Average Error: 34.8 → 9.3
Time: 6.1s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -9.19678115322534318 \cdot 10^{150}:\\ \;\;\;\;0.5 \cdot \frac{c}{b} - 0.66666666666666663 \cdot \frac{b}{a}\\ \mathbf{elif}\;b \le 5.0355868398843843 \cdot 10^{-134}:\\ \;\;\;\;\frac{\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3}}{a}\\ \mathbf{elif}\;b \le 8.0911772821281571 \cdot 10^{46}:\\ \;\;\;\;\frac{\frac{0 + 3 \cdot \left(a \cdot c\right)}{\left(-b\right) - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array}\]
\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}
↓
\begin{array}{l}
\mathbf{if}\;b \le -9.19678115322534318 \cdot 10^{150}:\\
\;\;\;\;0.5 \cdot \frac{c}{b} - 0.66666666666666663 \cdot \frac{b}{a}\\

\mathbf{elif}\;b \le 5.0355868398843843 \cdot 10^{-134}:\\
\;\;\;\;\frac{\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3}}{a}\\

\mathbf{elif}\;b \le 8.0911772821281571 \cdot 10^{46}:\\
\;\;\;\;\frac{\frac{0 + 3 \cdot \left(a \cdot c\right)}{\left(-b\right) - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}}{3 \cdot a}\\

\mathbf{else}:\\
\;\;\;\;-0.5 \cdot \frac{c}{b}\\

\end{array}
double f(double a, double b, double c) {
        double r110206 = b;
        double r110207 = -r110206;
        double r110208 = r110206 * r110206;
        double r110209 = 3.0;
        double r110210 = a;
        double r110211 = r110209 * r110210;
        double r110212 = c;
        double r110213 = r110211 * r110212;
        double r110214 = r110208 - r110213;
        double r110215 = sqrt(r110214);
        double r110216 = r110207 + r110215;
        double r110217 = r110216 / r110211;
        return r110217;
}
↓
double f(double a, double b, double c) {
        double r110218 = b;
        double r110219 = -9.196781153225343e+150;
        bool r110220 = r110218 <= r110219;
        double r110221 = 0.5;
        double r110222 = c;
        double r110223 = r110222 / r110218;
        double r110224 = r110221 * r110223;
        double r110225 = 0.6666666666666666;
        double r110226 = a;
        double r110227 = r110218 / r110226;
        double r110228 = r110225 * r110227;
        double r110229 = r110224 - r110228;
        double r110230 = 5.035586839884384e-134;
        bool r110231 = r110218 <= r110230;
        double r110232 = -r110218;
        double r110233 = r110218 * r110218;
        double r110234 = 3.0;
        double r110235 = r110234 * r110226;
        double r110236 = r110235 * r110222;
        double r110237 = r110233 - r110236;
        double r110238 = sqrt(r110237);
        double r110239 = r110232 + r110238;
        double r110240 = r110239 / r110234;
        double r110241 = r110240 / r110226;
        double r110242 = 8.091177282128157e+46;
        bool r110243 = r110218 <= r110242;
        double r110244 = 0.0;
        double r110245 = r110226 * r110222;
        double r110246 = r110234 * r110245;
        double r110247 = r110244 + r110246;
        double r110248 = r110232 - r110238;
        double r110249 = r110247 / r110248;
        double r110250 = r110249 / r110235;
        double r110251 = -0.5;
        double r110252 = r110251 * r110223;
        double r110253 = r110243 ? r110250 : r110252;
        double r110254 = r110231 ? r110241 : r110253;
        double r110255 = r110220 ? r110229 : r110254;
        return r110255;
}

Error

Bits error versus a

Bits error versus b

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 < -9.196781153225343e+150

    1. Initial program 62.9

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]

    2. Taylor expanded around -inf 2.3

      \[\leadsto \color{blue}{0.5 \cdot \frac{c}{b} - 0.66666666666666663 \cdot \frac{b}{a}}\]

    if -9.196781153225343e+150 < b < 5.035586839884384e-134

    1. Initial program 11.6

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]
    2. Using strategy rm

    3. Applied associate-/r*11.6

      \[\leadsto \color{blue}{\frac{\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3}}{a}}\]

    if 5.035586839884384e-134 < b < 8.091177282128157e+46

    1. Initial program 37.0

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]
    2. Using strategy rm

    3. Applied flip-+37.0

      \[\leadsto \frac{\color{blue}{\frac{\left(-b\right) \cdot \left(-b\right) - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} \cdot \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{\left(-b\right) - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}}}{3 \cdot a}\]

    4. Simplified15.8

      \[\leadsto \frac{\frac{\color{blue}{0 + 3 \cdot \left(a \cdot c\right)}}{\left(-b\right) - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}}{3 \cdot a}\]

    if 8.091177282128157e+46 < b

    1. Initial program 56.9

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]

    2. Taylor expanded around inf 5.0

      \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b}}\]
  3. Recombined 4 regimes into one program.

  4. Final simplification9.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -9.19678115322534318 \cdot 10^{150}:\\ \;\;\;\;0.5 \cdot \frac{c}{b} - 0.66666666666666663 \cdot \frac{b}{a}\\ \mathbf{elif}\;b \le 5.0355868398843843 \cdot 10^{-134}:\\ \;\;\;\;\frac{\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3}}{a}\\ \mathbf{elif}\;b \le 8.0911772821281571 \cdot 10^{46}:\\ \;\;\;\;\frac{\frac{0 + 3 \cdot \left(a \cdot c\right)}{\left(-b\right) - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2020034 
(FPCore (a b c)
  :name "Cubic critical"
  :precision binary64
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3 a) c)))) (* 3 a)))