Average Error: 34.2 → 10.0
Time: 5.7s
Precision: 64
\[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b_2 \le -9.66563711558993385 \cdot 10^{-69}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le 9.1585291365273219 \cdot 10^{122}:\\ \;\;\;\;\frac{-b_2}{a} - \frac{\sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a}\\ \end{array}\]
\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}
↓
\begin{array}{l}
\mathbf{if}\;b_2 \le -9.66563711558993385 \cdot 10^{-69}:\\
\;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\

\mathbf{elif}\;b_2 \le 9.1585291365273219 \cdot 10^{122}:\\
\;\;\;\;\frac{-b_2}{a} - \frac{\sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\\

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

\end{array}
double f(double a, double b_2, double c) {
        double r88915 = b_2;
        double r88916 = -r88915;
        double r88917 = r88915 * r88915;
        double r88918 = a;
        double r88919 = c;
        double r88920 = r88918 * r88919;
        double r88921 = r88917 - r88920;
        double r88922 = sqrt(r88921);
        double r88923 = r88916 - r88922;
        double r88924 = r88923 / r88918;
        return r88924;
}
↓
double f(double a, double b_2, double c) {
        double r88925 = b_2;
        double r88926 = -9.665637115589934e-69;
        bool r88927 = r88925 <= r88926;
        double r88928 = -0.5;
        double r88929 = c;
        double r88930 = r88929 / r88925;
        double r88931 = r88928 * r88930;
        double r88932 = 9.158529136527322e+122;
        bool r88933 = r88925 <= r88932;
        double r88934 = -r88925;
        double r88935 = a;
        double r88936 = r88934 / r88935;
        double r88937 = r88925 * r88925;
        double r88938 = r88935 * r88929;
        double r88939 = r88937 - r88938;
        double r88940 = sqrt(r88939);
        double r88941 = r88940 / r88935;
        double r88942 = r88936 - r88941;
        double r88943 = -2.0;
        double r88944 = r88925 / r88935;
        double r88945 = r88943 * r88944;
        double r88946 = r88933 ? r88942 : r88945;
        double r88947 = r88927 ? r88931 : r88946;
        return r88947;
}

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 < -9.665637115589934e-69

    1. Initial program 53.8

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

    2. Taylor expanded around -inf 8.6

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

    if -9.665637115589934e-69 < b_2 < 9.158529136527322e+122

    1. Initial program 13.0

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

    3. Applied div-sub13.0

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

    if 9.158529136527322e+122 < b_2

    1. Initial program 53.1

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

    3. Applied div-inv53.2

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

    5. Applied un-div-inv53.1

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

    7. Applied clear-num53.2

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

    8. Taylor expanded around 0 3.2

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

  4. Final simplification10.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -9.66563711558993385 \cdot 10^{-69}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le 9.1585291365273219 \cdot 10^{122}:\\ \;\;\;\;\frac{-b_2}{a} - \frac{\sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2020036 +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))