RNG 1.3.4

Python3 API for the C++ Random library

Random Number Generator: RNG Storm Engine

Python API for the C++ Random library.

RNG is not suitable for cryptography, but it could be perfect for other random stuff like data science, experimental programming, A.I. and games.

Recommended Installation: $ pip install RNG

Support this project: https://www.patreon.com/brokencode

Number Types, Precision & Size:

• Float: Python float -> double at the C++ layer.

  • Min Float: -1.7976931348623157e+308
  • Max Float: 1.7976931348623157e+308
  • Min Below Zero: -5e-324
  • Min Above Zero: 5e-324

• Integer: Python int -> long long at the C++ layer.

  • Input & Output Range: (-2**63, 2**63) or approximately +/- 9.2 billion billion.
  • Min Integer: -9223372036854775807
  • Max Integer: 9223372036854775807

Random Binary Function

• bernoulli(ratio_of_truth: float) -> bool
  • Bernoulli distribution.
  • @param ratio_of_truth :: the probability of True as a decimal. Expected input range: [0.0, 1.0], clamped.
  • @return :: True or False

Random Integer Functions

• random_int(left_limit: int, right_limit: int) -> int
  • Flat uniform distribution.
  • 20x faster than random.randint()
  • @param left_limit :: input A.
  • @param right_limit :: input B.
  • @return :: random integer in the inclusive range [A, B] or [B, A] if B < A
• random_below(upper_bound: int) -> int
  • Flat uniform distribution.
  • @param upper_bound :: inout A
  • @return :: random integer in exclusive range [0, A) or (A, 0] if A < 0
• binomial(number_of_trials: int, probability: float) -> int
  • Based on the idea of flipping a coin and counting how many heads come up after some number of flips.
  • @param number_of_trials :: how many times to flip a coin.
  • @param probability :: how likely heads will be flipped. 0.5 is a fair coin. 1.0 is a double headed coin.
  • @return :: count of how many heads came up.
• negative_binomial(trial_successes: int, probability: float) -> int
  • Based on the idea of flipping a coin as long as it takes to succeed.
  • @param trial_successes :: the required number of heads flipped to succeed.
  • @param probability :: how likely heads will be flipped. 0.50 is a fair coin.
  • @return :: the count of how many tails came up before the required number of heads.
• geometric(probability: float) -> int
  • Same as random_negative_binomial(1, probability).
• poisson(mean: float) -> int
  • @param mean :: sets the average output of the function.
  • @return :: random integer, poisson distribution centered on the mean.

Random Floating Point Functions

• generate_canonical() -> float
  • Evenly distributes real values of maximum precision.
  • @return :: random Float in range {0.0, 1.0} biclusive. The spec defines the output range to be [0.0, 1.0).
    • biclusive: feature/bug rendering the exclusivity of this function a bit more mysterious than desired. This is a known compiler bug.
• random_float(left_limit: float, right_limit: float) -> float
  • Suffers from the same biclusive feature/bug noted for generate_canonical().
  • @param left_limit :: input A
  • @param right_limit :: input B
  • @return :: random Float in range {A, B} biclusive. The spec defines the output range to be [A, B).
• normalvariate(mean: float, std_dev: float) -> float
  • @param mean :: sets the average output of the function.
  • @param std_dev :: standard deviation. Specifies spread of data from the mean.
• lognormvariate(log_mean: float, log_deviation: float) -> float
  • @param log_mean :: sets the log of the mean of the function.
  • @param log_deviation :: log of the standard deviation. Specifies spread of data from the mean.
• exponential(lambda_rate: float) -> float
  • Produces random non-negative floating-point values, distributed according to probability density function.
  • @param lambda_rate :: λ constant rate of a random event per unit of time/distance.
  • @return :: The time/distance until the next random event. For example, this distribution describes the time between the clicks of a Geiger counter or the distance between point mutations in a DNA strand.
• gammavariate(shape: float, scale: float) -> float
  • Generalization of the exponential distribution.
  • Produces random positive floating-point values, distributed according to probability density function.
  • @param shape :: α the number of independent exponentially distributed random variables.
  • @param scale :: β the scale factor or the mean of each of the distributed random variables.
  • @return :: the sum of α independent exponentially distributed random variables, each of which has a mean of β.
• weibullvariate(shape: float, scale: float) -> float
  • Generalization of the exponential distribution.
  • Similar to the gamma distribution but uses a closed form distribution function.
  • Popular in reliability and survival analysis.
• extreme_value(location: float, scale: float) -> float
  • Based on Extreme Value Theory.
  • Used for statistical models of the magnitude of earthquakes and volcanoes.
• chi_squared(degrees_of_freedom: float) -> float
  • Used with the Chi Squared Test and Null Hypotheses to test if sample data fits an expected distribution.
• cauchy(location: float, scale: float) -> float
  • @param location :: It specifies the location of the peak. The default value is 0.0.
  • @param scale :: It represents the half-width at half-maximum. The default value is 1.0.
  • @return :: Continuous Distribution.
• fisher_f(degrees_of_freedom_1: float, degrees_of_freedom_2: float) -> float
  • F distributions often arise when comparing ratios of variances.
• student_t(degrees_of_freedom: float) -> float
  • T distribution. Same as a normal distribution except it uses the sample standard deviation rather than the population standard deviation.
  • As degrees_of_freedom goes to infinity it converges with the normal distribution.

Engines

• mersenne_twister_engine: internal only
  • Implements 64 bit Mersenne twister algorithm. Default engine on most systems.
• linear_congruential_engine: internal only
  • Implements linear congruential algorithm.
• subtract_with_carry_engine: internal only
  • Implements a subtract-with-carry (lagged Fibonacci) algorithm.
• storm_engine: internal only
  • RNG: Custom Engine
  • Default Standard

Engine Adaptors

Engine adaptors generate pseudo-random numbers using another random number engine as entropy source. They are generally used to alter the spectral characteristics of the underlying engine.

• discard_block_engine: internal only
  • Discards some output of a random number engine.
• independent_bits_engine: internal only
  • Packs the output of a random number engine into blocks of a specified number of bits.
• shuffle_order_engine: internal only
  • Delivers the output of a random number engine in a different order.

Seeds & Entropy Source

• random_device: internal only
  • Non-deterministic uniform random bit generator, although implementations are allowed to implement random_device using a pseudo-random number engine if there is no support for non-deterministic random number generation.
• seed_seq: internal only
  • General-purpose bias-eliminating scrambled seed sequence generator.

Distribution & Performance Test Suite

• distribution_timer(func: staticmethod, *args, **kwargs) -> None
  • For statistical analysis of non-deterministic numeric functions.
  • @param func :: Function method or lambda to analyze. func(*args, **kwargs)
  • @optional_kw num_cycles :: Total number of samples for distribution analysis.
  • @optional_kw post_processor :: Used to scale a large set of data into a smaller set of groupings.
• quick_test(n=10000) -> None
  • Runs a battery of tests for every random distribution function in the module.
  • @param n :: the total number of samples to collect for each test. Default: 10,000

Development Log

RNG 1.3.4

• Storm Update 3.1.1

RNG 1.3.3

• Installer script update

RNG 1.3.2

• Minor Bug Fix

RNG 1.3.1

• Test Update

RNG 1.3.1

• Fixed Typos

RNG 1.3.0

• Storm Update

RNG 1.2.5

• Low level clean up

RNG 1.2.4

• Minor Typos Fixed

RNG 1.2.3

• Documentation Update
• Test Update
• Bug Fixes

RNG 1.0.0 - 1.2.2, internal

• API Changes:
  • randint changed to random_int
  • randbelow changed to random_below
  • random changed to generate_canonical
  • uniform changed to random_float

RNG 0.2.3

• Bug Fixes

RNG 0.2.2

• discrete() removed.

RNG 0.2.1

• minor typos
• discrete() depreciated.

RNG 0.2.0

• Major Rebuild.

RNG 0.1.22

• The RNG Storm Engine is now the default standard.
• Experimental Vortex Engine added for testing.

RNG 0.1.21 beta

• Small update to the testing suite.

RNG 0.1.20 beta

• Changed default inputs for random_int and random_below to sane values.
  • random_int(left_limit=1, right_limit=20) down from -2**63, 2**63 - 1
  • random_below(upper_bound=10) down from 2**63 - 1

RNG 0.1.19 beta

• Broke some fixed typos, for a change of pace.

RNG 0.1.18 beta

• Fixed some typos.

RNG 0.1.17 beta

• Major Refactoring.
• New primary engine: Hurricane.
• Experimental engine Typhoon added: random_below() only.

RNG 0.1.16 beta

• Internal Engine Performance Tuning.

RNG 0.1.15 beta

• Engine Testing.

RNG 0.1.14 beta

• Fixed a few typos.

RNG 0.1.13 beta

• Fixed a few typos.

RNG 0.1.12 beta

• Major Test Suite Upgrade.
• Major Bug Fixes.
  • Removed several 'foot-guns' in prep for fuzz testing in future releases.

RNG 0.1.11 beta

• Fixed small bug in the install script.

RNG 0.1.10 beta

• Fixed some typos.

RNG 0.1.9 beta

• Fixed some typos.

RNG 0.1.8 beta

• Fixed some typos.
• More documentation added.

RNG 0.1.7 beta

• The random_floating_point function renamed to random_float.
• The function c_rand() has been removed as well as all the cruft it required.
• Major Documentation Upgrade.
• Fixed an issue where keyword arguments would fail to propagate. Both, positional args and kwargs now work as intended.
• Added this Dev Log.

RNG 0.0.6 alpha

• Minor ABI changes.

RNG 0.0.5 alpha

• Tests redesigned slightly for Float functions.

RNG 0.0.4 alpha

• Random Float Functions Implemented.

RNG 0.0.3 alpha

• Random Integer Functions Implemented.

RNG 0.0.2 alpha

• Random Bool Function Implemented.

RNG 0.0.1 pre-alpha

• Planning & Design.

Distribution and Performance Test Suite

Quick Test: RNG Storm Engine

Round Trip Numeric Limits:
 Min Integer: -9223372036854775807
 Max Integer:  9223372036854775807
 Min Float: -1.7976931348623157e+308
 Max Float:  1.7976931348623157e+308
 Min Below Zero: -5e-324
 Min Above Zero:  5e-324


Binary Tests

Output Analysis: bernoulli(0.3333333333333333)
Typical Timing: 63 ± 1 ns
Statistics of 1000 Samples:
 Minimum: False
 Median: False
 Maximum: True
 Mean: 0.297
 Std Deviation: 0.4571651780264984
Distribution of 10000 Samples:
 False: 67.62%
 True: 32.38%


Integer Tests

Base Case
Output Analysis: Random.randint(1, 6)
Typical Timing: 1157 ± 11 ns
Statistics of 1000 Samples:
 Minimum: 1
 Median: 3
 Maximum: 6
 Mean: 3.476
 Std Deviation: 1.7083156448216303
Distribution of 10000 Samples:
 1: 16.54%
 2: 17.11%
 3: 16.54%
 4: 16.56%
 5: 17.0%
 6: 16.25%

Output Analysis: random_int(1, 6)
Typical Timing: 63 ± 1 ns
Statistics of 1000 Samples:
 Minimum: 1
 Median: 3
 Maximum: 6
 Mean: 3.515
 Std Deviation: 1.7363150030423022
Distribution of 10000 Samples:
 1: 16.53%
 2: 16.88%
 3: 17.16%
 4: 16.64%
 5: 16.52%
 6: 16.27%

Base Case
Output Analysis: Random.randrange(6)
Typical Timing: 813 ± 10 ns
Statistics of 1000 Samples:
 Minimum: 0
 Median: 2
 Maximum: 5
 Mean: 2.494
 Std Deviation: 1.7195699813967797
Distribution of 10000 Samples:
 0: 17.14%
 1: 16.5%
 2: 17.16%
 3: 16.13%
 4: 16.59%
 5: 16.48%

Output Analysis: random_below(6)
Typical Timing: 63 ± 1 ns
Statistics of 1000 Samples:
 Minimum: 0
 Median: 3
 Maximum: 5
 Mean: 2.559
 Std Deviation: 1.7371057931330884
Distribution of 10000 Samples:
 0: 16.39%
 1: 16.79%
 2: 16.69%
 3: 16.73%
 4: 16.21%
 5: 17.19%

Output Analysis: binomial(4, 0.5)
Typical Timing: 157 ± 7 ns
Statistics of 1000 Samples:
 Minimum: 0
 Median: 2
 Maximum: 4
 Mean: 2.023
 Std Deviation: 1.0185816160271637
Distribution of 10000 Samples:
 0: 6.25%
 1: 24.33%
 2: 37.79%
 3: 25.19%
 4: 6.44%

Output Analysis: negative_binomial(5, 0.75)
Typical Timing: 125 ± 3 ns
Statistics of 1000 Samples:
 Minimum: 0
 Median: 1
 Maximum: 12
 Mean: 1.69
 Std Deviation: 1.5932477642973295
Distribution of 10000 Samples:
 0: 23.56%
 1: 29.49%
 2: 22.08%
 3: 13.13%
 4: 6.8%
 5: 2.99%
 6: 1.23%
 7: 0.4%
 8: 0.24%
 9: 0.05%
 10: 0.02%
 12: 0.01%

Output Analysis: geometric(0.75)
Typical Timing: 63 ± 1 ns
Statistics of 1000 Samples:
 Minimum: 0
 Median: 0
 Maximum: 5
 Mean: 0.35
 Std Deviation: 0.6985613702374245
Distribution of 10000 Samples:
 0: 75.37%
 1: 18.49%
 2: 4.46%
 3: 1.15%
 4: 0.36%
 5: 0.12%
 6: 0.05%

Output Analysis: poisson(4.5)
Typical Timing: 94 ± 5 ns
Statistics of 1000 Samples:
 Minimum: 0
 Median: 4
 Maximum: 12
 Mean: 4.501
 Std Deviation: 2.1171871626007657
Distribution of 10000 Samples:
 0: 1.14%
 1: 4.56%
 2: 11.55%
 3: 17.14%
 4: 18.65%
 5: 16.61%
 6: 13.05%
 7: 8.39%
 8: 4.83%
 9: 2.37%
 10: 1.08%
 11: 0.39%
 12: 0.19%
 13: 0.03%
 14: 0.01%
 15: 0.01%


Floating Point Tests

Base Case
Output Analysis: Random.random()
Typical Timing: 32 ± 8 ns
Statistics of 1000 Samples:
 Minimum: 8.071296181855203e-05
 Median: (0.502924505186587, 0.5032775656862833)
 Maximum: 0.9988170135684447
 Mean: 0.5016736438385434
 Std Deviation: 0.28790699393489927
Post-processor Distribution of 10000 Samples using round method:
 0: 49.41%
 1: 50.59%

Output Analysis: generate_canonical()
Typical Timing: 32 ± 8 ns
Statistics of 1000 Samples:
 Minimum: 0.0003245331885518794
 Median: (0.516565971390449, 0.5170143603852831)
 Maximum: 0.9990982713581824
 Mean: 0.5145717546453246
 Std Deviation: 0.2884522610970008
Post-processor Distribution of 10000 Samples using round method:
 0: 49.92%
 1: 50.08%

Output Analysis: random_float(0.0, 10.0)
Typical Timing: 32 ± 8 ns
Statistics of 1000 Samples:
 Minimum: 0.0236416469945347
 Median: (4.842279802871613, 4.848817506632073)
 Maximum: 9.987421014269959
 Mean: 4.927751085416743
 Std Deviation: 2.8771478669049926
Post-processor Distribution of 10000 Samples using floor method:
 0: 10.6%
 1: 10.11%
 2: 10.21%
 3: 9.71%
 4: 9.99%
 5: 10.03%
 6: 9.74%
 7: 9.92%
 8: 9.87%
 9: 9.82%

Base Case
Output Analysis: Random.expovariate(1.0)
Typical Timing: 313 ± 7 ns
Statistics of 1000 Samples:
 Minimum: 0.0003062234170290055
 Median: (0.6996618370665845, 0.6997616759526261)
 Maximum: 7.159552971567411
 Mean: 1.0107104433923666
 Std Deviation: 0.9756097675287609
Post-processor Distribution of 10000 Samples using floor_mod_10 method:
 0: 63.02%
 1: 23.79%
 2: 8.19%
 3: 3.35%
 4: 1.07%
 5: 0.28%
 6: 0.18%
 7: 0.07%
 8: 0.04%
 9: 0.01%

Output Analysis: expovariate(1.0)
Typical Timing: 63 ± 1 ns
Statistics of 1000 Samples:
 Minimum: 0.0013738515219833946
 Median: (0.6659945404580686, 0.6679321003850357)
 Maximum: 6.184527605936165
 Mean: 0.9831307297176581
 Std Deviation: 0.9597396044531891
Post-processor Distribution of 10000 Samples using floor_mod_10 method:
 0: 63.1%
 1: 23.67%
 2: 8.52%
 3: 3.09%
 4: 0.96%
 5: 0.4%
 6: 0.17%
 7: 0.06%
 8: 0.03%

Base Case
Output Analysis: Random.gammavariate(1.0, 1.0)
Typical Timing: 469 ± 7 ns
Statistics of 1000 Samples:
 Minimum: 0.0006982739611174922
 Median: (0.6926420900298924, 0.6938024016111628)
 Maximum: 6.103667741783101
 Mean: 0.9617465094971319
 Std Deviation: 0.925375520445067
Post-processor Distribution of 10000 Samples using floor_mod_10 method:
 0: 63.19%
 1: 23.67%
 2: 8.08%
 3: 3.22%
 4: 1.21%
 5: 0.29%
 6: 0.18%
 7: 0.12%
 8: 0.03%
 9: 0.01%

Output Analysis: gammavariate(1.0, 1.0)
Typical Timing: 63 ± 4 ns
Statistics of 1000 Samples:
 Minimum: 0.0008031925002940128
 Median: (0.760013069220213, 0.7620212798008952)
 Maximum: 8.144418057675459
 Mean: 1.0445620110501181
 Std Deviation: 1.0166196642405472
Post-processor Distribution of 10000 Samples using floor_mod_10 method:
 0: 62.8%
 1: 23.32%
 2: 8.81%
 3: 3.09%
 4: 1.13%
 5: 0.59%
 6: 0.16%
 7: 0.07%
 8: 0.03%

Base Case
Output Analysis: Random.weibullvariate(1.0, 1.0)
Typical Timing: 407 ± 8 ns
Statistics of 1000 Samples:
 Minimum: 0.0033810095040254385
 Median: (0.7112308469171622, 0.7112703171484719)
 Maximum: 9.488506314169522
 Mean: 0.9923975403950522
 Std Deviation: 0.9769718629875843
Post-processor Distribution of 10000 Samples using floor_mod_10 method:
 0: 63.45%
 1: 23.04%
 2: 8.38%
 3: 3.17%
 4: 1.12%
 5: 0.54%
 6: 0.22%
 7: 0.07%
 9: 0.01%

Output Analysis: weibullvariate(1.0, 1.0)
Typical Timing: 94 ± 7 ns
Statistics of 1000 Samples:
 Minimum: 0.0001835254356867791
 Median: (0.6847916183943324, 0.685445483931424)
 Maximum: 7.664982901108874
 Mean: 0.9834642066050773
 Std Deviation: 0.9490050143193098
Post-processor Distribution of 10000 Samples using floor_mod_10 method:
 0: 62.75%
 1: 23.34%
 2: 8.95%
 3: 3.1%
 4: 1.19%
 5: 0.41%
 6: 0.15%
 7: 0.08%
 8: 0.03%

Output Analysis: extreme_value(0.0, 1.0)
Typical Timing: 63 ± 8 ns
Statistics of 1000 Samples:
 Minimum: -1.8690754939017922
 Median: (0.44928813309284127, 0.4495894525798417)
 Maximum: 8.274834739511434
 Mean: 0.6297141879838543
 Std Deviation: 1.3075096388384637
Post-processor Distribution of 10000 Samples using round method:
 -3: 0.01%
 -2: 1.1%
 -1: 17.95%
 0: 34.93%
 1: 25.37%
 2: 12.63%
 3: 5.05%
 4: 1.77%
 5: 0.75%
 6: 0.29%
 7: 0.09%
 8: 0.04%
 9: 0.01%
 10: 0.01%

Base Case
Output Analysis: Random.gauss(5.0, 2.0)
Typical Timing: 563 ± 8 ns
Statistics of 1000 Samples:
 Minimum: -1.8217604282017907
 Median: (5.051563263666794, 5.056609353790892)
 Maximum: 11.366999578488945
 Mean: 5.022749094201779
 Std Deviation: 1.9888992063128028
Post-processor Distribution of 10000 Samples using round method:
 -2: 0.06%
 -1: 0.27%
 0: 1.01%
 1: 2.71%
 2: 6.03%
 3: 12.23%
 4: 18.01%
 5: 20.12%
 6: 17.19%
 7: 12.14%
 8: 6.31%
 9: 2.82%
 10: 0.86%
 11: 0.22%
 12: 0.02%

Output Analysis: normalvariate(5.0, 2.0)
Typical Timing: 94 ± 1 ns
Statistics of 1000 Samples:
 Minimum: -2.0429195708720473
 Median: (4.901235259005797, 4.907550179190664)
 Maximum: 10.669609948445181
 Mean: 4.94351243779702
 Std Deviation: 1.9964278474382608
Post-processor Distribution of 10000 Samples using round method:
 -2: 0.04%
 -1: 0.21%
 0: 0.92%
 1: 2.79%
 2: 6.39%
 3: 11.72%
 4: 17.37%
 5: 20.17%
 6: 17.63%
 7: 12.32%
 8: 6.33%
 9: 2.94%
 10: 0.92%
 11: 0.21%
 12: 0.04%

Base Case
Output Analysis: Random.lognormvariate(1.6, 0.25)
Typical Timing: 813 ± 20 ns
Statistics of 1000 Samples:
 Minimum: 2.4909789931331665
 Median: (4.9469536651658785, 4.948132695702176)
 Maximum: 13.298525089985707
 Mean: 5.087805333829117
 Std Deviation: 1.2981416549832467
Post-processor Distribution of 10000 Samples using round method:
 2: 0.31%
 3: 8.0%
 4: 27.38%
 5: 30.97%
 6: 19.58%
 7: 9.05%
 8: 3.15%
 9: 1.1%
 10: 0.32%
 11: 0.09%
 12: 0.01%
 13: 0.04%

Output Analysis: lognormvariate(1.6, 0.25)
Typical Timing: 94 ± 7 ns
Statistics of 1000 Samples:
 Minimum: 2.4428465738204808
 Median: (4.9702179719562, 4.986770005889918)
 Maximum: 10.899243194201839
 Mean: 5.138117941614769
 Std Deviation: 1.2772123749368882
Post-processor Distribution of 10000 Samples using round method:
 2: 0.33%
 3: 8.03%
 4: 26.61%
 5: 31.9%
 6: 19.59%
 7: 8.79%
 8: 3.13%
 9: 1.21%
 10: 0.31%
 11: 0.09%
 13: 0.01%

Output Analysis: chi_squared(1.0)
Typical Timing: 125 ± 6 ns
Statistics of 1000 Samples:
 Minimum: 4.844684729641287e-08
 Median: (0.4759465801667526, 0.4795442615373534)
 Maximum: 11.409981115784303
 Mean: 0.9842604379661425
 Std Deviation: 1.3307195042368343
Post-processor Distribution of 10000 Samples using floor_mod_10 method:
 0: 67.52%
 1: 16.61%
 2: 7.66%
 3: 3.63%
 4: 2.25%
 5: 1.13%
 6: 0.63%
 7: 0.33%
 8: 0.12%
 9: 0.12%

Output Analysis: cauchy(0.0, 1.0)
Typical Timing: 63 ± 8 ns
Statistics of 1000 Samples:
 Minimum: -1972.6916185138157
 Median: (0.011961584087712716, 0.025839640114348472)
 Maximum: 759.9098882259025
 Mean: -2.3953059757615565
 Std Deviation: 69.10559085334823
Post-processor Distribution of 10000 Samples using floor_mod_10 method:
 0: 26.19%
 1: 11.39%
 2: 5.78%
 3: 4.13%
 4: 2.97%
 5: 3.27%
 6: 3.63%
 7: 5.82%
 8: 11.71%
 9: 25.11%

Output Analysis: fisher_f(8.0, 8.0)
Typical Timing: 188 ± 8 ns
Statistics of 1000 Samples:
 Minimum: 0.11290449448637255
 Median: (0.9539784476590794, 0.9545294808989871)
 Maximum: 15.141314152453434
 Mean: 1.296483358429577
 Std Deviation: 1.1955157250860309
Post-processor Distribution of 10000 Samples using floor_mod_10 method:
 0: 50.5%
 1: 32.68%
 2: 10.04%
 3: 3.43%
 4: 1.79%
 5: 0.77%
 6: 0.42%
 7: 0.22%
 8: 0.09%
 9: 0.06%

Output Analysis: student_t(8.0)
Typical Timing: 157 ± 8 ns
Statistics of 1000 Samples:
 Minimum: -4.090486444127238
 Median: (-0.07053637383717018, -0.069951340813996)
 Maximum: 4.475852462387794
 Mean: -0.056553151681631296
 Std Deviation: 1.127945410932898
Post-processor Distribution of 10000 Samples using round method:
 -7: 0.02%
 -6: 0.02%
 -5: 0.05%
 -4: 0.31%
 -3: 1.36%
 -2: 6.6%
 -1: 23.17%
 0: 36.64%
 1: 23.39%
 2: 6.57%
 3: 1.48%
 4: 0.31%
 5: 0.05%
 6: 0.02%
 7: 0.01%


=========================================================================
Total Test Time: 0.6052 seconds