Average Error: 19.4 → 7.6
Time: 6.2s
Precision: 64
\[\begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -4.297522851756149307625287590446857172218 \cdot 10^{130} \lor \neg \left(b \le 1.900113883782617364045683497909180234101 \cdot 10^{53}\right):\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{2 \cdot c}{2 \cdot \left(\frac{a}{\sqrt{b}} \cdot \frac{c}{{b}^{\frac{1}{2}}}\right) - 2 \cdot b}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \end{array}\\ \mathbf{elif}\;b \ge 0.0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - e^{\log \left(\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array}\]
\begin{array}{l}
\mathbf{if}\;b \ge 0.0:\\
\;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\

\mathbf{else}:\\
\;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\

\end{array}
↓
\begin{array}{l}
\mathbf{if}\;b \le -4.297522851756149307625287590446857172218 \cdot 10^{130} \lor \neg \left(b \le 1.900113883782617364045683497909180234101 \cdot 10^{53}\right):\\
\;\;\;\;\begin{array}{l}
\mathbf{if}\;b \ge 0.0:\\
\;\;\;\;\frac{2 \cdot c}{2 \cdot \left(\frac{a}{\sqrt{b}} \cdot \frac{c}{{b}^{\frac{1}{2}}}\right) - 2 \cdot b}\\

\mathbf{else}:\\
\;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\

\end{array}\\

\mathbf{elif}\;b \ge 0.0:\\
\;\;\;\;\frac{2 \cdot c}{\left(-b\right) - e^{\log \left(\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)}}\\

\mathbf{else}:\\
\;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\

\end{array}
double f(double a, double b, double c) {
        double r34986 = b;
        double r34987 = 0.0;
        bool r34988 = r34986 >= r34987;
        double r34989 = 2.0;
        double r34990 = c;
        double r34991 = r34989 * r34990;
        double r34992 = -r34986;
        double r34993 = r34986 * r34986;
        double r34994 = 4.0;
        double r34995 = a;
        double r34996 = r34994 * r34995;
        double r34997 = r34996 * r34990;
        double r34998 = r34993 - r34997;
        double r34999 = sqrt(r34998);
        double r35000 = r34992 - r34999;
        double r35001 = r34991 / r35000;
        double r35002 = r34992 + r34999;
        double r35003 = r34989 * r34995;
        double r35004 = r35002 / r35003;
        double r35005 = r34988 ? r35001 : r35004;
        return r35005;
}
↓
double f(double a, double b, double c) {
        double r35006 = b;
        double r35007 = -4.2975228517561493e+130;
        bool r35008 = r35006 <= r35007;
        double r35009 = 1.9001138837826174e+53;
        bool r35010 = r35006 <= r35009;
        double r35011 = !r35010;
        bool r35012 = r35008 || r35011;
        double r35013 = 0.0;
        bool r35014 = r35006 >= r35013;
        double r35015 = 2.0;
        double r35016 = c;
        double r35017 = r35015 * r35016;
        double r35018 = a;
        double r35019 = sqrt(r35006);
        double r35020 = r35018 / r35019;
        double r35021 = 0.5;
        double r35022 = pow(r35006, r35021);
        double r35023 = r35016 / r35022;
        double r35024 = r35020 * r35023;
        double r35025 = r35015 * r35024;
        double r35026 = 2.0;
        double r35027 = r35026 * r35006;
        double r35028 = r35025 - r35027;
        double r35029 = r35017 / r35028;
        double r35030 = 1.0;
        double r35031 = r35016 / r35006;
        double r35032 = r35006 / r35018;
        double r35033 = r35031 - r35032;
        double r35034 = r35030 * r35033;
        double r35035 = r35014 ? r35029 : r35034;
        double r35036 = -r35006;
        double r35037 = r35006 * r35006;
        double r35038 = 4.0;
        double r35039 = r35038 * r35018;
        double r35040 = r35039 * r35016;
        double r35041 = r35037 - r35040;
        double r35042 = sqrt(r35041);
        double r35043 = log(r35042);
        double r35044 = exp(r35043);
        double r35045 = r35036 - r35044;
        double r35046 = r35017 / r35045;
        double r35047 = r35036 + r35042;
        double r35048 = r35015 * r35018;
        double r35049 = r35047 / r35048;
        double r35050 = r35014 ? r35046 : r35049;
        double r35051 = r35012 ? r35035 : r35050;
        return r35051;
}

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 < -4.2975228517561493e+130 or 1.9001138837826174e+53 < b

    1. Initial program 34.6

      \[\begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array}\]

    2. Taylor expanded around inf 21.6

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{2 \cdot c}{\color{blue}{2 \cdot \frac{a \cdot c}{b} - 2 \cdot b}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array}\]
    3. Using strategy rm

    4. Applied add-sqr-sqrt21.6

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{2 \cdot c}{2 \cdot \frac{a \cdot c}{\color{blue}{\sqrt{b} \cdot \sqrt{b}}} - 2 \cdot b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array}\]

    5. Applied times-frac19.3

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{2 \cdot c}{2 \cdot \color{blue}{\left(\frac{a}{\sqrt{b}} \cdot \frac{c}{\sqrt{b}}\right)} - 2 \cdot b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array}\]

    6. Simplified19.3

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{2 \cdot c}{2 \cdot \left(\frac{a}{\sqrt{b}} \cdot \color{blue}{\frac{c}{{b}^{\frac{1}{2}}}}\right) - 2 \cdot b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array}\]

    7. Taylor expanded around -inf 5.9

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{2 \cdot c}{2 \cdot \left(\frac{a}{\sqrt{b}} \cdot \frac{c}{{b}^{\frac{1}{2}}}\right) - 2 \cdot b}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot \frac{a \cdot c}{b} - 2 \cdot b}{2 \cdot a}\\ \end{array}\]

    8. Taylor expanded around 0 3.7

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{2 \cdot c}{2 \cdot \left(\frac{a}{\sqrt{b}} \cdot \frac{c}{{b}^{\frac{1}{2}}}\right) - 2 \cdot b}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \frac{c}{b} - 1 \cdot \frac{b}{a}\\ \end{array}\]

    9. Simplified3.7

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{2 \cdot c}{2 \cdot \left(\frac{a}{\sqrt{b}} \cdot \frac{c}{{b}^{\frac{1}{2}}}\right) - 2 \cdot b}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \end{array}\]

    if -4.2975228517561493e+130 < b < 1.9001138837826174e+53

    1. Initial program 9.0

      \[\begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array}\]
    2. Using strategy rm

    3. Applied add-exp-log10.4

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \color{blue}{e^{\log \left(\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification7.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -4.297522851756149307625287590446857172218 \cdot 10^{130} \lor \neg \left(b \le 1.900113883782617364045683497909180234101 \cdot 10^{53}\right):\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{2 \cdot c}{2 \cdot \left(\frac{a}{\sqrt{b}} \cdot \frac{c}{{b}^{\frac{1}{2}}}\right) - 2 \cdot b}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \end{array}\\ \mathbf{elif}\;b \ge 0.0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - e^{\log \left(\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array}\]

Reproduce

herbie shell --seed 2019362 
(FPCore (a b c)
  :name "jeff quadratic root 2"
  :precision binary64
  (if (>= b 0.0) (/ (* 2 c) (- (- b) (sqrt (- (* b b) (* (* 4 a) c))))) (/ (+ (- b) (sqrt (- (* b b) (* (* 4 a) c)))) (* 2 a))))