Doubling time breaks at tiny rates and at negative ones

Adam Zettel · September 24, 2026 · written with an agent

ln (1 + r) with r = 1 0^{- 12} : 1 + r rounds first

An attempt to break the program in the post "Doubling time" (an adversarial review), posted as four tests that fail. At a rate of $1 0^{- 12}$ a year it is off by a relative $9 \times 1 0^{- 5}$ ; at $1 0^{- 16}$ it says the money never doubles; at $- 5%$ it says the money doubled 13.5 years ago; and the monthly formula has the same flaw once the additions are small enough. The program above the four tests is the reviewed one, with the formula for interest added $n$ times a year and its tests added.

Open in the workbench Download .vivOutputs below are from the site build’s run; Run all runs it again in your browser.

Look inside this program

Atlas
The program’s map: every function and which calls which.
There’s no recorded run of this program yet.

This post copies the program it reviews, adds the formula for interest added $n$ times a year with its tests, and adds four tests at the bottom, one for each problem found. The site runs those four tests on the reviewed program and checks that each one fails there. To solve this problem, change the program so that all its tests pass.

The problems were found by giving doubling_time rates far from the 3% and 7% the original tests use, and comparing with $ln 2/ r$ , which for a small rate $r$ is off by a relative error of only about $r /2$ .

The small-rate failure comes from rounding; the formula is exact. A double carries about 16 digits, so the $1 0^{- 12}$ in $1 + 1 0^{- 12}$ is stored as $1.00009 \times 1 0^{- 12}$ , and the logarithm of the sum is off in its fifth digit. At $1 0^{- 16}$ the sum rounds to exactly 1, the logarithm is 0, and the division gives infinity. The usual cure is log1p, a logarithm of $1 + x$ that never forms $1 + x$ .

The program never checks for a negative rate: money that shrinks never doubles, and the formula returns a negative number of years instead of saying so.

Your task

Write these 3 functions:

doubling_time(rate): Years to double at a yearly rate, with interest added once a year.
rule_of_70(rate): The rule of 70: divide 70 by the rate written as a percentage.
doubling_time_n(rate, times): Years to double when the yearly rate is split into times smaller additions.

Make these tests pass 4 of 12 fail

at 7% a year the doubling time is 10.24 years (passes)
after that long the amount is exactly 2 (passes)
the rule of 70 is 2.4% off at 7% and 0.5% off at 3% (passes)
the base of the logarithm does not matter: base 10 gives the same time (passes)
compounded once a year, the n-times formula gives the yearly doubling time to 1e-12 (passes)
compounded monthly at 7%, the money doubles in 9.931 years (passes)
more additions a year double it sooner, but never sooner than ln 2 / r (passes)
ten thousand additions a year is ln 2 / r plus r/(2n) of it, the next term of the series (passes)
at a rate of one part in a trillion a year, the doubling time is ln 2 / r to one part in a billion assert_near: |0.999911 - 1| = 8.88927e-05 > 1e-09 (fails)
at a rate of 1e-16 a year the doubling time is finite, 6.9e15 years assert_near: |inf - 1| = inf > 1e-09 (and 1 more) (fails)
money shrinking 5% a year never doubles: the doubling time is infinite assert_gt: -13.5134 <= 1e+300 (fails)
a billion additions a year double it in ln 2 / r, to one part in a billion assert_near: |1 - 1| = 8.27054e-08 > 1e-09 (fails)

Printed

t_double: 10.2448

t_monthly: 9.93096

t_continuous: 9.9021

double_digits: 15.9546

stored_part: 1.00009e-12

tiny_rate_error: 8.88927e-05

shrinking_time: -13.5134

Growth at 7% a year, compounding and not

1 + 0.07 t

Doubling time breaks at tiny rates and at negative ones

Your task

Solutions 0

Forks 1

History 4 versions · adversarial review, fix, extend