Average Error: 44.3 → 10.8
Time: 15.6s
Precision: 64
\[1.11022 \cdot 10^{-16} \lt a \lt 9.0072 \cdot 10^{15} \land 1.11022 \cdot 10^{-16} \lt b \lt 9.0072 \cdot 10^{15} \land 1.11022 \cdot 10^{-16} \lt c \lt 9.0072 \cdot 10^{15}\]
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le 0.0357101210953405454:\\ \;\;\;\;\frac{\frac{b \cdot b - \left(b \cdot b - 3 \cdot \left(a \cdot c\right)\right)}{\left(-b\right) - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array}\]
\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}
↓
\begin{array}{l}
\mathbf{if}\;b \le 0.0357101210953405454:\\
\;\;\;\;\frac{\frac{b \cdot b - \left(b \cdot b - 3 \cdot \left(a \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 r72588 = b;
        double r72589 = -r72588;
        double r72590 = r72588 * r72588;
        double r72591 = 3.0;
        double r72592 = a;
        double r72593 = r72591 * r72592;
        double r72594 = c;
        double r72595 = r72593 * r72594;
        double r72596 = r72590 - r72595;
        double r72597 = sqrt(r72596);
        double r72598 = r72589 + r72597;
        double r72599 = r72598 / r72593;
        return r72599;
}
↓
double f(double a, double b, double c) {
        double r72600 = b;
        double r72601 = 0.035710121095340545;
        bool r72602 = r72600 <= r72601;
        double r72603 = r72600 * r72600;
        double r72604 = 3.0;
        double r72605 = a;
        double r72606 = c;
        double r72607 = r72605 * r72606;
        double r72608 = r72604 * r72607;
        double r72609 = r72603 - r72608;
        double r72610 = r72603 - r72609;
        double r72611 = -r72600;
        double r72612 = r72604 * r72605;
        double r72613 = r72612 * r72606;
        double r72614 = r72603 - r72613;
        double r72615 = sqrt(r72614);
        double r72616 = r72611 - r72615;
        double r72617 = r72610 / r72616;
        double r72618 = r72617 / r72612;
        double r72619 = -0.5;
        double r72620 = r72606 / r72600;
        double r72621 = r72619 * r72620;
        double r72622 = r72602 ? r72618 : r72621;
        return r72622;
}

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 < 0.035710121095340545

    1. Initial program 21.3

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

    3. Applied flip-+21.2

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

    4. Simplified20.2

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

    if 0.035710121095340545 < b

    1. Initial program 47.1

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

    2. Taylor expanded around inf 9.6

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

  4. Final simplification10.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le 0.0357101210953405454:\\ \;\;\;\;\frac{\frac{b \cdot b - \left(b \cdot b - 3 \cdot \left(a \cdot c\right)\right)}{\left(-b\right) - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2019195 
(FPCore (a b c)
  :name "Cubic critical, medium range"
  :pre (and (< 1.1102230246251565e-16 a 9007199254740992.0) (< 1.1102230246251565e-16 b 9007199254740992.0) (< 1.1102230246251565e-16 c 9007199254740992.0))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))