Average Error: 20.0 → 6.6
Time: 15.1s
Precision: 64
\[\begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -5.503931209509311823152460653836804887804 \cdot 10^{150}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array}\\ \mathbf{elif}\;b \le 7.613007054465526370435384505937457344391 \cdot 10^{97}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{\left(-b\right) - \left|\sqrt[3]{b \cdot b - \left(4 \cdot a\right) \cdot c}\right| \cdot \sqrt{\sqrt[3]{b \cdot b - \left(4 \cdot a\right) \cdot c}}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array}\\ \mathbf{elif}\;b \ge 0.0:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{\frac{\left({b}^{2} - {b}^{2}\right) + 4 \cdot \left(a \cdot c\right)}{2}} \cdot \left(\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)\\ \end{array}\]
\begin{array}{l}
\mathbf{if}\;b \ge 0.0:\\
\;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\

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

\end{array}
↓
\begin{array}{l}
\mathbf{if}\;b \le -5.503931209509311823152460653836804887804 \cdot 10^{150}:\\
\;\;\;\;\begin{array}{l}
\mathbf{if}\;b \ge 0.0:\\
\;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\

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

\end{array}\\

\mathbf{elif}\;b \le 7.613007054465526370435384505937457344391 \cdot 10^{97}:\\
\;\;\;\;\begin{array}{l}
\mathbf{if}\;b \ge 0.0:\\
\;\;\;\;\frac{\left(-b\right) - \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}}}{2 \cdot a}\\

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

\end{array}\\

\mathbf{elif}\;b \ge 0.0:\\
\;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\

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

\end{array}
double f(double a, double b, double c) {
        double r34978 = b;
        double r34979 = 0.0;
        bool r34980 = r34978 >= r34979;
        double r34981 = -r34978;
        double r34982 = r34978 * r34978;
        double r34983 = 4.0;
        double r34984 = a;
        double r34985 = r34983 * r34984;
        double r34986 = c;
        double r34987 = r34985 * r34986;
        double r34988 = r34982 - r34987;
        double r34989 = sqrt(r34988);
        double r34990 = r34981 - r34989;
        double r34991 = 2.0;
        double r34992 = r34991 * r34984;
        double r34993 = r34990 / r34992;
        double r34994 = r34991 * r34986;
        double r34995 = r34981 + r34989;
        double r34996 = r34994 / r34995;
        double r34997 = r34980 ? r34993 : r34996;
        return r34997;
}
↓
double f(double a, double b, double c) {
        double r34998 = b;
        double r34999 = -5.503931209509312e+150;
        bool r35000 = r34998 <= r34999;
        double r35001 = 0.0;
        bool r35002 = r34998 >= r35001;
        double r35003 = 1.0;
        double r35004 = c;
        double r35005 = r35004 / r34998;
        double r35006 = a;
        double r35007 = r34998 / r35006;
        double r35008 = r35005 - r35007;
        double r35009 = r35003 * r35008;
        double r35010 = -1.0;
        double r35011 = r35010 * r35005;
        double r35012 = r35002 ? r35009 : r35011;
        double r35013 = 7.613007054465526e+97;
        bool r35014 = r34998 <= r35013;
        double r35015 = -r34998;
        double r35016 = r34998 * r34998;
        double r35017 = 4.0;
        double r35018 = r35017 * r35006;
        double r35019 = r35018 * r35004;
        double r35020 = r35016 - r35019;
        double r35021 = cbrt(r35020);
        double r35022 = fabs(r35021);
        double r35023 = sqrt(r35021);
        double r35024 = r35022 * r35023;
        double r35025 = r35015 - r35024;
        double r35026 = 2.0;
        double r35027 = r35026 * r35006;
        double r35028 = r35025 / r35027;
        double r35029 = r35026 * r35004;
        double r35030 = sqrt(r35020);
        double r35031 = r35015 + r35030;
        double r35032 = r35029 / r35031;
        double r35033 = r35002 ? r35028 : r35032;
        double r35034 = 2.0;
        double r35035 = pow(r34998, r35034);
        double r35036 = r35035 - r35035;
        double r35037 = r35006 * r35004;
        double r35038 = r35017 * r35037;
        double r35039 = r35036 + r35038;
        double r35040 = r35039 / r35026;
        double r35041 = r35004 / r35040;
        double r35042 = r35015 - r35030;
        double r35043 = r35041 * r35042;
        double r35044 = r35002 ? r35009 : r35043;
        double r35045 = r35014 ? r35033 : r35044;
        double r35046 = r35000 ? r35012 : r35045;
        return r35046;
}

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 < -5.503931209509312e+150

    1. Initial program 37.5

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

    2. Taylor expanded around inf 37.5

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

    3. Taylor expanded around 0 37.5

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

    4. Simplified37.5

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

    6. Applied associate-/l*37.5

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

    7. Taylor expanded around -inf 1.5

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

    if -5.503931209509312e+150 < b < 7.613007054465526e+97

    1. Initial program 8.5

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

    3. Applied add-cube-cbrt8.7

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

    4. Applied sqrt-prod8.7

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

    5. Simplified8.7

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

    if 7.613007054465526e+97 < b

    1. Initial program 47.4

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

    2. Taylor expanded around inf 10.2

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

    3. Taylor expanded around 0 3.8

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

    4. Simplified3.8

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

    6. Applied flip-+3.8

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

    7. Applied associate-/r/3.8

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

    8. Simplified3.8

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{\frac{\left({b}^{2} - {b}^{2}\right) + 4 \cdot \left(a \cdot c\right)}{2}} \cdot \left(\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)\\ \end{array}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification6.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -5.503931209509311823152460653836804887804 \cdot 10^{150}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array}\\ \mathbf{elif}\;b \le 7.613007054465526370435384505937457344391 \cdot 10^{97}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{\left(-b\right) - \left|\sqrt[3]{b \cdot b - \left(4 \cdot a\right) \cdot c}\right| \cdot \sqrt{\sqrt[3]{b \cdot b - \left(4 \cdot a\right) \cdot c}}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array}\\ \mathbf{elif}\;b \ge 0.0:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{\frac{\left({b}^{2} - {b}^{2}\right) + 4 \cdot \left(a \cdot c\right)}{2}} \cdot \left(\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019297 
(FPCore (a b c)
  :name "jeff quadratic root 1"
  :precision binary64
  (if (>= b 0.0) (/ (- (- b) (sqrt (- (* b b) (* (* 4 a) c)))) (* 2 a)) (/ (* 2 c) (+ (- b) (sqrt (- (* b b) (* (* 4 a) c)))))))