Average Error: 34.3 → 9.8
Time: 17.9s
Precision: 64
\[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b_2 \le -7.359940312872037386934109274309219747139 \cdot 10^{54}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le -1.394325716879235550939300459885270666443 \cdot 10^{-154}:\\ \;\;\;\;\frac{a \cdot c}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2} \cdot \frac{1}{a}\\ \mathbf{elif}\;b_2 \le 1920982614230223.5:\\ \;\;\;\;\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\\ \end{array}\]
\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}
↓
\begin{array}{l}
\mathbf{if}\;b_2 \le -7.359940312872037386934109274309219747139 \cdot 10^{54}:\\
\;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\

\mathbf{elif}\;b_2 \le -1.394325716879235550939300459885270666443 \cdot 10^{-154}:\\
\;\;\;\;\frac{a \cdot c}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2} \cdot \frac{1}{a}\\

\mathbf{elif}\;b_2 \le 1920982614230223.5:\\
\;\;\;\;\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\\

\mathbf{else}:\\
\;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\\

\end{array}
double f(double a, double b_2, double c) {
        double r61957 = b_2;
        double r61958 = -r61957;
        double r61959 = r61957 * r61957;
        double r61960 = a;
        double r61961 = c;
        double r61962 = r61960 * r61961;
        double r61963 = r61959 - r61962;
        double r61964 = sqrt(r61963);
        double r61965 = r61958 - r61964;
        double r61966 = r61965 / r61960;
        return r61966;
}
↓
double f(double a, double b_2, double c) {
        double r61967 = b_2;
        double r61968 = -7.359940312872037e+54;
        bool r61969 = r61967 <= r61968;
        double r61970 = -0.5;
        double r61971 = c;
        double r61972 = r61971 / r61967;
        double r61973 = r61970 * r61972;
        double r61974 = -1.3943257168792356e-154;
        bool r61975 = r61967 <= r61974;
        double r61976 = a;
        double r61977 = r61976 * r61971;
        double r61978 = r61967 * r61967;
        double r61979 = r61978 - r61977;
        double r61980 = sqrt(r61979);
        double r61981 = r61980 - r61967;
        double r61982 = r61977 / r61981;
        double r61983 = 1.0;
        double r61984 = r61983 / r61976;
        double r61985 = r61982 * r61984;
        double r61986 = 1920982614230223.5;
        bool r61987 = r61967 <= r61986;
        double r61988 = -r61967;
        double r61989 = r61988 - r61980;
        double r61990 = r61989 / r61976;
        double r61991 = 0.5;
        double r61992 = r61991 * r61972;
        double r61993 = 2.0;
        double r61994 = r61967 / r61976;
        double r61995 = r61993 * r61994;
        double r61996 = r61992 - r61995;
        double r61997 = r61987 ? r61990 : r61996;
        double r61998 = r61975 ? r61985 : r61997;
        double r61999 = r61969 ? r61973 : r61998;
        return r61999;
}

Error

Bits error versus a

Bits error versus b_2

Bits error versus c

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 4 regimes

  2. if b_2 < -7.359940312872037e+54

    1. Initial program 57.6

      \[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]

    2. Taylor expanded around -inf 3.8

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{c}{b_2}}\]

    if -7.359940312872037e+54 < b_2 < -1.3943257168792356e-154

    1. Initial program 39.5

      \[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
    2. Using strategy rm

    3. Applied div-inv39.6

      \[\leadsto \color{blue}{\left(\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}\right) \cdot \frac{1}{a}}\]
    4. Using strategy rm

    5. Applied flip--39.6

      \[\leadsto \color{blue}{\frac{\left(-b_2\right) \cdot \left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c} \cdot \sqrt{b_2 \cdot b_2 - a \cdot c}}{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}} \cdot \frac{1}{a}\]

    6. Simplified18.5

      \[\leadsto \frac{\color{blue}{0 + a \cdot c}}{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}} \cdot \frac{1}{a}\]

    7. Simplified18.5

      \[\leadsto \frac{0 + a \cdot c}{\color{blue}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}} \cdot \frac{1}{a}\]

    if -1.3943257168792356e-154 < b_2 < 1920982614230223.5

    1. Initial program 12.3

      \[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
    2. Using strategy rm

    3. Applied div-inv12.4

      \[\leadsto \color{blue}{\left(\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}\right) \cdot \frac{1}{a}}\]
    4. Using strategy rm

    5. Applied *-un-lft-identity12.4

      \[\leadsto \color{blue}{\left(1 \cdot \left(\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}\right)\right)} \cdot \frac{1}{a}\]

    6. Applied associate-*l*12.4

      \[\leadsto \color{blue}{1 \cdot \left(\left(\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}\right) \cdot \frac{1}{a}\right)}\]

    7. Simplified12.3

      \[\leadsto 1 \cdot \color{blue}{\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}}\]

    if 1920982614230223.5 < b_2

    1. Initial program 34.0

      \[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]

    2. Taylor expanded around inf 6.8

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}}\]
  3. Recombined 4 regimes into one program.

  4. Final simplification9.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -7.359940312872037386934109274309219747139 \cdot 10^{54}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le -1.394325716879235550939300459885270666443 \cdot 10^{-154}:\\ \;\;\;\;\frac{a \cdot c}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2} \cdot \frac{1}{a}\\ \mathbf{elif}\;b_2 \le 1920982614230223.5:\\ \;\;\;\;\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2019208 
(FPCore (a b_2 c)
  :name "NMSE problem 3.2.1"
  :precision binary64
  (/ (- (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))