libcore: add N(0,1) and Exp(1) distributions to core::rand.

Sample from the normal and exponential distributions using the Ziggurat
algorithm.

diff --git a/src/etc/ziggurat_tables.py b/src/etc/ziggurat_tables.py
new file mode 100755 (executable)
index 0000000..c8f8730
--- /dev/null
+++ b/src/etc/ziggurat_tables.py
@@ -0,0 +1,121 @@
+#!/usr/bin/env python
+# xfail-license
+
+# This creates the tables used for distributions implemented using the
+# ziggurat algorithm in `core::rand::distributions;`. They are
+# (basically) the tables as used in the ZIGNOR variant (Doornik 2005).
+# They are changed rarely, so the generated file should be checked in
+# to git.
+#
+# It creates 3 tables: X as in the paper, F which is f(x_i), and
+# F_DIFF which is f(x_i) - f(x_{i-1}). The latter two are just cached
+# values which is not done in that paper (but is done in other
+# variants). Note that the adZigR table is unnecessary because of
+# algebra.
+#
+# It is designed to be compatible with Python 2 and 3.
+
+from math import exp, sqrt, log, floor
+import random
+
+# The order should match the return value of `tables`
+TABLE_NAMES = ['X', 'F', 'F_DIFF']
+
+# The actual length of the table is 1 more, to stop
+# index-out-of-bounds errors. This should match the bitwise operation
+# to find `i` in `zigurrat` in `libstd/rand/mod.rs`. Also the *_R and
+# *_V constants below depend on this value.
+TABLE_LEN = 256
+
+# equivalent to `zigNorInit` in Doornik2005, but generalised to any
+# distribution. r = dR, v = dV, f = probability density function,
+# f_inv = inverse of f
+def tables(r, v, f, f_inv):
+    # compute the x_i
+    xvec = [0]*(TABLE_LEN+1)
+
+    xvec[0] = v / f(r)
+    xvec[1] = r
+
+    for i in range(2, TABLE_LEN):
+        last = xvec[i-1]
+        xvec[i] = f_inv(v / last + f(last))
+
+    # cache the f's
+    fvec = [0]*(TABLE_LEN+1)
+    fdiff = [0]*(TABLE_LEN+1)
+    for i in range(TABLE_LEN+1):
+        fvec[i] = f(xvec[i])
+        if i > 0:
+            fdiff[i] = fvec[i] - fvec[i-1]
+
+    return xvec, fvec, fdiff
+
+# Distributions
+# N(0, 1)
+def norm_f(x):
+    return exp(-x*x/2.0)
+def norm_f_inv(y):
+    return sqrt(-2.0*log(y))
+
+NORM_R = 3.6541528853610088
+NORM_V = 0.00492867323399
+
+NORM = tables(NORM_R, NORM_V,
+              norm_f, norm_f_inv)
+
+# Exp(1)
+def exp_f(x):
+    return exp(-x)
+def exp_f_inv(y):
+    return -log(y)
+
+EXP_R = 7.69711747013104972
+EXP_V = 0.0039496598225815571993
+
+EXP = tables(EXP_R, EXP_V,
+             exp_f, exp_f_inv)
+
+
+# Output the tables/constants/types
+
+def render_static(name, type, value):
+    # no space or
+    return 'pub static %s: %s =%s;\n' % (name, type, value)
+
+# static `name`: [`type`, .. `len(values)`] =
+#     [values[0], ..., values[3],
+#      values[4], ..., values[7],
+#      ... ];
+def render_table(name, values):
+    rows = []
+    # 4 values on each row
+    for i in range(0, len(values), 4):
+        row = values[i:i+4]
+        rows.append(', '.join('%.18f' % f for f in row))
+
+    rendered = '\n    [%s]' % ',\n     '.join(rows)
+    return render_static(name, '[f64, .. %d]' % len(values), rendered)
+
+
+with open('ziggurat_tables.rs', 'w') as f:
+    f.write('''// Copyright 2013 The Rust Project Developers. See the COPYRIGHT
+// file at the top-level directory of this distribution and at
+// http://rust-lang.org/COPYRIGHT.
+//
+// Licensed under the Apache License, Version 2.0 <LICENSE-APACHE or
+// http://www.apache.org/licenses/LICENSE-2.0> or the MIT license
+// <LICENSE-MIT or http://opensource.org/licenses/MIT>, at your
+// option. This file may not be copied, modified, or distributed
+// except according to those terms.
+
+// Tables for distributions which are sampled using the ziggurat
+// algorithm. Autogenerated by `ziggurat_tables.py`.
+
+pub type ZigTable = &\'static [f64, .. %d];
+'''  % (TABLE_LEN + 1))
+    for name, tables, r in [('NORM', NORM, NORM_R),
+                            ('EXP', EXP, EXP_R)]:
+        f.write(render_static('ZIG_%s_R' % name, 'f64', ' %.18f' % r))
+        for (tabname, table) in zip(TABLE_NAMES, tables):
+            f.write(render_table('ZIG_%s_%s' % (name, tabname), table))

diff --git a/src/libcore/rand.rs b/src/libcore/rand.rs
index 9fa099cabbfe9941d1e5579a08817272d1844aa2..94ba5ba31b756412b59d46371cc9b1b25fb23d95 100644 (file)
--- a/src/libcore/rand.rs
+++ b/src/libcore/rand.rs
@@ -16,6 +16,9 @@
 means that often a simple call to `rand::random()` or `rng.gen()` will
 suffice, but sometimes an annotation is required, e.g. `rand::random::<float>()`.
 
+See the `distributions` submodule for sampling random numbers from
+distributions like normal and exponential.
+
 # Examples
 ~~~
 use core::rand::RngUtil;
@@ -47,6 +50,9 @@ fn main () {
 use vec;
 use libc::size_t;
 
+#[path="rand/distributions.rs"]
+pub mod distributions;
+
 /// A type that can be randomly generated using an Rng
 pub trait Rand {
     fn rand<R: Rng>(rng: &R) -> Self;

diff --git a/src/libcore/rand/distributions.rs b/src/libcore/rand/distributions.rs
new file mode 100644 (file)
index 0000000..a644f60
--- /dev/null
+++ b/src/libcore/rand/distributions.rs
@@ -0,0 +1,148 @@
+// Copyright 2013 The Rust Project Developers. See the COPYRIGHT
+// file at the top-level directory of this distribution and at
+// http://rust-lang.org/COPYRIGHT.
+//
+// Licensed under the Apache License, Version 2.0 <LICENSE-APACHE or
+// http://www.apache.org/licenses/LICENSE-2.0> or the MIT license
+// <LICENSE-MIT or http://opensource.org/licenses/MIT>, at your
+// option. This file may not be copied, modified, or distributed
+// except according to those terms.
+
+//! Sampling from random distributions
+
+// Some implementations use the Ziggurat method
+// https://en.wikipedia.org/wiki/Ziggurat_algorithm
+//
+// The version used here is ZIGNOR [Doornik 2005, "An Improved
+// Ziggurat Method to Generate Normal Random Samples"] which is slower
+// (about double, it generates an extra random number) than the
+// canonical version [Marsaglia & Tsang 2000, "The Ziggurat Method for
+// Generating Random Variables"], but more robust. If one wanted, one
+// could implement VIZIGNOR the ZIGNOR paper for more speed.
+
+use prelude::*;
+use rand::{Rng,Rand};
+
+mod ziggurat_tables;
+
+// inlining should mean there is no performance penalty for this
+#[inline(always)]
+fn ziggurat<R:Rng>(rng: &R,
+                   center_u: bool,
+                   X: ziggurat_tables::ZigTable,
+                   F: ziggurat_tables::ZigTable,
+                   F_DIFF: ziggurat_tables::ZigTable,
+                   pdf: &'static fn(f64) -> f64, // probability density function
+                   zero_case: &'static fn(&R, f64) -> f64) -> f64 {
+    loop {
+        let u = if center_u {2.0 * rng.gen() - 1.0} else {rng.gen()};
+        let i: uint = rng.gen::<uint>() & 0xff;
+        let x = u * X[i];
+
+        let test_x = if center_u {f64::abs(x)} else {x};
+
+        // algebraically equivalent to |u| < X[i+1]/X[i] (or u < X[i+1]/X[i])
+        if test_x < X[i + 1] {
+            return x;
+        }
+        if i == 0 {
+            return zero_case(rng, u);
+        }
+        // algebraically equivalent to f1 + DRanU()*(f0 - f1) < 1
+        if F[i+1] + F_DIFF[i+1] * rng.gen() < pdf(x) {
+            return x;
+        }
+    }
+}
+
+/// A wrapper around an `f64` to generate N(0, 1) random numbers (a.k.a.  a
+/// standard normal, or Gaussian). Multiplying the generated values by the
+/// desired standard deviation `sigma` then adding the desired mean `mu` will
+/// give N(mu, sigma^2) distributed random numbers.
+///
+/// Note that this has to be unwrapped before use as an `f64` (using either
+/// `*` or `cast::transmute` is safe).
+///
+/// # Example
+///
+/// ~~~
+/// use core::rand::distributions::StandardNormal;
+///
+/// fn main() {
+///     let normal = 2.0 + (*rand::random::<StandardNormal>()) * 3.0;
+///     println(fmt!("%f is from a N(2, 9) distribution", normal))
+/// }
+/// ~~~
+pub struct StandardNormal(f64);
+
+impl Rand for StandardNormal {
+    fn rand<R:Rng>(rng: &R) -> StandardNormal {
+        #[inline(always)]
+        fn pdf(x: f64) -> f64 {
+            f64::exp((-x*x/2.0) as f64) as f64
+        }
+        #[inline(always)]
+        fn zero_case<R:Rng>(rng: &R, u: f64) -> f64 {
+            // compute a random number in the tail by hand
+
+            // strange initial conditions, because the loop is not
+            // do-while, so the condition should be true on the first
+            // run, they get overwritten anyway (0 < 1, so these are
+            // good).
+            let mut x = 1.0, y = 0.0;
+
+            // XXX infinities?
+            while -2.0*y < x * x {
+                x = f64::ln(rng.gen()) / ziggurat_tables::ZIG_NORM_R;
+                y = f64::ln(rng.gen());
+            }
+            if u < 0.0 {x-ziggurat_tables::ZIG_NORM_R} else {ziggurat_tables::ZIG_NORM_R-x}
+        }
+
+        StandardNormal(ziggurat(
+            rng,
+            true, // this is symmetric
+            &ziggurat_tables::ZIG_NORM_X,
+            &ziggurat_tables::ZIG_NORM_F, &ziggurat_tables::ZIG_NORM_F_DIFF,
+            pdf, zero_case))
+    }
+}
+
+/// A wrapper around an `f64` to generate Exp(1) random numbers. Dividing by
+/// the desired rate `lambda` will give Exp(lambda) distributed random
+/// numbers.
+///
+/// Note that this has to be unwrapped before use as an `f64` (using either
+/// `*` or `cast::transmute` is safe).
+///
+/// # Example
+///
+/// ~~~
+/// use core::rand::distributions::Exp1;
+///
+/// fn main() {
+///     let exp2 = (*rand::random::<Exp1>()) * 0.5;
+///     println(fmt!("%f is from a Exp(2) distribution", exp2));
+/// }
+/// ~~~
+pub struct Exp1(f64);
+
+// This could be done via `-f64::ln(rng.gen::<f64>())` but that is slower.
+impl Rand for Exp1 {
+    #[inline]
+    fn rand<R:Rng>(rng: &R) -> Exp1 {
+        #[inline(always)]
+        fn pdf(x: f64) -> f64 {
+            f64::exp(-x)
+        }
+        #[inline(always)]
+        fn zero_case<R:Rng>(rng: &R, _u: f64) -> f64 {
+            ziggurat_tables::ZIG_EXP_R - f64::ln(rng.gen())
+        }
+
+        Exp1(ziggurat(rng, false,
+                      &ziggurat_tables::ZIG_EXP_X,
+                      &ziggurat_tables::ZIG_EXP_F, &ziggurat_tables::ZIG_EXP_F_DIFF,
+                      pdf, zero_case))
+    }
+}

diff --git a/src/libcore/rand/ziggurat_tables.rs b/src/libcore/rand/ziggurat_tables.rs
new file mode 100644 (file)
index 0000000..aca2457
--- /dev/null
+++ b/src/libcore/rand/ziggurat_tables.rs
@@ -0,0 +1,412 @@
+// Copyright 2013 The Rust Project Developers. See the COPYRIGHT
+// file at the top-level directory of this distribution and at
+// http://rust-lang.org/COPYRIGHT.
+//
+// Licensed under the Apache License, Version 2.0 <LICENSE-APACHE or
+// http://www.apache.org/licenses/LICENSE-2.0> or the MIT license
+// <LICENSE-MIT or http://opensource.org/licenses/MIT>, at your
+// option. This file may not be copied, modified, or distributed
+// except according to those terms.
+
+// Tables for distributions which are sampled using the ziggurat
+// algorithm. Autogenerated by `ziggurat_tables.py`.
+
+pub type ZigTable = &'static [f64, .. 257];
+pub static ZIG_NORM_R: f64 = 3.654152885361008796;
+pub static ZIG_NORM_X: [f64, .. 257] =
+    [3.910757959537090045, 3.654152885361008796, 3.449278298560964462, 3.320244733839166074,
+     3.224575052047029100, 3.147889289517149969, 3.083526132001233044, 3.027837791768635434,
+     2.978603279880844834, 2.934366867207854224, 2.894121053612348060, 2.857138730872132548,
+     2.822877396825325125, 2.790921174000785765, 2.760944005278822555, 2.732685359042827056,
+     2.705933656121858100, 2.680514643284522158, 2.656283037575502437, 2.633116393630324570,
+     2.610910518487548515, 2.589575986706995181, 2.569035452680536569, 2.549221550323460761,
+     2.530075232158516929, 2.511544441625342294, 2.493583041269680667, 2.476149939669143318,
+     2.459208374333311298, 2.442725318198956774, 2.426670984935725972, 2.411018413899685520,
+     2.395743119780480601, 2.380822795170626005, 2.366237056715818632, 2.351967227377659952,
+     2.337996148795031370, 2.324308018869623016, 2.310888250599850036, 2.297723348901329565,
+     2.284800802722946056, 2.272108990226823888, 2.259637095172217780, 2.247375032945807760,
+     2.235313384928327984, 2.223443340090905718, 2.211756642882544366, 2.200245546609647995,
+     2.188902771624720689, 2.177721467738641614, 2.166695180352645966, 2.155817819875063268,
+     2.145083634046203613, 2.134487182844320152, 2.124023315687815661, 2.113687150684933957,
+     2.103474055713146829, 2.093379631137050279, 2.083399693996551783, 2.073530263516978778,
+     2.063767547809956415, 2.054107931648864849, 2.044547965215732788, 2.035084353727808715,
+     2.025713947862032960, 2.016433734904371722, 2.007240830558684852, 1.998132471356564244,
+     1.989106007615571325, 1.980158896898598364, 1.971288697931769640, 1.962493064942461896,
+     1.953769742382734043, 1.945116560006753925, 1.936531428273758904, 1.928012334050718257,
+     1.919557336591228847, 1.911164563769282232, 1.902832208548446369, 1.894558525668710081,
+     1.886341828534776388, 1.878180486290977669, 1.870072921069236838, 1.862017605397632281,
+     1.854013059758148119, 1.846057850283119750, 1.838150586580728607, 1.830289919680666566,
+     1.822474540091783224, 1.814703175964167636, 1.806974591348693426, 1.799287584547580199,
+     1.791640986550010028, 1.784033659547276329, 1.776464495522344977, 1.768932414909077933,
+     1.761436365316706665, 1.753975320315455111, 1.746548278279492994, 1.739154261283669012,
+     1.731792314050707216, 1.724461502945775715, 1.717160915015540690, 1.709889657069006086,
+     1.702646854797613907, 1.695431651932238548, 1.688243209434858727, 1.681080704722823338,
+     1.673943330923760353, 1.666830296159286684, 1.659740822855789499, 1.652674147080648526,
+     1.645629517902360339, 1.638606196773111146, 1.631603456932422036, 1.624620582830568427,
+     1.617656869570534228, 1.610711622367333673, 1.603784156023583041, 1.596873794420261339,
+     1.589979870021648534, 1.583101723393471438, 1.576238702733332886, 1.569390163412534456,
+     1.562555467528439657, 1.555733983466554893, 1.548925085471535512, 1.542128153226347553,
+     1.535342571438843118, 1.528567729435024614, 1.521803020758293101, 1.515047842773992404,
+     1.508301596278571965, 1.501563685112706548, 1.494833515777718391, 1.488110497054654369,
+     1.481394039625375747, 1.474683555695025516, 1.467978458615230908, 1.461278162507407830,
+     1.454582081885523293, 1.447889631277669675, 1.441200224845798017, 1.434513276002946425,
+     1.427828197027290358, 1.421144398672323117, 1.414461289772464658, 1.407778276843371534,
+     1.401094763676202559, 1.394410150925071257, 1.387723835686884621, 1.381035211072741964,
+     1.374343665770030531, 1.367648583594317957, 1.360949343030101844, 1.354245316759430606,
+     1.347535871177359290, 1.340820365893152122, 1.334098153216083604, 1.327368577624624679,
+     1.320630975217730096, 1.313884673146868964, 1.307128989027353860, 1.300363230327433728,
+     1.293586693733517645, 1.286798664489786415, 1.279998415710333237, 1.273185207661843732,
+     1.266358287014688333, 1.259516886060144225, 1.252660221891297887, 1.245787495544997903,
+     1.238897891102027415, 1.231990574742445110, 1.225064693752808020, 1.218119375481726552,
+     1.211153726239911244, 1.204166830140560140, 1.197157747875585931, 1.190125515422801650,
+     1.183069142678760732, 1.175987612011489825, 1.168879876726833800, 1.161744859441574240,
+     1.154581450355851802, 1.147388505416733873, 1.140164844363995789, 1.132909248648336975,
+     1.125620459211294389, 1.118297174115062909, 1.110938046009249502, 1.103541679420268151,
+     1.096106627847603487, 1.088631390649514197, 1.081114409698889389, 1.073554065787871714,
+     1.065948674757506653, 1.058296483326006454, 1.050595664586207123, 1.042844313139370538,
+     1.035040439828605274, 1.027181966030751292, 1.019266717460529215, 1.011292417434978441,
+     1.003256679539591412, 0.995156999629943084, 0.986990747093846266, 0.978755155288937750,
+     0.970447311058864615, 0.962064143217605250, 0.953602409875572654, 0.945058684462571130,
+     0.936429340280896860, 0.927710533396234771, 0.918898183643734989, 0.909987953490768997,
+     0.900975224455174528, 0.891855070726792376, 0.882622229578910122, 0.873271068082494550,
+     0.863795545546826915, 0.854189171001560554, 0.844444954902423661, 0.834555354079518752,
+     0.824512208745288633, 0.814306670128064347, 0.803929116982664893, 0.793369058833152785,
+     0.782615023299588763, 0.771654424216739354, 0.760473406422083165, 0.749056662009581653,
+     0.737387211425838629, 0.725446140901303549, 0.713212285182022732, 0.700661841097584448,
+     0.687767892786257717, 0.674499822827436479, 0.660822574234205984, 0.646695714884388928,
+     0.632072236375024632, 0.616896989996235545, 0.601104617743940417, 0.584616766093722262,
+     0.567338257040473026, 0.549151702313026790, 0.529909720646495108, 0.509423329585933393,
+     0.487443966121754335, 0.463634336771763245, 0.437518402186662658, 0.408389134588000746,
+     0.375121332850465727, 0.335737519180459465, 0.286174591747260509, 0.215241895913273806,
+     0.000000000000000000];
+pub static ZIG_NORM_F: [f64, .. 257] =
+    [0.000477467764586655, 0.001260285930498598, 0.002609072746106363, 0.004037972593371872,
+     0.005522403299264754, 0.007050875471392110, 0.008616582769422917, 0.010214971439731100,
+     0.011842757857943104, 0.013497450601780807, 0.015177088307982072, 0.016880083152595839,
+     0.018605121275783350, 0.020351096230109354, 0.022117062707379922, 0.023902203305873237,
+     0.025705804008632656, 0.027527235669693315, 0.029365939758230111, 0.031221417192023690,
+     0.033093219458688698, 0.034980941461833073, 0.036884215688691151, 0.038802707404656918,
+     0.040736110656078753, 0.042684144916619378, 0.044646552251446536, 0.046623094902089664,
+     0.048613553216035145, 0.050617723861121788, 0.052635418276973649, 0.054666461325077916,
+     0.056710690106399467, 0.058767952921137984, 0.060838108349751806, 0.062921024437977854,
+     0.065016577971470438, 0.067124653828023989, 0.069245144397250269, 0.071377949059141965,
+     0.073522973714240991, 0.075680130359194964, 0.077849336702372207, 0.080030515814947509,
+     0.082223595813495684, 0.084428509570654661, 0.086645194450867782, 0.088873592068594229,
+     0.091113648066700734, 0.093365311913026619, 0.095628536713353335, 0.097903279039215627,
+     0.100189498769172020, 0.102487158942306270, 0.104796225622867056, 0.107116667775072880,
+     0.109448457147210021, 0.111791568164245583, 0.114145977828255210, 0.116511665626037014,
+     0.118888613443345698, 0.121276805485235437, 0.123676228202051403, 0.126086870220650349,
+     0.128508722280473636, 0.130941777174128166, 0.133386029692162844, 0.135841476571757352,
+     0.138308116449064322, 0.140785949814968309, 0.143274978974047118, 0.145775208006537926,
+     0.148286642733128721, 0.150809290682410169, 0.153343161060837674, 0.155888264725064563,
+     0.158444614156520225, 0.161012223438117663, 0.163591108232982951, 0.166181285765110071,
+     0.168782774801850333, 0.171395595638155623, 0.174019770082499359, 0.176655321444406654,
+     0.179302274523530397, 0.181960655600216487, 0.184630492427504539, 0.187311814224516926,
+     0.190004651671193070, 0.192709036904328807, 0.195425003514885592, 0.198152586546538112,
+     0.200891822495431333, 0.203642749311121501, 0.206405406398679298, 0.209179834621935651,
+     0.211966076307852941, 0.214764175252008499, 0.217574176725178370, 0.220396127481011589,
+     0.223230075764789593, 0.226076071323264877, 0.228934165415577484, 0.231804410825248525,
+     0.234686861873252689, 0.237581574432173676, 0.240488605941449107, 0.243408015423711988,
+     0.246339863502238771, 0.249284212419516704, 0.252241126056943765, 0.255210669955677150,
+     0.258192911338648023, 0.261187919133763713, 0.264195763998317568, 0.267216518344631837,
+     0.270250256366959984, 0.273297054069675804, 0.276356989296781264, 0.279430141762765316,
+     0.282516593084849388, 0.285616426816658109, 0.288729728483353931, 0.291856585618280984,
+     0.294997087801162572, 0.298151326697901342, 0.301319396102034120, 0.304501391977896274,
+     0.307697412505553769, 0.310907558127563710, 0.314131931597630143, 0.317370638031222396,
+     0.320623784958230129, 0.323891482377732021, 0.327173842814958593, 0.330470981380537099,
+     0.333783015832108509, 0.337110066638412809, 0.340452257045945450, 0.343809713148291340,
+     0.347182563958251478, 0.350570941482881204, 0.353974980801569250, 0.357394820147290515,
+     0.360830600991175754, 0.364282468130549597, 0.367750569780596226, 0.371235057669821344,
+     0.374736087139491414, 0.378253817247238111, 0.381788410875031348, 0.385340034841733958,
+     0.388908860020464597, 0.392495061461010764, 0.396098818517547080, 0.399720314981931668,
+     0.403359739222868885, 0.407017284331247953, 0.410693148271983222, 0.414387534042706784,
+     0.418100649839684591, 0.421832709231353298, 0.425583931339900579, 0.429354541031341519,
+     0.433144769114574058, 0.436954852549929273, 0.440785034667769915, 0.444635565397727750,
+     0.448506701509214067, 0.452398706863882505, 0.456311852680773566, 0.460246417814923481,
+     0.464202689050278838, 0.468180961407822172, 0.472181538469883255, 0.476204732721683788,
+     0.480250865911249714, 0.484320269428911598, 0.488413284707712059, 0.492530263646148658,
+     0.496671569054796314, 0.500837575128482149, 0.505028667945828791, 0.509245245998136142,
+     0.513487720749743026, 0.517756517232200619, 0.522052074674794864, 0.526374847174186700,
+     0.530725304406193921, 0.535103932383019565, 0.539511234259544614, 0.543947731192649941,
+     0.548413963257921133, 0.552910490428519918, 0.557437893621486324, 0.561996775817277916,
+     0.566587763258951771, 0.571211506738074970, 0.575868682975210544, 0.580559996103683473,
+     0.585286179266300333, 0.590047996335791969, 0.594846243770991268, 0.599681752622167719,
+     0.604555390700549533, 0.609468064928895381, 0.614420723892076803, 0.619414360609039205,
+     0.624450015550274240, 0.629528779928128279, 0.634651799290960050, 0.639820277456438991,
+     0.645035480824251883, 0.650298743114294586, 0.655611470583224665, 0.660975147780241357,
+     0.666391343912380640, 0.671861719900766374, 0.677388036222513090, 0.682972161648791376,
+     0.688616083008527058, 0.694321916130032579, 0.700091918140490099, 0.705928501336797409,
+     0.711834248882358467, 0.717811932634901395, 0.723864533472881599, 0.729995264565802437,
+     0.736207598131266683, 0.742505296344636245, 0.748892447223726720, 0.755373506511754500,
+     0.761953346841546475, 0.768637315803334831, 0.775431304986138326, 0.782341832659861902,
+     0.789376143571198563, 0.796542330428254619, 0.803849483176389490, 0.811307874318219935,
+     0.818929191609414797, 0.826726833952094231, 0.834716292992930375, 0.842915653118441077,
+     0.851346258465123684, 0.860033621203008636, 0.869008688043793165, 0.878309655816146839,
+     0.887984660763399880, 0.898095921906304051, 0.908726440060562912, 0.919991505048360247,
+     0.932060075968990209, 0.945198953453078028, 0.959879091812415930, 0.977101701282731328,
+     1.000000000000000000];
+pub static ZIG_NORM_F_DIFF: [f64, .. 257] =
+    [0.000000000000000000, 0.000782818165911943, 0.001348786815607765, 0.001428899847265509,
+     0.001484430705892882, 0.001528472172127356, 0.001565707298030807, 0.001598388670308183,
+     0.001627786418212004, 0.001654692743837703, 0.001679637706201265, 0.001702994844613767,
+     0.001725038123187510, 0.001745974954326004, 0.001765966477270568, 0.001785140598493315,
+     0.001803600702759419, 0.001821431661060659, 0.001838704088536796, 0.001855477433793579,
+     0.001871802266665008, 0.001887722003144375, 0.001903274226858077, 0.001918491715965767,
+     0.001933403251421835, 0.001948034260540625, 0.001962407334827158, 0.001976542650643127,
+     0.001990458313945481, 0.002004170645086643, 0.002017694415851860, 0.002031043048104267,
+     0.002044228781321551, 0.002057262814738517, 0.002070155428613822, 0.002082916088226049,
+     0.002095553533492583, 0.002108075856553551, 0.002120490569226280, 0.002132804661891696,
+     0.002145024655099026, 0.002157156644953973, 0.002169206343177243, 0.002181179112575302,
+     0.002193079998548175, 0.002204913757158977, 0.002216684880213121, 0.002228397617726446,
+     0.002240055998106505, 0.002251663846325885, 0.002263224800326716, 0.002274742325862292,
+     0.002286219729956393, 0.002297660173134250, 0.002309066680560787, 0.002320442152205823,
+     0.002331789372137141, 0.002343111017035562, 0.002354409664009627, 0.002365687797781804,
+     0.002376947817308683, 0.002388192041889739, 0.002399422716815966, 0.002410642018598946,
+     0.002421852059823287, 0.002433054893654529, 0.002444252518034679, 0.002455446879594508,
+     0.002466639877306970, 0.002477833365903986, 0.002489029159078809, 0.002500229032490808,
+     0.002511434726590794, 0.002522647949281448, 0.002533870378427505, 0.002545103664226889,
+     0.002556349431455662, 0.002567609281597438, 0.002578884794865288, 0.002590177532127119,
+     0.002601489036740262, 0.002612820836305291, 0.002624174444343735, 0.002635551361907296,
+     0.002646953079123743, 0.002658381076686089, 0.002669836827288052, 0.002681321797012387,
+     0.002692837446676144, 0.002704385233135737, 0.002715966610556786, 0.002727583031652520,
+     0.002739235948893221, 0.002750926815690169, 0.002762657087557796, 0.002774428223256353,
+     0.002786241685917290, 0.002798098944155558, 0.002810001473169871, 0.002821950755833219,
+     0.002833948283778004, 0.002845995558475284, 0.002858094092312607, 0.002870245409671041,
+     0.002882451048004164, 0.002894712558920987, 0.002907031509275432, 0.002919409482262880,
+     0.002931848078526783, 0.002944348917277934, 0.002956913637427061, 0.002969543898733384,
+     0.002982241382970874, 0.002995007795115689, 0.003007844864553855, 0.003020754346314269,
+     0.003033738022328147, 0.003046797702715820, 0.003059935227105459, 0.003073152465984053,
+     0.003086451322084072, 0.003099833731808721, 0.003113301666695822, 0.003126857134927052,
+     0.003140502182881588, 0.003154238896738770, 0.003168069404132778, 0.003181995875862154,
+     0.003196020527657495, 0.003210145622009941, 0.003224373470066433, 0.003238706433592253,
+     0.003253146927007733, 0.003267697419501892, 0.003282360437226572, 0.003297138565578506,
+     0.003312034451571411, 0.003327050806304299, 0.003342190407532641, 0.003357456102345890,
+     0.003372850809960137, 0.003388377524629727, 0.003404039318688046, 0.003419839345721265,
+     0.003435780843885239, 0.003451867139373843, 0.003468101650046629, 0.003484487889225119,
+     0.003501029469670069, 0.003517730107746697, 0.003534593627793237, 0.003551623966702611,
+     0.003568825178730639, 0.003586201440546166, 0.003603757056536316, 0.003621496464384588,
+     0.003639424240937217, 0.003657545108379068, 0.003675863940735269, 0.003694385770723563,
+     0.003713115796977806, 0.003732059391668707, 0.003751222108547281, 0.003770609691440940,
+     0.003790228083232539, 0.003810083435355216, 0.003830182117840641, 0.003850530729957835,
+     0.003871136111486317, 0.003892005354668437, 0.003913145816891062, 0.003934565134149914,
+     0.003956271235355358, 0.003978272357543333, 0.004000577062061084, 0.004023194251800533,
+     0.004046133189565926, 0.004069403517661885, 0.004093015278800460, 0.004116978938436600,
+     0.004141305408647655, 0.004166006073685835, 0.004191092817346642, 0.004216578052307351,
+     0.004242474751606884, 0.004268796482457593, 0.004295557442594244, 0.004322772499391836,
+     0.004350457232007221, 0.004378627976825644, 0.004407301876525049, 0.004436496933105327,
+     0.004466232065271192, 0.004496527170598785, 0.004527403192966406, 0.004558882195791591,
+     0.004590987441673855, 0.004623743479123199, 0.004657176237135574, 0.004691313128472929,
+     0.004726183162616859, 0.004761817069491636, 0.004798247435199299, 0.004835508851176451,
+     0.004873638078381815, 0.004912674228345848, 0.004952658963181422, 0.004993636716962402,
+     0.005035654941235035, 0.005078764377854039, 0.005123019362831771, 0.005168478165478940,
+     0.005215203367812893, 0.005263262290042703, 0.005312727468930079, 0.005363677197016692,
+     0.005416196132139284, 0.005470375988385734, 0.005526316321746716, 0.005584125426278286,
+     0.005643921359735682, 0.005705833121505521, 0.005770002010457520, 0.005836583196307310,
+     0.005905747545561058, 0.005977683752542928, 0.006052600837980204, 0.006130731092920838,
+     0.006212333565464245, 0.006297698213369562, 0.006387150879090475, 0.006481059288027780,
+     0.006579840329791975, 0.006683968961788356, 0.006793989182803495, 0.006910527673723577,
+     0.007034310911336661, 0.007166186857056056, 0.007307152748134871, 0.007458391141830445,
+     0.007621317291194862, 0.007797642342679434, 0.007989459040836144, 0.008199360125510702,
+     0.008430605346682607, 0.008687362737884952, 0.008975066840784529, 0.009300967772353674,
+     0.009675004947253041, 0.010111261142904171, 0.010630518154258861, 0.011265064987797335,
+     0.012068570920629962, 0.013138877484087819, 0.014680138359337902, 0.017222609470315398,
+     0.022898298717268672];
+pub static ZIG_EXP_R: f64 = 7.697117470131050077;
+pub static ZIG_EXP_X: [f64, .. 257] =
+    [8.697117470131052741, 7.697117470131050077, 6.941033629377212577, 6.478378493832569696,
+     6.144164665772472667, 5.882144315795399869, 5.666410167454033697, 5.482890627526062488,
+     5.323090505754398016, 5.181487281301500047, 5.054288489981304089, 4.938777085901250530,
+     4.832939741025112035, 4.735242996601741083, 4.644491885420085175, 4.559737061707351380,
+     4.480211746528421912, 4.405287693473573185, 4.334443680317273007, 4.267242480277365857,
+     4.203313713735184365, 4.142340865664051464, 4.084051310408297830, 4.028208544647936762,
+     3.974606066673788796, 3.923062500135489739, 3.873417670399509127, 3.825529418522336744,
+     3.779270992411667862, 3.734528894039797375, 3.691201090237418825, 3.649195515760853770,
+     3.608428813128909507, 3.568825265648337020, 3.530315889129343354, 3.492837654774059608,
+     3.456332821132760191, 3.420748357251119920, 3.386035442460300970, 3.352149030900109405,
+     3.319047470970748037, 3.286692171599068679, 3.255047308570449882, 3.224079565286264160,
+     3.193757903212240290, 3.164053358025972873, 3.134938858084440394, 3.106389062339824481,
+     3.078380215254090224, 3.050890016615455114, 3.023897504455676621, 2.997382949516130601,
+     2.971327759921089662, 2.945714394895045718, 2.920526286512740821, 2.895747768600141825,
+     2.871364012015536371, 2.847360965635188812, 2.823725302450035279, 2.800444370250737780,
+     2.777506146439756574, 2.754899196562344610, 2.732612636194700073, 2.710636095867928752,
+     2.688959688741803689, 2.667573980773266573, 2.646469963151809157, 2.625639026797788489,
+     2.605072938740835564, 2.584763820214140750, 2.564704126316905253, 2.544886627111869970,
+     2.525304390037828028, 2.505950763528594027, 2.486819361740209455, 2.467904050297364815,
+     2.449198932978249754, 2.430698339264419694, 2.412396812688870629, 2.394289099921457886,
+     2.376370140536140596, 2.358635057409337321, 2.341079147703034380, 2.323697874390196372,
+     2.306486858283579799, 2.289441870532269441, 2.272558825553154804, 2.255833774367219213,
+     2.239262898312909034, 2.222842503111036816, 2.206569013257663858, 2.190438966723220027,
+     2.174449009937774679, 2.158595893043885994, 2.142876465399842001, 2.127287671317368289,
+     2.111826546019042183, 2.096490211801715020, 2.081275874393225145, 2.066180819490575526,
+     2.051202409468584786, 2.036338080248769611, 2.021585338318926173, 2.006941757894518563,
+     1.992404978213576650, 1.977972700957360441, 1.963642687789548313, 1.949412758007184943,
+     1.935280786297051359, 1.921244700591528076, 1.907302480018387536, 1.893452152939308242,
+     1.879691795072211180, 1.866019527692827973, 1.852433515911175554, 1.838931967018879954,
+     1.825513128903519799, 1.812175288526390649, 1.798916770460290859, 1.785735935484126014,
+     1.772631179231305643, 1.759600930889074766, 1.746643651946074405, 1.733757834985571566,
+     1.720942002521935299, 1.708194705878057773, 1.695514524101537912, 1.682900062917553896,
+     1.670349953716452118, 1.657862852574172763, 1.645437439303723659, 1.633072416535991334,
+     1.620766508828257901, 1.608518461798858379, 1.596327041286483395, 1.584191032532688892,
+     1.572109239386229707, 1.560080483527888084, 1.548103603714513499, 1.536177455041032092,
+     1.524300908219226258, 1.512472848872117082, 1.500692176842816750, 1.488957805516746058,
+     1.477268661156133867, 1.465623682245745352, 1.454021818848793446, 1.442462031972012504,
+     1.430943292938879674, 1.419464582769983219, 1.408024891569535697, 1.396623217917042137,
+     1.385258568263121992, 1.373929956328490576, 1.362636402505086775, 1.351376933258335189,
+     1.340150580529504643, 1.328956381137116560, 1.317793376176324749, 1.306660610415174117,
+     1.295557131686601027, 1.284481990275012642, 1.273434238296241139, 1.262412929069615330,
+     1.251417116480852521, 1.240445854334406572, 1.229498195693849105, 1.218573192208790124,
+     1.207669893426761121, 1.196787346088403092, 1.185924593404202199, 1.175080674310911677,
+     1.164254622705678921, 1.153445466655774743, 1.142652227581672841, 1.131873919411078511,
+     1.121109547701330200, 1.110358108727411031, 1.099618588532597308, 1.088889961938546813,
+     1.078171191511372307, 1.067461226479967662, 1.056759001602551429, 1.046063435977044209,
+     1.035373431790528542, 1.024687873002617211, 1.014005623957096480, 1.003325527915696735,
+     0.992646405507275897, 0.981967053085062602, 0.971286240983903260, 0.960602711668666509,
+     0.949915177764075969, 0.939222319955262286, 0.928522784747210395, 0.917815182070044311,
+     0.907098082715690257, 0.896370015589889935, 0.885629464761751528, 0.874874866291025066,
+     0.864104604811004484, 0.853317009842373353, 0.842510351810368485, 0.831682837734273206,
+     0.820832606554411814, 0.809957724057418282, 0.799056177355487174, 0.788125868869492430,
+     0.777164609759129710, 0.766170112735434672, 0.755139984181982249, 0.744071715500508102,
+     0.732962673584365398, 0.721810090308756203, 0.710611050909655040, 0.699362481103231959,
+     0.688061132773747808, 0.676703568029522584, 0.665286141392677943, 0.653804979847664947,
+     0.642255960424536365, 0.630634684933490286, 0.618936451394876075, 0.607156221620300030,
+     0.595288584291502887, 0.583327712748769489, 0.571267316532588332, 0.559100585511540626,
+     0.546820125163310577, 0.534417881237165604, 0.521885051592135052, 0.509211982443654398,
+     0.496388045518671162, 0.483401491653461857, 0.470239275082169006, 0.456886840931420235,
+     0.443327866073552401, 0.429543940225410703, 0.415514169600356364, 0.401214678896277765,
+     0.386617977941119573, 0.371692145329917234, 0.356399760258393816, 0.340696481064849122,
+     0.324529117016909452, 0.307832954674932158, 0.290527955491230394, 0.272513185478464703,
+     0.253658363385912022, 0.233790483059674731, 0.212671510630966620, 0.189958689622431842,
+     0.165127622564187282, 0.137304980940012589, 0.104838507565818778, 0.063852163815001570,
+     0.000000000000000000];
+pub static ZIG_EXP_F: [f64, .. 257] =
+    [0.000167066692307963, 0.000454134353841497, 0.000967269282327174, 0.001536299780301573,
+     0.002145967743718907, 0.002788798793574076, 0.003460264777836904, 0.004157295120833797,
+     0.004877655983542396, 0.005619642207205489, 0.006381905937319183, 0.007163353183634991,
+     0.007963077438017043, 0.008780314985808977, 0.009614413642502212, 0.010464810181029981,
+     0.011331013597834600, 0.012212592426255378, 0.013109164931254991, 0.014020391403181943,
+     0.014945968011691148, 0.015885621839973156, 0.016839106826039941, 0.017806200410911355,
+     0.018786700744696024, 0.019780424338009740, 0.020787204072578114, 0.021806887504283581,
+     0.022839335406385240, 0.023884420511558174, 0.024942026419731787, 0.026012046645134221,
+     0.027094383780955803, 0.028188948763978646, 0.029295660224637411, 0.030414443910466622,
+     0.031545232172893622, 0.032687963508959555, 0.033842582150874358, 0.035009037697397431,
+     0.036187284781931443, 0.037377282772959382, 0.038578995503074871, 0.039792391023374139,
+     0.041017441380414840, 0.042254122413316254, 0.043502413568888197, 0.044762297732943289,
+     0.046033761076175184, 0.047316792913181561, 0.048611385573379504, 0.049917534282706379,
+     0.051235237055126281, 0.052564494593071685, 0.053905310196046080, 0.055257689676697030,
+     0.056621641283742870, 0.057997175631200659, 0.059384305633420280, 0.060783046445479660,
+     0.062193415408541036, 0.063615431999807376, 0.065049117786753805, 0.066494496385339816,
+     0.067951593421936643, 0.069420436498728783, 0.070901055162371843, 0.072393480875708752,
+     0.073897746992364746, 0.075413888734058410, 0.076941943170480517, 0.078481949201606435,
+     0.080033947542319905, 0.081597980709237419, 0.083174093009632397, 0.084762330532368146,
+     0.086362741140756927, 0.087975374467270231, 0.089600281910032886, 0.091237516631040197,
+     0.092887133556043569, 0.094549189376055873, 0.096223742550432825, 0.097910853311492213,
+     0.099610583670637132, 0.101322997425953631, 0.103048160171257702, 0.104786139306570145,
+     0.106537004050001632, 0.108300825451033755, 0.110077676405185357, 0.111867631670056283,
+     0.113670767882744286, 0.115487163578633506, 0.117316899211555525, 0.119160057175327641,
+     0.121016721826674792, 0.122886979509545108, 0.124770918580830933, 0.126668629437510671,
+     0.128580204545228199, 0.130505738468330773, 0.132445327901387494, 0.134399071702213602,
+     0.136367070926428829, 0.138349428863580176, 0.140346251074862399, 0.142357645432472146,
+     0.144383722160634720, 0.146424593878344889, 0.148480375643866735, 0.150551185001039839,
+     0.152637142027442801, 0.154738369384468027, 0.156854992369365148, 0.158987138969314129,
+     0.161134939917591952, 0.163298528751901734, 0.165478041874935922, 0.167673618617250081,
+     0.169885401302527550, 0.172113535315319977, 0.174358169171353411, 0.176619454590494829,
+     0.178897546572478278, 0.181192603475496261, 0.183504787097767436, 0.185834262762197083,
+     0.188181199404254262, 0.190545769663195363, 0.192928149976771296, 0.195328520679563189,
+     0.197747066105098818, 0.200183974691911210, 0.202639439093708962, 0.205113656293837654,
+     0.207606827724221982, 0.210119159388988230, 0.212650861992978224, 0.215202151075378628,
+     0.217773247148700472, 0.220364375843359439, 0.222975768058120111, 0.225607660116683956,
+     0.228260293930716618, 0.230933917169627356, 0.233628783437433291, 0.236345152457059560,
+     0.239083290262449094, 0.241843469398877131, 0.244625969131892024, 0.247431075665327543,
+     0.250259082368862240, 0.253110290015629402, 0.255985007030415324, 0.258883549749016173,
+     0.261806242689362922, 0.264753418835062149, 0.267725419932044739, 0.270722596799059967,
+     0.273745309652802915, 0.276793928448517301, 0.279868833236972869, 0.282970414538780746,
+     0.286099073737076826, 0.289255223489677693, 0.292439288161892630, 0.295651704281261252,
+     0.298892921015581847, 0.302163400675693528, 0.305463619244590256, 0.308794066934560185,
+     0.312155248774179606, 0.315547685227128949, 0.318971912844957239, 0.322428484956089223,
+     0.325917972393556354, 0.329440964264136438, 0.332998068761809096, 0.336589914028677717,
+     0.340217149066780189, 0.343880444704502575, 0.347580494621637148, 0.351318016437483449,
+     0.355093752866787626, 0.358908472948750001, 0.362762973354817997, 0.366658079781514379,
+     0.370594648435146223, 0.374573567615902381, 0.378595759409581067, 0.382662181496010056,
+     0.386773829084137932, 0.390931736984797384, 0.395136981833290435, 0.399390684475231350,
+     0.403694012530530555, 0.408048183152032673, 0.412454465997161457, 0.416914186433003209,
+     0.421428728997616908, 0.425999541143034677, 0.430628137288459167, 0.435316103215636907,
+     0.440065100842354173, 0.444876873414548846, 0.449753251162755330, 0.454696157474615836,
+     0.459707615642138023, 0.464789756250426511, 0.469944825283960310, 0.475175193037377708,
+     0.480483363930454543, 0.485871987341885248, 0.491343869594032867, 0.496901987241549881,
+     0.502549501841348056, 0.508289776410643213, 0.514126393814748894, 0.520063177368233931,
+     0.526104213983620062, 0.532253880263043655, 0.538516872002862246, 0.544898237672440056,
+     0.551403416540641733, 0.558038282262587892, 0.564809192912400615, 0.571723048664826150,
+     0.578787358602845359, 0.586010318477268366, 0.593400901691733762, 0.600968966365232560,
+     0.608725382079622346, 0.616682180915207878, 0.624852738703666200, 0.633251994214366398,
+     0.641896716427266423, 0.650805833414571433, 0.660000841079000145, 0.669506316731925177,
+     0.679350572264765806, 0.689566496117078431, 0.700192655082788606, 0.711274760805076456,
+     0.722867659593572465, 0.735038092431424039, 0.747868621985195658, 0.761463388849896838,
+     0.775956852040116218, 0.791527636972496285, 0.808421651523009044, 0.826993296643051101,
+     0.847785500623990496, 0.871704332381204705, 0.900469929925747703, 0.938143680862176477,
+     1.000000000000000000];
+pub static ZIG_EXP_F_DIFF: [f64, .. 257] =
+    [0.000000000000000000, 0.000287067661533533, 0.000513134928485678, 0.000569030497974398,
+     0.000609667963417335, 0.000642831049855169, 0.000671465984262828, 0.000697030342996893,
+     0.000720360862708599, 0.000741986223663093, 0.000762263730113694, 0.000781447246315807,
+     0.000799724254382053, 0.000817237547791934, 0.000834098656693235, 0.000850396538527769,
+     0.000866203416804620, 0.000881578828420777, 0.000896572504999613, 0.000911226471926952,
+     0.000925576608509206, 0.000939653828282008, 0.000953484986066785, 0.000967093584871414,
+     0.000980500333784669, 0.000993723593313716, 0.001006779734568374, 0.001019683431705467,
+     0.001032447902101660, 0.001045085105172934, 0.001057605908173612, 0.001070020225402434,
+     0.001082337135821582, 0.001094564983022843, 0.001106711460658764, 0.001118783685829211,
+     0.001130788262427001, 0.001142731336065933, 0.001154618641914802, 0.001166455546523074,
+     0.001178247084534012, 0.001189997991027938, 0.001201712730115490, 0.001213395520299268,
+     0.001225050357040701, 0.001236681032901414, 0.001248291155571943, 0.001259884164055092,
+     0.001271463343231895, 0.001283031837006378, 0.001294592660197942, 0.001306148709326875,
+     0.001317702772419903, 0.001329257537945404, 0.001340815602974395, 0.001352379480650950,
+     0.001363951607045839, 0.001375534347457789, 0.001387130002219621, 0.001398740812059381,
+     0.001410368963061376, 0.001422016591266340, 0.001433685786946429, 0.001445378598586011,
+     0.001457097036596827, 0.001468843076792140, 0.001480618663643060, 0.001492425713336909,
+     0.001504266116655995, 0.001516141741693663, 0.001528054436422108, 0.001540006031125918,
+     0.001551998340713470, 0.001564033166917514, 0.001576112300394977, 0.001588237522735750,
+     0.001600410608388780, 0.001612633326513305, 0.001624907442762655, 0.001637234721007311,
+     0.001649616925003372, 0.001662055820012304, 0.001674553174376953, 0.001687110761059388,
+     0.001699730359144919, 0.001712413755316500, 0.001725162745304071, 0.001737979135312442,
+     0.001750864743431488, 0.001763821401032123, 0.001776850954151601, 0.001789955264870927,
+     0.001803136212688003, 0.001816395695889220, 0.001829735632922019, 0.001843157963772116,
+     0.001856664651347151, 0.001870257682870316, 0.001883939071285826, 0.001897710856679738,
+     0.001911575107717528, 0.001925533923102574, 0.001939589433056721, 0.001953743800826108,
+     0.001967999224215228, 0.001982357937151347, 0.001996822211282223, 0.002011394357609747,
+     0.002026076728162574, 0.002040871717710169, 0.002055781765521847, 0.002070809357173103,
+     0.002085957026402963, 0.002101227357025226, 0.002116622984897121, 0.002132146599948981,
+     0.002147800948277823, 0.002163588834309782, 0.002179513123034188, 0.002195576742314159,
+     0.002211782685277469, 0.002228134012792427, 0.002244633856033434, 0.002261285419141418,
+     0.002278091981983449, 0.002295056903017983, 0.002312183622271174, 0.002329475664429648,
+     0.002346936642057179, 0.002364570258941101, 0.002382380313575932, 0.002400370702791893,
+     0.002418545425535629, 0.002436908586812392, 0.002455464401797752, 0.002474217200128692,
+     0.002493171430384328, 0.002512331664766249, 0.002531702603989994, 0.002551289082400404,
+     0.002571096073321844, 0.002591128694658967, 0.002611392214760672, 0.002631892058563845,
+     0.002652633814032662, 0.002673623238910738, 0.002694866267805934, 0.002716369019626269,
+     0.002738137805389534, 0.002760179136428037, 0.002782499733014893, 0.002805106533435520,
+     0.002828006703534697, 0.002851207646767162, 0.002874717014785921, 0.002898542718600849,
+     0.002922692940346749, 0.002947176145699226, 0.002972001096982591, 0.002997176867015228,
+     0.003022712853742948, 0.003048618795714386, 0.003074904788455568, 0.003101581301807876,
+     0.003128659198296080, 0.003156149752600867, 0.003184064672214937, 0.003212416119368622,
+     0.003241216734320596, 0.003270479660111680, 0.003300218568896729, 0.003330447689969929,
+     0.003361181839619420, 0.003392436452949343, 0.003424227617828290, 0.003456572111131984,
+     0.003489487437467131, 0.003522991870580083, 0.003557104497672658, 0.003591845266868621,
+     0.003627235038102472, 0.003663295637722386, 0.003700049917134574, 0.003737521815846301,
+     0.003775736429304177, 0.003814720081962375, 0.003854500406067995, 0.003895106426696382,
+     0.003936568653631844, 0.003978919180756157, 0.004022191793678687, 0.004066422086428989,
+     0.004111647588127876, 0.004157907900659452, 0.004205244848493050, 0.004253702641940915,
+     0.004303328055299205, 0.004354170621502118, 0.004406282845128784, 0.004459720435841752,
+     0.004514542564613699, 0.004570812145417769, 0.004628596145424491, 0.004687965927177740,
+     0.004748997626717266, 0.004811772572194672, 0.004876377748206484, 0.004942906311860507,
+     0.005011458167522187, 0.005082140608288488, 0.005155069033533799, 0.005230367753417398,
+     0.005308170893076836, 0.005388623411430704, 0.005471882252147620, 0.005558117647517014,
+     0.005647514599798176, 0.005740274569295156, 0.005836617404105682, 0.005936783553485037,
+     0.006041036615386131, 0.006149666279423593, 0.006262991739818591, 0.006381365669577810,
+     0.006505178868201678, 0.006634865721946159, 0.006770910649812723, 0.006913855752425535,
+     0.007064309938019209, 0.007222959874423007, 0.007390583214465396, 0.007568064673498798,
+     0.007756415714389786, 0.007956798835585532, 0.008170557788458321, 0.008399255510700199,
+     0.008644722212900025, 0.008909116987305010, 0.009195007664428712, 0.009505475652925033,
+     0.009844255532840629, 0.010215923852312625, 0.010626158965710175, 0.011082105722287849,
+     0.011592898788496009, 0.012170432837851575, 0.012830529553771619, 0.013594766864701180,
+     0.014493463190219380, 0.015570784932380066, 0.016894014550512759, 0.018571645120042057,
+     0.020792203980939394, 0.023918831757214210, 0.028765597544542998, 0.037673750936428774,
+     0.061856319137823523];