Enhance Precision of Fixed-Point Square Root Computations

The current square root calculation for fixed-point numbers cannot
 handle values less than 1 accurately. To improve precision,
replace the existing method with the digit-by-digit calculation
method. This approach, combined with offset manipulation, will
minimize precision loss during the square root calculation.

src/fixed.c

@@ -9,42 +9,71 @@
 #define uint32_lo(i) ((i) & 0xffff)
 #define uint32_hi(i) ((i) >> 16)
 #define uint32_carry16 ((1) << 16)
+/* Check interval
+ * For any variable interval checking:
+ *     if (x > minx - epsilon && x < minx + epsilon) ...
+ * it is faster to use bit operation:
+ *     if ((signed)((x - (minx - epsilon)) | ((minx + epsilon) - x)) > 0) ...
+ */
+#define CHECK_INTERVAL(x, minx, epsilon) \
+    ((int32_t) ((x - (minx - epsilon)) | (minx + epsilon - x)) > 0)
 
 twin_fixed_t twin_fixed_sqrt(twin_fixed_t a)
 {
-    twin_fixed_t max = a, min = 0;
-
-    while (max > min) {
-        twin_fixed_t mid = (max + min) >> 1;
-        if (mid >= 181 * TWIN_FIXED_ONE) {
-            max = mid - 1;
-            continue;
-        }
-        twin_fixed_t sqr = twin_fixed_mul(mid, mid);
-        if (sqr == a)
-            return mid;
-        if (sqr < a)
-            min = mid + 1;
-        else
-            max = mid - 1;
+    if (a <= 0)
+        return 0;
+
+    if (CHECK_INTERVAL(a, TWIN_FIXED_ONE, (1 << (8 - 1))))
+        return TWIN_FIXED_ONE;
+
+    /* count leading zero */
+    int offset = 0;
+    for (twin_fixed_t i = a; !(0x40000000 & i); i <<= 1) {
+        ++offset;
     }
-    return (max + min) >> 1;
+    /* shift left 'a' to expand more digit for sqrt precision */
+    offset &= ~1;
+    a <<= offset;
+    /* calculate the digits need to shift back */
+    offset >>= 1;
+    offset -= (16 >> 1);
+    /* Use digit-by-digit calculation to compute square root */
+    twin_fixed_t z = 0;
+    for (twin_fixed_t m = 1UL << ((31 - __builtin_clz(a)) & ~1UL); m; m >>= 2) {
+        int b = z + m;
+        z >>= 1;
+        if (a >= b)
+            a -= b, z += m;
+    }
+    /* shift back the expanded digits */
+    return (offset >= 0) ? z >> offset : z << (-offset);
 }
 
 twin_sfixed_t _twin_sfixed_sqrt(twin_sfixed_t as)
 {
-    twin_dfixed_t max = as, min = 0;
-    twin_dfixed_t a = twin_sfixed_to_dfixed(as);
-
-    while (max > min) {
-        twin_dfixed_t mid = (max + min) >> 1;
-        twin_dfixed_t sqr = mid * mid;
-        if (sqr == a)
-            return (twin_sfixed_t) mid;
-        if (sqr < a)
-            min = mid + 1;
-        else
-            max = mid - 1;
+    if (as <= 0)
+        return 0;
+    if (CHECK_INTERVAL(as, TWIN_SFIXED_ONE, (1 << (2 - 1))))
+        return TWIN_SFIXED_ONE;
+
+    int offset = 0;
+    for (twin_sfixed_t i = as; !(0x4000 & i); i <<= 1) {
+        ++offset;
     }
-    return (twin_sfixed_t) ((max + min) >> 1);
+    offset &= ~1;
+    as <<= offset;
+
+    offset >>= 1;
+    offset -= (4 >> 1);
+
+    twin_sfixed_t z = 0;
+    for (twin_sfixed_t m = 1UL << ((31 - __builtin_clz(as)) & ~1UL); m;
+         m >>= 2) {
+        int16_t b = z + m;
+        z >>= 1;
+        if (as >= b)
+            as -= b, z += m;
+    }
+
+    return (offset >= 0) ? z >> offset : z << (-offset);
 }