import java.util.*;
import java.util.zip.*;
import java.util.List;
import java.util.regex.*;
import java.util.concurrent.*;
import java.util.concurrent.atomic.*;
import java.util.concurrent.locks.*;
import java.util.function.*;
import javax.swing.*;
import javax.swing.event.*;
import javax.swing.text.*;
import javax.swing.table.*;
import java.io.*;
import java.net.*;
import java.lang.reflect.*;
import java.lang.ref.*;
import java.lang.management.*;
import java.security.*;
import java.security.spec.*;
import java.awt.*;
import java.awt.event.*;
import java.awt.image.*;
import java.awt.geom.*;
import javax.imageio.*;
import java.math.*;
import java.time.Duration;
// converts the index of a correct guess in a probabilistic
// list to a score between 0 and 1

// properties:
// -index 0 gets score 1
// -higher indices get lower scores
// -index -1 (not found) gets score 0
class main {

static double scoreForIndexInProbabilisticList(int index) {
  return index < 0 ? 0 : distanceToProbability(index);
}
// properties:
// -yields a proper probability (0 < p <= 1)
// -distance 0 has probability 1
// -function is monotonically decreasing
//    (greater distance = lower probability)
// -negative distances are interpreted as positive
static double distanceToProbability(double distance) {
  return genericDistanceToProbability(distance);
}



static double distanceToProbability(double a, double b) {
  return genericDistanceToProbability(a, b);
}




// properties:
// -yields a proper probability (0 < p <= 1)
// -distance 0 has probability 1
// -function is monotonically decreasing
//    (greater distance = lower probability)
// -negative distances are interpreted as positive
static double genericDistanceToProbability(double distance) {
  return 1/(1+abs(distance));
}

static double genericDistanceToProbability(double a, double b) {
  return genericDistanceToProbability(a-b);
}




static float abs(float f) { return Math.abs(f); }
static int abs(int i) { return Math.abs(i); }
static double abs(double d) { return Math.abs(d); }





}