When X-rays are diffracted by a helical structure, the resulting diffraction pattern breaks into a series of horizontal “layer lines.” Cochran, Crick, and Vand showed mathematically that the diffracted amplitude on the n<\/em>-th layer line is proportional to a Bessel function of the first kind, order n<\/em>:<\/p>\n J<\/em>n<\/em><\/sub>(2\u03c0\u2009r\u2009R<\/em>)<\/p>\n where r<\/em> is the radius of the helix and R<\/em> is the distance out from the center (the meridian) in the diffraction pattern.<\/p><\/blockquote>\n The citation for this paragraph is a paper from 1952 [2] that amazingly is behind a paywall.<\/p>\n [1] Watson and Crick didn’t “discover DNA” as is commonly said. DNA was discovered in 1878. Watson and Crick discovered the structure<\/em> of DNA in 1953.<\/p>\n [2] Cochran, W., Crick, F. H. C., & Vand, V. (1952). “The Structure of Synthetic Polypeptides. I. The Transform of Atoms on a Helix.” Acta Crystallographica, 5(5), 581\u2013586.<\/p>\n","protected":false},"excerpt":{"rendered":" I was reading a book on the history of the discovery of the structure of DNA [1] and was surprised by a few passing references to Bessel functions. According to Claude, When X-rays are diffracted by a helical structure, the resulting diffraction pattern breaks into a series of horizontal “layer lines.” Cochran, Crick, and Vand […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"_acf_changed":false,"footnotes":""},"categories":[15],"tags":[166,129],"class_list":["post-247593","post","type-post","status-publish","format-standard","hentry","category-science","tag-math","tag-special-functions"],"acf":[],"aioseo_notices":[],"aioseo_head":"\n\t\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\n","aioseo_head_json":{"title":"DNA and Bessel functions","description":"I was reading a book on the history of the discovery of the structure of DNA [1] and was surprised by a few passing references to Bessel functions. According to Claude, When X-rays are diffracted by a helical structure, the resulting diffraction pattern breaks into a series of horizontal \"layer lines.\" Cochran, Crick, and Vand","canonical_url":"https:\/\/www.johndcook.com\/blog\/2026\/08\/09\/dna-and-bessel-functions\/","robots":"max-image-preview:large","keywords":"math,special functions","webmasterTools":{"miscellaneous":""},"schema":null,"og:locale":"en_US","og:site_name":"John D. Cook | Applied Mathematics Consulting","og:type":"article","og:title":"DNA and Bessel functions","og:description":"I was reading a book on the history of the discovery of the structure of DNA [1] and was surprised by a few passing references to Bessel functions. According to Claude, When X-rays are diffracted by a helical structure, the resulting diffraction pattern breaks into a series of horizontal "layer lines." Cochran, Crick, and Vand","og:url":"https:\/\/www.johndcook.com\/blog\/2026\/08\/09\/dna-and-bessel-functions\/","article:published_time":"2026-08-09T17:17:42+00:00","article:modified_time":"2026-08-09T17:18:48+00:00","twitter:card":"summary","twitter:title":"DNA and Bessel functions","twitter:description":"I was reading a book on the history of the discovery of the structure of DNA [1] and was surprised by a few passing references to Bessel functions. According to Claude, When X-rays are diffracted by a helical structure, the resulting diffraction pattern breaks into a series of horizontal "layer lines." Cochran, Crick, and Vand","twitter:image":"https:\/\/www.johndcook.com\/blog\/wp-content\/uploads\/2022\/05\/twittercard.png"},"aioseo_meta_data":{"post_id":"247593","title":null,"description":null,"keywords":null,"keyphrases":{"focus":{"keyphrase":"","score":0,"analysis":{"keyphraseInTitle":{"score":0,"maxScore":9,"error":1}}},"additional":[]},"primary_term":null,"canonical_url":null,"og_title":null,"og_description":null,"og_object_type":"default","og_image_type":"default","og_image_url":null,"og_image_width":null,"og_image_height":null,"og_image_custom_url":null,"og_image_custom_fields":null,"og_video":"","og_custom_url":null,"og_article_section":null,"og_article_tags":null,"twitter_use_og":false,"twitter_card":"default","twitter_image_type":"default","twitter_image_url":null,"twitter_image_custom_url":null,"twitter_image_custom_fields":null,"twitter_title":null,"twitter_description":null,"schema":{"blockGraphs":[],"customGraphs":[],"default":{"data":{"Article":[],"Course":[],"Dataset":[],"FAQPage":[],"Movie":[],"Person":[],"Product":[],"ProductReview":[],"Car":[],"Recipe":[],"Service":[],"SoftwareApplication":[],"WebPage":[]},"graphName":"Article","isEnabled":true},"graphs":[]},"schema_type":"default","schema_type_options":null,"pillar_content":false,"robots_default":true,"robots_noindex":false,"robots_noarchive":false,"robots_nosnippet":false,"robots_nofollow":false,"robots_noimageindex":false,"robots_noodp":false,"robots_notranslate":false,"robots_max_snippet":"-1","robots_max_videopreview":"-1","robots_max_imagepreview":"large","priority":null,"frequency":"default","location":null,"local_seo":null,"breadcrumb_settings":null,"limit_modified_date":false,"created":"2026-08-09 16:59:20","updated":"2026-08-09 21:16:58","ai":{"faqs":[],"keyPoints":[],"schemas":[],"titles":[],"descriptions":[],"socialPosts":{"email":{"subject":"","preview":"","content":""},"linkedin":[],"twitter":[],"facebook":[],"instagram":[]}},"seo_analyzer_scan_date":null},"aioseo_breadcrumb":" At the end of my post on how not to calculate cosine<\/a> I said that the first step in calculating cosine, particularly cosine of a large number, would be to do range reduction. This post will present a simple range reduction method by Cody and Waite that is adequate for moderately large arguments.<\/p>\n If you want to compute the sine or cosine of an angle\u00a0x<\/em> you could start by reducing\u00a0x<\/em> mod 2\u03c0 since that would not change the result. However, accurately reducing a number mod 2\u03c0 is not trivial; that’s why range reduction is an area of algorithm development.<\/p>\n Even better would be to reduce x<\/em> mod \u03c0\/2. Reducing to a smaller range means that power series method, and other methods such as rational approximation, will be more efficient.<\/p>\n So suppose you can find an integer k<\/em> such that<\/p>\n x<\/em> \u2212 k<\/em> \u03c0\/2 = y<\/em><\/p>\n where 0 \u2264 y<\/em> \u2264 \u03c0\/2. Then sin(x<\/em>) is \u00b1sin(y<\/em>) or \u00b1cos(y<\/em>), depending on k<\/em> mod 4 equals 0, 1, 2, or 3.<\/p>\n Now let’s set\u00a0x<\/em> = 500. Then k<\/em> = 318 because that’s the multiple of \u03c0\/2 we need to subtract to bring x<\/em> into range, and the sine of x<\/em> should be the negative of the sine of the reduced value y<\/em> because 318 = 2 mod 4.<\/p>\n The following code computes sin(x<\/em>) with naive range reduction<\/p>\n and when x<\/em> = 500 the error is on the order of 1.7 \u00d7 10\u221214<\/sup>.<\/p>\n The value of k<\/em> above is fine, but we’d like to calculate y<\/em> more accurately. The following code is much better.<\/p>\n This will compute sin(500) to full machine precision. What kind of magic is this?<\/p>\n The trick is that the exact value of C1 + C2 equals \u03c0\/2 to more precision than is possible in a single float [1]. You can confirm, with bc<\/em> or some other extended precision software, that the difference between C1 + C2 and \u03c0\/2 is roughly 2\u221288<\/sup>, while the limit of float precision is 2\u221252<\/sup>.<\/p>\n If we compute<\/p>\n then we’re doing the same calculation as we will get a more accurate result, provided\u00a0x<\/em> isn’t too large.<\/p>\n You can use the following code to play around and see how large x<\/em>\u00a0can be before errors start to creep in. For small enough x<\/em>, like 500, the Cody and Waite sine returns full precision. For larger x<\/em> it’s better than naive sine but does not return full precision. And for large enough x<\/em> it completely breaks down.<\/p>\n Now this may seem circular since we’re using The Cody and Waite algorithm is inadequate for large\u00a0x<\/em>, but it’s a good place to begin studying range reduction.\u00a0It shows there are clever ways of squeezing out more precision than seems possible.<\/p>\n <\/p>\n [1] The numerator n<\/em>1<\/sub> of C1 is \u230a230<\/sup> \u03c0\/2\u230b. The numerator n<\/em>2<\/sub> of C2 is the solution to<\/p>\n 286\u221230<\/sup> n<\/em>1<\/sub> + n<\/em>2<\/sub> = \u230a286<\/sup> \u03c0\/2\u230b.<\/p>\n","protected":false},"excerpt":{"rendered":" At the end of my post on how not to calculate cosine I said that the first step in calculating cosine, particularly cosine of a large number, would be to do range reduction. This post will present a simple range reduction method by Cody and Waite that is adequate for moderately large arguments. If you […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"_acf_changed":false,"footnotes":""},"categories":[5],"tags":[],"class_list":["post-247580","post","type-post","status-publish","format-standard","hentry","category-computing"],"acf":[],"aioseo_notices":[],"aioseo_head":"\n\t\t\n\t\n\t\n\t\n\t\n\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\n","aioseo_head_json":{"title":"Simple range reduction algorithm by Cody and Waite","description":"A simple range reduction method. Not the state of the art, but simple and better than naive range reduction.","canonical_url":"https:\/\/www.johndcook.com\/blog\/2026\/08\/09\/simple-range-reduction\/","robots":"max-image-preview:large","keywords":"","webmasterTools":{"miscellaneous":""},"schema":null,"og:locale":"en_US","og:site_name":"John D. Cook | Applied Mathematics Consulting","og:type":"article","og:title":"Simple range reduction algorithm by Cody and Waite","og:description":"A simple range reduction method. Not the state of the art, but simple and better than naive range reduction.","og:url":"https:\/\/www.johndcook.com\/blog\/2026\/08\/09\/simple-range-reduction\/","article:published_time":"2026-08-09T16:46:59+00:00","article:modified_time":"2026-08-09T21:16:33+00:00","twitter:card":"summary","twitter:title":"Simple range reduction algorithm by Cody and Waite","twitter:description":"A simple range reduction method. Not the state of the art, but simple and better than naive range reduction.","twitter:image":"https:\/\/www.johndcook.com\/blog\/wp-content\/uploads\/2022\/05\/twittercard.png"},"aioseo_meta_data":{"post_id":"247580","title":"Simple range reduction algorithm by Cody and Waite","description":"A simple range reduction method. Not the state of the art, but simple and better than naive range reduction.","keywords":null,"keyphrases":{"focus":{"keyphrase":"","score":0,"analysis":{"keyphraseInTitle":{"score":0,"maxScore":9,"error":1}}},"additional":[]},"primary_term":null,"canonical_url":null,"og_title":null,"og_description":null,"og_object_type":"default","og_image_type":"default","og_image_url":null,"og_image_width":null,"og_image_height":null,"og_image_custom_url":null,"og_image_custom_fields":null,"og_video":"","og_custom_url":null,"og_article_section":null,"og_article_tags":null,"twitter_use_og":false,"twitter_card":"default","twitter_image_type":"default","twitter_image_url":null,"twitter_image_custom_url":null,"twitter_image_custom_fields":null,"twitter_title":null,"twitter_description":null,"schema":{"blockGraphs":[],"customGraphs":[],"default":{"data":{"Article":[],"Course":[],"Dataset":[],"FAQPage":[],"Movie":[],"Person":[],"Product":[],"ProductReview":[],"Car":[],"Recipe":[],"Service":[],"SoftwareApplication":[],"WebPage":[]},"graphName":"Article","isEnabled":true},"graphs":[]},"schema_type":"default","schema_type_options":null,"pillar_content":false,"robots_default":true,"robots_noindex":false,"robots_noarchive":false,"robots_nosnippet":false,"robots_nofollow":false,"robots_noimageindex":false,"robots_noodp":false,"robots_notranslate":false,"robots_max_snippet":"-1","robots_max_videopreview":"-1","robots_max_imagepreview":"large","priority":null,"frequency":"default","location":null,"local_seo":null,"breadcrumb_settings":null,"limit_modified_date":false,"created":"2026-08-09 02:17:58","updated":"2026-08-09 21:16:58","ai":{"faqs":[],"keyPoints":[],"schemas":[],"titles":[],"descriptions":[],"socialPosts":{"email":{"subject":"","preview":"","content":""},"linkedin":[],"twitter":[],"facebook":[],"instagram":[]}},"seo_analyzer_scan_date":null},"aioseo_breadcrumb":" I have a program that shares files between my laptop and my phone. It works well, except for apostrophes.<\/p>\n When I type an apostrophe Apparently the phone turns the apostrophe (U+0027) into a right single quote (U+2019), then bungles bytes in the UTF-8 encoding of U+2019 as three Windows-1252 characters. The bytes E28099hex<\/sub> are interpreted as When I type I have a program that shares files between my laptop and my phone. It works well, except for apostrophes. When I type an apostrophe ‘ on my laptop, it becomes \u00e2\u20ac™ on my phone. And when I type ‘s on my phone, it becomes \u75f4 on my laptop. Apparently the phone turns the apostrophe (U+0027) […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"_acf_changed":false,"footnotes":""},"categories":[5],"tags":[135],"class_list":["post-247576","post","type-post","status-publish","format-standard","hentry","category-computing","tag-unicode"],"acf":[],"aioseo_notices":[],"aioseo_head":"\n\t\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\n","aioseo_head_json":{"title":"Corrupted apostrophes","description":"I have a program that shares files between my laptop and my phone. It works well, except for apostrophes. When I type an apostrophe ' on my laptop, it becomes \u00e2\u20ac\u2122 on my phone. And when I type 's on my phone, it becomes \u75f4 on my laptop. Apparently the phone turns the apostrophe (U+0027)","canonical_url":"https:\/\/www.johndcook.com\/blog\/2026\/08\/07\/corrupted-apostrophes\/","robots":"max-image-preview:large","keywords":"unicode","webmasterTools":{"miscellaneous":""},"schema":null,"og:locale":"en_US","og:site_name":"John D. Cook | Applied Mathematics Consulting","og:type":"article","og:title":"Corrupted apostrophes","og:description":"Autopsy of how apostrophes are corrupted between my phone and laptop.","og:url":"https:\/\/www.johndcook.com\/blog\/2026\/08\/07\/corrupted-apostrophes\/","article:published_time":"2026-08-08T01:20:25+00:00","article:modified_time":"2026-08-08T01:20:25+00:00","twitter:card":"summary","twitter:title":"Corrupted apostrophes","twitter:description":"Autopsy of how apostrophes are corrupted between my phone and laptop.","twitter:image":"https:\/\/www.johndcook.com\/blog\/wp-content\/uploads\/2022\/05\/twittercard.png"},"aioseo_meta_data":{"post_id":"247576","title":null,"description":null,"keywords":null,"keyphrases":{"focus":[],"additional":[]},"primary_term":null,"canonical_url":null,"og_title":null,"og_description":"Autopsy of how apostrophes are corrupted between my phone and laptop.","og_object_type":"default","og_image_type":"default","og_image_url":null,"og_image_width":null,"og_image_height":null,"og_image_custom_url":null,"og_image_custom_fields":null,"og_video":"","og_custom_url":null,"og_article_section":null,"og_article_tags":null,"twitter_use_og":false,"twitter_card":"default","twitter_image_type":"default","twitter_image_url":null,"twitter_image_custom_url":null,"twitter_image_custom_fields":null,"twitter_title":null,"twitter_description":null,"schema":{"blockGraphs":[],"customGraphs":[],"default":{"data":{"Article":[],"Course":[],"Dataset":[],"FAQPage":[],"Movie":[],"Person":[],"Product":[],"ProductReview":[],"Car":[],"Recipe":[],"Service":[],"SoftwareApplication":[],"WebPage":[]},"graphName":"Article","isEnabled":true},"graphs":[]},"schema_type":"default","schema_type_options":null,"pillar_content":false,"robots_default":true,"robots_noindex":false,"robots_noarchive":false,"robots_nosnippet":false,"robots_nofollow":false,"robots_noimageindex":false,"robots_noodp":false,"robots_notranslate":false,"robots_max_snippet":"-1","robots_max_videopreview":"-1","robots_max_imagepreview":"large","priority":null,"frequency":"default","location":null,"local_seo":null,"breadcrumb_settings":null,"limit_modified_date":false,"created":"2026-08-08 00:46:32","updated":"2026-08-09 13:20:55","ai":{"faqs":[],"keyPoints":[],"schemas":[],"titles":[],"descriptions":[],"socialPosts":{"email":{"subject":"","preview":"","content":""},"linkedin":[],"twitter":[],"facebook":[],"instagram":[]}},"seo_analyzer_scan_date":null},"aioseo_breadcrumb":" Calculus professors with no experience in numerical computing will tell students that computers calculate trig functions with power series. They don’t. I worked on the implementation of trig functions in hardware, and I can assure you we didn’t just use power series.<\/p>\n Power series are an excellent way to calculate functions near the center of the series<\/em>, such as computing sine for small angles<\/a>. But the further you get from the center, the less useful power series are.<\/p>\n Let’s suppose you want to calculate cos(200) using the power series for cosine. The\u00a0n<\/em>th term of that series is<\/p>\n (\u22121)n<\/em><\/sup> x<\/em>2n<\/em><\/sup> \/ (2n<\/em>)!<\/p>\n This is an alternating series, and so the error in truncating the series after n<\/em> terms is bounded by the size of the n<\/em>+1 term, if<\/em> you’ve gone far enough out in the series that the terms are monotonically decreasing in absolute value.<\/p>\n To calculate cos(200) to machine precision, i.e. with an error of less than 2\u221252<\/sup>, we’d need to sum the series up to n<\/em> where<\/p>\n | 2002n<\/em>+2<\/sup> \/ (2n<\/em> + 2)! | < 2\u221252<\/sup><\/p>\n Actually, that will ensure that the absolute<\/em> error is small enough, but not that the relative<\/em> error is small enough; if the value of cos(200) is small, we’d need more terms. Let’s ignore that and assume we’re only concerned with absolute error.<\/p>\n Turns out we’d need 287 terms. That’s a lot of terms. But you might say “That’s fine. I’m not in a hurry, and it’s just more work for the computer, not for me.” OK, so let’s try.<\/p>\n This prints -3.6840358571084123e+67. You may suspect the answer is incorrect since values of cosine are on the order of 1, not on the order of 1067<\/sup>. Something went spectacularly bad. On closer inspection, it’s remarkable the code didn’t crash.<\/p>\n If you changed When we don’t overflow, we have a different problem: catastrophic cancellation. You can’t calculate a number between \u22121 and 1 as an alternating sum of numbers as large as 1080<\/sup>. You’d need more than 80 + 16 = 96 decimal places of precision to compute the sum accurately, and floating point only gives you between 15 and 16 decimal places of precision.<\/p>\n So how would<\/em> you calculate cos(200)? The first step would be to use some sort of range reduction on 200. You could reduce 200 mod 2\u03c0 to get a smaller number to work with.<\/p>\n Using a power series to compute the cosine of 5.221255477432827 is feasible, but not optimal. There’s also another problem: the naive range reduction above loses some precision.<\/p>\n The error is small, but it’s still an order of magnitude larger than machine precision. You can’t simply reduce n<\/em> mod 2\u03c0 with ordinary float division because the integer part of n<\/em> \/ 2\u03c0 pushes some digits of precision off the right end. I intend to write about how range reduction works in future posts.<\/p>\n Update<\/strong>: See this post<\/a> for a simple range reduction algorithm that is fine for values of x<\/em> such as 200, but not adequate for much larger values.<\/p>\n","protected":false},"excerpt":{"rendered":" Calculus professors with no experience in numerical computing will tell students that computers calculate trig functions with power series. They don’t. I worked on the implementation of trig functions in hardware, and I can assure you we didn’t just use power series. Power series are an excellent way to calculate functions near the center of […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"_acf_changed":false,"footnotes":""},"categories":[5,9],"tags":[],"class_list":["post-247572","post","type-post","status-publish","format-standard","hentry","category-computing","category-math"],"acf":[],"aioseo_notices":[],"aioseo_head":"\n\t\t\n\t\n\t\n\t\n\t\n\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\n","aioseo_head_json":{"title":"How not to calculate cosine","description":"Calculus professors with no experience in numerical computing will tell students that computers calculate trig functions with power series. They don't.","canonical_url":"https:\/\/www.johndcook.com\/blog\/2026\/08\/07\/how-not-to-calculate-cos\/","robots":"max-image-preview:large","keywords":"","webmasterTools":{"miscellaneous":""},"schema":null,"og:locale":"en_US","og:site_name":"John D. Cook | Applied Mathematics Consulting","og:type":"article","og:title":"How not to calculate cosine","og:description":"Calculus professors with no experience in numerical computing will tell students that computers calculate trig functions with power series. They don't.","og:url":"https:\/\/www.johndcook.com\/blog\/2026\/08\/07\/how-not-to-calculate-cos\/","article:published_time":"2026-08-07T15:11:10+00:00","article:modified_time":"2026-08-09T16:52:29+00:00","twitter:card":"summary","twitter:title":"How not to calculate cosine","twitter:description":"Calculus professors with no experience in numerical computing will tell students that computers calculate trig functions with power series. 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\n
Range reduction mod \u03c0\/2<\/h2>\n
from math import *\r\n\r\ndef reduced_sin(x, k):\r\n match k % 4:\r\n case 0: return sin(x)\r\n case 1: return cos(x)\r\n case 2: return -sin(x)\r\n case 3: return -cos(x)\r\n<\/pre>\n
Naive range reduction<\/h2>\n
def naive_sin(x):\r\n k = floor(x \/ (pi\/2))\r\n y = x % (pi\/2)\r\n return reduced_sin(y, k)\r\n<\/pre>\n
Better range reduction<\/h2>\n
def Cody_Waite_sin(x):\r\n C1 = 1686629713 \/ 2**30\r\n C2 = 4701928774853425 \/ 2**86\r\n\r\n k = floor(x \/ (pi\/2))\r\n y = (x - k*C1) - k*C2\r\n return reduced_sin(y, k)\r\n<\/pre>\n
y = x - k*(C1 + C2)<\/pre>\n
naive_sin<\/code> and will get the same error. But if we compute<\/p>\ny = (x - k*C1) - k*C2<\/pre>\n
def compare(x):\r\n y0 = naive_sin(x) \r\n y1 = Cody_Waite_sin(x)\r\n y2 = sin(x)\r\n print(\"Naive error: \", y2 - y0)\r\n print(\"Cody Waite error:\", y2 - y1)\r\n<\/pre>\n
math.sin<\/code> as our gold standard. However, this function is calling the sine function on your CPU, which is using sophisticated range reduction to compute its result accurately down to the last bit, assuming you run the code on a computer that’s less than 40 years old.<\/p>\n'<\/code> on my laptop, it becomes \u00e2\u20ac™<\/code> on my phone. And when I type 's<\/code> on my phone, it becomes \u75f4<\/code> on my laptop.<\/p>\n\u00e2<\/code> (E2hex<\/sub>), \u20ac<\/code> (80hex<\/sub>), and ™<\/code> (99hex<\/sub>).<\/p>\n's<\/code> on my phone, it is encoded as two Windows-1252 characters 92hex<\/sub> and 73hex<\/sub>. Then by the time the text appears on my laptop, the bytes 9273hex<\/sub> are interpreted as a Shift-JIS encoding of the CJK character \u75f4<\/code> (U+75F4).<\/p>\n","protected":false},"excerpt":{"rendered":"from math import *\r\n\r\ns = 0\r\nfor n in range(288):\r\n s += (-1)**n * 200**(2*n) \/ factorial(2*n)\r\nprint(s)\r\n<\/pre>\n
200<\/code> to 200.0<\/code> above, the code would crash. Calculating 200.0**(2*n)<\/code> overflows when n<\/em> = 67. But when we calculate 200**(2*n)<\/code>, the result is an integer. And we’re dividing by factorial(2*n)<\/code>, which is also an integer. Both of these integers become too large to fit in a float, but their ratio<\/em> has a maximum value of around 1080<\/sup>, smaller than the maximum float, which is on the order of 10308<\/sup>.<\/p>\n>>> from math import cos, pi\r\n>>> x = 200 % (2*pi)\r\n>>> x\r\n5.221255477432827\r\n<\/pre>\n
>>> cos(x)\r\n0.48718767500701254\r\n>>> cos(x) - cos(200)\r\n6.661338147750939e-15\r\n<\/pre>\n
![](\"https:\/\/www.johndcook.com\/cos200factorial1.png\")
<\/p>\n![](\"https:\/\/www.johndcook.com\/cos200factorial2.png\")
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