import java.io.*; import java.util.*; import java.util.function.IntUnaryOperator; import java.util.function.LongUnaryOperator; import java.util.stream.Collectors; public class Main { static In in = new FastIn(); static Out out = new Out(false); static final long inf = 0x1fffffffffffffffL; static final int iinf = 0x3fffffff; static final double eps = 1e-9; static long mod = 998244353; void solve() { class Sec { final int l, r; final double y; final double as; final double bs; public Sec(int l, int r, double y, double as, double bs) { this.l = l; this.r = r; this.y = y; this.as = as; this.bs = bs; } Sec merge(Sec o) { double y = Math.max(1, Math.sqrt((as + o.as) / (bs + o.bs))); return new Sec(l, o.r, y, as + o.as, bs + o.bs); } } int n = in.nextInt(); long[] a = in.nextLongArray(n); long[] b = in.nextLongArray(n); List ys = new ArrayList<>(); for (int i = n - 1; i >= 0; i--) { double y = Math.max(1, Math.sqrt((double)a[i] / b[i])); Sec sec = new Sec(i, i + 1, y, a[i], b[i]); while (!ys.isEmpty()) { Sec s = ys.get(ys.size() - 1); if (sec.y < s.y) { break; } sec = sec.merge(s); ys.remove(ys.size() - 1); } ys.add(sec); } double ans = 0; for (Sec sec : ys) { ans += sec.as / sec.y + sec.bs * sec.y; } out.println(ans); } public static void main(String... args) { new Main().solve(); out.flush(); } } class FastIn extends In { private final BufferedInputStream reader = new BufferedInputStream(System.in); private final byte[] buffer = new byte[0x10000]; private int i = 0; private int length = 0; public int read() { if (i == length) { i = 0; try { length = reader.read(buffer); } catch (IOException ignored) { } if (length == -1) { return 0; } } if (length <= i) { throw new RuntimeException(); } return buffer[i++]; } String next() { StringBuilder builder = new StringBuilder(); int b = read(); while (b < '!' || '~' < b) { b = read(); } while ('!' <= b && b <= '~') { builder.appendCodePoint(b); b = read(); } return builder.toString(); } String nextLine() { StringBuilder builder = new StringBuilder(); int b = read(); while (b != 0 && b != '\r' && b != '\n') { builder.appendCodePoint(b); b = read(); } if (b == '\r') { read(); } return builder.toString(); } int nextInt() { long val = nextLong(); if ((int)val != val) { throw new NumberFormatException(); } return (int)val; } long nextLong() { int b = read(); while (b < '!' || '~' < b) { b = read(); } boolean neg = false; if (b == '-') { neg = true; b = read(); } long n = 0; int c = 0; while ('0' <= b && b <= '9') { n = n * 10 + b - '0'; b = read(); c++; } if (c == 0 || c >= 2 && n == 0) { throw new NumberFormatException(); } return neg ? -n : n; } } class In { private final BufferedReader reader = new BufferedReader(new InputStreamReader(System.in), 0x10000); private StringTokenizer tokenizer; String next() { try { while (tokenizer == null || !tokenizer.hasMoreTokens()) { tokenizer = new StringTokenizer(reader.readLine()); } } catch (IOException ignored) { } return tokenizer.nextToken(); } int nextInt() { return Integer.parseInt(next()); } long nextLong() { return Long.parseLong(next()); } double nextDouble() { return Double.parseDouble(next()); } char[] nextCharArray() { return next().toCharArray(); } String[] nextStringArray(int n) { String[] s = new String[n]; for (int i = 0; i < n; i++) { s[i] = next(); } return s; } char[][] nextCharGrid(int n, int m) { char[][] a = new char[n][m]; for (int i = 0; i < n; i++) { a[i] = next().toCharArray(); } return a; } int[] nextIntArray(int n) { int[] a = new int[n]; for (int i = 0; i < n; i++) { a[i] = nextInt(); } return a; } int[] nextIntArray(int n, IntUnaryOperator op) { int[] a = new int[n]; for (int i = 0; i < n; i++) { a[i] = op.applyAsInt(nextInt()); } return a; } int[][] nextIntMatrix(int h, int w) { int[][] a = new int[h][w]; for (int i = 0; i < h; i++) { a[i] = nextIntArray(w); } return a; } long[] nextLongArray(int n) { long[] a = new long[n]; for (int i = 0; i < n; i++) { a[i] = nextLong(); } return a; } long[] nextLongArray(int n, LongUnaryOperator op) { long[] a = new long[n]; for (int i = 0; i < n; i++) { a[i] = op.applyAsLong(nextLong()); } return a; } long[][] nextLongMatrix(int h, int w) { long[][] a = new long[h][w]; for (int i = 0; i < h; i++) { a[i] = nextLongArray(w); } return a; } List> nextEdges(int n, int m, boolean directed) { List> res = new ArrayList<>(); for (int i = 0; i < n; i++) { res.add(new ArrayList<>()); } for (int i = 0; i < m; i++) { int u = nextInt() - 1; int v = nextInt() - 1; res.get(u).add(v); if (!directed) { res.get(v).add(u); } } return res; } } class Out { private final PrintWriter out = new PrintWriter(System.out); private final PrintWriter err = new PrintWriter(System.err); boolean autoFlush = false; boolean enableDebug; Out(boolean enableDebug) { this.enableDebug = enableDebug; } void println(Object... args) { if (args == null || args.getClass() != Object[].class) { args = new Object[] {args}; } out.println(Arrays.stream(args).map(obj -> { Class clazz = obj == null ? null : obj.getClass(); return clazz == Double.class ? String.format("%.10f", obj) : clazz == byte[].class ? Arrays.toString((byte[])obj) : clazz == short[].class ? Arrays.toString((short[])obj) : clazz == int[].class ? Arrays.toString((int[])obj) : clazz == long[].class ? Arrays.toString((long[])obj) : clazz == char[].class ? Arrays.toString((char[])obj) : clazz == float[].class ? Arrays.toString((float[])obj) : clazz == double[].class ? Arrays.toString((double[])obj) : clazz == boolean[].class ? Arrays.toString((boolean[])obj) : obj instanceof Object[] ? Arrays.deepToString((Object[])obj) : String.valueOf(obj); }).collect(Collectors.joining(" "))); if (autoFlush) { out.flush(); } } void debug(Object... args) { if (!enableDebug) { return; } if (args == null || args.getClass() != Object[].class) { args = new Object[] {args}; } err.println(Arrays.stream(args).map(obj -> { Class clazz = obj == null ? null : obj.getClass(); return clazz == Double.class ? String.format("%.10f", obj) : clazz == byte[].class ? Arrays.toString((byte[])obj) : clazz == short[].class ? Arrays.toString((short[])obj) : clazz == int[].class ? Arrays.toString((int[])obj) : clazz == long[].class ? Arrays.toString((long[])obj) : clazz == char[].class ? Arrays.toString((char[])obj) : clazz == float[].class ? Arrays.toString((float[])obj) : clazz == double[].class ? Arrays.toString((double[])obj) : clazz == boolean[].class ? Arrays.toString((boolean[])obj) : obj instanceof Object[] ? Arrays.deepToString((Object[])obj) : String.valueOf(obj); }).collect(Collectors.joining(" "))); err.flush(); } void println(char[] s) { out.println(String.valueOf(s)); if (autoFlush) { out.flush(); } } void println(int[] a) { StringJoiner joiner = new StringJoiner(" "); for (int i : a) { joiner.add(Integer.toString(i)); } out.println(joiner); if (autoFlush) { out.flush(); } } void println(long[] a) { StringJoiner joiner = new StringJoiner(" "); for (long i : a) { joiner.add(Long.toString(i)); } out.println(joiner); if (autoFlush) { out.flush(); } } void flush() { err.flush(); out.flush(); } }