Re: Fw: Error in documentation for tan(), and poor implementation of sin() and cos()

       Dear Stefan,

I confirm your values, that we missed in our comparison [1], with maximal error of 268 ulps
for sin, 179 ulps for cos, and 268 ulps for tan:

zimmerma@thym:~/svn/tbd/20/src/binary64$ ./doit.glibc sin 1000
Checking sin with glibc-2.43/build and rndn
NEW sin 0 -1 0x1.2419db13f80ap+50 [268] [2.68e+02] 267.972 267.9719864551183
zimmerma@thym:~/svn/tbd/20/src/binary64$ ./doit.glibc cos 1000
Checking cos with glibc-2.43/build and rndn
NEW cos 0 -1 0x1.8577cec54ab8p+47 [179] [1.79e+02] 178.982 178.9813243034122
zimmerma@thym:~/svn/tbd/20/src/binary64$ ./doit.glibc tan 1000
Checking tan with glibc-2.43/build and rndn
NEW tan 0 -1 0x1.2419db13f80ap+50 [268] [2.68e+02] 267.972 267.9719864551183

Paul Zimmermann

[1] https://members.loria.fr/PZimmermann/papers/accuracy.pdf

> From: "Stefan Kanthak" <[email protected]>
> Cc: <[email protected]>
> Date: Tue, 7 Jul 2026 22:57:30 +0200
> Organization: Me, myself & IT
> 
> "Carlos O'Donell" <[email protected]> wrote:
> 
> > On 7/7/26 10:15 AM, Stefan Kanthak wrote:
> >> Hi maintainers,
> >>
> >> <https://sourceware.org/glibc/manual/latest/html_node/Trig-Functions.html>
> >> states:
> >
> > Stefan,
> >
> > Thank you for raising this issue.
> >
> > May you please report this to [email protected] or report a bug
> > in bugzilla with component glibc?
> >
> > Please see:
> > https://sourceware.org/glibc/wiki/FilingBug
> 
> This page only shows "Dein Browser wird geprüft!" and stalls.
> So my two part bug report goes here, per mail.
> 
> PART 1
> ~~~~~~
> 
> <https://sourceware.org/glibc/manual/latest/html_node/Trig-Functions.html>
> states:
> 
> | Mathematically, the tangent function has singularities at odd multiples
> | of pi/2. If the argument x is too close to one of these singularities,
> | tan will signal overflow.
> 
> There exists but no (double-precision) floating-point number for which tan()
> overflows or cos() yields 0, so signalling overflow respectively underflow
> is (or would be) wrong!
> 
> Shown by William Kahan MANY years ago, 6381956970095103 * 2**797 is THE
> double-precision floating-point number closest to a multiple of pi/2 -- it
> is about 4.687165924254627611e-19 or 0x1.14AE72E6BA22Fp-61 smaller than pi/2.
> 
> >From the identities
>     tan(x) = sin(x) / cos(x),
>     sin(x + pi) = cos(x + pi/2),
>     cos(x + pi) = -sin(x + pi/2)
> and the expansion of the Madhava-Newton series for sin(x) and cos(x),
>     cos(x) = x**0 / 0! - x**2 / 2! + ...
>            = 1 - x**2 / 2 + ...
>     sin(x) = x**1 / 1! - x**3 / 3! + ...
>            = x - x**3 / 6 + ...
> follows for double-precision floating-point numbers r,
> r**2 / 2 + 1 = 1 or |r| < 2**-25.5 ~ 2.107342425544701589e-8e-8,
>     cos(r) = -1
>     sin(r) = r
>     tan(r) = -r
> as well as
>     cos(pi/2 ± r) = -sin ±r
>     sin(pi/2 ± r) = 1
>     tan(pi/2 ± r) = -1 / ±r
> 
> The maximum absolute value for the double-precision tan() is therefore about
> 2.133485385753703844e+18, i.e. 290 orders of magnitude beyond overflow!
> 
> Evaluation of the maximum/minimum values for long double is left as an
> exercise.
> 
> 
> PART 2
> ~~~~~~
> 
> >> <https://sourceware.org/glibc/manual/latest/html_node/Errors-in-Math-Functions.html>
> >> states:
> >>
> >> | . Each function with a floating-point result behaves as if it computes an
> >> |   infinite-precision result that is within a few ulp of the mathematically
> >> |   correct value of the function [...]
> >>
> >> How much are "a few ulp"? Does 179 count as "few"?
> >
> > Generally <10 ULP. No, 179 does not count as a few. Though there are known outliers
> > that are more than 10 ULP.
> >
> > Please report them as bugs.
> 
> <https://godbolt.org/noscript/z/M9Ks87Wx9> is a slightly bigger demonstration -- it
> feeds integers which are near integral multiples of pi, pi/2 or pi/4 and exactly
> representable as double-precision floating-point numbers to cos(), sin() and tan()
> and prints a line if the value computed during runtime, i.e. by GLIBC, differs by
> more than 2 ULP from the value computed during compile time, i.e. by GCC.
> 
> It shows errors of 269 ULP for 72 sin/tan, 262 ULP for 66 tan, and 179 ULP for 66 cos!
> 
>   61 tan
>       65398140378926      -68524021915772.6328  -0x1.f293efe103e51p+45  -0x1.f293efe103e4fp+45      -68524021915772.6172
>   62 cos
>           74357078147863  -7.42638965257211217e-15  -0x1.0b905acac1b53p-47  -0x1.0b905acac1b51p-47  -7.42638965257210901e-15
>   62 tan            74357078147863       134654932852015.438   0x1.e9df2dc274bdcp+46    0x1.e9df2dc274bep+46
> 134654932852015.5
>   63 sin
>          139755218526789  -7.16703280049355271e-15  -0x1.023835bd45532p-47  -0x1.023835bd45536p-47  -7.16703280049355902e-15
>   63 tan           139755218526789   7.16703280049355271e-15   0x1.023835bd45532p-47   0x1.023835bd45536p-47
> 7.16703280049355902e-15
>   65 sin           148714156295726   1.48527793051442243e-14   0x1.0b905acac1b53p-46   0x1.0b905acac1b51p-46
> 1.4852779305144218e-14
>   65 tan
>         148714156295726  -1.48527793051442243e-14  -0x1.0b905acac1b53p-46  -0x1.0b905acac1b51p-46   -1.4852779305144218e-14
>   66 cos           214112296674652   2.59356852078558913e-16   0x1.2b04a1af8c415p-52   0x1.2b04a1af8c362p-52
> 2.59356852078550088e-16
>   66 tan           214112296674652          3855691461342618   0x1.b65763fd56b34p+51   0x1.b65763fd56c3ap+51
> 3855691461342749
>   67 sin           279510437053578   1.43340656009871054e-14   0x1.023835bd45532p-46   0x1.023835bd45536p-46
> 1.4334065600987118e-14
>   67 tan           279510437053578   1.43340656009871054e-14   0x1.023835bd45532p-46   0x1.023835bd45536p-46
> 1.4334065600987118e-14
>   68 sin
>        288469374822515  -7.68574650465067005e-15  -0x1.14e87fd83e173p-47  -0x1.14e87fd83e16cp-47    -7.685746504650659e-15
>   68 tan
>        288469374822515  -7.68574650465067005e-15  -0x1.14e87fd83e173p-47  -0x1.14e87fd83e16cp-47    -7.685746504650659e-15
>   69 sin           428224593349304   5.18713704157117826e-16   0x1.2b04a1af8c415p-51   0x1.2b04a1af8c362p-51
> 5.18713704157100176e-16
>   69 tan
>          428224593349304  -5.18713704157117826e-16  -0x1.2b04a1af8c415p-51  -0x1.2b04a1af8c362p-51  -5.18713704157100176e-16
>   70 sin           567979811876093   6.64831909633643459e-15   0x1.df0fd744991e1p-48   0x1.df0fd744991ffp-48
> 6.64831909633645826e-15
>   70 tan           567979811876093   6.64831909633643459e-15   0x1.df0fd744991e1p-48   0x1.df0fd744991ffp-48
> 6.64831909633645826e-15
>   71 sin
>          856449186698608  -1.03742740831423565e-15  -0x1.2b04a1af8c415p-50  -0x1.2b04a1af8c362p-50  -1.03742740831420035e-15
>   71 tan
>          856449186698608  -1.03742740831423565e-15  -0x1.2b04a1af8c415p-50  -0x1.2b04a1af8c362p-50  -1.03742740831420035e-15
>   72 sin          1284673780047912   1.55614111247135338e-15   0x1.c086f2875261fp-50   0x1.c086f28752513p-50
> 1.55614111247130053e-15
>   72 tan
>         1284673780047912  -1.55614111247135338e-15  -0x1.c086f2875261fp-50  -0x1.c086f28752513p-50  -1.55614111247130053e-15
> 
> >> With -DLIBM, i.e. sin() and cos() evaluated during runtime,
> >> <https://godbolt.org/noscript/z/x33TPr6vE> yields the following results:
> >>
> >>   5.31937264832654142e+255  -4.68716592425461995e-19  -0x1.14ae72e6ba227p-61
> >>            214112296674652   2.59356852078558913e-16   0x1.2b04a1af8c415p-52
> >>                                                                      ~~~
> >>             74357078147863  -7.42638965257211217e-15  -0x1.0b905acac1b53p-47
> >>             65398140378926   1.45934224530656633e-14   0x1.06e4484403842p-46
> >>
> >>            139755218526789  -7.16703280049355271e-15  -0x1.023835bd45532p-47
> >>            428224593349304   5.18713704157117826e-16   0x1.2b04a1af8c415p-51
> >>                                                                      ~~~
> >>            856449186698608  -1.03742740831423565e-15  -0x1.2b04a1af8c415p-50
> >>                                                                      ~~~
> >>
> >> Without -DLIBM, i.e. when GCC evaluates the functions at compile time, the
> >> results are:
> >>
> >>   5.31937264832654142e+255  -4.68716592425462765e-19  -0x1.14ae72e6ba22fp-61
> >>            214112296674652   2.59356852078550088e-16   0x1.2b04a1af8c362p-52
> >>                                                                      ~~~
> >>             74357078147863  -7.42638965257210901e-15  -0x1.0b905acac1b51p-47
> >>             65398140378926   1.45934224530656665e-14   0x1.06e4484403843p-46
> >>
> >>            139755218526789  -7.16703280049355902e-15  -0x1.023835bd45536p-47
> >>            428224593349304   5.18713704157100176e-16   0x1.2b04a1af8c362p-51
> >>                                                                      ~~~
> >>            856449186698608  -1.03742740831420035e-15  -0x1.2b04a1af8c362p-50
> >>                                                                      ~~~
> >>
> >> The values computed by GCC are correct, the underlined values computed by
> >> GLIBC are 179 ULP off!
> >
> > Thank you.
> >
> > -- 
> > Cheers,
> > Carlos.
> 
>