Average Error: 33.7 → 10.9
Time: 8.5s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -1.98276540088900058 \cdot 10^{134}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \mathbf{elif}\;b \le 1.1860189201379418 \cdot 10^{-161}:\\ \;\;\;\;\left(\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b\right) \cdot \frac{\frac{1}{2}}{a}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array}\]
\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 -1.98276540088900058 \cdot 10^{134}:\\
\;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\

\mathbf{elif}\;b \le 1.1860189201379418 \cdot 10^{-161}:\\
\;\;\;\;\left(\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b\right) \cdot \frac{\frac{1}{2}}{a}\\

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

\end{array}
double f(double a, double b, double c) {
        double r79838 = b;
        double r79839 = -r79838;
        double r79840 = r79838 * r79838;
        double r79841 = 4.0;
        double r79842 = a;
        double r79843 = c;
        double r79844 = r79842 * r79843;
        double r79845 = r79841 * r79844;
        double r79846 = r79840 - r79845;
        double r79847 = sqrt(r79846);
        double r79848 = r79839 + r79847;
        double r79849 = 2.0;
        double r79850 = r79849 * r79842;
        double r79851 = r79848 / r79850;
        return r79851;
}
↓
double f(double a, double b, double c) {
        double r79852 = b;
        double r79853 = -1.9827654008890006e+134;
        bool r79854 = r79852 <= r79853;
        double r79855 = 1.0;
        double r79856 = c;
        double r79857 = r79856 / r79852;
        double r79858 = a;
        double r79859 = r79852 / r79858;
        double r79860 = r79857 - r79859;
        double r79861 = r79855 * r79860;
        double r79862 = 1.1860189201379418e-161;
        bool r79863 = r79852 <= r79862;
        double r79864 = r79852 * r79852;
        double r79865 = 4.0;
        double r79866 = r79858 * r79856;
        double r79867 = r79865 * r79866;
        double r79868 = r79864 - r79867;
        double r79869 = sqrt(r79868);
        double r79870 = r79869 - r79852;
        double r79871 = 1.0;
        double r79872 = 2.0;
        double r79873 = r79871 / r79872;
        double r79874 = r79873 / r79858;
        double r79875 = r79870 * r79874;
        double r79876 = -1.0;
        double r79877 = r79876 * r79857;
        double r79878 = r79863 ? r79875 : r79877;
        double r79879 = r79854 ? r79861 : r79878;
        return r79879;
}

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

Original33.7
Target21.0
Herbie10.9
\[\begin{array}{l} \mathbf{if}\;b \lt 0.0:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if b < -1.9827654008890006e+134

    1. Initial program 56.8

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

    2. Simplified56.8

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

    3. Taylor expanded around -inf 3.1

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

    4. Simplified3.1

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

    if -1.9827654008890006e+134 < b < 1.1860189201379418e-161

    1. Initial program 10.3

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

    2. Simplified10.3

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

    4. Applied *-un-lft-identity10.3

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

    5. Applied div-inv10.3

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

    6. Applied times-frac10.5

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

    7. Simplified10.5

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

    if 1.1860189201379418e-161 < b

    1. Initial program 49.6

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

    2. Simplified49.6

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

    3. Taylor expanded around inf 13.7

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

  4. Final simplification10.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -1.98276540088900058 \cdot 10^{134}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \mathbf{elif}\;b \le 1.1860189201379418 \cdot 10^{-161}:\\ \;\;\;\;\left(\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b\right) \cdot \frac{\frac{1}{2}}{a}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2020047 
(FPCore (a b c)
  :name "quadp (p42, positive)"
  :precision binary64

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

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