Average Error: 20.3 → 7.1
Time: 19.9s
Precision: 64
\[\begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -5.617913947565299992326164335754974391576 \cdot 10^{116}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \left(\left(\left(\sqrt[3]{\frac{a}{b}} \cdot \sqrt[3]{\frac{a}{b}}\right) \cdot \left(2 \cdot c\right)\right) \cdot \sqrt[3]{\frac{a}{b}} - b\right)}{2 \cdot a}\\ \end{array}\\ \mathbf{elif}\;b \le 3.038903409991338138548211857189252856935 \cdot 10^{107}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}} \cdot \sqrt{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}{2 \cdot a}\\ \end{array}\\ \mathbf{elif}\;b \ge 0.0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \left(b - \left(2 \cdot c\right) \cdot \frac{a}{b}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} + \left(-b\right)}{2 \cdot a}\\ \end{array}\]
\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 -5.617913947565299992326164335754974391576 \cdot 10^{116}:\\
\;\;\;\;\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) + \left(\left(\left(\sqrt[3]{\frac{a}{b}} \cdot \sqrt[3]{\frac{a}{b}}\right) \cdot \left(2 \cdot c\right)\right) \cdot \sqrt[3]{\frac{a}{b}} - b\right)}{2 \cdot a}\\

\end{array}\\

\mathbf{elif}\;b \le 3.038903409991338138548211857189252856935 \cdot 10^{107}:\\
\;\;\;\;\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{\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}\\

\end{array}\\

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

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

\end{array}
double f(double a, double b, double c) {
        double r1123500 = b;
        double r1123501 = 0.0;
        bool r1123502 = r1123500 >= r1123501;
        double r1123503 = 2.0;
        double r1123504 = c;
        double r1123505 = r1123503 * r1123504;
        double r1123506 = -r1123500;
        double r1123507 = r1123500 * r1123500;
        double r1123508 = 4.0;
        double r1123509 = a;
        double r1123510 = r1123508 * r1123509;
        double r1123511 = r1123510 * r1123504;
        double r1123512 = r1123507 - r1123511;
        double r1123513 = sqrt(r1123512);
        double r1123514 = r1123506 - r1123513;
        double r1123515 = r1123505 / r1123514;
        double r1123516 = r1123506 + r1123513;
        double r1123517 = r1123503 * r1123509;
        double r1123518 = r1123516 / r1123517;
        double r1123519 = r1123502 ? r1123515 : r1123518;
        return r1123519;
}
↓
double f(double a, double b, double c) {
        double r1123520 = b;
        double r1123521 = -5.6179139475653e+116;
        bool r1123522 = r1123520 <= r1123521;
        double r1123523 = 0.0;
        bool r1123524 = r1123520 >= r1123523;
        double r1123525 = 2.0;
        double r1123526 = c;
        double r1123527 = r1123525 * r1123526;
        double r1123528 = -r1123520;
        double r1123529 = r1123520 * r1123520;
        double r1123530 = 4.0;
        double r1123531 = a;
        double r1123532 = r1123530 * r1123531;
        double r1123533 = r1123532 * r1123526;
        double r1123534 = r1123529 - r1123533;
        double r1123535 = sqrt(r1123534);
        double r1123536 = r1123528 - r1123535;
        double r1123537 = r1123527 / r1123536;
        double r1123538 = r1123531 / r1123520;
        double r1123539 = cbrt(r1123538);
        double r1123540 = r1123539 * r1123539;
        double r1123541 = r1123540 * r1123527;
        double r1123542 = r1123541 * r1123539;
        double r1123543 = r1123542 - r1123520;
        double r1123544 = r1123528 + r1123543;
        double r1123545 = r1123525 * r1123531;
        double r1123546 = r1123544 / r1123545;
        double r1123547 = r1123524 ? r1123537 : r1123546;
        double r1123548 = 3.038903409991338e+107;
        bool r1123549 = r1123520 <= r1123548;
        double r1123550 = sqrt(r1123535);
        double r1123551 = r1123550 * r1123550;
        double r1123552 = r1123528 + r1123551;
        double r1123553 = r1123552 / r1123545;
        double r1123554 = r1123524 ? r1123537 : r1123553;
        double r1123555 = r1123527 * r1123538;
        double r1123556 = r1123520 - r1123555;
        double r1123557 = r1123528 - r1123556;
        double r1123558 = r1123527 / r1123557;
        double r1123559 = r1123535 + r1123528;
        double r1123560 = r1123559 / r1123545;
        double r1123561 = r1123524 ? r1123558 : r1123560;
        double r1123562 = r1123549 ? r1123554 : r1123561;
        double r1123563 = r1123522 ? r1123547 : r1123562;
        return r1123563;
}

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 < -5.6179139475653e+116

    1. Initial program 52.0

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

    2. Taylor expanded around -inf 10.5

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

    3. Simplified3.8

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

    5. Applied add-cube-cbrt3.8

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

    6. Applied associate-*r*3.8

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

    if -5.6179139475653e+116 < b < 3.038903409991338e+107

    1. Initial program 9.4

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

    3. Applied add-sqr-sqrt9.4

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

    4. Applied sqrt-prod9.5

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

    if 3.038903409991338e+107 < b

    1. Initial program 31.2

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

    2. Taylor expanded around inf 6.4

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

    3. Simplified2.4

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

  4. Final simplification7.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -5.617913947565299992326164335754974391576 \cdot 10^{116}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \left(\left(\left(\sqrt[3]{\frac{a}{b}} \cdot \sqrt[3]{\frac{a}{b}}\right) \cdot \left(2 \cdot c\right)\right) \cdot \sqrt[3]{\frac{a}{b}} - b\right)}{2 \cdot a}\\ \end{array}\\ \mathbf{elif}\;b \le 3.038903409991338138548211857189252856935 \cdot 10^{107}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}} \cdot \sqrt{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}{2 \cdot a}\\ \end{array}\\ \mathbf{elif}\;b \ge 0.0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \left(b - \left(2 \cdot c\right) \cdot \frac{a}{b}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} + \left(-b\right)}{2 \cdot a}\\ \end{array}\]

Reproduce

herbie shell --seed 2019171 
(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))))