Average Error: 43.8 → 11.2
Time: 20.5s
Precision: 64
\[1.1102230246251565 \cdot 10^{-16} \lt a \lt 9007199254740992.0 \land 1.1102230246251565 \cdot 10^{-16} \lt b \lt 9007199254740992.0 \land 1.1102230246251565 \cdot 10^{-16} \lt c \lt 9007199254740992.0\]
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le 0.014244826613571458:\\ \;\;\;\;\frac{\frac{\frac{\left(\left(a \cdot -4\right) \cdot c + b \cdot b\right) \cdot \sqrt{\left(a \cdot -4\right) \cdot c + b \cdot b} - b \cdot \left(b \cdot b\right)}{\left(b \cdot b + \left(\left(a \cdot -4\right) \cdot c + b \cdot b\right)\right) + b \cdot \sqrt{\left(a \cdot -4\right) \cdot c + b \cdot b}}}{a}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2 \cdot \frac{c}{b}}{2}\\ \end{array}\]
\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}
↓
\begin{array}{l}
\mathbf{if}\;b \le 0.014244826613571458:\\
\;\;\;\;\frac{\frac{\frac{\left(\left(a \cdot -4\right) \cdot c + b \cdot b\right) \cdot \sqrt{\left(a \cdot -4\right) \cdot c + b \cdot b} - b \cdot \left(b \cdot b\right)}{\left(b \cdot b + \left(\left(a \cdot -4\right) \cdot c + b \cdot b\right)\right) + b \cdot \sqrt{\left(a \cdot -4\right) \cdot c + b \cdot b}}}{a}}{2}\\

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

\end{array}
double f(double a, double b, double c) {
        double r997616 = b;
        double r997617 = -r997616;
        double r997618 = r997616 * r997616;
        double r997619 = 4.0;
        double r997620 = a;
        double r997621 = r997619 * r997620;
        double r997622 = c;
        double r997623 = r997621 * r997622;
        double r997624 = r997618 - r997623;
        double r997625 = sqrt(r997624);
        double r997626 = r997617 + r997625;
        double r997627 = 2.0;
        double r997628 = r997627 * r997620;
        double r997629 = r997626 / r997628;
        return r997629;
}
↓
double f(double a, double b, double c) {
        double r997630 = b;
        double r997631 = 0.014244826613571458;
        bool r997632 = r997630 <= r997631;
        double r997633 = a;
        double r997634 = -4.0;
        double r997635 = r997633 * r997634;
        double r997636 = c;
        double r997637 = r997635 * r997636;
        double r997638 = r997630 * r997630;
        double r997639 = r997637 + r997638;
        double r997640 = sqrt(r997639);
        double r997641 = r997639 * r997640;
        double r997642 = r997630 * r997638;
        double r997643 = r997641 - r997642;
        double r997644 = r997638 + r997639;
        double r997645 = r997630 * r997640;
        double r997646 = r997644 + r997645;
        double r997647 = r997643 / r997646;
        double r997648 = r997647 / r997633;
        double r997649 = 2.0;
        double r997650 = r997648 / r997649;
        double r997651 = -2.0;
        double r997652 = r997636 / r997630;
        double r997653 = r997651 * r997652;
        double r997654 = r997653 / r997649;
        double r997655 = r997632 ? r997650 : r997654;
        return r997655;
}

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 < 0.014244826613571458

    1. Initial program 21.9

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]

    2. Simplified21.9

      \[\leadsto \color{blue}{\frac{\frac{\sqrt{b \cdot b - \left(c \cdot a\right) \cdot 4} - b}{a}}{2}}\]
    3. Using strategy rm

    4. Applied flip3--22.0

      \[\leadsto \frac{\frac{\color{blue}{\frac{{\left(\sqrt{b \cdot b - \left(c \cdot a\right) \cdot 4}\right)}^{3} - {b}^{3}}{\sqrt{b \cdot b - \left(c \cdot a\right) \cdot 4} \cdot \sqrt{b \cdot b - \left(c \cdot a\right) \cdot 4} + \left(b \cdot b + \sqrt{b \cdot b - \left(c \cdot a\right) \cdot 4} \cdot b\right)}}}{a}}{2}\]

    5. Simplified21.4

      \[\leadsto \frac{\frac{\frac{\color{blue}{\sqrt{\left(a \cdot -4\right) \cdot c + b \cdot b} \cdot \left(\left(a \cdot -4\right) \cdot c + b \cdot b\right) - \left(b \cdot b\right) \cdot b}}{\sqrt{b \cdot b - \left(c \cdot a\right) \cdot 4} \cdot \sqrt{b \cdot b - \left(c \cdot a\right) \cdot 4} + \left(b \cdot b + \sqrt{b \cdot b - \left(c \cdot a\right) \cdot 4} \cdot b\right)}}{a}}{2}\]

    6. Simplified21.3

      \[\leadsto \frac{\frac{\frac{\sqrt{\left(a \cdot -4\right) \cdot c + b \cdot b} \cdot \left(\left(a \cdot -4\right) \cdot c + b \cdot b\right) - \left(b \cdot b\right) \cdot b}{\color{blue}{\left(\left(\left(a \cdot -4\right) \cdot c + b \cdot b\right) + b \cdot b\right) + b \cdot \sqrt{\left(a \cdot -4\right) \cdot c + b \cdot b}}}}{a}}{2}\]

    if 0.014244826613571458 < b

    1. Initial program 46.6

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]

    2. Simplified46.6

      \[\leadsto \color{blue}{\frac{\frac{\sqrt{b \cdot b - \left(c \cdot a\right) \cdot 4} - b}{a}}{2}}\]

    3. Taylor expanded around inf 10.0

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

  4. Final simplification11.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le 0.014244826613571458:\\ \;\;\;\;\frac{\frac{\frac{\left(\left(a \cdot -4\right) \cdot c + b \cdot b\right) \cdot \sqrt{\left(a \cdot -4\right) \cdot c + b \cdot b} - b \cdot \left(b \cdot b\right)}{\left(b \cdot b + \left(\left(a \cdot -4\right) \cdot c + b \cdot b\right)\right) + b \cdot \sqrt{\left(a \cdot -4\right) \cdot c + b \cdot b}}}{a}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2 \cdot \frac{c}{b}}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019143 
(FPCore (a b c)
  :name "Quadratic roots, 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) (* (* 4 a) c)))) (* 2 a)))