Average Error: 28.4 → 16.5
Time: 24.0s
Precision: 64
\[1.053671212772350866701172186984739043147 \cdot 10^{-8} \lt a \lt 94906265.62425155937671661376953125 \land 1.053671212772350866701172186984739043147 \cdot 10^{-8} \lt b \lt 94906265.62425155937671661376953125 \land 1.053671212772350866701172186984739043147 \cdot 10^{-8} \lt c \lt 94906265.62425155937671661376953125\]
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le 710.3936434177851424465188756585121154785:\\ \;\;\;\;\frac{\frac{\frac{\left(b \cdot b - \left(a \cdot c\right) \cdot 3\right) - b \cdot b}{b + \sqrt{b \cdot b - \left(a \cdot c\right) \cdot 3}}}{a}}{3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array}\]
\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 710.3936434177851424465188756585121154785:\\
\;\;\;\;\frac{\frac{\frac{\left(b \cdot b - \left(a \cdot c\right) \cdot 3\right) - b \cdot b}{b + \sqrt{b \cdot b - \left(a \cdot c\right) \cdot 3}}}{a}}{3}\\

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

\end{array}
double f(double a, double b, double c) {
        double r92444 = b;
        double r92445 = -r92444;
        double r92446 = r92444 * r92444;
        double r92447 = 3.0;
        double r92448 = a;
        double r92449 = r92447 * r92448;
        double r92450 = c;
        double r92451 = r92449 * r92450;
        double r92452 = r92446 - r92451;
        double r92453 = sqrt(r92452);
        double r92454 = r92445 + r92453;
        double r92455 = r92454 / r92449;
        return r92455;
}
↓
double f(double a, double b, double c) {
        double r92456 = b;
        double r92457 = 710.3936434177851;
        bool r92458 = r92456 <= r92457;
        double r92459 = r92456 * r92456;
        double r92460 = a;
        double r92461 = c;
        double r92462 = r92460 * r92461;
        double r92463 = 3.0;
        double r92464 = r92462 * r92463;
        double r92465 = r92459 - r92464;
        double r92466 = r92465 - r92459;
        double r92467 = sqrt(r92465);
        double r92468 = r92456 + r92467;
        double r92469 = r92466 / r92468;
        double r92470 = r92469 / r92460;
        double r92471 = r92470 / r92463;
        double r92472 = r92461 / r92456;
        double r92473 = -0.5;
        double r92474 = r92472 * r92473;
        double r92475 = r92458 ? r92471 : r92474;
        return r92475;
}

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 < 710.3936434177851

    1. Initial program 17.2

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

    2. Simplified17.2

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

    4. Applied flip--17.2

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

    5. Simplified16.2

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

    6. Simplified16.2

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

    if 710.3936434177851 < b

    1. Initial program 35.7

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

    2. Simplified35.7

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

    3. Taylor expanded around inf 16.7

      \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b}}\]

    4. Simplified16.7

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

  4. Final simplification16.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le 710.3936434177851424465188756585121154785:\\ \;\;\;\;\frac{\frac{\frac{\left(b \cdot b - \left(a \cdot c\right) \cdot 3\right) - b \cdot b}{b + \sqrt{b \cdot b - \left(a \cdot c\right) \cdot 3}}}{a}}{3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array}\]

Reproduce

herbie shell --seed 2019194 
(FPCore (a b c)
  :name "Cubic critical, narrow range"
  :pre (and (< 1.0536712127723509e-08 a 94906265.62425156) (< 1.0536712127723509e-08 b 94906265.62425156) (< 1.0536712127723509e-08 c 94906265.62425156))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))