Average Error: 28.6 → 15.1
Time: 4.7s
Precision: 64
\[1.053671212772350866701172186984739043147 \cdot 10^{-8} \lt a \lt 94906265.62425155937671661376953125 \land 1.053671212772350866701172186984739043147 \cdot 10^{-8} \lt b \lt 94906265.62425155937671661376953125 \land 1.053671212772350866701172186984739043147 \cdot 10^{-8} \lt c \lt 94906265.62425155937671661376953125\]
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \le -1.450958961586127673982441615985194971472 \cdot 10^{-5}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(b, b, -\left(b \cdot b - \left(3 \cdot a\right) \cdot c\right)\right)}{\left(-b\right) - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array}\]
\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}
↓
\begin{array}{l}
\mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \le -1.450958961586127673982441615985194971472 \cdot 10^{-5}:\\
\;\;\;\;\frac{\frac{\mathsf{fma}\left(b, b, -\left(b \cdot b - \left(3 \cdot a\right) \cdot c\right)\right)}{\left(-b\right) - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}}{3 \cdot a}\\

\mathbf{else}:\\
\;\;\;\;-0.5 \cdot \frac{c}{b}\\

\end{array}
double f(double a, double b, double c) {
        double r96981 = b;
        double r96982 = -r96981;
        double r96983 = r96981 * r96981;
        double r96984 = 3.0;
        double r96985 = a;
        double r96986 = r96984 * r96985;
        double r96987 = c;
        double r96988 = r96986 * r96987;
        double r96989 = r96983 - r96988;
        double r96990 = sqrt(r96989);
        double r96991 = r96982 + r96990;
        double r96992 = r96991 / r96986;
        return r96992;
}
↓
double f(double a, double b, double c) {
        double r96993 = b;
        double r96994 = -r96993;
        double r96995 = r96993 * r96993;
        double r96996 = 3.0;
        double r96997 = a;
        double r96998 = r96996 * r96997;
        double r96999 = c;
        double r97000 = r96998 * r96999;
        double r97001 = r96995 - r97000;
        double r97002 = sqrt(r97001);
        double r97003 = r96994 + r97002;
        double r97004 = r97003 / r96998;
        double r97005 = -1.4509589615861277e-05;
        bool r97006 = r97004 <= r97005;
        double r97007 = -r97001;
        double r97008 = fma(r96993, r96993, r97007);
        double r97009 = r96994 - r97002;
        double r97010 = r97008 / r97009;
        double r97011 = r97010 / r96998;
        double r97012 = -0.5;
        double r97013 = r96999 / r96993;
        double r97014 = r97012 * r97013;
        double r97015 = r97006 ? r97011 : r97014;
        return r97015;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

  1. Split input into 2 regimes

  2. if (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)) < -1.4509589615861277e-05

    1. Initial program 17.1

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

    3. Applied flip-+17.0

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

    4. Simplified16.2

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

    if -1.4509589615861277e-05 < (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a))

    1. Initial program 39.2

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

    2. Taylor expanded around inf 14.0

      \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b}}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification15.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \le -1.450958961586127673982441615985194971472 \cdot 10^{-5}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(b, b, -\left(b \cdot b - \left(3 \cdot a\right) \cdot c\right)\right)}{\left(-b\right) - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2020002 +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)))