⚡️ Speed up function bisection_method by 20%

[codeflash-ai]

📄 20% (0.20x) speedup for bisection_method in src/numpy_pandas/numerical_methods.py

⏱️ Runtime : 234 microseconds → 195 microseconds (best of 319 runs)

📝 Explanation and details

The optimization eliminates redundant function evaluations by caching the function values at the interval endpoints.

Key Changes:

• Pre-compute initial function values: fa = f(a) and fb = f(b) are calculated once at the start
• Cache updates during iteration: When updating interval boundaries (a = c or b = c), the corresponding cached values (fa = fc or fb = fc) are updated instead of recalculating f(a) in each iteration
• Replace f(a) calls: The condition if f(a) * fc < 0: now uses the cached fa value: if fa * fc < 0:

Why This Is Faster:
In the original code, f(a) was evaluated in every loop iteration (line showing 24.5% of total time). The optimized version eliminates these repeated calls by maintaining cached function values that are only updated when the interval boundaries change. This is particularly effective when the function f is computationally expensive.

Performance Profile:
The line profiler shows the optimization saves ~145,000 nanoseconds (from 236,000 to 91,000 ns) on the condition check alone. Test results indicate 20-40% speedups for cases requiring many iterations (high precision, large intervals, slow-converging functions), while simple cases show minimal overhead from the additional variable assignments.

Best Use Cases:
Most effective for computationally expensive functions, high-precision requirements, or large search intervals where the bisection method performs many iterations.

✅ Correctness verification report:

Test	Status
⚙️ Existing Unit Tests	🔘 None Found
🌀 Generated Regression Tests	✅ 57 Passed
⏪ Replay Tests	🔘 None Found
🔎 Concolic Coverage Tests	✅ 3 Passed
📊 Tests Coverage	100.0%

🌀 Generated Regression Tests and Runtime

import math  # used for math functions in test cases
# function to test
# (copied from prompt)
from typing import Callable

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

# unit tests

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

def test_linear_root():
    # f(x) = x, root at 0
    codeflash_output = bisection_method(lambda x: x, -1, 1); root = codeflash_output # 834ns -> 875ns (4.69% slower)

def test_quadratic_root():
    # f(x) = x^2 - 2, root at sqrt(2)
    codeflash_output = bisection_method(lambda x: x**2 - 2, 0, 2); root = codeflash_output # 7.88μs -> 5.92μs (33.1% faster)

def test_negative_root():
    # f(x) = x + 3, root at -3
    codeflash_output = bisection_method(lambda x: x + 3, -5, 0); root = codeflash_output # 6.54μs -> 5.38μs (21.7% faster)

def test_root_with_custom_epsilon():
    # f(x) = x^2 - 4, root at 2, with looser epsilon
    codeflash_output = bisection_method(lambda x: x**2 - 4, 1, 3, epsilon=1e-4); root = codeflash_output # 1.25μs -> 1.29μs (3.25% slower)

def test_root_with_custom_max_iter():
    # f(x) = x^2 - 9, root at 3, but with low max_iter (should be less accurate)
    codeflash_output = bisection_method(lambda x: x**2 - 9, 2, 4, epsilon=1e-15, max_iter=5); root = codeflash_output # 1.21μs -> 1.21μs (0.083% slower)

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

def test_function_same_sign_raises():
    # f(x) = x^2 + 1, always positive, should raise
    with pytest.raises(ValueError):
        bisection_method(lambda x: x**2 + 1, -1, 1) # 833ns -> 833ns (0.000% faster)

def test_function_zero_at_endpoint_a():
    # f(x) = x, root at 0, a=0, b=1
    codeflash_output = bisection_method(lambda x: x, 0, 1); root = codeflash_output # 12.0μs -> 10.5μs (14.4% faster)

def test_function_zero_at_endpoint_b():
    # f(x) = x-1, root at 1, a=0, b=1
    codeflash_output = bisection_method(lambda x: x-1, 0, 1); root = codeflash_output # 6.42μs -> 5.12μs (25.2% faster)


def test_function_with_extremely_small_interval():
    # f(x) = x, interval very close to root
    codeflash_output = bisection_method(lambda x: x, -1e-12, 1e-12); root = codeflash_output # 1.12μs -> 1.08μs (3.78% faster)

def test_function_with_extremely_large_interval():
    # f(x) = x - 1, interval (-1e6, 1e6)
    codeflash_output = bisection_method(lambda x: x - 1, -1e6, 1e6); root = codeflash_output # 9.00μs -> 7.04μs (27.8% faster)

def test_function_with_non_integer_root():
    # f(x) = x^2 - 3, root at sqrt(3)
    codeflash_output = bisection_method(lambda x: x**2 - 3, 1, 2); root = codeflash_output # 8.42μs -> 6.29μs (33.8% faster)

def test_function_with_discontinuity():
    # f(x) = 1/x, root at 0 (but undefined at 0), interval (-1, 1)
    # Should converge to a value close to 0, but may never reach abs(fc) < epsilon
    codeflash_output = bisection_method(lambda x: x, -1, 1); root = codeflash_output # 791ns -> 833ns (5.04% slower)

def test_function_with_opposite_signs_but_no_root():
    # f(x) = 1/x, interval (-1, 1), but function is undefined at 0
    # Should not raise, but may return a value close to 0
    codeflash_output = bisection_method(lambda x: x, -1, 1); root = codeflash_output # 667ns -> 750ns (11.1% slower)

def test_function_with_flat_region():
    # f(x) = 0 for all x, interval (-1, 1)
    codeflash_output = bisection_method(lambda x: 0, -1, 1); root = codeflash_output # 875ns -> 916ns (4.48% slower)

def test_function_with_non_monotonic_behavior():
    # f(x) = sin(x), root at 0, interval (-1, 1)
    codeflash_output = bisection_method(math.sin, -1, 1); root = codeflash_output # 1.33μs -> 1.21μs (10.4% faster)

def test_function_with_large_max_iter_and_tight_epsilon():
    # f(x) = x^2 - 2, root at sqrt(2), tight epsilon and large max_iter
    codeflash_output = bisection_method(lambda x: x**2 - 2, 0, 2, epsilon=1e-15, max_iter=500); root = codeflash_output # 12.5μs -> 8.92μs (40.2% faster)

def test_function_with_small_max_iter_and_large_epsilon():
    # f(x) = x^2 - 2, root at sqrt(2), loose epsilon and small max_iter
    codeflash_output = bisection_method(lambda x: x**2 - 2, 0, 2, epsilon=0.5, max_iter=2); root = codeflash_output # 1.79μs -> 1.71μs (4.86% faster)

def test_function_with_extremely_small_epsilon():
    # f(x) = x - 1e-8, root at 1e-8
    codeflash_output = bisection_method(lambda x: x - 1e-8, 0, 1e-6, epsilon=1e-12); root = codeflash_output # 4.25μs -> 3.58μs (18.6% faster)

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

def test_large_interval_and_iterations():
    # f(x) = x - 123456, root at 123456, large interval
    codeflash_output = bisection_method(lambda x: x - 123456, 0, 1e6, epsilon=1e-8, max_iter=100); root = codeflash_output # 7.96μs -> 6.67μs (19.4% faster)

def test_large_max_iter_convergence():
    # f(x) = x^2 - 1e6, root at 1000, large interval and max_iter
    codeflash_output = bisection_method(lambda x: x**2 - 1e6, 0, 2000, epsilon=1e-8, max_iter=1000); root = codeflash_output # 1.54μs -> 1.62μs (5.11% slower)

def test_high_precision_large_interval():
    # f(x) = x - 0.123456789, root at 0.123456789, large interval and tight epsilon
    codeflash_output = bisection_method(lambda x: x - 0.123456789, -1e3, 1e3, epsilon=1e-12, max_iter=200); root = codeflash_output # 6.88μs -> 6.08μs (13.0% faster)

def test_many_iterations_for_slow_converging_function():
    # f(x) = x^5 - 1, root at 1, slow convergence near root
    codeflash_output = bisection_method(lambda x: x**5 - 1, 0, 2, epsilon=1e-10, max_iter=1000); root = codeflash_output # 1.33μs -> 1.42μs (5.86% slower)

def test_large_scale_with_sin_function():
    # f(x) = sin(x), root at pi, interval (3, 4), high precision
    codeflash_output = bisection_method(math.sin, 3, 4, epsilon=1e-12, max_iter=1000); root = codeflash_output # 6.29μs -> 5.17μs (21.8% faster)

def test_large_scale_with_exponential_function():
    # f(x) = exp(x) - 1000, root at ln(1000), interval (0, 10)
    codeflash_output = bisection_method(lambda x: math.exp(x) - 1000, 0, 10, epsilon=1e-10, max_iter=100); root = codeflash_output # 10.5μs -> 7.71μs (36.7% faster)

def test_large_scale_with_negative_interval():
    # f(x) = x^2 - 100, root at -10, interval (-100, 0)
    codeflash_output = bisection_method(lambda x: x**2 - 100, -100, 0, epsilon=1e-8, max_iter=100); root = codeflash_output # 9.50μs -> 7.12μs (33.3% faster)

def test_large_scale_with_small_numbers():
    # f(x) = x - 1e-9, root at 1e-9, interval (0, 1e-6)
    codeflash_output = bisection_method(lambda x: x - 1e-9, 0, 1e-6, epsilon=1e-12, max_iter=100); root = codeflash_output # 3.71μs -> 3.17μs (17.1% faster)

def test_large_scale_with_large_numbers():
    # f(x) = x - 1e8, root at 1e8, interval (0, 1e9)
    codeflash_output = bisection_method(lambda x: x - 1e8, 0, 1e9, epsilon=1e-6, max_iter=100); root = codeflash_output # 7.25μs -> 6.04μs (20.0% faster)
# codeflash_output is used to check that the output of the original code is the same as that of the optimized code.
#------------------------------------------------
from typing import Callable

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

# unit tests

# --- BASIC TEST CASES ---

def test_basic_quadratic_root():
    # Test root of f(x) = x^2 - 4 in [0, 3], root is 2
    codeflash_output = bisection_method(lambda x: x**2 - 4, 0, 3); root = codeflash_output # 9.04μs -> 6.50μs (39.1% faster)

def test_basic_negative_root():
    # Test root of f(x) = x^2 - 4 in [-3, 0], root is -2
    codeflash_output = bisection_method(lambda x: x**2 - 4, -3, 0); root = codeflash_output # 9.21μs -> 6.54μs (40.8% faster)

def test_basic_linear_function():
    # Test root of f(x) = x - 5 in [0, 10], root is 5
    codeflash_output = bisection_method(lambda x: x - 5, 0, 10); root = codeflash_output # 917ns -> 958ns (4.28% slower)

def test_basic_trigonometric_function():
    # Test root of f(x) = sin(x) in [3, 4], root is pi (~3.14159)
    import math
    codeflash_output = bisection_method(math.sin, 3, 4); root = codeflash_output # 6.21μs -> 5.29μs (17.3% faster)

def test_basic_with_custom_epsilon():
    # Test root with a larger epsilon
    codeflash_output = bisection_method(lambda x: x - 1, 0, 2, epsilon=1e-2); root = codeflash_output # 1.00μs -> 1.00μs (0.000% faster)

def test_basic_with_custom_max_iter():
    # Test root with a very small max_iter (should not converge exactly)
    codeflash_output = bisection_method(lambda x: x - 1, 0, 2, epsilon=1e-10, max_iter=2); root = codeflash_output # 958ns -> 958ns (0.000% faster)

# --- EDGE TEST CASES ---

def test_edge_no_sign_change_raises():
    # f(x) = x^2 + 1 has no root in [-1, 1], should raise ValueError
    with pytest.raises(ValueError):
        bisection_method(lambda x: x**2 + 1, -1, 1) # 833ns -> 875ns (4.80% slower)

def test_edge_root_at_endpoint_a():
    # Root at endpoint a: f(x) = x in [0, 1], root is 0
    codeflash_output = bisection_method(lambda x: x, 0, 1); root = codeflash_output # 11.8μs -> 10.5μs (12.4% faster)

def test_edge_root_at_endpoint_b():
    # Root at endpoint b: f(x) = x - 1 in [0, 1], root is 1
    codeflash_output = bisection_method(lambda x: x - 1, 0, 1); root = codeflash_output # 6.50μs -> 5.17μs (25.8% faster)


def test_edge_small_interval():
    # Very small interval containing root: f(x) = x in [-1e-12, 1e-12]
    codeflash_output = bisection_method(lambda x: x, -1e-12, 1e-12); root = codeflash_output # 1.08μs -> 1.12μs (3.73% slower)

def test_edge_epsilon_larger_than_interval():
    # Epsilon larger than interval: should return midpoint
    codeflash_output = bisection_method(lambda x: x, -1, 1, epsilon=2); root = codeflash_output # 1.17μs -> 1.17μs (0.000% faster)

def test_edge_max_iter_exceeded():
    # f(x) = x in [0, 1], with very small epsilon and low max_iter
    codeflash_output = bisection_method(lambda x: x, 0, 1, epsilon=1e-20, max_iter=1); root = codeflash_output # 1.42μs -> 1.42μs (0.000% faster)

def test_edge_function_with_discontinuity():
    # f(x) = 1/x in [-1, 1], root at 0, but discontinuous at 0
    codeflash_output = bisection_method(lambda x: x, -1, 1); root = codeflash_output # 791ns -> 791ns (0.000% faster)

def test_edge_non_float_inputs():
    # Test with integer a and b
    codeflash_output = bisection_method(lambda x: x - 2, 0, 4); root = codeflash_output # 1.00μs -> 958ns (4.38% faster)

def test_edge_function_returns_zero_at_midpoint():
    # f(x) = x - 2 in [0, 4], midpoint is 2, should return after first iteration
    codeflash_output = bisection_method(lambda x: x - 2, 0, 4); root = codeflash_output # 833ns -> 792ns (5.18% faster)

# --- LARGE SCALE TEST CASES ---

def test_large_scale_high_precision():
    # Test with very small epsilon and large interval
    codeflash_output = bisection_method(lambda x: x - 123456, 0, 2*123456, epsilon=1e-12, max_iter=100); root = codeflash_output # 1.38μs -> 1.42μs (2.96% slower)

def test_large_scale_many_iterations():
    # Test with large max_iter, should converge before reaching max_iter
    codeflash_output = bisection_method(lambda x: x - 1, 0, 2, epsilon=1e-15, max_iter=1000); root = codeflash_output # 1.04μs -> 1.08μs (3.88% slower)

def test_large_scale_performance():
    # Test with a function that is slow to converge
    # f(x) = x^2 - 1 in [0, 2], root at 1
    codeflash_output = bisection_method(lambda x: x**2 - 1, 0, 2, epsilon=1e-15, max_iter=1000); root = codeflash_output # 1.33μs -> 1.29μs (3.17% faster)

def test_large_scale_extreme_values():
    # Test with large interval values
    a, b = -1e9, 1e9
    codeflash_output = bisection_method(lambda x: x, a, b, epsilon=1e-6); root = codeflash_output # 1.08μs -> 1.17μs (7.20% slower)

def test_large_scale_small_epsilon_large_interval():
    # Test with large interval and tiny epsilon
    codeflash_output = bisection_method(lambda x: x - 1e6, 0, 2e6, epsilon=1e-12, max_iter=1000); root = codeflash_output # 1.33μs -> 1.25μs (6.72% faster)

def test_large_scale_boundary_behavior():
    # Test root near the boundary of a large interval
    codeflash_output = bisection_method(lambda x: x - 999999, 999998, 1000000, epsilon=1e-10); root = codeflash_output # 1.04μs -> 1.12μs (7.47% slower)

# --- DETERMINISM TEST CASES ---

def test_determinism_same_inputs_same_result():
    # Multiple calls with same inputs must return same result
    codeflash_output = bisection_method(lambda x: x - 7, 0, 10); r1 = codeflash_output # 7.38μs -> 6.08μs (21.2% faster)
    codeflash_output = bisection_method(lambda x: x - 7, 0, 10); r2 = codeflash_output # 5.79μs -> 4.54μs (27.5% faster)

def test_determinism_different_epsilon():
    # Different epsilon should affect result precision
    codeflash_output = bisection_method(lambda x: x - 7, 0, 10, epsilon=1e-2); r1 = codeflash_output # 2.58μs -> 2.33μs (10.7% faster)
    codeflash_output = bisection_method(lambda x: x - 7, 0, 10, epsilon=1e-10); r2 = codeflash_output # 5.83μs -> 4.54μs (28.4% faster)

# --- CLEAN CODE/READABILITY TEST CASES ---

def test_readability_documentation_example():
    # Example from documentation: f(x) = x^3 - x - 2 in [1, 2], root is ~1.521
    def cubic(x): return x**3 - x - 2
    codeflash_output = bisection_method(cubic, 1, 2); root = codeflash_output # 8.96μs -> 6.58μs (36.1% faster)

# --- ERROR HANDLING TEST CASES ---

def test_error_max_iter_zero():
    # max_iter = 0 should return midpoint
    codeflash_output = bisection_method(lambda x: x - 1, 0, 2, max_iter=0); root = codeflash_output # 792ns -> 792ns (0.000% faster)

def test_error_epsilon_zero():
    # epsilon = 0 should converge to root within max_iter
    codeflash_output = bisection_method(lambda x: x - 1, 0, 2, epsilon=0, max_iter=10); root = codeflash_output # 3.29μs -> 2.83μs (16.2% faster)

def test_error_large_max_iter():
    # Large max_iter should not break the function
    codeflash_output = bisection_method(lambda x: x - 1, 0, 2, max_iter=1000); root = codeflash_output # 1.00μs -> 1.04μs (4.03% slower)

# --- NON-DETERMINISTIC FUNCTION TEST CASE ---

def test_non_deterministic_function():
    # If function is non-deterministic, results may vary. Should still run.
    import random
    def noisy(x): return x - 1 + random.uniform(-1e-12, 1e-12)
    codeflash_output = bisection_method(noisy, 0, 2); root = codeflash_output # 2.42μs -> 2.50μs (3.32% 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)

🔎 Concolic Coverage Tests and Runtime

Test File::Test Function	Original ⏱️	Optimized ⏱️	Speedup
codeflash_concolic_mw8xqeap/tmpa4i9_pec/test_concolic_coverage.py::test_bisection_method	1.83μs	1.75μs	4.80%✅
codeflash_concolic_mw8xqeap/tmpa4i9_pec/test_concolic_coverage.py::test_bisection_method_2	1.04μs	1.08μs	-3.79%⚠️
codeflash_concolic_mw8xqeap/tmpa4i9_pec/test_concolic_coverage.py::test_bisection_method_3	917ns	958ns	-4.28%⚠️

To edit these changes git checkout codeflash/optimize-bisection_method-mh16w20a and push.