Cross track distance ill conditioned about nvector

[pbrod]

No, the following example shows that np.arcsin delivers double precision both for small and large angles, but np.arcos solution gives less than double precision for most of the angles (even though the precision is ok for most applications) :

>>> import numpy as np
>>> theta = np.logspace(-50, 0)*np.pi/2
>>> sin_theta = np.sin(theta)
>>> t1 = np.arcsin(sin_theta)
>>> t2 = np.arcos(-sin_theta)-np.pi/2
>>> np.abs(t1-theta)/theta
array([0.0000000e+00, 0.0000000e+00, 0.0000000e+00, 0.0000000e+00,
   0.0000000e+00, 0.0000000e+00, 0.0000000e+00, 0.0000000e+00,
   0.0000000e+00, 0.0000000e+00, 0.0000000e+00, 0.0000000e+00,
   0.0000000e+00, 0.0000000e+00, 0.0000000e+00, 0.0000000e+00,
   0.0000000e+00, 0.0000000e+00, 0.0000000e+00, 0.0000000e+00,
   0.0000000e+00, 0.0000000e+00, 0.0000000e+00, 0.0000000e+00,
   0.0000000e+00, 0.0000000e+00, 0.0000000e+00, 0.0000000e+00,
   0.0000000e+00, 0.0000000e+00, 0.0000000e+00, 0.0000000e+00,
   0.0000000e+00, 0.0000000e+00, 0.0000000e+00, 0.0000000e+00,
   0.0000000e+00, 0.0000000e+00, 0.0000000e+00, 0.0000000e+00,
   0.0000000e+00, 0.0000000e+00, 0.0000000e+00, 0.0000000e+00,
   0.0000000e+00, 0.0000000e+00, 0.0000000e+00, 1.8090627e-16,
   0.0000000e+00, 0.0000000e+00])

>>> np.abs(t2-theta)/theta
array([3.17768732e-07, 8.41285987e-08, 2.99824097e-08, 2.51113553e-07,
   3.65447084e-08, 6.71601921e-08, 1.13440618e-08, 6.73537798e-09,
   2.06761534e-08, 3.76403350e-09, 2.41976409e-09, 9.60252863e-10,
   1.34828226e-09, 3.70225909e-10, 4.83429429e-10, 1.75196678e-10,
   8.52621110e-11, 6.57949754e-11, 1.75711340e-10, 4.99285763e-12,
   8.42401296e-11, 4.23587289e-11, 3.14708133e-11, 1.77221187e-11,
   4.08716375e-12, 8.06297736e-12, 3.40705458e-12, 4.44254594e-13,
   1.95394667e-13, 1.00059067e-12, 7.88776657e-14, 4.48770241e-13,
   1.34252409e-13, 1.45198665e-13, 1.68480928e-14, 1.21205710e-14,
   4.80642950e-14, 2.70152645e-14, 1.37805543e-14, 1.01907463e-14,
   5.30814909e-15, 2.65430892e-15, 8.29544787e-16, 1.62960687e-15,
   1.85198952e-16, 3.47278700e-16, 4.34136709e-16, 1.80906270e-16,
   2.26152806e-16, 0.00000000e+00])

[oysstu]

Aha, I see. I assumed that the comment was related to footnote 2 in Gade (2010) regarding the numerical precision of the inverse trigonometric functions near the endpoints of their domains due to the steep slope.

I can confirm your findings on Linux, where presumably numpy uses the trigonometric functions of gcc/glibc. It looks like the asin and acos functions in glibc are implemented in terms of the atan machine instruction. The division by x in acos may be the cause for the numerical issues, but then again using long double does not help so perhaps not.
asin = atan (x / sqrt(1 - x^2))
acos = atan (sqrt(1 - x^2) / x)

import numpy as np

theta = np.logspace(-50, 0, dtype=np.float64)*np.pi/2
cos_theta = np.cos(theta)
sin_theta = np.sin(theta)

t1 = np.arcsin(sin_theta)
t2 = np.arccos(-sin_theta) - np.pi/2
t3 = np.arctan2(sin_theta, cos_theta)

print('theta\n', theta)
print('t1\n', t1)
print('t2\n', t2)
print('t3\n', t3)

Output:

theta
 [1.57079633e-50 1.64637226e-49 1.72558439e-48 1.80860766e-47
 1.89562545e-46 1.98682993e-45 2.08242254e-44 2.18261442e-43
 2.28762684e-42 2.39769174e-41 2.51305220e-40 2.63396302e-39
 2.76069123e-38 2.89351674e-37 3.03273290e-36 3.17864719e-35
 3.33158186e-34 3.49187471e-33 3.65987975e-32 3.83596803e-31
 4.02052848e-30 4.21396870e-29 4.41671594e-28 4.62921799e-27
 4.85194418e-26 5.08538642e-25 5.33006030e-24 5.58650620e-23
 5.85529052e-22 6.13700690e-21 6.43227754e-20 6.74175457e-19
 7.06612151e-18 7.40609477e-17 7.76242520e-16 8.13589980e-15
 8.52734344e-14 8.93762066e-13 9.36763760e-12 9.81834400e-11
 1.02907353e-09 1.07858548e-08 1.13047961e-07 1.18487053e-06
 1.24187836e-05 1.30162902e-04 1.36425448e-03 1.42989303e-02
 1.49868967e-01 1.57079633e+00]
t1
 [1.57079633e-50 1.64637226e-49 1.72558439e-48 1.80860766e-47
 1.89562545e-46 1.98682993e-45 2.08242254e-44 2.18261442e-43
 2.28762684e-42 2.39769174e-41 2.51305220e-40 2.63396302e-39
 2.76069123e-38 2.89351674e-37 3.03273290e-36 3.17864719e-35
 3.33158186e-34 3.49187471e-33 3.65987975e-32 3.83596803e-31
 4.02052848e-30 4.21396870e-29 4.41671594e-28 4.62921799e-27
 4.85194418e-26 5.08538642e-25 5.33006030e-24 5.58650620e-23
 5.85529052e-22 6.13700690e-21 6.43227754e-20 6.74175457e-19
 7.06612151e-18 7.40609477e-17 7.76242520e-16 8.13589980e-15
 8.52734344e-14 8.93762066e-13 9.36763760e-12 9.81834400e-11
 1.02907353e-09 1.07858548e-08 1.13047961e-07 1.18487053e-06
 1.24187836e-05 1.30162902e-04 1.36425448e-03 1.42989303e-02
 1.49868967e-01 1.57079633e+00]
t2
 [0.00000000e+00 0.00000000e+00 0.00000000e+00 0.00000000e+00
 0.00000000e+00 0.00000000e+00 0.00000000e+00 0.00000000e+00
 0.00000000e+00 0.00000000e+00 0.00000000e+00 0.00000000e+00
 0.00000000e+00 0.00000000e+00 0.00000000e+00 0.00000000e+00
 0.00000000e+00 0.00000000e+00 0.00000000e+00 0.00000000e+00
 0.00000000e+00 0.00000000e+00 0.00000000e+00 0.00000000e+00
 0.00000000e+00 0.00000000e+00 0.00000000e+00 0.00000000e+00
 0.00000000e+00 0.00000000e+00 0.00000000e+00 0.00000000e+00
 0.00000000e+00 2.22044605e-16 8.88178420e-16 8.21565038e-15
 8.52651283e-14 8.93729535e-13 9.36761779e-12 9.81834614e-11
 1.02907349e-09 1.07858549e-08 1.13047961e-07 1.18487053e-06
 1.24187836e-05 1.30162902e-04 1.36425448e-03 1.42989303e-02
 1.49868967e-01 1.57079633e+00]
t3
 [1.57079633e-50 1.64637226e-49 1.72558439e-48 1.80860766e-47
 1.89562545e-46 1.98682993e-45 2.08242254e-44 2.18261442e-43
 2.28762684e-42 2.39769174e-41 2.51305220e-40 2.63396302e-39
 2.76069123e-38 2.89351674e-37 3.03273290e-36 3.17864719e-35
 3.33158186e-34 3.49187471e-33 3.65987975e-32 3.83596803e-31
 4.02052848e-30 4.21396870e-29 4.41671594e-28 4.62921799e-27
 4.85194418e-26 5.08538642e-25 5.33006030e-24 5.58650620e-23
 5.85529052e-22 6.13700690e-21 6.43227754e-20 6.74175457e-19
 7.06612151e-18 7.40609477e-17 7.76242520e-16 8.13589980e-15
 8.52734344e-14 8.93762066e-13 9.36763760e-12 9.81834400e-11
 1.02907353e-09 1.07858548e-08 1.13047961e-07 1.18487053e-06
 1.24187836e-05 1.30162902e-04 1.36425448e-03 1.42989303e-02
 1.49868967e-01 1.57079633e+00]