added linear regression square root approximation

numerical_methods/linear_regression_sqrt.c

@@ -0,0 +1,104 @@
+#include <math.h>
+#include <stdint.h>
+#include <stdio.h>
+#include <stdlib.h>
+#include <string.h>
+
+/**
+ * Calculates the best-fit line (y = alpha*x + beta) for the function sqrt(x)
+ * over the interval [1.0, 2.0] using Least Squares Linear Regression.
+ */
+int compute_alpha_beta()
+{
+    int N = 1000;  // Number of sample points for the regression
+    float x_start = 1.0;
+    float x_end = 2.0;  // The approximation is optimized for this range
+    float x, y;
+
+    // Summation variables for the Least Squares formula
+    float sx = 0.0;   // Sum of x
+    float sy = 0.0;   // Sum of y (sqrt(x))
+    float sxx = 0.0;  // Sum of x squared
+    float sxy = 0.0;  // Sum of x * y
+
+    for (int i = 0; i < N; i++)
+    {
+        // Step through the range [1, 2]
+        x = x_start + (x_end - x_start) * ((float)i / N);
+        y = sqrt(x);  // Get the actual target value
+
+        sx += x;
+        sy += y;
+        sxx += x * x;
+        sxy += y * x;
+    }
+
+    // Standard Linear Regression formulas to find Slope (alpha) and Intercept
+    // (beta)
+    float alpha = ((N * sxy - sx * sy) / (N * sxx - sx * sx));
+    float beta = ((sy - alpha * sx) / N);
+
+    printf("--- Float Results ---\n");
+    printf("Alpha normal = %f\n", alpha);
+    printf("Beta normal = %f\n", beta);
+
+    // Type-punning: Printing the raw IEEE-754 hex bits of the float values.
+    // This allows us to see how these numbers are stored in memory/registers.
+    printf("Alpha hex normal = 0x%X\n", *((int*)(&alpha)));
+    printf("Beta hex normal  = 0x%X\n\n", *((int*)(&beta)));
+
+    // --- Fixed Point Conversion (16.16 Format) ---
+    // We scale the values by 2^16 (65536) to perform "decimal math" using
+    // integers. This is vital for systems without a Floating Point Unit (FPU).
+    int alpha_fixed = (int)(alpha * 65536);
+    int beta_fixed = (int)(beta * 65536);
+
+    printf("--- 16.16 Fixed Point Results ---\n");
+    printf("Alpha fixed point = %d\n", alpha_fixed);
+    printf("Beta fixed point  = %d\n", beta_fixed);
+    printf("Alpha hex fixed   = 0x%X\n", alpha_fixed);
+    printf("Beta hex fixed    = 0x%X\n", beta_fixed);
+
+    return 0;
+}
+
+/**
+ * Fast Square Root Approximation using the pre-computed alpha/beta
+ * coefficients. This replaces a heavy sqrt() call with 1 multiply and 1
+ * addition. Good for graphics, shaders, and real-time simulations.
+ */
+int alpha_beta_approximations(float x)
+{
+    // These constants are derived from the [1, 2] range regression above.
+    float alpha = 0.411859f;
+    float beta = 0.601160f;
+
+    // Linear Approximation: y = mx + b
+    // Note: This is highly accurate between 1.0 and 2.0, but error increases
+    // outside that range.
+    float result = (alpha * x) + beta;
+
+    printf("Approximated result = %4f\n", result);
+    return 0;
+}
+
+int main()
+{
+    float x;
+
+    // 1. Show the math behind the coefficients
+    compute_alpha_beta();
+
+    // 2. Test the approximation
+    printf(
+        "\nNote: Approximation is optimized for inputs between 1.0 and 2.0\n");
+    printf("Enter number to approximate sqrt: ");
+
+    if (scanf("%f", &x) == 1)
+    {
+        alpha_beta_approximations(x);
+        printf("Actual sqrt value   = %4f\n", sqrt(x));
+    }
+
+    return 0;
+}