Average Error: 1.6 → 1.6
Time: 20.5s
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 r732473 = b;
        double r732474 = -r732473;
        double r732475 = r732473 * r732473;
        double r732476 = 4.0;
        double r732477 = /* ERROR: no posit support in C */;
        double r732478 = a;
        double r732479 = c;
        double r732480 = r732478 * r732479;
        double r732481 = r732477 * r732480;
        double r732482 = r732475 - r732481;
        double r732483 = sqrt(r732482);
        double r732484 = r732474 + r732483;
        double r732485 = 2.0;
        double r732486 = /* ERROR: no posit support in C */;
        double r732487 = r732486 * r732478;
        double r732488 = r732484 / r732487;
        return r732488;
}
↓
double f(double a, double b, double c) {
        double r732489 = b;
        double r732490 = r732489 * r732489;
        double r732491 = c;
        double r732492 = a;
        double r732493 = r732491 * r732492;
        double r732494 = 4.0;
        double r732495 = r732493 * r732494;
        double r732496 = r732490 - r732495;
        double r732497 = sqrt(r732496);
        double r732498 = r732497 - r732489;
        double r732499 = 2.0;
        double r732500 = r732498 / r732499;
        double r732501 = r732500 / r732492;
        return r732501;
}

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)))