Average Error: 34.2 → 8.6
Time: 32.0s
Precision: 64
\[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b_2 \le -5.99946224548089213456388959139204668765 \cdot 10^{73}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le -4.296272708131498829504916428849430668856 \cdot 10^{-127}:\\ \;\;\;\;\frac{\frac{a \cdot c + \left(b_2 \cdot b_2 - b_2 \cdot b_2\right)}{a}}{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}\\ \mathbf{elif}\;b_2 \le 5.748502676649782580252214156933339561376 \cdot 10^{143}:\\ \;\;\;\;\frac{-b_2}{a} - \frac{\sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a}\\ \end{array}\]
\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}
↓
\begin{array}{l}
\mathbf{if}\;b_2 \le -5.99946224548089213456388959139204668765 \cdot 10^{73}:\\
\;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\

\mathbf{elif}\;b_2 \le -4.296272708131498829504916428849430668856 \cdot 10^{-127}:\\
\;\;\;\;\frac{\frac{a \cdot c + \left(b_2 \cdot b_2 - b_2 \cdot b_2\right)}{a}}{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}\\

\mathbf{elif}\;b_2 \le 5.748502676649782580252214156933339561376 \cdot 10^{143}:\\
\;\;\;\;\frac{-b_2}{a} - \frac{\sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\\

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

\end{array}
double f(double a, double b_2, double c) {
        double r2765767 = b_2;
        double r2765768 = -r2765767;
        double r2765769 = r2765767 * r2765767;
        double r2765770 = a;
        double r2765771 = c;
        double r2765772 = r2765770 * r2765771;
        double r2765773 = r2765769 - r2765772;
        double r2765774 = sqrt(r2765773);
        double r2765775 = r2765768 - r2765774;
        double r2765776 = r2765775 / r2765770;
        return r2765776;
}
↓
double f(double a, double b_2, double c) {
        double r2765777 = b_2;
        double r2765778 = -5.999462245480892e+73;
        bool r2765779 = r2765777 <= r2765778;
        double r2765780 = -0.5;
        double r2765781 = c;
        double r2765782 = r2765781 / r2765777;
        double r2765783 = r2765780 * r2765782;
        double r2765784 = -4.296272708131499e-127;
        bool r2765785 = r2765777 <= r2765784;
        double r2765786 = a;
        double r2765787 = r2765786 * r2765781;
        double r2765788 = r2765777 * r2765777;
        double r2765789 = r2765788 - r2765788;
        double r2765790 = r2765787 + r2765789;
        double r2765791 = r2765790 / r2765786;
        double r2765792 = -r2765777;
        double r2765793 = r2765788 - r2765787;
        double r2765794 = sqrt(r2765793);
        double r2765795 = r2765792 + r2765794;
        double r2765796 = r2765791 / r2765795;
        double r2765797 = 5.748502676649783e+143;
        bool r2765798 = r2765777 <= r2765797;
        double r2765799 = r2765792 / r2765786;
        double r2765800 = r2765794 / r2765786;
        double r2765801 = r2765799 - r2765800;
        double r2765802 = -2.0;
        double r2765803 = r2765777 / r2765786;
        double r2765804 = r2765802 * r2765803;
        double r2765805 = r2765798 ? r2765801 : r2765804;
        double r2765806 = r2765785 ? r2765796 : r2765805;
        double r2765807 = r2765779 ? r2765783 : r2765806;
        return r2765807;
}

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 < -5.999462245480892e+73

    1. Initial program 58.3

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

    2. Taylor expanded around -inf 3.4

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

    if -5.999462245480892e+73 < b_2 < -4.296272708131499e-127

    1. Initial program 40.0

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

    3. Applied div-inv40.0

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

    5. Applied flip--40.0

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

    6. Applied associate-*l/40.0

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

    7. Simplified14.7

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

    if -4.296272708131499e-127 < b_2 < 5.748502676649783e+143

    1. Initial program 11.3

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

    3. Applied div-sub11.2

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

    if 5.748502676649783e+143 < b_2

    1. Initial program 59.9

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

    3. Applied div-inv59.9

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

    5. Applied flip--63.9

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

    6. Applied associate-*l/63.9

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

    7. Simplified63.9

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

    8. Taylor expanded around 0 62.6

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

    9. Taylor expanded around 0 2.2

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

  4. Final simplification8.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -5.99946224548089213456388959139204668765 \cdot 10^{73}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le -4.296272708131498829504916428849430668856 \cdot 10^{-127}:\\ \;\;\;\;\frac{\frac{a \cdot c + \left(b_2 \cdot b_2 - b_2 \cdot b_2\right)}{a}}{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}\\ \mathbf{elif}\;b_2 \le 5.748502676649782580252214156933339561376 \cdot 10^{143}:\\ \;\;\;\;\frac{-b_2}{a} - \frac{\sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2019200 +o rules:numerics
(FPCore (a b_2 c)
  :name "NMSE problem 3.2.1"
  (/ (- (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))