Average Error: 34.0 → 10.2
Time: 18.9s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -1.006124725233072906597672755451758607334 \cdot 10^{153}:\\ \;\;\;\;\left(\frac{c}{b} - \frac{b}{a}\right) \cdot 1\\ \mathbf{elif}\;b \le 5.668416736491797065158030167390678793472 \cdot 10^{-35}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{a \cdot 2} - \frac{b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -1\\ \end{array}\]
\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 -1.006124725233072906597672755451758607334 \cdot 10^{153}:\\
\;\;\;\;\left(\frac{c}{b} - \frac{b}{a}\right) \cdot 1\\

\mathbf{elif}\;b \le 5.668416736491797065158030167390678793472 \cdot 10^{-35}:\\
\;\;\;\;\frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{a \cdot 2} - \frac{b}{a \cdot 2}\\

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

\end{array}
double f(double a, double b, double c) {
        double r2014811 = b;
        double r2014812 = -r2014811;
        double r2014813 = r2014811 * r2014811;
        double r2014814 = 4.0;
        double r2014815 = a;
        double r2014816 = r2014814 * r2014815;
        double r2014817 = c;
        double r2014818 = r2014816 * r2014817;
        double r2014819 = r2014813 - r2014818;
        double r2014820 = sqrt(r2014819);
        double r2014821 = r2014812 + r2014820;
        double r2014822 = 2.0;
        double r2014823 = r2014822 * r2014815;
        double r2014824 = r2014821 / r2014823;
        return r2014824;
}
↓
double f(double a, double b, double c) {
        double r2014825 = b;
        double r2014826 = -1.0061247252330729e+153;
        bool r2014827 = r2014825 <= r2014826;
        double r2014828 = c;
        double r2014829 = r2014828 / r2014825;
        double r2014830 = a;
        double r2014831 = r2014825 / r2014830;
        double r2014832 = r2014829 - r2014831;
        double r2014833 = 1.0;
        double r2014834 = r2014832 * r2014833;
        double r2014835 = 5.668416736491797e-35;
        bool r2014836 = r2014825 <= r2014835;
        double r2014837 = r2014825 * r2014825;
        double r2014838 = 4.0;
        double r2014839 = r2014830 * r2014828;
        double r2014840 = r2014838 * r2014839;
        double r2014841 = r2014837 - r2014840;
        double r2014842 = sqrt(r2014841);
        double r2014843 = 2.0;
        double r2014844 = r2014830 * r2014843;
        double r2014845 = r2014842 / r2014844;
        double r2014846 = r2014825 / r2014844;
        double r2014847 = r2014845 - r2014846;
        double r2014848 = -1.0;
        double r2014849 = r2014829 * r2014848;
        double r2014850 = r2014836 ? r2014847 : r2014849;
        double r2014851 = r2014827 ? r2014834 : r2014850;
        return r2014851;
}

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 < -1.0061247252330729e+153

    1. Initial program 63.7

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

    3. Applied div-inv63.7

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

    4. Taylor expanded around -inf 2.0

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

    5. Simplified2.0

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

    if -1.0061247252330729e+153 < b < 5.668416736491797e-35

    1. Initial program 13.9

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

    3. Applied clear-num14.1

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

    4. Simplified14.1

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

    6. Applied *-un-lft-identity14.1

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

    7. Applied add-cube-cbrt14.1

      \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{1} \cdot \sqrt[3]{1}\right) \cdot \sqrt[3]{1}}}{1 \cdot \frac{a \cdot 2}{\sqrt{b \cdot b - \left(a \cdot c\right) \cdot 4} - b}}\]

    8. Applied times-frac14.1

      \[\leadsto \color{blue}{\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{1} \cdot \frac{\sqrt[3]{1}}{\frac{a \cdot 2}{\sqrt{b \cdot b - \left(a \cdot c\right) \cdot 4} - b}}}\]

    9. Simplified14.1

      \[\leadsto \color{blue}{1} \cdot \frac{\sqrt[3]{1}}{\frac{a \cdot 2}{\sqrt{b \cdot b - \left(a \cdot c\right) \cdot 4} - b}}\]

    10. Simplified13.9

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

    12. Applied div-sub14.0

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

    if 5.668416736491797e-35 < b

    1. Initial program 54.6

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

    2. Taylor expanded around inf 7.2

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

  4. Final simplification10.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -1.006124725233072906597672755451758607334 \cdot 10^{153}:\\ \;\;\;\;\left(\frac{c}{b} - \frac{b}{a}\right) \cdot 1\\ \mathbf{elif}\;b \le 5.668416736491797065158030167390678793472 \cdot 10^{-35}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{a \cdot 2} - \frac{b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -1\\ \end{array}\]

Reproduce

herbie shell --seed 2019169 +o rules:numerics
(FPCore (a b c)
  :name "Quadratic roots, full range"
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)))