Calculate e^x and ln(x)

Question

Given a floating point x (x<100), return e^x and ln(x). The first 6 decimal places of the number have to be right, but any others after do not have to be correct.

You cannot have any "magic" constants explicitly stated (e.x. a=1.35914 since 1.35914*2 ~= e), but you can calculate them. Only use +, -, *, and / for arithmetic operators.

If x is less than or equal to 0, output ERROR instead of the intended value of ln(x).

Test Cases:

input: 2
output: 7.389056 0.693147
input: 0.25
output: 1.284025 -1.386294
input: 2.718281828
output: 15.154262 0.999999  (for this to output correctly, don't round to 6 places)
input: -0.1
output: 0.904837 ERROR

Shortest code wins.

Answer 1

PHP 109 107 bytes

<?for($g=1e4;--$g;$l=$y/$g+abs($y=$x>1?1-1/$x:$x-1)*$l)$e=1+$e/$g*$x.=fgets(STDIN);echo"$e ",$x>0?$l:ERROR;

is a fairly straight-forward calculation. I use a nested form of the sum of inverse factorials, which not only increases the convergence rate, but also allows for exponentiation at the same time:

is slightly more complicated. All convergent series seem to work for or , but not both (Newton's iteration does not have this limitation, but requires the calculation of each step). This isn't really a problem, though, given the log identity:

This means that if, for example, the iteration you're using only works on and , you can use the multiplicative inverse of and negate the result. Because I was using a nested identity for , I also chose to use a nested identity for :

where

Or equivalently, as demonstrated by Paul Walls' implementation:

I define the case as (which is necessarily negative), using the absolute value for the inner product, and then allowing a bare value in the fraction to correct the sign.

Sample I/O:

$ echo 2 | php exp_ln.php
7.3890560989307 0.69314718055995

$ echo 0.25 | php exp_ln.php
1.2840254166877 -1.3862943611199

$ echo 2.718281828 | php exp_ln.php
15.154262234523 0.99999999983113

$ echo -0.1 | php exp_ln.php
0.90483741803596 ERROR

Perl 95 93 bytes

$e=1-$e/$_*($x.=<>),$l=$y/$_+$l*abs($y=$x>1?1/$x-1:1-$x)for-1e4..-1;print"$e ",$x>0?$l:ERROR;

Nearly identical to the PHP solution above with a few negated terms, to allow iteration from -1e4..-1.

Both 2 byte improvements due to Paul Walls.

Answer 2

Haskell, 166 (89 without I/O)

s=1e-7
e x|x>s=1/e(-x)|x>0=1+x|y<-e$x+s=y-y*s
l x|x<1=0-l(1/x)|1>0=sum$map(s/)[1,1+s..x]
n x|x<0="ERROR"|1>0=show$l x
main=interact$unwords.(\x->[show$e x,n x]).read

Takes an alternative approach: we use that

∂/∂x e^x = e^x

and solve the differential equation numerically, with a simple euler method. Similarly, use

ln x = ₁∫^x 1/x d‌x

and calculate the integral with the rectangular method.

Example:

$ echo 2 | ./exp-and-log
7.389056835370484 0.6931472554471929
$ echo 0.25 | ./exp-and-log
1.284025432730133 -1.3862944228194163
$ echo 2.718281828 | ./exp-and-log
15.15426430695358 1.0000000576151333
$ echo -0.1 | ./exp-and-log
0.9048374135134576 ERROR

It's quite amazingly inefficient, in fact it uses about 4 GB of memory even for these examples (since it's non-tail–recursive... you need to compile (in GHC) with -with-rtsopts=-K2G so it even accepts such a ridiculous stack size).

Answer 3

JavaScript, 103 101 99 97 93 90

This implementation is based on primo's comprehensive description of the algorithm he used.

for(e=l=x=prompt(g=1e5);--g;y=x-1,l=(x>1?y/=x:-y)*l+y/g)
    e=1+e/g*x;
alert([e,x>0?l:"ERROR"])

Edit:
- Trying to catch Perl. Stole a byte back from primo by copying his branch. :)
- 2 more bytes courtesy of primo.
- Finally, caught primo's version! With primo's help, of course... :)
- Simplified the assignment to L. Shaved 3 more bytes.

Answer 4

Python 2 (168 char)

basic implementation of power series

I need to learn a golfing language =/

I increased the bound to 1e-14 (twice from 1e-7) since some values were off a bit in 6 decimal places, works well for input from 1e-5 to 100 (slows down at input approaches 0)

x=input();t,r,i=1,0,1.
while abs(t)>=1e-14:t,r,i=t*x/i,r+t,i+1
if x<=0:s='Error'
else:z=(x-1.)/(x+1);t,s,i=2*z,0,1
while abs(t/i)>=1e-14:t,s,i=t*z*z,s+t/i,i+2
print r,s

Answer 5

Surprised at no one using APL which has built-in exponentiation and natural logarithm, I will submit one.

(*x),⍟x←⎕