Each of the number classes declares its mathematical operations in the
corresponding include file. For example, if your code operates with
objects of type cl_I, it should #include <cl_integer.h>.
Here is how to create number objects "from nothing".
cl_I objects are most easily constructed from C integers and from
strings. See section 3.4 Conversions.
cl_RA objects can be constructed from strings. The syntax
for rational numbers is described in section 5.1 Internal and printed representation.
Another standard way to produce a rational number is through application
of `operator /' or `recip' on integers.
cl_F objects with low precision are most easily constructed from
C `float' and `double'. See section 3.4 Conversions.
To construct a cl_F with high precision, you can use the conversion
from `const char *', but you have to specify the desired precision
within the string. (See section 5.1 Internal and printed representation.)
Example:
cl_F e = "0.271828182845904523536028747135266249775724709369996e+1_40";
will set `e' to the given value, with a precision of 40 decimal digits.
The programmatic way to construct a cl_F with high precision is
through the cl_float conversion function, see
section 4.11.1 Conversion to floating-point numbers. For example, to compute
e to 40 decimal places, first construct 1.0 to 40 decimal places
and then apply the exponential function:
cl_float_format_t precision = cl_float_format(40); cl_F e = exp(cl_float(1,precision));
Non-real cl_N objects are normally constructed through the function
cl_N complex (const cl_R& realpart, const cl_R& imagpart)
See section 4.4 Elementary complex functions.
Each of the classes cl_N, cl_R, cl_RA, cl_I,
cl_F, cl_SF, cl_FF, cl_DF, cl_LF
defines the following operations:
type operator + (const type&, const type&)
type operator - (const type&, const type&)
type operator - (const type&)
type plus1 (const type& x)
x + 1.
type minus1 (const type& x)
x - 1.
type operator * (const type&, const type&)
type square (const type& x)
x * x.
Each of the classes cl_N, cl_R, cl_RA,
cl_F, cl_SF, cl_FF, cl_DF, cl_LF
defines the following operations:
type operator / (const type&, const type&)
type recip (const type&)
The class cl_I doesn't define a `/' operation because
in the C/C++ language this operator, applied to integral types,
denotes the `floor' or `truncate' operation (which one of these,
is implementation dependent). (See section 4.6 Rounding functions)
Instead, cl_I defines an "exact quotient" function:
cl_I exquo (const cl_I& x, const cl_I& y)
y divides x, and returns the quotient x/y.
The following exponentiation functions are defined:
cl_I expt_pos (const cl_I& x, const cl_I& y)
cl_RA expt_pos (const cl_RA& x, const cl_I& y)
y must be > 0. Returns x^y.
cl_RA expt (const cl_RA& x, const cl_I& y)
cl_R expt (const cl_R& x, const cl_I& y)
cl_N expt (const cl_N& x, const cl_I& y)
x^y.
Each of the classes cl_R, cl_RA, cl_I,
cl_F, cl_SF, cl_FF, cl_DF, cl_LF
defines the following operation:
type abs (const type& x)
x.
This is x if x >= 0, and -x if x <= 0.
The class cl_N implements this as follows:
cl_R abs (const cl_N x)
x.
Each of the classes cl_N, cl_R, cl_RA, cl_I,
cl_F, cl_SF, cl_FF, cl_DF, cl_LF
defines the following operation:
type signum (const type& x)
x, in the same number format as x.
This is defined as x / abs(x) if x is non-zero, and
x if x is zero. If x is real, the value is either
0 or 1 or -1.
Each of the classes cl_RA, cl_I defines the following operations:
cl_I numerator (const type& x)
x.
cl_I denominator (const type& x)
x.
The numerator and denominator of a rational number are normalized in such a way that they have no factor in common and the denominator is positive.
The class cl_N defines the following operation:
cl_N complex (const cl_R& a, const cl_R& b)
a+bi, that is, the complex number with
real part a and imaginary part b.
Each of the classes cl_N, cl_R defines the following operations:
cl_R realpart (const type& x)
x.
cl_R imagpart (const type& x)
x.
type conjugate (const type& x)
x.
We have the relations
x = complex(realpart(x), imagpart(x))
conjugate(x) = complex(realpart(x), -imagpart(x))
Each of the classes cl_N, cl_R, cl_RA, cl_I,
cl_F, cl_SF, cl_FF, cl_DF, cl_LF
defines the following operations:
bool operator == (const type&, const type&)
bool operator != (const type&, const type&)
uint32 cl_equal_hashcode (const type&)
==. This hash code depends on the number's value,
not its type or precision.
cl_boolean zerop (const type& x)
x == 0
Each of the classes cl_R, cl_RA, cl_I,
cl_F, cl_SF, cl_FF, cl_DF, cl_LF
defines the following operations:
cl_signean cl_compare (const type& x, const type& y)
x and y. Returns +1 if x>y,
-1 if x<y, 0 if x=y.
bool operator <= (const type&, const type&)
bool operator < (const type&, const type&)
bool operator >= (const type&, const type&)
bool operator > (const type&, const type&)
cl_boolean minusp (const type& x)
x < 0
cl_boolean plusp (const type& x)
x > 0
type max (const type& x, const type& y)
x and y.
type min (const type& x, const type& y)
x and y.
When a floating point number and a rational number are compared, the float
is first converted to a rational number using the function rational.
Since a floating point number actually represents an interval of real numbers,
the result might be surprising.
For example, (cl_F)(cl_R)"1/3" == (cl_R)"1/3" returns false because
there is no floating point number whose value is exactly 1/3.
When a real number is to be converted to an integer, there is no "best" rounding. The desired rounding function depends on the application. The Common Lisp and ISO Lisp standards offer four rounding functions:
floor(x)
x.
ceiling(x)
x.
truncate(x)
x (inclusive) the one nearest to x.
round(x)
x. If x is exactly halfway between two
integers, choose the even one.
These functions have different advantages:
floor and ceiling are translation invariant:
floor(x+n) = floor(x) + n and ceiling(x+n) = ceiling(x) + n
for every x and every integer n.
On the other hand, truncate and round are symmetric:
truncate(-x) = -truncate(x) and round(-x) = -round(x),
and furthermore round is unbiased: on the "average", it rounds
down exactly as often as it rounds up.
The functions are related like this:
ceiling(m/n) = floor((m+n-1)/n) = floor((m-1)/n)+1
for rational numbers m/n (m, n integers, n>0), and
truncate(x) = sign(x) * floor(abs(x))
Each of the classes cl_R, cl_RA,
cl_F, cl_SF, cl_FF, cl_DF, cl_LF
defines the following operations:
cl_I floor1 (const type& x)
floor(x).
cl_I ceiling1 (const type& x)
ceiling(x).
cl_I truncate1 (const type& x)
truncate(x).
cl_I round1 (const type& x)
round(x).
Each of the classes cl_R, cl_RA, cl_I,
cl_F, cl_SF, cl_FF, cl_DF, cl_LF
defines the following operations:
cl_I floor1 (const type& x, const type& y)
floor(x/y).
cl_I ceiling1 (const type& x, const type& y)
ceiling(x/y).
cl_I truncate1 (const type& x, const type& y)
truncate(x/y).
cl_I round1 (const type& x, const type& y)
round(x/y).
These functions are called `floor1', ... here instead of `floor', ..., because on some systems, system dependent include files define `floor' and `ceiling' as macros.
In many cases, one needs both the quotient and the remainder of a division. It is more efficient to compute both at the same time than to perform two divisions, one for quotient and the next one for the remainder. The following functions therefore return a structure containing both the quotient and the remainder. The suffix `2' indicates the number of "return values". The remainder is defined as follows:
quotient = floor(x),
remainder = x - quotient,
quotient = floor(x,y),
remainder = x - quotient*y,
and similarly for the other three operations.
Each of the classes cl_R, cl_RA,
cl_F, cl_SF, cl_FF, cl_DF, cl_LF
defines the following operations:
struct type_div_t { cl_I quotient; type remainder; };
type_div_t floor2 (const type& x)
type_div_t ceiling2 (const type& x)
type_div_t truncate2 (const type& x)
type_div_t round2 (const type& x)
Each of the classes cl_R, cl_RA, cl_I,
cl_F, cl_SF, cl_FF, cl_DF, cl_LF
defines the following operations:
struct type_div_t { cl_I quotient; type remainder; };
type_div_t floor2 (const type& x, const type& y)
type_div_t ceiling2 (const type& x, const type& y)
type_div_t truncate2 (const type& x, const type& y)
type_div_t round2 (const type& x, const type& y)
Sometimes, one wants the quotient as a floating-point number (of the same format as the argument, if the argument is a float) instead of as an integer. The prefix `f' indicates this.
Each of the classes
cl_F, cl_SF, cl_FF, cl_DF, cl_LF
defines the following operations:
type ffloor (const type& x)
type fceiling (const type& x)
type ftruncate (const type& x)
type fround (const type& x)
and similarly for class cl_R, but with return type cl_F.
The class cl_R defines the following operations:
cl_F ffloor (const type& x, const type& y)
cl_F fceiling (const type& x, const type& y)
cl_F ftruncate (const type& x, const type& y)
cl_F fround (const type& x, const type& y)
These functions also exist in versions which return both the quotient and the remainder. The suffix `2' indicates this.
Each of the classes
cl_F, cl_SF, cl_FF, cl_DF, cl_LF
defines the following operations:
struct type_fdiv_t { type quotient; type remainder; };
type_fdiv_t ffloor2 (const type& x)
type_fdiv_t fceiling2 (const type& x)
type_fdiv_t ftruncate2 (const type& x)
type_fdiv_t fround2 (const type& x)
and similarly for class cl_R, but with quotient type cl_F.
The class cl_R defines the following operations:
struct type_fdiv_t { cl_F quotient; cl_R remainder; };
type_fdiv_t ffloor2 (const type& x, const type& y)
type_fdiv_t fceiling2 (const type& x, const type& y)
type_fdiv_t ftruncate2 (const type& x, const type& y)
type_fdiv_t fround2 (const type& x, const type& y)
Other applications need only the remainder of a division. The remainder of `floor' and `ffloor' is called `mod' (abbreviation of "modulo"). The remainder `truncate' and `ftruncate' is called `rem' (abbreviation of "remainder").
mod(x,y) = floor2(x,y).remainder = x - floor(x/y)*y
rem(x,y) = truncate2(x,y).remainder = x - truncate(x/y)*y
If x and y are both >= 0, mod(x,y) = rem(x,y) >= 0.
In general, mod(x,y) has the sign of y or is zero,
and rem(x,y) has the sign of x or is zero.
The classes cl_R, cl_I define the following operations:
type mod (const type& x, const type& y)
type rem (const type& x, const type& y)
Each of the classes cl_R,
cl_F, cl_SF, cl_FF, cl_DF, cl_LF
defines the following operation:
type sqrt (const type& x)
x must be >= 0. This function returns the square root of x,
normalized to be >= 0. If x is the square of a rational number,
sqrt(x) will be a rational number, else it will return a
floating-point approximation.
The classes cl_RA, cl_I define the following operation:
cl_boolean sqrtp (const type& x, type* root)
x is a perfect square. If so, it returns true
and the exact square root in *root, else it returns false.
Furthermore, for integers, similarly:
cl_boolean isqrt (const type& x, type* root)
x should be >= 0. This function sets *root to
floor(sqrt(x)) and returns the same value as sqrtp:
the boolean value (expt(*root,2) == x).
For nth roots, the classes cl_RA, cl_I
define the following operation:
cl_boolean rootp (const type& x, const cl_I& n, type* root)
x must be >= 0. n must be > 0.
This tests whether x is an nth power of a rational number.
If so, it returns true and the exact root in *root, else it returns
false.
The only square root function which accepts negative numbers is the one
for class cl_N:
cl_N sqrt (const cl_N& z)
z, as defined by the formula
sqrt(z) = exp(log(z)/2). Conversion to a floating-point type
or to a complex number are done if necessary. The range of the result is the
right half plane realpart(sqrt(z)) >= 0
including the positive imaginary axis and 0, but excluding
the negative imaginary axis.
The result is an exact number only if z is an exact number.
The transcendental functions return an exact result if the argument
is exact and the result is exact as well. Otherwise they must return
inexact numbers even if the argument is exact.
For example, cos(0) = 1 returns the rational number 1.
cl_R exp (const cl_R& x)
cl_N exp (const cl_N& x)
x. This is e^x where
e is the base of the natural logarithms. The range of the result
is the entire complex plane excluding 0.
cl_R ln (const cl_R& x)
x must be > 0. Returns the (natural) logarithm of x.
cl_N log (const cl_N& x)
x is real and positive,
this is ln(x). In general, log(x) = log(abs(x)) + i*phase(x).
The range of the result is the strip in the complex plane
-pi < imagpart(log(x)) <= pi.
cl_R phase (const cl_N& x)
x in its polar representation as a
complex number. That is, phase(x) = atan(realpart(x),imagpart(x)).
This is also the imaginary part of log(x).
The range of the result is the interval -pi < phase(x) <= pi.
The result will be an exact number only if zerop(x) or
if x is real and positive.
cl_R log (const cl_R& a, const cl_R& b)
a and b must be > 0. Returns the logarithm of a with
respect to base b. log(a,b) = ln(a)/ln(b).
The result can be exact only if a = 1 or if a and b
are both rational.
cl_N log (const cl_N& a, const cl_N& b)
a with respect to base b.
log(a,b) = log(a)/log(b).
cl_N expt (const cl_N& x, const cl_N& y)
x^y = exp(y*log(x)).
The constant e = exp(1) = 2.71828... is returned by the following functions:
cl_F cl_exp1 (cl_float_format_t f)
f.
cl_F cl_exp1 (const cl_F& y)
y.
cl_F cl_exp1 (void)
cl_default_float_format.
cl_R sin (const cl_R& x)
sin(x). The range of the result is the interval
-1 <= sin(x) <= 1.
cl_N sin (const cl_N& z)
sin(z). The range of the result is the entire complex plane.
cl_R cos (const cl_R& x)
cos(x). The range of the result is the interval
-1 <= cos(x) <= 1.
cl_N cos (const cl_N& x)
cos(z). The range of the result is the entire complex plane.
struct cl_cos_sin_t { cl_R cos; cl_R sin; };
cl_cos_sin_t cl_cos_sin (const cl_R& x)
sin(x) and cos(x). This is more efficient than
computing them separately. The relation cos^2 + sin^2 = 1 will
hold only approximately.
cl_R tan (const cl_R& x)
cl_N tan (const cl_N& x)
tan(x) = sin(x)/cos(x).
cl_N cis (const cl_R& x)
cl_N cis (const cl_N& x)
exp(i*x). The name `cis' means "cos + i sin", because
e^(i*x) = cos(x) + i*sin(x).
cl_N asin (const cl_N& z)
arcsin(z). This is defined as
arcsin(z) = log(iz+sqrt(1-z^2))/i and satisfies
arcsin(-z) = -arcsin(z).
The range of the result is the strip in the complex domain
-pi/2 <= realpart(arcsin(z)) <= pi/2, excluding the numbers
with realpart = -pi/2 and imagpart < 0 and the numbers
with realpart = pi/2 and imagpart > 0.
cl_N acos (const cl_N& z)
arccos(z). This is defined as
arccos(z) = pi/2 - arcsin(z) = log(z+i*sqrt(1-z^2))/i
and satisfies arccos(-z) = pi - arccos(z).
The range of the result is the strip in the complex domain
0 <= realpart(arcsin(z)) <= pi, excluding the numbers
with realpart = 0 and imagpart < 0 and the numbers
with realpart = pi and imagpart > 0.
cl_R atan (const cl_R& x, const cl_R& y)
x+iy. This is atan(y/x) if x>0. The range of
the result is the interval -pi < atan(x,y) <= pi. The result will
be an exact number only if x > 0 and y is the exact 0.
WARNING: In Common Lisp, this function is called as (atan y x),
with reversed order of arguments.
cl_R atan (const cl_R& x)
arctan(x). This is the same as atan(1,x). The range
of the result is the interval -pi/2 < atan(x) < pi/2. The result
will be an exact number only if x is the exact 0.
cl_N atan (const cl_N& z)
arctan(z). This is defined as
arctan(z) = (log(1+iz)-log(1-iz)) / 2i and satisfies
arctan(-z) = -arctan(z). The range of the result is
the strip in the complex domain
-pi/2 <= realpart(arctan(z)) <= pi/2, excluding the numbers
with realpart = -pi/2 and imagpart >= 0 and the numbers
with realpart = pi/2 and imagpart <= 0.
Archimedes' constant pi = 3.14... is returned by the following functions:
cl_F cl_pi (cl_float_format_t f)
f.
cl_F cl_pi (const cl_F& y)
y.
cl_F cl_pi (void)
cl_default_float_format.
cl_R sinh (const cl_R& x)
sinh(x).
cl_N sinh (const cl_N& z)
sinh(z). The range of the result is the entire complex plane.
cl_R cosh (const cl_R& x)
cosh(x). The range of the result is the interval
cosh(x) >= 1.
cl_N cosh (const cl_N& z)
cosh(z). The range of the result is the entire complex plane.
struct cl_cosh_sinh_t { cl_R cosh; cl_R sinh; };
cl_cosh_sinh_t cl_cosh_sinh (const cl_R& x)
sinh(x) and cosh(x). This is more efficient than
computing them separately. The relation cosh^2 - sinh^2 = 1 will
hold only approximately.
cl_R tanh (const cl_R& x)
cl_N tanh (const cl_N& x)
tanh(x) = sinh(x)/cosh(x).
cl_N asinh (const cl_N& z)
arsinh(z). This is defined as
arsinh(z) = log(z+sqrt(1+z^2)) and satisfies
arsinh(-z) = -arsinh(z).
The range of the result is the strip in the complex domain
-pi/2 <= imagpart(arsinh(z)) <= pi/2, excluding the numbers
with imagpart = -pi/2 and realpart > 0 and the numbers
with imagpart = pi/2 and realpart < 0.
cl_N acosh (const cl_N& z)
arcosh(z). This is defined as
arcosh(z) = 2*log(sqrt((z+1)/2)+sqrt((z-1)/2)).
The range of the result is the half-strip in the complex domain
-pi < imagpart(arcosh(z)) <= pi, realpart(arcosh(z)) >= 0,
excluding the numbers with realpart = 0 and -pi < imagpart < 0.
cl_N atanh (const cl_N& z)
artanh(z). This is defined as
artanh(z) = (log(1+z)-log(1-z)) / 2 and satisfies
artanh(-z) = -artanh(z). The range of the result is
the strip in the complex domain
-pi/2 <= imagpart(artanh(z)) <= pi/2, excluding the numbers
with imagpart = -pi/2 and realpart <= 0 and the numbers
with imagpart = pi/2 and realpart >= 0.
Euler's constant C = 0.577... is returned by the following functions:
cl_F cl_eulerconst (cl_float_format_t f)
f.
cl_F cl_eulerconst (const cl_F& y)
y.
cl_F cl_eulerconst (void)
cl_default_float_format.
Catalan's constant G = 0.915... is returned by the following functions:
cl_F cl_catalanconst (cl_float_format_t f)
f.
cl_F cl_catalanconst (const cl_F& y)
y.
cl_F cl_catalanconst (void)
cl_default_float_format.
Riemann's zeta function at an integral point s>1 is returned by the
following functions:
cl_F cl_zeta (int s, cl_float_format_t f)
s as a float of format f.
cl_F cl_zeta (int s, const cl_F& y)
s in the float format of y.
cl_F cl_zeta (int s)
s as a float of format
cl_default_float_format.
Integers, when viewed as in two's complement notation, can be thought as infinite bit strings where the bits' values eventually are constant. For example,
17 = ......00010001
-6 = ......11111010
The logical operations view integers as such bit strings and operate on each of the bit positions in parallel.
cl_I lognot (const cl_I& x)
cl_I operator ~ (const cl_I& x)
~x in C. This is the same as -1-x.
cl_I logand (const cl_I& x, const cl_I& y)
cl_I operator & (const cl_I& x, const cl_I& y)
x & y in C.
cl_I logior (const cl_I& x, const cl_I& y)
cl_I operator | (const cl_I& x, const cl_I& y)
x | y in C.
cl_I logxor (const cl_I& x, const cl_I& y)
cl_I operator ^ (const cl_I& x, const cl_I& y)
x ^ y in C.
cl_I logeqv (const cl_I& x, const cl_I& y)
~(x ^ y) in C.
cl_I lognand (const cl_I& x, const cl_I& y)
~(x & y) in C.
cl_I lognor (const cl_I& x, const cl_I& y)
~(x | y) in C.
cl_I logandc1 (const cl_I& x, const cl_I& y)
~x & y in C.
cl_I logandc2 (const cl_I& x, const cl_I& y)
x & ~y in C.
cl_I logorc1 (const cl_I& x, const cl_I& y)
~x | y in C.
cl_I logorc2 (const cl_I& x, const cl_I& y)
x | ~y in C.
These operations are all available though the function
cl_I boole (cl_boole op, const cl_I& x, const cl_I& y)
where op must have one of the 16 values (each one stands for a function
which combines two bits into one bit): boole_clr, boole_set,
boole_1, boole_2, boole_c1, boole_c2,
boole_and, boole_ior, boole_xor, boole_eqv,
boole_nand, boole_nor, boole_andc1, boole_andc2,
boole_orc1, boole_orc2.
Other functions that view integers as bit strings:
cl_boolean logtest (const cl_I& x, const cl_I& y)
x and y, i.e. if
logand(x,y) != 0.
cl_boolean logbitp (const cl_I& n, const cl_I& x)
nth bit (from the right) of x is set.
Bit 0 is the least significant bit.
uintL logcount (const cl_I& x)
x, if x >= 0, or
the number of zero bits in x, if x < 0.
The following functions operate on intervals of bits in integers. The type
struct cl_byte { uintL size; uintL position; };
represents the bit interval containing the bits
position...position+size-1 of an integer.
The constructor cl_byte(size,position) constructs a cl_byte.
cl_I ldb (const cl_I& n, const cl_byte& b)
n described by the bit interval b
and returns them as a nonnegative integer with b.size bits.
cl_boolean ldb_test (const cl_I& n, const cl_byte& b)
b is set in
n.
cl_I dpb (const cl_I& newbyte, const cl_I& n, const cl_byte& b)
n, with the bits described by the bit interval b
replaced by newbyte. Only the lowest b.size bits of
newbyte are relevant.
The functions ldb and dpb implicitly shift. The following
functions are their counterparts without shifting:
cl_I mask_field (const cl_I& n, const cl_byte& b)
b
copied from the corresponding bits in n, the other bits zero.
cl_I deposit_field (const cl_I& newbyte, const cl_I& n, const cl_byte& b)
b
come from newbyte and the other bits come from n.
The following relations hold:
ldb (n, b) = mask_field(n, b) >> b.position,
dpb (newbyte, n, b) = deposit_field (newbyte << b.position, n, b),
deposit_field(newbyte,n,b) = n ^ mask_field(n,b) ^ mask_field(new_byte,b).
The following operations on integers as bit strings are efficient shortcuts for common arithmetic operations:
cl_boolean oddp (const cl_I& x)
x is 1. Equivalent to
mod(x,2) != 0.
cl_boolean evenp (const cl_I& x)
x is 0. Equivalent to
mod(x,2) == 0.
cl_I operator << (const cl_I& x, const cl_I& n)
x by n bits to the left. n should be >=0.
Equivalent to x * expt(2,n).
cl_I operator >> (const cl_I& x, const cl_I& n)
x by n bits to the right. n should be >=0.
Bits shifted out to the right are thrown away.
Equivalent to floor(x / expt(2,n)).
cl_I ash (const cl_I& x, const cl_I& y)
x by y bits to the left (if y>=0) or
by -y bits to the right (if y<=0). In other words, this
returns floor(x * expt(2,y)).
uintL integer_length (const cl_I& x)
x
in two's complement notation. This is the smallest n >= 0 such that
-2^n <= x < 2^n. If x > 0, this is the unique n > 0 such that
2^(n-1) <= x < 2^n.
uintL ord2 (const cl_I& x)
x must be non-zero. This function returns the number of 0 bits at the
right of x in two's complement notation. This is the largest n >= 0
such that 2^n divides x.
uintL power2p (const cl_I& x)
x must be > 0. This function checks whether x is a power of 2.
If x = 2^(n-1), it returns n. Else it returns 0.
(See also the function logp.)
uint32 gcd (uint32 a, uint32 b)
cl_I gcd (const cl_I& a, const cl_I& b)
a and b,
normalized to be >= 0.
cl_I xgcd (const cl_I& a, const cl_I& b, cl_I* u, cl_I* v)
g of
a and b and at the same time the representation of g
as an integral linear combination of a and b:
u and v with u*a+v*b = g, g >= 0.
u and v will be normalized to be of smallest possible absolute
value, in the following sense: If a and b are non-zero, and
abs(a) != abs(b), u and v will satisfy the inequalities
abs(u) <= abs(b)/(2*g), abs(v) <= abs(a)/(2*g).
cl_I lcm (const cl_I& a, const cl_I& b)
a and b,
normalized to be >= 0.
cl_boolean logp (const cl_I& a, const cl_I& b, cl_RA* l)
cl_boolean logp (const cl_RA& a, const cl_RA& b, cl_RA* l)
a must be > 0. b must be >0 and != 1. If log(a,b) is
rational number, this function returns true and sets *l = log(a,b), else
it returns false.
cl_I factorial (uintL n)
n must be a small integer >= 0. This function returns the factorial
n! = 1*2*...*n.
cl_I doublefactorial (uintL n)
n must be a small integer >= 0. This function returns the
doublefactorial n!! = 1*3*...*n or
n!! = 2*4*...*n, respectively.
cl_I binomial (uintL n, uintL k)
n and k must be small integers >= 0. This function returns the
binomial coefficient
(n choose k) = n! / k! (n-k)!
for 0 <= k <= n, 0 else.
Recall that a floating-point number consists of a sign s, an
exponent e and a mantissa m. The value of the number is
(-1)^s * 2^e * m.
Each of the classes
cl_F, cl_SF, cl_FF, cl_DF, cl_LF
defines the following operations.
type scale_float (const type& x, sintL delta)
type scale_float (const type& x, const cl_I& delta)
x*2^delta. This is more efficient than an explicit multiplication
because it copies x and modifies the exponent.
The following functions provide an abstract interface to the underlying representation of floating-point numbers.
sintL float_exponent (const type& x)
e of x.
For x = 0.0, this is 0. For x non-zero, this is the unique
integer with 2^(e-1) <= abs(x) < 2^e.
sintL float_radix (const type& x)
2.
type float_sign (const type& x)
s of x as a float. The value is 1 for
x >= 0, -1 for x < 0.
uintL float_digits (const type& x)
x, including the hidden bit. The value only depends on the type
of x, not on its value.
uintL float_precision (const type& x)
x. Since denormalized numbers are not supported,
this is the same as float_digits(x) if x is non-zero, and
0 if x = 0.
The complete internal representation of a float is encoded in the type
cl_decoded_float (or cl_decoded_sfloat, cl_decoded_ffloat,
cl_decoded_dfloat, cl_decoded_lfloat, respectively), defined by
struct cl_decoded_typefloat {
type mantissa; cl_I exponent; type sign;
};
and returned by the function
cl_decoded_typefloat decode_float (const type& x)
x non-zero, this returns (-1)^s, e, m with
x = (-1)^s * 2^e * m and 0.5 <= m < 1.0. For x = 0,
it returns (-1)^s=1, e=0, m=0.
e is the same as returned by the function float_exponent.
A complete decoding in terms of integers is provided as type
struct cl_idecoded_float {
cl_I mantissa; cl_I exponent; cl_I sign;
};
by the following function:
cl_idecoded_float integer_decode_float (const type& x)
x non-zero, this returns (-1)^s, e, m with
x = (-1)^s * 2^e * m and m an integer with float_digits(x)
bits. For x = 0, it returns (-1)^s=1, e=0, m=0.
WARNING: The exponent e is not the same as the one returned by
the functions decode_float and float_exponent.
Some other function, implemented only for class cl_F:
cl_F float_sign (const cl_F& x, const cl_F& y)
y and whose sign is that of x. If x is
zero, it is treated as positive. Same for y.
The type cl_float_format_t describes a floating-point format.
cl_float_format_t cl_float_format (uintL n)
n
decimal digits in the mantissa (after the decimal point).
cl_float_format_t cl_float_format (const cl_F& x)
x.
cl_float_format_t cl_default_float_format
To convert a real number to a float, each of the types
cl_R, cl_F, cl_I, cl_RA,
int, unsigned int, float, double
defines the following operations:
cl_F cl_float (const type&x, cl_float_format_t f)
x as a float of format f.
cl_F cl_float (const type&x, const cl_F& y)
x in the float format of y.
cl_F cl_float (const type&x)
x as a float of format cl_default_float_format if
it is an exact number, or x itself if it is already a float.
Of course, converting a number to a float can lose precision.
Every floating-point format has some characteristic numbers:
cl_F most_positive_float (cl_float_format_t f)
f.
cl_F most_negative_float (cl_float_format_t f)
f.
cl_F least_positive_float (cl_float_format_t f)
f.
cl_F least_negative_float (cl_float_format_t f)
f.
cl_F float_epsilon (cl_float_format_t f)
1+e != 1.
cl_F float_negative_epsilon (cl_float_format_t f)
1-e != 1.
Each of the classes cl_R, cl_RA, cl_F
defines the following operation:
cl_RA rational (const type& x)
x as an exact number. If x is already
an exact number, this is x. If x is a floating-point number,
the value is a rational number whose denominator is a power of 2.
In order to convert back, say, (cl_F)(cl_R)"1/3" to 1/3, there is
the function
cl_RA rationalize (const cl_R& x)
x is a floating-point number, it actually represents an interval
of real numbers, and this function returns the rational number with
smallest denominator (and smallest numerator, in magnitude)
which lies in this interval.
If x is already an exact number, this function returns x.
If x is any float, one has
cl_float(rational(x),x) = x
cl_float(rationalize(x),x) = x
A random generator is a machine which produces (pseudo-)random numbers.
The include file <cl_random.h> defines a class cl_random_state
which contains the state of a random generator. If you make a copy
of the random number generator, the original one and the copy will produce
the same sequence of random numbers.
The following functions return (pseudo-)random numbers in different formats. Calling one of these modifies the state of the random number generator in a complicated but deterministic way.
cl_random_state cl_default_random_state
contains a default random number generator. It is used when the functions
below are called without cl_random_state argument.
uint32 random32 (cl_random_state& randomstate)
uint32 random32 ()
cl_I random_I (cl_random_state& randomstate, const cl_I& n)
cl_I random_I (const cl_I& n)
n must be an integer > 0. This function returns a random integer x
in the range 0 <= x < n.
cl_F random_F (cl_random_state& randomstate, const cl_F& n)
cl_F random_F (const cl_F& n)
n must be a float > 0. This function returns a random floating-point
number of the same format as n in the range 0 <= x < n.
cl_R random_R (cl_random_state& randomstate, const cl_R& n)
cl_R random_R (const cl_R& n)
random_I if n is an integer and like random_F
if n is a float.
The modifying C/C++ operators +=, -=, *=, /=,
&=, |=, ^=, <<=, >>=
are not available by default because their
use tends to make programs unreadable. It is trivial to get away without
them. However, if you feel that you absolutely need these operators
to get happy, then add
#define WANT_OBFUSCATING_OPERATORS
to the beginning of your source files, before the inclusion of any CLN include files. This flag will enable the following operators:
For the classes cl_N, cl_R, cl_RA,
cl_F, cl_SF, cl_FF, cl_DF, cl_LF:
type& operator += (type&, const type&)
type& operator -= (type&, const type&)
type& operator *= (type&, const type&)
type& operator /= (type&, const type&)
For the class cl_I:
type& operator += (type&, const type&)
type& operator -= (type&, const type&)
type& operator *= (type&, const type&)
type& operator &= (type&, const type&)
type& operator |= (type&, const type&)
type& operator ^= (type&, const type&)
type& operator <<= (type&, const type&)
type& operator >>= (type&, const type&)
For the classes cl_N, cl_R, cl_RA, cl_I,
cl_F, cl_SF, cl_FF, cl_DF, cl_LF:
type& operator ++ (type& x)
++x.
void operator ++ (type& x, int)
x++.
type& operator -- (type& x)
--x.
void operator -- (type& x, int)
x--.
Note that by using these obfuscating operators, you wouldn't gain efficiency: In CLN `x += y;' is exactly the same as `x = x+y;', not more efficient.
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