Average Error: 33.8 → 10.2
Time: 20.1s
Precision: 64
\[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b_2 \le -2.569494919068124572690421335939486791404 \cdot 10^{-64}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le 2.8653816703769607550753035783606354728 \cdot 10^{117}:\\ \;\;\;\;\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{b_2}{a} \cdot -2\\ \end{array}\]
\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}
↓
\begin{array}{l}
\mathbf{if}\;b_2 \le -2.569494919068124572690421335939486791404 \cdot 10^{-64}:\\
\;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\

\mathbf{elif}\;b_2 \le 2.8653816703769607550753035783606354728 \cdot 10^{117}:\\
\;\;\;\;\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\\

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

\end{array}
double f(double a, double b_2, double c) {
        double r3283421 = b_2;
        double r3283422 = -r3283421;
        double r3283423 = r3283421 * r3283421;
        double r3283424 = a;
        double r3283425 = c;
        double r3283426 = r3283424 * r3283425;
        double r3283427 = r3283423 - r3283426;
        double r3283428 = sqrt(r3283427);
        double r3283429 = r3283422 - r3283428;
        double r3283430 = r3283429 / r3283424;
        return r3283430;
}
↓
double f(double a, double b_2, double c) {
        double r3283431 = b_2;
        double r3283432 = -2.5694949190681246e-64;
        bool r3283433 = r3283431 <= r3283432;
        double r3283434 = -0.5;
        double r3283435 = c;
        double r3283436 = r3283435 / r3283431;
        double r3283437 = r3283434 * r3283436;
        double r3283438 = 2.865381670376961e+117;
        bool r3283439 = r3283431 <= r3283438;
        double r3283440 = -r3283431;
        double r3283441 = r3283431 * r3283431;
        double r3283442 = a;
        double r3283443 = r3283442 * r3283435;
        double r3283444 = r3283441 - r3283443;
        double r3283445 = sqrt(r3283444);
        double r3283446 = r3283440 - r3283445;
        double r3283447 = r3283446 / r3283442;
        double r3283448 = r3283431 / r3283442;
        double r3283449 = -2.0;
        double r3283450 = r3283448 * r3283449;
        double r3283451 = r3283439 ? r3283447 : r3283450;
        double r3283452 = r3283433 ? r3283437 : r3283451;
        return r3283452;
}

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 < -2.5694949190681246e-64

    1. Initial program 53.1

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

    2. Taylor expanded around -inf 9.1

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

    if -2.5694949190681246e-64 < b_2 < 2.865381670376961e+117

    1. Initial program 13.1

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

    3. Applied div-inv13.2

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

    5. Applied un-div-inv13.1

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

    if 2.865381670376961e+117 < b_2

    1. Initial program 52.0

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

    3. Applied clear-num52.1

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

    4. Taylor expanded around 0 3.1

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

  4. Final simplification10.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -2.569494919068124572690421335939486791404 \cdot 10^{-64}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le 2.8653816703769607550753035783606354728 \cdot 10^{117}:\\ \;\;\;\;\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{b_2}{a} \cdot -2\\ \end{array}\]

Reproduce

herbie shell --seed 2019169 
(FPCore (a b_2 c)
  :name "NMSE problem 3.2.1"
  (/ (- (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))