Average Error: 1.7 → 0.4
Time: 16.0s
Precision: 64
\[\frac{\left(\left(-b_2\right) - \left(\sqrt{\left(\left(b_2 \cdot b_2\right) - \left(a \cdot c\right)\right)}\right)\right)}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b_2 \le -0.091796875:\\ \;\;\;\;\frac{\frac{a \cdot c + \left(\left(-b_2\right) + b_2\right) \cdot \left(\left(-b_2\right) + \left(-b_2\right)\right)}{a}}{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\\ \end{array}\]
\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 -0.091796875:\\
\;\;\;\;\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 r628756 = b_2;
        double r628757 = -r628756;
        double r628758 = r628756 * r628756;
        double r628759 = a;
        double r628760 = c;
        double r628761 = r628759 * r628760;
        double r628762 = r628758 - r628761;
        double r628763 = sqrt(r628762);
        double r628764 = r628757 - r628763;
        double r628765 = r628764 / r628759;
        return r628765;
}
↓
double f(double a, double b_2, double c) {
        double r628766 = b_2;
        double r628767 = -0.091796875;
        bool r628768 = r628766 <= r628767;
        double r628769 = a;
        double r628770 = c;
        double r628771 = r628769 * r628770;
        double r628772 = -r628766;
        double r628773 = r628772 + r628766;
        double r628774 = r628772 + r628772;
        double r628775 = r628773 * r628774;
        double r628776 = r628771 + r628775;
        double r628777 = r628776 / r628769;
        double r628778 = r628766 * r628766;
        double r628779 = r628778 - r628771;
        double r628780 = sqrt(r628779);
        double r628781 = r628772 + r628780;
        double r628782 = r628777 / r628781;
        double r628783 = r628772 - r628780;
        double r628784 = r628783 / r628769;
        double r628785 = r628768 ? r628782 : r628784;
        return r628785;
}

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 < -0.091796875

    1. Initial program 3.5

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

    3. Applied p16-flip--3.0

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

    4. Applied associate-/l/3.1

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

    5. Simplified0.7

      \[\leadsto \frac{\color{blue}{\left(\frac{\left(a \cdot c\right)}{\left(\left(\frac{\left(-b_2\right)}{b_2}\right) \cdot \left(\frac{\left(-b_2\right)}{\left(-b_2\right)}\right)\right)}\right)}}{\left(a \cdot \left(\frac{\left(-b_2\right)}{\left(\sqrt{\left(\left(b_2 \cdot b_2\right) - \left(a \cdot c\right)\right)}\right)}\right)\right)}\]
    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 -0.091796875 < 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

    \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -0.091796875:\\ \;\;\;\;\frac{\frac{a \cdot c + \left(\left(-b_2\right) + b_2\right) \cdot \left(\left(-b_2\right) + \left(-b_2\right)\right)}{a}}{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2019104 
(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))