Average Error: 33.3 → 10.4
Time: 22.0s
Precision: 64
\[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b_2 \le -5.961198324014865 \cdot 10^{-88}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le 6.384705165981893 \cdot 10^{+101}:\\ \;\;\;\;\frac{-b_2}{a} - \frac{\sqrt{b_2 \cdot b_2 - c \cdot a}}{a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{b_2}{a}, -2, \frac{\frac{1}{2}}{\frac{b_2}{c}}\right)\\ \end{array}\]
\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}
↓
\begin{array}{l}
\mathbf{if}\;b_2 \le -5.961198324014865 \cdot 10^{-88}:\\
\;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\

\mathbf{elif}\;b_2 \le 6.384705165981893 \cdot 10^{+101}:\\
\;\;\;\;\frac{-b_2}{a} - \frac{\sqrt{b_2 \cdot b_2 - c \cdot a}}{a}\\

\mathbf{else}:\\
\;\;\;\;\mathsf{fma}\left(\frac{b_2}{a}, -2, \frac{\frac{1}{2}}{\frac{b_2}{c}}\right)\\

\end{array}
double f(double a, double b_2, double c) {
        double r1326912 = b_2;
        double r1326913 = -r1326912;
        double r1326914 = r1326912 * r1326912;
        double r1326915 = a;
        double r1326916 = c;
        double r1326917 = r1326915 * r1326916;
        double r1326918 = r1326914 - r1326917;
        double r1326919 = sqrt(r1326918);
        double r1326920 = r1326913 - r1326919;
        double r1326921 = r1326920 / r1326915;
        return r1326921;
}
↓
double f(double a, double b_2, double c) {
        double r1326922 = b_2;
        double r1326923 = -5.961198324014865e-88;
        bool r1326924 = r1326922 <= r1326923;
        double r1326925 = -0.5;
        double r1326926 = c;
        double r1326927 = r1326926 / r1326922;
        double r1326928 = r1326925 * r1326927;
        double r1326929 = 6.384705165981893e+101;
        bool r1326930 = r1326922 <= r1326929;
        double r1326931 = -r1326922;
        double r1326932 = a;
        double r1326933 = r1326931 / r1326932;
        double r1326934 = r1326922 * r1326922;
        double r1326935 = r1326926 * r1326932;
        double r1326936 = r1326934 - r1326935;
        double r1326937 = sqrt(r1326936);
        double r1326938 = r1326937 / r1326932;
        double r1326939 = r1326933 - r1326938;
        double r1326940 = r1326922 / r1326932;
        double r1326941 = -2.0;
        double r1326942 = 0.5;
        double r1326943 = r1326922 / r1326926;
        double r1326944 = r1326942 / r1326943;
        double r1326945 = fma(r1326940, r1326941, r1326944);
        double r1326946 = r1326930 ? r1326939 : r1326945;
        double r1326947 = r1326924 ? r1326928 : r1326946;
        return r1326947;
}

Error

Bits error versus a

Bits error versus b_2

Bits error versus c

Derivation

  1. Split input into 3 regimes

  2. if b_2 < -5.961198324014865e-88

    1. Initial program 51.4

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

    2. Taylor expanded around -inf 9.8

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

    if -5.961198324014865e-88 < b_2 < 6.384705165981893e+101

    1. Initial program 13.1

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

    3. Applied div-inv13.3

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

    5. Applied un-div-inv13.1

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

    7. Applied div-sub13.1

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

    if 6.384705165981893e+101 < b_2

    1. Initial program 43.8

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

    3. Applied div-inv43.9

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

    5. Applied un-div-inv43.8

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

    6. Taylor expanded around inf 3.7

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

    7. Simplified3.7

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{b_2}{a}, -2, \frac{\frac{1}{2}}{\frac{b_2}{c}}\right)}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification10.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -5.961198324014865 \cdot 10^{-88}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le 6.384705165981893 \cdot 10^{+101}:\\ \;\;\;\;\frac{-b_2}{a} - \frac{\sqrt{b_2 \cdot b_2 - c \cdot a}}{a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{b_2}{a}, -2, \frac{\frac{1}{2}}{\frac{b_2}{c}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019154 +o rules:numerics
(FPCore (a b_2 c)
  :name "NMSE problem 3.2.1"
  (/ (- (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))