Average Error: 28.4 → 16.3
Time: 18.5s
Precision: 64
\[1.0536712127723509 \cdot 10^{-08} \lt a \lt 94906265.62425156 \land 1.0536712127723509 \cdot 10^{-08} \lt b \lt 94906265.62425156 \land 1.0536712127723509 \cdot 10^{-08} \lt c \lt 94906265.62425156\]
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le 2297.8344311922956:\\ \;\;\;\;\frac{\frac{\frac{\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)} \cdot \mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right) - \left(b \cdot b\right) \cdot b}{\mathsf{fma}\left(b, b + \sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)}, \mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)\right)}}{a}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2 \cdot \frac{c}{b}}{2}\\ \end{array}\]
\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 2297.8344311922956:\\
\;\;\;\;\frac{\frac{\frac{\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)} \cdot \mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right) - \left(b \cdot b\right) \cdot b}{\mathsf{fma}\left(b, b + \sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)}, \mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)\right)}}{a}}{2}\\

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

\end{array}
double f(double a, double b, double c) {
        double r1376926 = b;
        double r1376927 = -r1376926;
        double r1376928 = r1376926 * r1376926;
        double r1376929 = 4.0;
        double r1376930 = a;
        double r1376931 = r1376929 * r1376930;
        double r1376932 = c;
        double r1376933 = r1376931 * r1376932;
        double r1376934 = r1376928 - r1376933;
        double r1376935 = sqrt(r1376934);
        double r1376936 = r1376927 + r1376935;
        double r1376937 = 2.0;
        double r1376938 = r1376937 * r1376930;
        double r1376939 = r1376936 / r1376938;
        return r1376939;
}
↓
double f(double a, double b, double c) {
        double r1376940 = b;
        double r1376941 = 2297.8344311922956;
        bool r1376942 = r1376940 <= r1376941;
        double r1376943 = -4.0;
        double r1376944 = a;
        double r1376945 = r1376943 * r1376944;
        double r1376946 = c;
        double r1376947 = r1376940 * r1376940;
        double r1376948 = fma(r1376945, r1376946, r1376947);
        double r1376949 = sqrt(r1376948);
        double r1376950 = r1376949 * r1376948;
        double r1376951 = r1376947 * r1376940;
        double r1376952 = r1376950 - r1376951;
        double r1376953 = r1376940 + r1376949;
        double r1376954 = fma(r1376940, r1376953, r1376948);
        double r1376955 = r1376952 / r1376954;
        double r1376956 = r1376955 / r1376944;
        double r1376957 = 2.0;
        double r1376958 = r1376956 / r1376957;
        double r1376959 = -2.0;
        double r1376960 = r1376946 / r1376940;
        double r1376961 = r1376959 * r1376960;
        double r1376962 = r1376961 / r1376957;
        double r1376963 = r1376942 ? r1376958 : r1376962;
        return r1376963;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

  1. Split input into 2 regimes

  2. if b < 2297.8344311922956

    1. Initial program 17.7

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

    2. Simplified17.6

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

    4. Applied flip3--17.7

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

    5. Simplified17.1

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

    6. Simplified17.1

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

    if 2297.8344311922956 < b

    1. Initial program 37.1

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

    2. Simplified37.1

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

    3. Taylor expanded around inf 15.7

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

  4. Final simplification16.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le 2297.8344311922956:\\ \;\;\;\;\frac{\frac{\frac{\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)} \cdot \mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right) - \left(b \cdot b\right) \cdot b}{\mathsf{fma}\left(b, b + \sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)}, \mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)\right)}}{a}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2 \cdot \frac{c}{b}}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019162 +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 a) c)))) (* 2 a)))