Average Error: 1.8 → 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 r1677453 = b_2;
        double r1677454 = -r1677453;
        double r1677455 = r1677453 * r1677453;
        double r1677456 = a;
        double r1677457 = c;
        double r1677458 = r1677456 * r1677457;
        double r1677459 = r1677455 - r1677458;
        double r1677460 = sqrt(r1677459);
        double r1677461 = r1677454 - r1677460;
        double r1677462 = r1677461 / r1677456;
        return r1677462;
}
↓
double f(double a, double b_2, double c) {
        double r1677463 = b_2;
        double r1677464 = -r1677463;
        double r1677465 = r1677463 * r1677463;
        double r1677466 = /*Error: no posit support in C */;
        double r1677467 = a;
        double r1677468 = c;
        double r1677469 = /*Error: no posit support in C */;
        double r1677470 = /*Error: no posit support in C */;
        double r1677471 = sqrt(r1677470);
        double r1677472 = r1677464 - r1677471;
        double r1677473 = r1677472 / r1677467;
        return r1677473;
}

Error

Bits error versus a

Bits error versus b_2

Bits error versus c

Derivation

  1. Initial program 1.8

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

  3. Applied introduce-quire1.8

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

  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 2019146 +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))