Average Error: 33.7 → 8.4
Time: 15.6s
Precision: 64
\[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b_2 \le -3.234164035284793 \cdot 10^{+22}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le -6.3209183644448 \cdot 10^{-115}:\\ \;\;\;\;\frac{\frac{c \cdot a}{\sqrt{b_2 \cdot b_2 - c \cdot a} - b_2}}{a}\\ \mathbf{elif}\;b_2 \le 2.026128983134594 \cdot 10^{+103}:\\ \;\;\;\;\frac{1}{a} \cdot \left(\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - c \cdot a}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - \frac{b_2}{a} \cdot 2\\ \end{array}\]
\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}
↓
\begin{array}{l}
\mathbf{if}\;b_2 \le -3.234164035284793 \cdot 10^{+22}:\\
\;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\

\mathbf{elif}\;b_2 \le -6.3209183644448 \cdot 10^{-115}:\\
\;\;\;\;\frac{\frac{c \cdot a}{\sqrt{b_2 \cdot b_2 - c \cdot a} - b_2}}{a}\\

\mathbf{elif}\;b_2 \le 2.026128983134594 \cdot 10^{+103}:\\
\;\;\;\;\frac{1}{a} \cdot \left(\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - c \cdot a}\right)\\

\mathbf{else}:\\
\;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - \frac{b_2}{a} \cdot 2\\

\end{array}
double f(double a, double b_2, double c) {
        double r3366714 = b_2;
        double r3366715 = -r3366714;
        double r3366716 = r3366714 * r3366714;
        double r3366717 = a;
        double r3366718 = c;
        double r3366719 = r3366717 * r3366718;
        double r3366720 = r3366716 - r3366719;
        double r3366721 = sqrt(r3366720);
        double r3366722 = r3366715 - r3366721;
        double r3366723 = r3366722 / r3366717;
        return r3366723;
}
↓
double f(double a, double b_2, double c) {
        double r3366724 = b_2;
        double r3366725 = -3.234164035284793e+22;
        bool r3366726 = r3366724 <= r3366725;
        double r3366727 = -0.5;
        double r3366728 = c;
        double r3366729 = r3366728 / r3366724;
        double r3366730 = r3366727 * r3366729;
        double r3366731 = -6.3209183644448e-115;
        bool r3366732 = r3366724 <= r3366731;
        double r3366733 = a;
        double r3366734 = r3366728 * r3366733;
        double r3366735 = r3366724 * r3366724;
        double r3366736 = r3366735 - r3366734;
        double r3366737 = sqrt(r3366736);
        double r3366738 = r3366737 - r3366724;
        double r3366739 = r3366734 / r3366738;
        double r3366740 = r3366739 / r3366733;
        double r3366741 = 2.026128983134594e+103;
        bool r3366742 = r3366724 <= r3366741;
        double r3366743 = 1.0;
        double r3366744 = r3366743 / r3366733;
        double r3366745 = -r3366724;
        double r3366746 = r3366745 - r3366737;
        double r3366747 = r3366744 * r3366746;
        double r3366748 = 0.5;
        double r3366749 = r3366748 * r3366729;
        double r3366750 = r3366724 / r3366733;
        double r3366751 = 2.0;
        double r3366752 = r3366750 * r3366751;
        double r3366753 = r3366749 - r3366752;
        double r3366754 = r3366742 ? r3366747 : r3366753;
        double r3366755 = r3366732 ? r3366740 : r3366754;
        double r3366756 = r3366726 ? r3366730 : r3366755;
        return r3366756;
}

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 4 regimes

  2. if b_2 < -3.234164035284793e+22

    1. Initial program 55.9

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

    2. Taylor expanded around -inf 4.6

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

    if -3.234164035284793e+22 < b_2 < -6.3209183644448e-115

    1. Initial program 38.3

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

    3. Applied flip--38.4

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

    4. Simplified15.5

      \[\leadsto \frac{\frac{\color{blue}{0 + a \cdot c}}{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}}{a}\]

    5. Simplified15.5

      \[\leadsto \frac{\frac{0 + a \cdot c}{\color{blue}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}}{a}\]

    if -6.3209183644448e-115 < b_2 < 2.026128983134594e+103

    1. Initial program 11.3

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

    3. Applied div-inv11.4

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

    if 2.026128983134594e+103 < b_2

    1. Initial program 45.2

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

    2. Taylor expanded around inf 3.2

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}}\]
  3. Recombined 4 regimes into one program.

  4. Final simplification8.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -3.234164035284793 \cdot 10^{+22}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le -6.3209183644448 \cdot 10^{-115}:\\ \;\;\;\;\frac{\frac{c \cdot a}{\sqrt{b_2 \cdot b_2 - c \cdot a} - b_2}}{a}\\ \mathbf{elif}\;b_2 \le 2.026128983134594 \cdot 10^{+103}:\\ \;\;\;\;\frac{1}{a} \cdot \left(\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - c \cdot a}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - \frac{b_2}{a} \cdot 2\\ \end{array}\]

Reproduce

herbie shell --seed 2019165 
(FPCore (a b_2 c)
  :name "NMSE problem 3.2.1"
  (/ (- (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))