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Landon Curt Noll
102
help/appr
102
help/appr
@@ -5,11 +5,11 @@ SYNOPSIS
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appr(x [,y [,z]])
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TYPES
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x real, complex, matrix, list
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y real
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z integer
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x real, complex, matrix, list
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y real
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z integer
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return same type as x except that complex x may return a real number
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return same type as x except that complex x may return a real number
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DESCRIPTION
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Return the approximate value of x as specified by a specific error
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@@ -24,77 +24,77 @@ DESCRIPTION
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In the following it is assumed y is nonzero and x is not a multiple of y.
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For real x:
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appr(x,y,z) is either the nearest multiple of y greater
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than x or the nearest multiple of y less than x. Thus, if
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we write a = appr(x,y,z) and r = x - a, then a/y is an integer
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and abs(r) < abs(y). If r > 0, we say x has been "rounded down"
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to a; if r < 0, the rounding is "up". For particular x and y,
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whether the rounding is down or up is determined by z.
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appr(x,y,z) is either the nearest multiple of y greater
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than x or the nearest multiple of y less than x. Thus, if
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we write a = appr(x,y,z) and r = x - a, then a/y is an integer
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and abs(r) < abs(y). If r > 0, we say x has been "rounded down"
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to a; if r < 0, the rounding is "up". For particular x and y,
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whether the rounding is down or up is determined by z.
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Only the 5 lowest bits of z are used, so we may assume z has been
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replaced by its value modulo 32. The type of rounding depends on
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z as follows:
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Only the 5 lowest bits of z are used, so we may assume z has been
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replaced by its value modulo 32. The type of rounding depends on
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z as follows:
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z = 0 round down or up according as y is positive or negative,
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sgn(r) = sgn(y)
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z = 0 round down or up according as y is positive or negative,
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sgn(r) = sgn(y)
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z = 1 round up or down according as y is positive or negative,
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sgn(r) = -sgn(y)
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z = 1 round up or down according as y is positive or negative,
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sgn(r) = -sgn(y)
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z = 2 round towards zero, sgn(r) = sgn(x)
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z = 2 round towards zero, sgn(r) = sgn(x)
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z = 3 round away from zero, sgn(r) = -sgn(x)
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z = 3 round away from zero, sgn(r) = -sgn(x)
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z = 4 round down, r > 0
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z = 4 round down, r > 0
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z = 5 round up, r < 0
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z = 5 round up, r < 0
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z = 6 round towards or from zero according as y is positive or
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negative, sgn(r) = sgn(x/y)
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z = 6 round towards or from zero according as y is positive or
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negative, sgn(r) = sgn(x/y)
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z = 7 round from or towards zero according as y is positive or
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negative, sgn(r) = -sgn(x/y)
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z = 7 round from or towards zero according as y is positive or
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negative, sgn(r) = -sgn(x/y)
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z = 8 a/y is even
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z = 8 a/y is even
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z = 9 a/y is odd
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z = 9 a/y is odd
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z = 10 a/y is even or odd according as x/y is positive or negative
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z = 10 a/y is even or odd according as x/y is positive or negative
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z = 11 a/y is odd or even according as x/y is positive or negative
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z = 11 a/y is odd or even according as x/y is positive or negative
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z = 12 a/y is even or odd according as y is positive or negative
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z = 12 a/y is even or odd according as y is positive or negative
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z = 13 a/y is odd or even according as y is positive or negative
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z = 13 a/y is odd or even according as y is positive or negative
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z = 14 a/y is even or odd according as x is positive or negative
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z = 14 a/y is even or odd according as x is positive or negative
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z = 15 a/y is odd or even according as x is positive or negative
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z = 15 a/y is odd or even according as x is positive or negative
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z = 16 to 31 abs(r) <= abs(y)/2; if there is a unique multiple
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of y that is nearest x, appr(x,y,z) is that multiple of y
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and then abs(r) < abs(y)/2. If x is midway between
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successive multiples of y, then abs(r) = abs(y)/2 and
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the value of a is as given by appr(x, y, z-16).
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z = 16 to 31 abs(r) <= abs(y)/2; if there is a unique multiple
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of y that is nearest x, appr(x,y,z) is that multiple of y
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and then abs(r) < abs(y)/2. If x is midway between
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successive multiples of y, then abs(r) = abs(y)/2 and
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the value of a is as given by appr(x, y, z-16).
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Matrix or List x:
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appr(x,y,z) returns the matrix or list indexed in the same way as x,
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in which each element t has been replaced by appr(t,y,z).
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appr(x,y,z) returns the matrix or list indexed in the same way as x,
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in which each element t has been replaced by appr(t,y,z).
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Complex x:
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Returns appr(re(x), y, z) + appr(im(x), y, z) * 1i
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Returns appr(re(x), y, z) + appr(im(x), y, z) * 1i
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PROPERTIES
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If appr(x,y,z) != x, then abs(x - appr(x,y,z)) < abs(y).
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If appr(x,y,z) != x, then abs(x - appr(x,y,z)) < abs(y).
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If appr(x,y,z) != x and 16 <= z <= 31, abs(x - appr(x,y,z)) <= abs(y)/2.
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If appr(x,y,z) != x and 16 <= z <= 31, abs(x - appr(x,y,z)) <= abs(y)/2.
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For z = 0, 1, 4, 5, 16, 17, 20 or 21, and any integer n,
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appr(x + n*y, y, z) = appr(x, y, z) + n * y.
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For z = 0, 1, 4, 5, 16, 17, 20 or 21, and any integer n,
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appr(x + n*y, y, z) = appr(x, y, z) + n * y.
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If y is nonzero, appr(x,y,8)/y = an odd integer n only if x = n * y.
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If y is nonzero, appr(x,y,8)/y = an odd integer n only if x = n * y.
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EXAMPLE
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; print appr(-5.44,0.1,0), appr(5.44,0.1,0), appr(5.7,1,0), appr(-5.7,1,0)
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@@ -159,7 +159,7 @@ SEE ALSO
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##
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## Calc is distributed in the hope that it will be useful, but WITHOUT
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## ANY WARRANTY; without even the implied warranty of MERCHANTABILITY
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## or FITNESS FOR A PARTICULAR PURPOSE. See the GNU Lesser General
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## or FITNESS FOR A PARTICULAR PURPOSE. See the GNU Lesser General
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## Public License for more details.
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##
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## A copy of version 2.1 of the GNU Lesser General Public License is
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@@ -167,8 +167,8 @@ SEE ALSO
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## received a copy with calc; if not, write to Free Software Foundation, Inc.
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## 51 Franklin Street, Fifth Floor, Boston, MA 02110-1301, USA.
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##
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## Under source code control: 1994/09/25 17:18:21
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## File existed as early as: 1994
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## Under source code control: 1994/09/25 17:18:21
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## File existed as early as: 1994
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##
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## chongo <was here> /\oo/\ http://www.isthe.com/chongo/
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## Share and enjoy! :-) http://www.isthe.com/chongo/tech/comp/calc/
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## chongo <was here> /\oo/\ http://www.isthe.com/chongo/
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## Share and enjoy! :-) http://www.isthe.com/chongo/tech/comp/calc/
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