Average Error: 1.6 → 1.5
Time: 30.6s
Precision: 64
\[\frac{\left(\left(-b\right) - \left(\sqrt{\left(\left(b \cdot b\right) - \left(\left(4\right) \cdot \left(a \cdot c\right)\right)\right)}\right)\right)}{\left(\left(2\right) \cdot a\right)}\]
↓
\[\frac{\left(\left(-b\right) - \left(\sqrt{\left(\left(\mathsf{qms}\left(\left(\left(b \cdot b\right)\right), \left(\left(4\right) \cdot a\right), c\right)\right)\right)}\right)\right)}{\left(\left(2\right) \cdot a\right)}\]
\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{\left(\left(-b\right) - \left(\sqrt{\left(\left(\mathsf{qms}\left(\left(\left(b \cdot b\right)\right), \left(\left(4\right) \cdot a\right), c\right)\right)\right)}\right)\right)}{\left(\left(2\right) \cdot a\right)}
double f(double a, double b, double c) {
        double r1121662 = b;
        double r1121663 = -r1121662;
        double r1121664 = r1121662 * r1121662;
        double r1121665 = 4.0;
        double r1121666 = /* ERROR: no posit support in C */;
        double r1121667 = a;
        double r1121668 = c;
        double r1121669 = r1121667 * r1121668;
        double r1121670 = r1121666 * r1121669;
        double r1121671 = r1121664 - r1121670;
        double r1121672 = sqrt(r1121671);
        double r1121673 = r1121663 - r1121672;
        double r1121674 = 2.0;
        double r1121675 = /* ERROR: no posit support in C */;
        double r1121676 = r1121675 * r1121667;
        double r1121677 = r1121673 / r1121676;
        return r1121677;
}
↓
double f(double a, double b, double c) {
        double r1121678 = b;
        double r1121679 = -r1121678;
        double r1121680 = r1121678 * r1121678;
        double r1121681 = /*Error: no posit support in C */;
        double r1121682 = 4.0;
        double r1121683 = /* ERROR: no posit support in C */;
        double r1121684 = a;
        double r1121685 = r1121683 * r1121684;
        double r1121686 = c;
        double r1121687 = /*Error: no posit support in C */;
        double r1121688 = /*Error: no posit support in C */;
        double r1121689 = sqrt(r1121688);
        double r1121690 = r1121679 - r1121689;
        double r1121691 = 2.0;
        double r1121692 = /* ERROR: no posit support in C */;
        double r1121693 = r1121692 * r1121684;
        double r1121694 = r1121690 / r1121693;
        return r1121694;
}

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 -p16-rgt-identity-expand1.6

    \[\leadsto \frac{\left(\color{blue}{\left(\left(-b\right) - \left(0.0\right)\right)} - \left(\sqrt{\left(\left(b \cdot b\right) - \left(\left(4\right) \cdot \left(a \cdot c\right)\right)\right)}\right)\right)}{\left(\left(2\right) \cdot a\right)}\]

  4. Applied associate--l-1.6

    \[\leadsto \frac{\color{blue}{\left(\left(-b\right) - \left(\frac{\left(0.0\right)}{\left(\sqrt{\left(\left(b \cdot b\right) - \left(\left(4\right) \cdot \left(a \cdot c\right)\right)\right)}\right)}\right)\right)}}{\left(\left(2\right) \cdot a\right)}\]

  5. Simplified1.5

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

  7. Applied introduce-quire1.5

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

  8. Applied insert-quire-fdp-sub1.5

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

  9. Final simplification1.5

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

Reproduce

herbie shell --seed 2019163 +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)))