Release calc version 2.12.6.4

Fixed a man page warning about ./myfile where the leading dot
was mistook for an nroff macro.  Thanks goes to David Haller
<dnh at opensuse dot org> for providing the patch.

Improved gen_v1(h,n) in lucas.cal for cases where h is not a
multiple of 3. Optimized the search for v(1) when h is a
multiple of 3.

Fixed a Makefile problem, reported by Doug Hays <doughays6 at gmail
dot com>, where if a macOS user set BINDIR, LIBDIR, CALC_SHAREDIR
or INCDIR in the top section, their values will be overwritten by
the Darwin specific section.

CHANGES

@@ -9,6 +9,19 @@ The following are the changes from calc version 2.12.6.1 to date:
     Fixed an C code indenting issue that was reported by Thomas Walter
     <th dot walter42 at gmx dot de> in zfunc.c.
 
+    Fixed a man page warning about ./myfile where the leading dot
+    was mistook for an nroff macro.  Thanks goes to David Haller
+    <dnh at opensuse dot org> for providing the patch.
+
+    Improved gen_v1(h,n) in lucas.cal for cases where h is not a
+    multiple of 3. Optimized the search for v(1) when h is a
+    multiple of 3.
+
+    Fixed a Makefile problem, reported by Doug Hays <doughays6 at gmail
+    dot com>, where if a macOS user set BINDIR, LIBDIR, CALC_SHAREDIR
+    or INCDIR in the top section, their values will be overwritten by
+    the Darwin specific section.
+
 
 The following are the changes from calc version 2.12.6.0 to 2.12.6.0:
 

Makefile.ship

@@ -23,7 +23,7 @@
 #	 READLINE_LIB= -lreadline
 #	 READLINE_EXTRAS= -lhistory -lncurses
 #
-# Copyright (C) 1999-2017  Landon Curt Noll
+# Copyright (C) 1999-2018  Landon Curt Noll
 #
 # Calc is open software; you can redistribute it and/or modify it under
 # the terms of version 2.1 of the GNU Lesser General Public License
@@ -564,14 +564,30 @@ HAVE_UNUSED=
 #
 #	INCDIR= /dev/env/DJDIR/include
 #
-# If in doubt, set:
+# If in doubt, for non-macOS hosts set:
 #
 #	INCDIR= /usr/include
 #
+# However, if you are on macOS then set:
+#
+#	INCDIR= /usr/local/include
+#if 0	/* start of skip for non-Gnu makefiles */
+ifeq ($(target),Darwin)
 
+# default INCDIR for macOS
+INCDIR= /usr/local/include
+
+else
+#endif	/* end of skip for non-Gnu makefiles */
+
+# default INCDIR for non-macOS
+INCDIR= /usr/include
 #INCDIR= /usr/local/include
 #INCDIR= /dev/env/DJDIR/include
-INCDIR= /usr/include
+
+#if 0	/* start of skip for non-Gnu makefiles */
+endif
+#endif	/* end of skip for non-Gnu makefiles */
 
 # Where to install calc related things
 #
@@ -602,23 +618,71 @@ INCDIR= /usr/include
 #	LIBDIR= /dev/env/DJDIR/lib
 #	CALC_SHAREDIR= /dev/env/DJDIR/share/calc
 #
-# If in doubt, set:
+# If in doubt, for non-macOS hosts set:
 #
 #	BINDIR= /usr/bin
 #	LIBDIR= /usr/lib
 #	CALC_SHAREDIR= /usr/share/calc
 #
+# However, if you are on macOS then set:
+#
+#	BINDIR= /usr/local/bin
+#	LIBDIR= /usr/local/lib
+#	CALC_SHAREDIR= /usr/local/share/calc
+#
+# 	NOTE: Starting with macOS El Capitan OS X 10.11, root by default
+# 	      could not mkdir under system locations, so macOS must now
+# 	      use the /usr/local tree.
+
+#if 0	/* start of skip for non-Gnu makefiles */
+ifeq ($(target),Darwin)
+
+# default BINDIR for macOS
+BINDIR= /usr/local/bin
+
+else
+#endif	/* end of skip for non-Gnu makefiles */
+
+# default BINDIR for non-macOS
+BINDIR= /usr/bin
 #BINDIR= /usr/local/bin
 #BINDIR= /dev/env/DJDIR/bin
-BINDIR= /usr/bin
 
+#if 0	/* start of skip for non-Gnu makefiles */
+endif
+
+ifeq ($(target),Darwin)
+
+# default LIBDIR for macOS
+LIBDIR= /usr/local/lib
+
+else
+#endif	/* end of skip for non-Gnu makefiles */
+
+# default LIBDIR for non-macOS
+LIBDIR= /usr/lib
 #LIBDIR= /usr/local/lib
 #LIBDIR= /dev/env/DJDIR/lib
-LIBDIR= /usr/lib
 
+#if 0	/* start of skip for non-Gnu makefiles */
+endif
+
+ifeq ($(target),Darwin)
+
+# default CALC_SHAREDIR for macOS
+CALC_SHAREDIR= /usr/local/share/calc
+
+else
+#endif	/* end of skip for non-Gnu makefiles */
+
+# default CALC_SHAREDIR for non-macOS
+CALC_SHAREDIR= /usr/share/calc
 #CALC_SHAREDIR= /usr/local/lib/calc
 #CALC_SHAREDIR= /dev/env/DJDIR/share/calc
-CALC_SHAREDIR= /usr/share/calc
+
+#if 0	/* start of skip for non-Gnu makefiles */
+endif
+#endif	/* end of skip for non-Gnu makefiles */
 
 # NOTE: Do not set CALC_INCDIR to /usr/include or /usr/local/include!!!
 #	Always be sure that the CALC_INCDIR path ends in /calc to avoid
@@ -990,7 +1054,7 @@ EXT=
 
 # The default calc versions
 #
-VERSION= 2.12.6.3
+VERSION= 2.12.6.4
 
 # Names of shared libraries with versions
 #
@@ -1263,16 +1327,6 @@ LDCONFIG:=
 # DARWIN_ARCH= -arch ppc		# PPC binary
 # DARWIN_ARCH= -arch x86_64		# native 64-bit binary
 DARWIN_ARCH=				# native binary
-#
-# Starting with El Capitan OS X 10.11, root by default could not
-# mkdir under system locations, so we now use the /usr/local tree.
-#
-OPTDIR:= /usr/local
-BINDIR:= /${OPTDIR}/bin
-LIBDIR:= /${OPTDIR}/lib
-CALC_SHAREDIR:= /${OPTDIR}/share
-CALC_INCDIR:= /${OPTDIR}/include
-SCRIPTDIR:= ${BINDIR}/cscript
 endif
 
 ##################

cal/lucas.cal

@@ -28,6 +28,10 @@
  * For a general tutorial on how to find a new largest known prime, see:
  *
  *	http://www.isthe.com/chongo/tech/math/prime/prime-tutorial.pdf
+ *
+ * Also see the reference code, available both C and go:
+ *
+ *	https://github.com/arcetri/goprime
  */
 
 /*
@@ -154,6 +158,12 @@
  *
  * Finally, one should eliminate all values of 'h*2^n-1' where
  * 'h*2^n+1' is divisible by a small primes.
+ *
+ * NOTE: Today, for world record sized h*2^n-1 primes, one might
+ *	 search for factors < 2^46 or more.  By excluding h*2^n-1
+ *	 with prime factors < 2^46, where h*2^n-1 is a bit larger
+ *	 than the largest known prime, one may exclude about 96.5%
+ *	 of candidates that have "small" prime factors.
  */
 
 pprod256 = 0;	/* product of  "primes up to 256" / "primes up to 46" */
@@ -805,84 +815,154 @@ rodseth_xhn(x, h, n)
  *
  * The above distribution was found to hold fairly well over many values of
  * odd h that are also a multiple of 3 and for many values of n where h < 2^n.
+ *
  * For example for in a sample size of 1000000 numbers of the form h*2^n-1
  * where h is an odd multiple of 3, 12996351 <= h <= 13002351,
  * 4331116 <= n <= 4332116, these are the smallest v(1) values that were found:
  *
- *  smallest percentage
+ *  smallest  percentage
- *	v(1) used
+ *	v(1)     used
- *  -------------------
+ *  --------  ---------
- *         3 40.0000%
+ *	  3   40.0000 %
- *         5 25.6833%
+ *	  5   25.6833 %
- *         9 11.6924%
+ *	  9   11.6924 %
- *        11 10.4528%
+ *	 11   10.4528 %
- *        15 4.8048%
+ *	 15    4.8048 %
- *        17 2.3458%
+ *	 17    2.3458 %
- *        21 1.6568%
+ *	 21    1.3734 %
- *        29 1.2814%
+ *	 29    1.0527 %
- *        27 0.6906%
+ *	 20    0.8595 %
- *        35 0.4529%
+ *	 27    0.5758 %
- *        39 0.3140%
+ *	 35    0.4420 %
- *        41 0.1737%
+ *	 36    0.2433 %
- *        31 0.1413%
+ *	 39    0.1779 %
- *        45 0.1173%
+ *	 41    0.0885 %
- *        51 0.0526%
+ *	 45    0.0571 %
- *        55 0.0350%
+ *	 32    0.0337 %
- *        49 0.0332%
+ *	 51    0.0289 %
- *        59 0.0218%
+ *	 44    0.0205 %
- *        69 0.0099%
+ *	 49    0.0176 %
- *        65 0.0085%
+ *	 56    0.0137 %
- *        71 0.0073%
+ *	 59    0.0108 %
- *        57 0.0062%
+ *	 57    0.0053 %
- *        85 0.0048%
+ *	 65    0.0047 %
- *        81 0.0044%
+ *	 55    0.0045 %
- *        95 0.0028%
+ *	 69    0.0031 %
- *        99 0.0017%
+ *	 71    0.0024 %
- *        77 0.0009%
+ *	 66    0.0011 %
- *        53 0.0008%
+ *	 95    0.0008 %
- *        67 0.0004%
+ *	 81    0.0008 %
- *       105 0.0004%
+ *	 77    0.0006 %
- *       111 0.0004%
+ *	 72    0.0005 %
- *       125 0.0004%
+ *	 99    0.0004 %
- *        87 0.0003%
+ *	 80    0.0003 %
- *       101 0.0002%
+ *	 74    0.0003 %
- *        83 0.0001%
+ *	 84    0.0002 %
- *       109 0.0001%
+ *	 67    0.0002 %
- *       121 0.0001%
+ *	 87    0.0001 %
- *       129 0.0001%
+ *	104    0.0001 %
+ *	129    0.0001 %
  *
+ * However, a case can be made for considering only odd values for v(1) candidates.
  * When h * 2^n-1 is prime and h is an odd multiple of 3, a smallest v(1) that
- * is even is extremely rate.  Of the list of 127287 known primes of the form
+ * is even is extremely rate.  Of the list of 146553 known primes of the form
- * h*2^n-1 when h was a multiple of 3, none has an smallest v(1) that was even.
+ * h*2^n-1 when h is an odd a multiple of 3, none has an smallest v(1) that was even.
  *
- * About 1 out of 1000000 cases when h is a multiple of 3 use v(1) > 127 as the
+ * See:
- * smallest value of v(1).
+ *
+ *	https://github.com/arcetri/verified-prime
+ *
+ * for that list of 146553 known primes of the form h*2^n-1.
+ *
+ * That same example for in a sample size of 1000000 numbers of the form h*2^n-1
+ * where h is an odd multiple of 3, 12996351 <= h <= 13002351,
+ * 4331116 <= n <= 4332116, these are the smallest odd v(1) values that were found:
+ *
+ *  smallest    percentage
+ *  odd v(1)       used
+ *  --------    ---------
+ *	  3	40.0000 %
+ *	  5	25.6833 %
+ *	  9	11.6924 %
+ *	 11	10.4528 %
+ *	 15	 4.8048 %
+ *	 17	 2.3458 %
+ *	 21	 1.6568 %
+ *	 29	 1.6174 %
+ *	 35	 0.4529 %
+ *	 27	 0.3546 %
+ *	 39	 0.3470 %
+ *	 41	 0.2159 %
+ *	 45	 0.1173 %
+ *	 31	 0.0661 %
+ *	 51	 0.0619 %
+ *	 55	 0.0419 %
+ *	 59	 0.0250 %
+ *	 49	 0.0170 %
+ *	 69	 0.0110 %
+ *	 65	 0.0098 %
+ *	 71	 0.0078 %
+ *	 85	 0.0048 %
+ *	 81	 0.0044 %
+ *	 95	 0.0038 %
+ *	 99	 0.0021 %
+ *	125	 0.0009 %
+ *	 57	 0.0007 %
+ *	111	 0.0005 %
+ *	 77	 0.0003 %
+ *	165	 0.0003 %
+ *	155	 0.0002 %
+ *	129	 0.0002 %
+ *	101	 0.0002 %
+ *	 53	 0.0001 %
+ *
+ * Moreover when evaluating odd candidates for v(1), one may cache Jacobi symbol
+ * evaluations to reduce the number of Jacobi symbol evaluations to a minimum.
+ * For example, if one tests 5 and finds that the 2nd case fails:
+ *
+ *	jacobi(5+2, h*2^n-1) != -1
+ *
+ * Then if one is later testing 9, the Jacobi symbol value for the first 1st case:
+ *
+ *	jacobi(7-2, h*2^n-1)
+ *
+ * is already known.
+ *
+ * The hit rate in the cache improves (thus fewer Jacobi symbols need evaluating)
+ * if we sort the above "smallest odd v(1) values" in numerical order.
+ * Without Jacobi symbol value caching, it requires on average
+ * 4.851377 Jacobi symbol evaluations.  With Jacobi symbol value caching
+ * cacheing, an averare of 4.348820 Jacobi symbol evaluations is needed.
  *
  * Given this information, when odd h is a multiple of 3 we try, in order,
- * these sorted values of X:
+ * these sorted odd values of X:
  *
- *  3, 5, 9, 11, 15, 17, 21, 27, 29, 31, 35, 39, 41, 45, 49, 51, 53, 55, 57, 59,
+ *	3, 5, 9, 11, 15, 17, 21, 29, 27, 35, 39, 41, 31, 45, 51, 55, 49, 59,
- *  65, 67, 69, 71, 77, 81, 83, 85, 87, 95, 99, 101, 105, 109, 111, 121, 125
+ *	69, 65, 71, 57, 85, 81, 95, 99, 77, 53, 67, 125, 111, 105, 87, 129,
+ *	101, 83, 165, 155, 149, 141, 121, 109
  *
  * And stop on the first value of X where:
  *
  *      jacobi(X-2, h*2^n-1) == 1
  *      jacobi(X+2, h*2^n-1) == -1
  *
- * If no value in that list works, we start simple search starting with X = 129
+ * Less than 1 case out of 1000000 will not be satisifed by the above sorted list.
+ * If no value in that list works, we start simple search starting with X = 167
  * and incrementing by 2 until a value of X is found.
  *
- * The x_tbl[] matrix contains those common values of X to try in order.
+ * The x_tbl[] matrix contains those values of X to try in order.
- * If all x_tbl_len fail to satisfy Ref4 condition 1, then we begin a
+ * If all x_tbl_len fail to satisfy Ref4 condition 1 (this happens less than
- * linear search at next_x until we find a proper X value.
+ * 1 in 1000000 cases), then we begin a linear search of odd values starting at
+ * next_x until we find a proper X value.
  */
-x_tbl_len = 38;
+x_tbl_len = 42;
 mat x_tbl[x_tbl_len];
 x_tbl = {
-    3, 5, 9, 11, 15, 17, 21, 27, 29, 31, 35, 39, 41, 45, 49, 51, 53, 55, 57, 59,
+    3, 5, 9, 11, 15, 17, 21, 29, 27, 35, 39, 41, 31, 45, 51, 55, 49, 59,
-    65, 67, 69, 71, 77, 81, 83, 85, 87, 95, 99, 101, 105, 109, 111, 121, 125
+    69, 65, 71, 57, 85, 81, 95, 99, 77, 53, 67, 125, 111, 105, 87, 129,
+    101, 83, 165, 155, 149, 141, 121, 109
 };
-next_x = 129;
+next_x = 167;	/* must be 2 more than the largest value in x_tbl[] */
 
 /*
  * gen_v1 - compute the v(1) for a given h*2^n-1 if we can
@@ -940,12 +1020,22 @@ next_x = 129;
  *
  *	u(2) = v(h)				 (NOTE: some call this u(2))
  *
- * so we simply return
+ * so we can always return
  *
  *	v(1) = alpha^1 + alpha^(-1)
  *	     = (2+sqrt(3)) + (2-sqrt(3))
  *	     = 4
  *
+ * In 40% of the cases when h is not a multiple of 3, 3 is a valid value
+ * for v(1).  We can test if 3 is a valid value for v(1) in this case:
+ *
+ *	if jacobi(1, h*2^n-1) == 1 and jacobi(5, h*2^n-1) == -1, then
+ *	    v(1) = 3
+ *	else
+ *	    v(1) = 4
+ *
+ * HOTE: The above "if then else" works only of h is not a multiple of 3.
+ *
  ***
  *
  * Case 2:	(h mod 3 == 0)
@@ -1049,6 +1139,10 @@ gen_v1(h, n)
 {
 	local x;	/* potential v(1) to test */
 	local i;	/* x_tbl index */
+	local v1m2;	/* X-2 1st case */
+	local v1p2;	/* X+2 2nd case */
+	local testval;			/* h*2^n-1 - value we are testing if prime */
+	local mat cached_v1[next_x];	/* cached Jacobi symbol values or 0 */
 
 	/*
 	 * check arg types
@@ -1081,14 +1175,24 @@ gen_v1(h, n)
 	 * check for Case 1:      (h mod 3 != 0)
 	 */
 	if (h % 3 != 0) {
-		/* v(1) is easy to compute */
+		if (rodseth_xhn(3, h, n) == 1) {
-		return 4;
+			/* 40% of the time, 3 works when h mod 3 != 0 */
+			return 3;
+		} else {
+			/* otherwise 4 always works when h mod 3 != 0 */
+			return 4;
+		}
 	}
 
 	/*
 	 * What follow is Case 2:      (h mod 3 == 0)
 	 */
 
+	/*
+	 * clear cache
+	 */
+	matfill(cached_v1, 0);
+
 	/*
 	 * We will look for x that satisfies conditions in Ref4, condition 1:
 	 *
@@ -1100,26 +1204,51 @@ gen_v1(h, n)
 	 *	 to the next odd value, the now jacobi(X-2, h*2^n-1)
 	 *	 does not need to be re-evaluted.
 	 */
+	testval = h*2^n-1;
 	for (i=0; i < x_tbl_len; ++i) {
 
 		/*
-		 * test Ref4 condition 1:
+		 * obtain the next test candidate
 		 */
 		x = x_tbl[i];
-		if (rodseth_xhn(x, h, n) == 1) {
 
-			/*
+		/*
-			 * found a x that satisfies Ref4 condition 1
+		 * Check x for condition 1 part 1
-			 */
+		 *
-			ldebug("gen_v1", "h= " + str(h) + " n= " + str(n) +
+		 *	jacobi(x-2, h*2^n-1) == 1
-				" v1= " + str(x) + " using tbl[ " +
+		 */
-				str(i) + " ]");
+		v1m2 = x-2;
-			return x;
+		if (cached_v1[v1m2] == 0) {
+			cached_v1[v1m2] = jacobi(v1m2, testval);
 		}
+		if (cached_v1[v1m2] != 1) {
+			continue;
+		}
+
+		/*
+		 * Check x for condition 1 part 2
+		 *
+		 *	jacobi(x+2, h*2^n-1) == -1
+		 */
+		v1p2 = x+2;
+		if (cached_v1[v1p2] == 0) {
+			cached_v1[v1p2] = jacobi(v1p2, testval);
+		}
+		if (cached_v1[v1p2] != -1) {
+			continue;
+		}
+
+		/*
+		 * found a x that satisfies Ref4 condition 1
+		 */
+		ldebug("gen_v1", "h= " + str(h) + " n= " + str(n) +
+			" v1= " + str(x) + " using tbl[ " +
+			str(i) + " ]");
+		return x;
 	}
 
 	/*
-	 * We are in that rare case (about 1 in 835 000) where none of the
+	 * We are in that rare case (less than 1 in 1 000 000) where none of the
 	 * common X values satisfy Ref4 condition 1.  We start a linear search
 	 * of odd vules at next_x from here on.
 	 */

custom/Makefile

@@ -348,7 +348,7 @@ EXT=
 
 # The default calc versions
 #
-VERSION= 2.12.6.3
+VERSION= 2.12.6.4
 
 # Names of shared libraries with versions
 #
@@ -628,16 +628,6 @@ LDCONFIG:=
 # DARWIN_ARCH= -arch ppc		# PPC binary
 # DARWIN_ARCH= -arch x86_64		# native 64-bit binary
 DARWIN_ARCH=				# native binary
-#
-# Starting with El Capitan OS X 10.11, root by default could not
-# mkdir under system locations, so we now use the /usr/local tree.
-#
-OPTDIR:= /usr/local
-BINDIR:= /${OPTDIR}/bin
-LIBDIR:= /${OPTDIR}/lib
-CALC_SHAREDIR:= /${OPTDIR}/share
-CALC_INCDIR:= /${OPTDIR}/include
-SCRIPTDIR:= ${BINDIR}/cscript
 endif
 
 ##################

custom/Makefile.head

@@ -348,7 +348,7 @@ EXT=
 
 # The default calc versions
 #
-VERSION= 2.12.6.3
+VERSION= 2.12.6.4
 
 # Names of shared libraries with versions
 #

version.c

@@ -45,7 +45,7 @@ static char *program;
 #define MAJOR_VER	2	/* major library version */
 #define MINOR_VER	12	/* minor library version */
 #define MAJOR_PATCH	6	/* major software level under library version */
-#define MINOR_PATCH	3	/* minor software level or 0 if not patched */
+#define MINOR_PATCH	4	/* minor software level or 0 if not patched */
 
 
 /*