Average Error: 1.7 → 1.7
Time: 15.7s
Precision: 64
\[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
↓
\[\frac{\sqrt{b_2 \cdot b_2 - c \cdot a} - b_2}{a}\]
double f(double a, double b_2, double c) {
        double r833206 = b_2;
        double r833207 = -r833206;
        double r833208 = r833206 * r833206;
        double r833209 = a;
        double r833210 = c;
        double r833211 = r833209 * r833210;
        double r833212 = r833208 - r833211;
        double r833213 = sqrt(r833212);
        double r833214 = r833207 + r833213;
        double r833215 = r833214 / r833209;
        return r833215;
}
↓
double f(double a, double b_2, double c) {
        double r833216 = b_2;
        double r833217 = r833216 * r833216;
        double r833218 = c;
        double r833219 = a;
        double r833220 = r833218 * r833219;
        double r833221 = r833217 - r833220;
        double r833222 = sqrt(r833221);
        double r833223 = r833222 - r833216;
        double r833224 = r833223 / r833219;
        return r833224;
}


\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}
↓
\frac{\sqrt{b_2 \cdot b_2 - c \cdot a} - b_2}{a}

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 2019102 +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))