119 lines
3.8 KiB
Go
119 lines
3.8 KiB
Go
// Copyright 2009 The Go Authors. All rights reserved.
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// Use of this source code is governed by a BSD-style
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// license that can be found in the LICENSE file.
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// Multiprecision decimal numbers.
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// For floating-point formatting only; not general purpose.
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// Only operations are assign and (binary) left/right shift.
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// Can do binary floating point in multiprecision decimal precisely
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// because 2 divides 10; cannot do decimal floating point
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// in multiprecision binary precisely.
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package decimal
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type floatInfo struct {
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mantbits uint
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expbits uint
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bias int
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}
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var float32info = floatInfo{23, 8, -127}
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var float64info = floatInfo{52, 11, -1023}
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// roundShortest rounds d (= mant * 2^exp) to the shortest number of digits
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// that will let the original floating point value be precisely reconstructed.
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func roundShortest(d *decimal, mant uint64, exp int, flt *floatInfo) {
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// If mantissa is zero, the number is zero; stop now.
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if mant == 0 {
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d.nd = 0
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return
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}
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// Compute upper and lower such that any decimal number
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// between upper and lower (possibly inclusive)
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// will round to the original floating point number.
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// We may see at once that the number is already shortest.
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//
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// Suppose d is not denormal, so that 2^exp <= d < 10^dp.
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// The closest shorter number is at least 10^(dp-nd) away.
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// The lower/upper bounds computed below are at distance
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// at most 2^(exp-mantbits).
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//
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// So the number is already shortest if 10^(dp-nd) > 2^(exp-mantbits),
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// or equivalently log2(10)*(dp-nd) > exp-mantbits.
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// It is true if 332/100*(dp-nd) >= exp-mantbits (log2(10) > 3.32).
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minexp := flt.bias + 1 // minimum possible exponent
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if exp > minexp && 332*(d.dp-d.nd) >= 100*(exp-int(flt.mantbits)) {
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// The number is already shortest.
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return
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}
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// d = mant << (exp - mantbits)
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// Next highest floating point number is mant+1 << exp-mantbits.
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// Our upper bound is halfway between, mant*2+1 << exp-mantbits-1.
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upper := new(decimal)
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upper.Assign(mant*2 + 1)
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upper.Shift(exp - int(flt.mantbits) - 1)
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// d = mant << (exp - mantbits)
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// Next lowest floating point number is mant-1 << exp-mantbits,
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// unless mant-1 drops the significant bit and exp is not the minimum exp,
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// in which case the next lowest is mant*2-1 << exp-mantbits-1.
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// Either way, call it mantlo << explo-mantbits.
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// Our lower bound is halfway between, mantlo*2+1 << explo-mantbits-1.
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var mantlo uint64
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var explo int
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if mant > 1<<flt.mantbits || exp == minexp {
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mantlo = mant - 1
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explo = exp
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} else {
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mantlo = mant*2 - 1
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explo = exp - 1
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}
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lower := new(decimal)
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lower.Assign(mantlo*2 + 1)
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lower.Shift(explo - int(flt.mantbits) - 1)
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// The upper and lower bounds are possible outputs only if
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// the original mantissa is even, so that IEEE round-to-even
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// would round to the original mantissa and not the neighbors.
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inclusive := mant%2 == 0
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// Now we can figure out the minimum number of digits required.
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// Walk along until d has distinguished itself from upper and lower.
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for i := 0; i < d.nd; i++ {
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l := byte('0') // lower digit
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if i < lower.nd {
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l = lower.d[i]
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}
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m := d.d[i] // middle digit
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u := byte('0') // upper digit
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if i < upper.nd {
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u = upper.d[i]
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}
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// Okay to round down (truncate) if lower has a different digit
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// or if lower is inclusive and is exactly the result of rounding
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// down (i.e., and we have reached the final digit of lower).
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okdown := l != m || inclusive && i+1 == lower.nd
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// Okay to round up if upper has a different digit and either upper
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// is inclusive or upper is bigger than the result of rounding up.
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okup := m != u && (inclusive || m+1 < u || i+1 < upper.nd)
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// If it's okay to do either, then round to the nearest one.
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// If it's okay to do only one, do it.
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switch {
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case okdown && okup:
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d.Round(i + 1)
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return
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case okdown:
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d.RoundDown(i + 1)
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return
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case okup:
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d.RoundUp(i + 1)
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return
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}
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}
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}
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