Average Error: 34.2 → 10.1
Time: 5.3s
Precision: 64
\[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b_2 \le -4.01157973271056712 \cdot 10^{-81}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le 1.3176462918432122 \cdot 10^{99}:\\ \;\;\;\;\frac{1}{\frac{a}{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\\ \end{array}\]
\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}
↓
\begin{array}{l}
\mathbf{if}\;b_2 \le -4.01157973271056712 \cdot 10^{-81}:\\
\;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\

\mathbf{elif}\;b_2 \le 1.3176462918432122 \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 r87718 = b_2;
        double r87719 = -r87718;
        double r87720 = r87718 * r87718;
        double r87721 = a;
        double r87722 = c;
        double r87723 = r87721 * r87722;
        double r87724 = r87720 - r87723;
        double r87725 = sqrt(r87724);
        double r87726 = r87719 - r87725;
        double r87727 = r87726 / r87721;
        return r87727;
}
↓
double f(double a, double b_2, double c) {
        double r87728 = b_2;
        double r87729 = -4.011579732710567e-81;
        bool r87730 = r87728 <= r87729;
        double r87731 = -0.5;
        double r87732 = c;
        double r87733 = r87732 / r87728;
        double r87734 = r87731 * r87733;
        double r87735 = 1.3176462918432122e+99;
        bool r87736 = r87728 <= r87735;
        double r87737 = 1.0;
        double r87738 = a;
        double r87739 = -r87728;
        double r87740 = r87728 * r87728;
        double r87741 = r87738 * r87732;
        double r87742 = r87740 - r87741;
        double r87743 = sqrt(r87742);
        double r87744 = r87739 - r87743;
        double r87745 = r87738 / r87744;
        double r87746 = r87737 / r87745;
        double r87747 = 0.5;
        double r87748 = r87747 * r87733;
        double r87749 = 2.0;
        double r87750 = r87728 / r87738;
        double r87751 = r87749 * r87750;
        double r87752 = r87748 - r87751;
        double r87753 = r87736 ? r87746 : r87752;
        double r87754 = r87730 ? r87734 : r87753;
        return r87754;
}

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 < -4.011579732710567e-81

    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 9.4

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

    if -4.011579732710567e-81 < b_2 < 1.3176462918432122e+99

    1. Initial program 12.9

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

    3. Applied clear-num13.0

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

    if 1.3176462918432122e+99 < b_2

    1. Initial program 46.8

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

    2. Taylor expanded around inf 3.7

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

  4. Final simplification10.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -4.01157973271056712 \cdot 10^{-81}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le 1.3176462918432122 \cdot 10^{99}:\\ \;\;\;\;\frac{1}{\frac{a}{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2020060 +o rules:numerics
(FPCore (a b_2 c)
  :name "NMSE problem 3.2.1"
  :precision binary64
  (/ (- (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))