cases when number functions produce 0: clarify docs and fix (atan 0 x)
for positive exact x; also clarify docs on some cases when divide-by-zero exception is raised
This commit is contained in:
Matthew Flatt
parent
3638ea4963
commit
d6d5c914f7
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@ -201,46 +201,58 @@ otherwise.}
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@; ----------------------------------------
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@section{Arithmetic}
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@defproc[(+ [z number?] ...) number?]{ Returns the sum of the
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@racket[z]s, adding pairwise from left to right. If no arguments are
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provided, the result is @racket[0].
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@defproc[(+ [z number?] ...) number?]{
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Returns the sum of the @racket[z]s, adding pairwise from left to
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right. If no arguments are provided, the result is @racket[0].
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@mz-examples[(+ 1 2) (+ 1.0 2+3i 5) (+)]}
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@defproc*[([(- [z number?]) number?]
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[(- [z number?] [w number?] ...+) number?])]{
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When no @racket[w]s are supplied, returns @racket[(- 0 z)].
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When no @racket[w]s are supplied, returns @racket[(- 0 z)].
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Otherwise, returns the subtraction of the @racket[w]s from @racket[z]
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working pairwise from left to right.}
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@mz-examples[(- 5 3.0) (- 1) (- 2+7i 1 3)]
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@defproc[(* [z number?] ...) number?]{ Returns the product of the
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@racket[z]s, multiplying pairwise from left to right. If no arguments are
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provided, the result is @racket[1].}
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@defproc[(* [z number?] ...) number?]{
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@mz-examples[(* 2 3) (* 8.0 9) (* 1+2i 3+4i)]
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Returns the product of the @racket[z]s, multiplying pairwise from left
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to right. If no arguments are provided, the result is
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@racket[1]. Multiplying any number by exact @racket[0] produces exact
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@racket[0].
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@mz-examples[(* 2 3) (* 8.0 9) (* 1+2i 3+4i)]}
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@defproc*[([(/ [z number?]) number?]
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[(/ [z number?] [w number?] ...+) number?])]{
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When no @racket[w]s are supplied, returns @racket[(/ 1 z)].
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Otherwise, returns the division @racket[z] by the var[w]s
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working pairwise from left to right.}
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@mz-examples[(/ 3 4) (/ 81 3 3) (/ 10.0) (/ 1+2i 3+4i)]
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When no @racket[w]s are supplied, returns @racket[(/ 1 z)].
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Otherwise, returns the division @racket[z] by the @racket[w]s working
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pairwise from left to right.
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If @racket[z] is exact @racket[0] and no @racket[w] is exact
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@racket[0], then the result is exact @racket[0]. If any @racket[w] is
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exact @racket[0], the @exnraise[exn:fail:contract:divide-by-zero].
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@mz-examples[(/ 3 4) (/ 81 3 3) (/ 10.0) (/ 1+2i 3+4i)]}
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@defproc[(quotient [n integer?] [m integer?]) integer?]{ Returns
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@racket[(truncate (/ n m))].}
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@defproc[(quotient [n integer?] [m integer?]) integer?]{
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@mz-examples[(quotient 10 3) (quotient -10.0 3) (quotient +inf.0 3)]
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Returns @racket[(truncate (/ n m))].
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@mz-examples[(quotient 10 3) (quotient -10.0 3) (quotient +inf.0 3)]}
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@defproc[(remainder [n integer?] [m integer?]) integer?]{ Returns
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@racket[_q] with the same sign as @racket[n] such that
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@defproc[(remainder [n integer?] [m integer?]) integer?]{
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Returns @racket[_q] with the same sign as @racket[n] such that
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@itemize[
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@ -250,20 +262,26 @@ otherwise.}
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]
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If @racket[m] is exact @racket[0], the
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@exnraise[exn:fail:contract:divide-by-zero].
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@mz-examples[(remainder 10 3) (remainder -10.0 3) (remainder 10.0 -3) (remainder -10 -3) (remainder +inf.0 3)]}
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@defproc[(quotient/remainder [n integer?] [m integer?]) (values integer? integer?)]{ Returns
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@racket[(values (quotient n m) (remainder n m))], but the combination may be computed
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more efficiently than separate calls to @racket[quotient] and @racket[remainder].
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@defproc[(quotient/remainder [n integer?] [m integer?]) (values integer? integer?)]{
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Returns @racket[(values (quotient n m) (remainder n m))], but the
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combination may be computed more efficiently than separate calls to
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@racket[quotient] and @racket[remainder].
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@mz-examples[
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(quotient/remainder 10 3)
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]}
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@defproc[(modulo [n integer?] [m integer?]) integer?]{ Returns
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@racket[_q] with the same sign as @racket[m] where
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@defproc[(modulo [n integer?] [m integer?]) integer?]{
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Returns @racket[_q] with the same sign as @racket[m] where
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@itemize[
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@ -273,6 +291,9 @@ otherwise.}
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]
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If @racket[m] is exact @racket[0], the
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@exnraise[exn:fail:contract:divide-by-zero].
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@mz-examples[(modulo 10 3) (modulo -10.0 3) (modulo 10.0 -3) (modulo -10 -3) (modulo +inf.0 3)]}
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@ -285,72 +306,88 @@ otherwise.}
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@mz-examples[(abs 1.0) (abs -1)]}
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@defproc[(max [x real?] ...+) real?]{ Returns the largest of the
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@racket[x]s, or @racket[+nan.0] if any @racket[x] is @racket[+nan.0].
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If any @racket[x] is inexact, the result is coerced to inexact.
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@defproc[(max [x real?] ...+) real?]{
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Returns the largest of the @racket[x]s, or @racket[+nan.0] if any
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@racket[x] is @racket[+nan.0]. If any @racket[x] is inexact, the
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result is coerced to inexact.
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@mz-examples[(max 1 3 2) (max 1 3 2.0)]}
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@defproc[(min [x real?] ...+) real?]{ Returns the smallest of the
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@racket[x]s, or @racket[+nan.0] if any @racket[x] is @racket[+nan.0].
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If any @racket[x] is inexact, the result is coerced to inexact.
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@defproc[(min [x real?] ...+) real?]{
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Returns the smallest of the @racket[x]s, or @racket[+nan.0] if any
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@racket[x] is @racket[+nan.0]. If any @racket[x] is inexact, the
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result is coerced to inexact.
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@mz-examples[(min 1 3 2) (min 1 3 2.0)]}
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@defproc[(gcd [n integer?] ...) integer?]{ Returns the
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@as-index{greatest common divisor} (a non-negative number) of the
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@racket[n]s. If no arguments are provided, the result is
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@racket[0]. If all arguments are zero, the result is zero.
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@defproc[(gcd [n integer?] ...) integer?]{
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Returns the @as-index{greatest common divisor} (a non-negative
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number) of the @racket[n]s. If no arguments are provided, the result
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is @racket[0]. If all arguments are zero, the result is zero.
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@mz-examples[(gcd 10) (gcd 12 81.0)]}
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@defproc[(lcm [n integer?] ...) integer?]{ Returns the @as-index{least
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common multiple} (a non-negative number) of the @racket[n]s. If no
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arguments are provided, the result is @racket[1]. If any argument is
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zero, the result is zero.
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@defproc[(lcm [n integer?] ...) integer?]{
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Returns the @as-index{least common multiple} (a non-negative number)
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of the @racket[n]s. If no arguments are provided, the result is
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@racket[1]. If any argument is zero, the result is zero; furthermore,
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if any argument is exact @racket[0], the result is exact @racket[0].
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@mz-examples[(lcm 10) (lcm 3 4.0)]}
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@defproc[(round [x real?]) integer?]{ Returns the integer closest to
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@racket[x], resolving ties in favor of an even number.
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@defproc[(round [x real?]) integer?]{
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Returns the integer closest to @racket[x], resolving ties in favor of
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an even number.
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@mz-examples[(round 17/4) (round -17/4) (round 2.5) (round -2.5)]}
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@defproc[(floor [x real?]) integer?]{ Returns the largest integer is that
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is no more than @racket[x].
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@defproc[(floor [x real?]) integer?]{
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Returns the largest integer is that is no more than @racket[x].
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@mz-examples[(floor 17/4) (floor -17/4) (floor 2.5) (floor -2.5)]}
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@defproc[(ceiling [x real?]) integer?]{ Returns the smallest integer is
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that is at least as large as @racket[x].
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@defproc[(ceiling [x real?]) integer?]{
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Returns the smallest integer is that is at least as large as
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@racket[x].
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@mz-examples[(ceiling 17/4) (ceiling -17/4) (ceiling 2.5) (ceiling -2.5)]}
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@defproc[(truncate [x real?]) integer?]{ Returns the integer farthest
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from @racket[0] that is no closer to @racket[0] than @racket[x].
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@defproc[(truncate [x real?]) integer?]{
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Returns the integer farthest from @racket[0] that is no closer to
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@racket[0] than @racket[x].
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@mz-examples[(truncate 17/4) (truncate -17/4) (truncate 2.5) (truncate -2.5)]}
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@defproc[(numerator [q rational?]) integer?]{
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Coreces @racket[q] to an exact number, finds the numerator of the number
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expressed in its simplest fractional form, and returns this number
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coerced to the exactness of @racket[q].
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Coreces @racket[q] to an exact number, finds the numerator of the
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number expressed in its simplest fractional form, and returns this
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number coerced to the exactness of @racket[q].
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@mz-examples[(numerator 5) (numerator 34/8) (numerator 2.3)]}
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@defproc[(denominator [q rational?]) integer?]{
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Coreces @racket[q] to an exact number, finds the numerator of the number
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expressed in its simplest fractional form, and returns this number
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coerced to the exactness of @racket[q].
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Coreces @racket[q] to an exact number, finds the numerator of the
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number expressed in its simplest fractional form, and returns this
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number coerced to the exactness of @racket[q].
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@mz-examples[(denominator 5) (denominator 34/8) (denominator 2.3)]}
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@ -358,10 +395,10 @@ otherwise.}
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@defproc[(rationalize [x real?] [tolerance real?]) real?]{
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Among the real numbers within @racket[(abs tolerance)] of @racket[x],
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returns the one corresponding to an exact number whose
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@racket[denominator] is smallest. If multiple integers are within
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@racket[tolerance] of @racket[x], the one closest to @racket[0] is
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used.
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returns the one corresponding to an exact number whose
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@racket[denominator] is smallest. If multiple integers are within
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@racket[tolerance] of @racket[x], the one closest to @racket[0] is
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used.
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@mz-examples[
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(rationalize 1/4 1/10)
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@ -414,46 +451,58 @@ used.
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@; ------------------------------------------------------------------------
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@section{Powers and Roots}
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@defproc[(sqrt [z number?]) number?]{ Returns the principal
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@as-index{square root} of @racket[z]. The result is exact if
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@racket[z] is exact and @racket[z]'s square root is rational. See
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also @racket[integer-sqrt].
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@defproc[(sqrt [z number?]) number?]{
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Returns the principal @as-index{square root} of @racket[z]. The
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result is exact if @racket[z] is exact and @racket[z]'s square root
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is rational. See also @racket[integer-sqrt].
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@mz-examples[(sqrt 4/9) (sqrt 2) (sqrt -1)]}
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@defproc[(integer-sqrt [n integer?]) complex?]{ Returns @racket[(floor
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(sqrt n))] for positive @racket[n]. For negative @racket[n], the result is
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@racket[(* (integer-sqrt (- n)) 0+i)].
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@defproc[(integer-sqrt [n integer?]) complex?]{
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Returns @racket[(floor (sqrt n))] for positive @racket[n]. For
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negative @racket[n], the result is @racket[(* (integer-sqrt (- n))
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0+i)].
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@mz-examples[(integer-sqrt 4.0) (integer-sqrt 5)]}
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@defproc[(integer-sqrt/remainder [n integer?])
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(values integer? integer?)]{
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Returns @racket[(integer-sqrt n)] and @racket[(- n (expt
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Returns @racket[(integer-sqrt n)] and @racket[(- n (expt
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(integer-sqrt n) 2))].
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@mz-examples[(integer-sqrt/remainder 4.0) (integer-sqrt/remainder 5)]}
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@defproc[(expt [z number?] [w number?]) number?]{ Returns @racket[z]
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raised to the power of @racket[w]. If @racket[w] is exact @racket[0],
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the result is @racket[1]. If @racket[z] is exact @racket[0] and
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@racket[w] is negative, the @exnraise[exn:fail:contract].
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@defproc[(expt [z number?] [w number?]) number?]{
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Returns @racket[z] raised to the power of @racket[w]. If @racket[w] is
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exact @racket[0], the result is exact @racket[1]. If @racket[z] is
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exact @racket[0] and @racket[w] is negative, the
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@exnraise[exn:fail:contract:divide-by-zero].
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@mz-examples[(expt 2 3) (expt 4 0.5) (expt +inf.0 0)]}
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@defproc[(exp [z number?]) number?]{ Returns Euler's number raised to the
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power of @racket[z]. The result is normally inexact, but it is
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@racket[1] when @racket[z] is an exact @racket[0].
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@defproc[(exp [z number?]) number?]{
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Returns Euler's number raised to the power of @racket[z]. The result
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is normally inexact, but it is exact @racket[1] when @racket[z] is an
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exact @racket[0].
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@mz-examples[(exp 1) (exp 2+3i) (exp 0)]}
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@defproc[(log [z number?]) number?]{ Returns the natural logarithm of
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@racket[z]. The result is normally inexact, but it is
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@racket[0] when @racket[z] is an exact @racket[1]. When @racket[z]
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is exact @racket[0], @exnraise[exn:fail:contract:divide-by-zero].
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@defproc[(log [z number?]) number?]{
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Returns the natural logarithm of @racket[z]. The result is normally
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inexact, but it is exact @racket[0] when @racket[z] is an exact
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@racket[1]. When @racket[z] is exact @racket[0],
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@exnraise[exn:fail:contract:divide-by-zero].
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@mz-examples[(log (exp 1)) (log 2+3i) (log 1)]}
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@ -461,31 +510,42 @@ used.
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@; ------------------------------------------------------------------------
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@section{Trignometric Functions}
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@defproc[(sin [z number?]) number?]{ Returns the sine of @racket[z], where
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@racket[z] is in radians.
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@defproc[(sin [z number?]) number?]{
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Returns the sine of @racket[z], where @racket[z] is in radians. The
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result is normally inexact, but it is exact @racket[0] if @racket[z]
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is exact @scheme[0].
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@mz-examples[(sin 3.14159) (sin 1+05.i)]}
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@defproc[(cos [z number?]) number?]{ Returns the cosine of @racket[z],
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where @racket[z] is in radians.
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@defproc[(cos [z number?]) number?]{
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Returns the cosine of @racket[z], where @racket[z] is in radians.
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@mz-examples[(cos 3.14159) (cos 1+05.i)]}
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@defproc[(tan [z number?]) number?]{ Returns the tangent of @racket[z],
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where @racket[z] is in radians.
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@defproc[(tan [z number?]) number?]{
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Returns the tangent of @racket[z], where @racket[z] is in radians. The
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result is normally inexact, but it is exact @racket[0] if @racket[z]
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is exact @scheme[0].
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@mz-examples[(tan 0.7854) (tan 1+05.i)]}
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@defproc[(asin [z number?]) number?]{ Returns the arcsin in radians of @racket[z].
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@defproc[(asin [z number?]) number?]{
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Returns the arcsin in radians of @racket[z]. The result is normally
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inexact, but it is exact @racket[0] if @racket[z] is exact @scheme[0].
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@mz-examples[(asin 0.25) (asin 1+05.i)]}
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@defproc[(acos [z number?]) number?]{ Returns the arccosine in radians
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of @racket[z].
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@defproc[(acos [z number?]) number?]{
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Returns the arccosine in radians of @racket[z].
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@mz-examples[(acos 0.25) (acos 1+05.i)]}
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@ -494,48 +554,59 @@ used.
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[(atan [y real?] [x real?]) number?])]{
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In the one-argument case, returns the arctangent of the inexact
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approximation of @racket[z], except that the result is an exact
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@racket[0] for an exact @racket[0] argument.
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approximation of @racket[z], except that the result is an exact
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@racket[0] for an exact @racket[0] argument.
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In the two-argument case, the result is roughly the same as @racket[(/
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(exact->inexact y) (exact->inexact x))], but the signs of @racket[y]
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and @racket[x] determine the quadrant of the result. Moreover, a
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suitable angle is returned when @racket[y] divided by @racket[x]
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produces @racket[+nan.0] in the case that neither @racket[y] nor
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@racket[x] is @racket[+nan.0].
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(exact->inexact y) (exact->inexact x))], but the signs of @racket[y]
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and @racket[x] determine the quadrant of the result. Moreover, a
|
||||
suitable angle is returned when @racket[y] divided by @racket[x]
|
||||
produces @racket[+nan.0] in the case that neither @racket[y] nor
|
||||
@racket[x] is @racket[+nan.0]. Finally, if @racket[x] is exact
|
||||
@racket[0] and @racket[y] is an exact positive number, the result is
|
||||
exact @racket[0]. If both @racket[x] and @racket[y] are exact
|
||||
@racket[0], the @exnraise[exn:fail:contract:divide-by-zero].
|
||||
|
||||
@mz-examples[(atan 0.5) (atan 2 1) (atan -2 -1) (atan 1+05.i) (atan +inf.0 -inf.0)]}
|
||||
|
||||
@; ------------------------------------------------------------------------
|
||||
@section{Complex Numbers}
|
||||
|
||||
@defproc[(make-rectangular [x real?] [y real?]) number?]{ Returns
|
||||
@racket[(+ x (* y 0+1i))].
|
||||
@defproc[(make-rectangular [x real?] [y real?]) number?]{
|
||||
|
||||
Returns @racket[(+ x (* y 0+1i))].
|
||||
|
||||
@mz-examples[(make-rectangular 3 4.0)]}
|
||||
|
||||
|
||||
@defproc[(make-polar [magnitude real?] [angle real?]) number?]{ Returns
|
||||
@racket[(+ (* magnitude (cos angle)) (* magnitude (sin angle) 0+1i))].
|
||||
@defproc[(make-polar [magnitude real?] [angle real?]) number?]{
|
||||
|
||||
Returns @racket[(+ (* magnitude (cos angle)) (* magnitude (sin angle)
|
||||
0+1i))].
|
||||
|
||||
@mz-examples[#:eval math-eval
|
||||
(make-polar 10 (* pi 1/2))
|
||||
(make-polar 10 (* pi 1/4))]}
|
||||
|
||||
|
||||
@defproc[(real-part [z number?]) real?]{ Returns the real part of
|
||||
the complex number @racket[z] in rectangle coordinates.
|
||||
@defproc[(real-part [z number?]) real?]{
|
||||
|
||||
Returns the real part of the complex number @racket[z] in rectangle
|
||||
coordinates.
|
||||
|
||||
@mz-examples[(real-part 3+4i) (real-part 5.0)]}
|
||||
|
||||
|
||||
@defproc[(imag-part [z number?]) real?]{ Returns the imaginary part of
|
||||
the complex number @racket[z] in rectangle coordinates.
|
||||
@defproc[(imag-part [z number?]) real?]{
|
||||
|
||||
Returns the imaginary part of the complex number @racket[z] in
|
||||
rectangle coordinates.
|
||||
|
||||
@mz-examples[(imag-part 3+4i) (imag-part 5.0) (imag-part 5.0+0.0i)]}
|
||||
|
||||
|
||||
@defproc[(magnitude [z number?]) (and/c real? (not/c negative?))]{
|
||||
|
||||
Returns the magnitude of the complex number @racket[z] in polar
|
||||
coordinates.
|
||||
|
||||
|
|
|
|||
|
|
@ -1778,6 +1778,10 @@
|
|||
(test 100.0 round (* 100 (tan (atan 1))))
|
||||
(test 100.0-0.0i z-round (* 100 (tan (+ +0.0i (atan 1)))))
|
||||
(test 0.0 atan 0.0 0)
|
||||
(test 0 atan 0 1)
|
||||
(test 0 atan 0 (expt 2 100))
|
||||
(test 0.0 atan 0 1.0)
|
||||
(test 314.0 round (* 100 (atan 0 -1)))
|
||||
(err/rt-test (atan 0 0) exn:fail:contract:divide-by-zero?)
|
||||
(test 1024.0 round (expt 2.0 10.0))
|
||||
(test 1024.0 round (expt -2.0 10.0))
|
||||
|
|
|
|||
|
|
@ -2024,10 +2024,15 @@ atan_prim (int argc, Scheme_Object *argv[])
|
|||
|
||||
n2 = argv[1];
|
||||
|
||||
if ((n1 == zeroi) && (n2 == zeroi)) {
|
||||
scheme_raise_exn(MZEXN_FAIL_CONTRACT_DIVIDE_BY_ZERO,
|
||||
"atan: undefined for 0 and 0");
|
||||
ESCAPED_BEFORE_HERE;
|
||||
if (n1 == zeroi) {
|
||||
if (n2 == zeroi) {
|
||||
scheme_raise_exn(MZEXN_FAIL_CONTRACT_DIVIDE_BY_ZERO,
|
||||
"atan: undefined for 0 and 0");
|
||||
ESCAPED_BEFORE_HERE;
|
||||
}
|
||||
if ((SCHEME_INTP(n2) && (SCHEME_INT_VAL(n2) > 0))
|
||||
|| (SCHEME_BIGNUMP(n2) && (SCHEME_BIGPOS(n2))))
|
||||
return zeroi;
|
||||
}
|
||||
|
||||
if (SCHEME_INTP(n2))
|
||||
|
|
|
|||
Loading…
Reference in New Issue
Block a user