Average Error: 34.7 → 10.2
Time: 11.4s
Precision: 64
\[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b_2 \le -4.6537017569063518 \cdot 10^{-82}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le 1.0479007947857462 \cdot 10^{99}:\\ \;\;\;\;\frac{1}{\frac{a}{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\\ \end{array}\]
\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}
↓
\begin{array}{l}
\mathbf{if}\;b_2 \le -4.6537017569063518 \cdot 10^{-82}:\\
\;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\

\mathbf{elif}\;b_2 \le 1.0479007947857462 \cdot 10^{99}:\\
\;\;\;\;\frac{1}{\frac{a}{\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 r98608 = b_2;
        double r98609 = -r98608;
        double r98610 = r98608 * r98608;
        double r98611 = a;
        double r98612 = c;
        double r98613 = r98611 * r98612;
        double r98614 = r98610 - r98613;
        double r98615 = sqrt(r98614);
        double r98616 = r98609 - r98615;
        double r98617 = r98616 / r98611;
        return r98617;
}
↓
double f(double a, double b_2, double c) {
        double r98618 = b_2;
        double r98619 = -4.653701756906352e-82;
        bool r98620 = r98618 <= r98619;
        double r98621 = -0.5;
        double r98622 = c;
        double r98623 = r98622 / r98618;
        double r98624 = r98621 * r98623;
        double r98625 = 1.0479007947857462e+99;
        bool r98626 = r98618 <= r98625;
        double r98627 = 1.0;
        double r98628 = a;
        double r98629 = -r98618;
        double r98630 = r98618 * r98618;
        double r98631 = r98628 * r98622;
        double r98632 = r98630 - r98631;
        double r98633 = sqrt(r98632);
        double r98634 = r98629 - r98633;
        double r98635 = r98628 / r98634;
        double r98636 = r98627 / r98635;
        double r98637 = 0.5;
        double r98638 = r98637 * r98623;
        double r98639 = 2.0;
        double r98640 = r98618 / r98628;
        double r98641 = r98639 * r98640;
        double r98642 = r98638 - r98641;
        double r98643 = r98626 ? r98636 : r98642;
        double r98644 = r98620 ? r98624 : r98643;
        return r98644;
}

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 < -4.653701756906352e-82

    1. Initial program 52.8

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

    2. Taylor expanded around -inf 9.1

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

    if -4.653701756906352e-82 < b_2 < 1.0479007947857462e+99

    1. Initial program 13.4

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

    3. Applied clear-num13.5

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

    if 1.0479007947857462e+99 < b_2

    1. Initial program 47.6

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

    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 simplification10.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -4.6537017569063518 \cdot 10^{-82}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le 1.0479007947857462 \cdot 10^{99}:\\ \;\;\;\;\frac{1}{\frac{a}{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2020065 
(FPCore (a b_2 c)
  :name "NMSE problem 3.2.1"
  :precision binary64
  (/ (- (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))