# Overview

Atlassian Sourcetree is a free Git and Mercurial client for Windows.

Atlassian Sourcetree is a free Git and Mercurial client for Mac.

I N T R O D U C T I O N ======================= Computable real numbers x are interpreted as (potentially) infinite fractions in base 2 that are specified through a rule for computation of an integer a with |(2^k)*x - a| <= 1 for any k>=0. The internal data structure should not be accessed. The interface for the outside world is as follows: The type CREAL is a supertype of the type RATIONAL. (CREAL-P x), for an object x, returns T if x is of type CREAL, otherwise NIL. (APPROX-R x k), for a CREAL x and an integer k>=0, returns an integer a with |(2^k)*x - a| < 1. (MAKE-REAL fun) returns the real number given by fun. Here fun is a function taking an argument k, that computes a as above. CREALs are output by print etc. as a decimal fraction. The error hereby is at most one unit in the last digit that was output. The number of decimal digits after the decimal point is defined through the dynamic variable *PRINT-PREC*. For comparison operations etc. a precision threshold is used. It is defined through the dynamic variable *CREAL-TOLERANCE*. Its value should be a nonnegative integer n, meaning that numbers are considered equal if they differ by at most 2^(-n). E X P O R T E D F U N C T I O N S A N D C O N S T A N T S =============================================================== The following functions, constants and variables are exported. (The package is named "REALS".) CREAL type type of the computable real numbers CREAL-P object function tests for type CREAL *PRINT-PREC* variable specifies precision of output *CREAL-TOLERANCE* variable precision threshold for comparison APPROX-R x:creal k:int>=0 function returns approximation of x to k digits MAKE-REAL function function creates object of type CREAL RAW-APPROX-R x:creal function returns 3 values a,n,s with: if a = 0: |x| <= 2^(-n), s = 0 and n >= *CREAL-TOLERANCE* else: a0 integer > 4, n0 integer >=0, s = +1 or -1, and sign(x) = s, (a-1)*2^(-n) <= |x| <= (a+1)*2^(-n) PRINT-R x:creal k:int>=0 &optional (flag t) function outputs x with k decimal digits. If flag is true, first a newline. +R {creal}* function computes the sum of the arguments -R creal {creal}* function computes negative or difference *R {creal}* function computes the product of the arguments /R creal {creal}* function computes reciprocal or quotient SQRT-R creal function computes the square root LOG2-R constant log(2) PI-R constant pi 2PI-R constant 2*pi PI/2-R constant pi/2 PI/4-R constant pi/4 LOG-R x:creal &optional b:creal function computes the logarithm of n in base b; default is the natural logarithm EXP-R creal function computes the exponential function EXPT-R x:creal y:creal function computes x^y SIN-R creal function computes the sine COS-R creal function computes the cosine TAN-R creal function computes the tangent ATAN-R x:creal &optional y:creal function computes the arctangent of x or the phase angle of (x,y) ASH-R x:creal n:int function computes x * 2^n ROUND-R x:creal &optional y:creal function computes two values q (integer) and r (creal) with x = q*y + r and |r|<=|y|/2 according to the precision specified by *CREAL-TOLERANCE* FLOOR-R x:creal &optional y:creal function like ROUND-R, corresponding to floor CEILING-R x:creal &optional y:creal function like ROUND-R, corresponding to ceiling TRUNCATE-R x:creal &optional y:creal function like ROUND-R, corresponding to truncate In the code, exported symbols at their location of definition are emphasized in UPPERCASE.