Average Error: 33.6 → 9.8
Time: 17.7s
Precision: 64
\[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b_2 \le -3.396811349079212 \cdot 10^{+61}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - \frac{b_2}{a} \cdot 2\\ \mathbf{elif}\;b_2 \le 1.3659668388152999 \cdot 10^{-67}:\\ \;\;\;\;\frac{\sqrt{b_2 \cdot b_2 - c \cdot a} - b_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b_2} \cdot \frac{-1}{2}\\ \end{array}\]
\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}
↓
\begin{array}{l}
\mathbf{if}\;b_2 \le -3.396811349079212 \cdot 10^{+61}:\\
\;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - \frac{b_2}{a} \cdot 2\\

\mathbf{elif}\;b_2 \le 1.3659668388152999 \cdot 10^{-67}:\\
\;\;\;\;\frac{\sqrt{b_2 \cdot b_2 - c \cdot a} - b_2}{a}\\

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

\end{array}
double f(double a, double b_2, double c) {
        double r724516 = b_2;
        double r724517 = -r724516;
        double r724518 = r724516 * r724516;
        double r724519 = a;
        double r724520 = c;
        double r724521 = r724519 * r724520;
        double r724522 = r724518 - r724521;
        double r724523 = sqrt(r724522);
        double r724524 = r724517 + r724523;
        double r724525 = r724524 / r724519;
        return r724525;
}
↓
double f(double a, double b_2, double c) {
        double r724526 = b_2;
        double r724527 = -3.396811349079212e+61;
        bool r724528 = r724526 <= r724527;
        double r724529 = 0.5;
        double r724530 = c;
        double r724531 = r724530 / r724526;
        double r724532 = r724529 * r724531;
        double r724533 = a;
        double r724534 = r724526 / r724533;
        double r724535 = 2.0;
        double r724536 = r724534 * r724535;
        double r724537 = r724532 - r724536;
        double r724538 = 1.3659668388152999e-67;
        bool r724539 = r724526 <= r724538;
        double r724540 = r724526 * r724526;
        double r724541 = r724530 * r724533;
        double r724542 = r724540 - r724541;
        double r724543 = sqrt(r724542);
        double r724544 = r724543 - r724526;
        double r724545 = r724544 / r724533;
        double r724546 = -0.5;
        double r724547 = r724531 * r724546;
        double r724548 = r724539 ? r724545 : r724547;
        double r724549 = r724528 ? r724537 : r724548;
        return r724549;
}

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 < -3.396811349079212e+61

    1. Initial program 37.6

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

    2. Simplified37.6

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

    4. Applied div-inv37.7

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

    5. Taylor expanded around -inf 4.3

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

    if -3.396811349079212e+61 < b_2 < 1.3659668388152999e-67

    1. Initial program 13.8

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

    2. Simplified13.8

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

    if 1.3659668388152999e-67 < b_2

    1. Initial program 53.0

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

    2. Simplified53.0

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

    3. Taylor expanded around inf 8.1

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

  4. Final simplification9.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -3.396811349079212 \cdot 10^{+61}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - \frac{b_2}{a} \cdot 2\\ \mathbf{elif}\;b_2 \le 1.3659668388152999 \cdot 10^{-67}:\\ \;\;\;\;\frac{\sqrt{b_2 \cdot b_2 - c \cdot a} - b_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b_2} \cdot \frac{-1}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019130 
(FPCore (a b_2 c)
  :name "quad2p (problem 3.2.1, positive)"
  (/ (+ (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))