Linus/variant lut

docs/optimization_lut_exp.md

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+# 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<int>(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**:
+
+```
+<paste machine info here>
+```
+
+**Frequency**: <e.g., 2 GHz>
+
+| N    | W(n) [Flops] | T(n) [Cycles] | Performance [Flops/Cycle] | Diff vs Baseline |
+| ---- | ------------ | ------------- | ------------------------- | ---------------- |
+| 50   |              |               |                           |                  |
+| 100  |              |               |                           |                  |
+| 200  |              |               |                           |                  |
+| 500  |              |               |                           |                  |
+| 1000 |              |               |                           |                  |
+
+### ARM (Apple)
+
+**Machine**:
+
+```
+<paste machine info here>
+```
+
+**Frequency**: <e.g., 3.2 GHz>
+
+| 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

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 <algorithm>
+#include <cmath>
+#include <fstream>
+#include <iostream>
+#include <math.h> // keep — forces code uses unqualified sqrt/exp
+#include <random>
+#include <tuple>
+#include <vector>
+
+#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<int>(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<double> velocity_gen(v0_mean, v0_std_dev);
+  uniform_real_distribution<double> y_gen(0.3, BOARD_WIDTH - 0.3);
+  uniform_real_distribution<double> x_gen_left(2.0, BOARD_LENGTH / 2.0);
+  uniform_real_distribution<double> 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<tuple<int, int, double, double, double, double, int>> samples;
+
+  if (dump_path) {
+    samples.reserve(static_cast<size_t>(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;
+}