⚡️ Speed up function bisection_method by 25%

[codeflash-ai]

📄 25% (0.25x) speedup for bisection_method in src/numpy_pandas/np_opts.py

⏱️ Runtime : 142 microseconds → 114 microseconds (best of 779 runs)

📝 Explanation and details

The optimized code achieves a 24% speedup by caching function evaluations to eliminate redundant computations. Here are the key optimizations:

1. Pre-compute and cache endpoint evaluations:

• Original: Computes f(a) and f(b) every time they're needed (in validation and loop comparisons)
• Optimized: Computes fa = f(a) and fb = f(b) once at the start and maintains these cached values

2. Maintain cached values through iterations:

• Original: Always calls f(a) in the comparison if f(a) * fc < 0:
• Optimized: Uses the cached fa value and updates it when a changes: a, fa = c, fc

Performance Analysis from Line Profiler:
The most significant improvement is in the comparison line (if f(a) * fc < 0:):

• Original: 878 hits, 429,000ns total (488.6ns per hit) - 26.8% of total time
• Optimized: 878 hits, 149,000ns total (169.7ns per hit) - 11.3% of total time
• 65% reduction in time for this critical operation

Why This Works:
In bisection method, the interval endpoints a and b change infrequently relative to how often their function values are accessed. By caching f(a) and f(b), the algorithm avoids redundant function evaluations. Each iteration only requires one new function evaluation f(c) instead of potentially re-evaluating f(a).

Test Case Performance:
The optimization is particularly effective for:

• Complex functions requiring more computation time (quadratic, cubic functions show 30-45% speedup)
• High precision cases with many iterations (small epsilon values show 17-42% speedup)
• Functions with expensive evaluation where caching provides maximum benefit

The optimization maintains identical numerical behavior while reducing computational overhead through intelligent caching of intermediate results.

✅ Correctness verification report:

Test	Status
⚙️ Existing Unit Tests	🔘 None Found
🌀 Generated Regression Tests	✅ 48 Passed
⏪ Replay Tests	🔘 None Found
🔎 Concolic Coverage Tests	✅ 3 Passed
📊 Tests Coverage	100.0%

🌀 Generated Regression Tests and Runtime

import math
import time
# function to test
from typing import Callable

# imports
import pytest  # used for our unit tests
from src.numpy_pandas.np_opts import bisection_method

# unit tests

# -------------------- BASIC TEST CASES --------------------

def test_basic_linear_root():
    # f(x) = x, root at 0
    codeflash_output = bisection_method(lambda x: x, -1, 1); root = codeflash_output # 542ns -> 542ns (0.000% faster)

def test_basic_quadratic_root():
    # f(x) = x^2 - 4, roots at -2 and 2
    codeflash_output = bisection_method(lambda x: x**2 - 4, 0, 3); root = codeflash_output # 6.88μs -> 4.96μs (38.6% faster)

def test_basic_negative_interval():
    # f(x) = x^2 - 4, roots at -2 and 2, test negative interval
    codeflash_output = bisection_method(lambda x: x**2 - 4, -3, 0); root = codeflash_output # 7.17μs -> 5.17μs (38.7% faster)

def test_basic_nonzero_epsilon():
    # f(x) = x^3 - x, root at 0
    codeflash_output = bisection_method(lambda x: x**3 - x, -0.5, 0.5, epsilon=1e-5); root = codeflash_output # 792ns -> 833ns (4.92% slower)

def test_basic_custom_max_iter():
    # f(x) = x^2 - 2, root at sqrt(2)
    codeflash_output = bisection_method(lambda x: x**2 - 2, 0, 2, epsilon=1e-6, max_iter=10); root = codeflash_output # 2.79μs -> 2.17μs (28.9% faster)

# -------------------- EDGE TEST CASES --------------------

def test_edge_root_at_endpoint_a():
    # f(x) = x, root at 0, interval [0, 1]
    codeflash_output = bisection_method(lambda x: x, 0, 1); root = codeflash_output # 8.46μs -> 7.17μs (18.0% faster)

def test_edge_root_at_endpoint_b():
    # f(x) = x - 1, root at 1, interval [0, 1]
    codeflash_output = bisection_method(lambda x: x - 1, 0, 1); root = codeflash_output # 4.00μs -> 3.38μs (18.5% faster)

def test_edge_function_flat_near_root():
    # f(x) = (x-1)^5, root at 1, very flat near root
    codeflash_output = bisection_method(lambda x: (x-1)**5, 0, 2, epsilon=1e-8); root = codeflash_output # 709ns -> 792ns (10.5% slower)

def test_edge_function_with_multiple_roots():
    # f(x) = x^3 - x, roots at -1, 0, 1; interval [-2, -0.5] should find -1
    codeflash_output = bisection_method(lambda x: x**3 - x, -2, -0.5); root = codeflash_output # 6.38μs -> 4.88μs (30.8% faster)

def test_edge_no_sign_change_raises():
    # f(x) = x^2 + 1, always positive
    with pytest.raises(ValueError):
        bisection_method(lambda x: x**2 + 1, -1, 1) # 416ns -> 416ns (0.000% faster)

def test_edge_zero_length_interval_root():
    # f(x) = x, interval [0, 0]
    codeflash_output = bisection_method(lambda x: x, 0, 0); root = codeflash_output # 500ns -> 500ns (0.000% faster)

def test_edge_zero_length_interval_no_root():
    # f(x) = x-1, interval [0, 0], no root at 0
    with pytest.raises(ValueError):
        bisection_method(lambda x: x-1, 0, 0) # 333ns -> 375ns (11.2% slower)

def test_edge_small_epsilon_high_precision():
    # f(x) = x^2 - 2, root at sqrt(2), very small epsilon
    codeflash_output = bisection_method(lambda x: x**2 - 2, 1, 2, epsilon=1e-14, max_iter=200); root = codeflash_output # 8.58μs -> 6.04μs (42.1% faster)

def test_edge_large_epsilon_low_precision():
    # f(x) = x^2 - 2, root at sqrt(2), large epsilon
    codeflash_output = bisection_method(lambda x: x**2 - 2, 1, 2, epsilon=0.1); root = codeflash_output # 1.50μs -> 1.38μs (9.09% faster)

def test_edge_max_iter_exceeded():
    # f(x) = x^2 - 2, root at sqrt(2), but max_iter too small
    codeflash_output = bisection_method(lambda x: x**2 - 2, 1, 2, epsilon=1e-15, max_iter=1); root = codeflash_output # 875ns -> 875ns (0.000% faster)

def test_edge_function_with_discontinuity():
    # f(x) = 1 if x < 0 else -1, discontinuous at 0
    def f(x):
        return 1 if x < 0 else -1
    codeflash_output = bisection_method(f, -1, 1, epsilon=1e-10); root = codeflash_output # 10.5μs -> 8.21μs (28.4% faster)

# -------------------- LARGE SCALE TEST CASES --------------------

def test_large_scale_many_iterations():
    # f(x) = x - 0.123456789, interval [0, 1], tiny epsilon
    codeflash_output = bisection_method(lambda x: x - 0.123456789, 0, 1, epsilon=1e-12, max_iter=1000); root = codeflash_output # 4.79μs -> 3.92μs (22.4% faster)

def test_large_scale_long_interval():
    # f(x) = x - 500, interval [0, 1000], root at 500
    codeflash_output = bisection_method(lambda x: x - 500, 0, 1000, epsilon=1e-8); root = codeflash_output # 791ns -> 792ns (0.126% slower)

def test_large_scale_performance():
    # f(x) = x^2 - 10^6, root at 1000, interval [0, 2000]
    start = time.time()
    codeflash_output = bisection_method(lambda x: x**2 - 1_000_000, 0, 2000, epsilon=1e-8); root = codeflash_output # 917ns -> 1.00μs (8.30% slower)
    duration = time.time() - start

def test_large_scale_small_interval_high_precision():
    # f(x) = x - 1e-6, interval [0, 2e-6], tiny epsilon
    codeflash_output = bisection_method(lambda x: x - 1e-6, 0, 2e-6, epsilon=1e-12); root = codeflash_output # 708ns -> 708ns (0.000% faster)

def test_large_scale_many_roots_choose_correct_interval():
    # f(x) = sin(100x), many roots in [0, 2*pi], pick interval for one root
    # sin(100x) = 0 at x = k*pi/100, test for x in [0.03, 0.04] (contains root at pi/100 ~ 0.0314159)
    codeflash_output = bisection_method(lambda x: math.sin(100*x), 0.03, 0.04, epsilon=1e-10); root = codeflash_output # 5.71μs -> 4.17μs (37.0% faster)

# -------------------- DETERMINISM TEST CASE --------------------

def test_deterministic_output():
    # f(x) = x^2 - 3, interval [1, 2], run twice, should get same result
    codeflash_output = bisection_method(lambda x: x**2 - 3, 1, 2); root1 = codeflash_output # 6.12μs -> 4.50μs (36.1% faster)
    codeflash_output = bisection_method(lambda x: x**2 - 3, 1, 2); root2 = codeflash_output # 5.58μs -> 3.83μs (45.7% 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
import time
# function to test
from typing import Callable

# imports
import pytest  # used for our unit tests
from src.numpy_pandas.np_opts import bisection_method

# unit tests

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

def test_basic_linear_root():
    # f(x) = x, root at 0
    codeflash_output = bisection_method(lambda x: x, -1, 1); root = codeflash_output # 500ns -> 500ns (0.000% faster)

def test_basic_quadratic_root():
    # f(x) = x^2 - 4, roots at -2 and 2
    codeflash_output = bisection_method(lambda x: x**2 - 4, 0, 5); root1 = codeflash_output # 7.00μs -> 4.96μs (41.2% faster)
    codeflash_output = bisection_method(lambda x: x**2 - 4, -5, 0); root2 = codeflash_output # 6.29μs -> 4.42μs (42.4% faster)

def test_basic_cubic_root():
    # f(x) = x^3, root at 0
    codeflash_output = bisection_method(lambda x: x**3, -1, 1); root = codeflash_output # 625ns -> 667ns (6.30% slower)

def test_basic_nonzero_epsilon():
    # f(x) = x - 0.5, root at 0.5, with larger epsilon
    codeflash_output = bisection_method(lambda x: x - 0.5, 0, 1, epsilon=1e-3); root = codeflash_output # 667ns -> 708ns (5.79% slower)

def test_basic_max_iter_hit():
    # f(x) = x, root at 0, but with a very low max_iter
    codeflash_output = bisection_method(lambda x: x, -1, 1, epsilon=1e-20, max_iter=1); root = codeflash_output # 500ns -> 500ns (0.000% faster)

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

def test_edge_root_at_endpoint_a():
    # f(x) = x, root at a = 0
    codeflash_output = bisection_method(lambda x: x, 0, 1); root = codeflash_output # 8.46μs -> 7.17μs (18.0% faster)

def test_edge_root_at_endpoint_b():
    # f(x) = x - 1, root at b = 1
    codeflash_output = bisection_method(lambda x: x - 1, 0, 1); root = codeflash_output # 4.00μs -> 3.38μs (18.5% faster)

def test_edge_function_same_sign_raises():
    # f(x) = x^2 + 1, always positive on [-1, 1]
    with pytest.raises(ValueError):
        bisection_method(lambda x: x**2 + 1, -1, 1) # 416ns -> 416ns (0.000% faster)

def test_edge_discontinuous_function():
    # f(x) = 1/x, root at 0 (discontinuity at 0)
    # Should raise ValueError if endpoints do not bracket root
    with pytest.raises(ValueError):
        bisection_method(lambda x: 1/x, 0.1, 1) # 459ns -> 541ns (15.2% slower)
    # Should work if endpoints bracket root
    codeflash_output = bisection_method(lambda x: x, -1, 1); root = codeflash_output # 458ns -> 458ns (0.000% faster)

def test_edge_epsilon_larger_than_interval():
    # f(x) = x, root at 0, but epsilon larger than interval
    codeflash_output = bisection_method(lambda x: x, -0.1, 0.1, epsilon=1); root = codeflash_output # 708ns -> 708ns (0.000% faster)

def test_edge_zero_interval():
    # a == b, should raise ValueError unless f(a) == 0
    with pytest.raises(ValueError):
        bisection_method(lambda x: x, 1, 1) # 333ns -> 375ns (11.2% slower)
    # If root is exactly at a == b
    codeflash_output = bisection_method(lambda x: x - 1, 1, 1); root = codeflash_output # 500ns -> 500ns (0.000% faster)

def test_edge_negative_interval():
    # a > b, should still work if signs are correct
    codeflash_output = bisection_method(lambda x: x, 1, -1); root = codeflash_output # 458ns -> 417ns (9.83% faster)

def test_edge_non_callable_function():
    # Passing a non-callable as f should raise TypeError
    with pytest.raises(TypeError):
        bisection_method(42, -1, 1) # 375ns -> 417ns (10.1% slower)

def test_edge_non_float_inputs():
    # Should work with int as well as float
    codeflash_output = bisection_method(lambda x: x, -1, 1); root = codeflash_output # 458ns -> 500ns (8.40% slower)

def test_edge_multiple_roots_in_interval():
    # f(x) = x*(x-1)*(x+1), roots at -1, 0, 1
    codeflash_output = bisection_method(lambda x: x*(x-1)*(x+1), -0.5, 0.5); root = codeflash_output # 791ns -> 791ns (0.000% faster)

def test_edge_function_with_flat_region():
    # f(x) = (x-1)^3, root at 1, flat near root
    codeflash_output = bisection_method(lambda x: (x-1)**3, 0, 2); root = codeflash_output # 708ns -> 708ns (0.000% faster)

def test_edge_fails_if_no_root():
    # f(x) = x^2 + 1, always positive
    with pytest.raises(ValueError):
        bisection_method(lambda x: x**2 + 1, -10, 10) # 459ns -> 541ns (15.2% slower)

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

def test_large_scale_high_precision():
    # f(x) = x - 1e-6, root at 1e-6, with high precision
    codeflash_output = bisection_method(lambda x: x - 1e-6, 0, 1e-5, epsilon=1e-12, max_iter=1000); root = codeflash_output # 3.12μs -> 2.67μs (17.2% faster)

def test_large_scale_many_iterations():
    # f(x) = x - 1e-8, root at 1e-8, requires many iterations
    codeflash_output = bisection_method(lambda x: x - 1e-8, 0, 1, epsilon=1e-12, max_iter=1000); root = codeflash_output # 4.50μs -> 3.50μs (28.6% faster)

def test_large_scale_performance():
    # f(x) = x^2 - 2, root at sqrt(2), large interval
    start = time.time()
    codeflash_output = bisection_method(lambda x: x**2 - 2, 0, 2, epsilon=1e-12, max_iter=1000); root = codeflash_output # 7.50μs -> 5.42μs (38.5% faster)
    elapsed = time.time() - start

def test_large_scale_small_epsilon():
    # f(x) = x - 0.123456789, root at 0.123456789, very small epsilon
    codeflash_output = bisection_method(lambda x: x - 0.123456789, 0, 1, epsilon=1e-15, max_iter=1000); root = codeflash_output # 5.00μs -> 4.25μs (17.6% faster)

def test_large_scale_max_iter_exceeded():
    # f(x) = x, root at 0, but very small epsilon and low max_iter
    codeflash_output = bisection_method(lambda x: x, -1, 1, epsilon=1e-20, max_iter=5); root = codeflash_output # 541ns -> 542ns (0.185% slower)



from src.numpy_pandas.np_opts import bisection_method
import pytest

def test_bisection_method():
    bisection_method(((x := [0.0, 0.0, 0.0]), lambda *a: x.pop(0) if len(x) > 1 else x[0])[1], 0.0, 0.0, epsilon=0.0, max_iter=1)

def test_bisection_method_2():
    bisection_method(lambda *a: 0.0, 0.0, 0.0, epsilon=0.5, max_iter=1)

def test_bisection_method_3():
    with pytest.raises(ValueError, match='Function\\ must\\ have\\ opposite\\ signs\\ at\\ endpoints'):
        bisection_method(lambda *a: 2.0, float('inf'), 0.0, epsilon=0.0, max_iter=0)

To edit these changes git checkout codeflash/optimize-bisection_method-mdpb3rmo and push.