Average Error: 28.6 → 16.5
Time: 15.5s
Precision: 64
\[1.053671212772350866701172186984739043147 \cdot 10^{-8} \lt a \lt 94906265.62425155937671661376953125 \land 1.053671212772350866701172186984739043147 \cdot 10^{-8} \lt b \lt 94906265.62425155937671661376953125 \land 1.053671212772350866701172186984739043147 \cdot 10^{-8} \lt c \lt 94906265.62425155937671661376953125\]
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le 183.1843483503574816495529375970363616943:\\ \;\;\;\;\frac{\frac{\frac{\left(b \cdot b - \left(3 \cdot a\right) \cdot c\right) - b \cdot b}{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} + b}}{3}}{a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array}\]
\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 183.1843483503574816495529375970363616943:\\
\;\;\;\;\frac{\frac{\frac{\left(b \cdot b - \left(3 \cdot a\right) \cdot c\right) - b \cdot b}{\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 r38562 = b;
        double r38563 = -r38562;
        double r38564 = r38562 * r38562;
        double r38565 = 3.0;
        double r38566 = a;
        double r38567 = r38565 * r38566;
        double r38568 = c;
        double r38569 = r38567 * r38568;
        double r38570 = r38564 - r38569;
        double r38571 = sqrt(r38570);
        double r38572 = r38563 + r38571;
        double r38573 = r38572 / r38567;
        return r38573;
}
↓
double f(double a, double b, double c) {
        double r38574 = b;
        double r38575 = 183.18434835035748;
        bool r38576 = r38574 <= r38575;
        double r38577 = r38574 * r38574;
        double r38578 = 3.0;
        double r38579 = a;
        double r38580 = r38578 * r38579;
        double r38581 = c;
        double r38582 = r38580 * r38581;
        double r38583 = r38577 - r38582;
        double r38584 = r38583 - r38577;
        double r38585 = sqrt(r38583);
        double r38586 = r38585 + r38574;
        double r38587 = r38584 / r38586;
        double r38588 = r38587 / r38578;
        double r38589 = r38588 / r38579;
        double r38590 = -0.5;
        double r38591 = r38581 / r38574;
        double r38592 = r38590 * r38591;
        double r38593 = r38576 ? r38589 : r38592;
        return r38593;
}

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 < 183.18434835035748

    1. Initial program 15.9

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

    2. Simplified15.9

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

    4. Applied flip--15.9

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

    5. Simplified14.9

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

    if 183.18434835035748 < b

    1. Initial program 35.0

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

    2. Simplified35.0

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

    3. Taylor expanded around inf 17.2

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

  4. Final simplification16.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le 183.1843483503574816495529375970363616943:\\ \;\;\;\;\frac{\frac{\frac{\left(b \cdot b - \left(3 \cdot a\right) \cdot c\right) - b \cdot b}{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} + b}}{3}}{a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2019305 
(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)))