Average Error: 19.7 → 8.5
Time: 41.3s
Precision: 64
\[\begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{2.0 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4.0 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4.0 \cdot a\right) \cdot c}}{2.0 \cdot a}\\ \end{array}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -2.0519886074223697 \cdot 10^{+155}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{2.0 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4.0 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(2.0 \cdot \frac{a \cdot c}{b} - b\right) + \left(-b\right)}{2.0 \cdot a}\\ \end{array}\\ \mathbf{elif}\;b \le 5.714064955010435 \cdot 10^{+100}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{2.0 \cdot c}{\left(-b\right) - \sqrt{\sqrt{b \cdot b - \left(4.0 \cdot a\right) \cdot c}} \cdot \sqrt{\sqrt{b \cdot b - \left(4.0 \cdot a\right) \cdot c}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(4.0 \cdot a\right) \cdot c} + \left(-b\right)}{2.0 \cdot a}\\ \end{array}\\ \mathbf{elif}\;b \ge 0.0:\\ \;\;\;\;\frac{2.0 \cdot c}{\left(-b\right) - \left(b - 2.0 \cdot \frac{a \cdot c}{b}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(4.0 \cdot a\right) \cdot c} + \left(-b\right)}{2.0 \cdot a}\\ \end{array}\]
\begin{array}{l}
\mathbf{if}\;b \ge 0.0:\\
\;\;\;\;\frac{2.0 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4.0 \cdot a\right) \cdot c}}\\

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

\end{array}
↓
\begin{array}{l}
\mathbf{if}\;b \le -2.0519886074223697 \cdot 10^{+155}:\\
\;\;\;\;\begin{array}{l}
\mathbf{if}\;b \ge 0.0:\\
\;\;\;\;\frac{2.0 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4.0 \cdot a\right) \cdot c}}\\

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

\end{array}\\

\mathbf{elif}\;b \le 5.714064955010435 \cdot 10^{+100}:\\
\;\;\;\;\begin{array}{l}
\mathbf{if}\;b \ge 0.0:\\
\;\;\;\;\frac{2.0 \cdot c}{\left(-b\right) - \sqrt{\sqrt{b \cdot b - \left(4.0 \cdot a\right) \cdot c}} \cdot \sqrt{\sqrt{b \cdot b - \left(4.0 \cdot a\right) \cdot c}}}\\

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

\end{array}\\

\mathbf{elif}\;b \ge 0.0:\\
\;\;\;\;\frac{2.0 \cdot c}{\left(-b\right) - \left(b - 2.0 \cdot \frac{a \cdot c}{b}\right)}\\

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

\end{array}
double f(double a, double b, double c) {
        double r1076251 = b;
        double r1076252 = 0.0;
        bool r1076253 = r1076251 >= r1076252;
        double r1076254 = 2.0;
        double r1076255 = c;
        double r1076256 = r1076254 * r1076255;
        double r1076257 = -r1076251;
        double r1076258 = r1076251 * r1076251;
        double r1076259 = 4.0;
        double r1076260 = a;
        double r1076261 = r1076259 * r1076260;
        double r1076262 = r1076261 * r1076255;
        double r1076263 = r1076258 - r1076262;
        double r1076264 = sqrt(r1076263);
        double r1076265 = r1076257 - r1076264;
        double r1076266 = r1076256 / r1076265;
        double r1076267 = r1076257 + r1076264;
        double r1076268 = r1076254 * r1076260;
        double r1076269 = r1076267 / r1076268;
        double r1076270 = r1076253 ? r1076266 : r1076269;
        return r1076270;
}
↓
double f(double a, double b, double c) {
        double r1076271 = b;
        double r1076272 = -2.0519886074223697e+155;
        bool r1076273 = r1076271 <= r1076272;
        double r1076274 = 0.0;
        bool r1076275 = r1076271 >= r1076274;
        double r1076276 = 2.0;
        double r1076277 = c;
        double r1076278 = r1076276 * r1076277;
        double r1076279 = -r1076271;
        double r1076280 = r1076271 * r1076271;
        double r1076281 = 4.0;
        double r1076282 = a;
        double r1076283 = r1076281 * r1076282;
        double r1076284 = r1076283 * r1076277;
        double r1076285 = r1076280 - r1076284;
        double r1076286 = sqrt(r1076285);
        double r1076287 = r1076279 - r1076286;
        double r1076288 = r1076278 / r1076287;
        double r1076289 = r1076282 * r1076277;
        double r1076290 = r1076289 / r1076271;
        double r1076291 = r1076276 * r1076290;
        double r1076292 = r1076291 - r1076271;
        double r1076293 = r1076292 + r1076279;
        double r1076294 = r1076276 * r1076282;
        double r1076295 = r1076293 / r1076294;
        double r1076296 = r1076275 ? r1076288 : r1076295;
        double r1076297 = 5.714064955010435e+100;
        bool r1076298 = r1076271 <= r1076297;
        double r1076299 = sqrt(r1076286);
        double r1076300 = r1076299 * r1076299;
        double r1076301 = r1076279 - r1076300;
        double r1076302 = r1076278 / r1076301;
        double r1076303 = r1076286 + r1076279;
        double r1076304 = r1076303 / r1076294;
        double r1076305 = r1076275 ? r1076302 : r1076304;
        double r1076306 = r1076271 - r1076291;
        double r1076307 = r1076279 - r1076306;
        double r1076308 = r1076278 / r1076307;
        double r1076309 = r1076275 ? r1076308 : r1076304;
        double r1076310 = r1076298 ? r1076305 : r1076309;
        double r1076311 = r1076273 ? r1076296 : r1076310;
        return r1076311;
}

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 < -2.0519886074223697e+155

    1. Initial program 64.0

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

    2. Taylor expanded around -inf 11.8

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

    if -2.0519886074223697e+155 < b < 5.714064955010435e+100

    1. Initial program 8.5

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

    3. Applied add-sqr-sqrt8.5

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

    4. Applied sqrt-prod8.6

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

    if 5.714064955010435e+100 < b

    1. Initial program 29.5

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

    2. Taylor expanded around inf 6.8

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

  4. Final simplification8.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -2.0519886074223697 \cdot 10^{+155}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{2.0 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4.0 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(2.0 \cdot \frac{a \cdot c}{b} - b\right) + \left(-b\right)}{2.0 \cdot a}\\ \end{array}\\ \mathbf{elif}\;b \le 5.714064955010435 \cdot 10^{+100}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{2.0 \cdot c}{\left(-b\right) - \sqrt{\sqrt{b \cdot b - \left(4.0 \cdot a\right) \cdot c}} \cdot \sqrt{\sqrt{b \cdot b - \left(4.0 \cdot a\right) \cdot c}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(4.0 \cdot a\right) \cdot c} + \left(-b\right)}{2.0 \cdot a}\\ \end{array}\\ \mathbf{elif}\;b \ge 0.0:\\ \;\;\;\;\frac{2.0 \cdot c}{\left(-b\right) - \left(b - 2.0 \cdot \frac{a \cdot c}{b}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(4.0 \cdot a\right) \cdot c} + \left(-b\right)}{2.0 \cdot a}\\ \end{array}\]

Reproduce

herbie shell --seed 2019165 
(FPCore (a b c)
  :name "jeff quadratic root 2"
  (if (>= b 0.0) (/ (* 2.0 c) (- (- b) (sqrt (- (* b b) (* (* 4.0 a) c))))) (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a))))