package interval_intel

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

Every function, say cos, come in three flavors:

  • fcos which is an implementation of cos that is correct (contrary to the standard functions, see below) and which result lies inside the interval defined by the following two funtions.
  • Low.cos a lower bound on the (true) value of cos.
  • High.cos an upper bound on the (true) value of cos.

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

  • tan speed (10000000 calls):2.380149
  • ftan speed (10000000 calls):2.528158
  • cos speed (10000000 calls):1.804113
  • fcos speed (10000000 calls):2.076129
  • sin speed (10000000 calls):1.844116
  • fsin speed (10000000 calls):1.972123
  • +. speed (10000000 calls):0.604037
  • fadd speed (10000000 calls):0.980062
  • -. speed (10000000 calls):0.644040
  • fsub speed (10000000 calls):0.980061
  • *. speed (10000000 calls):0.604038
  • fmul speed (10000000 calls):0.980061
  • /. speed (10000000 calls):0.992062
  • fdiv speed (10000000 calls):1.424089
  • ** speed (10000000 calls):15.420964
  • pow speed (10000000 calls):4.528283
  • mod_float speed (10000000 calls):1.996125
  • fmod speed (10000000 calls):1.236077

Intel 980X Linux 64 bits

  • tan speed (10000000 calls):0.896056
  • ftan speed (10000000 calls):0.472029
  • cos speed (10000000 calls):0.520033
  • fcos speed (10000000 calls):0.400025
  • sin speed (10000000 calls):0.524033
  • fsin speed (10000000 calls):0.400025
  • +. speed (10000000 calls):0.068005
  • fadd speed (10000000 calls):0.124008
  • -. speed (10000000 calls):0.068004
  • fsub speed (10000000 calls):0.120008
  • *. speed (10000000 calls):0.068004
  • fmul speed (10000000 calls):0.128008
  • /. speed (10000000 calls):0.096006
  • fdiv speed (10000000 calls):0.156010
  • ** speed (10000000 calls):0.668041
  • pow speed (10000000 calls):1.028064
  • mod_float speed (10000000 calls):0.224014
  • fmod speed (10000000 calls):0.152010

Rounding down and up standard functions

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

module Low : sig ... end

Functions rounding down their results.

module High : sig ... end

Functions rounding up their results.

Improving standard functions

val ffloat : int -> float
val ffloat_high : int -> float
  • deprecated Use High.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

  • deprecated Use Low.float
val fadd : float -> float -> float
val fadd_low : float -> float -> float
  • deprecated Use Low.( +. )
val fadd_high : float -> float -> float

Floating point addition in nearest, low and high mode

  • deprecated Use High.( +. )
val fsub : float -> float -> float
val fsub_low : float -> float -> float
  • deprecated Use Low.( -. )
val fsub_high : float -> float -> float

Floating point substraction in nearest, low and high mode

  • deprecated Use High.( -. )
val fmul : float -> float -> float
val fmul_low : float -> float -> float
  • deprecated Use Low.( *. )
val fmul_high : float -> float -> float

Floating point multiplication in nearest, low and high mode

  • deprecated Use High.( *. )
val fdiv : float -> float -> float
val fdiv_low : float -> float -> float
  • deprecated Use Low.( /. )
val fdiv_high : float -> float -> float

Floating point division in nearest, low and high mode

  • deprecated Use High.( /. )
val fmod : float -> float -> float

Modulo (result is supposed to be exact)

val fsqrt : float -> float
val fsqrt_low : float -> float
  • deprecated Use Low.sqrt
val fsqrt_high : float -> float

Floating point square root in nearest, low and high mode

  • deprecated Use High.sqrt
val fexp : float -> float
val fexp_low : float -> float
  • deprecated Use Low.exp
val fexp_high : float -> float

Floating point exponential in nearest, low and high mode

  • deprecated Use High.exp
val flog : float -> float
val flog_low : float -> float
  • deprecated Use Low.log
val flog_high : float -> float

Floating point log in nearest, low and high mode

  • deprecated Use High.log
val flog_pow : float -> float -> float
val flog_pow_low : float -> float -> float
  • deprecated Use Low.pow
val flog_pow_high : float -> float -> float

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

  • deprecated Use High.pow
val fpow : float -> float -> float
val fpow_low : float -> float -> float
  • deprecated Use Low.( ** )
val fpow_high : float -> float -> float

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

  • deprecated Use High.( ** )
val fsin : float -> float
val fsin_low : float -> float
  • deprecated Use Low.sin
val fsin_high : float -> float

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

  • deprecated Use High.sin
val fcos : float -> float
val fcos_low : float -> float
  • deprecated Use Low.cos
val fcos_high : float -> float

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

  • deprecated Use High.cos
val ftan : float -> float
val ftan_low : float -> float
  • deprecated Use Low.tan
val ftan_high : float -> float

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

  • deprecated Use High.tan
val fatan : float -> float -> float
val fatan_low : float -> float -> float
  • deprecated Use [Low.atan2 y x] instead of [fatan_low x y]
val fatan_high : float -> float -> float

fatan x y computes atan2 y x.

  • deprecated Use [High.atan2 y x] instead of [fatan_high x y]
val facos : float -> float
val facos_low : float -> float
  • deprecated Use Low.acos
val facos_high : float -> float

arc-cosine functions

  • deprecated Use High.acos
val fasin : float -> float
val fasin_low : float -> float
  • deprecated Use Low.asin
val fasin_high : float -> float

arc-sinus functions

  • deprecated Use High.asin
val fsinh : float -> float
val fsinh_low : float -> float
  • deprecated Use Low.sinh
val fsinh_high : float -> float

Computes sinh(x)

  • deprecated Use High.sinh
val fcosh : float -> float
val fcosh_low : float -> float
  • deprecated Use Low.cosh
val fcosh_high : float -> float

Computes cosh(x)

  • deprecated Use High.cosh
val ftanh : float -> float
val ftanh_low : float -> float
  • deprecated Use Low.tanh
val ftanh_high : float -> float

Computes tanh(x)

  • deprecated Use High.tanh
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 /..