Improve MathUtils.asin(), acos() (#6332)

The average and largest error are significantly better for this version (2-3 orders of magnitude), and it can be slightly faster.

gdx/src/com/badlogic/gdx/math/MathUtils.java

@@ -108,16 +108,40 @@ static public float atan2 (float y, float x) {
 		return y < 0f ? atan - PI : atan;
 	}
 
-	/** Returns acos in radians, less accurate than Math.acos but may be faster. */
-	static public float acos (float a) {
-		return 1.5707963267948966f - (a * (1f + (a *= a) * (-0.141514171442891431f + a * -0.719110791477959357f)))
-			/ (1f + a * (-0.439110389941411144f + a * -0.471306172023844527f));
+	/** Returns acos in radians; less accurate than Math.acos but may be faster. Average error of 0.00002845 radians
+	 * (0.0016300649 degrees), largest error of 0.000067548 radians (0.0038702153 degrees). This implementation does not
+	 * return NaN if given an out-of-range input (Math.acos does return NaN), unless the input is NaN.
+	 * @param a acos is defined only when a is between -1f and 1f, inclusive
+	 * @return between {@code 0} and {@code PI} when a is in the defined range */
+	static public float acos(float a) {
+		float a2 = a * a;  // a squared
+		float a3 = a * a2; // a cubed
+		if (a >= 0f) {
+			return (float) Math.sqrt(1f - a) *
+					(1.5707288f - 0.2121144f * a + 0.0742610f * a2 - 0.0187293f * a3);
+		}
+		else {
+			return 3.14159265358979323846f - (float) Math.sqrt(1f + a) *
+					(1.5707288f + 0.2121144f * a + 0.0742610f * a2 + 0.0187293f * a3);
+		}
 	}
 
-	/** Returns asin in radians, less accurate than Math.asin but may be faster. */
-	static public float asin (float a) {
-		return (a * (1f + (a *= a) * (-0.141514171442891431f + a * -0.719110791477959357f)))
-			/ (1f + a * (-0.439110389941411144f + a * -0.471306172023844527f));
+	/** Returns asin in radians; less accurate than Math.asin but may be faster. Average error of 0.000028447 radians
+	 * (0.0016298931 degrees), largest error of 0.000067592 radians (0.0038727364 degrees). This implementation does not
+	 * return NaN if given an out-of-range input (Math.asin does return NaN), unless the input is NaN.
+	 * @param a asin is defined only when a is between -1f and 1f, inclusive
+	 * @return between {@code -HALF_PI} and {@code HALF_PI} when a is in the defined range */
+	static public float asin(float a) {
+		float a2 = a * a;  // a squared
+		float a3 = a * a2; // a cubed
+		if (a >= 0f) {
+			return 1.5707963267948966f - (float) Math.sqrt(1f - a) *
+					(1.5707288f - 0.2121144f * a + 0.0742610f * a2 - 0.0187293f * a3);
+		}
+		else {
+			return -1.5707963267948966f + (float) Math.sqrt(1f + a) *
+					(1.5707288f + 0.2121144f * a + 0.0742610f * a2 + 0.0187293f * a3);
+		}
 	}
 
 	// ---