Average Error: 20.2 → 6.9
Time: 18.4s
Precision: 64
\[\begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -7.406561173130011889031964813046982775699 \cdot 10^{149}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{\frac{c \cdot \left(a \cdot 4\right)}{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \left(\frac{a}{b} \cdot \left(2 \cdot c\right) - b\right)}\\ \end{array}\\ \mathbf{elif}\;b \le 2.216005534325192969068322454808916428103 \cdot 10^{86}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\sqrt{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}} \cdot \sqrt{\sqrt{\sqrt[3]{b \cdot b - c \cdot \left(a \cdot 4\right)} \cdot \sqrt[3]{b \cdot b - c \cdot \left(a \cdot 4\right)}} \cdot \sqrt{\sqrt[3]{b \cdot b - c \cdot \left(a \cdot 4\right)}}} + \left(-b\right)}\\ \end{array}\\ \mathbf{elif}\;b \ge 0.0:\\ \;\;\;\;\frac{\left(-b\right) - \left(b - \frac{a}{b} \cdot \left(2 \cdot c\right)\right)}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}}\\ \end{array}\]
\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 -7.406561173130011889031964813046982775699 \cdot 10^{149}:\\
\;\;\;\;\begin{array}{l}
\mathbf{if}\;b \ge 0.0:\\
\;\;\;\;\frac{\frac{c \cdot \left(a \cdot 4\right)}{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}}{2 \cdot a}\\

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

\end{array}\\

\mathbf{elif}\;b \le 2.216005534325192969068322454808916428103 \cdot 10^{86}:\\
\;\;\;\;\begin{array}{l}
\mathbf{if}\;b \ge 0.0:\\
\;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}}{2 \cdot a}\\

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

\end{array}\\

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

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

\end{array}
double f(double a, double b, double c) {
        double r1474343 = b;
        double r1474344 = 0.0;
        bool r1474345 = r1474343 >= r1474344;
        double r1474346 = -r1474343;
        double r1474347 = r1474343 * r1474343;
        double r1474348 = 4.0;
        double r1474349 = a;
        double r1474350 = r1474348 * r1474349;
        double r1474351 = c;
        double r1474352 = r1474350 * r1474351;
        double r1474353 = r1474347 - r1474352;
        double r1474354 = sqrt(r1474353);
        double r1474355 = r1474346 - r1474354;
        double r1474356 = 2.0;
        double r1474357 = r1474356 * r1474349;
        double r1474358 = r1474355 / r1474357;
        double r1474359 = r1474356 * r1474351;
        double r1474360 = r1474346 + r1474354;
        double r1474361 = r1474359 / r1474360;
        double r1474362 = r1474345 ? r1474358 : r1474361;
        return r1474362;
}
↓
double f(double a, double b, double c) {
        double r1474363 = b;
        double r1474364 = -7.406561173130012e+149;
        bool r1474365 = r1474363 <= r1474364;
        double r1474366 = 0.0;
        bool r1474367 = r1474363 >= r1474366;
        double r1474368 = c;
        double r1474369 = a;
        double r1474370 = 4.0;
        double r1474371 = r1474369 * r1474370;
        double r1474372 = r1474368 * r1474371;
        double r1474373 = r1474363 * r1474363;
        double r1474374 = r1474373 - r1474372;
        double r1474375 = sqrt(r1474374);
        double r1474376 = r1474375 - r1474363;
        double r1474377 = r1474372 / r1474376;
        double r1474378 = 2.0;
        double r1474379 = r1474378 * r1474369;
        double r1474380 = r1474377 / r1474379;
        double r1474381 = r1474378 * r1474368;
        double r1474382 = -r1474363;
        double r1474383 = r1474369 / r1474363;
        double r1474384 = r1474383 * r1474381;
        double r1474385 = r1474384 - r1474363;
        double r1474386 = r1474382 + r1474385;
        double r1474387 = r1474381 / r1474386;
        double r1474388 = r1474367 ? r1474380 : r1474387;
        double r1474389 = 2.216005534325193e+86;
        bool r1474390 = r1474363 <= r1474389;
        double r1474391 = r1474382 - r1474375;
        double r1474392 = r1474391 / r1474379;
        double r1474393 = sqrt(r1474375);
        double r1474394 = cbrt(r1474374);
        double r1474395 = r1474394 * r1474394;
        double r1474396 = sqrt(r1474395);
        double r1474397 = sqrt(r1474394);
        double r1474398 = r1474396 * r1474397;
        double r1474399 = sqrt(r1474398);
        double r1474400 = r1474393 * r1474399;
        double r1474401 = r1474400 + r1474382;
        double r1474402 = r1474381 / r1474401;
        double r1474403 = r1474367 ? r1474392 : r1474402;
        double r1474404 = r1474363 - r1474384;
        double r1474405 = r1474382 - r1474404;
        double r1474406 = r1474405 / r1474379;
        double r1474407 = r1474382 + r1474375;
        double r1474408 = r1474381 / r1474407;
        double r1474409 = r1474367 ? r1474406 : r1474408;
        double r1474410 = r1474390 ? r1474403 : r1474409;
        double r1474411 = r1474365 ? r1474388 : r1474410;
        return r1474411;
}

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 < -7.406561173130012e+149

    1. Initial program 37.8

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

    2. Taylor expanded around -inf 7.0

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

    3. Simplified1.6

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

    5. Applied add-sqr-sqrt1.6

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

    6. Applied sqrt-prod1.6

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

    8. Applied flip--1.6

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

    9. Simplified1.6

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{\frac{\color{blue}{0 + \left(4 \cdot a\right) \cdot c}}{\left(-b\right) + \sqrt{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}} \cdot \sqrt{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \left(\left(2 \cdot c\right) \cdot \frac{a}{b} - b\right)}\\ \end{array}\]

    10. Simplified1.6

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{\frac{0 + \left(4 \cdot a\right) \cdot c}{\color{blue}{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \left(\left(2 \cdot c\right) \cdot \frac{a}{b} - b\right)}\\ \end{array}\]

    if -7.406561173130012e+149 < b < 2.216005534325193e+86

    1. Initial program 9.0

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

    3. Applied add-sqr-sqrt9.0

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

    4. Applied sqrt-prod9.1

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

    6. Applied add-cube-cbrt9.1

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

    7. Applied sqrt-prod9.1

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

    if 2.216005534325193e+86 < b

    1. Initial program 44.1

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

    2. Taylor expanded around inf 9.6

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

    3. Simplified4.1

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

  4. Final simplification6.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -7.406561173130011889031964813046982775699 \cdot 10^{149}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{\frac{c \cdot \left(a \cdot 4\right)}{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \left(\frac{a}{b} \cdot \left(2 \cdot c\right) - b\right)}\\ \end{array}\\ \mathbf{elif}\;b \le 2.216005534325192969068322454808916428103 \cdot 10^{86}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\sqrt{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}} \cdot \sqrt{\sqrt{\sqrt[3]{b \cdot b - c \cdot \left(a \cdot 4\right)} \cdot \sqrt[3]{b \cdot b - c \cdot \left(a \cdot 4\right)}} \cdot \sqrt{\sqrt[3]{b \cdot b - c \cdot \left(a \cdot 4\right)}}} + \left(-b\right)}\\ \end{array}\\ \mathbf{elif}\;b \ge 0.0:\\ \;\;\;\;\frac{\left(-b\right) - \left(b - \frac{a}{b} \cdot \left(2 \cdot c\right)\right)}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019170 
(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)))))))