Average Error: 32.8 → 10.0
Time: 25.4s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -3.063397748446981 \cdot 10^{+71}:\\ \;\;\;\;\frac{\left(\frac{c}{b} - \frac{b}{a}\right) \cdot 2}{2}\\ \mathbf{elif}\;b \le 3.1295384133612364 \cdot 10^{-73}:\\ \;\;\;\;\frac{\frac{1}{a} \cdot \left(\sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)} - b\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{c}{b} \cdot -2}{2}\\ \end{array}\]
\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 -3.063397748446981 \cdot 10^{+71}:\\
\;\;\;\;\frac{\left(\frac{c}{b} - \frac{b}{a}\right) \cdot 2}{2}\\

\mathbf{elif}\;b \le 3.1295384133612364 \cdot 10^{-73}:\\
\;\;\;\;\frac{\frac{1}{a} \cdot \left(\sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)} - b\right)}{2}\\

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

\end{array}
double f(double a, double b, double c) {
        double r1584695 = b;
        double r1584696 = -r1584695;
        double r1584697 = r1584695 * r1584695;
        double r1584698 = 4.0;
        double r1584699 = a;
        double r1584700 = r1584698 * r1584699;
        double r1584701 = c;
        double r1584702 = r1584700 * r1584701;
        double r1584703 = r1584697 - r1584702;
        double r1584704 = sqrt(r1584703);
        double r1584705 = r1584696 + r1584704;
        double r1584706 = 2.0;
        double r1584707 = r1584706 * r1584699;
        double r1584708 = r1584705 / r1584707;
        return r1584708;
}
↓
double f(double a, double b, double c) {
        double r1584709 = b;
        double r1584710 = -3.063397748446981e+71;
        bool r1584711 = r1584709 <= r1584710;
        double r1584712 = c;
        double r1584713 = r1584712 / r1584709;
        double r1584714 = a;
        double r1584715 = r1584709 / r1584714;
        double r1584716 = r1584713 - r1584715;
        double r1584717 = 2.0;
        double r1584718 = r1584716 * r1584717;
        double r1584719 = r1584718 / r1584717;
        double r1584720 = 3.1295384133612364e-73;
        bool r1584721 = r1584709 <= r1584720;
        double r1584722 = 1.0;
        double r1584723 = r1584722 / r1584714;
        double r1584724 = r1584709 * r1584709;
        double r1584725 = 4.0;
        double r1584726 = r1584712 * r1584714;
        double r1584727 = r1584725 * r1584726;
        double r1584728 = r1584724 - r1584727;
        double r1584729 = sqrt(r1584728);
        double r1584730 = r1584729 - r1584709;
        double r1584731 = r1584723 * r1584730;
        double r1584732 = r1584731 / r1584717;
        double r1584733 = -2.0;
        double r1584734 = r1584713 * r1584733;
        double r1584735 = r1584734 / r1584717;
        double r1584736 = r1584721 ? r1584732 : r1584735;
        double r1584737 = r1584711 ? r1584719 : r1584736;
        return r1584737;
}

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 3 regimes

  2. if b < -3.063397748446981e+71

    1. Initial program 38.6

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

    2. Simplified38.5

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

    4. Applied div-inv38.6

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

    5. Taylor expanded around -inf 4.7

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

    6. Simplified4.7

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

    if -3.063397748446981e+71 < b < 3.1295384133612364e-73

    1. Initial program 13.0

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

    2. Simplified13.0

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

    4. Applied div-inv13.2

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

    if 3.1295384133612364e-73 < b

    1. Initial program 52.3

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

    2. Simplified52.3

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

    3. Taylor expanded around inf 9.0

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

  4. Final simplification10.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -3.063397748446981 \cdot 10^{+71}:\\ \;\;\;\;\frac{\left(\frac{c}{b} - \frac{b}{a}\right) \cdot 2}{2}\\ \mathbf{elif}\;b \le 3.1295384133612364 \cdot 10^{-73}:\\ \;\;\;\;\frac{\frac{1}{a} \cdot \left(\sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)} - b\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{c}{b} \cdot -2}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019152 
(FPCore (a b c)
  :name "Quadratic roots, full range"
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4 a) c)))) (* 2 a)))