Average Error: 1.7 → 1.7
Time: 14.1s
Precision: 64
\[\frac{\left(\frac{\left(-b_2\right)}{\left(\sqrt{\left(\left(b_2 \cdot b_2\right) - \left(a \cdot c\right)\right)}\right)}\right)}{a}\]
↓
\[\frac{\sqrt{b_2 \cdot b_2 - c \cdot a} - b_2}{a}\]
\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{b_2 \cdot b_2 - c \cdot a} - b_2}{a}
double f(double a, double b_2, double c) {
        double r851625 = b_2;
        double r851626 = -r851625;
        double r851627 = r851625 * r851625;
        double r851628 = a;
        double r851629 = c;
        double r851630 = r851628 * r851629;
        double r851631 = r851627 - r851630;
        double r851632 = sqrt(r851631);
        double r851633 = r851626 + r851632;
        double r851634 = r851633 / r851628;
        return r851634;
}
↓
double f(double a, double b_2, double c) {
        double r851635 = b_2;
        double r851636 = r851635 * r851635;
        double r851637 = c;
        double r851638 = a;
        double r851639 = r851637 * r851638;
        double r851640 = r851636 - r851639;
        double r851641 = sqrt(r851640);
        double r851642 = r851641 - r851635;
        double r851643 = r851642 / r851638;
        return r851643;
}

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. Final simplification1.7

    \[\leadsto \frac{\sqrt{b_2 \cdot b_2 - c \cdot a} - b_2}{a}\]

Reproduce

herbie shell --seed 2019112 +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))