<html><head><meta http-equiv="Content-Type" content="text/html; charset=utf-8"></head><body style="word-wrap: break-word; -webkit-nbsp-mode: space; line-break: after-white-space;" class=""><br class=""><div><br class=""><blockquote type="cite" class=""><div class="">On Jan 16, 2020, at 7:10 PM, Fred J. Tydeman <<a href="mailto:tydeman@tybor.com" class="">tydeman@tybor.com</a>> wrote:</div><br class="Apple-interchange-newline"><div class=""><span style="caret-color: rgb(0, 0, 0); font-family: Helvetica; font-size: 12px; font-style: normal; font-variant-caps: normal; font-weight: normal; letter-spacing: normal; text-align: start; text-indent: 0px; text-transform: none; white-space: normal; word-spacing: 0px; -webkit-text-stroke-width: 0px; text-decoration: none; float: none; display: inline !important;" class="">Attached is the latest version of my range error paper.</span><br style="caret-color: rgb(0, 0, 0); font-family: Helvetica; font-size: 12px; font-style: normal; font-variant-caps: normal; font-weight: normal; letter-spacing: normal; text-align: start; text-indent: 0px; text-transform: none; white-space: normal; word-spacing: 0px; -webkit-text-stroke-width: 0px; text-decoration: none;" class=""><br style="caret-color: rgb(0, 0, 0); font-family: Helvetica; font-size: 12px; font-style: normal; font-variant-caps: normal; font-weight: normal; letter-spacing: normal; text-align: start; text-indent: 0px; text-transform: none; white-space: normal; word-spacing: 0px; -webkit-text-stroke-width: 0px; text-decoration: none;" class=""><br style="caret-color: rgb(0, 0, 0); font-family: Helvetica; font-size: 12px; font-style: normal; font-variant-caps: normal; font-weight: normal; letter-spacing: normal; text-align: start; text-indent: 0px; text-transform: none; white-space: normal; word-spacing: 0px; -webkit-text-stroke-width: 0px; text-decoration: none;" class=""><span style="caret-color: rgb(0, 0, 0); font-family: Helvetica; font-size: 12px; font-style: normal; font-variant-caps: normal; font-weight: normal; letter-spacing: normal; text-align: start; text-indent: 0px; text-transform: none; white-space: normal; word-spacing: 0px; -webkit-text-stroke-width: 0px; text-decoration: none; float: none; display: inline !important;" class="">---</span><br style="caret-color: rgb(0, 0, 0); font-family: Helvetica; font-size: 12px; font-style: normal; font-variant-caps: normal; font-weight: normal; letter-spacing: normal; text-align: start; text-indent: 0px; text-transform: none; white-space: normal; word-spacing: 0px; -webkit-text-stroke-width: 0px; text-decoration: none;" class=""><span style="caret-color: rgb(0, 0, 0); font-family: Helvetica; font-size: 12px; font-style: normal; font-variant-caps: normal; font-weight: normal; letter-spacing: normal; text-align: start; text-indent: 0px; text-transform: none; white-space: normal; word-spacing: 0px; -webkit-text-stroke-width: 0px; text-decoration: none; float: none; display: inline !important;" class="">Fred J. Tydeman Tydeman Consulting</span><br style="caret-color: rgb(0, 0, 0); font-family: Helvetica; font-size: 12px; font-style: normal; font-variant-caps: normal; font-weight: normal; letter-spacing: normal; text-align: start; text-indent: 0px; text-transform: none; white-space: normal; word-spacing: 0px; -webkit-text-stroke-width: 0px; text-decoration: none;" class=""><a href="mailto:tydeman@tybor.com" style="font-family: Helvetica; font-size: 12px; font-style: normal; font-variant-caps: normal; font-weight: normal; letter-spacing: normal; orphans: auto; text-align: start; text-indent: 0px; text-transform: none; white-space: normal; widows: auto; word-spacing: 0px; -webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px;" class="">tydeman@tybor.com</a><span style="caret-color: rgb(0, 0, 0); font-family: Helvetica; font-size: 12px; font-style: normal; font-variant-caps: normal; font-weight: normal; letter-spacing: normal; text-align: start; text-indent: 0px; text-transform: none; white-space: normal; word-spacing: 0px; -webkit-text-stroke-width: 0px; text-decoration: none; float: none; display: inline !important;" class=""><span class="Apple-converted-space"> </span> Testing, numerics, programming</span><br style="caret-color: rgb(0, 0, 0); font-family: Helvetica; font-size: 12px; font-style: normal; font-variant-caps: normal; font-weight: normal; letter-spacing: normal; text-align: start; text-indent: 0px; text-transform: none; white-space: normal; word-spacing: 0px; -webkit-text-stroke-width: 0px; text-decoration: none;" class=""><span style="caret-color: rgb(0, 0, 0); font-family: Helvetica; font-size: 12px; font-style: normal; font-variant-caps: normal; font-weight: normal; letter-spacing: normal; text-align: start; text-indent: 0px; text-transform: none; white-space: normal; word-spacing: 0px; -webkit-text-stroke-width: 0px; text-decoration: none; float: none; display: inline !important;" class="">+1 (702) 608-6093 Vice-chair of PL22.11 (ANSI "C")</span><br style="caret-color: rgb(0, 0, 0); font-family: Helvetica; font-size: 12px; font-style: normal; font-variant-caps: normal; font-weight: normal; letter-spacing: normal; text-align: start; text-indent: 0px; text-transform: none; white-space: normal; word-spacing: 0px; -webkit-text-stroke-width: 0px; text-decoration: none;" class=""><span style="caret-color: rgb(0, 0, 0); font-family: Helvetica; font-size: 12px; font-style: normal; font-variant-caps: normal; font-weight: normal; letter-spacing: normal; text-align: start; text-indent: 0px; text-transform: none; white-space: normal; word-spacing: 0px; -webkit-text-stroke-width: 0px; text-decoration: none; float: none; display: inline !important;" class="">Sample C99+FPCE tests:<span class="Apple-converted-space"> </span></span><a href="http://www.tybor.com/" style="font-family: Helvetica; font-size: 12px; font-style: normal; font-variant-caps: normal; font-weight: normal; letter-spacing: normal; orphans: auto; text-align: start; text-indent: 0px; text-transform: none; white-space: normal; widows: auto; word-spacing: 0px; -webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px;" class="">http://www.tybor.com</a><br style="caret-color: rgb(0, 0, 0); font-family: Helvetica; font-size: 12px; font-style: normal; font-variant-caps: normal; font-weight: normal; letter-spacing: normal; text-align: start; text-indent: 0px; text-transform: none; white-space: normal; word-spacing: 0px; -webkit-text-stroke-width: 0px; text-decoration: none;" class=""><span style="caret-color: rgb(0, 0, 0); font-family: Helvetica; font-size: 12px; font-style: normal; font-variant-caps: normal; font-weight: normal; letter-spacing: normal; text-align: start; text-indent: 0px; text-transform: none; white-space: normal; word-spacing: 0px; -webkit-text-stroke-width: 0px; text-decoration: none; float: none; display: inline !important;" class="">Savers sleep well, investors eat well, spenders work forever.</span><br style="caret-color: rgb(0, 0, 0); font-family: Helvetica; font-size: 12px; font-style: normal; font-variant-caps: normal; font-weight: normal; letter-spacing: normal; text-align: start; text-indent: 0px; text-transform: none; white-space: normal; word-spacing: 0px; -webkit-text-stroke-width: 0px; text-decoration: none;" class=""><br style="caret-color: rgb(0, 0, 0); font-family: Helvetica; font-size: 12px; font-style: normal; font-variant-caps: normal; font-weight: normal; letter-spacing: normal; text-align: start; text-indent: 0px; text-transform: none; white-space: normal; word-spacing: 0px; -webkit-text-stroke-width: 0px; text-decoration: none;" class=""><p style="caret-color: rgb(0, 0, 0); font-family: Helvetica; font-size: 12px; font-style: normal; font-variant-caps: normal; font-weight: normal; letter-spacing: normal; text-align: start; text-indent: 0px; text-transform: none; white-space: normal; word-spacing: 0px; -webkit-text-stroke-width: 0px; text-decoration: none;" class=""><br class=""><b class="">Submitter:</b>Fred Tydeman<br class=""><b class="">Submission Date:</b><span class="Apple-converted-space"> </span>2019-10-??<br class=""><b class="">Document:</b><span class="Apple-converted-space"> </span>WG14 Nxxxx<br class=""><b class="">Reference Documents:</b></p><p style="caret-color: rgb(0, 0, 0); font-family: Helvetica; font-size: 12px; font-style: normal; font-variant-caps: normal; font-weight: normal; letter-spacing: normal; text-align: start; text-indent: 0px; text-transform: none; white-space: normal; word-spacing: 0px; -webkit-text-stroke-width: 0px; text-decoration: none;" class="">Summary</p><p style="caret-color: rgb(0, 0, 0); font-family: Helvetica; font-size: 12px; font-style: normal; font-variant-caps: normal; font-weight: normal; letter-spacing: normal; text-align: start; text-indent: 0px; text-transform: none; white-space: normal; word-spacing: 0px; -webkit-text-stroke-width: 0px; text-decoration: none;" class="">The standard has a few incorrect, missing, or ambiguous statements about range errors in some of the math functions.</p><p style="caret-color: rgb(0, 0, 0); font-family: Helvetica; font-size: 12px; font-style: normal; font-variant-caps: normal; font-weight: normal; letter-spacing: normal; text-align: start; text-indent: 0px; text-transform: none; white-space: normal; word-spacing: 0px; -webkit-text-stroke-width: 0px; text-decoration: none;" class="">Does "too large" mean close to +infinity or +/-infinity?</p><p style="caret-color: rgb(0, 0, 0); font-family: Helvetica; font-size: 12px; font-style: normal; font-variant-caps: normal; font-weight: normal; letter-spacing: normal; text-align: start; text-indent: 0px; text-transform: none; white-space: normal; word-spacing: 0px; -webkit-text-stroke-width: 0px; text-decoration: none;" class="">Does "too small" mean close to -infinity or +/-zero?</p><p style="caret-color: rgb(0, 0, 0); font-family: Helvetica; font-size: 12px; font-style: normal; font-variant-caps: normal; font-weight: normal; letter-spacing: normal; text-align: start; text-indent: 0px; text-transform: none; white-space: normal; word-spacing: 0px; -webkit-text-stroke-width: 0px; text-decoration: none;" class="">Some implementations incorrectly treat f(infinity) as a range error based upon the text: A range error occurs if [the magnitude of] x is too large. While this is covered by 7.12.1#5 about overflows, the CFP group believes it is better to be explicit in each math function where it matters.</p><p style="caret-color: rgb(0, 0, 0); font-family: Helvetica; font-size: 12px; font-style: normal; font-variant-caps: normal; font-weight: normal; letter-spacing: normal; text-align: start; text-indent: 0px; text-transform: none; white-space: normal; word-spacing: 0px; -webkit-text-stroke-width: 0px; text-decoration: none;" class="">Possible Technical Corrigendum:</p><ul style="caret-color: rgb(0, 0, 0); font-family: Helvetica; font-size: 12px; font-style: normal; font-variant-caps: normal; font-weight: normal; letter-spacing: normal; text-align: start; text-indent: 0px; text-transform: none; white-space: normal; word-spacing: 0px; -webkit-text-stroke-width: 0px; text-decoration: none;" class=""><li class="">The asin functions<p class="">Add to 7.12.4.2#2</p><blockquote style="display: block; font-family: "Times New Roman"; background-color: lightgray; margin: 1em 40px;" class=""><ins style="font-weight: bold; background-color: rgb(160, 255, 160);" class="">A range error occurs if nonzero x is too close to zero.</ins></blockquote></li><li class="">The atan functions<p class="">Add to 7.12.4.3#2</p><blockquote style="display: block; font-family: "Times New Roman"; background-color: lightgray; margin: 1em 40px;" class=""><ins style="font-weight: bold; background-color: rgb(160, 255, 160);" class="">A range error occurs if nonzero x is too close to zero.</ins></blockquote></li><li class="">The atan2 functions<p class="">Add to 7.12.4.4#2</p><blockquote style="display: block; font-family: "Times New Roman"; background-color: lightgray; margin: 1em 40px;" class=""><ins style="font-weight: bold; background-color: rgb(160, 255, 160);" class="">A range error occurs if x is positive and nonzero y/x is too close to zero.</ins></blockquote></li><li class="">The sin functions<p class="">Add to 7.12.4.6#2</p><blockquote style="display: block; font-family: "Times New Roman"; background-color: lightgray; margin: 1em 40px;" class=""><ins style="font-weight: bold; background-color: rgb(160, 255, 160);" class="">A range error occurs if nonzero x is too close to zero.</ins></blockquote></li><li class="">The tan functions<p class="">Add to 7.12.4.7#2</p><blockquote style="display: block; font-family: "Times New Roman"; background-color: lightgray; margin: 1em 40px;" class=""><ins style="font-weight: bold; background-color: rgb(160, 255, 160);" class="">A range error occurs if nonzero x is too close to zero.</ins></blockquote></li><li class="">The asinpi functions<p class="">Change 7.12.4.9#2</p><blockquote style="display: block; font-family: "Times New Roman"; background-color: lightgray; margin: 1em 40px;" class="">A range error occurs if the magnitude of nonzero x is too small.</blockquote>to:<blockquote style="display: block; font-family: "Times New Roman"; background-color: lightgray; margin: 1em 40px;" class="">A range error occurs if<span class="Apple-converted-space"> </span><del style="font-weight: bold; background-color: rgb(255, 160, 160);" class="">the magnitude</del><span class="Apple-converted-space"> </span>nonzero x is too<span class="Apple-converted-space"> </span><del style="font-weight: bold; background-color: rgb(255, 160, 160);" class="">small</del><ins style="font-weight: bold; background-color: rgb(160, 255, 160);" class="">close to zero</ins>.</blockquote></li><li class="">The atanpi functions<p class="">Change 7.12.4.10#2</p><blockquote style="display: block; font-family: "Times New Roman"; background-color: lightgray; margin: 1em 40px;" class="">A range error occurs if the magnitude of nonzero x is too small.</blockquote>to:<blockquote style="display: block; font-family: "Times New Roman"; background-color: lightgray; margin: 1em 40px;" class="">A range error occurs if<span class="Apple-converted-space"> </span><del style="font-weight: bold; background-color: rgb(255, 160, 160);" class="">the magnitude</del><span class="Apple-converted-space"> </span>nonzero x is too<span class="Apple-converted-space"> </span><del style="font-weight: bold; background-color: rgb(255, 160, 160);" class="">small</del><ins style="font-weight: bold; background-color: rgb(160, 255, 160);" class="">close to zero</ins>.</blockquote></li><li class="">The atan2pi functions<p class="">Change 7.12.4.11#2</p><blockquote style="display: block; font-family: "Times New Roman"; background-color: lightgray; margin: 1em 40px;" class="">A range error occurs if x is positive and the magnitude of nonzero y/x is too close to small.</blockquote>to:<blockquote style="display: block; font-family: "Times New Roman"; background-color: lightgray; margin: 1em 40px;" class="">A range error occurs if x is positive and<span class="Apple-converted-space"> </span><del style="font-weight: bold; background-color: rgb(255, 160, 160);" class="">the magnitude of</del>nonzero y/x is too close to<span class="Apple-converted-space"> </span><del style="font-weight: bold; background-color: rgb(255, 160, 160);" class="">small</del><ins style="font-weight: bold; background-color: rgb(160, 255, 160);" class="">zero</ins>.</blockquote></li><li class="">The sinpi functions<p class="">Add to 7.12.4.13#2</p><blockquote style="display: block; font-family: "Times New Roman"; background-color: lightgray; margin: 1em 40px;" class=""><ins style="font-weight: bold; background-color: rgb(160, 255, 160);" class="">A range error occurs if nonzero x is too close to zero.</ins></blockquote></li><li class="">The tanpi functions<p class="">Add to 7.12.4.14#2</p><blockquote style="display: block; font-family: "Times New Roman"; background-color: lightgray; margin: 1em 40px;" class=""><ins style="font-weight: bold; background-color: rgb(160, 255, 160);" class="">A range error occurs if nonzero x is too close to zero.</ins></blockquote></li><li class="">The asinh functions<p class="">Add to 7.12.5.2#2</p><blockquote style="display: block; font-family: "Times New Roman"; background-color: lightgray; margin: 1em 40px;" class=""><ins style="font-weight: bold; background-color: rgb(160, 255, 160);" class="">A range error occurs if nonzero x is too close to zero.</ins></blockquote></li><li class="">The atanh functions<p class="">Add to 7.12.5.3#2</p><blockquote style="display: block; font-family: "Times New Roman"; background-color: lightgray; margin: 1em 40px;" class=""><ins style="font-weight: bold; background-color: rgb(160, 255, 160);" class="">A range error occurs if nonzero x is too close to zero.</ins></blockquote></li><li class="">The cosh functions<p class="">Change 7.12.5.4#2 from</p><blockquote style="display: block; font-family: "Times New Roman"; background-color: lightgray; margin: 1em 40px;" class="">A range error occurs if the magnitude of x is too large.</blockquote><p class="">to</p><blockquote style="display: block; font-family: "Times New Roman"; background-color: lightgray; margin: 1em 40px;" class="">A range error occurs if<span class="Apple-converted-space"> </span><del style="font-weight: bold; background-color: rgb(255, 160, 160);" class="">the magnitude of</del><span class="Apple-converted-space"> </span><ins style="font-weight: bold; background-color: rgb(160, 255, 160);" class="">(positive or negative) finite</ins><span class="Apple-converted-space"> </span>x is too large.</blockquote></li></ul></div></blockquote>“the magnitude of” cannot be deleted here.<br class=""><blockquote type="cite" class=""><div class=""><ul style="caret-color: rgb(0, 0, 0); font-family: Helvetica; font-size: 12px; font-style: normal; font-variant-caps: normal; font-weight: normal; letter-spacing: normal; text-align: start; text-indent: 0px; text-transform: none; white-space: normal; word-spacing: 0px; -webkit-text-stroke-width: 0px; text-decoration: none;" class=""><li class="">The sinh functions<p class="">Change 7.12.5.5#2 from</p><blockquote style="display: block; font-family: "Times New Roman"; background-color: lightgray; margin: 1em 40px;" class="">A range error occurs if the magnitude of x is too large.</blockquote><p class="">to</p><blockquote style="display: block; font-family: "Times New Roman"; background-color: lightgray; margin: 1em 40px;" class="">A range error occurs if<span class="Apple-converted-space"> </span><del style="font-weight: bold; background-color: rgb(255, 160, 160);" class="">the magnitude of</del><span class="Apple-converted-space"> </span><ins style="font-weight: bold; background-color: rgb(160, 255, 160);" class="">(positive or negative) finite</ins><span class="Apple-converted-space"> </span>x is too large<span class="Apple-converted-space"> </span><ins style="font-weight: bold; background-color: rgb(160, 255, 160);" class="">or if nonzero x is too close to zero</ins>.</blockquote></li></ul></div></blockquote>“the magnitude of” cannot be deleted here.<br class=""><blockquote type="cite" class=""><ul class=""></ul></blockquote><blockquote type="cite" class=""><div class=""><ul style="caret-color: rgb(0, 0, 0); font-family: Helvetica; font-size: 12px; font-style: normal; font-variant-caps: normal; font-weight: normal; letter-spacing: normal; text-align: start; text-indent: 0px; text-transform: none; white-space: normal; word-spacing: 0px; -webkit-text-stroke-width: 0px; text-decoration: none;" class=""><li class="">The tanh functions<p class="">Add to 7.12.5.6#2</p><blockquote style="display: block; font-family: "Times New Roman"; background-color: lightgray; margin: 1em 40px;" class=""><ins style="font-weight: bold; background-color: rgb(160, 255, 160);" class="">A range error occurs if nonzero x is too close to zero.</ins></blockquote></li><li class="">The exp functions<p class="">Change 7.12.6.1#2 from</p><blockquote style="display: block; font-family: "Times New Roman"; background-color: lightgray; margin: 1em 40px;" class="">A range error occurs if the magnitude of x is too large.</blockquote><p class="">to</p><blockquote style="display: block; font-family: "Times New Roman"; background-color: lightgray; margin: 1em 40px;" class="">A range error occurs if<span class="Apple-converted-space"> </span><del style="font-weight: bold; background-color: rgb(255, 160, 160);" class="">the magnitude of</del><span class="Apple-converted-space"> </span><ins style="font-weight: bold; background-color: rgb(160, 255, 160);" class="">positive finite</ins><span class="Apple-converted-space"> </span>x is too large<span class="Apple-converted-space"> </span><ins style="font-weight: bold; background-color: rgb(160, 255, 160);" class="">or if negative finite x is too large</ins>.</blockquote></li></ul></div></blockquote><div class="">“the magnitude of” is needed in the negative cases:</div><div class=""><br class=""></div><div class=""><span class="Apple-tab-span" style="white-space:pre"> </span>• A range error occurs if positive finite x is too large or if the magnitude of negative finite x is too large.</div><blockquote type="cite" class=""><div class=""><ul style="caret-color: rgb(0, 0, 0); font-family: Helvetica; font-size: 12px; font-style: normal; font-variant-caps: normal; font-weight: normal; letter-spacing: normal; text-align: start; text-indent: 0px; text-transform: none; white-space: normal; word-spacing: 0px; -webkit-text-stroke-width: 0px; text-decoration: none;" class=""><li class="">The exp10 functions<p class="">Change 7.12.6.2#2 from</p><blockquote style="display: block; font-family: "Times New Roman"; background-color: lightgray; margin: 1em 40px;" class="">A range error occurs if the magnitude of x is too large.</blockquote><p class="">to</p><blockquote style="display: block; font-family: "Times New Roman"; background-color: lightgray; margin: 1em 40px;" class="">A range error occurs if<span class="Apple-converted-space"> </span><del style="font-weight: bold; background-color: rgb(255, 160, 160);" class="">the magnitude of</del><span class="Apple-converted-space"> </span><ins style="font-weight: bold; background-color: rgb(160, 255, 160);" class="">positive finite</ins><span class="Apple-converted-space"> </span>x is too large<span class="Apple-converted-space"> </span><ins style="font-weight: bold; background-color: rgb(160, 255, 160);" class="">or if negative finite x is too large</ins>.</blockquote></li></ul></div></blockquote><div class=""><div class="">“the magnitude of” is needed in the negative cases:</div><div class=""><br class=""></div><div class=""><span class="Apple-tab-span" style="white-space: pre;"> </span>• A range error occurs if positive finite x is too large or if the magnitude of negative finite x is too large.</div></div><blockquote type="cite" class=""><ul class=""></ul></blockquote><blockquote type="cite" class=""><div class=""><ul style="caret-color: rgb(0, 0, 0); font-family: Helvetica; font-size: 12px; font-style: normal; font-variant-caps: normal; font-weight: normal; letter-spacing: normal; text-align: start; text-indent: 0px; text-transform: none; white-space: normal; word-spacing: 0px; -webkit-text-stroke-width: 0px; text-decoration: none;" class=""><li class="">The exp10m1 functions<p class="">Change 7.12.6.3#2 from</p><blockquote style="display: block; font-family: "Times New Roman"; background-color: lightgray; margin: 1em 40px;" class="">A range error occurs if finite x is too large or if the magnitude of nonzero x is too small.</blockquote><p class="">to</p><blockquote style="display: block; font-family: "Times New Roman"; background-color: lightgray; margin: 1em 40px;" class="">A range error occurs if<span class="Apple-converted-space"> </span><ins style="font-weight: bold; background-color: rgb(160, 255, 160);" class="">positive</ins><span class="Apple-converted-space"> </span>finite x is too large or if<span class="Apple-converted-space"> </span><del style="font-weight: bold; background-color: rgb(255, 160, 160);" class="">the magnitude of</del><span class="Apple-converted-space"> </span><ins style="font-weight: bold; background-color: rgb(160, 255, 160);" class="">positive</ins><span class="Apple-converted-space"> </span>nonzero x is too<span class="Apple-converted-space"> </span><del style="font-weight: bold; background-color: rgb(255, 160, 160);" class="">small</del><ins style="font-weight: bold; background-color: rgb(160, 255, 160);" class="">close to zero or if negative nonzero x is too close to zero</ins>.</blockquote></li></ul></div></blockquote>Reconsidering, I think it’s ok to say “… if nonzero x is too close to zero.” See tgamma.<br class=""><blockquote type="cite" class=""><div class=""><ul style="caret-color: rgb(0, 0, 0); font-family: Helvetica; font-size: 12px; font-style: normal; font-variant-caps: normal; font-weight: normal; letter-spacing: normal; text-align: start; text-indent: 0px; text-transform: none; white-space: normal; word-spacing: 0px; -webkit-text-stroke-width: 0px; text-decoration: none;" class=""><li class="">The exp2 functions<p class="">Change 7.12.6.4#2 from</p><blockquote style="display: block; font-family: "Times New Roman"; background-color: lightgray; margin: 1em 40px;" class="">A range error occurs if the magnitude of x is too large.</blockquote><p class="">to</p><blockquote style="display: block; font-family: "Times New Roman"; background-color: lightgray; margin: 1em 40px;" class="">A range error occurs if<span class="Apple-converted-space"> </span><del style="font-weight: bold; background-color: rgb(255, 160, 160);" class="">the magnitude of</del><span class="Apple-converted-space"> </span><ins style="font-weight: bold; background-color: rgb(160, 255, 160);" class="">positive finite</ins><span class="Apple-converted-space"> </span>x is too large<span class="Apple-converted-space"> </span><ins style="font-weight: bold; background-color: rgb(160, 255, 160);" class="">or if negative finite x is too large</ins>.</blockquote></li></ul></div></blockquote><div class="">“the magnitude of” is needed in the negative cases:</div><div class=""><br class=""></div><div class=""><span class="Apple-tab-span" style="white-space: pre;"> </span>• A range error occurs if positive finite x is too large or if the magnitude of negative finite x is too large.</div><blockquote type="cite" class=""><div class=""><ul style="caret-color: rgb(0, 0, 0); font-family: Helvetica; font-size: 12px; font-style: normal; font-variant-caps: normal; font-weight: normal; letter-spacing: normal; text-align: start; text-indent: 0px; text-transform: none; white-space: normal; word-spacing: 0px; -webkit-text-stroke-width: 0px; text-decoration: none;" class=""><li class="">The exp2m1 functions<p class="">Change 7.12.6.5#2 from</p><blockquote style="display: block; font-family: "Times New Roman"; background-color: lightgray; margin: 1em 40px;" class="">A range error occurs if the magnitude of x is too large or if the magnitude of nonzero x is too small.</blockquote><p class="">to</p><blockquote style="display: block; font-family: "Times New Roman"; background-color: lightgray; margin: 1em 40px;" class="">A range error occurs if<span class="Apple-converted-space"> </span><del style="font-weight: bold; background-color: rgb(255, 160, 160);" class="">the magnitude of</del><span class="Apple-converted-space"> </span><ins style="font-weight: bold; background-color: rgb(160, 255, 160);" class="">positive finite</ins><span class="Apple-converted-space"> </span>x is too large or if<span class="Apple-converted-space"> </span><del style="font-weight: bold; background-color: rgb(255, 160, 160);" class="">the magnitude of</del><ins style="font-weight: bold; background-color: rgb(160, 255, 160);" class="">positive</ins><span class="Apple-converted-space"> </span>nonzero x is too<span class="Apple-converted-space"> </span><del style="font-weight: bold; background-color: rgb(255, 160, 160);" class="">small</del><ins style="font-weight: bold; background-color: rgb(160, 255, 160);" class="">close to zero or if negative nonzero x is too close to zero</ins>.</blockquote></li></ul></div></blockquote>I think it’s ok to say “… if nonzero x is too close to zero.”<br class=""><blockquote type="cite" class=""><div class=""><ul style="caret-color: rgb(0, 0, 0); font-family: Helvetica; font-size: 12px; font-style: normal; font-variant-caps: normal; font-weight: normal; letter-spacing: normal; text-align: start; text-indent: 0px; text-transform: none; white-space: normal; word-spacing: 0px; -webkit-text-stroke-width: 0px; text-decoration: none;" class=""><li class="">The expm1 functions<p class="">Change 7.12.6.6#2 from</p><blockquote style="display: block; font-family: "Times New Roman"; background-color: lightgray; margin: 1em 40px;" class="">A range error occurs if positive x is too large.252)</blockquote><p class="">to</p><blockquote style="display: block; font-family: "Times New Roman"; background-color: lightgray; margin: 1em 40px;" class="">A range error occurs if positive<span class="Apple-converted-space"> </span><ins style="font-weight: bold; background-color: rgb(160, 255, 160);" class="">finite</ins><span class="Apple-converted-space"> </span>x is too large<span class="Apple-converted-space"> </span><ins style="font-weight: bold; background-color: rgb(160, 255, 160);" class="">or if positive nonzero x is too close to zero or if negavie nonzero x is too close to zero</ins>.252)</blockquote></li></ul></div></blockquote>I think it’s ok to say “… if nonzero x is too close to zero."<br class=""><blockquote type="cite" class=""><div class=""><ul style="caret-color: rgb(0, 0, 0); font-family: Helvetica; font-size: 12px; font-style: normal; font-variant-caps: normal; font-weight: normal; letter-spacing: normal; text-align: start; text-indent: 0px; text-transform: none; white-space: normal; word-spacing: 0px; -webkit-text-stroke-width: 0px; text-decoration: none;" class=""><li class="">The ldexp functions<p class="">Change 7.12.6.9#2 from</p><blockquote style="display: block; font-family: "Times New Roman"; background-color: lightgray; margin: 1em 40px;" class="">A range error may occur.</blockquote><p class="">to</p><blockquote style="display: block; font-family: "Times New Roman"; background-color: lightgray; margin: 1em 40px;" class="">A range error<span class="Apple-converted-space"> </span><del style="font-weight: bold; background-color: rgb(255, 160, 160);" class="">may occur</del><ins style="font-weight: bold; background-color: rgb(160, 255, 160);" class="">occurs for some finite x, depending on p</ins>.</blockquote></li><li class="">The log10p1 function<p class="">Change 7.12.6.13#2</p><blockquote style="display: block; font-family: "Times New Roman"; background-color: lightgray; margin: 1em 40px;" class="">A range error occurs if the magnitude of nonzero x is too small.</blockquote>to:<blockquote style="display: block; font-family: "Times New Roman"; background-color: lightgray; margin: 1em 40px;" class="">A range error occurs if<span class="Apple-converted-space"> </span><del style="font-weight: bold; background-color: rgb(255, 160, 160);" class="">the magnitude</del><span class="Apple-converted-space"> </span>positive nonzero x is too<span class="Apple-converted-space"> </span><del style="font-weight: bold; background-color: rgb(255, 160, 160);" class="">small</del><ins style="font-weight: bold; background-color: rgb(160, 255, 160);" class="">close to zero or if negative nonzero x is too close to zero</ins>.</blockquote></li></ul></div></blockquote>I think it’s ok to say “… if nonzero x is too close to zero."<br class=""><blockquote type="cite" class=""><div class=""><ul style="caret-color: rgb(0, 0, 0); font-family: Helvetica; font-size: 12px; font-style: normal; font-variant-caps: normal; font-weight: normal; letter-spacing: normal; text-align: start; text-indent: 0px; text-transform: none; white-space: normal; word-spacing: 0px; -webkit-text-stroke-width: 0px; text-decoration: none;" class=""><li class="">The log1p and logp1 functions<p class="">Add to 7.12.6.14#2</p><blockquote style="display: block; font-family: "Times New Roman"; background-color: lightgray; margin: 1em 40px;" class=""><ins style="font-weight: bold; background-color: rgb(160, 255, 160);" class="">A range error occurs if positive nonzero x is too close to zero or if negative nonzero x is too close to zero.</ins></blockquote></li></ul></div></blockquote>I think it’s ok to say “… if nonzero x is too close to zero."<br class=""><blockquote type="cite" class=""><div class=""><ul style="caret-color: rgb(0, 0, 0); font-family: Helvetica; font-size: 12px; font-style: normal; font-variant-caps: normal; font-weight: normal; letter-spacing: normal; text-align: start; text-indent: 0px; text-transform: none; white-space: normal; word-spacing: 0px; -webkit-text-stroke-width: 0px; text-decoration: none;" class=""><li class="">The log2p1 functions<p class="">Add to 7.12.6.16#2</p><blockquote style="display: block; font-family: "Times New Roman"; background-color: lightgray; margin: 1em 40px;" class=""><ins style="font-weight: bold; background-color: rgb(160, 255, 160);" class="">A range error occurs if positive nonzero x is too close to zero or it negative nonzero x is too close to zero.</ins></blockquote></li></ul></div></blockquote>I think it’s ok to say “… if nonzero x is too close to zero."<br class=""><blockquote type="cite" class=""><div class=""><ul style="caret-color: rgb(0, 0, 0); font-family: Helvetica; font-size: 12px; font-style: normal; font-variant-caps: normal; font-weight: normal; letter-spacing: normal; text-align: start; text-indent: 0px; text-transform: none; white-space: normal; word-spacing: 0px; -webkit-text-stroke-width: 0px; text-decoration: none;" class=""><li class="">The scalbn and scalbln functions<p class="">Change 7.12.6.19#2 from</p><blockquote style="display: block; font-family: "Times New Roman"; background-color: lightgray; margin: 1em 40px;" class="">A range error may occur.</blockquote><p class="">to</p><blockquote style="display: block; font-family: "Times New Roman"; background-color: lightgray; margin: 1em 40px;" class="">A range error<span class="Apple-converted-space"> </span><del style="font-weight: bold; background-color: rgb(255, 160, 160);" class="">may occur</del><span class="Apple-converted-space"> </span><ins style="font-weight: bold; background-color: rgb(160, 255, 160);" class="">occurs for some finite x, depending on n</ins>.</blockquote></li><li class="">The compoundn functions<p class="">Change 7.12.7.2#2 from</p><blockquote style="display: block; font-family: "Times New Roman"; background-color: lightgray; margin: 1em 40px;" class="">A range error may occur if n is too large, depending on x.</blockquote><p class="">to</p><blockquote style="display: block; font-family: "Times New Roman"; background-color: lightgray; margin: 1em 40px;" class="">A range error<span class="Apple-converted-space"> </span><del style="font-weight: bold; background-color: rgb(255, 160, 160);" class="">may occur</del><ins style="font-weight: bold; background-color: rgb(160, 255, 160);" class="">occurs</ins><span class="Apple-converted-space"> </span>if<span class="Apple-converted-space"> </span><ins style="font-weight: bold; background-color: rgb(160, 255, 160);" class="">positive finite x</ins><span class="Apple-converted-space"> </span><del style="font-weight: bold; background-color: rgb(255, 160, 160);" class="">n</del><span class="Apple-converted-space"> </span>is too large<span class="Apple-converted-space"> </span><ins style="font-weight: bold; background-color: rgb(160, 255, 160);" class="">or if negative x is too near -1</ins>, depending on<span class="Apple-converted-space"> </span><ins style="font-weight: bold; background-color: rgb(160, 255, 160);" class="">n</ins><del style="font-weight: bold; background-color: rgb(255, 160, 160);" class="">x</del>.</blockquote></li><li class="">The hypot functions<p class="">Change 7.12.7.4#2 from</p><blockquote style="display: block; font-family: "Times New Roman"; background-color: lightgray; margin: 1em 40px;" class="">A range error may occur.</blockquote><p class="">to</p><blockquote style="display: block; font-family: "Times New Roman"; background-color: lightgray; margin: 1em 40px;" class="">A range error<span class="Apple-converted-space"> </span><del style="font-weight: bold; background-color: rgb(255, 160, 160);" class="">may occur</del><ins style="font-weight: bold; background-color: rgb(160, 255, 160);" class="">occurs for some finite arguments</ins>.</blockquote></li><li class="">The pow functions<p class="">Change 7.12.7.5#2 from</p><blockquote style="display: block; font-family: "Times New Roman"; background-color: lightgray; margin: 1em 40px;" class="">A range error may occur.</blockquote><p class="">to</p><blockquote style="display: block; font-family: "Times New Roman"; background-color: lightgray; margin: 1em 40px;" class="">A range error<span class="Apple-converted-space"> </span><del style="font-weight: bold; background-color: rgb(255, 160, 160);" class="">may occur</del><ins style="font-weight: bold; background-color: rgb(160, 255, 160);" class="">ocurs if (positive or negative) nonzero finite x is too large or too near zero, depending on y</ins>.</blockquote></li></ul></div></blockquote><div class=""><span class="Apple-tab-span" style="white-space:pre"> </span>• A range error occurs if the magnitude of (positive or negative) nonzero finite x is too large or too near zero, depending on y.</div><blockquote type="cite" class=""><div class=""><ul style="caret-color: rgb(0, 0, 0); font-family: Helvetica; font-size: 12px; font-style: normal; font-variant-caps: normal; font-weight: normal; letter-spacing: normal; text-align: start; text-indent: 0px; text-transform: none; white-space: normal; word-spacing: 0px; -webkit-text-stroke-width: 0px; text-decoration: none;" class=""><li class="">The pown functions<p class="">Change 7.12.7.6#2 from</p><blockquote style="display: block; font-family: "Times New Roman"; background-color: lightgray; margin: 1em 40px;" class="">A range error may occur.</blockquote><p class="">to</p><blockquote style="display: block; font-family: "Times New Roman"; background-color: lightgray; margin: 1em 40px;" class="">A range error<span class="Apple-converted-space"> </span><del style="font-weight: bold; background-color: rgb(255, 160, 160);" class="">may occur</del><ins style="font-weight: bold; background-color: rgb(160, 255, 160);" class="">occurs if (positive or negative) nonzero finite x if too large or too near zero, depending on n</ins>.</blockquote></li></ul></div></blockquote><span class="Apple-tab-span" style="white-space:pre"> </span>• A range error ocurs if the magnitude of (positive or negative) nonzero finite x is too large or too near zero, depending on n.<br class=""><br class=""><blockquote type="cite" class=""><div class=""><ul style="text-align: start; text-indent: 0px;" class=""><li class="">The powr functions<p class="">Change 7.12.7.7#2 from</p><blockquote style="display: block; font-family: "Times New Roman"; background-color: lightgray; margin: 1em 40px;" class="">A range error may occur.</blockquote><p class="">to</p><blockquote style="display: block; font-family: "Times New Roman"; background-color: lightgray; margin: 1em 40px;" class="">A range error<span class="Apple-converted-space"> </span><del style="font-weight: bold; background-color: rgb(255, 160, 160);" class="">may occur</del><ins style="font-weight: bold; background-color: rgb(160, 255, 160);" class="">occurs if positive nonzero finite x is too large or too near zero, depending on y</ins>.</blockquote></li><li class="" style="caret-color: rgb(0, 0, 0); font-family: Helvetica; font-size: 12px; font-style: normal; font-variant-caps: normal; font-weight: normal; letter-spacing: normal; text-transform: none; white-space: normal; word-spacing: 0px; text-decoration: none; -webkit-text-stroke-width: 0px;">The rootn functions<p class="">Change 7.12.7.8#2 from</p><blockquote style="display: block; font-family: "Times New Roman"; background-color: lightgray; margin: 1em 40px;" class="">A range error may occur if n is -1.</blockquote><p class="">to</p><blockquote style="display: block; font-family: "Times New Roman"; background-color: lightgray; margin: 1em 40px;" class="">A range error<span class="Apple-converted-space"> </span><del style="font-weight: bold; background-color: rgb(255, 160, 160);" class="">may occur</del><ins style="font-weight: bold; background-color: rgb(160, 255, 160);" class="">occurs if (positive or negative) nonzero finite x is too large or too near zero and</ins><span class="Apple-converted-space"> </span><del style="font-weight: bold; background-color: rgb(255, 160, 160);" class="">if</del><span class="Apple-converted-space"> </span>n is -1.</blockquote></li></ul></div></blockquote><div class=""><span class="Apple-tab-span" style="white-space:pre"> </span>• A range error occurs if the magnitude of (positive or negative) nonzero finite x is too large or too near zero and if n is -1.</div><blockquote type="cite" class=""><div class=""><ul style="text-align: start; text-indent: 0px;" class=""><li class="" style="caret-color: rgb(0, 0, 0); font-family: Helvetica; font-size: 12px; font-style: normal; font-variant-caps: normal; font-weight: normal; letter-spacing: normal; text-transform: none; white-space: normal; word-spacing: 0px; text-decoration: none; -webkit-text-stroke-width: 0px;">The erf functions<p class="">Add to 7.12.8.1#2</p><blockquote style="display: block; font-family: "Times New Roman"; background-color: lightgray; margin: 1em 40px;" class=""><ins style="font-weight: bold; background-color: rgb(160, 255, 160);" class="">A range error occurs if nonzero x is too close to zero.</ins></blockquote></li><li class="" style="caret-color: rgb(0, 0, 0); font-family: Helvetica; font-size: 12px; font-style: normal; font-variant-caps: normal; font-weight: normal; letter-spacing: normal; text-transform: none; white-space: normal; word-spacing: 0px; text-decoration: none; -webkit-text-stroke-width: 0px;">The erfc functions<p class="">Change 7.12.8.2#2</p><blockquote style="display: block; font-family: "Times New Roman"; background-color: lightgray; margin: 1em 40px;" class="">A range error occurs if positive x is too large.</blockquote><p class="">to:</p><blockquote style="display: block; font-family: "Times New Roman"; background-color: lightgray; margin: 1em 40px;" class="">A range error occurs if positive<span class="Apple-converted-space"> </span><ins style="font-weight: bold; background-color: rgb(160, 255, 160);" class="">finite</ins><span class="Apple-converted-space"> </span>x is too large.</blockquote></li><li class="" style="caret-color: rgb(0, 0, 0); font-family: Helvetica; font-size: 12px; font-style: normal; font-variant-caps: normal; font-weight: normal; letter-spacing: normal; text-transform: none; white-space: normal; word-spacing: 0px; text-decoration: none; -webkit-text-stroke-width: 0px;">The lgamma functions<p class="">Change 7.12.8.3#2</p><blockquote style="display: block; font-family: "Times New Roman"; background-color: lightgray; margin: 1em 40px;" class="">A range error occurs if positive x is too large.</blockquote><p class="">to:</p><blockquote style="display: block; font-family: "Times New Roman"; background-color: lightgray; margin: 1em 40px;" class="">A range error occurs if positive<span class="Apple-converted-space"> </span><ins style="font-weight: bold; background-color: rgb(160, 255, 160);" class="">finite</ins><span class="Apple-converted-space"> </span>x is too large.</blockquote></li><li class="" style="caret-color: rgb(0, 0, 0); font-family: Helvetica; font-size: 12px; font-style: normal; font-variant-caps: normal; font-weight: normal; letter-spacing: normal; text-transform: none; white-space: normal; word-spacing: 0px; text-decoration: none; -webkit-text-stroke-width: 0px;">The tgamma functions<p class="">Change 7.12.8.4#2</p><blockquote style="display: block; font-family: "Times New Roman"; background-color: lightgray; margin: 1em 40px;" class="">A range error occurs if the magnitude of x is too large and may occur if the magnitude of x is too small.</blockquote><p class="">to:</p><blockquote style="display: block; font-family: "Times New Roman"; background-color: lightgray; margin: 1em 40px;" class="">A range error occurs<span class="Apple-converted-space"> </span><ins style="font-weight: bold; background-color: rgb(160, 255, 160);" class="">for some negative finite x, and</ins><span class="Apple-converted-space"> </span>if<span class="Apple-converted-space"> </span><del style="font-weight: bold; background-color: rgb(255, 160, 160);" class="">the magnitude of</del><ins style="font-weight: bold; background-color: rgb(160, 255, 160);" class="">positive finite</ins><span class="Apple-converted-space"> </span>x is too large<ins style="font-weight: bold; background-color: rgb(160, 255, 160);" class="">,</ins><span class="Apple-converted-space"> </span>and<span class="Apple-converted-space"> </span><del style="font-weight: bold; background-color: rgb(255, 160, 160);" class="">may occur</del><span class="Apple-converted-space"> </span>if<span class="Apple-converted-space"> </span><del style="font-weight: bold; background-color: rgb(255, 160, 160);" class="">the magnitude of</del><span class="Apple-converted-space"> </span>nonzero x is too<span class="Apple-converted-space"> </span><del style="font-weight: bold; background-color: rgb(255, 160, 160);" class="">small</del><ins style="font-weight: bold; background-color: rgb(160, 255, 160);" class="">close to zero</ins>.</blockquote></li><li class="" style="caret-color: rgb(0, 0, 0); font-family: Helvetica; font-size: 12px; font-style: normal; font-variant-caps: normal; font-weight: normal; letter-spacing: normal; text-transform: none; white-space: normal; word-spacing: 0px; text-decoration: none; -webkit-text-stroke-width: 0px;">The fmod functions<p class="">Add to 7.12.10.1#2</p><blockquote style="display: block; font-family: "Times New Roman"; background-color: lightgray; margin: 1em 40px;" class=""><ins style="font-weight: bold; background-color: rgb(160, 255, 160);" class="">A range error occurs if x is finite, both x and y are nonzero, and either is too close to zero.</ins></blockquote></li><li class="" style="caret-color: rgb(0, 0, 0); font-family: Helvetica; font-size: 12px; font-style: normal; font-variant-caps: normal; font-weight: normal; letter-spacing: normal; text-transform: none; white-space: normal; word-spacing: 0px; text-decoration: none; -webkit-text-stroke-width: 0px;">The remainder functions<p class="">Add to 7.12.10.2#2</p><blockquote style="display: block; font-family: "Times New Roman"; background-color: lightgray; margin: 1em 40px;" class=""><ins style="font-weight: bold; background-color: rgb(160, 255, 160);" class="">A range error occurs if x is finite, both x and y are nonzero, and either is too close to zero.</ins></blockquote></li><li class="" style="caret-color: rgb(0, 0, 0); font-family: Helvetica; font-size: 12px; font-style: normal; font-variant-caps: normal; font-weight: normal; letter-spacing: normal; text-transform: none; white-space: normal; word-spacing: 0px; text-decoration: none; -webkit-text-stroke-width: 0px;">The fdim functions<p class="">Change 7.12.12.1#2</p><blockquote style="display: block; font-family: "Times New Roman"; background-color: lightgray; margin: 1em 40px;" class="">A range error may occur.</blockquote><p class="">to:</p><blockquote style="display: block; font-family: "Times New Roman"; background-color: lightgray; margin: 1em 40px;" class="">A range error<span class="Apple-converted-space"> </span><del style="font-weight: bold; background-color: rgb(255, 160, 160);" class="">may occur</del><ins style="font-weight: bold; background-color: rgb(160, 255, 160);" class="">occurs if positive x-y is too large or too small</ins>.</blockquote></li><li class="" style="caret-color: rgb(0, 0, 0); font-family: Helvetica; font-size: 12px; font-style: normal; font-variant-caps: normal; font-weight: normal; letter-spacing: normal; text-transform: none; white-space: normal; word-spacing: 0px; text-decoration: none; -webkit-text-stroke-width: 0px;">The fma functions<p class="">Change 7.12.13.1#2</p><blockquote style="display: block; font-family: "Times New Roman"; background-color: lightgray; margin: 1em 40px;" class="">A range error may occur.</blockquote><p class="">to:</p><blockquote style="display: block; font-family: "Times New Roman"; background-color: lightgray; margin: 1em 40px;" class="">A range error<span class="Apple-converted-space"> </span><del style="font-weight: bold; background-color: rgb(255, 160, 160);" class="">may occur</del><ins style="font-weight: bold; background-color: rgb(160, 255, 160);" class="">occurs for some finite arguments</ins>.</blockquote></li></ul><span style="caret-color: rgb(0, 0, 0); font-family: Helvetica; font-size: 12px; font-style: normal; font-variant-caps: normal; font-weight: normal; letter-spacing: normal; text-align: start; text-indent: 0px; text-transform: none; white-space: normal; word-spacing: 0px; -webkit-text-stroke-width: 0px; text-decoration: none; float: none; display: inline !important;" class="">_______________________________________________</span><br style="caret-color: rgb(0, 0, 0); font-family: Helvetica; font-size: 12px; font-style: normal; font-variant-caps: normal; font-weight: normal; letter-spacing: normal; text-align: start; text-indent: 0px; text-transform: none; white-space: normal; word-spacing: 0px; -webkit-text-stroke-width: 0px; text-decoration: none;" class=""><span style="caret-color: rgb(0, 0, 0); font-family: Helvetica; font-size: 12px; font-style: normal; font-variant-caps: normal; font-weight: normal; letter-spacing: normal; text-align: start; text-indent: 0px; text-transform: none; white-space: normal; word-spacing: 0px; -webkit-text-stroke-width: 0px; text-decoration: none; float: none; display: inline !important;" class="">Cfp-interest mailing list</span><br style="caret-color: rgb(0, 0, 0); font-family: Helvetica; font-size: 12px; font-style: normal; font-variant-caps: normal; font-weight: normal; letter-spacing: normal; text-align: start; text-indent: 0px; text-transform: none; white-space: normal; word-spacing: 0px; -webkit-text-stroke-width: 0px; text-decoration: none;" class=""><a href="mailto:Cfp-interest@oakapple.net" style="font-family: Helvetica; font-size: 12px; font-style: normal; font-variant-caps: normal; font-weight: normal; letter-spacing: normal; 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