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From: Keith Thompson <Keith.S.Thompson+u@gmail.com>
Newsgroups: comp.lang.c,comp.arch
Subject: Re: Radians Or Degrees?
Date: Sat, 16 Mar 2024 16:19:11 -0700
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mitchalsup@aol.com (MitchAlsup1) writes:
> Michael S wrote:
>> On Sat, 16 Mar 2024 01:16:25 +0000
>> mitchalsup@aol.com (MitchAlsup1) wrote:
>>> Keith Thompson wrote:
>>> > I can see how computing sin(x) with high precision for
>>> > "reasonable" values of x would be useful, but does any of that
>>> > benefit from being able to compute sin(2^53) accurately?
>>> Because accurate argument reduction reduces the burden on the
>>> programmer to remain within his sandbox.
>
>> Not really.
>
> Say you are a programmer and you receive a value like 2^53 from an
> Input read and you wan the most accurate possible SIN( of that ).

I can't think of a scenario where that would be useful (other than just
doing it for the sake of doing it).

If 2^53 represents a physical quantity, how likely is the actual value
to be known within ±π (+/i pi for those who prefer ASCII)?

If you can get better precision without too much extra cost, that's
great.  I don't know enough to have an opinion about what the best
tradeoff is, but I presume it's going to be different depending on the
application.

Here's a C program that shows how precise sin(2^53) can be for types
float, double, and long double (I used gcc and glibc).  The nextafter
functions are used to compute the nearest representable number.  For
long double, the value of sin() changes by about 1 part in 1600, which
seems decent, but it's not nearly as precise as for values around 1.0.
For float and double, the imprecision of the argument is enough to make
the result practically meaningless.

#include <math.h>
#include <stdio.h>
#include <limits.h>
#include <float.h>
int main(void) {
    {
        printf("float (%zu bits, %d mantissa bits)\n", CHAR_BIT * sizeof (float), FLT_MANT_DIG);
        const float x = (float)(1LL<<53);
        const float y = nextafterf(x, x*2);
        printf("%.8f %.8f\n", x, sinf(x));
        printf("%.8f %.8f\n", y, sinf(y));
    }
    putchar('\n');
    {
        printf("double (%zu bits, %d mantissa bits)\n", CHAR_BIT * sizeof (double), DBL_MANT_DIG);
        const double x = (double)(1LL<<53);
        const double y = nextafter(x, x*2);
        printf("%.8f %.8f\n", x, sin(x));
        printf("%.8f %.8f\n", y, sin(y));
    }
    putchar('\n');
    {
        printf("long double (%zu bits, %d mantissa bits)\n", CHAR_BIT * sizeof (long double), LDBL_MANT_DIG);
        const long double x = (long double)(1LL<<53);
        const long double y = nextafterl(x, x*2);
        printf("%.8Lf %.8Lf\n", x, sinl(x));
        printf("%.8Lf %.8Lf\n", y, sinl(y));
    }
}

Output:

float (32 bits, 24 mantissa bits)
9007199254740992.00000000 -0.84892595
9007200328482816.00000000 -0.34159181

double (64 bits, 53 mantissa bits)
9007199254740992.00000000 -0.84892596
9007199254740994.00000000 -0.12729655

long double (128 bits, 64 mantissa bits)
9007199254740992.00000000 -0.84892596
9007199254740992.00097656 -0.84944168

-- 
Keith Thompson (The_Other_Keith) Keith.S.Thompson+u@gmail.com
Working, but not speaking, for Medtronic
void Void(void) { Void(); } /* The recursive call of the void */