Numerical improvements to `Phi_approx` by jachymb · Pull Request #3450 · stan-dev/math

jachymb · 2026-10-12T20:40:48Z

AI use disclosure: This was developed with the help of Claude Code (Claude Opus 5.5, Claude Fable 5.1).

Includes a merge of PR #3446, which is not yet merged; until it is, the diff shown here includes its changes.

Summary

Phi_approx(x) is $\sigma(u)$ with $u = \gamma x^3 + \beta x$ and σ the inverse logit, the logistic approximation to the standard normal CDF of Bowling et al. (2009). This PR makes two independent numerical improvements to it.

1. A derivative that does not cancel in the upper tail

The reverse-mode derivative f * (1 - f) * u' subtracts a number close to 1: it is 3.5% off at x = 7 and exactly 0 for x ≳ 7.11 (true value 8.7e-21 at x = 8). The var, var_value (now Eigen array expressions) and OpenCL overloads use

$$ \frac{d}{dx},\Phi_{\text{approx}}(x) = \sigma(u),\sigma(-u),u' = \frac{e,u'}{(1 + e)^2}, \qquad e = e^{-|u|}. $$

Forward mode has the same defect through the fvar overload of inv_logit, fixed in the inv_logit branch that this one merges; its forward-mode tail test is here.

2. The coefficients as named constants, at double precision

The two coefficients were literals repeated in five files. They are now PHI_APPROX_CUBIC_COEF and PHI_APPROX_LINEAR_COEF in stan/math/prim/fun/constants.hpp, used by every overload. The OpenCL value path, until now a device-function source string with the digits in it, is a kernel-generator expression built from the constants (stan/math/opencl/prim/Phi_approx.hpp), the same way the OpenCL reverse-mode overload builds its derivative. Their values are the pair that minimises the maximum absolute deviation from Φ over the real line, solved from the equioscillation conditions at 50 digits and rounded to double precision:

$$ \gamma = 0.07056409656566716, \qquad \beta = 1.597602874240784, \qquad \max_x |\Phi_{\text{approx}}(x) - \Phi(x)| = 1.40313 \times 10^{-4}, $$

attained at x = ±0.4816, ±1.4764 and ±2.6876. The paper prints this pair rounded to 0.07056 and 1.5976, with a maximum deviation of 1.41448e-4; the rounding costs accuracy for nothing, since a double holds the full pair at the same price. The Stan Functions Reference prints the rounded formula and needs a follow-up docs change.

Tests

References are mpmath values (60 digits) computed from the double values of the new constants, in the new header test/unit/math/mix/fun/Phi_approx_tail_refs.hpp, shared by the mix and OpenCL tests; tolerances are 1e-13 relative, floored at the smallest normal double.

Derivative (improvement 1):

PhiApproxScalarDerivativeTails and PhiApproxDerivativeTails in test/unit/math/mix/fun/Phi_approx_test.cpp: the reverse-mode derivative at x = −40 … 40 for var, var_value<VectorXd>, Matrix<var> and std::vector<var>. They fail on develop: at x = 7, 8 and 10 from the cancellation (the derivative is 3.5% off, then exactly 0), elsewhere only from the coefficient change.
PhiApproxFwdDerivativeTails in the same file: first and second derivatives at the same points through fvar<double>, fvar<fvar<double>> and fvar<var>. It fails on develop and needs the inv_logit branch.
OpenCL: rev_tail_derivative_against_references in test/unit/math/opencl/rev/Phi_approx_test.cpp.

Coefficients (improvement 2):

The reference tables above are computed for the new coefficients, so every tail test also pins them.
The existing prim test, which checks Phi_approx against Phi within 0.00014, passes with the new coefficients.

Side Effects

Improvement 1, benchmarked against develop in one process, -O3 -DNDEBUG, on MinGW (MSYS2 g++ 15.2) and glibc (WSL g++ 12.2), with the published coefficients: time, this branch / develop, var_value 0.25–1.00, other paths 0.73–1.06. Arena bytes per call are unchanged (64 for a scalar var, 24 000 for Matrix<var> and 48 144 for var_value at N = 1000). OpenCL was not timed; its reverse pass gains one exp per element.

Improvement 2 is a behaviour change: every Phi_approx value moves, by up to 1.14e-6 absolute (at x ≈ ±1.44) and, in the lower tail, by about 4.1e-6 |x|³ relative (2e-3 at x = −8, 4e-2 at x = −21, just before the value underflows); the maximum deviation from Φ falls from 1.4145e-4 to 1.4031e-4.

Release notes

Two numerical improvements to Phi_approx: a derivative that stays accurate in the upper tail, and double-precision minimax coefficients.

Checklist

Copyright holder: Jáchym Barvínek jachymb@gmail.com

The copyright holder is typically you or your assignee, such as a university or company. By submitting this pull request, the copyright holder is agreeing to the license the submitted work under the following licenses:
- Code: BSD 3-clause (https://opensource.org/licenses/BSD-3-Clause)
- Documentation: CC-BY 4.0 (https://creativecommons.org/licenses/by/4.0/)
the basic tests are passing
- unit tests pass (to run, use: ./runTests.py test/unit)
- header checks pass, (make test-headers)
- dependencies checks pass, (make test-math-dependencies)
- docs build, (make doxygen)
- code passes the built in C++ standards checks (make cpplint)
the code is written in idiomatic C++ and changes are documented in the doxygen
the new changes are tested

…-precision minimax optimum

…le-derivatives

… forward-mode derivatives

… constants

stan-buildbot · 2026-10-14T18:01:38Z

Name	Old Result	New Result	Ratio	Performance change( 1 - new / old )
stat_comp_benchmarks/benchmarks/gp_regr/gp_regr.stan	0.23	0.23	1.0	-0.33% slower
stat_comp_benchmarks/benchmarks/gp_regr/gen_gp_data.stan	0.06	0.06	1.03	2.83% faster
stat_comp_benchmarks/benchmarks/low_dim_gauss_mix/low_dim_gauss_mix.stan	6.36	6.34	1.0	0.16% faster
stat_comp_benchmarks/benchmarks/low_dim_corr_gauss/low_dim_corr_gauss.stan	0.02	0.02	0.92	-9.27% slower
stat_comp_benchmarks/benchmarks/irt_2pl/irt_2pl.stan	8.28	8.26	1.0	0.14% faster
stat_comp_benchmarks/benchmarks/gp_pois_regr/gp_pois_regr.stan	4.47	4.48	1.0	-0.05% slower
stat_comp_benchmarks/benchmarks/sir/sir.stan	169.15	168.17	1.01	0.58% faster
stat_comp_benchmarks/benchmarks/garch/garch.stan	0.92	0.89	1.03	3.15% faster
stat_comp_benchmarks/benchmarks/arma/arma.stan	0.61	0.61	1.0	-0.32% slower
stat_comp_benchmarks/benchmarks/pkpd/one_comp_mm_elim_abs.stan	42.97	43.11	1.0	-0.32% slower
stat_comp_benchmarks/benchmarks/pkpd/sim_one_comp_mm_elim_abs.stan	0.58	0.61	0.96	-4.67% slower
stat_comp_benchmarks/benchmarks/low_dim_gauss_mix_collapse/low_dim_gauss_mix_collapse.stan	20.99	21.06	1.0	-0.35% slower
stat_comp_benchmarks/benchmarks/eight_schools/eight_schools.stan	0.11	0.11	1.01	0.6% faster
stat_comp_benchmarks/benchmarks/arK/arK.stan	3.18	3.18	1.0	-0.03% slower
performance.compilation	391.58	396.4	0.99	-1.23% slower
Mean result: 0.9947308782973309

Jenkins Console Log
Jenkins Build Stages
Commit hash: 017f091b63e5f89b01de871d2a7ead2d78127331

Machine information

Distributor ID:	Ubuntu
Description:	Ubuntu 20.04.3 LTS
Release:	20.04
Codename:	focal

CPU:

Architecture:                            x86_64
CPU op-mode(s):                          32-bit, 64-bit
Byte Order:                              Little Endian
Address sizes:                           43 bits physical, 48 bits virtual
CPU(s):                                  256
On-line CPU(s) list:                     0-255
Thread(s) per core:                      2
Core(s) per socket:                      64
Socket(s):                               2
NUMA node(s):                            2
Vendor ID:                               AuthenticAMD
CPU family:                              23
Model:                                   49
Model name:                              AMD EPYC 7742 64-Core Processor
Stepping:                                0
Frequency boost:                         enabled
CPU MHz:                                 1494.456
CPU max MHz:                             3416.0681
CPU min MHz:                             1500.0000
BogoMIPS:                                4491.85
Virtualization:                          AMD-V
L1d cache:                               4 MiB
L1i cache:                               4 MiB
L2 cache:                                64 MiB
L3 cache:                                512 MiB
NUMA node0 CPU(s):                       0-63,128-191
NUMA node1 CPU(s):                       64-127,192-255
Vulnerability Gather data sampling:      Not affected
Vulnerability Indirect target selection: Not affected
Vulnerability Itlb multihit:             Not affected
Vulnerability L1tf:                      Not affected
Vulnerability Mds:                       Not affected
Vulnerability Meltdown:                  Not affected
Vulnerability Mmio stale data:           Not affected
Vulnerability Old microcode:             Not affected
Vulnerability Reg file data sampling:    Not affected
Vulnerability Retbleed:                  Mitigation; untrained return thunk; SMT enabled with STIBP protection
Vulnerability Spec rstack overflow:      Mitigation; Safe RET
Vulnerability Spec store bypass:         Mitigation; Speculative Store Bypass disabled via prctl
Vulnerability Spectre v1:                Mitigation; usercopy/swapgs barriers and __user pointer sanitization
Vulnerability Spectre v2:                Mitigation; Retpolines; IBPB conditional; STIBP always-on; RSB filling; PBRSB-eIBRS Not affected; BHI Not affected
Vulnerability Srbds:                     Not affected
Vulnerability Tsa:                       Not affected
Vulnerability Tsx async abort:           Not affected
Vulnerability Vmscape:                   Mitigation; IBPB before exit to userspace
Flags:                                   fpu vme de pse tsc msr pae mce cx8 apic sep mtrr pge mca cmov pat pse36 clflush mmx fxsr sse sse2 ht syscall nx mmxext fxsr_opt pdpe1gb rdtscp lm constant_tsc rep_good nopl xtopology nonstop_tsc cpuid extd_apicid aperfmperf rapl pni pclmulqdq monitor ssse3 fma cx16 sse4_1 sse4_2 x2apic movbe popcnt aes xsave avx f16c rdrand lahf_lm cmp_legacy svm extapic cr8_legacy abm sse4a misalignsse 3dnowprefetch osvw ibs skinit wdt tce topoext perfctr_core perfctr_nb bpext perfctr_llc mwaitx cpb cat_l3 cdp_l3 hw_pstate ssbd mba ibrs ibpb stibp vmmcall fsgsbase bmi1 avx2 smep bmi2 cqm rdt_a rdseed adx smap clflushopt clwb sha_ni xsaveopt xsavec xgetbv1 xsaves cqm_llc cqm_occup_llc cqm_mbm_total cqm_mbm_local clzero irperf xsaveerptr rdpru wbnoinvd amd_ppin arat npt lbrv svm_lock nrip_save tsc_scale vmcb_clean flushbyasid decodeassists pausefilter pfthreshold avic v_vmsave_vmload vgif v_spec_ctrl umip rdpid overflow_recov succor smca sev sev_es

G++:

g++ (Ubuntu 9.4.0-1ubuntu1~20.04) 9.4.0
Copyright (C) 2019 Free Software Foundation, Inc.
This is free software; see the source for copying conditions.  There is NO
warranty; not even for MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.

Clang:

clang version 10.0.0-4ubuntu1 
Target: x86_64-pc-linux-gnu
Thread model: posix
InstalledDir: /usr/bin

jachymb · 2026-10-17T14:25:20Z

Marking as draft until #3446 is merged.

stan-buildbot · 2026-10-17T23:21:00Z

Name	Old Result	New Result	Ratio	Performance change( 1 - new / old )
stat_comp_benchmarks/benchmarks/gp_regr/gen_gp_data.stan	0.04	0.04	1.02	1.69% faster
stat_comp_benchmarks/benchmarks/gp_regr/gp_regr.stan	0.17	0.17	0.99	-0.71% slower
stat_comp_benchmarks/benchmarks/garch/garch.stan	0.88	0.86	1.02	1.81% faster
stat_comp_benchmarks/benchmarks/low_dim_corr_gauss/low_dim_corr_gauss.stan	0.01	0.01	0.96	-3.78% slower
stat_comp_benchmarks/benchmarks/eight_schools/eight_schools.stan	0.11	0.11	1.0	-0.2% slower
stat_comp_benchmarks/benchmarks/low_dim_gauss_mix/low_dim_gauss_mix.stan	8.09	8.24	0.98	-1.94% slower
stat_comp_benchmarks/benchmarks/gp_pois_regr/gp_pois_regr.stan	3.66	3.7	0.99	-1.08% slower
stat_comp_benchmarks/benchmarks/pkpd/one_comp_mm_elim_abs.stan	33.38	32.99	1.01	1.15% faster
stat_comp_benchmarks/benchmarks/pkpd/sim_one_comp_mm_elim_abs.stan	0.39	0.38	1.01	0.55% faster
stat_comp_benchmarks/benchmarks/irt_2pl/irt_2pl.stan	6.82	6.81	1.0	0.09% faster
stat_comp_benchmarks/benchmarks/sir/sir.stan	168.05	176.09	0.95	-4.79% slower
stat_comp_benchmarks/benchmarks/arK/arK.stan	3.27	3.23	1.01	1.32% faster
stat_comp_benchmarks/benchmarks/arma/arma.stan	0.6	0.61	0.99	-0.75% slower
stat_comp_benchmarks/benchmarks/low_dim_gauss_mix_collapse/low_dim_gauss_mix_collapse.stan	25.82	26.3	0.98	-1.87% slower
performance.compilation	410.94	402.2	1.02	2.13% faster
Mean result: 0.9961487971322112

Jenkins Console Log
Jenkins Build Stages
Commit hash: 6a335dd02ea35aac291d2cb571d03092a8b16ad7

Machine information

Distributor ID:	Ubuntu
Description:	Ubuntu 20.04.3 LTS
Release:	20.04
Codename:	focal

CPU:

Architecture:                            x86_64
CPU op-mode(s):                          32-bit, 64-bit
Byte Order:                              Little Endian
Address sizes:                           52 bits physical, 57 bits virtual
CPU(s):                                  192
On-line CPU(s) list:                     0-191
Thread(s) per core:                      2
Core(s) per socket:                      48
Socket(s):                               2
NUMA node(s):                            2
Vendor ID:                               AuthenticAMD
CPU family:                              25
Model:                                   17
Model name:                              AMD EPYC 9474F 48-Core Processor
Stepping:                                1
Frequency boost:                         enabled
CPU MHz:                                 1497.639
CPU max MHz:                             4114.4229
CPU min MHz:                             1500.0000
BogoMIPS:                                7189.39
Virtualization:                          AMD-V
L1d cache:                               3 MiB
L1i cache:                               3 MiB
L2 cache:                                96 MiB
L3 cache:                                512 MiB
NUMA node0 CPU(s):                       0-47,96-143
NUMA node1 CPU(s):                       48-95,144-191
Vulnerability Gather data sampling:      Not affected
Vulnerability Indirect target selection: Not affected
Vulnerability Itlb multihit:             Not affected
Vulnerability L1tf:                      Not affected
Vulnerability Mds:                       Not affected
Vulnerability Meltdown:                  Not affected
Vulnerability Mmio stale data:           Not affected
Vulnerability Old microcode:             Not affected
Vulnerability Reg file data sampling:    Not affected
Vulnerability Retbleed:                  Not affected
Vulnerability Spec rstack overflow:      Mitigation; Safe RET
Vulnerability Spec store bypass:         Mitigation; Speculative Store Bypass disabled via prctl
Vulnerability Spectre v1:                Mitigation; usercopy/swapgs barriers and __user pointer sanitization
Vulnerability Spectre v2:                Mitigation; Enhanced / Automatic IBRS; IBPB conditional; STIBP always-on; PBRSB-eIBRS Not affected; BHI Not affected
Vulnerability Srbds:                     Not affected
Vulnerability Tsa:                       Mitigation; Clear CPU buffers
Vulnerability Tsx async abort:           Not affected
Vulnerability Vmscape:                   Mitigation; IBPB before exit to userspace
Flags:                                   fpu vme de pse tsc msr pae mce cx8 apic sep mtrr pge mca cmov pat pse36 clflush mmx fxsr sse sse2 ht syscall nx mmxext fxsr_opt pdpe1gb rdtscp lm constant_tsc rep_good amd_lbr_v2 nopl xtopology nonstop_tsc cpuid extd_apicid aperfmperf rapl pni pclmulqdq monitor ssse3 fma cx16 pcid sse4_1 sse4_2 x2apic movbe popcnt aes xsave avx f16c rdrand lahf_lm cmp_legacy svm extapic cr8_legacy abm sse4a misalignsse 3dnowprefetch osvw ibs skinit wdt tce topoext perfctr_core perfctr_nb bpext perfctr_llc mwaitx cpb cat_l3 cdp_l3 hw_pstate ssbd mba perfmon_v2 ibrs ibpb stibp ibrs_enhanced vmmcall fsgsbase bmi1 avx2 smep bmi2 erms invpcid cqm rdt_a avx512f avx512dq rdseed adx smap avx512ifma clflushopt clwb avx512cd sha_ni avx512bw avx512vl xsaveopt xsavec xgetbv1 xsaves cqm_llc cqm_occup_llc cqm_mbm_total cqm_mbm_local user_shstk avx512_bf16 clzero irperf xsaveerptr rdpru wbnoinvd amd_ppin cppc arat npt lbrv svm_lock nrip_save tsc_scale vmcb_clean flushbyasid decodeassists pausefilter pfthreshold avic v_vmsave_vmload vgif x2avic v_spec_ctrl vnmi avx512vbmi umip pku ospke avx512_vbmi2 gfni vaes vpclmulqdq avx512_vnni avx512_bitalg avx512_vpopcntdq la57 rdpid overflow_recov succor smca fsrm flush_l1d debug_swap

G++:

g++ (Ubuntu 9.4.0-1ubuntu1~20.04) 9.4.0
Copyright (C) 2019 Free Software Foundation, Inc.
This is free software; see the source for copying conditions.  There is NO
warranty; not even for MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.

Clang:

clang version 10.0.0-4ubuntu1 
Target: x86_64-pc-linux-gnu
Thread model: posix
InstalledDir: /usr/bin

The Eigen::Matrix<var> overload computed exp(-|x|) with a vectorized pass over a heap copy of the values, stored it, and divided again in the reverse pass: 1.07-1.30 of develop on glibc. Now one loop computes the value and the derivative e / (1 + e)^2 per element and stores the derivative, as suggested in the review of stan-dev#3446: 0.94-1.11 on glibc and 1.00-1.07 on MinGW, within the noise of identical code; values within 2 ulp of develop, +8 arena bytes per element as before.

…le-derivatives

stan-buildbot · 2026-10-20T18:45:25Z

Name	Old Result	New Result	Ratio	Performance change( 1 - new / old )
stat_comp_benchmarks/benchmarks/gp_regr/gen_gp_data.stan	0.04	0.04	1.0	-0.2% slower
stat_comp_benchmarks/benchmarks/gp_regr/gp_regr.stan	0.16	0.16	1.0	-0.19% slower
stat_comp_benchmarks/benchmarks/garch/garch.stan	0.88	0.87	1.01	0.81% faster
stat_comp_benchmarks/benchmarks/low_dim_corr_gauss/low_dim_corr_gauss.stan	0.01	0.01	0.96	-4.39% slower
stat_comp_benchmarks/benchmarks/eight_schools/eight_schools.stan	0.11	0.1	1.04	3.96% faster
stat_comp_benchmarks/benchmarks/low_dim_gauss_mix/low_dim_gauss_mix.stan	8.08	8.13	0.99	-0.62% slower
stat_comp_benchmarks/benchmarks/gp_pois_regr/gp_pois_regr.stan	3.72	3.68	1.01	1.02% faster
stat_comp_benchmarks/benchmarks/pkpd/one_comp_mm_elim_abs.stan	33.01	32.65	1.01	1.08% faster
stat_comp_benchmarks/benchmarks/pkpd/sim_one_comp_mm_elim_abs.stan	0.38	0.39	0.99	-0.86% slower
stat_comp_benchmarks/benchmarks/irt_2pl/irt_2pl.stan	6.79	6.8	1.0	-0.19% slower
stat_comp_benchmarks/benchmarks/sir/sir.stan	174.14	167.42	1.04	3.86% faster
stat_comp_benchmarks/benchmarks/arK/arK.stan	3.23	3.21	1.01	0.56% faster
stat_comp_benchmarks/benchmarks/arma/arma.stan	0.6	0.63	0.96	-4.35% slower
stat_comp_benchmarks/benchmarks/low_dim_gauss_mix_collapse/low_dim_gauss_mix_collapse.stan	25.88	25.85	1.0	0.12% faster
performance.compilation	390.62	376.28	1.04	3.67% faster
Mean result: 1.0034333153279384

Jenkins Console Log
Jenkins Build Stages
Commit hash: a2236819d562837e97a9bfe97dc9ebd6d8613729

Machine information

Distributor ID:	Ubuntu
Description:	Ubuntu 20.04.3 LTS
Release:	20.04
Codename:	focal

CPU:

Architecture:                            x86_64
CPU op-mode(s):                          32-bit, 64-bit
Byte Order:                              Little Endian
Address sizes:                           52 bits physical, 57 bits virtual
CPU(s):                                  192
On-line CPU(s) list:                     0-191
Thread(s) per core:                      2
Core(s) per socket:                      48
Socket(s):                               2
NUMA node(s):                            2
Vendor ID:                               AuthenticAMD
CPU family:                              25
Model:                                   17
Model name:                              AMD EPYC 9474F 48-Core Processor
Stepping:                                1
Frequency boost:                         enabled
CPU MHz:                                 1497.853
CPU max MHz:                             4114.4229
CPU min MHz:                             1500.0000
BogoMIPS:                                7189.39
Virtualization:                          AMD-V
L1d cache:                               3 MiB
L1i cache:                               3 MiB
L2 cache:                                96 MiB
L3 cache:                                512 MiB
NUMA node0 CPU(s):                       0-47,96-143
NUMA node1 CPU(s):                       48-95,144-191
Vulnerability Gather data sampling:      Not affected
Vulnerability Indirect target selection: Not affected
Vulnerability Itlb multihit:             Not affected
Vulnerability L1tf:                      Not affected
Vulnerability Mds:                       Not affected
Vulnerability Meltdown:                  Not affected
Vulnerability Mmio stale data:           Not affected
Vulnerability Old microcode:             Not affected
Vulnerability Reg file data sampling:    Not affected
Vulnerability Retbleed:                  Not affected
Vulnerability Spec rstack overflow:      Mitigation; Safe RET
Vulnerability Spec store bypass:         Mitigation; Speculative Store Bypass disabled via prctl
Vulnerability Spectre v1:                Mitigation; usercopy/swapgs barriers and __user pointer sanitization
Vulnerability Spectre v2:                Mitigation; Enhanced / Automatic IBRS; IBPB conditional; STIBP always-on; PBRSB-eIBRS Not affected; BHI Not affected
Vulnerability Srbds:                     Not affected
Vulnerability Tsa:                       Mitigation; Clear CPU buffers
Vulnerability Tsx async abort:           Not affected
Vulnerability Vmscape:                   Mitigation; IBPB before exit to userspace
Flags:                                   fpu vme de pse tsc msr pae mce cx8 apic sep mtrr pge mca cmov pat pse36 clflush mmx fxsr sse sse2 ht syscall nx mmxext fxsr_opt pdpe1gb rdtscp lm constant_tsc rep_good amd_lbr_v2 nopl xtopology nonstop_tsc cpuid extd_apicid aperfmperf rapl pni pclmulqdq monitor ssse3 fma cx16 pcid sse4_1 sse4_2 x2apic movbe popcnt aes xsave avx f16c rdrand lahf_lm cmp_legacy svm extapic cr8_legacy abm sse4a misalignsse 3dnowprefetch osvw ibs skinit wdt tce topoext perfctr_core perfctr_nb bpext perfctr_llc mwaitx cpb cat_l3 cdp_l3 hw_pstate ssbd mba perfmon_v2 ibrs ibpb stibp ibrs_enhanced vmmcall fsgsbase bmi1 avx2 smep bmi2 erms invpcid cqm rdt_a avx512f avx512dq rdseed adx smap avx512ifma clflushopt clwb avx512cd sha_ni avx512bw avx512vl xsaveopt xsavec xgetbv1 xsaves cqm_llc cqm_occup_llc cqm_mbm_total cqm_mbm_local user_shstk avx512_bf16 clzero irperf xsaveerptr rdpru wbnoinvd amd_ppin cppc arat npt lbrv svm_lock nrip_save tsc_scale vmcb_clean flushbyasid decodeassists pausefilter pfthreshold avic v_vmsave_vmload vgif x2avic v_spec_ctrl vnmi avx512vbmi umip pku ospke avx512_vbmi2 gfni vaes vpclmulqdq avx512_vnni avx512_bitalg avx512_vpopcntdq la57 rdpid overflow_recov succor smca fsrm flush_l1d debug_swap

G++:

g++ (Ubuntu 9.4.0-1ubuntu1~20.04) 9.4.0
Copyright (C) 2019 Free Software Foundation, Inc.
This is free software; see the source for copying conditions.  There is NO
warranty; not even for MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.

Clang:

clang version 10.0.0-4ubuntu1 
Target: x86_64-pc-linux-gnu
Thread model: posix
InstalledDir: /usr/bin

Jachym.Barvinek added 9 commits October 12, 2026 11:25

Tests: Phi_approx derivative in the upper tail

a335aa2

Fix Phi_approx reverse-mode derivative cancellation in the upper tail

198d939

Tests: inv_logit and log_inv_logit_diff derivatives at large arguments

c305f3b

Accurate inv_logit and log_inv_logit_diff derivatives at large arguments

2eeebdc

Phi_approx: name the coefficients in constants.hpp and use the double…

36694eb

…-precision minimax optimum

Move the Phi_approx forward-mode test to the Phi_approx branch

32e48bd

Merge bugfix/inv-logit-stable-derivatives into bugfix/phi-approx-stab…

0c06fa7

…le-derivatives

Tests: Phi_approx tail references shared by the mix and OpenCL tests,…

5b4e614

… forward-mode derivatives

Phi_approx: OpenCL value path as a kernel-generator expression of the…

d2742ff

… constants

avehtari added the numerics Numerical issues label Oct 13, 2026

jachymb added 2 commits October 13, 2026 15:49

Merge branch 'develop' into bugfix/inv-logit-stable-derivatives

549b4b5

Merge branch 'develop' into bugfix/phi-approx-stable-derivatives

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jachymb marked this pull request as draft October 17, 2026 14:25

jachymb added 3 commits October 20, 2026 01:07

Re-run CI: Windows runner lost communication in mix tests 5

de0c943

Merge bugfix/inv-logit-stable-derivatives into bugfix/phi-approx-stab…

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…le-derivatives

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Numerical improvements to `Phi_approx` - #3450

Numerical improvements to `Phi_approx`#3450
jachymb wants to merge 14 commits into
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jachymb commented Oct 12, 2026

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jachymb commented Oct 12, 2026

Summary

1. A derivative that does not cancel in the upper tail

2. The coefficients as named constants, at double precision

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stan-buildbot commented Oct 14, 2026

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jachymb commented Oct 17, 2026

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