Average Error: 1.6 → 1.7
Time: 31.3s
Precision: 64
\[\frac{\left(\left(-b\right) - \left(\sqrt{\left(\left(b \cdot b\right) - \left(\left(4\right) \cdot \left(a \cdot c\right)\right)\right)}\right)\right)}{\left(\left(2\right) \cdot a\right)}\]
↓
\[\frac{1.0}{\frac{2 \cdot a}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}\]
\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{1.0}{\frac{2 \cdot a}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}
double f(double a, double b, double c) {
        double r399973 = b;
        double r399974 = -r399973;
        double r399975 = r399973 * r399973;
        double r399976 = 4.0;
        double r399977 = /* ERROR: no posit support in C */;
        double r399978 = a;
        double r399979 = c;
        double r399980 = r399978 * r399979;
        double r399981 = r399977 * r399980;
        double r399982 = r399975 - r399981;
        double r399983 = sqrt(r399982);
        double r399984 = r399974 - r399983;
        double r399985 = 2.0;
        double r399986 = /* ERROR: no posit support in C */;
        double r399987 = r399986 * r399978;
        double r399988 = r399984 / r399987;
        return r399988;
}
↓
double f(double a, double b, double c) {
        double r399989 = 1.0;
        double r399990 = 2.0;
        double r399991 = a;
        double r399992 = r399990 * r399991;
        double r399993 = b;
        double r399994 = -r399993;
        double r399995 = r399993 * r399993;
        double r399996 = 4.0;
        double r399997 = r399996 * r399991;
        double r399998 = c;
        double r399999 = r399997 * r399998;
        double r400000 = r399995 - r399999;
        double r400001 = sqrt(r400000);
        double r400002 = r399994 - r400001;
        double r400003 = r399992 / r400002;
        double r400004 = r399989 / r400003;
        return r400004;
}

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*1.6

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

  5. Applied p16-*-un-lft-identity1.6

    \[\leadsto \frac{\left(\left(-b\right) - \color{blue}{\left(\left(1.0\right) \cdot \left(\sqrt{\left(\left(b \cdot b\right) - \left(\left(\left(4\right) \cdot a\right) \cdot c\right)\right)}\right)\right)}\right)}{\left(\left(2\right) \cdot a\right)}\]

  6. Applied p16-*-un-lft-identity1.6

    \[\leadsto \frac{\left(\color{blue}{\left(\left(1.0\right) \cdot \left(-b\right)\right)} - \left(\left(1.0\right) \cdot \left(\sqrt{\left(\left(b \cdot b\right) - \left(\left(\left(4\right) \cdot a\right) \cdot c\right)\right)}\right)\right)\right)}{\left(\left(2\right) \cdot a\right)}\]

  7. Applied distribute-lft-out--1.6

    \[\leadsto \frac{\color{blue}{\left(\left(1.0\right) \cdot \left(\left(-b\right) - \left(\sqrt{\left(\left(b \cdot b\right) - \left(\left(\left(4\right) \cdot a\right) \cdot c\right)\right)}\right)\right)\right)}}{\left(\left(2\right) \cdot a\right)}\]

  8. Applied associate-/l*1.7

    \[\leadsto \color{blue}{\frac{\left(1.0\right)}{\left(\frac{\left(\left(2\right) \cdot a\right)}{\left(\left(-b\right) - \left(\sqrt{\left(\left(b \cdot b\right) - \left(\left(\left(4\right) \cdot a\right) \cdot c\right)\right)}\right)\right)}\right)}}\]

  9. Final simplification1.7

    \[\leadsto \frac{1.0}{\frac{2 \cdot a}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}\]

Reproduce

herbie shell --seed 2019156 +o rules:numerics
(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)))