A000329

BigDecimalInterval.cs

using System;
using System.Collections.Generic;
using System.Numerics;
using System.Text;

namespace Sandbox
{
    public class Rational
    {
        public Rational(BigInteger num, BigInteger denom)
        {
            if (denom.IsZero) throw new ArgumentOutOfRangeException("denom");

            if (denom < 0) { num = -num; denom = -denom; }

            var g = _Gcd(num, denom);
            Num = num / g;
            Denom = denom / g;
        }

        public BigInteger Num { get; private set; }
        public BigInteger Denom { get; private set; }

        public IEnumerable<Rational> Approximations
        {
            get
            {
                var continuedFrac = new List<BigInteger>();
                var sgn = Num < 0 ? -1 : 1;
                var n = sgn * Num;
                var d = Denom;
                while (d > 0)
                {
                    continuedFrac.Add(n / d);
                    // n/d := 1 / (n/d % 1) = d / (n % d)
                    var tmp = d;
                    d = n % d;
                    n = tmp;

                    // TODO This can be improved
                    Rational approx = continuedFrac[continuedFrac.Count - 1];
                    for (int i = continuedFrac.Count - 2; i >= 0; i--) approx = continuedFrac[i] + 1 / approx;
                    yield return sgn * approx;
                }
            }
        }

        public Interval Bound(BigInteger targetDenominator)
        {
            var approximations = Approximations.GetEnumerator();
            if (!approximations.MoveNext()) return this;

            var a = approximations.Current;
            Rational b = a;
            while (approximations.MoveNext())
            {
                a = b;
                b = approximations.Current;
                if (b.Denom > targetDenominator)
                {
                    break;
                }
            }

            return a < b ? new Interval(a, b) : new Interval(b, a);
        }


        public override string ToString() { return string.Format(Denom == 1 ? "{0}" : "{0}/{1}", Num, Denom); }
        public override int GetHashCode() { return Num.GetHashCode() * 37 + Denom.GetHashCode(); }
        public override bool Equals(object obj) { return obj is Rational && (this == (Rational)obj); }

        public static implicit operator Rational(int i) { return new Rational(i, 1); }
        public static implicit operator Rational(long i) { return new Rational(i, 1); }
        public static implicit operator Rational(BigInteger i) { return new Rational(i, 1); }

        public static implicit operator double(Rational r) { return (double)r.Num / (double)r.Denom; }

        public static Rational operator -(Rational a) { return new Rational(-a.Num, a.Denom); }

        public static Rational operator +(Rational a, Rational b) { return new Rational(a.Num * b.Denom + a.Denom * b.Num, a.Denom * b.Denom)$
        public static Rational operator -(Rational a, Rational b) { return new Rational(a.Num * b.Denom - a.Denom * b.Num, a.Denom * b.Denom)$
        public static Rational operator *(Rational a, Rational b) { return new Rational(a.Num * b.Num, a.Denom * b.Denom); }
        public static Rational operator /(Rational a, Rational b) { return new Rational(a.Num * b.Denom, a.Denom * b.Num); }

        public static bool operator <(Rational a, Rational b) { return a.Num * b.Denom < a.Denom * b.Num; }
        public static bool operator >(Rational a, Rational b) { return a.Num * b.Denom > a.Denom * b.Num; }
        public static bool operator <=(Rational a, Rational b) { return a.Num * b.Denom <= a.Denom * b.Num; }
        public static bool operator >=(Rational a, Rational b) { return a.Num * b.Denom >= a.Denom * b.Num; }
        public static bool operator ==(Rational a, Rational b) { return a.Num == b.Num && a.Denom == b.Denom; }
        public static bool operator !=(Rational a, Rational b) { return !(a == b); }

        private static BigInteger _Gcd(BigInteger a, BigInteger b)
        {
            if (a < 0) a = -a;
            if (b < 0) b = -b;
            while (!a.IsZero)
            {
                var tmp = b % a;
                b = a;
                a = tmp;
            }
            return b;
        }
    }

    // https://arxiv.org/pdf/0708.3721.pdf
    class BigDecimalInterval
    {
        static readonly BigInteger Precision = BigInteger.Pow(2, 256);
        static readonly Rational Epsilon = new Rational(1, Precision);

        public BigDecimalInterval(Rational r) : this(r, Precision)
        {
        }

        private BigDecimalInterval(Rational r, BigInteger precision)
        {
            _Precision = precision;
            if (r.Denom <= precision)
            {
                Lower = Upper = r;
                return;
            }

            // TODO Should probably prefer continuants over treating the precision as a desired denominator
            r = r * precision;
            if (r.Denom == 1)
            {
                Lower = Upper = r;
            }
            else if (r.Num > 0)
            {
                var floor = r.Num / r.Denom;
                Lower = new Rational(floor, precision);
                Upper = new Rational(floor + 1, precision);
            }
            else
            {
                var ceil = -((-r.Num) / r.Denom);
                Lower = new Rational(ceil - 1, precision);
                Upper = new Rational(ceil, precision);
            }
        }

        public BigDecimalInterval(Rational lower, Rational upper) : this(lower, upper, Precision)
        {
        }

        private BigDecimalInterval(Rational lower, Rational upper, BigInteger precision)
        {
            if (lower > upper) throw new ArgumentOutOfRangeException();

            _Precision = precision;
            Lower = new BigDecimalInterval(lower, precision).Lower;
            Upper = new BigDecimalInterval(upper, precision).Upper;
        }

        private BigInteger _Precision { get; set; }
        public Rational Lower { get; private set; }
        public Rational Upper { get; private set; }

        public override string ToString() => string.Format("[{0} ~= {1}, {2} ~= {3} width {4}]", Lower, (double)Lower, Upper, (double)Upper, (double)(Upper - Lower));

        #region Operators

        public static implicit operator BigDecimalInterval(int r) => new BigDecimalInterval(r);
        public static implicit operator BigDecimalInterval(long r) => new BigDecimalInterval(r);
        public static implicit operator BigDecimalInterval(BigInteger r) => new BigDecimalInterval(r);
        public static implicit operator BigDecimalInterval(Rational r) => new BigDecimalInterval(r);

        public static BigDecimalInterval operator -(BigDecimalInterval a) => new BigDecimalInterval(-a.Upper, -a.Lower, a._Precision);

        public static BigDecimalInterval operator +(BigDecimalInterval a, BigDecimalInterval b) => new BigDecimalInterval(a.Lower + b.Lower, a.Upper + b.Upper, Max(a._Precision, b._Precision));
        public static BigDecimalInterval operator -(BigDecimalInterval a, BigDecimalInterval b) => new BigDecimalInterval(a.Lower - b.Upper, a.Upper - b.Lower, Max(a._Precision, b._Precision));
        // Assumes a > 0
        public static BigDecimalInterval operator *(int a, BigDecimalInterval b) => new BigDecimalInterval(a * b.Lower, a * b.Upper, b._Precision);
        public static BigDecimalInterval operator /(BigDecimalInterval a, int b) => new BigDecimalInterval(a.Lower / b, a.Upper / b, a._Precision);

        #endregion

        private static BigInteger Max(BigInteger a, BigInteger b) => a > b ? a : b;

        public static BigDecimalInterval Atan(Rational x)
        {
            // atan(x) = \sum_{i=0}^\infty (-1)^i x^{2i+1} / (2i+1)
            Rational lower = x;
            Rational upper = x;

            var intermediatePrecision = Precision * Precision;
            var epsilon = new Rational(1, intermediatePrecision);

            BigDecimalInterval sum = 0;
            BigDecimalInterval term = new BigDecimalInterval(x, intermediatePrecision);
            var x2 = x * x;
            for (int i = 0; true; i++)
            {
                sum += term / (2 * i + 1);
                term = new BigDecimalInterval(term.Upper * -x2, term.Lower * -x2, intermediatePrecision);

                if (i % 2 == 0) upper = sum.Upper;
                else lower = sum.Lower;

                if (term.Lower.Abs() <= epsilon && term.Upper.Abs() <= epsilon) break;
            }

            return new BigDecimalInterval(lower, upper, Precision);
        }

        private static readonly BigDecimalInterval Pi = 16 * Atan(new Rational(1, 5)) - 4 * Atan(new Rational(1, 239));

        private static BigDecimalInterval Sin(Rational x)
        {
            if (x < 0) return -Sin(-x);

            // sin(x) = sum_{i=0}^\infty (-1)^{i} x^{2i+1} / (2i+1)!
            Rational lower = x;
            Rational upper = x;

            var intermediatePrecision = Precision * Precision;
            var epsilon = new Rational(1, intermediatePrecision);

            BigDecimalInterval term = new BigDecimalInterval(x, intermediatePrecision);
            BigDecimalInterval sum = term;
            var x2 = x * x;
            for (int i = 1; true; i++)
            {
                var m = x2 / -((2 * i) * (2 * i + 1));
                term = new BigDecimalInterval(term.Upper * m, term.Lower * m, intermediatePrecision);
                sum += term;

                if (i % 2 == 0) upper = sum.Upper;
                else lower = sum.Lower;

                if (term.Lower.Abs() <= epsilon && term.Upper.Abs() <= epsilon) break;
            }

            return new BigDecimalInterval(lower, upper, Precision);
        }

        private static BigDecimalInterval Cos(Rational x)
        {
            if (x < 0) x = -x;

            // cos(x) = sum_{i=0}^\infty (-1)^i x^{2i} / (2i)!
            Rational lower = 1;
            Rational upper = 1;

            var intermediatePrecision = Precision * Precision;
            var epsilon = new Rational(1, intermediatePrecision);

            BigDecimalInterval term = new BigDecimalInterval(1, intermediatePrecision);
            BigDecimalInterval sum = term;
            var x2 = x * x;
            for (int i = 1; true; i++)
            {
                var m = x2 / -((2 * i - 1) * (2 * i));
                term = new BigDecimalInterval(term.Upper * m, term.Lower * m, intermediatePrecision);
                sum += term;

                if (i % 2 == 0) upper = sum.Upper;
                else lower = sum.Lower;

                if (term.Lower.Abs() <= epsilon && term.Upper.Abs() <= epsilon) break;
            }

            return new BigDecimalInterval(lower, upper, Precision);
        }

        public static BigDecimalInterval Tan(BigDecimalInterval x)
        {
            if (x.Upper < 0) return -Tan(-x);

            if (x.Lower >= Pi.Lower / -2 && x.Upper <= Pi.Lower / 2)
            {
                var lb = Sin(x.Lower).Lower / Cos(x.Lower).Upper;
                var ub = Sin(x.Upper).Upper / Cos(x.Upper).Lower;
                return new BigDecimalInterval(lb, ub);
            }

            throw new ArgumentOutOfRangeException(nameof(x));
        }

        internal static void Main()
        {
            BigDecimalInterval x = 1;
            for (int i = 1; i < 1001; i++)
            {
                // Tan is fine for -pi/2 to pi/2.
                while (x.Upper > Pi.Lower / 2) x -= Pi;
                while (x.Lower < Pi.Lower / -2) x += Pi;

                x = Tan(x);

                double lb = x.Lower;
                double ub = x.Upper;
                double avg = (lb + ub) / 2;
                if (lb == ub) Console.WriteLine("{0}: {1}", i, lb);
                else
                {
                    var sl = lb.ToString("F16");
                    var su = ub.ToString("F16");
                    var sa = avg.ToString("F16");
                    var sb = new StringBuilder();
                    for (int j = 0; j < sa.Length; j++)
                    {
                        if (j < sl.Length && j < su.Length && sl[j] == sa[j] && su[j] == sa[j]) sb.Append(sa[j]);
                        else { sb.Append('(').Append(sa[j]).Append(')'); break; }
                    }

                    Console.WriteLine("{0}: {1}", i, sb);
                }
            }
        }
    }
}

Processed results.txt

1: 1.5574077246549
2: 74.685933398765
3: -0.863518854878059
4: -1.1698563550598700
5: -2.3590377341888900
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988: -0.2327(5)
989: -0.2370(4)
990: -0.2415(9)
991: -0.246(4)
992: -0.2515(1)
993: -0.2569(5)
994: -0.2627(6)
995: -0.2689(8)
996: -0.2756(6)
997: -0.2828(6)
998: -0.2906(6)
999: -0.2991(3)
1000: -0.3083(8)