2022-09-28 18:04:21 +02:00
<?xml version="1.0" encoding="utf-8"?>
< !DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN"
"http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd">
< html xmlns = "http://www.w3.org/1999/xhtml" lang = "en" xml:lang = "en" >
< head >
2023-11-30 19:15:51 +01:00
<!-- 2023 - 11 - 30 Thu 18:15 -->
2022-09-28 18:04:21 +02:00
< meta http-equiv = "Content-Type" content = "text/html;charset=utf-8" / >
< meta name = "viewport" content = "width=device-width, initial-scale=1" / >
2023-11-30 01:19:47 +01:00
< title > ‎ < / title >
2022-09-28 18:04:21 +02:00
< meta name = "generator" content = "Org mode" / >
< meta name = "author" content = "TREX CoE" / >
< style type = "text/css" >
<!-- /* --> <![CDATA[/*> <!-- */
.title { text-align: center;
margin-bottom: .2em; }
.subtitle { text-align: center;
font-size: medium;
font-weight: bold;
margin-top:0; }
.todo { font-family: monospace; color: red; }
.done { font-family: monospace; color: green; }
.priority { font-family: monospace; color: orange; }
.tag { background-color: #eee; font-family: monospace;
padding: 2px; font-size: 80%; font-weight: normal; }
.timestamp { color: #bebebe; }
.timestamp-kwd { color: #5f9ea0; }
.org-right { margin-left: auto; margin-right: 0px; text-align: right; }
.org-left { margin-left: 0px; margin-right: auto; text-align: left; }
.org-center { margin-left: auto; margin-right: auto; text-align: center; }
.underline { text-decoration: underline; }
#postamble p, #preamble p { font-size: 90%; margin: .2em; }
p.verse { margin-left: 3%; }
pre {
border: 1px solid #ccc;
box-shadow: 3px 3px 3px #eee;
padding: 8pt;
font-family: monospace;
overflow: auto;
margin: 1.2em;
}
pre.src {
position: relative;
overflow: visible;
padding-top: 1.2em;
}
pre.src:before {
display: none;
position: absolute;
background-color: white;
top: -10px;
right: 10px;
padding: 3px;
border: 1px solid black;
}
pre.src:hover:before { display: inline;}
/* Languages per Org manual */
pre.src-asymptote:before { content: 'Asymptote'; }
pre.src-awk:before { content: 'Awk'; }
pre.src-C:before { content: 'C'; }
/* pre.src-C++ doesn't work in CSS */
pre.src-clojure:before { content: 'Clojure'; }
pre.src-css:before { content: 'CSS'; }
pre.src-D:before { content: 'D'; }
pre.src-ditaa:before { content: 'ditaa'; }
pre.src-dot:before { content: 'Graphviz'; }
pre.src-calc:before { content: 'Emacs Calc'; }
pre.src-emacs-lisp:before { content: 'Emacs Lisp'; }
pre.src-fortran:before { content: 'Fortran'; }
pre.src-gnuplot:before { content: 'gnuplot'; }
pre.src-haskell:before { content: 'Haskell'; }
pre.src-hledger:before { content: 'hledger'; }
pre.src-java:before { content: 'Java'; }
pre.src-js:before { content: 'Javascript'; }
pre.src-latex:before { content: 'LaTeX'; }
pre.src-ledger:before { content: 'Ledger'; }
pre.src-lisp:before { content: 'Lisp'; }
pre.src-lilypond:before { content: 'Lilypond'; }
pre.src-lua:before { content: 'Lua'; }
pre.src-matlab:before { content: 'MATLAB'; }
pre.src-mscgen:before { content: 'Mscgen'; }
pre.src-ocaml:before { content: 'Objective Caml'; }
pre.src-octave:before { content: 'Octave'; }
pre.src-org:before { content: 'Org mode'; }
pre.src-oz:before { content: 'OZ'; }
pre.src-plantuml:before { content: 'Plantuml'; }
pre.src-processing:before { content: 'Processing.js'; }
pre.src-python:before { content: 'Python'; }
pre.src-R:before { content: 'R'; }
pre.src-ruby:before { content: 'Ruby'; }
pre.src-sass:before { content: 'Sass'; }
pre.src-scheme:before { content: 'Scheme'; }
pre.src-screen:before { content: 'Gnu Screen'; }
pre.src-sed:before { content: 'Sed'; }
pre.src-sh:before { content: 'shell'; }
pre.src-sql:before { content: 'SQL'; }
pre.src-sqlite:before { content: 'SQLite'; }
/* additional languages in org.el's org-babel-load-languages alist */
pre.src-forth:before { content: 'Forth'; }
pre.src-io:before { content: 'IO'; }
pre.src-J:before { content: 'J'; }
pre.src-makefile:before { content: 'Makefile'; }
pre.src-maxima:before { content: 'Maxima'; }
pre.src-perl:before { content: 'Perl'; }
pre.src-picolisp:before { content: 'Pico Lisp'; }
pre.src-scala:before { content: 'Scala'; }
pre.src-shell:before { content: 'Shell Script'; }
pre.src-ebnf2ps:before { content: 'ebfn2ps'; }
/* additional language identifiers per "defun org-babel-execute"
in ob-*.el */
pre.src-cpp:before { content: 'C++'; }
pre.src-abc:before { content: 'ABC'; }
pre.src-coq:before { content: 'Coq'; }
pre.src-groovy:before { content: 'Groovy'; }
/* additional language identifiers from org-babel-shell-names in
ob-shell.el: ob-shell is the only babel language using a lambda to put
the execution function name together. */
pre.src-bash:before { content: 'bash'; }
pre.src-csh:before { content: 'csh'; }
pre.src-ash:before { content: 'ash'; }
pre.src-dash:before { content: 'dash'; }
pre.src-ksh:before { content: 'ksh'; }
pre.src-mksh:before { content: 'mksh'; }
pre.src-posh:before { content: 'posh'; }
/* Additional Emacs modes also supported by the LaTeX listings package */
pre.src-ada:before { content: 'Ada'; }
pre.src-asm:before { content: 'Assembler'; }
pre.src-caml:before { content: 'Caml'; }
pre.src-delphi:before { content: 'Delphi'; }
pre.src-html:before { content: 'HTML'; }
pre.src-idl:before { content: 'IDL'; }
pre.src-mercury:before { content: 'Mercury'; }
pre.src-metapost:before { content: 'MetaPost'; }
pre.src-modula-2:before { content: 'Modula-2'; }
pre.src-pascal:before { content: 'Pascal'; }
pre.src-ps:before { content: 'PostScript'; }
pre.src-prolog:before { content: 'Prolog'; }
pre.src-simula:before { content: 'Simula'; }
pre.src-tcl:before { content: 'tcl'; }
pre.src-tex:before { content: 'TeX'; }
pre.src-plain-tex:before { content: 'Plain TeX'; }
pre.src-verilog:before { content: 'Verilog'; }
pre.src-vhdl:before { content: 'VHDL'; }
pre.src-xml:before { content: 'XML'; }
pre.src-nxml:before { content: 'XML'; }
/* add a generic configuration mode; LaTeX export needs an additional
(add-to-list 'org-latex-listings-langs '(conf " ")) in .emacs */
pre.src-conf:before { content: 'Configuration File'; }
table { border-collapse:collapse; }
caption.t-above { caption-side: top; }
caption.t-bottom { caption-side: bottom; }
td, th { vertical-align:top; }
th.org-right { text-align: center; }
th.org-left { text-align: center; }
th.org-center { text-align: center; }
td.org-right { text-align: right; }
td.org-left { text-align: left; }
td.org-center { text-align: center; }
dt { font-weight: bold; }
.footpara { display: inline; }
.footdef { margin-bottom: 1em; }
.figure { padding: 1em; }
.figure p { text-align: center; }
2022-11-18 14:16:53 +01:00
.equation-container {
display: table;
text-align: center;
width: 100%;
}
.equation {
vertical-align: middle;
}
.equation-label {
display: table-cell;
text-align: right;
vertical-align: middle;
}
2022-09-28 18:04:21 +02:00
.inlinetask {
padding: 10px;
border: 2px solid gray;
margin: 10px;
background: #ffffcc;
}
#org-div-home-and-up
{ text-align: right; font-size: 70%; white-space: nowrap; }
textarea { overflow-x: auto; }
.linenr { font-size: smaller }
.code-highlighted { background-color: #ffff00; }
.org-info-js_info-navigation { border-style: none; }
#org-info-js_console-label
{ font-size: 10px; font-weight: bold; white-space: nowrap; }
.org-info-js_search-highlight
{ background-color: #ffff00; color: #000000; font-weight: bold; }
.org-svg { width: 90%; }
/*]]>*/-->
< / style >
< link rel = "stylesheet" title = "Standard" href = "qmckl.css" type = "text/css" / >
< script type = "text/javascript" src = "org-info.js" >
/**
*
* @source: org-info.js
*
* @licstart The following is the entire license notice for the
* JavaScript code in org-info.js.
*
2022-11-18 14:16:53 +01:00
* Copyright (C) 2012-2020 Free Software Foundation, Inc.
2022-09-28 18:04:21 +02:00
*
*
* The JavaScript code in this tag is free software: you can
* redistribute it and/or modify it under the terms of the GNU
* General Public License (GNU GPL) as published by the Free Software
* Foundation, either version 3 of the License, or (at your option)
* any later version. The code is distributed WITHOUT ANY WARRANTY;
* without even the implied warranty of MERCHANTABILITY or FITNESS
* FOR A PARTICULAR PURPOSE. See the GNU GPL for more details.
*
* As additional permission under GNU GPL version 3 section 7, you
* may distribute non-source (e.g., minimized or compacted) forms of
* that code without the copy of the GNU GPL normally required by
* section 4, provided you include this license notice and a URL
* through which recipients can access the Corresponding Source.
*
* @licend The above is the entire license notice
* for the JavaScript code in org-info.js.
*
*/
< / script >
< script type = "text/javascript" >
/*
@licstart The following is the entire license notice for the
JavaScript code in this tag.
2022-11-18 14:16:53 +01:00
Copyright (C) 2012-2020 Free Software Foundation, Inc.
2022-09-28 18:04:21 +02:00
The JavaScript code in this tag is free software: you can
redistribute it and/or modify it under the terms of the GNU
General Public License (GNU GPL) as published by the Free Software
Foundation, either version 3 of the License, or (at your option)
any later version. The code is distributed WITHOUT ANY WARRANTY;
without even the implied warranty of MERCHANTABILITY or FITNESS
FOR A PARTICULAR PURPOSE. See the GNU GPL for more details.
As additional permission under GNU GPL version 3 section 7, you
may distribute non-source (e.g., minimized or compacted) forms of
that code without the copy of the GNU GPL normally required by
section 4, provided you include this license notice and a URL
through which recipients can access the Corresponding Source.
@licend The above is the entire license notice
for the JavaScript code in this tag.
*/
<!-- /* --> <![CDATA[/*> <!-- */
org_html_manager.set("TOC_DEPTH", "4");
org_html_manager.set("LINK_HOME", "index.html");
org_html_manager.set("LINK_UP", "");
org_html_manager.set("LOCAL_TOC", "1");
org_html_manager.set("VIEW_BUTTONS", "0");
org_html_manager.set("MOUSE_HINT", "underline");
org_html_manager.set("FIXED_TOC", "0");
org_html_manager.set("TOC", "1");
org_html_manager.set("VIEW", "info");
org_html_manager.setup(); // activate after the parameters are set
/*]]>*///-->
< / script >
< script type = "text/javascript" >
/*
@licstart The following is the entire license notice for the
JavaScript code in this tag.
2022-11-18 14:16:53 +01:00
Copyright (C) 2012-2020 Free Software Foundation, Inc.
2022-09-28 18:04:21 +02:00
The JavaScript code in this tag is free software: you can
redistribute it and/or modify it under the terms of the GNU
General Public License (GNU GPL) as published by the Free Software
Foundation, either version 3 of the License, or (at your option)
any later version. The code is distributed WITHOUT ANY WARRANTY;
without even the implied warranty of MERCHANTABILITY or FITNESS
FOR A PARTICULAR PURPOSE. See the GNU GPL for more details.
As additional permission under GNU GPL version 3 section 7, you
may distribute non-source (e.g., minimized or compacted) forms of
that code without the copy of the GNU GPL normally required by
section 4, provided you include this license notice and a URL
through which recipients can access the Corresponding Source.
@licend The above is the entire license notice
for the JavaScript code in this tag.
*/
<!-- /* --> <![CDATA[/*> <!-- */
function CodeHighlightOn(elem, id)
{
var target = document.getElementById(id);
if(null != target) {
elem.cacheClassElem = elem.className;
elem.cacheClassTarget = target.className;
target.className = "code-highlighted";
elem.className = "code-highlighted";
}
}
function CodeHighlightOff(elem, id)
{
var target = document.getElementById(id);
if(elem.cacheClassElem)
elem.className = elem.cacheClassElem;
if(elem.cacheClassTarget)
target.className = elem.cacheClassTarget;
}
/*]]>*///-->
< / script >
< script type = "text/x-mathjax-config" >
MathJax.Hub.Config({
displayAlign: "center",
displayIndent: "0em",
"HTML-CSS": { scale: 100,
linebreaks: { automatic: "false" },
webFont: "TeX"
},
SVG: {scale: 100,
linebreaks: { automatic: "false" },
font: "TeX"},
NativeMML: {scale: 100},
TeX: { equationNumbers: {autoNumber: "AMS"},
MultLineWidth: "85%",
TagSide: "right",
TagIndent: ".8em"
}
});
< / script >
< script type = "text/javascript"
src="https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.0/MathJax.js?config=TeX-AMS_HTML">< / script >
< / head >
< body >
< div id = "org-div-home-and-up" >
< a accesskey = "h" href = "" > UP < / a >
|
< a accesskey = "H" href = "index.html" > HOME < / a >
< / div > < div id = "content" >
< div id = "table-of-contents" >
< h2 > Table of Contents< / h2 >
< div id = "text-table-of-contents" >
< ul >
2023-11-30 19:15:51 +01:00
< li > < a href = "#orga7758b7" > 1. Control of the numerical precision< / a > < / li >
< li > < a href = "#orgaef4cc2" > 2. Precision< / a > < / li >
< li > < a href = "#org3556767" > 3. Range< / a > < / li >
< li > < a href = "#org05dde97" > 4. Helper functions< / a >
2023-11-28 17:09:55 +01:00
< ul >
2023-11-30 19:15:51 +01:00
< li > < a href = "#orgbaeb19d" > 4.1. Epsilon< / a > < / li >
< li > < a href = "#org4b9c21d" > 4.2. Testing the number of unchanged bits< / a > < / li >
2023-11-30 01:19:47 +01:00
< / ul >
< / li >
2023-11-30 19:15:51 +01:00
< li > < a href = "#orga14c0b4" > 5. Approximate functions< / a >
2023-11-30 01:19:47 +01:00
< ul >
2023-11-30 19:15:51 +01:00
< li > < a href = "#org42fca28" > 5.1. Exponential< / a > < / li >
2023-11-28 17:09:55 +01:00
< / ul >
< / li >
2022-09-28 18:04:21 +02:00
< / ul >
< / div >
< / div >
2023-11-30 01:19:47 +01:00
< p >
3+TITLE: Numerical precision
< / p >
2022-09-28 18:04:21 +02:00
2023-11-30 19:15:51 +01:00
< div id = "outline-container-orga7758b7" class = "outline-2" >
< h2 id = "orga7758b7" > < span class = "section-number-2" > 1< / span > Control of the numerical precision< / h2 >
2022-09-28 18:04:21 +02:00
< div class = "outline-text-2" id = "text-1" >
< p >
Controlling numerical precision enables optimizations. Here, the
default parameters determining the target numerical precision and
range are defined. Following the IEEE Standard for Floating-Point
Arithmetic (IEEE 754),
2023-11-28 17:09:55 +01:00
< i > precision< / i > refers to the number of significand bits (including the
sign bit) and < i > range< / i > refers to the number of exponent bits.
2022-09-28 18:04:21 +02:00
< / p >
2023-11-30 19:15:51 +01:00
< table id = "orgc22a454" border = "2" cellspacing = "0" cellpadding = "6" rules = "groups" frame = "hsides" >
2022-09-28 18:04:21 +02:00
< colgroup >
< col class = "org-left" / >
< col class = "org-right" / >
< / colgroup >
< tbody >
< tr >
< td class = "org-left" > < code > QMCKL_DEFAULT_PRECISION< / code > < / td >
< td class = "org-right" > 53< / td >
< / tr >
< tr >
< td class = "org-left" > < code > QMCKL_DEFAULT_RANGE< / code > < / td >
< td class = "org-right" > 11< / td >
< / tr >
< / tbody >
< / table >
< div class = "org-src-container" >
< pre class = "src src-c" > < span style = "color: #a020f0;" > typedef< / span > < span style = "color: #a020f0;" > struct< / span > < span style = "color: #228b22;" > qmckl_numprec_struct< / span > {
< span style = "color: #228b22;" > uint32_t< / span > < span style = "color: #a0522d;" > precision< / span > ;
< span style = "color: #228b22;" > uint32_t< / span > < span style = "color: #a0522d;" > range< / span > ;
} < span style = "color: #228b22;" > qmckl_numprec_struct< / span > ;
< / pre >
< / div >
< p >
The following functions set and get the required precision and
range. < code > precision< / code > is an integer between 2 and 53, and < code > range< / code > is an
integer between 2 and 11.
< / p >
< p >
The setter functions functions return a new context as a 64-bit
integer. The getter functions return the value, as a 32-bit
integer. The update functions return < code > QMCKL_SUCCESS< / code > or
< code > QMCKL_FAILURE< / code > .
< / p >
< / div >
< / div >
2023-11-30 19:15:51 +01:00
< div id = "outline-container-orgaef4cc2" class = "outline-2" >
< h2 id = "orgaef4cc2" > < span class = "section-number-2" > 2< / span > Precision< / h2 >
2022-09-28 18:04:21 +02:00
< div class = "outline-text-2" id = "text-2" >
< p >
< code > qmckl_context_set_numprec_precision< / code > modifies the parameter for the
numerical precision in the context.
< / p >
< div class = "org-src-container" >
< pre class = "src src-c" > < span style = "color: #228b22;" > qmckl_exit_code< / span > < span style = "color: #0000ff;" > qmckl_set_numprec_precision< / span > (< span style = "color: #a020f0;" > const< / span > < span style = "color: #228b22;" > qmckl_context< / span > < span style = "color: #a0522d;" > context< / span > , < span style = "color: #a020f0;" > const< / span > < span style = "color: #228b22;" > int< / span > < span style = "color: #a0522d;" > precision< / span > ) {
< span style = "color: #a020f0;" > if< / span > (qmckl_context_check(context) == QMCKL_NULL_CONTEXT)
< span style = "color: #a020f0;" > return< / span > QMCKL_INVALID_CONTEXT;
< span style = "color: #a020f0;" > if< / span > (precision < 2) {
< span style = "color: #a020f0;" > return< / span > qmckl_failwith(context,
QMCKL_INVALID_ARG_2,
< span style = "color: #8b2252;" > "qmckl_update_numprec_precision"< / span > ,
< span style = "color: #8b2252;" > "precision < 2"< / span > );
}
< span style = "color: #a020f0;" > if< / span > (precision > 53) {
< span style = "color: #a020f0;" > return< / span > qmckl_failwith(context,
QMCKL_INVALID_ARG_2,
< span style = "color: #8b2252;" > "qmckl_update_numprec_precision"< / span > ,
< span style = "color: #8b2252;" > "precision > 53"< / span > );
}
< span style = "color: #228b22;" > qmckl_context_struct< / span > * < span style = "color: #a020f0;" > const< / span > < span style = "color: #a0522d;" > ctx< / span > = (< span style = "color: #228b22;" > qmckl_context_struct< / span > *) context;
2022-11-18 14:16:53 +01:00
/* < span style = "color: #b22222;" > This should be always true because the context is valid< / span > */
2022-09-28 18:04:21 +02:00
assert (ctx != < span style = "color: #008b8b;" > NULL< / span > );
qmckl_lock(context);
{
ctx-> numprec.precision = (< span style = "color: #228b22;" > uint32_t< / span > ) precision;
}
qmckl_unlock(context);
< span style = "color: #a020f0;" > return< / span > QMCKL_SUCCESS;
}
< / pre >
< / div >
< div class = "org-src-container" >
< pre class = "src src-f90" > < span style = "color: #a020f0;" > interface< / span >
< span style = "color: #228b22;" > integer< / span > (qmckl_exit_code) < span style = "color: #a020f0;" > function< / span > < span style = "color: #0000ff;" > qmckl_set_numprec_precision< / span > (context, precision) < span style = "color: #a020f0;" > bind< / span > (C)
< span style = "color: #a020f0;" > use< / span > , < span style = "color: #a020f0;" > intrinsic< / span > :: < span style = "color: #0000ff;" > iso_c_binding< / span >
< span style = "color: #a020f0;" > import< / span >
< span style = "color: #228b22;" > integer< / span > (qmckl_context), < span style = "color: #a020f0;" > intent< / span > (in), < span style = "color: #a020f0;" > value< / span > ::< span style = "color: #a0522d;" > context< / span >
< span style = "color: #228b22;" > integer< / span > (< span style = "color: #008b8b;" > c_int32_t< / span > ), < span style = "color: #a020f0;" > intent< / span > (in), < span style = "color: #a020f0;" > value< / span > ::< span style = "color: #a0522d;" > precision< / span >
< span style = "color: #a020f0;" > end function< / span > < span style = "color: #0000ff;" > qmckl_set_numprec_precision< / span >
< span style = "color: #a020f0;" > end interface< / span >
< / pre >
< / div >
< p >
< code > qmckl_get_numprec_precision< / code > returns the value of the numerical precision in the context.
< / p >
< div class = "org-src-container" >
< pre class = "src src-c" > < span style = "color: #228b22;" > int< / span > < span style = "color: #0000ff;" > qmckl_get_numprec_precision< / span > (< span style = "color: #a020f0;" > const< / span > < span style = "color: #228b22;" > qmckl_context< / span > < span style = "color: #a0522d;" > context< / span > ) {
< span style = "color: #a020f0;" > if< / span > (qmckl_context_check(context) == QMCKL_NULL_CONTEXT) {
< span style = "color: #a020f0;" > return< / span > qmckl_failwith(context,
QMCKL_INVALID_CONTEXT,
< span style = "color: #8b2252;" > "qmckl_get_numprec_precision"< / span > ,
< span style = "color: #8b2252;" > ""< / span > );
}
< span style = "color: #a020f0;" > const< / span > < span style = "color: #228b22;" > qmckl_context_struct< / span > * < span style = "color: #a020f0;" > const< / span > < span style = "color: #a0522d;" > ctx< / span > = (< span style = "color: #228b22;" > qmckl_context_struct< / span > *) context;
< span style = "color: #a020f0;" > return< / span > ctx-> numprec.precision;
}
< / pre >
< / div >
< div class = "org-src-container" >
< pre class = "src src-f90" > < span style = "color: #a020f0;" > interface< / span >
< span style = "color: #228b22;" > integer< / span > (qmckl_exit_code) < span style = "color: #a020f0;" > function< / span > < span style = "color: #0000ff;" > qmckl_get_numprec_precision< / span > (context) < span style = "color: #a020f0;" > bind< / span > (C)
< span style = "color: #a020f0;" > use< / span > , < span style = "color: #a020f0;" > intrinsic< / span > :: < span style = "color: #0000ff;" > iso_c_binding< / span >
< span style = "color: #a020f0;" > import< / span >
< span style = "color: #228b22;" > integer< / span > (qmckl_context), < span style = "color: #a020f0;" > intent< / span > (in), < span style = "color: #a020f0;" > value< / span > ::< span style = "color: #a0522d;" > context< / span >
< span style = "color: #a020f0;" > end function< / span > < span style = "color: #0000ff;" > qmckl_get_numprec_precision< / span >
< span style = "color: #a020f0;" > end interface< / span >
< / pre >
< / div >
< / div >
< / div >
2023-11-30 19:15:51 +01:00
< div id = "outline-container-org3556767" class = "outline-2" >
< h2 id = "org3556767" > < span class = "section-number-2" > 3< / span > Range< / h2 >
2022-09-28 18:04:21 +02:00
< div class = "outline-text-2" id = "text-3" >
< p >
< code > qmckl_set_numprec_range< / code > modifies the parameter for the numerical
range in a given context.
< / p >
< div class = "org-src-container" >
< pre class = "src src-c" > < span style = "color: #228b22;" > qmckl_exit_code< / span > < span style = "color: #0000ff;" > qmckl_set_numprec_range< / span > (< span style = "color: #a020f0;" > const< / span > < span style = "color: #228b22;" > qmckl_context< / span > < span style = "color: #a0522d;" > context< / span > , < span style = "color: #a020f0;" > const< / span > < span style = "color: #228b22;" > int< / span > < span style = "color: #a0522d;" > range< / span > ) {
< span style = "color: #a020f0;" > if< / span > (qmckl_context_check(context) == QMCKL_NULL_CONTEXT)
< span style = "color: #a020f0;" > return< / span > QMCKL_INVALID_CONTEXT;
< span style = "color: #a020f0;" > if< / span > (range < 2) {
< span style = "color: #a020f0;" > return< / span > qmckl_failwith(context,
QMCKL_INVALID_ARG_2,
< span style = "color: #8b2252;" > "qmckl_set_numprec_range"< / span > ,
< span style = "color: #8b2252;" > "range < 2"< / span > );
}
< span style = "color: #a020f0;" > if< / span > (range > 11) {
< span style = "color: #a020f0;" > return< / span > qmckl_failwith(context,
QMCKL_INVALID_ARG_2,
< span style = "color: #8b2252;" > "qmckl_set_numprec_range"< / span > ,
< span style = "color: #8b2252;" > "range > 11"< / span > );
}
< span style = "color: #228b22;" > qmckl_context_struct< / span > * < span style = "color: #a020f0;" > const< / span > < span style = "color: #a0522d;" > ctx< / span > = (< span style = "color: #228b22;" > qmckl_context_struct< / span > *) context;
2022-11-18 14:16:53 +01:00
/* < span style = "color: #b22222;" > This should be always true because the context is valid< / span > */
2022-09-28 18:04:21 +02:00
assert (ctx != < span style = "color: #008b8b;" > NULL< / span > );
qmckl_lock(context);
{
ctx-> numprec.range = (< span style = "color: #228b22;" > uint32_t< / span > ) range;
}
qmckl_unlock(context);
< span style = "color: #a020f0;" > return< / span > QMCKL_SUCCESS;
}
< / pre >
< / div >
< div class = "org-src-container" >
< pre class = "src src-f90" > < span style = "color: #a020f0;" > interface< / span >
< span style = "color: #228b22;" > integer< / span > (qmckl_exit_code) < span style = "color: #a020f0;" > function< / span > < span style = "color: #0000ff;" > qmckl_set_numprec_range< / span > (context, range) < span style = "color: #a020f0;" > bind< / span > (C)
< span style = "color: #a020f0;" > use< / span > , < span style = "color: #a020f0;" > intrinsic< / span > :: < span style = "color: #0000ff;" > iso_c_binding< / span >
< span style = "color: #a020f0;" > import< / span >
< span style = "color: #228b22;" > integer< / span > (qmckl_context), < span style = "color: #a020f0;" > intent< / span > (in), < span style = "color: #a020f0;" > value< / span > ::< span style = "color: #a0522d;" > context< / span >
< span style = "color: #228b22;" > integer< / span > (< span style = "color: #008b8b;" > c_int32_t< / span > ), < span style = "color: #a020f0;" > intent< / span > (in), < span style = "color: #a020f0;" > value< / span > ::< span style = "color: #a0522d;" > range< / span >
< span style = "color: #a020f0;" > end function< / span > < span style = "color: #0000ff;" > qmckl_set_numprec_range< / span >
< span style = "color: #a020f0;" > end interface< / span >
< / pre >
< / div >
< p >
< code > qmckl_get_numprec_range< / code > returns the value of the numerical range in the context.
< / p >
< div class = "org-src-container" >
< pre class = "src src-c" > < span style = "color: #228b22;" > int< / span > < span style = "color: #0000ff;" > qmckl_get_numprec_range< / span > (< span style = "color: #a020f0;" > const< / span > < span style = "color: #228b22;" > qmckl_context< / span > < span style = "color: #a0522d;" > context< / span > ) {
< span style = "color: #a020f0;" > if< / span > (qmckl_context_check(context) == QMCKL_NULL_CONTEXT) {
< span style = "color: #a020f0;" > return< / span > qmckl_failwith(context,
QMCKL_INVALID_CONTEXT,
< span style = "color: #8b2252;" > "qmckl_get_numprec_range"< / span > ,
< span style = "color: #8b2252;" > ""< / span > );
}
< span style = "color: #a020f0;" > const< / span > < span style = "color: #228b22;" > qmckl_context_struct< / span > * < span style = "color: #a020f0;" > const< / span > < span style = "color: #a0522d;" > ctx< / span > = (< span style = "color: #228b22;" > qmckl_context_struct< / span > *) context;
< span style = "color: #a020f0;" > return< / span > ctx-> numprec.range;
}
< / pre >
< / div >
< / div >
< / div >
2023-11-30 19:15:51 +01:00
< div id = "outline-container-org05dde97" class = "outline-2" >
< h2 id = "org05dde97" > < span class = "section-number-2" > 4< / span > Helper functions< / h2 >
2022-09-28 18:04:21 +02:00
< div class = "outline-text-2" id = "text-4" >
2023-11-30 01:19:47 +01:00
< / div >
2023-11-30 19:15:51 +01:00
< div id = "outline-container-orgbaeb19d" class = "outline-3" >
< h3 id = "orgbaeb19d" > < span class = "section-number-3" > 4.1< / span > Epsilon< / h3 >
2023-11-30 01:19:47 +01:00
< div class = "outline-text-3" id = "text-4-1" >
2022-09-28 18:04:21 +02:00
< p >
< code > qmckl_get_numprec_epsilon< / code > returns \(\epsilon = 2^{1-n}\) where < code > n< / code > is the precision.
We need to remove the sign bit from the precision.
< / p >
< div class = "org-src-container" >
< pre class = "src src-c" > < span style = "color: #228b22;" > double< / span > < span style = "color: #0000ff;" > qmckl_get_numprec_epsilon< / span > (< span style = "color: #a020f0;" > const< / span > < span style = "color: #228b22;" > qmckl_context< / span > < span style = "color: #a0522d;" > context< / span > ) {
2023-11-30 01:19:47 +01:00
< span style = "color: #a020f0;" > if< / span > (qmckl_context_check(context) == QMCKL_NULL_CONTEXT)
< span style = "color: #a020f0;" > return< / span > QMCKL_INVALID_CONTEXT;
< span style = "color: #a020f0;" > const< / span > < span style = "color: #228b22;" > qmckl_context_struct< / span > * < span style = "color: #a020f0;" > const< / span > < span style = "color: #a0522d;" > ctx< / span > = (< span style = "color: #228b22;" > qmckl_context_struct< / span > *) context;
< span style = "color: #a020f0;" > const< / span > < span style = "color: #228b22;" > int< / span > < span style = "color: #a0522d;" > precision< / span > = ctx-> numprec.precision;
2023-11-28 17:09:55 +01:00
< span style = "color: #a020f0;" > return< / span > 1. / (< span style = "color: #228b22;" > double< / span > ) ( ((< span style = "color: #228b22;" > uint64_t< / span > ) 1) < < (precision-2));
2022-09-28 18:04:21 +02:00
}
< / pre >
< / div >
< / div >
< / div >
2023-11-30 19:15:51 +01:00
< div id = "outline-container-org4b9c21d" class = "outline-3" >
< h3 id = "org4b9c21d" > < span class = "section-number-3" > 4.2< / span > Testing the number of unchanged bits< / h3 >
2023-11-30 01:19:47 +01:00
< div class = "outline-text-3" id = "text-4-2" >
< p >
To test that a given approximation keeps a given number of bits
unchanged, we need a function that returns the number of unchanged
bits in the range, and in the precision.
< / p >
< p >
For this, we first count by how many units in the last place (ulps) two
numbers differ.
< / p >
< div class = "org-src-container" >
< pre class = "src src-c" > < span style = "color: #228b22;" > int64_t< / span > < span style = "color: #0000ff;" > countUlpDifference_64< / span > (< span style = "color: #228b22;" > double< / span > < span style = "color: #a0522d;" > a< / span > , < span style = "color: #228b22;" > double< / span > < span style = "color: #a0522d;" > b< / span > ) {
< span style = "color: #a020f0;" > union< / span > < span style = "color: #228b22;" > int_or_float< / span > {
< span style = "color: #228b22;" > int64_t< / span > < span style = "color: #a0522d;" > i< / span > ;
< span style = "color: #228b22;" > double< / span > < span style = "color: #a0522d;" > f< / span > ;
} < span style = "color: #a0522d;" > x< / span > , < span style = "color: #a0522d;" > y< / span > ;
x.f = a;
y.f = b;
// < span style = "color: #b22222;" > Handle sign bit discontinuity: if the signs are different and either value is not zero< / span >
< span style = "color: #a020f0;" > if< / span > ((x.i < 0) != (y.i < 0) & & (x.f != 0.0) & & (y.f != 0.0)) {
// < span style = "color: #b22222;" > Use the absolute values and add the distance to zero for both numbers< / span >
< span style = "color: #228b22;" > int64_t< / span > < span style = "color: #a0522d;" > distanceToZeroForX< / span > = x.i < 0 ? INT64_MAX + x.i : INT64_MAX - x.i;
< span style = "color: #228b22;" > int64_t< / span > < span style = "color: #a0522d;" > distanceToZeroForY< / span > = y.i < 0 ? INT64_MAX + y.i : INT64_MAX - y.i;
< span style = "color: #a020f0;" > return< / span > distanceToZeroForX + distanceToZeroForY;
}
// < span style = "color: #b22222;" > Calculate the difference in their binary representations< / span >
< span style = "color: #228b22;" > int64_t< / span > < span style = "color: #a0522d;" > result< / span > = x.i - y.i;
result = result > 0 ? result : -result;
< span style = "color: #a020f0;" > return< / span > result;
}
< / pre >
< / div >
< div class = "org-src-container" >
< pre class = "src src-c" > < span style = "color: #228b22;" > int32_t< / span > < span style = "color: #0000ff;" > qmckl_test_precision_64< / span > (< span style = "color: #228b22;" > double< / span > < span style = "color: #a0522d;" > a< / span > , < span style = "color: #228b22;" > double< / span > < span style = "color: #a0522d;" > b< / span > ) {
< span style = "color: #228b22;" > int64_t< / span > < span style = "color: #a0522d;" > diff< / span > = countUlpDifference_64(a,b);
< span style = "color: #a020f0;" > if< / span > (diff == 0) < span style = "color: #a020f0;" > return< / span > 53;
< span style = "color: #228b22;" > int32_t< / span > < span style = "color: #a0522d;" > result< / span > = 53;
< span style = "color: #a020f0;" > for< / span > (< span style = "color: #228b22;" > int< / span > < span style = "color: #a0522d;" > i< / span > =0 ; i< 53 & & diff != 0 ; ++i) {
diff > > = 1;
result--;
}
< span style = "color: #a020f0;" > return< / span > result;
}
< / pre >
< / div >
< div class = "org-src-container" >
< pre class = "src src-c" > < span style = "color: #228b22;" > int32_t< / span > < span style = "color: #0000ff;" > qmckl_test_precision_32< / span > (< span style = "color: #228b22;" > float< / span > < span style = "color: #a0522d;" > a< / span > , < span style = "color: #228b22;" > float< / span > < span style = "color: #a0522d;" > b< / span > ) {
< span style = "color: #a020f0;" > return< / span > qmckl_test_precision_64( (< span style = "color: #228b22;" > double< / span > ) a, (< span style = "color: #228b22;" > double< / span > ) b );
}
< / pre >
< / div >
< / div >
< / div >
< / div >
2023-11-30 19:15:51 +01:00
< div id = "outline-container-orga14c0b4" class = "outline-2" >
< h2 id = "orga14c0b4" > < span class = "section-number-2" > 5< / span > Approximate functions< / h2 >
2023-11-28 17:09:55 +01:00
< div class = "outline-text-2" id = "text-5" >
< / div >
2023-11-30 19:15:51 +01:00
< div id = "outline-container-org42fca28" class = "outline-3" >
< h3 id = "org42fca28" > < span class = "section-number-3" > 5.1< / span > Exponential< / h3 >
2023-11-28 17:09:55 +01:00
< div class = "outline-text-3" id = "text-5-1" >
< p >
Fast exponential function, adapted from Johan Rade's implementation
(< a href = "https://gist.github.com/jrade/293a73f89dfef51da6522428c857802d" > https://gist.github.com/jrade/293a73f89dfef51da6522428c857802d< / a > ). It
is based on Schraudolph's paper:
< / p >
< p >
N. Schraudolph, "A Fast, Compact Approximation of the Exponential Function",
< i > Neural Computation< / i > < b > 11< / b > , 853–
(available at < a href = "https://nic.schraudolph.org/pubs/Schraudolph99.pdf" > https://nic.schraudolph.org/pubs/Schraudolph99.pdf< / a > )
< / p >
< div class = "org-src-container" >
< pre class = "src src-c" > < span style = "color: #228b22;" > float< / span > < span style = "color: #0000ff;" > fastExpf< / span > (< span style = "color: #228b22;" > float< / span > < span style = "color: #a0522d;" > x< / span > )
{
< span style = "color: #a020f0;" > const< / span > < span style = "color: #228b22;" > float< / span > < span style = "color: #a0522d;" > a< / span > = 12102203.0;
< span style = "color: #a020f0;" > const< / span > < span style = "color: #228b22;" > float< / span > < span style = "color: #a0522d;" > b< / span > = 1064986816.0;
x = a * x + b;
< span style = "color: #a020f0;" > const< / span > < span style = "color: #228b22;" > float< / span > < span style = "color: #a0522d;" > c< / span > = 8388608.0;
< span style = "color: #a020f0;" > const< / span > < span style = "color: #228b22;" > float< / span > < span style = "color: #a0522d;" > d< / span > = 2139095040.0;
< span style = "color: #a020f0;" > if< / span > (x < c || x > d)
x = (x < c) ? 0.0f : d;
< span style = "color: #228b22;" > uint32_t< / span > < span style = "color: #a0522d;" > n< / span > = (< span style = "color: #228b22;" > uint32_t< / span > ) x;
memcpy(& x, & n, 4);
< span style = "color: #a020f0;" > return< / span > x;
}
< span style = "color: #228b22;" > double< / span > < span style = "color: #0000ff;" > fastExp< / span > (< span style = "color: #228b22;" > double< / span > < span style = "color: #a0522d;" > x< / span > )
{
< span style = "color: #a020f0;" > const< / span > < span style = "color: #228b22;" > double< / span > < span style = "color: #a0522d;" > a< / span > = 6497320848556798.0;
< span style = "color: #a020f0;" > const< / span > < span style = "color: #228b22;" > double< / span > < span style = "color: #a0522d;" > b< / span > = 4606985713057410560.0;
x = a * x + b;
< span style = "color: #a020f0;" > const< / span > < span style = "color: #228b22;" > double< / span > < span style = "color: #a0522d;" > c< / span > = 4503599627370496.0;
< span style = "color: #a020f0;" > const< / span > < span style = "color: #228b22;" > double< / span > < span style = "color: #a0522d;" > d< / span > = 9218868437227405312.0;
< span style = "color: #a020f0;" > if< / span > (x < c || x > d)
x = (x < c) ? 0.0 : d;
< span style = "color: #228b22;" > uint64_t< / span > < span style = "color: #a0522d;" > n< / span > = (< span style = "color: #228b22;" > uint64_t< / span > ) x;
memcpy(& x, & n, 8);
< span style = "color: #a020f0;" > return< / span > x;
}
< / pre >
< / div >
< / div >
< / div >
< / div >
2022-09-28 18:04:21 +02:00
< / div >
< div id = "postamble" class = "status" >
< p class = "author" > Author: TREX CoE< / p >
2023-11-30 19:15:51 +01:00
< p class = "date" > Created: 2023-11-30 Thu 18:15< / p >
2022-09-28 18:04:21 +02:00
< p class = "validation" > < a href = "http://validator.w3.org/check?uri=referer" > Validate< / a > < / p >
< / div >
< / body >
< / html >
