Average Error: 43.8 → 11.2
Time: 25.3s
Precision: 64
\[1.1102230246251565 \cdot 10^{-16} \lt a \lt 9007199254740992.0 \land 1.1102230246251565 \cdot 10^{-16} \lt b \lt 9007199254740992.0 \land 1.1102230246251565 \cdot 10^{-16} \lt c \lt 9007199254740992.0\]
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le 0.002027355084314286:\\ \;\;\;\;\frac{\frac{\left(\left(a \cdot c\right) \cdot -3 + b \cdot b\right) \cdot \sqrt{\left(a \cdot c\right) \cdot -3 + b \cdot b} - b \cdot \left(b \cdot b\right)}{\left(\left(a \cdot c\right) \cdot -3 + b \cdot b\right) + \left(b \cdot b + b \cdot \sqrt{\left(a \cdot c\right) \cdot -3 + b \cdot b}\right)}}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b}\\ \end{array}\]
\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 0.002027355084314286:\\
\;\;\;\;\frac{\frac{\left(\left(a \cdot c\right) \cdot -3 + b \cdot b\right) \cdot \sqrt{\left(a \cdot c\right) \cdot -3 + b \cdot b} - b \cdot \left(b \cdot b\right)}{\left(\left(a \cdot c\right) \cdot -3 + b \cdot b\right) + \left(b \cdot b + b \cdot \sqrt{\left(a \cdot c\right) \cdot -3 + b \cdot b}\right)}}{a \cdot 3}\\

\mathbf{else}:\\
\;\;\;\;\frac{-1}{2} \cdot \frac{c}{b}\\

\end{array}
double f(double a, double b, double c) {
        double r3456668 = b;
        double r3456669 = -r3456668;
        double r3456670 = r3456668 * r3456668;
        double r3456671 = 3.0;
        double r3456672 = a;
        double r3456673 = r3456671 * r3456672;
        double r3456674 = c;
        double r3456675 = r3456673 * r3456674;
        double r3456676 = r3456670 - r3456675;
        double r3456677 = sqrt(r3456676);
        double r3456678 = r3456669 + r3456677;
        double r3456679 = r3456678 / r3456673;
        return r3456679;
}
↓
double f(double a, double b, double c) {
        double r3456680 = b;
        double r3456681 = 0.002027355084314286;
        bool r3456682 = r3456680 <= r3456681;
        double r3456683 = a;
        double r3456684 = c;
        double r3456685 = r3456683 * r3456684;
        double r3456686 = -3.0;
        double r3456687 = r3456685 * r3456686;
        double r3456688 = r3456680 * r3456680;
        double r3456689 = r3456687 + r3456688;
        double r3456690 = sqrt(r3456689);
        double r3456691 = r3456689 * r3456690;
        double r3456692 = r3456680 * r3456688;
        double r3456693 = r3456691 - r3456692;
        double r3456694 = r3456680 * r3456690;
        double r3456695 = r3456688 + r3456694;
        double r3456696 = r3456689 + r3456695;
        double r3456697 = r3456693 / r3456696;
        double r3456698 = 3.0;
        double r3456699 = r3456683 * r3456698;
        double r3456700 = r3456697 / r3456699;
        double r3456701 = -0.5;
        double r3456702 = r3456684 / r3456680;
        double r3456703 = r3456701 * r3456702;
        double r3456704 = r3456682 ? r3456700 : r3456703;
        return r3456704;
}

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 < 0.002027355084314286

    1. Initial program 20.7

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

    2. Simplified20.7

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

    4. Applied flip3--20.8

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

    5. Simplified20.0

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

    6. Simplified20.0

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

    if 0.002027355084314286 < b

    1. Initial program 46.2

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

    2. Simplified46.2

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

    3. Taylor expanded around inf 10.6

      \[\leadsto \frac{\color{blue}{\frac{-3}{2} \cdot \frac{a \cdot c}{b}}}{3 \cdot a}\]

    4. Taylor expanded around 0 10.3

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{c}{b}}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification11.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le 0.002027355084314286:\\ \;\;\;\;\frac{\frac{\left(\left(a \cdot c\right) \cdot -3 + b \cdot b\right) \cdot \sqrt{\left(a \cdot c\right) \cdot -3 + b \cdot b} - b \cdot \left(b \cdot b\right)}{\left(\left(a \cdot c\right) \cdot -3 + b \cdot b\right) + \left(b \cdot b + b \cdot \sqrt{\left(a \cdot c\right) \cdot -3 + b \cdot b}\right)}}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2019138 
(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 a) c)))) (* 3 a)))