Average Error: 34.0 → 6.7
Time: 12.0s
Precision: 64
\[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b_2 \le -6.256826279459623886155459132503749770426 \cdot 10^{138}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le -5.489520038969419318018107519878377265481 \cdot 10^{-285}:\\ \;\;\;\;\frac{c}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}\\ \mathbf{elif}\;b_2 \le 84394753974248718336:\\ \;\;\;\;\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\\ \end{array}\]
\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}
↓
\begin{array}{l}
\mathbf{if}\;b_2 \le -6.256826279459623886155459132503749770426 \cdot 10^{138}:\\
\;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\

\mathbf{elif}\;b_2 \le -5.489520038969419318018107519878377265481 \cdot 10^{-285}:\\
\;\;\;\;\frac{c}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}\\

\mathbf{elif}\;b_2 \le 84394753974248718336:\\
\;\;\;\;\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 r83627 = b_2;
        double r83628 = -r83627;
        double r83629 = r83627 * r83627;
        double r83630 = a;
        double r83631 = c;
        double r83632 = r83630 * r83631;
        double r83633 = r83629 - r83632;
        double r83634 = sqrt(r83633);
        double r83635 = r83628 - r83634;
        double r83636 = r83635 / r83630;
        return r83636;
}
↓
double f(double a, double b_2, double c) {
        double r83637 = b_2;
        double r83638 = -6.256826279459624e+138;
        bool r83639 = r83637 <= r83638;
        double r83640 = -0.5;
        double r83641 = c;
        double r83642 = r83641 / r83637;
        double r83643 = r83640 * r83642;
        double r83644 = -5.489520038969419e-285;
        bool r83645 = r83637 <= r83644;
        double r83646 = r83637 * r83637;
        double r83647 = a;
        double r83648 = r83647 * r83641;
        double r83649 = r83646 - r83648;
        double r83650 = sqrt(r83649);
        double r83651 = r83650 - r83637;
        double r83652 = r83641 / r83651;
        double r83653 = 8.439475397424872e+19;
        bool r83654 = r83637 <= r83653;
        double r83655 = -r83637;
        double r83656 = r83655 - r83650;
        double r83657 = r83656 / r83647;
        double r83658 = 0.5;
        double r83659 = r83658 * r83642;
        double r83660 = 2.0;
        double r83661 = r83637 / r83647;
        double r83662 = r83660 * r83661;
        double r83663 = r83659 - r83662;
        double r83664 = r83654 ? r83657 : r83663;
        double r83665 = r83645 ? r83652 : r83664;
        double r83666 = r83639 ? r83643 : r83665;
        return r83666;
}

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 < -6.256826279459624e+138

    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 1.8

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

    if -6.256826279459624e+138 < b_2 < -5.489520038969419e-285

    1. Initial program 35.1

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

    3. Applied flip--35.1

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

    4. Simplified16.0

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

    5. Simplified16.0

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

    7. Applied *-un-lft-identity16.0

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

    8. Applied *-un-lft-identity16.0

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

    9. Applied times-frac16.0

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

    10. Simplified16.0

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

    11. Simplified7.5

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

    if -5.489520038969419e-285 < b_2 < 8.439475397424872e+19

    1. Initial program 9.5

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

    if 8.439475397424872e+19 < b_2

    1. Initial program 34.5

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

    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 simplification6.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -6.256826279459623886155459132503749770426 \cdot 10^{138}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le -5.489520038969419318018107519878377265481 \cdot 10^{-285}:\\ \;\;\;\;\frac{c}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}\\ \mathbf{elif}\;b_2 \le 84394753974248718336:\\ \;\;\;\;\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2019350 
(FPCore (a b_2 c)
  :name "NMSE problem 3.2.1"
  :precision binary64
  (/ (- (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))