Average Error: 33.3 → 5.9
Time: 31.3s
Precision: 64
\[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b_2 \le -6.909589459766455 \cdot 10^{+149}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le -1.1973583843642258 \cdot 10^{-287}:\\ \;\;\;\;\frac{c}{\sqrt{b_2 \cdot b_2 - a \cdot c} + \left(-b_2\right)}\\ \mathbf{elif}\;b_2 \le 1.0185417924042504 \cdot 10^{+119}:\\ \;\;\;\;\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b_2} \cdot \frac{1}{2} - \frac{b_2}{a} \cdot 2\\ \end{array}\]
\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}
↓
\begin{array}{l}
\mathbf{if}\;b_2 \le -6.909589459766455 \cdot 10^{+149}:\\
\;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\

\mathbf{elif}\;b_2 \le -1.1973583843642258 \cdot 10^{-287}:\\
\;\;\;\;\frac{c}{\sqrt{b_2 \cdot b_2 - a \cdot c} + \left(-b_2\right)}\\

\mathbf{elif}\;b_2 \le 1.0185417924042504 \cdot 10^{+119}:\\
\;\;\;\;\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\\

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

\end{array}
double f(double a, double b_2, double c) {
        double r1179711 = b_2;
        double r1179712 = -r1179711;
        double r1179713 = r1179711 * r1179711;
        double r1179714 = a;
        double r1179715 = c;
        double r1179716 = r1179714 * r1179715;
        double r1179717 = r1179713 - r1179716;
        double r1179718 = sqrt(r1179717);
        double r1179719 = r1179712 - r1179718;
        double r1179720 = r1179719 / r1179714;
        return r1179720;
}
↓
double f(double a, double b_2, double c) {
        double r1179721 = b_2;
        double r1179722 = -6.909589459766455e+149;
        bool r1179723 = r1179721 <= r1179722;
        double r1179724 = -0.5;
        double r1179725 = c;
        double r1179726 = r1179725 / r1179721;
        double r1179727 = r1179724 * r1179726;
        double r1179728 = -1.1973583843642258e-287;
        bool r1179729 = r1179721 <= r1179728;
        double r1179730 = r1179721 * r1179721;
        double r1179731 = a;
        double r1179732 = r1179731 * r1179725;
        double r1179733 = r1179730 - r1179732;
        double r1179734 = sqrt(r1179733);
        double r1179735 = -r1179721;
        double r1179736 = r1179734 + r1179735;
        double r1179737 = r1179725 / r1179736;
        double r1179738 = 1.0185417924042504e+119;
        bool r1179739 = r1179721 <= r1179738;
        double r1179740 = r1179735 - r1179734;
        double r1179741 = r1179740 / r1179731;
        double r1179742 = 0.5;
        double r1179743 = r1179726 * r1179742;
        double r1179744 = r1179721 / r1179731;
        double r1179745 = 2.0;
        double r1179746 = r1179744 * r1179745;
        double r1179747 = r1179743 - r1179746;
        double r1179748 = r1179739 ? r1179741 : r1179747;
        double r1179749 = r1179729 ? r1179737 : r1179748;
        double r1179750 = r1179723 ? r1179727 : r1179749;
        return r1179750;
}

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 < -6.909589459766455e+149

    1. Initial program 62.5

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

    2. Taylor expanded around -inf 1.5

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

    if -6.909589459766455e+149 < b_2 < -1.1973583843642258e-287

    1. Initial program 34.4

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

    2. Taylor expanded around inf 34.4

      \[\leadsto \frac{\left(-b_2\right) - \sqrt{\color{blue}{{b_2}^{2} - a \cdot c}}}{a}\]

    3. Simplified34.4

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

    5. Applied div-inv34.4

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

    7. Applied flip--34.5

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

    8. Applied associate-*l/34.5

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

    9. Simplified13.8

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

    10. Taylor expanded around 0 6.9

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

    if -1.1973583843642258e-287 < b_2 < 1.0185417924042504e+119

    1. Initial program 8.4

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

    2. Taylor expanded around inf 8.4

      \[\leadsto \frac{\left(-b_2\right) - \sqrt{\color{blue}{{b_2}^{2} - a \cdot c}}}{a}\]

    3. Simplified8.4

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

    if 1.0185417924042504e+119 < b_2

    1. Initial program 50.9

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

    2. Taylor expanded around inf 3.5

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

  4. Final simplification5.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -6.909589459766455 \cdot 10^{+149}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le -1.1973583843642258 \cdot 10^{-287}:\\ \;\;\;\;\frac{c}{\sqrt{b_2 \cdot b_2 - a \cdot c} + \left(-b_2\right)}\\ \mathbf{elif}\;b_2 \le 1.0185417924042504 \cdot 10^{+119}:\\ \;\;\;\;\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b_2} \cdot \frac{1}{2} - \frac{b_2}{a} \cdot 2\\ \end{array}\]

Reproduce

herbie shell --seed 2019119 
(FPCore (a b_2 c)
  :name "quad2m (problem 3.2.1, negative)"
  (/ (- (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))