The Simple Tait One Rep Max calculator computes the approximate one rep max using the Simple Tait formula adjusted for RPE.
INSTRUCTIONS: Enter the following:
- (WT) Weight: The weight lifted during the exercise (must be greater than 0). As the formula is linear in weight the unit does not matter. The unit of the output will always match that of the input.
- (Reps) The number of repetitions completed (must be at least 1).
- (RIR) Reps In Reserve. This is how many reps you had left in the tank at the end of the set. If you are more familiar with RPE, this is the equivalent of 10 - RPE. Default is set to 0, this assumes you went to failure.
- (a) This parameter controls the growth rate of the multiplier. The default value is set to 0.03939979. More information about this parameter and the process used to determine the default can be found below.
- (b) This parameter determines how quickly diminishing returns dominate the function as reps increase. The default value is set to 0.03695154. More information about this parameter and the process used to determine the default can be found below.
Max weight (1RM): The calculator returns the estimated one rep max. The unit is the same as unit you used for the number you entered in the Weight input.
The Math / Science behind the Tait Family of Estimators:
The Simple Tait Estimator is a member of the Tait Family of one rep max estimators. The Tait Family of estimators are estimators that have the following structure: Weight*(1 + a*Reps/(1 + b*Reps)). Here a and b can be chosen to adjust the exact properties of the estimator. This structure was designed using principles from mathematical economics, and is grounded in physiology. The formulation of this estimator was guided by two key physiological properties. First, in the low-repetition range, the strength multiplier is approximately linear with respect to the number of repetitions. This aligns with the observation that the effort required to add an additional repetition in this range remains relatively constant. For example, the difficulty of adding a third repetition to a set of two is comparable to adding a fourth repetition to a set of three. Second, at higher repetition ranges, the estimator accounts for diminishing returns, characterized by concavity in the function. This reflects the reduced physiological significance of additional repetitions at higher volumes. For instance, the difference in maximal strength estimation between completing 29 and 30 repetitions is negligible compared to differences observed at lower repetition ranges.
There is however one obvious problem with this structure; When one rep is performed, the estimated max is higher than the the actual max that was performed. Three fixes were divised for this problem:
- Normalising the estimator: Weight*((1 + a*Reps/(1 + b*Reps))/(1 + a/(1 + b)))
- Fully translating the estimator: Weight*(1 + a*(Reps -1)/(1 + b*(Reps -1)))
- Only translating the numerator of the estimator: Weight*(1 + a*(Reps -1)/(1 + b*Reps))
The Simple Tait Estimator uses the final method.
Finally, it should be noted that very little research appears to have been conducted into the mathematical models underlying not only strength curves—such as those used in estimators—but also other fundamental aspects of fitness and exercise science. This lack of focus is surprising, considering the potential insights that might be gained by treating fitness as a dynamic system and analyzing it through rigorous mathematical modeling, akin to approaches used in fields like mathematical economics and quantitative finance. For instance, long-term strength gain could be conceptualized as analogous to the stock market, where performance is influenced by a multitude of variables, many of which interact in complex and unpredictable ways. Given the impossibility of incorporating all these variables into a comprehensive model, a random walk framework seems logical, as it accounts for the stochastic and inherently uncertain nature of adaptation and progress.
That’s why I’m proposing the creation of a new field, which I’m calling mathematical fitness. As far as I can tell, this concept doesn’t exist yet as its own distinct area of research. The closest thing I could find was Quantified Self and Fitness Informatics, but these focus much more on data analysis than on building fundamental models. So, I think mathematical fitness could still stand on its own as a new field.
I don’t know whether anyone will ever pick this up, or if research in this area would even turn out to be useful. But honestly, I think it could be really fun.
Notes on how the default values for a and b were found:
I originally wanted to estimate a and b using my own workout data. The issue, though, is that I don’t test my one-rep max very often, so there wasn’t much data available for my actual maxes. On top of that, I tend to train mostly in the 4 to 6 rep range, which means I didn’t have much data for other rep ranges either. Because of this, I decided to estimate a and b using existing one-rep max estimators. I tried two methods for this.
The first approach involved creating a hybrid function that combines the properties of the Lombardi and Brzycki formulas. Brzycki was chosen because it’s approximately linear at lower reps and widely used as a one-rep max estimator. Lombardi, on the other hand, was chosen because it captures diminishing returns at higher reps, even though it tends to overestimate for low reps. To combine the two, I used Brzycki for reps below about 7.6 (in my code, I used a more specific value where Brzycki and Lombardi are nearly equal) and Lombardi for anything higher. This hybrid function balances Brzycki’s reliability at low reps with Lombardi’s tapering effect at higher reps, creating a well-rounded estimate.
The second method involved fitting the formula to the average of all the one-rep max estimators listed on the Wikipedia page for one-repetition maximum. This approach gives a broad and balanced estimate by averaging the behaviors of many well-regarded formulas.
For both the hybrid function and the average function, I optimized a and b using R’s built-in optim function. The optimization was done over a rep range of 1 to 12. While this range might seem arbitrary, I found that expanding it didn’t have much impact on the results. After running both optimizations, I found that the parameters derived from the hybrid function produced slightly more conservative one-rep max estimates. This felt more in line with what I was looking for, so I chose the hybrid-based values for a and b as the default.
That said, these parameters aren’t fixed. If you have your own workout data or specific training goals, you can tweak a and b to better suit your needs. If you have enough personal data, you could even use R’s optim function to optimize the formula for your unique performance. Feel free to experiment with these values and adjust them to match your training preferences! If you would like to play around with my R code feel free to reach out via my vcalc profile.
Weight Lifting and Body Building Formulas
- Brzycki: Computes the approximate one-rep maximum weight one can lift using the Brzycki formula.
- Epley 1 Rep Max Formula: Computes the one repetition max using the Epley formula, the weight lifted and the number of repetitions.
- Landers Max Weight: Computes the approximate one-rep maximum weight one can lift using the Lander's formula, weight lifted and number of repetitions.
- Lombardi One Rep Maximum: Estimates the maximum amount of force that can be generated in one maximal lift based on the reps done on a lesser weight.
- Mayhew Max Weight: Computes the approximate one repetition maximum weight one can lift using the Mayhew formula.
- O'Conner one Rep Max Weight: Computes the approximate maximum weight one can lift using the O'Conner formula.
- Wathen Max Weight: Computes the approximate maximum one repetition weight one can lift using the Wathen formula.
- DAPRE Progression for Endurance: Computes the amount of weight you should lift for a given set to build endurance based on the target weight.
- DAPRE Progression for Endurance: Set 4 New Target Weight: Computes the target weight change (if any) for set four of a DAPRE Endurance workout.
- DAPRE Progression for Strength: Computes the amount of weight you should lift for a given set to build strength based on the target weight.
- DAPRE Progression for Strength: Set 4 New Target Weight: Computes the target weight change (if any) for set four of a DAPRE Strength workout
- Macronutrient Breakdown for Building Muscle
- IPF GL Coefficient Points Calculator: Computes the points associated with a comparative weight lifted between weight lifters based on the weight of the lifter (w), the weight lifted (x), the type of life and the gender of the lifter, all corrected by the International Powerlifting Federation Good Lift (IPF GL) Coefficient formula.
- IPF Good Lift Coefficient: Computes the comparative weight coefficient between weight lifters based on the weight of the lifter (x), the type of lift and the gender of the lifter, all combined in the IPF GL Coefficient formula.
- Simple Tait One Rep Max: Computes the approximate one rep max using the Simple Tait formula adjusted for RPE.
- Wilks Coefficient: Computes the comparative weight lifted coefficient between weight lifters based on the weight of the lifter and the gender of the lifter, all combined by the Wilks Coefficient formula.
- Wilks Coefficient Applied: Computes Wilks Coefficient formula value and applies it to Good Lift weight.