_N_a_m_e _A_p_p_e_a_r_s _o_n _P_a_g_e _D_e_s_c_r_i_p_t_i_o_n _E_r_r_o_r _B_o_u_n_d _(_U_L_P_s_)
acos acos.3 inverse trigonometric function 3 acosh acosh.3 inverse hyperbolic function 3 asin asin.3 inverse trigonometric function 3 asinh asinh.3 inverse hyperbolic function 3 atan atan.3 inverse trigonometric function 1 atanh atanh.3 inverse hyperbolic function 3 atan2 atan2.3 inverse trigonometric function 2 cbrt sqrt.3 cube root 1 ceil ceil.3 integer no less than 0 copysign ieee.3 copy sign bit 0 cos cos.3 trigonometric function 1 cosh cosh.3 hyperbolic function 3 erf erf.3 error function ??? erfc erf.3 complementary error function ??? exp exp.3 exponential 1 expm1 exp.3 exp(x)-1 1 fabs fabs.3 absolute value 0 finite ieee.3 test for finity 0 floor floor.3 integer no greater than 0 fmod fmod.3 remainder ??? hypot hypot.3 Euclidean distance 1 ilogb ieee.3 exponent extraction 0 isinf isinf.3 test for infinity 0 isnan isnan.3 test for not-a-number 0 j0 j0.3 Bessel function ??? j1 j0.3 Bessel function ??? jn j0.3 Bessel function ??? lgamma lgamma.3 log gamma function ??? log exp.3 natural logarithm 1 log10 exp.3 logarithm to base 10 3 log1p exp.3 log(1+x) 1 nan nan.3 return quiet _N_a_N 0 nextafter ieee.3 next representable number 0 pow exp.3 exponential x**y 60-500 remainder ieee.3 remainder 0 rint rint.3 round to nearest integer 0 scalbn ieee.3 exponent adjustment 0 sin sin.3 trigonometric function 1 sinh sinh.3 hyperbolic function 3 sqrt sqrt.3 square root 1 tan tan.3 trigonometric function 3 tanh tanh.3 hyperbolic function 3 trunc trunc.3 nearest integral value 3 y0 j0.3 Bessel function ??? y1 j0.3 Bessel function ??? yn j0.3 Bessel function ???
_N_a_m_e _V_a_l_u_e _D_e_s_c_r_i_p_t_i_o_n
M_E 2.7182818284590452354 e M_LOG2E 1.4426950408889634074 log 2e M_LOG10E 0.43429448190325182765 log 10e M_LN2 0.69314718055994530942 log e2 M_LN10 2.30258509299404568402 log e10 M_PI 3.14159265358979323846 pi M_PI_2 1.57079632679489661923 pi/2 M_PI_4 0.78539816339744830962 pi/4 M_1_PI 0.31830988618379067154 1/pi M_2_PI 0.63661977236758134308 2/pi M_2_SQRTPI 1.12837916709551257390 2/sqrt(pi) M_SQRT2 1.41421356237309504880 sqrt(2) M_SQRT1_2 0.70710678118654752440 1/sqrt(2)
DDEECC VVAAXX--1111 DD__ffllooaattiinngg--ppooiinntt::
This is the format for which the original math library _l_i_b_m was developed, and to which this manual is still principally dedicated. It is _t_h_e double-precision format for the PDP-11 and the earlier VAX-11 machines; VAX-11s after 1983 were provided with an optional "G" format closer to the IEEE double-precision format. The earlier DEC MicroVAXs have no D format, only G double-precision. (Why? Why not?)
Properties of D_floating-point:
Wordsize: 64 bits, 8 bytes.
Radix: Binary.
Precision: 56
sig.
bits, roughly like 17
sig.
decimals.
If x and x' are consecutive positive D_floating-point
numbers (they differ by 1 _u_l_p), then
1.3e-17 < 0.5**56 < (x'-x)/x 0.5**55 < 2.8e-17.
Range: Overflow threshold = 2.0**127 = 1.7e38. Underflow threshold = 0.5**128 = 2.9e-39. NOTE: THIS RANGE IS COMPARATIVELY NARROW.Overflow customarily stops computation.
Except for its narrow range, D_floating-point is one of the better computer arithmetics designed in the 1960's. Its properties are reflected fairly faithfully in the elementary functions for a VAX distributed in 4.3 BSD. They over/underflow only if their results have to lie out of range or very nearly so, and then they behave much as any rational arithmetic operation that over/underflowed would behave. Similarly, expressions like log(0) and atanh(1) behave like 1/0; and sqrt(-3) and acos(3) behave like 0/0; they all produce reserved operands and/or stop computation! The situation is described in more detail in manual pages. _T_h_i_s _r_e_s_p_o_n_s_e _s_e_e_m_s _e_x_c_e_s_s_i_v_e_l_y _p_u_n_i_t_i_v_e_, _s_o _i_t _i_s _d_e_s_t_i_n_e_d _t_o _b_e _r_e_p_l_a_c_e_d _a_t _s_o_m_e _t_i_m_e _i_n _t_h_e _f_o_r_e_s_e_e_a_b_l_e _f_u_t_u_r_e _b_y _a _m_o_r_e _f_l_e_x_i_b_l_e _b_u_t _s_t_i_l_l _u_n_i_f_o_r_m _s_c_h_e_m_e _b_e_i_n_g _d_e_v_e_l_o_p_e_d _t_o _h_a_n_d_l_e _a_l_l _f_l_o_a_t_i_n_g_-_p_o_i_n_t _a_r_i_t_h_m_e_t_i_c _e_x_c_e_p_t_i_o_n_s _n_e_a_t_l_y_.
How do the functions in 4.3 BSD's new _l_i_b_m for UNIX compare with their counterparts in DEC's VAX/VMS library? Some of the VMS functions are a little faster, some are a little more accurate, some are more puritanical about exceptions (like pow(0.0,0.0) and atan2(0.0,0.0)), and most occupy much more memory than their counterparts in _l_i_b_m. The VMS codes interpolate in large table to achieve speed and accuracy; the _l_i_b_m codes use tricky formulas compact enough that all of them may some day fit into a ROM.
More important, DEC regards the VMS codes as proprietary and guards them zealously against unauthorized use. But the _l_i_b_m codes in 4.3 BSD are intended for the public domain; they may be copied freely provided their provenance is always acknowledged, and provided users assist the authors in their researches by reporting experience with the codes. Therefore no user of UNIX on a machine whose arithmetic resembles VAX D_floating-point need use anything worse than the new _l_i_b_m.
IIEEEEEE SSTTAANNDDAARRDD 775544 FFllooaattiinngg--PPooiinntt AArriitthhmmeettiicc::
This standard is on its way to becoming more widely adopted
than any other design for computer arithmetic.
VLSI chips that conform to some version of that standard have been
produced by a host of manufacturers, among them ...
Intel i8087, i80287 National Semiconductor 32081 Motorola 68881 Weitek WTL-1032, ... , -1165 Zilog Z8070 Western Electric (AT&T) WE32106.Other implementations range from software, done thoroughly in the Apple Macintosh, through VLSI in the Hewlett-Packard 9000 series, to the ELXSI 6400 running ECL at 3 Megaflops. Several other companies have adopted the formats of IEEE 754 without, alas, adhering to the standard's way of handling rounding and exceptions like over/underflow. The DEC VAX G_floating-point format is very similar to the IEEE 754 Double format, so similar that the C programs for the IEEE versions of most of the elementary functions listed above could easily be converted to run on a MicroVAX, though nobody has volunteered to do that yet.
The codes in 4.3 BSD's _l_i_b_m for machines that conform to IEEE 754 are intended primarily for the National Semi. 32081 and WTL 1164/65. To use these codes with the Intel or Zilog chips, or with the Apple Macintosh or ELXSI 6400, is to forego the use of better codes provided (perhaps freely) by those companies and designed by some of the authors of the codes above. Except for _a_t_a_n, _c_b_r_t, _e_r_f, _e_r_f_c, _h_y_p_o_t, _j_0_-_j_n, _l_g_a_m_m_a, _p_o_w and _y_0_-_y_n, the Motorola 68881 has all the functions in _l_i_b_m on chip, and faster and more accurate; it, Apple, the i8087, Z8070 and WE32106 all use 64 sig. bits. The main virtue of 4.3 BSD's _l_i_b_m codes is that they are intended for the public domain; they may be copied freely provided their provenance is always acknowledged, and provided users assist the authors in their researches by reporting experience with the codes. Therefore no user of UNIX on a machine that conforms to IEEE 754 need use anything worse than the new _l_i_b_m.
Properties of IEEE 754 Double-Precision:
Wordsize: 64 bits, 8 bytes.
Radix: Binary.
Precision: 53
sig.
bits, roughly like 16
sig.
decimals.
If x and x' are consecutive positive Double-Precision
numbers (they differ by 1 _u_l_p), then
1.1e-16 < 0.5**53 < (x'-x)/x 0.5**52 < 2.3e-16.
Range: Overflow threshold = 2.0**1024 = 1.8e308 Underflow threshold = 0.5**1022 = 2.2e-308Overflow goes by default to a signed Infinity.
Exception Default Result_______________________
Invalid Operation _N_a_N, or FALSE Overflow ±Infinity Divide by Zero ±Infinity Underflow Gradual Underflow Inexact Rounded value NOTE: An Exception is not an Error unless handled badly. What makes a class of exceptions exceptional is that no single default response can be satisfactory in every instance. On the other hand, if a default response will serve most instances satisfactorily, the unsatisfactory instances cannot justify aborting computation every time the exception occurs.
For each kind of floating-point exception, IEEE 754 provides a Flag that is raised each time its exception is signaled, and stays raised until the program resets it. Programs may also test, save and restore a flag. Thus, IEEE 754 provides three ways by which programs may cope with exceptions for which the default result might be unsatisfactory:
At the option of an implementor conforming to IEEE 754, other ways to cope with exceptions may be provided:
The crucial problem for exception handling is the problem of Scope, and the problem's solution is understood, but not enough manpower was available to implement it fully in time to be distributed in 4.3 BSD's _l_i_b_m. Ideally, each elementary function should act as if it were indivisible, or atomic, in the sense that ...
Ideally, every programmer should be able _c_o_n_v_e_n_i_e_n_t_l_y to turn a debugged subprogram into one that appears atomic to its users. But simulating all three characteristics of an atomic function is still a tedious affair, entailing hosts of tests and saves-restores; work is under way to ameliorate the inconvenience.
Meanwhile, the functions in _l_i_b_m are only approximately atomic. They signal no inappropriate exception except possibly ... Over/Underflow when a result, if properly computed, might have lain barely within range, and Inexact in _c_b_r_t, _h_y_p_o_t, _l_o_g_1_0 and _p_o_w when it happens to be exact, thanks to fortuitous cancellation of errors. Otherwise, ... Invalid Operation is signaled only when any result but _N_a_N would probably be misleading. Overflow is signaled only when the exact result would be finite but beyond the overflow threshold. Divide-by-Zero is signaled only when a function takes exactly infinite values at finite operands. Underflow is signaled only when the exact result would be nonzero but tinier than the underflow threshold. Inexact is signaled only when greater range or precision would be needed to represent the exact result.