Average Error: 33.6 → 10.0
Time: 19.2s
Precision: 64
\[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b_2 \le -2.4515142736560382 \cdot 10^{+31}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le -6.822071483797687 \cdot 10^{-95}:\\ \;\;\;\;\frac{a}{\frac{a}{\frac{c}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}}\\ \mathbf{elif}\;b_2 \le 0.17389787404847717:\\ \;\;\;\;\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b_2} \cdot \frac{1}{2} - 2 \cdot \frac{b_2}{a}\\ \end{array}\]
\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}
↓
\begin{array}{l}
\mathbf{if}\;b_2 \le -2.4515142736560382 \cdot 10^{+31}:\\
\;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\

\mathbf{elif}\;b_2 \le -6.822071483797687 \cdot 10^{-95}:\\
\;\;\;\;\frac{a}{\frac{a}{\frac{c}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}}\\

\mathbf{elif}\;b_2 \le 0.17389787404847717:\\
\;\;\;\;\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} - 2 \cdot \frac{b_2}{a}\\

\end{array}
double f(double a, double b_2, double c) {
        double r1014615 = b_2;
        double r1014616 = -r1014615;
        double r1014617 = r1014615 * r1014615;
        double r1014618 = a;
        double r1014619 = c;
        double r1014620 = r1014618 * r1014619;
        double r1014621 = r1014617 - r1014620;
        double r1014622 = sqrt(r1014621);
        double r1014623 = r1014616 - r1014622;
        double r1014624 = r1014623 / r1014618;
        return r1014624;
}
↓
double f(double a, double b_2, double c) {
        double r1014625 = b_2;
        double r1014626 = -2.4515142736560382e+31;
        bool r1014627 = r1014625 <= r1014626;
        double r1014628 = -0.5;
        double r1014629 = c;
        double r1014630 = r1014629 / r1014625;
        double r1014631 = r1014628 * r1014630;
        double r1014632 = -6.822071483797687e-95;
        bool r1014633 = r1014625 <= r1014632;
        double r1014634 = a;
        double r1014635 = r1014625 * r1014625;
        double r1014636 = r1014634 * r1014629;
        double r1014637 = r1014635 - r1014636;
        double r1014638 = sqrt(r1014637);
        double r1014639 = r1014638 - r1014625;
        double r1014640 = r1014629 / r1014639;
        double r1014641 = r1014634 / r1014640;
        double r1014642 = r1014634 / r1014641;
        double r1014643 = 0.17389787404847717;
        bool r1014644 = r1014625 <= r1014643;
        double r1014645 = -r1014625;
        double r1014646 = r1014645 - r1014638;
        double r1014647 = r1014646 / r1014634;
        double r1014648 = 0.5;
        double r1014649 = r1014630 * r1014648;
        double r1014650 = 2.0;
        double r1014651 = r1014625 / r1014634;
        double r1014652 = r1014650 * r1014651;
        double r1014653 = r1014649 - r1014652;
        double r1014654 = r1014644 ? r1014647 : r1014653;
        double r1014655 = r1014633 ? r1014642 : r1014654;
        double r1014656 = r1014627 ? r1014631 : r1014655;
        return r1014656;
}

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 < -2.4515142736560382e+31

    1. Initial program 55.8

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

    2. Taylor expanded around -inf 4.3

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

    if -2.4515142736560382e+31 < b_2 < -6.822071483797687e-95

    1. Initial program 39.5

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

    3. Applied flip--39.5

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

    4. Simplified14.0

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

    5. Simplified14.0

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

    7. Applied *-un-lft-identity14.0

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

    8. Applied times-frac11.4

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

    9. Applied associate-/l*19.9

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

    if -6.822071483797687e-95 < b_2 < 0.17389787404847717

    1. Initial program 14.2

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

    if 0.17389787404847717 < b_2

    1. Initial program 29.8

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

    2. Taylor expanded around inf 7.3

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

  4. Final simplification10.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -2.4515142736560382 \cdot 10^{+31}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le -6.822071483797687 \cdot 10^{-95}:\\ \;\;\;\;\frac{a}{\frac{a}{\frac{c}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}}\\ \mathbf{elif}\;b_2 \le 0.17389787404847717:\\ \;\;\;\;\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b_2} \cdot \frac{1}{2} - 2 \cdot \frac{b_2}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2019168 
(FPCore (a b_2 c)
  :name "quad2m (problem 3.2.1, negative)"
  (/ (- (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))