Fw: Error in documentation for tan(), and poor implementation of sin() and cos()
Stefan Kanthak
stefan.kanthak@nexgo.de
Tue Jul 7 20:57:30 GMT 2026
"Carlos O'Donell" <carlos@redhat.com> 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 libc-alpha@sourceware.org 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.
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