Hypercalc — The Calculator That Doesn't Overflow
Hypercalc is an opensource interpreted calculator program designed to calculate extremely large numbers (such as your phone number raised to the power of the factorial of the Gross world product) without overflowing.
It stores and manipulates numbers using a levelindex format; as such it can go far beyond the limits of bc, dc, MACSYMA/maxima,Mathematica and Maple, all of which use a bigint library. For example, Hypercalc can tell you whether 128^{481024} is larger than 8^{88888}.
or
Contents
Overview: Versions and Features
HyperCalc JavaScript by Kenny TM~ Chan
NonIntuitive Results When Working With Huge Numbers
Overview: Versions and Features
There have been three manifestations of Hypercalc:
 The original Palm Pilot version, no longer maintained
 The Perl version (because Perl rules!) for the terminal/console in UNIX, Linux, Mac OS and Cygwin systems. Source code is here
 The excellent HyperCalc Javacript, which was translated from the Perl by by Kenny TM~ Chan.
All versions of Hypercalc use an internal representation similar to levelindex.
The Perl and JavaScript versions provide command history (input and result substitution, as in Maxima). Other features vary as follows:

The Perl and JavaScript versions are made available under a free (libre) GPL license, but with no warranty or support.
Background: Avoiding Overflow
The primary advantage of Hypercalc is that it does not "overflow": for large numbers, its range is far greater than handheld calculators, calculator apps for phones, numeric libraries like gmp, or maths software like Mathematica. Here is a brief comparison (more on my floatingpoint formats page):

I began exploring very large numbers such as 2^{65536} in the early 1970's using a Texas Instruments SR50 calculator, and had to manually take logarithms, extract fractional parts and compute mantissas, etc. I made my own BIGNUM library in assembly language for the Apple II, and again on later machines. Such an approach is limited by computer memory (on my Large Numbers page I refer to this as the class2 limit).
I always wanted a portable calculator that could do my hugenumbers problems, and the Palm Pilot was the first device that really made this possible. I created the Palm OS HyperCalc in October 1998, and got it working within about a week.
The screen on my Pilot cracked, and I could see that the platform wouldn't last too long. More importantly, I wanted to be able to copy and paste numbers and results to other files while working on my web pages. So I created the vastly more powerful Perl version in the summer of 1999. I have maintained and expanded it greatly over the years, adding extended precision (up to 295 digits) later in 1999, the BASIC interpreter late in 2005, base60 formatting in late 2007, uncertainty calculation in 2011, and so on.
In 2004, Kenny TM~ Chan, at the time a member of the maths club at Yuen Long Merchants Association secondary school (元朗商會中學) in Hong Kong, found Hypercalc and implemented the JavaScript version. This version is described briefly in its own section below.
Hypercalc Perl
The Perl version is the latest and most capable version. The source code is here.
Here is a sample interaction session:
Hypercalc is distributed under the terms and conditions of the GNU General Public License, version 2, June 1991. Type 'help gpl' at the Hypercalc prompt for details. Go ahead  just TRY to make me overflow! _ _ _ . . ._ _ ._ _ ._  _      ) (`  ( ,  ( ~ ~ 7 ~ ~' ~ ~ `~` ~ ~ ' mrob.com/hypercalc Enter expressions, or type 'help' for help. C1 = 27^86! R1 = 10 ^ ( 3.4677786443013 x 10 ^ 130 ) C2 = scale=50 Note: For all values less than 23, factorial will give only 31 digits of accuracy; and for values less than 136.032, it will give fewer than the requested 50 digits. C2 = c1 C1: 27^86! C2: (27^86!) R2 = 10 ^ ( 3.4677786443012627135848832197820460548430862081954 x 10 ^ 130 ) C3 =
There is extensive builtin help, accessed by typing help at theHypercalc prompt. After an initial introductory help page, just
hit
HyperCalc JavaScript by Kenny TM~ Chan
To use HyperCalc from your web browser, go here: HyperCalc JavaScript. There is a detailed manual in PDF format: HyperCalc JavaScript manual
NonIntuitive Results When Working With Huge Numbers
If you spend a while exploring the ranges of huge numbers HyperCalc can handle, you will probably start noticing some paradoxical results and might even start to think the calculator is giving wrong answers.
For example, try calculating 27 to the power of googolplex (a googolplex is 10 to the power of googol and a googol is 10^{100}). Try:
27^(10^(10^100))
and Hypercalc gives:
10^(10^(1 x 10^100))
It appears that it thinks that
27^(10^(10^100)) = 10^(10^(10^100))
This is clearly wrong — and it doesn't even seem to be a good approximation. What's going on?
Let's try calculating the correct answer ourselves. We need to express the answer as 10 to the power of 10 to the power of something, because that's the standard format the calculator is using, and we're going to see how much of an error it made. So, we want to compute
27^{1010100}
as a "tower" of powers of 10. The first step is express the power of 27 as a power of 10 with a product in the exponent, using the formula x^{y} = 10^{(log(x) . y)} :
27^{1010100} = 10^{(log1027 . 1010100)}
log_{10}27 is about 1.43, so we have
27^{1010100} = 10^{1.43 . 1010100}
Now we have a base of 10 but the exponent still needs work. The next step is to express the product as a sum in the nexthigher exponent; this time the formula we use is x ^{.} y = 10^{(log(x) + log(y))} :
10^{1.43 . 1010100} = 10^{10(log101.43 + 10100)}
log_{10}1.43 is about 0.155, and if we add this to 10^{100} we get
10^{10(0.155 + 10100)} = 10^{101000...000.155}
= 10^{10(1.000...000155 . 10100)}
where there are 94 more 0's in place of each of the "...". So our final answer is:
27^{1010100} = 10^{10(1.000...000155 . 10100)}
Now that we've expressed the value of 27^googolplex precisely enough to see the calculator's error — and look how small the error is! The calculator would need to have at least 104 digits of precision to be able to handle the value "1.000...000155" accurately — but it only has 16 digits of accuracy. Those 16 digits are taken up by the 1 and the first fifteen 0's — so when the calculator gets to the step where we're adding 0.155 to 1.0^{.}10^{100}, it just rounds off the answer to 1.0×10^{100} — and produces the answer we saw when we performed the calculation:
10^(10^(1.00 x 10^100)) = 10^{101.00 × 10100}
The original Palm version of Hypercalc had a calculatorlike display, a short, wide rectangle giving enough room to show one line of text with about 30 or 40 characters. Given this limited display area, even if it did have the necessary 104 digits of precision, it wouldn't have room to print the whole 104 digits on the screen, so the answer displayed would still look the same.
More to the point, no matter how many digits of accuracy we try to display, there's always going to be another even bigger number that we won't be able to handle. For example, Hypercalc would need slightly over a million digits of accuracy to distinguish
27^{10101000000} from 10^{10101000000}
and if we just add one more 10 to that tower of exponents, all hope of avoiding roundoff is lost!
For more on this issue, see my discussion of the "power tower paradox", and the Class3 Numbers and Class4 Numbers sections of my large numbers pages.
This page was written in the "embarrassingly readable" markup language RHTF, and was last updated on 2018 Aug 27. s.11