feat(internal/librarian): removes library.output when tidy if can be derived (#3218)

When running `tidy`, remove `output` from library if can be derived from
default. At generation, output is filled with default if empty
[here](https://github.com/googleapis/librarian/blob/e36a73fdd000edab456b5ed9cbc70623b80596d7/internal/librarian/generate.go#L212).

For #3139

Author: zhumin8

internal/librarian/tidy.go

@@ -51,9 +51,19 @@ func RunTidy() error {
 	if err := validateLibraries(cfg); err != nil {
 		return err
 	}
+	for _, lib := range cfg.Libraries {
+		if lib.Output != "" && len(lib.Channels) == 1 && isDerivableOutput(cfg, lib) {
+			lib.Output = ""
+		}
+	}
 	return yaml.Write(librarianConfigPath, formatConfig(cfg))
 }
 
+func isDerivableOutput(cfg *config.Config, lib *config.Library) bool {
+	derivedOutput := defaultOutput(cfg.Language, lib.Channels[0].Path, cfg.Default.Output)
+	return lib.Output == derivedOutput
+}
+
 func validateLibraries(cfg *config.Config) error {
 	var (
 		errs      []error

internal/librarian/tidy_test.go

@@ -150,3 +150,35 @@ libraries:
 		t.Errorf("expected %v, got %v", errDuplicateLibraryName, err)
 	}
 }
+
+func TestTidy_DerivableOutput(t *testing.T) {
+	tempDir := t.TempDir()
+	t.Chdir(tempDir)
+	configPath := filepath.Join(tempDir, librarianConfigPath)
+	configContent := `
+language: rust
+default:
+  output: generated/
+libraries:
+  - name: google-cloud-secretmanager-v1
+    output: generated/cloud/secretmanager/v1
+    channels:
+      - path: google/cloud/secretmanager/v1
+`
+	if err := os.WriteFile(configPath, []byte(configContent), 0644); err != nil {
+		t.Fatal(err)
+	}
+	if err := RunTidy(); err != nil {
+		t.Fatal(err)
+	}
+	cfg, err := yaml.Read[config.Config](configPath)
+	if err != nil {
+		t.Fatal(err)
+	}
+	if len(cfg.Libraries) != 1 {
+		t.Fatalf("expected 1 library, got %d", len(cfg.Libraries))
+	}
+	if cfg.Libraries[0].Output != "" {
+		t.Errorf("expected output to be empty, got %q", cfg.Libraries[0].Output)
+	}
+}