Average Error: 28.5 → 16.3
Time: 33.4s
Precision: 64
\[1.0536712127723509 \cdot 10^{-08} \lt a \lt 94906265.62425156 \land 1.0536712127723509 \cdot 10^{-08} \lt b \lt 94906265.62425156 \land 1.0536712127723509 \cdot 10^{-08} \lt c \lt 94906265.62425156\]
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le 1647.131571019072:\\ \;\;\;\;\frac{\frac{\frac{\sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)} \cdot \mathsf{fma}\left(a \cdot c, -4, b \cdot b\right) - \left(b \cdot b\right) \cdot b}{\mathsf{fma}\left(b, \sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)}, b \cdot b + \mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)\right)}}{a}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-2 \cdot \frac{\frac{1}{\frac{\sqrt{b}}{a}}}{\frac{\sqrt{b}}{c}}}{a}}{2}\\ \end{array}\]
\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 1647.131571019072:\\
\;\;\;\;\frac{\frac{\frac{\sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)} \cdot \mathsf{fma}\left(a \cdot c, -4, b \cdot b\right) - \left(b \cdot b\right) \cdot b}{\mathsf{fma}\left(b, \sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)}, b \cdot b + \mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)\right)}}{a}}{2}\\

\mathbf{else}:\\
\;\;\;\;\frac{\frac{-2 \cdot \frac{\frac{1}{\frac{\sqrt{b}}{a}}}{\frac{\sqrt{b}}{c}}}{a}}{2}\\

\end{array}
double f(double a, double b, double c) {
        double r1188891 = b;
        double r1188892 = -r1188891;
        double r1188893 = r1188891 * r1188891;
        double r1188894 = 4.0;
        double r1188895 = a;
        double r1188896 = r1188894 * r1188895;
        double r1188897 = c;
        double r1188898 = r1188896 * r1188897;
        double r1188899 = r1188893 - r1188898;
        double r1188900 = sqrt(r1188899);
        double r1188901 = r1188892 + r1188900;
        double r1188902 = 2.0;
        double r1188903 = r1188902 * r1188895;
        double r1188904 = r1188901 / r1188903;
        return r1188904;
}
↓
double f(double a, double b, double c) {
        double r1188905 = b;
        double r1188906 = 1647.131571019072;
        bool r1188907 = r1188905 <= r1188906;
        double r1188908 = a;
        double r1188909 = c;
        double r1188910 = r1188908 * r1188909;
        double r1188911 = -4.0;
        double r1188912 = r1188905 * r1188905;
        double r1188913 = fma(r1188910, r1188911, r1188912);
        double r1188914 = sqrt(r1188913);
        double r1188915 = r1188914 * r1188913;
        double r1188916 = r1188912 * r1188905;
        double r1188917 = r1188915 - r1188916;
        double r1188918 = r1188912 + r1188913;
        double r1188919 = fma(r1188905, r1188914, r1188918);
        double r1188920 = r1188917 / r1188919;
        double r1188921 = r1188920 / r1188908;
        double r1188922 = 2.0;
        double r1188923 = r1188921 / r1188922;
        double r1188924 = -2.0;
        double r1188925 = 1.0;
        double r1188926 = sqrt(r1188905);
        double r1188927 = r1188926 / r1188908;
        double r1188928 = r1188925 / r1188927;
        double r1188929 = r1188926 / r1188909;
        double r1188930 = r1188928 / r1188929;
        double r1188931 = r1188924 * r1188930;
        double r1188932 = r1188931 / r1188908;
        double r1188933 = r1188932 / r1188922;
        double r1188934 = r1188907 ? r1188923 : r1188933;
        return r1188934;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

  1. Split input into 2 regimes

  2. if b < 1647.131571019072

    1. Initial program 17.3

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

    2. Simplified17.2

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

    4. Applied flip3--17.3

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

    5. Simplified16.7

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

    6. Simplified16.7

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

    if 1647.131571019072 < b

    1. Initial program 36.6

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

    2. Simplified36.6

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

    3. Taylor expanded around inf 16.0

      \[\leadsto \frac{\frac{\color{blue}{-2 \cdot \frac{a \cdot c}{b}}}{a}}{2}\]
    4. Using strategy rm

    5. Applied clear-num16.0

      \[\leadsto \frac{\frac{-2 \cdot \color{blue}{\frac{1}{\frac{b}{a \cdot c}}}}{a}}{2}\]
    6. Using strategy rm

    7. Applied add-sqr-sqrt16.1

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

    8. Applied times-frac16.1

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

    9. Applied associate-/r*16.1

      \[\leadsto \frac{\frac{-2 \cdot \color{blue}{\frac{\frac{1}{\frac{\sqrt{b}}{a}}}{\frac{\sqrt{b}}{c}}}}{a}}{2}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification16.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le 1647.131571019072:\\ \;\;\;\;\frac{\frac{\frac{\sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)} \cdot \mathsf{fma}\left(a \cdot c, -4, b \cdot b\right) - \left(b \cdot b\right) \cdot b}{\mathsf{fma}\left(b, \sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)}, b \cdot b + \mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)\right)}}{a}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-2 \cdot \frac{\frac{1}{\frac{\sqrt{b}}{a}}}{\frac{\sqrt{b}}{c}}}{a}}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019149 +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)))