Average Error: 32.8 → 10.3
Time: 21.3s
Precision: 64
\[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b_2 \le -1.6239127264630285 \cdot 10^{-63}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le 7.052614559736995 \cdot 10^{+62}:\\ \;\;\;\;\frac{1}{a} \cdot \left(\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - c \cdot a}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{b_2}{a}, -2, \frac{\frac{c}{b_2}}{2}\right)\\ \end{array}\]
\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}
↓
\begin{array}{l}
\mathbf{if}\;b_2 \le -1.6239127264630285 \cdot 10^{-63}:\\
\;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\

\mathbf{elif}\;b_2 \le 7.052614559736995 \cdot 10^{+62}:\\
\;\;\;\;\frac{1}{a} \cdot \left(\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - c \cdot a}\right)\\

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

\end{array}
double f(double a, double b_2, double c) {
        double r875790 = b_2;
        double r875791 = -r875790;
        double r875792 = r875790 * r875790;
        double r875793 = a;
        double r875794 = c;
        double r875795 = r875793 * r875794;
        double r875796 = r875792 - r875795;
        double r875797 = sqrt(r875796);
        double r875798 = r875791 - r875797;
        double r875799 = r875798 / r875793;
        return r875799;
}
↓
double f(double a, double b_2, double c) {
        double r875800 = b_2;
        double r875801 = -1.6239127264630285e-63;
        bool r875802 = r875800 <= r875801;
        double r875803 = -0.5;
        double r875804 = c;
        double r875805 = r875804 / r875800;
        double r875806 = r875803 * r875805;
        double r875807 = 7.052614559736995e+62;
        bool r875808 = r875800 <= r875807;
        double r875809 = 1.0;
        double r875810 = a;
        double r875811 = r875809 / r875810;
        double r875812 = -r875800;
        double r875813 = r875800 * r875800;
        double r875814 = r875804 * r875810;
        double r875815 = r875813 - r875814;
        double r875816 = sqrt(r875815);
        double r875817 = r875812 - r875816;
        double r875818 = r875811 * r875817;
        double r875819 = r875800 / r875810;
        double r875820 = -2.0;
        double r875821 = 2.0;
        double r875822 = r875805 / r875821;
        double r875823 = fma(r875819, r875820, r875822);
        double r875824 = r875808 ? r875818 : r875823;
        double r875825 = r875802 ? r875806 : r875824;
        return r875825;
}

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 < -1.6239127264630285e-63

    1. Initial program 52.2

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

    2. Taylor expanded around -inf 8.6

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

    if -1.6239127264630285e-63 < b_2 < 7.052614559736995e+62

    1. Initial program 13.9

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

    3. Applied clear-num14.0

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

    5. Applied div-inv14.1

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

    6. Applied add-cube-cbrt14.1

      \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{1} \cdot \sqrt[3]{1}\right) \cdot \sqrt[3]{1}}}{a \cdot \frac{1}{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}}\]

    7. Applied times-frac14.1

      \[\leadsto \color{blue}{\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{a} \cdot \frac{\sqrt[3]{1}}{\frac{1}{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}}}\]

    8. Simplified14.1

      \[\leadsto \color{blue}{\frac{1}{a}} \cdot \frac{\sqrt[3]{1}}{\frac{1}{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}}\]

    9. Simplified14.0

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

    if 7.052614559736995e+62 < b_2

    1. Initial program 38.0

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

    2. Taylor expanded around inf 4.6

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

    3. Simplified4.6

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

  4. Final simplification10.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -1.6239127264630285 \cdot 10^{-63}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le 7.052614559736995 \cdot 10^{+62}:\\ \;\;\;\;\frac{1}{a} \cdot \left(\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - c \cdot a}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{b_2}{a}, -2, \frac{\frac{c}{b_2}}{2}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019158 +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))