/// <summary>
/// Calculation of
/// b = Φ(h+t)·exp(h·t) - Φ(h-t)·exp(-h·t)
/// exp(-(h²+t²)/2)
/// = --------------- · [ Y(h+t) - Y(h-t) ]
/// √(2π)
/// with
/// Y(z) := Φ(z)/φ(z)
/// using an expansion of Y(h+t)-Y(h-t) for small t to twelvth order in t.
/// Theoretically accurate to (better than) precision ε = 2.23E-16 when h<=0 and t < τ with τ :=
/// 2·ε^(1/16) ≈ 0.21.
/// The main bottleneck for precision is the coefficient a:=1+h·Y(h) when |h|>1 .
/// Smalltexpansions the of normalised black call.
/// Y(h) := Φ(h)/φ(h) = √(π/2)·erfcx(-h/√2)
/// a := 1+h·Y(h) --- Note that due to h<0, and h·Y(h) -> -1 (from above) as h -> -∞, we also have that a>
/// 0 and a -> 0 as h -> -∞
/// w := t² , h2 := h²
/// </summary>
/// <param name="h">The h.</param>
/// <param name="t">The t.</param>
/// <returns></returns>
private double SmalltexpansionOfNormalisedBlackCall(double h, double t)
{
var a = 1 + h * (0.5 * SqrtTwoPi) * Erfc.ErfcxCody(-OneOverSqrtTwo * h);
var w = t * t;
var h2 = h * h;
var expansion = 2 * t * (a + w * ((-1 + 3 * a + a * h2) / 6 + w *
((-7 + 15 * a + h2 * (-1 + 10 * a + a * h2)) / 120 + w *
((-57 + 105 * a +
h2 * (-18 + 105 * a + h2 * (-1 + 21 * a + a * h2))) / 5040 +
w * ((-561 + 945 * a +
h2 * (-285 + 1260 * a +
h2 * (-33 + 378 * a + h2 * (-1 + 36 * a + a * h2)))
) / 362880 + w *
((-6555 + 10395 * a +
h2 * (-4680 + 17325 * a +
h2 * (-840 + 6930 * a +
h2 * (-52 + 990 * a +
h2 * (-1 + 55 * a + a * h2))))) /
39916800 + (-89055 + 135135 * a +
h2 * (-82845 + 270270 * a +
h2 * (-20370 + 135135 * a +
h2 * (-1926 + 25740 * a +
h2 * (-75 + 2145 * a +
h2 * (-1 + 78 * a +
a * h2)))))) *
w / 6227020800.0))))));
var b = OneOverSqrtTwoPi * Math.Exp(-0.5 * (h * h + t * t)) * expansion;
return(Math.Max(b, 0.0));
}
//
// Introduced on 2017-02-18
//
// b(x,s) = Φ(x/s+s/2)·exp(x/2) - Φ(x/s-s/2)·exp(-x/2)
// = Φ(h+t)·exp(x/2) - Φ(h-t)·exp(-x/2)
// = ½ · exp(-u²-v²) · [ erfcx(u-v) - erfcx(u+v) ]
// = ½ · [ exp(x/2)·erfc(u-v) - exp(-x/2)·erfc(u+v) ]
// = ½ · [ exp(x/2)·erfc(u-v) - exp(-u²-v²)·erfcx(u+v) ]
// = ½ · [ exp(-u²-v²)·erfcx(u-v) - exp(-x/2)·erfc(u+v) ]
// with
// h = x/s , t = s/2 ,
// and
// u = -h/√2 and v = t/√2 .
//
// Cody's erfc() and erfcx() functions each, for some values of their argument, involve the evaluation
// of the exponential function exp(). The normalised Black function requires additional evaluation(s)
// of the exponential function irrespective of which of the above formulations is used. However, the total
// number of exponential function evaluations can be minimised by a judicious choice of one of the above
// formulations depending on the input values and the branch logic in Cody's erfc() and erfcx().
//
private double NormalisedBlackCallWithOptimalUseOfCodysFunctions(double x, double s)
{
const double codysThreshold = 0.46875;
var h = x / s;
var t = 0.5 * s;
var q1 = -OneOverSqrtTwo * (h + t);
var q2 = -OneOverSqrtTwo * (h - t);
double twoB;
if (q1 < codysThreshold)
{
if (q2 < codysThreshold)
{
twoB = Math.Exp(0.5 * x) * Erfc.ErfcCody(q1) -
Math.Exp(-0.5 * x) * Erfc.ErfcCody(q2);
}
else
{
twoB = Math.Exp(0.5 * x) * Erfc.ErfcCody(q1) -
Math.Exp(-0.5 * (h * h + t * t)) * Erfc.ErfcCody(q2);
}
}
else
{
if (q2 < codysThreshold)
{
twoB = Math.Exp(-0.5 * (h * h + t * t)) * Erfc.ErfcxCody(q1) -
Math.Exp(-0.5 * x) * Erfc.ErfcCody(q2);
}
else
{
twoB = Math.Exp(-0.5 * (h * h + t * t)) *
(Erfc.ErfcxCody(q1) - Erfc.ErfcxCody(q2));
}
}
return(Math.Max(0.5 * twoB, 0.0));
}
public BuiltInFunctions()
{
// Text
Functions["len"] = new Len();
Functions["lower"] = new Lower();
Functions["upper"] = new Upper();
Functions["left"] = new Left();
Functions["right"] = new Right();
Functions["mid"] = new Mid();
Functions["replace"] = new Replace();
Functions["rept"] = new Rept();
Functions["substitute"] = new Substitute();
Functions["concatenate"] = new Concatenate();
Functions["concat"] = new Concat();
Functions["textjoin"] = new Textjoin();
Functions["char"] = new CharFunction();
Functions["exact"] = new Exact();
Functions["find"] = new Find();
Functions["fixed"] = new Fixed();
Functions["proper"] = new Proper();
Functions["search"] = new Search();
Functions["text"] = new Text.Text();
Functions["t"] = new T();
Functions["hyperlink"] = new Hyperlink();
Functions["value"] = new Value(CultureInfo.CurrentCulture);
Functions["trim"] = new Trim();
Functions["clean"] = new Clean();
Functions["unicode"] = new Unicode();
Functions["unichar"] = new Unichar();
Functions["numbervalue"] = new NumberValue();
Functions["dollar"] = new Dollar();
// Numbers
Functions["int"] = new CInt();
// Math
Functions["aggregate"] = new Aggregate();
Functions["abs"] = new Abs();
Functions["asin"] = new Asin();
Functions["asinh"] = new Asinh();
Functions["acot"] = new Acot();
Functions["acoth"] = new Acoth();
Functions["cos"] = new Cos();
Functions["cot"] = new Cot();
Functions["coth"] = new Coth();
Functions["cosh"] = new Cosh();
Functions["csc"] = new Csc();
Functions["csch"] = new Csch();
Functions["power"] = new Power();
Functions["gcd"] = new Gcd();
Functions["lcm"] = new Lcm();
Functions["sec"] = new Sec();
Functions["sech"] = new SecH();
Functions["sign"] = new Sign();
Functions["sqrt"] = new Sqrt();
Functions["sqrtpi"] = new SqrtPi();
Functions["pi"] = new Pi();
Functions["product"] = new Product();
Functions["ceiling"] = new Ceiling();
Functions["ceiling.precise"] = new CeilingPrecise();
Functions["ceiling.math"] = new CeilingMath();
Functions["iso.ceiling"] = new IsoCeiling();
Functions["combin"] = new Combin();
Functions["combina"] = new Combina();
Functions["permut"] = new Permut();
Functions["permutationa"] = new Permutationa();
Functions["count"] = new Count();
Functions["counta"] = new CountA();
Functions["countblank"] = new CountBlank();
Functions["countif"] = new CountIf();
Functions["countifs"] = new CountIfs();
Functions["fact"] = new Fact();
Functions["factdouble"] = new FactDouble();
Functions["floor"] = new Floor();
Functions["floor.precise"] = new FloorPrecise();
Functions["floor.math"] = new FloorMath();
Functions["radians"] = new Radians();
Functions["roman"] = new Roman();
Functions["sin"] = new Sin();
Functions["sinh"] = new Sinh();
Functions["sum"] = new Sum();
Functions["sumif"] = new SumIf();
Functions["sumifs"] = new SumIfs();
Functions["sumproduct"] = new SumProduct();
Functions["sumsq"] = new Sumsq();
Functions["sumxmy2"] = new Sumxmy2();
Functions["sumx2my2"] = new SumX2mY2();
Functions["sumx2py2"] = new SumX2pY2();
Functions["seriessum"] = new Seriessum();
Functions["stdev"] = new Stdev();
Functions["stdeva"] = new Stdeva();
Functions["stdevp"] = new StdevP();
Functions["stdevpa"] = new Stdevpa();
Functions["stdev.s"] = new StdevDotS();
Functions["stdev.p"] = new StdevDotP();
Functions["subtotal"] = new Subtotal();
Functions["exp"] = new Exp();
Functions["log"] = new Log();
Functions["log10"] = new Log10();
Functions["ln"] = new Ln();
Functions["max"] = new Max();
Functions["maxa"] = new Maxa();
Functions["median"] = new Median();
Functions["min"] = new Min();
Functions["mina"] = new Mina();
Functions["mod"] = new Mod();
Functions["mode"] = new Mode();
Functions["mode.sngl"] = new ModeSngl();
Functions["mround"] = new Mround();
Functions["multinomial"] = new Multinomial();
Functions["average"] = new Average();
Functions["averagea"] = new AverageA();
Functions["averageif"] = new AverageIf();
Functions["averageifs"] = new AverageIfs();
Functions["round"] = new Round();
Functions["rounddown"] = new Rounddown();
Functions["roundup"] = new Roundup();
Functions["rand"] = new Rand();
Functions["randbetween"] = new RandBetween();
Functions["rank"] = new Rank();
Functions["rank.eq"] = new RankEq();
Functions["rank.avg"] = new RankAvg();
Functions["percentile"] = new Percentile();
Functions["percentile.inc"] = new PercentileInc();
Functions["percentile.exc"] = new PercentileExc();
Functions["quartile"] = new Quartile();
Functions["quartile.inc"] = new QuartileInc();
Functions["quartile.exc"] = new QuartileExc();
Functions["percentrank"] = new Percentrank();
Functions["percentrank.inc"] = new PercentrankInc();
Functions["percentrank.exc"] = new PercentrankExc();
Functions["quotient"] = new Quotient();
Functions["trunc"] = new Trunc();
Functions["tan"] = new Tan();
Functions["tanh"] = new Tanh();
Functions["atan"] = new Atan();
Functions["atan2"] = new Atan2();
Functions["atanh"] = new Atanh();
Functions["acos"] = new Acos();
Functions["acosh"] = new Acosh();
Functions["covar"] = new Covar();
Functions["covariance.p"] = new CovarianceP();
Functions["covariance.s"] = new CovarianceS();
Functions["var"] = new Var();
Functions["vara"] = new Vara();
Functions["var.s"] = new VarDotS();
Functions["varp"] = new VarP();
Functions["varpa"] = new Varpa();
Functions["var.p"] = new VarDotP();
Functions["large"] = new Large();
Functions["small"] = new Small();
Functions["degrees"] = new Degrees();
Functions["odd"] = new Odd();
Functions["even"] = new Even();
// Information
Functions["isblank"] = new IsBlank();
Functions["isnumber"] = new IsNumber();
Functions["istext"] = new IsText();
Functions["isnontext"] = new IsNonText();
Functions["iserror"] = new IsError();
Functions["iserr"] = new IsErr();
Functions["error.type"] = new ErrorType();
Functions["iseven"] = new IsEven();
Functions["isodd"] = new IsOdd();
Functions["islogical"] = new IsLogical();
Functions["isna"] = new IsNa();
Functions["na"] = new Na();
Functions["n"] = new N();
Functions["type"] = new TypeFunction();
// Logical
Functions["if"] = new If();
Functions["ifs"] = new Ifs();
Functions["maxifs"] = new MaxIfs();
Functions["minifs"] = new MinIfs();
Functions["iferror"] = new IfError();
Functions["ifna"] = new IfNa();
Functions["not"] = new Not();
Functions["and"] = new And();
Functions["or"] = new Or();
Functions["true"] = new True();
Functions["false"] = new False();
Functions["switch"] = new Switch();
Functions["xor"] = new Xor();
// Reference and lookup
Functions["address"] = new Address();
Functions["hlookup"] = new HLookup();
Functions["vlookup"] = new VLookup();
Functions["lookup"] = new Lookup();
Functions["match"] = new Match();
Functions["row"] = new Row();
Functions["rows"] = new Rows();
Functions["column"] = new Column();
Functions["columns"] = new Columns();
Functions["choose"] = new Choose();
Functions["index"] = new RefAndLookup.Index();
Functions["indirect"] = new Indirect();
Functions["offset"] = new Offset();
// Date
Functions["date"] = new Date();
Functions["datedif"] = new DateDif();
Functions["today"] = new Today();
Functions["now"] = new Now();
Functions["day"] = new Day();
Functions["month"] = new Month();
Functions["year"] = new Year();
Functions["time"] = new Time();
Functions["hour"] = new Hour();
Functions["minute"] = new Minute();
Functions["second"] = new Second();
Functions["weeknum"] = new Weeknum();
Functions["weekday"] = new Weekday();
Functions["days"] = new Days();
Functions["days360"] = new Days360();
Functions["yearfrac"] = new Yearfrac();
Functions["edate"] = new Edate();
Functions["eomonth"] = new Eomonth();
Functions["isoweeknum"] = new IsoWeekNum();
Functions["workday"] = new Workday();
Functions["workday.intl"] = new WorkdayIntl();
Functions["networkdays"] = new Networkdays();
Functions["networkdays.intl"] = new NetworkdaysIntl();
Functions["datevalue"] = new DateValue();
Functions["timevalue"] = new TimeValue();
// Database
Functions["dget"] = new Dget();
Functions["dcount"] = new Dcount();
Functions["dcounta"] = new DcountA();
Functions["dmax"] = new Dmax();
Functions["dmin"] = new Dmin();
Functions["dsum"] = new Dsum();
Functions["daverage"] = new Daverage();
Functions["dvar"] = new Dvar();
Functions["dvarp"] = new Dvarp();
//Finance
Functions["cumipmt"] = new Cumipmt();
Functions["cumprinc"] = new Cumprinc();
Functions["dollarde"] = new DollarDe();
Functions["dollarfr"] = new DollarFr();
Functions["ddb"] = new Ddb();
Functions["effect"] = new Effect();
Functions["fvschedule"] = new FvSchedule();
Functions["pduration"] = new Pduration();
Functions["rri"] = new Rri();
Functions["pmt"] = new Pmt();
Functions["ppmt"] = new Ppmt();
Functions["ipmt"] = new Ipmt();
Functions["ispmt"] = new IsPmt();
Functions["pv"] = new Pv();
Functions["fv"] = new Fv();
Functions["npv"] = new Npv();
Functions["rate"] = new Rate();
Functions["nper"] = new Nper();
Functions["nominal"] = new Nominal();
Functions["irr"] = new Irr();
Functions["mirr"] = new Mirr();
Functions["xirr"] = new Xirr();
Functions["sln"] = new Sln();
Functions["syd"] = new Syd();
Functions["xnpv"] = new Xnpv();
Functions["coupdays"] = new Coupdays();
Functions["coupdaysnc"] = new Coupdaysnc();
Functions["coupdaybs"] = new Coupdaybs();
Functions["coupnum"] = new Coupnum();
Functions["coupncd"] = new Coupncd();
Functions["couppcd"] = new Couppcd();
Functions["price"] = new Price();
Functions["yield"] = new Yield();
Functions["yieldmat"] = new Yieldmat();
Functions["duration"] = new Duration();
Functions["disc"] = new Disc();
//Engineering
Functions["bitand"] = new BitAnd();
Functions["bitor"] = new BitOr();
Functions["bitxor"] = new BitXor();
Functions["bitlshift"] = new BitLshift();
Functions["bitrshift"] = new BitRshift();
Functions["convert"] = new ConvertFunction();
Functions["bin2dec"] = new Bin2Dec();
Functions["bin2hex"] = new Bin2Hex();
Functions["bin2oct"] = new Bin2Oct();
Functions["dec2bin"] = new Dec2Bin();
Functions["dec2hex"] = new Dec2Hex();
Functions["dec2oct"] = new Dec2Oct();
Functions["hex2bin"] = new Hex2Bin();
Functions["hex2dec"] = new Hex2Dec();
Functions["hex2oct"] = new Hex2Oct();
Functions["oct2bin"] = new Oct2Bin();
Functions["oct2dec"] = new Oct2Dec();
Functions["oct2hex"] = new Oct2Hex();
Functions["delta"] = new Delta();
Functions["erf"] = new Erf();
Functions["erf.precise"] = new ErfPrecise();
Functions["erfc"] = new Erfc();
Functions["erfc.precise"] = new ErfcPrecise();
Functions["besseli"] = new BesselI();
Functions["besselj"] = new BesselJ();
Functions["besselk"] = new BesselK();
Functions["bessely"] = new BesselY();
}