Average Error: 1.7 → 1.7
Time: 1.7m
Precision: 64
\[\frac{\left(\left(-b_2\right) - \left(\sqrt{\left(\left(b_2 \cdot b_2\right) - \left(a \cdot c\right)\right)}\right)\right)}{a}\]
↓
\[\frac{\left(-b_2\right) - \sqrt{\left(\mathsf{qms}\left(\left(\left(b_2 \cdot b_2\right)\right), a, c\right)\right)}}{a}\]
\frac{\left(\left(-b_2\right) - \left(\sqrt{\left(\left(b_2 \cdot b_2\right) - \left(a \cdot c\right)\right)}\right)\right)}{a}
↓
\frac{\left(-b_2\right) - \sqrt{\left(\mathsf{qms}\left(\left(\left(b_2 \cdot b_2\right)\right), a, c\right)\right)}}{a}
double f(double a, double b_2, double c) {
        double r1380785 = b_2;
        double r1380786 = -r1380785;
        double r1380787 = r1380785 * r1380785;
        double r1380788 = a;
        double r1380789 = c;
        double r1380790 = r1380788 * r1380789;
        double r1380791 = r1380787 - r1380790;
        double r1380792 = sqrt(r1380791);
        double r1380793 = r1380786 - r1380792;
        double r1380794 = r1380793 / r1380788;
        return r1380794;
}
↓
double f(double a, double b_2, double c) {
        double r1380795 = b_2;
        double r1380796 = -r1380795;
        double r1380797 = r1380795 * r1380795;
        double r1380798 = /*Error: no posit support in C */;
        double r1380799 = a;
        double r1380800 = c;
        double r1380801 = /*Error: no posit support in C */;
        double r1380802 = /*Error: no posit support in C */;
        double r1380803 = sqrt(r1380802);
        double r1380804 = r1380796 - r1380803;
        double r1380805 = r1380804 / r1380799;
        return r1380805;
}

Error

Bits error versus a

Bits error versus b_2

Bits error versus c

Derivation

  1. Initial program 1.7

    \[\frac{\left(\left(-b_2\right) - \left(\sqrt{\left(\left(b_2 \cdot b_2\right) - \left(a \cdot c\right)\right)}\right)\right)}{a}\]
  2. Using strategy rm

  3. Applied introduce-quire1.7

    \[\leadsto \frac{\left(\left(-b_2\right) - \left(\sqrt{\left(\color{blue}{\left(\left(\left(b_2 \cdot b_2\right)\right)\right)} - \left(a \cdot c\right)\right)}\right)\right)}{a}\]

  4. Applied insert-quire-fdp-sub1.7

    \[\leadsto \frac{\left(\left(-b_2\right) - \left(\sqrt{\color{blue}{\left(\left(\mathsf{qms}\left(\left(\left(b_2 \cdot b_2\right)\right), a, c\right)\right)\right)}}\right)\right)}{a}\]

  5. Final simplification1.7

    \[\leadsto \frac{\left(-b_2\right) - \sqrt{\left(\mathsf{qms}\left(\left(\left(b_2 \cdot b_2\right)\right), a, c\right)\right)}}{a}\]

Reproduce

herbie shell --seed 2019144 +o rules:numerics
(FPCore (a b_2 c)
  :name "quad2m (problem 3.2.1, negative)"
  (/.p16 (-.p16 (neg.p16 b_2) (sqrt.p16 (-.p16 (*.p16 b_2 b_2) (*.p16 a c)))) a))