diff --git a/src/lib.rs b/src/lib.rs
index 934ea29..52cfe02 100644
--- a/src/lib.rs
+++ b/src/lib.rs
@@ -224,16 +224,32 @@ impl<T: Clone + Float> Complex<T> {
             if self.re.is_sign_positive() {
                 // simple positive real √r, and copy `im` for its sign
                 Self::new(self.re.sqrt(), self.im)
-            } else if self.im.is_sign_positive() {
-                // √(r e^(iπ)) = √r e^(iπ/2) = i√r
-                Self::new(T::zero(), self.re.abs().sqrt())
             } else {
+                // √(r e^(iπ)) = √r e^(iπ/2) = i√r
                 // √(r e^(-iπ)) = √r e^(-iπ/2) = -i√r
-                Self::new(T::zero(), -self.re.abs().sqrt())
+                let re = T::zero();
+                let im = (-self.re).sqrt();
+                if self.im.is_sign_positive() {
+                    Self::new(re, im)
+                } else {
+                    Self::new(re, -im)
+                }
+            }
+        } else if self.re.is_zero() {
+            // √(r e^(iπ/2)) = √r e^(iπ/4) = √(r/2) + i√(r/2)
+            // √(r e^(-iπ/2)) = √r e^(-iπ/4) = √(r/2) - i√(r/2)
+            let one = T::one();
+            let two = one + one;
+            let x = (self.im.abs() / two).sqrt();
+            if self.im.is_sign_positive() {
+                Self::new(x, x)
+            } else {
+                Self::new(x, -x)
             }
         } else {
             // formula: sqrt(r e^(it)) = sqrt(r) e^(it/2)
-            let two = T::one() + T::one();
+            let one = T::one();
+            let two = one + one;
             let (r, theta) = self.to_polar();
             Self::from_polar(&(r.sqrt()), &(theta / two))
         }
@@ -1748,8 +1764,8 @@ mod test {
                 assert!(close(c.conj().sqrt(), c.sqrt().conj()));
                 // for this branch, -pi/2 <= arg(sqrt(z)) <= pi/2
                 assert!(
-                    -f64::consts::PI / 2.0 <= c.sqrt().arg()
-                        && c.sqrt().arg() <= f64::consts::PI / 2.0
+                    -f64::consts::FRAC_PI_2 <= c.sqrt().arg()
+                        && c.sqrt().arg() <= f64::consts::FRAC_PI_2
                 );
                 // sqrt(z) * sqrt(z) = z
                 assert!(close(c.sqrt() * c.sqrt(), c));
@@ -1760,11 +1776,29 @@ mod test {
         fn test_sqrt_real() {
             for n in (0..100).map(f64::from) {
                 // √(n² + 0i) = n + 0i
-                assert_eq!(Complex64::new(n * n, 0.0).sqrt(), Complex64::new(n, 0.0));
+                let n2 = n * n;
+                assert_eq!(Complex64::new(n2, 0.0).sqrt(), Complex64::new(n, 0.0));
                 // √(-n² + 0i) = 0 + ni
-                assert_eq!(Complex64::new(-n * n, 0.0).sqrt(), Complex64::new(0.0, n));
+                assert_eq!(Complex64::new(-n2, 0.0).sqrt(), Complex64::new(0.0, n));
                 // √(-n² - 0i) = 0 - ni
-                assert_eq!(Complex64::new(-n * n, -0.0).sqrt(), Complex64::new(0.0, -n));
+                assert_eq!(Complex64::new(-n2, -0.0).sqrt(), Complex64::new(0.0, -n));
+            }
+        }
+
+        #[test]
+        fn test_sqrt_imag() {
+            for n in (0..100).map(f64::from) {
+                // √(0 + n²i) = n e^(iπ/4)
+                let n2 = n * n;
+                assert!(close(
+                    Complex64::new(0.0, n2).sqrt(),
+                    Complex64::from_polar(&n, &(f64::consts::FRAC_PI_4))
+                ));
+                // √(0 - n²i) = n e^(-iπ/4)
+                assert!(close(
+                    Complex64::new(0.0, -n2).sqrt(),
+                    Complex64::from_polar(&n, &(-f64::consts::FRAC_PI_4))
+                ));
             }
         }