FAQ¶
The following is a list of answers to frequently asked questions. For questions not answered here or elsewhere in the documentation, please e-mail Peter Lindstrom.
Questions answered in this FAQ:
- Can zfp compress vector fields?
- Should I declare a 2D array as zfp::array1d a(nx * ny, rate)?
- How can I initialize a zfp compressed array from disk?
- Can I use zfp to represent dense linear algebra matrices?
- Can zfp compress logically regular but geometrically irregular data?
- Does zfp handle infinities, NaNs,and subnormal floating-point numbers?
- Can zfp handle data with some missing values?
- Can I use zfp to store integer data?
- Can I compress 32-bit integers using zfp?
- Why does zfp corrupt memory if my allocated buffer is too small?
- Are zfp compressed streams portable across platforms?
- How can I achieve finer rate granularity?
- Can I generate progressive zfp streams?
- How do I initialize the decompressor?
- Must I use the same parameters during compression and decompression?
- Do strides have to match during compression and decompression?
- Why does zfp sometimes not respect my error tolerance?
- Why is the actual rate sometimes not what I requested?
- Can zfp perform compression in place?
- How should I set the precision to bound the relative error?
- Does zfp support lossless compression?
- Why is my actual, measured error so much smaller than the tolerance?
Q1: Can zfp compress vector fields?
I have a 2D vector field
double velocity[ny][nx][2];
of dimensions nx × ny. Can I use a 3D zfp array to store this as:
array3d velocity(2, nx, ny, rate);
A: Although this could be done, zfp assumes that consecutive values are related. The two velocity components (vx, vy) are almost suredly independent and would not be correlated. This will severely hurt the compression rate or quality. Instead, consider storing vx and vy as two separate 2D scalar arrays:
array2d vx(nx, ny, rate);
array2d vy(nx, ny, rate);
or as
array2d velocity[2] = {array2d(nx, ny, rate), array2d(nx, ny, rate)};
Q2: Should I declare a 2D array as zfp::array1d a(nx * ny, rate)?
I have a 2D scalar field of dimensions nx × ny that I allocate as
double* a = new double[nx * ny];
and index as
a[x + nx * y]
Should I use a corresponding zfp array
array1d a(nx * ny, rate);
to store my data in compressed form?
A: Although this is certainly possible, if the scalar field exhibits coherence in both spatial dimensions, then far better results can be achieved by using a 2D array:
array2d a(nx, ny, rate);
Although both compressed arrays can be indexed as above, the 2D array can exploit smoothness in both dimensions and improve the quality dramatically for the same rate.
Since zfp 0.5.2, proxy pointers are also available that act much like
the flat double*.
Q3: How can I initialize a zfp compressed array from disk?
I have a large, uncompressed, 3D data set:
double a[nz][ny][nx];
stored on disk that I would like to read into a compressed array. This data set will not fit in memory uncompressed. What is the best way of doing this?
A: Using a zfp array:
array3d a(nx, ny, nz, rate);
the most straightforward (but perhaps not best) way is to read one floating-point value at a time and copy it into the array:
for (uint z = 0; z < nz; z++)
for (uint y = 0; y < ny; y++)
for (uint x = 0; x < nx; x++) {
double f;
if (fread(&f, sizeof(f), 1, file) == 1)
a(x, y, z) = f;
else {
// handle I/O error
}
}
Note, however, that if the array cache is not large enough, then this may compress blocks before they have been completely filled. Therefore it is recommended that the cache holds at least one complete layer of blocks, i.e. (nx / 4) × (ny / 4) blocks in the example above.
To avoid inadvertent evictions of partially initialized blocks, it is better to buffer four layers of nx × ny values each at a time, when practical, and to completely initialize one block after another, which is facilitated using zfp’s iterators:
double* buffer = new double[nx * ny * 4];
int zmin = -4;
for (zfp::array3d::iterator it = a.begin(); it != a.end(); it++) {
int x = it.i();
int y = it.j();
int z = it.k();
if (z > zmin + 3) {
// read another layer of blocks
if (fread(buffer, sizeof(*buffer), nx * ny * 4, file) != nx * ny * 4) {
// handle I/O error
}
zmin += 4;
}
a(x, y, z) = buffer[x + nx * (y + ny * (z - zmin))];
}
Iterators have been available since zfp 0.5.2.
Q4: Can I use zfp to represent dense linear algebra matrices?
A: Yes, but your mileage may vary. Dense matrices, unlike smooth scalar fields, rarely exhibit correlation between adjacent rows and columns. Thus, the quality or compression ratio may suffer.
Q5: Can zfp compress logically regular but geometrically irregular data?
My data is logically structured but irregularly sampled, e.g. it is rectilinear, curvilinear, or Lagrangian, or uses an irregular spacing of quadrature points. Can I still use zfp to compress it?
A: Yes, as long as the data is (or can be) represented as a logical multidimensional array, though your mileage may vary. zfp has been designed for uniformly sampled data, and compression will in general suffer the more irregular the sampling is.
Q6: Does zfp handle infinities, NaNs,and subnormal floating-point numbers?
A: Only finite, valid floating-point values are currently supported. If a
block contains a NaN or an infinity, undefined behavior is invoked due to the
C math function frexp() being undefined for non-numbers. Subnormal
numbers are, however, handled correctly.
Q7: Can zfp handle data with some missing values?
My data has some missing values that are flagged by very large numbers, e.g. 1e30. Is that OK?
A: Although all finite numbers are “correctly” handled, such large sentinel values are likely to pollute nearby values, because all values within a block are expressed with respect to a common largest exponent. The presence of very large values may result in complete loss of precision of nearby, valid numbers. Currently no solution to this problem is available, but future versions of zfp will likely support a bit mask to tag values that should be excluded from compression.
Q8: Can I use zfp to store integer data?
Can I use zfp to store integer data such as 8-bit quantized images or 16-bit digital elevation models?
A: Yes (as of version 0.4.0), but the data has to be promoted to 32-bit signed
integers first. This should be done one block at a time using an appropriate
zfp_promote_*_to_int32() function call (see zfp.h). Future
versions of zfp may provide a high-level interface that automatically
performs promotion and demotion.
Note that the promotion functions shift the low-precision integers into the most significant bits of 31-bit (not 32-bit) integers and also convert unsigned to signed integers. Do use these functions rather than simply casting 8-bit integers to 32 bits to avoid wasting compressed bits to encode leading zeros. Moreover, in fixed-precision mode, set the precision relative to the precision of the (unpromoted) source data.
As of version 0.5.1, integer data is supported both by the low-level API and
high-level calls zfp_compress() and zfp_decompress().
Q9: Can I compress 32-bit integers using zfp?
I have some 32-bit integer data. Can I compress it using zfp’s 32-bit integer support?
A: Maybe. zfp compression of 32-bit and 64-bit integers requires that each integer f have magnitude |f| < 230 and |f| < 262, respectively. To handle signed integers that span the entire range −231 ≤ x < 231, or unsigned integers 0 ≤ x < 232, the data has to be promoted to 64 bits first.
As with floating-point data, the integers should ideally represent a quantized continuous function rather than, say, categorical data or set of indices. Depending on compression settings and data range, the integers may or may not be losslessly compressed. If fixed-precision mode is used, the integers may be stored at less precision than requested. See Q21 for more details on precision and lossless compression.
Q10: Why does zfp corrupt memory if my allocated buffer is too small?
Why does zfp corrupt memory rather than return an error code if not enough memory is allocated for the compressed data?
A: This is for performance reasons. zfp was primarily designed for fast random access to fixed-rate compressed arrays, where checking for buffer overruns is unnecessary. Adding a test for every compressed byte output would significantly compromise performance.
One way around this problem (when not in fixed-rate mode) is to use the
maxbits parameter in conjunction with the maximum precision or
maximum absolute error parameters to limit the size of compressed blocks.
Finally, the function zfp_stream_maximum_size() returns a conservative
buffer size that is guaranteed to be large enough to hold the compressed data
and the optional header.
Q11: Are zfp compressed streams portable across platforms?
Are zfp compressed streams portable across platforms? Are there, for example, endianness issues?
A: Yes, zfp can write portable compressed streams. To ensure portability across different endian platforms, the bit stream must however be written in increments of single bytes on big endian processors (e.g. PowerPC, SPARC), which is achieved by compiling zfp with an 8-bit (single-byte) word size:
-DBIT_STREAM_WORD_TYPE=uint8
See BIT_STREAM_WORD_TYPE. Note that on little endian processors
(e.g. Intel x86-64 and AMD64), the word size does not affect the bit stream
produced, and thus the default word size may be used. By default, zfp uses
a word size of 64 bits, which results in the coarsest rate granularity but
fastest (de)compression. If cross-platform portability is not needed, then
the maximum word size is recommended (but see also Q12).
When using 8-bit words, zfp produces a compressed stream that is byte order independent, i.e. the exact same compressed sequence of bytes is generated on little and big endian platforms. When decompressing such streams, floating-point and integer values are recovered in the native byte order of the machine performing decompression. The decompressed values can be used immediately without the need for byte swapping and without having to worry about the byte order of the computer that generated the compressed stream.
Finally, zfp assumes that the floating-point format conforms to IEEE 754. Issues may arise on architectures that do not support IEEE floating point.
Q12: How can I achieve finer rate granularity?
A: For d-dimensional arrays, zfp supports a granularity of 8 / 4d
bits, i.e. the rate can be specified in increments of a fraction of a bit for
2D and 3D arrays. Such fine rate selection is always available for sequential
compression (e.g. when calling zfp_compress()).
Unlike in sequential compression, zfp’s compressed arrays require random access writes, which are supported only at the granularity of whole words. By default, a word is 64 bits, which gives a rate granularity of 64 / 4d in d dimensions, i.e. 16 bits in 1D, 4 bits in 2D, and 1 bit in 3D.
To achieve finer granularity, recompile zfp with a smaller (but as large as possible) stream word size, e.g.:
-DBIT_STREAM_WORD_TYPE=uint8
gives the finest possible granularity, but at the expense of (de)compression
speed. See BIT_STREAM_WORD_TYPE.
Q13: Can I generate progressive zfp streams?
A: Yes, but it requires some coding effort. There is currently no high-level
support for progressive zfp streams. To implement progressive fixed-rate
streams, the fixed-length bit streams should be interleaved among the blocks
that make up an array. For instance, if a 3D array uses 1024 bits per block,
then those 1024 bits could be broken down into, say, 16 pieces of 64 bits
each, resulting in 16 discrete quality settings. By storing the blocks
interleaved such that the first 64 bits of all blocks are contiguous,
followed by the next 64 bits of all blocks, etc., one can achieve progressive
decompression by setting the zfp_stream.maxbits parameter (see
zfp_stream_set_params()) to the number of bits per block received so
far.
To enable interleaving of blocks, zfp must first be compiled with:
-DBIT_STREAM_STRIDED
to enable strided bit stream access. In the example above, if the stream word size is 64 bits and there are n blocks, then:
stream_set_stride(stream, m, n);
implies that after every m 64-bit words have been decoded, the bit stream is advanced by m × n words to the next set of m 64-bit words associated with the block.
Q14: How do I initialize the decompressor?
A: The zfp_stream and zfp_field objects usually need to
be initialized with the same values as they had during compression (but see
Q15 for exceptions).
These objects hold the compression mode and parameters, and field data like
the scalar type and dimensions. By default, these parameters are not stored
with the compressed stream (the “codestream”) and prior to zfp 0.5.0 had to
be maintained separately by the application.
Since version 0.5.0, functions exist for reading and writing a 12- to 19-byte header that encodes compression and field parameters. For applications that wish to embed only the compression parameters, e.g. when the field dimensions are already known, there are separate functions that encode and decode this information independently.
Q15: Must I use the same parameters during compression and decompression?
A: Not necessarily. It is possible to use more tightly constrained
zfp_stream parameters during decompression than were used during
compression. For instance, one may use a larger
zfp_stream.minbits, smaller zfp_stream.maxbits,
smaller zfp_stream.maxprec, or larger zfp_stream.minexp
during decompression to process fewer compressed bits than are stored, and to
decompress the array more quickly at a lower precision. This may be useful
in situations where the precision and accuracy requirements are not known a
priori, thus forcing conservative settings during compression, or when the
compressed stream is used for multiple purposes. For instance, visualization
usually has less stringent precision requirements than quantitative data
analysis. This feature of decompressing to a lower precision is particularly
useful when the stream is stored progressively (see Q13).
Note that one may not use less constrained parameters during decompression,
e.g. one cannot ask for more than zfp_stream.maxprec bits of
precision when decompressing.
Currently float arrays have a different compressed representation from compressed double arrays due to differences in exponent width. It is not possible to compress a double array and then decompress (demote) the result to floats, for instance. Future versions of the zfp codec may use a unified representation that does allow this.
Q16: Do strides have to match during compression and decompression?
A: No. For instance, a 2D vector field:
float in[ny][nx][2];
could be compressed as two scalar fields with strides sx = 2,
sy = 2 × nx, and with pointers &in[0][0][0] and
&in[0][0][1] to the first value of each scalar field. These two
scalar fields can later be decompressed as non-interleaved fields:
float out[2][ny][nx];
using strides sx = 1, sy = nx and pointers &out[0][0][0]
and &out[1][0][0].
Q17: Why does zfp sometimes not respect my error tolerance?
A: zfp does not store each floating-point value independently, but represents a group of values (4, 16, or 64 values, depending on dimensionality) as linear combinations like averages by evaluating arithmetic expressions. Just like in uncompressed IEEE floating-point arithmetic, both representation error and roundoff error in the least significant bit(s) often occur.
To illustrate this, consider compressing the following 1D array of four floats:
float f[4] = { 1, 1e-1, 1e-2, 1e-3 };
using the zfp command-line tool:
zfp -f -1 4 -a 0 -i input.dat -o output.dat
In spite of an error tolerance of zero, the reconstructed values are:
float g[4] = { 1, 1e-1, 9.999998e-03, 9.999946e-04 };
with a (computed) maximum error of 5.472e-9. Because f[3] = 1e-3 can only be approximately represented in radix-2 floating-point, the actual error is even smaller: 5.424e-9. This reconstruction error is primarily due to zfp’s block-floating-point representation, which expresses the four values in a block relative to a single, common binary exponent. Such exponent alignment occurs also in regular IEEE floating-point operations like addition. For instance,:
float x = (f[0] + f[3]) - 1;
should of course result in x = f[3] = 1e-3, but due to exponent
alignment a few of the least significant bits of f[3] are lost in the
addition, giving a result of x = 1.0000467e-3 and a roundoff error
of 4.668e-8. Similarly,:
float sum = f[0] + f[1] + f[2] + f[3];
should return sum = 1.111, but is computed as 1.1110000610. Moreover,
the value 1.111 cannot even be represented exactly in (radix-2) floating-point;
the closest float is 1.1109999. Thus the computed error:
float error = sum - 1.111f;
which itself has some roundoff error, is 1.192e-7.
Phew! Note how the error introduced by zfp (5.472e-9) is in fact one to two orders of magnitude smaller than the roundoff errors (4.668e-8 and 1.192e-7) introduced by IEEE floating-point in these computations. This lower error is in part due to zfp’s use of 30-bit significands compared to IEEE’s 24-bit single-precision significands. Note that data sets with a large dynamic range, e.g. where adjacent values differ a lot in magnitude, are more susceptible to representation errors.
The moral of the story is that error tolerances smaller than machine epsilon (relative to the data range) cannot always be satisfied by zfp. Nor are such tolerances necessarily meaningful for representing floating-point data that originated in floating-point arithmetic expressions, since accumulated roundoff errors are likely to swamp compression errors. Because such roundoff errors occur frequently in floating-point arithmetic, insisting on lossless compression on the grounds of accuracy is tenuous at best.
Q18: Why is the actual rate sometimes not what I requested?
A: In principle, zfp allows specifying the size of a compressed block in
increments of single bits, thus allowing very fine-grained tuning of the
bit rate. There are, however, cases when the desired rate does not exactly
agree with the effective rate, and users are encouraged to check the return
value of zfp_stream_set_rate(), which gives the actual rate.
There are several reasons why the requested rate may not be honored. First, the rate is specified in bits/value, while zfp always represents a block of 4d values in d dimensions, i.e. using N = 4d × rate bits. N must be an integer number of bits, which constrains the actual rate to be a multiple of 1 / 4d. The actual rate is computed by rounding 4d times the desired rate.
Second, if the array dimensions are not multiples of four, then zfp pads the dimensions to the next higher multiple of four. Thus, the total number of bits for a 2D array of dimensions nx × ny is computed in terms of the number of blocks bx × by:
bitsize = (4 * bx) * (4 * by) * rate
where nx ≤ 4 × bx < nx + 4 and ny ≤ 4 × by < ny + 4. When amortizing bitsize over the nx × ny values, a slightly higher rate than requested may result.
Third, to support updating compressed blocks, as is needed by zfp’s compressed array classes, the user may request write random access to the fixed-rate stream. To support this, each block must be aligned on a stream word boundary (see Q12), and therefore the rate when write random access is requested must be a multiple of wordsize / 4d bits. By default wordsize = 64 bits.
Fourth, for floating-point data, each block must hold at least the common exponent and one additional bit, which places a lower bound on the rate.
Finally, the user may optionally include a header with each array. Although
the header is small, it must be accounted for in the rate. The function
zfp_stream_maximum_size() conservatively includes space for a header,
for instance.
Q19: Can zfp perform compression in place?
Because the compressed data tends to be far smaller than the uncompressed data, it is natural to ask if the compressed stream can overwrite the uncompressed array to avoid having to allocate separate storage for the compressed stream. zfp does allow for the possibility of such in-place compression, but with several caveats and restrictions:
- A bitstream must be created whose buffer points to the beginning of uncompressed (and to be compressed) storage.
- The array must be compressed using zfp’s low-level API. In particular, the data must already be partitioned and organized into contiguous blocks so that all values of a block can be pulled out once and then replaced with the corresponding shorter compressed representation.
- No one compressed block can occupy more space than its corresponding uncompressed block so that the not-yet compressed data is not overwritten. This is usually easily accomplished in fixed-rate mode, although the expert interface also allows guarding against this in all modes using the
zfp_stream.maxbitsparameter. This parameter should be set tomaxbits = 4^d * 8 * sizeof(type), where d is the array dimensionality (1, 2, or 3) and where type is the scalar type of the uncompressed data.- No header information may be stored in the compressed stream.
In-place decompression can also be achieved, but in addition to the above constraints requires even more care:
- The data must be decompressed in reverse block order, so that the last block is decompressed first to the end of the block array. This requires the user to maintain a pointer to uncompressed storage and to seek via
stream_rseek()to the proper location in the compressed stream where the block is stored.- The space allocated to the compressed stream must be large enough to also hold the uncompressed data.
An example is provided that shows how in-place compression can be done.
Q20: How should I set the precision to bound the relative error?
In general, zfp cannot bound the point-wise relative error due to its use of a block-floating-point representation, in which all values within a block are represented in relation to a single common exponent. For a high enough dynamic range within a block there may simply not be enough precision available to guard against loss. For instance, a block containing the values 20 = 1 and 2-n would require a precision of n + 3 bits to represent losslessly, and zfp uses at most 64-bit integers to represent values. Thus, if n ≥ 62, then 2-n is replaced with 0, which is a 100% relative error. Note that such loss also occurs when, for instance, 20 and 2-n are added using floating-point arithmetic (see also Q17).
It is, however, possible to bound the error relative to the largest (in magnitude) value, fmax, within a block, which if the magnitude of values does not change too rapidly may serve as a reasonable proxy for point-wise relative errors.
One might then ask if using zfp’s fixed-precision mode with p bits of precision ensures that the block-wise relative error is at most 2-p × fmax. This is, unfortunately, not the case, because the requested precision, p, is ensured only for the transform coefficients. During the inverse transform of these quantized coefficients the quantization error may amplify. That being said, it is possible to derive a bound on the error in terms of p that would allow choosing an appropriate precision. Such a bound is derived below.
Let
emax = floor(log2(fmax))
be the largest base-2 exponent within a block. For transform coefficient precision, p, one can show that the maximum absolute error, err, is bounded by:
err <= k(d) * (2^emax / 2^p) <= k(d) * (fmax / 2^p)
Here k(d) is a constant that depends on the data dimensionality d:
k(d) = 20 * (15/4)^(d-1)
so that in 1D, 2D, and 3D we have:
k(1) = 20
k(2) = 125
k(3) = 1125/4
Thus, to guarantee n bits of accuracy in the decompressed data, we need to choose a higher precision, p, for the transform coefficients:
p(n, d) = n + ceil(log2(k(d))) = n + 2 * d + 3
so that
p(n, 1) = n + 5
p(n, 2) = n + 7
p(n, 3) = n + 9
This p value should be used in the call to
zfp_stream_set_precision().
Note, again, that some values in the block may have leading zeros when expressed relative to 2emax, and these leading zeros are counted toward the n-bit precision. Using decimal to illustrate this, suppose we used 4-digit precision for a 1D block containing these four values:
-1.41421e+1 ~ -1.414e+1 = -1414 * (10^1 / 1000)
+2.71828e-1 ~ +0.027e+1 = +27 * (10^1 / 1000)
+3.14159e-6 ~ +0.000e+1 = 0 * (10^1 / 1000)
+1.00000e+0 ~ +0.100e+1 = +100 * (10^1 / 1000)
with the values in the middle column aligned to the common base-10 exponent +1, and with the values on the right expressed as scaled integers. These are all represented using four digits of precision, but some of those digits are leading zeros.
Q21: Does zfp support lossless compression?
Yes, and no. For integer data, zfp can with few exceptions ensure lossless compression. For a given n-bit integer type (n = 32 or n = 64), consider compressing p-bit signed integer data, with the sign bit counting toward the precision. In other words, there are exactly 2p possible signed integers. If the integers are unsigned, then subtract 2p-1 first so that they range from −2p-1 to 2p-1 - 1.
Lossless compression is achieved by first promoting the p-bit integers to n - 1 bits (see Q8) such that all integer values fall in [−230, +230), when n = 32, or in [−262, +262), when n = 64. In other words, the p-bit integers first need to be shifted left by n - p - 1 bits. After promotion, the data should be compressed in zfp’s fixed-precision mode using:
q = p + 4 * d + 1
bits of precision to ensure no loss, where d is the data dimensionality (1 ≤ d ≤ 3). Consequently, the p-bit data can be losslessly compressed as long as p ≤ n - 4 × d - 1. The table below lists the maximum precision p that can be losslessly compressed using 32- and 64-bit integer types.
d n=32 n=64 1 27 59 2 23 55 3 19 51
Although lossless compression is possible as long as the precision constraint is met, the precision needed to guarantee no loss is generally much higher than the precision intrinsic in the uncompressed data, making lossless compression via zfp not competitive with compressors designed for lossless compression. Lossy integer compression with zfp can, on the other hand, work fairly well by using fewer than q bits of precision.
Furthermore, the minimum precision, q, given above is often larger than what is necessary in practice. There are worst-case inputs that do require such large q values, but they are quite rare.
The reason for expanded precision, i.e., why q > p, is that zfp’s decorrelating transform computes averages of integers, and this transform is applied d times in d dimensions. Each average of two p-bit numbers requires p + 1 bits to avoid loss, and each transform can be thought of involving up to four such averaging operations.
For floating-point data, fully lossless compression with zfp is unlikely, albeit possible. If the dynamic range is low or varies slowly such that values within a 4d block have the same or similar exponent, then the precision gained by discarding the 8 or 11 bits of the common floating-point exponents can offset the precision lost in the decorrelating transform. For instance, if all values in a block have the same exponent, then lossless compression is obtained using q = 26 + 4 × d ≤ 32 bits of precision for single-precision data and q = 55 + 4 × d ≤ 64 bits of precision for double-precision data. Of course, the constraint imposed by the available integer precision n implies that lossless compression of such data is possible only in 1D for single-precision data and only in 1D and 2D for double-precision data.
Q22: Why is my actual, measured error so much smaller than the tolerance?
For two reasons. The way zfp bounds the absolute error in
fixed-accuracy mode is by keeping all transform
coefficient bits whose place value exceeds the tolerance while discarding the
less significant bits. Each such bit has a place value that is a power of
two, and therefore the tolerance must first be rounded down to the next
smaller power of two, which itself will introduce some slack. This possibly
lower, effective tolerance is returned by the
zfp_stream_set_accuracy() call.
Second, the quantized coefficients are then put through an inverse transform. This linear transform will combine signed quantization errors that, in the worst case, may cause them to add up and increase the error, even though the average (RMS) error remains the same, i.e. some errors cancel while others compound. For d-dimensional data, d such inverse transforms are applied, with the possibility of errors cascading across transforms. To account for the worst possible case, zfp has to conservatively lower its internal error tolerance further, once for each of the d transform passes.
Unless the data is highly oscillatory or noisy, the error is not likely to be magnified much, leaving an observed error in the decompressed data that is much lower than the prescribed tolerance. In practice, the observed maximum error tends to be about 4-8 times lower than the error tolerance for 3D data, while the difference is smaller for 2D and 1D data.
We recommend experimenting with tolerances and evaluating what error levels are appropriate for each application, e.g. by starting with a low, conservative tolerance and successively doubling it. The distribution of errors produced by zfp is approximately Gaussian, so even if the maximum error may seem large at an individual grid point, most errors tend to be much smaller and tightly clustered around zero.