Average Error: 34.8 → 10.1
Time: 19.7s
Precision: 64
\[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b_2 \le -3.680329042988888396603264581948851078331 \cdot 10^{148}:\\ \;\;\;\;\frac{\frac{c}{b_2}}{2} - \frac{b_2}{a} \cdot 2\\ \mathbf{elif}\;b_2 \le 4.612990823111230552052602417245542305295 \cdot 10^{-104}:\\ \;\;\;\;\left(\sqrt{b_2 \cdot b_2 - c \cdot a} - b_2\right) \cdot \frac{1}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \end{array}\]
\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}
↓
\begin{array}{l}
\mathbf{if}\;b_2 \le -3.680329042988888396603264581948851078331 \cdot 10^{148}:\\
\;\;\;\;\frac{\frac{c}{b_2}}{2} - \frac{b_2}{a} \cdot 2\\

\mathbf{elif}\;b_2 \le 4.612990823111230552052602417245542305295 \cdot 10^{-104}:\\
\;\;\;\;\left(\sqrt{b_2 \cdot b_2 - c \cdot a} - b_2\right) \cdot \frac{1}{a}\\

\mathbf{else}:\\
\;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\

\end{array}
double f(double a, double b_2, double c) {
        double r734536 = b_2;
        double r734537 = -r734536;
        double r734538 = r734536 * r734536;
        double r734539 = a;
        double r734540 = c;
        double r734541 = r734539 * r734540;
        double r734542 = r734538 - r734541;
        double r734543 = sqrt(r734542);
        double r734544 = r734537 + r734543;
        double r734545 = r734544 / r734539;
        return r734545;
}
↓
double f(double a, double b_2, double c) {
        double r734546 = b_2;
        double r734547 = -3.6803290429888884e+148;
        bool r734548 = r734546 <= r734547;
        double r734549 = c;
        double r734550 = r734549 / r734546;
        double r734551 = 2.0;
        double r734552 = r734550 / r734551;
        double r734553 = a;
        double r734554 = r734546 / r734553;
        double r734555 = r734554 * r734551;
        double r734556 = r734552 - r734555;
        double r734557 = 4.6129908231112306e-104;
        bool r734558 = r734546 <= r734557;
        double r734559 = r734546 * r734546;
        double r734560 = r734549 * r734553;
        double r734561 = r734559 - r734560;
        double r734562 = sqrt(r734561);
        double r734563 = r734562 - r734546;
        double r734564 = 1.0;
        double r734565 = r734564 / r734553;
        double r734566 = r734563 * r734565;
        double r734567 = -0.5;
        double r734568 = r734567 * r734550;
        double r734569 = r734558 ? r734566 : r734568;
        double r734570 = r734548 ? r734556 : r734569;
        return r734570;
}

Error

Bits error versus a

Bits error versus b_2

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 < -3.6803290429888884e+148

    1. Initial program 62.1

      \[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]

    2. Simplified62.1

      \[\leadsto \color{blue}{\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{a}}\]

    3. Taylor expanded around -inf 2.3

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}}\]

    4. Simplified2.3

      \[\leadsto \color{blue}{\frac{\frac{c}{b_2}}{2} - 2 \cdot \frac{b_2}{a}}\]

    if -3.6803290429888884e+148 < b_2 < 4.6129908231112306e-104

    1. Initial program 12.1

      \[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]

    2. Simplified12.1

      \[\leadsto \color{blue}{\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{a}}\]
    3. Using strategy rm

    4. Applied div-inv12.3

      \[\leadsto \color{blue}{\left(\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2\right) \cdot \frac{1}{a}}\]

    if 4.6129908231112306e-104 < b_2

    1. Initial program 52.7

      \[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]

    2. Simplified52.7

      \[\leadsto \color{blue}{\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{a}}\]

    3. Taylor expanded around inf 9.8

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{c}{b_2}}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification10.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -3.680329042988888396603264581948851078331 \cdot 10^{148}:\\ \;\;\;\;\frac{\frac{c}{b_2}}{2} - \frac{b_2}{a} \cdot 2\\ \mathbf{elif}\;b_2 \le 4.612990823111230552052602417245542305295 \cdot 10^{-104}:\\ \;\;\;\;\left(\sqrt{b_2 \cdot b_2 - c \cdot a} - b_2\right) \cdot \frac{1}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019170 
(FPCore (a b_2 c)
  :name "quad2p (problem 3.2.1, positive)"
  (/ (+ (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))