Admittedly it's kind of late for 1979, but I feel like this can be done better using exact differentiation. You don't need to take the symbolic derivative; just

Admittedly it's kind of late for 1979, but I feel like this can be done better using exact differentiation. You don't need to take the symbolic derivative; just use the dual numbers to evaluate the derivative at a single point. Then the secant method becomes Newton's method. Also, assuming that a second guess isn't provided, how does the calculator choose one? I'd imagine there are numerical precision problems that make choosing one hard. [edit] Wait, is it using this algorithm? "False position": https://en.wikipedia.org/wiki/Regula_falsi https://en.wikipedia.org/wiki/Regula_falsi . I thought it was using something primitive like the "secant method": https://en.wikipedia.org/wiki/Secant_method https://en.wikipedia.org/wiki/Secant_method . But the article suggests that it tries to be more clever than that.