Average Error: 1.6 → 1.5
Time: 25.6s
Precision: 64
\[\frac{\left(\left(-b\right) - \left(\sqrt{\left(\left(b \cdot b\right) - \left(\left(4\right) \cdot \left(a \cdot c\right)\right)\right)}\right)\right)}{\left(\left(2\right) \cdot a\right)}\]
↓
\[\frac{\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2}}{a}\]
\frac{\left(\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{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2}}{a}
double f(double a, double b, double c) {
        double r1329869 = b;
        double r1329870 = -r1329869;
        double r1329871 = r1329869 * r1329869;
        double r1329872 = 4.0;
        double r1329873 = /* ERROR: no posit support in C */;
        double r1329874 = a;
        double r1329875 = c;
        double r1329876 = r1329874 * r1329875;
        double r1329877 = r1329873 * r1329876;
        double r1329878 = r1329871 - r1329877;
        double r1329879 = sqrt(r1329878);
        double r1329880 = r1329870 - r1329879;
        double r1329881 = 2.0;
        double r1329882 = /* ERROR: no posit support in C */;
        double r1329883 = r1329882 * r1329874;
        double r1329884 = r1329880 / r1329883;
        return r1329884;
}
↓
double f(double a, double b, double c) {
        double r1329885 = b;
        double r1329886 = -r1329885;
        double r1329887 = r1329885 * r1329885;
        double r1329888 = 4.0;
        double r1329889 = a;
        double r1329890 = c;
        double r1329891 = r1329889 * r1329890;
        double r1329892 = r1329888 * r1329891;
        double r1329893 = r1329887 - r1329892;
        double r1329894 = sqrt(r1329893);
        double r1329895 = r1329886 - r1329894;
        double r1329896 = 2.0;
        double r1329897 = r1329895 / r1329896;
        double r1329898 = r1329897 / r1329889;
        return r1329898;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

  1. Initial program 1.6

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

  3. Applied associate-/r*1.5

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

  4. Final simplification1.5

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

Reproduce

herbie shell --seed 2019128 
(FPCore (a b c)
  :name "quadm (p42, negative)"
  (/.p16 (-.p16 (neg.p16 b) (sqrt.p16 (-.p16 (*.p16 b b) (*.p16 (real->posit16 4) (*.p16 a c))))) (*.p16 (real->posit16 2) a)))