Average Error: 1.7 → 1.7
Time: 13.4s
Precision: 64
\[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
↓
\[\frac{\sqrt{b_2 \cdot b_2 - c \cdot a} - b_2}{a}\]
double f(double a, double b_2, double c) {
        double r835918 = b_2;
        double r835919 = -r835918;
        double r835920 = r835918 * r835918;
        double r835921 = a;
        double r835922 = c;
        double r835923 = r835921 * r835922;
        double r835924 = r835920 - r835923;
        double r835925 = sqrt(r835924);
        double r835926 = r835919 + r835925;
        double r835927 = r835926 / r835921;
        return r835927;
}
↓
double f(double a, double b_2, double c) {
        double r835928 = b_2;
        double r835929 = r835928 * r835928;
        double r835930 = c;
        double r835931 = a;
        double r835932 = r835930 * r835931;
        double r835933 = r835929 - r835932;
        double r835934 = sqrt(r835933);
        double r835935 = r835934 - r835928;
        double r835936 = r835935 / r835931;
        return r835936;
}


\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}
↓
\frac{\sqrt{b_2 \cdot b_2 - c \cdot a} - b_2}{a}

Error

Bits error versus a

Bits error versus b_2

Bits error versus c

Derivation

  1. Initial program 1.7

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

  2. Simplified1.7

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

  3. Final simplification1.7

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

Reproduce

herbie shell --seed 2019101 +o rules:numerics
(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))