Average Error: 34.2 → 10.4
Time: 5.3s
Precision: 64
\[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b_2 \le -4.4270058556435274 \cdot 10^{-117}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le 2.49922826628406174 \cdot 10^{84}:\\ \;\;\;\;\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\\ \end{array}\]
\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}
↓
\begin{array}{l}
\mathbf{if}\;b_2 \le -4.4270058556435274 \cdot 10^{-117}:\\
\;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\

\mathbf{elif}\;b_2 \le 2.49922826628406174 \cdot 10^{84}:\\
\;\;\;\;\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 r17585 = b_2;
        double r17586 = -r17585;
        double r17587 = r17585 * r17585;
        double r17588 = a;
        double r17589 = c;
        double r17590 = r17588 * r17589;
        double r17591 = r17587 - r17590;
        double r17592 = sqrt(r17591);
        double r17593 = r17586 - r17592;
        double r17594 = r17593 / r17588;
        return r17594;
}
↓
double f(double a, double b_2, double c) {
        double r17595 = b_2;
        double r17596 = -4.4270058556435274e-117;
        bool r17597 = r17595 <= r17596;
        double r17598 = -0.5;
        double r17599 = c;
        double r17600 = r17599 / r17595;
        double r17601 = r17598 * r17600;
        double r17602 = 2.4992282662840617e+84;
        bool r17603 = r17595 <= r17602;
        double r17604 = -r17595;
        double r17605 = r17595 * r17595;
        double r17606 = a;
        double r17607 = r17606 * r17599;
        double r17608 = r17605 - r17607;
        double r17609 = sqrt(r17608);
        double r17610 = r17604 - r17609;
        double r17611 = r17610 / r17606;
        double r17612 = 0.5;
        double r17613 = r17612 * r17600;
        double r17614 = 2.0;
        double r17615 = r17595 / r17606;
        double r17616 = r17614 * r17615;
        double r17617 = r17613 - r17616;
        double r17618 = r17603 ? r17611 : r17617;
        double r17619 = r17597 ? r17601 : r17618;
        return r17619;
}

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 < -4.4270058556435274e-117

    1. Initial program 51.5

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

    2. Taylor expanded around -inf 11.0

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

    if -4.4270058556435274e-117 < b_2 < 2.4992282662840617e+84

    1. Initial program 12.4

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

    3. Applied div-inv12.5

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

    5. Applied un-div-inv12.4

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

    if 2.4992282662840617e+84 < b_2

    1. Initial program 43.1

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

    2. Taylor expanded around inf 4.1

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

  4. Final simplification10.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -4.4270058556435274 \cdot 10^{-117}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le 2.49922826628406174 \cdot 10^{84}:\\ \;\;\;\;\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2020056 
(FPCore (a b_2 c)
  :name "quad2m (problem 3.2.1, negative)"
  :precision binary64
  (/ (- (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))