.\" Copyright (c) 1985, 1991 Regents of the University of California. .\" All rights reserved. .\" .\" Redistribution and use in source and binary forms, with or without .\" modification, are permitted provided that the following conditions .\" are met: .\" 1. Redistributions of source code must retain the above copyright .\" notice, this list of conditions and the following disclaimer. .\" 2. Redistributions in binary form must reproduce the above copyright .\" notice, this list of conditions and the following disclaimer in the .\" documentation and/or other materials provided with the distribution. .\" 3. All advertising materials mentioning features or use of this software .\" must display the following acknowledgement: .\" This product includes software developed by the University of .\" California, Berkeley and its contributors. .\" 4. Neither the name of the University nor the names of its contributors .\" may be used to endorse or promote products derived from this software .\" without specific prior written permission. .\" .\" THIS SOFTWARE IS PROVIDED BY THE REGENTS AND CONTRIBUTORS ``AS IS'' AND .\" ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE .\" IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE .\" ARE DISCLAIMED. IN NO EVENT SHALL THE REGENTS OR CONTRIBUTORS BE LIABLE .\" FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL .\" DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS .\" OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) .\" HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT .\" LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY .\" OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF .\" SUCH DAMAGE. .\" .\" from: @(#) exp. 3 6.12 (Berkeley) 7/31/91 .\" %FreeBSD: src/lib/msun/man/exp. 3, v 1.9. 2.4 2001/12/17 10:08:36 ru Exp % .\" .\" $FreeBSD: doc/ja_JP.eucJP/man/man3/exp. 3, v 1.6 2002/11/03 06:38:28 horikawa Exp $ .Dd July 31, 1991 .Dt EXP 3 .Os .Sh ¸íĪ .Nm exp , .Nm expf , .Nm exp10 , .Nm exp10f , .Nm expm1 , .Nm expm1f , .Nm log , .Nm logf , .Nm log10 , .Nm log10f , .Nm log1p , .Nm log1pf , .Nm pow , .Nm powf .Nd Áö¼öÇÔ¼ö, ´ë¼ö°ü°è, ´©½Â ÇÔ¼ö .Sh ÇÁ·Î±×·¥ ¶óÀ̺귯¸® .Lb libm .Sh ¼­½Ä .In math.h .Ft double .Fn exp "double x" .Ft float .Fn expf "float x" .Ft double .Fn expm1 "double x" .Ft float .Fn expm1f "float x" .Ft double .Fn log "double x" .Ft float .Fn logf "float x" .Ft double .Fn log10 "double x" .Ft float .Fn log10f "float x" .Ft double .Fn log1p "double x" .Ft float .Fn log1pf "float x" .Ft double .Fn pow "double x" "double y" .Ft float .Fn powf "float x" "float y" .Sh ÇØ¼³ .Fn exp ÇÔ¼ö¿Í .Fn expf ÇÔ¼ö´Â, ÁöÁ¤ÀÇ Àμö .Fa x ÀÇ Áö¼öÄ¡¸¦ °è»êÇÕ´Ï´Ù. .Pp .Fn expm1 ÇÔ¼ö¿Í .Fn expm1f ÇÔ¼ö´Â, ÀÛÀº Àμö .Fa x ¿¡ ºÙ¾î Á¶Â÷ Á¤È®ÇϰÔ, exp(x)\-1 ÀÇ °ªÀ» °è»êÇÕ´Ï´Ù. .Pp .Fn log ÇÔ¼ö¿Í .Fn logf ÇÔ¼ö´Â Àμö .Fa x ÀÇ ÀÚ¿¬´ë¼öÀÇ °ªÀ» °è»êÇÕ´Ï´Ù. .Pp .Fn log10 ÇÔ¼ö¿Í .Fn log10f ÇÔ¼ö´Â, ±âÀú 10 ¿¡ ´ëÇØ Àμö .Fa x ÀÇ ´ë¼ö¸¦ °è»êÇÕ´Ï´Ù. .Pp .Fn log1p ÇÔ¼ö¿Í .Fn log1pf ÇÔ¼ö´Â, ÀÛÀº Àμö .Fa x ¿¡ ºÙ¾î Á¶Â÷ log(1+x)ÀÇ °ªÀ» Á¤È®ÇÏ°Ô °è»êÇÕ´Ï´Ù. .Pp .Fn pow ÇÔ¼ö¿Í .Fn powf ÇÔ¼ö´Â, Áö¼ö .Ar y ¿¡ ´ëÇØ¼­ .Ar x ÀÇ °ªÀ» °è»êÇÕ´Ï´Ù. .Sh ¿ÀÂ÷ (¶ó¿îµå ¿ÀÇÁµîÀ» (À§ÇØ)¶§¹®¿¡¿¡ ÀÇÇÑ´Ù) .Fn exp x , .Fn log x , .Fn expm1 x ¹× .Fn log1p x (Àº)´Â, 1 .Em ulp ³»ºÎ¿¡¼­ Á¤È®ÇÕ´Ï´Ù. .Fn log10 x (Àº)´Â 2 .Em ulps ³»ºÎ¿¡¼­ Á¤È®ÇÕ´Ï´Ù. 1 .Em ulp ÇÏ .Em ¸¶Áö¸· .Em Àå¼Ò ¿¡¼­ÀÇ 1 .Em ´ÜÀ§ ÀÔ´Ï´Ù. .Fn pow x y ÀÇ ¿ÀÂ÷´Â, Å©±â°¡ Áß°£ Á¤µµ ¶§´Â ¾à 2 .Em ulps ¹Ì¸¸ÀÔ´Ï´Ù. .Fn pow x y ÇÏÁö¸¸ ¿À¹öÇ÷οì/¾ð´õ Ç÷οì(underflow) ÇØ ±ÍÀÇÄ¡¿¡ °¡±î¿öÁö¸é(ÀÚ) Áõ´ëÇÕ´Ï´Ù. ¸¶Áö¸·¿¡´Â, °ÅÀÇ ¸ðµç ºñÆ®°¡ ºÎµ¿ ¼Ò¼öÁ¡ Áö¼ö Çʵ忡 ÀÇÇØ Â÷ÁöÇÒ ¼ö ÀÖ¾î ¾ø¾îÁý´Ï´Ù. À̰ÍÀº .Tn "VAX D" ±×·³ 8 ºñÆ®À̸ç, IEEE 754 Double ¿¡¼­´Â 11 ºñÆ®ÀÔ´Ï´Ù. ÀÌ¿Í °°ÀÌ Å« °áÇÌÀº ½ÃÇè¿¡ ÀÇÇØ ¹àÇôÁö°í ÀÖÁö ¾Ê½À´Ï´Ù. °üÂûµÈ ÃÖ¾ÇÀÇ ¿ÀÂ÷´Â, .Tn "VAX D" ¿¡ ´ëÇØ¼­´Â 20 .Em ulps ¹Ì¸¸. .Tn IEEE 754 Double ¿¡ ´ëÇØ¼­´Â 300 .Em ulps ÀÔ´Ï´Ù. .Fn pow Áß°£ Á¤µµÀÇ °ªÀÌ, .Fn pow integer integer ÇÏÁö¸¸ Á¤È®ÇÏ°Ô µÇ±â À§Çؼ­(¶§¹®¿¡) ÃæºÐÇÕ´Ï´Ù. ÃÖÈÄ´Â, .Tn VAX ¿¡ ´ëÇØ¼­´Â 2**56 º¸´Ù Ä¿Á®, .Tn IEEE 754 ¿¡ ´ëÇØ¼­´Â 2**53 º¸´Ù Ä¿Áý´Ï´Ù. .Sh ¹Ýȯ°ª ÀÌ·¯ÇÑ ÇÔ¼ö´Â, ¿ÀÂ÷°¡ ¹ß»ýÇÒ±î Àμö°¡ ¹üÀ§¿ÜÀÇ °æ¿ì¸¦ Á¦¿ÜÇØ, ÀûÀýÇÑ °è»êÀ» µ¹·ÁÁÝ´Ï´Ù. ÇÔ¼ö .Fn exp , .Fn expm1 , .Fn pow (Àº)´Â, °è»êµÈ °ªÀÌ ¿À¹öÇ÷οì ÇÏ´ÂÁö ¾î¶²Áö °ËÃâÇØ, ±Û·Î¹ú º¯¼ö .Va errno (À»)¸¦ .Er ERANGE (À¸)·Î ¼³Á¤ÇØ, .Tn VAX ¶Ç´Â .Tn Tahoe ±×¸®°í ¿¹¾à ¿ÀÆÛ·£µå fault¸¦ ¹ß»ý½Ãŵ´Ï´Ù. ÇÔ¼ö .Fn pow x y (Àº)´Â, .Fa x < 0 Àΰ¡ .Fa y ÇÏÁö¸¸ Á¤¼ö°¡ ¾Æ´Ñ°¡ üũÇÕ´Ï´Ù. ÀÌ °æ¿ì, ±Û·Î¹ú º¯¼ö .Va errno (À»)¸¦ .Er EDOM (À¸)·Î ¼³Á¤ÇØ, .Tn VAX (¿Í)°ú .Tn Tahoe ±×¸®°í, ¿¹¾à ¿ÀÆÛ·£µå fault¸¦ »ý¼ºÇÕ´Ï´Ù. .Fa x > 0 ÀÌ ¾Æ´Ñ °æ¿ìÀÇ log ¿Í .Fa x > \-1 (ÀÌ)°¡ ¾Æ´Ñ °æ¿ìÀÇ .Fn log1p (Àº)´Â, .Tn VAX (¿Í)°ú .Tn Tahoe ±×·³, .Va errno ÇÏÁö¸¸ .Er EDOM (À¸)·Î ¼³Á¤µÇ¾î ¿¹¾à ¿ÀÆÛ·£µå fault°¡ µ¹·ÁÁÖ¾îÁý´Ï´Ù. .Sh ÁÖ ÇÔ¼ö exp(x)\-1 ¿Í log(1+x)´Â, Hewlett Packard .Tn HP Ns \-71B ¹× .Tn APPLE ÀÇ Macintosh ÀÇ .Tn BASIC ±×·³ expm1 ¿Í logp1 ·Î ºÒ·Á Pascal ¿¡¼­´Â .Tn EXP1 (¿Í)°ú .Tn LN1 (À¸)·Î ºÒ·Á .Tn APPLE Macintosh ÀÇ C ¿¡¼­´Â exp1 ¿Í log1 ·Î ºÒ¸³´Ï´Ù. ÀÌ °æ¿ì, À̰͵éÀº ((1+x) **n\-1) /x Áï expm1(n\(**log1p(x)) /x ÀÇ ÀçÁ¤Àû °è»êÀÌ x °¡ ÀÛÀº °æ¿ì¿¡ Á¤È®ÇÏ°Ô µÇ´Â °ÍÀ» È®ÀÎÇϱâ À§Çؼ­ °®ÃçÁ® ÀÖ½À´Ï´Ù. À̰͵éÀº Á¤È®ÇÑ ¿ª½Ö°î¼± ÇÔ¼öµµ Á¦°øÇÕ´Ï´Ù. .Pp ÇÔ¼ö .Fn pow x 0 (Àº)´Â, ¸ðµç x ¿¡ ´ëÇØ x**0 = 1 À» µ¹·ÁÁÝ´Ï´Ù. x ¿¡´Â, x = 0, .if n \ infinity .if t \ \(if .Tn ( VAX ¿¡´Â ¾ø½À´Ï´Ù), .Em NaN .Tn ( VAX ÀÇ ¿¹¾à ¿ÀÆÛ·£µå)°¡ Æ÷ÇԵ˴ϴÙ. pow ³²À» ¼ö Àִ±îÁöÀÇ ½ÇÀåÀº, ÀÌ·¯ÇÑ ¸ðµç °æ¿ì ¶Ç´Â ¸î°³ÀÇ °æ¿ì¿¡, x**0 ¸¦ ¹ÌÁ¤µµ¸®·Î¼­ ¿Ô´ø ÀûÀÌ ÀÖ½À´Ï´Ù. À̰ÍÀÌ ¹Ýµå½Ã x**0 = 1 À» µ¹·ÁÁÖ´Â ÀÌÀ¯ÀÔ´Ï´Ù. .Bl -enum -width indent .It x**0 ¸¦ °è»êÇϱâ Àü¿¡ x °¡ 0 (¶Ç´Â ¹«ÇÑ ¶Ç´Â \*(Na)ÀÎÁö¸¦ ½ÃÇèÇÏ´Â ÇÁ·Î±×·¥Àº 0**0 = 1 ÀÎÁö ¾î¶²Áö¸¦ ¹è·ÁÇÒ ¼ö ¾ø½À´Ï´Ù. 0**0 ÇÏÁö¸¸ ¹«È¿¶ó´Â Á¡¿¡ Á¿ìµÇ´Â ÇÁ·Î±×·¥Àº, ½ÄÀÇ ÀÇ¹Ì¿Í ¹«È¿ÀÎ °æ¿ì, ½ÄÀÇ °á°ú°¡ ÄÄÇ»ÅÍ ½Ã½ºÅÛ ¸¶´Ù º¯È­Çϱâ À§ÇØ(¶§¹®¿¡) Àǽɽº·¯¿î °ÍÀÔ´Ï´Ù. .It »ê¼ú ÅØ½ºÆ® (¿¹¸¦ µé¾î, Sigler ÀÇ °Í)¿¡´Â, x = 0 À» Æ÷ÇÔÇØ ¸ðµç x ¿¡ ´ëÇØ x**0 = 1 À» Á¤ÀÇÇÏ´Â °ÍÀÌ ÀÖ½À´Ï´Ù. À̰ÍÀº, a[0] À», a[0]\(**0**0 ¸¦ ¹«È¿·Î¼­ °ÅÀýÇÏ´Â °ÍÀÌ ¾Æ´Ï¶ó, ´ÙÇ×½Ä .Bd -literal -offset -indent p(x) = a[0]\(**x**0 + a[1]\(**x**1 + a[2]\(**x**2 +...+ a[n]\(**x**n .Ed .Pp ÀÇ °ªÀ» a[0] À¸·Î¼­ ¹Þ¾ÆµéÀÌ´Â °üÇà°ú ȣȯ¼ºÀÌ ÀÖ½À´Ï´Ù. .It ¾î³Î¸®½ºÆ®´Â, x ¿Í y °¡ µ¶¸³¿¡ 0 ¿¡ °¡±î¿öÁú ¶§, x**y °¡ ƯÁ¤ÀÇ °ª¿¡ °¡±î¿öÁöµç°¡ ¶Ç´Â ¾Æ¹«°Íµµ °¡±î¿öÁöÁö ¾Ê´Â °Íó·³ °ü·ÃµÇÁö ¾Ê°í, 0**0 = 1 À» ¹Þ¾ÆµéÀÔ´Ï´Ù. 0**0 = 1 À» ¹Þ¾ÆµéÀÌ´Â ÀÌÀ¯´Â ´ÙÀ½°ú °°½À´Ï´Ù. .Bd -filled -offset indent x(z)¿Í y(z)°¡ z = 0 ºÎ±ÙÀÇ z ·Î ºÐ¼®ÀûÀÎ ÇÔ¼öÀ̸ç (´©½ÂÀÇ ¿¬¼ÓÀ¸·Î È®Àå °¡´É), °Ô´Ù°¡ x(0) = y(0) = 0 ÀÇ °æ¿ì, z \(-> 0 ¶§ x(z) **y(z) \(-> 1 ÀÔ´Ï´Ù. .Ed .It 0**0 = 1 ÀÇ °æ¿ì, .if n \ infinity**0 = 1/0**0 = 1 .if t \ \(if**0 = 1/0**0 = 1 ÀÔ´Ï´Ù. ±× ´ÙÀ½¿¡,\*(Na**0 = 1 À̱⵵ ÇÕ´Ï´Ù. ¸ðµç À¯ÇÑ ¹× ¹«ÇÑÇÑ x ¿¡ ´ëÇØ, Áï x ¿¡ µ¶¸³¿¡ x**0 = 1 À̱⠶§¹®ÀÔ´Ï´Ù. .El .Sh °ü·Ã Ç׸ñ .Xr math 3 .Sh ¿ª»ç .Fn exp , .Fn log ¹× .Fn pow ÇÔ¼ö´Â .At v6 ±×¸®°í µîÀåÇß½À´Ï´Ù. .Fn log10 ÇÔ¼ö´Â .At v7 ±×¸®°í ³ªÅ¸³µ½À´Ï´Ù. .Fn log1p ÇÔ¼ö¿Í .Fn expm1 ÇÔ¼ö´Â .Bx 4.3 ±×¸®°í µîÀåÇß½À´Ï´Ù.