Notebooks · lib/stats/tables

The constants of lib/stats/special.viv's series

Generated by lib/stats/make_tables.py with mpmath 1.3 at 60 digits, each rounded once to the nearest double.

ZETA_M1[k - 2] is zeta(k) - 1 for k = 2 to 65: the coefficients of the Taylor series of log Gamma about 1 and 2.
PSI_ROOT_HI + PSI_ROOT_LO is the positive zero of the digamma function to twice double precision, and PSI_ROOT_TAYLOR[n - 1] is psi^(n)(x0) / n!, the Taylor coefficients of psi about that zero.

Used with from stats.tables import ....

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[1]lines 17–138

ZETA_M1 = [
    0.6449340668482264,
    0.2020569031595943,
    0.08232323371113819,
    0.03692775514336993,
    0.01734306198444914,
    0.008349277381922827,
    0.00407735619794434,
    0.0020083928260822143,
    0.0009945751278180853,
    0.0004941886041194645,
    0.0002460865533080483,
    0.00012271334757848915,
    6.124813505870483e-05,
    3.058823630702049e-05,
    1.528225940865187e-05,
    7.637197637899763e-06,
    3.81729326499984e-06,
    1.908212716553939e-06,
    9.539620338727962e-07,
    4.769329867878064e-07,
    2.38450502727733e-07,
    1.1921992596531106e-07,
    5.960818905125948e-08,
    2.980350351465228e-08,
    1.4901554828365043e-08,
    7.45071178983543e-09,
    3.725334024788457e-09,
    1.862659723513049e-09,
    9.313274324196682e-10,
    4.656629065033784e-10,
    2.3283118336765053e-10,
    1.164155017270052e-10,
    5.820772087902701e-11,
    2.9103850444971e-11,
    1.4551921891041985e-11,
    7.275959835057482e-12,
    3.637979547378651e-12,
    1.818989650307066e-12,
    9.094947840263888e-13,
    4.547473783042154e-13,
    2.2737368458246524e-13,
    1.136868407680228e-13,
    5.684341987627585e-14,
    2.842170976889302e-14,
    1.4210854828031608e-14,
    7.105427395210853e-15,
    3.552713691337114e-15,
    1.7763568435791204e-15,
    8.881784210930816e-16,
    4.440892103143813e-16,
    2.220446050798042e-16,
    1.1102230251410661e-16,
    5.551115124845481e-17,
    2.775557562136124e-17,
    1.3877787809725232e-17,
    6.938893904544153e-18,
    3.4694469521659225e-18,
    1.7347234760475765e-18,
    8.673617380119933e-19,
    4.336808690020651e-19,
    2.16840434499722e-19,
    1.0842021724942414e-19,
    5.421010862456646e-20,
    2.710505431223469e-20,
]

PSI_ROOT_HI = 1.4616321449683622
PSI_ROOT_LO = 9.549995429965697e-17
PSI_ROOT_TAYLOR = [
    0.9676722454476212,
    -0.4427631689835921,
    0.258499760955651,
    -0.16394270544240652,
    0.10782405069126237,
    -0.07219956125645471,
    0.04880428816414311,
    -0.03316112647484736,
    0.022597648232218104,
    -0.01542476590494896,
    0.010538791616612175,
    -0.007204534386356869,
    0.004926781395729853,
    -0.003369801655439328,
    0.002305126326734928,
    -0.0015769367714301972,
    0.0010788252019162967,
    -0.0007380709389960052,
    0.000504953265834602,
    -0.0003454680251063077,
    0.00023635601564027053,
    -0.00016170622091974803,
    0.0001106337276874741,
    -7.569179582195066e-05,
    5.178575795222081e-05,
    -3.5430070947659604e-05,
    2.424006611860132e-05,
    -1.6584242271854135e-05,
    1.134638458466385e-05,
    -7.762817668462094e-06,
    5.3110609208898636e-06,
    -3.6336507898010456e-06,
]
EULER_GAMMA = 0.5772156649015329
LN_SQRT_2PI = 0.9189385332046728
SQRT_PI = 1.772453850905516
SQRT_2PI = 2.5066282746310007
SQRT_2 = 1.4142135623730951

test "the tables against closed forms" {
    # zeta(2) = pi^2 / 6 and zeta(4) = pi^4 / 90
    assert_near(ZETA_M1[0], pi * pi / 6.0 - 1.0, 4.0e-16)
    assert_near(ZETA_M1[2], pi * pi * pi * pi / 90.0 - 1.0, 4.0e-16)
    # the positive zero of digamma, 1.46163214496836234126... (OEIS A030169)
    assert_near(PSI_ROOT_HI + PSI_ROOT_LO, 1.4616321449683623, 4.0e-16)
    # sqrt(2 pi) and sqrt(2)
    assert_near(SQRT_2PI, sqrt(2.0 * pi), 1.0e-15)          # one rounding of 2 pi and one of the root
    assert_near(SQRT_2, sqrt(2.0), 0.0)
}

# SPDX-License-Identifier: MIT OR Apache-2.0
# SPDX-FileCopyrightText: 2026 The Viv authors

ZETA_M1 = [
    0.6449340668482264,
    0.2020569031595943,
    0.08232323371113819,
    0.03692775514336993,
    0.01734306198444914,
    0.008349277381922827,
    0.00407735619794434,
    0.0020083928260822143,
    0.0009945751278180853,
    0.0004941886041194645,
    0.0002460865533080483,
    0.00012271334757848915,
    6.124813505870483e-05,
    3.058823630702049e-05,
    1.528225940865187e-05,
    7.637197637899763e-06,
    3.81729326499984e-06,
    1.908212716553939e-06,
    9.539620338727962e-07,
    4.769329867878064e-07,
    2.38450502727733e-07,
    1.1921992596531106e-07,
    5.960818905125948e-08,
    2.980350351465228e-08,
    1.4901554828365043e-08,
    7.45071178983543e-09,
    3.725334024788457e-09,
    1.862659723513049e-09,
    9.313274324196682e-10,
    4.656629065033784e-10,
    2.3283118336765053e-10,
    1.164155017270052e-10,
    5.820772087902701e-11,
    2.9103850444971e-11,
    1.4551921891041985e-11,
    7.275959835057482e-12,
    3.637979547378651e-12,
    1.818989650307066e-12,
    9.094947840263888e-13,
    4.547473783042154e-13,
    2.2737368458246524e-13,
    1.136868407680228e-13,
    5.684341987627585e-14,
    2.842170976889302e-14,
    1.4210854828031608e-14,
    7.105427395210853e-15,
    3.552713691337114e-15,
    1.7763568435791204e-15,
    8.881784210930816e-16,
    4.440892103143813e-16,
    2.220446050798042e-16,
    1.1102230251410661e-16,
    5.551115124845481e-17,
    2.775557562136124e-17,
    1.3877787809725232e-17,
    6.938893904544153e-18,
    3.4694469521659225e-18,
    1.7347234760475765e-18,
    8.673617380119933e-19,
    4.336808690020651e-19,
    2.16840434499722e-19,
    1.0842021724942414e-19,
    5.421010862456646e-20,
    2.710505431223469e-20,
]

PSI_ROOT_HI = 1.4616321449683622
PSI_ROOT_LO = 9.549995429965697e-17
PSI_ROOT_TAYLOR = [
    0.9676722454476212,
    -0.4427631689835921,
    0.258499760955651,
    -0.16394270544240652,
    0.10782405069126237,
    -0.07219956125645471,
    0.04880428816414311,
    -0.03316112647484736,
    0.022597648232218104,
    -0.01542476590494896,
    0.010538791616612175,
    -0.007204534386356869,
    0.004926781395729853,
    -0.003369801655439328,
    0.002305126326734928,
    -0.0015769367714301972,
    0.0010788252019162967,
    -0.0007380709389960052,
    0.000504953265834602,
    -0.0003454680251063077,
    0.00023635601564027053,
    -0.00016170622091974803,
    0.0001106337276874741,
    -7.569179582195066e-05,
    5.178575795222081e-05,
    -3.5430070947659604e-05,
    2.424006611860132e-05,
    -1.6584242271854135e-05,
    1.134638458466385e-05,
    -7.762817668462094e-06,
    5.3110609208898636e-06,
    -3.6336507898010456e-06,
]
EULER_GAMMA = 0.5772156649015329
LN_SQRT_2PI = 0.9189385332046728
SQRT_PI = 1.772453850905516
SQRT_2PI = 2.5066282746310007
SQRT_2 = 1.4142135623730951

test "the tables against closed forms" {
    # zeta(2) = pi^2 / 6 and zeta(4) = pi^4 / 90
    assert_near(ZETA_M1[0], pi * pi / 6.0 - 1.0, 4.0e-16)
    assert_near(ZETA_M1[2], pi * pi * pi * pi / 90.0 - 1.0, 4.0e-16)
    # the positive zero of digamma, 1.46163214496836234126... (OEIS A030169)
    assert_near(PSI_ROOT_HI + PSI_ROOT_LO, 1.4616321449683623, 4.0e-16)
    # sqrt(2 pi) and sqrt(2)
    assert_near(SQRT_2PI, sqrt(2.0 * pi), 1.0e-15)          # one rounding of 2 pi and one of the root
    assert_near(SQRT_2, sqrt(2.0), 0.0)
}

# SPDX-License-Identifier: MIT OR Apache-2.0
# SPDX-FileCopyrightText: 2026 The Viv authors