{"quadrature_checks":[{"n":0,"quadrature":1.5707963267948966,"closed_or_recurrence":1.5707963267948966,"relative_error":0},{"n":1,"quadrature":0.9999999999999999,"closed_or_recurrence":1,"relative_error":1.1102230246251565e-16},{"n":2,"quadrature":0.7853981633974483,"closed_or_recurrence":0.7853981633974483,"relative_error":0},{"n":3,"quadrature":0.6666666666666666,"closed_or_recurrence":0.6666666666666666,"relative_error":0},{"n":4,"quadrature":0.5890486225480862,"closed_or_recurrence":0.5890486225480862,"relative_error":0},{"n":5,"quadrature":0.5333333333333333,"closed_or_recurrence":0.5333333333333333,"relative_error":0},{"n":6,"quadrature":0.4908738521234053,"closed_or_recurrence":0.4908738521234052,"relative_error":2.220446049250313e-16},{"n":7,"quadrature":0.4571428571428572,"closed_or_recurrence":0.45714285714285713,"relative_error":2.220446049250313e-16},{"n":8,"quadrature":0.4295146206079795,"closed_or_recurrence":0.4295146206079795,"relative_error":0},{"n":9,"quadrature":0.40634920634920646,"closed_or_recurrence":0.4063492063492063,"relative_error":4.440892098500626e-16},{"n":10,"quadrature":0.3865631585471816,"closed_or_recurrence":0.3865631585471816,"relative_error":0},{"n":11,"quadrature":0.36940836940836935,"closed_or_recurrence":0.36940836940836935,"relative_error":0},{"n":12,"quadrature":0.35434956200158296,"closed_or_recurrence":0.35434956200158313,"relative_error":4.440892098500626e-16},{"n":13,"quadrature":0.3409923409923409,"closed_or_recurrence":0.34099234099234094,"relative_error":1.1102230246251565e-16},{"n":14,"quadrature":0.32903887900146994,"closed_or_recurrence":0.32903887900147005,"relative_error":3.3306690738754696e-16},{"n":15,"quadrature":0.31825951825951815,"closed_or_recurrence":0.3182595182595182,"relative_error":2.220446049250313e-16},{"n":16,"quadrature":0.30847394906387804,"closed_or_recurrence":0.30847394906387815,"relative_error":3.3306690738754696e-16},{"n":17,"quadrature":0.29953837012660534,"closed_or_recurrence":0.2995383701266054,"relative_error":2.220446049250313e-16},{"n":18,"quadrature":0.2913365074492182,"closed_or_recurrence":0.29133650744921824,"relative_error":2.220446049250313e-16},{"n":19,"quadrature":0.28377319275152085,"closed_or_recurrence":0.2837731927515209,"relative_error":2.220446049250313e-16},{"n":20,"quadrature":0.2767696820767573,"closed_or_recurrence":0.27676968207675734,"relative_error":2.220446049250313e-16},{"n":21,"quadrature":0.270260183572877,"closed_or_recurrence":0.27026018357287707,"relative_error":2.220446049250313e-16},{"n":22,"quadrature":0.2641892419823592,"closed_or_recurrence":0.26418924198235927,"relative_error":2.220446049250313e-16},{"n":23,"quadrature":0.2585097408088389,"closed_or_recurrence":0.25850974080883893,"relative_error":2.220446049250313e-16},{"n":24,"quadrature":0.2531813568997609,"closed_or_recurrence":0.25318135689976096,"relative_error":2.220446049250313e-16},{"n":25,"quadrature":0.2481693511764853,"closed_or_recurrence":0.24816935117648536,"relative_error":2.220446049250313e-16},{"n":26,"quadrature":0.24344361240361626,"closed_or_recurrence":0.2434436124036163,"relative_error":2.220446049250313e-16},{"n":27,"quadrature":0.23897789372550435,"closed_or_recurrence":0.2389778937255044,"relative_error":2.220446049250313e-16},{"n":28,"quadrature":0.23474919767491567,"closed_or_recurrence":0.23474919767491573,"relative_error":2.220446049250313e-16},{"n":29,"quadrature":0.23073727670048697,"closed_or_recurrence":0.23073727670048702,"relative_error":2.220446049250313e-16},{"n":30,"quadrature":0.2269242244190851,"closed_or_recurrence":0.2269242244190852,"relative_error":3.3306690738754696e-16},{"n":50,"quadrature":0.17636142880336086,"closed_or_recurrence":0.1763614288033609,"relative_error":1.1102230246251565e-16},{"n":80,"quadrature":0.13968758532771516,"closed_or_recurrence":0.13968758532771527,"relative_error":7.771561172376096e-16},{"n":-0.9,"quadrature":10.676724666240105,"closed_or_recurrence":10.676724666240025,"relative_error":7.549516567451064e-15},{"n":-0.5,"quadrature":2.622057554292121,"closed_or_recurrence":2.62205755429212,"relative_error":4.440892098500626e-16},{"n":0.5,"quadrature":1.1981402347355916,"closed_or_recurrence":1.1981402347355918,"relative_error":2.220446049250313e-16},{"n":2.3,"quadrature":0.7434388969002591,"closed_or_recurrence":0.7434388969002661,"relative_error":9.43689570931383e-15}],"max_quadrature_relative_error":9.43689570931383e-15,"product_brackets_checked":10000,"adjacent_identities_checked":79,"binomial_brackets_checked":80,"table":[{"n":1,"two_Pn":2.6666666666666665,"absolute_error":0.4749259869231266,"relative_error":0.15117363684322493},{"n":2,"two_Pn":2.8444444444444446,"absolute_error":0.29714820914534856,"relative_error":0.09458521263277309},{"n":5,"two_Pn":3.002175954556907,"absolute_error":0.139416699032886,"relative_error":0.044377713601277735},{"n":10,"two_Pn":3.067703806643499,"absolute_error":0.07388884694629416,"relative_error":0.02351955046172638},{"n":100,"two_Pn":3.133787490628162,"absolute_error":0.00780516296163114,"relative_error":0.0024844605339627845},{"n":1000,"two_Pn":3.1408077460303927,"absolute_error":0.0007849075594004162,"relative_error":0.0002498438358975452},{"n":10000,"two_Pn":3.14151411868192,"absolute_error":0.00007853490787290696,"relative_error":0.000024998437586520872}],"n_times_pi_error_at_10000":0.7853490787290696,"asymptotic_target_pi_over_4":0.7853981633974483,"figures":["01-roadmap.png","02-sine-areas.png","03-squeeze.png","04-product-error.png","05-gaussian-limit.png","06-binomial.png"]}