std::tan() extremely slow after updating glibc

Question

I have a C++ program that calls lots of trig functions. It has been running fine for more than a year. I recently installed gcc-4.8, and in the same go, updated glibc. This resulted in my program slowing down by almost a factor x1000. Using gdb I discovered that the cause of the slowdown was a call to std::tan(). When the argument is either pi or pi/2, the function takes very long to return.

Here's an MWE that reproduces the problem if compiled without optimization (the real program has the same problem both with and without the -O2 flag).

#include <cmath>

int main() {
    double pi = 3.141592653589793;
    double approxPi = 3.14159;
    double ret = 0.;

    for(int i = 0; i < 100000; ++i) ret = std::tan(pi); //Very slow
    for(int i = 0; i < 100000; ++i) ret = std::tan(approxPi); //Not slow
}

Here's a sample backtrace from gdb (obtained after interrupting the program randomly with Ctrl+c). Starting from the call to tan, the backtrace is the same in the MWE and my real program.

#0  0x00007ffff7b1d048 in __mul (p=32, z=0x7fffffffc740, y=0x7fffffffcb30, x=0x7fffffffc890) at ../sysdeps/ieee754/dbl-64/mpa.c:458
#1  __mul (x=0x7fffffffc890, y=0x7fffffffcb30, z=0x7fffffffc740, p=32) at ../sysdeps/ieee754/dbl-64/mpa.c:443
#2  0x00007ffff7b1e348 in cc32 (p=32, y=0x7fffffffc4a0, x=0x7fffffffbf60) at ../sysdeps/ieee754/dbl-64/sincos32.c:111
#3  __c32 (x=<optimized out>, y=0x7fffffffcf50, z=0x7fffffffd0a0, p=32) at ../sysdeps/ieee754/dbl-64/sincos32.c:128
#4  0x00007ffff7b1e170 in __mptan (x=<optimized out>, mpy=0x7fffffffd690, p=32) at ../sysdeps/ieee754/dbl-64/mptan.c:57
#5  0x00007ffff7b45b46 in tanMp (x=<optimized out>) at ../sysdeps/ieee754/dbl-64/s_tan.c:503
#6  __tan_avx (x=<optimized out>) at ../sysdeps/ieee754/dbl-64/s_tan.c:488
#7  0x00000000004005b8 in main ()

I've tried running the code (both the MWE and the real program) on four different systems. Two of them are in clusters where I run my code. Two are my laptops. The MWE runs without issues on one of the clusters and one laptop. I checked which version of libm.so.6 each system uses in case that's relevant. The following list shows the system description (taken from cat /etc/*-release), whether the CPU is 32 or 64 bit, whether the MWE is slow, and finally the output of running /lib/libc.so.6 and cat /proc/cpuinfo.

• SUSE Linux Enterprise Server 11 (x86_64), 64 bit, using libm-2.11.1.so (MWE is fast)

GNU C Library stable release version 2.11.1 (20100118), by Roland McGrath et al.
Copyright (C) 2009 Free Software Foundation, Inc.
This is free software; see the source for copying conditions.
There is NO warranty; not even for MERCHANTABILITY or FITNESS FOR A
PARTICULAR PURPOSE.
Configured for x86_64-suse-linux.
Compiled by GNU CC version 4.3.4 [gcc-4_3-branch revision 152973].
Compiled on a Linux 2.6.32 system on 2012-04-12.
Available extensions:
        crypt add-on version 2.1 by Michael Glad and others
        GNU Libidn by Simon Josefsson
        Native POSIX Threads Library by Ulrich Drepper et al
        BIND-8.2.3-T5B
For bug reporting instructions, please see:
<http://www.gnu.org/software/libc/bugs.html>.

processor       : 0
vendor_id       : GenuineIntel
cpu family      : 6
model           : 63
model name      : Intel(R) Xeon(R) CPU E5-2680 v3 @ 2.50GHz
stepping        : 2
microcode       : 53
cpu MHz         : 1200.000
cache size      : 30720 KB
physical id     : 0
siblings        : 24
core id         : 0
cpu cores       : 12
apicid          : 0
initial apicid  : 0
fpu             : yes
fpu_exception   : yes
cpuid level     : 15
wp              : yes
flags           : fpu vme de pse tsc msr pae mce cx8 apic sep mtrr pge mca cmov pat pse36 clflush dts acpi mmx fxsr sse sse2 ss ht tm pbe syscall nx pdpe1gb rdtscp lm constant_tsc arch_perfmon pebs bts rep_good xtopology nonstop_tsc aperfmperf pni pclmulqdq dtes64 monitor ds_cpl vmx smx est tm2 ssse3 fma cx16 xtpr pdcm pcid dca sse4_1 sse4_2 x2apic movbe popcnt tsc_deadline_timer aes xsave avx f16c rdrand lahf_lm abm ida arat epb xsaveopt pln pts dts tpr_shadow vnmi flexpriority ept vpid fsgsbase bmi1 avx2 smep bmi2 erms invpcid
bogomips        : 5000.05
clflush size    : 64
cache_alignment : 64
address sizes   : 46 bits physical, 48 bits virtual
power management:

• CentOS release 6.7 (Final), 64 bit, using libm-2.12.so (MWE is slow)

GNU C Library stable release version 2.12, by Roland McGrath et al.
Copyright (C) 2010 Free Software Foundation, Inc.
This is free software; see the source for copying conditions.
There is NO warranty; not even for MERCHANTABILITY or FITNESS FOR A
PARTICULAR PURPOSE.
Compiled by GNU CC version 4.4.7 20120313 (Red Hat 4.4.7-16).
Compiled on a Linux 2.6.32 system on 2015-09-22.
Available extensions:
        The C stubs add-on version 2.1.2.
        crypt add-on version 2.1 by Michael Glad and others
        GNU Libidn by Simon Josefsson
        Native POSIX Threads Library by Ulrich Drepper et al
        BIND-8.2.3-T5B
        RT using linux kernel aio
libc ABIs: UNIQUE IFUNC
For bug reporting instructions, please see:
<http://www.gnu.org/software/libc/bugs.html>.

processor       : 0
vendor_id       : GenuineIntel
cpu family      : 6
model           : 26
model name      : Intel(R) Xeon(R) CPU           E5507  @ 2.27GHz
stepping        : 5
cpu MHz         : 1596.000
cache size      : 4096 KB
physical id     : 0
siblings        : 4
core id         : 0
cpu cores       : 4
apicid          : 0
initial apicid  : 0
fpu             : yes
fpu_exception   : yes
cpuid level     : 11
wp              : yes
flags           : fpu vme de pse tsc msr pae mce cx8 apic sep mtrr pge mca cmov pat pse36 clflush dts acpi mmx fxsr sse sse2 ss ht tm pbe syscall nx rdtscp lm constant_tsc arch_perfmon pebs bts rep_good xtopology nonstop_tsc aperfmperf pni dtes64 monitor ds_cpl vmx est tm2 ssse3 cx16 xtpr pdcm dca sse4_1 sse4_2 popcnt lahf_lm tpr_shadow vnmi flexpriority ept vpid
bogomips        : 4533.16
clflush size    : 64
cache_alignment : 64
address sizes   : 40 bits physical, 48 bits virtual
power management:

• Ubuntu precise (12.04.5 LTS), 64 bit, using libm-2.15.so (my first laptop, MWE is slow)

GNU C Library (Ubuntu EGLIBC 2.15-0ubuntu10.15) stable release version 2.15, by Roland McGrath et al.
Copyright (C) 2012 Free Software Foundation, Inc.
This is free software; see the source for copying conditions.
There is NO warranty; not even for MERCHANTABILITY or FITNESS FOR A
PARTICULAR PURPOSE.
Compiled by GNU CC version 4.6.3.
Compiled on a Linux 3.2.79 system on 2016-05-26.
Available extensions:
    crypt add-on version 2.1 by Michael Glad and others
    GNU Libidn by Simon Josefsson
    Native POSIX Threads Library by Ulrich Drepper et al
    BIND-8.2.3-T5B
libc ABIs: UNIQUE IFUNC
For bug reporting instructions, please see:
<http://www.debian.org/Bugs/>.

processor   : 0
vendor_id   : GenuineIntel
cpu family  : 6
model       : 42
model name  : Intel(R) Core(TM) i7-2620M CPU @ 2.70GHz
stepping    : 7
microcode   : 0x1a
cpu MHz     : 800.000
cache size  : 4096 KB
physical id : 0
siblings    : 4
core id     : 0
cpu cores   : 2
apicid      : 0
initial apicid  : 0
fpu     : yes
fpu_exception   : yes
cpuid level : 13
wp      : yes
flags       : fpu vme de pse tsc msr pae mce cx8 apic sep mtrr pge mca cmov pat pse36 clflush dts acpi mmx fxsr sse sse2 ss ht tm pbe syscall nx rdtscp lm constant_tsc arch_perfmon pebs bts rep_good nopl xtopology nonstop_tsc aperfmperf pni pclmulqdq dtes64 monitor ds_cpl vmx smx est tm2 ssse3 cx16 xtpr pdcm pcid sse4_1 sse4_2 x2apic popcnt tsc_deadline_timer aes xsave avx lahf_lm ida arat epb xsaveopt pln pts dtherm tpr_shadow vnmi flexpriority ept vpid
bogomips    : 5387.59
clflush size    : 64
cache_alignment : 64
address sizes   : 36 bits physical, 48 bits virtual
power management:

• Ubuntu precise (12.04.5 LTS), 32 bit, using libm-2.15.so (my second laptop, MWE is fast)

GNU C Library (Ubuntu EGLIBC 2.15-0ubuntu10.12) stable release version 2.15, by Roland McGrath et al.
Copyright (C) 2012 Free Software Foundation, Inc.
This is free software; see the source for copying conditions.
There is NO warranty; not even for MERCHANTABILITY or FITNESS FOR A
PARTICULAR PURPOSE.
Compiled by GNU CC version 4.6.3.
Compiled on a Linux 3.2.68 system on 2015-03-26.
Available extensions:
    crypt add-on version 2.1 by Michael Glad and others
    GNU Libidn by Simon Josefsson
    Native POSIX Threads Library by Ulrich Drepper et al
    BIND-8.2.3-T5B
libc ABIs: UNIQUE IFUNC
For bug reporting instructions, please see:
<http://www.debian.org/Bugs/>.

processor    : 0
vendor_id    : GenuineIntel
cpu family    : 6
model        : 15
model name    : Intel(R) Core(TM)2 Duo CPU     T5800  @ 2.00GHz
stepping    : 13
microcode    : 0xa3
cpu MHz        : 800.000
cache size    : 2048 KB
physical id    : 0
siblings    : 2
core id        : 0
cpu cores    : 2
apicid        : 0
initial apicid    : 0
fdiv_bug    : no
hlt_bug        : no
f00f_bug    : no
coma_bug    : no
fpu        : yes
fpu_exception    : yes
cpuid level    : 10
wp        : yes
flags        : fpu vme de pse tsc msr pae mce cx8 apic sep mtrr pge mca cmov pat pse36 clflush dts acpi mmx fxsr sse sse2 ss ht tm pbe nx lm constant_tsc arch_perfmon pebs bts aperfmperf pni dtes64 monitor ds_cpl est tm2 ssse3 cx16 xtpr pdcm lahf_lm dtherm
bogomips    : 3989.79
clflush size    : 64
cache_alignment    : 64
address sizes    : 36 bits physical, 48 bits virtual
power management:

I hope I have managed to provide sufficient background info. These are my questions.

1. Why did std::tan() turn slow?
2. Is there a way to restore it to normal speed?

I would very much prefer a solution that does not require installing/replacing a bunch of libraries. That might work on my laptop, but I don't have the necessary permissions on the cluster nodes.

Update #1: I removed my observation about passing constants to tan as it was explained by Sam Varshavchik. I added the output of running /lib/libc.so.6 to my system list. Also added a fourth system. As for timing, here's the output of running time ./mwe with the pi loop (approxPi commented out).

real    0m11.483s
user    0m11.465s
sys 0m0.004s

Here it is with the approxPi loop (pi commented out).

real    0m0.011s
user    0m0.008s
sys 0m0.000s

Update #2: For each system, added whether the CPU is 32 or 64 bit as well as the output of cat /proc/cpuinfo for the first core.

Answer 1

Accuracy for transcendental functions (things like trigonometric functions and exponentials) has always a problem¹.

Why some trig function calls are slower than others

For many arguments to trigonometric functions there is a fast approximation that produces a highly accurate result for most arguments. However, for certain arguments the approximation can be quite drastically wrong. As such, more precise methods need to be employed, but these take much longer (as you've noticed).

Why might the new library be slower now

For a long time Intel made misleading claims about the accuracy of it's float versions of trigonmetric functions, saying they were much more accurate than they really were². So much so, that glibc used to just have sin(double) as a wrapper around fsin(float)³. You have likely upgraded to a version of glibc that has rectified this mistake. I can't speak for AMD's libm, but it is likely still relying on incorrect claims of accuracy around the float versions of the trigonometric functions^4,5.

What to do

If you want speed and aren't too fussed about accuracy then use the float version of tan (ftan). Otherwise, if you need accuracy then you're stuck using the slower methods. Best you can do is cache the result of tan(pi) and tan(pi/2) and use the precomputed values when you think you might need them.