I am supposed to show that two algorithms execute identical statements in identical order. One is a tail recursive version of the other. They are written in Eiffel.

```
tail_rec(x:instance_type):result_type is
local
y:instance_type;
do
if b(x) then
Result :=c(x)
else
y:=d(x);
Result:=tail_rec(y)
end
end
```

Then the non-tail recursive version.

```
non_rec(x:instance_type):result_type is
do
from until b(x) loop
x:= d(x)
end;
Result:= c(x)
end
```

where b(x), c(x), and d(x) are any functions of type BOOLEAN, result_type, and instance_type respectively.

How are these two algorithms similar and how do they execute identical statements in identical order?

Answer:

By substituting all flow-of-control constructs with `goto`

s, (essentially turning the code from `Eiffel`

into pseudocode,) and allowing `if`

statements to execute only `goto`

s, it can be shown that both functions end up consisting of the exact same set of instructions.

Let me begin by copying the original `tail_rec`

here for convenience:

```
tail_rec(x:instance_type):result_type is
local
y:instance_type;
do
if b(x) then
Result := c(x)
else
y := d(x);
Result := tail_rec(y)
end
end
```

First, get rid of Eiffel's silly `Result :=`

construct and replace it with `return`

for convenience. (Otherwise, we are going to have to add more `goto`

s, and frankly, the fewer the better.)

```
tail_rec(x:instance_type):result_type is
local
y:instance_type;
do
if b(x) then
return c(x)
end
y := d(x);
return tail_rec(y)
end
```

Substitute the `if`

-`then`

-`end`

with `if`

-`then goto`

:

```
tail_rec(x:instance_type):result_type is
local
y:instance_type;
do
if not b(x) then goto label1
return c(x)
label1:
y := d(x);
return tail_rec(y)
end
```

Replace tail recursion with another `goto`

:

```
tail_rec(x:instance_type):result_type is
do
label0:
if not b(x) then goto label1
return c(x)
label1:
x := d(x);
goto label0
end
```

Replace `if not b(x)`

with `if b(x)`

:

```
tail_rec(x:instance_type):result_type is
do
label0:
if b(x) then goto label1
x := d(x);
goto label0
label1:
return c(x)
end
```

Similar transformations to `tail_rec`

should turn it into the exact same thing.

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