RE: [boost] Math constants - Some suggested constants

From: Paul A. Bristow (boost_at_[hidden])
Date: 2001-11-06 15:27:55

Here are some suggested constants, in three groups:

1  Basic math constants.  The absolute minimum?
2  More, and some derived math constants.
3  More obscure (but essential for some people) math constants.

I have indicated some rationale for including each one,
drawing on the 'authority' of 3 texts:

1 Knuth, Art of Computer Programming,
2 John Hart, Computer Approximations, and
3 Stephen Moshier, Math Functions, mathlib
4 Morris, ACM 715 & Brown CDFlib collection of math functions used in
statistics.
as a guide to what has been used in typical math work.

Although there was some discussion and many disagreements,
in the end I have generally stuck to the naming scheme I proposed earlier.
I still prefer ln to log, but C99 seems to use log for ln and log10,
so I am persauded to reluctantly follow this.
I have yet to find a nicer way of describing reciprocal than one_div_...
Any ideas? Is reciprocal_sqrt_2 better than one_div_sqrt_2 ?

All the sqrt, log, trig and pow functions are liable to be inaccurate,
(seriously inaccurate at times)
and all lead to run-time code which is most unlikely to be optimised.
I indicate this by saying for example 'avoid log(pi)'.

Please object (Boosters are really, really expert at this!)
and suggest others (your pet constant!) that are missing.
Is the .hpp naming scheme sensible, if boring?

1  Basic math_constants.hpp

pi
log_pi - avoid log(pi)
sqrt_2 - avoid sqrt(2.)
sqrt_3
sqrt_10
cbrt_2 - avoid pow  (but follow cbrt C99 naming convention, even if ugly!)
cbrt_3 - avoid pow
fourth_root_2 - avoid pow
log_2 - avoid log(2.)
log_3
log_4 - avoid log(4.)
log_10 - avoid log(10.)
log_e - avoid log(e)
log_pi - avoid log(pi)
log_two_pi - avoid log(2.*pi)
one_div_log_2
one_div_log_10
one_div_pi - accuracy.
sqrt(1_div_pi) - avoid sqrt
sqrt_10 - avoid sqrt(10.)
sqrt_pi - avoid sqrt == gamma_half
log_sqrt_two_pi - avoid log and sqrt
log10_2 - avoid possible log and *
log10_e -

2  Additional in math_constants2.hpp

Some of these may be calculated accurately by the compiler,
but are good for clear programs,
and avoid any run-time code for debug and/or less-optimised code.

half_pi - convenient and commonly used (even if compiler should get it
exact).
quarter_pi - convenient and commonly used (even if compiler should get it
exact).
two_pi - convenient and commonly used(even if compiler should get it exact).
half_sqrt_2  - convenient and commonly used(even if compiler should get it
exact).
one_div_sqrt_2 -
one_div_sqrt_two_pi
two_pow_two_thirds - avoid pow
two_div_pi - commonly used(even if compiler should get it exactly from
one_div_pi).
three_quarters_pi - for accuracy (or three_pi_div_4?)
one_div_pi -
pi_squared - accuracy
log_2_div_2 - avoid log(2)
e_squared - accuracy
one_div_e
pi_div_180
two_pow_2_div_3 - avoid pow
log_log_2 - avoid log, twice!
phi - golden ratio
euler

a handful of real constants (1. NOT 1 - useful)
one
two
three
four
ten
half
third - accuracy?
two_thirds - accuracy?
quarter

3  Obscure in math_constants3.hpp

cbrt_10
sqrt_5
sqrt_32
sin_0
cos_0
tan_0
sin_1
cos_1
tan_1
sin_half
cos_half
tan_half
log_euler
one_div_euler
one_div_log_euler
gamma_third
gamma_two_thirds
gamma_sixth
gamma_min_at -- 1.46...
gamma_min -- 0.88560...
e_pow_euler - avoid pow
e_pow_quarter_pi
e_pow_pi
pi_pow_e
zeta_three ?

Paul

Dr Paul A Bristow, hetp Chromatography
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Kendal, Cumbria
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