Average Error: 20.1 → 6.4
Time: 38.5s
Precision: 64
\[\begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4.0 \cdot a\right) \cdot c}}{2.0 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2.0 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4.0 \cdot a\right) \cdot c}}\\ \end{array}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -3.3646042166231086 \cdot 10^{+152}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;1.0 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \mathbf{else}:\\ \;\;\;\;-1.0 \cdot \frac{c}{b}\\ \end{array}\\ \mathbf{elif}\;b \le 7.935917722613779 \cdot 10^{+99}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{\sqrt[3]{b \cdot b - c \cdot \left(a \cdot 4.0\right)} \cdot \sqrt[3]{b \cdot b - c \cdot \left(a \cdot 4.0\right)}} \cdot \sqrt{\sqrt[3]{b \cdot b - c \cdot \left(a \cdot 4.0\right)}}}{a \cdot 2.0}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2.0}{\sqrt{b \cdot b - c \cdot \left(a \cdot 4.0\right)} + \left(-b\right)}\\ \end{array}\\ \mathbf{elif}\;b \ge 0.0:\\ \;\;\;\;1.0 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2.0}{\sqrt{\left(\sqrt[3]{b \cdot b - c \cdot \left(a \cdot 4.0\right)} \cdot \sqrt[3]{b \cdot b - c \cdot \left(a \cdot 4.0\right)}\right) \cdot \sqrt[3]{b \cdot b - c \cdot \left(a \cdot 4.0\right)}} + \left(-b\right)}\\ \end{array}\]
\begin{array}{l}
\mathbf{if}\;b \ge 0.0:\\
\;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4.0 \cdot a\right) \cdot c}}{2.0 \cdot a}\\

\mathbf{else}:\\
\;\;\;\;\frac{2.0 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4.0 \cdot a\right) \cdot c}}\\

\end{array}
↓
\begin{array}{l}
\mathbf{if}\;b \le -3.3646042166231086 \cdot 10^{+152}:\\
\;\;\;\;\begin{array}{l}
\mathbf{if}\;b \ge 0.0:\\
\;\;\;\;1.0 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\

\mathbf{else}:\\
\;\;\;\;-1.0 \cdot \frac{c}{b}\\

\end{array}\\

\mathbf{elif}\;b \le 7.935917722613779 \cdot 10^{+99}:\\
\;\;\;\;\begin{array}{l}
\mathbf{if}\;b \ge 0.0:\\
\;\;\;\;\frac{\left(-b\right) - \sqrt{\sqrt[3]{b \cdot b - c \cdot \left(a \cdot 4.0\right)} \cdot \sqrt[3]{b \cdot b - c \cdot \left(a \cdot 4.0\right)}} \cdot \sqrt{\sqrt[3]{b \cdot b - c \cdot \left(a \cdot 4.0\right)}}}{a \cdot 2.0}\\

\mathbf{else}:\\
\;\;\;\;\frac{c \cdot 2.0}{\sqrt{b \cdot b - c \cdot \left(a \cdot 4.0\right)} + \left(-b\right)}\\

\end{array}\\

\mathbf{elif}\;b \ge 0.0:\\
\;\;\;\;1.0 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\

\mathbf{else}:\\
\;\;\;\;\frac{c \cdot 2.0}{\sqrt{\left(\sqrt[3]{b \cdot b - c \cdot \left(a \cdot 4.0\right)} \cdot \sqrt[3]{b \cdot b - c \cdot \left(a \cdot 4.0\right)}\right) \cdot \sqrt[3]{b \cdot b - c \cdot \left(a \cdot 4.0\right)}} + \left(-b\right)}\\

\end{array}
double f(double a, double b, double c) {
        double r1602281 = b;
        double r1602282 = 0.0;
        bool r1602283 = r1602281 >= r1602282;
        double r1602284 = -r1602281;
        double r1602285 = r1602281 * r1602281;
        double r1602286 = 4.0;
        double r1602287 = a;
        double r1602288 = r1602286 * r1602287;
        double r1602289 = c;
        double r1602290 = r1602288 * r1602289;
        double r1602291 = r1602285 - r1602290;
        double r1602292 = sqrt(r1602291);
        double r1602293 = r1602284 - r1602292;
        double r1602294 = 2.0;
        double r1602295 = r1602294 * r1602287;
        double r1602296 = r1602293 / r1602295;
        double r1602297 = r1602294 * r1602289;
        double r1602298 = r1602284 + r1602292;
        double r1602299 = r1602297 / r1602298;
        double r1602300 = r1602283 ? r1602296 : r1602299;
        return r1602300;
}
↓
double f(double a, double b, double c) {
        double r1602301 = b;
        double r1602302 = -3.3646042166231086e+152;
        bool r1602303 = r1602301 <= r1602302;
        double r1602304 = 0.0;
        bool r1602305 = r1602301 >= r1602304;
        double r1602306 = 1.0;
        double r1602307 = c;
        double r1602308 = r1602307 / r1602301;
        double r1602309 = a;
        double r1602310 = r1602301 / r1602309;
        double r1602311 = r1602308 - r1602310;
        double r1602312 = r1602306 * r1602311;
        double r1602313 = -1.0;
        double r1602314 = r1602313 * r1602308;
        double r1602315 = r1602305 ? r1602312 : r1602314;
        double r1602316 = 7.935917722613779e+99;
        bool r1602317 = r1602301 <= r1602316;
        double r1602318 = -r1602301;
        double r1602319 = r1602301 * r1602301;
        double r1602320 = 4.0;
        double r1602321 = r1602309 * r1602320;
        double r1602322 = r1602307 * r1602321;
        double r1602323 = r1602319 - r1602322;
        double r1602324 = cbrt(r1602323);
        double r1602325 = r1602324 * r1602324;
        double r1602326 = sqrt(r1602325);
        double r1602327 = sqrt(r1602324);
        double r1602328 = r1602326 * r1602327;
        double r1602329 = r1602318 - r1602328;
        double r1602330 = 2.0;
        double r1602331 = r1602309 * r1602330;
        double r1602332 = r1602329 / r1602331;
        double r1602333 = r1602307 * r1602330;
        double r1602334 = sqrt(r1602323);
        double r1602335 = r1602334 + r1602318;
        double r1602336 = r1602333 / r1602335;
        double r1602337 = r1602305 ? r1602332 : r1602336;
        double r1602338 = r1602325 * r1602324;
        double r1602339 = sqrt(r1602338);
        double r1602340 = r1602339 + r1602318;
        double r1602341 = r1602333 / r1602340;
        double r1602342 = r1602305 ? r1602312 : r1602341;
        double r1602343 = r1602317 ? r1602337 : r1602342;
        double r1602344 = r1602303 ? r1602315 : r1602343;
        return r1602344;
}

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 < -3.3646042166231086e+152

    1. Initial program 38.4

      \[\begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4.0 \cdot a\right) \cdot c}}{2.0 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2.0 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4.0 \cdot a\right) \cdot c}}\\ \end{array}\]

    2. Taylor expanded around inf 38.4

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{\left(-b\right) - \color{blue}{\left(b - 2.0 \cdot \frac{a \cdot c}{b}\right)}}{2.0 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2.0 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4.0 \cdot a\right) \cdot c}}\\ \end{array}\]

    3. Taylor expanded around 0 38.4

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\color{blue}{1.0 \cdot \frac{c}{b} - 1.0 \cdot \frac{b}{a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2.0 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4.0 \cdot a\right) \cdot c}}\\ \end{array}\]

    4. Simplified38.4

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\color{blue}{1.0 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2.0 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4.0 \cdot a\right) \cdot c}}\\ \end{array}\]
    5. Using strategy rm

    6. Applied associate-/l*38.4

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;1.0 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2.0}{\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4.0 \cdot a\right) \cdot c}}{c}}\\ \end{array}\]

    7. Simplified38.4

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;1.0 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \mathbf{else}:\\ \;\;\;\;\color{blue}{\frac{2.0}{\frac{\sqrt{b \cdot b - \left(a \cdot c\right) \cdot 4.0} - b}{c}}}\\ \end{array}\]

    8. Taylor expanded around -inf 1.7

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;1.0 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \mathbf{else}:\\ \;\;\;\;-1.0 \cdot \frac{c}{b}\\ \end{array}\]

    if -3.3646042166231086e+152 < b < 7.935917722613779e+99

    1. Initial program 8.3

      \[\begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4.0 \cdot a\right) \cdot c}}{2.0 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2.0 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4.0 \cdot a\right) \cdot c}}\\ \end{array}\]
    2. Using strategy rm

    3. Applied add-cube-cbrt8.5

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{\color{blue}{\left(\sqrt[3]{b \cdot b - \left(4.0 \cdot a\right) \cdot c} \cdot \sqrt[3]{b \cdot b - \left(4.0 \cdot a\right) \cdot c}\right) \cdot \sqrt[3]{b \cdot b - \left(4.0 \cdot a\right) \cdot c}}}}{2.0 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2.0 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4.0 \cdot a\right) \cdot c}}\\ \end{array}\]

    4. Applied sqrt-prod8.5

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{\left(-b\right) - \color{blue}{\sqrt{\sqrt[3]{b \cdot b - \left(4.0 \cdot a\right) \cdot c} \cdot \sqrt[3]{b \cdot b - \left(4.0 \cdot a\right) \cdot c}} \cdot \sqrt{\sqrt[3]{b \cdot b - \left(4.0 \cdot a\right) \cdot c}}}}{2.0 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2.0 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4.0 \cdot a\right) \cdot c}}\\ \end{array}\]

    if 7.935917722613779e+99 < b

    1. Initial program 46.7

      \[\begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4.0 \cdot a\right) \cdot c}}{2.0 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2.0 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4.0 \cdot a\right) \cdot c}}\\ \end{array}\]

    2. Taylor expanded around inf 10.7

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{\left(-b\right) - \color{blue}{\left(b - 2.0 \cdot \frac{a \cdot c}{b}\right)}}{2.0 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2.0 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4.0 \cdot a\right) \cdot c}}\\ \end{array}\]

    3. Taylor expanded around 0 3.2

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\color{blue}{1.0 \cdot \frac{c}{b} - 1.0 \cdot \frac{b}{a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2.0 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4.0 \cdot a\right) \cdot c}}\\ \end{array}\]

    4. Simplified3.2

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\color{blue}{1.0 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2.0 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4.0 \cdot a\right) \cdot c}}\\ \end{array}\]
    5. Using strategy rm

    6. Applied add-cube-cbrt3.2

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;1.0 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2.0 \cdot c}{\color{blue}{\left(-b\right)} + \sqrt{\left(\sqrt[3]{b \cdot b - \left(4.0 \cdot a\right) \cdot c} \cdot \sqrt[3]{b \cdot b - \left(4.0 \cdot a\right) \cdot c}\right) \cdot \sqrt[3]{b \cdot b - \left(4.0 \cdot a\right) \cdot c}}}\\ \end{array}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification6.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -3.3646042166231086 \cdot 10^{+152}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;1.0 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \mathbf{else}:\\ \;\;\;\;-1.0 \cdot \frac{c}{b}\\ \end{array}\\ \mathbf{elif}\;b \le 7.935917722613779 \cdot 10^{+99}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{\sqrt[3]{b \cdot b - c \cdot \left(a \cdot 4.0\right)} \cdot \sqrt[3]{b \cdot b - c \cdot \left(a \cdot 4.0\right)}} \cdot \sqrt{\sqrt[3]{b \cdot b - c \cdot \left(a \cdot 4.0\right)}}}{a \cdot 2.0}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2.0}{\sqrt{b \cdot b - c \cdot \left(a \cdot 4.0\right)} + \left(-b\right)}\\ \end{array}\\ \mathbf{elif}\;b \ge 0.0:\\ \;\;\;\;1.0 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2.0}{\sqrt{\left(\sqrt[3]{b \cdot b - c \cdot \left(a \cdot 4.0\right)} \cdot \sqrt[3]{b \cdot b - c \cdot \left(a \cdot 4.0\right)}\right) \cdot \sqrt[3]{b \cdot b - c \cdot \left(a \cdot 4.0\right)}} + \left(-b\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019165 
(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)))))))