The Twin Paradox

One of the most fascinating predictions of the special theory of relativity is that time passes more slowly for a moving object. The effect is so tiny that we normally never notice it, but it becomes appreciable when something moves at a speed close to the speed of light (299,792,458 meters per second).

This time dilation is often illustrated with the twin paradox. Twins A and B start out together; A stays on Earth while B boards a spaceship that travels at nearly the speed of light. After 50 years B returns to Earth, and A, now 50 years older, is shocked to find that B has aged only a few years.

The times experienced by the two twins are related by

$$t_B = \gamma, t_A$$

where $t_B$ is the time experienced by B on the spaceship and $t_A$ is the time experienced by A on Earth. The conversion factor $\gamma$ is defined as

$$\gamma = \sqrt{1 - \frac{v^2}{c^2}}$$

where $v$ is the average speed of the spaceship and $c$ is the speed of light, both measured in the same units.

For example, if $v$ is 259,620,268 meters per second, then $\gamma \approx 0.5$, so while 2 years pass for the twin on Earth, only 1 year passes for the twin on the spaceship.

Given $t_A$ and $t_B$, write a program that computes the average speed of the spaceship needed to produce this difference.

The input consists of several test cases. Each test case is a single line containing two real numbers $t_A$ and $t_B$ separated by a space. It is always true that $t_A \ge t_B$, and both values lie between $1.0$ and $100.0$.

The last line of the input contains two zeros; this line is not processed.

For each test case, print the average speed of the spaceship on its own line. The speed is expressed in units of the speed of light ($c$) and must be printed to three decimal places.