Average Error: 1.6 → 1.6
Time: 27.2s
Precision: 64
\[\frac{\left(\frac{\left(-b\right)}{\left(\sqrt{\left(\left(b \cdot b\right) - \left(\left(4\right) \cdot \left(a \cdot c\right)\right)\right)}\right)}\right)}{\left(\left(2\right) \cdot a\right)}\]
↓
\[\frac{\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{2}}{a}\]
\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 - c \cdot \left(a \cdot 4\right)} - b}{2}}{a}
double f(double a, double b, double c) {
        double r1741495 = b;
        double r1741496 = -r1741495;
        double r1741497 = r1741495 * r1741495;
        double r1741498 = 4.0;
        double r1741499 = /* ERROR: no posit support in C */;
        double r1741500 = a;
        double r1741501 = c;
        double r1741502 = r1741500 * r1741501;
        double r1741503 = r1741499 * r1741502;
        double r1741504 = r1741497 - r1741503;
        double r1741505 = sqrt(r1741504);
        double r1741506 = r1741496 + r1741505;
        double r1741507 = 2.0;
        double r1741508 = /* ERROR: no posit support in C */;
        double r1741509 = r1741508 * r1741500;
        double r1741510 = r1741506 / r1741509;
        return r1741510;
}
↓
double f(double a, double b, double c) {
        double r1741511 = b;
        double r1741512 = r1741511 * r1741511;
        double r1741513 = c;
        double r1741514 = a;
        double r1741515 = 4.0;
        double r1741516 = r1741514 * r1741515;
        double r1741517 = r1741513 * r1741516;
        double r1741518 = r1741512 - r1741517;
        double r1741519 = sqrt(r1741518);
        double r1741520 = r1741519 - r1741511;
        double r1741521 = 2.0;
        double r1741522 = r1741520 / r1741521;
        double r1741523 = r1741522 / r1741514;
        return r1741523;
}

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. Using strategy rm

  6. Applied associate-*l*1.6

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

  7. Final simplification1.6

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

Reproduce

herbie shell --seed 2019135 +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)))