Average Error: 34.5 → 10.5
Time: 4.4s
Precision: 64
\[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b_2 \le -4.78285893492843261 \cdot 10^{-126}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le 3.1323104234385293 \cdot 10^{111}:\\ \;\;\;\;\left(\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}\right) \cdot \frac{1}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\\ \end{array}\]
\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}
↓
\begin{array}{l}
\mathbf{if}\;b_2 \le -4.78285893492843261 \cdot 10^{-126}:\\
\;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\

\mathbf{elif}\;b_2 \le 3.1323104234385293 \cdot 10^{111}:\\
\;\;\;\;\left(\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}\right) \cdot \frac{1}{a}\\

\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 r72353 = b_2;
        double r72354 = -r72353;
        double r72355 = r72353 * r72353;
        double r72356 = a;
        double r72357 = c;
        double r72358 = r72356 * r72357;
        double r72359 = r72355 - r72358;
        double r72360 = sqrt(r72359);
        double r72361 = r72354 - r72360;
        double r72362 = r72361 / r72356;
        return r72362;
}
↓
double f(double a, double b_2, double c) {
        double r72363 = b_2;
        double r72364 = -4.7828589349284326e-126;
        bool r72365 = r72363 <= r72364;
        double r72366 = -0.5;
        double r72367 = c;
        double r72368 = r72367 / r72363;
        double r72369 = r72366 * r72368;
        double r72370 = 3.1323104234385293e+111;
        bool r72371 = r72363 <= r72370;
        double r72372 = -r72363;
        double r72373 = r72363 * r72363;
        double r72374 = a;
        double r72375 = r72374 * r72367;
        double r72376 = r72373 - r72375;
        double r72377 = sqrt(r72376);
        double r72378 = r72372 - r72377;
        double r72379 = 1.0;
        double r72380 = r72379 / r72374;
        double r72381 = r72378 * r72380;
        double r72382 = 0.5;
        double r72383 = r72382 * r72368;
        double r72384 = 2.0;
        double r72385 = r72363 / r72374;
        double r72386 = r72384 * r72385;
        double r72387 = r72383 - r72386;
        double r72388 = r72371 ? r72381 : r72387;
        double r72389 = r72365 ? r72369 : r72388;
        return r72389;
}

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 < -4.7828589349284326e-126

    1. Initial program 51.3

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

    2. Taylor expanded around -inf 11.3

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

    if -4.7828589349284326e-126 < b_2 < 3.1323104234385293e+111

    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 div-inv12.1

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

    if 3.1323104234385293e+111 < b_2

    1. Initial program 49.7

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

    2. Taylor expanded around inf 3.4

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

  4. Final simplification10.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -4.78285893492843261 \cdot 10^{-126}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le 3.1323104234385293 \cdot 10^{111}:\\ \;\;\;\;\left(\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}\right) \cdot \frac{1}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2020047 +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))