Average Error: 34.6 → 10.6
Time: 1.8m
Precision: 64
\[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b_2 \le -2.766818940874854722177248139872145176232 \cdot 10^{100}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\\ \mathbf{elif}\;b_2 \le 7.923524897992036987166355557663274472861 \cdot 10^{-153}:\\ \;\;\;\;\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b_2} \cdot \frac{-1}{2}\\ \end{array}\]
\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}
↓
\begin{array}{l}
\mathbf{if}\;b_2 \le -2.766818940874854722177248139872145176232 \cdot 10^{100}:\\
\;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\\

\mathbf{elif}\;b_2 \le 7.923524897992036987166355557663274472861 \cdot 10^{-153}:\\
\;\;\;\;\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - 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 r758428 = b_2;
        double r758429 = -r758428;
        double r758430 = r758428 * r758428;
        double r758431 = a;
        double r758432 = c;
        double r758433 = r758431 * r758432;
        double r758434 = r758430 - r758433;
        double r758435 = sqrt(r758434);
        double r758436 = r758429 + r758435;
        double r758437 = r758436 / r758431;
        return r758437;
}
↓
double f(double a, double b_2, double c) {
        double r758438 = b_2;
        double r758439 = -2.7668189408748547e+100;
        bool r758440 = r758438 <= r758439;
        double r758441 = 0.5;
        double r758442 = c;
        double r758443 = r758442 / r758438;
        double r758444 = r758441 * r758443;
        double r758445 = 2.0;
        double r758446 = a;
        double r758447 = r758438 / r758446;
        double r758448 = r758445 * r758447;
        double r758449 = r758444 - r758448;
        double r758450 = 7.923524897992037e-153;
        bool r758451 = r758438 <= r758450;
        double r758452 = r758438 * r758438;
        double r758453 = r758446 * r758442;
        double r758454 = r758452 - r758453;
        double r758455 = sqrt(r758454);
        double r758456 = r758455 - r758438;
        double r758457 = r758456 / r758446;
        double r758458 = -0.5;
        double r758459 = r758443 * r758458;
        double r758460 = r758451 ? r758457 : r758459;
        double r758461 = r758440 ? r758449 : r758460;
        return r758461;
}

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 < -2.7668189408748547e+100

    1. Initial program 47.1

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

    2. Simplified47.1

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

    3. Taylor expanded around -inf 4.0

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

    if -2.7668189408748547e+100 < b_2 < 7.923524897992037e-153

    1. Initial program 10.8

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

    2. Simplified10.8

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

    4. Applied div-inv10.9

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

    6. Applied un-div-inv10.8

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

    if 7.923524897992037e-153 < b_2

    1. Initial program 50.5

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

    2. Simplified50.5

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

    3. Taylor expanded around inf 12.7

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

  4. Final simplification10.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -2.766818940874854722177248139872145176232 \cdot 10^{100}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\\ \mathbf{elif}\;b_2 \le 7.923524897992036987166355557663274472861 \cdot 10^{-153}:\\ \;\;\;\;\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b_2} \cdot \frac{-1}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019200 
(FPCore (a b_2 c)
  :name "quad2p (problem 3.2.1, positive)"
  (/ (+ (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))