\documentclass{article}
\usepackage{axiom}
\begin{document}
\title{\$SPAD/src/input eval.input}
\author{The Axiom Team}
\maketitle
\begin{abstract}
\end{abstract}
\eject
\tableofcontents
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\section{License}
<<license>>=
--Copyright The Numerical Algorithms Group Limited 1991.
@
<<*>>=
<<license>>

-- This file demonstrate the various eval's available on EXPR, and the
-- handling of formal derivatives.
-- Lines starting with --** are comments indicating that the final syntax
-- will be different.
 
)cl all
--** This line will be optional interactively, since the a := f(x**2)
--** will prompt you if you don't declare f this way.
f := operator 'f
a := f(x**2)
b := differentiate(a,x,2) + f 5
-- This is the 'variable' evaluation, similar to what's available on
-- polynomials:
eval(b, x = x + y)
-- This is the 'kernel' evaluation, allowing you to specify special
-- values. Only the specified value f 5 is affected, not the others:
eval(b, f 5 = 1)
-- This is the 'operator' evaluation, allowing you to specify an actual
-- function for a formal one. ALL the values of f are affected.
--** will eventually use the +-> notation in the eval statement
foo(u:EXPR INT):EXPR INT == exp u
-- So what is b if f were the exponential function?
-- Notice that the formal derivatives will be computed properly now:
c := eval(b, 'f, foo)
-- We can also use that evaluation on 'system' operators, which allows
-- us to replace an actual function by a formal one:
oof(u:EXPR INT):EXPR INT == f u
eval(c, 'exp, oof)
-- It is also possible to give f a derivative without replacing it by
-- a 'concrete' function:
f'(u:EXPR INT):EXPR INT == f u
-- this will make f differentiate like an exponential:
derivative(f,f')
b
--** The coercion is needed to avoid an interpreter bug.
--** This will just be eval(b) eventually:
eval(b, x = x::(EXPR INT))
differentiate(%, x)
-- This is the 'operator/power' evaluation: suppose that we know that
-- f squared is the exponential, but we do not want to replace f(u) by
-- sqrt(exp u). It is still possible to eliminate higher powers of f
-- in the following way:
a3 := a * a * a
foo
eval(a3,'f,2,foo)
-- Several 'operator' evaluations can be carried out simultaneously:
g := operator 'g
bar(u:EXPR INT):EXPR INT == sin(u) + cos(2*u)
a + g a
eval(%,['f,'g],[foo,bar])
a3 + g a
-- The grand finale: by now the effect of the following should be clear:
eval(%,['f,'g],[2,1],[foo,bar])
@
\eject
\begin{thebibliography}{99}
\bibitem{1} nothing
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\end{document}