I'm looking for a way to measure temperatures in range of 40 to -200 °C (100 to −320 °F). The usual DS18B20 digital thermometer works only down to -55 °C (-67 °F).
A precision of 2 °C would be good enough.
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Answer is simple: thermocouple.
Thermocouple type K (chromel - alumel) works from -200°C and has very linear characteristics. It's very popular and cheap, many multimeters use it as a temperature sensor. Type T will be better for such low temperatures, but it is hard to buy and not as cheap as K.
Another solution is a platinum resistive sensor, like Pt100, but it is not as linear as thermocouple.