#version 430
// Mixed-precision Runge-Kutta 4th order compute shader
// Dynamically adjusts precision based on numerical requirements and performance
layout(local_size_x = 256, local_size_y = 1, local_size_z = 1) in;
// Solution vectors with different precision levels
layout(std430, binding = 0) buffer YBuffer {
float y[]; // Current solution (mixed precision storage)
};
layout(std430, binding = 1) buffer K1Buffer {
float k1[]; // RK4 stage 1
};
layout(std430, binding = 2) buffer K2Buffer {
float k2[]; // RK4 stage 2
};
layout(std430, binding = 3) buffer K3Buffer {
float k3[]; // RK4 stage 3
};
layout(std430, binding = 4) buffer K4Buffer {
float k4[]; // RK4 stage 4
};
layout(std430, binding = 5) buffer ResultBuffer {
float result[]; // Final result
};
// Precision control
layout(std430, binding = 6) buffer PrecisionMaskBuffer {
int precision_mask[]; // 0=half, 1=single, 2=double emulation
};
layout(std430, binding = 7) buffer ErrorEstimateBuffer {
float error_estimate[]; // Error estimates for precision adaptation
};
// Performance metrics
layout(std430, binding = 8) buffer PerformanceBuffer {
float performance_metrics[]; // [execution_time, memory_bandwidth, cache_misses]
};
uniform float t;
uniform float h;
uniform int n;
uniform float precision_threshold_low; // Switch to lower precision
uniform float precision_threshold_high; // Switch to higher precision
uniform bool adaptive_precision; // Enable adaptive precision control
uniform int rk4_stage; // Which RK4 stage to compute (1-4, 5=combine)
// Shared memory for performance optimization
shared float shared_y[256];
shared float shared_k[256];
// Half precision emulation helpers
float to_half_precision(float value) {
// Emulate half precision by reducing mantissa bits
const float scale = 1024.0; // 2^10 for half precision range
return floor(value * scale) / scale;
}
// Double precision emulation using two floats
vec2 to_double_precision(float value) {
float hi = value;
float lo = value - hi;
return vec2(hi, lo);
}
vec2 add_double(vec2 a, vec2 b) {
float s = a.x + b.x;
float v = s - a.x;
float e = (a.x - (s - v)) + (b.x - v);
return vec2(s, e + a.y + b.y);
}
vec2 mul_double(vec2 a, vec2 b) {
float p = a.x * b.x;
float e = p - a.x * b.x;
return vec2(p, ((a.x * b.y) + (a.y * b.x)) + e);
}
// ODE function with precision-aware computation
float ode_function_precise(float time, float state_val, int precision_level) {
// Example: harmonic oscillator dy/dt = -y
float derivative = -state_val;
if (precision_level == 0) {
// Half precision
return to_half_precision(derivative);
} else if (precision_level == 1) {
// Single precision (native)
return derivative;
} else {
// Double precision emulation
vec2 state_double = to_double_precision(state_val);
vec2 neg_one = vec2(-1.0, 0.0);
vec2 result_double = mul_double(neg_one, state_double);
return result_double.x + result_double.y;
}
}
// Adaptive precision control
int update_precision_level(uint index, float current_error, int current_precision) {
if (!adaptive_precision) return current_precision;
// Get performance metrics
float exec_time = performance_metrics[index * 3 + 0];
float memory_bw = performance_metrics[index * 3 + 1];
float cache_misses = performance_metrics[index * 3 + 2];
// Performance pressure factor
float perf_pressure = (exec_time / 1000.0) + (cache_misses / 1000.0) +
max(0.0, memory_bw - 0.8) * 5.0;
int new_precision = current_precision;
// Increase precision if error is too high and performance allows
if (current_error > precision_threshold_high && perf_pressure < 0.5) {
new_precision = min(2, current_precision + 1);
}
// Decrease precision if error is low or performance pressure is high
else if (current_error < precision_threshold_low || perf_pressure > 1.0) {
new_precision = max(0, current_precision - 1);
}
return new_precision;
}
void main() {
uint index = gl_GlobalInvocationID.x;
uint tid = gl_LocalInvocationID.x;
if (index >= n) return;
// Load current precision level
int precision_level = precision_mask[index];
// Load current state into shared memory
shared_y[tid] = y[index];
barrier();
if (rk4_stage <= 4) {
// RK4 stage computation
float state_val = shared_y[tid];
float stage_time = t;
// Adjust time for different stages
if (rk4_stage == 2 || rk4_stage == 3) {
stage_time += h * 0.5;
} else if (rk4_stage == 4) {
stage_time += h;
}
// Add previous stage contributions
if (rk4_stage == 2) {
state_val += 0.5 * h * k1[index];
} else if (rk4_stage == 3) {
state_val += 0.5 * h * k2[index];
} else if (rk4_stage == 4) {
state_val += h * k3[index];
}
// Compute derivative with precision-aware calculation
float derivative = ode_function_precise(stage_time, state_val, precision_level);
// Store result in appropriate k buffer
if (rk4_stage == 1) {
k1[index] = derivative;
} else if (rk4_stage == 2) {
k2[index] = derivative;
} else if (rk4_stage == 3) {
k3[index] = derivative;
} else if (rk4_stage == 4) {
k4[index] = derivative;
}
} else if (rk4_stage == 5) {
// Final combination stage
float y_val = shared_y[tid];
float k1_val = k1[index];
float k2_val = k2[index];
float k3_val = k3[index];
float k4_val = k4[index];
float final_result;
if (precision_level == 0) {
// Half precision computation
float rk_sum = to_half_precision(k1_val +
2.0 * to_half_precision(k2_val) +
2.0 * to_half_precision(k3_val) +
to_half_precision(k4_val));
final_result = to_half_precision(y_val + to_half_precision(h / 6.0) * rk_sum);
} else if (precision_level == 1) {
// Single precision computation
final_result = y_val + (h / 6.0) * (k1_val + 2.0 * k2_val + 2.0 * k3_val + k4_val);
} else {
// Double precision emulation
vec2 y_double = to_double_precision(y_val);
vec2 h_sixth = to_double_precision(h / 6.0);
vec2 k_sum = add_double(add_double(to_double_precision(k1_val),
mul_double(vec2(2.0, 0.0), to_double_precision(k2_val))),
add_double(mul_double(vec2(2.0, 0.0), to_double_precision(k3_val)),
to_double_precision(k4_val)));
vec2 increment = mul_double(h_sixth, k_sum);
vec2 result_double = add_double(y_double, increment);
final_result = result_double.x + result_double.y;
}
result[index] = final_result;
// Estimate local truncation error for precision adaptation
float error_est = abs(final_result - y_val) / max(abs(final_result), 1e-12);
error_estimate[index] = error_est;
// Update precision level for next iteration
int new_precision = update_precision_level(index, error_est, precision_level);
precision_mask[index] = new_precision;
}
}