Average Error: 33.6 → 10.3
Time: 19.0s
Precision: 64
\[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b_2 \le -1.264659490877098 \cdot 10^{-67}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le 0.17389787404847717:\\ \;\;\;\;\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-2, \frac{b_2}{a}, \frac{c}{b_2} \cdot \frac{1}{2}\right)\\ \end{array}\]
\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}
↓
\begin{array}{l}
\mathbf{if}\;b_2 \le -1.264659490877098 \cdot 10^{-67}:\\
\;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\

\mathbf{elif}\;b_2 \le 0.17389787404847717:\\
\;\;\;\;\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\\

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

\end{array}
double f(double a, double b_2, double c) {
        double r812809 = b_2;
        double r812810 = -r812809;
        double r812811 = r812809 * r812809;
        double r812812 = a;
        double r812813 = c;
        double r812814 = r812812 * r812813;
        double r812815 = r812811 - r812814;
        double r812816 = sqrt(r812815);
        double r812817 = r812810 - r812816;
        double r812818 = r812817 / r812812;
        return r812818;
}
↓
double f(double a, double b_2, double c) {
        double r812819 = b_2;
        double r812820 = -1.264659490877098e-67;
        bool r812821 = r812819 <= r812820;
        double r812822 = -0.5;
        double r812823 = c;
        double r812824 = r812823 / r812819;
        double r812825 = r812822 * r812824;
        double r812826 = 0.17389787404847717;
        bool r812827 = r812819 <= r812826;
        double r812828 = -r812819;
        double r812829 = r812819 * r812819;
        double r812830 = a;
        double r812831 = r812830 * r812823;
        double r812832 = r812829 - r812831;
        double r812833 = sqrt(r812832);
        double r812834 = r812828 - r812833;
        double r812835 = r812834 / r812830;
        double r812836 = -2.0;
        double r812837 = r812819 / r812830;
        double r812838 = 0.5;
        double r812839 = r812824 * r812838;
        double r812840 = fma(r812836, r812837, r812839);
        double r812841 = r812827 ? r812835 : r812840;
        double r812842 = r812821 ? r812825 : r812841;
        return r812842;
}

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 < -1.264659490877098e-67

    1. Initial program 53.0

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

    2. Taylor expanded around -inf 8.0

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

    if -1.264659490877098e-67 < b_2 < 0.17389787404847717

    1. Initial program 15.0

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

    3. Applied div-inv15.1

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

    5. Applied un-div-inv15.0

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

    if 0.17389787404847717 < b_2

    1. Initial program 29.8

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

    3. Applied div-inv29.9

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

    4. Taylor expanded around inf 7.3

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

    5. Simplified7.3

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

  4. Final simplification10.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -1.264659490877098 \cdot 10^{-67}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le 0.17389787404847717:\\ \;\;\;\;\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-2, \frac{b_2}{a}, \frac{c}{b_2} \cdot \frac{1}{2}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019168 +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))