Bisection Calculator

The Poisson Calculator recovers λ from a target probability with a higher-or-lower search: guess a rate, compare, keep the half of the window that must hold the answer. This page is that same search set loose on any expression you type. Enter f(x) — or a full equation f(x) = g(x), which is solved as f − g = 0 — plus an interval, and it scans for sign flips, then bisects every bracket it finds, narrating each halving and plotting the window as it shrinks. Type something without an x and it simply evaluates the expression — a plain calculator with the same grammar.

The grammar covers + − × ÷ ^, parentheses, implicit multiplication (2x, 2(x+1)), the constants pi, e, tau, and the usual scientific functions (sin, ln, log, sqrt, abs, min, pow, …). Angles are radians.

Seeded from the engine's worked example, written out as raw math: a fair Total quote says P(over 2.5) = 0.5, i.e. P(X ≤ 2) = 0.5, and for a Poisson count that is e^−λ(1 + λ + λ²/2) = 0.5. The root the search converges on is the same λ_total ≈ 2.6743 the Odds Generation engine recovers — and the same answer the Poisson page's inverse solve returns.

It is honest about the two classic traps: a sign flip across a pole (like 1/(x − 2)) pinches down to a point where |f| explodes instead of vanishing — reported as a jump, not a root — and a curve that touches zero without crossing (like x² + 0.05, or x² between samples) gives bisection nothing to grab. Both have example chips.

↑/↓ steps an interval bound, Shift steps bigger.

Bisection facts

f continuous on [a, b] and f(a)·f(b) < 0 ⇒ a root lies inside (intermediate-value theorem)
After k halvings the window is (b − a)/2^k wide — the midpoint misses the root by at most (b − a)/2^k+1
Halvings to a tolerance τ: ⌈log₂((b − a)/τ)⌉ — here ⌈log₂(8/1e-6)⌉ = 23

Inputs

Functionf(x) or f(x) = g(x)

parsed as f(x) = exp(-x)·(1 + x + x^2/2) - 0.5 — the two sides moved onto one, solving f(x) = 0

+ − × ÷ ^ ( ) · juxtaposition (2x) · x, pi, e, tau · sin cos tan asin acos atan sinh cosh tanh exp ln log log2 sqrt cbrt abs floor ceil round sign min max pow atan2 · P Pcdf Ptail malay prob fact choose

Tryexamples

Interval starta

Left end of the search window.

|a| ≤ 10¹² and a < b

Interval endb

Right end of the search window.

|b| ≤ 10¹² and b > a

Toleranceτ

Stop once the window is this tight either side of its midpoint — the root's error bound.

Evaluate f at a point

Probex

Plug any x into the parsed f — handy for sanity-checking a root by hand.

LHS	exp(-x)·(1 + x + x^2/2) at x = 2.6743	0.499941
RHS	0.5 at x = 2.6743	0.500000
f	f(2.6743) = exp(-x)·(1 + x + x^2/2) - 0.5LHS − RHS — zero exactly where the equation balances	-5.910e-5

Scan the interval — 129 samples hunting sign flips

·	sampled 129 points, one every 0.0625, across [0, 8]two neighbours with opposite signs trap a crossing between them — roots closer together than one spacing can hide	1 bracket
1	f flips + → − between x = 2.625 and x = 2.6875f(2.625) = 0.012172, f(2.6875) = -0.003309	bracket 1

Roots

root 1 — closed from bracket [2.625, 2.6875] in 15 halvingsf(root) = 1.099e-7

2.6740599

Watch the bracket shrink

step 15/15

window [2.6740589, 2.6740608] · width 1.91e-6 · root ≈ 2.6740599

The halvings, narrated

0	start with the scan's bracket: [2.625, 2.6875], width 0.062500f(2.625) = 0.012172 and f(2.6875) = -0.003309 — opposite signs, so a crossing is trapped inside
1	window [2.6250000, 2.6875000] → midpoint 2.6562500f(mid) = 0.004402 — same sign as the positive low end, so the crossing sits above the midpoint	▲ keep upper half
2	window [2.6562500, 2.6875000] → midpoint 2.6718750f(mid) = 0.000539 — same sign as the positive low end, so the crossing sits above the midpoint	▲ keep upper half
3	window [2.6718750, 2.6875000] → midpoint 2.6796875f(mid) = -0.001387 — opposite sign to the positive low end, so the crossing sits below the midpoint	▼ keep lower half
4	window [2.6718750, 2.6796875] → midpoint 2.6757813f(mid) = -0.000424 — opposite sign to the positive low end, so the crossing sits below the midpoint	▼ keep lower half
5	window [2.6718750, 2.6757813] → midpoint 2.6738281f(mid) = 5.726e-5 — same sign as the positive low end, so the crossing sits above the midpoint	▲ keep upper half
6	window [2.6738281, 2.6757813] → midpoint 2.6748047f(mid) = -0.000184 — opposite sign to the positive low end, so the crossing sits below the midpoint	▼ keep lower half
7	window [2.6738281, 2.6748047] → midpoint 2.6743164f(mid) = -6.315e-5 — opposite sign to the positive low end, so the crossing sits below the midpoint	▼ keep lower half
8	window [2.6738281, 2.6743164] → midpoint 2.6740723f(mid) = -2.947e-6 — opposite sign to the positive low end, so the crossing sits below the midpoint	▼ keep lower half
9	window [2.6738281, 2.6740723] → midpoint 2.6739502f(mid) = 2.715e-5 — same sign as the positive low end, so the crossing sits above the midpoint	▲ keep upper half
10	window [2.6739502, 2.6740723] → midpoint 2.6740112f(mid) = 1.210e-5 — same sign as the positive low end, so the crossing sits above the midpoint	▲ keep upper half
11	window [2.6740112, 2.6740723] → midpoint 2.6740417f(mid) = 4.578e-6 — same sign as the positive low end, so the crossing sits above the midpoint	▲ keep upper half
12	window [2.6740417, 2.6740723] → midpoint 2.6740570f(mid) = 8.155e-7 — same sign as the positive low end, so the crossing sits above the midpoint	▲ keep upper half
13	window [2.6740570, 2.6740723] → midpoint 2.6740646f(mid) = -1.066e-6 — opposite sign to the positive low end, so the crossing sits below the midpoint	▼ keep lower half
14	window [2.6740570, 2.6740646] → midpoint 2.6740608f(mid) = -1.252e-7 — opposite sign to the positive low end, so the crossing sits below the midpoint	▼ keep lower half
15	window [2.6740570, 2.6740608] → midpoint 2.6740589f(mid) = 3.451e-7 — same sign as the positive low end, so the crossing sits above the midpoint	▲ keep upper half
✓	window is 1.91e-6 wide — at most the 1e-6 tolerance either side of the midpointcheck: f(2.6740599) = 1.099e-7	2.6740599