Add pi.awk program in MPFR chapter.

5 files changed, 350 insertions, 146 deletions

diff --git a/awklib/eg/prog/pi.awk b/awklib/eg/prog/pi.awk
new file mode 100644
index 00000000..3297beff
--- /dev/null
+++ b/awklib/eg/prog/pi.awk
@@ -0,0 +1,18 @@
+# pi.awk --- compute the digits of pi
+#
+# Katie Wasserman, katie@wass.net
+# August 2014
+
+BEGIN {
+    digits = 100000
+    two = 2 * 10 ^ digits
+    pi = two
+    for (m = digits * 4; m > 0; --m) {
+        d = m * 2 + 1
+        x = pi * m
+        div(x, d, result)
+        pi = result["quotient"]
+        pi = pi + two
+    }
+    print pi
+}
diff --git a/doc/ChangeLog b/doc/ChangeLog
index 89015ce3..552ece4e 100644
--- a/doc/ChangeLog
+++ b/doc/ChangeLog
@@ -2,6 +2,11 @@
 
 	* gawktexi.in: More fixes after reading through the MS.
 
+	Unrelated:
+
+	* gawktexi.in: Add Katie Wasserman's program to compute
+	the digits of PI.
+
 2014-09-30         Arnold D. Robbins     <arnold@skeeve.com>
 
 	* gawktexi.in: More fixes after reading through the MS.
diff --git a/doc/gawk.info b/doc/gawk.info
index dfa21a7b..d6faeeea 100644
--- a/doc/gawk.info
+++ b/doc/gawk.info
@@ -22381,6 +22381,39 @@ user-defined function:
          return 0.0
      }
 
+   The following example program, contributed by Katie Wasserman, uses
+`div()' to compute the digits of pi to as many places as you choose to
+set:
+
+     # pi.awk --- compute the digits of pi
+
+     BEGIN {
+         digits = 100000
+         two = 2 * 10 ^ digits
+         pi = two
+         for (m = digits * 4; m > 0; --m) {
+             d = m * 2 + 1
+             x = pi * m
+             div(x, d, result)
+             pi = result["quotient"]
+             pi = pi + two
+         }
+         print pi
+     }
+
+   When asked about the algorithm used, Katie replied:
+
+     It's not that well known but it's not that obscure either.  It's
+     Euler's modification to Newton's method for calculating pi.  Take
+     a look at lines (23) - (25) here:
+     `http://mathworld.wolfram.com/PiFormulas.htm'.
+
+     The algorithm I wrote simply expands the multiply by 2 and works
+     from the innermost expression outwards.  I used this to program HP
+     calculators because it's quite easy to modify for tiny memory
+     devices with smallish word sizes. See
+     `http://www.hpmuseum.org/cgi-sys/cgiwrap/hpmuseum/articles.cgi?read=899'.
+
    ---------- Footnotes ----------
 
    (1) Weisstein, Eric W.  `Sylvester's Sequence'. From MathWorld--A
@@ -34676,151 +34709,151 @@ Node: Setting the rounding mode895668
 Ref: table-gawk-rounding-modes896032
 Ref: Setting the rounding mode-Footnote-1899486
 Node: Arbitrary Precision Integers899665
-Ref: Arbitrary Precision Integers-Footnote-1903448
-Node: POSIX Floating Point Problems903597
-Ref: POSIX Floating Point Problems-Footnote-1907473
-Node: Floating point summary907511
-Node: Dynamic Extensions909703
-Node: Extension Intro911255
-Node: Plugin License912521
-Node: Extension Mechanism Outline913318
-Ref: figure-load-extension913746
-Ref: figure-register-new-function915226
-Ref: figure-call-new-function916230
-Node: Extension API Description918216
-Node: Extension API Functions Introduction919666
-Node: General Data Types924502
-Ref: General Data Types-Footnote-1930189
-Node: Memory Allocation Functions930488
-Ref: Memory Allocation Functions-Footnote-1933318
-Node: Constructor Functions933414
-Node: Registration Functions935148
-Node: Extension Functions935833
-Node: Exit Callback Functions938129
-Node: Extension Version String939377
-Node: Input Parsers940027
-Node: Output Wrappers949842
-Node: Two-way processors954358
-Node: Printing Messages956562
-Ref: Printing Messages-Footnote-1957639
-Node: Updating `ERRNO'957791
-Node: Requesting Values958531
-Ref: table-value-types-returned959259
-Node: Accessing Parameters960217
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-Node: Symbol table by name961962
-Node: Symbol table by cookie963942
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-Node: Cached values968144
-Ref: Cached values-Footnote-1971648
-Node: Array Manipulation971739
-Ref: Array Manipulation-Footnote-1972837
-Node: Array Data Types972876
-Ref: Array Data Types-Footnote-1975533
-Node: Array Functions975625
-Node: Flattening Arrays979479
-Node: Creating Arrays986366
-Node: Extension API Variables991133
-Node: Extension Versioning991769
-Node: Extension API Informational Variables993670
-Node: Extension API Boilerplate994758
-Node: Finding Extensions998574
-Node: Extension Example999134
-Node: Internal File Description999906
-Node: Internal File Ops1003973
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-Node: Using Internal File Ops1015771
-Ref: Using Internal File Ops-Footnote-11018154
-Node: Extension Samples1018427
-Node: Extension Sample File Functions1019951
-Node: Extension Sample Fnmatch1027553
-Node: Extension Sample Fork1029035
-Node: Extension Sample Inplace1030248
-Node: Extension Sample Ord1031923
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-Ref: table-readdir-file-types1033615
-Node: Extension Sample Revout1034426
-Node: Extension Sample Rev2way1035017
-Node: Extension Sample Read write array1035758
-Node: Extension Sample Readfile1037697
-Node: Extension Sample Time1038792
-Node: Extension Sample API Tests1040141
-Node: gawkextlib1040632
-Node: Extension summary1043282
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-Node: Language History1047686
-Node: V7/SVR3.11049343
-Node: SVR41051524
-Node: POSIX1052969
-Node: BTL1054358
-Node: POSIX/GNU1055092
-Node: Feature History1060721
-Node: Common Extensions1073812
-Node: Ranges and Locales1075136
-Ref: Ranges and Locales-Footnote-11079775
-Ref: Ranges and Locales-Footnote-21079802
-Ref: Ranges and Locales-Footnote-31080036
-Node: Contributors1080257
-Node: History summary1085797
-Node: Installation1087166
-Node: Gawk Distribution1088122
-Node: Getting1088606
-Node: Extracting1089430
-Node: Distribution contents1091072
-Node: Unix Installation1096842
-Node: Quick Installation1097459
-Node: Additional Configuration Options1099890
-Node: Configuration Philosophy1101630
-Node: Non-Unix Installation1103981
-Node: PC Installation1104439
-Node: PC Binary Installation1105765
-Node: PC Compiling1107613
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-Node: PC Testing1110739
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-Node: VMS Installation1117351
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-Node: VMS Installation Details1121107
-Node: VMS Running1123359
-Node: VMS GNV1126200
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-Node: Other Versions1131369
-Node: Installation summary1137582
-Node: Notes1138638
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-Node: Additions1140285
-Node: Accessing The Source1141210
-Node: Adding Code1142646
-Node: New Ports1148818
-Node: Derived Files1153299
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-Node: Implementation Limitations1160124
-Node: Extension Design1161372
-Node: Old Extension Problems1162526
-Ref: Old Extension Problems-Footnote-11164043
-Node: Extension New Mechanism Goals1164100
-Ref: Extension New Mechanism Goals-Footnote-11167460
-Node: Extension Other Design Decisions1167649
-Node: Extension Future Growth1169757
-Node: Old Extension Mechanism1170593
-Node: Notes summary1172355
-Node: Basic Concepts1173541
-Node: Basic High Level1174222
-Ref: figure-general-flow1174494
-Ref: figure-process-flow1175093
-Ref: Basic High Level-Footnote-11178322
-Node: Basic Data Typing1178507
-Node: Glossary1181835
-Node: Copying1206993
-Node: GNU Free Documentation License1244549
-Node: Index1269685
+Ref: Arbitrary Precision Integers-Footnote-1904575
+Node: POSIX Floating Point Problems904724
+Ref: POSIX Floating Point Problems-Footnote-1908600
+Node: Floating point summary908638
+Node: Dynamic Extensions910830
+Node: Extension Intro912382
+Node: Plugin License913648
+Node: Extension Mechanism Outline914445
+Ref: figure-load-extension914873
+Ref: figure-register-new-function916353
+Ref: figure-call-new-function917357
+Node: Extension API Description919343
+Node: Extension API Functions Introduction920793
+Node: General Data Types925629
+Ref: General Data Types-Footnote-1931316
+Node: Memory Allocation Functions931615
+Ref: Memory Allocation Functions-Footnote-1934445
+Node: Constructor Functions934541
+Node: Registration Functions936275
+Node: Extension Functions936960
+Node: Exit Callback Functions939256
+Node: Extension Version String940504
+Node: Input Parsers941154
+Node: Output Wrappers950969
+Node: Two-way processors955485
+Node: Printing Messages957689
+Ref: Printing Messages-Footnote-1958766
+Node: Updating `ERRNO'958918
+Node: Requesting Values959658
+Ref: table-value-types-returned960386
+Node: Accessing Parameters961344
+Node: Symbol Table Access962575
+Node: Symbol table by name963089
+Node: Symbol table by cookie965069
+Ref: Symbol table by cookie-Footnote-1969208
+Node: Cached values969271
+Ref: Cached values-Footnote-1972775
+Node: Array Manipulation972866
+Ref: Array Manipulation-Footnote-1973964
+Node: Array Data Types974003
+Ref: Array Data Types-Footnote-1976660
+Node: Array Functions976752
+Node: Flattening Arrays980606
+Node: Creating Arrays987493
+Node: Extension API Variables992260
+Node: Extension Versioning992896
+Node: Extension API Informational Variables994797
+Node: Extension API Boilerplate995885
+Node: Finding Extensions999701
+Node: Extension Example1000261
+Node: Internal File Description1001033
+Node: Internal File Ops1005100
+Ref: Internal File Ops-Footnote-11016758
+Node: Using Internal File Ops1016898
+Ref: Using Internal File Ops-Footnote-11019281
+Node: Extension Samples1019554
+Node: Extension Sample File Functions1021078
+Node: Extension Sample Fnmatch1028680
+Node: Extension Sample Fork1030162
+Node: Extension Sample Inplace1031375
+Node: Extension Sample Ord1033050
+Node: Extension Sample Readdir1033886
+Ref: table-readdir-file-types1034742
+Node: Extension Sample Revout1035553
+Node: Extension Sample Rev2way1036144
+Node: Extension Sample Read write array1036885
+Node: Extension Sample Readfile1038824
+Node: Extension Sample Time1039919
+Node: Extension Sample API Tests1041268
+Node: gawkextlib1041759
+Node: Extension summary1044409
+Node: Extension Exercises1048091
+Node: Language History1048813
+Node: V7/SVR3.11050470
+Node: SVR41052651
+Node: POSIX1054096
+Node: BTL1055485
+Node: POSIX/GNU1056219
+Node: Feature History1061848
+Node: Common Extensions1074939
+Node: Ranges and Locales1076263
+Ref: Ranges and Locales-Footnote-11080902
+Ref: Ranges and Locales-Footnote-21080929
+Ref: Ranges and Locales-Footnote-31081163
+Node: Contributors1081384
+Node: History summary1086924
+Node: Installation1088293
+Node: Gawk Distribution1089249
+Node: Getting1089733
+Node: Extracting1090557
+Node: Distribution contents1092199
+Node: Unix Installation1097969
+Node: Quick Installation1098586
+Node: Additional Configuration Options1101017
+Node: Configuration Philosophy1102757
+Node: Non-Unix Installation1105108
+Node: PC Installation1105566
+Node: PC Binary Installation1106892
+Node: PC Compiling1108740
+Ref: PC Compiling-Footnote-11111761
+Node: PC Testing1111866
+Node: PC Using1113042
+Node: Cygwin1117157
+Node: MSYS1117980
+Node: VMS Installation1118478
+Node: VMS Compilation1119270
+Ref: VMS Compilation-Footnote-11120492
+Node: VMS Dynamic Extensions1120550
+Node: VMS Installation Details1122234
+Node: VMS Running1124486
+Node: VMS GNV1127327
+Node: VMS Old Gawk1128056
+Node: Bugs1128526
+Node: Other Versions1132496
+Node: Installation summary1138709
+Node: Notes1139765
+Node: Compatibility Mode1140630
+Node: Additions1141412
+Node: Accessing The Source1142337
+Node: Adding Code1143773
+Node: New Ports1149945
+Node: Derived Files1154426
+Ref: Derived Files-Footnote-11159901
+Ref: Derived Files-Footnote-21159935
+Ref: Derived Files-Footnote-31160531
+Node: Future Extensions1160645
+Node: Implementation Limitations1161251
+Node: Extension Design1162499
+Node: Old Extension Problems1163653
+Ref: Old Extension Problems-Footnote-11165170
+Node: Extension New Mechanism Goals1165227
+Ref: Extension New Mechanism Goals-Footnote-11168587
+Node: Extension Other Design Decisions1168776
+Node: Extension Future Growth1170884
+Node: Old Extension Mechanism1171720
+Node: Notes summary1173482
+Node: Basic Concepts1174668
+Node: Basic High Level1175349
+Ref: figure-general-flow1175621
+Ref: figure-process-flow1176220
+Ref: Basic High Level-Footnote-11179449
+Node: Basic Data Typing1179634
+Node: Glossary1182962
+Node: Copying1208120
+Node: GNU Free Documentation License1245676
+Node: Index1270812
 
 End Tag Table
diff --git a/doc/gawk.texi b/doc/gawk.texi
index ebaafbf3..5315422f 100644
--- a/doc/gawk.texi
+++ b/doc/gawk.texi
@@ -31004,6 +31004,80 @@ function div(numerator, denominator, result,    i)
 @c endfile
 @end example
 
+The following example program, contributed by Katie Wasserman,
+uses @code{div()} to
+compute the digits of @value{PI} to as many places as you
+choose to set:
+
+@example
+@c file eg/prog/pi.awk
+# pi.awk --- compute the digits of pi
+@c endfile
+@c endfile
+@ignore
+@c file eg/prog/pi.awk
+#
+# Katie Wasserman, katie@@wass.net
+# August 2014
+@c endfile
+@end ignore
+@c file eg/prog/pi.awk
+
+BEGIN @{
+    digits = 100000
+    two = 2 * 10 ^ digits
+    pi = two
+    for (m = digits * 4; m > 0; --m) @{
+        d = m * 2 + 1
+        x = pi * m
+        div(x, d, result)
+        pi = result["quotient"]
+        pi = pi + two
+    @}
+    print pi
+@}
+@c endfile
+@end example
+
+@ignore
+Date: Wed, 20 Aug 2014 10:19:11 -0400
+To: arnold@skeeve.com
+From: Katherine Wasserman <katie@wass.net>
+Subject: Re: computation of digits of pi?
+
+Arnold,
+
+>The program that you sent to compute the digits of pi using div(). Is
+>that some standard algorithm that every math student knows? If so,
+>what's it called?
+
+It's not that well known but it's not that obscure either
+
+It's Euler's modification to Newton's method for calculating pi.
+
+Take a look at lines (23) - (25)  here: http://mathworld.wolfram.com/PiFormulas.htm
+
+The algorithm I wrote simply expands the multiply by 2 and works from the innermost expression outwards.  I used this to program HP calculators because it's quite easy to modify for tiny memory devices with smallish word sizes.   
+
+http://www.hpmuseum.org/cgi-sys/cgiwrap/hpmuseum/articles.cgi?read=899
+
+-Katie
+@end ignore
+
+When asked about the algorithm used, Katie replied:
+
+@quotation
+It's not that well known but it's not that obscure either.
+It's Euler's modification to Newton's method for calculating pi.
+Take a look at lines (23) - (25) here: @uref{http://mathworld.wolfram.com/PiFormulas.htm}.
+
+The algorithm I wrote simply expands the multiply by 2 and works from
+the innermost expression outwards.  I used this to program HP calculators
+because it's quite easy to modify for tiny memory devices with smallish
+word sizes. See
+@uref{http://www.hpmuseum.org/cgi-sys/cgiwrap/hpmuseum/articles.cgi?read=899}.
+@end quotation
+
 @node POSIX Floating Point Problems
 @section Standards Versus Existing Practice
 
diff --git a/doc/gawktexi.in b/doc/gawktexi.in
index 96c0a76c..2600011f 100644
--- a/doc/gawktexi.in
+++ b/doc/gawktexi.in
@@ -30098,6 +30098,80 @@ function div(numerator, denominator, result,    i)
 @c endfile
 @end example
 
+The following example program, contributed by Katie Wasserman,
+uses @code{div()} to
+compute the digits of @value{PI} to as many places as you
+choose to set:
+
+@example
+@c file eg/prog/pi.awk
+# pi.awk --- compute the digits of pi
+@c endfile
+@c endfile
+@ignore
+@c file eg/prog/pi.awk
+#
+# Katie Wasserman, katie@@wass.net
+# August 2014
+@c endfile
+@end ignore
+@c file eg/prog/pi.awk
+
+BEGIN @{
+    digits = 100000
+    two = 2 * 10 ^ digits
+    pi = two
+    for (m = digits * 4; m > 0; --m) @{
+        d = m * 2 + 1
+        x = pi * m
+        div(x, d, result)
+        pi = result["quotient"]
+        pi = pi + two
+    @}
+    print pi
+@}
+@c endfile
+@end example
+
+@ignore
+Date: Wed, 20 Aug 2014 10:19:11 -0400
+To: arnold@skeeve.com
+From: Katherine Wasserman <katie@wass.net>
+Subject: Re: computation of digits of pi?
+
+Arnold,
+
+>The program that you sent to compute the digits of pi using div(). Is
+>that some standard algorithm that every math student knows? If so,
+>what's it called?
+
+It's not that well known but it's not that obscure either
+
+It's Euler's modification to Newton's method for calculating pi.
+
+Take a look at lines (23) - (25)  here: http://mathworld.wolfram.com/PiFormulas.htm
+
+The algorithm I wrote simply expands the multiply by 2 and works from the innermost expression outwards.  I used this to program HP calculators because it's quite easy to modify for tiny memory devices with smallish word sizes.   
+
+http://www.hpmuseum.org/cgi-sys/cgiwrap/hpmuseum/articles.cgi?read=899
+
+-Katie
+@end ignore
+
+When asked about the algorithm used, Katie replied:
+
+@quotation
+It's not that well known but it's not that obscure either.
+It's Euler's modification to Newton's method for calculating pi.
+Take a look at lines (23) - (25) here: @uref{http://mathworld.wolfram.com/PiFormulas.htm}.
+
+The algorithm I wrote simply expands the multiply by 2 and works from
+the innermost expression outwards.  I used this to program HP calculators
+because it's quite easy to modify for tiny memory devices with smallish
+word sizes. See
+@uref{http://www.hpmuseum.org/cgi-sys/cgiwrap/hpmuseum/articles.cgi?read=899}.
+@end quotation
+
 @node POSIX Floating Point Problems
 @section Standards Versus Existing Practice