C# BigDecimal.Ulp方法代码示例

        public static BigDecimal Hypot(int n, BigDecimal x)
        {
            /* compute n^2+x^2 in infinite precision
                */
            BigDecimal z = (new BigDecimal(n)).Pow(2).Add(x.Pow(2));

            /* Truncate to the precision set by x. Absolute error = in z (square of the result) is |2*x*xerr|,
                * where the error is 1/2 of the ulp. Two intermediate protection digits.
                * zerr is a signed value, but used only in conjunction with err2prec(), so this feature does not harm.
                */
            double zerr = x.ToDouble()*x.Ulp().ToDouble();
            var mc = new MathContext(2 + ErrorToPrecision(z.ToDouble(), zerr));

            /* Pull square root */
            z = Sqrt(z.Round(mc));

            /* Final rounding. Absolute error in the square root is x*xerr/z, where zerr holds 2*x*xerr.
                */
            mc = new MathContext(ErrorToPrecision(z.ToDouble(), 0.5*zerr/z.ToDouble()));
            return z.Round(mc);
        }

        public static BigDecimal Log(BigDecimal x)
        {
            /* the value is undefined if x is negative.
                */
            if (x.CompareTo(BigDecimal.Zero) < 0)
                throw new ArithmeticException("Cannot take log of negative " + x);
            if (x.CompareTo(BigDecimal.One) == 0) {
                /* log 1. = 0. */
                return ScalePrecision(BigDecimal.Zero, x.Precision - 1);
            }
            if (System.Math.Abs(x.ToDouble() - 1.0) <= 0.3) {
                /* The standard Taylor series around x=1, z=0, z=x-1. Abramowitz-Stegun 4.124.
                        * The absolute error is err(z)/(1+z) = err(x)/x.
                        */
                BigDecimal z = ScalePrecision(x.Subtract(BigDecimal.One), 2);
                BigDecimal zpown = z;
                double eps = 0.5*x.Ulp().ToDouble()/System.Math.Abs(x.ToDouble());
                BigDecimal resul = z;
                for (int k = 2;; k++) {
                    zpown = MultiplyRound(zpown, z);
                    BigDecimal c = DivideRound(zpown, k);
                    if (k%2 == 0)
                        resul = resul.Subtract(c);
                    else
                        resul = resul.Add(c);
                    if (System.Math.Abs(c.ToDouble()) < eps)
                        break;
                }
                var mc = new MathContext(ErrorToPrecision(resul.ToDouble(), eps));
                return resul.Round(mc);
            } else {
                double xDbl = x.ToDouble();
                double xUlpDbl = x.Ulp().ToDouble();

                /* Map log(x) = log root[r](x)^r = r*log( root[r](x)) with the aim
                        * to move roor[r](x) near to 1.2 (that is, below the 0.3 appearing above), where log(1.2) is roughly 0.2.
                        */
                var r = (int) (System.Math.Log(xDbl)/0.2);

                /* Since the actual requirement is a function of the value 0.3 appearing above,
                        * we avoid the hypothetical case of endless recurrence by ensuring that r >= 2.
                        */
                r = System.Math.Max(2, r);

                /* Compute r-th root with 2 additional digits of precision
                        */
                BigDecimal xhighpr = ScalePrecision(x, 2);
                BigDecimal resul = Root(r, xhighpr);
                resul = Log(resul).Multiply(new BigDecimal(r));

                /* error propagation: log(x+errx) = log(x)+errx/x, so the absolute error
                        * in the result equals the relative error in the input, xUlpDbl/xDbl .
                        */
                var mc = new MathContext(ErrorToPrecision(resul.ToDouble(), xUlpDbl/xDbl));
                return resul.Round(mc);
            }
        }

        public static BigDecimal Exp(BigDecimal x)
        {
            /* To calculate the value if x is negative, use exp(-x) = 1/exp(x)
                */
            if (x.CompareTo(BigDecimal.Zero) < 0) {
                BigDecimal invx = Exp(x.Negate());
                /* Relative error in inverse of invx is the same as the relative errror in invx.
                        * This is used to define the precision of the result.
                        */
                var mc = new MathContext(invx.Precision);
                return BigDecimal.One.Divide(invx, mc);
            }
            if (x.CompareTo(BigDecimal.Zero) == 0) {
                /* recover the valid number of digits from x.ulp(), if x hits the
                        * zero. The x.precision() is 1 then, and does not provide this information.
                        */
                return ScalePrecision(BigDecimal.One, -(int) (System.Math.Log10(x.Ulp().ToDouble())));
            }
            /* Push the number in the Taylor expansion down to a small
                        * value where TAYLOR_NTERM terms will do. If x<1, the n-th term is of the order
                        * x^n/n!, and equal to both the absolute and relative error of the result
                        * since the result is close to 1. The x.ulp() sets the relative and absolute error
                        * of the result, as estimated from the first Taylor term.
                        * We want x^TAYLOR_NTERM/TAYLOR_NTERM! < x.ulp, which is guaranteed if
                        * x^TAYLOR_NTERM < TAYLOR_NTERM*(TAYLOR_NTERM-1)*...*x.ulp.
                        */
            double xDbl = x.ToDouble();
            double xUlpDbl = x.Ulp().ToDouble();
            if (System.Math.Pow(xDbl, TaylorNterm) < TaylorNterm*(TaylorNterm - 1.0)*(TaylorNterm - 2.0)*xUlpDbl) {
                /* Add TAYLOR_NTERM terms of the Taylor expansion (Euler's sum formula)
                                */
                BigDecimal resul = BigDecimal.One;

                /* x^i */
                BigDecimal xpowi = BigDecimal.One;

                /* i factorial */
                BigInteger ifac = BigInteger.One;

                /* TAYLOR_NTERM terms to be added means we move x.ulp() to the right
                                * for each power of 10 in TAYLOR_NTERM, so the addition won't add noise beyond
                                * what's already in x.
                                */
                var mcTay = new MathContext(ErrorToPrecision(1d, xUlpDbl/TaylorNterm));
                for (int i = 1; i <= TaylorNterm; i++) {
                    ifac = ifac.Multiply(BigInteger.ValueOf(i));
                    xpowi = xpowi.Multiply(x);
                    BigDecimal c = xpowi.Divide(new BigDecimal(ifac), mcTay);
                    resul = resul.Add(c);
                    if (System.Math.Abs(xpowi.ToDouble()) < i &&
                        System.Math.Abs(c.ToDouble()) < 0.5*xUlpDbl)
                        break;
                }
                /* exp(x+deltax) = exp(x)(1+deltax) if deltax is <<1. So the relative error
                                * in the result equals the absolute error in the argument.
                                */
                var mc = new MathContext(ErrorToPrecision(xUlpDbl/2d));
                return resul.Round(mc);
            } else {
                /* Compute exp(x) = (exp(0.1*x))^10. Division by 10 does not lead
                                * to loss of accuracy.
                                */
                var exSc = (int) (1.0 - System.Math.Log10(TaylorNterm*(TaylorNterm - 1.0)*(TaylorNterm - 2.0)*xUlpDbl
                                                          /System.Math.Pow(xDbl, TaylorNterm))/(TaylorNterm - 1.0));
                BigDecimal xby10 = x.ScaleByPowerOfTen(-exSc);
                BigDecimal expxby10 = Exp(xby10);

                /* Final powering by 10 means that the relative error of the result
                                * is 10 times the relative error of the base (First order binomial expansion).
                                * This looses one digit.
                                */
                var mc = new MathContext(expxby10.Precision - exSc);
                /* Rescaling the powers of 10 is done in chunks of a maximum of 8 to avoid an invalid operation
                                * response by the BigDecimal.pow library or integer overflow.
                                */
                while (exSc > 0) {
                    int exsub = System.Math.Min(8, exSc);
                    exSc -= exsub;
                    var mctmp = new MathContext(expxby10.Precision - exsub + 2);
                    int pex = 1;
                    while (exsub-- > 0)
                        pex *= 10;
                    expxby10 = expxby10.Pow(pex, mctmp);
                }
                return expxby10.Round(mc);
            }
        }

        public static BigDecimal Cos(BigDecimal x)
        {
            if (x.CompareTo(BigDecimal.Zero) < 0)
                return Cos(x.Negate());
            if (x.CompareTo(BigDecimal.Zero) == 0)
                return BigDecimal.One;
            /* reduce modulo 2pi
                        */
            BigDecimal res = Mod2Pi(x);
            double errpi = 0.5*System.Math.Abs(x.Ulp().ToDouble());
            var mc = new MathContext(2 + ErrorToPrecision(3.14159, errpi));
            BigDecimal p = PiRound(mc);
            mc = new MathContext(x.Precision);
            if (res.CompareTo(p) > 0) {
                /* pi<x<=2pi: cos(x)= - cos(x-pi)
                                */
                return Cos(SubtractRound(res, p)).Negate();
            }
            if (res.Multiply(BigDecimal.ValueOf(2)).CompareTo(p) > 0) {
                /* pi/2<x<=pi: cos(x)= -cos(pi-x)
                                */
                return Cos(SubtractRound(p, res)).Negate();
            }
            /* for the range 0<=x<Pi/2 one could use cos(2x)= 1-2*sin^2(x)
                                * to split this further, or use the cos up to pi/4 and the sine higher up.
                                        throw new ProviderException("Not implemented: cosine ") ;
                                */
            if (res.Multiply(BigDecimal.ValueOf(4)).CompareTo(p) > 0) {
                /* x>pi/4: cos(x) = sin(pi/2-x)
                                        */
                return Sin(SubtractRound(p.Divide(BigDecimal.ValueOf(2)), res));
            }
            /* Simple Taylor expansion, sum_{i=0..infinity} (-1)^(..)res^(2i)/(2i)! */
            BigDecimal resul = BigDecimal.One;

            /* x^i */
            BigDecimal xpowi = BigDecimal.One;

            /* 2i factorial */
            BigInteger ifac = BigInteger.One;

            /* The absolute error in the result is the error in x^2/2 which is x times the error in x.
                                        */
            double xUlpDbl = 0.5*res.Ulp().ToDouble()*res.ToDouble();

            /* The error in the result is set by the error in x^2/2 itself, xUlpDbl.
                                        * We need at most k terms to push x^(2k+1)/(2k+1)! below this value.
                                        * x^(2k) < xUlpDbl; (2k)*log(x) < log(xUlpDbl);
                                        */
            int k = (int) (System.Math.Log(xUlpDbl)/System.Math.Log(res.ToDouble()))/2;
            var mcTay = new MathContext(ErrorToPrecision(1d, xUlpDbl/k));
            for (int i = 1;; i++) {
                /* TBD: at which precision will 2*i-1 or 2*i overflow?
                                                */
                ifac = ifac.Multiply(BigInteger.ValueOf((2*i - 1)));
                ifac = ifac.Multiply(BigInteger.ValueOf((2*i)));
                xpowi = xpowi.Multiply(res).Multiply(res).Negate();
                BigDecimal corr = xpowi.Divide(new BigDecimal(ifac), mcTay);
                resul = resul.Add(corr);
                if (corr.Abs().ToDouble() < 0.5*xUlpDbl)
                    break;
            }
            /* The error in the result is governed by the error in x itself.
                                        */
            mc = new MathContext(ErrorToPrecision(resul.ToDouble(), xUlpDbl));
            return resul.Round(mc);
        }

        public static BigDecimal Cot(BigDecimal x)
        {
            if (x.CompareTo(BigDecimal.Zero) == 0) {
                throw new ArithmeticException("Cannot take cot of zero " + x);
            }
            if (x.CompareTo(BigDecimal.Zero) < 0) {
                return Cot(x.Negate()).Negate();
            }
            /* reduce modulo pi
                        */
            BigDecimal res = ModPi(x);

            /* absolute error in the result is err(x)/sin^2(x) to lowest order
                        */
            double xDbl = res.ToDouble();
            double xUlpDbl = x.Ulp().ToDouble()/2d;
            double eps = xUlpDbl/2d/System.Math.Pow(System.Math.Sin(xDbl), 2d);

            BigDecimal xhighpr = ScalePrecision(res, 2);
            BigDecimal xhighprSq = MultiplyRound(xhighpr, xhighpr);

            var mc = new MathContext(ErrorToPrecision(xhighpr.ToDouble(), eps));
            BigDecimal resul = BigDecimal.One.Divide(xhighpr, mc);

            /* x^(2i-1) */
            BigDecimal xpowi = xhighpr;

            var b = new Bernoulli();

            /* 2^(2i) */
            var fourn = BigInteger.Parse("4");
            /* (2i)! */
            BigInteger fac = BigInteger.One;

            for (int i = 1;; i++) {
                Rational f = b[2*i];
                fac = fac.Multiply(BigInteger.ValueOf((2*i))).Multiply(BigInteger.ValueOf((2*i - 1)));
                f = f.Multiply(fourn).Divide(fac);
                BigDecimal c = MultiplyRound(xpowi, f);
                if (i%2 == 0)
                    resul = resul.Add(c);
                else
                    resul = resul.Subtract(c);
                if (System.Math.Abs(c.ToDouble()) < 0.1*eps)
                    break;

                fourn = fourn.ShiftLeft(2);
                xpowi = MultiplyRound(xpowi, xhighprSq);
            }
            mc = new MathContext(ErrorToPrecision(resul.ToDouble(), eps));
            return resul.Round(mc);
        }

        public static BigDecimal Tan(BigDecimal x)
        {
            if (x.CompareTo(BigDecimal.Zero) == 0)
                return BigDecimal.Zero;
            if (x.CompareTo(BigDecimal.Zero) < 0) {
                return Tan(x.Negate()).Negate();
            }
            /* reduce modulo pi
                        */
            BigDecimal res = ModPi(x);

            /* absolute error in the result is err(x)/cos^2(x) to lowest order
                        */
            double xDbl = res.ToDouble();
            double xUlpDbl = x.Ulp().ToDouble()/2d;
            double eps = xUlpDbl/2d/System.Math.Pow(System.Math.Cos(xDbl), 2d);

            if (xDbl > 0.8) {
                /* tan(x) = 1/cot(x) */
                BigDecimal co = Cot(x);
                var mc = new MathContext(ErrorToPrecision(1d/co.ToDouble(), eps));
                return BigDecimal.One.Divide(co, mc);
            } else {
                BigDecimal xhighpr = ScalePrecision(res, 2);
                BigDecimal xhighprSq = MultiplyRound(xhighpr, xhighpr);

                BigDecimal resul = xhighpr.Plus();

                /* x^(2i+1) */
                BigDecimal xpowi = xhighpr;

                var b = new Bernoulli();

                /* 2^(2i) */
                BigInteger fourn = BigInteger.ValueOf(4);
                /* (2i)! */
                BigInteger fac = BigInteger.ValueOf(2);

                for (int i = 2;; i++) {
                    Rational f = b[2*i].Abs();
                    fourn = fourn.ShiftLeft(2);
                    fac = fac.Multiply(BigInteger.ValueOf((2*i))).Multiply(BigInteger.ValueOf((2*i - 1)));
                    f = f.Multiply(fourn).Multiply(fourn.Subtract(BigInteger.One)).Divide(fac);
                    xpowi = MultiplyRound(xpowi, xhighprSq);
                    BigDecimal c = MultiplyRound(xpowi, f);
                    resul = resul.Add(c);
                    if (System.Math.Abs(c.ToDouble()) < 0.1*eps)
                        break;
                }
                var mc = new MathContext(ErrorToPrecision(resul.ToDouble(), eps));
                return resul.Round(mc);
            }
        }

 public void UlpZero()
 {
     String a = "0";
     int aScale = 2;
     BigDecimal aNumber = new BigDecimal(BigInteger.Parse(a), aScale);
     BigDecimal result = aNumber.Ulp();
     String res = "0.01";
     int resScale = 2;
     Assert.AreEqual(res, result.ToString(), "incorrect value");
     Assert.AreEqual(resScale, result.Scale, "incorrect scale");
 }

        public static BigDecimal Sin(BigDecimal x)
        {
            if (x.CompareTo(BigDecimal.Zero) < 0)
                return Sin(x.Negate()).Negate();
            if (x.CompareTo(BigDecimal.Zero) == 0)
                return BigDecimal.Zero;
            /* reduce modulo 2pi
                        */
            BigDecimal res = Mod2Pi(x);
            double errpi = 0.5*System.Math.Abs(x.Ulp().ToDouble());
            var mc = new MathContext(2 + ErrorToPrecision(3.14159, errpi));
            BigDecimal p = PiRound(mc);
            mc = new MathContext(x.Precision);
            if (res.CompareTo(p) > 0) {
                /* pi<x<=2pi: sin(x)= - sin(x-pi)
                                */
                return Sin(SubtractRound(res, p)).Negate();
            }
            if (res.Multiply(BigDecimal.ValueOf(2)).CompareTo(p) > 0) {
                /* pi/2<x<=pi: sin(x)= sin(pi-x)
                                */
                return Sin(SubtractRound(p, res));
            }
            /* for the range 0<=x<Pi/2 one could use sin(2x)=2sin(x)cos(x)
                                * to split this further. Here, use the sine up to pi/4 and the cosine higher up.
                                */
            if (res.Multiply(BigDecimal.ValueOf(4)).CompareTo(p) > 0) {
                /* x>pi/4: sin(x) = cos(pi/2-x)
                                        */
                return Cos(SubtractRound(p.Divide(BigDecimal.ValueOf(2)), res));
            }
            /* Simple Taylor expansion, sum_{i=1..infinity} (-1)^(..)res^(2i+1)/(2i+1)! */
            BigDecimal resul = res;

            /* x^i */
            BigDecimal xpowi = res;

            /* 2i+1 factorial */
            BigInteger ifac = BigInteger.One;

            /* The error in the result is set by the error in x itself.
                                        */
            double xUlpDbl = res.Ulp().ToDouble();

            /* The error in the result is set by the error in x itself.
                                        * We need at most k terms to squeeze x^(2k+1)/(2k+1)! below this value.
                                        * x^(2k+1) < x.ulp; (2k+1)*log10(x) < -x.precision; 2k*log10(x)< -x.precision;
                                        * 2k*(-log10(x)) > x.precision; 2k*log10(1/x) > x.precision
                                        */
            int k = (int) (res.Precision/System.Math.Log10(1.0/res.ToDouble()))/2;
            var mcTay = new MathContext(ErrorToPrecision(res.ToDouble(), xUlpDbl/k));
            for (int i = 1;; i++) {
                /* TBD: at which precision will 2*i or 2*i+1 overflow?
                                                */
                ifac = ifac.Multiply(BigInteger.ValueOf(2*i));
                ifac = ifac.Multiply(BigInteger.ValueOf((2*i + 1)));
                xpowi = xpowi.Multiply(res).Multiply(res).Negate();
                BigDecimal corr = xpowi.Divide(new BigDecimal(ifac), mcTay);
                resul = resul.Add(corr);
                if (corr.Abs().ToDouble() < 0.5*xUlpDbl)
                    break;
            }
            /* The error in the result is set by the error in x itself.
                                        */
            mc = new MathContext(res.Precision);
            return resul.Round(mc);
        }

 public static BigDecimal SubtractRound(BigDecimal x, BigDecimal y)
 {
     BigDecimal resul = x.Subtract(y);
     // The estimation of the absolute error in the result is |err(y)|+|err(x)|
     double errR = System.Math.Abs(y.Ulp().ToDouble()/2d) + System.Math.Abs(x.Ulp().ToDouble()/2d);
     var mc = new MathContext(ErrorToPrecision(resul.ToDouble(), errR));
     return resul.Round(mc);
 }

        public static BigDecimal Root(int n, BigDecimal x)
        {
            if (x.CompareTo(BigDecimal.Zero) < 0)
                throw new ArithmeticException("negative argument " + x + " of root");
            if (n <= 0)
                throw new ArithmeticException("negative power " + n + " of root");

            if (n == 1)
                return x;

            /* start the computation from a double precision estimate */
            var s = new BigDecimal(System.Math.Pow(x.ToDouble(), 1.0/n));

            /* this creates nth with nominal precision of 1 digit
                */
            var nth = new BigDecimal(n);

            /* Specify an internal accuracy within the loop which is
                * slightly larger than what is demanded by 'eps' below.
                */
            BigDecimal xhighpr = ScalePrecision(x, 2);
            var mc = new MathContext(2 + x.Precision);

            /* Relative accuracy of the result is eps.
                */
            double eps = x.Ulp().ToDouble()/(2*n*x.ToDouble());
            for (;;) {
                /* s = s -(s/n-x/n/s^(n-1)) = s-(s-x/s^(n-1))/n; test correction s/n-x/s for being
                        * smaller than the precision requested. The relative correction is (1-x/s^n)/n,
                        */
                BigDecimal c = xhighpr.Divide(s.Pow(n - 1), mc);
                c = s.Subtract(c);
                var locmc = new MathContext(c.Precision);
                c = c.Divide(nth, locmc);
                s = s.Subtract(c);
                if (System.Math.Abs(c.ToDouble()/s.ToDouble()) < eps)
                    break;
            }
            return s.Round(new MathContext(ErrorToPrecision(eps)));
        }

        public static BigDecimal PowRound(BigDecimal x, Rational q)
        {
            /** Special cases: x^1=x and x^0 = 1
                */
            if (q.CompareTo(BigInteger.One) == 0)
                return x;
            if (q.Sign == 0)
                return BigDecimal.One;
            if (q.IsInteger) {
                /* We are sure that the denominator is positive here, because normalize() has been
                        * called during constrution etc.
                        */
                return PowRound(x, q.Numerator);
            }
            /* Refuse to operate on the general negative basis. The integer q have already been handled above.
                        */
            if (x.CompareTo(BigDecimal.Zero) < 0)
                throw new ArithmeticException("Cannot power negative " + x);
            if (q.IsIntegerFraction) {
                /* Newton method with first estimate in double precision.
                                * The disadvantage of this first line here is that the result must fit in the
                                * standard range of double precision numbers exponents.
                                */
                double estim = System.Math.Pow(x.ToDouble(), q.ToDouble());
                var res = new BigDecimal(estim);

                /* The error in x^q is q*x^(q-1)*Delta(x).
                                * The relative error is q*Delta(x)/x, q times the relative error of x.
                                */
                var reserr = new BigDecimal(0.5*q.Abs().ToDouble()
                                            *x.Ulp().Divide(x.Abs(), MathContext.Decimal64).ToDouble());

                /* The main point in branching the cases above is that this conversion
                                * will succeed for numerator and denominator of q.
                                */
                int qa = q.Numerator.ToInt32();
                int qb = q.Denominator.ToInt32();

                /* Newton iterations. */
                BigDecimal xpowa = PowRound(x, qa);
                for (;;) {
                    /* numerator and denominator of the Newton term.  The major
                                        * disadvantage of this implementation is that the updates of the powers
                                        * of the new estimate are done in full precision calling BigDecimal.pow(),
                                        * which becomes slow if the denominator of q is large.
                                        */
                    BigDecimal nu = res.Pow(qb).Subtract(xpowa);
                    BigDecimal de = MultiplyRound(res.Pow(qb - 1), q.Denominator);

                    /* estimated correction */
                    BigDecimal eps = nu.Divide(de, MathContext.Decimal64);

                    BigDecimal err = res.Multiply(reserr, MathContext.Decimal64);
                    int precDiv = 2 + ErrorToPrecision(eps, err);
                    if (precDiv <= 0) {
                        /* The case when the precision is already reached and any precision
                                                * will do. */
                        eps = nu.Divide(de, MathContext.Decimal32);
                    } else {
                        eps = nu.Divide(de, new MathContext(precDiv));
                    }

                    res = SubtractRound(res, eps);
                    /* reached final precision if the relative error fell below reserr,
                                        * |eps/res| < reserr
                                        */
                    if (eps.Divide(res, MathContext.Decimal64).Abs().CompareTo(reserr) < 0) {
                        /* delete the bits of extra precision kept in this
                                                * working copy.
                                                */
                        return res.Round(new MathContext(ErrorToPrecision(reserr.ToDouble())));
                    }
                }
            }
            /* The error in x^q is q*x^(q-1)*Delta(x) + Delta(q)*x^q*log(x).
                                * The relative error is q/x*Delta(x) + Delta(q)*log(x). Convert q to a floating point
                                * number such that its relative error becomes negligible: Delta(q)/q << Delta(x)/x/log(x) .
                                */
            int precq = 3 + ErrorToPrecision((x.Ulp().Divide(x, MathContext.Decimal64)).ToDouble()
                                             /System.Math.Log(x.ToDouble()));

            /* Perform the actual calculation as exponentiation of two floating point numbers.
                                */
            return Pow(x, q.ToBigDecimal(new MathContext(precq)));
        }

        public static BigDecimal Pow(BigDecimal x, BigDecimal y)
        {
            if (x.CompareTo(BigDecimal.Zero) < 0)
                throw new ArithmeticException("Cannot power negative " + x);
            if (x.CompareTo(BigDecimal.Zero) == 0)
                return BigDecimal.Zero;
            /* return x^y = exp(y*log(x)) ;
                        */
            BigDecimal logx = Log(x);
            BigDecimal ylogx = y.Multiply(logx);
            BigDecimal resul = Exp(ylogx);

            /* The estimation of the relative error in the result is |log(x)*err(y)|+|y*err(x)/x|
                        */
            double errR = System.Math.Abs(logx.ToDouble())*y.Ulp().ToDouble()/2d
                          + System.Math.Abs(y.ToDouble()*x.Ulp().ToDouble()/2d/x.ToDouble());
            var mcR = new MathContext(ErrorToPrecision(1.0, errR));
            return resul.Round(mcR);
        }

        public static BigDecimal Mod2Pi(BigDecimal x)
        {
            /* write x= 2*pi*k+r with the precision in r defined by the precision of x and not
                * compromised by the precision of 2*pi, so the ulp of 2*pi*k should match the ulp of x.
                * First get a guess of k to figure out how many digits of 2*pi are needed.
                */
            var k = (int) (0.5*x.ToDouble()/System.Math.PI);

            /* want to have err(2*pi*k)< err(x)=0.5*x.ulp, so err(pi) = err(x)/(4k) with two safety digits
                */
            double err2pi;
            if (k != 0)
                err2pi = 0.25*System.Math.Abs(x.Ulp().ToDouble()/k);
            else
                err2pi = 0.5*System.Math.Abs(x.Ulp().ToDouble());
            var mc = new MathContext(2 + ErrorToPrecision(6.283, err2pi));
            BigDecimal twopi = PiRound(mc).Multiply(new BigDecimal(2));

            /* Delegate the actual operation to the BigDecimal class, which may return
                * a negative value of x was negative .
                */
            BigDecimal res = x.Remainder(twopi);
            if (res.CompareTo(BigDecimal.Zero) < 0)
                res = res.Add(twopi);

            /* The actual precision is set by the input value, its absolute value of x.ulp()/2.
                */
            mc = new MathContext(ErrorToPrecision(res.ToDouble(), x.Ulp().ToDouble()/2d));
            return res.Round(mc);
        }

        public static BigDecimal Asin(BigDecimal x)
        {
            if (x.CompareTo(BigDecimal.One) > 0 ||
                x.CompareTo(BigDecimal.One.Negate()) < 0) {
                throw new ArithmeticException("Out of range argument " + x + " of asin");
            }
            if (x.CompareTo(BigDecimal.Zero) == 0)
                return BigDecimal.Zero;
            if (x.CompareTo(BigDecimal.One) == 0) {
                /* arcsin(1) = pi/2
                        */
                double errpi = System.Math.Sqrt(x.Ulp().ToDouble());
                var mc = new MathContext(ErrorToPrecision(3.14159, errpi));
                return PiRound(mc).Divide(new BigDecimal(2));
            }
            if (x.CompareTo(BigDecimal.Zero) < 0) {
                return Asin(x.Negate()).Negate();
            }
            if (x.ToDouble() > 0.7) {
                BigDecimal xCompl = BigDecimal.One.Subtract(x);
                double xDbl = x.ToDouble();
                double xUlpDbl = x.Ulp().ToDouble()/2d;
                double eps = xUlpDbl/2d/System.Math.Sqrt(1d - System.Math.Pow(xDbl, 2d));

                BigDecimal xhighpr = ScalePrecision(xCompl, 3);
                BigDecimal xhighprV = DivideRound(xhighpr, 4);

                BigDecimal resul = BigDecimal.One;

                /* x^(2i+1) */
                BigDecimal xpowi = BigDecimal.One;

                /* i factorial */
                BigInteger ifacN = BigInteger.One;
                BigInteger ifacD = BigInteger.One;

                for (int i = 1;; i++) {
                    ifacN = ifacN.Multiply(BigInteger.ValueOf((2*i - 1)));
                    ifacD = ifacD.Multiply(BigInteger.ValueOf(i));
                    if (i == 1)
                        xpowi = xhighprV;
                    else
                        xpowi = MultiplyRound(xpowi, xhighprV);
                    BigDecimal c = DivideRound(MultiplyRound(xpowi, ifacN),
                        ifacD.Multiply(BigInteger.ValueOf((2*i + 1))));
                    resul = resul.Add(c);
                    /* series started 1+x/12+... which yields an estimate of the sum's error
                                */
                    if (System.Math.Abs(c.ToDouble()) < xUlpDbl/120d)
                        break;
                }
                /* sqrt(2*z)*(1+...)
                        */
                xpowi = Sqrt(xhighpr.Multiply(new BigDecimal(2)));
                resul = MultiplyRound(xpowi, resul);

                var mc = new MathContext(resul.Precision);
                BigDecimal pihalf = PiRound(mc).Divide(new BigDecimal(2));

                mc = new MathContext(ErrorToPrecision(resul.ToDouble(), eps));
                return pihalf.Subtract(resul, mc);
            } else {
                /* absolute error in the result is err(x)/sqrt(1-x^2) to lowest order
                        */
                double xDbl = x.ToDouble();
                double xUlpDbl = x.Ulp().ToDouble()/2d;
                double eps = xUlpDbl/2d/System.Math.Sqrt(1d - System.Math.Pow(xDbl, 2d));

                BigDecimal xhighpr = ScalePrecision(x, 2);
                BigDecimal xhighprSq = MultiplyRound(xhighpr, xhighpr);

                BigDecimal resul = xhighpr.Plus();

                /* x^(2i+1) */
                BigDecimal xpowi = xhighpr;

                /* i factorial */
                BigInteger ifacN = BigInteger.One;
                BigInteger ifacD = BigInteger.One;

                for (int i = 1;; i++) {
                    ifacN = ifacN.Multiply(BigInteger.ValueOf((2*i - 1)));
                    ifacD = ifacD.Multiply(BigInteger.ValueOf((2*i)));
                    xpowi = MultiplyRound(xpowi, xhighprSq);
                    BigDecimal c = DivideRound(MultiplyRound(xpowi, ifacN),
                        ifacD.Multiply(BigInteger.ValueOf((2*i + 1))));
                    resul = resul.Add(c);
                    if (System.Math.Abs(c.ToDouble()) < 0.1*eps)
                        break;
                }
                var mc = new MathContext(ErrorToPrecision(resul.ToDouble(), eps));
                return resul.Round(mc);
            }
        }

        public static BigDecimal ModPi(BigDecimal x)
        {
            /* write x= pi*k+r with the precision in r defined by the precision of x and not
                * compromised by the precision of pi, so the ulp of pi*k should match the ulp of x.
                * First get a guess of k to figure out how many digits of pi are needed.
                */
            var k = (int) (x.ToDouble()/System.Math.PI);

            /* want to have err(pi*k)< err(x)=x.ulp/2, so err(pi) = err(x)/(2k) with two safety digits
                */
            double errpi;
            if (k != 0)
                errpi = 0.5*System.Math.Abs(x.Ulp().ToDouble()/k);
            else
                errpi = 0.5*System.Math.Abs(x.Ulp().ToDouble());
            var mc = new MathContext(2 + ErrorToPrecision(3.1416, errpi));
            BigDecimal onepi = PiRound(mc);
            BigDecimal pihalf = onepi.Divide(new BigDecimal(2));

            /* Delegate the actual operation to the BigDecimal class, which may return
                * a negative value of x was negative .
                */
            BigDecimal res = x.Remainder(onepi);
            if (res.CompareTo(pihalf) > 0)
                res = res.Subtract(onepi);
            else if (res.CompareTo(pihalf.Negate()) < 0)
                res = res.Add(onepi);

            /* The actual precision is set by the input value, its absolute value of x.ulp()/2.
                */
            mc = new MathContext(ErrorToPrecision(res.ToDouble(), x.Ulp().ToDouble()/2d));
            return res.Round(mc);
        }