کد:
{ *********************************************************************** }
{ }
{ Delphi / Kylix Cross-Platform Runtime Library }
{ }
{ Copyright (c) 1996, 2001 Borland Software Corporation }
{ }
{ *********************************************************************** }
unit Math;
{ This unit contains high-performance arithmetic, trigonometric, logarithmic,
statistical, financial calculation and FPU routines which supplement the math
routines that are part of the Delphi language or System unit.
References:
1) P.J. Plauger, "The Standard C Library", Prentice-Hall, 1992, Ch. 7.
2) W.J. Cody, Jr., and W. Waite, "Software Manual For the Elementary
Functions", Prentice-Hall, 1980.
3) Namir Shammas, "C/C++ Mathematical Algorithms for Scientists and Engineers",
McGraw-Hill, 1995, Ch 8.
4) H.T. Lau, "A Numerical Library in C for Scientists and Engineers",
CRC Press, 1994, Ch. 6.
5) "Pentium(tm) Processor User's Manual, Volume 3: Architecture
and Programming Manual", Intel, 1994
Some of the functions, concepts or constants in this unit were provided by
Earl F. Glynn (www.efg2.com) and Ray Lischner (www.tempest-sw.com)
All angle parameters and results of trig functions are in radians.
Most of the following trig and log routines map directly to Intel 80387 FPU
floating point machine instructions. Input domains, output ranges, and
error handling are determined largely by the FPU hardware.
Routines coded in assembler favor the Pentium FPU pipeline architecture.
}
{$N+,S-}
interface
uses SysUtils, Types;
const { Ranges of the IEEE floating point types, including denormals }
MinSingle = 1.5e-45;
MaxSingle = 3.4e+38;
MinDouble = 5.0e-324;
MaxDouble = 1.7e+308;
MinExtended = 3.4e-4932;
MaxExtended = 1.1e+4932;
MinComp = -9.223372036854775807e+18;
MaxComp = 9.223372036854775807e+18;
{ The following constants should not be used for comparison, only
assignments. For comparison please use the IsNan and IsInfinity functions
provided below. }
NaN = 0.0 / 0.0;
(*$EXTERNALSYM NaN*)
(*$HPPEMIT 'static const Extended NaN = 0.0 / 0.0;'*)
Infinity = 1.0 / 0.0;
(*$EXTERNALSYM Infinity*)
(*$HPPEMIT 'static const Extended Infinity = 1.0 / 0.0;'*)
NegInfinity = -1.0 / 0.0;
(*$EXTERNALSYM NegInfinity*)
(*$HPPEMIT 'static const Extended NegInfinity = -1.0 / 0.0;'*)
{ Trigonometric functions }
function ArcCos(const X: Extended): Extended; { IN: |X| <= 1 OUT: [0..PI] radians }
function ArcSin(const X: Extended): Extended; { IN: |X| <= 1 OUT: [-PI/2..PI/2] radians }
{ ArcTan2 calculates ArcTan(Y/X), and returns an angle in the correct quadrant.
IN: |Y| < 2^64, |X| < 2^64, X <> 0 OUT: [-PI..PI] radians }
function ArcTan2(const Y, X: Extended): Extended;
{ SinCos is 2x faster than calling Sin and Cos separately for the same angle }
procedure SinCos(const Theta: Extended; var Sin, Cos: Extended) register;
function Tan(const X: Extended): Extended;
function Cotan(const X: Extended): Extended; { 1 / tan(X), X <> 0 }
function Secant(const X: Extended): Extended; { 1 / cos(X) }
function Cosecant(const X: Extended): Extended; { 1 / sin(X) }
function Hypot(const X, Y: Extended): Extended; { Sqrt(X**2 + Y**2) }
{ Angle unit conversion routines }
function RadToDeg(const Radians: Extended): Extended; { Degrees := Radians * 180 / PI }
function RadToGrad(const Radians: Extended): Extended; { Grads := Radians * 200 / PI }
function RadToCycle(const Radians: Extended): Extended;{ Cycles := Radians / 2PI }
function DegToRad(const Degrees: Extended): Extended; { Radians := Degrees * PI / 180}
function DegToGrad(const Degrees: Extended): Extended;
function DegToCycle(const Degrees: Extended): Extended;
function GradToRad(const Grads: Extended): Extended; { Radians := Grads * PI / 200 }
function GradToDeg(const Grads: Extended): Extended;
function GradToCycle(const Grads: Extended): Extended;
function CycleToRad(const Cycles: Extended): Extended; { Radians := Cycles * 2PI }
function CycleToDeg(const Cycles: Extended): Extended;
function CycleToGrad(const Cycles: Extended): Extended;
{ Hyperbolic functions and inverses }
function Cot(const X: Extended): Extended; { alias for Cotan }
function Sec(const X: Extended): Extended; { alias for Secant }
function Csc(const X: Extended): Extended; { alias for Cosecant }
function Cosh(const X: Extended): Extended;
function Sinh(const X: Extended): Extended;
function Tanh(const X: Extended): Extended;
function CotH(const X: Extended): Extended;
function SecH(const X: Extended): Extended;
function CscH(const X: Extended): Extended;
function ArcCot(const X: Extended): Extended; { IN: X <> 0 }
function ArcSec(const X: Extended): Extended; { IN: X <> 0 }
function ArcCsc(const X: Extended): Extended; { IN: X <> 0 }
function ArcCosh(const X: Extended): Extended; { IN: X >= 1 }
function ArcSinh(const X: Extended): Extended;
function ArcTanh(const X: Extended): Extended; { IN: |X| <= 1 }
function ArcCotH(const X: Extended): Extended; { IN: X <> 0 }
function ArcSecH(const X: Extended): Extended; { IN: X <> 0 }
function ArcCscH(const X: Extended): Extended; { IN: X <> 0 }
{ Logarithmic functions }
function LnXP1(const X: Extended): Extended; { Ln(X + 1), accurate for X near zero }
function Log10(const X: Extended): Extended; { Log base 10 of X }
function Log2(const X: Extended): Extended; { Log base 2 of X }
function LogN(const Base, X: Extended): Extended; { Log base N of X }
{ Exponential functions }
{ IntPower: Raise base to an integral power. Fast. }
function IntPower(const Base: Extended; const Exponent: Integer): Extended register;
{ Power: Raise base to any power.
For fractional exponents, or |exponents| > MaxInt, base must be > 0. }
function Power(const Base, Exponent: Extended): Extended;
{ Miscellaneous Routines }
{ Frexp: Separates the mantissa and exponent of X. }
procedure Frexp(const X: Extended; var Mantissa: Extended; var Exponent: Integer) register;
{ Ldexp: returns X*2**P }
function Ldexp(const X: Extended; const P: Integer): Extended register;
{ Ceil: Smallest integer >= X, |X| < MaxInt }
function Ceil(const X: Extended):Integer;
{ Floor: Largest integer <= X, |X| < MaxInt }
function Floor(const X: Extended): Integer;
{ Poly: Evaluates a uniform polynomial of one variable at value X.
The coefficients are ordered in increasing powers of X:
Coefficients[0] + Coefficients[1]*X + ... + Coefficients[N]*(X**N) }
function Poly(const X: Extended; const Coefficients: array of Double): Extended;
{-----------------------------------------------------------------------
Statistical functions.
Common commercial spreadsheet macro names for these statistical and
financial functions are given in the comments preceding each function.
-----------------------------------------------------------------------}
{ Mean: Arithmetic average of values. (AVG): SUM / N }
function Mean(const Data: array of Double): Extended;
{ Sum: Sum of values. (SUM) }
function Sum(const Data: array of Double): Extended register;
function SumInt(const Data: array of Integer): Integer register;
function SumOfSquares(const Data: array of Double): Extended;
procedure SumsAndSquares(const Data: array of Double;
var Sum, SumOfSquares: Extended) register;
{ MinValue: Returns the smallest signed value in the data array (MIN) }
function MinValue(const Data: array of Double): Double;
function MinIntValue(const Data: array of Integer): Integer;
function Min(const A, B: Integer): Integer; overload;
function Min(const A, B: Int64): Int64; overload;
function Min(const A, B: Single): Single; overload;
function Min(const A, B: Double): Double; overload;
function Min(const A, B: Extended): Extended; overload;
{ MaxValue: Returns the largest signed value in the data array (MAX) }
function MaxValue(const Data: array of Double): Double;
function MaxIntValue(const Data: array of Integer): Integer;
function Max(const A, B: Integer): Integer; overload;
function Max(const A, B: Int64): Int64; overload;
function Max(const A, B: Single): Single; overload;
function Max(const A, B: Double): Double; overload;
function Max(const A, B: Extended): Extended; overload;
{ Standard Deviation (STD): Sqrt(Variance). aka Sample Standard Deviation }
function StdDev(const Data: array of Double): Extended;
{ MeanAndStdDev calculates Mean and StdDev in one call. }
procedure MeanAndStdDev(const Data: array of Double; var Mean, StdDev: Extended);
{ Population Standard Deviation (STDP): Sqrt(PopnVariance).
Used in some business and financial calculations. }
function PopnStdDev(const Data: array of Double): Extended;
{ Variance (VARS): TotalVariance / (N-1). aka Sample Variance }
function Variance(const Data: array of Double): Extended;
{ Population Variance (VAR or VARP): TotalVariance/ N }
function PopnVariance(const Data: array of Double): Extended;
{ Total Variance: SUM(i=1,N)[(X(i) - Mean)**2] }
function TotalVariance(const Data: array of Double): Extended;
{ Norm: The Euclidean L2-norm. Sqrt(SumOfSquares) }
function Norm(const Data: array of Double): Extended;
{ MomentSkewKurtosis: Calculates the core factors of statistical analysis:
the first four moments plus the coefficients of skewness and kurtosis.
M1 is the Mean. M2 is the Variance.
Skew reflects symmetry of distribution: M3 / (M2**(3/2))
Kurtosis reflects flatness of distribution: M4 / Sqr(M2) }
procedure MomentSkewKurtosis(const Data: array of Double;
var M1, M2, M3, M4, Skew, Kurtosis: Extended);
{ RandG produces random numbers with Gaussian distribution about the mean.
Useful for simulating data with sampling errors. }
function RandG(Mean, StdDev: Extended): Extended;
{-----------------------------------------------------------------------
General/Misc use functions
-----------------------------------------------------------------------}
{ Extreme testing }
// Like an infinity, a NaN double value has an exponent of 7FF, but the NaN
// values have a fraction field that is not 0.
function IsNan(const AValue: Double): Boolean; overload;
function IsNan(const AValue: Single): Boolean; overload;
function IsNan(const AValue: Extended): Boolean; overload;
// Like a NaN, an infinity double value has an exponent of 7FF, but the
// infinity values have a fraction field of 0. Infinity values can be positive
// or negative, which is specified in the high-order, sign bit.
function IsInfinite(const AValue: Double): Boolean;
{ Simple sign testing }
type
TValueSign = -1..1;
const
NegativeValue = Low(TValueSign);
ZeroValue = 0;
PositiveValue = High(TValueSign);
function Sign(const AValue: Integer): TValueSign; overload;
function Sign(const AValue: Int64): TValueSign; overload;
function Sign(const AValue: Double): TValueSign; overload;
{ CompareFloat & SameFloat: If epsilon is not given (or is zero) we will
attempt to compute a reasonable one based on the precision of the floating
point type used. }
function CompareValue(const A, B: Extended; Epsilon: Extended = 0): TValueRelationship; overload;
function CompareValue(const A, B: Double; Epsilon: Double = 0): TValueRelationship; overload;
function CompareValue(const A, B: Single; Epsilon: Single = 0): TValueRelationship; overload;
function CompareValue(const A, B: Integer): TValueRelationship; overload;
function CompareValue(const A, B: Int64): TValueRelationship; overload;
function SameValue(const A, B: Extended; Epsilon: Extended = 0): Boolean; overload;
function SameValue(const A, B: Double; Epsilon: Double = 0): Boolean; overload;
function SameValue(const A, B: Single; Epsilon: Single = 0): Boolean; overload;
{ IsZero: These will return true if the given value is zero (or very very very
close to it). }
function IsZero(const A: Extended; Epsilon: Extended = 0): Boolean; overload;
function IsZero(const A: Double; Epsilon: Double = 0): Boolean; overload;
function IsZero(const A: Single; Epsilon: Single = 0): Boolean; overload;
{ Easy to use conditional functions }
function IfThen(AValue: Boolean; const ATrue: Integer; const AFalse: Integer = 0): Integer; overload;
function IfThen(AValue: Boolean; const ATrue: Int64; const AFalse: Int64 = 0): Int64; overload;
function IfThen(AValue: Boolean; const ATrue: Double; const AFalse: Double = 0.0): Double; overload;
{ Various random functions }
function RandomRange(const AFrom, ATo: Integer): Integer;
function RandomFrom(const AValues: array of Integer): Integer; overload;
function RandomFrom(const AValues: array of Int64): Int64; overload;
function RandomFrom(const AValues: array of Double): Double; overload;
{ Range testing functions }
function InRange(const AValue, AMin, AMax: Integer): Boolean; overload;
function InRange(const AValue, AMin, AMax: Int64): Boolean; overload;
function InRange(const AValue, AMin, AMax: Double): Boolean; overload;
{ Range truncation functions }
function EnsureRange(const AValue, AMin, AMax: Integer): Integer; overload;
function EnsureRange(const AValue, AMin, AMax: Int64): Int64; overload;
function EnsureRange(const AValue, AMin, AMax: Double): Double; overload;
{ 16 bit integer division and remainder in one operation }
procedure DivMod(Dividend: Integer; Divisor: Word;
var Result, Remainder: Word);
{ Round to a specific digit or power of ten }
{ ADigit has a valid range of 37 to -37. Here are some valid examples
of ADigit values...
3 = 10^3 = 1000 = thousand's place
2 = 10^2 = 100 = hundred's place
1 = 10^1 = 10 = ten's place
-1 = 10^-1 = 1/10 = tenth's place
-2 = 10^-2 = 1/100 = hundredth's place
-3 = 10^-3 = 1/1000 = thousandth's place }
type
TRoundToRange = -37..37;
function RoundTo(const AValue: Double; const ADigit: TRoundToRange): Double;
{ This variation of the RoundTo function follows the asymmetric arithmetic
rounding algorithm (if Frac(X) < .5 then return X else return X + 1). This
function defaults to rounding to the hundredth's place (cents). }
function SimpleRoundTo(const AValue: Double; const ADigit: TRoundToRange = -2): Double;
{-----------------------------------------------------------------------
Financial functions. Standard set from Quattro Pro.
Parameter conventions:
From the point of view of A, amounts received by A are positive and
amounts disbursed by A are negative (e.g. a borrower's loan repayments
are regarded by the borrower as negative).
Interest rates are per payment period. 11% annual percentage rate on a
loan with 12 payments per year would be (11 / 100) / 12 = 0.00916667
-----------------------------------------------------------------------}
type
TPaymentTime = (ptEndOfPeriod, ptStartOfPeriod);
{ Double Declining Balance (DDB) }
function DoubleDecliningBalance(const Cost, Salvage: Extended;
Life, Period: Integer): Extended;
{ Future Value (FVAL) }
function FutureValue(const Rate: Extended; NPeriods: Integer; const Payment,
PresentValue: Extended; PaymentTime: TPaymentTime): Extended;
{ Interest Payment (IPAYMT) }
function InterestPayment(const Rate: Extended; Period, NPeriods: Integer;
const PresentValue, FutureValue: Extended; PaymentTime: TPaymentTime): Extended;
{ Interest Rate (IRATE) }
function InterestRate(NPeriods: Integer; const Payment, PresentValue,
FutureValue: Extended; PaymentTime: TPaymentTime): Extended;
{ Internal Rate of Return. (IRR) Needs array of cash flows. }
function InternalRateOfReturn(const Guess: Extended;
const CashFlows: array of Double): Extended;
{ Number of Periods (NPER) }
function NumberOfPeriods(const Rate: Extended; Payment: Extended;
const PresentValue, FutureValue: Extended; PaymentTime: TPaymentTime): Extended;
{ Net Present Value. (NPV) Needs array of cash flows. }
function NetPresentValue(const Rate: Extended; const CashFlows: array of Double;
PaymentTime: TPaymentTime): Extended;
{ Payment (PAYMT) }
function Payment(Rate: Extended; NPeriods: Integer; const PresentValue,
FutureValue: Extended; PaymentTime: TPaymentTime): Extended;
{ Period Payment (PPAYMT) }
function PeriodPayment(const Rate: Extended; Period, NPeriods: Integer;
const PresentValue, FutureValue: Extended; PaymentTime: TPaymentTime): Extended;
{ Present Value (PVAL) }
function PresentValue(const Rate: Extended; NPeriods: Integer;
const Payment, FutureValue: Extended; PaymentTime: TPaymentTime): Extended;
{ Straight Line depreciation (SLN) }
function SLNDepreciation(const Cost, Salvage: Extended; Life: Integer): Extended;
{ Sum-of-Years-Digits depreciation (SYD) }
function SYDDepreciation(const Cost, Salvage: Extended; Life, Period: Integer): Extended;
type
EInvalidArgument = class(EMathError) end;
{-----------------------------------------------------------------------
FPU exception/precision/rounding management
The following functions allow you to control the behavior of the FPU. With
them you can control what constutes an FPU exception, what the default
precision is used and finally how rounding is handled by the FPU.
-----------------------------------------------------------------------}
type
TFPURoundingMode = (rmNearest, rmDown, rmUp, rmTruncate);
{ Return the current rounding mode }
function GetRoundMode: TFPURoundingMode;
{ Set the rounding mode and return the old mode }
function SetRoundMode(const RoundMode: TFPURoundingMode): TFPURoundingMode;
type
TFPUPrecisionMode = (pmSingle, pmReserved, pmDouble, pmExtended);
{ Return the current precision control mode }
function GetPrecisionMode: TFPUPrecisionMode;
{ Set the precision control mode and return the old one }
function SetPrecisionMode(const Precision: TFPUPrecisionMode): TFPUPrecisionMode;
type
TFPUException = (exInvalidOp, exDenormalized, exZeroDivide,
exOverflow, exUnderflow, exPrecision);
TFPUExceptionMask = set of TFPUException;
{ Return the exception mask from the control word.
Any element set in the mask prevents the FPU from raising that kind of
exception. Instead, it returns its best attempt at a value, often NaN or an
infinity. The value depends on the operation and the current rounding mode. }
function GetExceptionMask: TFPUExceptionMask;
{ Set a new exception mask and return the old one }
function SetExceptionMask(const Mask: TFPUExceptionMask): TFPUExceptionMask;
{ Clear any pending exception bits in the status word }
procedure ClearExceptions(RaisePending: Boolean = True);
implementation
uses SysConst;
procedure DivMod(Dividend: Integer; Divisor: Word;
var Result, Remainder: Word);
asm
PUSH EBX
MOV EBX,EDX
MOV EDX,EAX
SHR EDX,16
DIV BX
MOV EBX,Remainder
MOV [ECX],AX
MOV [EBX],DX
POP EBX
end;
function RoundTo(const AValue: Double; const ADigit: TRoundToRange): Double;
var
LFactor: Double;
begin
LFactor := IntPower(10, ADigit);
Result := Round(AValue / LFactor) * LFactor;
end;
function SimpleRoundTo(const AValue: Double; const ADigit: TRoundToRange = -2): Double;
var
LFactor: Double;
begin
LFactor := IntPower(10, ADigit);
Result := Trunc((AValue / LFactor) + 0.5) * LFactor;
end;
function Annuity2(const R: Extended; N: Integer; PaymentTime: TPaymentTime;
var CompoundRN: Extended): Extended; Forward;
function Compound(const R: Extended; N: Integer): Extended; Forward;
function RelSmall(const X, Y: Extended): Boolean; Forward;
type
TPoly = record
Neg, Pos, DNeg, DPos: Extended
end;
const
MaxIterations = 15;
procedure ArgError(const Msg: string);
begin
raise EInvalidArgument.Create(Msg);
end;
function DegToRad(const Degrees: Extended): Extended; { Radians := Degrees * PI / 180 }
begin
Result := Degrees * (PI / 180);
end;
function RadToDeg(const Radians: Extended): Extended; { Degrees := Radians * 180 / PI }
begin
Result := Radians * (180 / PI);
end;
function GradToRad(const Grads: Extended): Extended; { Radians := Grads * PI / 200 }
begin
Result := Grads * (PI / 200);
end;
function RadToGrad(const Radians: Extended): Extended; { Grads := Radians * 200 / PI}
begin
Result := Radians * (200 / PI);
end;
function CycleToRad(const Cycles: Extended): Extended; { Radians := Cycles * 2PI }
begin
Result := Cycles * (2 * PI);
end;
function RadToCycle(const Radians: Extended): Extended;{ Cycles := Radians / 2PI }
begin
Result := Radians / (2 * PI);
end;
function DegToGrad(const Degrees: Extended): Extended;
begin
Result := RadToGrad(DegToRad(Degrees));
end;
function DegToCycle(const Degrees: Extended): Extended;
begin
Result := RadToCycle(DegToRad(Degrees));
end;
function GradToDeg(const Grads: Extended): Extended;
begin
Result := RadToDeg(GradToRad(Grads));
end;
function GradToCycle(const Grads: Extended): Extended;
begin
Result := RadToCycle(GradToRad(Grads));
end;
function CycleToDeg(const Cycles: Extended): Extended;
begin
Result := RadToDeg(CycleToRad(Cycles));
end;
function CycleToGrad(const Cycles: Extended): Extended;
begin
Result := RadToGrad(CycleToRad(Cycles));
end;
function LnXP1(const X: Extended): Extended;
{ Return ln(1 + X). Accurate for X near 0. }
asm
FLDLN2
MOV AX,WORD PTR X+8 { exponent }
FLD X
CMP AX,$3FFD { .4225 }
JB @@1
FLD1
FADD
FYL2X
JMP @@2
@@1:
FYL2XP1
@@2:
FWAIT
end;
{ Invariant: Y >= 0 & Result*X**Y = X**I. Init Y = I and Result = 1. }
{function IntPower(X: Extended; I: Integer): Extended;
var
Y: Integer;
begin
Y := Abs(I);
Result := 1.0;
while Y > 0 do begin
while not Odd(Y) do
begin
Y := Y shr 1;
X := X * X
end;
Dec(Y);
Result := Result * X
end;
if I < 0 then Result := 1.0 / Result
end;
}
function IntPower(const Base: Extended; const Exponent: Integer): Extended;
asm
mov ecx, eax
cdq
fld1 { Result := 1 }
xor eax, edx
sub eax, edx { eax := Abs(Exponent) }
jz @@3
fld Base
jmp @@2
@@1: fmul ST, ST { X := Base * Base }
@@2: shr eax,1
jnc @@1
fmul ST(1),ST { Result := Result * X }
jnz @@1
fstp st { pop X from FPU stack }
cmp ecx, 0
jge @@3
fld1
fdivrp { Result := 1 / Result }
@@3:
fwait
end;
function Compound(const R: Extended; N: Integer): Extended;
{ Return (1 + R)**N. }
begin
Result := IntPower(1.0 + R, N)
end;
function Annuity2(const R: Extended; N: Integer; PaymentTime: TPaymentTime;
var CompoundRN: Extended): Extended;
{ Set CompoundRN to Compound(R, N),
return (1+Rate*PaymentTime)*(Compound(R,N)-1)/R;
}
begin
if R = 0.0 then
begin
CompoundRN := 1.0;
Result := N;
end
else
begin
{ 6.1E-5 approx= 2**-14 }
if Abs(R) < 6.1E-5 then
begin
CompoundRN := Exp(N * LnXP1(R));
Result := N*(1+(N-1)*R/2);
end
else
begin
CompoundRN := Compound(R, N);
Result := (CompoundRN-1) / R
end;
if PaymentTime = ptStartOfPeriod then
Result := Result * (1 + R);
end;
end; {Annuity2}
procedure PolyX(const A: array of Double; X: Extended; var Poly: TPoly);
{ Compute A[0] + A[1]*X + ... + A[N]*X**N and X * its derivative.
Accumulate positive and negative terms separately. }
var
I: Integer;
Neg, Pos, DNeg, DPos: Extended;
begin
Neg := 0.0;
Pos := 0.0;
DNeg := 0.0;
DPos := 0.0;
for I := High(A) downto Low(A) do
begin
DNeg := X * DNeg + Neg;
Neg := Neg * X;
DPos := X * DPos + Pos;
Pos := Pos * X;
if A[I] >= 0.0 then
Pos := Pos + A[I]
else
Neg := Neg + A[I]
end;
Poly.Neg := Neg;
Poly.Pos := Pos;
Poly.DNeg := DNeg * X;
Poly.DPos := DPos * X;
end; {PolyX}
function RelSmall(const X, Y: Extended): Boolean;
{ Returns True if X is small relative to Y }
const
C1: Double = 1E-15;
C2: Double = 1E-12;
begin
Result := Abs(X) < (C1 + C2 * Abs(Y))
end;
{ Math functions. }
function ArcCos(const X: Extended): Extended;
begin
Result := ArcTan2(Sqrt(1 - X * X), X);
end;
function ArcSin(const X: Extended): Extended;
begin
Result := ArcTan2(X, Sqrt(1 - X * X))
end;
function ArcTan2(const Y, X: Extended): Extended;
asm
FLD Y
FLD X
FPATAN
FWAIT
end;
function Tan(const X: Extended): Extended;
{ Tan := Sin(X) / Cos(X) }
asm
FLD X
FPTAN
FSTP ST(0) { FPTAN pushes 1.0 after result }
FWAIT
end;
function CoTan(const X: Extended): Extended;
{ CoTan := Cos(X) / Sin(X) = 1 / Tan(X) }
asm
FLD X
FPTAN
FDIVRP
FWAIT
end;
function Secant(const X: Extended): Extended;
{ Secant := 1 / Cos(X) }
asm
FLD X
FCOS
FLD1
FDIVRP
FWAIT
end;
function Cosecant(const X: Extended): Extended;
{ Cosecant := 1 / Sin(X) }
asm
FLD X
FSIN
FLD1
FDIVRP
FWAIT
end;
function Hypot(const X, Y: Extended): Extended;
{ formula: Sqrt(X*X + Y*Y)
implemented as: |Y|*Sqrt(1+Sqr(X/Y)), |X| < |Y| for greater precision
var
Temp: Extended;
begin
X := Abs(X);
Y := Abs(Y);
if X > Y then
begin
Temp := X;
X := Y;
Y := Temp;
end;
if X = 0 then
Result := Y
else // Y > X, X <> 0, so Y > 0
Result := Y * Sqrt(1 + Sqr(X/Y));
end;
}
asm
FLD Y
FABS
FLD X
FABS
FCOM
FNSTSW AX
TEST AH,$45
JNZ @@1 // if ST > ST(1) then swap
FXCH ST(1) // put larger number in ST(1)
@@1: FLDZ
FCOMP
FNSTSW AX
TEST AH,$40 // if ST = 0, return ST(1)
JZ @@2
FSTP ST // eat ST(0)
JMP @@3
@@2: FDIV ST,ST(1) // ST := ST / ST(1)
FMUL ST,ST // ST := ST * ST
FLD1
FADD // ST := ST + 1
FSQRT // ST := Sqrt(ST)
FMUL // ST(1) := ST * ST(1); Pop ST
@@3: FWAIT
end;
procedure SinCos(const Theta: Extended; var Sin, Cos: Extended);
asm
FLD Theta
FSINCOS
FSTP tbyte ptr [edx] // Cos
FSTP tbyte ptr [eax] // Sin
FWAIT
end;
{ Extract exponent and mantissa from X }
procedure Frexp(const X: Extended; var Mantissa: Extended; var Exponent: Integer);
{ Mantissa ptr in EAX, Exponent ptr in EDX }
asm
FLD X
PUSH EAX
MOV dword ptr [edx], 0 { if X = 0, return 0 }
FTST
FSTSW AX
FWAIT
SAHF
JZ @@Done
FXTRACT // ST(1) = exponent, (pushed) ST = fraction
FXCH
// The FXTRACT instruction normalizes the fraction 1 bit higher than
// wanted for the definition of frexp() so we need to tweak the result
// by scaling the fraction down and incrementing the exponent.
FISTP dword ptr [edx]
FLD1
FCHS
FXCH
FSCALE // scale fraction
INC dword ptr [edx] // exponent biased to match
FSTP ST(1) // discard -1, leave fraction as TOS
@@Done:
POP EAX
FSTP tbyte ptr [eax]
FWAIT
end;
function Ldexp(const X: Extended; const P: Integer): Extended;
{ Result := X * (2^P) }
asm
PUSH EAX
FILD dword ptr [ESP]
FLD X
FSCALE
POP EAX
FSTP ST(1)
FWAIT
end;
function Ceil(const X: Extended): Integer;
begin
Result := Integer(Trunc(X));
if Frac(X) > 0 then
Inc(Result);
end;
function Floor(const X: Extended): Integer;
begin
Result := Integer(Trunc(X));
if Frac(X) < 0 then
Dec(Result);
end;
{ Conversion of bases: Log.b(X) = Log.a(X) / Log.a(b) }
function Log10(const X: Extended): Extended;
{ Log.10(X) := Log.2(X) * Log.10(2) }
asm
FLDLG2 { Log base ten of 2 }
FLD X
FYL2X
FWAIT
end;
function Log2(const X: Extended): Extended;
asm
FLD1
FLD X
FYL2X
FWAIT
end;
function LogN(const Base, X: Extended): Extended;
{ Log.N(X) := Log.2(X) / Log.2(N) }
asm
FLD1
FLD X
FYL2X
FLD1
FLD Base
FYL2X
FDIV
FWAIT
end;
function Poly(const X: Extended; const Coefficients: array of Double): Extended;
{ Horner's method }
var
I: Integer;
begin
Result := Coefficients[High(Coefficients)];
for I := High(Coefficients)-1 downto Low(Coefficients) do
Result := Result * X + Coefficients[I];
end;
function Power(const Base, Exponent: Extended): Extended;
begin
if Exponent = 0.0 then
Result := 1.0 { n**0 = 1 }
else if (Base = 0.0) and (Exponent > 0.0) then
Result := 0.0 { 0**n = 0, n > 0 }
else if (Frac(Exponent) = 0.0) and (Abs(Exponent) <= MaxInt) then
Result := IntPower(Base, Integer(Trunc(Exponent)))
else
Result := Exp(Exponent * Ln(Base))
end;
{ Hyperbolic functions }
function Cosh(const X: Extended): Extended;
begin
if IsZero(X) then
Result := 1
else
Result := (Exp(X) + Exp(-X)) / 2;
end;
function Sinh(const X: Extended): Extended;
begin
if IsZero(X) then
Result := 0
else
Result := (Exp(X) - Exp(-X)) / 2;
end;
function Tanh(const X: Extended): Extended;
begin
if IsZero(X) then
Result := 0
else
Result := SinH(X) / CosH(X);
end;
function ArcCosh(const X: Extended): Extended;
begin
Result := Ln(X + Sqrt((X - 1) / (X + 1)) * (X + 1));
end;
function ArcSinh(const X: Extended): Extended;
begin
Result := Ln(X + Sqrt((X * X) + 1));
end;
function ArcTanh(const X: Extended): Extended;
begin
if SameValue(X, 1) then
Result := Infinity
else if SameValue(X, -1) then
Result := NegInfinity
else
Result := 0.5 * Ln((1 + X) / (1 - X));
end;
function Cot(const X: Extended): Extended;
begin
Result := CoTan(X);
end;
function Sec(const X: Extended): Extended;
begin
Result := Secant(X);
end;
function Csc(const X: Extended): Extended;
begin
Result := Cosecant(X);
end;
function CotH(const X: Extended): Extended;
begin
Result := 1 / TanH(X);
end;
function SecH(const X: Extended): Extended;
begin
Result := 1 / CosH(X);
end;
function CscH(const X: Extended): Extended;
begin
Result := 1 / SinH(X);
end;
function ArcCot(const X: Extended): Extended;
begin
if IsZero(X) then
Result := PI / 2
else
Result := ArcTan(1 / X);
end;
function ArcSec(const X: Extended): Extended;
begin
if IsZero(X) then
Result := Infinity
else
Result := ArcCos(1 / X);
end;
function ArcCsc(const X: Extended): Extended;
begin
if IsZero(X) then
Result := Infinity
else
Result := ArcSin(1 / X);
end;
function ArcCotH(const X: Extended): Extended;
begin
if SameValue(X, 1) then
Result := Infinity
else if SameValue(X, -1) then
Result := NegInfinity
else
Result := 0.5 * Ln((X + 1) / (X - 1));
end;
function ArcSecH(const X: Extended): Extended;
begin
if IsZero(X) then
Result := Infinity
else if SameValue(X, 1) then
Result := 0
else
Result := Ln((Sqrt(1 - X * X) + 1) / X);
end;
function ArcCscH(const X: Extended): Extended;
begin
Result := Ln(Sqrt(1 + (1 / (X * X)) + (1 / X)));
end;
function IsNan(const AValue: Single): Boolean;
begin
Result := ((PLongWord(@AValue)^ and $7F800000) = $7F800000) and
((PLongWord(@AValue)^ and $007FFFFF) <> $00000000);
end;
function IsNan(const AValue: Double): Boolean;
begin
Result := ((PInt64(@AValue)^ and $7FF0000000000000) = $7FF0000000000000) and
((PInt64(@AValue)^ and $000FFFFFFFFFFFFF) <> $0000000000000000);
end;
function IsNan(const AValue: Extended): Boolean;
type
TExtented = packed record
Mantissa: Int64;
Exponent: Word;
end;
PExtended = ^TExtented;
begin
Result := ((PExtended(@AValue)^.Exponent and $7FFF) = $7FFF) and
((PExtended(@AValue)^.Mantissa and $7FFFFFFFFFFFFFFF) <> 0);
end;
function IsInfinite(const AValue: Double): Boolean;
begin
Result := ((PInt64(@AValue)^ and $7FF0000000000000) = $7FF0000000000000) and
((PInt64(@AValue)^ and $000FFFFFFFFFFFFF) = $0000000000000000);
end;
{ Statistical functions }
function Mean(const Data: array of Double): Extended;
begin
Result := SUM(Data) / (High(Data) - Low(Data) + 1);
end;
function MinValue(const Data: array of Double): Double;
var
I: Integer;
begin
Result := Data[Low(Data)];
for I := Low(Data) + 1 to High(Data) do
if Result > Data[I] then
Result := Data[I];
end;
function MinIntValue(const Data: array of Integer): Integer;
var
I: Integer;
begin
Result := Data[Low(Data)];
for I := Low(Data) + 1 to High(Data) do
if Result > Data[I] then
Result := Data[I];
end;
function Min(const A, B: Integer): Integer;
begin
if A < B then
Result := A
else
Result := B;
end;
function Min(const A, B: Int64): Int64;
begin
if A < B then
Result := A
else
Result := B;
end;
function Min(const A, B: Single): Single;
begin
if A < B then
Result := A
else
Result := B;
end;
function Min(const A, B: Double): Double;
begin
if A < B then
Result := A
else
Result := B;
end;
function Min(const A, B: Extended): Extended;
begin
if A < B then
Result := A
else
Result := B;
end;
function MaxValue(const Data: array of Double): Double;
var
I: Integer;
begin
Result := Data[Low(Data)];
for I := Low(Data) + 1 to High(Data) do
if Result < Data[I] then
Result := Data[I];
end;
function MaxIntValue(const Data: array of Integer): Integer;
var
I: Integer;
begin
Result := Data[Low(Data)];
for I := Low(Data) + 1 to High(Data) do
if Result < Data[I] then
Result := Data[I];
end;
function Max(const A, B: Integer): Integer;
begin
if A > B then
Result := A
else
Result := B;
end;
function Max(const A, B: Int64): Int64;
begin
if A > B then
Result := A
else
Result := B;
end;
function Max(const A, B: Single): Single;
begin
if A > B then
Result := A
else
Result := B;
end;
function Max(const A, B: Double): Double;
begin
if A > B then
Result := A
else
Result := B;
end;
function Max(const A, B: Extended): Extended;
begin
if A > B then
Result := A
else
Result := B;
end;
function Sign(const AValue: Integer): TValueSign;
begin
Result := ZeroValue;
if AValue < 0 then
Result := NegativeValue
else if AValue > 0 then
Result := PositiveValue;
end;
function Sign(const AValue: Int64): TValueSign;
begin
Result := ZeroValue;
if AValue < 0 then
Result := NegativeValue
else if AValue > 0 then
Result := PositiveValue;
end;
function Sign(const AValue: Double): TValueSign;
begin
if ((PInt64(@AValue)^ and $7FFFFFFFFFFFFFFF) = $0000000000000000) then
Result := ZeroValue
else if ((PInt64(@AValue)^ and $8000000000000000) = $8000000000000000) then
Result := NegativeValue
else
Result := PositiveValue;
end;
const
FuzzFactor = 1000;
ExtendedResolution = 1E-19 * FuzzFactor;
DoubleResolution = 1E-15 * FuzzFactor;
SingleResolution = 1E-7 * FuzzFactor;
function CompareValue(const A, B: Extended; Epsilon: Extended): TValueRelationship;
begin
if SameValue(A, B, Epsilon) then
Result := EqualsValue
else if A < B then
Result := LessThanValue
else
Result := GreaterThanValue;
end;
function CompareValue(const A, B: Double; Epsilon: Double): TValueRelationship;
begin
if SameValue(A, B, Epsilon) then
Result := EqualsValue
else if A < B then
Result := LessThanValue
else
Result := GreaterThanValue;
end;
function CompareValue(const A, B: Single; Epsilon: Single): TValueRelationship;
begin
if SameValue(A, B, Epsilon) then
Result := EqualsValue
else if A < B then
Result := LessThanValue
else
Result := GreaterThanValue;
end;
function CompareValue(const A, B: Integer): TValueRelationship;
begin
if A = B then
Result := EqualsValue
else if A < B then
Result := LessThanValue
else
Result := GreaterThanValue;
end;
function CompareValue(const A, B: Int64): TValueRelationship;
begin
if A = B then
Result := EqualsValue
else if A < B then
Result := LessThanValue
else
Result := GreaterThanValue;
end;
function SameValue(const A, B: Extended; Epsilon: Extended): Boolean;
begin
if Epsilon = 0 then
Epsilon := Max(Min(Abs(A), Abs(B)) * ExtendedResolution, ExtendedResolution);
if A > B then
Result := (A - B) <= Epsilon
else
Result := (B - A) <= Epsilon;
end;
function SameValue(const A, B: Double; Epsilon: Double): Boolean;
begin
if Epsilon = 0 then
Epsilon := Max(Min(Abs(A), Abs(B)) * DoubleResolution, DoubleResolution);
if A > B then
Result := (A - B) <= Epsilon
else
Result := (B - A) <= Epsilon;
end;
function SameValue(const A, B: Single; Epsilon: Single): Boolean;
begin
if Epsilon = 0 then
Epsilon := Max(Min(Abs(A), Abs(B)) * SingleResolution, SingleResolution);
if A > B then
Result := (A - B) <= Epsilon
else
Result := (B - A) <= Epsilon;
end;
function IsZero(const A: Extended; Epsilon: Extended): Boolean;
begin
if Epsilon = 0 then
Epsilon := ExtendedResolution;
Result := Abs(A) <= Epsilon;
end;
function IsZero(const A: Double; Epsilon: Double): Boolean;
begin
if Epsilon = 0 then
Epsilon := DoubleResolution;
Result := Abs(A) <= Epsilon;
end;
function IsZero(const A: Single; Epsilon: Single): Boolean;
begin
if Epsilon = 0 then
Epsilon := SingleResolution;
Result := Abs(A) <= Epsilon;
end;
function IfThen(AValue: Boolean; const ATrue: Integer; const AFalse: Integer): Integer;
begin
if AValue then
Result := ATrue
else
Result := AFalse;
end;
function IfThen(AValue: Boolean; const ATrue: Int64; const AFalse: Int64): Int64;
begin
if AValue then
Result := ATrue
else
Result := AFalse;
end;
function IfThen(AValue: Boolean; const ATrue: Double; const AFalse: Double): Double;
begin
if AValue then
Result := ATrue
else
Result := AFalse;
end;
function RandomRange(const AFrom, ATo: Integer): Integer;
begin
if AFrom > ATo then
Result := Random(AFrom - ATo) + ATo
else
Result := Random(ATo - AFrom) + AFrom;
end;
function RandomFrom(const AValues: array of Integer): Integer;
begin
Result := AValues[Random(High(AValues) + 1)];
end;
function RandomFrom(const AValues: array of Int64): Int64;
begin
Result := AValues[Random(High(AValues) + 1)];
end;
function RandomFrom(const AValues: array of Double): Double;
begin
Result := AValues[Random(High(AValues) + 1)];
end;
{ Range testing functions }
function InRange(const AValue, AMin, AMax: Integer): Boolean;
begin
Result := (AValue >= AMin) and (AValue <= AMax);
end;
function InRange(const AValue, AMin, AMax: Int64): Boolean;
begin
Result := (AValue >= AMin) and (AValue <= AMax);
end;
function InRange(const AValue, AMin, AMax: Double): Boolean;
begin
Result := (AValue >= AMin) and (AValue <= AMax);
end;
{ Range truncation functions }
function EnsureRange(const AValue, AMin, AMax: Integer): Integer;
begin
Result := AValue;
assert(AMin <= AMax);
if Result < AMin then
Result := AMin;
if Result > AMax then
Result := AMax;
end;
function EnsureRange(const AValue, AMin, AMax: Int64): Int64;
begin
Result := AValue;
assert(AMin <= AMax);
if Result < AMin then
Result := AMin;
if Result > AMax then
Result := AMax;
end;
function EnsureRange(const AValue, AMin, AMax: Double): Double;
begin
Result := AValue;
assert(AMin <= AMax);
if Result < AMin then
Result := AMin;
if Result > AMax then
Result := AMax;
end;
procedure MeanAndStdDev(const Data: array of Double; var Mean, StdDev: Extended);
var
S: Extended;
N,I: Integer;
begin
N := High(Data)- Low(Data) + 1;
if N = 1 then
begin
Mean := Data[0];
StdDev := Data[0];
Exit;
end;
Mean := Sum(Data) / N;
S := 0; // sum differences from the mean, for greater accuracy
for I := Low(Data) to High(Data) do
S := S + Sqr(Mean - Data[I]);
StdDev := Sqrt(S / (N - 1));
end;
procedure MomentSkewKurtosis(const Data: array of Double;
var M1, M2, M3, M4, Skew, Kurtosis: Extended);
var
Sum, SumSquares, SumCubes, SumQuads, OverN, Accum, M1Sqr, S2N, S3N: Extended;
I: Integer;
begin
OverN := 1 / (High(Data) - Low(Data) + 1);
Sum := 0;
SumSquares := 0;
SumCubes := 0;
SumQuads := 0;
for I := Low(Data) to High(Data) do
begin
Sum := Sum + Data[I];
Accum := Sqr(Data[I]);
SumSquares := SumSquares + Accum;
Accum := Accum*Data[I];
SumCubes := SumCubes + Accum;
SumQuads := SumQuads + Accum*Data[I];
end;
M1 := Sum * OverN;
M1Sqr := Sqr(M1);
S2N := SumSquares * OverN;
S3N := SumCubes * OverN;
M2 := S2N - M1Sqr;
M3 := S3N - (M1 * 3 * S2N) + 2*M1Sqr*M1;
M4 := (SumQuads * OverN) - (M1 * 4 * S3N) + (M1Sqr*6*S2N - 3*Sqr(M1Sqr));
Skew := M3 * Power(M2, -3/2); // = M3 / Power(M2, 3/2)
Kurtosis := M4 / Sqr(M2);
end;
function Norm(const Data: array of Double): Extended;
begin
Result := Sqrt(SumOfSquares(Data));
end;
function PopnStdDev(const Data: array of Double): Extended;
begin
Result := Sqrt(PopnVariance(Data))
end;
function PopnVariance(const Data: array of Double): Extended;
begin
Result := TotalVariance(Data) / (High(Data) - Low(Data) + 1)
end;
function RandG(Mean, StdDev: Extended): Extended;
{ Marsaglia-Bray algorithm }
var
U1, S2: Extended;
begin
repeat
U1 := 2*Random - 1;
S2 := Sqr(U1) + Sqr(2*Random-1);
until S2 < 1;
Result := Sqrt(-2*Ln(S2)/S2) * U1 * StdDev + Mean;
end;
function StdDev(const Data: array of Double): Extended;
begin
Result := Sqrt(Variance(Data))
end;
procedure RaiseOverflowError; forward;
function SumInt(const Data: array of Integer): Integer;
asm // IN: EAX = ptr to Data, EDX = High(Data) = Count - 1
// loop unrolled 4 times, 5 clocks per loop, 1.2 clocks per datum
PUSH EBX
MOV ECX, EAX // ecx = ptr to data
MOV EBX, EDX
XOR EAX, EAX
AND EDX, not 3
AND EBX, 3
SHL EDX, 2
JMP @Vector.Pointer[EBX*4]
@Vector:
DD @@1
DD @@2
DD @@3
DD @@4
@@4:
ADD EAX, [ECX+12+EDX]
JO RaiseOverflowError
@@3:
ADD EAX, [ECX+8+EDX]
JO RaiseOverflowError
@@2:
ADD EAX, [ECX+4+EDX]
JO RaiseOverflowError
@@1:
ADD EAX, [ECX+EDX]
JO RaiseOverflowError
SUB EDX,16
JNS @@4
POP EBX
end;
procedure RaiseOverflowError;
begin
raise EIntOverflow.Create(SIntOverflow);
end;
function SUM(const Data: array of Double): Extended;
asm // IN: EAX = ptr to Data, EDX = High(Data) = Count - 1
// Uses 4 accumulators to minimize read-after-write delays and loop overhead
// 5 clocks per loop, 4 items per loop = 1.2 clocks per item
FLDZ
MOV ECX, EDX
FLD ST(0)
AND EDX, not 3
FLD ST(0)
AND ECX, 3
FLD ST(0)
SHL EDX, 3 // count * sizeof(Double) = count * 8
JMP @Vector.Pointer[ECX*4]
@Vector:
DD @@1
DD @@2
DD @@3
DD @@4
@@4: FADD qword ptr [EAX+EDX+24] // 1
FXCH ST(3) // 0
@@3: FADD qword ptr [EAX+EDX+16] // 1
FXCH ST(2) // 0
@@2: FADD qword ptr [EAX+EDX+8] // 1
FXCH ST(1) // 0
@@1: FADD qword ptr [EAX+EDX] // 1
FXCH ST(2) // 0
SUB EDX, 32
JNS @@4
FADDP ST(3),ST // ST(3) := ST + ST(3); Pop ST
FADD // ST(1) := ST + ST(1); Pop ST
FADD // ST(1) := ST + ST(1); Pop ST
FWAIT
end;
function SumOfSquares(const Data: array of Double): Extended;
var
I: Integer;
begin
Result := 0.0;
for I := Low(Data) to High(Data) do
Result := Result + Sqr(Data[I]);
end;
procedure SumsAndSquares(const Data: array of Double; var Sum, SumOfSquares: Extended);
asm // IN: EAX = ptr to Data
// EDX = High(Data) = Count - 1
// ECX = ptr to Sum
// Est. 17 clocks per loop, 4 items per loop = 4.5 clocks per data item
FLDZ // init Sum accumulator
PUSH ECX
MOV ECX, EDX
FLD ST(0) // init Sqr1 accum.
AND EDX, not 3
FLD ST(0) // init Sqr2 accum.
AND ECX, 3
FLD ST(0) // init/simulate last data item left in ST
SHL EDX, 3 // count * sizeof(Double) = count * 8
JMP @Vector.Pointer[ECX*4]
@Vector:
DD @@1
DD @@2
DD @@3
DD @@4
@@4: FADD // Sqr2 := Sqr2 + Sqr(Data4); Pop Data4
FLD qword ptr [EAX+EDX+24] // Load Data1
FADD ST(3),ST // Sum := Sum + Data1
FMUL ST,ST // Data1 := Sqr(Data1)
@@3: FLD qword ptr [EAX+EDX+16] // Load Data2
FADD ST(4),ST // Sum := Sum + Data2
FMUL ST,ST // Data2 := Sqr(Data2)
FXCH // Move Sqr(Data1) into ST(0)
FADDP ST(3),ST // Sqr1 := Sqr1 + Sqr(Data1); Pop Data1
@@2: FLD qword ptr [EAX+EDX+8] // Load Data3
FADD ST(4),ST // Sum := Sum + Data3
FMUL ST,ST // Data3 := Sqr(Data3)
FXCH // Move Sqr(Data2) into ST(0)
FADDP ST(3),ST // Sqr1 := Sqr1 + Sqr(Data2); Pop Data2
@@1: FLD qword ptr [EAX+EDX] // Load Data4
FADD ST(4),ST // Sum := Sum + Data4
FMUL ST,ST // Sqr(Data4)
FXCH // Move Sqr(Data3) into ST(0)
FADDP ST(3),ST // Sqr1 := Sqr1 + Sqr(Data3); Pop Data3
SUB EDX,32
JNS @@4
FADD // Sqr2 := Sqr2 + Sqr(Data4); Pop Data4
POP ECX
FADD // Sqr1 := Sqr2 + Sqr1; Pop Sqr2
FXCH // Move Sum1 into ST(0)
MOV EAX, SumOfSquares
FSTP tbyte ptr [ECX] // Sum := Sum1; Pop Sum1
FSTP tbyte ptr [EAX] // SumOfSquares := Sum1; Pop Sum1
FWAIT
end;
function TotalVariance(const Data: array of Double): Extended;
var
Sum, SumSquares: Extended;
begin
SumsAndSquares(Data, Sum, SumSquares);
Result := SumSquares - Sqr(Sum)/(High(Data) - Low(Data) + 1);
end;
function Variance(const Data: array of Double): Extended;
begin
Result := TotalVariance(Data) / (High(Data) - Low(Data))
end;
{ Depreciation functions. }
function DoubleDecliningBalance(const Cost, Salvage: Extended; Life, Period: Integer): Extended;
{ dv := cost * (1 - 2/life)**(period - 1)
DDB = (2/life) * dv
if DDB > dv - salvage then DDB := dv - salvage
if DDB < 0 then DDB := 0
}
var
DepreciatedVal, Factor: Extended;
begin
Result := 0;
if (Period < 1) or (Life < Period) or (Life < 1) or (Cost <= Salvage) then
Exit;
{depreciate everything in period 1 if life is only one or two periods}
if ( Life <= 2 ) then
begin
if ( Period = 1 ) then
DoubleDecliningBalance:=Cost-Salvage
else
DoubleDecliningBalance:=0; {all depreciation occurred in first period}
exit;
end;
Factor := 2.0 / Life;
DepreciatedVal := Cost * IntPower((1.0 - Factor), Period - 1);
{DepreciatedVal is Cost-(sum of previous depreciation results)}
Result := Factor * DepreciatedVal;
{Nominal computed depreciation for this period. The rest of the
function applies limits to this nominal value. }
{Only depreciate until total depreciation equals cost-salvage.}
if Result > DepreciatedVal - Salvage then
Result := DepreciatedVal - Salvage;
{No more depreciation after salvage value is reached. This is mostly a nit.
If Result is negative at this point, it's very close to zero.}
if Result < 0.0 then
Result := 0.0;
end;
function SLNDepreciation(const Cost, Salvage: Extended; Life: Integer): Extended;
{ Spreads depreciation linearly over life. }
begin
if Life < 1 then ArgError('SLNDepreciation');
Result := (Cost - Salvage) / Life
end;
function SYDDepreciation(const Cost, Salvage: Extended; Life, Period: Integer): Extended;
{ SYD = (cost - salvage) * (life - period + 1) / (life*(life + 1)/2) }
{ Note: life*(life+1)/2 = 1+2+3+...+life "sum of years"
The depreciation factor varies from life/sum_of_years in first period = 1
downto 1/sum_of_years in last period = life.
Total depreciation over life is cost-salvage.}
var
X1, X2: Extended;
begin
Result := 0;
if (Period < 1) or (Life < Period) or (Cost <= Salvage) then Exit;
X1 := 2 * (Life - Period + 1);
X2 := Life * (Life + 1);
Result := (Cost - Salvage) * X1 / X2
end;
{ Discounted cash flow functions. }
function InternalRateOfReturn(const Guess: Extended; const CashFlows: array of Double): Extended;
{
Use Newton's method to solve NPV = 0, where NPV is a polynomial in
x = 1/(1+rate). Split the coefficients into negative and postive sets:
neg + pos = 0, so pos = -neg, so -neg/pos = 1
Then solve:
log(-neg/pos) = 0
Let t = log(1/(1+r) = -LnXP1(r)
then r = exp(-t) - 1
Iterate on t, then use the last equation to compute r.
}
var
T, Y: Extended;
Poly: TPoly;
K, Count: Integer;
function ConditionP(const CashFlows: array of Double): Integer;
{ Guarantees existence and uniqueness of root. The sign of payments
must change exactly once, the net payout must be always > 0 for
first portion, then each payment must be >= 0.
Returns: 0 if condition not satisfied, > 0 if condition satisfied
and this is the index of the first value considered a payback. }
var
X: Double;
I, K: Integer;
begin
K := High(CashFlows);
while (K >= 0) and (CashFlows[K] >= 0.0) do Dec(K);
Inc(K);
if K > 0 then
begin
X := 0.0;
I := 0;
while I < K do
begin
X := X + CashFlows[I];
if X >= 0.0 then
begin
K := 0;
Break;
end;
Inc(I)
end
end;
ConditionP := K
end;
begin
InternalRateOfReturn := 0;
K := ConditionP(CashFlows);
if K < 0 then ArgError('InternalRateOfReturn');
if K = 0 then
begin
if Guess <= -1.0 then ArgError('InternalRateOfReturn');
T := -LnXP1(Guess)
end else
T := 0.0;
for Count := 1 to MaxIterations do
begin
PolyX(CashFlows, Exp(T), Poly);
if Poly.Pos <= Poly.Neg then ArgError('InternalRateOfReturn');
if (Poly.Neg >= 0.0) or (Poly.Pos <= 0.0) then
begin
InternalRateOfReturn := -1.0;
Exit;
end;
with Poly do
Y := Ln(-Neg / Pos) / (DNeg / Neg - DPos / Pos);
T := T - Y;
if RelSmall(Y, T) then
begin
InternalRateOfReturn := Exp(-T) - 1.0;
Exit;
end
end;
ArgError('InternalRateOfReturn');
end;
function NetPresentValue(const Rate: Extended; const CashFlows: array of Double;
PaymentTime: TPaymentTime): Extended;
{ Caution: The sign of NPV is reversed from what would be expected for standard
cash flows!}
var
rr: Extended;
I: Integer;
begin
if Rate <= -1.0 then ArgError('NetPresentValue');
rr := 1/(1+Rate);
result := 0;
for I := High(CashFlows) downto Low(CashFlows) do
result := rr * result + CashFlows[I];
if PaymentTime = ptEndOfPeriod then result := rr * result;
end;
{ Annuity functions. }
{---------------
From the point of view of A, amounts received by A are positive and
amounts disbursed by A are negative (e.g. a borrower's loan repayments
are regarded by the borrower as negative).
Given interest rate r, number of periods n:
compound(r, n) = (1 + r)**n "Compounding growth factor"
annuity(r, n) = (compound(r, n)-1) / r "Annuity growth factor"
Given future value fv, periodic payment pmt, present value pv and type
of payment (start, 1 , or end of period, 0) pmtTime, financial variables satisfy:
fv = -pmt*(1 + r*pmtTime)*annuity(r, n) - pv*compound(r, n)
For fv, pv, pmt:
C := compound(r, n)
A := (1 + r*pmtTime)*annuity(r, n)
Compute both at once in Annuity2.
if C > 1E16 then A = C/r, so:
fv := meaningless
pv := -pmt*(pmtTime+1/r)
pmt := -pv*r/(1 + r*pmtTime)
else
fv := -pmt(1+r*pmtTime)*A - pv*C
pv := (-pmt(1+r*pmtTime)*A - fv)/C
pmt := (-pv*C-fv)/((1+r*pmtTime)*A)
---------------}
function PaymentParts(Period, NPeriods: Integer; Rate, PresentValue,
FutureValue: Extended; PaymentTime: TPaymentTime; var IntPmt: Extended):
Extended;
var
Crn:extended; { =Compound(Rate,NPeriods) }
Crp:extended; { =Compound(Rate,Period-1) }
Arn:extended; { =Annuity2(...) }
begin
if Rate <= -1.0 then ArgError('PaymentParts');
Crp:=Compound(Rate,Period-1);
Arn:=Annuity2(Rate,NPeriods,PaymentTime,Crn);
IntPmt:=(FutureValue*(Crp-1)-PresentValue*(Crn-Crp))/Arn;
PaymentParts:=(-FutureValue-PresentValue)*Crp/Arn;
end;
function FutureValue(const Rate: Extended; NPeriods: Integer; const Payment,
PresentValue: Extended; PaymentTime: TPaymentTime): Extended;
var
Annuity, CompoundRN: Extended;
begin
if Rate <= -1.0 then ArgError('FutureValue');
Annuity := Annuity2(Rate, NPeriods, PaymentTime, CompoundRN);
if CompoundRN > 1.0E16 then ArgError('FutureValue');
FutureValue := -Payment * Annuity - PresentValue * CompoundRN
end;
function InterestPayment(const Rate: Extended; Period, NPeriods: Integer;
const PresentValue, FutureValue: Extended; PaymentTime: TPaymentTime): Extended;
var
Crp:extended; { compound(rate,period-1)}
Crn:extended; { compound(rate,nperiods)}
Arn:extended; { annuityf(rate,nperiods)}
begin
if (Rate <= -1.0)
or (Period < 1) or (Period > NPeriods) then ArgError('InterestPayment');
Crp:=Compound(Rate,Period-1);
Arn:=Annuity2(Rate,Nperiods,PaymentTime,Crn);
InterestPayment:=(FutureValue*(Crp-1)-PresentValue*(Crn-Crp))/Arn;
end;
function InterestRate(NPeriods: Integer; const Payment, PresentValue,
FutureValue: Extended; PaymentTime: TPaymentTime): Extended;
{
Given:
First and last payments are non-zero and of opposite signs.
Number of periods N >= 2.
Convert data into cash flow of first, N-1 payments, last with
first < 0, payment > 0, last > 0.
Compute the IRR of this cash flow:
0 = first + pmt*x + pmt*x**2 + ... + pmt*x**(N-1) + last*x**N
where x = 1/(1 + rate).
Substitute x = exp(t) and apply Newton's method to
f(t) = log(pmt*x + ... + last*x**N) / -first
which has a unique root given the above hypotheses.
}
var
X, Y, Z, First, Pmt, Last, T, ET, EnT, ET1: Extended;
Count: Integer;
Reverse: Boolean;
function LostPrecision(X: Extended): Boolean;
asm
XOR EAX, EAX
MOV BX,WORD PTR X+8
INC EAX
AND EBX, $7FF0
JZ @@1
CMP EBX, $7FF0
JE @@1
XOR EAX,EAX
@@1:
end;
begin
Result := 0;
if NPeriods <= 0 then ArgError('InterestRate');
Pmt := Payment;
if PaymentTime = ptEndOfPeriod then
begin
X := PresentValue;
Y := FutureValue + Payment
end
else
begin
X := PresentValue + Payment;
Y := FutureValue
end;
First := X;
Last := Y;
Reverse := False;
if First * Payment > 0.0 then
begin
Reverse := True;
T := First;
First := Last;
Last := T
end;
if first > 0.0 then
begin
First := -First;
Pmt := -Pmt;
Last := -Last
end;
if (First = 0.0) or (Last < 0.0) then ArgError('InterestRate');
T := 0.0; { Guess at solution }
for Count := 1 to MaxIterations do
begin
EnT := Exp(NPeriods * T);
if {LostPrecision(EnT)} ent=(ent+1) then
begin
Result := -Pmt / First;
if Reverse then
Result := Exp(-LnXP1(Result)) - 1.0;
Exit;
end;
ET := Exp(T);
ET1 := ET - 1.0;
if ET1 = 0.0 then
begin
X := NPeriods;
Y := X * (X - 1.0) / 2.0
end
else
begin
X := ET * (Exp((NPeriods - 1) * T)-1.0) / ET1;
Y := (NPeriods * EnT - ET - X * ET) / ET1
end;
Z := Pmt * X + Last * EnT;
Y := Ln(Z / -First) / ((Pmt * Y + Last * NPeriods *EnT) / Z);
T := T - Y;
if RelSmall(Y, T) then
begin
if not Reverse then T := -T;
InterestRate := Exp(T)-1.0;
Exit;
end
end;
ArgError('InterestRate');
end;
function NumberOfPeriods(const Rate: Extended; Payment: Extended;
const PresentValue, FutureValue: Extended; PaymentTime: TPaymentTime): Extended;
{ If Rate = 0 then nper := -(pv + fv) / pmt
else cf := pv + pmt * (1 + rate*pmtTime) / rate
nper := LnXP1(-(pv + fv) / cf) / LnXP1(rate) }
var
PVRPP: Extended; { =PV*Rate+Payment } {"initial cash flow"}
T: Extended;
begin
if Rate <= -1.0 then ArgError('NumberOfPeriods');
{whenever both Payment and PaymentTime are given together, the PaymentTime has the effect
of modifying the effective Payment by the interest accrued on the Payment}
if ( PaymentTime=ptStartOfPeriod ) then
Payment:=Payment*(1+Rate);
{if the payment exactly matches the interest accrued periodically on the
presentvalue, then an infinite number of payments are going to be
required to effect a change from presentvalue to futurevalue. The
following catches that specific error where payment is exactly equal,
but opposite in sign to the interest on the present value. If PVRPP
("initial cash flow") is simply close to zero, the computation will
be numerically unstable, but not as likely to cause an error.}
PVRPP:=PresentValue*Rate+Payment;
if PVRPP=0 then ArgError('NumberOfPeriods');
{ 6.1E-5 approx= 2**-14 }
if ( ABS(Rate)<6.1E-5 ) then
Result:=-(PresentValue+FutureValue)/PVRPP
else
begin
{starting with the initial cash flow, each compounding period cash flow
should result in the current value approaching the final value. The
following test combines a number of simultaneous conditions to ensure
reasonableness of the cashflow before computing the NPER.}
T:= -(PresentValue+FutureValue)*Rate/PVRPP;
if T<=-1.0 then ArgError('NumberOfPeriods');
Result := LnXP1(T) / LnXP1(Rate)
end;
NumberOfPeriods:=Result;
end;
function Payment(Rate: Extended; NPeriods: Integer; const PresentValue,
FutureValue: Extended; PaymentTime: TPaymentTime): Extended;
var
Annuity, CompoundRN: Extended;
begin
if Rate <= -1.0 then ArgError('Payment');
Annuity := Annuity2(Rate, NPeriods, PaymentTime, CompoundRN);
if CompoundRN > 1.0E16 then
Payment := -PresentValue * Rate / (1 + Integer(PaymentTime) * Rate)
else
Payment := (-PresentValue * CompoundRN - FutureValue) / Annuity
end;
function PeriodPayment(const Rate: Extended; Period, NPeriods: Integer;
const PresentValue, FutureValue: Extended; PaymentTime: TPaymentTime): Extended;
var
Junk: Extended;
begin
if (Rate <= -1.0) or (Period < 1) or (Period > NPeriods) then ArgError('PeriodPayment');
PeriodPayment := PaymentParts(Period, NPeriods, Rate, PresentValue,
FutureValue, PaymentTime, Junk);
end;
function PresentValue(const Rate: Extended; NPeriods: Integer; const Payment,
FutureValue: Extended; PaymentTime: TPaymentTime): Extended;
var
Annuity, CompoundRN: Extended;
begin
if Rate <= -1.0 then ArgError('PresentValue');
Annuity := Annuity2(Rate, NPeriods, PaymentTime, CompoundRN);
if CompoundRN > 1.0E16 then
PresentValue := -(Payment / Rate * Integer(PaymentTime) * Payment)
else
PresentValue := (-Payment * Annuity - FutureValue) / CompoundRN
end;
function GetRoundMode: TFPURoundingMode;
begin
Result := TFPURoundingMode((Get8087CW shr 10) and 3);
end;
function SetRoundMode(const RoundMode: TFPURoundingMode): TFPURoundingMode;
var
CtlWord: Word;
begin
CtlWord := Get8087CW;
Set8087CW((CtlWord and $F3FF) or (Ord(RoundMode) shl 10));
Result := TFPURoundingMode((CtlWord shr 10) and 3);
end;
function GetPrecisionMode: TFPUPrecisionMode;
begin
Result := TFPUPrecisionMode((Get8087CW shr 8) and 3);
end;
function SetPrecisionMode(const Precision: TFPUPrecisionMode): TFPUPrecisionMode;
var
CtlWord: Word;
begin
CtlWord := Get8087CW;
Set8087CW((CtlWord and $FCFF) or (Ord(Precision) shl 8));
Result := TFPUPrecisionMode((CtlWord shr 8) and 3);
end;
function GetExceptionMask: TFPUExceptionMask;
begin
Byte(Result) := Get8087CW and $3F;
end;
function SetExceptionMask(const Mask: TFPUExceptionMask): TFPUExceptionMask;
var
CtlWord: Word;
begin
CtlWord := Get8087CW;
Set8087CW( (CtlWord and $FFC0) or Byte(Mask) );
Byte(Result) := CtlWord and $3F;
end;
procedure ClearExceptions(RaisePending: Boolean);
asm
cmp al, 0
jz @@clear
fwait
@@clear:
fnclex
end;
end.
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