Average Error: 33.4 → 10.0
Time: 20.6s
Precision: 64
\[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b_2 \le -5.6179897107029005 \cdot 10^{+143}:\\ \;\;\;\;\mathsf{fma}\left(-2, \left(\frac{b_2}{a}\right), \left(\frac{1}{2} \cdot \frac{c}{b_2}\right)\right)\\ \mathbf{elif}\;b_2 \le 5.046853365273247 \cdot 10^{-144}:\\ \;\;\;\;\frac{1}{a} \cdot \left(\sqrt{b_2 \cdot b_2 - c \cdot a} - b_2\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \end{array}\]
\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}
↓
\begin{array}{l}
\mathbf{if}\;b_2 \le -5.6179897107029005 \cdot 10^{+143}:\\
\;\;\;\;\mathsf{fma}\left(-2, \left(\frac{b_2}{a}\right), \left(\frac{1}{2} \cdot \frac{c}{b_2}\right)\right)\\

\mathbf{elif}\;b_2 \le 5.046853365273247 \cdot 10^{-144}:\\
\;\;\;\;\frac{1}{a} \cdot \left(\sqrt{b_2 \cdot b_2 - c \cdot a} - b_2\right)\\

\mathbf{else}:\\
\;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\

\end{array}
double f(double a, double b_2, double c) {
        double r585871 = b_2;
        double r585872 = -r585871;
        double r585873 = r585871 * r585871;
        double r585874 = a;
        double r585875 = c;
        double r585876 = r585874 * r585875;
        double r585877 = r585873 - r585876;
        double r585878 = sqrt(r585877);
        double r585879 = r585872 + r585878;
        double r585880 = r585879 / r585874;
        return r585880;
}
↓
double f(double a, double b_2, double c) {
        double r585881 = b_2;
        double r585882 = -5.6179897107029005e+143;
        bool r585883 = r585881 <= r585882;
        double r585884 = -2.0;
        double r585885 = a;
        double r585886 = r585881 / r585885;
        double r585887 = 0.5;
        double r585888 = c;
        double r585889 = r585888 / r585881;
        double r585890 = r585887 * r585889;
        double r585891 = fma(r585884, r585886, r585890);
        double r585892 = 5.046853365273247e-144;
        bool r585893 = r585881 <= r585892;
        double r585894 = 1.0;
        double r585895 = r585894 / r585885;
        double r585896 = r585881 * r585881;
        double r585897 = r585888 * r585885;
        double r585898 = r585896 - r585897;
        double r585899 = sqrt(r585898);
        double r585900 = r585899 - r585881;
        double r585901 = r585895 * r585900;
        double r585902 = -0.5;
        double r585903 = r585902 * r585889;
        double r585904 = r585893 ? r585901 : r585903;
        double r585905 = r585883 ? r585891 : r585904;
        return r585905;
}

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 < -5.6179897107029005e+143

    1. Initial program 57.1

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

    2. Simplified57.1

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

    3. Taylor expanded around -inf 2.1

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

    4. Simplified2.1

      \[\leadsto \color{blue}{\mathsf{fma}\left(-2, \left(\frac{b_2}{a}\right), \left(\frac{1}{2} \cdot \frac{c}{b_2}\right)\right)}\]

    if -5.6179897107029005e+143 < b_2 < 5.046853365273247e-144

    1. Initial program 10.3

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

    2. Simplified10.3

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

    4. Applied clear-num10.4

      \[\leadsto \color{blue}{\frac{1}{\frac{a}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}}\]
    5. Using strategy rm

    6. Applied associate-/r/10.4

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

    if 5.046853365273247e-144 < b_2

    1. Initial program 49.8

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

    2. Simplified49.8

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

    3. Taylor expanded around inf 11.6

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{c}{b_2}}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification10.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -5.6179897107029005 \cdot 10^{+143}:\\ \;\;\;\;\mathsf{fma}\left(-2, \left(\frac{b_2}{a}\right), \left(\frac{1}{2} \cdot \frac{c}{b_2}\right)\right)\\ \mathbf{elif}\;b_2 \le 5.046853365273247 \cdot 10^{-144}:\\ \;\;\;\;\frac{1}{a} \cdot \left(\sqrt{b_2 \cdot b_2 - c \cdot a} - b_2\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019132 +o rules:numerics
(FPCore (a b_2 c)
  :name "quad2p (problem 3.2.1, positive)"
  (/ (+ (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))