Average Error: 33.0 → 10.3
Time: 21.4s
Precision: 64
\[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b_2 \le -6.615151909502748 \cdot 10^{-87}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le 3.5387363548079373 \cdot 10^{+99}:\\ \;\;\;\;\frac{-b_2}{a} - \frac{\sqrt{b_2 \cdot b_2 - c \cdot a}}{a}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a}\\ \end{array}\]
\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}
↓
\begin{array}{l}
\mathbf{if}\;b_2 \le -6.615151909502748 \cdot 10^{-87}:\\
\;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\

\mathbf{elif}\;b_2 \le 3.5387363548079373 \cdot 10^{+99}:\\
\;\;\;\;\frac{-b_2}{a} - \frac{\sqrt{b_2 \cdot b_2 - c \cdot a}}{a}\\

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

\end{array}
double f(double a, double b_2, double c) {
        double r534840 = b_2;
        double r534841 = -r534840;
        double r534842 = r534840 * r534840;
        double r534843 = a;
        double r534844 = c;
        double r534845 = r534843 * r534844;
        double r534846 = r534842 - r534845;
        double r534847 = sqrt(r534846);
        double r534848 = r534841 - r534847;
        double r534849 = r534848 / r534843;
        return r534849;
}
↓
double f(double a, double b_2, double c) {
        double r534850 = b_2;
        double r534851 = -6.615151909502748e-87;
        bool r534852 = r534850 <= r534851;
        double r534853 = -0.5;
        double r534854 = c;
        double r534855 = r534854 / r534850;
        double r534856 = r534853 * r534855;
        double r534857 = 3.5387363548079373e+99;
        bool r534858 = r534850 <= r534857;
        double r534859 = -r534850;
        double r534860 = a;
        double r534861 = r534859 / r534860;
        double r534862 = r534850 * r534850;
        double r534863 = r534854 * r534860;
        double r534864 = r534862 - r534863;
        double r534865 = sqrt(r534864);
        double r534866 = r534865 / r534860;
        double r534867 = r534861 - r534866;
        double r534868 = -2.0;
        double r534869 = r534850 / r534860;
        double r534870 = r534868 * r534869;
        double r534871 = r534858 ? r534867 : r534870;
        double r534872 = r534852 ? r534856 : r534871;
        return r534872;
}

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 < -6.615151909502748e-87

    1. Initial program 52.0

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

    2. Taylor expanded around -inf 10.0

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

    if -6.615151909502748e-87 < b_2 < 3.5387363548079373e+99

    1. Initial program 12.7

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

    3. Applied div-inv12.8

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

    5. Applied un-div-inv12.7

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

    7. Applied div-sub12.7

      \[\leadsto \color{blue}{\frac{-b_2}{a} - \frac{\sqrt{b_2 \cdot b_2 - a \cdot c}}{a}}\]

    if 3.5387363548079373e+99 < b_2

    1. Initial program 44.5

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

    3. Applied *-un-lft-identity44.5

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

    4. Applied associate-/l*44.6

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

    5. Taylor expanded around 0 3.9

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

  4. Final simplification10.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -6.615151909502748 \cdot 10^{-87}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le 3.5387363548079373 \cdot 10^{+99}:\\ \;\;\;\;\frac{-b_2}{a} - \frac{\sqrt{b_2 \cdot b_2 - c \cdot a}}{a}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2019133 +o rules:numerics
(FPCore (a b_2 c)
  :name "quad2m (problem 3.2.1, negative)"
  (/ (- (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))