Improve color_wheel so it blends more between colors

Fix: improved accuracy of generated rainbow colors for LEDs

src/leds.c

@@ -40,6 +40,7 @@
 #define W(c) ((uint8_t) ((c) >> 24))
 #define RGB(r, g, b) ((r) << 16 | (g) << 8 | (b))
 #define RGBW(r, g, b, w) ((w) << 24 | (r) << 16 | (g) << 8 | (b))
+#define TAU (2.0 * M_PI)
 
 static const uint32_t colors[] = {
     0x00000000,  // BLACK
@@ -99,18 +100,21 @@ static uint32_t color_blend(uint32_t color1, uint32_t color2, float blend) {
     return RGBW(r, g, b, w);
 }
 
-// Borrowed from WLED
+// This function is an approximation of `sqrtf(sinf(x) * 0.5 + 0.5)`
+inline static float taylor_series(float zero_to_tau) {
+    float x = (-0.5 * zero_to_tau) + 2.4;
+    float x3 = x * x * x;
+    float x5 = x3 * x * x;
+    float r = fabs(x - (x3 / 6.0) + (x5 / 120.0));
+    return fminf(r, 1.0f);
+}
+
 static uint32_t color_wheel(uint8_t pos) {
-    pos = 255 - pos;
-    if (pos < 85) {
-        return ((uint32_t) (255 - pos * 3) << 16) | ((uint32_t) 0 << 8) | (pos * 3);
-    } else if (pos < 170) {
-        pos -= 85;
-        return ((uint32_t) 0 << 16) | ((uint32_t) (pos * 3) << 8) | (255 - pos * 3);
-    } else {
-        pos -= 170;
-        return ((uint32_t) (pos * 3) << 16) | ((uint32_t) (255 - pos * 3) << 8) | 0;
-    }
+    float norm = (float) pos / 255.0;
+    float r = taylor_series((norm + 0.0 / 3.0) * TAU);
+    float b = taylor_series((norm + 1.0 / 3.0) * TAU);
+    float g = taylor_series((norm + 2.0 / 3.0) * TAU);
+    return ((uint8_t) (r * 255) << 16) | ((uint8_t) (g * 255) << 8) | (uint8_t) (b * 255);
 }
 
 static void sattolo_shuffle(uint32_t seed, uint8_t *array, uint8_t length) {