Average Error: 33.7 → 9.1
Time: 17.7s
Precision: 64
\[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b_2 \le -1.6806111715441095 \cdot 10^{-29}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le -2.5349830112643849 \cdot 10^{-83}:\\ \;\;\;\;\frac{\frac{a \cdot c}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}{a}\\ \mathbf{elif}\;b_2 \le 5.9911994584698608 \cdot 10^{103}:\\ \;\;\;\;\frac{1}{\frac{a}{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\\ \end{array}\]
\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}
↓
\begin{array}{l}
\mathbf{if}\;b_2 \le -1.6806111715441095 \cdot 10^{-29}:\\
\;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\

\mathbf{elif}\;b_2 \le -2.5349830112643849 \cdot 10^{-83}:\\
\;\;\;\;\frac{\frac{a \cdot c}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}{a}\\

\mathbf{elif}\;b_2 \le 5.9911994584698608 \cdot 10^{103}:\\
\;\;\;\;\frac{1}{\frac{a}{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}}\\

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

\end{array}
double f(double a, double b_2, double c) {
        double r25479 = b_2;
        double r25480 = -r25479;
        double r25481 = r25479 * r25479;
        double r25482 = a;
        double r25483 = c;
        double r25484 = r25482 * r25483;
        double r25485 = r25481 - r25484;
        double r25486 = sqrt(r25485);
        double r25487 = r25480 - r25486;
        double r25488 = r25487 / r25482;
        return r25488;
}
↓
double f(double a, double b_2, double c) {
        double r25489 = b_2;
        double r25490 = -1.6806111715441095e-29;
        bool r25491 = r25489 <= r25490;
        double r25492 = -0.5;
        double r25493 = c;
        double r25494 = r25493 / r25489;
        double r25495 = r25492 * r25494;
        double r25496 = -2.534983011264385e-83;
        bool r25497 = r25489 <= r25496;
        double r25498 = a;
        double r25499 = r25498 * r25493;
        double r25500 = r25489 * r25489;
        double r25501 = r25500 - r25499;
        double r25502 = sqrt(r25501);
        double r25503 = r25502 - r25489;
        double r25504 = r25499 / r25503;
        double r25505 = r25504 / r25498;
        double r25506 = 5.991199458469861e+103;
        bool r25507 = r25489 <= r25506;
        double r25508 = 1.0;
        double r25509 = -r25489;
        double r25510 = r25509 - r25502;
        double r25511 = r25498 / r25510;
        double r25512 = r25508 / r25511;
        double r25513 = 0.5;
        double r25514 = r25513 * r25494;
        double r25515 = 2.0;
        double r25516 = r25489 / r25498;
        double r25517 = r25515 * r25516;
        double r25518 = r25514 - r25517;
        double r25519 = r25507 ? r25512 : r25518;
        double r25520 = r25497 ? r25505 : r25519;
        double r25521 = r25491 ? r25495 : r25520;
        return r25521;
}

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 < -1.6806111715441095e-29

    1. Initial program 54.8

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

    2. Taylor expanded around -inf 6.6

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

    if -1.6806111715441095e-29 < b_2 < -2.534983011264385e-83

    1. Initial program 32.9

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

    3. Applied flip--33.0

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

    4. Simplified16.9

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

    5. Simplified16.9

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

    if -2.534983011264385e-83 < b_2 < 5.991199458469861e+103

    1. Initial program 12.0

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

    3. Applied clear-num12.1

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

    if 5.991199458469861e+103 < b_2

    1. Initial program 48.7

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

    2. Taylor expanded around inf 4.1

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

  4. Final simplification9.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -1.6806111715441095 \cdot 10^{-29}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le -2.5349830112643849 \cdot 10^{-83}:\\ \;\;\;\;\frac{\frac{a \cdot c}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}{a}\\ \mathbf{elif}\;b_2 \le 5.9911994584698608 \cdot 10^{103}:\\ \;\;\;\;\frac{1}{\frac{a}{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2019198 
(FPCore (a b_2 c)
  :name "quad2m (problem 3.2.1, negative)"
  (/ (- (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))