Transcript TS 18661 Floating-point extensions to C Interchange and

TS 18661 Part 4
Supplementary Functions
WG 14 N1797
2014-04-07
Math functions
IEC 60559:2011 specifies and recommends these
math functions:
exp exp2 exp10
[−∞, +∞]
expm1 exp2m1 exp10m1
[−∞, +∞]
log log2 log10
[0, +∞]
logp1=log1p log2p1 log10p1 [−1, +∞]
hypot(x, y)
[−∞, +∞] × [−∞, +∞]
rSqrt = 1/√x
[0, +∞]
compound(x, n) = (1 + x)n
[−1, +∞] × Z
Math functions (2)
rootn(x, n) = x1/n
pown(x, n) = xn
pow(x, y) = xy
powr(x, y) = xy
sin cos tan
sinPi(x) = sin(π × x) and
cosPi(x) = cos(π × x)
tanPi(x) = tan(π × x)
[−∞, +∞] × Z
[−∞, +∞] × Z
[−∞, +∞] × [−∞, +∞]
[0, +∞] × [−∞, +∞]
(−∞, +∞)
(−∞, +∞)
[−∞, +∞]
Math functions (3)
atan2Pi(y, x)
asin acos
atan
atan2(y, x)
sinh cosh tanh
asinh
acosh
atanh
[−∞, +∞] × [−∞, +∞]
[−1, +1]
[−∞, +∞]
[−∞, +∞] × [−∞, +∞]
[−∞, +∞]
[−∞, +∞]
[+1, +∞]
[−1, +1]
Math function binding
• Some IEC 60559 math functions already in C11
• TS adds the rest, in Library 7.12 Mathematics and
Annex F
• Also, for completeness
tanpi
asinpi acospi
[−∞, +∞]
[−1, +1]
• TS does not require IEC 60559-specified correct
rounding
• Names with cr prefixes reserved for correctly rounded
verisons, e.g., crsin for correctly rounded sin function
Math function binding (2)
• Added tgmath macros for new functions
• Reserved names for complex versions of new
functions, for binary floating types
Math function names
• Added logp1 equivalent to log1p
– For consistency with log2p1 and log10p1
– And to avoid the confusing log21p and log101p
• Used compoundn for compound(x, n)
– Because of existing compound(x, y) extensions
– Fits with scalbn(x, n) and others
• Otherwise used IEC 60559 names, without
camelCase (IEC 60559 does not require using
its names)
Math function special cases
• IEC 60559 and C11 Annex F treat special cases
the same
• New functions follow same principles
• TS follows C11 style for specifying math errors
in 7.12
Sum reductions
IEC 60559:2011 specifies and recommends sum
reduction operations on vectors p and q of
length n:
sum(p, n)
Σi=1,npi
dot(p, q, n)
Σi=1,npi × qi
sumSquare(p, n)
Σi=1,npi2
sumAbs(p, n)
Σi=1,n|pi|
Scaled products
IEC 60559 specifies and recommends scaled
product reduction operations: compute without
over/underflow
pr = scaled product and sf = scale factor
such that
result product = pr × radixsf
scaledProd(p, n)
scaledProdSum(p, q, n)
scaledProdDiff(p, q, n)
∏i=1,npi
∏i=1,n(pi + qi)
∏i=1,n(pi – qi)
Reduction function names
IEC 60559
sum
dot
sumSquare
sumAbs
scaledProd
scaledProdSum
scaledProdDiff
TS 16881-4
reduc_sum
reduc_sumprod
reduc_sumsq
reduc_sumabs
scaled_prod
scaled_prodsum
scaled_proddiff
Reduction function interfaces
double reduc_sum ( size_t n,
const double p[static n] );
double scaled_prod ( size_t n,
const double p[static n],
intmax_t * restrict sfptr );
Arrays indexed 0 to n - 1
IEC 60559 reductions
• Result values not fully specified like other IEC
60559 operations
• Implementation can (re)order operations and
use extra range and precision, for speed and
accuracy
• Must avoid over/underflow, except if final
result of sum reduction deserves
over/underflow
Reduction special cases
• Follows general principles for special cases, e.g.,
– reduc_sum(n, p) returns a NaN if any member of
array p is a NaN.
– reduc_sum(n, p) returns a NaN and raises the
“invalid” floating-point exception if any two members
of array p are infinities with different signs.
– Otherwise, reduc_sum(n, p) returns ±∞ if the
members of p include one or more infinities ±∞ (with
the same sign).
Reduction special cases (2)
• For scaled product:
– scaled_prod(n, p, sfptr) returns a NaN if any member of
array p is a NaN.
– scaled_prod(n, p, sfptr) returns a NaN and raises the
“invalid” floating-point exception if any two members of
array p are a zero and an infinity.
– Otherwise, scaled_prod(n, p, sfptr) returns an infinity if
any member of array p is an infinity.
– Otherwise, scaled_prod(n, p, sfptr) returns a zero if any
member of array p is a zero.
– Otherwise, scaled_prod(n, p, sfptr) returns a NaN and
raises the “invalid” floating-point exception if the scale
factor is outside the range of the intmax_t type.