How to get higher precision for sine using the Arb library?

I found the Arb library, which should be able to compute very high precision values of sine, given enough time. However, I am unable to.

Trying the sine example, I could get the predicted output.

However, when I tried to increase the precision by increasing the number of bits from 4096 to 32768, I was unable to:

Using    64 bits, sin(x) = [+/- 2.67e+859]  Using   128 bits, sin(x) = [+/- 1.30e+840]  Using   256 bits, sin(x) = [+/- 3.60e+801]  Using   512 bits, sin(x) = [+/- 3.01e+724]  Using  1024 bits, sin(x) = [+/- 2.18e+570]  Using  2048 bits, sin(x) = [+/- 1.22e+262]  Using  4096 bits, sin(x) = [-0.7190842207 +/- 1.20e-11]  Using  8192 bits, sin(x) = [-0.7190842207 +/- 1.20e-11]  Using 16384 bits, sin(x) = [-0.7190842207 +/- 1.20e-11]  Using 32768 bits, sin(x) = [-0.7190842207 +/- 1.20e-11]  

The given example has x = 2016.1.

With x = 0.1, we get the following output:

Using    64 bits, sin(x) = [0.09983341665 +/- 3.18e-12]  Using   128 bits, sin(x) = [0.09983341665 +/- 3.18e-12]  Using   256 bits, sin(x) = [0.09983341665 +/- 3.18e-12]  Using   512 bits, sin(x) = [0.09983341665 +/- 3.18e-12]  Using  1024 bits, sin(x) = [0.09983341665 +/- 3.18e-12]  Using  2048 bits, sin(x) = [0.09983341665 +/- 3.18e-12]  Using  4096 bits, sin(x) = [0.09983341665 +/- 3.18e-12]  Using  8192 bits, sin(x) = [0.09983341665 +/- 3.18e-12]  Using 16384 bits, sin(x) = [0.09983341665 +/- 3.18e-12]  Using 32768 bits, sin(x) = [0.09983341665 +/- 3.18e-12]  

This precision seems to be less than even the sin function of math.h.

I would like to increase the precision to say e-40 (or any other precision). I would be most grateful if someone could guide me 🙏🏼🙏🏼.

The code:

#include "arb.h"    void arb_sin_naive(arb_t res, const arb_t x, slong prec)  {      arb_t s, t, u, tol;      slong k;      arb_init(s); arb_init(t); arb_init(u); arb_init(tol);        arb_one(tol);      arb_mul_2exp_si(tol, tol, -prec);  /* tol = 2^-prec */        for (k = 0; ; k++)      {          arb_pow_ui(t, x, 2 * k + 1, prec);          arb_fac_ui(u, 2 * k + 1, prec);          arb_div(t, t, u, prec);  /* t = x^(2k+1) / (2k+1)! */            arb_abs(u, t);          if (arb_le(u, tol))   /* if |t| <= 2^-prec */          {              arb_add_error(s, u);    /* add |t| to the radius and stop */              break;          }            if (k % 2 == 0)              arb_add(s, s, t, prec);          else              arb_sub(s, s, t, prec);        }        arb_set(res, s);      arb_clear(s); arb_clear(t); arb_clear(u); arb_clear(tol);  }    int main()  {      arb_t x, y;      slong prec;      arb_init(x); arb_init(y);        for (prec = 64; prec <= 32768 ; prec *= 2)      {          arb_set_str(x, "0.1", prec);          arb_sin_naive(y, x, prec);          printf("Using %5ld bits, sin(x) = ", prec);          arb_printn(y, 10, 0); printf("\n");      }        arb_clear(x); arb_clear(y);  }  

https://stackoverflow.com/questions/66540401/how-to-get-higher-precision-for-sine-using-the-arb-library March 09, 2021 at 11:29AM