Average Error: 33.7 → 8.8
Time: 17.2s
Precision: 64
\[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -1.6806111715441095 \cdot 10^{-29}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \le -8.10634488243571495 \cdot 10^{-168}:\\ \;\;\;\;\frac{\frac{1}{\frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}{4 \cdot \left(a \cdot c\right)}}}{2 \cdot a}\\ \mathbf{elif}\;b \le 5.47518920665104056 \cdot 10^{101}:\\ \;\;\;\;\frac{1}{\frac{2 \cdot a}{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \end{array}\]
\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 -1.6806111715441095 \cdot 10^{-29}:\\
\;\;\;\;-1 \cdot \frac{c}{b}\\

\mathbf{elif}\;b \le -8.10634488243571495 \cdot 10^{-168}:\\
\;\;\;\;\frac{\frac{1}{\frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}{4 \cdot \left(a \cdot c\right)}}}{2 \cdot a}\\

\mathbf{elif}\;b \le 5.47518920665104056 \cdot 10^{101}:\\
\;\;\;\;\frac{1}{\frac{2 \cdot a}{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}}\\

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

\end{array}
double f(double a, double b, double c) {
        double r59810 = b;
        double r59811 = -r59810;
        double r59812 = r59810 * r59810;
        double r59813 = 4.0;
        double r59814 = a;
        double r59815 = c;
        double r59816 = r59814 * r59815;
        double r59817 = r59813 * r59816;
        double r59818 = r59812 - r59817;
        double r59819 = sqrt(r59818);
        double r59820 = r59811 - r59819;
        double r59821 = 2.0;
        double r59822 = r59821 * r59814;
        double r59823 = r59820 / r59822;
        return r59823;
}
↓
double f(double a, double b, double c) {
        double r59824 = b;
        double r59825 = -1.6806111715441095e-29;
        bool r59826 = r59824 <= r59825;
        double r59827 = -1.0;
        double r59828 = c;
        double r59829 = r59828 / r59824;
        double r59830 = r59827 * r59829;
        double r59831 = -8.106344882435715e-168;
        bool r59832 = r59824 <= r59831;
        double r59833 = 1.0;
        double r59834 = r59824 * r59824;
        double r59835 = 4.0;
        double r59836 = a;
        double r59837 = r59836 * r59828;
        double r59838 = r59835 * r59837;
        double r59839 = r59834 - r59838;
        double r59840 = sqrt(r59839);
        double r59841 = r59840 - r59824;
        double r59842 = r59841 / r59838;
        double r59843 = r59833 / r59842;
        double r59844 = 2.0;
        double r59845 = r59844 * r59836;
        double r59846 = r59843 / r59845;
        double r59847 = 5.4751892066510406e+101;
        bool r59848 = r59824 <= r59847;
        double r59849 = -r59824;
        double r59850 = r59849 - r59840;
        double r59851 = r59845 / r59850;
        double r59852 = r59833 / r59851;
        double r59853 = 1.0;
        double r59854 = r59824 / r59836;
        double r59855 = r59829 - r59854;
        double r59856 = r59853 * r59855;
        double r59857 = r59848 ? r59852 : r59856;
        double r59858 = r59832 ? r59846 : r59857;
        double r59859 = r59826 ? r59830 : r59858;
        return r59859;
}

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
Target20.6
Herbie8.8
\[\begin{array}{l} \mathbf{if}\;b \lt 0.0:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array}\]

Derivation

  1. Split input into 4 regimes

  2. if b < -1.6806111715441095e-29

    1. Initial program 54.8

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

    2. Taylor expanded around -inf 6.6

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

    if -1.6806111715441095e-29 < b < -8.106344882435715e-168

    1. Initial program 28.0

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

    3. Applied flip--28.0

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

    4. Simplified18.2

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

    5. Simplified18.2

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

    7. Applied clear-num18.2

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

    8. Simplified18.2

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

    if -8.106344882435715e-168 < b < 5.4751892066510406e+101

    1. Initial program 10.2

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

    3. Applied clear-num10.3

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

    if 5.4751892066510406e+101 < b

    1. Initial program 48.1

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

    2. Taylor expanded around inf 4.2

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

    3. Simplified4.2

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

  4. Final simplification8.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -1.6806111715441095 \cdot 10^{-29}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \le -8.10634488243571495 \cdot 10^{-168}:\\ \;\;\;\;\frac{\frac{1}{\frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}{4 \cdot \left(a \cdot c\right)}}}{2 \cdot a}\\ \mathbf{elif}\;b \le 5.47518920665104056 \cdot 10^{101}:\\ \;\;\;\;\frac{1}{\frac{2 \cdot a}{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019198 
(FPCore (a b c)
  :name "quadm (p42, negative)"

  :herbie-target
  (if (< b 0.0) (/ c (* a (/ (+ (- b) (sqrt (- (* b b) (* 4.0 (* a c))))) (* 2.0 a)))) (/ (- (- b) (sqrt (- (* b b) (* 4.0 (* a c))))) (* 2.0 a)))

  (/ (- (- b) (sqrt (- (* b b) (* 4.0 (* a c))))) (* 2.0 a)))