### Riegel formula

An empirical way to estimate the race time is to use the time and the distance of a recent run in the so-called Riegel formula. Pete Riegel proposed the following formula in 1977:

t = tin * (d/din)1.06

with tin and din respectively the time and the distance of the known performance and t is the estimated time to travel the distance d.

### Dave Cameron's Model

This model uses the top 10 times in the world at various distances to compute performances across distances. In his article Cameron states that to obtain the following formulas he used non-linear regression methods:

t = (tin/din) * d *(13.49681 - 0.000030363 * din + 835.7114/din0.7905) / (13.49681 - 0.000030363 * d + 835.7114/d0.7905)

with tin and din respectively the time [s] and the distance [m] of the known performance and t [s] is the estimated time to travel the distance d [m].

### Estimate based on VO2 Max

VO2 Max (also maximal oxygen consumption or maximal aerobic capacity) is the maximum rate of oxygen consumption as measured during incremental exercise.
The estimate based on the VO2 Max first computes the VO2 Max, given the performance entered, using the Daniels and Gilbert VO2 Max formula:

VO2 Max = (-4.60 + 0.182258 * v + 0.000104 * v2) / (0.8 + 0.1894393 * e-0.012778*tin + 0.2989558 * e-0.1932605*tin)

with v = din/tin, where din the distance in meters and tin the time in seconds of the performance entered. VO2max is expressed in millilitres of oxygen per kilogram of body mass per minute (e.g., ml/(kg·min)).
The same formula is then used in reverse to estimate the times by using the same VO2 Max found at the beginning. There is no analytic solution to solve this problem, but the solution can be found numerically (we use the Newton's Method).