Average Error: 28.3 → 16.9
Time: 15.4s
Precision: 64
\[1.053671212772350866701172186984739043147 \cdot 10^{-8} \lt a \lt 94906265.62425155937671661376953125 \land 1.053671212772350866701172186984739043147 \cdot 10^{-8} \lt b \lt 94906265.62425155937671661376953125 \land 1.053671212772350866701172186984739043147 \cdot 10^{-8} \lt c \lt 94906265.62425155937671661376953125\]
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le 22041.35648233499887282960116863250732422:\\ \;\;\;\;\frac{\frac{\frac{b \cdot b - \left(\left(3 \cdot a\right) \cdot c + b \cdot b\right)}{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} + b}}{3}}{a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array}\]
\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}
↓
\begin{array}{l}
\mathbf{if}\;b \le 22041.35648233499887282960116863250732422:\\
\;\;\;\;\frac{\frac{\frac{b \cdot b - \left(\left(3 \cdot a\right) \cdot c + b \cdot b\right)}{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} + b}}{3}}{a}\\

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

\end{array}
double f(double a, double b, double c) {
        double r55720 = b;
        double r55721 = -r55720;
        double r55722 = r55720 * r55720;
        double r55723 = 3.0;
        double r55724 = a;
        double r55725 = r55723 * r55724;
        double r55726 = c;
        double r55727 = r55725 * r55726;
        double r55728 = r55722 - r55727;
        double r55729 = sqrt(r55728);
        double r55730 = r55721 + r55729;
        double r55731 = r55730 / r55725;
        return r55731;
}
↓
double f(double a, double b, double c) {
        double r55732 = b;
        double r55733 = 22041.356482335;
        bool r55734 = r55732 <= r55733;
        double r55735 = r55732 * r55732;
        double r55736 = 3.0;
        double r55737 = a;
        double r55738 = r55736 * r55737;
        double r55739 = c;
        double r55740 = r55738 * r55739;
        double r55741 = r55740 + r55735;
        double r55742 = r55735 - r55741;
        double r55743 = r55735 - r55740;
        double r55744 = sqrt(r55743);
        double r55745 = r55744 + r55732;
        double r55746 = r55742 / r55745;
        double r55747 = r55746 / r55736;
        double r55748 = r55747 / r55737;
        double r55749 = -0.5;
        double r55750 = r55739 / r55732;
        double r55751 = r55749 * r55750;
        double r55752 = r55734 ? r55748 : r55751;
        return r55752;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 2 regimes

  2. if b < 22041.356482335

    1. Initial program 19.9

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

    2. Simplified19.9

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

    4. Applied flip--20.0

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

    5. Simplified19.0

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

    if 22041.356482335 < b

    1. Initial program 38.7

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

    2. Simplified38.7

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

    3. Taylor expanded around inf 14.5

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

    4. Taylor expanded around 0 14.4

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

  4. Final simplification16.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le 22041.35648233499887282960116863250732422:\\ \;\;\;\;\frac{\frac{\frac{b \cdot b - \left(\left(3 \cdot a\right) \cdot c + b \cdot b\right)}{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} + b}}{3}}{a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2019235 
(FPCore (a b c)
  :name "Cubic critical, narrow range"
  :precision binary64
  :pre (and (< 1.05367121277235087e-8 a 94906265.6242515594) (< 1.05367121277235087e-8 b 94906265.6242515594) (< 1.05367121277235087e-8 c 94906265.6242515594))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3 a) c)))) (* 3 a)))