Average Error: 34.4 → 6.8
Time: 8.5s
Precision: 64
\[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -1.5688227236985301 \cdot 10^{105}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \le 3.119187438943242 \cdot 10^{-255}:\\ \;\;\;\;\frac{1}{\frac{0.5}{c} \cdot \left(\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b\right)}\\ \mathbf{elif}\;b \le 6.74838527698993 \cdot 10^{90}:\\ \;\;\;\;\frac{1}{\frac{2 \cdot a}{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{b}{a}\\ \end{array}\]
\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}
↓
\begin{array}{l}
\mathbf{if}\;b \le -1.5688227236985301 \cdot 10^{105}:\\
\;\;\;\;-1 \cdot \frac{c}{b}\\

\mathbf{elif}\;b \le 3.119187438943242 \cdot 10^{-255}:\\
\;\;\;\;\frac{1}{\frac{0.5}{c} \cdot \left(\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b\right)}\\

\mathbf{elif}\;b \le 6.74838527698993 \cdot 10^{90}:\\
\;\;\;\;\frac{1}{\frac{2 \cdot a}{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}}\\

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

\end{array}
double f(double a, double b, double c) {
        double r83870 = b;
        double r83871 = -r83870;
        double r83872 = r83870 * r83870;
        double r83873 = 4.0;
        double r83874 = a;
        double r83875 = c;
        double r83876 = r83874 * r83875;
        double r83877 = r83873 * r83876;
        double r83878 = r83872 - r83877;
        double r83879 = sqrt(r83878);
        double r83880 = r83871 - r83879;
        double r83881 = 2.0;
        double r83882 = r83881 * r83874;
        double r83883 = r83880 / r83882;
        return r83883;
}
↓
double f(double a, double b, double c) {
        double r83884 = b;
        double r83885 = -1.56882272369853e+105;
        bool r83886 = r83884 <= r83885;
        double r83887 = -1.0;
        double r83888 = c;
        double r83889 = r83888 / r83884;
        double r83890 = r83887 * r83889;
        double r83891 = 3.119187438943242e-255;
        bool r83892 = r83884 <= r83891;
        double r83893 = 1.0;
        double r83894 = 0.5;
        double r83895 = r83894 / r83888;
        double r83896 = r83884 * r83884;
        double r83897 = 4.0;
        double r83898 = a;
        double r83899 = r83898 * r83888;
        double r83900 = r83897 * r83899;
        double r83901 = r83896 - r83900;
        double r83902 = sqrt(r83901);
        double r83903 = r83902 - r83884;
        double r83904 = r83895 * r83903;
        double r83905 = r83893 / r83904;
        double r83906 = 6.74838527698993e+90;
        bool r83907 = r83884 <= r83906;
        double r83908 = 2.0;
        double r83909 = r83908 * r83898;
        double r83910 = -r83884;
        double r83911 = r83910 - r83902;
        double r83912 = r83909 / r83911;
        double r83913 = r83893 / r83912;
        double r83914 = r83884 / r83898;
        double r83915 = r83887 * r83914;
        double r83916 = r83907 ? r83913 : r83915;
        double r83917 = r83892 ? r83905 : r83916;
        double r83918 = r83886 ? r83890 : r83917;
        return r83918;
}

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

Target

Original34.4
Target21.3
Herbie6.8
\[\begin{array}{l} \mathbf{if}\;b \lt 0.0:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array}\]

Derivation

  1. Split input into 4 regimes

  2. if b < -1.56882272369853e+105

    1. Initial program 60.4

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

    2. Taylor expanded around -inf 2.5

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

    if -1.56882272369853e+105 < b < 3.119187438943242e-255

    1. Initial program 31.0

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

    3. Applied flip--31.1

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

    4. Simplified16.3

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

    5. Simplified16.3

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

    7. Applied *-un-lft-identity16.3

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

    9. Applied clear-num16.4

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

    10. Simplified15.6

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

    11. Taylor expanded around 0 9.7

      \[\leadsto \frac{1}{\color{blue}{\frac{0.5}{c}} \cdot \left(\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b\right)}\]

    if 3.119187438943242e-255 < b < 6.74838527698993e+90

    1. Initial program 8.3

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

    3. Applied clear-num8.4

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

    if 6.74838527698993e+90 < b

    1. Initial program 45.7

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

    3. Applied flip--62.7

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

    4. Simplified61.8

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

    5. Simplified61.8

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

    7. Applied *-un-lft-identity61.8

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

    8. Taylor expanded around 0 4.6

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

  4. Final simplification6.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -1.5688227236985301 \cdot 10^{105}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \le 3.119187438943242 \cdot 10^{-255}:\\ \;\;\;\;\frac{1}{\frac{0.5}{c} \cdot \left(\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b\right)}\\ \mathbf{elif}\;b \le 6.74838527698993 \cdot 10^{90}:\\ \;\;\;\;\frac{1}{\frac{2 \cdot a}{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{b}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2020042 
(FPCore (a b c)
  :name "quadm (p42, negative)"
  :precision binary64

  :herbie-target
  (if (< b 0.0) (/ c (* a (/ (+ (- b) (sqrt (- (* b b) (* 4 (* a c))))) (* 2 a)))) (/ (- (- b) (sqrt (- (* b b) (* 4 (* a c))))) (* 2 a)))

  (/ (- (- b) (sqrt (- (* b b) (* 4 (* a c))))) (* 2 a)))