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| author | zimmerma <zimmerma@280ebfd0-de03-0410-8827-d642c229c3f4> | 2021-02-03 17:52:27 +0000 |
|---|---|---|
| committer | zimmerma <zimmerma@280ebfd0-de03-0410-8827-d642c229c3f4> | 2021-02-03 17:52:27 +0000 |
| commit | 989e335afe8a4824ff64ef1269f880b3dbb6d81a (patch) | |
| tree | 403efca4fb1c6b078c78f3d399704973f043a7dc /src | |
| parent | a1d8f0d2108482d3c5679cd25b27ca5c240b5be9 (diff) | |
| download | mpfr-989e335afe8a4824ff64ef1269f880b3dbb6d81a.tar.gz | |
[src/atan2u.c] fixed error analysis
git-svn-id: https://scm.gforge.inria.fr/anonscm/svn/mpfr/trunk@14356 280ebfd0-de03-0410-8827-d642c229c3f4
Diffstat (limited to 'src')
| -rw-r--r-- | src/atan2u.c | 10 |
1 files changed, 6 insertions, 4 deletions
diff --git a/src/atan2u.c b/src/atan2u.c index 0d13a94fa..c7497a4da 100644 --- a/src/atan2u.c +++ b/src/atan2u.c @@ -311,13 +311,15 @@ mpfr_atan2u (mpfr_ptr z, mpfr_srcptr y, mpfr_srcptr x, unsigned long u, } MPFR_SET_SIGN (tmp, 1); expt = MPFR_GET_EXP(tmp); - /* |tmp - y/x| <= e1 := 1/2*ulp(tmp) = 2^(expt-prec-1) */ + /* |tmp - |y/x|| <= e1 := 1/2*ulp(tmp) = 2^(expt-prec-1) */ inex |= mpfr_atanu (tmp, tmp, u, MPFR_RNDN); - /* the derivative of atan(t) is 1/(1+t^2), thus if tmp2 is the new value - of tmp, we have |tmp2 - atan(y/x)| <= 1/2*ulp(tmp2) + e1/(1+tmp^2) */ + /* the derivative of atan(t) is 1/(1+t^2), thus that of atanu(t) is + u/(1+t^2)/(2*pi), and if tmp2 is the new value of tmp, we have + |tmp2 - atanu|y/x|| <= 1/2*ulp(tmp2) + e1*u/(1+tmp^2)/4 */ e = (expt < 1) ? 0 : expt - 1; /* max(1,|tmp|) >= 2^e thus 1/(1+tmp^2) <= 2^(-2*e) */ - e = expt - 2 * e; + e = expt - 2 * e + MPFR_INT_CEIL_LOG2(u) - 2; + /* now e1*u/(1+tmp^2)/4 <= 2^(e-prec-1) */ /* |tmp2 - atanu(y/x)| <= 1/2*ulp(tmp2) + 2^(e-prec-1) */ e = (e < MPFR_GET_EXP(tmp)) ? MPFR_GET_EXP(tmp) : e; /* |tmp2 - atanu(y/x)| <= 2^(e-prec) */ |