Average Error: 28.6 → 16.4
Time: 7.0s
Precision: 64
\[1.05367121277235087 \cdot 10^{-8} \lt a \lt 94906265.6242515594 \land 1.05367121277235087 \cdot 10^{-8} \lt b \lt 94906265.6242515594 \land 1.05367121277235087 \cdot 10^{-8} \lt c \lt 94906265.6242515594\]
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le 3187.18097597923543:\\ \;\;\;\;\frac{\frac{b \cdot b - \mathsf{fma}\left(b, b, \left(3 \cdot a\right) \cdot c\right)}{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} + b}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(a \cdot -1.5\right) \cdot c}{\left(3 \cdot a\right) \cdot b}\\ \end{array}\]
\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}
↓
\begin{array}{l}
\mathbf{if}\;b \le 3187.18097597923543:\\
\;\;\;\;\frac{\frac{b \cdot b - \mathsf{fma}\left(b, b, \left(3 \cdot a\right) \cdot c\right)}{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} + b}}{3 \cdot a}\\

\mathbf{else}:\\
\;\;\;\;\frac{\left(a \cdot -1.5\right) \cdot c}{\left(3 \cdot a\right) \cdot b}\\

\end{array}
double f(double a, double b, double c) {
        double r80983 = b;
        double r80984 = -r80983;
        double r80985 = r80983 * r80983;
        double r80986 = 3.0;
        double r80987 = a;
        double r80988 = r80986 * r80987;
        double r80989 = c;
        double r80990 = r80988 * r80989;
        double r80991 = r80985 - r80990;
        double r80992 = sqrt(r80991);
        double r80993 = r80984 + r80992;
        double r80994 = r80993 / r80988;
        return r80994;
}
↓
double f(double a, double b, double c) {
        double r80995 = b;
        double r80996 = 3187.1809759792354;
        bool r80997 = r80995 <= r80996;
        double r80998 = r80995 * r80995;
        double r80999 = 3.0;
        double r81000 = a;
        double r81001 = r80999 * r81000;
        double r81002 = c;
        double r81003 = r81001 * r81002;
        double r81004 = fma(r80995, r80995, r81003);
        double r81005 = r80998 - r81004;
        double r81006 = r80998 - r81003;
        double r81007 = sqrt(r81006);
        double r81008 = r81007 + r80995;
        double r81009 = r81005 / r81008;
        double r81010 = r81009 / r81001;
        double r81011 = -1.5;
        double r81012 = r81000 * r81011;
        double r81013 = r81012 * r81002;
        double r81014 = r81001 * r80995;
        double r81015 = r81013 / r81014;
        double r81016 = r80997 ? r81010 : r81015;
        return r81016;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

  1. Split input into 2 regimes

  2. if b < 3187.1809759792354

    1. Initial program 18.3

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]

    2. Simplified18.3

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

    4. Applied flip--18.3

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

    5. Simplified17.5

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

    if 3187.1809759792354 < b

    1. Initial program 37.4

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]

    2. Simplified37.4

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

    3. Taylor expanded around inf 15.5

      \[\leadsto \frac{\color{blue}{-1.5 \cdot \frac{a \cdot c}{b}}}{3 \cdot a}\]
    4. Using strategy rm

    5. Applied *-un-lft-identity15.5

      \[\leadsto \frac{-1.5 \cdot \frac{a \cdot c}{\color{blue}{1 \cdot b}}}{3 \cdot a}\]

    6. Applied times-frac15.5

      \[\leadsto \frac{-1.5 \cdot \color{blue}{\left(\frac{a}{1} \cdot \frac{c}{b}\right)}}{3 \cdot a}\]

    7. Applied associate-*r*15.4

      \[\leadsto \frac{\color{blue}{\left(-1.5 \cdot \frac{a}{1}\right) \cdot \frac{c}{b}}}{3 \cdot a}\]

    8. Simplified15.4

      \[\leadsto \frac{\color{blue}{\left(a \cdot -1.5\right)} \cdot \frac{c}{b}}{3 \cdot a}\]
    9. Using strategy rm

    10. Applied associate-*r/15.5

      \[\leadsto \frac{\color{blue}{\frac{\left(a \cdot -1.5\right) \cdot c}{b}}}{3 \cdot a}\]

    11. Applied associate-/l/15.5

      \[\leadsto \color{blue}{\frac{\left(a \cdot -1.5\right) \cdot c}{\left(3 \cdot a\right) \cdot b}}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification16.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le 3187.18097597923543:\\ \;\;\;\;\frac{\frac{b \cdot b - \mathsf{fma}\left(b, b, \left(3 \cdot a\right) \cdot c\right)}{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} + b}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(a \cdot -1.5\right) \cdot c}{\left(3 \cdot a\right) \cdot b}\\ \end{array}\]

Reproduce

herbie shell --seed 2020047 +o rules:numerics
(FPCore (a b c)
  :name "Cubic critical, narrow range"
  :precision binary64
  :pre (and (< 1.0536712127723509e-08 a 94906265.62425156) (< 1.0536712127723509e-08 b 94906265.62425156) (< 1.0536712127723509e-08 c 94906265.62425156))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3 a) c)))) (* 3 a)))