Average Error: 43.4 → 11.5
Time: 16.0s
Precision: 64
\[1.1102230246251565 \cdot 10^{-16} \lt a \lt 9007199254740992.0 \land 1.1102230246251565 \cdot 10^{-16} \lt b \lt 9007199254740992.0 \land 1.1102230246251565 \cdot 10^{-16} \lt c \lt 9007199254740992.0\]
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le 0.0012535322255036849:\\ \;\;\;\;\frac{\frac{\sqrt[3]{\sqrt{\mathsf{fma}\left(b, b, \left(a \cdot -4\right) \cdot c\right)} - b}}{\sqrt{a}} \cdot \frac{\sqrt[3]{\sqrt{\mathsf{fma}\left(b, b, \left(a \cdot -4\right) \cdot c\right)} - b} \cdot \sqrt[3]{\sqrt{\mathsf{fma}\left(b, b, \left(a \cdot -4\right) \cdot c\right)} - b}}{\sqrt{a}}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2 \cdot \frac{c}{b}}{2}\\ \end{array}\]
\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}
↓
\begin{array}{l}
\mathbf{if}\;b \le 0.0012535322255036849:\\
\;\;\;\;\frac{\frac{\sqrt[3]{\sqrt{\mathsf{fma}\left(b, b, \left(a \cdot -4\right) \cdot c\right)} - b}}{\sqrt{a}} \cdot \frac{\sqrt[3]{\sqrt{\mathsf{fma}\left(b, b, \left(a \cdot -4\right) \cdot c\right)} - b} \cdot \sqrt[3]{\sqrt{\mathsf{fma}\left(b, b, \left(a \cdot -4\right) \cdot c\right)} - b}}{\sqrt{a}}}{2}\\

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

\end{array}
double f(double a, double b, double c) {
        double r1572909 = b;
        double r1572910 = -r1572909;
        double r1572911 = r1572909 * r1572909;
        double r1572912 = 4.0;
        double r1572913 = a;
        double r1572914 = r1572912 * r1572913;
        double r1572915 = c;
        double r1572916 = r1572914 * r1572915;
        double r1572917 = r1572911 - r1572916;
        double r1572918 = sqrt(r1572917);
        double r1572919 = r1572910 + r1572918;
        double r1572920 = 2.0;
        double r1572921 = r1572920 * r1572913;
        double r1572922 = r1572919 / r1572921;
        return r1572922;
}
↓
double f(double a, double b, double c) {
        double r1572923 = b;
        double r1572924 = 0.0012535322255036849;
        bool r1572925 = r1572923 <= r1572924;
        double r1572926 = a;
        double r1572927 = -4.0;
        double r1572928 = r1572926 * r1572927;
        double r1572929 = c;
        double r1572930 = r1572928 * r1572929;
        double r1572931 = fma(r1572923, r1572923, r1572930);
        double r1572932 = sqrt(r1572931);
        double r1572933 = r1572932 - r1572923;
        double r1572934 = cbrt(r1572933);
        double r1572935 = sqrt(r1572926);
        double r1572936 = r1572934 / r1572935;
        double r1572937 = r1572934 * r1572934;
        double r1572938 = r1572937 / r1572935;
        double r1572939 = r1572936 * r1572938;
        double r1572940 = 2.0;
        double r1572941 = r1572939 / r1572940;
        double r1572942 = -2.0;
        double r1572943 = r1572929 / r1572923;
        double r1572944 = r1572942 * r1572943;
        double r1572945 = r1572944 / r1572940;
        double r1572946 = r1572925 ? r1572941 : r1572945;
        return r1572946;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

  1. Split input into 2 regimes

  2. if b < 0.0012535322255036849

    1. Initial program 20.2

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]

    2. Simplified20.2

      \[\leadsto \color{blue}{\frac{\frac{\sqrt{\mathsf{fma}\left(b, b, \left(a \cdot -4\right) \cdot c\right)} - b}{a}}{2}}\]
    3. Using strategy rm

    4. Applied add-sqr-sqrt20.2

      \[\leadsto \frac{\frac{\sqrt{\mathsf{fma}\left(b, b, \left(a \cdot -4\right) \cdot c\right)} - b}{\color{blue}{\sqrt{a} \cdot \sqrt{a}}}}{2}\]

    5. Applied add-cube-cbrt20.2

      \[\leadsto \frac{\frac{\color{blue}{\left(\sqrt[3]{\sqrt{\mathsf{fma}\left(b, b, \left(a \cdot -4\right) \cdot c\right)} - b} \cdot \sqrt[3]{\sqrt{\mathsf{fma}\left(b, b, \left(a \cdot -4\right) \cdot c\right)} - b}\right) \cdot \sqrt[3]{\sqrt{\mathsf{fma}\left(b, b, \left(a \cdot -4\right) \cdot c\right)} - b}}}{\sqrt{a} \cdot \sqrt{a}}}{2}\]

    6. Applied times-frac20.2

      \[\leadsto \frac{\color{blue}{\frac{\sqrt[3]{\sqrt{\mathsf{fma}\left(b, b, \left(a \cdot -4\right) \cdot c\right)} - b} \cdot \sqrt[3]{\sqrt{\mathsf{fma}\left(b, b, \left(a \cdot -4\right) \cdot c\right)} - b}}{\sqrt{a}} \cdot \frac{\sqrt[3]{\sqrt{\mathsf{fma}\left(b, b, \left(a \cdot -4\right) \cdot c\right)} - b}}{\sqrt{a}}}}{2}\]

    if 0.0012535322255036849 < b

    1. Initial program 45.7

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]

    2. Simplified45.7

      \[\leadsto \color{blue}{\frac{\frac{\sqrt{\mathsf{fma}\left(b, b, \left(a \cdot -4\right) \cdot c\right)} - b}{a}}{2}}\]

    3. Taylor expanded around inf 10.6

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

  4. Final simplification11.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le 0.0012535322255036849:\\ \;\;\;\;\frac{\frac{\sqrt[3]{\sqrt{\mathsf{fma}\left(b, b, \left(a \cdot -4\right) \cdot c\right)} - b}}{\sqrt{a}} \cdot \frac{\sqrt[3]{\sqrt{\mathsf{fma}\left(b, b, \left(a \cdot -4\right) \cdot c\right)} - b} \cdot \sqrt[3]{\sqrt{\mathsf{fma}\left(b, b, \left(a \cdot -4\right) \cdot c\right)} - b}}{\sqrt{a}}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2 \cdot \frac{c}{b}}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019168 +o rules:numerics
(FPCore (a b c)
  :name "Quadratic roots, medium range"
  :pre (and (< 1.1102230246251565e-16 a 9007199254740992.0) (< 1.1102230246251565e-16 b 9007199254740992.0) (< 1.1102230246251565e-16 c 9007199254740992.0))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4 a) c)))) (* 2 a)))