Average Error: 34.0 → 10.5
Time: 34.2s
Precision: 64
\[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b_2 \le -1.229142930221511 \cdot 10^{-57}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le 2.6656023684116586 \cdot 10^{+55}:\\ \;\;\;\;\left(-\frac{b_2}{a}\right) - \sqrt{b_2 \cdot b_2 - c \cdot a} \cdot \frac{1}{a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{b_2}{a}, -2, \frac{\frac{1}{2}}{\frac{b_2}{c}}\right)\\ \end{array}\]
\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}
↓
\begin{array}{l}
\mathbf{if}\;b_2 \le -1.229142930221511 \cdot 10^{-57}:\\
\;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\

\mathbf{elif}\;b_2 \le 2.6656023684116586 \cdot 10^{+55}:\\
\;\;\;\;\left(-\frac{b_2}{a}\right) - \sqrt{b_2 \cdot b_2 - c \cdot a} \cdot \frac{1}{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 r2011800 = b_2;
        double r2011801 = -r2011800;
        double r2011802 = r2011800 * r2011800;
        double r2011803 = a;
        double r2011804 = c;
        double r2011805 = r2011803 * r2011804;
        double r2011806 = r2011802 - r2011805;
        double r2011807 = sqrt(r2011806);
        double r2011808 = r2011801 - r2011807;
        double r2011809 = r2011808 / r2011803;
        return r2011809;
}
↓
double f(double a, double b_2, double c) {
        double r2011810 = b_2;
        double r2011811 = -1.229142930221511e-57;
        bool r2011812 = r2011810 <= r2011811;
        double r2011813 = -0.5;
        double r2011814 = c;
        double r2011815 = r2011814 / r2011810;
        double r2011816 = r2011813 * r2011815;
        double r2011817 = 2.6656023684116586e+55;
        bool r2011818 = r2011810 <= r2011817;
        double r2011819 = a;
        double r2011820 = r2011810 / r2011819;
        double r2011821 = -r2011820;
        double r2011822 = r2011810 * r2011810;
        double r2011823 = r2011814 * r2011819;
        double r2011824 = r2011822 - r2011823;
        double r2011825 = sqrt(r2011824);
        double r2011826 = 1.0;
        double r2011827 = r2011826 / r2011819;
        double r2011828 = r2011825 * r2011827;
        double r2011829 = r2011821 - r2011828;
        double r2011830 = -2.0;
        double r2011831 = 0.5;
        double r2011832 = r2011810 / r2011814;
        double r2011833 = r2011831 / r2011832;
        double r2011834 = fma(r2011820, r2011830, r2011833);
        double r2011835 = r2011818 ? r2011829 : r2011834;
        double r2011836 = r2011812 ? r2011816 : r2011835;
        return r2011836;
}

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 < -1.229142930221511e-57

    1. Initial program 53.7

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

    3. Applied div-sub54.1

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

    4. Taylor expanded around -inf 8.2

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

    if -1.229142930221511e-57 < b_2 < 2.6656023684116586e+55

    1. Initial program 14.5

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

    3. Applied div-sub14.5

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

    5. Applied div-inv14.6

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

    if 2.6656023684116586e+55 < b_2

    1. Initial program 37.1

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

    2. Taylor expanded around inf 6.2

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

    3. Simplified6.2

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

  4. Final simplification10.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -1.229142930221511 \cdot 10^{-57}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le 2.6656023684116586 \cdot 10^{+55}:\\ \;\;\;\;\left(-\frac{b_2}{a}\right) - \sqrt{b_2 \cdot b_2 - c \cdot a} \cdot \frac{1}{a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{b_2}{a}, -2, \frac{\frac{1}{2}}{\frac{b_2}{c}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019141 +o rules:numerics
(FPCore (a b_2 c)
  :name "NMSE problem 3.2.1"
  (/ (- (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))