Average Error: 34.0 → 10.5
Time: 19.5s
Precision: 64
\[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b_2 \le -9.332433396832084322962138528577137922234 \cdot 10^{-58}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le 3.038903409991338138548211857189252856935 \cdot 10^{107}:\\ \;\;\;\;\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{b_2}{a} \cdot -2\\ \end{array}\]
\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}
↓
\begin{array}{l}
\mathbf{if}\;b_2 \le -9.332433396832084322962138528577137922234 \cdot 10^{-58}:\\
\;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\

\mathbf{elif}\;b_2 \le 3.038903409991338138548211857189252856935 \cdot 10^{107}:\\
\;\;\;\;\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\\

\mathbf{else}:\\
\;\;\;\;\frac{b_2}{a} \cdot -2\\

\end{array}
double f(double a, double b_2, double c) {
        double r3256646 = b_2;
        double r3256647 = -r3256646;
        double r3256648 = r3256646 * r3256646;
        double r3256649 = a;
        double r3256650 = c;
        double r3256651 = r3256649 * r3256650;
        double r3256652 = r3256648 - r3256651;
        double r3256653 = sqrt(r3256652);
        double r3256654 = r3256647 - r3256653;
        double r3256655 = r3256654 / r3256649;
        return r3256655;
}
↓
double f(double a, double b_2, double c) {
        double r3256656 = b_2;
        double r3256657 = -9.332433396832084e-58;
        bool r3256658 = r3256656 <= r3256657;
        double r3256659 = -0.5;
        double r3256660 = c;
        double r3256661 = r3256660 / r3256656;
        double r3256662 = r3256659 * r3256661;
        double r3256663 = 3.038903409991338e+107;
        bool r3256664 = r3256656 <= r3256663;
        double r3256665 = -r3256656;
        double r3256666 = r3256656 * r3256656;
        double r3256667 = a;
        double r3256668 = r3256667 * r3256660;
        double r3256669 = r3256666 - r3256668;
        double r3256670 = sqrt(r3256669);
        double r3256671 = r3256665 - r3256670;
        double r3256672 = r3256671 / r3256667;
        double r3256673 = r3256656 / r3256667;
        double r3256674 = -2.0;
        double r3256675 = r3256673 * r3256674;
        double r3256676 = r3256664 ? r3256672 : r3256675;
        double r3256677 = r3256658 ? r3256662 : r3256676;
        return r3256677;
}

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 < -9.332433396832084e-58

    1. Initial program 53.5

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

    2. Taylor expanded around -inf 8.7

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

    if -9.332433396832084e-58 < b_2 < 3.038903409991338e+107

    1. Initial program 14.0

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

    if 3.038903409991338e+107 < b_2

    1. Initial program 49.1

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

    3. Applied flip--63.2

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

    4. Simplified62.2

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

    5. Simplified62.2

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

    6. Taylor expanded around 0 3.7

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

  4. Final simplification10.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -9.332433396832084322962138528577137922234 \cdot 10^{-58}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le 3.038903409991338138548211857189252856935 \cdot 10^{107}:\\ \;\;\;\;\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{b_2}{a} \cdot -2\\ \end{array}\]

Reproduce

herbie shell --seed 2019171 
(FPCore (a b_2 c)
  :name "NMSE problem 3.2.1"
  (/ (- (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))