Improved log1p implementation for complex inputs (#107100)

This PR improves the implementation of `torch.log1p` for complex inputs as mentioned in issue #107022. The new implementation is based on the insights provided in numpy/numpy#22611 (comment).

Pull Request resolved: #107100
Approved by: https://github.com/lezcano

c10/util/complex_math.h

@@ -291,6 +291,11 @@ C10_HOST_DEVICE inline c10::complex<T> atanh(const c10::complex<T>& x) {
 
 template <typename T>
 C10_HOST_DEVICE inline c10::complex<T> log1p(const c10::complex<T>& z) {
+#if defined(__APPLE__) || defined(__MACOSX)
+  // For Mac, the new implementation yielded a high relative error. Falling back
+  // to the old version for now.
+  // See https://github.com/numpy/numpy/pull/22611#issuecomment-1667945354
+
   // log1p(z) = log(1 + z)
   // Let's define 1 + z = r * e ^ (i * a), then we have
   // log(r * e ^ (i * a)) = log(r) + i * a
@@ -316,6 +321,19 @@ C10_HOST_DEVICE inline c10::complex<T> log1p(const c10::complex<T>& z) {
     T z0 = std::hypot(x + 1, y);
     return {std::log(z0), theta};
   }
+#else
+  // Based on https://github.com/numpy/numpy/pull/22611#issuecomment-1667945354
+  c10::complex<T> u = z + T(1);
+  if (u == T(1)) {
+    return z;
+  } else {
+    auto log_u = log(u);
+    if (u - T(1) == z) {
+      return log_u;
+    }
+    return log_u * (z / (u - T(1)));
+  }
+#endif
 }
 
 template <typename T>