Average Error: 1.7 → 1.7
Time: 24.3s
Precision: 64
\[\frac{\left(\frac{\left(-b_2\right)}{\left(\sqrt{\left(\left(b_2 \cdot b_2\right) - \left(a \cdot c\right)\right)}\right)}\right)}{a}\]
↓
\[\frac{\sqrt{\left(\mathsf{qms}\left(\left(\left(b_2 \cdot b_2\right)\right), c, a\right)\right)} - b_2}{a}\]
\frac{\left(\frac{\left(-b_2\right)}{\left(\sqrt{\left(\left(b_2 \cdot b_2\right) - \left(a \cdot c\right)\right)}\right)}\right)}{a}
↓
\frac{\sqrt{\left(\mathsf{qms}\left(\left(\left(b_2 \cdot b_2\right)\right), c, a\right)\right)} - b_2}{a}
double f(double a, double b_2, double c) {
        double r890277 = b_2;
        double r890278 = -r890277;
        double r890279 = r890277 * r890277;
        double r890280 = a;
        double r890281 = c;
        double r890282 = r890280 * r890281;
        double r890283 = r890279 - r890282;
        double r890284 = sqrt(r890283);
        double r890285 = r890278 + r890284;
        double r890286 = r890285 / r890280;
        return r890286;
}
↓
double f(double a, double b_2, double c) {
        double r890287 = b_2;
        double r890288 = r890287 * r890287;
        double r890289 = /*Error: no posit support in C */;
        double r890290 = c;
        double r890291 = a;
        double r890292 = /*Error: no posit support in C */;
        double r890293 = /*Error: no posit support in C */;
        double r890294 = sqrt(r890293);
        double r890295 = r890294 - r890287;
        double r890296 = r890295 / r890291;
        return r890296;
}

Error

Bits error versus a

Bits error versus b_2

Bits error versus c

Derivation

  1. Initial program 1.7

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

  2. Simplified1.7

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

  4. Applied introduce-quire1.7

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

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

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

  6. Final simplification1.7

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

Reproduce

herbie shell --seed 2019158 +o rules:numerics
(FPCore (a b_2 c)
  :name "quad2p (problem 3.2.1, positive)"
  (/.p16 (+.p16 (neg.p16 b_2) (sqrt.p16 (-.p16 (*.p16 b_2 b_2) (*.p16 a c)))) a))