Average Error: 44.1 → 11.2
Time: 14.3s
Precision: 64
\[1.1102230246251565 \cdot 10^{-16} \lt a \lt 9007199254740992.0 \land 1.1102230246251565 \cdot 10^{-16} \lt b \lt 9007199254740992.0 \land 1.1102230246251565 \cdot 10^{-16} \lt c \lt 9007199254740992.0\]
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le 0.10455593216828488:\\ \;\;\;\;\frac{\frac{\frac{\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)} \cdot \mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right) - \left(b \cdot b\right) \cdot b}{\mathsf{fma}\left(b, b + \sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)}, \mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)\right)}}{a}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2 \cdot \frac{c}{b}}{2}\\ \end{array}\]
\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 0.10455593216828488:\\
\;\;\;\;\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 r1520913 = b;
        double r1520914 = -r1520913;
        double r1520915 = r1520913 * r1520913;
        double r1520916 = 4.0;
        double r1520917 = a;
        double r1520918 = r1520916 * r1520917;
        double r1520919 = c;
        double r1520920 = r1520918 * r1520919;
        double r1520921 = r1520915 - r1520920;
        double r1520922 = sqrt(r1520921);
        double r1520923 = r1520914 + r1520922;
        double r1520924 = 2.0;
        double r1520925 = r1520924 * r1520917;
        double r1520926 = r1520923 / r1520925;
        return r1520926;
}
↓
double f(double a, double b, double c) {
        double r1520927 = b;
        double r1520928 = 0.10455593216828488;
        bool r1520929 = r1520927 <= r1520928;
        double r1520930 = -4.0;
        double r1520931 = a;
        double r1520932 = r1520930 * r1520931;
        double r1520933 = c;
        double r1520934 = r1520927 * r1520927;
        double r1520935 = fma(r1520932, r1520933, r1520934);
        double r1520936 = sqrt(r1520935);
        double r1520937 = r1520936 * r1520935;
        double r1520938 = r1520934 * r1520927;
        double r1520939 = r1520937 - r1520938;
        double r1520940 = r1520927 + r1520936;
        double r1520941 = fma(r1520927, r1520940, r1520935);
        double r1520942 = r1520939 / r1520941;
        double r1520943 = r1520942 / r1520931;
        double r1520944 = 2.0;
        double r1520945 = r1520943 / r1520944;
        double r1520946 = -2.0;
        double r1520947 = r1520933 / r1520927;
        double r1520948 = r1520946 * r1520947;
        double r1520949 = r1520948 / r1520944;
        double r1520950 = r1520929 ? r1520945 : r1520949;
        return r1520950;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

  1. Split input into 2 regimes

  2. if b < 0.10455593216828488

    1. Initial program 23.0

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

    2. Simplified23.0

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

    4. Applied flip3--23.0

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

    5. Simplified22.3

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

    6. Simplified22.4

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

    if 0.10455593216828488 < b

    1. Initial program 47.3

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

    2. Simplified47.2

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

    3. Taylor expanded around inf 9.5

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

  4. Final simplification11.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le 0.10455593216828488:\\ \;\;\;\;\frac{\frac{\frac{\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)} \cdot \mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right) - \left(b \cdot b\right) \cdot b}{\mathsf{fma}\left(b, b + \sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)}, \mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)\right)}}{a}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2 \cdot \frac{c}{b}}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019162 +o rules:numerics
(FPCore (a b c)
  :name "Quadratic roots, 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)))