Module Interval_intel.Fpu

Access to low level floating point functions. THIS LIBRARY ONLY WORKS FOR INTEL PROCESSORS.

Every function, say cos, come in three flavors:

Almost all low level functions are implemented using the x87 functions and x87 rounding modes. There are unfortunately a few problems to understand. The x87 is supposed to be able to return a nearest value and a upper and a lower bound for each elementary operation it can perform. This is not always true. Some functions such as cos(), sin() or tan() are not properly implemented everywhere.

For example, for the angle a= 1.570 796 326 794 896 557 998 981 734 272 092 580 795 288 085 937 5 the following values are computed for cos(a), by (1) the MPFI library (with 128 bits precision), (2) the x87 in low mode, (3) the x87 in nearest mode (default value for the C and OCaml library on 32 bits linux), (4) the x87 in high mode, (5) the SSE2 implementation (default value for the C and OCaml library on 64 bits linux):

(1) 6.123 233 995 736 765 886 130 329 661 375 001 464 640 377 798 836e-17

(2) 6.123 031 769 111 885 058 461 925 285 082 049 859 451 216 355 021e-17

(3) 6.123 031 769 111 886 291 057 089 692 912 995 815 277 099 609 375e-17

(4) 6.123 031 769 111 886 291 057 089 692 912 995 815 277 099 609 375e-17

(5) 6.123 233 995 736 766 035 868 820 147 291 983 023 128 460 623 387e-17

The upper bound (4) computed by the x87 is clearly incorrect, as it is lower than the correct value computed by the MPFI library.

The value computed by the SSE2 (5) is much more precise than the one computed by the x87. Unfortunately, there is no way to get an upper and lower bound value, and we are thus stuck with the x87 for computing these (sometimes incorrect) bounds.

The problem here is that the value computed by the standard, C-lib (or OCaml) cos function doesn't always lie in the lower/upper bound interval returned by the x87 functions, and this can be a very serious problem when executing Branch and Bound algorithms which expect the mid-value to be inside the lower/upper interval.

We solved the problem by rewritting the trigonometric functions in order to make them both consistant and correct. We used the following property: when -pi/4<=a<=pi/4 the rounding in 64 bits of the 80 bits low/std/high value returned by the x87 are correct. Moreover, when 0<a<2**53 then (a mod (2Pi_low)) and (a mod (2Pi_high)) are in the same quadrant. Last, (a mod Pi/2_High) <= (a mod Pi/2) <= (a mod Pi/2_Low). With this implementation, the lower and upper bounds are properly set and they are always lower (resp. higher) than the value computed by the standard cos functions on 32 and 64 bits architecture. This rewritting has been done in assembly language and is quite efficient.

Keep in mind that values returned by the standard (C-lib or OCaml) cos(), sin() or tan() functions are still different on 32 and 64 bits architecture. If you want to have a program which behaves exactly in the same way on both architectures, you can use the Fpu module fcos, fsin or ftan functions which always return the same values on all architectures, or even use the Fpu_rename or Fpu_rename_all modules to transparently rename the floating point functions.

The functions are quite efficient (see below). However, they have a serious disadvantage compared to their standard counterparts. When the compiler compiles instruction ''a+.b'', the code of the operation is inlined, while when it compiles ''(fadd a b)'', the compiler generates a function call, which is expensive.

Intel Atom 230 Linux 32 bits

Intel 980X Linux 64 bits

Rounding down and up standard functions

The following sub-modules RoundDown and RoundUp implement the same functions but with different roundings (down for RoundDown and up for RoundUp).

module RoundDown : sig ... end

Functions rounding down their results.

module RoundUp : sig ... end

Functions rounding up their results.

module Low = RoundDown
module High = RoundUp

Improving standard functions

val ffloat : int -> float
val ffloat_high : int -> float
val ffloat_low : int -> float

float() functions. The float function is exact on 32 bits machine but not on 64 bits machine with ints larger than 53 bits

val fadd : float -> float -> float
val fadd_low : float -> float -> float
val fadd_high : float -> float -> float

Floating point addition in nearest, low and high mode

val fsub : float -> float -> float
val fsub_low : float -> float -> float
val fsub_high : float -> float -> float

Floating point substraction in nearest, low and high mode

val fmul : float -> float -> float
val fmul_low : float -> float -> float
val fmul_high : float -> float -> float

Floating point multiplication in nearest, low and high mode

val fdiv : float -> float -> float
val fdiv_low : float -> float -> float
val fdiv_high : float -> float -> float

Floating point division in nearest, low and high mode

val fmod : float -> float -> float

Modulo (result is supposed to be exact)

val fexp : float -> float
val fexp_low : float -> float
val fexp_high : float -> float

Floating point exponential in nearest, low and high mode

val flog : float -> float
val flog_low : float -> float
val flog_high : float -> float

Floating point log in nearest, low and high mode

val flog_pow : float -> float -> float
val flog_pow_low : float -> float -> float
val flog_pow_high : float -> float -> float

Computes x^y for 0 < x < infinity and neg_infinity < y < infinity

val fpow : float -> float -> float
val fpow_low : float -> float -> float
val fpow_high : float -> float -> float

Computes x^y expanded to its mathematical limit when it exists

val fsin : float -> float
val fsin_low : float -> float
val fsin_high : float -> float

Computes sin(x) for x in ]-2^63, 2^63[

val fcos : float -> float
val fcos_low : float -> float
val fcos_high : float -> float

Computes cos(x) for x in ]-2^63, 2^63[

val ftan : float -> float
val ftan_low : float -> float
val ftan_high : float -> float

Computes tan(x) for x in ]-2^63, 2^63[

val fatan : float -> float -> float
val fatan_low : float -> float -> float
val fatan_high : float -> float -> float

fatan x y computes atan2 y x.

val facos : float -> float
val facos_low : float -> float
val facos_high : float -> float

arc-cosine functions

val fasin : float -> float
val fasin_low : float -> float
val fasin_high : float -> float

arc-sinus functions

val fsinh : float -> float
val fsinh_low : float -> float
val fsinh_high : float -> float

Computes sinh(x)

val fcosh : float -> float
val fcosh_low : float -> float
val fcosh_high : float -> float

Computes cosh(x)

val ftanh : float -> float
val ftanh_low : float -> float
val ftanh_high : float -> float

Computes tanh(x)

val is_neg : float -> bool

is_neg x returns if x has its sign bit set (true for -0.).

Overriding standard functions

module Rename : sig ... end

Aliases floating point functions to their "constant" counterparts, except for "ordinary functions".

module Rename_all : sig ... end

Aliases floating point functions to their "constant" counterparts, including +., -., *. and /..