When Do Extra Evaluations per Step Pay Off in Few-Step Sampling?

Few-step samplers for diffusion and flow models spend a fixed budget of network function evaluations (NFE). A multi-stage solver such as Heun spends some of them inside a step to correct it; a first-order solver spends each on a new step. We ask when an extra evaluation is worth more inside a step. Measured on the forward path of a reference sample, the error of one step is a term shared by all solvers plus a correction from the extra evaluations; for Euler the shared term is the step's share of the remaining time times the model's error in predicting the clean sample. An Euler step is a convex combination of the current state and this prediction, so it is non-expansive whenever the prediction is. The correction grows with the ratio $h/k$ of the step length to the distance of its end to the clean image, where the sampling equation becomes stiff. Extra evaluations should therefore pay off only on short steps away from the image, and their cost inside long steps should shrink as the budget grows. A solver menu graph records the one-step error of eight solvers on every interval of a fine grid and plans the timestep and solver of every step for an exact budget by dynamic programming. On three undistilled models, one Heun step never beats two Euler steps once $h/k>5$. On GenEval at 10 NFE, re-placing Euler steps raises the score by up to 0.107, while solvers that spend evaluations inside a step lose 0.031 on SDXL and 0.090 on Qwen-Image on average; at 15 NFE the loss falls to 0.010 and 0.028. On FLUX.1-dev and SD3.5-Large, five Heun steps score 0.05 below ten Euler steps, both placed by the native scheduler. An Euler schedule calibrated on a few prompts compares favourably with Align Your Steps and GITS on SDXL at 10 NFE, and paths planned on Qwen-Image carry over to Qwen-Image-Edit, where they bring edits closer to the 50-step result. On the distilled Z-Image-Turbo, whose graph has almost no window, two planned Euler steps score as high as its default eight. In few-step sampling, evaluations should thus go first to placing first-order steps, and into a step only where the step is short relative to its distance from the clean image.