public void Subtract(DiyFp other)
{
Debug.Assert(Exponent == other.Exponent);
Debug.Assert(Significand >= other.Significand);
Significand -= other.Significand;
}
public static DiyFp Minus(DiyFp a, DiyFp b)
{
var result = a;
result.Subtract(b);
return(result);
}
public static DiyFp Normalize(DiyFp a)
{
var result = a;
result.Normalize();
return(result);
}
public static DiyFp Times(DiyFp a, DiyFp b)
{
var result = a;
result.Multiply(b);
return(result);
}
public void NormalizedBoundaries(out DiyFp minus, out DiyFp plus)
{
Debug.Assert(Value() > 0.0);
var v = AsDiyFp();
plus = DiyFp.Normalize(new DiyFp((v.Significand << 1) + 1, v.Exponent - 1));
minus = IsLowerBoundaryCloser() ?
new DiyFp((v.Significand << 2) - 1, v.Exponent - 2) :
new DiyFp((v.Significand << 1) - 1, v.Exponent - 1);
minus.Significand = minus.Significand << (minus.Exponent - plus.Exponent);
minus.Exponent = plus.Exponent;
}
// Returns a cached power-of-ten with a binary exponent in the range
// [min_exponent; max_exponent] (boundaries included).
private static void GetCachedPowerForBinaryExponentRange(Int32 min_exponent, Int32 max_exponent, out DiyFp power, out Int32 decimal_exponent)
{
const Double D_1_LOG2_10 = 0.30102999566398114; // 1 / lg(10)
var kQ = DiyFp.SignificantSize;
var k = Math.Ceiling((min_exponent + kQ - 1) * D_1_LOG2_10);
var index = (CACHED_POWERS_OFFSET + (Int32)(k) - 1) / DECIMAL_EXPONENT_DISTANCE + 1;
var cached_power = CACHED_POWERS[index];
Debug.Assert(min_exponent <= cached_power.binary_exponent);
Debug.Assert(cached_power.binary_exponent <= max_exponent);
decimal_exponent = cached_power.decimal_exponent;
power = new DiyFp(cached_power.significand, cached_power.binary_exponent);
}
public void Multiply(DiyFp other)
{
// Simply "emulates" a 128 bit multiplication.
// However: the resulting number only contains 64 bits. The least
// significant 64 bits are only used for rounding the most significant 64 bits.
const UInt64 M32 = 0xFFFFFFFFU;
var a = Significand >> 32;
var b = Significand & M32;
var c = other.Significand >> 32;
var d = other.Significand & M32;
var ac = a * c;
var bc = b * c;
var ad = a * d;
var bd = b * d;
var tmp = (bd >> 32) + (ad & M32) + (bc & M32);
// By adding 1U << 31 to tmp we round the final result.
// Halfway cases will be round up.
tmp += 1U << 31;
Exponent += other.Exponent + 64;
Significand = ac + (ad >> 32) + (bc >> 32) + (tmp >> 32);
}
private static Boolean DigitGen(DiyFp low,
DiyFp w,
DiyFp high,
Span <Byte> buffer,
out Int32 length,
out Int32 kappa)
{
Debug.Assert(low.Exponent == w.Exponent && w.Exponent == high.Exponent);
Debug.Assert(low.Significand + 1 <= high.Significand - 1);
Debug.Assert(MINIMAL_TARGET_EXPONENT <= w.Exponent && w.Exponent <= MAXIMAL_TARGET_EXPONENT);
// low, w and high are imprecise, but by less than one ulp (unit in the last place).
// If we remove (resp. add) 1 ulp from low (resp. high) we are certain that
// the new numbers are outside of the interval we want the final representation to lie in.
// Inversely adding (resp. removing) 1 ulp from low (resp. high) would yield
// numbers that are certain to lie in the interval. We will use this fact later on.
// We will now start by generating the digits within the uncertain
// interval. Later we will weed out representations that lie outside the safe
// interval and thus _might_ lie outside the correct interval.
UInt64 unit = 1;
var too_low = new DiyFp(low.Significand - unit, low.Exponent);
var too_high = new DiyFp(high.Significand + unit, high.Exponent);
// too_low and too_high are guaranteed to lie outside the interval we want the generated number in.
var unsafe_interval = DiyFp.Minus(too_high, too_low);
// We now cut the input number into two parts: the integral digits and the
// fractionals. We will not write any decimal separator though, but adapt
// kappa instead.
// Reminder: we are currently computing the digits (stored inside the buffer)
// such that: too_low < buffer * 10^kappa < too_high
// We use too_high for the digit_generation and stop as soon as possible.
// If we stop early we effectively round down.
var one = new DiyFp(1UL << -w.Exponent, w.Exponent);
// Division by one is a shift.
var integrals = (UInt32)(too_high.Significand >> -one.Exponent);
// Modulo by one is an and.
var fractionals = too_high.Significand & (one.Significand - 1);
BiggestPowerTen(integrals, DiyFp.SignificantSize - (-one.Exponent), out var divisor, out var divisor_exponent_plus_one);
kappa = divisor_exponent_plus_one;
length = 0;
// Loop invariant: buffer = too_high / 10^kappa (integer division)
// The invariant holds for the first iteration: kappa has been initialized
// with the divisor exponent + 1. And the divisor is the biggest power of ten
// that is smaller than integrals.
while (kappa > 0)
{
var digit = integrals / divisor;
Debug.Assert(digit <= 9);
buffer[length] = (Byte)('0' + digit);
++length;
integrals %= divisor;
--kappa;
// Note that kappa now equals the exponent of the divisor and that the
// invariant thus holds again.
var rest = ((UInt64)(integrals) << -one.Exponent) + fractionals;
// Invariant: too_high = buffer * 10^kappa + DiyFp(rest, one.e())
// Reminder: unsafe_interval.e() == one.e()
if (rest < unsafe_interval.Significand)
{
// Rounding down (by not emitting the remaining digits) yields a number
// that lies within the unsafe interval.
return(RoundWeed(buffer,
length,
DiyFp.Minus(too_high, w).Significand,
unsafe_interval.Significand,
rest,
(UInt64)(divisor) << -one.Exponent,
unit));
}
divisor /= 10;
}
// The integrals have been generated. We are at the point of the decimal
// separator. In the following loop we simply multiply the remaining digits by
// 10 and divide by one. We just need to pay attention to multiply associated
// data (like the interval or 'unit'), too.
// Note that the multiplication by 10 does not overflow, because w.e >= -60
// and thus one.e >= -60.
Debug.Assert(one.Exponent >= -60);
Debug.Assert(fractionals < one.Significand);
Debug.Assert(0xFFFFFFFF_FFFFFFFF / 10 >= one.Significand);
while (true)
{
fractionals *= 10;
unit *= 10;
unsafe_interval.Significand = unsafe_interval.Significand * 10;
// Integer division by one.
var digit = (Int32)(fractionals >> -one.Exponent);
Debug.Assert(digit <= 9);
buffer[length] = (Byte)('0' + digit);
++length;
fractionals &= one.Significand - 1; // Modulo by one.
--kappa;
if (fractionals < unsafe_interval.Significand)
{
return(RoundWeed(buffer,
length,
DiyFp.Minus(too_high, w).Significand *unit,
unsafe_interval.Significand,
fractionals,
one.Significand,
unit));
}
}
}
private static Boolean Grisu3(Double v, Boolean single_precision, Span <Byte> buffer, out Int32 length, out Int32 decimal_exponent)
{
var w = new DiyDouble(v).AsNormalizedDiyFp();
// boundary_minus and boundary_plus are the boundaries between v and its
// closest floating-point neighbors. Any number strictly between
// boundary_minus and boundary_plus will round to v when convert to a double.
// Grisu3 will never output representations that lie exactly on a boundary.
DiyFp boundary_minus, boundary_plus;
if (single_precision)
{
new DiySingle((Single)v).NormalizedBoundaries(out boundary_minus, out boundary_plus);
}
else
{
new DiyDouble(v).NormalizedBoundaries(out boundary_minus, out boundary_plus);
}
Debug.Assert(boundary_plus.Exponent == w.Exponent);
var ten_mk_minimal_binary_exponent = MINIMAL_TARGET_EXPONENT - (w.Exponent + DiyFp.SignificantSize);
var ten_mk_maximal_binary_exponent = MAXIMAL_TARGET_EXPONENT - (w.Exponent + DiyFp.SignificantSize);
// Note that ten_mk is only an approximation of 10^-k. A DiyFp only contains a
// 64 bit significand and ten_mk is thus only precise up to 64 bits.
GetCachedPowerForBinaryExponentRange(ten_mk_minimal_binary_exponent,
ten_mk_maximal_binary_exponent,
out var ten_mk,
out var mk);
Debug.Assert(MINIMAL_TARGET_EXPONENT <= w.Exponent + ten_mk.Exponent + DiyFp.SignificantSize);
Debug.Assert(MAXIMAL_TARGET_EXPONENT >= w.Exponent + ten_mk.Exponent + DiyFp.SignificantSize);
// The DiyFp.Times procedure rounds its result, and ten_mk is approximated
// too. The variable scaled_w (as well as scaled_boundary_minus/plus) are now
// off by a small amount.
// In fact: scaled_w - w*10^k < 1ulp (unit in the last place) of scaled_w.
// In other words: let f = scaled_w.f() and e = scaled_w.e(), then (f-1) * 2^e < w*10^k < (f+1) * 2^e
var scaled_w = DiyFp.Times(w, ten_mk);
Debug.Assert(scaled_w.Exponent == boundary_plus.Exponent + ten_mk.Exponent + DiyFp.SignificantSize);
// In theory it would be possible to avoid some recomputations by computing
// the difference between w and boundary_minus/plus (a power of 2) and to
// compute scaled_boundary_minus/plus by subtracting/adding from
// scaled_w. However the code becomes much less readable and the speed enhancements are not terriffic.
var scaled_boundary_minus = DiyFp.Times(boundary_minus, ten_mk);
var scaled_boundary_plus = DiyFp.Times(boundary_plus, ten_mk);
// DigitGen will generate the digits of scaled_w. Therefore we have
// v == (double) (scaled_w * 10^-mk).
// Set decimal_exponent == -mk and pass it to DigitGen. If scaled_w is not an
// integer than it will be updated. For instance if scaled_w == 1.23 then
// the buffer will be filled with "123" und the decimal_exponent will be decreased by 2.
var result = DigitGen(scaled_boundary_minus,
scaled_w,
scaled_boundary_plus,
buffer,
out length,
out var kappa);
decimal_exponent = -mk + kappa;
return(result);
}
