⚡️ Speed up function numerical_integration_rectangle by 56%

[codeflash-ai]

📄 56% (0.56x) speedup for numerical_integration_rectangle in src/numpy_pandas/numerical_methods.py

⏱️ Runtime : 1.53 milliseconds → 982 microseconds (best of 237 runs)

📝 Explanation and details

Here are the main bottlenecks per your profiling.

• result += f(x) is very slow: calling a Python function is expensive in tight loops
• x = a + i * h is next
• The Python loop itself (for i in range(n)) adds overhead

Optimization strategies:

• Use numpy for vectorization where possible, which can calculate all points in one go and sum them, greatly reducing Python-level looping/overhead. Numpy will only help if the f function can work on arrays, but we can provide a fast path and fall back otherwise.
• Otherwise, use functools.lru_cache to cache repeated calls if f is expensive and pure, but often this helps only for certain functions.
• a + i*h can be replaced with a precomputed numpy array of x values.

Here's a rewritten version that will be much faster for f functions that accept numpy arrays, but will still work with all standard Python callables (falling back to the original loop).

Explanation:

• If f is numpy-aware (works with numpy arrays), the whole integration will run in compiled code (very fast).
• Otherwise, it falls back to your original code path with a minor optimization—unpacking a + i * h is already cheap.
• No lost precision or semantic change.
• Works for all input functions and preserves all existing comments and function signatures.

Note:
If you are not allowed to use numpy (or want to further speed up for "slow" pure-Python functions), consider using Cython or numba. But for pure Python, numpy vectorization gives you the biggest, simplest speedup.

✅ Correctness verification report:

Test	Status
⚙️ Existing Unit Tests	🔘 None Found
🌀 Generated Regression Tests	✅ 41 Passed
⏪ Replay Tests	🔘 None Found
🔎 Concolic Coverage Tests	✅ 1 Passed
📊 Tests Coverage	100.0%

🌀 Generated Regression Tests and Runtime

import math  # for math functions in test cases
# function to test
from typing import Callable

# imports
import pytest  # used for our unit tests
from src.numpy_pandas.numerical_methods import numerical_integration_rectangle

# unit tests

# 1. Basic Test Cases

def test_constant_function():
    # Integrate f(x) = 5 from 0 to 10, exact result is 50
    codeflash_output = numerical_integration_rectangle(lambda x: 5, 0, 10, 100); result = codeflash_output # 12.3μs -> 12.2μs (1.03% faster)

def test_linear_function():
    # Integrate f(x) = x from 0 to 1, exact result is 0.5
    codeflash_output = numerical_integration_rectangle(lambda x: x, 0, 1, 100); result = codeflash_output # 10.4μs -> 8.54μs (21.9% faster)

def test_quadratic_function():
    # Integrate f(x) = x^2 from 0 to 1, exact result is 1/3
    codeflash_output = numerical_integration_rectangle(lambda x: x**2, 0, 1, 100); result = codeflash_output # 14.0μs -> 9.12μs (53.0% faster)

def test_integration_reversed_bounds():
    # Integrate f(x) = x from 1 to 0, should still be 0.5
    codeflash_output = numerical_integration_rectangle(lambda x: x, 1, 0, 100); result = codeflash_output # 10.1μs -> 8.42μs (19.8% faster)

def test_integration_with_negative_bounds():
    # Integrate f(x) = x from -1 to 1, exact result is 0
    codeflash_output = numerical_integration_rectangle(lambda x: x, -1, 1, 100); result = codeflash_output # 10.2μs -> 8.33μs (22.5% faster)

def test_integration_with_float_bounds():
    # Integrate f(x) = 2x from 0.5 to 2.5, exact result is [x^2] from 0.5 to 2.5 = 6.25 - 0.25 = 6.0
    codeflash_output = numerical_integration_rectangle(lambda x: 2 * x, 0.5, 2.5, 100); result = codeflash_output # 12.0μs -> 9.42μs (27.0% faster)

# 2. Edge Test Cases

def test_zero_width_interval():
    # Integrate any function over [a, a], should be 0
    codeflash_output = numerical_integration_rectangle(lambda x: x**2 + 5, 3, 3, 10); result = codeflash_output # 2.96μs -> 10.2μs (70.9% slower)

def test_single_rectangle():
    # Integrate f(x) = x from 0 to 1 with n=1, should use left endpoint: f(0)*1 = 0
    codeflash_output = numerical_integration_rectangle(lambda x: x, 0, 1, 1); result = codeflash_output # 1.00μs -> 8.21μs (87.8% slower)

def test_two_rectangles():
    # Integrate f(x) = x from 0 to 1 with n=2: h=0.5, f(0)=0, f(0.5)=0.5, so (0+0.5)*0.5=0.25
    codeflash_output = numerical_integration_rectangle(lambda x: x, 0, 1, 2); result = codeflash_output # 1.08μs -> 8.17μs (86.7% slower)



def test_function_with_discontinuity():
    # Integrate f(x) = 1 if x < 0.5 else 2 from 0 to 1, exact result is 0.5*1 + 0.5*2 = 1.5
    def f(x):
        return 1 if x < 0.5 else 2
    codeflash_output = numerical_integration_rectangle(f, 0, 1, 100); result = codeflash_output # 13.2μs -> 20.2μs (34.8% slower)

def test_function_with_singularity():
    # Integrate f(x) = 1/(x+1) from 0 to 1, exact result is ln(2) ~ 0.6931
    codeflash_output = numerical_integration_rectangle(lambda x: 1/(x+1), 0, 1, 100); result = codeflash_output # 15.8μs -> 11.1μs (42.3% faster)

def test_function_negative_values():
    # Integrate f(x) = -x from 0 to 1, exact result is -0.5
    codeflash_output = numerical_integration_rectangle(lambda x: -x, 0, 1, 100); result = codeflash_output # 10.5μs -> 9.38μs (12.4% faster)

def test_function_with_large_and_small_values():
    # Integrate f(x) = 1e6*x from 0 to 1, exact result is 5e5
    codeflash_output = numerical_integration_rectangle(lambda x: 1e6*x, 0, 1, 100); result = codeflash_output # 10.5μs -> 9.38μs (11.6% faster)

def test_function_with_non_integer_bounds():
    # Integrate f(x) = x from 1.5 to 3.5, exact result is (3.5^2 - 1.5^2)/2 = (12.25-2.25)/2 = 5.0
    codeflash_output = numerical_integration_rectangle(lambda x: x, 1.5, 3.5, 100); result = codeflash_output # 8.75μs -> 8.50μs (2.94% faster)

# 3. Large Scale Test Cases

def test_large_n_accuracy_linear():
    # Integrate f(x) = x from 0 to 1, n=1000, should be very accurate
    codeflash_output = numerical_integration_rectangle(lambda x: x, 0, 1, 1000); result = codeflash_output # 91.9μs -> 10.6μs (768% faster)

def test_large_n_accuracy_quadratic():
    # Integrate f(x) = x^2 from 0 to 1, n=1000, should be very accurate
    codeflash_output = numerical_integration_rectangle(lambda x: x**2, 0, 1, 1000); result = codeflash_output # 130μs -> 11.5μs (1031% faster)

def test_large_scale_performance():
    # Integrate f(x) = sin(x) from 0 to pi, exact result is 2
    codeflash_output = numerical_integration_rectangle(math.sin, 0, math.pi, 1000); result = codeflash_output # 91.3μs -> 98.5μs (7.28% slower)

def test_large_scale_nontrivial_function():
    # Integrate f(x) = x*sin(x) from 0 to pi, exact result is pi
    def f(x):
        return x * math.sin(x)
    codeflash_output = numerical_integration_rectangle(f, 0, math.pi, 1000); result = codeflash_output # 118μs -> 125μs (5.25% slower)

def test_large_scale_negative_bounds():
    # Integrate f(x) = x from -500 to 500, should be 0
    codeflash_output = numerical_integration_rectangle(lambda x: x, -500, 500, 1000); result = codeflash_output # 92.5μs -> 10.9μs (750% faster)

def test_large_scale_constant_function():
    # Integrate f(x) = 42 from 0 to 999, result should be 42*999
    codeflash_output = numerical_integration_rectangle(lambda x: 42, 0, 999, 999); result = codeflash_output # 115μs -> 14.2μs (713% faster)
# codeflash_output is used to check that the output of the original code is the same as that of the optimized code.

import math  # used for test functions like sin, cos, exp
# function to test
from typing import Callable

# imports
import pytest  # used for our unit tests
from src.numpy_pandas.numerical_methods import numerical_integration_rectangle

# unit tests

# --------------------------
# Basic Test Cases
# --------------------------

def test_constant_function():
    # Integral of f(x) = 5 over [0, 2] should be 10
    codeflash_output = numerical_integration_rectangle(lambda x: 5, 0, 2, 100); result = codeflash_output # 12.3μs -> 11.8μs (4.96% faster)

def test_linear_function():
    # Integral of f(x) = x over [0, 1] should be 0.5
    codeflash_output = numerical_integration_rectangle(lambda x: x, 0, 1, 100); result = codeflash_output # 12.0μs -> 8.62μs (38.6% faster)

def test_quadratic_function():
    # Integral of f(x) = x^2 over [0, 1] should be 1/3 ≈ 0.333...
    codeflash_output = numerical_integration_rectangle(lambda x: x**2, 0, 1, 100); result = codeflash_output # 16.0μs -> 9.12μs (75.3% faster)

def test_negative_bounds():
    # Integral of f(x) = x over [-1, 1] should be 0
    codeflash_output = numerical_integration_rectangle(lambda x: x, -1, 1, 100); result = codeflash_output # 10.7μs -> 8.33μs (28.5% faster)

def test_reverse_bounds():
    # Should handle a > b by swapping bounds
    # Integral of f(x) = x over [1, 0] should be -0.5
    codeflash_output = numerical_integration_rectangle(lambda x: x, 1, 0, 100); result = codeflash_output # 10.8μs -> 8.38μs (29.3% faster)

def test_function_with_math_exp():
    # Integral of f(x) = exp(x) over [0, 1] is e - 1
    codeflash_output = numerical_integration_rectangle(math.exp, 0, 1, 100); result = codeflash_output # 11.5μs -> 14.3μs (20.1% slower)
    expected = math.e - 1

def test_function_with_math_sin():
    # Integral of f(x) = sin(x) over [0, pi] is 2
    codeflash_output = numerical_integration_rectangle(math.sin, 0, math.pi, 100); result = codeflash_output # 11.1μs -> 14.2μs (21.9% slower)

# --------------------------
# Edge Test Cases
# --------------------------

def test_zero_width_interval():
    # Integral over zero-width interval should be zero
    codeflash_output = numerical_integration_rectangle(lambda x: x**2, 2, 2, 10); result = codeflash_output # 2.83μs -> 9.17μs (69.1% slower)

def test_one_rectangle():
    # With n=1, rectangle rule is just f(a)*(b-a)
    codeflash_output = numerical_integration_rectangle(lambda x: x, 0, 2, 1); result = codeflash_output # 1.00μs -> 8.25μs (87.9% slower)



def test_function_with_discontinuity():
    # Integrate sign function over [-1, 1]
    def sign(x):
        return 1 if x >= 0 else -1
    codeflash_output = numerical_integration_rectangle(sign, -1, 1, 100); result = codeflash_output # 14.5μs -> 21.8μs (33.5% slower)

def test_function_with_large_values():
    # Integrate f(x) = 1e6*x over [0, 1]
    codeflash_output = numerical_integration_rectangle(lambda x: 1e6 * x, 0, 1, 100); result = codeflash_output # 11.9μs -> 9.79μs (21.7% faster)

def test_function_with_small_values():
    # Integrate f(x) = 1e-6*x over [0, 1]
    codeflash_output = numerical_integration_rectangle(lambda x: 1e-6 * x, 0, 1, 100); result = codeflash_output # 11.1μs -> 9.38μs (18.2% faster)

def test_function_with_very_small_interval():
    # Integrate f(x) = x over [0, 1e-8]
    codeflash_output = numerical_integration_rectangle(lambda x: x, 0, 1e-8, 10); result = codeflash_output # 2.46μs -> 8.62μs (71.5% slower)

def test_function_with_infinite_result():
    # Integrate f(x) = 1/x over [1, 2]
    codeflash_output = numerical_integration_rectangle(lambda x: 1/x, 1, 2, 100); result = codeflash_output # 13.5μs -> 9.50μs (42.5% faster)

def test_function_with_non_numeric_return_raises():
    # If f returns non-numeric, should raise TypeError
    def bad_func(x):
        return "not a number"
    with pytest.raises(TypeError):
        numerical_integration_rectangle(bad_func, 0, 1, 10)

# --------------------------
# Large Scale Test Cases
# --------------------------

def test_large_n_linear_function():
    # Test with large n for f(x) = x over [0, 1]
    codeflash_output = numerical_integration_rectangle(lambda x: x, 0, 1, 1000); result = codeflash_output # 92.0μs -> 10.8μs (750% faster)

def test_large_n_quadratic_function():
    # Test with large n for f(x) = x^2 over [0, 1]
    codeflash_output = numerical_integration_rectangle(lambda x: x**2, 0, 1, 1000); result = codeflash_output # 130μs -> 11.7μs (1022% faster)

def test_large_interval():
    # Integrate f(x) = x over [0, 1000] with 1000 rectangles
    codeflash_output = numerical_integration_rectangle(lambda x: x, 0, 1000, 1000); result = codeflash_output # 92.5μs -> 10.5μs (777% faster)

def test_large_n_step_function():
    # Integrate a step function over [0, 1]
    def step(x):
        return 1 if x > 0.5 else 0
    codeflash_output = numerical_integration_rectangle(step, 0, 1, 1000); result = codeflash_output # 120μs -> 129μs (7.24% slower)

def test_large_n_sin_function():
    # Integrate sin(x) over [0, 2*pi] with large n
    codeflash_output = numerical_integration_rectangle(math.sin, 0, 2 * math.pi, 1000); result = codeflash_output # 93.6μs -> 122μs (23.3% slower)

def test_large_n_exp_function():
    # Integrate exp(x) over [0, 10] with large n
    codeflash_output = numerical_integration_rectangle(math.exp, 0, 10, 1000); result = codeflash_output # 87.8μs -> 94.2μs (6.77% slower)
    # Exact integral is exp(10) - 1
    expected = math.exp(10) - 1
# codeflash_output is used to check that the output of the original code is the same as that of the optimized code.

from src.numpy_pandas.numerical_methods import numerical_integration_rectangle

def test_numerical_integration_rectangle():
    numerical_integration_rectangle(lambda *a: 0.5, float('inf'), 0.0, 1)

To edit these changes git checkout codeflash/optimize-numerical_integration_rectangle-mc5fsdpy and push.