Average Error: 34.0 → 8.9
Time: 5.6s
Precision: 64
\[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b_2 \le -8.87226991740961 \cdot 10^{30}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le -3.6437072157644369 \cdot 10^{-156}:\\ \;\;\;\;\frac{\frac{a}{\frac{e^{\log \left(\sqrt{b_2 \cdot b_2 - a \cdot c}\right)} - b_2}{c}}}{a}\\ \mathbf{elif}\;b_2 \le 1.9541965478093634 \cdot 10^{85}:\\ \;\;\;\;\left(\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}\right) \cdot \frac{1}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2 \cdot b_2}{a}\\ \end{array}\]
\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}
↓
\begin{array}{l}
\mathbf{if}\;b_2 \le -8.87226991740961 \cdot 10^{30}:\\
\;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\

\mathbf{elif}\;b_2 \le -3.6437072157644369 \cdot 10^{-156}:\\
\;\;\;\;\frac{\frac{a}{\frac{e^{\log \left(\sqrt{b_2 \cdot b_2 - a \cdot c}\right)} - b_2}{c}}}{a}\\

\mathbf{elif}\;b_2 \le 1.9541965478093634 \cdot 10^{85}:\\
\;\;\;\;\left(\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}\right) \cdot \frac{1}{a}\\

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

\end{array}
double f(double a, double b_2, double c) {
        double r19693 = b_2;
        double r19694 = -r19693;
        double r19695 = r19693 * r19693;
        double r19696 = a;
        double r19697 = c;
        double r19698 = r19696 * r19697;
        double r19699 = r19695 - r19698;
        double r19700 = sqrt(r19699);
        double r19701 = r19694 - r19700;
        double r19702 = r19701 / r19696;
        return r19702;
}
↓
double f(double a, double b_2, double c) {
        double r19703 = b_2;
        double r19704 = -8.87226991740961e+30;
        bool r19705 = r19703 <= r19704;
        double r19706 = -0.5;
        double r19707 = c;
        double r19708 = r19707 / r19703;
        double r19709 = r19706 * r19708;
        double r19710 = -3.643707215764437e-156;
        bool r19711 = r19703 <= r19710;
        double r19712 = a;
        double r19713 = r19703 * r19703;
        double r19714 = r19712 * r19707;
        double r19715 = r19713 - r19714;
        double r19716 = sqrt(r19715);
        double r19717 = log(r19716);
        double r19718 = exp(r19717);
        double r19719 = r19718 - r19703;
        double r19720 = r19719 / r19707;
        double r19721 = r19712 / r19720;
        double r19722 = r19721 / r19712;
        double r19723 = 1.9541965478093634e+85;
        bool r19724 = r19703 <= r19723;
        double r19725 = -r19703;
        double r19726 = r19725 - r19716;
        double r19727 = 1.0;
        double r19728 = r19727 / r19712;
        double r19729 = r19726 * r19728;
        double r19730 = -2.0;
        double r19731 = r19730 * r19703;
        double r19732 = r19731 / r19712;
        double r19733 = r19724 ? r19729 : r19732;
        double r19734 = r19711 ? r19722 : r19733;
        double r19735 = r19705 ? r19709 : r19734;
        return r19735;
}

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 < -8.87226991740961e+30

    1. Initial program 56.7

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

    2. Taylor expanded around -inf 5.1

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

    if -8.87226991740961e+30 < b_2 < -3.643707215764437e-156

    1. Initial program 34.4

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

    3. Applied flip--34.5

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

    4. Simplified16.8

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

    5. Simplified16.8

      \[\leadsto \frac{\frac{0 + a \cdot c}{\color{blue}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}}{a}\]
    6. Using strategy rm

    7. Applied *-un-lft-identity16.8

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

    8. Applied associate-/r*16.8

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

    9. Simplified12.7

      \[\leadsto \frac{\color{blue}{\frac{a}{\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{c}}}}{a}\]
    10. Using strategy rm

    11. Applied add-exp-log15.4

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

    if -3.643707215764437e-156 < b_2 < 1.9541965478093634e+85

    1. Initial program 11.5

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

    3. Applied div-inv11.6

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

    if 1.9541965478093634e+85 < b_2

    1. Initial program 43.6

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

    3. Applied flip--63.0

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

    4. Simplified62.2

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

    5. Simplified62.2

      \[\leadsto \frac{\frac{0 + a \cdot c}{\color{blue}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}}{a}\]
    6. Using strategy rm

    7. Applied *-un-lft-identity62.2

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

    8. Applied associate-/r*62.2

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

    9. Simplified62.0

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

    10. Taylor expanded around 0 4.0

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

  4. Final simplification8.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -8.87226991740961 \cdot 10^{30}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le -3.6437072157644369 \cdot 10^{-156}:\\ \;\;\;\;\frac{\frac{a}{\frac{e^{\log \left(\sqrt{b_2 \cdot b_2 - a \cdot c}\right)} - b_2}{c}}}{a}\\ \mathbf{elif}\;b_2 \le 1.9541965478093634 \cdot 10^{85}:\\ \;\;\;\;\left(\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}\right) \cdot \frac{1}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2 \cdot b_2}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2020039 
(FPCore (a b_2 c)
  :name "quad2m (problem 3.2.1, negative)"
  :precision binary64
  (/ (- (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))