Average Error: 34.4 → 9.0
Time: 7.1s
Precision: 64
\[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b_2 \le -1.857238265713216596268581045781308602833 \cdot 10^{109}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\\ \mathbf{elif}\;b_2 \le 8.706117685651469092807044052871735370705 \cdot 10^{-130}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt[3]{-b_2} \cdot \sqrt[3]{-b_2}, \sqrt[3]{-b_2}, \sqrt{b_2 \cdot b_2 - a \cdot c}\right)}{a}\\ \mathbf{elif}\;b_2 \le 4.211987902224083571861816259903906920885 \cdot 10^{89}:\\ \;\;\;\;\frac{0 + a \cdot c}{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}} \cdot \frac{1}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \end{array}\]
\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}
↓
\begin{array}{l}
\mathbf{if}\;b_2 \le -1.857238265713216596268581045781308602833 \cdot 10^{109}:\\
\;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\\

\mathbf{elif}\;b_2 \le 8.706117685651469092807044052871735370705 \cdot 10^{-130}:\\
\;\;\;\;\frac{\mathsf{fma}\left(\sqrt[3]{-b_2} \cdot \sqrt[3]{-b_2}, \sqrt[3]{-b_2}, \sqrt{b_2 \cdot b_2 - a \cdot c}\right)}{a}\\

\mathbf{elif}\;b_2 \le 4.211987902224083571861816259903906920885 \cdot 10^{89}:\\
\;\;\;\;\frac{0 + a \cdot c}{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}} \cdot \frac{1}{a}\\

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

\end{array}
double f(double a, double b_2, double c) {
        double r23863 = b_2;
        double r23864 = -r23863;
        double r23865 = r23863 * r23863;
        double r23866 = a;
        double r23867 = c;
        double r23868 = r23866 * r23867;
        double r23869 = r23865 - r23868;
        double r23870 = sqrt(r23869);
        double r23871 = r23864 + r23870;
        double r23872 = r23871 / r23866;
        return r23872;
}
↓
double f(double a, double b_2, double c) {
        double r23873 = b_2;
        double r23874 = -1.8572382657132166e+109;
        bool r23875 = r23873 <= r23874;
        double r23876 = 0.5;
        double r23877 = c;
        double r23878 = r23877 / r23873;
        double r23879 = r23876 * r23878;
        double r23880 = 2.0;
        double r23881 = a;
        double r23882 = r23873 / r23881;
        double r23883 = r23880 * r23882;
        double r23884 = r23879 - r23883;
        double r23885 = 8.706117685651469e-130;
        bool r23886 = r23873 <= r23885;
        double r23887 = -r23873;
        double r23888 = cbrt(r23887);
        double r23889 = r23888 * r23888;
        double r23890 = r23873 * r23873;
        double r23891 = r23881 * r23877;
        double r23892 = r23890 - r23891;
        double r23893 = sqrt(r23892);
        double r23894 = fma(r23889, r23888, r23893);
        double r23895 = r23894 / r23881;
        double r23896 = 4.2119879022240836e+89;
        bool r23897 = r23873 <= r23896;
        double r23898 = 0.0;
        double r23899 = r23898 + r23891;
        double r23900 = r23887 - r23893;
        double r23901 = r23899 / r23900;
        double r23902 = 1.0;
        double r23903 = r23902 / r23881;
        double r23904 = r23901 * r23903;
        double r23905 = -0.5;
        double r23906 = r23905 * r23878;
        double r23907 = r23897 ? r23904 : r23906;
        double r23908 = r23886 ? r23895 : r23907;
        double r23909 = r23875 ? r23884 : r23908;
        return r23909;
}

Error

Bits error versus a

Bits error versus b_2

Bits error versus c

Derivation

  1. Split input into 4 regimes

  2. if b_2 < -1.8572382657132166e+109

    1. Initial program 50.1

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

    2. Taylor expanded around -inf 3.6

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

    if -1.8572382657132166e+109 < b_2 < 8.706117685651469e-130

    1. Initial program 11.4

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

    3. Applied add-cube-cbrt11.7

      \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{-b_2} \cdot \sqrt[3]{-b_2}\right) \cdot \sqrt[3]{-b_2}} + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]

    4. Applied fma-def11.7

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\sqrt[3]{-b_2} \cdot \sqrt[3]{-b_2}, \sqrt[3]{-b_2}, \sqrt{b_2 \cdot b_2 - a \cdot c}\right)}}{a}\]

    if 8.706117685651469e-130 < b_2 < 4.2119879022240836e+89

    1. Initial program 39.7

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

    3. Applied flip-+39.7

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

    4. Simplified15.8

      \[\leadsto \frac{\frac{\color{blue}{0 + a \cdot c}}{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}}{a}\]
    5. Using strategy rm

    6. Applied div-inv15.8

      \[\leadsto \color{blue}{\frac{0 + a \cdot c}{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}} \cdot \frac{1}{a}}\]

    if 4.2119879022240836e+89 < b_2

    1. Initial program 59.4

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

    2. Taylor expanded around inf 2.5

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{c}{b_2}}\]
  3. Recombined 4 regimes into one program.

  4. Final simplification9.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -1.857238265713216596268581045781308602833 \cdot 10^{109}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\\ \mathbf{elif}\;b_2 \le 8.706117685651469092807044052871735370705 \cdot 10^{-130}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt[3]{-b_2} \cdot \sqrt[3]{-b_2}, \sqrt[3]{-b_2}, \sqrt{b_2 \cdot b_2 - a \cdot c}\right)}{a}\\ \mathbf{elif}\;b_2 \le 4.211987902224083571861816259903906920885 \cdot 10^{89}:\\ \;\;\;\;\frac{0 + a \cdot c}{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}} \cdot \frac{1}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019347 +o rules:numerics
(FPCore (a b_2 c)
  :name "quad2p (problem 3.2.1, positive)"
  :precision binary64
  (/ (+ (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))