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Trigonometric functions precision #22

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MarekKnapek opened this issue Nov 7, 2020 · 1 comment
Open

Trigonometric functions precision #22

MarekKnapek opened this issue Nov 7, 2020 · 1 comment

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@MarekKnapek
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Hi, I like your library, but I cannot find any information about precision guarantees.

In my program, I need a precomputed lookup table of some angles. Before I discovered your library I computed the values at run-time and copy & pasted them into my source. After I discovered your library I wanted to assert, that my precomputed table values match constexpr computation from your library. And this test failed.

I noticed, that your computation are sometimes off by one at last significant place. So, I wanted to ask what precision do you guarantee? Is this a bug or is this (sometimes off by one at ULP) intended behavior? Or maybe it is not possible to compute trigonometric functions precisely at compile-time at all?

I created a little test program, it computes sin and cos of all angles from -15 to +15 degrees, stepping by one degree. I compiled it on Ubuntu 18.04 using GCC/G++ 10, and on Windows using Visual Studio 2019 (19.27.29112).

Best regards, Marek.

The program:

#include <cmath>
#include <cstdio>
#include <numbers>

#include <gcem.hpp>

int main()
{
	static constexpr auto const deg_to_rad = [](double const& deg) -> double { return deg * (std::numbers::pi_v<double> / 180.0); };
	for(int i = -15; i != 15 + 1; ++i)
	{
		std::printf("std::cos (%+3d deg) == %.30f (%a)\n", i, std::cos(deg_to_rad(i)), std::cos(deg_to_rad(i)));
		std::printf("gcem::cos(%+3d deg) == %.30f (%a)%s\n", i, gcem::cos(deg_to_rad(i)), gcem::cos(deg_to_rad(i)), std::cos(deg_to_rad(i)) == gcem::cos(deg_to_rad(i)) ? "" : " bug");
	}
	for(int i = -15; i != 15 + 1; ++i)
	{
		std::printf("std::sin (%+3d deg) == %.30f (%a)\n", i, std::sin(deg_to_rad(i)), std::sin(deg_to_rad(i)));
		std::printf("gcem::sin(%+3d deg) == %.30f (%a)%s\n", i, gcem::sin(deg_to_rad(i)), gcem::sin(deg_to_rad(i)), std::sin(deg_to_rad(i)) == gcem::sin(deg_to_rad(i)) ? "" : " bug");
	}
}

Compile on Ubuntu:

g++-10 -std=c++20 gcem_trig_check.cpp

Output on Ubuntu:

std::cos (-15 deg) == 0.965925826289068312213714762038 (0x1.ee8dd4748bf15p-1)
gcem::cos(-15 deg) == 0.965925826289068312213714762038 (0x1.ee8dd4748bf15p-1)
std::cos (-14 deg) == 0.970295726275996472942608761514 (0x1.f0ca99f79ba25p-1)
gcem::cos(-14 deg) == 0.970295726275996583964911224029 (0x1.f0ca99f79ba26p-1) bug
std::cos (-13 deg) == 0.974370064785235245885530730447 (0x1.f2e0a214e870fp-1)
gcem::cos(-13 deg) == 0.974370064785235245885530730447 (0x1.f2e0a214e870fp-1)
std::cos (-12 deg) == 0.978147600733805688832944724709 (0x1.f4cfc327a008p-1)
gcem::cos(-12 deg) == 0.978147600733805577810642262193 (0x1.f4cfc327a007fp-1) bug
std::cos (-11 deg) == 0.981627183447663975712771389226 (0x1.f697d6938b6c2p-1)
gcem::cos(-11 deg) == 0.981627183447663975712771389226 (0x1.f697d6938b6c2p-1)
std::cos (-10 deg) == 0.984807753012208020315654266597 (0x1.f838b8c811c17p-1)
gcem::cos(-10 deg) == 0.984807753012208131337956729112 (0x1.f838b8c811c18p-1) bug
std::cos ( -9 deg) == 0.987688340595137770350220307591 (0x1.f9b24942fe45cp-1)
gcem::cos( -9 deg) == 0.987688340595137659327917845076 (0x1.f9b24942fe45bp-1) bug
std::cos ( -8 deg) == 0.990268068741570361979142944620 (0x1.fb046a930947ap-1)
gcem::cos( -8 deg) == 0.990268068741570473001445407135 (0x1.fb046a930947bp-1) bug
std::cos ( -7 deg) == 0.992546151641321983127852490725 (0x1.fc2f025a23e8bp-1)
gcem::cos( -7 deg) == 0.992546151641322094150154953240 (0x1.fc2f025a23e8cp-1) bug
std::cos ( -6 deg) == 0.994521895368273289861349439889 (0x1.fd31f94f867c6p-1)
gcem::cos( -6 deg) == 0.994521895368273289861349439889 (0x1.fd31f94f867c6p-1)
std::cos ( -5 deg) == 0.996194698091745545198705258372 (0x1.fe0d3b41815a2p-1)
gcem::cos( -5 deg) == 0.996194698091745434176402795856 (0x1.fe0d3b41815a1p-1) bug
std::cos ( -4 deg) == 0.997564050259824197652847033169 (0x1.fec0b7170fff6p-1)
gcem::cos( -4 deg) == 0.997564050259824197652847033169 (0x1.fec0b7170fff6p-1)
std::cos ( -3 deg) == 0.998629534754573833232882407174 (0x1.ff4c5ed12e61dp-1)
gcem::cos( -3 deg) == 0.998629534754573833232882407174 (0x1.ff4c5ed12e61dp-1)
std::cos ( -2 deg) == 0.999390827019095762118183756684 (0x1.ffb0278bf0567p-1)
gcem::cos( -2 deg) == 0.999390827019095651095881294168 (0x1.ffb0278bf0566p-1) bug
std::cos ( -1 deg) == 0.999847695156391269577511593525 (0x1.ffec097f5af8ap-1)
gcem::cos( -1 deg) == 0.999847695156391158555209131009 (0x1.ffec097f5af89p-1) bug
std::cos ( +0 deg) == 1.000000000000000000000000000000 (0x1p+0)
gcem::cos( +0 deg) == 1.000000000000000000000000000000 (0x1p+0)
std::cos ( +1 deg) == 0.999847695156391269577511593525 (0x1.ffec097f5af8ap-1)
gcem::cos( +1 deg) == 0.999847695156391158555209131009 (0x1.ffec097f5af89p-1) bug
std::cos ( +2 deg) == 0.999390827019095762118183756684 (0x1.ffb0278bf0567p-1)
gcem::cos( +2 deg) == 0.999390827019095651095881294168 (0x1.ffb0278bf0566p-1) bug
std::cos ( +3 deg) == 0.998629534754573833232882407174 (0x1.ff4c5ed12e61dp-1)
gcem::cos( +3 deg) == 0.998629534754573833232882407174 (0x1.ff4c5ed12e61dp-1)
std::cos ( +4 deg) == 0.997564050259824197652847033169 (0x1.fec0b7170fff6p-1)
gcem::cos( +4 deg) == 0.997564050259824197652847033169 (0x1.fec0b7170fff6p-1)
std::cos ( +5 deg) == 0.996194698091745545198705258372 (0x1.fe0d3b41815a2p-1)
gcem::cos( +5 deg) == 0.996194698091745434176402795856 (0x1.fe0d3b41815a1p-1) bug
std::cos ( +6 deg) == 0.994521895368273289861349439889 (0x1.fd31f94f867c6p-1)
gcem::cos( +6 deg) == 0.994521895368273289861349439889 (0x1.fd31f94f867c6p-1)
std::cos ( +7 deg) == 0.992546151641321983127852490725 (0x1.fc2f025a23e8bp-1)
gcem::cos( +7 deg) == 0.992546151641322094150154953240 (0x1.fc2f025a23e8cp-1) bug
std::cos ( +8 deg) == 0.990268068741570361979142944620 (0x1.fb046a930947ap-1)
gcem::cos( +8 deg) == 0.990268068741570473001445407135 (0x1.fb046a930947bp-1) bug
std::cos ( +9 deg) == 0.987688340595137770350220307591 (0x1.f9b24942fe45cp-1)
gcem::cos( +9 deg) == 0.987688340595137659327917845076 (0x1.f9b24942fe45bp-1) bug
std::cos (+10 deg) == 0.984807753012208020315654266597 (0x1.f838b8c811c17p-1)
gcem::cos(+10 deg) == 0.984807753012208131337956729112 (0x1.f838b8c811c18p-1) bug
std::cos (+11 deg) == 0.981627183447663975712771389226 (0x1.f697d6938b6c2p-1)
gcem::cos(+11 deg) == 0.981627183447663975712771389226 (0x1.f697d6938b6c2p-1)
std::cos (+12 deg) == 0.978147600733805688832944724709 (0x1.f4cfc327a008p-1)
gcem::cos(+12 deg) == 0.978147600733805577810642262193 (0x1.f4cfc327a007fp-1) bug
std::cos (+13 deg) == 0.974370064785235245885530730447 (0x1.f2e0a214e870fp-1)
gcem::cos(+13 deg) == 0.974370064785235245885530730447 (0x1.f2e0a214e870fp-1)
std::cos (+14 deg) == 0.970295726275996472942608761514 (0x1.f0ca99f79ba25p-1)
gcem::cos(+14 deg) == 0.970295726275996583964911224029 (0x1.f0ca99f79ba26p-1) bug
std::cos (+15 deg) == 0.965925826289068312213714762038 (0x1.ee8dd4748bf15p-1)
gcem::cos(+15 deg) == 0.965925826289068312213714762038 (0x1.ee8dd4748bf15p-1)
std::sin (-15 deg) == -0.258819045102520739476403832668 (-0x1.0907dc193069p-2)
gcem::sin(-15 deg) == -0.258819045102520739476403832668 (-0x1.0907dc193069p-2)
std::sin (-14 deg) == -0.241921895599667730047954705697 (-0x1.ef74bf2e4b91dp-3)
gcem::sin(-14 deg) == -0.241921895599667730047954705697 (-0x1.ef74bf2e4b91dp-3)
std::sin (-13 deg) == -0.224951054343865003426472526371 (-0x1.ccb3236cdc675p-3)
gcem::sin(-13 deg) == -0.224951054343864975670896910742 (-0x1.ccb3236cdc674p-3) bug
std::sin (-12 deg) == -0.207911690817759342575499204031 (-0x1.a9cd9ac4258f6p-3)
gcem::sin(-12 deg) == -0.207911690817759314819923588402 (-0x1.a9cd9ac4258f5p-3) bug
std::sin (-11 deg) == -0.190808995376544804356555573577 (-0x1.86c6ddd76624fp-3)
gcem::sin(-11 deg) == -0.190808995376544804356555573577 (-0x1.86c6ddd76624fp-3)
std::sin (-10 deg) == -0.173648177666930331186634361984 (-0x1.63a1a7e0b7389p-3)
gcem::sin(-10 deg) == -0.173648177666930358942209977613 (-0x1.63a1a7e0b738ap-3) bug
std::sin ( -9 deg) == -0.156434465040230868959625354364 (-0x1.4060b67a85375p-3)
gcem::sin( -9 deg) == -0.156434465040230868959625354364 (-0x1.4060b67a85375p-3)
std::sin ( -8 deg) == -0.139173100960065437847745783984 (-0x1.1d06c968d9e19p-3)
gcem::sin( -8 deg) == -0.139173100960065465603321399612 (-0x1.1d06c968d9e1ap-3) bug
std::sin ( -7 deg) == -0.121869343405147476100403025612 (-0x1.f32d44c4f62d3p-4)
gcem::sin( -7 deg) == -0.121869343405147489978190833426 (-0x1.f32d44c4f62d4p-4) bug
std::sin ( -6 deg) == -0.104528463267653470847307062286 (-0x1.ac2609b3c576cp-4)
gcem::sin( -6 deg) == -0.104528463267653456969519254471 (-0x1.ac2609b3c576bp-4) bug
std::sin ( -5 deg) == -0.087155742747658165869850677154 (-0x1.64fd6b8c28102p-4)
gcem::sin( -5 deg) == -0.087155742747658151992062869340 (-0x1.64fd6b8c28101p-4) bug
std::sin ( -4 deg) == -0.069756473744125302438590097154 (-0x1.1db8f6d6a5128p-4)
gcem::sin( -4 deg) == -0.069756473744125302438590097154 (-0x1.1db8f6d6a5128p-4)
std::sin ( -3 deg) == -0.052335956242943834637593170100 (-0x1.acbc748efc90ep-5)
gcem::sin( -3 deg) == -0.052335956242943834637593170100 (-0x1.acbc748efc90ep-5)
std::sin ( -2 deg) == -0.034899496702500969191884649945 (-0x1.1de58c9f7dc27p-5)
gcem::sin( -2 deg) == -0.034899496702500969191884649945 (-0x1.1de58c9f7dc27p-5)
std::sin ( -1 deg) == -0.017452406437283511653202339176 (-0x1.1df0b2b89dd1ep-6)
gcem::sin( -1 deg) == -0.017452406437283511653202339176 (-0x1.1df0b2b89dd1ep-6)
std::sin ( +0 deg) == 0.000000000000000000000000000000 (0x0p+0)
gcem::sin( +0 deg) == 0.000000000000000000000000000000 (0x0p+0)
std::sin ( +1 deg) == 0.017452406437283511653202339176 (0x1.1df0b2b89dd1ep-6)
gcem::sin( +1 deg) == 0.017452406437283511653202339176 (0x1.1df0b2b89dd1ep-6)
std::sin ( +2 deg) == 0.034899496702500969191884649945 (0x1.1de58c9f7dc27p-5)
gcem::sin( +2 deg) == 0.034899496702500969191884649945 (0x1.1de58c9f7dc27p-5)
std::sin ( +3 deg) == 0.052335956242943834637593170100 (0x1.acbc748efc90ep-5)
gcem::sin( +3 deg) == 0.052335956242943834637593170100 (0x1.acbc748efc90ep-5)
std::sin ( +4 deg) == 0.069756473744125302438590097154 (0x1.1db8f6d6a5128p-4)
gcem::sin( +4 deg) == 0.069756473744125302438590097154 (0x1.1db8f6d6a5128p-4)
std::sin ( +5 deg) == 0.087155742747658165869850677154 (0x1.64fd6b8c28102p-4)
gcem::sin( +5 deg) == 0.087155742747658151992062869340 (0x1.64fd6b8c28101p-4) bug
std::sin ( +6 deg) == 0.104528463267653470847307062286 (0x1.ac2609b3c576cp-4)
gcem::sin( +6 deg) == 0.104528463267653456969519254471 (0x1.ac2609b3c576bp-4) bug
std::sin ( +7 deg) == 0.121869343405147476100403025612 (0x1.f32d44c4f62d3p-4)
gcem::sin( +7 deg) == 0.121869343405147489978190833426 (0x1.f32d44c4f62d4p-4) bug
std::sin ( +8 deg) == 0.139173100960065437847745783984 (0x1.1d06c968d9e19p-3)
gcem::sin( +8 deg) == 0.139173100960065465603321399612 (0x1.1d06c968d9e1ap-3) bug
std::sin ( +9 deg) == 0.156434465040230868959625354364 (0x1.4060b67a85375p-3)
gcem::sin( +9 deg) == 0.156434465040230868959625354364 (0x1.4060b67a85375p-3)
std::sin (+10 deg) == 0.173648177666930331186634361984 (0x1.63a1a7e0b7389p-3)
gcem::sin(+10 deg) == 0.173648177666930358942209977613 (0x1.63a1a7e0b738ap-3) bug
std::sin (+11 deg) == 0.190808995376544804356555573577 (0x1.86c6ddd76624fp-3)
gcem::sin(+11 deg) == 0.190808995376544804356555573577 (0x1.86c6ddd76624fp-3)
std::sin (+12 deg) == 0.207911690817759342575499204031 (0x1.a9cd9ac4258f6p-3)
gcem::sin(+12 deg) == 0.207911690817759314819923588402 (0x1.a9cd9ac4258f5p-3) bug
std::sin (+13 deg) == 0.224951054343865003426472526371 (0x1.ccb3236cdc675p-3)
gcem::sin(+13 deg) == 0.224951054343864975670896910742 (0x1.ccb3236cdc674p-3) bug
std::sin (+14 deg) == 0.241921895599667730047954705697 (0x1.ef74bf2e4b91dp-3)
gcem::sin(+14 deg) == 0.241921895599667730047954705697 (0x1.ef74bf2e4b91dp-3)
std::sin (+15 deg) == 0.258819045102520739476403832668 (0x1.0907dc193069p-2)
gcem::sin(+15 deg) == 0.258819045102520739476403832668 (0x1.0907dc193069p-2)

Compile on Windows:

cl /std:c++latest gcem_trig_test.cpp

Output on Windows:

std::cos (-15 deg) == 0.965925826289068312213714762038 (0x1.ee8dd4748bf15p-1)
gcem::cos(-15 deg) == 0.965925826289068312213714762038 (0x1.ee8dd4748bf15p-1)
std::cos (-14 deg) == 0.970295726275996472942608761514 (0x1.f0ca99f79ba25p-1)
gcem::cos(-14 deg) == 0.970295726275996583964911224029 (0x1.f0ca99f79ba26p-1) bug
std::cos (-13 deg) == 0.974370064785235245885530730447 (0x1.f2e0a214e870fp-1)
gcem::cos(-13 deg) == 0.974370064785235245885530730447 (0x1.f2e0a214e870fp-1)
std::cos (-12 deg) == 0.978147600733805688832944724709 (0x1.f4cfc327a0080p-1)
gcem::cos(-12 deg) == 0.978147600733805577810642262193 (0x1.f4cfc327a007fp-1) bug
std::cos (-11 deg) == 0.981627183447663975712771389226 (0x1.f697d6938b6c2p-1)
gcem::cos(-11 deg) == 0.981627183447663975712771389226 (0x1.f697d6938b6c2p-1)
std::cos (-10 deg) == 0.984807753012208020315654266597 (0x1.f838b8c811c17p-1)
gcem::cos(-10 deg) == 0.984807753012208131337956729112 (0x1.f838b8c811c18p-1) bug
std::cos ( -9 deg) == 0.987688340595137770350220307591 (0x1.f9b24942fe45cp-1)
gcem::cos( -9 deg) == 0.987688340595137659327917845076 (0x1.f9b24942fe45bp-1) bug
std::cos ( -8 deg) == 0.990268068741570361979142944620 (0x1.fb046a930947ap-1)
gcem::cos( -8 deg) == 0.990268068741570473001445407135 (0x1.fb046a930947bp-1) bug
std::cos ( -7 deg) == 0.992546151641321983127852490725 (0x1.fc2f025a23e8bp-1)
gcem::cos( -7 deg) == 0.992546151641322094150154953240 (0x1.fc2f025a23e8cp-1) bug
std::cos ( -6 deg) == 0.994521895368273289861349439889 (0x1.fd31f94f867c6p-1)
gcem::cos( -6 deg) == 0.994521895368273289861349439889 (0x1.fd31f94f867c6p-1)
std::cos ( -5 deg) == 0.996194698091745545198705258372 (0x1.fe0d3b41815a2p-1)
gcem::cos( -5 deg) == 0.996194698091745434176402795856 (0x1.fe0d3b41815a1p-1) bug
std::cos ( -4 deg) == 0.997564050259824197652847033169 (0x1.fec0b7170fff6p-1)
gcem::cos( -4 deg) == 0.997564050259824197652847033169 (0x1.fec0b7170fff6p-1)
std::cos ( -3 deg) == 0.998629534754573833232882407174 (0x1.ff4c5ed12e61dp-1)
gcem::cos( -3 deg) == 0.998629534754573833232882407174 (0x1.ff4c5ed12e61dp-1)
std::cos ( -2 deg) == 0.999390827019095762118183756684 (0x1.ffb0278bf0567p-1)
gcem::cos( -2 deg) == 0.999390827019095651095881294168 (0x1.ffb0278bf0566p-1) bug
std::cos ( -1 deg) == 0.999847695156391269577511593525 (0x1.ffec097f5af8ap-1)
gcem::cos( -1 deg) == 0.999847695156391158555209131009 (0x1.ffec097f5af89p-1) bug
std::cos ( +0 deg) == 1.000000000000000000000000000000 (0x1.0000000000000p+0)
gcem::cos( +0 deg) == 1.000000000000000000000000000000 (0x1.0000000000000p+0)
std::cos ( +1 deg) == 0.999847695156391269577511593525 (0x1.ffec097f5af8ap-1)
gcem::cos( +1 deg) == 0.999847695156391158555209131009 (0x1.ffec097f5af89p-1) bug
std::cos ( +2 deg) == 0.999390827019095762118183756684 (0x1.ffb0278bf0567p-1)
gcem::cos( +2 deg) == 0.999390827019095651095881294168 (0x1.ffb0278bf0566p-1) bug
std::cos ( +3 deg) == 0.998629534754573833232882407174 (0x1.ff4c5ed12e61dp-1)
gcem::cos( +3 deg) == 0.998629534754573833232882407174 (0x1.ff4c5ed12e61dp-1)
std::cos ( +4 deg) == 0.997564050259824197652847033169 (0x1.fec0b7170fff6p-1)
gcem::cos( +4 deg) == 0.997564050259824197652847033169 (0x1.fec0b7170fff6p-1)
std::cos ( +5 deg) == 0.996194698091745545198705258372 (0x1.fe0d3b41815a2p-1)
gcem::cos( +5 deg) == 0.996194698091745434176402795856 (0x1.fe0d3b41815a1p-1) bug
std::cos ( +6 deg) == 0.994521895368273289861349439889 (0x1.fd31f94f867c6p-1)
gcem::cos( +6 deg) == 0.994521895368273289861349439889 (0x1.fd31f94f867c6p-1)
std::cos ( +7 deg) == 0.992546151641321983127852490725 (0x1.fc2f025a23e8bp-1)
gcem::cos( +7 deg) == 0.992546151641322094150154953240 (0x1.fc2f025a23e8cp-1) bug
std::cos ( +8 deg) == 0.990268068741570361979142944620 (0x1.fb046a930947ap-1)
gcem::cos( +8 deg) == 0.990268068741570473001445407135 (0x1.fb046a930947bp-1) bug
std::cos ( +9 deg) == 0.987688340595137770350220307591 (0x1.f9b24942fe45cp-1)
gcem::cos( +9 deg) == 0.987688340595137659327917845076 (0x1.f9b24942fe45bp-1) bug
std::cos (+10 deg) == 0.984807753012208020315654266597 (0x1.f838b8c811c17p-1)
gcem::cos(+10 deg) == 0.984807753012208131337956729112 (0x1.f838b8c811c18p-1) bug
std::cos (+11 deg) == 0.981627183447663975712771389226 (0x1.f697d6938b6c2p-1)
gcem::cos(+11 deg) == 0.981627183447663975712771389226 (0x1.f697d6938b6c2p-1)
std::cos (+12 deg) == 0.978147600733805688832944724709 (0x1.f4cfc327a0080p-1)
gcem::cos(+12 deg) == 0.978147600733805577810642262193 (0x1.f4cfc327a007fp-1) bug
std::cos (+13 deg) == 0.974370064785235245885530730447 (0x1.f2e0a214e870fp-1)
gcem::cos(+13 deg) == 0.974370064785235245885530730447 (0x1.f2e0a214e870fp-1)
std::cos (+14 deg) == 0.970295726275996472942608761514 (0x1.f0ca99f79ba25p-1)
gcem::cos(+14 deg) == 0.970295726275996583964911224029 (0x1.f0ca99f79ba26p-1) bug
std::cos (+15 deg) == 0.965925826289068312213714762038 (0x1.ee8dd4748bf15p-1)
gcem::cos(+15 deg) == 0.965925826289068312213714762038 (0x1.ee8dd4748bf15p-1)
std::sin (-15 deg) == -0.258819045102520739476403832668 (-0x1.0907dc1930690p-2)
gcem::sin(-15 deg) == -0.258819045102520739476403832668 (-0x1.0907dc1930690p-2)
std::sin (-14 deg) == -0.241921895599667730047954705697 (-0x1.ef74bf2e4b91dp-3)
gcem::sin(-14 deg) == -0.241921895599667730047954705697 (-0x1.ef74bf2e4b91dp-3)
std::sin (-13 deg) == -0.224951054343865003426472526371 (-0x1.ccb3236cdc675p-3)
gcem::sin(-13 deg) == -0.224951054343864975670896910742 (-0x1.ccb3236cdc674p-3) bug
std::sin (-12 deg) == -0.207911690817759342575499204031 (-0x1.a9cd9ac4258f6p-3)
gcem::sin(-12 deg) == -0.207911690817759314819923588402 (-0x1.a9cd9ac4258f5p-3) bug
std::sin (-11 deg) == -0.190808995376544804356555573577 (-0x1.86c6ddd76624fp-3)
gcem::sin(-11 deg) == -0.190808995376544804356555573577 (-0x1.86c6ddd76624fp-3)
std::sin (-10 deg) == -0.173648177666930331186634361984 (-0x1.63a1a7e0b7389p-3)
gcem::sin(-10 deg) == -0.173648177666930358942209977613 (-0x1.63a1a7e0b738ap-3) bug
std::sin ( -9 deg) == -0.156434465040230868959625354364 (-0x1.4060b67a85375p-3)
gcem::sin( -9 deg) == -0.156434465040230868959625354364 (-0x1.4060b67a85375p-3)
std::sin ( -8 deg) == -0.139173100960065437847745783984 (-0x1.1d06c968d9e19p-3)
gcem::sin( -8 deg) == -0.139173100960065465603321399612 (-0x1.1d06c968d9e1ap-3) bug
std::sin ( -7 deg) == -0.121869343405147476100403025612 (-0x1.f32d44c4f62d3p-4)
gcem::sin( -7 deg) == -0.121869343405147489978190833426 (-0x1.f32d44c4f62d4p-4) bug
std::sin ( -6 deg) == -0.104528463267653470847307062286 (-0x1.ac2609b3c576cp-4)
gcem::sin( -6 deg) == -0.104528463267653456969519254471 (-0x1.ac2609b3c576bp-4) bug
std::sin ( -5 deg) == -0.087155742747658165869850677154 (-0x1.64fd6b8c28102p-4)
gcem::sin( -5 deg) == -0.087155742747658151992062869340 (-0x1.64fd6b8c28101p-4) bug
std::sin ( -4 deg) == -0.069756473744125302438590097154 (-0x1.1db8f6d6a5128p-4)
gcem::sin( -4 deg) == -0.069756473744125302438590097154 (-0x1.1db8f6d6a5128p-4)
std::sin ( -3 deg) == -0.052335956242943834637593170100 (-0x1.acbc748efc90ep-5)
gcem::sin( -3 deg) == -0.052335956242943834637593170100 (-0x1.acbc748efc90ep-5)
std::sin ( -2 deg) == -0.034899496702500969191884649945 (-0x1.1de58c9f7dc27p-5)
gcem::sin( -2 deg) == -0.034899496702500969191884649945 (-0x1.1de58c9f7dc27p-5)
std::sin ( -1 deg) == -0.017452406437283511653202339176 (-0x1.1df0b2b89dd1ep-6)
gcem::sin( -1 deg) == -0.017452406437283511653202339176 (-0x1.1df0b2b89dd1ep-6)
std::sin ( +0 deg) == 0.000000000000000000000000000000 (0x0.0000000000000p+0)
gcem::sin( +0 deg) == 0.000000000000000000000000000000 (0x0.0000000000000p+0)
std::sin ( +1 deg) == 0.017452406437283511653202339176 (0x1.1df0b2b89dd1ep-6)
gcem::sin( +1 deg) == 0.017452406437283511653202339176 (0x1.1df0b2b89dd1ep-6)
std::sin ( +2 deg) == 0.034899496702500969191884649945 (0x1.1de58c9f7dc27p-5)
gcem::sin( +2 deg) == 0.034899496702500969191884649945 (0x1.1de58c9f7dc27p-5)
std::sin ( +3 deg) == 0.052335956242943834637593170100 (0x1.acbc748efc90ep-5)
gcem::sin( +3 deg) == 0.052335956242943834637593170100 (0x1.acbc748efc90ep-5)
std::sin ( +4 deg) == 0.069756473744125302438590097154 (0x1.1db8f6d6a5128p-4)
gcem::sin( +4 deg) == 0.069756473744125302438590097154 (0x1.1db8f6d6a5128p-4)
std::sin ( +5 deg) == 0.087155742747658165869850677154 (0x1.64fd6b8c28102p-4)
gcem::sin( +5 deg) == 0.087155742747658151992062869340 (0x1.64fd6b8c28101p-4) bug
std::sin ( +6 deg) == 0.104528463267653470847307062286 (0x1.ac2609b3c576cp-4)
gcem::sin( +6 deg) == 0.104528463267653456969519254471 (0x1.ac2609b3c576bp-4) bug
std::sin ( +7 deg) == 0.121869343405147476100403025612 (0x1.f32d44c4f62d3p-4)
gcem::sin( +7 deg) == 0.121869343405147489978190833426 (0x1.f32d44c4f62d4p-4) bug
std::sin ( +8 deg) == 0.139173100960065437847745783984 (0x1.1d06c968d9e19p-3)
gcem::sin( +8 deg) == 0.139173100960065465603321399612 (0x1.1d06c968d9e1ap-3) bug
std::sin ( +9 deg) == 0.156434465040230868959625354364 (0x1.4060b67a85375p-3)
gcem::sin( +9 deg) == 0.156434465040230868959625354364 (0x1.4060b67a85375p-3)
std::sin (+10 deg) == 0.173648177666930331186634361984 (0x1.63a1a7e0b7389p-3)
gcem::sin(+10 deg) == 0.173648177666930358942209977613 (0x1.63a1a7e0b738ap-3) bug
std::sin (+11 deg) == 0.190808995376544804356555573577 (0x1.86c6ddd76624fp-3)
gcem::sin(+11 deg) == 0.190808995376544804356555573577 (0x1.86c6ddd76624fp-3)
std::sin (+12 deg) == 0.207911690817759342575499204031 (0x1.a9cd9ac4258f6p-3)
gcem::sin(+12 deg) == 0.207911690817759314819923588402 (0x1.a9cd9ac4258f5p-3) bug
std::sin (+13 deg) == 0.224951054343865003426472526371 (0x1.ccb3236cdc675p-3)
gcem::sin(+13 deg) == 0.224951054343864975670896910742 (0x1.ccb3236cdc674p-3) bug
std::sin (+14 deg) == 0.241921895599667730047954705697 (0x1.ef74bf2e4b91dp-3)
gcem::sin(+14 deg) == 0.241921895599667730047954705697 (0x1.ef74bf2e4b91dp-3)
std::sin (+15 deg) == 0.258819045102520739476403832668 (0x1.0907dc1930690p-2)
gcem::sin(+15 deg) == 0.258819045102520739476403832668 (0x1.0907dc1930690p-2)
@AuroraDysis
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@MarekKnapek Morwenn/static_math#19 provides a better solution (but not good enough).

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@AuroraDysis @MarekKnapek