Average Error: 28.6 → 16.4
Time: 6.9s
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 r143489 = b;
        double r143490 = -r143489;
        double r143491 = r143489 * r143489;
        double r143492 = 3.0;
        double r143493 = a;
        double r143494 = r143492 * r143493;
        double r143495 = c;
        double r143496 = r143494 * r143495;
        double r143497 = r143491 - r143496;
        double r143498 = sqrt(r143497);
        double r143499 = r143490 + r143498;
        double r143500 = r143499 / r143494;
        return r143500;
}
↓
double f(double a, double b, double c) {
        double r143501 = b;
        double r143502 = 3187.1809759792354;
        bool r143503 = r143501 <= r143502;
        double r143504 = r143501 * r143501;
        double r143505 = 3.0;
        double r143506 = a;
        double r143507 = r143505 * r143506;
        double r143508 = c;
        double r143509 = r143507 * r143508;
        double r143510 = fma(r143501, r143501, r143509);
        double r143511 = r143504 - r143510;
        double r143512 = r143504 - r143509;
        double r143513 = sqrt(r143512);
        double r143514 = r143513 + r143501;
        double r143515 = r143511 / r143514;
        double r143516 = r143515 / r143507;
        double r143517 = -1.5;
        double r143518 = r143506 * r143517;
        double r143519 = r143518 * r143508;
        double r143520 = r143507 * r143501;
        double r143521 = r143519 / r143520;
        double r143522 = r143503 ? r143516 : r143521;
        return r143522;
}

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)))