Average Error: 34.4 → 10.0
Time: 16.8s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -2.297624534318876743725099723501638614139 \cdot 10^{152}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \mathbf{elif}\;b \le 8.703667783082919749023199154845924676168 \cdot 10^{-52}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} - \frac{b}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array}\]
\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 -2.297624534318876743725099723501638614139 \cdot 10^{152}:\\
\;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\

\mathbf{elif}\;b \le 8.703667783082919749023199154845924676168 \cdot 10^{-52}:\\
\;\;\;\;\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} - \frac{b}{2 \cdot a}\\

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

\end{array}
double f(double a, double b, double c) {
        double r40646 = b;
        double r40647 = -r40646;
        double r40648 = r40646 * r40646;
        double r40649 = 4.0;
        double r40650 = a;
        double r40651 = r40649 * r40650;
        double r40652 = c;
        double r40653 = r40651 * r40652;
        double r40654 = r40648 - r40653;
        double r40655 = sqrt(r40654);
        double r40656 = r40647 + r40655;
        double r40657 = 2.0;
        double r40658 = r40657 * r40650;
        double r40659 = r40656 / r40658;
        return r40659;
}
↓
double f(double a, double b, double c) {
        double r40660 = b;
        double r40661 = -2.2976245343188767e+152;
        bool r40662 = r40660 <= r40661;
        double r40663 = 1.0;
        double r40664 = c;
        double r40665 = r40664 / r40660;
        double r40666 = a;
        double r40667 = r40660 / r40666;
        double r40668 = r40665 - r40667;
        double r40669 = r40663 * r40668;
        double r40670 = 8.70366778308292e-52;
        bool r40671 = r40660 <= r40670;
        double r40672 = r40660 * r40660;
        double r40673 = 4.0;
        double r40674 = r40673 * r40666;
        double r40675 = r40674 * r40664;
        double r40676 = r40672 - r40675;
        double r40677 = sqrt(r40676);
        double r40678 = 2.0;
        double r40679 = r40678 * r40666;
        double r40680 = r40677 / r40679;
        double r40681 = r40660 / r40679;
        double r40682 = r40680 - r40681;
        double r40683 = -1.0;
        double r40684 = r40683 * r40665;
        double r40685 = r40671 ? r40682 : r40684;
        double r40686 = r40662 ? r40669 : r40685;
        return r40686;
}

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 < -2.2976245343188767e+152

    1. Initial program 63.3

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

    2. Simplified63.3

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

    3. Taylor expanded around -inf 2.2

      \[\leadsto \color{blue}{1 \cdot \frac{c}{b} - 1 \cdot \frac{b}{a}}\]

    4. Simplified2.2

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

    if -2.2976245343188767e+152 < b < 8.70366778308292e-52

    1. Initial program 13.2

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

    2. Simplified13.2

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

    4. Applied div-sub13.2

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

    if 8.70366778308292e-52 < b

    1. Initial program 54.1

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

    2. Simplified54.1

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

    3. Taylor expanded around inf 7.9

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

  4. Final simplification10.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -2.297624534318876743725099723501638614139 \cdot 10^{152}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \mathbf{elif}\;b \le 8.703667783082919749023199154845924676168 \cdot 10^{-52}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} - \frac{b}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2019209 +o rules:numerics
(FPCore (a b c)
  :name "Quadratic roots, full range"
  :precision binary64
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4 a) c)))) (* 2 a)))