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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.

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I assume you mean to exclude exponentiation of entirely (not just for the the calculation of e itself)? 2**(x*1.442695) for example seems a bit too easy. –  primo Nov 25 '12 at 18:04
@primo: yes, there is no exponentiation whatsoever, but things like product([2]*3) is ok. I guess I should rule out "magic constants" in the case of 1.442695. –  beary605 Nov 25 '12 at 18:35
You might want to also specify a valid range for the input - any approximation can lose accuracy if x becomes too large or too small. –  arshajii Nov 26 '12 at 0:08
@A.R.S.: added a boundary. Thanks. –  beary605 Nov 26 '12 at 0:14
No one uses APL!? Seriously? –  TwiNight Dec 25 '12 at 19:45
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5 Answers

up vote 3 down vote accepted

JavaScript, 103 101 99 97 93 90

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


- 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.

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It's somewhat humorous that defining Math.abs inline is actually shorter than using the built-in. –  primo Nov 27 '12 at 7:39
As far as catching perl goes, would it help if I pointed out that the initial values of e and l don't really matter? –  primo Dec 2 '12 at 6:58
@primo Nice catch! –  Paul Walls Dec 2 '12 at 8:13
I've got 93, by combining the y and l definitions: l=(x>1?y=1-1/x:-(y=x-1))*l+y/g –  primo Dec 14 '12 at 7:29
Very elegant. I probably spent over an hour trying to figure out a way to exploit the relationship between x<1 and y<0. You sir, are a genius! :) –  Paul Walls Dec 14 '12 at 9:37
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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 :


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 89 bytes

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

Nearly identical to the PHP solution above, with a slightly larger iteration (65535 down to 0).


  • Both 2 byte improvements due to Paul Walls.
  • Four more bytes saved in Perl by (ab)using $?, which is stored internally as an unsigned short, and by using $_ to save parentheses in abs.
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+1 lovely explanation –  David Carraher Nov 26 '12 at 12:13
For x = 1e-4, I get php "1.0001000050002 -8.9909564381939" and perl "1.00010000500017 -8.99099322429857". Should be like "1.000100005 -9.21034037195" Anyways nice job taking advantage of ln x = -ln 1/x to use a nested expression –  miles t Nov 26 '12 at 13:49
@milest x=1e-5 will be off by even more, because it's the same calculation as x=100000. As it is, x=1e-2 and x=100 are accurate to full floating point precision, and x=1e-3 and x=1000 are accurate to about 5 digits. If more accuracy is needed, the arbitrary 1e4 bound that I've chosen can be increased accordingly. –  primo Nov 26 '12 at 14:25
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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)

while abs(t)>=1e-14:t,r,i=t*x/i,r+t,i+1
if x<=0:s='Error'
while abs(t/i)>=1e-14:t,s,i=t*z*z,s+t/i,i+2
print r,s
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Doesn't python support m=1e-7? –  dmckee Nov 25 '12 at 20:53
yes it does but 1e-7 can be thought of as 1*10^-7 which might be thought of as exponentiation, so I'm not really sure if its allowed here –  miles t Nov 25 '12 at 20:59
I will allow for scientific notation. –  beary605 Nov 26 '12 at 0:02
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Haskell, 166 (89 without I/O)

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 ex = ex

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

ln x = 1x 1/x d‌x

and calculate the integral with the rectangular method.


$ 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).

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

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Only use +, -, *, and / for arithmetic operators.? –  beary605 Dec 27 '12 at 17:12
Oh didn't see that! Was falling back to Taylor expansion (which I thought would still favor APL due to built-in factorial, reciprocal and polynomial functions) when I found out the function breaks when calculating e^99 using 154 terms 'cuz 99^154>2^1024. Given up –  TwiNight Dec 27 '12 at 18:27
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