public virtual Number Calculate(BigDecimal num)
{
if (num == null)
{
return -0;
}
return num.Negate();
}
public static BigDecimal /*!*/ Negate(RubyContext /*!*/ context, BigDecimal /*!*/ self)
{
return(BigDecimal.Negate(GetConfig(context), self));
}
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 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 Cbrt(BigDecimal x)
{
if (x.CompareTo(BigDecimal.Zero) < 0)
return Root(3, x.Negate()).Negate();
return Root(3, x);
}
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 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 Number operator -(Number a)
{
return(BigDecimal.Negate(new BigDecimal.Config(), a.value));
}
public void TestTruncateOnAllArithmeticOperations()
{
var savePrecision = BigDecimal.Precision;
BigDecimal mod1 = BigDecimal.Parse("3141592653589793238462643383279502");
BigDecimal mod2 = BigDecimal.Parse("27182818284590452");
BigDecimal neg1 = BigDecimal.Parse("-3.141592653589793238462643383279502884197169399375105820974944592307816406286208998628034825342117067982148086513282306647");
BigDecimal lrg1 = BigDecimal.Parse("3.141592653589793238462643383279502884197169399375105820974944592307816406286208998628034825342117067982148086513282306647");
BigDecimal lrg2 = BigDecimal.Parse("2.718281828459045235360287471352662497757247093699959574966967");
var expected1 = "5.859874482";
var expected2 = "0.4233108251";
var expected3 = "8.5397342226";
var expected4 = "0.8652559794";
var expected5 = "9.869604401";
var expected6 = "148.4131591";
var expected7 = "8003077319547306";
var expected8 = "-3.1415926535";
var expected9 = "3";
var expected10 = "4";
var expected11 = "3.1415926535";
var actual1 = "";
var actual2 = "";
var actual3 = "";
var actual4 = "";
var actual5 = "";
var actual6 = "";
var actual7 = "";
var actual8 = "";
var actual9 = "";
var actual10 = "";
var actual11 = "";
try
{
BigDecimal.Precision = 10;
BigDecimal.AlwaysTruncate = true;
TestContext.WriteLine($"E = {BigDecimal.E}");
TestContext.WriteLine($"{new BigDecimal(lrg1.Mantissa, lrg1.Exponent)}");
TestContext.WriteLine($"{new BigDecimal(lrg2.Mantissa, lrg2.Exponent)}");
BigDecimal result1 = BigDecimal.Add(lrg1, lrg2);
BigDecimal result2 = BigDecimal.Subtract(lrg1, lrg2);
BigDecimal result3 = BigDecimal.Multiply(lrg1, lrg2);
BigDecimal result4 = BigDecimal.Divide(lrg2, lrg1);
BigDecimal result5 = BigDecimal.Pow(lrg1, 2);
BigDecimal result6 = BigDecimal.Exp(new BigInteger(5));
BigDecimal result7 = BigDecimal.Mod(mod1, mod2);
BigDecimal result8 = BigDecimal.Negate(lrg1);
BigDecimal result9 = BigDecimal.Floor(lrg1);
BigDecimal result10 = BigDecimal.Ceiling(lrg1);
BigDecimal result11 = BigDecimal.Abs(lrg1);
actual1 = result1.ToString();
actual2 = result2.ToString();
actual3 = result3.ToString();
actual4 = result4.ToString();
actual5 = result5.ToString();
actual6 = result6.ToString();
actual7 = result7.ToString();
actual8 = result8.ToString();
actual9 = result9.ToString();
actual10 = result10.ToString();
actual11 = result11.ToString();
}
finally
{
BigDecimal.Precision = savePrecision;
BigDecimal.AlwaysTruncate = false;
}
Assert.AreEqual(expected1, actual1, $"Test Truncate On All Arithmetic Operations - #1: ");
Assert.AreEqual(expected2, actual2, $"Test Truncate On All Arithmetic Operations - #2: ");
Assert.AreEqual(expected3, actual3, $"Test Truncate On All Arithmetic Operations - #3: ");
Assert.AreEqual(expected4, actual4, $"Test Truncate On All Arithmetic Operations - #4: ");
Assert.AreEqual(expected5, actual5, $"Test Truncate On All Arithmetic Operations - #5: ");
StringAssert.StartsWith(expected6, actual6, $"Test Truncate On All Arithmetic Operations - #6: ");
Assert.AreEqual(expected7, actual7, $"Test Truncate On All Arithmetic Operations - #7: ");
Assert.AreEqual(expected8, actual8, $"Test Truncate On All Arithmetic Operations - #8: ");
Assert.AreEqual(expected9, actual9, $"Test Truncate On All Arithmetic Operations - #9: ");
Assert.AreEqual(expected10, actual10, $"Test Truncate On All Arithmetic Operations - #10: ");
Assert.AreEqual(expected11, actual11, $"Test Truncate On All Arithmetic Operations - #11: ");
Assert.AreEqual(5000, BigDecimal.Precision, "Restore Precision to 5000");
}
public void Negate()
{
BigDecimal negate1 = new BigDecimal(value2, 7);
Assert.IsTrue(negate1.Negate().ToString().Equals("-1233.4560000"), "the negate of 1233.4560000 is not -1233.4560000");
negate1 = BigDecimal.Parse("-23465839");
Assert.IsTrue(negate1.Negate().ToString().Equals("23465839"), "the negate of -23465839 is not 23465839");
negate1 = new BigDecimal(-3.456E6);
Assert.IsTrue(negate1.Negate().Negate().Equals(negate1), "the negate of -3.456E6 is not 3.456E6");
}