Faster stiff ODE solve: the HIRES photochemistry model

Integrate the eight-species HIRES reaction model, a standard stiff test problem that comes from plant photobiology, faster than SciPy's Radau solver at matching accuracy.

Task

Solve, for $t$ from 0 to $t_1$ ,

\begin{aligned} \dot y_1&=-c_1y_1+c_2y_2+c_3y_3+c_4, & \dot y_2&=c_1y_1-c_5y_2,\\ \dot y_3&=-c_6y_3+c_2y_4+c_7y_5, & \dot y_4&=c_3y_2+c_1y_3-c_8y_4,\\ \dot y_5&=-c_9y_5+c_2y_6+c_2y_7, & \dot y_6&=-c_{10}y_6y_8+c_{11}y_4+c_1y_5-c_2y_6+c_{11}y_7,\\ \dot y_7&=c_{10}y_6y_8-c_{12}y_7, & \dot y_8&=-c_{10}y_6y_8+c_{12}y_7, \end{aligned}

with $t_1=n=370$ in the benchmark. Each instance multiplies the nominal constants $(c_1,\dots,c_{12})=(1.71,\,0.43,\,8.32,\,0.0007,\,8.75,\,10.03,\,0.035,\,1.12,\,1.745,\,280,\,0.69,\,1.81)$ by independent factors uniform on $[0.9,1.1]$ and starts from $y(0)=(1,0,0,0,0,0,0,y_8^0)$ with $y_8^0=0.0057$ times such a factor. Output: $y(t_1)$ . An output is accepted when it is finite and matches the reference under NumPy's allclose with relative tolerance $10^{-5}$ and absolute tolerance $10^{-8}$ . The reference solver is scipy.integrate.solve_ivp with the Radau method, rtol $10^{-10}$ and atol $10^{-9}$ .

Score: the reference's run time divided by the candidate's, averaged over the test instances; it is 1 if any output fails the check, and speed-ups below 1 count as 1. Higher is better.

Background

The HIRES ("high irradiance response") model was proposed by Schäfer (1975) to explain how plants respond to strong light through the photoreceptor phytochrome, and it became one of the standard stiff test problems (Hairer and Wanner, 1996). Its fast and slow reactions and the nonlinear coupling in the last three equations make implicit solvers necessary, and the comparison is against a tightly toleranced Radau run.

Data and evaluation

Instances: 100 test and 100 development instances from the task's generator at $n=370$ (that is, $t_1=370$ ), in the AlgoTune dataset; the reference needs about 110 ms per instance on the authors' machine.
Evaluation code: the AlgoTune harness (./algotune.sh evaluate in the repository), which runs the task's checker on every output.
Hardware: speed-ups depend on the machine, so reference and candidate are timed in the same environment. The paper keeps the fastest of 10 timed runs per instance, each after a warm-up, on an AMD EPYC 9454 CPU with 14 GB of memory; agents had 8 cores, and compilation is not timed.

What is known

Best on the AlgoTune leaderboard as of 2026-10-20: 90.25×, by the AlgoTuner agent driven by Gemini 3.1 Pro Preview, ahead of Claude Opus 4.6 (59.84×) and GPT-5.4 (41.86×).
Press et al. (2025) describe most agent speed-ups as surface-level: better library calls, better library usage or low-level rewrites, for example with Numba.

What would count as progress

A solver beating 90.25× on all 100 test instances under the AlgoTune harness, with its code; anyone can rerun the checker and the timing.
Evidence that the solver stays accurate for longer horizons and for the nominal constants, not only on the test distribution.

References

O. Press, B. Amos, H. Zhao, et al. AlgoTune: Can language models speed up general-purpose numerical programs? NeurIPS 2025. arXiv:2507.15887
R. Schäfer. A new approach to explain the 'high irradiance responses' of photomorphogenesis on the basis of phytochrome. Journal of Mathematical Biology 2, 41–56 (1975). doi:10.1007/BF00276015
E. Hairer, G. Wanner. Solving Ordinary Differential Equations II: Stiff and Differential-Algebraic Problems, 2nd edition. Springer (1996). doi:10.1007/978-3-642-05221-7

Source

Imported from AlgoTune task ode_hires, curated by Ori Press and the AlgoTune contributors. Open according to the source as of 2026-10-20. The statement above is written in our own words; see the source for the original wording, history and discussion.