Where does this error come from: 45541212*10^100 = 455412120000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000 but 45541212e+100 = 455412119999999983555897104077443290740751549803401114245293369909540524601237768792085383454306712577638400 Shouldent they be the same? Why has the e notation severe rounding errors that the other has not? Python 3.12 code: import math
import re
from typing import Any, Union, Callable

from PySide6.QtWidgets import (
    QApplication, QMainWindow, QWidget, QVBoxLayout, QHBoxLayout,
    QLineEdit, QTextEdit, QScrollBar, QPushButton, QLabel, QFrame,
    QGridLayout
)
from PySide6.QtGui import QFont
from PySide6.QtCore import Qt, QSize

def convert_angle(func: Callable[[float], float], x: float, use_degrees: bool) -> float:
    """Apply trig function with optional degree conversion."""
    return func(math.radians(x) if use_degrees else x)

def kilograms_to_pounds(kg: float) -> float:
    """Convert kilograms to pounds."""
    return kg / 0.45359237

def pounds_to_kilograms(lb: float) -> float:
    """Convert pounds to kilograms."""
    return lb * 0.45359237

def celsius_to_kelvin(c: float) -> float:
    """Convert Celsius to Kelvin."""
    return c + 273.15

def kelvin_to_celsius(k: float) -> float:
    """Convert Kelvin to Celsius."""
    return k - 273.15

def celsius_to_fahrenheit(c: float) -> float:
    """Convert Celsius to Fahrenheit."""
    return c * 9 / 5 + 32

def fahrenheit_to_celsius(f: float) -> float:
    """Convert Fahrenheit to Celsius."""
    return (f - 32) * 5 / 9

def capitalize_sentence(sentence: str) -> str:
    """Capitalize first letter of each word."""
    return ' '.join(word.capitalize() for word in sentence.split())

def remove_duplicates(text: str) -> str:
    """Remove duplicate words and empty lines."""
    words = ' '.join(line.strip() for line in text.splitlines() if line.strip()).split()
    return ' '.join(dict.fromkeys(words))

def get_safe_globals(use_degrees: bool, variables: dict) -> dict:
    """Provide safe globals for expression evaluation."""
    trig_funcs = {'sin', 'cos', 'tan', 'asin', 'acos', 'atan'}
    safe_globals = {
        "sqrt": math.sqrt,
        "log": math.log10,
        "ln": math.log,
        "pi": math.pi,
        "e": math.e,
        "lb": kilograms_to_pounds,
        "kg": pounds_to_kilograms,
        "C": kelvin_to_celsius,
        "K": celsius_to_kelvin,
        "F": celsius_to_fahrenheit,
        "CF": fahrenheit_to_celsius,
        "capitalize": capitalize_sentence,
        "duplicates": remove_duplicates,
    }
    for func in trig_funcs:
        safe_globals[func] = lambda x, f=func: convert_angle(getattr(math, f), x, use_degrees)
    safe_globals.update(variables)
    return safe_globals

def preprocess_text_functions(expression: str) -> str:
    """Preprocess text functions with argument quotes."""
    def add_quotes(match):
        func, args = match.groups()
        return f'{func}("{args}")' if func in ['capitalize', 'duplicates'] else match.group(0)
    return re.sub(r'\b(capitalize|duplicates)\s*\(\s*([^)]+)\s*\)', add_quotes, expression)

def validate_inverse_trig(expression: str, safe_globals: dict) -> Union[str, None]:
    """Check inverse trig function domain validity."""
    for func, arg in re.findall(r"(\b\w+\b)\s*\(\s*([^)]+)\s*\)", expression):
        if func in ["asin", "acos"]:
            try:
                value = eval(arg, {"__builtins__": None}, safe_globals)
                if abs(value) > 1:
                    return f"Error: {func}(x) is undefined for |x| > 1"
            except Exception as e:
                return f"Error: {e}"
    return None

def safe_evaluate(expression: str, safe_globals: dict, use_e_notation: bool) -> Union[str, Any]:
    """Safely evaluate expression and format result."""
    try:
        result = eval(expression, {"builtins": None}, safe_globals)
        if isinstance(result, (int, float)):
            if use_e_notation:
                result = f"{result:.6e}"
            else:
                if isinstance(result, float):
                    result = f"{result:.15f}".rstrip('0').rstrip('.')
                else:
                    result = str(result)
        return result
    except Exception as e:
        return f"Error: {e}"

def evaluate_expression(expression: str, variables: dict, use_degrees: bool = False, use_e_notation: bool = False) -> Union[float, str]:
    """Evaluate math expression considering modes."""
    assignment_match = re.match(r'^\s*([a-zA-Z_]\w*)\s*=\s*(.+)$', expression)
    if assignment_match:
        var_name, var_expr = assignment_match.groups()
        safe_globals = get_safe_globals(use_degrees, variables)
        var_value = safe_evaluate(var_expr, safe_globals, use_e_notation)
        if isinstance(var_value, (int, float)):
            variables[var_name] = var_value
            return f"{var_name} = {var_value}"
        return var_value
    else:
        safe_globals = get_safe_globals(use_degrees, variables)
        expression = expression.replace("^", "**")
        expression = preprocess_text_functions(expression)
        error = validate_inverse_trig(expression, safe_globals)
        if error:
            return error
        return safe_evaluate(expression, safe_globals, use_e_notation)

class CalculatorGUI(QMainWindow):
    """PySide6-based calculator application."""

    def __init__(self):
        super().__init__()
        self.use_degrees = False
        self.use_e_notation = False
        self.variables = {}
        self.history_entries = []
        self.setup_window()
        self.create_widgets()
        self.layout_widgets()
        self.connect_signals()

    def setup_window(self) -> None:
        """Set main window properties."""
        self.setWindowTitle("Calculator")
        self.setGeometry(100, 100, 1280, 720)

    def create_widgets(self) -> None:
        """Initialize GUI components."""
        font = QFont()
        font.setPointSize(12)
        font.setFamily('Literata')

        self.entry = QLineEdit()
        self.entry.setFont(font)
        self.entry.setPlaceholderText('Enter expression')
        self.entry.setClearButtonEnabled(True)
        self.entry.returnPressed.connect(self.calculate)

        self.history_text = QTextEdit()
        self.history_text.setFont(font)
        self.history_text.setReadOnly(True)

        self.scrollbar = QScrollBar(Qt.Vertical)
        self.history_text.setVerticalScrollBar(self.scrollbar)

        self.search_button = QPushButton("🔍", self.history_text)
        self.search_button.setFixedSize(24, 24)
        self.search_button.setToolTip("Search History")
        self.search_button.setStyleSheet("""
            QPushButton {
                border: none;
                background-color: rgba(255, 255, 255, 200);
                font-size: 14px;
                border-radius: 12px;
            }
            QPushButton:hover {
                background-color: rgba(0, 120, 215, 200);
                color: white;
            }
        """)
        self.search_button.clicked.connect(self.toggle_search)

        self.search_bar = QLineEdit(self.history_text)
        self.search_bar.setPlaceholderText("Search...")
        self.search_bar.setFixedHeight(24)
        self.search_bar.setVisible(False)
        self.search_bar.setStyleSheet("""
            QLineEdit {
                border: 1px solid gray;
                border-radius: 4px;
                padding: 2px 4px;
                background-color: white;
            }
        """)
        self.search_bar.textChanged.connect(self.filter_history)
        self.search_bar.setClearButtonEnabled(True)

        self.function_buttons = {
            "Functions": ["sqrt", "sin", "cos", "tan", "asin", "acos", "atan", "log", "ln"],
            "Conversions": ["lb", "kg", "K", "C", "F", "CF"],
            "Text": ["capitalize", "duplicates"],
            "Constants": ["pi", "e"],
            "Settings": []
        }
        self.function_frame = QFrame()
        self.create_function_buttons(font)

    def create_function_buttons(self, font: QFont) -> None:
        """Create buttons for calculator functions."""
        layout = QGridLayout()
        categories = [("Functions", 3), ("Conversions", 2), ("Text", 2), ("Constants", 2), ("Settings", 1)]
        starting_col = 0

        for category, col_span in categories:
            label = QLabel(category)
            label.setFont(font)
            layout.addWidget(label, 0, starting_col, 1, col_span, Qt.AlignCenter)

            buttons = self.function_buttons.get(category, [])
            if category == "Functions":
                for i, func in enumerate(buttons):
                    button = QPushButton(func)
                    button.setFont(font)
                    button.clicked.connect(lambda _, f=func: self.insert_function(f, False))
                    layout.addWidget(button, (i // 3) + 1, starting_col + (i % 3))

            elif category == "Settings":
                self.mode_button = QPushButton("Radians")
                self.mode_button.setFont(font)
                self.mode_button.clicked.connect(self.toggle_mode)
                layout.addWidget(self.mode_button, 1, starting_col, 1, 1)

                self.e_notation_button = QPushButton("E Notation: Off")
                self.e_notation_button.setFont(font)
                self.e_notation_button.clicked.connect(self.toggle_e_notation)
                layout.addWidget(self.e_notation_button, 2, starting_col, 1, 1)

            elif category == "Conversions":
                for idx, func in enumerate(buttons):
                    display_text = {
                        "lb": "lb(kg)", "kg": "kg(lb)", "K": "K(°C)",
                        "C": "°C(K)", "F": "F(°C)", "CF": "°C(F)"
                    }.get(func, func)
                    button = QPushButton(display_text)
                    button.setFont(font)
                    button.clicked.connect(lambda _, f=func: self.insert_function(f, False))
                    row, col = 1 + idx // 2, starting_col + idx % 2
                    layout.addWidget(button, row, col)

            else:
                num_buttons = len(buttons)
                num_rows = math.ceil(num_buttons / col_span)

                for idx, func in enumerate(buttons):
                    button = QPushButton(func)
                    button.setFont(font)
                    button.clicked.connect(lambda _, f=func, c=(category == "Constants"): self.insert_function(f, c))
                    row, col = 1 + idx // col_span, starting_col + idx % col_span
                    layout.addWidget(button, row, col)

                for extra in range(col_span * num_rows - num_buttons):
                    layout.addWidget(QLabel(""), 1 + extra // col_span, starting_col + extra % col_span)

            starting_col += col_span

        self.function_frame.setLayout(layout)

    def layout_widgets(self) -> None:
        """Arrange widgets in the main window."""
        central_widget = QWidget()
        main_layout = QVBoxLayout()

        main_layout.addWidget(self.entry)

        history_layout = QHBoxLayout()
        history_layout.addWidget(self.history_text)
        history_layout.addWidget(self.scrollbar)
        main_layout.addLayout(history_layout)

        main_layout.addWidget(self.function_frame)

        central_widget.setLayout(main_layout)
        self.setCentralWidget(central_widget)

    def connect_signals(self) -> None:
        """Connect widget signals."""
        QApplication.instance().focusChanged.connect(self.on_focus_changed)

    def toggle_mode(self) -> None:
        """Toggle between degrees and radians."""
        self.use_degrees = not self.use_degrees
        self.mode_button.setText("Degrees" if self.use_degrees else "Radians")

    def toggle_e_notation(self) -> None:
        """Toggle scientific notation for results."""
        self.use_e_notation = not self.use_e_notation
        self.e_notation_button.setText(f"E Notation: {'On' if self.use_e_notation else 'Off'}")

    def insert_function(self, function: str, is_constant: bool = False) -> None:
        """Insert function or constant in entry."""
        current_text = self.entry.text()
        cursor_position = self.entry.cursorPosition()
        if is_constant:
            new_text = f"{current_text[:cursor_position]}{function}"
            new_cursor_position = cursor_position + len(function)
        else:
            new_text = f"{current_text[:cursor_position]}{function}() {current_text[cursor_position:]}"
            new_cursor_position = cursor_position + len(function) + 1
        self.entry.setText(new_text)
        self.entry.setCursorPosition(new_cursor_position)
        self.entry.setFocus()

    def calculate(self) -> None:
        """Evaluate entered expression."""
        expression = self.entry.text().strip()
        if expression:
            result = evaluate_expression(
                expression, self.variables, use_degrees=self.use_degrees, use_e_notation=self.use_e_notation
            )
            self.display_result(expression, result)
            self.entry.clear()

    def display_result(self, expression: str, result: Union[float, str]) -> None:
        """Show result in history."""
        mode = "D" if self.use_degrees else "R"
        if re.match(r'^\s*[a-zA-Z_]\w*\s*=\s*.+$', expression):
            history_entry = f"[{mode}] {result}"
        else:
            history_entry = f"[{mode}] {expression} = {result}"
        self.history_entries.append(history_entry)
        self.refresh_history()
        self.history_text.verticalScrollBar().setValue(
            self.history_text.verticalScrollBar().maximum()
        )

    def toggle_search(self) -> None:
        """Toggle search bar visibility."""
        is_visible = self.search_bar.isVisible()
        self.search_bar.setVisible(not is_visible)
        if not is_visible:
            self.search_bar.setFocus()
            self.search_bar.setText("")
        else:
            self.search_bar.clear()
        self.update_search_bar_position()
        self.refresh_history()

    def filter_history(self, text: str) -> None:
        """Filter history based on search input."""
        self.refresh_history(text)

    def refresh_history(self, filter_text: str = "") -> None:
        """Refresh history view with optional filtering."""
        self.history_text.clear()
        for entry in self.history_entries:
            if filter_text.lower() in entry.lower():
                self.history_text.append(entry)

    def resizeEvent(self, event) -> None:
        """Handle window resize events."""
        super().resizeEvent(event)
        self.update_search_bar_position()

    def update_search_bar_position(self) -> None:
        """Reposition search button and bar, moving search button left when search bar is open."""
        margin_right = 5
        margin_top = 5
        spacing = 5
    
        history_width = self.history_text.viewport().width()
    
        search_bar_width = int(history_width * 0.25)
        total_width = self.search_button.width() + spacing + search_bar_width
    
        if total_width > history_width:
            search_bar_width = history_width - self.search_button.width() - spacing - margin_right
    
        if self.search_bar.isVisible():
            self.search_button.move(
                history_width - total_width - margin_right,
                margin_top
            )
            self.search_bar.setFixedWidth(search_bar_width)
            self.search_bar.move(
                self.search_button.x() + self.search_button.width() + spacing,
                self.search_button.y()
            )
        else:
            self.search_button.move(
                history_width - self.search_button.width() - margin_right,
                margin_top
            )

    def on_focus_changed(self, old: QWidget, new: QWidget) -> None:
        """Auto-hide search if focus is lost."""
        if self.search_bar.isVisible() and not self.search_bar.text().strip() and new != self.search_bar:
            self.toggle_search()

    def showEvent(self, event) -> None:
        """Position search button on showing the window."""
        super().showEvent(event)
        self.update_search_bar_position()

def main() -> None:
    """Start the calculator application."""
    app = QApplication([])
    calculator = CalculatorGUI()
    calculator.show()
    app.exec()

if __name__ == "__main__":
    main()