Average Error: 1.8 → 1.8
Time: 14.5s
Precision: 64
\[\frac{\left(\frac{\left(-b_2\right)}{\left(\sqrt{\left(\left(b_2 \cdot b_2\right) - \left(a \cdot c\right)\right)}\right)}\right)}{a}\]
↓
\[\frac{\sqrt{b_2 \cdot b_2 - c \cdot a} - b_2}{a}\]
\frac{\left(\frac{\left(-b_2\right)}{\left(\sqrt{\left(\left(b_2 \cdot b_2\right) - \left(a \cdot c\right)\right)}\right)}\right)}{a}
↓
\frac{\sqrt{b_2 \cdot b_2 - c \cdot a} - b_2}{a}
double f(double a, double b_2, double c) {
        double r899884 = b_2;
        double r899885 = -r899884;
        double r899886 = r899884 * r899884;
        double r899887 = a;
        double r899888 = c;
        double r899889 = r899887 * r899888;
        double r899890 = r899886 - r899889;
        double r899891 = sqrt(r899890);
        double r899892 = r899885 + r899891;
        double r899893 = r899892 / r899887;
        return r899893;
}
↓
double f(double a, double b_2, double c) {
        double r899894 = b_2;
        double r899895 = r899894 * r899894;
        double r899896 = c;
        double r899897 = a;
        double r899898 = r899896 * r899897;
        double r899899 = r899895 - r899898;
        double r899900 = sqrt(r899899);
        double r899901 = r899900 - r899894;
        double r899902 = r899901 / r899897;
        return r899902;
}

Error

Bits error versus a

Bits error versus b_2

Bits error versus c

Derivation

  1. Initial program 1.8

    \[\frac{\left(\frac{\left(-b_2\right)}{\left(\sqrt{\left(\left(b_2 \cdot b_2\right) - \left(a \cdot c\right)\right)}\right)}\right)}{a}\]

  2. Simplified1.8

    \[\leadsto \color{blue}{\frac{\left(\left(\sqrt{\left(\left(b_2 \cdot b_2\right) - \left(c \cdot a\right)\right)}\right) - b_2\right)}{a}}\]

  3. Final simplification1.8

    \[\leadsto \frac{\sqrt{b_2 \cdot b_2 - c \cdot a} - b_2}{a}\]

Reproduce

herbie shell --seed 2019165 
(FPCore (a b_2 c)
  :name "quad2p (problem 3.2.1, positive)"
  (/.p16 (+.p16 (neg.p16 b_2) (sqrt.p16 (-.p16 (*.p16 b_2 b_2) (*.p16 a c)))) a))