 15, 103);
				return text.Made();
			}

			/// An f64 by `format` (G or R: the shortest text that reads back as the same number; `nan`, `inf`, `-inf`).
			FormatDouble(f64 value, str format) : str
			{
				if (value != value)
					return "nan";
				if (value - value != 0.0)
					return value > 0.0 ? "inf" : "-inf";
				u8 letter = Letter(format);
				i32 precision = Precision(format);
				NumberText text = new NumberText();
				if (Float64.Bits(value) < 0)
					text.Add(45);
				if ((letter == 71 || letter == 103) && precision <= 0 || letter == 82 || letter == 114)
				{
					General(text, Shortest(value), 0, 17, letter == 103 ? 103 : 71);
					return text.Made();
				}
				if (!Digits(text, ExactDigits(value), letter, precision, 15, 15))
					throw new FormatError("bad number format: " + format);
				return text.Made();
			}

			/// An f32 by `format` (no format, G or R: the shortest text that reads back as the same f32).
			FormatSingle(f32 value, str format) : str
			{
				if (value != value)
					return "nan";
				if (value - value != 0.0)
					return value > 0.0 ? "inf" : "-inf";
				u8 letter = Letter(format);
				i32 precision = Precision(format);
				NumberText text = new NumberText();
				if (Float32.Bits(value) >= 2147483648)
					text.Add(45);
				if ((letter == 71 || letter == 103) && precision <= 0 || letter == 82 || letter == 114)
				{
					General(text, ShortestSingle(value), 0, 9, letter == 103 ? 103 : 71);
					return text.Made();
				}
				if (!Digits(text, ExactDigits(value), letter, precision, 7, 7))
					throw new FormatError("bad number format: " + format);
				return text.Made();
			}

			/// An f16 by `format` (no format, G or R: the shortest text that reads back as the same f16).
			FormatHalf(f16 value, str format) : str
			{
				u16 bits = Float16.Bits(value);
				bool negative = bits >= 32768;
				i32 biased = (i32)(bits >> 10) & 31;
				u64 mantissa = (u64)bits & 1023;
				if (biased == 31)
					return mantissa != 0 ? "nan" : negative ? "-inf" : "inf";
				i32 e2 = -24;
				if (biased != 0)
				{
					mantissa |= 1024;
					e2 = biased - 25;
				}
				u8 letter = Letter(format);
				i32 precision = Precision(format);
				NumberText text = new NumberText();
				if (negative)
					text.Add(45);
				if ((letter == 71 || letter == 103) && precision <= 0 || letter == 82 || letter == 114)
				{
					General(text, ShortestOf(mantissa, e2, biased > 1 && mantissa == 1024, 5), 0, 5, letter == 103 ? 103 : 71);
					return text.Made();
				}
				if (!Digits(text, ExactOf(mantissa, e2, negative), letter, precision, 3, 3))
					throw new FormatError("bad number format: " + format);
				return text.Made();
			}

			/// An f128 or f256 (its IEEE bits; `width` 128 or 256) by `format`: no format, G or R give the shortest text
			/// that reads back as the same number; the fixed formats show up to 33 (f128) or 71 (f256) digits.
			FormatWide(u256 bits, i32 width, str format) : str
			{
				i32 exponentBits = width == 128 ? 15 : 19;
				i32 fractionBits = width - 1 - exponentBits;
				i32 bias = (1 << exponentBits - 1) - 1;
				i32 exponentMask = (1 << exponentBits) - 1;
				bool negative = bits >> (u256)(width - 1) != 0;
				i32 biased = (i32)(bits >> (u256)fractionBits & (u256)exponentMask);
				u256 hidden = 1 << (u256)fractionBits;
				u256 mantissa = bits & hidden - 1;
				if (biased == exponentMask)
					return mantissa != 0 ? "nan" : negative ? "-inf" : "inf";
				i32 e2 = 1 - bias - fractionBits;
				if (biased != 0)
				{
					mantissa |= hidden;
					e2 = biased - bias - fractionBits;
				}
				u8 letter = Letter(format);
				i32 precision = Precision(format);
				i32 most = width == 128 ? 36 : 72;
				i32 exact = width == 128 ? 33 : 71;
				NumberText text = new NumberText();
				if (negative)
					text.Add(45);
				if ((letter == 71 || letter == 103) && precision <= 0 || letter == 82 || letter == 114)
				{
					General(text, ShortestWide(mantissa, e2, biased > 1 && mantissa == hidden, most), 0, most, letter == 103 ? 103 : 71);
					return text.Made();
				}
				if (!Digits(text, ExactWide(mantissa, e2, negative), letter, precision, exact, exact))
					throw new FormatError("bad number format: " + format);
				return text.Made();
			}
	}
}