Floating Point and IEEE 754

Why 0.1 + 0.2 is not 0.3 — bit layout, rounding, ULP spacing, cancellation and NaN, shown on real bit patterns

1 / 27

steps

Act 1 of 8·The Anomaly

step 1

every language, same answer

>>>0.1 + 0.2

0.30000000000000004

>>>0.1 + 0.2 === 0.3

false

what the two values really are, to 20 digits

0.1 + 0.20.30000000000000004441

0.30.29999999999999998890

difference5.551115123125783e-17

not a bug in any one language — it is the hardware format nearly all of them share

Speednormal

Open a console in almost any language and add 0.1 to 0.2. You will not get 0.3. This is the single most reported non-bug in programming history, and the answer is not "computers are imprecise" — it is a specific, understandable consequence of the format the hardware uses.

double (binary64)

Sign1 bit

Exponent11 bits · bias 1023

Mantissa52 bits + implicit 1

Roundingnearest, ties to even

ULP at 12⁻⁵² ≈ 2.2e−16

Safe intsup to 2⁵³ − 1

Gotchascancellation · NaN

Active Recall

Question 1 / 3

Why can a double not store 0.1 exactly?

Under the Hood

Loading visualizer…

DNSresolving host

TCPconnecting

TLShandshake

HTTPrequesting

DSAverse

Floating Point and IEEE 754

Floating Point and IEEE 754