Average Error: 33.8 → 6.7
Time: 9.4s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -8.301687926884188663878043402578250574713 \cdot 10^{98}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \mathbf{elif}\;b \le \frac{-1077853067741081}{1.365609355853794155331553646739713596855 \cdot 10^{244}}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{elif}\;b \le 6.208057797080499790536355473922963434123 \cdot 10^{104}:\\ \;\;\;\;\frac{\frac{1}{\frac{1}{2}} \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array}\]
\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 -8.301687926884188663878043402578250574713 \cdot 10^{98}:\\
\;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\

\mathbf{elif}\;b \le \frac{-1077853067741081}{1.365609355853794155331553646739713596855 \cdot 10^{244}}:\\
\;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\

\mathbf{elif}\;b \le 6.208057797080499790536355473922963434123 \cdot 10^{104}:\\
\;\;\;\;\frac{\frac{1}{\frac{1}{2}} \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\

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

\end{array}
double f(double a, double b, double c) {
        double r48681 = b;
        double r48682 = -r48681;
        double r48683 = r48681 * r48681;
        double r48684 = 4.0;
        double r48685 = a;
        double r48686 = r48684 * r48685;
        double r48687 = c;
        double r48688 = r48686 * r48687;
        double r48689 = r48683 - r48688;
        double r48690 = sqrt(r48689);
        double r48691 = r48682 + r48690;
        double r48692 = 2.0;
        double r48693 = r48692 * r48685;
        double r48694 = r48691 / r48693;
        return r48694;
}
↓
double f(double a, double b, double c) {
        double r48695 = b;
        double r48696 = -8.301687926884189e+98;
        bool r48697 = r48695 <= r48696;
        double r48698 = 1.0;
        double r48699 = c;
        double r48700 = r48699 / r48695;
        double r48701 = a;
        double r48702 = r48695 / r48701;
        double r48703 = r48700 - r48702;
        double r48704 = r48698 * r48703;
        double r48705 = -1077853067741081.0;
        double r48706 = 1.3656093558537942e+244;
        double r48707 = r48705 / r48706;
        bool r48708 = r48695 <= r48707;
        double r48709 = -r48695;
        double r48710 = r48695 * r48695;
        double r48711 = 4.0;
        double r48712 = r48711 * r48701;
        double r48713 = r48712 * r48699;
        double r48714 = r48710 - r48713;
        double r48715 = sqrt(r48714);
        double r48716 = r48709 + r48715;
        double r48717 = 2.0;
        double r48718 = r48717 * r48701;
        double r48719 = r48716 / r48718;
        double r48720 = 6.2080577970805e+104;
        bool r48721 = r48695 <= r48720;
        double r48722 = 1.0;
        double r48723 = r48698 / r48717;
        double r48724 = r48722 / r48723;
        double r48725 = r48724 * r48699;
        double r48726 = r48709 - r48715;
        double r48727 = r48725 / r48726;
        double r48728 = -1.0;
        double r48729 = r48728 * r48700;
        double r48730 = r48721 ? r48727 : r48729;
        double r48731 = r48708 ? r48719 : r48730;
        double r48732 = r48697 ? r48704 : r48731;
        return r48732;
}

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 4 regimes

  2. if b < -8.301687926884189e+98

    1. Initial program 46.2

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

    2. Taylor expanded around -inf 3.6

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

    3. Simplified3.6

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

    if -8.301687926884189e+98 < b < -7.892835993842436e-230

    1. Initial program 8.1

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

    if -7.892835993842436e-230 < b < 6.2080577970805e+104

    1. Initial program 29.9

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

    3. Applied flip-+30.0

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

    4. Simplified15.8

      \[\leadsto \frac{\frac{\color{blue}{0 + 4 \cdot \left(a \cdot c\right)}}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}{2 \cdot a}\]
    5. Using strategy rm

    6. Applied *-un-lft-identity15.8

      \[\leadsto \frac{\frac{0 + 4 \cdot \left(a \cdot c\right)}{\color{blue}{1 \cdot \left(\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)}}}{2 \cdot a}\]

    7. Applied *-un-lft-identity15.8

      \[\leadsto \frac{\frac{\color{blue}{1 \cdot \left(0 + 4 \cdot \left(a \cdot c\right)\right)}}{1 \cdot \left(\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)}}{2 \cdot a}\]

    8. Applied times-frac15.8

      \[\leadsto \frac{\color{blue}{\frac{1}{1} \cdot \frac{0 + 4 \cdot \left(a \cdot c\right)}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}}{2 \cdot a}\]

    9. Applied associate-/l*15.9

      \[\leadsto \color{blue}{\frac{\frac{1}{1}}{\frac{2 \cdot a}{\frac{0 + 4 \cdot \left(a \cdot c\right)}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}}}\]

    10. Simplified15.3

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

    11. Taylor expanded around 0 9.8

      \[\leadsto \frac{\frac{1}{1}}{\color{blue}{\frac{0.5}{c}} \cdot \left(\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)}\]

    12. Simplified9.8

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

    14. Applied associate-/r*9.5

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

    15. Simplified9.4

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

    if 6.2080577970805e+104 < b

    1. Initial program 59.4

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

    2. Taylor expanded around inf 2.8

      \[\leadsto \color{blue}{-1 \cdot \frac{c}{b}}\]
  3. Recombined 4 regimes into one program.

  4. Final simplification6.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -8.301687926884188663878043402578250574713 \cdot 10^{98}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \mathbf{elif}\;b \le \frac{-1077853067741081}{1.365609355853794155331553646739713596855 \cdot 10^{244}}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{elif}\;b \le 6.208057797080499790536355473922963434123 \cdot 10^{104}:\\ \;\;\;\;\frac{\frac{1}{\frac{1}{2}} \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2019304 
(FPCore (a b c)
  :name "Quadratic roots, full range"
  :precision binary64
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4 a) c)))) (* 2 a)))