Average Error: 28.4 → 16.6
Time: 18.8s
Precision: 64
\[1.053671212772350866701172186984739043147 \cdot 10^{-8} \lt a \lt 94906265.62425155937671661376953125 \land 1.053671212772350866701172186984739043147 \cdot 10^{-8} \lt b \lt 94906265.62425155937671661376953125 \land 1.053671212772350866701172186984739043147 \cdot 10^{-8} \lt c \lt 94906265.62425155937671661376953125\]
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le 245.5646346995640669774729758501052856445:\\ \;\;\;\;\frac{\frac{\frac{\left(b \cdot b - \left(a \cdot c\right) \cdot 4\right) \cdot \sqrt{b \cdot b - \left(a \cdot c\right) \cdot 4} - \left(b \cdot b\right) \cdot b}{\mathsf{fma}\left(\sqrt{b \cdot b - \left(a \cdot c\right) \cdot 4}, b, b \cdot b - \left(a \cdot c\right) \cdot 4\right) + b \cdot b}}{a}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2 \cdot \frac{c}{b}}{2}\\ \end{array}\]
\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 245.5646346995640669774729758501052856445:\\
\;\;\;\;\frac{\frac{\frac{\left(b \cdot b - \left(a \cdot c\right) \cdot 4\right) \cdot \sqrt{b \cdot b - \left(a \cdot c\right) \cdot 4} - \left(b \cdot b\right) \cdot b}{\mathsf{fma}\left(\sqrt{b \cdot b - \left(a \cdot c\right) \cdot 4}, b, b \cdot b - \left(a \cdot c\right) \cdot 4\right) + b \cdot b}}{a}}{2}\\

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

\end{array}
double f(double a, double b, double c) {
        double r1745920 = b;
        double r1745921 = -r1745920;
        double r1745922 = r1745920 * r1745920;
        double r1745923 = 4.0;
        double r1745924 = a;
        double r1745925 = r1745923 * r1745924;
        double r1745926 = c;
        double r1745927 = r1745925 * r1745926;
        double r1745928 = r1745922 - r1745927;
        double r1745929 = sqrt(r1745928);
        double r1745930 = r1745921 + r1745929;
        double r1745931 = 2.0;
        double r1745932 = r1745931 * r1745924;
        double r1745933 = r1745930 / r1745932;
        return r1745933;
}
↓
double f(double a, double b, double c) {
        double r1745934 = b;
        double r1745935 = 245.56463469956407;
        bool r1745936 = r1745934 <= r1745935;
        double r1745937 = r1745934 * r1745934;
        double r1745938 = a;
        double r1745939 = c;
        double r1745940 = r1745938 * r1745939;
        double r1745941 = 4.0;
        double r1745942 = r1745940 * r1745941;
        double r1745943 = r1745937 - r1745942;
        double r1745944 = sqrt(r1745943);
        double r1745945 = r1745943 * r1745944;
        double r1745946 = r1745937 * r1745934;
        double r1745947 = r1745945 - r1745946;
        double r1745948 = fma(r1745944, r1745934, r1745943);
        double r1745949 = r1745948 + r1745937;
        double r1745950 = r1745947 / r1745949;
        double r1745951 = r1745950 / r1745938;
        double r1745952 = 2.0;
        double r1745953 = r1745951 / r1745952;
        double r1745954 = -2.0;
        double r1745955 = r1745939 / r1745934;
        double r1745956 = r1745954 * r1745955;
        double r1745957 = r1745956 / r1745952;
        double r1745958 = r1745936 ? r1745953 : r1745957;
        return r1745958;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

  1. Split input into 2 regimes

  2. if b < 245.56463469956407

    1. Initial program 15.8

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

    2. Simplified15.8

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

    4. Applied flip3--15.9

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

    5. Simplified15.2

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

    6. Simplified15.2

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

    if 245.56463469956407 < b

    1. Initial program 34.8

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

    2. Simplified34.8

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

    3. Taylor expanded around inf 17.4

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

  4. Final simplification16.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le 245.5646346995640669774729758501052856445:\\ \;\;\;\;\frac{\frac{\frac{\left(b \cdot b - \left(a \cdot c\right) \cdot 4\right) \cdot \sqrt{b \cdot b - \left(a \cdot c\right) \cdot 4} - \left(b \cdot b\right) \cdot b}{\mathsf{fma}\left(\sqrt{b \cdot b - \left(a \cdot c\right) \cdot 4}, b, b \cdot b - \left(a \cdot c\right) \cdot 4\right) + b \cdot b}}{a}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2 \cdot \frac{c}{b}}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019192 +o rules:numerics
(FPCore (a b c)
  :name "Quadratic roots, narrow range"
  :pre (and (< 1.0536712127723509e-08 a 94906265.62425156) (< 1.0536712127723509e-08 b 94906265.62425156) (< 1.0536712127723509e-08 c 94906265.62425156))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)))