feat(optimizer): improve repeat_p_error speed

concrete-optimizer/src/noise_estimator/p_error.rs

@@ -4,6 +4,45 @@ pub fn combine_errors(p_error1: f64, p_error2: f64) -> f64 {
 }
 
 pub fn repeat_p_error(p_error: f64, count: u64) -> f64 {
+    if p_error * count as f64 > 1. {
+        iterative_repeat_p_error(p_error, count)
+    } else {
+        binomial_decomposition_repeat_p_error(p_error, count)
+    }
+}
+
+// (1 - global_p_error) = (1 - p_error)^count
+// global_p_error = 1 - (1-p)^N = 1 - (1 - N p + N(N-1)/2 p^2 - N(N-1)(N-2)/(2*3) p^3...)
+// global_p_error = N p - N(N-1)/2 p^2 + N(N-1)(N-2)/(2*3) p^3...
+fn binomial_decomposition_repeat_p_error(p_error: f64, count: u64) -> f64 {
+    // This guarantees abs(factor) is decreasing
+    // Without that, factors grow and lose precision
+    assert!(p_error * (count as f64) <= 1.0);
+
+    let mut global_p_error = 0.0;
+
+    let mut factor = -1.0;
+
+    for i in 1..=count {
+        factor *= -p_error * (count - i + 1) as f64 / i as f64;
+
+        let new_global_p_error = global_p_error + factor;
+
+        #[allow(clippy::float_cmp)]
+        //if factor is too small to make a difference
+        if new_global_p_error == global_p_error {
+            // abs(factor) is decreasing and factor sign alternates
+            // so the remaining series is bounded (in absolute value) by abs(factor) which makes no difference
+            break;
+        }
+
+        global_p_error = new_global_p_error;
+    }
+
+    global_p_error
+}
+
+fn iterative_repeat_p_error(p_error: f64, count: u64) -> f64 {
     let mut global_p_error = 0.0;
 
     for _ in 0..count {
@@ -12,3 +51,29 @@ pub fn repeat_p_error(p_error: f64, count: u64) -> f64 {
 
     global_p_error
 }
+
+#[cfg(test)]
+mod tests {
+    use super::*;
+
+    #[allow(clippy::float_cmp)]
+    fn assert_eq_both_repeat_p_error(p_error: f64, count: u64) {
+        let iterative = iterative_repeat_p_error(p_error, count);
+        let binomial = binomial_decomposition_repeat_p_error(p_error, count);
+
+        assert!(((iterative - binomial) / (iterative + binomial)).abs() < 0.000_000_1);
+    }
+
+    #[test]
+    #[allow(clippy::float_cmp)]
+    fn test_repeat_p_error() {
+        assert_eq!(repeat_p_error(0.5, 1), 0.5);
+        assert_eq!(repeat_p_error(0.5, 2), 0.75);
+
+        assert_eq_both_repeat_p_error(0.00001, 10000);
+
+        assert_eq_both_repeat_p_error(0.001, 100);
+
+        assert_eq_both_repeat_p_error(0.000_000_000_01, 100);
+    }
+}