diff --git a/books/bookvolbib.pamphlet b/books/bookvolbib.pamphlet index 2b24ac7..602047a 100644 --- a/books/bookvolbib.pamphlet +++ b/books/bookvolbib.pamphlet @@ -3201,5 +3201,29 @@ Greve, David A.; McClurg, Jedidiah R.\\ in Lecture Notes in Computer Science, Springer ISBN 978-3-540-85520-0 (2006) pp102-113 +\subsection{Numeric Tests} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + +\bibitem[Lef\'evre 06]{Lef06} Lef\'evre, Vincent; Stehl\'e, Damien; +Zimmermann, Paul\\ +``Worst Cases for the Exponential Function +in the IEEE-754r decimal64 Format''\\ +in Lecture Notes in Computer Science, Springer ISBN 978-3-540-85520-0 +(2006) pp114-125 + +\begin{adjustwidth}{2.5em}{0pt} +We searched for the worst cases for correct rounding of the +exponential function in the IEEE 754r decimal64 format, and computed +all the bad cases whose distance from a breakpoint (for all rounding +modes) is less than $10^{-15}$ ulp, and we give the worst ones. In +particular, the worst case for +$\vert{}x\vert{} \ge 3 x 10^{-11}$ is +\[ +exp(9.407822313572878x10^{-2} = 1.09864568206633850000000000000000278\ldots +\] +This work can be extended to other elementary functions in the decimal64 +format and allows the design of reasonably fast routines that will +evaluate these functions with correct rounding, at least in some situations. +\end{adjustwidth} + \end{thebibliography} \end{document} diff --git a/changelog b/changelog index 263703c..0ddadf3 100644 --- a/changelog +++ b/changelog @@ -1,3 +1,7 @@ +20140618 tpd src/axiom-website/patches.html 20140618.01.tpd.patch +20140618 tpd src/input/Makefile add src/input/exptest.input +20140618 tpd src/input/exptest.input test worst case exponential values +20140618 tpd books/bookvolbib add Lef06 20140617 tpd src/axiom-website/patches.html 20140617.01.tpd.patch 20140617 tpd books/bookvol10.1 add interval arithmetic 20140617 tpd books/bookvolbib add interval arithmetic diff --git a/src/axiom-website/patches.html b/src/axiom-website/patches.html index b76c633..0cf65d4 100644 --- a/src/axiom-website/patches.html +++ b/src/axiom-website/patches.html @@ -4476,6 +4476,8 @@ buglist wish 1011: sum(1/(k+a), k=1..n) by Gosper's method src/input/polygamma.input add a polygamma example 20140617.01.tpd.patch books/bookvol10.1, bookvolbib add interval arithmetic +20140618.01.tpd.patch +src/input/exptest.input test worst case exponential values diff --git a/src/input/Makefile.pamphlet b/src/input/Makefile.pamphlet index 394ffd6..a26026c 100644 --- a/src/input/Makefile.pamphlet +++ b/src/input/Makefile.pamphlet @@ -328,7 +328,7 @@ REGRESSTESTS= ackermann.regress \ evalex.regress eval.regress exdiff.regress \ exampleagcode.regress \ exint.regress exit.regress exlap.regress exlimit.regress \ - exp.regress \ + exp.regress exptest.regress \ expexpan.regress explim.regress expr1.regress exprode.regress \ expr.regress exseries.regress exsum.regress exprpoly.regress \ farray.regress ffdemo.regress fferr.regress ffieldbug.regress \ @@ -740,8 +740,8 @@ FILES= ${OUT}/ackermann.input \ ${OUT}/elt.input ${OUT}/en.input \ ${OUT}/eq.input ${OUT}/eqtbl.input ${OUT}/equation.input \ ${OUT}/errortrap.input \ - ${OUT}/eval.input ${OUT}/exit.input ${OUT}/exp.input \ - ${OUT}/expexpan.input \ + ${OUT}/eval.input ${OUT}/exit.input ${OUT}/exptest.input \ + ${OUT}/exp.input ${OUT}/expexpan.input \ ${OUT}/explim.input ${OUT}/explot2d.input ${OUT}/explot3d.input \ ${OUT}/expr1.input ${OUT}/exprpoly.input \ ${OUT}/farray.input ${OUT}/ffdemo.input \ @@ -1125,7 +1125,7 @@ DOCFILES= \ ${DOC}/exampleagcode.input.dvi \ ${DOC}/exint.input.dvi ${DOC}/exit.input.dvi \ ${DOC}/exlap.input.dvi ${DOC}/exlimit.input.dvi \ - ${DOC}/exp.input.dvi \ + ${DOC}/exp.input.dvi ${DOC}/exptest.input.dvi \ ${DOC}/expexpan.input.dvi ${DOC}/explim.input.dvi \ ${DOC}/explot2d.input.dvi ${DOC}/explot3d.input.dvi \ ${DOC}/expr1.input.dvi ${DOC}/expr.input.dvi \ diff --git a/src/input/exptest.input.pamphlet b/src/input/exptest.input.pamphlet new file mode 100644 index 0000000..7273a18 --- /dev/null +++ b/src/input/exptest.input.pamphlet @@ -0,0 +1,2387 @@ +\documentclass{article} +\usepackage{axiom} +\setlength{\textwidth}{400pt} +\begin{document} +\title{\$SPAD/src/input exptest.input} +\author{Timothy Daly} +\maketitle +\begin{abstract} +Worst Cases for the Exponential Function in the IEEE-754r decimal64 Format + +by Vincent Lef\'evre, Damien Stehl\'e, Paul Zimmermann + +We searched for the worst cases for correct rounding of the +exponential function in the IEEE 754r decimal64 format, and computed +all the bad cases whose distance from a breakpoint (for all rounding +modes) is less than $10^{-15}$ ulp, and we give the worst ones. In +particular, the worst case for +$\vert{}x\vert{} \ge 3 x 10^{-11}$ is +\[ +exp(9.407822313572878x10^{-2} = 1.09864568206633850000000000000000278\ldots +\] +This work can be extended to other elementary functions in the decimal64 +format and allows the design of reasonably fast routines that will +evaluate these functions with correct rounding, at least in some situations. + +\end{abstract} +\eject +\tableofcontents +\eject +\begin{chunk}{*} +)set break resume +)sys rm -f exptest.output +)spool exptest.output +)set message test on +)set message auto off +)clear all + +--S 1 of 288 +digits(50) +--R +--R +--R (1) 20 +--R Type: PositiveInteger +--E 1 + +--S 2 of 288 +t1:=exp(9.407822313572878E-2) +--R +--R +--R (2) 1.0986456820 6633850000 0000000000 0027805367 999050547 +--R Type: Float +--E 2 + +--S 3 of 288 +t1-1.09864568206633850000000000000000278 +--R +--R +--R (3) 0.5367999050 547 E -36 +--R Type: Float +--E 3 + +--S 4 of 288 +t2:=exp(9.999999999999995E-16) +--R +--R +--R (4) 1.0000000000 0000099999 9999999999 9999999999 999996667 +--R Type: Float +--E 4 + +--S 5 of 288 +t2-1.000000000000000999999999999999999999999999999666 +--R +--R +--R (5) 0.7 E -48 +--R Type: Float +--E 5 + +--S 6 of 288 +t3:=exp(0.5091077534282133) +--R +--R +--R (6) 1.6638060072 6150950000 0000000000 0492443217 490289874 +--R Type: Float +--E 6 + +--S 7 of 288 +t3-1.663806007261509500000000000000049 +--R +--R +--R (7) 0.2443217490 289874 E -33 +--R Type: Float +--E 7 + +--S 8 of 288 +t4:=exp(0.7906867968553504) +--R +--R +--R (8) 2.2049102317 7150949999 9999999999 9163630300 32777977 +--R Type: Float +--E 8 + +--S 9 of 288 +t4-2.204910231771509499999999999999916 +--R +--R +--R (9) 0.3630300327 77977 E -33 +--R Type: Float +--E 9 + +--S 10 of 288 +t5:=exp(0.001548443067391468) +--R +--R +--R (10) 1.0015496425 2437499999 9999999999 9265764835 714768111 +--R Type: Float +--E 10 + +--S 11 of 288 +t5-1.001549642524374999999999999999926 +--R +--R +--R (11) 0.5764835714 768111 E -33 +--R Type: Float +--E 11 + +--S 12 of 288 +t6:=exp(0.2953379504777270) +--R +--R +--R (12) 1.3435803455 8906700000 0000000000 0864880085 307257641 +--R Type: Float +--E 12 + +--S 13 of 288 +t6-1.343580345589067000000000000000086 +--R +--R +--R (13) 0.4880085307 257641 E -33 +--R Type: Float +--E 13 + +--S 14 of 288 +t7:=log(10.E385-10.E369/2) +--R +--R +--R (14) 888.7978458957 0163398094 4701508164 5828840251 7461071 +--R Type: Float +--E 14 + +--S 15 of 288 +t7-886.4952608027075882469 +--R +--R +--R (15) 2.3025850929 9404573404 4701508164 5828840251 746107 +--R Type: Float +--E 15 + +--S 16 of 288 +t8:=log(10.E-398/2) +--R +--R +--R (16) - 914.8194290991 9608186455 9839631150 7669857127 9111998 +--R Type: Float +--E 16 + +--S 17 of 288 +t8-(-917.12201419219012) +--R +--R +--R (17) 2.3025850929 9403813544 0160368849 2330142872 0888 +--R Type: Float +--E 17 + +--S 18 of 288 +t9:=log(1.0-10.0E-16/2) +--R +--R +--R (18) - 0.5000000000 0000012500 0000000000 0416679603 9339282205 E -15 +--R Type: Float +--E 18 + +--S 19 of 288 +t9-(-5.000000000000000127E-17) +--R +--R +--R (19) - 0.4500000000 0000012373 0000000000 0416679603 9339282204 E -15 +--R Type: Float +--E 19 + +--S 20 of 288 +t10:=log(1.0+10.E-15/2) +--R +--R +--R (20) 0.4999999999 9999875000 0000000004 1666668912 8749314454 E -14 +--R Type: Float +--E 20 + +--S 21 of 288 +t10-4.999999999999999750E-16 +--R +--R +--R (21) 0.4499999999 9999875250 0000000004 1666668912 8749314454 E -14 +--R Type: Float +--E 21 + +\end{chunk} + +Table 2 has all worst cases of the decimal64 exponential function for +$x \ge 10^{-9}$, whose distance from a breakpoint is less than +$5 x 10^{-17}$ ulp. The notation $d^k$ means that the digit $d$ +is repeated $k$ times. + +\begin{chunk}{*} + +--S 22 of 288 +table2ax:=6.581539478341669E-9 +--R +--R +--R (22) 0.6581539478 341669 E -8 +--R Type: Float +--E 22 + +--S 23 of 288 +t2a:=exp(table2ax) +--R +--R +--R (23) 1.0000000065 8153950000 0000000000 0177317374 05802621 +--R Type: Float +--E 23 + +--S 24 of 288 +table2ay:=1.0000000065815395000000000000000177 +--R +--R +--R (24) 1.0000000065 8153950000 0000000000 0177 +--R Type: Float +--E 24 + +--S 25 of 288 +t2a-table2ay +--R +--R +--R (25) 0.3173740580 2621 E -34 +--R Type: Float +--E 25 + +--S 26 of 288 +table2bx:=2.662858264545929E-8 +--R +--R +--R (26) 0.2662858264 545929 E -7 +--R Type: Float +--E 26 + +--S 27 of 288 +t2b:=exp(table2bx) +--R +--R +--R (27) 1.0000000266 2858300000 0000000000 0318906153 379112945 +--R Type: Float +--E 27 + +--S 28 of 288 +table2by:=1.0000000266285830000000000000000318 +--R +--R +--R (28) 1.0000000266 2858300000 0000000000 0318 +--R Type: Float +--E 28 + +--S 29 of 288 +t2b-table2by +--R +--R +--R (29) 0.9061533791 12945 E -34 +--R Type: Float +--E 29 + +--S 30 of 288 +table2cx:=3.639588333766983E-8 +--R +--R +--R (30) 0.3639588333 766983 E -7 +--R Type: Float +--E 30 + +--S 31 of 288 +t2c:=exp(table2cx) +--R +--R +--R (31) 1.0000000363 9588400000 0000000000 0240621754 275940933 +--R Type: Float +--E 31 + +--S 32 of 288 +table2cy:=1.0000000363958840000000000000000240 +--R +--R +--R (32) 1.0000000363 9588400000 0000000000 024 +--R Type: Float +--E 32 + +--S 33 of 288 +t2c-table2cy +--R +--R +--R (33) 0.6217542759 40933 E -34 +--R Type: Float +--E 33 + +--S 34 of 288 +table2dx:=6.036998017773271E-8 +--R +--R +--R (34) 0.6036998017 773271 E -7 +--R Type: Float +--E 34 + +--S 35 of 288 +t2d:=exp(table2dx) +--R +--R +--R (35) 1.0000000603 6998200000 0000000000 0379694922 010746078 +--R Type: Float +--E 35 + +--S 36 of 288 +table2dy:=1.0000000603699820000000000000000379 +--R +--R +--R (36) 1.0000000603 6998200000 0000000000 0379 +--R Type: Float +--E 36 + +--S 37 of 288 +t2d-table2dy +--R +--R +--R (37) 0.6949220107 46078 E -34 +--R Type: Float +--E 37 + +--S 38 of 288 +table2ex:=6.638670361402304E-7 +--R +--R +--R (38) 0.6638670361 402304 E -6 +--R Type: Float +--E 38 + +--S 39 of 288 +t2e:=exp(table2ex) +--R +--R +--R (39) 1.0000006638 6725649999 9999999999 9569713175 574234992 +--R Type: Float +--E 39 + +--S 40 of 288 +table2ey:=1.0000006638672564999999999999999569 +--R +--R +--R (40) 1.0000006638 6725649999 9999999999 9569 +--R Type: Float +--E 40 + +--S 41 of 288 +t2e-table2ey +--R +--R +--R (41) 0.7131755742 34992 E -34 +--R Type: Float +--E 41 + +--S 42 of 288 +table2fx:=9.366572213364879E-7 +--R +--R +--R (42) 0.9366572213 364879 E -6 +--R Type: Float +--E 42 + +--S 43 of 288 +t2f:=exp(table2fx) +--R +--R +--R (43) 1.0000009366 5765999999 9999999999 9883070511 809163402 +--R Type: Float +--E 43 + +--S 44 of 288 +table2fy:=1.0000009366576599999999999999999883 +--R +--R +--R (44) 1.0000009366 5765999999 9999999999 9883 +--R Type: Float +--E 44 + +--S 45 of 288 +t2f-table2fy +--R +--R +--R (45) 0.7051180916 34015 E -35 +--R Type: Float +--E 45 + +--S 46 of 288 +table2gx:=7.970613003079781E-6 +--R +--R +--R (46) 0.0000079706 1300307978 1 +--R Type: Float +--E 46 + +--S 47 of 288 +t2g:=exp(table2gx) +--R +--R +--R (47) 1.0000079706 4476850000 0000000000 0362815663 769936294 +--R Type: Float +--E 47 + +--S 48 of 288 +table2gy:=1.0000079706447685000000000000000362 +--R +--R +--R (48) 1.0000079706 4476850000 0000000000 0362 +--R Type: Float +--E 48 + +--S 49 of 288 +t2g-table2gy +--R +--R +--R (49) 0.8156637699 36294 E -34 +--R Type: Float +--E 49 + +--S 50 of 288 +table2hx:=3.089765552852523E-5 +--R +--R +--R (50) 0.0000308976 5552852523 +--R Type: Float +--E 50 + +--S 51 of 288 +t2h:=exp(table2hx) +--R +--R +--R (51) 1.0000308981 3286600000 0000000000 0241262089 465734127 +--R Type: Float +--E 51 + +--S 52 of 288 +table2hy:=1.0000308981328660000000000000000241 +--R +--R +--R (52) 1.0000308981 3286600000 0000000000 0241 +--R Type: Float +--E 52 + +--S 53 of 288 +t2h-table2hy +--R +--R +--R (53) 0.2620894657 34127 E -34 +--R Type: Float +--E 53 + +--S 54 of 288 +table2ix:=1.302531956641873E-4 +--R +--R +--R (54) 0.0001302531 956641873 +--R Type: Float +--E 54 + +--S 55 of 288 +t2i:=exp(table2ix) +--R +--R +--R (55) 1.0001302616 7898000000 0000000000 0079860366 294631982 +--R Type: Float +--E 55 + +--S 56 of 288 +table2iy:=1.00013026167898000000000000000000798 +--R +--R +--R (56) 1.0001302616 7898000000 0000000000 00798 +--R Type: Float +--E 56 + +--S 57 of 288 +t2i-table2iy +--R +--R +--R (57) 0.6036629463 19818 E -35 +--R Type: Float +--E 57 + +--S 58 of 288 +table2jx:=2.241856702421245E-4 +--R +--R +--R (58) 0.0002241856 702421245 +--R Type: Float +--E 58 + +--S 59 of 288 +t2j:=exp(table2jx) +--R +--R +--R (59) 1.0002242108 0172750000 0000000000 0118637733 688720889 +--R Type: Float +--E 59 + +--S 60 of 288 +table2jy:=1.0002242108017275000000000000000118 +--R +--R +--R (60) 1.0002242108 0172750000 0000000000 0118 +--R Type: Float +--E 60 + +--S 61 of 288 +t2j-table2jy +--R +--R +--R (61) 0.6377336887 20889 E -34 +--R Type: Float +--E 61 + +--S 62 of 288 +table2kx:=7.230293679121590E-4 +--R +--R +--R (62) 0.0007230293 67912159 +--R Type: Float +--E 62 + +--S 63 of 288 +t2k:=exp(table2kx) +--R +--R +--R (63) 1.0007232908 1665349999 9999999999 9912755178 441089582 +--R Type: Float +--E 63 + +--S 64 of 288 +table2ky:=1.00072329081665349999999999999999127 +--R +--R +--R (64) 1.0007232908 1665349999 9999999999 99127 +--R Type: Float +--E 64 + +--S 65 of 288 +t2k-table2ky +--R +--R +--R (65) 0.5517844108 9582 E -35 +--R Type: Float +--E 65 + +--S 66 of 288 +table2lx:=5.259640428979129E-3 +--R +--R +--R (66) 0.0052596404 28979129 +--R Type: Float +--E 66 + +--S 67 of 288 +t2l:=exp(table2lx) +--R +--R +--R (67) 1.0052734966 1990949999 9999999999 9739699064 455042098 +--R Type: Float +--E 67 + +--S 68 of 288 +table2ly:=1.0052734966199094999999999999999739 +--R +--R +--R (68) 1.0052734966 1990949999 9999999999 9739 +--R Type: Float +--E 68 + +--S 69 of 288 +t2l-table2ly +--R +--R +--R (69) 0.6990644550 42098 E -34 +--R Type: Float +--E 69 + +--S 70 of 288 +table2mx:=9.407822313572878E-2 +--R +--R +--R (70) 0.0940782231 3572878 +--R Type: Float +--E 70 + +--S 71 of 288 +t2m:=exp(table2mx) +--R +--R +--R (71) 1.0986456820 6633850000 0000000000 0027805367 999050547 +--R Type: Float +--E 71 + +--S 72 of 288 +table2my:=1.09864568206633850000000000000000278 +--R +--R +--R (72) 1.0986456820 6633850000 0000000000 00278 +--R Type: Float +--E 72 + +--S 73 of 288 +t2m-table2my +--R +--R +--R (73) 0.5367999050 547 E -36 +--R Type: Float +--E 73 + +--S 74 of 288 +table2nx:=1.267914924960933E-1 +--R +--R +--R (74) 0.1267914924 960933 +--R Type: Float +--E 74 + +--S 75 of 288 +t2n:=exp(table2nx) +--R +--R +--R (75) 1.1351802994 9284300000 0000000000 0070604048 847780277 +--R Type: Float +--E 75 + +--S 76 of 288 +table2ny:=1.13518029949284300000000000000000706 +--R +--R +--R (76) 1.1351802994 9284300000 0000000000 00706 +--R Type: Float +--E 76 + +--S 77 of 288 +t2n-table2ny +--R +--R +--R (77) 0.4048847780 277 E -36 +--R Type: Float +--E 77 + +--S 78 of 288 +table2ox:=5.091077534282133E-1 +--R +--R +--R (78) 0.5091077534 282133 +--R Type: Float +--E 78 + +--S 79 of 288 +t2o:=exp(table2ox) +--R +--R +--R (79) 1.6638060072 6150950000 0000000000 0492443217 490289874 +--R Type: Float +--E 79 + +--S 80 of 288 +table2oy:=1.6638060072615095000000000000000492 +--R +--R +--R (80) 1.6638060072 6150950000 0000000000 0492 +--R Type: Float +--E 80 + +--S 81 of 288 +t2o-table2oy +--R +--R +--R (81) 0.4432174902 89874 E -34 +--R Type: Float +--E 81 + +--S 82 of 288 +table2px:=3.359104074009002 +--R +--R +--R (82) 3.3591040740 09002 +--R Type: Float +--E 82 + +--S 83 of 288 +t2p:=exp(table2px) +--R +--R +--R (83) 28.7634094457 2687500000 0000000000 0904859838 42482194 +--R Type: Float +--E 83 + +--S 84 of 288 +table2py:=28.7634094457268750000000000000000904 +--R +--R +--R (84) 28.7634094457 2687500000 0000000000 0904 +--R Type: Float +--E 84 + +--S 85 of 288 +t2p-table2py +--R +--R +--R (85) 0.8598384248 2194 E -34 +--R Type: Float +--E 85 + +--S 86 of 288 +table2qx:=19.10511686234796 +--R +--R +--R (86) 19.1051168623 4796 +--R Type: Float +--E 86 + +--S 87 of 288 +t2q:=exp(table2qx) +--R +--R +--R (87) 198265353.8414981999 9999999999 9997354256 3518374229 1 +--R Type: Float +--E 87 + +--S 88 of 288 +table2qy:=1.9826535384149819999999999999999735E8 +--R +--R +--R (88) 198265353.8414981999 9999999999 999735 +--R Type: Float +--E 88 + +--S 89 of 288 +t2q-table2qy +--R +--R +--R (89) 0.4256351837 42291 E -26 +--R Type: Float +--E 89 + +--S 90 of 288 +table2rx:=294.9551257293143 +--R +--R +--R (90) 294.9551257293 143 +--R Type: Float +--E 90 + +--S 91 of 288 +t2r:=exp(table2rx) +--R +--R +--R (91) 0.1251363586 6597895000 0000000000 0010873376 9483401177 E 129 +--R Type: Float +--E 91 + +--S 92 of 288 +table2ry:=1.2513635866597895000000000000000108E128 +--R +--R +--R (92) 0.1251363586 6597895000 0000000000 00108 E 129 +--R Type: Float +--E 92 + +--S 93 of 288 +t2r-table2ry +--R +--R +--R (93) 0.7337694834 01177 E 94 +--R Type: Float +--E 93 + +--S 94 of 288 +table2sx:=587.9131381356093 +--R +--R +--R (94) 587.9131381356 093 +--R Type: Float +--E 94 + +--S 95 of 288 +t2s:=exp(table2sx) +--R +--R +--R (95) 0.2125356221 8255224999 9999999999 9959486081 8097015346 E 256 +--R Type: Float +--E 95 + +--S 96 of 288 +table2sy:=2.1253562218255224999999999999999594E255 +--R +--R +--R (96) 0.2125356221 8255224999 9999999999 99594 E 256 +--R Type: Float +--E 96 + +--S 97 of 288 +t2s-table2sy +--R +--R +--R (97) 0.8608180970 15346 E 221 +--R Type: Float +--E 97 + +\end{chunk} + +Table 3 is all the worst cases of the decimal64 exponential function for +$x \le -10^{-10}$, whose distance from a breakpoint is less than +$5 x 10^{-17}$ ulp. The notation $d^k$ means that the digit $d$ +is repeated $k$ times. + +\begin{chunk}{*} + +--S 98 of 288 +table3ax:=-2.090862502185853E-9 +--R +--R +--R (98) - 0.2090862502 185853 E -8 +--R Type: Float +--E 98 + +--S 99 of 288 +t3a:=exp(table3ax) +--R +--R +--R (99) 0.9999999979 0913750000 0000000000 0037126725 4778313484 +--R Type: Float +--E 99 + +--S 100 of 288 +table3ay:=0.99999999790913750000000000000000371 +--R +--R +--R (100) 0.9999999979 0913750000 0000000000 00371 +--R Type: Float +--E 100 + +--S 101 of 288 +t3a-table3ay +--R +--R +--R (101) 0.2672547783 1348 E -35 +--R Type: Float +--E 101 + +--S 102 of 288 +table3bx:=-3.803619857233762E-9 +--R +--R +--R (102) - 0.3803619857 233762 E -8 +--R Type: Float +--E 102 + +--S 103 of 288 +t3b:=exp(table3bx) +--R +--R +--R (103) 0.9999999961 9638014999 9999999999 9984113979 4786371347 +--R Type: Float +--E 103 + +--S 104 of 288 +table3by:=0.99999999619638014999999999999999841 +--R +--R +--R (104) 0.9999999961 9638014999 9999999999 99841 +--R Type: Float +--E 104 + +--S 105 of 288 +t3b-table3by +--R +--R +--R (105) 0.1397947863 7135 E -35 +--R Type: Float +--E 105 + +--S 106 of 288 +table3cx:=-7.170496225708008E-9 +--R +--R +--R (106) - 0.7170496225 708008 E -8 +--R Type: Float +--E 106 + +--S 107 of 288 +t3c:=exp(table3cx) +--R +--R +--R (107) 0.9999999928 2950380000 0000000000 0025224799 1338154446 +--R Type: Float +--E 107 + +--S 108 of 288 +table3cy:=0.99999999282950380000000000000000252 +--R +--R +--R (108) 0.9999999928 2950380000 0000000000 00252 +--R Type: Float +--E 108 + +--S 109 of 288 +t3c-table3cy +--R +--R +--R (109) 0.2479913381 5445 E -35 +--R Type: Float +--E 109 + +--S 110 of 288 +table3dx:=-9.362256793825926E-9 +--R +--R +--R (110) - 0.9362256793 825926 E -8 +--R Type: Float +--E 110 + +--S 111 of 288 +t3d:=exp(table3dx) +--R +--R +--R (111) 0.9999999906 3774324999 9999999999 9958090794 3389658595 +--R Type: Float +--E 111 + +--S 112 of 288 +table3dy:=0.99999999063774324999999999999999580 +--R +--R +--R (112) 0.9999999906 3774324999 9999999999 9958 +--R Type: Float +--E 112 + +--S 113 of 288 +t3d-table3dy +--R +--R +--R (113) 0.9079433896 58595 E -35 +--R Type: Float +--E 113 + +--S 114 of 288 +table3ex:=-4.024416580979643E-8 +--R +--R +--R (114) - 0.4024416580 979643 E -7 +--R Type: Float +--E 114 + +--S 115 of 288 +t3e:=exp(table3ex) +--R +--R +--R (115) 0.9999999597 5583500000 0000000000 0030839949 0551291379 +--R Type: Float +--E 115 + +--S 116 of 288 +table3ey:=0.99999995975583500000000000000000308 +--R +--R +--R (116) 0.9999999597 5583500000 0000000000 00308 +--R Type: Float +--E 116 + +--S 117 of 288 +t3e-table3ey +--R +--R +--R (117) 0.3994905512 91379 E -35 +--R Type: Float +--E 117 + +--S 118 of 288 +table3fx:=-6.306378165019860E-7 +--R +--R +--R (118) - 0.6306378165 01986 E -6 +--R Type: Float +--E 118 + +--S 119 of 288 +t3f:=exp(table3fx) +--R +--R +--R (119) 0.9999993693 6238235000 0000000000 0030198486 9980386153 +--R Type: Float +--E 119 + +--S 120 of 288 +table3fy:=0.99999936936238235000000000000000301 +--R +--R +--R (120) 0.9999993693 6238235000 0000000000 00301 +--R Type: Float +--E 120 + +--S 121 of 288 +t3f-table3fy +--R +--R +--R (121) 0.9848699803 86153 E -35 +--R Type: Float +--E 121 + +--S 122 of 288 +table3gx:=-7.720146779532548E-7 +--R +--R +--R (122) - 0.7720146779 532548 E -6 +--R Type: Float +--E 122 + +--S 123 of 288 +t3g:=exp(table3gx) +--R +--R +--R (123) 0.9999992279 8562004999 9999999999 9961275219 425019545 +--R Type: Float +--E 123 + +--S 124 of 288 +table3gy:=0.99999922798562004999999999999999612 +--R +--R +--R (124) 0.9999992279 8562004999 9999999999 99612 +--R Type: Float +--E 124 + +--S 125 of 288 +t3g-table3gy +--R +--R +--R (125) 0.7521942501 95449 E -35 +--R Type: Float +--E 125 + +--S 126 of 288 +table3hx:=-9.753167969712726E-7 +--R +--R +--R (126) - 0.9753167969 712726 E -6 +--R Type: Float +--E 126 + +--S 127 of 288 +t3h:=exp(table3hx) +--R +--R +--R (127) 0.9999990246 8367864999 9999999999 9991202359 4289724846 +--R Type: Float +--E 127 + +--S 128 of 288 +table3hy:=0.999999024683678649999999999999999120 +--R +--R +--R (128) 0.9999990246 8367864999 9999999999 99912 +--R Type: Float +--E 128 + +--S 129 of 288 +t3h-table3hy +--R +--R +--R (129) 0.2359428972 485 E -36 +--R Type: Float +--E 129 + +--S 130 of 288 +table3ix:=-5.911964024384330E-5 +--R +--R +--R (130) - 0.0000591196 402438433 +--R Type: Float +--E 130 + +--S 131 of 288 +t3i:=exp(table3ix) +--R +--R +--R (131) 0.9999408821 0728765000 0000000000 0038489594 1622665923 +--R Type: Float +--E 131 + +--S 132 of 288 +table3iy:=0.99994088210728765000000000000000384 +--R +--R +--R (132) 0.9999408821 0728765000 0000000000 00384 +--R Type: Float +--E 132 + +--S 133 of 288 +t3i-table3iy +--R +--R +--R (133) 0.8959416226 65923 E -35 +--R Type: Float +--E 133 + +--S 134 of 288 +table3jx:=-8.232272117182855E-5 +--R +--R +--R (134) - 0.0000823227 2117182855 +--R Type: Float +--E 134 + +--S 135 of 288 +t3j:=exp(table3jx) +--R +--R +--R (135) 0.9999176806 6725040000 0000000000 0031272715 7320394822 +--R Type: Float +--E 135 + +--S 136 of 288 +table3jy:=0.99991768066725040000000000000000312 +--R +--R +--R (136) 0.9999176806 6725040000 0000000000 00312 +--R Type: Float +--E 136 + +--S 137 of 288 +t3j-table3jy +--R +--R +--R (137) 0.7271573203 94822 E -35 +--R Type: Float +--E 137 + +--S 138 of 288 +table3kx:=-8.232461306131942E-5 +--R +--R +--R (138) - 0.0000823246 1306131942 +--R Type: Float +--E 138 + +--S 139 of 288 +t3k:=exp(table3kx) +--R +--R +--R (139) 0.9999176787 7551664999 9999999999 9955576707 8594207526 +--R Type: Float +--E 139 + +--S 140 of 288 +table3ky:=0.99991767877551664999999999999999555 +--R +--R +--R (140) 0.9999176787 7551664999 9999999999 99555 +--R Type: Float +--E 140 + +--S 141 of 288 +t3k-table3ky +--R +--R +--R (141) 0.7670785942 07526 E -35 +--R Type: Float +--E 141 + +--S 142 of 288 +table3lx:=-8.496743395712491E-2 +--R +--R +--R (142) - 0.0849674339 5712491 +--R Type: Float +--E 142 + +--S 143 of 288 +t3l:=exp(table3lx) +--R +--R +--R (143) 0.9185421971 9896054999 9999999999 9984320060 4468310134 +--R Type: Float +--E 143 + +--S 144 of 288 +table3ly:=0.91854219719896054999999999999999843 +--R +--R +--R (144) 0.9185421971 9896054999 9999999999 99843 +--R Type: Float +--E 144 + +--S 145 of 288 +t3l-table3ly +--R +--R +--R (145) 0.2006044683 1013 E -35 +--R Type: Float +--E 145 + +--S 146 of 288 +table3mx:=-9.250971335383380E-2 +--R +--R +--R (146) - 0.0925097133 538338 +--R Type: Float +--E 146 + +--S 147 of 288 +t3m:=exp(table3mx) +--R +--R +--R (147) 0.9116403558 3610989999 9999999999 9956380374 9072231327 +--R Type: Float +--E 147 + +--S 148 of 288 +table3my:=0.91164035583610989999999999999999563 +--R +--R +--R (148) 0.9116403558 3610989999 9999999999 99563 +--R Type: Float +--E 148 + +--S 149 of 288 +t3m-table3my +--R +--R +--R (149) 0.8037490722 31327 E -35 +--R Type: Float +--E 149 + +--S 150 of 288 +table3nx:=-9.337621398029658E-2 +--R +--R +--R (150) - 0.0933762139 8029658 +--R Type: Float +--E 150 + +--S 151 of 288 +t3n:=exp(table3nx) +--R +--R +--R (151) 0.9108507610 3826650000 0000000000 0040013766 8727220579 +--R Type: Float +--E 151 + +--S 152 of 288 +table3ny:=0.91085076103826650000000000000000400 +--R +--R +--R (152) 0.9108507610 3826650000 0000000000 004 +--R Type: Float +--E 152 + +--S 153 of 288 +t3n-table3ny +--R +--R +--R (153) 0.1376687272 2058 E -35 +--R Type: Float +--E 153 + +--S 154 of 288 +table3ox:=-9.341228128742237E-2 +--R +--R +--R (154) - 0.0934122812 8742237 +--R Type: Float +--E 154 + +--S 155 of 288 +t3o:=exp(table3ox) +--R +--R +--R (155) 0.9108179096 9655564999 9999999999 9995875465 54011791 +--R Type: Float +--E 155 + +--S 156 of 288 +table3oy:=0.910817909696555649999999999999999587 +--R +--R +--R (156) 0.9108179096 9655564999 9999999999 999587 +--R Type: Float +--E 156 + +--S 157 of 288 +t3o-table3oy +--R +--R +--R (157) 0.5465540117 91 E -36 +--R Type: Float +--E 157 + +--S 158 of 288 +table3px:=-9.998733949173545E-2 +--R +--R +--R (158) - 0.0999873394 9173545 +--R Type: Float +--E 158 + +--S 159 of 288 +t3p:=exp(table3px) +--R +--R +--R (159) 0.9048488738 1008650000 0000000000 0033076194 2534636856 +--R Type: Float +--E 159 + +--S 160 of 288 +table3py:=0.90484887381008650000000000000000330 +--R +--R +--R (160) 0.9048488738 1008650000 0000000000 0033 +--R Type: Float +--E 160 + +--S 161 of 288 +t3p-table3py +--R +--R +--R (161) 0.7619425346 36856 E -35 +--R Type: Float +--E 161 + +--S 162 of 288 +table3qx:=-1.452866822458144 +--R +--R +--R (162) - 1.4528668224 58144 +--R Type: Float +--E 162 + +--S 163 of 288 +t3q:=exp(table3qx) +--R +--R +--R (163) 0.2338987797 3141290000 0000000000 0041311743 3341591011 +--R Type: Float +--E 163 + +--S 164 of 288 +table3qy:=0.23389877973141290000000000000000413 +--R +--R +--R (164) 0.2338987797 3141290000 0000000000 00413 +--R Type: Float +--E 164 + +--S 165 of 288 +t3q-table3qy +--R +--R +--R (165) 0.1174333415 91011 E -35 +--R Type: Float +--E 165 + +--S 166 of 288 +table3rx:=-5.085363904672046 +--R +--R +--R (166) - 5.0853639046 72046 +--R Type: Float +--E 166 + +--S 167 of 288 +t3r:=exp(table3rx) +--R +--R +--R (167) 0.0061866353 3511597549 9999999999 9999774966 5221424755 88 +--R Type: Float +--E 167 + +--S 168 of 288 +table3ry:=6.1866353351159754999999999999999774E-3 +--R +--R +--R (168) 0.0061866353 3511597549 9999999999 9999774 +--R Type: Float +--E 168 + +--S 169 of 288 +t3r-table3ry +--R +--R +--R (169) 0.9665221424 75588 E -37 +--R Type: Float +--E 169 + +--S 170 of 288 +table3sx:=-5.815903811599861 +--R +--R +--R (170) - 5.8159038115 99861 +--R Type: Float +--E 170 + +--S 171 of 288 +t3s:=exp(table3sx) +--R +--R +--R (171) 0.0029797859 4494580450 0000000000 0000173080 4093500054 44 +--R Type: Float +--E 171 + +--S 172 of 288 +table3sy:=2.9797859449458045000000000000000173E-3 +--R +--R +--R (172) 0.0029797859 4494580450 0000000000 0000173 +--R Type: Float +--E 172 + +--S 173 of 288 +t3s-table3sy +--R +--R +--R (173) 0.8040935000 5444 E -38 +--R Type: Float +--E 173 + +--S 174 of 288 +table3tx:=-11.93382527979436 +--R +--R +--R (174) - 11.9338252797 9436 +--R Type: Float +--E 174 + +--S 175 of 288 +t3t:=exp(table3tx) +--R +--R +--R (175) 0.0000065645 5865261145 6999999999 9999999658 7484497146 90131 +--R Type: Float +--E 175 + +--S 176 of 288 +table3ty:=6.5645586526114569999999999999999658E-6 +--R +--R +--R (176) 0.0000065645 5865261145 6999999999 9999999658 +--R Type: Float +--E 176 + +--S 177 of 288 +t3t-table3ty +--R +--R +--R (177) 0.7484497146 90131 E -40 +--R Type: Float +--E 177 + +--S 178 of 288 +table3ux:=-46.84177248885496 +--R +--R +--R (178) - 46.8417724888 5496 +--R Type: Float +--E 178 + +--S 179 of 288 +t3u:=exp(table3ux) +--R +--R +--R (179) 0.4538127418 2205359999 9999999999 9976996689 5333158931 E -20 +--R Type: Float +--E 179 + +--S 180 of 288 +table3uy:=4.5381274182205359999999999999999769E-21 +--R +--R +--R (180) 0.4538127418 2205359999 9999999999 99769 E -20 +--R Type: Float +--E 180 + +--S 181 of 288 +t3u-table3uy +--R +--R +--R (181) 0.9668953331 58931 E -55 +--R Type: Float +--E 181 + +--S 182 of 288 +table3vx:=-84.88922783213444 +--R +--R +--R (182) - 84.8892278321 3444 +--R Type: Float +--E 182 + +--S 183 of 288 +t3v:=exp(table3vx) +--R +--R +--R (183) 0.1358553605 7843999891 2340715182 5158925719 9577954788 E -36 +--R Type: Float +--E 183 + +--S 184 of 288 +table3vy:=1.3599128388934695000000000000000266E-37 +--R +--R +--R (184) 0.1359912838 8934695000 0000000000 00266 E -36 +--R Type: Float +--E 184 + +--S 185 of 288 +t3v-table3vy +--R +--R +--R (185) - 0.1359233109 0695108765 9284817486 7674280042 2045212 E -39 +--R Type: Float +--E 185 + +--S 186 of 288 +table3wx:=-495.9839910528425 +--R +--R +--R (186) - 495.9839910528 425 +--R Type: Float +--E 186 + +--S 187 of 288 +t3w:=exp(table3wx) +--R +--R +--R (187) 0.3952661043 0311695000 0000000000 0037150791 9467288834 E -215 +--R Type: Float +--E 187 + +--S 188 of 288 +table3wy:=3.9526610430311695000000000000000371E-216 +--R +--R +--R (188) 0.3952661043 0311695000 0000000000 00371 E -215 +--R Type: Float +--E 188 + +--S 189 of 288 +t3w-table3wy +--R +--R +--R (189) 0.5079194672 88834 E -250 +--R Type: Float +--E 189 + +--S 190 of 288 +table3xx:=-524.2585830842744 +--R +--R +--R (190) - 524.2585830842 744 +--R Type: Float +--E 190 + +--S 191 of 288 +t3x:=exp(table3xx) +--R +--R +--R (191) 0.2076778963 8678450000 0000000000 0028745873 4089244785 E -227 +--R Type: Float +--E 191 + +--S 192 of 288 +table3xy:=2.0767789638678450000000000000000287E-228 +--R +--R +--R (192) 0.2076778963 8678450000 0000000000 00287 E -227 +--R Type: Float +--E 192 + +--S 193 of 288 +t3x-table3xy +--R +--R +--R (193) 0.4587340892 44785 E -262 +--R Type: Float +--E 193 + +\end{chunk} + +Table 4 is some bad cases of the exponential function in the decimal64 +format, for +\[e^+ = 4.999999999999999{\rm x}10^{-16} \le x < 10^{-9}\]. +At most two bad cases (the worst ones) are given per exponent. + +\begin{chunk}{*} + +--S 194 of 288 +table4ax:=6.000119998199928E-10 +--R +--R +--R (194) 0.6000119998 199928 E -9 +--R Type: Float +--E 194 + +--S 195 of 288 +t4a:=exp(table4ax) +--R +--R +--R (195) 1.0000000006 0001199999 9999999999 9956799459 99423978 +--R Type: Float +--E 195 + +--S 196 of 288 +table4ay:=1.00000000060001199999999999999999567 +--R +--R +--R (196) 1.0000000006 0001199999 9999999999 99567 +--R Type: Float +--E 196 + +--S 197 of 288 +t4a-table4ay +--R +--R +--R (197) 0.9945999423 97797 E -35 +--R Type: Float +--E 197 + +--S 198 of 288 +table4bx:=5.999879998200072E-10 +--R +--R +--R (198) 0.5999879998 200072 E -9 +--R Type: Float +--E 198 + +--S 199 of 288 +t4b:=exp(table4bx) +--R +--R +--R (199) 1.0000000005 9998800000 0000000000 0043199460 00575978 +--R Type: Float +--E 199 + +--S 200 of 288 +table4by:=1.00000000059998800000000000000000431 +--R +--R +--R (200) 1.0000000005 9998800000 0000000000 00431 +--R Type: Float +--E 200 + +--S 201 of 288 +t4b-table4by +--R +--R +--R (201) 0.9946000575 97797 E -35 +--R Type: Float +--E 201 + +--S 202 of 288 +table4cx:=1.039999999994592E-11 +--R +--R +--R (202) 0.1039999999 994592 E -10 +--R Type: Float +--E 202 + +--S 203 of 288 +t4c:=exp(table4cx) +--R +--R +--R (203) 1.0000000000 1039999999 9999999999 9996250453 333323584 +--R Type: Float +--E 203 + +--S 204 of 288 +table4cy:=1.000000000010399999999999999999999625 +--R +--R +--R (204) 1.0000000000 1039999999 9999999999 999625 +--R Type: Float +--E 204 + +--S 205 of 288 +t4c-table4cy +--R +--R +--R (205) 0.4533333235 84 E -37 +--R Type: Float +--E 205 + +--S 206 of 288 +table4dx:=1.019999999994798E-11 +--R +--R +--R (206) 0.1019999999 994798 E -10 +--R Type: Float +--E 206 + +--S 207 of 288 +t4d:=exp(table4dx) +--R +--R +--R (207) 1.0000000000 1019999999 9999999999 9996462639 99999098 +--R Type: Float +--E 207 + +--S 208 of 288 +table4dy:=1.000000000010199999999999999999999646 +--R +--R +--R (208) 1.0000000000 1019999999 9999999999 999646 +--R Type: Float +--E 208 + +--S 209 of 288 +t4d-table4dy +--R +--R +--R (209) 0.2639999990 98 E -36 +--R Type: Float +--E 209 + +--S 210 of 288 +table4ex:=1.199999999999280E-12 +--R +--R +--R (210) 0.1199999999 99928 E -11 +--R Type: Float +--E 210 + +--S 211 of 288 +t4e:=exp(table4ex) +--R +--R +--R (211) 1.0000000000 0119999999 9999999999 9999994239 999999998 +--R Type: Float +--E 211 + +--S 212 of 288 +table4ey:=1.000000000001199999999999999999999999423 +--R +--R +--R (212) 1.0000000000 0119999999 9999999999 999999423 +--R Type: Float +--E 212 + +--S 213 of 288 +t4e-table4ey +--R +--R +--R (213) 0.9999999998 3 E -39 +--R Type: Float +--E 213 + +--S 214 of 288 +table4fx:=1.099999999999395E-12 +--R +--R +--R (214) 0.1099999999 999395 E -11 +--R Type: Float +--E 214 + +--S 215 of 288 +t4f:=exp(table4fx) +--R +--R +--R (215) 1.0000000000 0109999999 9999999999 9999995563 333333332 +--R Type: Float +--E 215 + +--S 216 of 288 +table4fy:=1.000000000001099999999999999999999999556 +--R +--R +--R (216) 1.0000000000 0109999999 9999999999 999999556 +--R Type: Float +--E 216 + +--S 217 of 288 +t4f-table4fy +--R +--R +--R (217) 0.3333333332 E -39 +--R Type: Float +--E 217 + +--S 218 of 288 +table4gx:=1.399999999999902E-13 +--R +--R +--R (218) 0.1399999999 999902 E -12 +--R Type: Float +--E 218 + +--S 219 of 288 +t4g:=exp(table4gx) +--R +--R +--R (219) 1.0000000000 0013999999 9999999999 9999999990 853333333 +--R Type: Float +--E 219 + +--S 220 of 288 +table4gy:=1.000000000000139999999999999999999999999085 +--R +--R +--R (220) 1.0000000000 0013999999 9999999999 9999999990 85 +--R Type: Float +--E 220 + +--S 221 of 288 +t4g-table4gy +--R +--R +--R (221) 0.3333333 E -42 +--R Type: Float +--E 221 + +--S 222 of 288 +table4hx:=1.199999999999928E-13 +--R +--R +--R (222) 0.1199999999 999928 E -12 +--R Type: Float +--E 222 + +--S 223 of 288 +t4h:=exp(table4hx) +--R +--R +--R (223) 1.0000000000 0011999999 9999999999 9999999994 24 +--R Type: Float +--E 223 + +--S 224 of 288 +table4hy:=1.000000000000119999999999999999999999999423 +--R +--R +--R (224) 1.0000000000 0011999999 9999999999 9999999994 23 +--R Type: Float +--E 224 + +--S 225 of 288 +t4h-table4hy +--R +--R +--R (225) 0.1 E -41 +--R Type: Float +--E 225 + +--S 226 of 288 +table4ix:=2.999999999999955E-14 +--R +--R +--R (226) 0.2999999999 999955 E -13 +--R Type: Float +--E 226 + +--S 227 of 288 +t4i:=exp(table4ix) +--R +--R +--R (227) 1.0000000000 0002999999 9999999999 9999999999 91 +--R Type: Float +--E 227 + +--S 228 of 288 +table4iy:=1.00000000000002999999999999999999999999999099 +--R +--R +--R (228) 1.0000000000 0002999999 9999999999 9999999999 9099 +--R Type: Float +--E 228 + +--S 229 of 288 +t4i-table4iy +--R +--R +--R (229) 0.1 E -43 +--R Type: Float +--E 229 + +--S 230 of 288 +table4jx:=1.999999999999980E-14 +--R +--R +--R (230) 0.1999999999 99998 E -13 +--R Type: Float +--E 230 + +--S 231 of 288 +t4j:=exp(table4jx) +--R +--R +--R (231) 1.0000000000 0001999999 9999999999 9999999999 973333333 +--R Type: Float +--E 231 + +--S 232 of 288 +table4jy:=1.00000000000001999999999999999999999999999733 +--R +--R +--R (232) 1.0000000000 0001999999 9999999999 9999999999 9733 +--R Type: Float +--E 232 + +--S 233 of 288 +t4j-table4jy +--R +--R +--R (233) 0.33333 E -44 +--R Type: Float +--E 233 + +--S 234 of 288 +table4kx:=3.999999999999992E-15 +--R +--R +--R (234) 0.3999999999 999992 E -14 +--R Type: Float +--E 234 + +--S 235 of 288 +t4k:=exp(table4kx) +--R +--R +--R (235) 1.0000000000 0000399999 9999999999 9999999999 999786667 +--R Type: Float +--E 235 + +--S 236 of 288 +table4ky:=1.0000000000000039999999999999999999999999999786 +--R +--R +--R (236) 1.0000000000 0000399999 9999999999 9999999999 999786 +--R Type: Float +--E 236 + +--S 237 of 288 +t4k-table4ky +--R +--R +--R (237) 0.667 E -46 +--R Type: Float +--E 237 + +--S 238 of 288 +table4lx:=1.999999999999998E-15 +--R +--R +--R (238) 0.1999999999 999998 E -14 +--R Type: Float +--E 238 + +--S 239 of 288 +t4l:=exp(table4lx) +--R +--R +--R (239) 1.0000000000 0000199999 9999999999 9999999999 999973333 +--R Type: Float +--E 239 + +--S 240 of 288 +table4ly:=1.00000000000000199999999999999999999999999999733 +--R +--R +--R (240) 1.0000000000 0000199999 9999999999 9999999999 9999733 +--R Type: Float +--E 240 + +--S 241 of 288 +t4l-table4ly +--R +--R +--R (241) 0.33 E -47 +--R Type: Float +--E 241 + +--S 242 of 288 +table4mx:=9.999999999999995E-16 +--R +--R +--R (242) 0.9999999999 999995 E -15 +--R Type: Float +--E 242 + +--S 243 of 288 +t4m:=exp(table4mx) +--R +--R +--R (243) 1.0000000000 0000099999 9999999999 9999999999 999996667 +--R Type: Float +--E 243 + +--S 244 of 288 +table4my:=1.000000000000000999999999999999999999999999999666 +--R +--R +--R (244) 1.0000000000 0000099999 9999999999 9999999999 99999666 +--R Type: Float +--E 244 + +--S 245 of 288 +t4m-table4my +--R +--R +--R (245) 0.7 E -48 +--R Type: Float +--E 245 + +\end{chunk} + +Table 5 is some bad cases of the exponential function in the decimal64 +format, for +\[-10^{-10} < x \le b^- = -5.000000000000001{\rm x}10^{-17}\] +At most two bad cases (the worst ones) are given per exponent. + +\begin{chunk}{*} + +--S 246 of 288 +table5ax:=-1.020000000005202E-11 +--R +--R +--R (246) - 0.1020000000 005202 E -10 +--R Type: Float +--E 246 + +--S 247 of 288 +t5a:=exp(table5ax) +--R +--R +--R (247) 0.9999999999 8980000000 0000000000 0003537359 9999909797 +--R Type: Float +--E 247 + +--S 248 of 288 +table5ay:=0.999999999989800000000000000000000353 +--R +--R +--R (248) 0.9999999999 8980000000 0000000000 000353 +--R Type: Float +--E 248 + +--S 249 of 288 +t5a-table5ay +--R +--R +--R (249) 0.7359999990 9797 E -36 +--R Type: Float +--E 249 + +--S 250 of 288 +table5bx:=-1.000000000005000E-11 +--R +--R +--R (250) - 0.1000000000 005 E -10 +--R Type: Float +--E 250 + +--S 251 of 288 +t5b:=exp(table5bx) +--R +--R +--R (251) 0.9999999999 9000000000 0000000000 0003333333 333325 +--R Type: Float +--E 251 + +--S 252 of 288 +table5by:=0.999999999990000000000000000000000333 +--R +--R +--R (252) 0.9999999999 9000000000 0000000000 000333 +--R Type: Float +--E 252 + +--S 253 of 288 +t5b-table5by +--R +--R +--R (253) 0.3333333325 E -36 +--R Type: Float +--E 253 + +--S 254 of 288 +table5cx:=-1.100000000000605E-12 +--R +--R +--R (254) - 0.1100000000 000605 E -11 +--R Type: Float +--E 254 + +--S 255 of 288 +t5c:=exp(table5cx) +--R +--R +--R (255) 0.9999999999 9890000000 0000000000 0000004436 6666666654 +--R Type: Float +--E 255 + +--S 256 of 288 +table5cy:=0.999999999998900000000000000000000000443 +--R +--R +--R (256) 0.9999999999 9890000000 0000000000 000000443 +--R Type: Float +--E 256 + +--S 257 of 288 +t5c-table5cy +--R +--R +--R (257) 0.6666666665 4 E -39 +--R Type: Float +--E 257 + +--S 258 of 288 +table5dx:=-1.000000000000500E-12 +--R +--R +--R (258) - 0.1000000000 0005 E -11 +--R Type: Float +--E 258 + +--S 259 of 288 +t5d:=exp(table5dx) +--R +--R +--R (259) 0.9999999999 9900000000 0000000000 0000003333 3333333325 +--R Type: Float +--E 259 + +--S 260 of 288 +table5dy:=0.999999999999000000000000000000000000333 +--R +--R +--R (260) 0.9999999999 9900000000 0000000000 000000333 +--R Type: Float +--E 260 + +--S 261 of 288 +t5d-table5dy +--R +--R +--R (261) 0.3333333332 E -39 +--R Type: Float +--E 261 + +--S 262 of 288 +table5ex:=-1.200000000000072E-13 +--R +--R +--R (262) - 0.1200000000 000072 E -12 +--R Type: Float +--E 262 + +--S 263 of 288 +t5e:=exp(table5ex) +--R +--R +--R (263) 0.9999999999 9988000000 0000000000 0000000005 76 +--R Type: Float +--E 263 + +--S 264 of 288 +table5ey:=0.999999999999880000000000000000000000000575 +--R +--R +--R (264) 0.9999999999 9988000000 0000000000 0000000005 75 +--R Type: Float +--E 264 + +--S 265 of 288 +t5e-table5ey +--R +--R +--R (265) 0.1 E -41 +--R Type: Float +--E 265 + +--S 266 of 288 +table5fx:=-1.000000000000050E-13 +--R +--R +--R (266) - 0.1000000000 00005 E -12 +--R Type: Float +--E 266 + +--S 267 of 288 +t5f:=exp(table5fx) +--R +--R +--R (267) 0.9999999999 9990000000 0000000000 0000000003 3333333333 +--R Type: Float +--E 267 + +--S 268 of 288 +table5fy:=0.999999999999900000000000000000000000000333 +--R +--R +--R (268) 0.9999999999 9990000000 0000000000 0000000003 33 +--R Type: Float +--E 268 + +--S 269 of 288 +t5f-table5fy +--R +--R +--R (269) 0.3333333 E -42 +--R Type: Float +--E 269 + +--S 270 of 288 +table5gx:=-2.000000000000020E-14 +--R +--R +--R (270) - 0.2000000000 00002 E -13 +--R Type: Float +--E 270 + +--S 271 of 288 +t5g:=exp(table5gx) +--R +--R +--R (271) 0.9999999999 9998000000 0000000000 0000000000 0266666667 +--R Type: Float +--E 271 + +--S 272 of 288 +table5gy:=0.99999999999998000000000000000000000000000266 +--R +--R +--R (272) 0.9999999999 9998000000 0000000000 0000000000 0266 +--R Type: Float +--E 272 + +--S 273 of 288 +t5g-table5gy +--R +--R +--R (273) 0.666667 E -44 +--R Type: Float +--E 273 + +--S 274 of 288 +table5hx:=-1.000000000000005E-14 +--R +--R +--R (274) - 0.1000000000 000005 E -13 +--R Type: Float +--E 274 + +--S 275 of 288 +t5h:=exp(table5hx) +--R +--R +--R (275) 0.9999999999 9999000000 0000000000 0000000000 0033333333 +--R Type: Float +--E 275 + +--S 276 of 288 +table5hy:=0.999999999999990000000000000000000000000000333 +--R +--R +--R (276) 0.9999999999 9999000000 0000000000 0000000000 00333 +--R Type: Float +--E 276 + +--S 277 of 288 +t5h-table5hy +--R +--R +--R (277) 0.3333 E -45 +--R Type: Float +--E 277 + +--S 278 of 288 +table5ix:=-4.000000000000008E-15 +--R +--R +--R (278) - 0.4000000000 000008 E -14 +--R Type: Float +--E 278 + +--S 279 of 288 +t5i:=exp(table5ix) +--R +--R +--R (279) 0.9999999999 9999600000 0000000000 0000000000 0002133333 +--R Type: Float +--E 279 + +--S 280 of 288 +table5iy:=0.9999999999999960000000000000000000000000000213 +--R +--R +--R (280) 0.9999999999 9999600000 0000000000 0000000000 000213 +--R Type: Float +--E 280 + +--S 281 of 288 +t5i-table5iy +--R +--R +--R (281) 0.333 E -46 +--R Type: Float +--E 281 + +--S 282 of 288 +table5jx:=-2.000000000000002E-15 +--R +--R +--R (282) - 0.2000000000 000002 E -14 +--R Type: Float +--E 282 + +--S 283 of 288 +t5j:=exp(table5jx) +--R +--R +--R (283) 0.9999999999 9999800000 0000000000 0000000000 0000266667 +--R Type: Float +--E 283 + +--S 284 of 288 +table5jy:=0.99999999999999800000000000000000000000000000266 +--R +--R +--R (284) 0.9999999999 9999800000 0000000000 0000000000 0000266 +--R Type: Float +--E 284 + +--S 285 of 288 +t5j-table5jy +--R +--R +--R (285) 0.667 E -47 +--R Type: Float +--E 285 + +--S 286 of 288 +a1:=9.407822313572878E-2 +--R +--R +--R (286) 0.0940782231 3572878 +--R Type: Float +--E 286 + +--S 287 of 288 +a2:=exp(a1) +--R +--R +--R (287) 1.0986456820 6633850000 0000000000 0027805367 999050547 +--R Type: Float +--E 287 + +--S 288 of 288 +a2-1.09864568206633850000000000000000278 +--R +--R +--R (288) 0.5367999050 547 E -36 +--R Type: Float +--E 288 + + +)spool +)lisp (bye) + +\end{chunk} +\eject +\begin{thebibliography}{99} +\bibitem{1} nothing +\end{thebibliography} +\end{document}