2015-07-08 Paolo Carlini <paolo.carlini@oracle.com>

[official-gcc.git] / libgo / go / strconv / extfloat.go

// Copyright 2011 The Go Authors. All rights reserved.
// Use of this source code is governed by a BSD-style
// license that can be found in the LICENSE file.

package strconv

// An extFloat represents an extended floating-point number, with more
// precision than a float64. It does not try to save bits: the
// number represented by the structure is mant*(2^exp), with a negative
// sign if neg is true.
type extFloat struct {
        mant uint64
        exp  int
        neg  bool
}

// Powers of ten taken from double-conversion library.
// http://code.google.com/p/double-conversion/
const (
        firstPowerOfTen = -348
        stepPowerOfTen  = 8
)

var smallPowersOfTen = [...]extFloat{
        {1 << 63, -63, false},        // 1
        {0xa << 60, -60, false},      // 1e1
        {0x64 << 57, -57, false},     // 1e2
        {0x3e8 << 54, -54, false},    // 1e3
        {0x2710 << 50, -50, false},   // 1e4
        {0x186a0 << 47, -47, false},  // 1e5
        {0xf4240 << 44, -44, false},  // 1e6
        {0x989680 << 40, -40, false}, // 1e7
}

var powersOfTen = [...]extFloat{
        {0xfa8fd5a0081c0288, -1220, false}, // 10^-348
        {0xbaaee17fa23ebf76, -1193, false}, // 10^-340
        {0x8b16fb203055ac76, -1166, false}, // 10^-332
        {0xcf42894a5dce35ea, -1140, false}, // 10^-324
        {0x9a6bb0aa55653b2d, -1113, false}, // 10^-316
        {0xe61acf033d1a45df, -1087, false}, // 10^-308
        {0xab70fe17c79ac6ca, -1060, false}, // 10^-300
        {0xff77b1fcbebcdc4f, -1034, false}, // 10^-292
        {0xbe5691ef416bd60c, -1007, false}, // 10^-284
        {0x8dd01fad907ffc3c, -980, false},  // 10^-276
        {0xd3515c2831559a83, -954, false},  // 10^-268
        {0x9d71ac8fada6c9b5, -927, false},  // 10^-260
        {0xea9c227723ee8bcb, -901, false},  // 10^-252
        {0xaecc49914078536d, -874, false},  // 10^-244
        {0x823c12795db6ce57, -847, false},  // 10^-236
        {0xc21094364dfb5637, -821, false},  // 10^-228
        {0x9096ea6f3848984f, -794, false},  // 10^-220
        {0xd77485cb25823ac7, -768, false},  // 10^-212
        {0xa086cfcd97bf97f4, -741, false},  // 10^-204
        {0xef340a98172aace5, -715, false},  // 10^-196
        {0xb23867fb2a35b28e, -688, false},  // 10^-188
        {0x84c8d4dfd2c63f3b, -661, false},  // 10^-180
        {0xc5dd44271ad3cdba, -635, false},  // 10^-172
        {0x936b9fcebb25c996, -608, false},  // 10^-164
        {0xdbac6c247d62a584, -582, false},  // 10^-156
        {0xa3ab66580d5fdaf6, -555, false},  // 10^-148
        {0xf3e2f893dec3f126, -529, false},  // 10^-140
        {0xb5b5ada8aaff80b8, -502, false},  // 10^-132
        {0x87625f056c7c4a8b, -475, false},  // 10^-124
        {0xc9bcff6034c13053, -449, false},  // 10^-116
        {0x964e858c91ba2655, -422, false},  // 10^-108
        {0xdff9772470297ebd, -396, false},  // 10^-100
        {0xa6dfbd9fb8e5b88f, -369, false},  // 10^-92
        {0xf8a95fcf88747d94, -343, false},  // 10^-84
        {0xb94470938fa89bcf, -316, false},  // 10^-76
        {0x8a08f0f8bf0f156b, -289, false},  // 10^-68
        {0xcdb02555653131b6, -263, false},  // 10^-60
        {0x993fe2c6d07b7fac, -236, false},  // 10^-52
        {0xe45c10c42a2b3b06, -210, false},  // 10^-44
        {0xaa242499697392d3, -183, false},  // 10^-36
        {0xfd87b5f28300ca0e, -157, false},  // 10^-28
        {0xbce5086492111aeb, -130, false},  // 10^-20
        {0x8cbccc096f5088cc, -103, false},  // 10^-12
        {0xd1b71758e219652c, -77, false},   // 10^-4
        {0x9c40000000000000, -50, false},   // 10^4
        {0xe8d4a51000000000, -24, false},   // 10^12
        {0xad78ebc5ac620000, 3, false},     // 10^20
        {0x813f3978f8940984, 30, false},    // 10^28
        {0xc097ce7bc90715b3, 56, false},    // 10^36
        {0x8f7e32ce7bea5c70, 83, false},    // 10^44
        {0xd5d238a4abe98068, 109, false},   // 10^52
        {0x9f4f2726179a2245, 136, false},   // 10^60
        {0xed63a231d4c4fb27, 162, false},   // 10^68
        {0xb0de65388cc8ada8, 189, false},   // 10^76
        {0x83c7088e1aab65db, 216, false},   // 10^84
        {0xc45d1df942711d9a, 242, false},   // 10^92
        {0x924d692ca61be758, 269, false},   // 10^100
        {0xda01ee641a708dea, 295, false},   // 10^108
        {0xa26da3999aef774a, 322, false},   // 10^116
        {0xf209787bb47d6b85, 348, false},   // 10^124
        {0xb454e4a179dd1877, 375, false},   // 10^132
        {0x865b86925b9bc5c2, 402, false},   // 10^140
        {0xc83553c5c8965d3d, 428, false},   // 10^148
        {0x952ab45cfa97a0b3, 455, false},   // 10^156
        {0xde469fbd99a05fe3, 481, false},   // 10^164
        {0xa59bc234db398c25, 508, false},   // 10^172
        {0xf6c69a72a3989f5c, 534, false},   // 10^180
        {0xb7dcbf5354e9bece, 561, false},   // 10^188
        {0x88fcf317f22241e2, 588, false},   // 10^196
        {0xcc20ce9bd35c78a5, 614, false},   // 10^204
        {0x98165af37b2153df, 641, false},   // 10^212
        {0xe2a0b5dc971f303a, 667, false},   // 10^220
        {0xa8d9d1535ce3b396, 694, false},   // 10^228
        {0xfb9b7cd9a4a7443c, 720, false},   // 10^236
        {0xbb764c4ca7a44410, 747, false},   // 10^244
        {0x8bab8eefb6409c1a, 774, false},   // 10^252
        {0xd01fef10a657842c, 800, false},   // 10^260
        {0x9b10a4e5e9913129, 827, false},   // 10^268
        {0xe7109bfba19c0c9d, 853, false},   // 10^276
        {0xac2820d9623bf429, 880, false},   // 10^284
        {0x80444b5e7aa7cf85, 907, false},   // 10^292
        {0xbf21e44003acdd2d, 933, false},   // 10^300
        {0x8e679c2f5e44ff8f, 960, false},   // 10^308
        {0xd433179d9c8cb841, 986, false},   // 10^316
        {0x9e19db92b4e31ba9, 1013, false},  // 10^324
        {0xeb96bf6ebadf77d9, 1039, false},  // 10^332
        {0xaf87023b9bf0ee6b, 1066, false},  // 10^340
}

// floatBits returns the bits of the float64 that best approximates
// the extFloat passed as receiver. Overflow is set to true if
// the resulting float64 is ±Inf.
func (f *extFloat) floatBits(flt *floatInfo) (bits uint64, overflow bool) {
        f.Normalize()

        exp := f.exp + 63

        // Exponent too small.
        if exp < flt.bias+1 {
                n := flt.bias + 1 - exp
                f.mant >>= uint(n)
                exp += n
        }

        // Extract 1+flt.mantbits bits from the 64-bit mantissa.
        mant := f.mant >> (63 - flt.mantbits)
        if f.mant&(1<<(62-flt.mantbits)) != 0 {
                // Round up.
                mant += 1
        }

        // Rounding might have added a bit; shift down.
        if mant == 2<<flt.mantbits {
                mant >>= 1
                exp++
        }

        // Infinities.
        if exp-flt.bias >= 1<<flt.expbits-1 {
                // ±Inf
                mant = 0
                exp = 1<<flt.expbits - 1 + flt.bias
                overflow = true
        } else if mant&(1<<flt.mantbits) == 0 {
                // Denormalized?
                exp = flt.bias
        }
        // Assemble bits.
        bits = mant & (uint64(1)<<flt.mantbits - 1)
        bits |= uint64((exp-flt.bias)&(1<<flt.expbits-1)) << flt.mantbits
        if f.neg {
                bits |= 1 << (flt.mantbits + flt.expbits)
        }
        return
}

// AssignComputeBounds sets f to the floating point value
// defined by mant, exp and precision given by flt. It returns
// lower, upper such that any number in the closed interval
// [lower, upper] is converted back to the same floating point number.
func (f *extFloat) AssignComputeBounds(mant uint64, exp int, neg bool, flt *floatInfo) (lower, upper extFloat) {
        f.mant = mant
        f.exp = exp - int(flt.mantbits)
        f.neg = neg
        if f.exp <= 0 && mant == (mant>>uint(-f.exp))<<uint(-f.exp) {
                // An exact integer
                f.mant >>= uint(-f.exp)
                f.exp = 0
                return *f, *f
        }
        expBiased := exp - flt.bias

        upper = extFloat{mant: 2*f.mant + 1, exp: f.exp - 1, neg: f.neg}
        if mant != 1<<flt.mantbits || expBiased == 1 {
                lower = extFloat{mant: 2*f.mant - 1, exp: f.exp - 1, neg: f.neg}
        } else {
                lower = extFloat{mant: 4*f.mant - 1, exp: f.exp - 2, neg: f.neg}
        }
        return
}

// Normalize normalizes f so that the highest bit of the mantissa is
// set, and returns the number by which the mantissa was left-shifted.
func (f *extFloat) Normalize() (shift uint) {
        mant, exp := f.mant, f.exp
        if mant == 0 {
                return 0
        }
        if mant>>(64-32) == 0 {
                mant <<= 32
                exp -= 32
        }
        if mant>>(64-16) == 0 {
                mant <<= 16
                exp -= 16
        }
        if mant>>(64-8) == 0 {
                mant <<= 8
                exp -= 8
        }
        if mant>>(64-4) == 0 {
                mant <<= 4
                exp -= 4
        }
        if mant>>(64-2) == 0 {
                mant <<= 2
                exp -= 2
        }
        if mant>>(64-1) == 0 {
                mant <<= 1
                exp -= 1
        }
        shift = uint(f.exp - exp)
        f.mant, f.exp = mant, exp
        return
}

// Multiply sets f to the product f*g: the result is correctly rounded,
// but not normalized.
func (f *extFloat) Multiply(g extFloat) {
        fhi, flo := f.mant>>32, uint64(uint32(f.mant))
        ghi, glo := g.mant>>32, uint64(uint32(g.mant))

        // Cross products.
        cross1 := fhi * glo
        cross2 := flo * ghi

        // f.mant*g.mant is fhi*ghi << 64 + (cross1+cross2) << 32 + flo*glo
        f.mant = fhi*ghi + (cross1 >> 32) + (cross2 >> 32)
        rem := uint64(uint32(cross1)) + uint64(uint32(cross2)) + ((flo * glo) >> 32)
        // Round up.
        rem += (1 << 31)

        f.mant += (rem >> 32)
        f.exp = f.exp + g.exp + 64
}

var uint64pow10 = [...]uint64{
        1, 1e1, 1e2, 1e3, 1e4, 1e5, 1e6, 1e7, 1e8, 1e9,
        1e10, 1e11, 1e12, 1e13, 1e14, 1e15, 1e16, 1e17, 1e18, 1e19,
}

// AssignDecimal sets f to an approximate value mantissa*10^exp. It
// returns true if the value represented by f is guaranteed to be the
// best approximation of d after being rounded to a float64 or
// float32 depending on flt.
func (f *extFloat) AssignDecimal(mantissa uint64, exp10 int, neg bool, trunc bool, flt *floatInfo) (ok bool) {
        const uint64digits = 19
        const errorscale = 8
        errors := 0 // An upper bound for error, computed in errorscale*ulp.
        if trunc {
                // the decimal number was truncated.
                errors += errorscale / 2
        }

        f.mant = mantissa
        f.exp = 0
        f.neg = neg

        // Multiply by powers of ten.
        i := (exp10 - firstPowerOfTen) / stepPowerOfTen
        if exp10 < firstPowerOfTen || i >= len(powersOfTen) {
                return false
        }
        adjExp := (exp10 - firstPowerOfTen) % stepPowerOfTen

        // We multiply by exp%step
        if adjExp < uint64digits && mantissa < uint64pow10[uint64digits-adjExp] {
                // We can multiply the mantissa exactly.
                f.mant *= uint64pow10[adjExp]
                f.Normalize()
        } else {
                f.Normalize()
                f.Multiply(smallPowersOfTen[adjExp])
                errors += errorscale / 2
        }

        // We multiply by 10 to the exp - exp%step.
        f.Multiply(powersOfTen[i])
        if errors > 0 {
                errors += 1
        }
        errors += errorscale / 2

        // Normalize
        shift := f.Normalize()
        errors <<= shift

        // Now f is a good approximation of the decimal.
        // Check whether the error is too large: that is, if the mantissa
        // is perturbated by the error, the resulting float64 will change.
        // The 64 bits mantissa is 1 + 52 bits for float64 + 11 extra bits.
        //
        // In many cases the approximation will be good enough.
        denormalExp := flt.bias - 63
        var extrabits uint
        if f.exp <= denormalExp {
                // f.mant * 2^f.exp is smaller than 2^(flt.bias+1).
                extrabits = uint(63 - flt.mantbits + 1 + uint(denormalExp-f.exp))
        } else {
                extrabits = uint(63 - flt.mantbits)
        }

        halfway := uint64(1) << (extrabits - 1)
        mant_extra := f.mant & (1<<extrabits - 1)

        // Do a signed comparison here! If the error estimate could make
        // the mantissa round differently for the conversion to double,
        // then we can't give a definite answer.
        if int64(halfway)-int64(errors) < int64(mant_extra) &&
                int64(mant_extra) < int64(halfway)+int64(errors) {
                return false
        }
        return true
}

// Frexp10 is an analogue of math.Frexp for decimal powers. It scales
// f by an approximate power of ten 10^-exp, and returns exp10, so
// that f*10^exp10 has the same value as the old f, up to an ulp,
// as well as the index of 10^-exp in the powersOfTen table.
func (f *extFloat) frexp10() (exp10, index int) {
        // The constants expMin and expMax constrain the final value of the
        // binary exponent of f. We want a small integral part in the result
        // because finding digits of an integer requires divisions, whereas
        // digits of the fractional part can be found by repeatedly multiplying
        // by 10.
        const expMin = -60
        const expMax = -32
        // Find power of ten such that x * 10^n has a binary exponent
        // between expMin and expMax.
        approxExp10 := ((expMin+expMax)/2 - f.exp) * 28 / 93 // log(10)/log(2) is close to 93/28.
        i := (approxExp10 - firstPowerOfTen) / stepPowerOfTen
Loop:
        for {
                exp := f.exp + powersOfTen[i].exp + 64
                switch {
                case exp < expMin:
                        i++
                case exp > expMax:
                        i--
                default:
                        break Loop
                }
        }
        // Apply the desired decimal shift on f. It will have exponent
        // in the desired range. This is multiplication by 10^-exp10.
        f.Multiply(powersOfTen[i])

        return -(firstPowerOfTen + i*stepPowerOfTen), i
}

// frexp10Many applies a common shift by a power of ten to a, b, c.
func frexp10Many(a, b, c *extFloat) (exp10 int) {
        exp10, i := c.frexp10()
        a.Multiply(powersOfTen[i])
        b.Multiply(powersOfTen[i])
        return
}

// FixedDecimal stores in d the first n significant digits
// of the decimal representation of f. It returns false
// if it cannot be sure of the answer.
func (f *extFloat) FixedDecimal(d *decimalSlice, n int) bool {
        if f.mant == 0 {
                d.nd = 0
                d.dp = 0
                d.neg = f.neg
                return true
        }
        if n == 0 {
                panic("strconv: internal error: extFloat.FixedDecimal called with n == 0")
        }
        // Multiply by an appropriate power of ten to have a reasonable
        // number to process.
        f.Normalize()
        exp10, _ := f.frexp10()

        shift := uint(-f.exp)
        integer := uint32(f.mant >> shift)
        fraction := f.mant - (uint64(integer) << shift)
        ε := uint64(1) // ε is the uncertainty we have on the mantissa of f.

        // Write exactly n digits to d.
        needed := n        // how many digits are left to write.
        integerDigits := 0 // the number of decimal digits of integer.
        pow10 := uint64(1) // the power of ten by which f was scaled.
        for i, pow := 0, uint64(1); i < 20; i++ {
                if pow > uint64(integer) {
                        integerDigits = i
                        break
                }
                pow *= 10
        }
        rest := integer
        if integerDigits > needed {
                // the integral part is already large, trim the last digits.
                pow10 = uint64pow10[integerDigits-needed]
                integer /= uint32(pow10)
                rest -= integer * uint32(pow10)
        } else {
                rest = 0
        }

        // Write the digits of integer: the digits of rest are omitted.
        var buf [32]byte
        pos := len(buf)
        for v := integer; v > 0; {
                v1 := v / 10
                v -= 10 * v1
                pos--
                buf[pos] = byte(v + '0')
                v = v1
        }
        for i := pos; i < len(buf); i++ {
                d.d[i-pos] = buf[i]
        }
        nd := len(buf) - pos
        d.nd = nd
        d.dp = integerDigits + exp10
        needed -= nd

        if needed > 0 {
                if rest != 0 || pow10 != 1 {
                        panic("strconv: internal error, rest != 0 but needed > 0")
                }
                // Emit digits for the fractional part. Each time, 10*fraction
                // fits in a uint64 without overflow.
                for needed > 0 {
                        fraction *= 10
                        ε *= 10 // the uncertainty scales as we multiply by ten.
                        if 2*ε > 1<<shift {
                                // the error is so large it could modify which digit to write, abort.
                                return false
                        }
                        digit := fraction >> shift
                        d.d[nd] = byte(digit + '0')
                        fraction -= digit << shift
                        nd++
                        needed--
                }
                d.nd = nd
        }

        // We have written a truncation of f (a numerator / 10^d.dp). The remaining part
        // can be interpreted as a small number (< 1) to be added to the last digit of the
        // numerator.
        //
        // If rest > 0, the amount is:
        //    (rest<<shift | fraction) / (pow10 << shift)
        //    fraction being known with a ±ε uncertainty.
        //    The fact that n > 0 guarantees that pow10 << shift does not overflow a uint64.
        //
        // If rest = 0, pow10 == 1 and the amount is
        //    fraction / (1 << shift)
        //    fraction being known with a ±ε uncertainty.
        //
        // We pass this information to the rounding routine for adjustment.

        ok := adjustLastDigitFixed(d, uint64(rest)<<shift|fraction, pow10, shift, ε)
        if !ok {
                return false
        }
        // Trim trailing zeros.
        for i := d.nd - 1; i >= 0; i-- {
                if d.d[i] != '0' {
                        d.nd = i + 1
                        break
                }
        }
        return true
}

// adjustLastDigitFixed assumes d contains the representation of the integral part
// of some number, whose fractional part is num / (den << shift). The numerator
// num is only known up to an uncertainty of size ε, assumed to be less than
// (den << shift)/2.
//
// It will increase the last digit by one to account for correct rounding, typically
// when the fractional part is greater than 1/2, and will return false if ε is such
// that no correct answer can be given.
func adjustLastDigitFixed(d *decimalSlice, num, den uint64, shift uint, ε uint64) bool {
        if num > den<<shift {
                panic("strconv: num > den<<shift in adjustLastDigitFixed")
        }
        if 2*ε > den<<shift {
                panic("strconv: ε > (den<<shift)/2")
        }
        if 2*(num+ε) < den<<shift {
                return true
        }
        if 2*(num-ε) > den<<shift {
                // increment d by 1.
                i := d.nd - 1
                for ; i >= 0; i-- {
                        if d.d[i] == '9' {
                                d.nd--
                        } else {
                                break
                        }
                }
                if i < 0 {
                        d.d[0] = '1'
                        d.nd = 1
                        d.dp++
                } else {
                        d.d[i]++
                }
                return true
        }
        return false
}

// ShortestDecimal stores in d the shortest decimal representation of f
// which belongs to the open interval (lower, upper), where f is supposed
// to lie. It returns false whenever the result is unsure. The implementation
// uses the Grisu3 algorithm.
func (f *extFloat) ShortestDecimal(d *decimalSlice, lower, upper *extFloat) bool {
        if f.mant == 0 {
                d.nd = 0
                d.dp = 0
                d.neg = f.neg
                return true
        }
        if f.exp == 0 && *lower == *f && *lower == *upper {
                // an exact integer.
                var buf [24]byte
                n := len(buf) - 1
                for v := f.mant; v > 0; {
                        v1 := v / 10
                        v -= 10 * v1
                        buf[n] = byte(v + '0')
                        n--
                        v = v1
                }
                nd := len(buf) - n - 1
                for i := 0; i < nd; i++ {
                        d.d[i] = buf[n+1+i]
                }
                d.nd, d.dp = nd, nd
                for d.nd > 0 && d.d[d.nd-1] == '0' {
                        d.nd--
                }
                if d.nd == 0 {
                        d.dp = 0
                }
                d.neg = f.neg
                return true
        }
        upper.Normalize()
        // Uniformize exponents.
        if f.exp > upper.exp {
                f.mant <<= uint(f.exp - upper.exp)
                f.exp = upper.exp
        }
        if lower.exp > upper.exp {
                lower.mant <<= uint(lower.exp - upper.exp)
                lower.exp = upper.exp
        }

        exp10 := frexp10Many(lower, f, upper)
        // Take a safety margin due to rounding in frexp10Many, but we lose precision.
        upper.mant++
        lower.mant--

        // The shortest representation of f is either rounded up or down, but
        // in any case, it is a truncation of upper.
        shift := uint(-upper.exp)
        integer := uint32(upper.mant >> shift)
        fraction := upper.mant - (uint64(integer) << shift)

        // How far we can go down from upper until the result is wrong.
        allowance := upper.mant - lower.mant
        // How far we should go to get a very precise result.
        targetDiff := upper.mant - f.mant

        // Count integral digits: there are at most 10.
        var integerDigits int
        for i, pow := 0, uint64(1); i < 20; i++ {
                if pow > uint64(integer) {
                        integerDigits = i
                        break
                }
                pow *= 10
        }
        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)
                }
        }
}

// 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 *decimalSlice, 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
}