Average Error: 19.7 → 6.3
Time: 17.8s
Precision: 64
\[\begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -2.953923144342913498375256513451576421986 \cdot 10^{138}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{b \cdot -2 + \frac{2 \cdot a}{\frac{b}{c}}}\\ \end{array}\\ \mathbf{elif}\;b \le 1.482832280613327405571569362927236109326 \cdot 10^{130}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{\left(-b\right) + \sqrt[3]{\sqrt{b \cdot b - \left(4 \cdot c\right) \cdot a}} \cdot \left(\sqrt[3]{\sqrt{b \cdot b - \left(4 \cdot c\right) \cdot a}} \cdot \sqrt[3]{\sqrt{b \cdot b - \left(4 \cdot c\right) \cdot a}}\right)}\\ \end{array}\\ \mathbf{elif}\;b \ge 0.0:\\ \;\;\;\;\left(\frac{c}{b} - \frac{b}{a}\right) \cdot 1\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} + \left(-b\right)}\\ \end{array}\]
\begin{array}{l}
\mathbf{if}\;b \ge 0.0:\\
\;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\

\mathbf{else}:\\
\;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\

\end{array}
↓
\begin{array}{l}
\mathbf{if}\;b \le -2.953923144342913498375256513451576421986 \cdot 10^{138}:\\
\;\;\;\;\begin{array}{l}
\mathbf{if}\;b \ge 0.0:\\
\;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\

\mathbf{else}:\\
\;\;\;\;\frac{c \cdot 2}{b \cdot -2 + \frac{2 \cdot a}{\frac{b}{c}}}\\

\end{array}\\

\mathbf{elif}\;b \le 1.482832280613327405571569362927236109326 \cdot 10^{130}:\\
\;\;\;\;\begin{array}{l}
\mathbf{if}\;b \ge 0.0:\\
\;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\

\mathbf{else}:\\
\;\;\;\;\frac{c \cdot 2}{\left(-b\right) + \sqrt[3]{\sqrt{b \cdot b - \left(4 \cdot c\right) \cdot a}} \cdot \left(\sqrt[3]{\sqrt{b \cdot b - \left(4 \cdot c\right) \cdot a}} \cdot \sqrt[3]{\sqrt{b \cdot b - \left(4 \cdot c\right) \cdot a}}\right)}\\

\end{array}\\

\mathbf{elif}\;b \ge 0.0:\\
\;\;\;\;\left(\frac{c}{b} - \frac{b}{a}\right) \cdot 1\\

\mathbf{else}:\\
\;\;\;\;\frac{c \cdot 2}{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} + \left(-b\right)}\\

\end{array}
double f(double a, double b, double c) {
        double r32261 = b;
        double r32262 = 0.0;
        bool r32263 = r32261 >= r32262;
        double r32264 = -r32261;
        double r32265 = r32261 * r32261;
        double r32266 = 4.0;
        double r32267 = a;
        double r32268 = r32266 * r32267;
        double r32269 = c;
        double r32270 = r32268 * r32269;
        double r32271 = r32265 - r32270;
        double r32272 = sqrt(r32271);
        double r32273 = r32264 - r32272;
        double r32274 = 2.0;
        double r32275 = r32274 * r32267;
        double r32276 = r32273 / r32275;
        double r32277 = r32274 * r32269;
        double r32278 = r32264 + r32272;
        double r32279 = r32277 / r32278;
        double r32280 = r32263 ? r32276 : r32279;
        return r32280;
}
↓
double f(double a, double b, double c) {
        double r32281 = b;
        double r32282 = -2.9539231443429135e+138;
        bool r32283 = r32281 <= r32282;
        double r32284 = 0.0;
        bool r32285 = r32281 >= r32284;
        double r32286 = -r32281;
        double r32287 = r32281 * r32281;
        double r32288 = 4.0;
        double r32289 = a;
        double r32290 = r32288 * r32289;
        double r32291 = c;
        double r32292 = r32290 * r32291;
        double r32293 = r32287 - r32292;
        double r32294 = sqrt(r32293);
        double r32295 = r32286 - r32294;
        double r32296 = 2.0;
        double r32297 = r32296 * r32289;
        double r32298 = r32295 / r32297;
        double r32299 = r32291 * r32296;
        double r32300 = -2.0;
        double r32301 = r32281 * r32300;
        double r32302 = r32281 / r32291;
        double r32303 = r32297 / r32302;
        double r32304 = r32301 + r32303;
        double r32305 = r32299 / r32304;
        double r32306 = r32285 ? r32298 : r32305;
        double r32307 = 1.4828322806133274e+130;
        bool r32308 = r32281 <= r32307;
        double r32309 = r32288 * r32291;
        double r32310 = r32309 * r32289;
        double r32311 = r32287 - r32310;
        double r32312 = sqrt(r32311);
        double r32313 = cbrt(r32312);
        double r32314 = r32313 * r32313;
        double r32315 = r32313 * r32314;
        double r32316 = r32286 + r32315;
        double r32317 = r32299 / r32316;
        double r32318 = r32285 ? r32298 : r32317;
        double r32319 = r32291 / r32281;
        double r32320 = r32281 / r32289;
        double r32321 = r32319 - r32320;
        double r32322 = 1.0;
        double r32323 = r32321 * r32322;
        double r32324 = r32294 + r32286;
        double r32325 = r32299 / r32324;
        double r32326 = r32285 ? r32323 : r32325;
        double r32327 = r32308 ? r32318 : r32326;
        double r32328 = r32283 ? r32306 : r32327;
        return r32328;
}

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 3 regimes

  2. if b < -2.9539231443429135e+138

    1. Initial program 35.0

      \[\begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array}\]

    2. Taylor expanded around -inf 5.7

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\color{blue}{\frac{2 \cdot c}{2 \cdot \frac{a \cdot c}{b} - 2 \cdot b}}\\ \end{array}\]

    3. Simplified1.4

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\color{blue}{\frac{2 \cdot c}{b \cdot -2 + \frac{2 \cdot a}{\frac{b}{c}}}}\\ \end{array}\]

    if -2.9539231443429135e+138 < b < 1.4828322806133274e+130

    1. Initial program 8.1

      \[\begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array}\]
    2. Using strategy rm

    3. Applied add-cube-cbrt8.5

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\color{blue}{\left(-b\right) + \left(\sqrt[3]{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}} \cdot \sqrt[3]{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\right) \cdot \sqrt[3]{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}}\\ \end{array}\]

    4. Simplified8.5

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\color{blue}{\left(-b\right)} + \left(\sqrt[3]{\sqrt{b \cdot b - \left(c \cdot 4\right) \cdot a}} \cdot \sqrt[3]{\sqrt{b \cdot b - \left(c \cdot 4\right) \cdot a}}\right) \cdot \sqrt[3]{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}\\ \end{array}\]

    5. Simplified8.5

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \color{blue}{\left(\sqrt[3]{\sqrt{b \cdot b - \left(c \cdot 4\right) \cdot a}} \cdot \sqrt[3]{\sqrt{b \cdot b - \left(c \cdot 4\right) \cdot a}}\right) \cdot \sqrt[3]{\sqrt{b \cdot b - \left(c \cdot 4\right) \cdot a}}}}\\ \end{array}\]

    if 1.4828322806133274e+130 < b

    1. Initial program 56.1

      \[\begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array}\]

    2. Taylor expanded around inf 10.4

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{\left(-b\right) - \color{blue}{\left(b - 2 \cdot \frac{a \cdot c}{b}\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array}\]

    3. Simplified2.6

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{\left(-b\right) - \color{blue}{\left(b - \frac{2 \cdot a}{\frac{b}{c}}\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array}\]

    4. Taylor expanded around 0 2.5

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\color{blue}{1 \cdot \frac{c}{b} - 1 \cdot \frac{b}{a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array}\]

    5. Simplified2.5

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\color{blue}{\left(\frac{c}{b} - \frac{b}{a}\right) \cdot 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification6.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -2.953923144342913498375256513451576421986 \cdot 10^{138}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{b \cdot -2 + \frac{2 \cdot a}{\frac{b}{c}}}\\ \end{array}\\ \mathbf{elif}\;b \le 1.482832280613327405571569362927236109326 \cdot 10^{130}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{\left(-b\right) + \sqrt[3]{\sqrt{b \cdot b - \left(4 \cdot c\right) \cdot a}} \cdot \left(\sqrt[3]{\sqrt{b \cdot b - \left(4 \cdot c\right) \cdot a}} \cdot \sqrt[3]{\sqrt{b \cdot b - \left(4 \cdot c\right) \cdot a}}\right)}\\ \end{array}\\ \mathbf{elif}\;b \ge 0.0:\\ \;\;\;\;\left(\frac{c}{b} - \frac{b}{a}\right) \cdot 1\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} + \left(-b\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019174 
(FPCore (a b c)
  :name "jeff quadratic root 1"
  (if (>= b 0.0) (/ (- (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)) (/ (* 2.0 c) (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))))))