Riemann's Zeta Function - another approach (RPL)

[Gerson W. Barbosa]

(07-31-2017 05:00 PM)Dieter Wrote:  

(07-31-2017 03:59 PM)Gerson W. Barbosa Wrote:  I've tested it on Emu42 set to "Authentic Calculator Speed". The number of terms hasn't been adjusted yet, thus accuracy differences between both can be highlighted:

Hmmm... x=0 should return exactly –0,5 because the coefficients are designed accordingly. So I wonder why you get –0,500000000004. This would mean that one of the coefficients is off by 4 E-12. ?!?

BTW the optimized coefficient set has two slight changes in the last digit:
line 123: -8.47149 E-7
line 131: 3.42683395 E-4

I wonder what might have caused these differences. The coefficients were copied from your HP-41C listing:

 43 -1.276ᴇ-8            18 -1.276 E-8
 44 ×                    19 *
 45 7.05133ᴇ-6           20 7.05133 E-6
 46 -                    21 -
 47 ×                    22 *
 48 9.721157ᴇ-5          23 9.721157 E-5
 49 +                    24 +
 50 ×                    25 *
 51 3.4243368ᴇ-4         26 3.4243368 E-4
 52 -                    27 -
 53 ×                    28 *
 54 0.00484515482        29 4.84515482 E-3
 55 -                    30 -
 56 ×                    31 *
 57 0.07281584288        32 7.281584288 E-2
 58 +                    33 +
 59 ×                    34 *
 60 0.007215664988       35 7.215664988 E-3
 61 +                    36 +


123 -8.4715ᴇ-7          102 -8.4715 E-7
124 ×                   103 *
125 7.51334ᴇ-6          104 7.51334 E-6
126 -                   105 -
127 ×                   106 *
128 9.609657ᴇ-5         107 9.609657 E-5
129 +                   108 +
130 ×                   109 *
131 3.42683396ᴇ-4       110 3.42683396 E-4
132 -                   111 -
133 ×                   112 *
134 0.00484527616       113 4.84527616 E-3
135 -                   114 -
136 ×                   115 *
137 0.07281583446       116 7.281583446 E-2
138 +                   117 +
139 ×                   118 *
140 0.007215664464      119 7.215664464 E-3
141 +                   120 +

(07-31-2017 05:00 PM)Dieter Wrote:  And the error check for x<0 (SQRT in line 29) can be omitted here. ;-)

That was a blind copy, as you can see.

(07-31-2017 05:00 PM)Dieter Wrote:  But remember, the two polynomial approximations are intended for 10 digit calculators. While the approximation for 1<x≤2 has an error less than 1 unit in the 12th digit and thus is useable for the 42s, the one for 0≤x<1 may be off by 4 ULP, so here one more term would make sense. Looking at your examples, indeed the largest error between –1 and +2 is 2 ULP.

This is good enough for me, at least for the time being. I will only double the number of terms, which is an easy thing to do, and see what I get. I'll leave the 42S as it is so anyone interested can improve it, starting by deleting line 29 :-)

Thanks for your work on the two polynomial approximations which have made this fast enough for all arguments in the valid range.

Gerson.