Average Error: 34.5 → 10.6
Time: 4.6s
Precision: 64
\[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b_2 \le -1.279587145681289 \cdot 10^{-136}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le 9.9017234375838935 \cdot 10^{139}:\\ \;\;\;\;\left(\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}\right) \cdot \frac{1}{a}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a}\\ \end{array}\]
\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}
↓
\begin{array}{l}
\mathbf{if}\;b_2 \le -1.279587145681289 \cdot 10^{-136}:\\
\;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\

\mathbf{elif}\;b_2 \le 9.9017234375838935 \cdot 10^{139}:\\
\;\;\;\;\left(\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}\right) \cdot \frac{1}{a}\\

\mathbf{else}:\\
\;\;\;\;-2 \cdot \frac{b_2}{a}\\

\end{array}
double f(double a, double b_2, double c) {
        double r73730 = b_2;
        double r73731 = -r73730;
        double r73732 = r73730 * r73730;
        double r73733 = a;
        double r73734 = c;
        double r73735 = r73733 * r73734;
        double r73736 = r73732 - r73735;
        double r73737 = sqrt(r73736);
        double r73738 = r73731 - r73737;
        double r73739 = r73738 / r73733;
        return r73739;
}
↓
double f(double a, double b_2, double c) {
        double r73740 = b_2;
        double r73741 = -1.279587145681289e-136;
        bool r73742 = r73740 <= r73741;
        double r73743 = -0.5;
        double r73744 = c;
        double r73745 = r73744 / r73740;
        double r73746 = r73743 * r73745;
        double r73747 = 9.901723437583893e+139;
        bool r73748 = r73740 <= r73747;
        double r73749 = -r73740;
        double r73750 = r73740 * r73740;
        double r73751 = a;
        double r73752 = r73751 * r73744;
        double r73753 = r73750 - r73752;
        double r73754 = sqrt(r73753);
        double r73755 = r73749 - r73754;
        double r73756 = 1.0;
        double r73757 = r73756 / r73751;
        double r73758 = r73755 * r73757;
        double r73759 = -2.0;
        double r73760 = r73740 / r73751;
        double r73761 = r73759 * r73760;
        double r73762 = r73748 ? r73758 : r73761;
        double r73763 = r73742 ? r73746 : r73762;
        return r73763;
}

Error

Bits error versus a

Bits error versus b_2

Bits error versus c

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 3 regimes

  2. if b_2 < -1.279587145681289e-136

    1. Initial program 51.1

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

    2. Taylor expanded around -inf 12.2

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{c}{b_2}}\]

    if -1.279587145681289e-136 < b_2 < 9.901723437583893e+139

    1. Initial program 11.1

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

    3. Applied div-inv11.2

      \[\leadsto \color{blue}{\left(\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}\right) \cdot \frac{1}{a}}\]

    if 9.901723437583893e+139 < b_2

    1. Initial program 57.7

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

    3. Applied clear-num57.7

      \[\leadsto \color{blue}{\frac{1}{\frac{a}{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}}}\]

    4. Taylor expanded around 0 2.3

      \[\leadsto \color{blue}{-2 \cdot \frac{b_2}{a}}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification10.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -1.279587145681289 \cdot 10^{-136}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le 9.9017234375838935 \cdot 10^{139}:\\ \;\;\;\;\left(\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}\right) \cdot \frac{1}{a}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2020089 +o rules:numerics
(FPCore (a b_2 c)
  :name "NMSE problem 3.2.1"
  :precision binary64
  (/ (- (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))