Average Error: 1.7 → 1.7
Time: 15.4s
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 r989245 = b_2;
        double r989246 = -r989245;
        double r989247 = r989245 * r989245;
        double r989248 = a;
        double r989249 = c;
        double r989250 = r989248 * r989249;
        double r989251 = r989247 - r989250;
        double r989252 = sqrt(r989251);
        double r989253 = r989246 + r989252;
        double r989254 = r989253 / r989248;
        return r989254;
}
↓
double f(double a, double b_2, double c) {
        double r989255 = b_2;
        double r989256 = r989255 * r989255;
        double r989257 = c;
        double r989258 = a;
        double r989259 = r989257 * r989258;
        double r989260 = r989256 - r989259;
        double r989261 = sqrt(r989260);
        double r989262 = r989261 - r989255;
        double r989263 = r989262 / r989258;
        return r989263;
}

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 2019107 +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))