Average Error: 34.3 → 7.2
Time: 5.2s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -3.4397859828859872 \cdot 10^{117}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \mathbf{elif}\;b \le -1.0621980184587312 \cdot 10^{-219}:\\ \;\;\;\;\frac{e^{\log \left(\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)}}{2 \cdot a}\\ \mathbf{elif}\;b \le 1.66563346804975556 \cdot 10^{146}:\\ \;\;\;\;\frac{\left(\frac{1}{2} \cdot 4\right) \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array}\]
\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 -3.4397859828859872 \cdot 10^{117}:\\
\;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\

\mathbf{elif}\;b \le -1.0621980184587312 \cdot 10^{-219}:\\
\;\;\;\;\frac{e^{\log \left(\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)}}{2 \cdot a}\\

\mathbf{elif}\;b \le 1.66563346804975556 \cdot 10^{146}:\\
\;\;\;\;\frac{\left(\frac{1}{2} \cdot 4\right) \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 r50967 = b;
        double r50968 = -r50967;
        double r50969 = r50967 * r50967;
        double r50970 = 4.0;
        double r50971 = a;
        double r50972 = r50970 * r50971;
        double r50973 = c;
        double r50974 = r50972 * r50973;
        double r50975 = r50969 - r50974;
        double r50976 = sqrt(r50975);
        double r50977 = r50968 + r50976;
        double r50978 = 2.0;
        double r50979 = r50978 * r50971;
        double r50980 = r50977 / r50979;
        return r50980;
}
↓
double f(double a, double b, double c) {
        double r50981 = b;
        double r50982 = -3.439785982885987e+117;
        bool r50983 = r50981 <= r50982;
        double r50984 = 1.0;
        double r50985 = c;
        double r50986 = r50985 / r50981;
        double r50987 = a;
        double r50988 = r50981 / r50987;
        double r50989 = r50986 - r50988;
        double r50990 = r50984 * r50989;
        double r50991 = -1.0621980184587312e-219;
        bool r50992 = r50981 <= r50991;
        double r50993 = -r50981;
        double r50994 = r50981 * r50981;
        double r50995 = 4.0;
        double r50996 = r50995 * r50987;
        double r50997 = r50996 * r50985;
        double r50998 = r50994 - r50997;
        double r50999 = sqrt(r50998);
        double r51000 = r50993 + r50999;
        double r51001 = log(r51000);
        double r51002 = exp(r51001);
        double r51003 = 2.0;
        double r51004 = r51003 * r50987;
        double r51005 = r51002 / r51004;
        double r51006 = 1.6656334680497556e+146;
        bool r51007 = r50981 <= r51006;
        double r51008 = 1.0;
        double r51009 = r51008 / r51003;
        double r51010 = r51009 * r50995;
        double r51011 = r51010 * r50985;
        double r51012 = r50993 - r50999;
        double r51013 = r51011 / r51012;
        double r51014 = -1.0;
        double r51015 = r51014 * r50986;
        double r51016 = r51007 ? r51013 : r51015;
        double r51017 = r50992 ? r51005 : r51016;
        double r51018 = r50983 ? r50990 : r51017;
        return r51018;
}

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 < -3.439785982885987e+117

    1. Initial program 51.7

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

    2. Taylor expanded around -inf 2.9

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

    3. Simplified2.9

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

    if -3.439785982885987e+117 < b < -1.0621980184587312e-219

    1. Initial program 6.7

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

    3. Applied add-exp-log10.5

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

    if -1.0621980184587312e-219 < b < 1.6656334680497556e+146

    1. Initial program 32.0

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

    3. Applied flip-+32.0

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

    4. Simplified16.0

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

    6. Applied clear-num16.2

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

    7. Simplified15.0

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

    9. Applied associate-/l*15.1

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

    10. Simplified9.6

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

    12. Applied associate-/r*9.1

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

    13. Simplified9.0

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

    if 1.6656334680497556e+146 < b

    1. Initial program 63.3

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

    2. Taylor expanded around inf 1.8

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

  4. Final simplification7.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -3.4397859828859872 \cdot 10^{117}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \mathbf{elif}\;b \le -1.0621980184587312 \cdot 10^{-219}:\\ \;\;\;\;\frac{e^{\log \left(\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)}}{2 \cdot a}\\ \mathbf{elif}\;b \le 1.66563346804975556 \cdot 10^{146}:\\ \;\;\;\;\frac{\left(\frac{1}{2} \cdot 4\right) \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2020049 
(FPCore (a b c)
  :name "Quadratic roots, full range"
  :precision binary64
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4 a) c)))) (* 2 a)))