drygdryg/calc
1
0
Fork 0
mirror of https://github.com/lcn2/calc.git synced 2025-08-16 01:03:29 +03:00
Code Issues Packages Projects Releases Wiki Activity

fix and improve trigonometric help files

Browse Source 
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
This commit is contained in:
Landon Curt Noll
2023-09-24 12:52:14 -07:00
parent db80afb843
commit ab95e47c0a
30 changed files with 928 additions and 176 deletions
Download Patch File Download Diff File Expand all files Collapse all files

25
help/Makefile
View File

@@ -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 \
asinh assign atan atan2 atanh avercos aversin avg base base2 bernoulli \
bit blk blkcpy blkfree blocks bround btrunc calc_tty calclevel \
calcpath catalan ceil cfappr cfsim char cmdbuf cmp comb conj cos cosh \
cot coth count covercos coversin cp csc csch ctime d2dm d2dms d2g d2r \
delete den dereference det digit digits display dms2d dp epsilon \
errcount errmax errno error errsym estr euler eval exp fact factor \
fclose fcnt feof ferror fflush fgetc fgetfield fgetfile fgetline fgets \
fgetstr fib files floor fopen forall fpathopen fprintf fputc fputs \
fputstr frac free freebernoulli freeeuler freeglobals freeredc \
freestatics frem freopen fscan fscanf fseek fsize ftell g2d g2gm g2gms \
g2r gcd gcdrem gd getenv gms2g h2hm h2hms hash head highbit hmean \
acsc acsch address agd ahacovercos ahacoversin ahavercos ahaversin \
append appr arg argv arrow asec asech asin asinh assign atan atan2 \
atanh avercos aversin avg base base2 bernoulli bit blk blkcpy blkfree \
blocks bround btrunc calc_tty calclevel calcpath catalan ceil cfappr \
cfsim char cmdbuf cmp comb conj cos cosh cot coth count covercos \
coversin cp csc csch ctime d2dm d2dms d2g d2r delete den dereference \
det digit digits display dms2d dp epsilon errcount errmax errno error \
errsym estr euler eval exp fact factor fclose fcnt feof ferror fflush \
fgetc fgetfield fgetfile fgetline fgets fgetstr fib files floor fopen \
forall fpathopen fprintf fputc fputs fputstr frac free freebernoulli \
freeeuler freeglobals freeredc freestatics frem freopen fscan fscanf \
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 \

29
help/acos
View File

@@ -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:
cos(acos(x)) = x
and such that:
acos(x) = asrc(1/x)
This function is sometimes called arccos.
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
aversin, acoversin, covercos, acovercos
versin, coversin, vercos, covercos
aversin, acoversin, avercos, acovercos
haversin, hacoversin, havercos, hacovercos
ahaversin, hacoversin, havercos, ahacovercos
epsilon
## Copyright (C) 1999,2023 Landon Curt Noll

37
help/acot
View File

@@ -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:
cot(acot(x)) = x
and such that:
acot(x) = atan(1/x)
This function is sometimes called arccot.
EXAMPLE
; print acot(2, 1e-5), acot(2, 1e-10), acot(2, 1e-15), acot(2, 1e-20)
0.46365 0.463647609 0.463647609000806 0.46364760900080611621
; print acot(0), acot(0.5), acot(1)
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)
0.79039774680951249644-2.64919617780647114961i
; print acot(5+5i)
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
aversin, acoversin, covercos, acovercos
versin, coversin, vercos, covercos
aversin, acoversin, avercos, acovercos
haversin, hacoversin, havercos, hacovercos
ahaversin, hacoversin, havercos, ahacovercos
epsilon
## Copyright (C) 1999,2021,2023 Landon Curt Noll

23
help/acovercos
View File

@@ -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)
0 -1.57079632679489661923+2.47788873028847500485i 0.30760364953071124992+1.86416154415788242834i
; print acovercos(5), acovercos(5i)
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
aversin, acoversin, covercos, acovercos
versin, coversin, vercos, covercos
aversin, acoversin, avercos
haversin, hacoversin, havercos, hacovercos
ahaversin, hacoversin, havercos, ahacovercos
epsilon
## Copyright (C) 2023 Landon Curt Noll

23
help/acoversin
View File

@@ -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)
0 1.57079632679489661923-2.47788873028847500481i -0.30760364953071124992-1.86416154415788242834i
; print acoversin(5), acoversin(5i)
-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
aversin, acoversin, covercos, acovercos
versin, coversin, vercos, covercos
aversin, avercos, acovercos
haversin, hacoversin, havercos, hacovercos
ahaversin, hacoversin, havercos, ahacovercos
epsilon
## Copyright (C) 2023 Landon Curt Noll

40
help/acsc
View File

@@ -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:
csc(acsc(x)) = x
and such that:
acsc(x) = asin(1/x)
This function is sometimes called arccsc.
EXAMPLE
; print acsc(2, 1e-5), acsc(2, 1e-10), acsc(2, 1e-15), acsc(2, 1e-20)
0.5236 0.5235987756 0.523598775598299 0.52359877559829887308
; print acsc(0), acsc(0.5), acsc(1)
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)
0.79039774680951249644-2.64919617780647114961i
; print acsc(5+5i)
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
aversin, acoversin, covercos, acovercos
versin, coversin, vercos, covercos
aversin, acoversin, avercos, acovercos
haversin, hacoversin, havercos, hacovercos
ahaversin, hacoversin, havercos, ahacovercos
epsilon
## Copyright (C) 1999,2021,2023 Landon Curt Noll

80
help/ahacovercos Normal file
View File

@@ -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/

80
help/ahacoversin Normal file
View File

@@ -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/

81
help/ahavercos Normal file
View File

@@ -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/

81
help/ahaversin Normal file
View File

@@ -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/

37
help/asec
View File

@@ -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:
sec(asec(x)) = x
and such that:
asec(x) = acos(1/x)
This function is sometimes called arcsec.
EXAMPLE
; print asec(2, 1e-5), asec(2, 1e-10), asec(2, 1e-15), asec(2, 1e-20)
1.0472 1.0471975512 1.047197551196598 1.04719755119659774615
; print asec(0), asec(0.5), asec(1)
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)
0.79039774680951249644-2.64919617780647114961i
; print asec(5+5i)
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
aversin, acoversin, covercos, acovercos
versin, coversin, vercos, covercos
aversin, acoversin, avercos, acovercos
haversin, hacoversin, havercos, hacovercos
ahaversin, hacoversin, havercos, ahacovercos
epsilon
## Copyright (C) 1999,2021,2023 Landon Curt Noll

29
help/asin
View File

@@ -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:
sin(asin(x)) = x
and such that:
asin(x) = acsc(1/x)
This function is sometimes called arcsin.
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
aversin, acoversin, covercos, acovercos
versin, coversin, vercos, covercos
aversin, acoversin, avercos, acovercos
haversin, hacoversin, havercos, hacovercos
ahaversin, hacoversin, havercos, ahacovercos
epsilon
## Copyright (C) 1999,2021,2023 Landon Curt Noll

37
help/atan
View File

@@ -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:
tan(atan(x)) = x
and such that:
atan(x) = acot(1/x)
This function is sometimes called arctan.
EXAMPLE
; print atan(2, 1e-5), atan(2, 1e-10), atan(2, 1e-15), atan(2, 1e-20)
1.10715 1.1071487178 1.107148717794091 1.10714871779409050302
; print atan(0), atan(0.5), atan(1)
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)
0.79039774680951249644-2.64919617780647114961i
; print atan(5+5i)
~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
aversin, acoversin, covercos, acovercos
versin, coversin, vercos, covercos
aversin, acoversin, avercos, acovercos
haversin, hacoversin, havercos, hacovercos
ahaversin, hacoversin, havercos, ahacovercos
epsilon
## Copyright (C) 1999,2023 Landon Curt Noll

23
help/avercos
View File

@@ -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)
0 3.14159265358979323846-2.47788873028847500481i 1.26319267726418536931-1.86416154415788242834i
; print avercos(5), avercos(5i)
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
aversin, acoversin, covercos, acovercos
versin, coversin, vercos, covercos
aversin, acoversin, acovercos
haversin, hacoversin, havercos, hacovercos
ahaversin, hacoversin, havercos, ahacovercos
epsilon
## Copyright (C) 2023 Landon Curt Noll

23
help/aversin
View File

@@ -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)
0 2.47788873028847500485i 1.87839997632560786916+1.86416154415788242831i
; print aversin(5), aversin(5i)
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
acoversin
versin, coversin, vercos, covercos
acoversin, avercos, acovercos
haversin, hacoversin, havercos, hacovercos
ahaversin, hacoversin, havercos, ahacovercos
epsilon
## Copyright (C) 2023 Landon Curt Noll

18
help/cos
View File

@@ -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)
0.5 0 -1
; print cos(1/2), cos(5/7), cos(42/7)
0.87758256189037271612 0.75556134670069659847 0.96017028665036602055
; print cos(pi/6), cos(pi/2), cos(pi)
0.86602540378443864676 0 -1
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
aversin, acoversin, covercos, acovercos
versin, coversin, vercos, covercos
aversin, acoversin, avercos, acovercos
haversin, hacoversin, havercos, hacovercos
ahaversin, hacoversin, havercos, ahacovercos
epsilon
## Copyright (C) 1999,2021,2023 Landon Curt Noll

18
help/cot
View File

@@ -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)
3.7320508076 1.7320508076 0.5773502692 0
; print cot(pi/6), cot(pi/2), cot(pi)
1.73205080756887729353 0 -378303066284076833726.92240766640467212978
; print cot(1/2), cot(5/7), cot(42/7)
1.83048772171245191927 1.15339160419695060142 -3.43635300418012783207
; # NOTE: The huge value for cot(pi) is due to the fact that pi is not exact.
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
aversin, acoversin, covercos, acovercos
versin, coversin, vercos, covercos
aversin, acoversin, avercos, acovercos
haversin, hacoversin, havercos, hacovercos
ahaversin, hacoversin, havercos, ahacovercos
epsilon
## Copyright (C) 1999,2021,2023 Landon Curt Noll

16
help/covercos
View File

@@ -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)
1.19866933079506121546 1.41557185499305200807 1.40403764532306500605
; print covercos(1/2), covercos(5/7), covercos(42/7)
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)
1.5 2 1 0
; print covercos(pi/6), covercos(pi/2), covercos(pi)
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
aversin, acoversin, covercos, acovercos
versin, coversin, vercos
aversin, acoversin, avercos, acovercos
haversin, hacoversin, havercos, hacovercos
ahaversin, hacoversin, havercos, ahacovercos
epsilon
## Copyright (C) 2023 Landon Curt Noll

14
help/coversin
View File

@@ -19,8 +19,8 @@ DESCRIPTION
coversin(x) = 1 - sin(x)
EXAMPLE
; print coversin(0.2), coversin(3/7), coversin(-31)
0.80133066920493878454 0.58442814500694799193 0.59596235467693499395
; print coversin(1/2), coversin(5/7), coversin(42/7)
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)
0.5 0 1 2
; print coversin(pi/6), coversin(pi/2), coversin(pi)
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
aversin, acoversin, covercos, acovercos
versin, vercos, covercos
aversin, acoversin, avercos, acovercos
haversin, hacoversin, havercos, hacovercos
ahaversin, hacoversin, havercos, ahacovercos
epsilon
## Copyright (C) 2023 Landon Curt Noll

18
help/csc
View File

@@ -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)
2 1.1547005384 -1.1547005384
; print csc(pi/6), csc(pi/2), csc(pi)
2 1 378303066284076833726.92240766640467212978
; print csc(1/2), csc(5/7), csc(42/7)
2.08582964293348818577 1.52653601091884339347 -3.57889954725440563736
; # NOTE: The huge value for csc(pi) is because pi, as calculated, is not exact.
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
aversin, acoversin, covercos, acovercos
versin, coversin, vercos, covercos
aversin, acoversin, avercos, acovercos
haversin, hacoversin, havercos, hacovercos
ahaversin, hacoversin, havercos, ahacovercos
epsilon
## Copyright (C) 1999,2023 Landon Curt Noll

71
help/hacovercos Normal file
View File

@@ -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/

71
help/hacoversin Normal file
View File

@@ -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/

6
help/hash
View File

@@ -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
## File existed as early as: 1996
## Under source code control: 2023/09/24 11:12:16
## 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/

71
help/havercos Normal file
View File

@@ -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/

71
help/haversin Normal file
View File

@@ -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/

18
help/sec
View File

@@ -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)
1.1547005384 2 -1
; print sec(pi/6), sec(pi/2), sec(pi)
1.15470053837925152902 756606132568153667453.84481533280934425956 -1
; print sec(1/2), sec(5/7), sec(42/7)
1.13949392732454912231 1.3235192673191814545 1.04148192659510767648
; # NOTE: The huge value for sec(pi/2) is because pi, as calculated, is not exact.
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
aversin, acoversin, covercos, acovercos
versin, coversin, vercos, covercos
aversin, acoversin, avercos, acovercos
haversin, hacoversin, havercos, hacovercos
ahaversin, hacoversin, havercos, ahacovercos
epsilon
## Copyright (C) 1999,2023 Landon Curt Noll

16
help/sin
View File

@@ -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
aversin, acoversin, covercos, acovercos
versin, coversin, vercos, covercos
aversin, acoversin, avercos, acovercos
haversin, hacoversin, havercos, hacovercos
ahaversin, hacoversin, havercos, ahacovercos
epsilon
## Copyright (C) 1999,2021,2023 Landon Curt Noll

18
help/tan
View File

@@ -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)
0 0.5773502692 1.7320508076 0
; print tan(pi/6), tan(pi/2), tan(pi)
0.57735026918962576451 756606132568153667453.84481533280934425956 0
; print tan(1/2), tan(5/7), tan(42/7)
0.54630248984379051326 0.8670082185107029875 -0.29100619138474915705
; # NOTE: The huge value for tan(pi/2) is due to the fact that pi is not exact.
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
aversin, acoversin, covercos, acovercos
versin, coversin, vercos, covercos
aversin, acoversin, avercos, acovercos
haversin, hacoversin, havercos, hacovercos
ahaversin, hacoversin, havercos, ahacovercos
epsilon
## Copyright (C) 1999,2023 Landon Curt Noll

16
help/vercos
View File

@@ -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)
1.98006657784124163112 1.90956035167416667403 1.91474235780453127896
; print vercos(1/2), vercos(5/7), vercos(42/7)
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)
1.5 1 0 1
; print vercos(pi/6), vercos(pi/2), vercos(pi)
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
aversin, acoversin, covercos, acovercos
versin, coversin, covercos
aversin, acoversin, avercos, acovercos
haversin, hacoversin, havercos, hacovercos
ahaversin, hacoversin, havercos, ahacovercos
epsilon
## Copyright (C) 2023 Landon Curt Noll

14
help/versin
View File

@@ -19,8 +19,8 @@ DESCRIPTION
versin(x) = 1 - cos(x)
EXAMPLE
; print versin(0.2), versin(3/7), versin(-31)
0.01993342215875836888 0.09043964832583332597 0.08525764219546872104
; print versin(1/2), versin(5/7), versin(42/7)
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)
0.5 1 2 1
; print versin(pi/6), versin(pi/2), versin(pi)
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
aversin, acoversin, covercos, acovercos
coversin, vercos, covercos
aversin, acoversin, avercos, acovercos
haversin, hacoversin, havercos, hacovercos
ahaversin, hacoversin, havercos, ahacovercos
epsilon
## Copyright (C) 2023 Landon Curt Noll