[fricas-devel] Expressing erf as cumulative normal

[Slawomir Kolodynski]

I am learning FriCAS so please excuse my basic questions and point me to
better place to ask them is such exists.

FriCAS is able to evaluate some integrals in terms of the Gauss error
function (erf). For example

(1/(sqrt(2*%pi)))*integrate(exp(-t^2/2),t=%minusInfinity..x)

returns an expression involving erf. I would prefer such integrals to be
evaluated in terms of cumulative distribution function of the standard
normal distribution.
To achieve that I tried to define a rewriting rule

phirule: rule erf(%x)==2*N(x*sqrt(2))-1

hoping that applying that on the result of integration will express the
result in terms of N (the CDF of standard normal). However, I get

rule(erf(QUOTE(%x)),2*applyQuote(QUOTE(N),QUOTE(x)*sqrt(2))-1,[
QUOTE(N)]) is not a valid type.

when I try to define the rule.
What I am doing wrong when defining the rule?
Is defining such rule the best way to convert the result of integration so
that it is expressed in terms the CDF of standard normal rather than the
erf function?

Thanks,

Slawomir

[Waldek Hebisch]

Post by Slawomir Kolodynski
I am learning FriCAS so please excuse my basic questions and point me to
better place to ask them is such exists.
FriCAS is able to evaluate some integrals in terms of the Gauss error
function (erf). For example
(1/(sqrt(2*%pi)))*integrate(exp(-t^2/2),t=%minusInfinity..x)
returns an expression involving erf. I would prefer such integrals to be
evaluated in terms of cumulative distribution function of the standard
normal distribution.
To achieve that I tried to define a rewriting rule
phirule: rule erf(%x)==2*N(x*sqrt(2))-1
hoping that applying that on the result of integration will express the
result in terms of N (the CDF of standard normal). However, I get
rule(erf(QUOTE(%x)),2*applyQuote(QUOTE(N),QUOTE(x)*sqrt(2))-1,[
QUOTE(N)]) is not a valid type.
when I try to define the rule.
What I am doing wrong when defining the rule?

1) ':' means type declaration. You need ':=' for assignment
2) You need to declare 'N' befor use

Below is working version:

(1) -> N := operator 'N

(1) N
Type: BasicOperator
(2) -> phirule := rule erf(%x)==2*N(x*sqrt(2))-1

+-+
(2) erf(%x) == 2 N(x\|2 ) - 1
Type: RewriteRule(Integer,Integer,Expression(Integer))
(3) -> (1/(sqrt(2*%pi)))*integrate(exp(-t^2/2),t=%minusInfinity..x)

+-+ x +-+ +---+
(\|2 erf(----) + \|2 )\|%pi
+-+
\|2
(3) ----------------------------
+-----+
2 \|2 %pi
Type: Expression(Integer)
(4) -> phirule(%)

+-+ +-+ +---+
\|2 N(x\|2 )\|%pi
(4) ------------------
+-----+
\|2 %pi
Type: Expression(Integer)


Note that for FriCAS 'N' is an arbitrary functions with unknown
properties. The first line tells FriCAS that 'N' is being
used as function name (operator), otherwise FriCAS would complain
about undefined function.

[Riccardo GUIDA]

IIUC there is a misprint in both previous emails: %x in the LHS of the rule should be replaced by x as in the RHS. ric

(1) -> N := operator 'N

(1) N
Type: BasicOperator
(2) -> phirule := rule erf(%x)==2*N(x*sqrt(2))-1 -- BAD

┌─┐
(2) erf(%x) == 2 N(x\│2 ) - 1
Type: RewriteRule(Integer,Integer,Expression(Integer))
(3) -> phirule(erf(y))

┌─┐
(3) 2 N(x\│2 ) - 1
Type: Expression(Integer)
(4) -> phirule := rule erf(x)==2*N(x*sqrt(2))-1 -- OK

┌─┐
(4) erf(x) == 2 N(x\│2 ) - 1
Type: RewriteRule(Integer,Integer,Expression(Integer))
(5) -> phirule(erf(y))

┌─┐
(5) 2 N(y\│2 ) - 1
Type: Expression(Integer)