From 7ac4ce950bc08a5a63124fded75e7578143c5a32 Mon Sep 17 00:00:00 2001 From: Linus Baumberger Date: Wed, 20 May 2026 22:55:47 +0200 Subject: [PATCH] Linus/variant lut --- docs/optimization_lut_exp.md | 138 +++++++++++ variants/variant_lut_exp.cpp | 454 +++++++++++++++++++++++++++++++++++ 2 files changed, 592 insertions(+) create mode 100644 docs/optimization_lut_exp.md create mode 100644 variants/variant_lut_exp.cpp diff --git a/docs/optimization_lut_exp.md b/docs/optimization_lut_exp.md new file mode 100644 index 0000000..116b317 --- /dev/null +++ b/docs/optimization_lut_exp.md @@ -0,0 +1,138 @@ +# Optimization: LUT for exp() + +- **Author**: Linus +- **Predecessor**: variant_baseline.cpp +- **File(s)**: variants/variant_lut_exp.cpp +- **Status**: positive at small N (10-15%), tied at large N + +## What this changes + +Replaced the exp() calls in pedestrian_repulsion_dt and +obstacle_repulsion_dt with a precomputed lookup table over the argument +range [-40, 0]. The table stores exp at evenly spaced sample points and we +interpolate with a cubic Hermite spline between them. Below -40 the value +is below FP noise relative to the force accumulator, so we just return 0. + +The thing that makes this work cleanly: for exp(x), the derivative IS the +function value (d/dx exp(x) = exp(x)). So at every sample point we know +the exact slope without having to estimate it with a centered difference. +That gives O(h^4) accuracy with just 2 LUT loads per call. + +## Code (relevant snippet) + +```cpp +constexpr int EXP_LUT_SIZE = 69000; +constexpr double EXP_LUT_MIN = -40.0; +constexpr double EXP_LUT_MAX = 0.0; +constexpr double EXP_LUT_INV_STEP = (EXP_LUT_SIZE - 1) / (EXP_LUT_MAX - EXP_LUT_MIN); + +static double exp_lut[EXP_LUT_SIZE]; + +// Populated before main() via a static initializer. +struct ExpLutInit { + ExpLutInit() { + const double step = (EXP_LUT_MAX - EXP_LUT_MIN) / (EXP_LUT_SIZE - 1); + for (int i = 0; i < EXP_LUT_SIZE; i++) { + exp_lut[i] = std::exp(EXP_LUT_MIN + i * step); + } + } +}; +static ExpLutInit exp_lut_init_helper; + +inline double fast_exp(double x) { + if (x <= EXP_LUT_MIN) return 0.0; + if (x >= EXP_LUT_MAX) return 1.0; + double pos = (x - EXP_LUT_MIN) * EXP_LUT_INV_STEP; + int idx = static_cast(pos); + if (idx >= EXP_LUT_SIZE - 1) return 1.0; + double t = pos - idx; + double y0 = exp_lut[idx]; + double y1 = exp_lut[idx + 1]; + // For exp, slope = value. In normalized parameter space, slope = h * y. + constexpr double h_step = (EXP_LUT_MAX - EXP_LUT_MIN) / (EXP_LUT_SIZE - 1); + double t2 = t * t; + double t3 = t2 * t; + double h00 = 2.0*t3 - 3.0*t2 + 1.0; + double h10 = t3 - 2.0*t2 + t; + double h01 = -2.0*t3 + 3.0*t2; + double h11 = t3 - t2; + return y0 * (h00 + h_step * h10) + y1 * (h01 + h_step * h11); +} +``` + +Then in pedestrian_repulsion_dt and obstacle_repulsion_dt, the libm +`exp(...)` is replaced with `fast_exp(...)`. + +## Benchmark + +Conditions: min of 3 runs, N_ITER=1000, -O3 -march=native -flto. + +On my laptop (Intel ULTRA 9 185H @ 2.2 GHz, WSL Ubuntu): about +15% faster at N=50, 12% at N=100, 8% at N=200, then roughly tied at N=500 +and N=1000. The shrinking speedup matches a cache story (see Analysis). + + +### x86 + +**Machine**: + +``` + +``` + +**Frequency**: + +| N | W(n) [Flops] | T(n) [Cycles] | Performance [Flops/Cycle] | Diff vs Baseline | +| ---- | ------------ | ------------- | ------------------------- | ---------------- | +| 50 | | | | | +| 100 | | | | | +| 200 | | | | | +| 500 | | | | | +| 1000 | | | | | + +### ARM (Apple) + +**Machine**: + +``` + +``` + +**Frequency**: + +| N | W(n) [Flops] | T(n) [Cycles] | Performance [Flops/Cycle] | Diff vs Baseline | +| ---- | ------------ | ------------- | ------------------------- | ---------------- | +| 50 | | | | | +| 100 | | | | | +| 200 | | | | | +| 500 | | | | | +| 1000 | | | | | + +## Analysis + +The LUT replaces a libm exp (about 30 cycles) with a table lookup plus a +small cubic polynomial (about 18 cycles). At small N the table fits in L2 +and the speedup is real, around 10-15%. As N grows the pedestrian array +starts competing with the LUT for L2 cache, and the LUT lookups end +up in L3 instead, which is about as expensive as the libm exp we were +trying to replace. + +About verify: the LUT introduces a small per-call error, and chaos in the +simulation amplifies it over 4000 timesteps, but still passes our strict verification + +If we are more tolerant with the "error" its possible to use a much smaller LUT. Maybe we could try that out + +## Takeaway + +LUT for exp works at small N where the table fits in cache. At large N the +cache pressure kills the gain. A smaller LUT (or range reduction into a +tiny table) would probably do better across the whole N range without the +cache penalty. + +## References + +- Tang, P. T. P. (1989). "Table-driven implementation of the exponential + function in IEEE floating-point arithmetic." ACM TOMS 15(2), 144-157. + https://dl.acm.org/doi/10.1145/63522.214389 +- Cubic Hermite spline. Wikipedia. + https://en.wikipedia.org/wiki/Cubic_Hermite_spline diff --git a/variants/variant_lut_exp.cpp b/variants/variant_lut_exp.cpp new file mode 100644 index 0000000..45b7cdb --- /dev/null +++ b/variants/variant_lut_exp.cpp @@ -0,0 +1,454 @@ +// This file is copied from the variant_baseline and then modified +// The goal is that we use a look up table to precompute expensive calculations +// especially to precompute the exp calls +// we call exp(two_b * NEG_TWO_SIGMA_INV) +// which is always negativ +// if a value is smaller then -40 its so small that we can ignore it +// for the other values we will create a lookup table + +#include "../headers/board.hpp" +#include "../headers/constants.hpp" +#include "../headers/pedestrian.hpp" +#include "../headers/vector2d.hpp" + +#include +#include +#include +#include +#include // keep — forces code uses unqualified sqrt/exp +#include +#include +#include + +#define SEED 42 +#define BOARD_WIDTH 10.0 +#define BOARD_LENGTH 50.0 +#define WALL_EPS 1e-3 +#define BOARD_WIDTH_MINUS_WALL_EPS 9.999 + +#ifndef RANDOM_SEED +#define RANDOM_SEED 0 +#endif + +#ifndef BENCHMARK +#define BENCHMARK 0 +#endif + +using namespace std; + +// exp lookup table +constexpr int EXP_LUT_SIZE = 69000; +constexpr double EXP_LUT_MIN = -40.0; +constexpr double EXP_LUT_MAX = 0.0; +constexpr double EXP_LUT_INV_STEP = (EXP_LUT_SIZE - 1) / (EXP_LUT_MAX - EXP_LUT_MIN); + +static double exp_lut[EXP_LUT_SIZE]; + +struct ExpLutInit { + ExpLutInit() { + const double step = (EXP_LUT_MAX - EXP_LUT_MIN) / (EXP_LUT_SIZE - 1); + for (int i = 0; i < EXP_LUT_SIZE; i++) { + exp_lut[i] = std::exp(EXP_LUT_MIN + i * step); + } + } +}; +static ExpLutInit exp_lut_init_helper; +// i used the following things to get a better understanding of how i can solve that problem +// Tang, P. T. P. (1989). "Table-driven implementation of the exponential function in IEEE floating-point arithmetic." ACM TOMS 15(2), 144-157. +// https://dl.acm.org/doi/10.1145/63522.214389 +// Cubic Hermite spline. Wikipedia. +// https://en.wikipedia.org/wiki/Cubic_Hermite_spline +inline double fast_exp(double x) { + if (x <= EXP_LUT_MIN) + return 0.0; + if (x >= EXP_LUT_MAX) + return 1.0; + + double pos = (x - EXP_LUT_MIN) * EXP_LUT_INV_STEP; + int idx = static_cast(pos); + if (idx >= EXP_LUT_SIZE - 1) + return 1.0; + + double t = pos - idx; + double y0 = exp_lut[idx]; + double y1 = exp_lut[idx + 1]; + + // For exp(x), derivative equals function value. + // In normalized parameter space (t in [0,1] across width h), + // dy/dt = h * y. No centered differences needed - we know the exact slope. + constexpr double h_step = (EXP_LUT_MAX - EXP_LUT_MIN) / (EXP_LUT_SIZE - 1); + + double t2 = t * t; + double t3 = t2 * t; + + double h00 = 2.0 * t3 - 3.0 * t2 + 1.0; + double h10 = t3 - 2.0 * t2 + t; + double h01 = -2.0 * t3 + 3.0 * t2; + double h11 = t3 - t2; + + return y0 * (h00 + h_step * h10) + y1 * (h01 + h_step * h11); +} +// forces.cpp + +// Flop count: 2 ADDs, 5 MULTs, 1 SQRT +double w(Vector2d e, Vector2d f) { + if (e.dot(f) >= f.norm() * cos_phi) + return dt_neg; + else + return c_neg_times_dt; +} + +// Flop count: 2 ADDs, 4 MULTs +Vector2d acceleration_term_dt(const Pedestrian &alpha) { + + // # 2 MULTs + Vector2d desired_velocity = alpha.e * alpha.desired_v0; + + Vector2d actual_velocity = alpha.curr_vel; + + // # 2 ADDs, 2 MULTs + Vector2d f_alpha_0_dt = (desired_velocity - actual_velocity) * dt_times_tau_alpha_inv; + + return f_alpha_0_dt; +} + +// Flop count: 12 ADDs, 23 MULTs, 1 DIV, 4 SQRTs, 1 EXP +Vector2d pedestrian_repulsion_dt(const Pedestrian &alpha, const Pedestrian &beta) { + + // # 2 ADDs + // $\vec{r}_{\alpha\beta}$ + Vector2d r_ab = alpha.curr_pos - beta.curr_pos; + + // # 1 ADD, 2 MULTs, 1 SQRT + // $\Arrowvert \vec{r}_{\alpha \beta} \Arrowvert$ + double r_ab_norm = r_ab.norm(); + r_ab_norm = r_ab_norm > EPS ? r_ab_norm : EPS; // Avoid division by zero + + // $v_\beta \Delta t \vec{e}_\beta$ + Vector2d s_b_e = beta.s_b_e; + + // # 2 ADDs + // $\vec{r}_{\alpha\beta} - \vec{v}_{\beta}\varDelta_{t}\vec{e}_{\beta}$ + Vector2d r_ab2 = r_ab - s_b_e; + + // # 1 ADD, 2 MULTs, 1 SQRT + // $\Arrowvert \vec{r}_{\alpha \beta} - v_\beta \Delta t \vec{e}_\beta \Arrowvert$ + double r_ab2_norm = r_ab2.norm(); + r_ab2_norm = r_ab2_norm > EPS ? r_ab2_norm : EPS; // Avoid division by zero + + // # 1 ADD + // $\Arrowvert \vec{r}_{\alpha \beta} \Arrowvert + \Arrowvert \vec{r}_{\alpha \beta} - v_\beta \Delta t \vec{e}_\beta \Arrowvert$ + double r_ab_norm_sum = r_ab_norm + r_ab2_norm; + + // # 1 ADD, 1 MULTs, 1 SQRT + // + // $2b = \sqrt{(\Arrowvert \vec{r}_{\alpha \beta} \Arrowvert + \Arrowvert \vec{r}_{\alpha \beta} - v_\beta \Delta t \vec{e}_\beta \Arrowvert)^2 - (v_\beta \Delta t)^2}$ + // + double two_b_sq = r_ab_norm_sum * r_ab_norm_sum - beta.s_b_sq; + two_b_sq = two_b_sq > EPS_SQ ? two_b_sq : EPS_SQ; // Avoid sqrt of negative and avoid division by zero + double two_b = sqrt(two_b_sq); + + // # 5 MULTs, 1 DIV, 1 EXP + // + // $\frac{-V^0_{\alpha\beta}}{2\sigma}\,e^{-b/\sigma}\,\frac{\Arrowvert \vec{r}_{\alpha \beta} \Arrowvert + \Arrowvert \vec{r}_{\alpha \beta} - v_\beta \Delta t \vec{e}_\beta \Arrowvert}{2b\,\Arrowvert \vec{r}_{\alpha \beta} \Arrowvert\,\Arrowvert \vec{r}_{\alpha \beta} - v_\beta \Delta t \vec{e}_\beta \Arrowvert}$ + // + double scalar_mult_common = (r_ab_norm_sum / (two_b * r_ab_norm * r_ab2_norm)) * (NEG_V0_AB_div_TWO_SIGMA * fast_exp(two_b * NEG_TWO_SIGMA_INV)); + + // # 1 MULT + // + // $\frac{-V^0_{\alpha\beta}}{2\sigma}\,e^{-b/\sigma}\,\frac{\Arrowvert \vec{r}_{\alpha \beta} \Arrowvert + \Arrowvert \vec{r}_{\alpha \beta} - v_\beta \Delta t \vec{e}_\beta \Arrowvert}{2b\,\Arrowvert \vec{r}_{\alpha \beta} \Arrowvert}$ + // + double scalar_mult_r_ab = scalar_mult_common * r_ab2_norm; + + // # 1 MULT + // + // $\frac{-V^0_{\alpha\beta}}{2\sigma}\,e^{-b/\sigma}\,\frac{\Arrowvert \vec{r}_{\alpha \beta} \Arrowvert + \Arrowvert \vec{r}_{\alpha \beta} - v_\beta \Delta t \vec{e}_\beta \Arrowvert}{2b\,\Arrowvert \vec{r}_{\alpha \beta} - v_\beta \Delta t \vec{e}_\beta \Arrowvert}$ + // + double scalar_mult_r_ab2 = scalar_mult_common * r_ab_norm; + + // # 2 ADDs, 4 MULTs + // + // $\vec{-f}_{\alpha \beta}(\vec{r}_{\alpha \beta}) = \frac{-V^0_{\alpha \beta}}{2\sigma}e^{-b / \sigma} 2\grad_{\vec{r}_{\alpha \beta}}b $ + // + // + // $= \frac{-V^0_{\alpha\beta}}{2\sigma}\,e^{-b/\sigma}\,\frac{\Arrowvert \vec{r}_{\alpha \beta} \Arrowvert + \Arrowvert \vec{r}_{\alpha \beta} - v_\beta \Delta t \vec{e}_\beta \Arrowvert}{2b\,\Arrowvert \vec{r}_{\alpha \beta} \Arrowvert}\,\vec{r}_{\alpha \beta} + \frac{-V^0_{\alpha\beta}}{2\sigma}\,e^{-b/\sigma}\,\frac{\Arrowvert \vec{r}_{\alpha \beta} \Arrowvert + \Arrowvert \vec{r}_{\alpha \beta} - v_\beta \Delta t \vec{e}_\beta \Arrowvert}{2b\,\Arrowvert \vec{r}_{\alpha \beta} - + // v_\beta \Delta t \vec{e}_\beta \Arrowvert}\,(\vec{r}_{\alpha \beta} - v_\beta \Delta t \vec{e}_\beta)$ + // + // + // + // $2\grad_{\vec{r}_{\alpha \beta}}b = \frac{(\Arrowvert \vec{r}_{\alpha \beta} \Arrowvert + \Arrowvert \vec{r}_{\alpha \beta} - v_\beta \Delta t \vec{e}_\beta \Arrowvert)(\frac{\vec{r}_{\alpha \beta}}{\Arrowvert \vec{r}_{\alpha \beta} \Arrowvert} + \frac{\vec{r}_{\alpha \beta} - v_\beta \Delta t \vec{e}_\beta}{\Arrowvert \vec{r}_{\alpha \beta} - v_\beta \Delta t \vec{e}_\beta \Arrowvert})}{2b}$ + // + // + // $= \frac{\Arrowvert \vec{r}_{\alpha \beta} \Arrowvert + \Arrowvert \vec{r}_{\alpha \beta} - v_\beta Delta t \vec{e}_\beta \Arrowvert}{2b\,\Arrowvert \vec{r}_{\alpha \beta} \Arrowvert}\,\vec{r}_{\alpha \beta} + \frac{\Arrowvert \vec{r}_{\alpha \beta} \Arrowvert + \Arrowvert \vec{r}_{\alpha \beta} - v_\beta \Delta t \vec{e}_\beta \Arrowvert}{2b\,\Arrowvert \vec{r}_{\alpha \beta} - v_\beta \Delta t \vec{e}_\beta \Arrowvert}\,(\vec{r}_{\alpha \beta} - v_\beta \Delta t \vec{e}_\beta)$ + // + // + Vector2d f_alpha_beta_neg = r_ab * scalar_mult_r_ab + r_ab2 * scalar_mult_r_ab2; + + // # 2 ADDs, 7 MULTs, 1 SQRT + // + // $\vec{F}_{\alpha\beta}(\vec{e}_{\alpha}, \vec{r}_{\alpha \beta}).dt = w(\vec{e}_{\alpha}, -\vec{f}_{\alpha\beta}(\vec{r}_{\alpha \beta})).\vec{f}_{\alpha\beta}(\vec{r}_{\alpha \beta}).dt$ + // + Vector2d f_alpha_beta_dt = f_alpha_beta_neg * w(alpha.e, f_alpha_beta_neg); + + return f_alpha_beta_dt; +} + +// Flop count: 10 ADDs, 18 MULTs, 1 DIVs, 2 SQRTs, 1 EXP +Vector2d obstacle_repulsion_dt(const Pedestrian &alpha, const Obstacle &B) { + + // # 7 ADDs, 5 MULTs + // $\vec{r}_{\alpha B}$ + Vector2d r_aB = alpha.curr_pos - B.find_nearest_obstacle_point(alpha.curr_pos); + + // # 1 ADD, 2 MULTs, 1 SQRT + // $\Arrowvert \vec{r}_{\alpha B} \Arrowvert$ + double r_aB_norm = r_aB.norm(); + r_aB_norm = r_aB_norm > EPS ? r_aB_norm : EPS; // Avoid division by zero + + // # 2 MULTs, 1 DIV, 1 EXP + // + // $\frac{-U^0_{\alpha B}}{R} e^{- \Arrowvert \vec{r}_{\alpha B} \Arrowvert / R}$ + // + double scalar_mult = (NEG_U0_AB_div_R * fast_exp(r_aB_norm * NEG_R_INV)) / r_aB_norm; + + // # 2 MULTs + // + // $-\vec{F}_{\alpha B}(\vec{r}_{\alpha B}) = \frac{-U^0_{\alpha B}}{R} e^{- \Arrowvert \vec{r}_{\alpha B} \Arrowvert / R} \frac{\vec{r}_{\alpha B}}{\Arrowvert \vec{r}_{\alpha B} \Arrowvert}$ + // + Vector2d f_alpha_B_neg = r_aB * scalar_mult; + + // # 2 ADDs, 7 MULTs, 1 SQRT + // + // $\vec{F}_{\alpha B}(\vec{e}_{\alpha}, \vec{r}_{\alpha B}).dt = w(\vec{e}_{\alpha}, -\vec{F}_{\alpha B}(\vec{r}_{\alpha B})).\vec{F}_{\alpha B}(\vec{r}_{\alpha B}).dt$ + // + Vector2d f_alpha_B_dt = f_alpha_B_neg * w(alpha.e, f_alpha_B_neg); + + return f_alpha_B_dt; +} + +// social_force_baseline.cpp + +Pedestrian *init_pedestrians(int n_p) { +#if RANDOM_SEED + random_device rd; // Platform-provided random seed source. + mt19937 gen(rd()); // Non-reproducible initialization. +#else + mt19937 gen(SEED); // Reproducible initialization. +#endif + + normal_distribution velocity_gen(v0_mean, v0_std_dev); + uniform_real_distribution y_gen(0.3, BOARD_WIDTH - 0.3); + uniform_real_distribution x_gen_left(2.0, BOARD_LENGTH / 2.0); + uniform_real_distribution x_gen_right(BOARD_LENGTH / 2.0, BOARD_LENGTH - 2.0); + + Pedestrian *P = new Pedestrian[n_p]; + + int n_p_half = n_p / 2; + + // Group of pedestrians walking from left to right. + for (int i = 0; i < n_p_half; i++) { + double x = x_gen_left(gen); + double y = y_gen(gen); + + Vector2d start_pos = Vector2d(x, y); + Vector2d destination = Vector2d(BOARD_LENGTH + 30.0, y); + + Vector2d init_e = destination - start_pos; + double init_e_norm = init_e.norm(); + Vector2d init_e_unit = init_e / init_e_norm; + + // Reject near-zero desired speeds to avoid pedestrians that barely move, + // and reject negative speeds because they would reverse the intended + // walking direction. + double desired_v0; + do { + desired_v0 = velocity_gen(gen); + } while (desired_v0 <= 0.1); + + double v_max = 1.3 * desired_v0; + + P[i] = Pedestrian(WHITE, Vector2d(), start_pos, init_e_unit, v_max, desired_v0, destination); + } + + // Group of pedestrians walking from right to left. + for (int i = n_p_half; i < n_p; i++) { + double x = x_gen_right(gen); + double y = y_gen(gen); + + Vector2d start_pos = Vector2d(x, y); + Vector2d destination = Vector2d(-30.0, y); + + Vector2d init_e = destination - start_pos; + double init_e_norm = init_e.norm(); + Vector2d init_e_unit = init_e / init_e_norm; + + // Reject near-zero desired speeds to avoid pedestrians that barely move, + // and reject negative speeds because they would reverse the intended + // walking direction. + double desired_v0; + do { + desired_v0 = velocity_gen(gen); + } while (desired_v0 <= 0.1); + + double v_max = 1.3 * desired_v0; + + P[i] = Pedestrian(BLACK, Vector2d(), start_pos, init_e_unit, v_max, desired_v0, destination); + } + + return P; +} + +Board init_board_without_obstacles() { return Board(BOARD_LENGTH, BOARD_WIDTH, 0, nullptr); } + +int run_simulation(Pedestrian *pedestrians, Board &board, int n_p, int n_iter, const char *dump_path) { + +#if BENCHMARK + // Avoid compiler unused parameter warning + (void)dump_path; +#endif + +#if !BENCHMARK + const int dump_interval = 1; + + // Each sample stores: time step, pedestrian id, x, y, e.x, e.y, color. + vector> samples; + + if (dump_path) { + samples.reserve(static_cast(n_iter / dump_interval + 1) * n_p); + + // Store the initial state before the first simulation step. + for (int i = 0; i < n_p; i++) { + Pedestrian &p = pedestrians[i]; + samples.emplace_back(0, i, p.curr_pos.x, p.curr_pos.y, p.e.x, p.e.y, p.color); + } + } +#endif + + for (int t = 0; t < n_iter; t++) { + // Pass 1: Compute all forces from the current state. + for (int i = 0; i < n_p; i++) { + Pedestrian &p = pedestrians[i]; + + // # 2 ADDs, 4 MULTs + // + // $\vec{F}_\alpha(\vec{v}_\alpha, v^0_\alpha \vec{e}_\alpha) = \frac{1}{\tau_\alpha}(v^0_\alpha \vec{e}_\alpha - \vec{v}_\alpha)$ + // + Vector2d f_alpha_dt = acceleration_term_dt(p); + + // # (N-1) * (12 ADDs, 23 MULTs, 1 DIV, 4 SQRTs, 1 EXP) + // + (N-1) * 2 ADDs + // + // $\sum_{\beta} \vec{F}_{\alpha\beta}(\vec{e}_{\alpha},\vec{r}_{\alpha} - \vec{r}_{\beta})$ + // + for (int j = 0; j < n_p; j++) { + if (i == j) + continue; + + f_alpha_dt += pedestrian_repulsion_dt(p, pedestrians[j]); + } + + // # board.n_obstacles * (10 ADDs, 18 MULTs, 1 DIV, 2 SQRTs, 1 EXP) + // + board.n_obstacles * 2 ADDs + // + // $\sum_{B} \vec{F}_{\alpha B}(\vec{e}_{\alpha}, \vec{r}_{\alpha} - \vec{r}_{B})$ + // + for (int k = 0; k < board.n_obstacles; k++) { + f_alpha_dt += obstacle_repulsion_dt(p, board.obstacles[k]); + } + + // # 2 ADDs + // + // + // $\frac{d\vec{w}_\alpha}{dt} = \vec{F}_{\alpha}(t)$ + // + p.w += f_alpha_dt; + } + + // Pass 2: Update pedestrian states after all forces have been computed. + // # (7 ADDs, 16 MULTs, 2 DIVs, 3 SQRTs) * n_p + for (int i = 0; i < n_p; i++) { + Pedestrian &p = pedestrians[i]; + + // # 1 ADD, 2 MULTs, 1 SQRT + double w_norm = p.w.norm(); + + double w_norm2 = w_norm > EPS ? w_norm : EPS; // Avoid division by zero. + + // # 1 DIV + double correction_factor = std::min(1.0, p.v_max / w_norm2); + + // # 2 MULTs + p.curr_vel = p.w * correction_factor; + + // # 2 ADDs, 2 MULTs + // + // $\vec{r}_i^{t+1} = \vec{r}_i^t + \vec{v}_i^t dt$ + // + p.curr_pos = p.curr_pos + p.curr_vel * dt; + + // When a pedestrian reaches the end of the corridor, teleport them back + // to the opposite entrance while keeping the same y-coordinate. This + // maintains a constant pedestrian flow and makes lane formation easier to + // observe. + if (p.color == WHITE && p.curr_pos.x > (BOARD_LENGTH - 1.5)) { + p.curr_pos.x = 1.0; + p.destination.y = p.curr_pos.y; + p.w = Vector2d(p.desired_v0, 0.0); + p.curr_vel = p.w; + } + if (p.color == BLACK && p.curr_pos.x < 1.5) { + p.curr_pos.x = BOARD_LENGTH - 1.0; + p.destination.y = p.curr_pos.y; + p.w = Vector2d(-p.desired_v0, 0.0); + p.curr_vel = p.w; + } + + // Keep pedestrians inside the board boundaries. + p.curr_pos.y = clamp(p.curr_pos.y, WALL_EPS, BOARD_WIDTH_MINUS_WALL_EPS); + + // # 2 ADDs + Vector2d e = p.destination - p.curr_pos; + + // # 1 ADD, 2 MULTs, 1 DIV, 1 SQRT + double e_norm = e.norm(); + double e_norm2 = e_norm > EPS ? e_norm : EPS; // Avoid division by zero. + double e_norm_inv = 1.0 / e_norm2; + + // # 2 MULTs + p.e = e * e_norm_inv; + + // # 1 ADD, 2 MULTs + double velocity_norm_sq = p.curr_vel.dot(p.curr_vel); + + // # 1 SQRT + double velocity_norm = std::sqrt(velocity_norm_sq); + + // # 1 MULT + p.s_b_sq = velocity_norm_sq * delta_t_sq; + + // # 3 MULTs + p.s_b_e = p.e * (velocity_norm * delta_t); + +#if !BENCHMARK + // Store the current pedestrian state for visualization. + if (dump_path && (t % dump_interval == 0)) { + samples.emplace_back(t + 1, i, p.curr_pos.x, p.curr_pos.y, p.e.x, p.e.y, p.color); + } +#endif + } + } + +#if !BENCHMARK + // Export recorded simulation states to CSV. + if (dump_path) { + ofstream out(dump_path); + out << "time,id,x,y,p.e.x,p.e.y,color\n"; + + for (auto &[t, id, x, y, e_x, e_y, c] : samples) { + out << t << ',' << id << ',' << x << ',' << y << ',' << e_x << ',' << e_y << ',' << c << '\n'; + } + } +#endif + + return 0; +} \ No newline at end of file