Average Error: 32.7 → 10.3
Time: 19.7s
Precision: 64
\[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b_2 \le -2.3093766494757864 \cdot 10^{+77}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - \frac{b_2}{a} \cdot 2\\ \mathbf{elif}\;b_2 \le 1.2352405779465016 \cdot 10^{-131}:\\ \;\;\;\;\frac{1}{\frac{a}{\sqrt{b_2 \cdot b_2 - c \cdot a} - b_2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \end{array}\]
\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}
↓
\begin{array}{l}
\mathbf{if}\;b_2 \le -2.3093766494757864 \cdot 10^{+77}:\\
\;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - \frac{b_2}{a} \cdot 2\\

\mathbf{elif}\;b_2 \le 1.2352405779465016 \cdot 10^{-131}:\\
\;\;\;\;\frac{1}{\frac{a}{\sqrt{b_2 \cdot b_2 - c \cdot a} - b_2}}\\

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

\end{array}
double f(double a, double b_2, double c) {
        double r659326 = b_2;
        double r659327 = -r659326;
        double r659328 = r659326 * r659326;
        double r659329 = a;
        double r659330 = c;
        double r659331 = r659329 * r659330;
        double r659332 = r659328 - r659331;
        double r659333 = sqrt(r659332);
        double r659334 = r659327 + r659333;
        double r659335 = r659334 / r659329;
        return r659335;
}
↓
double f(double a, double b_2, double c) {
        double r659336 = b_2;
        double r659337 = -2.3093766494757864e+77;
        bool r659338 = r659336 <= r659337;
        double r659339 = 0.5;
        double r659340 = c;
        double r659341 = r659340 / r659336;
        double r659342 = r659339 * r659341;
        double r659343 = a;
        double r659344 = r659336 / r659343;
        double r659345 = 2.0;
        double r659346 = r659344 * r659345;
        double r659347 = r659342 - r659346;
        double r659348 = 1.2352405779465016e-131;
        bool r659349 = r659336 <= r659348;
        double r659350 = 1.0;
        double r659351 = r659336 * r659336;
        double r659352 = r659340 * r659343;
        double r659353 = r659351 - r659352;
        double r659354 = sqrt(r659353);
        double r659355 = r659354 - r659336;
        double r659356 = r659343 / r659355;
        double r659357 = r659350 / r659356;
        double r659358 = -0.5;
        double r659359 = r659358 * r659341;
        double r659360 = r659349 ? r659357 : r659359;
        double r659361 = r659338 ? r659347 : r659360;
        return r659361;
}

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 < -2.3093766494757864e+77

    1. Initial program 39.5

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

    2. Simplified39.5

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

    3. Taylor expanded around -inf 5.0

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

    if -2.3093766494757864e+77 < b_2 < 1.2352405779465016e-131

    1. Initial program 11.0

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

    2. Simplified11.0

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

    4. Applied clear-num11.1

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

    if 1.2352405779465016e-131 < b_2

    1. Initial program 50.3

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

    2. Simplified50.3

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

    3. Taylor expanded around -inf 50.3

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

    4. Simplified50.3

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

    5. Taylor expanded around inf 11.7

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

  4. Final simplification10.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -2.3093766494757864 \cdot 10^{+77}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - \frac{b_2}{a} \cdot 2\\ \mathbf{elif}\;b_2 \le 1.2352405779465016 \cdot 10^{-131}:\\ \;\;\;\;\frac{1}{\frac{a}{\sqrt{b_2 \cdot b_2 - c \cdot a} - b_2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019134 
(FPCore (a b_2 c)
  :name "quad2p (problem 3.2.1, positive)"
  (/ (+ (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))