Average Error: 1.6 → 0.4
Time: 15.2s
Precision: 64
\[\frac{\left(\left(-b_2\right) - \left(\sqrt{\left(\left(b_2 \cdot b_2\right) - \left(a \cdot c\right)\right)}\right)\right)}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b_2 \le 0.0343017578125:\\ \;\;\;\;\frac{\frac{a \cdot c + \left(\left(-b_2\right) + b_2\right) \cdot \left(\left(-b_2\right) + \left(-b_2\right)\right)}{a}}{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\\ \end{array}\]
\frac{\left(\left(-b_2\right) - \left(\sqrt{\left(\left(b_2 \cdot b_2\right) - \left(a \cdot c\right)\right)}\right)\right)}{a}
↓
\begin{array}{l}
\mathbf{if}\;b_2 \le 0.0343017578125:\\
\;\;\;\;\frac{\frac{a \cdot c + \left(\left(-b_2\right) + b_2\right) \cdot \left(\left(-b_2\right) + \left(-b_2\right)\right)}{a}}{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}\\

\mathbf{else}:\\
\;\;\;\;\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\\

\end{array}
double f(double a, double b_2, double c) {
        double r628768 = b_2;
        double r628769 = -r628768;
        double r628770 = r628768 * r628768;
        double r628771 = a;
        double r628772 = c;
        double r628773 = r628771 * r628772;
        double r628774 = r628770 - r628773;
        double r628775 = sqrt(r628774);
        double r628776 = r628769 - r628775;
        double r628777 = r628776 / r628771;
        return r628777;
}
↓
double f(double a, double b_2, double c) {
        double r628778 = b_2;
        double r628779 = 0.0343017578125;
        bool r628780 = r628778 <= r628779;
        double r628781 = a;
        double r628782 = c;
        double r628783 = r628781 * r628782;
        double r628784 = -r628778;
        double r628785 = r628784 + r628778;
        double r628786 = r628784 + r628784;
        double r628787 = r628785 * r628786;
        double r628788 = r628783 + r628787;
        double r628789 = r628788 / r628781;
        double r628790 = r628778 * r628778;
        double r628791 = r628790 - r628783;
        double r628792 = sqrt(r628791);
        double r628793 = r628784 + r628792;
        double r628794 = r628789 / r628793;
        double r628795 = r628784 - r628792;
        double r628796 = r628795 / r628781;
        double r628797 = r628780 ? r628794 : r628796;
        return r628797;
}

Error

Bits error versus a

Bits error versus b_2

Bits error versus c

Derivation

  1. Split input into 2 regimes

  2. if b_2 < 0.0343017578125

    1. Initial program 2.7

      \[\frac{\left(\left(-b_2\right) - \left(\sqrt{\left(\left(b_2 \cdot b_2\right) - \left(a \cdot c\right)\right)}\right)\right)}{a}\]
    2. Using strategy rm

    3. Applied p16-flip--2.4

      \[\leadsto \frac{\color{blue}{\left(\frac{\left(\left(\left(-b_2\right) \cdot \left(-b_2\right)\right) - \left(\left(\sqrt{\left(\left(b_2 \cdot b_2\right) - \left(a \cdot c\right)\right)}\right) \cdot \left(\sqrt{\left(\left(b_2 \cdot b_2\right) - \left(a \cdot c\right)\right)}\right)\right)\right)}{\left(\frac{\left(-b_2\right)}{\left(\sqrt{\left(\left(b_2 \cdot b_2\right) - \left(a \cdot c\right)\right)}\right)}\right)}\right)}}{a}\]

    4. Applied associate-/l/2.5

      \[\leadsto \color{blue}{\frac{\left(\left(\left(-b_2\right) \cdot \left(-b_2\right)\right) - \left(\left(\sqrt{\left(\left(b_2 \cdot b_2\right) - \left(a \cdot c\right)\right)}\right) \cdot \left(\sqrt{\left(\left(b_2 \cdot b_2\right) - \left(a \cdot c\right)\right)}\right)\right)\right)}{\left(a \cdot \left(\frac{\left(-b_2\right)}{\left(\sqrt{\left(\left(b_2 \cdot b_2\right) - \left(a \cdot c\right)\right)}\right)}\right)\right)}}\]

    5. Simplified0.6

      \[\leadsto \frac{\color{blue}{\left(\frac{\left(a \cdot c\right)}{\left(\left(\frac{\left(-b_2\right)}{b_2}\right) \cdot \left(\frac{\left(-b_2\right)}{\left(-b_2\right)}\right)\right)}\right)}}{\left(a \cdot \left(\frac{\left(-b_2\right)}{\left(\sqrt{\left(\left(b_2 \cdot b_2\right) - \left(a \cdot c\right)\right)}\right)}\right)\right)}\]
    6. Using strategy rm

    7. Applied associate-/r*0.5

      \[\leadsto \color{blue}{\frac{\left(\frac{\left(\frac{\left(a \cdot c\right)}{\left(\left(\frac{\left(-b_2\right)}{b_2}\right) \cdot \left(\frac{\left(-b_2\right)}{\left(-b_2\right)}\right)\right)}\right)}{a}\right)}{\left(\frac{\left(-b_2\right)}{\left(\sqrt{\left(\left(b_2 \cdot b_2\right) - \left(a \cdot c\right)\right)}\right)}\right)}}\]

    if 0.0343017578125 < b_2

    1. Initial program 0.4

      \[\frac{\left(\left(-b_2\right) - \left(\sqrt{\left(\left(b_2 \cdot b_2\right) - \left(a \cdot c\right)\right)}\right)\right)}{a}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le 0.0343017578125:\\ \;\;\;\;\frac{\frac{a \cdot c + \left(\left(-b_2\right) + b_2\right) \cdot \left(\left(-b_2\right) + \left(-b_2\right)\right)}{a}}{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2019112 +o rules:numerics
(FPCore (a b_2 c)
  :name "quad2m (problem 3.2.1, negative)"
  (/.p16 (-.p16 (neg.p16 b_2) (sqrt.p16 (-.p16 (*.p16 b_2 b_2) (*.p16 a c)))) a))