Average Error: 1.5 → 1.5
Time: 24.0s
Precision: 64
\[\frac{\left(\frac{\left(-b\right)}{\left(\sqrt{\left(\left(b \cdot b\right) - \left(\left(4\right) \cdot \left(a \cdot c\right)\right)\right)}\right)}\right)}{\left(\left(2\right) \cdot a\right)}\]
↓
\[\frac{\frac{\sqrt{b \cdot b - \left(c \cdot a\right) \cdot 4} - b}{2}}{a}\]
\frac{\left(\frac{\left(-b\right)}{\left(\sqrt{\left(\left(b \cdot b\right) - \left(\left(4\right) \cdot \left(a \cdot c\right)\right)\right)}\right)}\right)}{\left(\left(2\right) \cdot a\right)}
↓
\frac{\frac{\sqrt{b \cdot b - \left(c \cdot a\right) \cdot 4} - b}{2}}{a}
double f(double a, double b, double c) {
        double r1548888 = b;
        double r1548889 = -r1548888;
        double r1548890 = r1548888 * r1548888;
        double r1548891 = 4.0;
        double r1548892 = /* ERROR: no posit support in C */;
        double r1548893 = a;
        double r1548894 = c;
        double r1548895 = r1548893 * r1548894;
        double r1548896 = r1548892 * r1548895;
        double r1548897 = r1548890 - r1548896;
        double r1548898 = sqrt(r1548897);
        double r1548899 = r1548889 + r1548898;
        double r1548900 = 2.0;
        double r1548901 = /* ERROR: no posit support in C */;
        double r1548902 = r1548901 * r1548893;
        double r1548903 = r1548899 / r1548902;
        return r1548903;
}
↓
double f(double a, double b, double c) {
        double r1548904 = b;
        double r1548905 = r1548904 * r1548904;
        double r1548906 = c;
        double r1548907 = a;
        double r1548908 = r1548906 * r1548907;
        double r1548909 = 4.0;
        double r1548910 = r1548908 * r1548909;
        double r1548911 = r1548905 - r1548910;
        double r1548912 = sqrt(r1548911);
        double r1548913 = r1548912 - r1548904;
        double r1548914 = 2.0;
        double r1548915 = r1548913 / r1548914;
        double r1548916 = r1548915 / r1548907;
        return r1548916;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

  1. Initial program 1.5

    \[\frac{\left(\frac{\left(-b\right)}{\left(\sqrt{\left(\left(b \cdot b\right) - \left(\left(4\right) \cdot \left(a \cdot c\right)\right)\right)}\right)}\right)}{\left(\left(2\right) \cdot a\right)}\]

  2. Simplified1.5

    \[\leadsto \color{blue}{\frac{\left(\left(\sqrt{\left(\left(b \cdot b\right) - \left(\left(c \cdot a\right) \cdot \left(4\right)\right)\right)}\right) - b\right)}{\left(\left(2\right) \cdot a\right)}}\]
  3. Using strategy rm

  4. Applied associate-/r*1.5

    \[\leadsto \color{blue}{\frac{\left(\frac{\left(\left(\sqrt{\left(\left(b \cdot b\right) - \left(\left(c \cdot a\right) \cdot \left(4\right)\right)\right)}\right) - b\right)}{\left(2\right)}\right)}{a}}\]

  5. Final simplification1.5

    \[\leadsto \frac{\frac{\sqrt{b \cdot b - \left(c \cdot a\right) \cdot 4} - b}{2}}{a}\]

Reproduce

herbie shell --seed 2019130 +o rules:numerics
(FPCore (a b c)
  :name "quadp (p42, positive)"
  (/.p16 (+.p16 (neg.p16 b) (sqrt.p16 (-.p16 (*.p16 b b) (*.p16 (real->posit16 4) (*.p16 a c))))) (*.p16 (real->posit16 2) a)))