Average Error: 19.8 → 7.9
Time: 8.9s
Precision: 64
\[\begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -2.2724541866372811 \cdot 10^{165}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 \cdot \frac{a \cdot c}{b} - 2 \cdot b}{2}}{a}\\ \end{array}\\ \mathbf{elif}\;b \le 3.2649111998892948 \cdot 10^{111}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \left|\sqrt[3]{b \cdot b - \left(4 \cdot a\right) \cdot c}\right| \cdot \sqrt{\sqrt[3]{b \cdot b - \left(4 \cdot a\right) \cdot c}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{2}}{a}\\ \end{array}\\ \mathbf{elif}\;b \ge 0.0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \left(b - 2 \cdot \left(\frac{a}{\sqrt[3]{b} \cdot \sqrt[3]{b}} \cdot \frac{c}{\sqrt[3]{b}}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\sqrt{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}} \cdot \sqrt{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}} - b}{2}}{a}\\ \end{array}\]
\begin{array}{l}
\mathbf{if}\;b \ge 0.0:\\
\;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\

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

\end{array}
↓
\begin{array}{l}
\mathbf{if}\;b \le -2.2724541866372811 \cdot 10^{165}:\\
\;\;\;\;\begin{array}{l}
\mathbf{if}\;b \ge 0.0:\\
\;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\

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

\end{array}\\

\mathbf{elif}\;b \le 3.2649111998892948 \cdot 10^{111}:\\
\;\;\;\;\begin{array}{l}
\mathbf{if}\;b \ge 0.0:\\
\;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \left|\sqrt[3]{b \cdot b - \left(4 \cdot a\right) \cdot c}\right| \cdot \sqrt{\sqrt[3]{b \cdot b - \left(4 \cdot a\right) \cdot c}}}\\

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

\end{array}\\

\mathbf{elif}\;b \ge 0.0:\\
\;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \left(b - 2 \cdot \left(\frac{a}{\sqrt[3]{b} \cdot \sqrt[3]{b}} \cdot \frac{c}{\sqrt[3]{b}}\right)\right)}\\

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

\end{array}
double f(double a, double b, double c) {
        double r28261 = b;
        double r28262 = 0.0;
        bool r28263 = r28261 >= r28262;
        double r28264 = 2.0;
        double r28265 = c;
        double r28266 = r28264 * r28265;
        double r28267 = -r28261;
        double r28268 = r28261 * r28261;
        double r28269 = 4.0;
        double r28270 = a;
        double r28271 = r28269 * r28270;
        double r28272 = r28271 * r28265;
        double r28273 = r28268 - r28272;
        double r28274 = sqrt(r28273);
        double r28275 = r28267 - r28274;
        double r28276 = r28266 / r28275;
        double r28277 = r28267 + r28274;
        double r28278 = r28264 * r28270;
        double r28279 = r28277 / r28278;
        double r28280 = r28263 ? r28276 : r28279;
        return r28280;
}
↓
double f(double a, double b, double c) {
        double r28281 = b;
        double r28282 = -2.272454186637281e+165;
        bool r28283 = r28281 <= r28282;
        double r28284 = 0.0;
        bool r28285 = r28281 >= r28284;
        double r28286 = 2.0;
        double r28287 = c;
        double r28288 = r28286 * r28287;
        double r28289 = -r28281;
        double r28290 = r28281 * r28281;
        double r28291 = 4.0;
        double r28292 = a;
        double r28293 = r28291 * r28292;
        double r28294 = r28293 * r28287;
        double r28295 = r28290 - r28294;
        double r28296 = sqrt(r28295);
        double r28297 = r28289 - r28296;
        double r28298 = r28288 / r28297;
        double r28299 = r28292 * r28287;
        double r28300 = r28299 / r28281;
        double r28301 = r28286 * r28300;
        double r28302 = 2.0;
        double r28303 = r28302 * r28281;
        double r28304 = r28301 - r28303;
        double r28305 = r28304 / r28286;
        double r28306 = r28305 / r28292;
        double r28307 = r28285 ? r28298 : r28306;
        double r28308 = 3.264911199889295e+111;
        bool r28309 = r28281 <= r28308;
        double r28310 = cbrt(r28295);
        double r28311 = fabs(r28310);
        double r28312 = sqrt(r28310);
        double r28313 = r28311 * r28312;
        double r28314 = r28289 - r28313;
        double r28315 = r28288 / r28314;
        double r28316 = r28296 - r28281;
        double r28317 = r28316 / r28286;
        double r28318 = r28317 / r28292;
        double r28319 = r28285 ? r28315 : r28318;
        double r28320 = cbrt(r28281);
        double r28321 = r28320 * r28320;
        double r28322 = r28292 / r28321;
        double r28323 = r28287 / r28320;
        double r28324 = r28322 * r28323;
        double r28325 = r28286 * r28324;
        double r28326 = r28281 - r28325;
        double r28327 = r28289 - r28326;
        double r28328 = r28288 / r28327;
        double r28329 = sqrt(r28296);
        double r28330 = r28329 * r28329;
        double r28331 = r28330 - r28281;
        double r28332 = r28331 / r28286;
        double r28333 = r28332 / r28292;
        double r28334 = r28285 ? r28328 : r28333;
        double r28335 = r28309 ? r28319 : r28334;
        double r28336 = r28283 ? r28307 : r28335;
        return r28336;
}

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 < -2.272454186637281e+165

    1. Initial program 64.0

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

    2. Simplified64.0

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

    3. Taylor expanded around -inf 10.2

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

    if -2.272454186637281e+165 < b < 3.264911199889295e+111

    1. Initial program 9.2

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

    2. Simplified9.2

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

    4. Applied add-cube-cbrt9.4

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

    5. Applied sqrt-prod9.4

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

    6. Simplified9.4

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

    if 3.264911199889295e+111 < b

    1. Initial program 31.6

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

    2. Simplified31.6

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

    3. Taylor expanded around inf 6.2

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

    5. Applied add-cube-cbrt6.2

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

    6. Applied times-frac2.2

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

    8. Applied add-sqr-sqrt2.2

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

    9. Applied sqrt-prod2.2

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

  4. Final simplification7.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -2.2724541866372811 \cdot 10^{165}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 \cdot \frac{a \cdot c}{b} - 2 \cdot b}{2}}{a}\\ \end{array}\\ \mathbf{elif}\;b \le 3.2649111998892948 \cdot 10^{111}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \left|\sqrt[3]{b \cdot b - \left(4 \cdot a\right) \cdot c}\right| \cdot \sqrt{\sqrt[3]{b \cdot b - \left(4 \cdot a\right) \cdot c}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{2}}{a}\\ \end{array}\\ \mathbf{elif}\;b \ge 0.0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \left(b - 2 \cdot \left(\frac{a}{\sqrt[3]{b} \cdot \sqrt[3]{b}} \cdot \frac{c}{\sqrt[3]{b}}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\sqrt{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}} \cdot \sqrt{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}} - b}{2}}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2020047 
(FPCore (a b c)
  :name "jeff quadratic root 2"
  :precision binary64
  (if (>= b 0.0) (/ (* 2 c) (- (- b) (sqrt (- (* b b) (* (* 4 a) c))))) (/ (+ (- b) (sqrt (- (* b b) (* (* 4 a) c)))) (* 2 a))))