Average Error: 19.1 → 7.0
Time: 20.0s
Precision: 64
\[\begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -6.1701110130378705 \cdot 10^{+68}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;-\frac{c}{b}\\ \end{array}\\ \mathbf{elif}\;b \le 8.01638212637136 \cdot 10^{+101}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{\sqrt{b \cdot b - c \cdot \left(4 \cdot a\right)}} \cdot \sqrt{\sqrt{b \cdot b - c \cdot \left(4 \cdot a\right)}}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{\sqrt{b \cdot b - c \cdot \left(4 \cdot a\right)} + \left(-b\right)}\\ \end{array}\\ \mathbf{elif}\;b \ge 0:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\sqrt{\sqrt[3]{b \cdot b + \left(a \cdot c\right) \cdot -4}} \cdot \left|\sqrt[3]{b \cdot b + \left(a \cdot c\right) \cdot -4}\right| - b}{c}}\\ \end{array}\]
\begin{array}{l}
\mathbf{if}\;b \ge 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 -6.1701110130378705 \cdot 10^{+68}:\\
\;\;\;\;\begin{array}{l}
\mathbf{if}\;b \ge 0:\\
\;\;\;\;\frac{c}{b} - \frac{b}{a}\\

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

\end{array}\\

\mathbf{elif}\;b \le 8.01638212637136 \cdot 10^{+101}:\\
\;\;\;\;\begin{array}{l}
\mathbf{if}\;b \ge 0:\\
\;\;\;\;\frac{\left(-b\right) - \sqrt{\sqrt{b \cdot b - c \cdot \left(4 \cdot a\right)}} \cdot \sqrt{\sqrt{b \cdot b - c \cdot \left(4 \cdot a\right)}}}{2 \cdot a}\\

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

\end{array}\\

\mathbf{elif}\;b \ge 0:\\
\;\;\;\;\frac{c}{b} - \frac{b}{a}\\

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

\end{array}
double f(double a, double b, double c) {
        double r860318 = b;
        double r860319 = 0.0;
        bool r860320 = r860318 >= r860319;
        double r860321 = -r860318;
        double r860322 = r860318 * r860318;
        double r860323 = 4.0;
        double r860324 = a;
        double r860325 = r860323 * r860324;
        double r860326 = c;
        double r860327 = r860325 * r860326;
        double r860328 = r860322 - r860327;
        double r860329 = sqrt(r860328);
        double r860330 = r860321 - r860329;
        double r860331 = 2.0;
        double r860332 = r860331 * r860324;
        double r860333 = r860330 / r860332;
        double r860334 = r860331 * r860326;
        double r860335 = r860321 + r860329;
        double r860336 = r860334 / r860335;
        double r860337 = r860320 ? r860333 : r860336;
        return r860337;
}
↓
double f(double a, double b, double c) {
        double r860338 = b;
        double r860339 = -6.1701110130378705e+68;
        bool r860340 = r860338 <= r860339;
        double r860341 = 0.0;
        bool r860342 = r860338 >= r860341;
        double r860343 = c;
        double r860344 = r860343 / r860338;
        double r860345 = a;
        double r860346 = r860338 / r860345;
        double r860347 = r860344 - r860346;
        double r860348 = -r860344;
        double r860349 = r860342 ? r860347 : r860348;
        double r860350 = 8.01638212637136e+101;
        bool r860351 = r860338 <= r860350;
        double r860352 = -r860338;
        double r860353 = r860338 * r860338;
        double r860354 = 4.0;
        double r860355 = r860354 * r860345;
        double r860356 = r860343 * r860355;
        double r860357 = r860353 - r860356;
        double r860358 = sqrt(r860357);
        double r860359 = sqrt(r860358);
        double r860360 = r860359 * r860359;
        double r860361 = r860352 - r860360;
        double r860362 = 2.0;
        double r860363 = r860362 * r860345;
        double r860364 = r860361 / r860363;
        double r860365 = r860343 * r860362;
        double r860366 = r860358 + r860352;
        double r860367 = r860365 / r860366;
        double r860368 = r860342 ? r860364 : r860367;
        double r860369 = r860345 * r860343;
        double r860370 = -4.0;
        double r860371 = r860369 * r860370;
        double r860372 = r860353 + r860371;
        double r860373 = cbrt(r860372);
        double r860374 = sqrt(r860373);
        double r860375 = fabs(r860373);
        double r860376 = r860374 * r860375;
        double r860377 = r860376 - r860338;
        double r860378 = r860377 / r860343;
        double r860379 = r860362 / r860378;
        double r860380 = r860342 ? r860347 : r860379;
        double r860381 = r860351 ? r860368 : r860380;
        double r860382 = r860340 ? r860349 : r860381;
        return r860382;
}

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 < -6.1701110130378705e+68

    1. Initial program 26.5

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

    2. Taylor expanded around inf 26.5

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

    3. Simplified26.5

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

    4. Taylor expanded around inf 26.5

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

    6. Applied associate-/l*26.8

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

    7. Simplified26.8

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

    8. Taylor expanded around -inf 3.3

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array}\]

    if -6.1701110130378705e+68 < b < 8.01638212637136e+101

    1. Initial program 9.4

      \[\begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array}\]
    2. Using strategy rm

    3. Applied add-sqr-sqrt9.5

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

    if 8.01638212637136e+101 < b

    1. Initial program 44.8

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

    2. Taylor expanded around inf 10.1

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

    3. Simplified3.8

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

    4. Taylor expanded around inf 3.7

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

    6. Applied associate-/l*3.7

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

    7. Simplified3.7

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

    9. Applied add-cube-cbrt3.7

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\sqrt{\left(\sqrt[3]{b \cdot b + \left(a \cdot c\right) \cdot -4} \cdot \sqrt[3]{b \cdot b + \left(a \cdot c\right) \cdot -4}\right) \cdot \sqrt[3]{b \cdot b + \left(a \cdot c\right) \cdot -4}} - b}{c}}\\ \end{array}\]

    10. Applied sqrt-prod3.7

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\sqrt{\sqrt[3]{b \cdot b + \left(a \cdot c\right) \cdot -4} \cdot \sqrt[3]{b \cdot b + \left(a \cdot c\right) \cdot -4}} \cdot \sqrt{\sqrt[3]{b \cdot b + \left(a \cdot c\right) \cdot -4}} - b}{c}}\\ \end{array}\]

    11. Simplified3.7

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left|\sqrt[3]{b \cdot b + \left(a \cdot c\right) \cdot -4}\right| \cdot \sqrt{\sqrt[3]{b \cdot b + \left(a \cdot c\right) \cdot -4}} - b}{c}}\\ \end{array}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification7.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -6.1701110130378705 \cdot 10^{+68}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;-\frac{c}{b}\\ \end{array}\\ \mathbf{elif}\;b \le 8.01638212637136 \cdot 10^{+101}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{\sqrt{b \cdot b - c \cdot \left(4 \cdot a\right)}} \cdot \sqrt{\sqrt{b \cdot b - c \cdot \left(4 \cdot a\right)}}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{\sqrt{b \cdot b - c \cdot \left(4 \cdot a\right)} + \left(-b\right)}\\ \end{array}\\ \mathbf{elif}\;b \ge 0:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\sqrt{\sqrt[3]{b \cdot b + \left(a \cdot c\right) \cdot -4}} \cdot \left|\sqrt[3]{b \cdot b + \left(a \cdot c\right) \cdot -4}\right| - b}{c}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019133 
(FPCore (a b c)
  :name "jeff quadratic root 1"
  (if (>= b 0) (/ (- (- b) (sqrt (- (* b b) (* (* 4 a) c)))) (* 2 a)) (/ (* 2 c) (+ (- b) (sqrt (- (* b b) (* (* 4 a) c)))))))