Average Error: 33.8 → 10.1
Time: 4.5s
Precision: 64
\[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b_2 \le -1.369694371126339229257094016308893237032 \cdot 10^{-83}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le 2.678238127073728805877873599258558355989 \cdot 10^{53}:\\ \;\;\;\;\frac{1}{\frac{a}{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\\ \end{array}\]
\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}
↓
\begin{array}{l}
\mathbf{if}\;b_2 \le -1.369694371126339229257094016308893237032 \cdot 10^{-83}:\\
\;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\

\mathbf{elif}\;b_2 \le 2.678238127073728805877873599258558355989 \cdot 10^{53}:\\
\;\;\;\;\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 r91371 = b_2;
        double r91372 = -r91371;
        double r91373 = r91371 * r91371;
        double r91374 = a;
        double r91375 = c;
        double r91376 = r91374 * r91375;
        double r91377 = r91373 - r91376;
        double r91378 = sqrt(r91377);
        double r91379 = r91372 - r91378;
        double r91380 = r91379 / r91374;
        return r91380;
}
↓
double f(double a, double b_2, double c) {
        double r91381 = b_2;
        double r91382 = -1.3696943711263392e-83;
        bool r91383 = r91381 <= r91382;
        double r91384 = -0.5;
        double r91385 = c;
        double r91386 = r91385 / r91381;
        double r91387 = r91384 * r91386;
        double r91388 = 2.678238127073729e+53;
        bool r91389 = r91381 <= r91388;
        double r91390 = 1.0;
        double r91391 = a;
        double r91392 = -r91381;
        double r91393 = r91381 * r91381;
        double r91394 = r91391 * r91385;
        double r91395 = r91393 - r91394;
        double r91396 = sqrt(r91395);
        double r91397 = r91392 - r91396;
        double r91398 = r91391 / r91397;
        double r91399 = r91390 / r91398;
        double r91400 = 0.5;
        double r91401 = r91400 * r91386;
        double r91402 = 2.0;
        double r91403 = r91381 / r91391;
        double r91404 = r91402 * r91403;
        double r91405 = r91401 - r91404;
        double r91406 = r91389 ? r91399 : r91405;
        double r91407 = r91383 ? r91387 : r91406;
        return r91407;
}

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 < -1.3696943711263392e-83

    1. Initial program 53.2

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

    2. Taylor expanded around -inf 9.4

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

    if -1.3696943711263392e-83 < b_2 < 2.678238127073729e+53

    1. Initial program 12.8

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

    3. Applied clear-num12.9

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

    if 2.678238127073729e+53 < b_2

    1. Initial program 38.1

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

    2. Taylor expanded around inf 5.7

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

  4. Final simplification10.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -1.369694371126339229257094016308893237032 \cdot 10^{-83}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le 2.678238127073728805877873599258558355989 \cdot 10^{53}:\\ \;\;\;\;\frac{1}{\frac{a}{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2019362 +o rules:numerics
(FPCore (a b_2 c)
  :name "NMSE problem 3.2.1"
  :precision binary64
  (/ (- (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))