Formal power series
You are encouraged to solve this task according to the task description, using any language you may know.
A power series is an infinite sum of the form
The ai are called the coefficients of the series. Such sums can be added, multiplied etc., where the new coefficients of the powers of x are calculated according to the usual rules.
If one is not interested in evaluating such a series for particular values of x, or in other words, if convergence doesn't play a role, then such a collection of coefficients is called formal power series. It can be treated like a new kind of number.
Task: Implement formal power series as a numeric type. Operations should at least include addition, multiplication, division and additionally non-numeric operations like differentiation and integration (with an integration constant of zero). Take care that your implementation deals with the potentially infinite number of coefficients.
As an example, define the power series of sine and cosine in terms of each other using integration, as in
Goals: Demonstrate how the language handles new numeric types and delayed (or lazy) evaluation.
Ada
The Taylor series package is generic to be instantiated with any rational type implementation provided by the task Rational Arithmetic: <lang ada> with Generic_Rational;
generic
with package Rational_Numbers is new Generic_Rational (<>);
package Generic_Taylor_Series is
use Rational_Numbers; type Taylor_Series is array (Natural range <>) of Rational; function "+" (A : Taylor_Series) return Taylor_Series; function "-" (A : Taylor_Series) return Taylor_Series;
function "+" (A, B : Taylor_Series) return Taylor_Series; function "-" (A, B : Taylor_Series) return Taylor_Series; function "*" (A, B : Taylor_Series) return Taylor_Series;
function Integral (A : Taylor_Series) return Taylor_Series; function Differential (A : Taylor_Series) return Taylor_Series;
function Value (A : Taylor_Series; X : Rational) return Rational;
Zero : constant Taylor_Series := (0 => Rational_Numbers.Zero); One : constant Taylor_Series := (0 => Rational_Numbers.One);
end Generic_Taylor_Series; </lang> The package implementation: <lang ada> package body Generic_Taylor_Series is
function Normalize (A : Taylor_Series) return Taylor_Series is
begin
for Power in reverse A'Range loop
if A (Power) /= 0 then
return A (0..Power);
end if;
end loop;
return Zero;
end Normalize;
function "+" (A : Taylor_Series) return Taylor_Series is
begin
return A;
end "+";
function "-" (A : Taylor_Series) return Taylor_Series is
Result : Taylor_Series (A'Range);
begin
for Power in A'Range loop
Result (Power) := -A (Power);
end loop;
return Result;
end "-";
function "+" (A, B : Taylor_Series) return Taylor_Series is
begin
if A'Last > B'Last then
return B + A;
else
declare
Result : Taylor_Series (0..B'Last);
begin
for Power in A'Range loop
Result (Power) := A (Power) + B (Power);
end loop;
for Power in A'Last + 1..B'Last loop
Result (Power) := B (Power);
end loop;
return Normalize (Result);
end;
end if;
end "+";
function "-" (A, B : Taylor_Series) return Taylor_Series is
begin
return A + (-B);
end "-";
function "*" (A, B : Taylor_Series) return Taylor_Series is
Result : Taylor_Series (0..A'Last + B'Last);
begin
for I in A'Range loop
for J in B'Range loop
Result (I + J) := A (I) * B (J);
end loop;
end loop;
return Normalize (Result);
end "*";
function Integral (A : Taylor_Series) return Taylor_Series is
begin
if A = Zero then
return Zero;
else
declare
Result : Taylor_Series (0..A'Last + 1);
begin
for Power in A'Range loop
Result (Power + 1) := A (Power) / Number (Power + 1);
end loop;
Result (0) := Rational_Numbers.Zero;
return Result;
end;
end if;
end Integral;
function Differential (A : Taylor_Series) return Taylor_Series is
begin
if A'Length = 1 then
return Zero;
else
declare
Result : Taylor_Series (0..A'Last - 1);
begin
for Power in Result'Range loop
Result (Power) := A (Power + 1) * Number (Power);
end loop;
return Result;
end;
end if;
end Differential;
function Value (A : Taylor_Series; X : Rational) return Rational is
Sum : Rational := A (A'Last);
begin
for Power in reverse 0..A'Last - 1 loop
Sum := Sum * X + A (Power);
end loop;
return Sum;
end Value;
end Generic_Taylor_Series; </lang> The procedure Normalize is used to truncate the series when the coefficients are zero. The summation of a series (function Value) uses Horner scheme.
Test task
<lang ada> with Ada.Text_IO; use Ada.Text_IO;
with Generic_Taylor_Series; with Generic_Rational;
procedure Test_Taylor_Series is
package Integer_Rationals is new Generic_Rational (Integer);
package Integer_Taylor_Series is new Generic_Taylor_Series (Integer_Rationals);
use Integer_Taylor_Series;
-- Procedure to print a series
procedure Put (A : Taylor_Series) is
use Integer_Rationals;
procedure Put (A : Rational) is
begin
if Numerator (A) = 1 then
Put (" 1");
else
Put (Integer'Image (Numerator (A)));
end if;
if Denominator (A) /= 1 then
Put (" /");
Put (Integer'Image (Denominator (A)));
end if;
end Put;
begin
if A (0) /= 0 then
Put (A (0));
end if;
for Power in 1..A'Last loop
if A (Power) > 0 then
Put (" +");
Put (A (Power));
Put (" X **" & Integer'Image (Power));
elsif A (Power) < 0 then
Put (" -");
Put (abs A (Power));
Put (" X **" & Integer'Image (Power));
end if;
end loop;
end Put;
-- Cosine generator
function Cos (N : Natural) return Taylor_Series is
begin
if N = 0 then
return One;
else
return One - Integral (Integral (Cos (N - 1)));
end if;
end Cos;
begin
Put ("Cos ="); Put (Cos (5)); Put_Line (" ...");
Put ("Sin ="); Put (Integral (Cos (5))); Put_Line (" ...");
end Test_Taylor_Series; </lang> Sample output:
Cos = 1 - 1 / 2 X ** 2 + 1 / 24 X ** 4 - 1 / 720 X ** 6 + 1 / 40320 X ** 8 - 1 / 3628800 X ** 10 ... Sin = + 1 X ** 1 - 1 / 6 X ** 3 + 1 / 120 X ** 5 - 1 / 5040 X ** 7 + 1 / 362880X ** 9 - 1 / 39916800 X ** 11 ...
D
| <lang d>
module fps; import q ; // a module for Rational Number import std.format : sformat = format; import std.math, std.stdio ; interface Gene(U) { U coef(int n) ; } class Term(U) { U[] cache ;
Gene!(U) gene ;
this(Gene!(U) g) { gene = g ; }
U opIndex(int n) {
if(n < 0) return cast(U)0 ;
if(n >= cache.length)
for(int i = cache.length ; i <= n ; i++)
cache ~= gene.coef(i) ;
return cache[n] ;
}
} struct FPS(U) { alias Term!(U) UT ;
alias Gene!(U) UG ;
alias FPS!(U) UF ;
static int DispTerm = 12 ;
static string XVar = "x" ;
UT term ;
static UF opCall(UT t) { UF f ; f.term = t ; return f ;}
static UF opCall(U[] polynomial) {
UF f ;
f.term = new UT( new class() UG { U coef(int n) {
if (n < 0 || n >= polynomial.length )
return cast(U)0 ;
return polynomial[n] ;
};}) ;
return f ;
}
U inverseCoef(int n) {
U[] res = new U[n + 1] ;
res[0] = 1/term[0] ;
for(int i = 1 ; i <= n ; i++) {
res[i] = cast(U)0 ;
for(int j = 0 ; j < i ; j++) {
res[i] += term[i - j] * res[j] ;
}
res[i] = -res[0] * res[i] ;
}
return res[n] ;
}
UF opAdd(UF rhs) { return UF(new UT(new class() UG { U coef(int n) {
return term[n] + rhs.term[n] ;
}}));}
UF opSub(UF rhs) { return UF(new UT(new class() UG { U coef(int n) {
return term[n] - rhs.term[n] ;
}}));}
UF opMul(UF rhs) { return UF(new UT(new class() UG { U coef(int n) {
U res = cast(U) 0 ;
for(int i = 0 ; i <=n ; i++)
res += term[i]*rhs.term[n -i] ; // Cauchy product with rhs
return res ;
}}));}
UF opDiv(UF rhs) { return UF(new UT(new class() UG { U coef(int n) {
U res = cast(U) 0 ;
for(int i = 0 ; i <=n ; i++)
res += term[i]*rhs.inverseCoef(n -i) ; // Cauchy product with inverse of rhs
return res ;
}}));}
UF Diff() { return UF(new UT(new class() UG { U coef(int n) {
return term[n + 1] * (n + 1) ;
}}));}
UF Intg() { return UF(new UT(new class() UG { U coef(int n) {
if (n == 0) return cast(U)0 ;
return term[n-1]/n ;
}}));}
string toString(int dpTerm = -1) {
string s ;
if(dpTerm <= 0) dpTerm = DispTerm ;
U c = term[0] ;
if(c != cast(U)0) s ~= sformat("%s", c) ;
for(int i = 1 ; i < dpTerm ; i++)
if((c = term[i]) != cast(U)0) {
string t ;
if(c > 0 && s.length > 0)
t = "+" ;
if (c == cast(U)1)
t ~= XVar ;
else if ( c == cast(U)-1)
t ~= sformat("-%s", XVar) ;
else
t ~= sformat("%s%s", c, XVar) ;
if(i > 1) t ~= sformat("%s", i) ;
s ~= t ;
}
if(s.length == 0)
s = "0" ;
s ~= "+..." ;
return s ;
}
} alias Q!(long) QT ; alias FPS!(QT) RF ; void main() { QT.DenoLimit = 0 ; RF COS ;
RF SIN = COS.Intg ;
COS = RF([QT(1,1)]) - SIN.Intg ;
writefln("SIN(x) = ",SIN) ;
writefln("COS(x) = ",COS) ;
}</lang> |
Output:
SIN(x) = x-1/6x3+1/120x5-1/5040x7+1/362880x9-1/39916800x11+... COS(x) = 1-1/2x2+1/24x4-1/720x6+1/40320x8-1/3628800x10+...
The q module. Because of the limited precision of long, the denominator will quickly overflow when multiplied, but it is sufficient to show first few coefficients in this task.
| <lang d>
module q; import std.format : sformat = format; T gcd(T)(T u, T v) { // binary gcd algorithm if(u < 0) u = -u ;
if(v < 0) v = -v ;
if(u == 0 || v == 0) return u | v ;
int shift = 0 ;
for(; 0 == ((u|v) & 1) ; shift++) { u >>>= 1 ; v >>>= 1 ; }
while((u & 1) == 0) u >>>= 1 ;
while (v != 0) {
while((v & 1) == 0) v >>>= 1 ;
if (u <= v) v -= u ;
else { v = u - v ; u -= v ;}
v >>>= 1 ;
}
return u << shift ;
} struct Q(T) { static T DenoLimit = int.max ; alias Q!(T) QX ; T num, den ; static QX opCall(long n, long d) {
QX q ;
with(q) { num = n ; den = d; reduce ; }
return q ;
}
static QX opCall(real r) {
QX q ;
T multiple = 1_000_000 ;
with(q) { num = cast(T)(r*multiple) ; den = multiple; reduce ; }
return q ;
}
void reduce() {
T g = gcd(num, den) ;
if(g == 0 || den == 0)
throw new Exception(sformat("Zero denominator:%s/%s gcd:%s", num, den, g )) ;
if(num == 0) { den = 1 ; return ; }
num /= g ; den /= g ;
if(den < 0) { num = -num ; den = -den ;} // keep den positive
if(DenoLimit != 0 && den > DenoLimit) {
// prevent denominator overflow, buggy?
den /= 2;
num /= 2;
reduce ;
}
}
T toBaseType() {T result = num /den ; return result ; }
real toReal() { real result = num /cast(real)den ; return result ; }
real opCast() { return toReal ; }
T opCmp(QX rhs) { return num*rhs.den - den*rhs.num ; }
T opCmp(int rhs) { return num - den*rhs ; }
T opCmp(real rhs) { return cast(T)(num - den*rhs) ; }
bool opEquals(QX rhs) { return opCmp(rhs) == 0 ; }
bool opEquals(int rhs) { return opCmp(rhs) == 0 ; }
bool opEquals(real rhs) { return opCmp(rhs) == 0 ; }
QX opPostInc() { num += den ; reduce ; return *this ; }
QX opPostDec() { num -= den ; reduce ; return *this ; }
QX opAddAssign(QX rhs)
{ num = num*rhs.den + rhs.num*den ; den *= rhs.den ; reduce ; return *this ;}
QX opSubAssign(QX rhs)
{ num = num*rhs.den - rhs.num*den ; den *= rhs.den ; reduce ; return *this ;}
QX opMulAssign(QX rhs) { num *= rhs.num ; den *= rhs.den ; reduce ; return *this ;}
QX opDivAssign(QX rhs) { num *= rhs.den ; den *= rhs.num ; reduce ; return *this ;}
QX opAddAssign(int rhs) { num += rhs*den ; reduce ; return *this ; }
QX opSubAssign(int rhs) { num -= rhs*den ; reduce ; return *this ; }
QX opMulAssign(int rhs) { num *= rhs ; reduce ; return *this ; }
QX opDivAssign(int rhs) { den *= rhs ; reduce ; return *this ; }
QX opNeg() { return QX(-num, den) ;}
QX opPos() { return QX(num, den) ;}
QX opAdd(QX rhs) { return QX(num*rhs.den + rhs.num*den, den*rhs.den);}
QX opSub(QX rhs) { return QX(num*rhs.den - rhs.num*den, den*rhs.den);}
QX opMul(QX rhs) { return QX(num*rhs.num, den*rhs.den);}
QX opDiv(QX rhs) { return QX(num*rhs.den, den*rhs.num);}
QX opAdd(int rhs) { return QX(num + rhs*den, den);}
QX opSub(int rhs) { return QX(num - rhs*den, den);}
QX opSub_r(int lhs) { return QX(lhs*den - num, den);}
QX opMul(int rhs) { return QX(num*rhs, den);}
QX opDiv(int rhs) { return QX(num, den*rhs);}
QX opDiv_r(int lhs) { return QX(lhs*den, num);}
string toString() {
if(den == 1) return sformat("%s", num) ;
return sformat("%s/%s", num, den) ;
}
}</lang> |
Haskell
It's simpler to assume we are always dealing with an infinite list of coefficients. Mathematically, a finite power series can be generalized to an infinite power series with trailing zeros. <lang haskell>newtype Series a = S { coeffs :: [a] } deriving (Eq, Show) -- Invariant: coeffs must be an infinite list
instance Num a => Num (Series a) where
fromInteger n = S $ fromInteger n : repeat 0 negate (S fs) = S $ map negate fs S fs + S gs = S $ zipWith (+) fs gs S (f:ft) * S gs@(g:gt) = S $ f*g : coeffs (S ft * S gs + S (map (f*) gt))
instance Fractional a => Fractional (Series a) where
fromRational n = S $ fromRational n : repeat 0 S (f:ft) / S (g:gt) = S qs where qs = f/g : map (/g) (coeffs (S ft - S qs * S gt))
-- utility function to convert from a finite polynomial fromFiniteList xs = S (xs ++ repeat 0)
int (S fs) = S $ 0 : zipWith (/) fs [1..]
diff (S (_:ft)) = S $ zipWith (*) ft [1..]
sinx,cosx :: Series Rational sinx = int cosx cosx = 1 - int sinx
fiboS = 1 / fromFiniteList [1,-1,-1] </lang>
Output:
*Main> take 11 $ coeffs sinx [0 % 1,1 % 1,0 % 1,(-1) % 6,0 % 1,1 % 120,0 % 1,(-1) % 5040,0 % 1,1 % 362880,0 % 1] *Main> take 11 $ coeffs cosx [1 % 1,0 % 1,(-1) % 2,0 % 1,1 % 24,0 % 1,(-1) % 720,0 % 1,1 % 40320,0 % 1,(-1) % 3628800] *Main> take 11 $ coeffs $ sinx / cosx -- tangent [0 % 1,1 % 1,0 % 1,1 % 3,0 % 1,2 % 15,0 % 1,17 % 315,0 % 1,62 % 2835,0 % 1] *Main> take 11 $ map truncate $ coeffs $ fiboS -- some fibonaccis [1,1,2,3,5,8,13,21,34,55,89] *Main> take 11 $ coeffs $ fromFiniteList [1,5] * fromFiniteList [2,3,7] -- multiplying polynomials [2,13,22,35,0,0,0,0,0,0,0]
Java
Copied from the D example. Java has no generic numeric interface, and has no templates, so we cannot make it work for all numeric types at the same time. Because the Java library does not come with a Fraction class (and I am too lazy to implement one, although there is one in Apache Commons Math), here I will just hard-code it to use doubles (64-bit floating-point numbers) instead of fractions. It won't be as pretty, but it can be changed to Fraction or any other type trivially by substituting the types.
<lang java> import java.util.*;
interface Gene {
double coef(int n);
}
class Term {
private final List<Double> cache = new ArrayList<Double>(); private final Gene gene;
public Term(Gene g) { gene = g; }
public double get(int n) {
if (n < 0)
return 0;
else if (n >= cache.size())
for (int i = cache.size(); i <= n; i++)
cache.add(gene.coef(i));
return cache.get(n);
}
}
public class FormalPS {
private static final int DISP_TERM = 12; private static final String X_VAR = "x"; private Term term;
public FormalPS() { }
public void copyFrom(FormalPS foo) {
term = foo.term;
}
public FormalPS(Term t) {
term = t;
}
public FormalPS(final double[] polynomial) {
this(new Term(new Gene() {
public double coef(int n) {
if (n < 0 || n >= polynomial.length)
return 0;
else
return polynomial[n];
}
}));
}
public double inverseCoef(int n) {
double[] res = new double[n + 1];
res[0] = 1 / term.get(0);
for (int i = 1; i <= n; i++) {
res[i] = 0;
for (int j = 0; j < i; j++)
res[i] += term.get(i-j) * res[j];
res[i] *= -res[0];
}
return res[n];
}
public FormalPS add(final FormalPS rhs) {
return new FormalPS(new Term(new Gene() {
public double coef(int n) {
return term.get(n) + rhs.term.get(n);
}
}));
}
public FormalPS sub(final FormalPS rhs) {
return new FormalPS(new Term(new Gene() {
public double coef(int n) {
return term.get(n) - rhs.term.get(n);
}
}));
}
public FormalPS mul(final FormalPS rhs) {
return new FormalPS(new Term(new Gene() {
public double coef(int n) {
double res = 0;
for (int i = 0; i <= n; i++)
res += term.get(i) * rhs.term.get(n-i);
return res;
}
}));
}
public FormalPS div(final FormalPS rhs) {
return new FormalPS(new Term(new Gene() {
public double coef(int n) {
double res = 0;
for (int i = 0; i <= n; i++)
res += term.get(i) * rhs.term.get(n-i);
return res;
}
}));
}
public FormalPS diff() {
return new FormalPS(new Term(new Gene() {
public double coef(int n) {
return term.get(n+1) * (n+1);
}
}));
}
public FormalPS intg() {
return new FormalPS(new Term(new Gene() {
public double coef(int n) {
if (n == 0)
return 0;
else
return term.get(n-1) / n;
}
}));
}
public String toString() {
return toString(DISP_TERM);
}
public String toString(int dpTerm) {
StringBuffer s = new StringBuffer();
{
double c = term.get(0);
if (c != 0)
s.append(c);
}
for (int i = 1; i < dpTerm; i++) {
double c = term.get(i);
if (c != 0) {
if (c > 0 && s.length() > 0)
s.append("+");
if (c == 1)
s.append(X_VAR);
else if (c == -1)
s.append("-" + X_VAR);
else
s.append(c + X_VAR);
if (i > 1)
s.append(i);
}
}
if (s.length() == 0)
s.append("0");
s.append("+...");
return s.toString();
}
public static void main(String[] args) {
FormalPS cos = new FormalPS();
FormalPS sin = cos.intg();
cos.copyFrom(new FormalPS(new double[]{1}).sub(sin.intg()));
System.out.println("SIN(x) = " + sin);
System.out.println("COS(x) = " + cos);
}
} </lang> Output:
SIN(x) = x-0.16666666666666666x3+0.008333333333333333x5-1.984126984126984E-4x7+2.7557319223985893E-6x9-2.505210838544172E-8x11+... COS(x) = 1.0-0.5x2+0.041666666666666664x4-0.001388888888888889x6+2.48015873015873E-5x8-2.7557319223985894E-7x10+...
Tcl
Tcl doesn't arbitrary definitions of numbers without extension packages, so we'll represent these formal power series as objects, which are really just wrappers around a pair of functions: one determining how many terms there are in the series (possibly including "infinitely many") and the other producing the factor for a particular term.
This code makes extensive use of the fact that objects can have methods and variables independent of their class. This greatly reduces the requirement for singleton classes. <lang tcl>package require Tcl 8.6
oo::class create PowerSeries {
variable name
constructor {{body {}} args} {
# Use the body to adapt the methods of the _object_
oo::objdefine [self] $body
# Use the rest to configure variables in the object
foreach {var val} $args { set [my varname $var] $val }
# Guess the name if not already set
if {![info exists [my varname name]]} { set name [namespace tail [self]] }
}
method name {} {
return $name
}
method term i {
return 0
}
method limit {} {
return inf
}
# A pretty-printer, that prints the first $terms non-zero terms
method print {terms} {
set result "${name}(x) == " set limit [my limit] if {$limit == 0} { # Special case return $result[my term 0] } set tCount 0 for {set i 0} {$tCount<$terms && $i<=$limit} {incr i} { set t [my term $i] if {$t == 0} continue incr tCount set t [format %.4g $t]
if {$t eq "1" && $i != 0} {set t ""}
if {$i == 0} { append result "$t + " } elseif {$i == 1} { append result "${t}x + " } else { set p [string map { 0 \u2070 1 \u00b9 2 \u00b2 3 \u00b3 4 \u2074 5 \u2075 6 \u2076 7 \u2077 8 \u2078 9 \u2079 } $i] append result "${t}x$p + " } } return [string trimright $result "+ "]
}
# Evaluate (a prefix of) the series at a particular x
# The terms parameter gives the number; 5 is enough for show
method evaluate {x {terms 5}} {
set result 0 set limit [my limit] set tCount 0 for {set i 0} {$tCount<$terms && $i<=$limit} {incr i} { set t [my term $i] if {$t == 0} continue incr tCount set result [expr {$result + $t * ($x ** $i)}] } return $result
}
# Operations to build new sequences from old ones
method add {s} {
PowerSeries new { variable S1 S2 method limit {} {expr {max([$S1 limit],[$S2 limit])}} method term i { set t1 [expr {$i>[$S1 limit] ? 0 : [$S1 term $i]}] set t2 [expr {$i>[$S2 limit] ? 0 : [$S2 term $i]}] expr {$t1 + $t2} } } S1 [self] S2 $s name "$name+[$s name]"
}
method subtract {s} {
PowerSeries new { variable S1 S2 method limit {} {expr {max([$S1 limit],[$S2 limit])}} method term i { set t1 [expr {$i>[$S1 limit] ? 0 : [$S1 term $i]}] set t2 [expr {$i>[$S2 limit] ? 0 : [$S2 term $i]}] expr {$t1 - $t2} } } S1 [self] S2 $s name "$name-[$s name]"
} method integrate Template:Name "" {
if {$Name eq ""} {set Name "Integrate\[[my name]\]"} PowerSeries new { variable S limit method limit {} { if {[info exists limit]} {return $limit} try { return [expr {[$S limit] + 1}] } on error {} { # If the limit spirals out of control, it's infinite! return [set limit inf] } } method term i { if {$i == 0} {return 0} set t [$S term [expr {$i-1}]] expr {$t / double($i)} } } S [self] name $Name
} method differentiate Template:Name "" {
if {$Name eq ""} {set Name "Differentiate\[[my name]\]"} PowerSeries new { variable S method limit {} {expr {[$S limit] ? [$S limit] - 1 : 0}} method term i {expr {[incr i] * [$S term $i]}} } S [self] name $Name
}
# Special constructor for making constants
self method constant n {
PowerSeries new { variable n method limit {} {return 0} method term i {return $n} } n $n name $n
}
}
- Define the two power series in terms of each other
PowerSeries create cos ;# temporary dummy object... rename [cos integrate "sin"] sin cos destroy ;# remove the dummy to make way for the real one... rename [[PowerSeries constant 1] subtract [sin integrate]] cos</lang> Demonstrating: <lang tcl>% sin print 7 sin(x) == x + -0.1667x³ + 0.008333x⁵ + -0.0001984x⁷ + 2.756e-06x⁹ + -2.505e-08x¹¹ + 1.606e-10x¹³ % cos print 7 1-Integrate[sin](x) == 1 + -0.5x² + 0.04167x⁴ + -0.001389x⁶ + 2.48e-05x⁸ + -2.756e-07x¹⁰ + 2.088e-09x¹² % sin evaluate [expr acos(0)] 1.0000035425842861 % cos evaluate [expr acos(0)] 2.473727636463901e-5</lang>