Average Error: 1.7 → 1.7
Time: 17.4s
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 r867898 = b_2;
        double r867899 = -r867898;
        double r867900 = r867898 * r867898;
        double r867901 = a;
        double r867902 = c;
        double r867903 = r867901 * r867902;
        double r867904 = r867900 - r867903;
        double r867905 = sqrt(r867904);
        double r867906 = r867899 + r867905;
        double r867907 = r867906 / r867901;
        return r867907;
}
↓
double f(double a, double b_2, double c) {
        double r867908 = b_2;
        double r867909 = r867908 * r867908;
        double r867910 = c;
        double r867911 = a;
        double r867912 = r867910 * r867911;
        double r867913 = r867909 - r867912;
        double r867914 = sqrt(r867913);
        double r867915 = r867914 - r867908;
        double r867916 = r867915 / r867911;
        return r867916;
}

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 2019128 +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))