RE: HP48-HP50G : Error functions ERF(z), FRESNEL integrals, PROBIT, etc.

[Gil]

Please note that the last uploaded version 9f
has a "print" error at variable VERSIO.

Please change it

from
Version 9e
2023.12.14
campart
@hotmail.com

into
Version 9f
2023.12.14
campart
@hotmail.com

ERF "\\<< \"
1 Arg

.real 
x

(no limit for real)
.complex
- 
(
a b
)

-
 '
a+i*b
'
- '
r*e^(i\\Gh)
'
- '
r*EXP(i\\Gh)
'

(b\\=/0 &

|
a
| 
& 
|
b
| 
\\<=9)
\" DROP DTAG DUP DUP DUP DUP PUSH euler DUP DUP RE SWAP IM \\oo \\->NUM 9 .01 \\-> \\<-z0 \\<-z \\<-re \\<-im \\<-max.x \\<-max.z critic
\\<< arg.num inf.erf max.arg DROP -105 SF EVAL ABS \\oo ==
IF
THEN SIGN -105 CF '\\oo' EVAL *
END DUP DUP \"ERF

(
\" SWAP DROP \\<-z0 + \"
)
\" + \"'\" \"\" SREPL DROP DUP \"F\" \"F
c
\" SREPL DROP 4 ROLLD SWAP DUP TYPE 9 ==
IF
THEN DROP \\<-z
END
CASE \\<-re DUP 0 \\>= 1 -1 IFTE SWAP ABS '\\oo' EVAL ==
THEN 1 * DUP 1 SWAP - ROT 5 ROLL DROP2 UNROT
END DROP \\<-im 0 == \\<-re ABS critic \\<= AND
IF
THEN DROP '\\.S(-\\<-re,\\<-re,e^-X^2,X)/\\v/\\pi' \\->NUM
ELSE \\<-im 0 == \\<-re 0 \\<= AND
IF
THEN -1
ELSE 1
END SWAP \\<-im 0 ==
IF
THEN ABS
END complex
END \\<-im 0 ==
IF
THEN \\<-re ABS critic \\<=
IF
THEN 1 SWAP
ELSE
END 4 ROLL DROP DUP 1 SWAP - ROT * DUP 0 <
IF
THEN DUP NEG 1 + ROT DROP SWAP
END \\<-re ABS critic >
IF
THEN SWAP
END UNROT
ELSE ROT DROP SWAP UNROT
END
END \\<-im 0 \\=/
IF
THEN 4 ROLL DROP
END
\\>> 'IERR' PURGE POP
\\>>" "



" INV¦ "\\<< \"
1 Arg

.a real 
p
, |
p
| \\<= 1
<or equivalent
complex 
z 
= 
p

- 
(
a b
)

-
 '
a+i*b
'
- '
r*e^(i\\Gh)
'
- '
r*EXP(i\\Gh)
' 
>


\" DROP PUSH DUP DUP 0 .9065 \\-> p0 \\<-z0 p \\<-z critic
\\<< p0 euler DUP DUP '\\<-z' STO 'p' STO arg.num \\<-z0 'p0' STO IM 0 \\=/ p ABS 1 \\>= AND p RE ABS 1 > OR
IF
THEN POP p0 \"
Not
 OK as:
.Real 
|
p
|
 > 1
or
.Complex p
with 
|
p
|
 > 1\" DOERR
END -105 CF \"
Inv
ERF
\" \"
(
\" + p0 + \"
)
\" + \"'\" \"\" SREPL DROP p IM 0 \\=/
IF
THEN p inv\\166z
ELSE p RE DUP 'p' STO 0 \\>= 1 -1 IFTE p ABS 'p' STO
CASE p 1 ==
THEN '\\oo' EVAL
END p 0 ==
THEN 0
END 1 p - critic \\<=
THEN
\\<< 1 p - 0 .5 x UTPN 2 * -
\\>> 'x' 2 ROOT
END p '\\.S(-x,x,e^-X^2,X)/\\v/\\pi' - 'x' 2 ROOT
END * { x IERR } PURGE
END POP
\\>>
\\>>" "



" INV¦c "\\<< \"
1 Arg

.a real 
p
 in [0 2]
<or equivalent
complex 
z 
= 
p

- 
(
a b
)

-
 '
a+i*b
'
- '
r*e^(i\\Gh)
'


 - '
r*EXP(i\\Gh)
' 
>
\" DROP PUSH DUP DUP 0 .9065 \\-> p0 \\<-z0 p \\<-z critic
\\<< p0 euler DUP '\\<-z' STO 'p' STO arg.num \\<-z0 'p0' STO \"
Inv
ERF
c

\" \"
(
\" + p0 + \"
)
\" + \"'\" \"\" SREPL DROP p IM 0 \\=/ 1 p - ABS 1 \\>= AND p RE 0 < OR p RE 2 > OR
IF
THEN DROP POP p0 \"
Not
 OK as:
.Real p<0, >2
or
.Complex with
IM(p)\\=/0 &

|
1-p
|
 \\>= 1
\" DOERR
ELSE 1 p - IM 0 \\=/
IF
THEN 1 p - inv\\166z
ELSE -105 CF p RE DUP 'p' STO 1 \\<=
IF
THEN 1
ELSE 2 p - 'p' STO -1
END
CASE p 0 ==
THEN '\\oo' EVAL
END p 1 ==
THEN 0
END p critic \\<=
THEN
\\<< p 0 .5 x UTPN 2 * -
\\>> 'x' 2 ROOT
END 1 p - RE '\\.S(-x,x,e^-X^2,X)/\\v/\\pi' - 'x' 2 ROOT
END * { x IERR } PURGE
END
END POP
\\>>
\\>>" "



" PROBIT "\\<< \"
1 Arg

.a real 
p
 in [0 1]
<or equivalent
complex 
z 
= 
p

- 
(
a b
)

-
 '
a+i*b
'
- '
r*e^(i\\Gh)
'
- '
r*EXP(i\\Gh)
' 
>


\" DROP PUSH DUP DUP 0 \\-> p0 \\<-z0 p \\<-z
\\<< p0 euler DUP DUP '\\<-z' STO 'p' STO arg.num \\<-z0 'p0' STO \"Probit
\" \"
(
\" + p0 + \"
)
\" + \"'\" \"\" SREPL DROP p IM 0 \\=/ p RE 1 > OR p RE 0 < OR
IF
THEN DROP POP \"
Not
 OK as:
.Real p<0, >1
or
.Complex with
IM(p) \\=/ 0 or
RE(p) <0, >1
\" DOERR
END SWAP DROP -105 CF
CASE p 1 ==
THEN \\oo EVAL
END p 0 ==
THEN \\oo EVAL NEG
END p RE 2 * 1 - INV\\166 2 \\v/ * \\->NUM SWAP DROP
END POP
\\>>
\\>>" "



" ERFi "\\<< \"
1 Arg

.real 
x


|
x
|
 \\<=9
.Or complex
- 
(
a b
)

-
 '
a+i*b
'
- '
r*e^(i\\Gh)
'
- '
r*EXP(i\\Gh)
'

 
|
a
| 
& 
|
b
| 
\\<=7
\" DROP PUSH DUP DUP DUP DUP euler DUP DUP RE SWAP IM 9 7 0 \\-> \\<-z0 \\<-z \\<-re \\<-im \\<-max.x \\<-max.z \\<-inf1x
\\<< arg.num inf2x DUP \"OK\" SAME
IF
THEN DROP inf1x.erfi \\<-inf1x 1 ==
IF
THEN inf1x.end \"ERF
i

\" SWAP + \"(+\" \"(\" SREPL 5 ROLL 5 ROLL 3 DROPN SWAP
ELSE max.arg -105 SF EVAL ABS \\oo ==
IF
THEN SIGN -105 CF '\\oo' EVAL * '\\<-z0' STO
ELSE DROP
END \"ERF
i

\" \"
(
\" + \\<-z0 + \"
)
\" + \"'\" \"\" SREPL ROT DROP2
CASE \\<-z ABS '\\oo' EVAL ==
THEN \\<-z SIGN -105 CF \\oo EVAL *
END \\<-z i * \\->NUM ERF 4 ROLLD 3 DROPN i / \\->NUM
END DUP IM 0 == { RE } IFT POP
END
ELSE \"ERF
i

\" SWAP + SWAP 5 ROLL 5 ROLL 5 ROLL 3 DROPN
END POP
\\>>
\\>>" "



" iERFc "\\<< \"
1 Arg

.real 
x

(no limit)
.or complex
- 
(
a b
)

-
 '
a+i*b
'
- '
r*e^(i\\Gh)
'
- '
r*EXP(i\\Gh)
'

(b\\=/0 &

|
a
| 
or 
|
b
| 
\\<=9)
\" DROP PUSH DUP DUP DUP DUP euler DUP DUP RE SWAP IM \\oo \\->NUM 9 0 \\-> \\<-z0 \\<-z \\<-re \\<-im \\<-max.x \\<-max.z \\<-inf1x
\\<< arg.num max.arg \\->NUM ABS \\oo \\->NUM ==
IF
THEN SIGN -105 CF '\\oo' EVAL * '\\<-z0' STO
ELSE DROP
END \"
i
ERF
c

\" \"
(
\" + \\<-z0 + \"
)
\" + \"'\" \"\" SREPL ROT DROP2
CASE \\<-z \\->NUM '\\oo' EVAL \\->NUM ==
THEN \"Undef by Wolfram
as '0/0', or \\->0
(try 
i
ERF
c
(x) with
x=10 x=20 x=33)\"
END \\<-z \\->NUM '\\oo' EVAL NEG \\->NUM ==
IF
THEN -105 CF \\oo EVAL NEG DUP NEG
ELSE \\<-z ERF ROT 4 ROLLD 3 DROPN \\<-z * \\<-z SQ NEG EXP \\pi \\v/ / - \\->NUM DUP IM 0 == { RE } IFT DUP NEG
END DUP 0 ==
IF
THEN DROP
ELSE \"Wiki_French\" \\->TAG SWAP \"Wolfram\" \\->TAG
END
END POP
\\>>
\\>>" "



" ERFcx "\\<< \"
1 Arg

.real 
x 


|
x
|
 \\<=33 or
.complex
- 
(
a b
)

-
 '
a+i*b
'
- '
r*e^(i\\Gh)
'
- '
r*EXP(i\\Gh)
'

(b\\=/0 &

|
a
| 
or 
|
b
| 
\\<=9)
\" DROP PUSH DUP DUP DUP DUP euler DUP DUP RE SWAP IM 33 9 0 \\-> \\<-z0 \\<-z \\<-re \\<-im \\<-max.x \\<-max.z \\<-inf1x
\\<< arg.num max.arg -105 SF EVAL ABS \\oo ==
IF
THEN SIGN -105 CF '\\oo' EVAL * '\\<-z0' STO
ELSE DROP
END \"ERF
cx

\" \"
(
\" + \\<-z0 + \"
)
\" + \"'\" \"\" SREPL ROT DROP2
CASE \\<-z '\\oo' EVAL ==
THEN 0
END \\<-z '\\oo' EVAL NEG ==
THEN '\\oo' EVAL
END \\<-z ERF ROT 4 ROLLD 3 DROPN \\<-z SQ EXP \\->NUM * DUP IM 0 == { RE } IFT
END POP
\\>>
\\>>" "



" FADDEEVA "\\<< \"
1 Arg

.real 
x


|
x
|
 \\<=9 or
.complex
- 
(
a b
)

-
 '
a+i*b
'
- '
r*e^(i\\Gh)
'
- '
r*EXP(i\\Gh)
'

(b\\=/0 &

|
a
| 
or 
|
b
| 
\\<=9)
\" DROP PUSH DUP DUP DUP DUP euler DUP DUP RE SWAP IM 9 9 0 \\-> \\<-z0 \\<-z \\<-re \\<-im \\<-max.x \\<-max.z \\<-inf1x
\\<< arg.num max.arg -105 SF EVAL ABS \\oo ==
IF
THEN SIGN -105 CF '\\oo' EVAL * '\\<-z0' STO
ELSE DROP
END \"
w
_
F
aaddeva-Kramp
\" \"
(
\" + \\<-z0 + \"
)
\" + \"'\" \"\" SREPL ROT DROP2 \\<-z i NEG * \\->NUM ERF ROT \\<-z SQ NEG EXP * \\->NUM 4 ROLLD 3 DROPN DUP IM 0 == { RE } IFT POP
\\>>
\\>>" "



" DAWSON "\\<< \"
1 Arg

.real 
x


|
x
|
 \\<=9 or
.complex
- 
(
a b
)

-
 '
a+i*b
'
- '
r*e^(i\\Gh)
'
- '
r*EXP(i\\Gh)
'
|
a
| 
& 
|
b
| 
\\<=9
 

\" DROP PUSH DUP DUP DUP DUP euler DUP DUP RE SWAP IM 9 9 0 \\-> \\<-z0 \\<-z \\<-re \\<-im \\<-max.x \\<-max.z \\<-inf1x
\\<< arg.num max.arg -105 SF EVAL ABS \\oo ==
IF
THEN SIGN -105 CF '\\oo' EVAL * '\\<-z0' STO
ELSE DROP
END \"
F
_Dawson
\" \"
(
\" + \\<-z0 + \"
)
\" + \"'\" \"\" SREPL ROT DROP2 \\<-z ABS \\->NUM '\\oo' EVAL \\->NUM ==
IF
THEN 0
ELSE \\<-z ERFi \\<-z SQ NEG EXP \\pi \\v/ * 2 / * \\->NUM SWAP DROP DUP IM 0 == { RE } IFT
END POP
\\>>
\\>>" "



" euler "\\<< -2. CF -103. SF -105. SF DUP \\->STR \"EXP\" \"e^\" SREPL DROP \\-> e s
\\<< s \"e^(\" POS DUP \\-> p
\\<< 0. ==
IF
THEN e
ELSE s p 2. + s SIZE SUB \"'\" SWAP + \"\\pi\" \"1\" SREPL DROP \"i\" \"1\" SREPL 0. ==
IF
THEN DROP e
ELSE OBJ\\-> \\->NUM DUP FP DUP 0. \\=/ SWAP .5 \\=/ AND
IF
THEN DROP e
ELSE \\->Q \\pi * -105. CF i * EXP EVAL p 2. \\=/
IF
THEN s 1. p 2. - SUB \"'\" + OBJ\\-> *
END
END
END
END \\->NUM DUP IM 0. ==
IF
THEN RE
END
\\>>
\\>>
\\>>" "



" arg.num "\\<< -105 SF \\<-z0 TYPE 9 == \\<-z0 TYPE 1 == \\<-z0 IM 0 == AND OR
IF
THEN \\<-z0 \"
=\" + \\<-z DUP IM 0 == \\<-z RE FP 0 == AND { R\\->I } IFT + '\\<-z0' STO
END
\\>>" "



" complex "\\<< DUP IM 0 ==
IF
THEN RE 0 .5 ROT UTPN 2 *
ELSE C\\->R SWAP DUP 0 ==
IF
THEN .00000000001 SWAP
ELSE 1
END \\-> y x z
\\<< '\\.S(-x,x,e^-X^2,X)/\\v/\\pi' \\->NUM 0 .5 x UTPN 2 * 'e^-x^2/(2*\\pi*x)*(1-COS(2*x*y)+i*SIN(2*x*y))' \\->NUM '2/\\pi*e^-x^2' \\->NUM '\\GS(k=1,30,e^(-k^2/4)/(k^2+4*x^2)*(2*x*(1-COS(2*x*y)*COSH(k*y))+k*SIN(2*x*y)*SINH(k*y)+i*(2*x*SIN(2*x*y)*COSH(k*y)+k*COS(2​*x*y)*SINH(k*y))))' \\->NUM * + NEG DUP 4 ROLL SWAP - UNROT + SWAP z 0 ==
IF
THEN C\\->R SWAP DROP 0 SWAP R\\->C SWAP C\\->R SWAP DROP 1 SWAP R\\->C SWAP
END
\\>>
END
\\>>" "



" max.arg "\\<< \\<-im 0. ==
IF
THEN \\<-re ABS \\<-max.x >
IF
THEN message
END
ELSE \\<-re ABS \\<-max.z > \\<-im ABS \\<-max.z > OR
IF
THEN message
END
END
\\>>" "



" message "\\<< DROP2 POP \\<-max.x \\oo \\->NUM ==
IF
THEN \"
Not
 OK with
z=a+ib, b\\=/0
if 
|
a
|

or 
|
b
| 
>\" \\<-max.z R\\->I + DOERR
ELSE \"
Not
 OK with
.Real
if 
|
x
| 
>\" \\<-max.x DUP \\oo \\->NUM ==
IF
THEN DROP \"\\oo\"
ELSE R\\->I
END + \"
\\183Or z=a+ib &
b\\=/0 if 
|
a
|

or 
|
b
| 
>\" + \\<-max.z R\\->I + DOERR
END
\\>>" "



" inf.erf "\\<< \\<-re ABS \\oo \\->NUM == \\<-im ABS \\oo \\->NUM == OR
IF
THEN DROP \\<-im ABS \\oo \\->NUM == \\<-re ABS \\oo \\->NUM == AND
IF
THEN \"Undef\" DUP \\<-re 0. \\>= \"(\\oo\" \"(-\\oo\" IFTE \\<-im 0. \\>= \"+i\\oo)\" \"-i\\oo)\" IFTE +
ELSE
CASE \\<-re ABS \\oo \\->NUM ==
THEN \\<-re 0. \\>= 1. -1. IFTE DUP 1. SWAP - SWAP \"(\" \\<-re 0. \\>= \"\" \"-\" IFTE + \"\\oo\" + \\<-im 0. \\=/
IF
THEN \\<-im 0. \\>= \"+i\" \"-i\" IFTE + \\<-im ABS +
END \")\" +
END \"1\" \\<-im 0. \\>= \"-\" \"+\" IFTE + \"i\\oo\" + DUP \"1\" \"\" SREPL DROP \"-\" \"\" SREPL 0. ==
IF
THEN \"+\" \"-\" SREPL DROP
END \"(\" \\<-re 0. \\=/
IF
THEN \\<-re + \" \" +
END \\<-im 0. \\>= \"+\" \"-\" IFTE + \"i\\oo)\" +
END
END \"ERF
\" SWAP + \"(+\" \"(\" SREPL DROP DUP \"F\" \"F
c
\" SREPL DROP SWAP ROT 4. ROLL UNROT 6. ROLL 6. ROLL DROP2 POP KILL
END
\\>>" "



" inf2x "\\<< \\<-im ABS \\oo \\->NUM == \\<-re ABS \\oo \\->NUM == AND
IF
THEN \"Undef\" \\<-re 0. \\>= \"(\\oo\" \"(-\\oo\" IFTE \\<-im 0. \\>= \"+i\\oo)\" \"-i\\oo)\" IFTE +
ELSE \"OK\"
END
\\>>" "



" inf1x.erfi "\\<< \\<-re ABS \\oo \\->NUM == \\<-im ABS \\oo \\->NUM == OR
IF
THEN DROP 1. '\\<-inf1x' STO \\<-im ABS \\oo \\->NUM == \\<-re ABS \\oo \\->NUM == AND
IF
THEN \"Undef\" \\<-im 0. \\>= \"+i\\oo)\" \"-i\\oo)\" IFTE
ELSE
CASE \\<-im \\oo \\->NUM ==
THEN \"i\" \"+i\\oo)\"
END \\<-im NEG \\oo \\->NUM ==
THEN \"-i\" \"-i\\oo)\"
END \\<-re 0. \\>= \"\\oo\" \"-\\oo\" IFTE \\<-im 0. \\=/
IF
THEN \\<-im 0. > \"+i\" \"-i\" IFTE \\<-im ABS +
ELSE \"\"
END \")\" +
END
END
END
\\>>" "



" inf1x.end "\\<<
CASE \\<-re 0. ==
THEN \"(\"
END \\<-re \\oo \\->NUM ==
THEN \"(\\oo\"
END \\<-re NEG \\oo \\->NUM ==
THEN \"(-\\oo\"
END \"(\" \\<-re + \" \" +
END SWAP +
\\>>" "



" inv¦z "\\<< { 1 } 100 \"This last value can
be incr. \\->160 or +
if 
|
z
|
 \\-> 1
\" DROP \\-> z c max
\\<< 1 max
FOR k 0 0 k 1 -
FOR m c m 1 + GET c k 1 - m - 1 + GET * m 1 + m 2 * 1 + * / + EVAL
NEXT c SWAP \\->NUM + 'c' STO
NEXT 0 0 max
FOR k c k 1 + GET 2 k * 1 + / \\pi \\v/ 2 / z * 2 k * 1 + ^ * \\->NUM +
NEXT
\\>>
\\>>" "



" EXPL "\\<< \"ERF(z) ERFi(z): 
OK

\\-> z=(a b),(\\177\\oo b),
 (a \\177 \\oo),(\\177\\oo \\177\\oo) 
iERFc(z) ERFcx
FADDEEVA(z) DAWSON(z):
\\-> z 
still \\=/
 \\177\\oo(i) 
INV\\166 INV\\166c PROBIT: 
OK

To speed up \\-> change

max
 in 'inv\\166z' to 
60
\" 1 DISP 7 FREEZE
\\>>" "



" VERSIO "Version 9f
2023.12.14

campart
@hotmail.com"