Average Error: 43.4 → 11.5
Time: 15.3s
Precision: 64
\[1.1102230246251565 \cdot 10^{-16} \lt a \lt 9007199254740992.0 \land 1.1102230246251565 \cdot 10^{-16} \lt b \lt 9007199254740992.0 \land 1.1102230246251565 \cdot 10^{-16} \lt c \lt 9007199254740992.0\]
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le 0.0012535322255036849:\\ \;\;\;\;\frac{\frac{1}{\sqrt[3]{a} \cdot \sqrt[3]{a}} \cdot \frac{\sqrt{b \cdot b - \left(a \cdot c\right) \cdot 4} - b}{\sqrt[3]{a}}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2 \cdot \frac{c}{b}}{2}\\ \end{array}\]
\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 0.0012535322255036849:\\
\;\;\;\;\frac{\frac{1}{\sqrt[3]{a} \cdot \sqrt[3]{a}} \cdot \frac{\sqrt{b \cdot b - \left(a \cdot c\right) \cdot 4} - b}{\sqrt[3]{a}}}{2}\\

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

\end{array}
double f(double a, double b, double c) {
        double r1386438 = b;
        double r1386439 = -r1386438;
        double r1386440 = r1386438 * r1386438;
        double r1386441 = 4.0;
        double r1386442 = a;
        double r1386443 = r1386441 * r1386442;
        double r1386444 = c;
        double r1386445 = r1386443 * r1386444;
        double r1386446 = r1386440 - r1386445;
        double r1386447 = sqrt(r1386446);
        double r1386448 = r1386439 + r1386447;
        double r1386449 = 2.0;
        double r1386450 = r1386449 * r1386442;
        double r1386451 = r1386448 / r1386450;
        return r1386451;
}
↓
double f(double a, double b, double c) {
        double r1386452 = b;
        double r1386453 = 0.0012535322255036849;
        bool r1386454 = r1386452 <= r1386453;
        double r1386455 = 1.0;
        double r1386456 = a;
        double r1386457 = cbrt(r1386456);
        double r1386458 = r1386457 * r1386457;
        double r1386459 = r1386455 / r1386458;
        double r1386460 = r1386452 * r1386452;
        double r1386461 = c;
        double r1386462 = r1386456 * r1386461;
        double r1386463 = 4.0;
        double r1386464 = r1386462 * r1386463;
        double r1386465 = r1386460 - r1386464;
        double r1386466 = sqrt(r1386465);
        double r1386467 = r1386466 - r1386452;
        double r1386468 = r1386467 / r1386457;
        double r1386469 = r1386459 * r1386468;
        double r1386470 = 2.0;
        double r1386471 = r1386469 / r1386470;
        double r1386472 = -2.0;
        double r1386473 = r1386461 / r1386452;
        double r1386474 = r1386472 * r1386473;
        double r1386475 = r1386474 / r1386470;
        double r1386476 = r1386454 ? r1386471 : r1386475;
        return r1386476;
}

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 < 0.0012535322255036849

    1. Initial program 20.2

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

    2. Simplified20.2

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

    4. Applied add-cube-cbrt20.3

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

    5. Applied *-un-lft-identity20.3

      \[\leadsto \frac{\frac{\color{blue}{1 \cdot \left(\sqrt{b \cdot b - \left(c \cdot a\right) \cdot 4} - b\right)}}{\left(\sqrt[3]{a} \cdot \sqrt[3]{a}\right) \cdot \sqrt[3]{a}}}{2}\]

    6. Applied times-frac20.3

      \[\leadsto \frac{\color{blue}{\frac{1}{\sqrt[3]{a} \cdot \sqrt[3]{a}} \cdot \frac{\sqrt{b \cdot b - \left(c \cdot a\right) \cdot 4} - b}{\sqrt[3]{a}}}}{2}\]

    if 0.0012535322255036849 < b

    1. Initial program 45.7

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

    2. Simplified45.7

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

    3. Taylor expanded around inf 10.6

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

  4. Final simplification11.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le 0.0012535322255036849:\\ \;\;\;\;\frac{\frac{1}{\sqrt[3]{a} \cdot \sqrt[3]{a}} \cdot \frac{\sqrt{b \cdot b - \left(a \cdot c\right) \cdot 4} - b}{\sqrt[3]{a}}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2 \cdot \frac{c}{b}}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019168 
(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)))