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Date: Wed, 30 Jul 2025 05:52:31 +0000
Subject: [PATCH] =?UTF-8?q?=E2=9A=A1=EF=B8=8F=20Speed=20up=20function=20`b?=
 =?UTF-8?q?isection=5Fmethod`=20by=2021%?=
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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.
---
 src/numpy_pandas/numerical_methods.py | 8 ++++++--
 1 file changed, 6 insertions(+), 2 deletions(-)

diff --git a/src/numpy_pandas/numerical_methods.py b/src/numpy_pandas/numerical_methods.py
index ff90bb9..aad5fce 100644
--- a/src/numpy_pandas/numerical_methods.py
+++ b/src/numpy_pandas/numerical_methods.py
@@ -146,15 +146,19 @@ def bisection_method(
     epsilon: float = 1e-10,
     max_iter: int = 100,
 ) -> float:
-    if f(a) * f(b) > 0:
+    fa = f(a)
+    fb = f(b)
+    if fa * fb > 0:
         raise ValueError("Function must have opposite signs at endpoints")
     for _ in range(max_iter):
         c = (a + b) / 2
         fc = f(c)
         if abs(fc) < epsilon:
             return c
-        if f(a) * fc < 0:
+        if fa * fc < 0:
             b = c
+            fb = fc  # Only recalculate f(b)
         else:
             a = c
+            fa = fc  # Only recalculate f(a)
     return (a + b) / 2