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tokens: reduce the in equation ( 60 ). Thus, it can achieve arbitrary precision by increasing the bit length. However, the round off error will also propagate and accumulate through the extrapolation table in Table 1 . As the order of the GBS extrapolation increases, the round off will accumulate faster and faster. Thus the additional mantissa bits becomes less and less efficient in increasing the integration accuracy. Therefore, reducing the round off error in GBS extrapolation is also important in order to achieve the same precision with less mantissa bits and less CPU time. 3.1.1 Improvement on GBS extrapolation In the BS extrapolation, higher order methods can be obtained from lower order methods by extrapolation, as described in equation ( 52 ). However, the round off error will also propagate and accumulate through the extrapolation table. If the round off error from the base integration method is ε, −ε, ε, ... in the first column, then the round off error propagation with the extensively used step sequence n i = 2 i suggested by Deuflhard ( 1983 ) will behave as in Table 2 . As shown in the table, T 8, 8 with 16th order precision magnified the round off error to become 100 times larger. Thus, we will lose at least 2 significant digits from the extrapolation. This becomes more problematic for higher order extrapolations, as required by arbitrary precision integration methods with extremely low tolerance. Table 2. Round off error propagation in the Bulirsch–Stoer extrapolation with step sequence n i = 2 i . To reduce the round off error in the extrapolation process, there are two fine-tunings that can be performed on the extrapolations. The first is a step sequence choice. The step sequences n i = 1, 2, 3, 5, 8, 12, 17, 25, 36, 51, 73, ...(Fukushima 1996 ) is better than the extensively used sequence n i = 2 i , where the extrapolation coefficient reduces the round-off error propagation. Table 3 shows the error propagation with the new step sequence. We see that it performs much better than the original sequence suggested by Deuflhard ( 1983 ). Table 3. Round off error propagation with optimal step sequence in the BS extrapolation. The second method to reduce the round off error in the extrapolation process is to extrapolate the raw increment dξ of the integration instead of the integrated result ξ. Since the GBS extrapolation is linear, it is possible to do the following Because the raw increment dξ is usually tiny compared to the integrated results, the round of error in the first column of Table 1 is ∼eps × |dξ| instead of ∼eps × |ξ|. Then, the round off error accumulated from the extrapolation table will be smaller than the error accumulated from the direct extrapolation. We note, however, that the initial Δ T i , 1 should be the raw increment evaluated from the basic integrator, not from the subtraction of T i , 1 ( H ) − T i , 1 (0). The close value subtraction will introduce additional round off errors and will significantly degrade the