Average Error: 1.7 → 1.7
Time: 16.0s
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 r989240 = b_2;
        double r989241 = -r989240;
        double r989242 = r989240 * r989240;
        double r989243 = a;
        double r989244 = c;
        double r989245 = r989243 * r989244;
        double r989246 = r989242 - r989245;
        double r989247 = sqrt(r989246);
        double r989248 = r989241 + r989247;
        double r989249 = r989248 / r989243;
        return r989249;
}
↓
double f(double a, double b_2, double c) {
        double r989250 = b_2;
        double r989251 = r989250 * r989250;
        double r989252 = c;
        double r989253 = a;
        double r989254 = r989252 * r989253;
        double r989255 = r989251 - r989254;
        double r989256 = sqrt(r989255);
        double r989257 = r989256 - r989250;
        double r989258 = r989257 / r989253;
        return r989258;
}

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))