fix and improve trigonometric help files

The trigonometric file files have had several bug fixed.

The EXAMPLE sections of trigonometric file have
been made consistent.

Added these new help files in anticipation of
the code for these new (i.e., to be written) builtin functions:

    hacovercos hacoversin havercos haversin
    ahacovercos ahacoversin ahavercos ahaversin

help/Makefile

@@ -200,18 +200,19 @@ BLT_HELP_FILES= ${BLT_HELP_FILES_3} ${BLT_HELP_FILES_5} \
 # to keep this list in nice sorted order.
 #
 DETAIL_HELP= abs access acos acosh acot acoth acovercos acoversin \
-	acsc acsch address agd append appr arg argv arrow asec asech asin \
+	acsc acsch address agd ahacovercos ahacoversin ahavercos ahaversin \
-	asinh assign atan atan2 atanh avercos aversin avg base base2 bernoulli \
+	append appr arg argv arrow asec asech asin asinh assign atan atan2 \
-	bit blk blkcpy blkfree blocks bround btrunc calc_tty calclevel \
+	atanh avercos aversin avg base base2 bernoulli bit blk blkcpy blkfree \
-	calcpath catalan ceil cfappr cfsim char cmdbuf cmp comb conj cos cosh \
+	blocks bround btrunc calc_tty calclevel calcpath catalan ceil cfappr \
-	cot coth count covercos coversin cp csc csch ctime d2dm d2dms d2g d2r \
+	cfsim char cmdbuf cmp comb conj cos cosh cot coth count covercos \
-	delete den dereference det digit digits display dms2d dp epsilon \
+	coversin cp csc csch ctime d2dm d2dms d2g d2r delete den dereference \
-	errcount errmax errno error errsym estr euler eval exp fact factor \
+	det digit digits display dms2d dp epsilon errcount errmax errno error \
-	fclose fcnt feof ferror fflush fgetc fgetfield fgetfile fgetline fgets \
+	errsym estr euler eval exp fact factor fclose fcnt feof ferror fflush \
-	fgetstr fib files floor fopen forall fpathopen fprintf fputc fputs \
+	fgetc fgetfield fgetfile fgetline fgets fgetstr fib files floor fopen \
-	fputstr frac free freebernoulli freeeuler freeglobals freeredc \
+	forall fpathopen fprintf fputc fputs fputstr frac free freebernoulli \
-	freestatics frem freopen fscan fscanf fseek fsize ftell g2d g2gm g2gms \
+	freeeuler freeglobals freeredc freestatics frem freopen fscan fscanf \
-	g2r gcd gcdrem gd getenv gms2g h2hm h2hms hash head highbit hmean \
+	fseek fsize ftell g2d g2gm g2gms g2r gcd gcdrem gd getenv gms2g h2hm \
+	h2hms hacovercos hacoversin hash havercos haversin head highbit hmean \
 	hms2h hnrmod hypot ilog ilog10 ilog2 im indices inputlevel insert int \
 	inverse iroot isalnum isalpha isassoc isatty isblk iscntrl isconfig \
 	isdefined isdigit iserror iseven isfile isgraph ishash isident isint \

help/acos

@@ -11,18 +11,18 @@ TYPES
     return	real
 
 DESCRIPTION
-    Returns the inverse cosine of x to a multiple of eps with error less in
+    Returns the inverse trigonometric cosine of x to a multiple of eps with error less in
     absolute value than .75 * eps.
 
-    This function is sometimes called arccos, is such that:
+    This function is sometimes called arccos.
-
-	cos(acos(x)) = x
-
-    and such that:
-
-	acos(x) = asrc(1/x)
 
 EXAMPLE
+    ; print acos(0), acos(0.5), acos(1)
+    1.57079632679489661923 1.04719755119659774615 0
+
+    ; print acos(-0.5), acos(-1)
+    2.09439510239319549231 3.14159265358979323846
+
     ; print acos(.5, 1e-5), acos(.5, 1e-10), acos(.5, 1e-15), acos(.5, 1e-20)
     1.0472 1.0471975512 1.047197551196598 1.04719755119659774615
 
@@ -32,6 +32,13 @@ EXAMPLE
     ; print acos(5+5i)
     0.79039774680951249644-2.64919617780647114961i
 
+    ; pi = pi(1e-20)
+    ; print acos(pi/6), acos(pi/3)
+    1.01972674369545025075 0.30604210861326573755i
+
+    ; print acos(4*pi/3)
+    2.11099630859669864896i
+
 LIMITS
     0 < eps < 1
 
@@ -42,8 +49,10 @@ LINK LIBRARY
 SEE ALSO
     sin, cos, tan, cot, sec, csc
     asin, atan, acot, asec, acsc
-    versin, coversin, vercos, avercos
+    versin, coversin, vercos, covercos
-    aversin, acoversin, covercos, acovercos
+    aversin, acoversin, avercos, acovercos
+    haversin, hacoversin, havercos, hacovercos
+    ahaversin, hacoversin, havercos, ahacovercos
     epsilon
 
 ## Copyright (C) 1999,2023  Landon Curt Noll

help/acot

@@ -11,26 +11,33 @@ TYPES
     return	real
 
 DESCRIPTION
-    Returns the inverse cotangent of x to a multiple of eps with error less in
+    Returns the inverse trigonometric cotangent of x to a multiple of eps with error less in
     absolute value than .75 * eps.
 
-    This function is sometimes called arccot, is such that:
+    This function is sometimes called arccot.
-
-	cot(acot(x)) = x
-
-    and such that:
-
-	acot(x) = atan(1/x)
 
 EXAMPLE
-    ; print acot(2, 1e-5),  acot(2, 1e-10), acot(2, 1e-15), acot(2, 1e-20)
+    ; print acot(0), acot(0.5), acot(1)
-    0.46365 0.463647609 0.463647609000806 0.46364760900080611621
+    1.57079632679489661923 1.10714871779409050302 0.78539816339744830962
+
+    ; print acot(-0.5), acot(-1)
+    2.03444393579570273544 2.35619449019234492885
+
+    ; print acot(.5, 1e-5), acot(.5, 1e-10), acot(.5, 1e-15), acot(.5, 1e-20)
+    1.10715 1.1071487178 1.107148717794091 1.10714871779409050302
 
     ; print acot(5), acot(5i)
     0.19739555984988075837 -0.20273255405408219099i
 
-    ; print acos(5+5i)
+    ; print acot(5+5i)
-    0.79039774680951249644-2.64919617780647114961i
+    0.10065855418732038003-0.09932544936725086122i
+
+    ; pi = pi(1e-20)
+    ; print acot(pi/6), acot(pi/3)
+    1.08844841969387164375 0.76234753416487458792
+
+    ; print acot(4*pi/3)
+    0.23434608748833616077
 
 LIMITS
     0 < eps < 1
@@ -42,8 +49,10 @@ LINK LIBRARY
 SEE ALSO
     sin, cos, tan, cot, sec, csc
     asin, acos, atan, asec, acsc
-    versin, coversin, vercos, avercos
+    versin, coversin, vercos, covercos
-    aversin, acoversin, covercos, acovercos
+    aversin, acoversin, avercos, acovercos
+    haversin, hacoversin, havercos, hacovercos
+    ahaversin, hacoversin, havercos, ahacovercos
     epsilon
 
 ## Copyright (C) 1999,2021,2023  Landon Curt Noll

help/acovercos

@@ -19,11 +19,24 @@ DESCRIPTION
 	acovercos(x) = asin(x - 1)
 
 EXAMPLE
+    ; print acovercos(0), acovercos(0.5), acovercos(1)
+    1.57079632679489661923 0.52359877559829887308 0
+
+    ; print acovercos(-0.5), acovercos(-1)
+    -1.57079632679489661923+0.96242365011920689498i -1.57079632679489661923+1.31695789692481670863i
+
     ; print acovercos(.5, 1e-5), acovercos(.5, 1e-10), acovercos(.5, 1e-15), acovercos(.5, 1e-20)
     0.5236 0.5235987756 0.523598775598299 0.52359877559829887308
 
-    ; print acovercos(1), acovercos(-5), acovercos(2 + 3i)
+    ; print acovercos(5), acovercos(5i)
-    0 -1.57079632679489661923+2.47788873028847500485i 0.30760364953071124992+1.86416154415788242834i
+    1.57079632679489661923-2.06343706889556054673i -0.19379313655493218978+2.33097465304931241744i
+
+    ; print acovercos(5+5i)
+    0.66881657532592728975+2.5513216254756988588i
+
+    ; pi = pi(1e-20)
+    ; print acovercos(pi/6), acovercos(pi/3), acovercos(4*pi/3)
+    0.49655704344777825545 -0.04721509173238348552 1.57079632679489661923-1.82724307729474887791i
 
 LIMITS
     0 < eps < 1
@@ -35,8 +48,10 @@ LINK LIBRARY
 SEE ALSO
     sin, cos, tan, cot, sec, csc
     asin, acos, atan, acot, asec, acsc
-    versin, coversin, vercos, avercos
+    versin, coversin, vercos, covercos
-    aversin, acoversin, covercos, acovercos
+    aversin, acoversin, avercos
+    haversin, hacoversin, havercos, hacovercos
+    ahaversin, hacoversin, havercos, ahacovercos
     epsilon
 
 ## Copyright (C) 2023  Landon Curt Noll

help/acoversin

@@ -19,11 +19,24 @@ DESCRIPTION
 	acoversin(x) = asin(1 - x)
 
 EXAMPLE
+    ; print acoversin(0), acoversin(0.5), acoversin(1)
+    1.57079632679489661923 0.52359877559829887308 0
+
+    ; print acoversin(-0.5), acoversin(-1)
+    1.57079632679489661923-0.96242365011920689499i 1.57079632679489661923-1.31695789692481670863i
+
     ; print acoversin(.5, 1e-5), acoversin(.5, 1e-10), acoversin(.5, 1e-15), acoversin(.5, 1e-20)
     0.5236 0.5235987756 0.523598775598299 0.52359877559829887308
 
-    ; print acoversin(1), acoversin(-5), acoversin(2 + 3i)
+    ; print acoversin(5), acoversin(5i)
-    0 1.57079632679489661923-2.47788873028847500481i -0.30760364953071124992-1.86416154415788242834i
+    -1.57079632679489661923+2.06343706889556054673i 0.19379313655493218978-2.33097465304931241744i
+
+    ; print acoversin(5+5i)
+    -0.66881657532592728975-2.5513216254756988588i
+
+    ; pi = pi(1e-20)
+    ; print acoversin(pi/6), acoversin(pi/3), acoversin(4*pi/3)
+    0.49655704344777825545 -0.04721509173238348552 -1.57079632679489661923+1.82724307729474887792i
 
 LIMITS
     0 < eps < 1
@@ -35,8 +48,10 @@ LINK LIBRARY
 SEE ALSO
     sin, cos, tan, cot, sec, csc
     asin, acos, atan, acot, asec, acsc
-    versin, coversin, vercos, avercos
+    versin, coversin, vercos, covercos
-    aversin, acoversin, covercos, acovercos
+    aversin, avercos, acovercos
+    haversin, hacoversin, havercos, hacovercos
+    ahaversin, hacoversin, havercos, ahacovercos
     epsilon
 
 ## Copyright (C) 2023  Landon Curt Noll

help/acsc

@@ -11,26 +11,36 @@ TYPES
     return	real
 
 DESCRIPTION
-    Returns the inverse cosecant of x to a multiple of eps with error less in
+    Returns the inverse trigonometric cosecant of x to a multiple of eps with error less in
     absolute value than .75 * eps.
 
-    This function is sometimes called arccsc, is such that:
+    This function is sometimes called arccsc.
-
-	csc(acsc(x)) = x
-
-    and such that:
-
-	acsc(x) = asin(1/x)
 
 EXAMPLE
-    ; print acsc(2, 1e-5), acsc(2, 1e-10), acsc(2, 1e-15), acsc(2, 1e-20)
+    ; print acsc(0), acsc(0.5), acsc(1)
-    0.5236 0.5235987756 0.523598775598299 0.52359877559829887308
+    Error 10454 1.57079632679489661923-1.31695789692481670863i 1.57079632679489661923
+
+    ; print acsc(-0.5), acsc(-1)
+    -1.57079632679489661923+1.31695789692481670863i -1.57079632679489661923
+
+    ; print acsc(.5, 1e-5), acsc(.5, 1e-10), acsc(.5, 1e-15)
+    1.5708-1.31696i 1.5707963268-1.3169578969i 1.570796326794897-1.316957896924817i
+
+    ; print acsc(.5, 1e-20)
+    1.57079632679489661923-1.31695789692481670863i
 
     ; print acsc(5), acsc(5i)
     0.20135792079033079145 -0.19869011034924140648i
 
-    ; print acos(5+5i)
+    ; print acsc(5+5i)
-    0.79039774680951249644-2.64919617780647114961i
+    0.09966370285979516002-0.10033029811220500594i
+
+    ; pi = pi(1e-20)
+    ; print acsc(pi/6), acsc(pi/3)
+    1.57079632679489661923-1.26327680856656556834i 1.2694227170496051647
+
+    ; print acsc(4*pi/3)
+    0.24106031273407362824
 
 LIMITS
     0 < eps < 1
@@ -42,8 +52,10 @@ LINK LIBRARY
 SEE ALSO
     sin, cos, tan, cot, sec, csc
     asin, acos, atan, acot, asec
-    versin, coversin, vercos, avercos
+    versin, coversin, vercos, covercos
-    aversin, acoversin, covercos, acovercos
+    aversin, acoversin, avercos, acovercos
+    haversin, hacoversin, havercos, hacovercos
+    ahaversin, hacoversin, havercos, ahacovercos
     epsilon
 
 ## Copyright (C) 1999,2021,2023  Landon Curt Noll

help/ahacovercos

@@ -0,0 +1,80 @@
+NAME
+    ahacovercos - inverse half coversed trigonometric cosine
+
+SYNOPSIS
+    ahacovercos(x [,eps])
+
+TYPES
+    x		number (real or complex)
+    eps		0 < real < 1, defaults to epsilon()
+
+    return	number
+
+DESCRIPTION
+    Calculate the inverse half coversed trigonometric cosine of x to a multiple of eps with error less in
+    absolute value than .75 * eps.
+
+    This function is sometimes called ahacvc, or archacovercos, or archacovercosine, is equivalent to:
+
+	ahacovercos(x) = asin(2*x - 1)
+
+EXAMPLE
+    ; print ahacovercos(0), ahacovercos(0.5), ahacovercos(1)
+    -1.57079632679489661923 0 1.57079632679489661923
+
+    ; print ahacovercos(-0.5), ahacovercos(-1)
+    -1.57079632679489661923+1.31695789692481670863i -1.57079632679489661923+1.76274717403908605045i
+
+    ; print ahacovercos(.5, 1e-5), ahacovercos(.5, 1e-10), ahacovercos(.5, 1e-15), ahacovercos(.5, 1e-20)
+    0 0 0 0
+
+    ; print ahacovercos(5), ahacovercos(5i)
+    1.57079632679489661923-2.88727095035762068499i -0.09918204387849273625+3.00312524371312921749i
+
+    ; print ahacovercos(5+5i)
+    0.73144212314418701068+3.29254348147605425075i
+
+    ; pi = pi(1e-20)
+    ; print ahacovercos(pi/6), ahacovercos(pi/3)
+    0.04721509173238348552 1.57079632679489661923-0.4311526751624601302i
+
+    ; print ahacovercos(4*pi/3)
+    1.57079632679489661923-2.68696776930293700505i
+
+LIMITS
+    0 < eps < 1
+
+LINK LIBRARY
+    NUMBER *qahacovercos(NUMBER *x, NUMBER *eps)
+    COMPLEX *c_ahacovercos(COMPLEX *x, NUMBER *eps)
+
+SEE ALSO
+    sin, cos, tan, cot, sec, csc
+    asin, acos, atan, acot, asec, acsc
+    versin, coversin, vercos, covercos
+    aversin, acoversin, avercos, acovercos
+    haversin, hacoversin, havercos, hacovercos
+    ahaversin, hacoversin, havercos
+    epsilon
+
+## Copyright (C) 2023  Landon Curt Noll
+##
+## Calc is open software; you can redistribute it and/or modify it under
+## the terms of the version 2.1 of the GNU Lesser General Public License
+## as published by the Free Software Foundation.
+##
+## Calc is distributed in the hope that it will be useful, but WITHOUT
+## ANY WARRANTY; without even the implied warranty of MERCHANTABILITY
+## or FITNESS FOR A PARTICULAR PURPOSE.	 See the GNU Lesser General
+## Public License for more details.
+##
+## A copy of version 2.1 of the GNU Lesser General Public License is
+## distributed with calc under the filename COPYING-LGPL.  You should have
+## received a copy with calc; if not, write to Free Software Foundation, Inc.
+## 51 Franklin Street, Fifth Floor, Boston, MA  02110-1301, USA.
+##
+## Under source code control:	2023/09/24 11:16:08
+## File existed as early as:	2023
+##
+## chongo <was here> /\oo/\	http://www.isthe.com/chongo/
+## Share and enjoy!  :-)	http://www.isthe.com/chongo/tech/comp/calc/

help/ahacoversin

@@ -0,0 +1,80 @@
+NAME
+    ahacoversin - inverse half coversed trigonometric sine
+
+SYNOPSIS
+    ahacoversin(x [,eps])
+
+TYPES
+    x		number (real or complex)
+    eps		0 < real < 1, defaults to epsilon()
+
+    return	number
+
+DESCRIPTION
+    Calculate the inverse half coversed trigonometric sine of x to a multiple of eps with error less in
+    absolute value than .75 * eps.
+
+    This function is sometimes called ahacovers, or ahacvs, or archacoversin, is equivalent to:
+
+	ahacoversin(x) = asin(1 - 2*x)
+
+EXAMPLE
+    ; print ahacoversin(0), ahacoversin(0.5), ahacoversin(1)
+    1.57079632679489661923 0 -1.57079632679489661923
+
+    ; print ahacoversin(-0.5), ahacoversin(-1)
+    1.57079632679489661923-1.31695789692481670863i 1.57079632679489661923-1.76274717403908605046i
+
+    ; print ahacoversin(.5, 1e-5), ahacoversin(.5, 1e-10), ahacoversin(.5, 1e-15), ahacoversin(.5, 1e-20)
+    0 0 0 0
+
+    ; print ahacoversin(5), ahacoversin(5i)
+    -1.57079632679489661923+2.88727095035762068505i 0.09918204387849273625-3.00312524371312921749i
+
+    ; print ahacoversin(5+5i)
+    -0.73144212314418701068-3.29254348147605425075i
+
+    ; pi = pi(1e-20)
+    ; print ahacoversin(pi/6), ahacoversin(pi/3)
+    -0.04721509173238348552 -1.57079632679489661923+0.4311526751624601302i
+
+    ; print ahacoversin(4*pi/3)
+    -1.57079632679489661923+2.6869677693029370051i
+
+LIMITS
+    0 < eps < 1
+
+LINK LIBRARY
+    NUMBER *qahacoversin(NUMBER *x, NUMBER *eps)
+    COMPLEX *c_ahacoversin(COMPLEX *x, NUMBER *eps)
+
+SEE ALSO
+    sin, cos, tan, cot, sec, csc
+    asin, acos, atan, acot, asec, acsc
+    versin, coversin, vercos, covercos
+    aversin, acoversin, avercos, acovercos
+    haversin, hacoversin, havercos, hacovercos
+    ahaversin, havercos, ahacovercos
+    epsilon
+
+## Copyright (C) 2023  Landon Curt Noll
+##
+## Calc is open software; you can redistribute it and/or modify it under
+## the terms of the version 2.1 of the GNU Lesser General Public License
+## as published by the Free Software Foundation.
+##
+## Calc is distributed in the hope that it will be useful, but WITHOUT
+## ANY WARRANTY; without even the implied warranty of MERCHANTABILITY
+## or FITNESS FOR A PARTICULAR PURPOSE.	 See the GNU Lesser General
+## Public License for more details.
+##
+## A copy of version 2.1 of the GNU Lesser General Public License is
+## distributed with calc under the filename COPYING-LGPL.  You should have
+## received a copy with calc; if not, write to Free Software Foundation, Inc.
+## 51 Franklin Street, Fifth Floor, Boston, MA  02110-1301, USA.
+##
+## Under source code control:	2023/09/24 11:17:28
+## File existed as early as:	2023
+##
+## chongo <was here> /\oo/\	http://www.isthe.com/chongo/
+## Share and enjoy!  :-)	http://www.isthe.com/chongo/tech/comp/calc/

help/ahavercos

@@ -0,0 +1,81 @@
+NAME
+    ahavercos - inverse half versed trigonometric cosine
+
+SYNOPSIS
+    ahavercos(x [,eps])
+
+TYPES
+    x		number (real or complex)
+    eps		0 < real < 1, defaults to epsilon()
+
+    return	real
+
+DESCRIPTION
+    Returns the inverse half versed trigonometric cosine of x to a multiple of eps with error less in
+    absolute value than .75 * eps.
+
+    This function is sometimes called ahaverc, or archavercos, or archavercosine, is equivalent to:
+
+	ahavercos(x) = acos(2*x - 1)
+
+EXAMPLE
+    ; print ahavercos(0), ahavercos(0.5), ahavercos(1)
+    3.14159265358979323846 1.57079632679489661923 0
+
+    ; print ahavercos(-0.5), ahavercos(-1)
+    3.14159265358979323846-1.31695789692481670863i 3.14159265358979323846-1.76274717403908605046i
+
+    ; print ahavercos(.5, 1e-5), ahavercos(.5, 1e-10), ahavercos(.5, 1e-15), ahavercos(.5, 1e-20)
+    1.5708 1.5707963268 1.570796326794897 1.57079632679489661923
+
+    ; print ahavercos(5), ahavercos(5i)
+    2.88727095035762068505i 1.66997837067338935548-3.00312524371312921749i
+
+    ; print ahavercos(5+5i)
+    0.83935420365070960856-3.29254348147605425075i
+
+    ; pi = pi(1e-20)
+    ; print ahavercos(pi/6), ahavercos(pi/3)
+    1.52358123506251313372 0.4311526751624601302i
+
+    ; print ahavercos(4*pi/3)
+    2.6869677693029370051i
+
+LIMITS
+    0 < eps < 1
+
+LINK LIBRARY
+    NUMBER *qahavercos_or_NULL(NUMBER *q, NUMBER *epsilon);
+    NUMBER *qahavercos(NUMBER *q, NUMBER *epsilon);
+    COMPLEX *c_ahavercos(COMPLEX *c, NUMBER *epsilon);
+
+SEE ALSO
+    sin, cos, tan, cot, sec, csc
+    asin, acos, atan, acot, asec, acsc
+    versin, coversin, vercos, covercos
+    aversin, acoversin, avercos, acovercos
+    haversin, hacoversin, havercos, hacovercos
+    ahaversin, hacoversin, ahacovercos
+    epsilon
+
+## Copyright (C) 2023  Landon Curt Noll
+##
+## Calc is open software; you can redistribute it and/or modify it under
+## the terms of the version 2.1 of the GNU Lesser General Public License
+## as published by the Free Software Foundation.
+##
+## Calc is distributed in the hope that it will be useful, but WITHOUT
+## ANY WARRANTY; without even the implied warranty of MERCHANTABILITY
+## or FITNESS FOR A PARTICULAR PURPOSE.	 See the GNU Lesser General
+## Public License for more details.
+##
+## A copy of version 2.1 of the GNU Lesser General Public License is
+## distributed with calc under the filename COPYING-LGPL.  You should have
+## received a copy with calc; if not, write to Free Software Foundation, Inc.
+## 51 Franklin Street, Fifth Floor, Boston, MA  02110-1301, USA.
+##
+## Under source code control:	2023/09/24 11:18:04
+## File existed as early as:	2023
+##
+## chongo <was here> /\oo/\	http://www.isthe.com/chongo/
+## Share and enjoy!  :-)	http://www.isthe.com/chongo/tech/comp/calc/

help/ahaversin

@@ -0,0 +1,81 @@
+NAME
+    ahaversin - inverse half versed trigonometric sine
+
+SYNOPSIS
+    ahaversin(x [,eps])
+
+TYPES
+    x		number (real or complex)
+    eps		0 < real < 1, defaults to epsilon()
+
+    return	real
+
+DESCRIPTION
+    Returns the inverse half versed trigonometric sine of x to a multiple of eps with error less in
+    absolute value than .75 * eps.
+
+    This function is sometimes called ahavers, or archaversin, is equivalent to:
+
+	ahaversin(x) = acos(1 - 2*x)
+
+EXAMPLE
+    ; print ahaversin(0), ahaversin(0.5), ahaversin(1)
+    0 1.57079632679489661923 3.14159265358979323846
+
+    ; print ahaversin(-0.5), ahaversin(-1)
+    1.31695789692481670863i 1.76274717403908605045i
+
+    ; print ahaversin(.5, 1e-5), ahaversin(.5, 1e-10), ahaversin(.5, 1e-15), ahaversin(.5, 1e-20)
+    1.5708 1.5707963268 1.570796326794897 1.57079632679489661923
+
+    ; print ahaversin(5), ahaversin(5i)
+    3.14159265358979323846-2.88727095035762068499i 1.47161428291640388291+3.0031252437131292174i
+
+    ; print ahaversin(5+5i)
+    2.30223844993908362988+3.29254348147605425081i
+
+    ; pi = pi(1e-20)
+    ; print ahaversin(pi/6), ahaversin(pi/3)
+    1.61801141852728010475 3.14159265358979323846-0.4311526751624601302i
+
+    ; print ahaversin(4*pi/3)
+    3.14159265358979323846-2.68696776930293700505i
+
+LIMITS
+    0 < eps < 1
+
+LINK LIBRARY
+    NUMBER *qahaversin_or_NULL(NUMBER *q, NUMBER *epsilon);
+    NUMBER *qahaversin(NUMBER *q, NUMBER *epsilon);
+    COMPLEX *c_ahaversin(COMPLEX *c, NUMBER *epsilon);
+
+SEE ALSO
+    sin, cos, tan, cot, sec, csc
+    asin, acos, atan, acot, asec, acsc
+    versin, coversin, vercos, covercos
+    aversin, acoversin, avercos, acovercos
+    haversin, hacoversin, havercos, hacovercos
+    hacoversin, havercos, ahacovercos
+    epsilon
+
+## Copyright (C) 2023  Landon Curt Noll
+##
+## Calc is open software; you can redistribute it and/or modify it under
+## the terms of the version 2.1 of the GNU Lesser General Public License
+## as published by the Free Software Foundation.
+##
+## Calc is distributed in the hope that it will be useful, but WITHOUT
+## ANY WARRANTY; without even the implied warranty of MERCHANTABILITY
+## or FITNESS FOR A PARTICULAR PURPOSE.	 See the GNU Lesser General
+## Public License for more details.
+##
+## A copy of version 2.1 of the GNU Lesser General Public License is
+## distributed with calc under the filename COPYING-LGPL.  You should have
+## received a copy with calc; if not, write to Free Software Foundation, Inc.
+## 51 Franklin Street, Fifth Floor, Boston, MA  02110-1301, USA.
+##
+## Under source code control:	2023/09/24 11:19:54
+## File existed as early as:	2023
+##
+## chongo <was here> /\oo/\	http://www.isthe.com/chongo/
+## Share and enjoy!  :-)	http://www.isthe.com/chongo/tech/comp/calc/

help/asec

@@ -11,26 +11,33 @@ TYPES
     return	real
 
 DESCRIPTION
-    Returns the inverse secant of x to a multiple of eps with error less in
+    Returns the inverse trigonometric secant of x to a multiple of eps with error less in
     absolute value than .75 * eps.
 
-    This function is sometimes called arcsec, is such that:
+    This function is sometimes called arcsec.
-
-	sec(asec(x)) = x
-
-    and such that:
-
-	asec(x) = acos(1/x)
 
 EXAMPLE
-    ; print asec(2, 1e-5), asec(2, 1e-10), asec(2, 1e-15), asec(2, 1e-20)
+    ; print asec(0), asec(0.5), asec(1)
-    1.0472 1.0471975512 1.047197551196598 1.04719755119659774615
+    Error 10453 1.31695789692481670863i 0
+
+    ; print asec(-0.5), asec(-1)
+    3.14159265358979323846-1.31695789692481670863i 3.14159265358979323846
+
+    ; print asec(.5, 1e-5), asec(.5, 1e-10), asec(.5, 1e-15), asec(.5, 1e-20)
+    1.31695i 1.316957897i 1.316957896924816i 1.31695789692481670863i
 
     ; print asec(5), asec(5i)
     1.36943840600456582778 1.57079632679489661923+0.19869011034924140647i
 
-    ; print acos(5+5i)
+    ; print asec(5+5i)
-    0.79039774680951249644-2.64919617780647114961i
+    1.4711326239351014592+0.10033029811220500594i
+
+    ; pi = pi(1e-20)
+    ; print asec(pi/6), asec(pi/3)
+    1.26327680856656556836i 0.30137360974529145454
+
+    ; print asec(4*pi/3)
+    1.32973601406082299099
 
 LIMITS
     0 < eps < 1
@@ -42,8 +49,10 @@ LINK LIBRARY
 SEE ALSO
     sin, cos, tan, cot, sec, csc
     asin, acos, atan, acot, acsc
-    versin, coversin, vercos, avercos
+    versin, coversin, vercos, covercos
-    aversin, acoversin, covercos, acovercos
+    aversin, acoversin, avercos, acovercos
+    haversin, hacoversin, havercos, hacovercos
+    ahaversin, hacoversin, havercos, ahacovercos
     epsilon
 
 ## Copyright (C) 1999,2021,2023  Landon Curt Noll

help/asin

@@ -11,18 +11,18 @@ TYPES
     return	real
 
 DESCRIPTION
-    Returns the inverse sine of x to a multiple of eps with error less in
+    Returns the inverse trigonometric sine of x to a multiple of eps with error less in
     absolute value than .75 * eps.
 
-    This function is sometimes called arcsin, is such that:
+    This function is sometimes called arcsin.
-
-	sin(asin(x)) = x
-
-    and such that:
-
-	asin(x) = acsc(1/x)
 
 EXAMPLE
+    ; print asin(0), asin(0.5), asin(1)
+    0 0.52359877559829887308 1.57079632679489661923
+
+    ; print asin(-0.5), asin(-1)
+    -0.52359877559829887308 -1.57079632679489661923
+
     ; print asin(.5, 1e-5), asin(.5, 1e-10), asin(.5, 1e-15), asin(.5, 1e-20)
     0.5236 0.5235987756 0.523598775598299 0.52359877559829887308
 
@@ -32,6 +32,13 @@ EXAMPLE
     ; print asin(5+5i)
     0.78039857998538412279+2.64919617780647114961i
 
+    ; pi = pi(1e-20)
+    ; print asin(pi/6), asin(pi/3)
+    0.55106958309944636848 1.57079632679489661923-0.30604210861326573756i
+
+    ; print asin(4*pi/3)
+    1.57079632679489661923-2.11099630859669864896i
+
 LIMITS
     0 < eps < 1
 
@@ -42,8 +49,10 @@ LINK LIBRARY
 SEE ALSO
     sin, cos, tan, cot, sec, csc
     acos, atan, acot, asec, acsc
-    versin, coversin, vercos, avercos
+    versin, coversin, vercos, covercos
-    aversin, acoversin, covercos, acovercos
+    aversin, acoversin, avercos, acovercos
+    haversin, hacoversin, havercos, hacovercos
+    ahaversin, hacoversin, havercos, ahacovercos
     epsilon
 
 ## Copyright (C) 1999,2021,2023  Landon Curt Noll

help/atan

@@ -11,26 +11,33 @@ TYPES
     return	real
 
 DESCRIPTION
-    Returns the inverse tangent of x to a multiple of eps with error less in
+    Returns the inverse trigonometric tangent of x to a multiple of eps with error less in
     absolute value than .75 * eps.
 
-    This function is sometimes called arctan, is such that:
+    This function is sometimes called arctan.
-
-	tan(atan(x)) = x
-
-    and such that:
-
-	atan(x) = acot(1/x)
 
 EXAMPLE
-    ; print atan(2, 1e-5), atan(2, 1e-10), atan(2, 1e-15), atan(2, 1e-20)
+    ; print atan(0), atan(0.5), atan(1)
-    1.10715 1.1071487178 1.107148717794091 1.10714871779409050302
+    0 0.46364760900080611621 0.78539816339744830962
+
+    ; print atan(-0.5), atan(-1)
+    -0.46364760900080611621 -0.78539816339744830962
+
+    ; print atan(.5, 1e-5), atan(.5, 1e-10), atan(.5, 1e-15), atan(.5, 1e-20)
+    0.46365 0.463647609 0.463647609000806 0.46364760900080611621
 
     ; print atan(5), atan(5i)
     1.37340076694501586086 1.57079632679489661923+0.20273255405408219099i
 
-    ; print acos(5+5i)
+    ; print atan(5+5i)
-    0.79039774680951249644-2.64919617780647114961i
+    ~1.47013777260757623920+0.09932544936725086122i
+
+    ; pi = pi(1e-20)
+    ; print atan(pi/6), atan(pi/3)
+    0.48234790710102497548 0.80844879263002203131
+
+    ; print atan(4*pi/3)
+    1.33645023930656045846
 
 LIMITS
     0 < eps < 1
@@ -42,8 +49,10 @@ LINK LIBRARY
 SEE ALSO
     sin, cos, tan, cot, sec, csc
     asin, acos, acot, asec, acsc
-    versin, coversin, vercos, avercos
+    versin, coversin, vercos, covercos
-    aversin, acoversin, covercos, acovercos
+    aversin, acoversin, avercos, acovercos
+    haversin, hacoversin, havercos, hacovercos
+    ahaversin, hacoversin, havercos, ahacovercos
     epsilon
 
 ## Copyright (C) 1999,2023  Landon Curt Noll

help/avercos

@@ -19,11 +19,24 @@ DESCRIPTION
 	avercos(x) = acos(x - 1)
 
 EXAMPLE
+    ; print avercos(0), avercos(0.5), avercos(1)
+    3.14159265358979323846 2.09439510239319549231 1.57079632679489661923
+
+    ; print avercos(-0.5), avercos(-1)
+    3.14159265358979323846-0.96242365011920689499i 3.14159265358979323846-1.31695789692481670863i
+
     ; print avercos(.5, 1e-5), avercos(.5, 1e-10), avercos(.5, 1e-15), avercos(.5, 1e-20)
     2.0944 2.0943951024 2.094395102393195 2.09439510239319549231
 
-    ; print avercos(2), avercos(-5), avercos(2 + 3i)
+    ; print avercos(5), avercos(5i)
-    0 3.14159265358979323846-2.47788873028847500481i 1.26319267726418536931-1.86416154415788242834i
+    2.06343706889556054673i 1.76458946334982880901-2.33097465304931241744i
+
+    ; print avercos(5+5i)
+    0.90197975146896932949-2.5513216254756988588i
+
+    ; pi = pi(1e-20)
+    ; print avercos(pi/6), avercos(pi/3), avercos(4*pi/3)
+    2.06735337024267487468 1.52358123506251313372 1.82724307729474887792i
 
 LIMITS
     0 < eps < 1
@@ -36,8 +49,10 @@ LINK LIBRARY
 SEE ALSO
     sin, cos, tan, cot, sec, csc
     asin, acos, atan, acot, asec, acsc
-    versin, coversin, vercos, avercos
+    versin, coversin, vercos, covercos
-    aversin, acoversin, covercos, acovercos
+    aversin, acoversin, acovercos
+    haversin, hacoversin, havercos, hacovercos
+    ahaversin, hacoversin, havercos, ahacovercos
     epsilon
 
 ## Copyright (C) 2023  Landon Curt Noll

help/aversin

@@ -19,11 +19,24 @@ DESCRIPTION
 	aversin(x) = acos(1 - x)
 
 EXAMPLE
+    ; print aversin(0), aversin(0.5), aversin(1)
+    0 1.04719755119659774615 1.57079632679489661923
+
+    ; print aversin(-0.5), aversin(-1)
+    0.96242365011920689498i 1.31695789692481670863i
+
     ; print aversin(.5, 1e-5), aversin(.5, 1e-10), aversin(.5, 1e-15), aversin(.5, 1e-20)
     1.0472 1.0471975512 1.047197551196598 1.04719755119659774615
 
-    ; print aversin(0), aversin(-5), aversin(2 + 3i)
+    ; print aversin(5), aversin(5i)
-    0 2.47788873028847500485i 1.87839997632560786916+1.86416154415788242831i
+    3.14159265358979323846-2.06343706889556054673i 1.37700319023996442947+2.33097465304931241742i
+
+    ; print aversin(5+5i)
+    2.23961290212082390892+2.55132162547569885882i
+
+    ; pi = pi(1e-20)
+    ; print aversin(pi/6), aversin(pi/3), aversin(4*pi/3)
+    1.07423928334711836378 1.61801141852728010475 3.14159265358979323846-1.82724307729474887791i
 
 LIMITS
     0 < eps < 1
@@ -36,8 +49,10 @@ LINK LIBRARY
 SEE ALSO
     sin, cos, tan, cot, sec, csc
     asin, acos, atan, acot, asec, acsc
-    versin, coversin
+    versin, coversin, vercos, covercos
-    acoversin
+    acoversin, avercos, acovercos
+    haversin, hacoversin, havercos, hacovercos
+    ahaversin, hacoversin, havercos, ahacovercos
     epsilon
 
 ## Copyright (C) 2023  Landon Curt Noll

help/cos

@@ -11,10 +11,13 @@ TYPES
     return	number
 
 DESCRIPTION
-    Calculate the cosine of x to a multiple of eps with error less in
+    Calculate the trigonometric cosine of x to a multiple of eps with error less in
     absolute value than .75 * eps.
 
 EXAMPLE
+    ; print cos(1/2), cos(5/7), cos(42/7)
+    0.87758256189037271612 0.75556134670069659847 0.96017028665036602055
+
     ; print cos(1, 1e-5), cos(1, 1e-10), cos(1, 1e-15), cos(1, 1e-20)
     0.5403 0.5403023059 0.54030230586814 0.5403023058681397174
 
@@ -22,11 +25,8 @@ EXAMPLE
     -4.18963-9.10923i -4.189625691-9.1092278938i
 
     ; pi = pi(1e-20)
-    ; print cos(pi/3, 1e-10), cos(pi/2, 1e-10), cos(pi, 1e-10)
+    ; print cos(pi/6), cos(pi/2), cos(pi)
-    0.5 0 -1
+    0.86602540378443864676 0 -1
-
-    ; print cos(1/2), cos(5/7), cos(42/7)
-    0.87758256189037271612 0.75556134670069659847 0.96017028665036602055
 
 LIMITS
     0 < eps < 1
@@ -38,8 +38,10 @@ LINK LIBRARY
 SEE ALSO
     sin, tan, cot, sec, csc
     asin, acos, atan, acot, asec, acsc
-    versin, coversin, vercos, avercos
+    versin, coversin, vercos, covercos
-    aversin, acoversin, covercos, acovercos
+    aversin, acoversin, avercos, acovercos
+    haversin, hacoversin, havercos, hacovercos
+    ahaversin, hacoversin, havercos, ahacovercos
     epsilon
 
 ## Copyright (C) 1999,2021,2023  Landon Curt Noll

help/cot

@@ -11,7 +11,7 @@ TYPES
     return	real
 
 DESCRIPTION
-    Calculate the cotangent of x to a multiple of eps, with error less
+    Calculate the trigonometric cotangent of x to a multiple of eps, with error less
     in absolute value than .75 * eps.
 
     This function is equivalent to:
@@ -19,6 +19,9 @@ DESCRIPTION
 	cot(x) = cos(x) / sin(x)
 
 EXAMPLE
+    ; print cot(1/2), cot(5/7), cot(42/7)
+    1.83048772171245191927 1.15339160419695060142 -3.43635300418012783207
+
     ; print cot(1, 1e-5), cot(1, 1e-10), cot(1, 1e-15), cot(1, 1e-20)
     0.64209 0.6420926159 0.642092615934331 0.64209261593433070301
 
@@ -26,11 +29,10 @@ EXAMPLE
     ~-0.00373977357605613583-~0.99675796378381737782i ~-0.00373971037383300017-~0.99675779657435500069i
 
     ; pi = pi(1e-20)
-    ; print cot(pi/12, 1e-10), cot(pi/6, 1e-10), cot(pi/3, 1e-10), cot(pi/2, 1e-10)
+    ; print cot(pi/6), cot(pi/2), cot(pi)
-    3.7320508076 1.7320508076 0.5773502692 0
+    1.73205080756887729353 0 -378303066284076833726.92240766640467212978
 
-    ; print cot(1/2), cot(5/7), cot(42/7)
+    ; # NOTE: The huge value for cot(pi) is due to the fact that pi is not exact.
-    1.83048772171245191927 1.15339160419695060142 -3.43635300418012783207
 
 LIMITS
     0 < eps < 1
@@ -42,8 +44,10 @@ LINK LIBRARY
 SEE ALSO
     sin, cos, tan, sec, csc
     asin, acos, atan, acot, asec, acsc
-    versin, coversin, vercos, avercos
+    versin, coversin, vercos, covercos
-    aversin, acoversin, covercos, acovercos
+    aversin, acoversin, avercos, acovercos
+    haversin, hacoversin, havercos, hacovercos
+    ahaversin, hacoversin, havercos, ahacovercos
     epsilon
 
 ## Copyright (C) 1999,2021,2023  Landon Curt Noll

help/covercos

@@ -14,13 +14,13 @@ DESCRIPTION
     Calculate the coversed trigonometric cosine of x to a multiple of eps with error less in
     absolute value than .75 * eps.
 
-    This function is sometimes called cvc, is equivalent to:
+    This function is sometimes called cvc, or covercosine, is equivalent to:
 
 	covercos(x) = 1 + sin(x)
 
 EXAMPLE
-    ; print covercos(0.2), covercos(3/7), covercos(-31)
+    ; print covercos(1/2), covercos(5/7), covercos(42/7)
-    1.19866933079506121546 1.41557185499305200807 1.40403764532306500605
+    1.47942553860420300027 1.6550778971785185742 0.72058450180107412719
 
     ; print covercos(1, 1e-5), covercos(1, 1e-10), covercos(1, 1e-15), covercos(1, 1e-20)
     1.84147 1.8414709848 1.841470984807896 1.84147098480789650665
@@ -29,8 +29,8 @@ EXAMPLE
     10.1545-4.16891i 10.1544991469-4.16890696i
 
     ; pi = pi(1e-20)
-    ; print covercos(pi/6, 1e-10), covercos(pi/2, 1e-10), covercos(pi, 1e-10), covercos(3*pi/2, 1e-10)
+    ; print covercos(pi/6), covercos(pi/2), covercos(pi)
-    1.5 2 1 0
+    1.5 2 1
 
 LIMITS
     0 < eps < 1
@@ -42,8 +42,10 @@ LINK LIBRARY
 SEE ALSO
     sin, cos, tan, cot, sec, csc
     asin, acos, atan, acot, asec, acsc
-    versin, coversin, vercos, avercos
+    versin, coversin, vercos
-    aversin, acoversin, covercos, acovercos
+    aversin, acoversin, avercos, acovercos
+    haversin, hacoversin, havercos, hacovercos
+    ahaversin, hacoversin, havercos, ahacovercos
     epsilon
 
 ## Copyright (C) 2023  Landon Curt Noll

help/coversin

@@ -19,8 +19,8 @@ DESCRIPTION
 	coversin(x) = 1 - sin(x)
 
 EXAMPLE
-    ; print coversin(0.2), coversin(3/7), coversin(-31)
+    ; print coversin(1/2), coversin(5/7), coversin(42/7)
-    0.80133066920493878454 0.58442814500694799193 0.59596235467693499395
+    0.52057446139579699973 0.3449221028214814258 1.27941549819892587281
 
     ; print coversin(1, 1e-5), coversin(1, 1e-10), coversin(1, 1e-15), coversin(1, 1e-20)
     0.15853 0.1585290152 0.158529015192104 0.15852901519210349335
@@ -29,8 +29,8 @@ EXAMPLE
     -8.1545+4.16891i -8.1544991469+4.16890696i
 
     ; pi = pi(1e-20)
-    ; print coversin(pi/6, 1e-10), coversin(pi/2, 1e-10), coversin(pi, 1e-10), coversin(3*pi/2, 1e-10)
+    ; print coversin(pi/6), coversin(pi/2), coversin(pi)
-    0.5 0 1 2
+    0.5 0 1
 
 LIMITS
     0 < eps < 1
@@ -42,8 +42,10 @@ LINK LIBRARY
 SEE ALSO
     sin, cos, tan, cot, sec, csc
     asin, acos, atan, acot, asec, acsc
-    versin, coversin, vercos, avercos
+    versin, vercos, covercos
-    aversin, acoversin, covercos, acovercos
+    aversin, acoversin, avercos, acovercos
+    haversin, hacoversin, havercos, hacovercos
+    ahaversin, hacoversin, havercos, ahacovercos
     epsilon
 
 ## Copyright (C) 2023  Landon Curt Noll

help/csc

@@ -11,7 +11,7 @@ TYPES
     return	real
 
 DESCRIPTION
-    Calculate the cosecant of x to a multiple of eps, with error less
+    Calculate the trigonometric cosecant of x to a multiple of eps, with error less
     in absolute value than .75 * eps.
 
     This function is equivalent to:
@@ -19,6 +19,9 @@ DESCRIPTION
 	csc(x) = 1 / sin(x)
 
 EXAMPLE
+    ; print csc(1/2), csc(5/7), csc(42/7)
+    2.08582964293348818577 1.52653601091884339347 -3.57889954725440563736
+
     ; print csc(1, 1e-5), csc(1, 1e-10), csc(1, 1e-15), csc(1, 1e-20)
     1.1884 1.1883951058 1.188395105778121 1.18839510577812121626
 
@@ -26,11 +29,10 @@ EXAMPLE
     ~0.09047318155450436310+~0.04120099965201690801i ~0.09047320975303232503+~0.04120098628887626238i
 
     ; pi = pi(1e-20)
-    ; print csc(pi/6, 1e-10), csc(pi/3, 1e-10), csc(4*pi/3, 1e-10)
+    ; print csc(pi/6), csc(pi/2), csc(pi)
-    2 1.1547005384 -1.1547005384
+    2 1 378303066284076833726.92240766640467212978
 
-    ; print csc(1/2), csc(5/7), csc(42/7)
+    ; # NOTE: The huge value for csc(pi) is because pi, as calculated, is not exact.
-    2.08582964293348818577 1.52653601091884339347 -3.57889954725440563736
 
 LIMITS
     0 < eps < 1
@@ -42,8 +44,10 @@ LINK LIBRARY
 SEE ALSO
     sin, cos, tan, cot, sec
     asin, acos, atan, acot, asec, acsc
-    versin, coversin, vercos, avercos
+    versin, coversin, vercos, covercos
-    aversin, acoversin, covercos, acovercos
+    aversin, acoversin, avercos, acovercos
+    haversin, hacoversin, havercos, hacovercos
+    ahaversin, hacoversin, havercos, ahacovercos
     epsilon
 
 ## Copyright (C) 1999,2023  Landon Curt Noll

help/hacovercos

@@ -0,0 +1,71 @@
+NAME
+    hacovercos - half coversed trigonometric cosine
+
+SYNOPSIS
+    hacovercos(x [,eps])
+
+TYPES
+    x		number (real or complex)
+    eps		0 < real < 1, defaults to epsilon()
+
+    return	number
+
+DESCRIPTION
+    Calculate the half coversed trigonometric cosine of x to a multiple of eps with error less in
+    absolute value than .75 * eps.
+
+    This function is sometimes called hacvc, or hacovercosine, is equivalent to:
+
+	hacovercos(x) = covercos(x) / 2 = (1 + sin(x)) / 2
+
+EXAMPLE
+    ; print hacovercos(1/2), hacovercos(5/7), hacovercos(42/7)
+    ~0.73971276930210150014 0.8275389485892592871 ~0.36029225090053706360
+
+    ; print hacovercos(1, 1e-5), hacovercos(1, 1e-10), hacovercos(1, 1e-15), hacovercos(1, 1e-20)
+    0.920735 0.9207354924 0.920735492403948 ~0.92073549240394825332
+
+    ; print hacovercos(2 + 3i, 1e-5), hacovercos(2 + 3i, 1e-10)
+    5.07725-2.084455i 5.07724957345-2.08445348i
+
+    ; pi = pi(1e-20)
+    ; print hacovercos(pi/6), hacovercos(pi/2), hacovercos(pi)
+    0.75 1 0.5
+
+LIMITS
+    0 < eps < 1
+
+LINK LIBRARY
+    NUMBER *qhacovercos(NUMBER *x, NUMBER *eps)
+    COMPLEX *c_hacovercos(COMPLEX *x, NUMBER *eps)
+
+SEE ALSO
+    sin, cos, tan, cot, sec, csc
+    asin, acos, atan, acot, asec, acsc
+    versin, coversin, vercos, covercos
+    aversin, acoversin, avercos, acovercos
+    haversin, hacoversin, havercos
+    ahaversin, hacoversin, havercos, ahacovercos
+    epsilon
+
+## Copyright (C) 2023  Landon Curt Noll
+##
+## Calc is open software; you can redistribute it and/or modify it under
+## the terms of the version 2.1 of the GNU Lesser General Public License
+## as published by the Free Software Foundation.
+##
+## Calc is distributed in the hope that it will be useful, but WITHOUT
+## ANY WARRANTY; without even the implied warranty of MERCHANTABILITY
+## or FITNESS FOR A PARTICULAR PURPOSE.	 See the GNU Lesser General
+## Public License for more details.
+##
+## A copy of version 2.1 of the GNU Lesser General Public License is
+## distributed with calc under the filename COPYING-LGPL.  You should have
+## received a copy with calc; if not, write to Free Software Foundation, Inc.
+## 51 Franklin Street, Fifth Floor, Boston, MA  02110-1301, USA.
+##
+## Under source code control:	2023/09/24 11:10:08
+## File existed as early as:	2023
+##
+## chongo <was here> /\oo/\	http://www.isthe.com/chongo/
+## Share and enjoy!  :-)	http://www.isthe.com/chongo/tech/comp/calc/

help/hacoversin

@@ -0,0 +1,71 @@
+NAME
+    hacoversin - half coversed trigonometric sine
+
+SYNOPSIS
+    hacoversin(x [,eps])
+
+TYPES
+    x		number (real or complex)
+    eps		0 < real < 1, defaults to epsilon()
+
+    return	number
+
+DESCRIPTION
+    Calculate the half coversed trigonometric sine of x to a multiple of eps with error less in
+    absolute value than .75 * eps.
+
+    This function is sometimes called hacovers, or hacvs, is equivalent to:
+
+	hacoversin(x) = coversin(x) / 2 = (1 - sin(x)) / 2
+
+EXAMPLE
+    ; print hacoversin(1/2), hacoversin(5/7), hacoversin(42/7)
+    ~0.26028723069789849986 0.1724610514107407129 ~0.63970774909946293640
+
+    ; print hacoversin(1, 1e-5), hacoversin(1, 1e-10), hacoversin(1, 1e-15), hacoversin(1, 1e-20)
+    0.079265 0.0792645076 0.079264507596052 ~0.07926450759605174668
+
+    ; print hacoversin(2 + 3i, 1e-5), hacoversin(2 + 3i, 1e-10)
+    -4.07725+2.084455i -4.07724957345+2.08445348i
+
+    ; pi = pi(1e-20)
+    ; print hacoversin(pi/6), hacoversin(pi/2), hacoversin(pi)
+    0.25 0 0.5
+
+LIMITS
+    0 < eps < 1
+
+LINK LIBRARY
+    NUMBER *qhacoversin(NUMBER *x, NUMBER *eps)
+    COMPLEX *c_hacoversin(COMPLEX *x, NUMBER *eps)
+
+SEE ALSO
+    sin, cos, tan, cot, sec, csc
+    asin, acos, atan, acot, asec, acsc
+    versin, coversin, vercos, covercos
+    aversin, acoversin, avercos, acovercos
+    haversin, havercos, hacovercos
+    ahaversin, hacoversin, havercos, ahacovercos
+    epsilon
+
+## Copyright (C) 2023  Landon Curt Noll
+##
+## Calc is open software; you can redistribute it and/or modify it under
+## the terms of the version 2.1 of the GNU Lesser General Public License
+## as published by the Free Software Foundation.
+##
+## Calc is distributed in the hope that it will be useful, but WITHOUT
+## ANY WARRANTY; without even the implied warranty of MERCHANTABILITY
+## or FITNESS FOR A PARTICULAR PURPOSE.	 See the GNU Lesser General
+## Public License for more details.
+##
+## A copy of version 2.1 of the GNU Lesser General Public License is
+## distributed with calc under the filename COPYING-LGPL.  You should have
+## received a copy with calc; if not, write to Free Software Foundation, Inc.
+## 51 Franklin Street, Fifth Floor, Boston, MA  02110-1301, USA.
+##
+## Under source code control:	2023/09/24 11:11:11
+## File existed as early as:	2023
+##
+## chongo <was here> /\oo/\	http://www.isthe.com/chongo/
+## Share and enjoy!  :-)	http://www.isthe.com/chongo/tech/comp/calc/

help/hash

@@ -47,7 +47,7 @@ LINK LIBRARY
 SEE ALSO
     ishash, fnv, sha1
 
-## Copyright (C) 1999-2007,2014,2021  Landon Curt Noll
+## Copyright (C) 2023  Landon Curt Noll
 ##
 ## Calc is open software; you can redistribute it and/or modify it under
 ## the terms of the version 2.1 of the GNU Lesser General Public License
@@ -63,8 +63,8 @@ SEE ALSO
 ## received a copy with calc; if not, write to Free Software Foundation, Inc.
 ## 51 Franklin Street, Fifth Floor, Boston, MA  02110-1301, USA.
 ##
-## Under source code control:	1996/03/12 23:10:01
+## Under source code control:	2023/09/24 11:12:16
-## File existed as early as:	1996
+## File existed as early as:	2023
 ##
 ## chongo <was here> /\oo/\	http://www.isthe.com/chongo/
 ## Share and enjoy!  :-)	http://www.isthe.com/chongo/tech/comp/calc/

help/havercos

@@ -0,0 +1,71 @@
+NAME
+    havercos - half versed trigonometric cosine
+
+SYNOPSIS
+    havercos(x [,eps])
+
+TYPES
+    x		number (real or complex)
+    eps		0 < real < 1, defaults to epsilon()
+
+    return	number
+
+DESCRIPTION
+    Calculate the half versed trigonometric cosine of x to a multiple of eps with error less in
+    absolute value than .75 * eps.
+
+    This function is sometimes called haverc, or haverconsine, is equivalent to:
+
+	havercos(x) = vercos(x) / 2 = (1 + cos(x)) / 2
+
+EXAMPLE
+    ; print havercos(1/2), havercos(5/7), havercos(42/7)
+    0.93879128094518635806 ~0.87778067335034829924 ~0.98008514332518301028
+
+    ; print havercos(1, 1e-5), havercos(1, 1e-10), havercos(1, 1e-15), havercos(1, 1e-20)
+    0.77015 0.77015115295 0.77015115293407 0.7701511529340698587
+
+    ; print havercos(2 + 3i, 1e-5), havercos(2 + 3i, 1e-10)
+    -1.594815-4.554615i -1.5948128455-4.5546139469i
+
+    ; pi = pi(1e-20)
+    ; print havercos(pi/6), havercos(pi/2), havercos(pi)
+    0.93301270189221932338 0.5 0
+
+LIMITS
+    0 < eps < 1
+
+LINK LIBRARY
+    NUMBER *qhavercos(NUMBER *x, NUMBER *eps)
+    COMPLEX *c_havercos(COMPLEX *x, NUMBER *eps)
+
+SEE ALSO
+    sin, cos, tan, cot, sec, csc
+    asin, acos, atan, acot, asec, acsc
+    versin, coversin, vercos, covercos
+    aversin, acoversin, avercos, acovercos
+    haversin, hacoversin, hacovercos
+    ahaversin, hacoversin, havercos, ahacovercos
+    epsilon
+
+## Copyright (C) 2023  Landon Curt Noll
+##
+## Calc is open software; you can redistribute it and/or modify it under
+## the terms of the version 2.1 of the GNU Lesser General Public License
+## as published by the Free Software Foundation.
+##
+## Calc is distributed in the hope that it will be useful, but WITHOUT
+## ANY WARRANTY; without even the implied warranty of MERCHANTABILITY
+## or FITNESS FOR A PARTICULAR PURPOSE.	 See the GNU Lesser General
+## Public License for more details.
+##
+## A copy of version 2.1 of the GNU Lesser General Public License is
+## distributed with calc under the filename COPYING-LGPL.  You should have
+## received a copy with calc; if not, write to Free Software Foundation, Inc.
+## 51 Franklin Street, Fifth Floor, Boston, MA  02110-1301, USA.
+##
+## Under source code control:	2023/09/24 11:13:28
+## File existed as early as:	2023
+##
+## chongo <was here> /\oo/\	http://www.isthe.com/chongo/
+## Share and enjoy!  :-)	http://www.isthe.com/chongo/tech/comp/calc/

help/haversin

@@ -0,0 +1,71 @@
+NAME
+    haversin - half versed trigonometric sine
+
+SYNOPSIS
+    haversin(x [,eps])
+
+TYPES
+    x		number (real or complex)
+    eps		0 < real < 1, defaults to epsilon()
+
+    return	number
+
+DESCRIPTION
+    Calculate the half versed trigonometric sine of x to a multiple of eps with error less in
+    absolute value than .75 * eps.
+
+    This function is sometimes called havers, is equivalent to:
+
+	haversin(x) = versin(x) / 2 = (1 - cos(x)) / 2
+
+EXAMPLE
+    ; print haversin(1/2), haversin(5/7), haversin(42/7)
+    0.06120871905481364194 ~0.12221932664965170076 ~0.01991485667481698972
+
+    ; print haversin(1, 1e-5), haversin(1, 1e-10), haversin(1, 1e-15), haversin(1, 1e-20)
+    0.22985 0.22984884705 0.22984884706593 0.2298488470659301413
+
+    ; print haversin(2 + 3i, 1e-5), haversin(2 + 3i, 1e-10)
+    2.594815+4.554615i 2.5948128455+4.5546139469i
+
+    ; pi = pi(1e-20)
+    ; print haversin(pi/6), haversin(pi/2), haversin(pi)
+    0.06698729810778067662 0.5 1
+
+LIMITS
+    0 < eps < 1
+
+LINK LIBRARY
+    NUMBER *qhaversin(NUMBER *x, NUMBER *eps)
+    COMPLEX *c_haversin(COMPLEX *x, NUMBER *eps)
+
+SEE ALSO
+    sin, cos, tan, cot, sec, csc
+    asin, acos, atan, acot, asec, acsc
+    versin, coversin, vercos, covercos
+    aversin, acoversin, avercos, acovercos
+    hacoversin, havercos, hacovercos
+    ahaversin, hacoversin, havercos, ahacovercos
+    epsilon
+
+## Copyright (C) 2023  Landon Curt Noll
+##
+## Calc is open software; you can redistribute it and/or modify it under
+## the terms of the version 2.1 of the GNU Lesser General Public License
+## as published by the Free Software Foundation.
+##
+## Calc is distributed in the hope that it will be useful, but WITHOUT
+## ANY WARRANTY; without even the implied warranty of MERCHANTABILITY
+## or FITNESS FOR A PARTICULAR PURPOSE.	 See the GNU Lesser General
+## Public License for more details.
+##
+## A copy of version 2.1 of the GNU Lesser General Public License is
+## distributed with calc under the filename COPYING-LGPL.  You should have
+## received a copy with calc; if not, write to Free Software Foundation, Inc.
+## 51 Franklin Street, Fifth Floor, Boston, MA  02110-1301, USA.
+##
+## Under source code control:	2023/09/24 11:14:48
+## File existed as early as:	2023
+##
+## chongo <was here> /\oo/\	http://www.isthe.com/chongo/
+## Share and enjoy!  :-)	http://www.isthe.com/chongo/tech/comp/calc/

help/sec

@@ -11,7 +11,7 @@ TYPES
     return	real
 
 DESCRIPTION
-    Calculate the secant of x to a multiple of eps, with error less
+    Calculate the trigonometric secant of x to a multiple of eps, with error less
     in absolute value than .75 * eps.
 
     This function is equivalent to:
@@ -19,6 +19,9 @@ DESCRIPTION
 	sec(x) = 1 / cos(x)
 
 EXAMPLE
+    ; print sec(1/2), sec(5/7), sec(42/7)
+    1.13949392732454912231 1.3235192673191814545 1.04148192659510767648
+
     ; print sec(1, 1e-5), sec(1, 1e-10), sec(1, 1e-15), sec(1, 1e-20)
     1.85082 1.8508157177 1.850815717680926 1.85081571768092561791
 
@@ -26,11 +29,10 @@ EXAMPLE
     ~-0.04167497639869547021+~0.09061109101765280898i ~-0.04167496441100888150+~0.09061113719571288336i
 
     ; pi = pi(1e-20)
-    ; print 1e-10), sec(pi/6, 1e-10), sec(pi/3, 1e-10), sec(pi, 1e-10)
+    ; print sec(pi/6), sec(pi/2), sec(pi)
-    1.1547005384 2 -1
+    1.15470053837925152902 756606132568153667453.84481533280934425956 -1
 
-    ; print sec(1/2), sec(5/7), sec(42/7)
+    ; # NOTE: The huge value for sec(pi/2) is because pi, as calculated, is not exact.
-    1.13949392732454912231 1.3235192673191814545 1.04148192659510767648
 
 LIMITS
     0 < eps < 1
@@ -42,8 +44,10 @@ LINK LIBRARY
 SEE ALSO
     sin, cos, tan, cot, csc
     asin, acos, atan, acot, asec, acsc
-    versin, coversin, vercos, avercos
+    versin, coversin, vercos, covercos
-    aversin, acoversin, covercos, acovercos
+    aversin, acoversin, avercos, acovercos
+    haversin, hacoversin, havercos, hacovercos
+    ahaversin, hacoversin, havercos, ahacovercos
     epsilon
 
 ## Copyright (C) 1999,2023  Landon Curt Noll

help/sin

@@ -11,10 +11,13 @@ TYPES
     return	number
 
 DESCRIPTION
-    Calculate the sine of x to a multiple of eps with error less in
+    Calculate the trigonometric sine of x to a multiple of eps with error less in
     absolute value than .75 * eps.
 
 EXAMPLE
+    ; print sin(1/2), sin(5/7), sin(42/7)
+    0.47942553860420300027 0.6550778971785185742 -0.27941549819892587281
+
     ; print sin(1, 1e-5), sin(1, 1e-10), sin(1, 1e-15), sin(1, 1e-20)
     0.84147 0.8414709848 0.841470984807896 0.84147098480789650665
 
@@ -22,12 +25,9 @@ EXAMPLE
     9.1545-4.16891i 9.1544991469-4.16890696i
 
     ; pi = pi(1e-20)
-    ; print sin(pi/6, 1e-10), sin(pi/2, 1e-10), sin(pi, 1e-10)
+    ; print sin(pi/6), sin(pi/2), sin(pi)
     0.5 1 0
 
-    ; print sin(1/2), sin(5/7), sin(42/7)
-    0.47942553860420300027 0.6550778971785185742 -0.27941549819892587281
-
 LIMITS
     0 < eps < 1
 
@@ -38,8 +38,10 @@ LINK LIBRARY
 SEE ALSO
     cos, tan, cot, sec, csc
     asin, acos, atan, acot, asec, acsc
-    versin, coversin, vercos, avercos
+    versin, coversin, vercos, covercos
-    aversin, acoversin, covercos, acovercos
+    aversin, acoversin, avercos, acovercos
+    haversin, hacoversin, havercos, hacovercos
+    ahaversin, hacoversin, havercos, ahacovercos
     epsilon
 
 ## Copyright (C) 1999,2021,2023  Landon Curt Noll

help/tan

@@ -11,7 +11,7 @@ TYPES
     return	real
 
 DESCRIPTION
-    Calculate the tangent of x to a multiple of eps, with error less
+    Calculate the trigonometric tangent of x to a multiple of eps, with error less
     in absolute value than .75 * eps.
 
     This function is equivalent to:
@@ -19,6 +19,9 @@ DESCRIPTION
 	tan(x) = sin(x) / cos(x)
 
 EXAMPLE
+    ; print tan(1/2), tan(5/7), tan(42/7)
+    0.54630248984379051326 0.8670082185107029875 -0.29100619138474915705
+
     ; print tan(1, 1e-5), tan(1, 1e-10), tan(1, 1e-15), tan(1, 1e-20)
     1.55741 1.5574077247 1.557407724654902 1.55740772465490223051
 
@@ -26,11 +29,10 @@ EXAMPLE
     ~-0.00376408798745471014+~1.00323845857938817252i ~-0.00376402563894634508+~1.00323862734859967572i
 
     ; pi = pi(1e-20)
-    ; print tan(0, 1e-10), tan(pi/6, 1e-10), tan(pi/3, 1e-10), tan(pi, 1e-10)
+    ; print tan(pi/6), tan(pi/2), tan(pi)
-    0 0.5773502692 1.7320508076 0
+    0.57735026918962576451 756606132568153667453.84481533280934425956 0
 
-    ; print tan(1/2), tan(5/7), tan(42/7)
+    ; # NOTE: The huge value for tan(pi/2) is due to the fact that pi is not exact.
-    0.54630248984379051326 0.8670082185107029875 -0.29100619138474915705
 
 LIMITS
     0 < eps < 1
@@ -42,8 +44,10 @@ LINK LIBRARY
 SEE ALSO
     sin, cos, cot, sec, csc
     asin, acos, atan, acot, asec, acsc
-    versin, coversin, vercos, avercos
+    versin, coversin, vercos, covercos
-    aversin, acoversin, covercos, acovercos
+    aversin, acoversin, avercos, acovercos
+    haversin, hacoversin, havercos, hacovercos
+    ahaversin, hacoversin, havercos, ahacovercos
     epsilon
 
 ## Copyright (C) 1999,2023  Landon Curt Noll

help/vercos

@@ -14,13 +14,13 @@ DESCRIPTION
     Calculate the versed trigonometric cosine of x to a multiple of eps with error less in
     absolute value than .75 * eps.
 
-    This function is sometimes called verc, is equivalent to:
+    This function is sometimes called verc, or vercosin, is equivalent to:
 
 	vercos(x) = 1 + cos(x)
 
 EXAMPLE
-    ; print vercos(0.2), vercos(3/7), vercos(-31)
+    ; print vercos(1/2), vercos(5/7), vercos(42/7)
-    1.98006657784124163112 1.90956035167416667403 1.91474235780453127896
+    1.87758256189037271612 1.75556134670069659847 1.96017028665036602055
 
     ; print vercos(1, 1e-5), vercos(1, 1e-10), vercos(1, 1e-15), vercos(1, 1e-20)
     1.5403 1.5403023059 1.54030230586814 1.5403023058681397174
@@ -29,8 +29,8 @@ EXAMPLE
     -3.18963-9.10923i -3.189625691-9.1092278938i
 
     ; pi = pi(1e-20)
-    ; print vercos(pi/3, 1e-10), vercos(pi/2, 1e-10), vercos(pi, 1e-10), vercos(3*pi/2, 1e-10)
+    ; print vercos(pi/6), vercos(pi/2), vercos(pi)
-    1.5 1 0 1
+    1.86602540378443864676 1 0
 
 LIMITS
     0 < eps < 1
@@ -42,8 +42,10 @@ LINK LIBRARY
 SEE ALSO
     sin, cos, tan, cot, sec, csc
     asin, acos, atan, acot, asec, acsc
-    versin, coversin, vercos, avercos
+    versin, coversin, covercos
-    aversin, acoversin, covercos, acovercos
+    aversin, acoversin, avercos, acovercos
+    haversin, hacoversin, havercos, hacovercos
+    ahaversin, hacoversin, havercos, ahacovercos
     epsilon
 
 ## Copyright (C) 2023  Landon Curt Noll

help/versin

@@ -19,8 +19,8 @@ DESCRIPTION
 	versin(x) = 1 - cos(x)
 
 EXAMPLE
-    ; print versin(0.2), versin(3/7), versin(-31)
+    ; print versin(1/2), versin(5/7), versin(42/7)
-    0.01993342215875836888 0.09043964832583332597 0.08525764219546872104
+    0.12241743810962728388 0.24443865329930340153 0.03982971334963397945
 
     ; print versin(1, 1e-5), versin(1, 1e-10), versin(1, 1e-15), versin(1, 1e-20)
     0.4597 0.4596976941 0.45969769413186 0.4596976941318602826
@@ -29,8 +29,8 @@ EXAMPLE
     5.18963+9.10923i 5.189625691+9.1092278938i
 
     ; pi = pi(1e-20)
-    ; print versin(pi/3, 1e-10), versin(pi/2, 1e-10), versin(pi, 1e-10), versin(3*pi/2, 1e-10)
+    ; print versin(pi/6), versin(pi/2), versin(pi)
-    0.5 1 2 1
+    0.13397459621556135324 1 2
 
 LIMITS
     0 < eps < 1
@@ -42,8 +42,10 @@ LINK LIBRARY
 SEE ALSO
     sin, cos, tan, cot, sec, csc
     asin, acos, atan, acot, asec, acsc
-    versin, coversin, vercos, avercos
+    coversin, vercos, covercos
-    aversin, acoversin, covercos, acovercos
+    aversin, acoversin, avercos, acovercos
+    haversin, hacoversin, havercos, hacovercos
+    ahaversin, hacoversin, havercos, ahacovercos
     epsilon
 
 ## Copyright (C) 2023  Landon Curt Noll