Average Error: 1.7 → 1.7
Time: 21.4s
Precision: 64
\[\frac{\left(\left(-b_2\right) - \left(\sqrt{\left(\left(b_2 \cdot b_2\right) - \left(a \cdot c\right)\right)}\right)\right)}{a}\]
↓
\[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
\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{b_2 \cdot b_2 - a \cdot c}}{a}
double f(double a, double b_2, double c) {
        double r1671479 = b_2;
        double r1671480 = -r1671479;
        double r1671481 = r1671479 * r1671479;
        double r1671482 = a;
        double r1671483 = c;
        double r1671484 = r1671482 * r1671483;
        double r1671485 = r1671481 - r1671484;
        double r1671486 = sqrt(r1671485);
        double r1671487 = r1671480 - r1671486;
        double r1671488 = r1671487 / r1671482;
        return r1671488;
}
↓
double f(double a, double b_2, double c) {
        double r1671489 = b_2;
        double r1671490 = -r1671489;
        double r1671491 = r1671489 * r1671489;
        double r1671492 = a;
        double r1671493 = c;
        double r1671494 = r1671492 * r1671493;
        double r1671495 = r1671491 - r1671494;
        double r1671496 = sqrt(r1671495);
        double r1671497 = r1671490 - r1671496;
        double r1671498 = r1671497 / r1671492;
        return r1671498;
}

Error

Bits error versus a

Bits error versus b_2

Bits error versus c

Derivation

  1. Initial program 1.7

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

  2. Final simplification1.7

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

Reproduce

herbie shell --seed 2019132 +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))