Claiming that a 32-bit numeric format can outperform a 64-bit format is an example of BS claim.
Al 32-bit formats have the same number of points that are distributed over the real numbers, partitioning them in intervals.
The difference in the possible numeric formats is only in the position of the points, so the number of intervals is identical. When for a numeric format the intervals in a certain area are smaller, i.e. the precision is better, that means that in another area the intervals must be larger, so the precision must be worse. Which areas are more important depends on the problem that must be solved.
The posits are just floating-point numbers where the partitioning between the logarithmic part (the exponent) and the linear part (the fraction) is not fixed, but variable.
The posits close to 1 have more fraction bits, while the posits closer to 0 and to infinities have more exponent bits and less fraction bits.
For certain problems 32-bit posits can outperform 32-bit IEEE floats, but 32-bit posits cannot outperform any 64-bit numeric format, because the 64-bit format has billions times more intervals so a 32-bit format does not have any chance of approximating better a number.
Posits can outperform standard floats only at low precision, because in the formats with few bits there are not enough bits to reserve for the exponent, so the exponent range is small, which makes overflows and underflows very likely.
Posits can have a much larger exponent range, while maintaining a good precision close to 1, paying their extended exponent range with reduced precision towards 0 and infinities, i.e. posits have a gradual underflow and overflow, which are more gradual than the IEEE gradual underflow.
At double precision or higher precisions, the IEEE floats have enough exponent range that underflows and overflows become very unlikely, so posits no longer have any advantage.
The standard floating-point numbers have an almost constant relative approximation error over their exponent range.
For most serious numerical work this is the approximation property that is desired. Posits have a variable relative approximation error that becomes worse and worse for small and large numbers. This is normally undesirable. Complex physical models always have both very small numbers and very large numbers that must be approximated well, even better than the numbers close to 1 that are preferred by posits.
Nevertheless for small floating-point numbers, e.g. 16-bit FP, avoiding overflows and underflows becomes more important than the loss of precision at exponent range extremities, so posits are better.
32-bit floats are around the threshold where posits transition from being better to being worse that standard FP numbers.
Depending on the problem, 32-bit floats or 32-bit posits may be better. On the other hand, I have never seen any problem where 64-bit posits, with their worse relative errors, could be better than IEEE double precision.
Comments
Claiming that a 32-bit numeric format can outperform a 64-bit format is an example of BS claim.
Al 32-bit formats have the same number of points that are distributed over the real numbers, partitioning them in intervals.
The difference in the possible numeric formats is only in the position of the points, so the number of intervals is identical. When for a numeric format the intervals in a certain area are smaller, i.e. the precision is better, that means that in another area the intervals must be larger, so the precision must be worse. Which areas are more important depends on the problem that must be solved.
The posits are just floating-point numbers where the partitioning between the logarithmic part (the exponent) and the linear part (the fraction) is not fixed, but variable.
The posits close to 1 have more fraction bits, while the posits closer to 0 and to infinities have more exponent bits and less fraction bits.
For certain problems 32-bit posits can outperform 32-bit IEEE floats, but 32-bit posits cannot outperform any 64-bit numeric format, because the 64-bit format has billions times more intervals so a 32-bit format does not have any chance of approximating better a number.
Posits can outperform standard floats only at low precision, because in the formats with few bits there are not enough bits to reserve for the exponent, so the exponent range is small, which makes overflows and underflows very likely.
Posits can have a much larger exponent range, while maintaining a good precision close to 1, paying their extended exponent range with reduced precision towards 0 and infinities, i.e. posits have a gradual underflow and overflow, which are more gradual than the IEEE gradual underflow.
At double precision or higher precisions, the IEEE floats have enough exponent range that underflows and overflows become very unlikely, so posits no longer have any advantage.
The standard floating-point numbers have an almost constant relative approximation error over their exponent range.
For most serious numerical work this is the approximation property that is desired. Posits have a variable relative approximation error that becomes worse and worse for small and large numbers. This is normally undesirable. Complex physical models always have both very small numbers and very large numbers that must be approximated well, even better than the numbers close to 1 that are preferred by posits.
Nevertheless for small floating-point numbers, e.g. 16-bit FP, avoiding overflows and underflows becomes more important than the loss of precision at exponent range extremities, so posits are better.
32-bit floats are around the threshold where posits transition from being better to being worse that standard FP numbers.
Depending on the problem, 32-bit floats or 32-bit posits may be better. On the other hand, I have never seen any problem where 64-bit posits, with their worse relative errors, could be better than IEEE double precision.