Average Error: 33.7 → 8.4
Time: 17.2s
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 r888198 = b_2;
        double r888199 = -r888198;
        double r888200 = r888198 * r888198;
        double r888201 = a;
        double r888202 = c;
        double r888203 = r888201 * r888202;
        double r888204 = r888200 - r888203;
        double r888205 = sqrt(r888204);
        double r888206 = r888199 - r888205;
        double r888207 = r888206 / r888201;
        return r888207;
}
↓
double f(double a, double b_2, double c) {
        double r888208 = b_2;
        double r888209 = -3.234164035284793e+22;
        bool r888210 = r888208 <= r888209;
        double r888211 = -0.5;
        double r888212 = c;
        double r888213 = r888212 / r888208;
        double r888214 = r888211 * r888213;
        double r888215 = -6.3209183644448e-115;
        bool r888216 = r888208 <= r888215;
        double r888217 = a;
        double r888218 = r888212 * r888217;
        double r888219 = r888208 * r888208;
        double r888220 = r888219 - r888218;
        double r888221 = sqrt(r888220);
        double r888222 = r888221 - r888208;
        double r888223 = r888218 / r888222;
        double r888224 = r888223 / r888217;
        double r888225 = 2.026128983134594e+103;
        bool r888226 = r888208 <= r888225;
        double r888227 = 1.0;
        double r888228 = r888227 / r888217;
        double r888229 = -r888208;
        double r888230 = r888229 - r888221;
        double r888231 = r888228 * r888230;
        double r888232 = 0.5;
        double r888233 = r888232 * r888213;
        double r888234 = r888208 / r888217;
        double r888235 = 2.0;
        double r888236 = r888234 * r888235;
        double r888237 = r888233 - r888236;
        double r888238 = r888226 ? r888231 : r888237;
        double r888239 = r888216 ? r888224 : r888238;
        double r888240 = r888210 ? r888214 : r888239;
        return r888240;
}

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 "quad2m (problem 3.2.1, negative)"
  (/ (- (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))