Average Error: 33.3 → 9.1
Time: 17.2s
Precision: 64
\[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b_2 \le -1.2810366229709043 \cdot 10^{+70}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le 1.2889142225980239 \cdot 10^{-280}:\\ \;\;\;\;\frac{\frac{c \cdot a}{\sqrt{b_2 \cdot b_2 - c \cdot a} - b_2}}{a}\\ \mathbf{elif}\;b_2 \le 1.4483715500512764 \cdot 10^{+101}:\\ \;\;\;\;\frac{1}{a} \cdot \left(\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - c \cdot a}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - \frac{b_2}{a} \cdot 2\\ \end{array}\]
\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}
↓
\begin{array}{l}
\mathbf{if}\;b_2 \le -1.2810366229709043 \cdot 10^{+70}:\\
\;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\

\mathbf{elif}\;b_2 \le 1.2889142225980239 \cdot 10^{-280}:\\
\;\;\;\;\frac{\frac{c \cdot a}{\sqrt{b_2 \cdot b_2 - c \cdot a} - b_2}}{a}\\

\mathbf{elif}\;b_2 \le 1.4483715500512764 \cdot 10^{+101}:\\
\;\;\;\;\frac{1}{a} \cdot \left(\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - c \cdot a}\right)\\

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

\end{array}
double f(double a, double b_2, double c) {
        double r1635401 = b_2;
        double r1635402 = -r1635401;
        double r1635403 = r1635401 * r1635401;
        double r1635404 = a;
        double r1635405 = c;
        double r1635406 = r1635404 * r1635405;
        double r1635407 = r1635403 - r1635406;
        double r1635408 = sqrt(r1635407);
        double r1635409 = r1635402 - r1635408;
        double r1635410 = r1635409 / r1635404;
        return r1635410;
}
↓
double f(double a, double b_2, double c) {
        double r1635411 = b_2;
        double r1635412 = -1.2810366229709043e+70;
        bool r1635413 = r1635411 <= r1635412;
        double r1635414 = -0.5;
        double r1635415 = c;
        double r1635416 = r1635415 / r1635411;
        double r1635417 = r1635414 * r1635416;
        double r1635418 = 1.2889142225980239e-280;
        bool r1635419 = r1635411 <= r1635418;
        double r1635420 = a;
        double r1635421 = r1635415 * r1635420;
        double r1635422 = r1635411 * r1635411;
        double r1635423 = r1635422 - r1635421;
        double r1635424 = sqrt(r1635423);
        double r1635425 = r1635424 - r1635411;
        double r1635426 = r1635421 / r1635425;
        double r1635427 = r1635426 / r1635420;
        double r1635428 = 1.4483715500512764e+101;
        bool r1635429 = r1635411 <= r1635428;
        double r1635430 = 1.0;
        double r1635431 = r1635430 / r1635420;
        double r1635432 = -r1635411;
        double r1635433 = r1635432 - r1635424;
        double r1635434 = r1635431 * r1635433;
        double r1635435 = 0.5;
        double r1635436 = r1635435 * r1635416;
        double r1635437 = r1635411 / r1635420;
        double r1635438 = 2.0;
        double r1635439 = r1635437 * r1635438;
        double r1635440 = r1635436 - r1635439;
        double r1635441 = r1635429 ? r1635434 : r1635440;
        double r1635442 = r1635419 ? r1635427 : r1635441;
        double r1635443 = r1635413 ? r1635417 : r1635442;
        return r1635443;
}

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 4 regimes

  2. if b_2 < -1.2810366229709043e+70

    1. Initial program 56.8

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

    2. Taylor expanded around -inf 3.0

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

    if -1.2810366229709043e+70 < b_2 < 1.2889142225980239e-280

    1. Initial program 30.3

      \[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
    2. Using strategy rm

    3. Applied flip--30.4

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

    4. Simplified17.2

      \[\leadsto \frac{\frac{\color{blue}{0 + a \cdot c}}{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}}{a}\]

    5. Simplified17.2

      \[\leadsto \frac{\frac{0 + a \cdot c}{\color{blue}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}}{a}\]

    if 1.2889142225980239e-280 < b_2 < 1.4483715500512764e+101

    1. Initial program 8.9

      \[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
    2. Using strategy rm

    3. Applied div-inv9.1

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

    if 1.4483715500512764e+101 < b_2

    1. Initial program 43.7

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

    2. Taylor expanded around inf 3.7

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}}\]
  3. Recombined 4 regimes into one program.

  4. Final simplification9.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -1.2810366229709043 \cdot 10^{+70}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le 1.2889142225980239 \cdot 10^{-280}:\\ \;\;\;\;\frac{\frac{c \cdot a}{\sqrt{b_2 \cdot b_2 - c \cdot a} - b_2}}{a}\\ \mathbf{elif}\;b_2 \le 1.4483715500512764 \cdot 10^{+101}:\\ \;\;\;\;\frac{1}{a} \cdot \left(\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - c \cdot a}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - \frac{b_2}{a} \cdot 2\\ \end{array}\]

Reproduce

herbie shell --seed 2019154 
(FPCore (a b_2 c)
  :name "NMSE problem 3.2.1"
  (/ (- (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))