Boost C++ Libraries: Ticket #13606: sinc_pi incorrect taylor_0_bound

status changed; resolution set

John Maddock — Sun, 17 Jun 2018 18:21:08 GMT

status new → closed
resolution → fixed

You're quite right: fixed in https://github.com/boostorg/math/commit/658945d50822adae83d77083a559e785ce18609c along with better tests.

status changed; resolution deleted

anonymous — Mon, 18 Jun 2018 08:47:52 GMT

status closed → reopened
resolution fixed

Bounds is still incorrect. Should look like

Range0 for abs(x) in ( 0 , pow(eps, 1/2) ) result == 1.0, cause next Taylor x2/6 LESS than "epsilon" and 1.0-x2/6 == 1.0

For example (for double) if abs(x) == 1.0e-10. The x2 == 1.0e-20 and 1.0e-20/6 is less than double::epsilon, so 1.0 + 1.0e-20/6 == 1.0.

Range1 for abs(x) in ( pow(eps, 1/2) , pow(eps, 1/4) )

result == 1.0-x2/6, cause next Taylor x2*x2/120 LESS than "epsilon" even in Horner method. Important signs of x2*x2 is out of epsilon range.

Range2 for abs(x) in ( pow(eps, 1/4) , pow(eps, 1/6) )

result == 1.0 + x2 * (-1.0 + x2 / 20.0) / 6.0;, cause x^{6/7! LESS than "epsilon" even in Horner method.}

NOTE: We can use approximate bounds from above ignoring Taylor series coefficients. Calculation of exact bounds is not trivial, and should use exact analytical precision bounds of all operations.

anonymous — Mon, 18 Jun 2018 11:36:22 GMT

Apologies: you're quite correct that the powers of epsilon are all wrong, I confess when looking at this that it may be over-optimised in any case given that evaluation of sin(x) is usually pretty fast in any case. I'll come back to this shortly when I'm less rushed.

minorlogic@… — Mon, 18 Jun 2018 13:31:08 GMT

It only should optimize precision. Cause sin(x)/x is pretty fast and quite precise (it use hardware asm code on many platforms). sin(x)/x only should check x != 0. With float ranges close to zero, sin(x) strongly equal to x, and division of any floating point to itself must equal to 1.0. So sin(x)/x with x!= 0 is quite precise and save.

Within range (pow(eps, 1/2) , pow(eps, 1/6)) taylor expansion provide a small precision improvement. It is small comparing to direct sin(x)/x but can provide several bits of precision (difference of errors about 5%-10% on most platforms).

It is not easy, to predict precision loss from sin(x)/x. Different platforms and compilers provides different sin(x) precision, and its error grows after division by x. Using Taylor approximation we can provide exact solution with known error upper bound (compared to float epsilon).

From other hand, if we use sin(x) on most of "sinc" ranges, we can use it in whole range, and final error depends from sin(x) implementation.

For improvement of precision, ranges and errors should be carefully verified and tested ( against six(x)/x ) in ranges approximating "sinc".

For fast and save implementation i propose use only one branch and range check (0, pow(eps, 1/4) ) and expansion with 1.0 - x2/6.

if(abs(x) < eps_root_four) {

return 1.0 - x2/6;

}

return six(x)/x;

status changed; resolution set

John Maddock — Tue, 19 Jun 2018 17:27:58 GMT

status reopened → closed
resolution → fixed

See https://github.com/boostorg/math/commit/838dd9419374d43b3021a959522a005bef0df4a2