Average Error: 34.5 → 10.6
Time: 10.7s
Precision: 64
\[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b_2 \le -4.78285893492843261 \cdot 10^{-126}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le 3.6627135292415903 \cdot 10^{111}:\\ \;\;\;\;\left(\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}\right) \cdot \frac{1}{a}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a}\\ \end{array}\]
\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}
↓
\begin{array}{l}
\mathbf{if}\;b_2 \le -4.78285893492843261 \cdot 10^{-126}:\\
\;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\

\mathbf{elif}\;b_2 \le 3.6627135292415903 \cdot 10^{111}:\\
\;\;\;\;\left(\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}\right) \cdot \frac{1}{a}\\

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

\end{array}
double f(double a, double b_2, double c) {
        double r24166 = b_2;
        double r24167 = -r24166;
        double r24168 = r24166 * r24166;
        double r24169 = a;
        double r24170 = c;
        double r24171 = r24169 * r24170;
        double r24172 = r24168 - r24171;
        double r24173 = sqrt(r24172);
        double r24174 = r24167 - r24173;
        double r24175 = r24174 / r24169;
        return r24175;
}
↓
double f(double a, double b_2, double c) {
        double r24176 = b_2;
        double r24177 = -4.7828589349284326e-126;
        bool r24178 = r24176 <= r24177;
        double r24179 = -0.5;
        double r24180 = c;
        double r24181 = r24180 / r24176;
        double r24182 = r24179 * r24181;
        double r24183 = 3.6627135292415903e+111;
        bool r24184 = r24176 <= r24183;
        double r24185 = -r24176;
        double r24186 = r24176 * r24176;
        double r24187 = a;
        double r24188 = r24187 * r24180;
        double r24189 = r24186 - r24188;
        double r24190 = sqrt(r24189);
        double r24191 = r24185 - r24190;
        double r24192 = 1.0;
        double r24193 = r24192 / r24187;
        double r24194 = r24191 * r24193;
        double r24195 = -2.0;
        double r24196 = r24176 / r24187;
        double r24197 = r24195 * r24196;
        double r24198 = r24184 ? r24194 : r24197;
        double r24199 = r24178 ? r24182 : r24198;
        return r24199;
}

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 < -4.7828589349284326e-126

    1. Initial program 51.3

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

    2. Taylor expanded around -inf 11.3

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

    if -4.7828589349284326e-126 < b_2 < 3.6627135292415903e+111

    1. Initial program 12.0

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

    3. Applied div-inv12.1

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

    if 3.6627135292415903e+111 < b_2

    1. Initial program 49.7

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

    3. Applied flip--63.3

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

    4. Simplified62.3

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

    5. Simplified62.3

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

    6. Taylor expanded around 0 3.6

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

  4. Final simplification10.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -4.78285893492843261 \cdot 10^{-126}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le 3.6627135292415903 \cdot 10^{111}:\\ \;\;\;\;\left(\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}\right) \cdot \frac{1}{a}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2020047 
(FPCore (a b_2 c)
  :name "quad2m (problem 3.2.1, negative)"
  :precision binary64
  (/ (- (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))