Abstract
Floating-point arithmetic is a well-known and extremely efficient way of performing approximate computations over the real numbers. Although it requires some careful considerations, floating-point numbers are nowadays routinely used to prove mathematical theorems. Numerical computations have been applied in the context of formal proofs too, as illustrated by the CoqInterval library. But these computations do not benefit from the powerful floating-point units available in modern processors, since they are emulated inside the logic of the formal system. This paper experiments with the use of hardware floating-point numbers for numerically intensive proofs verified by the Coq proof assistant. This gives rise to various questions regarding the formalization, the implementation, the usability, and the level of trust. This approach has been applied to the CoqInterval and ValidSDP libraries, which demonstrates a speedup of at least one order of magnitude.
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Notes
For instance, the binary64 format makes room for \(2^{53}-2\) different NaN values.
The naivety of the approach can be seen in that both mantissas mx and my are aligned by shifting them left according to the smallest exponent min ex ey. A fast implementation would instead shift them right according to the largest exponent, as done in hardware. This is much trickier to get right and the improved performance is not required for our specification purpose.
The authors would like to thank Guillaume Bertholon who implemented the original version of primitive_float_infnan.
References
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Acknowledgements
We are grateful to Guillaume Bertholon for his early work on extending the kernel of Coq with hardware floating-point numbers. This work was partly supported by the European Research Council (ERC) under the European Union’s Horizon 2020 Research and Innovation Programme - Grant Agreement n\(^\circ \)101001995. This work was partly supported by the NuSCAP project (ANR-20-CE48-0014) of the French national research agency (ANR).
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Appendix A: Full Data for CoqInterval Benchmarks
Appendix A: Full Data for CoqInterval Benchmarks
In the table below, the string 452 stands for the CoqInterval version (i.e., 4.5.2). The source code for these benchmarks, along with guidelines to rerun them, can be found at the following URL: https://github.com/validsdp/benchs-primitive-floats.
Problems | 452-bigz-prec53 | 452-bigz-prec30 | 452-primfloat | 452-bigz-prec53-native | 452-bigz-prec30-native | 452-primfloat-native | |||||
|---|---|---|---|---|---|---|---|---|---|---|---|
Lemma | Time (s) | Time (s) | Speedup | Time (s) | Speedup | Time (s) | Speedup | Time (s) | Speedup | Time (s) | Speedup |
Chebyshev | 6.173 | N/A | 1.179 | 5.2x | 1.76 | 3.5x | N/A | 0.519 | 11.9x | ||
MT1 | 2.112 | 0.988 | 2.1x | 0.067 | 31.5x | 0.908 | 2.3x | 0.55 | 3.8x | 0.169 | 12.5x |
MT10 | 1.052 | 0.514 | 2.0x | 0.035 | 30.1x | 0.561 | 1.9x | 0.359 | 2.9x | 0.155 | 6.8x |
MT11 | 1.094 | 0.499 | 2.2x | 0.037 | 29.6x | 0.568 | 1.9x | 0.356 | 3.1x | 0.153 | 7.2x |
MT12 | 1.657 | 1.133 | 1.5x | 0.179 | 9.3x | 0.657 | 2.5x | 0.498 | 3.3x | 0.199 | 8.3x |
MT13 | 33.727 | 16.222 | 2.1x | 0.855 | 39.4x | 11.569 | 2.9x | 5.992 | 5.6x | 0.443 | 76.1x |
MT14 | 0.02 | 0.014 | 1.4x | 0.007 | 2.9x | 0.152 | 0.1x | 0.145 | 0.1x | 0.141 | 0.1x |
MT15 | 0.076 | 0.043 | 1.8x | 0.018 | 4.2x | 0.175 | 0.4x | 0.155 | 0.5x | 0.146 | 0.5x |
MT16 | 0.04 | 0.024 | 1.7x | 0.008 | 5.0x | 0.161 | 0.2x | 0.148 | 0.3x | 0.142 | 0.3x |
MT16__1 | 0.008 | 0.007 | 1.1x | 0.007 | 1.1x | 0.147 | 0.1x | 0.15 | 0.1x | 0.141 | 0.1x |
MT16__2 | 0.006 | 0.006 | 1.0x | 0.005 | 1.2x | 0.15 | 0.0x | 0.142 | 0.0x | 0.141 | 0.0x |
MT16__3 | 0.007 | 0.007 | 1.0x | 0.006 | 1.2x | 0.15 | 0.0x | 0.141 | 0.0x | 0.141 | 0.0x |
MT16__4 | 0.007 | 0.006 | 1.2x | 0.005 | 1.4x | 0.15 | 0.0x | 0.142 | 0.0x | 0.142 | 0.0x |
MT17 | 0.036 | 0.023 | 1.6x | 0.007 | 5.1x | 0.162 | 0.2x | 0.146 | 0.2x | 0.142 | 0.3x |
MT17’ | 0.036 | 0.023 | 1.6x | 0.007 | 5.1x | 0.162 | 0.2x | 0.146 | 0.2x | 0.141 | 0.3x |
MT18 | 0.413 | 0.225 | 1.8x | 0.027 | 15.3x | 0.287 | 1.4x | 0.214 | 1.9x | 0.147 | 2.8x |
MT18’ | 0.41 | 0.221 | 1.9x | 0.026 | 15.8x | 0.288 | 1.4x | 0.212 | 1.9x | 0.145 | 2.8x |
MT19 | 0.24 | 0.146 | 1.6x | 0.158 | 1.5x | 0.228 | 1.1x | 0.189 | 1.3x | 0.191 | 1.3x |
MT19’ | 0.24 | 0.147 | 1.6x | 0.022 | 10.9x | 0.23 | 1.0x | 0.194 | 1.2x | 0.154 | 1.6x |
MT2 | 1.33 | 0.786 | 1.7x | 0.119 | 11.2x | 0.598 | 2.2x | 0.425 | 3.1x | 0.184 | 7.2x |
MT20 | 2.881 | 1.529 | 1.9x | 0.112 | 25.7x | 1.001 | 2.9x | 0.622 | 4.6x | 0.168 | 17.1x |
MT20’ | 2.853 | 1.529 | 1.9x | 0.112 | 25.5x | 0.987 | 2.9x | 0.619 | 4.6x | 0.169 | 16.9x |
MT21 | 0.157 | 0.096 | 1.6x | 0.015 | 10.5x | 0.196 | 0.8x | 0.174 | 0.9x | 0.147 | 1.1x |
MT21’ | 0.156 | 0.097 | 1.6x | 0.09 | 1.7x | 0.197 | 0.8x | 0.174 | 0.9x | 0.17 | 0.9x |
Problems | 452-bigz-prec53 | 452-bigz-prec30 | 452-primfloat | 452-bigz-prec53-native | 452-bigz-prec30-native | 452-primfloat-native | |||||
|---|---|---|---|---|---|---|---|---|---|---|---|
Lemma | Time (s) | Time (s) | Speedup | Time (s) | Speedup | Time (s) | Speedup | Time (s) | Speedup | Time (s) | Speedup |
MT22 | 0.319 | 0.24 | 1.3x | 0.046 | 6.9x | 0.247 | 1.3x | 0.213 | 1.5x | 0.146 | 2.2x |
MT22’ | 0.32 | 0.241 | 1.3x | 0.035 | 9.1x | 0.251 | 1.3x | 0.212 | 1.5x | 0.147 | 2.2x |
MT23 | 0.513 | 0.429 | 1.2x | 0.083 | 6.2x | 0.303 | 1.7x | 0.269 | 1.9x | 0.163 | 3.1x |
MT23’ | 0.53 | 0.429 | 1.2x | 0.083 | 6.4x | 0.303 | 1.7x | 0.268 | 2.0x | 0.167 | 3.2x |
MT24 | 0.008 | 0.006 | 1.3x | 0.005 | 1.6x | 0.15 | 0.1x | 0.142 | 0.1x | 0.142 | 0.1x |
MT24’ | 0.007 | 0.006 | 1.2x | 0.005 | 1.4x | 0.149 | 0.0x | 0.141 | 0.0x | 0.141 | 0.0x |
MT25 | 0.031 | 0.024 | 1.3x | 0.015 | 2.1x | 0.172 | 0.2x | 0.161 | 0.2x | 0.157 | 0.2x |
MT3 | 0.335 | 0.16 | 2.1x | 0.017 | 19.7x | 0.273 | 1.2x | 0.206 | 1.6x | 0.148 | 2.3x |
MT4 | 0.345 | 0.168 | 2.1x | 0.018 | 19.2x | 0.285 | 1.2x | 0.21 | 1.6x | 0.148 | 2.3x |
MT5 | 0.03 | 0.021 | 1.4x | 0.012 | 2.5x | 0.308 | 0.1x | 0.295 | 0.1x | 0.286 | 0.1x |
MT6 | 0.064 | 0.04 | 1.6x | 0.016 | 4.0x | 0.323 | 0.2x | 0.3 | 0.2x | 0.291 | 0.2x |
MT7 | 0.01 | 0.008 | 1.2x | 0.006 | 1.7x | 0.152 | 0.1x | 0.142 | 0.1x | 0.141 | 0.1x |
MT8 | 0.34 | N/A | 0.034 | 10.0x | 0.261 | 1.3x | N/A | 0.155 | 2.2x | ||
MT9 | 0.47 | N/A | 0.076 | 6.2x | 0.299 | 1.6x | N/A | 0.169 | 2.8x | ||
RD | 0.014 | N/A | 0.011 | 1.3x | 0.527 | 0.0x | N/A | 0.492 | 0.0x | ||
Rump_Tucker | 117.45 | N/A | 6.012 | 19.5x | 29.703 | 4.0x | N/A | 1.593 | 73.7x | ||
abs_err_atan | 0.213 | 0.121 | 1.8x | 0.019 | 11.2x | 0.224 | 1.0x | 0.187 | 1.1x | 0.155 | 1.4x |
adaptiveLV | 0.011 | 0.014 | 0.8x | 0.011 | 1.0x | 0.155 | 0.1x | 0.524 | 0.0x | 0.148 | 0.1x |
arctan_0_1 | 0.039 | 0.033 | 1.2x | 0.016 | 2.4x | 0.168 | 0.2x | 0.154 | 0.3x | 0.151 | 0.3x |
bissect | 2.054 | N/A | 0.274 | 7.5x | 0.717 | 2.9x | N/A | 0.223 | 9.2x | ||
bug20120927 | 0.01 | N/A | 0.005 | 2.0x | 0.288 | 0.0x | N/A | 0.274 | 0.0x | ||
bug20140723_1 | 0.008 | N/A | 0.004 | 2.0x | 0.288 | 0.0x | N/A | 0.269 | 0.0x | ||
bug20140723_2 | 0.008 | N/A | 0.004 | 2.0x | 0.292 | 0.0x | N/A | 0.27 | 0.0x | ||
bug20140728 | 0.002 | 0.002 | 1.0x | 0.002 | 1.0x | 0.142 | 0.0x | 0.139 | 0.0x | 0.137 | 0.0x |
Problems | 452-bigz-prec53 | 452-bigz-prec30 | 452-primfloat | 452-bigz-prec53-native | 452-bigz-prec30-native | 452-primfloat-native | |||||
|---|---|---|---|---|---|---|---|---|---|---|---|
Lemma | Time (s) | Time (s) | Speedup | Time (s) | Speedup | Time (s) | Speedup | Time (s) | Speedup | Time (s) | Speedup |
bug20150924 | 0.002 | 0.002 | 1.0x | N/A | 0.144 | 0.0x | 0.139 | 0.0x | N/A | ||
bug20150925 | 0.003 | 0.003 | 1.0x | N/A | 0.144 | 0.0x | 0.138 | 0.0x | N/A | ||
butcher | 0.02 | 0.02 | 1.0x | 0.018 | 1.1x | 0.171 | 0.1x | 0.162 | 0.1x | 0.162 | 0.1x |
circle | 0.018 | 0.017 | 1.1x | 0.01 | 1.8x | 0.159 | 0.1x | 0.149 | 0.1x | 0.148 | 0.1x |
cos_cos_d2 | 0.97 | 0.644 | 1.5x | 0.087 | 11.1x | 0.458 | 2.1x | 0.362 | 2.7x | 0.178 | 5.4x |
cos_cos_d2__1 | 0.744 | 0.495 | 1.5x | 0.071 | 10.5x | 0.376 | 2.0x | 0.306 | 2.4x | 0.168 | 4.4x |
cos_cos_d2__2 | 0.756 | 0.499 | 1.5x | 0.075 | 10.1x | 0.414 | 1.8x | 0.318 | 2.4x | 0.176 | 4.3x |
cos_cos_d3 | 1.319 | 0.876 | 1.5x | 0.119 | 11.1x | 0.545 | 2.4x | 0.449 | 2.9x | 0.192 | 6.9x |
cos_cos_d3__1 | 0.981 | 0.652 | 1.5x | 0.093 | 10.5x | 0.465 | 2.1x | 0.373 | 2.6x | 0.19 | 5.2x |
cos_cos_d3__2 | 0.985 | 0.655 | 1.5x | 0.096 | 10.3x | 0.458 | 2.2x | 0.373 | 2.6x | 0.189 | 5.2x |
cos_cos_d4 | 1.562 | 1.036 | 1.5x | 0.141 | 11.1x | 0.607 | 2.6x | 0.478 | 3.3x | 0.201 | 7.8x |
cos_cos_d4__1 | 1.337 | 0.884 | 1.5x | 0.123 | 10.9x | 0.541 | 2.5x | 0.445 | 3.0x | 0.199 | 6.7x |
cos_cos_d4__2 | 1.222 | 0.814 | 1.5x | 0.118 | 10.4x | 0.522 | 2.3x | 0.412 | 3.0x | 0.203 | 6.0x |
cos_cos_d5 | 2.383 | 1.578 | 1.5x | 0.21 | 11.3x | 0.806 | 3.0x | 0.623 | 3.8x | 0.227 | 10.5x |
cos_cos_d5__1 | 1.823 | 1.197 | 1.5x | 0.164 | 11.1x | 0.666 | 2.7x | 0.53 | 3.4x | 0.221 | 8.2x |
cos_cos_d5__2 | 1.811 | 1.208 | 1.5x | 0.169 | 10.7x | 0.674 | 2.7x | 0.531 | 3.4x | 0.225 | 8.0x |
cos_cos_d6 | 2.322 | 1.52 | 1.5x | 0.207 | 11.2x | 0.792 | 2.9x | 0.611 | 3.8x | 0.233 | 10.0x |
cos_cos_d6__1 | 1.826 | 1.21 | 1.5x | 0.17 | 10.7x | 0.674 | 2.7x | 0.53 | 3.4x | 0.227 | 8.0x |
cos_cos_d6__2 | 1.597 | 1.068 | 1.5x | 0.157 | 10.2x | 0.62 | 2.6x | 0.497 | 3.2x | 0.23 | 6.9x |
cos_cos_d7 | 3.252 | 2.237 | 1.5x | 0.287 | 11.3x | 1.024 | 3.2x | 0.8 | 4.1x | 0.264 | 12.3x |
cos_cos_d7__1 | 2.211 | 1.458 | 1.5x | 0.204 | 10.8x | 0.766 | 2.9x | 0.597 | 3.7x | 0.245 | 9.0x |
cos_cos_d7__2 | 2.56 | 1.776 | 1.4x | 0.236 | 10.8x | 0.855 | 3.0x | 0.685 | 3.7x | 0.257 | 10.0x |
cos_cos_d8 | 3.274 | 2.26 | 1.4x | 0.293 | 11.2x | 1.037 | 3.2x | 0.809 | 4.0x | 0.288 | 11.4x |
cos_cos_d8__1 | 2.45 | 1.708 | 1.4x | 0.23 | 10.7x | 0.833 | 2.9x | 0.669 | 3.7x | 0.259 | 9.5x |
cos_cos_d8__2 | 2.462 | 1.632 | 1.5x | 0.233 | 10.6x | 0.836 | 2.9x | 0.649 | 3.8x | 0.265 | 9.3x |
example20071016_1 | 0.004 | 0.003 | 1.3x | 0.003 | 1.3x | 0.145 | 0.0x | 0.141 | 0.0x | 0.14 | 0.0x |
Problems | 452-bigz-prec53 | 452-bigz-prec30 | 452-primfloat | 452-bigz-prec53-native | 452-bigz-prec30-native | 452-primfloat-native | |||||
|---|---|---|---|---|---|---|---|---|---|---|---|
Lemma | Time (s) | Time (s) | Speedup | Time (s) | Speedup | Time (s) | Speedup | Time (s) | Speedup | Time (s) | Speedup |
example20071016_2 | 0.004 | 0.004 | 1.0x | 0.004 | 1.0x | 0.149 | 0.0x | 0.14 | 0.0x | 0.14 | 0.0x |
example20071016_3 | 0.009 | N/A | 0.005 | 1.8x | 0.293 | 0.0x | N/A | 0.7 | 0.0x | ||
example20071016_3’ | 0.005 | N/A | 0.009 | 0.6x | 0.148 | 0.0x | N/A | 0.144 | 0.0x | ||
example20071016_4 | 0.066 | 0.048 | 1.4x | 0.012 | 5.5x | 0.179 | 0.4x | 0.16 | 0.4x | 0.147 | 0.4x |
example20071016_5 | 0.07 | 0.056 | 1.2x | 0.022 | 3.2x | 0.181 | 0.4x | 0.17 | 0.4x | 0.158 | 0.4x |
example20071016_6 | 0.004 | 0.004 | 1.0x | 0.003 | 1.3x | 0.144 | 0.0x | 0.139 | 0.0x | 0.139 | 0.0x |
example20071016_7 | 0.067 | 0.05 | 1.3x | 0.018 | 3.7x | 0.179 | 0.4x | 0.167 | 0.4x | 0.157 | 0.4x |
example20071016_8 | 0.009 | 0.008 | 1.1x | N/A | 0.153 | 0.1x | 0.144 | 0.1x | N/A | ||
example20120205_1 | 0.002 | 0.002 | 1.0x | 0.002 | 1.0x | 0.141 | 0.0x | 0.14 | 0.0x | 0.137 | 0.0x |
example20120205_2 | 0.003 | 0.003 | 1.0x | N/A | 0.144 | 0.0x | 0.139 | 0.0x | N/A | ||
example20120205_3 | 0.003 | 0.003 | 1.0x | 0.003 | 1.0x | 0.144 | 0.0x | 0.141 | 0.0x | 0.138 | 0.0x |
example20120205_4 | 0.004 | 0.004 | 1.0x | 0.003 | 1.3x | 0.145 | 0.0x | 0.139 | 0.0x | 0.142 | 0.0x |
example20140221_1 | 13.612 | N/A | 0.759 | 17.9x | 3.752 | 3.6x | N/A | 0.386 | 35.3x | ||
example20140221_2 | 0.087 | N/A | 0.018 | 4.8x | 0.177 | 0.5x | N/A | 0.15 | 0.6x | ||
example20140610_1 | 13.331 | 8.219 | 1.6x | 0.658 | 20.3x | 4.138 | 3.2x | 2.674 | 5.0x | 0.413 | 32.3x |
example20140610_2 | 1.631 | 1.142 | 1.4x | 0.196 | 8.3x | 0.629 | 2.6x | 0.514 | 3.2x | 0.222 | 7.3x |
example20150105 | 0.516 | N/A | 0.068 | 7.6x | 0.305 | 1.7x | N/A | 0.177 | 2.9x | ||
example_ln_1 | 0.004 | 0.004 | 1.0x | 0.003 | 1.3x | 0.149 | 0.0x | 0.14 | 0.0x | 0.14 | 0.0x |
example_ln_2 | 0.008 | 0.006 | 1.3x | 0.005 | 1.6x | 0.295 | 0.0x | 0.281 | 0.0x | 0.278 | 0.0x |
exp_0_3 | 0.024 | 0.017 | 1.4x | 0.009 | 2.7x | 0.16 | 0.1x | 0.161 | 0.1x | 0.144 | 0.2x |
exp_cos_0_1 | 0.045 | 0.033 | 1.4x | 0.012 | 3.8x | 0.165 | 0.3x | 0.155 | 0.3x | 0.148 | 0.3x |
h_54_ln_2 | 0.01 | 0.008 | 1.2x | 0.006 | 1.7x | 0.155 | 0.1x | 0.143 | 0.1x | 0.143 | 0.1x |
Magnetism | 0.021 | 0.02 | 1.1x | 0.021 | 1.0x | 0.173 | 0.1x | 0.166 | 0.1x | 0.164 | 0.1x |
rel_err_geodesic | 13.374 | 8.293 | 1.6x | 0.687 | 19.5x | 4.395 | 3.0x | 2.691 | 5.0x | 0.424 | 31.5x |
rel_err_geodesic’ | 1.63 | 1.141 | 1.4x | 0.198 | 8.2x | 0.634 | 2.6x | 0.519 | 3.1x | 0.224 | 7.3x |
remez_sqrt | 0.07 | 0.057 | 1.2x | 0.02 | 3.5x | 0.184 | 0.4x | 0.171 | 0.4x | 0.159 | 0.4x |
x_ln1p_0_1 | 0.02 | 0.016 | 1.2x | 0.01 | 2.0x | 0.159 | 0.1x | 0.148 | 0.1x | 0.146 | 0.1x |
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Martin-Dorel, É., Melquiond, G. & Roux, P. Enabling Floating-Point Arithmetic in the Coq Proof Assistant. J Autom Reasoning 67, 33 (2023). https://doi.org/10.1007/s10817-023-09679-x
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DOI: https://doi.org/10.1007/s10817-023-09679-x