502 lines
15 KiB
Go
502 lines
15 KiB
Go
// Copyright 2011 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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package strconv
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import "math"
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// An extFloat represents an extended floating-point number, with more
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// precision than a float64. It does not try to save bits: the
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// number represented by the structure is mant*(2^exp), with a negative
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// sign if neg is true.
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type extFloat struct {
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mant uint64
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exp int
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neg bool
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}
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// Powers of ten taken from double-conversion library.
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// http://code.google.com/p/double-conversion/
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const (
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firstPowerOfTen = -348
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stepPowerOfTen = 8
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)
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var smallPowersOfTen = [...]extFloat{
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{1 << 63, -63, false}, // 1
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{0xa << 60, -60, false}, // 1e1
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{0x64 << 57, -57, false}, // 1e2
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{0x3e8 << 54, -54, false}, // 1e3
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{0x2710 << 50, -50, false}, // 1e4
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{0x186a0 << 47, -47, false}, // 1e5
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{0xf4240 << 44, -44, false}, // 1e6
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{0x989680 << 40, -40, false}, // 1e7
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}
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var powersOfTen = [...]extFloat{
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{0xfa8fd5a0081c0288, -1220, false}, // 10^-348
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{0xbaaee17fa23ebf76, -1193, false}, // 10^-340
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{0x8b16fb203055ac76, -1166, false}, // 10^-332
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{0xcf42894a5dce35ea, -1140, false}, // 10^-324
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{0x9a6bb0aa55653b2d, -1113, false}, // 10^-316
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{0xe61acf033d1a45df, -1087, false}, // 10^-308
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{0xab70fe17c79ac6ca, -1060, false}, // 10^-300
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{0xff77b1fcbebcdc4f, -1034, false}, // 10^-292
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{0xbe5691ef416bd60c, -1007, false}, // 10^-284
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{0x8dd01fad907ffc3c, -980, false}, // 10^-276
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{0xd3515c2831559a83, -954, false}, // 10^-268
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{0x9d71ac8fada6c9b5, -927, false}, // 10^-260
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{0xea9c227723ee8bcb, -901, false}, // 10^-252
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{0xaecc49914078536d, -874, false}, // 10^-244
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{0x823c12795db6ce57, -847, false}, // 10^-236
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{0xc21094364dfb5637, -821, false}, // 10^-228
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{0x9096ea6f3848984f, -794, false}, // 10^-220
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{0xd77485cb25823ac7, -768, false}, // 10^-212
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{0xa086cfcd97bf97f4, -741, false}, // 10^-204
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{0xef340a98172aace5, -715, false}, // 10^-196
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{0xb23867fb2a35b28e, -688, false}, // 10^-188
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{0x84c8d4dfd2c63f3b, -661, false}, // 10^-180
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{0xc5dd44271ad3cdba, -635, false}, // 10^-172
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{0x936b9fcebb25c996, -608, false}, // 10^-164
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{0xdbac6c247d62a584, -582, false}, // 10^-156
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{0xa3ab66580d5fdaf6, -555, false}, // 10^-148
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{0xf3e2f893dec3f126, -529, false}, // 10^-140
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{0xb5b5ada8aaff80b8, -502, false}, // 10^-132
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{0x87625f056c7c4a8b, -475, false}, // 10^-124
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{0xc9bcff6034c13053, -449, false}, // 10^-116
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{0x964e858c91ba2655, -422, false}, // 10^-108
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{0xdff9772470297ebd, -396, false}, // 10^-100
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{0xa6dfbd9fb8e5b88f, -369, false}, // 10^-92
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{0xf8a95fcf88747d94, -343, false}, // 10^-84
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{0xb94470938fa89bcf, -316, false}, // 10^-76
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{0x8a08f0f8bf0f156b, -289, false}, // 10^-68
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{0xcdb02555653131b6, -263, false}, // 10^-60
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{0x993fe2c6d07b7fac, -236, false}, // 10^-52
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{0xe45c10c42a2b3b06, -210, false}, // 10^-44
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{0xaa242499697392d3, -183, false}, // 10^-36
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{0xfd87b5f28300ca0e, -157, false}, // 10^-28
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{0xbce5086492111aeb, -130, false}, // 10^-20
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{0x8cbccc096f5088cc, -103, false}, // 10^-12
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{0xd1b71758e219652c, -77, false}, // 10^-4
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{0x9c40000000000000, -50, false}, // 10^4
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{0xe8d4a51000000000, -24, false}, // 10^12
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{0xad78ebc5ac620000, 3, false}, // 10^20
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{0x813f3978f8940984, 30, false}, // 10^28
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{0xc097ce7bc90715b3, 56, false}, // 10^36
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{0x8f7e32ce7bea5c70, 83, false}, // 10^44
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{0xd5d238a4abe98068, 109, false}, // 10^52
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{0x9f4f2726179a2245, 136, false}, // 10^60
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{0xed63a231d4c4fb27, 162, false}, // 10^68
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{0xb0de65388cc8ada8, 189, false}, // 10^76
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{0x83c7088e1aab65db, 216, false}, // 10^84
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{0xc45d1df942711d9a, 242, false}, // 10^92
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{0x924d692ca61be758, 269, false}, // 10^100
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{0xda01ee641a708dea, 295, false}, // 10^108
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{0xa26da3999aef774a, 322, false}, // 10^116
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{0xf209787bb47d6b85, 348, false}, // 10^124
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{0xb454e4a179dd1877, 375, false}, // 10^132
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{0x865b86925b9bc5c2, 402, false}, // 10^140
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{0xc83553c5c8965d3d, 428, false}, // 10^148
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{0x952ab45cfa97a0b3, 455, false}, // 10^156
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{0xde469fbd99a05fe3, 481, false}, // 10^164
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{0xa59bc234db398c25, 508, false}, // 10^172
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{0xf6c69a72a3989f5c, 534, false}, // 10^180
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{0xb7dcbf5354e9bece, 561, false}, // 10^188
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{0x88fcf317f22241e2, 588, false}, // 10^196
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{0xcc20ce9bd35c78a5, 614, false}, // 10^204
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{0x98165af37b2153df, 641, false}, // 10^212
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{0xe2a0b5dc971f303a, 667, false}, // 10^220
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{0xa8d9d1535ce3b396, 694, false}, // 10^228
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{0xfb9b7cd9a4a7443c, 720, false}, // 10^236
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{0xbb764c4ca7a44410, 747, false}, // 10^244
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{0x8bab8eefb6409c1a, 774, false}, // 10^252
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{0xd01fef10a657842c, 800, false}, // 10^260
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{0x9b10a4e5e9913129, 827, false}, // 10^268
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{0xe7109bfba19c0c9d, 853, false}, // 10^276
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{0xac2820d9623bf429, 880, false}, // 10^284
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{0x80444b5e7aa7cf85, 907, false}, // 10^292
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{0xbf21e44003acdd2d, 933, false}, // 10^300
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{0x8e679c2f5e44ff8f, 960, false}, // 10^308
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{0xd433179d9c8cb841, 986, false}, // 10^316
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{0x9e19db92b4e31ba9, 1013, false}, // 10^324
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{0xeb96bf6ebadf77d9, 1039, false}, // 10^332
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{0xaf87023b9bf0ee6b, 1066, false}, // 10^340
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}
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// floatBits returns the bits of the float64 that best approximates
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// the extFloat passed as receiver. Overflow is set to true if
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// the resulting float64 is ±Inf.
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func (f *extFloat) floatBits() (bits uint64, overflow bool) {
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flt := &float64info
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f.Normalize()
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exp := f.exp + 63
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// Exponent too small.
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if exp < flt.bias+1 {
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n := flt.bias + 1 - exp
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f.mant >>= uint(n)
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exp += n
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}
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// Extract 1+flt.mantbits bits.
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mant := f.mant >> (63 - flt.mantbits)
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if f.mant&(1<<(62-flt.mantbits)) != 0 {
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// Round up.
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mant += 1
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}
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// Rounding might have added a bit; shift down.
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if mant == 2<<flt.mantbits {
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mant >>= 1
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exp++
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}
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// Infinities.
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if exp-flt.bias >= 1<<flt.expbits-1 {
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goto overflow
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}
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// Denormalized?
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if mant&(1<<flt.mantbits) == 0 {
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exp = flt.bias
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}
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goto out
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overflow:
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// ±Inf
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mant = 0
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exp = 1<<flt.expbits - 1 + flt.bias
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overflow = true
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out:
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// Assemble bits.
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bits = mant & (uint64(1)<<flt.mantbits - 1)
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bits |= uint64((exp-flt.bias)&(1<<flt.expbits-1)) << flt.mantbits
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if f.neg {
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bits |= 1 << (flt.mantbits + flt.expbits)
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}
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return
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}
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// Assign sets f to the value of x.
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func (f *extFloat) Assign(x float64) {
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if x < 0 {
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x = -x
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f.neg = true
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}
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x, f.exp = math.Frexp(x)
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f.mant = uint64(x * float64(1<<64))
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f.exp -= 64
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}
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// AssignComputeBounds sets f to the value of x and returns
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// lower, upper such that any number in the closed interval
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// [lower, upper] is converted back to x.
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func (f *extFloat) AssignComputeBounds(x float64) (lower, upper extFloat) {
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// Special cases.
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bits := math.Float64bits(x)
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flt := &float64info
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neg := bits>>(flt.expbits+flt.mantbits) != 0
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expBiased := int(bits>>flt.mantbits) & (1<<flt.expbits - 1)
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mant := bits & (uint64(1)<<flt.mantbits - 1)
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if expBiased == 0 {
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// denormalized.
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f.mant = mant
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f.exp = 1 + flt.bias - int(flt.mantbits)
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} else {
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f.mant = mant | 1<<flt.mantbits
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f.exp = expBiased + flt.bias - int(flt.mantbits)
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}
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f.neg = neg
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upper = extFloat{mant: 2*f.mant + 1, exp: f.exp - 1, neg: f.neg}
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if mant != 0 || expBiased == 1 {
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lower = extFloat{mant: 2*f.mant - 1, exp: f.exp - 1, neg: f.neg}
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} else {
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lower = extFloat{mant: 4*f.mant - 1, exp: f.exp - 2, neg: f.neg}
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}
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return
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}
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// Normalize normalizes f so that the highest bit of the mantissa is
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// set, and returns the number by which the mantissa was left-shifted.
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func (f *extFloat) Normalize() uint {
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if f.mant == 0 {
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return 0
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}
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exp_before := f.exp
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for f.mant < (1 << 55) {
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f.mant <<= 8
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f.exp -= 8
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}
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for f.mant < (1 << 63) {
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f.mant <<= 1
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f.exp -= 1
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}
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return uint(exp_before - f.exp)
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}
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// Multiply sets f to the product f*g: the result is correctly rounded,
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// but not normalized.
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func (f *extFloat) Multiply(g extFloat) {
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fhi, flo := f.mant>>32, uint64(uint32(f.mant))
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ghi, glo := g.mant>>32, uint64(uint32(g.mant))
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// Cross products.
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cross1 := fhi * glo
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cross2 := flo * ghi
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// f.mant*g.mant is fhi*ghi << 64 + (cross1+cross2) << 32 + flo*glo
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f.mant = fhi*ghi + (cross1 >> 32) + (cross2 >> 32)
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rem := uint64(uint32(cross1)) + uint64(uint32(cross2)) + ((flo * glo) >> 32)
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// Round up.
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rem += (1 << 31)
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f.mant += (rem >> 32)
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f.exp = f.exp + g.exp + 64
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}
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var uint64pow10 = [...]uint64{
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1, 1e1, 1e2, 1e3, 1e4, 1e5, 1e6, 1e7, 1e8, 1e9,
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1e10, 1e11, 1e12, 1e13, 1e14, 1e15, 1e16, 1e17, 1e18, 1e19,
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}
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// AssignDecimal sets f to an approximate value of the decimal d. It
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// returns true if the value represented by f is guaranteed to be the
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// best approximation of d after being rounded to a float64.
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func (f *extFloat) AssignDecimal(d *decimal) (ok bool) {
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const uint64digits = 19
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const errorscale = 8
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mant10, digits := d.atou64()
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exp10 := d.dp - digits
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errors := 0 // An upper bound for error, computed in errorscale*ulp.
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if digits < d.nd {
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// the decimal number was truncated.
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errors += errorscale / 2
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}
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f.mant = mant10
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f.exp = 0
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f.neg = d.neg
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// Multiply by powers of ten.
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i := (exp10 - firstPowerOfTen) / stepPowerOfTen
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if exp10 < firstPowerOfTen || i >= len(powersOfTen) {
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return false
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}
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adjExp := (exp10 - firstPowerOfTen) % stepPowerOfTen
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// We multiply by exp%step
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if digits+adjExp <= uint64digits {
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// We can multiply the mantissa
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f.mant *= uint64(float64pow10[adjExp])
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f.Normalize()
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} else {
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f.Normalize()
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f.Multiply(smallPowersOfTen[adjExp])
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errors += errorscale / 2
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}
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// We multiply by 10 to the exp - exp%step.
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f.Multiply(powersOfTen[i])
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if errors > 0 {
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errors += 1
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}
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errors += errorscale / 2
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// Normalize
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shift := f.Normalize()
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errors <<= shift
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// Now f is a good approximation of the decimal.
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// Check whether the error is too large: that is, if the mantissa
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// is perturbated by the error, the resulting float64 will change.
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// The 64 bits mantissa is 1 + 52 bits for float64 + 11 extra bits.
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//
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// In many cases the approximation will be good enough.
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const denormalExp = -1023 - 63
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flt := &float64info
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var extrabits uint
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if f.exp <= denormalExp {
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extrabits = uint(63 - flt.mantbits + 1 + uint(denormalExp-f.exp))
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} else {
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extrabits = uint(63 - flt.mantbits)
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}
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halfway := uint64(1) << (extrabits - 1)
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mant_extra := f.mant & (1<<extrabits - 1)
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// Do a signed comparison here! If the error estimate could make
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// the mantissa round differently for the conversion to double,
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// then we can't give a definite answer.
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if int64(halfway)-int64(errors) < int64(mant_extra) &&
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int64(mant_extra) < int64(halfway)+int64(errors) {
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return false
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}
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return true
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}
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// Frexp10 is an analogue of math.Frexp for decimal powers. It scales
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// f by an approximate power of ten 10^-exp, and returns exp10, so
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// that f*10^exp10 has the same value as the old f, up to an ulp,
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// as well as the index of 10^-exp in the powersOfTen table.
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// The arguments expMin and expMax constrain the final value of the
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// binary exponent of f.
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func (f *extFloat) frexp10(expMin, expMax int) (exp10, index int) {
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// it is illegal to call this function with a too restrictive exponent range.
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if expMax-expMin <= 25 {
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panic("strconv: invalid exponent range")
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}
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// Find power of ten such that x * 10^n has a binary exponent
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// between expMin and expMax
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approxExp10 := -(f.exp + 100) * 28 / 93 // log(10)/log(2) is close to 93/28.
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i := (approxExp10 - firstPowerOfTen) / stepPowerOfTen
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Loop:
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for {
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exp := f.exp + powersOfTen[i].exp + 64
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switch {
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case exp < expMin:
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i++
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case exp > expMax:
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i--
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default:
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break Loop
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}
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}
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// Apply the desired decimal shift on f. It will have exponent
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// in the desired range. This is multiplication by 10^-exp10.
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f.Multiply(powersOfTen[i])
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return -(firstPowerOfTen + i*stepPowerOfTen), i
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}
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// frexp10Many applies a common shift by a power of ten to a, b, c.
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func frexp10Many(expMin, expMax int, a, b, c *extFloat) (exp10 int) {
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exp10, i := c.frexp10(expMin, expMax)
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a.Multiply(powersOfTen[i])
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b.Multiply(powersOfTen[i])
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return
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}
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// ShortestDecimal stores in d the shortest decimal representation of f
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// which belongs to the open interval (lower, upper), where f is supposed
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// to lie. It returns false whenever the result is unsure. The implementation
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// uses the Grisu3 algorithm.
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func (f *extFloat) ShortestDecimal(d *decimal, lower, upper *extFloat) bool {
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if f.mant == 0 {
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d.d[0] = '0'
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d.nd = 1
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d.dp = 0
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d.neg = f.neg
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}
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const minExp = -60
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const maxExp = -32
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upper.Normalize()
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// Uniformize exponents.
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if f.exp > upper.exp {
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f.mant <<= uint(f.exp - upper.exp)
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f.exp = upper.exp
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}
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if lower.exp > upper.exp {
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lower.mant <<= uint(lower.exp - upper.exp)
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lower.exp = upper.exp
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}
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exp10 := frexp10Many(minExp, maxExp, lower, f, upper)
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// Take a safety margin due to rounding in frexp10Many, but we lose precision.
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upper.mant++
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lower.mant--
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// The shortest representation of f is either rounded up or down, but
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// in any case, it is a truncation of upper.
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shift := uint(-upper.exp)
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integer := uint32(upper.mant >> shift)
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fraction := upper.mant - (uint64(integer) << shift)
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// How far we can go down from upper until the result is wrong.
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allowance := upper.mant - lower.mant
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// How far we should go to get a very precise result.
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targetDiff := upper.mant - f.mant
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// Count integral digits: there are at most 10.
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var integerDigits int
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for i, pow := range uint64pow10 {
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if uint64(integer) >= pow {
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integerDigits = i + 1
|
|
}
|
|
}
|
|
for i := 0; i < integerDigits; i++ {
|
|
pow := uint64pow10[integerDigits-i-1]
|
|
digit := integer / uint32(pow)
|
|
d.d[i] = byte(digit + '0')
|
|
integer -= digit * uint32(pow)
|
|
// evaluate whether we should stop.
|
|
if currentDiff := uint64(integer)<<shift + fraction; currentDiff < allowance {
|
|
d.nd = i + 1
|
|
d.dp = integerDigits + exp10
|
|
d.neg = f.neg
|
|
// Sometimes allowance is so large the last digit might need to be
|
|
// decremented to get closer to f.
|
|
return adjustLastDigit(d, currentDiff, targetDiff, allowance, pow<<shift, 2)
|
|
}
|
|
}
|
|
d.nd = integerDigits
|
|
d.dp = d.nd + exp10
|
|
d.neg = f.neg
|
|
|
|
// Compute digits of the fractional part. At each step fraction does not
|
|
// overflow. The choice of minExp implies that fraction is less than 2^60.
|
|
var digit int
|
|
multiplier := uint64(1)
|
|
for {
|
|
fraction *= 10
|
|
multiplier *= 10
|
|
digit = int(fraction >> shift)
|
|
d.d[d.nd] = byte(digit + '0')
|
|
d.nd++
|
|
fraction -= uint64(digit) << shift
|
|
if fraction < allowance*multiplier {
|
|
// We are in the admissible range. Note that if allowance is about to
|
|
// overflow, that is, allowance > 2^64/10, the condition is automatically
|
|
// true due to the limited range of fraction.
|
|
return adjustLastDigit(d,
|
|
fraction, targetDiff*multiplier, allowance*multiplier,
|
|
1<<shift, multiplier*2)
|
|
}
|
|
}
|
|
return false
|
|
}
|
|
|
|
// adjustLastDigit modifies d = x-currentDiff*ε, to get closest to
|
|
// d = x-targetDiff*ε, without becoming smaller than x-maxDiff*ε.
|
|
// It assumes that a decimal digit is worth ulpDecimal*ε, and that
|
|
// all data is known with a error estimate of ulpBinary*ε.
|
|
func adjustLastDigit(d *decimal, currentDiff, targetDiff, maxDiff, ulpDecimal, ulpBinary uint64) bool {
|
|
if ulpDecimal < 2*ulpBinary {
|
|
// Approximation is too wide.
|
|
return false
|
|
}
|
|
for currentDiff+ulpDecimal/2+ulpBinary < targetDiff {
|
|
d.d[d.nd-1]--
|
|
currentDiff += ulpDecimal
|
|
}
|
|
if currentDiff+ulpDecimal <= targetDiff+ulpDecimal/2+ulpBinary {
|
|
// we have two choices, and don't know what to do.
|
|
return false
|
|
}
|
|
if currentDiff < ulpBinary || currentDiff > maxDiff-ulpBinary {
|
|
// we went too far
|
|
return false
|
|
}
|
|
if d.nd == 1 && d.d[0] == '0' {
|
|
// the number has actually reached zero.
|
|
d.nd = 0
|
|
d.dp = 0
|
|
}
|
|
return true
|
|
}
|