Average Error: 1.6 → 1.6
Time: 20.7s
Precision: 64
\[\frac{\left(\frac{\left(-b\right)}{\left(\sqrt{\left(\left(b \cdot b\right) - \left(\left(4\right) \cdot \left(a \cdot c\right)\right)\right)}\right)}\right)}{\left(\left(2\right) \cdot a\right)}\]
↓
\[\frac{\frac{\sqrt{b \cdot b - \left(c \cdot a\right) \cdot 4} - b}{2}}{a}\]
\frac{\left(\frac{\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{\frac{\sqrt{b \cdot b - \left(c \cdot a\right) \cdot 4} - b}{2}}{a}
double f(double a, double b, double c) {
        double r732468 = b;
        double r732469 = -r732468;
        double r732470 = r732468 * r732468;
        double r732471 = 4.0;
        double r732472 = /* ERROR: no posit support in C */;
        double r732473 = a;
        double r732474 = c;
        double r732475 = r732473 * r732474;
        double r732476 = r732472 * r732475;
        double r732477 = r732470 - r732476;
        double r732478 = sqrt(r732477);
        double r732479 = r732469 + r732478;
        double r732480 = 2.0;
        double r732481 = /* ERROR: no posit support in C */;
        double r732482 = r732481 * r732473;
        double r732483 = r732479 / r732482;
        return r732483;
}
↓
double f(double a, double b, double c) {
        double r732484 = b;
        double r732485 = r732484 * r732484;
        double r732486 = c;
        double r732487 = a;
        double r732488 = r732486 * r732487;
        double r732489 = 4.0;
        double r732490 = r732488 * r732489;
        double r732491 = r732485 - r732490;
        double r732492 = sqrt(r732491);
        double r732493 = r732492 - r732484;
        double r732494 = 2.0;
        double r732495 = r732493 / r732494;
        double r732496 = r732495 / r732487;
        return r732496;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

  1. Initial program 1.6

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

  2. Simplified1.6

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

  4. Applied associate-/r*1.6

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

  5. Final simplification1.6

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

Reproduce

herbie shell --seed 2019107 +o rules:numerics
(FPCore (a b c)
  :name "quadp (p42, positive)"
  (/.p16 (+.p16 (neg.p16 b) (sqrt.p16 (-.p16 (*.p16 b b) (*.p16 (real->posit16 4) (*.p16 a c))))) (*.p16 (real->posit16 2) a)))