⚡️ Speed up function bisection_method by 21%

[codeflash-ai]

📄 21% (0.21x) speedup for bisection_method in src/numpy_pandas/numerical_methods.py

⏱️ Runtime : 140 microseconds → 116 microseconds (best of 363 runs)

📝 Explanation and details

The optimization eliminates redundant function evaluations by caching the function values at the interval endpoints (fa and fb).

Key Changes:

1. Pre-compute endpoint values: Store fa = f(a) and fb = f(b) at initialization
2. Cache updates: When updating interval bounds, reuse the already-computed fc value instead of recalculating f(a) or f(b)
3. Eliminate repeated evaluations: Replace f(a) * fc < 0 comparison with fa * fc < 0 using the cached value

Why This Creates a Speedup:
The original code calls f(a) in every iteration of the main loop (line with 25.9% of total time), even though f(a) doesn't change unless a is updated. The optimization reduces function calls from ~2 per iteration to ~1 per iteration by:

• Eliminating the f(a) call in the comparison f(a) * fc < 0
• Reusing the computed fc value when updating fa or fb

Performance Analysis:
The line profiler shows the original code spent 530,000 time units (25.9%) on f(a) * fc < 0 evaluations across 1,116 hits. The optimized version spends only 203,000 time units (10.7%) on fa * fc < 0 comparisons, nearly halving the time for this critical operation.

Test Case Performance:
The optimization is most effective for:

• Complex functions (39.4% speedup for x^2 - 2, 37.7% for x^5 - 32) where function evaluation is expensive
• Many iterations (22.2% speedup for high-precision cases) where the cumulative effect of avoiding redundant calls compounds
• Standard bisection scenarios (15-25% typical speedup) where the algorithm runs for multiple iterations

The optimization shows minimal or slight slowdowns only in edge cases with very few iterations where the overhead of variable assignments outweighs the savings.

✅ Correctness verification report:

Test	Status
⚙️ Existing Unit Tests	🔘 None Found
🌀 Generated Regression Tests	✅ 44 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.numerical_methods import bisection_method

# unit tests

# =========================
# 1. Basic Test Cases
# =========================

def test_linear_root():
    # f(x) = x - 3 has root at 3
    def f(x): return x - 3
    codeflash_output = bisection_method(f, 0, 5); root = codeflash_output # 4.79μs -> 4.17μs (15.0% faster)

def test_quadratic_root_positive():
    # f(x) = x^2 - 4 has roots at -2 and 2
    def f(x): return x**2 - 4
    codeflash_output = bisection_method(f, 0, 3); root = codeflash_output # 7.25μs -> 5.29μs (37.0% faster)

def test_quadratic_root_negative():
    # f(x) = x^2 - 4 has roots at -2 and 2
    def f(x): return x**2 - 4
    codeflash_output = bisection_method(f, -3, 0); root = codeflash_output # 7.25μs -> 5.12μs (41.5% faster)

def test_cubic_root():
    # f(x) = x^3 has root at 0
    def f(x): return x**3
    codeflash_output = bisection_method(f, -1, 1); root = codeflash_output # 750ns -> 792ns (5.30% slower)

def test_sine_root():
    # f(x) = sin(x) has root at 0, pi, 2pi, etc.
    def f(x): return math.sin(x)
    codeflash_output = bisection_method(f, 3, 4); root = codeflash_output # 5.08μs -> 4.04μs (25.8% faster)

def test_root_with_custom_epsilon():
    # Use a looser epsilon to test early stopping
    def f(x): return x - 7
    codeflash_output = bisection_method(f, 0, 10, epsilon=1e-2); root = codeflash_output # 2.04μs -> 1.88μs (8.91% faster)

def test_root_with_custom_max_iter():
    # Should still find the root within 50 iterations
    def f(x): return x - 1
    codeflash_output = bisection_method(f, 0, 2, max_iter=50); root = codeflash_output # 750ns -> 750ns (0.000% faster)

# =========================
# 2. Edge Test Cases
# =========================

def test_root_at_endpoint_a():
    # f(x) = x, root at a=0
    def f(x): return x
    codeflash_output = bisection_method(f, 0, 5); root = codeflash_output # 8.67μs -> 7.46μs (16.2% faster)

def test_root_at_endpoint_b():
    # f(x) = x - 5, root at b=5
    def f(x): return x - 5
    codeflash_output = bisection_method(f, 0, 5); root = codeflash_output # 4.29μs -> 3.71μs (15.7% faster)

def test_function_same_sign_raises():
    # f(x) = x^2 + 1, always positive, should raise ValueError
    def f(x): return x**2 + 1
    with pytest.raises(ValueError):
        bisection_method(f, -2, 2) # 500ns -> 541ns (7.58% slower)

def test_zero_interval():
    # a == b, and f(a) == 0, should return a
    def f(x): return x - 2
    codeflash_output = bisection_method(f, 2, 2); root = codeflash_output # 583ns -> 583ns (0.000% faster)

def test_zero_interval_not_root():
    # a == b, and f(a) != 0, but function should raise ValueError
    def f(x): return x + 1
    with pytest.raises(ValueError):
        bisection_method(f, 1, 1) # 375ns -> 417ns (10.1% slower)

def test_discontinuous_function():
    # f(x) = 1 if x < 0, -1 if x >= 0; root at x=0
    def f(x): return 1 if x < 0 else -1
    codeflash_output = bisection_method(f, -1, 1); root = codeflash_output # 10.7μs -> 8.67μs (23.1% faster)


def test_function_with_flat_region():
    # f(x) = 0 for x in [1,2], otherwise x-1
    def f(x): return 0 if 1 <= x <= 2 else x - 1
    codeflash_output = bisection_method(f, 0, 3); root = codeflash_output # 875ns -> 875ns (0.000% faster)


def test_function_with_infinite():
    # f(x) = inf for x < 0, -inf for x >= 0; root at 0
    def f(x): return float('inf') if x < 0 else float('-inf')
    codeflash_output = bisection_method(f, -1, 1); root = codeflash_output # 14.7μs -> 10.1μs (44.9% faster)

def test_max_iterations_exceeded():
    # f(x) = x, but with very small max_iter, should return midpoint
    def f(x): return x
    codeflash_output = bisection_method(f, -1, 1, epsilon=1e-20, max_iter=1); result = codeflash_output # 666ns -> 708ns (5.93% slower)

# =========================
# 3. Large Scale Test Cases
# =========================

def test_high_precision():
    # f(x) = x - 1e-7, root at 1e-7, very small epsilon
    def f(x): return x - 1e-7
    codeflash_output = bisection_method(f, 0, 1, epsilon=1e-12, max_iter=1000); root = codeflash_output # 4.25μs -> 3.54μs (20.0% faster)

def test_large_interval():
    # f(x) = x - 1, root at 1, but interval is very large
    def f(x): return x - 1
    codeflash_output = bisection_method(f, -1e6, 1e6); root = codeflash_output # 5.92μs -> 4.88μs (21.4% faster)

def test_large_max_iter():
    # f(x) = x - 0.5, root at 0.5, with large max_iter
    def f(x): return x - 0.5
    codeflash_output = bisection_method(f, 0, 1, epsilon=1e-15, max_iter=1000); root = codeflash_output # 708ns -> 750ns (5.60% slower)

def test_performance_large_scale():
    # f(x) = x^2 - 2, root at sqrt(2), with tight epsilon and large interval
    def f(x): return x**2 - 2
    start = time.time()
    codeflash_output = bisection_method(f, 0, 2, epsilon=1e-12, max_iter=1000); root = codeflash_output # 7.67μs -> 5.50μs (39.4% faster)
    elapsed = time.time() - start

def test_many_bisections():
    # f(x) = x - 0.123456789, root at 0.123456789, large max_iter and small epsilon
    def f(x): return x - 0.123456789
    codeflash_output = bisection_method(f, 0, 1, epsilon=1e-14, max_iter=1000); root = codeflash_output # 5.08μs -> 4.38μs (16.2% 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 mathematical functions
# function to test
from typing import Callable

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

# unit tests

# -------------------------------
# 1. 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 -> 583ns (14.2% slower)

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, 4); root = codeflash_output # 792ns -> 833ns (4.92% slower)

def test_basic_negative_interval():
    # f(x) = x^2 - 1, roots at -1 and 1, test negative interval
    codeflash_output = bisection_method(lambda x: x**2 - 1, -2, 0); root = codeflash_output # 750ns -> 708ns (5.93% faster)

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 # 708ns -> 750ns (5.60% slower)

def test_basic_trig_function():
    # f(x) = sin(x), root at 0
    codeflash_output = bisection_method(math.sin, -1, 1); root = codeflash_output # 750ns -> 791ns (5.18% slower)

# -------------------------------
# 2. Edge Test Cases
# -------------------------------

def test_error_on_same_sign_endpoints():
    # f(x) = x^2 + 1, always positive, should raise ValueError
    with pytest.raises(ValueError):
        bisection_method(lambda x: x**2 + 1, -1, 1) # 458ns -> 500ns (8.40% slower)

def test_root_at_endpoint():
    # f(x) = x, root at 0 (endpoint)
    codeflash_output = bisection_method(lambda x: x, 0, 2); root = codeflash_output # 8.58μs -> 7.50μs (14.5% faster)

def test_root_very_close_to_endpoint():
    # f(x) = x - 1e-12, root at 1e-12, interval [0, 1e-10]
    codeflash_output = bisection_method(lambda x: x - 1e-12, 0, 1e-10, epsilon=1e-15); root = codeflash_output # 2.25μs -> 2.00μs (12.5% faster)


def test_max_iter_limit():
    # f(x) = x, root at 0, but with very tight epsilon and low max_iter
    codeflash_output = bisection_method(lambda x: x, -1, 1, epsilon=1e-20, max_iter=2); root = codeflash_output # 667ns -> 667ns (0.000% faster)

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

def test_zero_width_interval_without_root():
    # f(x) = x - 1, interval [0, 0], no root at 0, should raise ValueError
    with pytest.raises(ValueError):
        bisection_method(lambda x: x - 1, 0, 0) # 375ns -> 416ns (9.86% slower)

def test_non_monotonic_function():
    # f(x) = cos(x), root at pi/2 ~ 1.5708 in [1, 2]
    codeflash_output = bisection_method(math.cos, 1, 2); root = codeflash_output # 3.83μs -> 3.38μs (13.6% faster)

def test_function_with_flat_slope_near_root():
    # f(x) = (x-1)^3, root at 1, flat slope near root
    codeflash_output = bisection_method(lambda x: (x-1)**3, 0, 2); root = codeflash_output # 833ns -> 875ns (4.80% slower)

# -------------------------------
# 3. Large Scale Test Cases
# -------------------------------

def test_large_interval():
    # f(x) = x - 1e6, root at 1e6, interval [0, 2e6]
    codeflash_output = bisection_method(lambda x: x - 1e6, 0, 2e6); root = codeflash_output # 750ns -> 791ns (5.18% slower)

def test_small_epsilon_large_interval():
    # f(x) = x - 500, root at 500, interval [0, 1000], tight epsilon
    codeflash_output = bisection_method(lambda x: x - 500, 0, 1000, epsilon=1e-12); root = codeflash_output # 959ns -> 1.00μs (4.10% slower)

def test_many_iterations_needed():
    # f(x) = x - 1e-6, root at 1e-6, interval [0, 1], tiny epsilon
    codeflash_output = bisection_method(lambda x: x - 1e-6, 0, 1, epsilon=1e-12, max_iter=100); root = codeflash_output # 4.58μs -> 3.75μs (22.2% faster)

def test_high_degree_polynomial_large_interval():
    # f(x) = x^5 - 32, root at 2, interval [0, 10]
    codeflash_output = bisection_method(lambda x: x**5 - 32, 0, 10); root = codeflash_output # 8.38μs -> 6.08μs (37.7% faster)

def test_large_numbers():
    # f(x) = x - 1e9, root at 1e9, interval [1e8, 2e9]
    codeflash_output = bisection_method(lambda x: x - 1e9, 1e8, 2e9); root = codeflash_output # 5.46μs -> 4.75μs (14.9% faster)

def test_large_negative_numbers():
    # f(x) = x + 1e8, root at -1e8, interval [-2e8, 0]
    codeflash_output = bisection_method(lambda x: x + 1e8, -2e8, 0); root = codeflash_output # 750ns -> 792ns (5.30% slower)

# -------------------------------
# Additional Edge/Robustness Cases
# -------------------------------




def test_function_with_almost_zero_epsilon():
    # f(x) = x - 0.25, root at 0.25, epsilon very small
    codeflash_output = bisection_method(lambda x: x - 0.25, 0, 1, epsilon=1e-15); root = codeflash_output # 1.08μs -> 1.12μs (3.64% slower)
# 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 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-mdpju0gs and push.