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Source for wiki FlonumsCowan version 17

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cowan

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127.11.51.1

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FlonumsCowan

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== Abstract ==

''Flonums'' are a subset of the inexact real numbers provided by a Scheme implementation.  In most Schemes, the flonums and the inexact reals are the same.  It is required that if two flonums are equal in the sense of `=`, they are also equal in the sense of `eqv?`.  That is, if 12.0f0 is a 32-bit inexact number, and 12.0 is a 64-bit inexact number, they cannot both be flonums.  In this situation, it is recommended that the 64-bit numbers be flonums.

== Rationale ==

Flonum arithmetic is already supported by many systems, mainly to remove type-dispatching overhead. Standardizing flonum arithmetic increases the portability of code that uses it. Standardizing the range of flonums would make flonum operations inefficient on some systems, which would defeat their purpose. Therefore, this SRFI specifies some of the semantics of flonums, but makes the range implementation-dependent.

The sources of the procedures in this SRFI are R7RS-small, [http://srfi.schemers.org/srfi-141/srfi-141.html SRFI 141], [http://www.r6rs.org/final/html/r6rs-lib/r6rs-lib-Z-H-12.html#node_sec_11.3 the R6RS flonum library], and the [http://pubs.opengroup.org/onlinepubs/9699919799/basedefs/math.h.html C99/Posix library] `<math.h>`, which should be available directly or indirectly to Scheme implementers.  (The C90 version of `<math.h>` lacks ``arcsinh``, ``arccosh``, ``arctanh``, ``erf``, and ``tgamma``.)

== Specification ==

Flonum operations must be at least as accurate as their generic counterparts applied to flonum arguments.  It is an error, except as otherwise noted, for an argument not to be a flonum.  In some cases, operations should be more accurate than their naive generic expansions because they have a smaller total roundoff error.  If the generic result is a non-real number, the result is `+nan.0` if the implementation supports that number, or an arbitrary flonum if not.

This SRFI uses ''x, y, z'' as
parameter names for flonum arguments, and ''ix, iy, iz''
as a name for integer-valued flonum arguments, i.e., flonums for which the
`integer?` predicate returns true.

=== Constants ===

The following C99 constants are provided as Scheme variables.

`fl-e`

Value of the mathematical constant ''e''.  (C99 `M_E`)

`fl-log2-e`

Value of `(fllog fl-e 2.0)`.  (C99 `M_LOG2E`)

`fl-log10-e`

Value of `(fllog fl-e 10.0)`.  (C99 `M_LOG10E`)

`fl-ln-2`

Value of `(fllog 2.0)`  (C99 `M_LN2`)

`fl-ln-10`

Value of `(fllog 10.0)`  (C99 `M_LN10`)

`fl-pi`

Value of the mathematical constant π.  (C99 `M_PI`)

`fl-pi/2`

Value of `(fl/ fl-pi 2.0)`  (C99 `M_PI_2`)

`fl-pi/4`

Value of `(fl/ fl-pi 4.0)`  (C99 `M_PI_4`)

`fl-1/pi`

Value of `(fl/ 1.0 fl-pi)`.  (C99 `M_1_PI`)

`fl-2/pi`

Value of `(fl/ 2.0 fl-pi)`.  (C99 `M_2_PI`)

`fl-2/sqrt-pi`

Value of `(fl/ 2.0 (flsqrt fl-pi))`.  (C99 `M_2_SQRTPI`)

`fl-sqrt-2`

Value of `(flsqrt 2.0)`.  (C99 `M_SQRT2`)

`fl-1/sqrt-2`

Value of `(fl/ 1,0 (flsqrt 2.0))`.  (C99 `M_SQRT1_2`)

`fl-maximum`

Value of the largest finite flonum.

`fl-fast-multiply-add`

Equal to `#t` if `(fl+* x y z)` is known to be faster than `(fl+ (fl* x y) z), or `#f` otherwise.  (C99 `FP_FAST_FMA`)

`fl-integer-exponent-zero`

Value of `(flinteger-binary-log 0)`.  (C99 `FP_ILOGB0`)

`fl-integer-exponent-nan`

Value of `(flinteger-binary-log +0.nan)`.  (C99 `FP_ILOGBNAN`)

== Constructors ==

`(flonum `''number''`)`

Returns the closest flonum equivalent to ''number'' in the sense of `=` and `<`.

`(fladjacent `''x y''`)`

Returns a flonum adjacent to ''x'' in the direction of ''y''.  Specifically: if ''x < y'', returns the smallest flonum larger than ''x''; if ''x > y'', returns the largest flonum smaller than ''x''; if ''x = y'', returns ''x''.  (C99 `nextafter`)

`(flcopysign `''x y''`)`

Returns a flonum whose magnitude is the magnitude of ''x'' and whose sign is the sign of ''y''.  (C99 `copysign`)

=== Predicates ===

`(flonum? `''obj''`)`

Returns `#t` if ''obj'' is a flonum and `#f` otherwise.

`(fl= `''x y z'' ...`)`

`(fl< `''x y z'' ...`)`

`(fl> `''x y z'' ...`)`

`(fl<= `''x y z'' ...`)`

`(fl>= `''x y z'' ...`)`

These procedures return `#t` if their arguments are (respectively): equal, monotonically increasing, monotonically decreasing, monotonically nondecreasing, or monotonically nonincreasing; they return `#f` otherwise. These predicates must be transitive.

`(flinteger? `''x''`)‌‌

`(flzero? `''x''`)`

`(flpositive? `''x''`)`

`(flnegative? `''x''`)`

`(flodd? `''ix''`)`

`(fleven? `''ix''`)`

`(flfinite? `''ix''`)`

`(flinfinite? `''ix''`)`

`(flnan? `''ix''`)`

These numerical predicates test a flonum for a particular property, returning `#t` or `#f`. The `flinteger?` procedure tests whether the flonum is an integer, `flzero?` tests whether it is `fl=?` to zero, `flpositive?` tests whether it is greater than zero, `flnegative?` tests whether it is less than zero, `flodd?` tests whether it is odd, `fleven?` tests whether it is even, `flfinite?` tests whether it is not an infinity and not a NaN, `flinfinite?` tests whether it is an infinity, and `flnan?` tests whether it is a NaN.

Note that `(flnegative? -0.0)<` must return `#f`;
otherwise it would lose the correspondence with
`(fl< -0.0 0.0)`, which is `#f`
according to IEEE 754.

=== Arithmetic ===

`(fl+ `''x''`)`

`(fl* `''x''`)`

Return the flonum sum or product of their flonum
arguments.  In general, they should return the flonum that best
approximates the mathematical sum or product.  (For implementations
that represent flonums using IEEE binary floating point, the
meaning of "best" is defined by the IEEE standards.)

`(fl+* `''x y z''`)`

Returns `(fl+ (fl* x y) z)`, possibly faster.  If the constant `fl-fast-fl+*`
is `#f`, it will definitely be faster.  (C99 `fma`)

`(fl- `''x y'' ...`)`

`(fl/ `''x y'' ...`)`

With two or more arguments, these procedures return the 
difference or quotient of their arguments, associating to the
left.  With one argument, however, they return the additive or
multiplicative inverse of their argument.  In general, they
should return the flonum that best approximates the mathematical
difference or quotient.  For undefined quotients, `fl/`
behaves as specified by the
IEEE standards.

For implementations that represent flonums
using IEEE binary floating point, the meaning of "best" is
reasonably well-defined by the IEEE standards.

`(flmax `''x'' ...`)`

`(flmin `''x'' ...`)`

Return the maximum/minimum argument.  If there are no arguments, these procedures return `+inf.0`/`-inf.0` if the implementation provides these numbers, and `fl-maximum` / its negation otherwise.

`(flabs `''x''`)`

Returns the absolute value of ''x''.

`(flabsdiff `''x y''`)`

Returns `(flabs (fl- x y))`.  (C99 `fdim`)

`(flsgn `''x''`)`

Returns `(flcopy-sign 1.0 x)`.

`(flnumerator `''x''`)`

`(fldenominator `''x''`)`

Returns the numerator/enominator of ``x``
as a flonum; the result is computed as if <i>x</i> was represented as
a fraction in lowest terms.  The denominator is always positive.  The
denominator of 0.0 is defined to be 1.0.

`(flfloor `''x''`)`

`(flceiling `''x''`)`

`(flround `''x''`)`

`(fltruncate `''x''`)`

These procedures return integral flonums for flonum arguments that are
not infinities or NaNs.  For such arguments, `flfloor` returns the
largest integral flonum not larger than ''x''.  The `flceiling`
procedure
returns the smallest integral flonum not smaller than ''x''.
The `fltruncate` procedure returns the integral flonum closest to ''x'' whose
absolute value is not larger than the absolute value of ''x''.
The `flround` procedure returns the closest integral flonum to ''x'',
rounding to even when ''x'' represents a number halfway between two integers.


Although infinities and NaNs are not integer objects, these procedures return
an infinity when given an infinity as an argument, and a NaN when
given a NaN.

=== Exponents and logarithms ===



=== Trigonometric functions ===

`(flsin `''x''`)`  (C99 `sin`)

`(flcos `''x''`)`  (C99 `cos`)

`(fltan `''x''`)`  (C99 `tan`)

`(flasin `''x''`)`  (C99 `asin`)

`(flacos `''x''`)`  (C99 `acos`)

`(flatan `''x'' [''y'']`)`  (C99 `atan` and `atan2`)

`(flsinh `''x''`)`  (C99 `sinh`)

`(flcosh `''x''`)`  (C99 `cosh`)

`(fltanh `''x''`)`  (C99 `tanh`)

`(flasinh `''x''`)`  (C99 `asinh`)

`(flacosh `''x''`)`  (C99 `acosh`)

`(flatanh `''x''`)`  (C99 `atanh`)

These are the usual trigonometric functions.  Note that if the result is not a real number, `+nan.0` is returned if available, or if not, an arbitrary flonum.  The `flatan` function, when passed two arguments, returns `(flatan (/ y x))` without requiring the use of complex numbers.
Implementations that use IEEE binary floating-point arithmetic 
should follow the relevant standards for these procedures.

||`flremquo`||`double      remquo(double, double, int *)`||returns two values, rounded remainder and low-order ''n'' bits of the quotient (''n'' is implementation-defined)


=== Integer division ===

The following procedures are the flonum counterparts of procedures from [http://srfi.schemers.org/srfi-141/srfi-141.html ​SRFI 141]:

{{{
flfloor/ flfloor-quotient flfloor-remainder
flceiling/ flceiling-quotient flceiling-remainder
fltruncate/ fltruncate-quotient fltruncate-remainder
flround/ flround-quotient flround-remainder
fleuclidean/ fleuclidean-quotient fleuclidean-remainder
flbalanced/ flbalanced-quotient flbalanced-remainder
}}}

They have the same arguments and semantics as their generic counterparts,
except that it is an error if the arguments are not flonums.

=== Special functions ===

||Scheme name||C signature||Comments||
||`flcomplementary-error-function`||`double      erfc(double)`||-||
||`flerror-function`||`double      erf(double)`||-||
||`flfirst-bessel`||`double      jn(int n, double)`||bessel function of the first kind, order n||
||`flgamma`||`double      tgamma(double)`||-||
||`fllog-gamma`||`double      lgamma(double)`||returns two values, log(|gamma(x)|) and sgn(gamma(x))||
||`flsecond-bessel`||`double      yn(int n, double)`||bessel function of the second kind, order ''n''||

=== Junk ===

{{{
#!html
Returns the principal square root of <i>fl</i>. For &minus;0.0,
<tt>flsqrt</tt> should return &minus;0.0; for other negative arguments,
the result may be a NaN or some unspecified flonum.</p>
<p>
</p>

<div align=left><tt>(<a name="node_idx_1046"></a>flexpt<i> <i>fl<sub>1</sub></i> <i>fl<sub>2</sub></i></i>)</tt><span style="margin-left: .5em">&zwnj;</span><span style="margin-left: .5em">&zwnj;</span>procedure&nbsp;</div>
<p>
Either <i>fl<sub>1</sub></i> should be non-negative, or, if <i>fl<sub>1</sub></i> is
negative, <i>fl<sub>2</sub></i> should be an integer object.
The <tt>flexpt</tt> procedure returns <i>fl<sub>1</sub></i> raised to the power <i>fl<sub>2</sub></i>.  If <i>fl<sub>1</sub></i> is
negative and <i>fl<sub>2</sub></i> is not an integer object, the result may be a
NaN, or may be some unspecified flonum.  If <i>fl<sub>1</sub></i> is zero, then
the result is zero.

}}}


=== Decoding flonums ===

decode-float float => significand, exponent, sign

scale-float float integer => scaled-float

float-radix float => float-radix

float-sign float-1 &optional float-2 => signed-float

float-digits float => digits1

float-precision float => digits2

integer-decode-float float => significand, exponent, integer-sign



Description:

decode-float computes three values that characterize float. The first value is of the same type as float and represents the significand. The second value represents the exponent to which the radix (notated in this description by b) must be raised to obtain the value that, when multiplied with the first result, produces the absolute value of float. If float is zero, any integer value may be returned, provided that the identity shown for scale-float holds. The third value is of the same type as float and is 1.0 if float is greater than or equal to zero or -1.0 otherwise.

decode-float divides float by an integral power of b so as to bring its value between 1/b (inclusive) and 1 (exclusive), and returns the quotient as the first value. If float is zero, however, the result equals the absolute value of float (that is, if there is a negative zero, its significand is considered to be a positive zero).

scale-float returns `(* float (expt (float b float) integer))`, where b is the radix of the floating-point representation. float is not necessarily between 1/b and 1.

float-radix returns the radix of float.

float-sign returns a number z such that z and float-1 have the same sign and also such that z and float-2 have the same absolute value. If float-2 is not supplied, its value is `(float 1 float-1)`. If an implementation has distinct representations for negative zero and positive zero, then `(float-sign -0.0) => -1.0`.

float-digits returns the number of radix b digits used in the representation of float (including any implicit digits, such as a ``hidden bit'').

float-precision returns the number of significant radix b digits present in float; if float is a float zero, then the result is an integer zero.

For normalized floats, the results of float-digits and float-precision are the same, but the precision is less than the number of representation digits for a denormalized or zero number.

integer-decode-float computes three values that characterize float - the significand scaled so as to be an integer, and the same last two values that are returned by decode-float. If float is zero, integer-decode-float returns zero as the first value. The second value bears the same relationship to the first value as for decode-float:

{{{
 (multiple-value-bind (signif expon sign)
                      (integer-decode-float f)
   (scale-float (float signif f) expon)) ==  (abs f)
}}}

=== NaN functions ===

The following functions are applicable not only to flonum NaNs but to all inexact real NaNs, which is why their names do not begin with `fl`.

`(make-nan `''payload''`)`

Returns a NaN, using the exact integer ''payload'' in an implementation-defined way to generate the payload bits.  In particular, the sign bit of the NaN is set from the sign of `payload`.  If the implementation does not support NaNs, it is an error.

`(nan-payload `''nan''`)`

Returns the payload of ''nan''.

`(nan-signaling? `''nan''`)`

Returns `#t` if ''nan'' is a signaling NaN, and `#f` otherwise.  This function is required because different floating-point processors implement the signaling bit in different ways: on most processors, the most significant bit of the payload is clear if the NaN is signaling, but on the PA-RISC and MIPS processors it is set.

`(nan= `''x y''`)`

Returns `#t` if ''x'' and ''y'' are both NaNs the same payload.

time

2016-09-15 02:30:10

version

17