Average Error: 33.5 → 10.4
Time: 57.9s
Precision: 64
\[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b_2 \le -5.691277786452672 \cdot 10^{-38}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le 1.502350718288979 \cdot 10^{+75}:\\ \;\;\;\;\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b_2} \cdot \frac{1}{2} - 2 \cdot \frac{b_2}{a}\\ \end{array}\]
\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}
↓
\begin{array}{l}
\mathbf{if}\;b_2 \le -5.691277786452672 \cdot 10^{-38}:\\
\;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\

\mathbf{elif}\;b_2 \le 1.502350718288979 \cdot 10^{+75}:\\
\;\;\;\;\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 r13723731 = b_2;
        double r13723732 = -r13723731;
        double r13723733 = r13723731 * r13723731;
        double r13723734 = a;
        double r13723735 = c;
        double r13723736 = r13723734 * r13723735;
        double r13723737 = r13723733 - r13723736;
        double r13723738 = sqrt(r13723737);
        double r13723739 = r13723732 - r13723738;
        double r13723740 = r13723739 / r13723734;
        return r13723740;
}
↓
double f(double a, double b_2, double c) {
        double r13723741 = b_2;
        double r13723742 = -5.691277786452672e-38;
        bool r13723743 = r13723741 <= r13723742;
        double r13723744 = -0.5;
        double r13723745 = c;
        double r13723746 = r13723745 / r13723741;
        double r13723747 = r13723744 * r13723746;
        double r13723748 = 1.502350718288979e+75;
        bool r13723749 = r13723741 <= r13723748;
        double r13723750 = -r13723741;
        double r13723751 = r13723741 * r13723741;
        double r13723752 = a;
        double r13723753 = r13723752 * r13723745;
        double r13723754 = r13723751 - r13723753;
        double r13723755 = sqrt(r13723754);
        double r13723756 = r13723750 - r13723755;
        double r13723757 = r13723756 / r13723752;
        double r13723758 = 0.5;
        double r13723759 = r13723746 * r13723758;
        double r13723760 = 2.0;
        double r13723761 = r13723741 / r13723752;
        double r13723762 = r13723760 * r13723761;
        double r13723763 = r13723759 - r13723762;
        double r13723764 = r13723749 ? r13723757 : r13723763;
        double r13723765 = r13723743 ? r13723747 : r13723764;
        return r13723765;
}

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 < -5.691277786452672e-38

    1. Initial program 54.0

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

    2. Taylor expanded around 0 54.0

      \[\leadsto \frac{\left(-b_2\right) - \sqrt{\color{blue}{{b_2}^{2} - a \cdot c}}}{a}\]

    3. Simplified54.0

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

    4. Taylor expanded around -inf 7.9

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

    if -5.691277786452672e-38 < b_2 < 1.502350718288979e+75

    1. Initial program 14.7

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

    2. Taylor expanded around 0 14.7

      \[\leadsto \frac{\left(-b_2\right) - \sqrt{\color{blue}{{b_2}^{2} - a \cdot c}}}{a}\]

    3. Simplified14.7

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

    if 1.502350718288979e+75 < b_2

    1. Initial program 40.8

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

    2. Taylor expanded around 0 40.8

      \[\leadsto \frac{\left(-b_2\right) - \sqrt{\color{blue}{{b_2}^{2} - a \cdot c}}}{a}\]

    3. Simplified40.8

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

    4. Taylor expanded around inf 4.5

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

  4. Final simplification10.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -5.691277786452672 \cdot 10^{-38}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le 1.502350718288979 \cdot 10^{+75}:\\ \;\;\;\;\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b_2} \cdot \frac{1}{2} - 2 \cdot \frac{b_2}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2019120 
(FPCore (a b_2 c)
  :name "NMSE problem 3.2.1"
  (/ (- (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))