Average Error: 33.9 → 11.4
Time: 7.6m
Precision: 64
\[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b_2 \le -5.776846430852749186156433035624395729394 \cdot 10^{100}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le 7.455592343308264166675918758902222662503 \cdot 10^{-170}:\\ \;\;\;\;\frac{\frac{1}{\frac{1}{c}}}{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\\ \end{array}\]
\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}
↓
\begin{array}{l}
\mathbf{if}\;b_2 \le -5.776846430852749186156433035624395729394 \cdot 10^{100}:\\
\;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\

\mathbf{elif}\;b_2 \le 7.455592343308264166675918758902222662503 \cdot 10^{-170}:\\
\;\;\;\;\frac{\frac{1}{\frac{1}{c}}}{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}\\

\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 r23620 = b_2;
        double r23621 = -r23620;
        double r23622 = r23620 * r23620;
        double r23623 = a;
        double r23624 = c;
        double r23625 = r23623 * r23624;
        double r23626 = r23622 - r23625;
        double r23627 = sqrt(r23626);
        double r23628 = r23621 - r23627;
        double r23629 = r23628 / r23623;
        return r23629;
}
↓
double f(double a, double b_2, double c) {
        double r23630 = b_2;
        double r23631 = -5.776846430852749e+100;
        bool r23632 = r23630 <= r23631;
        double r23633 = -0.5;
        double r23634 = c;
        double r23635 = r23634 / r23630;
        double r23636 = r23633 * r23635;
        double r23637 = 7.455592343308264e-170;
        bool r23638 = r23630 <= r23637;
        double r23639 = 1.0;
        double r23640 = r23639 / r23634;
        double r23641 = r23639 / r23640;
        double r23642 = -r23630;
        double r23643 = r23630 * r23630;
        double r23644 = a;
        double r23645 = r23644 * r23634;
        double r23646 = r23643 - r23645;
        double r23647 = sqrt(r23646);
        double r23648 = r23642 + r23647;
        double r23649 = r23641 / r23648;
        double r23650 = 0.5;
        double r23651 = r23650 * r23635;
        double r23652 = 2.0;
        double r23653 = r23630 / r23644;
        double r23654 = r23652 * r23653;
        double r23655 = r23651 - r23654;
        double r23656 = r23638 ? r23649 : r23655;
        double r23657 = r23632 ? r23636 : r23656;
        return r23657;
}

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 3 regimes

  2. if b_2 < -5.776846430852749e+100

    1. Initial program 59.8

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

    2. Taylor expanded around -inf 2.5

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

    if -5.776846430852749e+100 < b_2 < 7.455592343308264e-170

    1. Initial program 28.9

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

    3. Applied clear-num28.9

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

    5. Applied flip--29.1

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

    6. Applied associate-/r/29.1

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

    7. Applied associate-/r*29.1

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

    8. Simplified16.1

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

    10. Applied clear-num16.1

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

    11. Simplified11.2

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

    if 7.455592343308264e-170 < b_2

    1. Initial program 23.0

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

    2. Taylor expanded around inf 17.1

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

  4. Final simplification11.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -5.776846430852749186156433035624395729394 \cdot 10^{100}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le 7.455592343308264166675918758902222662503 \cdot 10^{-170}:\\ \;\;\;\;\frac{\frac{1}{\frac{1}{c}}}{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2019323 
(FPCore (a b_2 c)
  :name "quad2m (problem 3.2.1, negative)"
  :precision binary64
  (/ (- (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))