Average Error: 19.9 → 6.8
Time: 18.0s
Precision: 64
\[\begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -3.241187172550559408188235703536897569729 \cdot 10^{147}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 \cdot a}{\frac{b}{c}} + b \cdot -2}{2 \cdot a}\\ \end{array}\\ \mathbf{elif}\;b \le 2.349402930427110769075791183992464758725 \cdot 10^{98}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \left|\sqrt{\sqrt[3]{b \cdot b - \left(4 \cdot c\right) \cdot a}} \cdot \sqrt{\sqrt[3]{b \cdot b - \left(4 \cdot c\right) \cdot a}}\right| \cdot \sqrt{\sqrt[3]{b \cdot b - \left(4 \cdot a\right) \cdot c}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} + \left(-b\right)}{2 \cdot a}\\ \end{array}\\ \mathbf{elif}\;b \ge 0.0:\\ \;\;\;\;\frac{2 \cdot c}{\frac{2 \cdot a}{\frac{b}{c}} + b \cdot -2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{b \cdot b - \left(b \cdot b - \left(a \cdot c\right) \cdot 4\right)}{\left(-b\right) - \sqrt{b \cdot b - \left(a \cdot c\right) \cdot 4}}}{2 \cdot a}\\ \end{array}\]
\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 -3.241187172550559408188235703536897569729 \cdot 10^{147}:\\
\;\;\;\;\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{\frac{2 \cdot a}{\frac{b}{c}} + b \cdot -2}{2 \cdot a}\\

\end{array}\\

\mathbf{elif}\;b \le 2.349402930427110769075791183992464758725 \cdot 10^{98}:\\
\;\;\;\;\begin{array}{l}
\mathbf{if}\;b \ge 0.0:\\
\;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \left|\sqrt{\sqrt[3]{b \cdot b - \left(4 \cdot c\right) \cdot a}} \cdot \sqrt{\sqrt[3]{b \cdot b - \left(4 \cdot c\right) \cdot a}}\right| \cdot \sqrt{\sqrt[3]{b \cdot b - \left(4 \cdot a\right) \cdot c}}}\\

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

\end{array}\\

\mathbf{elif}\;b \ge 0.0:\\
\;\;\;\;\frac{2 \cdot c}{\frac{2 \cdot a}{\frac{b}{c}} + b \cdot -2}\\

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

\end{array}
double f(double a, double b, double c) {
        double r34975 = b;
        double r34976 = 0.0;
        bool r34977 = r34975 >= r34976;
        double r34978 = 2.0;
        double r34979 = c;
        double r34980 = r34978 * r34979;
        double r34981 = -r34975;
        double r34982 = r34975 * r34975;
        double r34983 = 4.0;
        double r34984 = a;
        double r34985 = r34983 * r34984;
        double r34986 = r34985 * r34979;
        double r34987 = r34982 - r34986;
        double r34988 = sqrt(r34987);
        double r34989 = r34981 - r34988;
        double r34990 = r34980 / r34989;
        double r34991 = r34981 + r34988;
        double r34992 = r34978 * r34984;
        double r34993 = r34991 / r34992;
        double r34994 = r34977 ? r34990 : r34993;
        return r34994;
}
↓
double f(double a, double b, double c) {
        double r34995 = b;
        double r34996 = -3.2411871725505594e+147;
        bool r34997 = r34995 <= r34996;
        double r34998 = 0.0;
        bool r34999 = r34995 >= r34998;
        double r35000 = 2.0;
        double r35001 = c;
        double r35002 = r35000 * r35001;
        double r35003 = -r34995;
        double r35004 = r34995 * r34995;
        double r35005 = 4.0;
        double r35006 = a;
        double r35007 = r35005 * r35006;
        double r35008 = r35007 * r35001;
        double r35009 = r35004 - r35008;
        double r35010 = sqrt(r35009);
        double r35011 = r35003 - r35010;
        double r35012 = r35002 / r35011;
        double r35013 = r35000 * r35006;
        double r35014 = r34995 / r35001;
        double r35015 = r35013 / r35014;
        double r35016 = -2.0;
        double r35017 = r34995 * r35016;
        double r35018 = r35015 + r35017;
        double r35019 = r35018 / r35013;
        double r35020 = r34999 ? r35012 : r35019;
        double r35021 = 2.3494029304271108e+98;
        bool r35022 = r34995 <= r35021;
        double r35023 = r35005 * r35001;
        double r35024 = r35023 * r35006;
        double r35025 = r35004 - r35024;
        double r35026 = cbrt(r35025);
        double r35027 = sqrt(r35026);
        double r35028 = r35027 * r35027;
        double r35029 = fabs(r35028);
        double r35030 = cbrt(r35009);
        double r35031 = sqrt(r35030);
        double r35032 = r35029 * r35031;
        double r35033 = r35003 - r35032;
        double r35034 = r35002 / r35033;
        double r35035 = r35010 + r35003;
        double r35036 = r35035 / r35013;
        double r35037 = r34999 ? r35034 : r35036;
        double r35038 = r35002 / r35018;
        double r35039 = r35006 * r35001;
        double r35040 = r35039 * r35005;
        double r35041 = r35004 - r35040;
        double r35042 = r35004 - r35041;
        double r35043 = sqrt(r35041);
        double r35044 = r35003 - r35043;
        double r35045 = r35042 / r35044;
        double r35046 = r35045 / r35013;
        double r35047 = r34999 ? r35038 : r35046;
        double r35048 = r35022 ? r35037 : r35047;
        double r35049 = r34997 ? r35020 : r35048;
        return r35049;
}

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 < -3.2411871725505594e+147

    1. Initial program 61.4

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

    2. Taylor expanded around -inf 10.6

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

    3. Simplified1.7

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

    if -3.2411871725505594e+147 < b < 2.3494029304271108e+98

    1. Initial program 8.9

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

    3. Applied add-cube-cbrt9.1

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

    4. Applied sqrt-prod9.1

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

    5. Simplified9.1

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

    7. Applied add-sqr-sqrt9.2

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

    8. Simplified9.2

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

    9. Simplified9.2

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

    if 2.3494029304271108e+98 < b

    1. Initial program 30.1

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

    2. Taylor expanded around inf 6.4

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

    3. Simplified2.5

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

    5. Applied flip-+2.5

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

    6. Simplified2.5

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

    7. Simplified2.5

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

  4. Final simplification6.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -3.241187172550559408188235703536897569729 \cdot 10^{147}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 \cdot a}{\frac{b}{c}} + b \cdot -2}{2 \cdot a}\\ \end{array}\\ \mathbf{elif}\;b \le 2.349402930427110769075791183992464758725 \cdot 10^{98}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \left|\sqrt{\sqrt[3]{b \cdot b - \left(4 \cdot c\right) \cdot a}} \cdot \sqrt{\sqrt[3]{b \cdot b - \left(4 \cdot c\right) \cdot a}}\right| \cdot \sqrt{\sqrt[3]{b \cdot b - \left(4 \cdot a\right) \cdot c}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} + \left(-b\right)}{2 \cdot a}\\ \end{array}\\ \mathbf{elif}\;b \ge 0.0:\\ \;\;\;\;\frac{2 \cdot c}{\frac{2 \cdot a}{\frac{b}{c}} + b \cdot -2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{b \cdot b - \left(b \cdot b - \left(a \cdot c\right) \cdot 4\right)}{\left(-b\right) - \sqrt{b \cdot b - \left(a \cdot c\right) \cdot 4}}}{2 \cdot a}\\ \end{array}\]

Reproduce

herbie shell --seed 2019195 
(FPCore (a b c)
  :name "jeff quadratic root 2"
  (if (>= b 0.0) (/ (* 2.0 c) (- (- b) (sqrt (- (* b b) (* (* 4.0 a) c))))) (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a))))