Average Error: 1.7 → 1.7
Time: 17.8s
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 r285330 = b_2;
        double r285331 = -r285330;
        double r285332 = r285330 * r285330;
        double r285333 = a;
        double r285334 = c;
        double r285335 = r285333 * r285334;
        double r285336 = r285332 - r285335;
        double r285337 = sqrt(r285336);
        double r285338 = r285331 + r285337;
        double r285339 = r285338 / r285333;
        return r285339;
}
↓
double f(double a, double b_2, double c) {
        double r285340 = b_2;
        double r285341 = r285340 * r285340;
        double r285342 = c;
        double r285343 = a;
        double r285344 = r285342 * r285343;
        double r285345 = r285341 - r285344;
        double r285346 = sqrt(r285345);
        double r285347 = r285346 - r285340;
        double r285348 = r285347 / r285343;
        return r285348;
}

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 2019154 
(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))