Average Error: 34.4 → 8.2
Time: 20.7s
Precision: 64
\[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b_2 \le -7.741777288939024183924384840560245543701 \cdot 10^{81}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le 4.744536474227931203452738058362502791987 \cdot 10^{-289}:\\ \;\;\;\;\frac{a \cdot \frac{c}{\sqrt{b_2 \cdot b_2 - c \cdot a} - b_2}}{a}\\ \mathbf{elif}\;b_2 \le 3.355858625783055094237525774982320834143 \cdot 10^{101}:\\ \;\;\;\;\frac{1}{a} \cdot \left(\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - c \cdot a}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{c}{b_2}}{2} - \frac{b_2 \cdot 2}{a}\\ \end{array}\]
\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}
↓
\begin{array}{l}
\mathbf{if}\;b_2 \le -7.741777288939024183924384840560245543701 \cdot 10^{81}:\\
\;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\

\mathbf{elif}\;b_2 \le 4.744536474227931203452738058362502791987 \cdot 10^{-289}:\\
\;\;\;\;\frac{a \cdot \frac{c}{\sqrt{b_2 \cdot b_2 - c \cdot a} - b_2}}{a}\\

\mathbf{elif}\;b_2 \le 3.355858625783055094237525774982320834143 \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{\frac{c}{b_2}}{2} - \frac{b_2 \cdot 2}{a}\\

\end{array}
double f(double a, double b_2, double c) {
        double r3004494 = b_2;
        double r3004495 = -r3004494;
        double r3004496 = r3004494 * r3004494;
        double r3004497 = a;
        double r3004498 = c;
        double r3004499 = r3004497 * r3004498;
        double r3004500 = r3004496 - r3004499;
        double r3004501 = sqrt(r3004500);
        double r3004502 = r3004495 - r3004501;
        double r3004503 = r3004502 / r3004497;
        return r3004503;
}
↓
double f(double a, double b_2, double c) {
        double r3004504 = b_2;
        double r3004505 = -7.741777288939024e+81;
        bool r3004506 = r3004504 <= r3004505;
        double r3004507 = -0.5;
        double r3004508 = c;
        double r3004509 = r3004508 / r3004504;
        double r3004510 = r3004507 * r3004509;
        double r3004511 = 4.744536474227931e-289;
        bool r3004512 = r3004504 <= r3004511;
        double r3004513 = a;
        double r3004514 = r3004504 * r3004504;
        double r3004515 = r3004508 * r3004513;
        double r3004516 = r3004514 - r3004515;
        double r3004517 = sqrt(r3004516);
        double r3004518 = r3004517 - r3004504;
        double r3004519 = r3004508 / r3004518;
        double r3004520 = r3004513 * r3004519;
        double r3004521 = r3004520 / r3004513;
        double r3004522 = 3.355858625783055e+101;
        bool r3004523 = r3004504 <= r3004522;
        double r3004524 = 1.0;
        double r3004525 = r3004524 / r3004513;
        double r3004526 = -r3004504;
        double r3004527 = r3004526 - r3004517;
        double r3004528 = r3004525 * r3004527;
        double r3004529 = 2.0;
        double r3004530 = r3004509 / r3004529;
        double r3004531 = r3004504 * r3004529;
        double r3004532 = r3004531 / r3004513;
        double r3004533 = r3004530 - r3004532;
        double r3004534 = r3004523 ? r3004528 : r3004533;
        double r3004535 = r3004512 ? r3004521 : r3004534;
        double r3004536 = r3004506 ? r3004510 : r3004535;
        return r3004536;
}

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 < -7.741777288939024e+81

    1. Initial program 58.6

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

    2. Taylor expanded around -inf 2.9

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

    if -7.741777288939024e+81 < b_2 < 4.744536474227931e-289

    1. Initial program 30.9

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

    3. Applied flip--30.9

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

    4. Simplified16.5

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

    5. Simplified16.5

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

    7. Applied *-un-lft-identity16.5

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

    8. Applied times-frac13.7

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

    9. Simplified13.7

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

    if 4.744536474227931e-289 < b_2 < 3.355858625783055e+101

    1. Initial program 9.1

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

    3. Applied div-inv9.2

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

    if 3.355858625783055e+101 < b_2

    1. Initial program 46.6

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

    2. Taylor expanded around inf 4.5

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

    3. Simplified4.6

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

  4. Final simplification8.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -7.741777288939024183924384840560245543701 \cdot 10^{81}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le 4.744536474227931203452738058362502791987 \cdot 10^{-289}:\\ \;\;\;\;\frac{a \cdot \frac{c}{\sqrt{b_2 \cdot b_2 - c \cdot a} - b_2}}{a}\\ \mathbf{elif}\;b_2 \le 3.355858625783055094237525774982320834143 \cdot 10^{101}:\\ \;\;\;\;\frac{1}{a} \cdot \left(\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - c \cdot a}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{c}{b_2}}{2} - \frac{b_2 \cdot 2}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2019172 
(FPCore (a b_2 c)
  :name "NMSE problem 3.2.1"
  (/ (- (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))