Average Error: 33.5 → 10.0
Time: 17.3s
Precision: 64
\[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b_2 \le -3.2092322739463293 \cdot 10^{-86}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le 2.891777552454845 \cdot 10^{+74}:\\ \;\;\;\;-\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} + b_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-2, b_2, \left(\frac{a}{\frac{b_2}{c}} \cdot \frac{1}{2}\right)\right)}{a}\\ \end{array}\]
\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}
↓
\begin{array}{l}
\mathbf{if}\;b_2 \le -3.2092322739463293 \cdot 10^{-86}:\\
\;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\

\mathbf{elif}\;b_2 \le 2.891777552454845 \cdot 10^{+74}:\\
\;\;\;\;-\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} + b_2}{a}\\

\mathbf{else}:\\
\;\;\;\;\frac{\mathsf{fma}\left(-2, b_2, \left(\frac{a}{\frac{b_2}{c}} \cdot \frac{1}{2}\right)\right)}{a}\\

\end{array}
double f(double a, double b_2, double c) {
        double r1987193 = b_2;
        double r1987194 = -r1987193;
        double r1987195 = r1987193 * r1987193;
        double r1987196 = a;
        double r1987197 = c;
        double r1987198 = r1987196 * r1987197;
        double r1987199 = r1987195 - r1987198;
        double r1987200 = sqrt(r1987199);
        double r1987201 = r1987194 - r1987200;
        double r1987202 = r1987201 / r1987196;
        return r1987202;
}
↓
double f(double a, double b_2, double c) {
        double r1987203 = b_2;
        double r1987204 = -3.2092322739463293e-86;
        bool r1987205 = r1987203 <= r1987204;
        double r1987206 = -0.5;
        double r1987207 = c;
        double r1987208 = r1987207 / r1987203;
        double r1987209 = r1987206 * r1987208;
        double r1987210 = 2.891777552454845e+74;
        bool r1987211 = r1987203 <= r1987210;
        double r1987212 = r1987203 * r1987203;
        double r1987213 = a;
        double r1987214 = r1987213 * r1987207;
        double r1987215 = r1987212 - r1987214;
        double r1987216 = sqrt(r1987215);
        double r1987217 = r1987216 + r1987203;
        double r1987218 = r1987217 / r1987213;
        double r1987219 = -r1987218;
        double r1987220 = -2.0;
        double r1987221 = r1987203 / r1987207;
        double r1987222 = r1987213 / r1987221;
        double r1987223 = 0.5;
        double r1987224 = r1987222 * r1987223;
        double r1987225 = fma(r1987220, r1987203, r1987224);
        double r1987226 = r1987225 / r1987213;
        double r1987227 = r1987211 ? r1987219 : r1987226;
        double r1987228 = r1987205 ? r1987209 : r1987227;
        return r1987228;
}

Error

Bits error versus a

Bits error versus b_2

Bits error versus c

Derivation

  1. Split input into 3 regimes

  2. if b_2 < -3.2092322739463293e-86

    1. Initial program 52.3

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

    2. Taylor expanded around -inf 9.4

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

    if -3.2092322739463293e-86 < b_2 < 2.891777552454845e+74

    1. Initial program 13.1

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

    3. Applied add-sqr-sqrt13.1

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

    4. Applied sqrt-prod13.3

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

    6. Applied neg-sub013.3

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

    7. Applied associate--l-13.3

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

    8. Simplified13.1

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

    if 2.891777552454845e+74 < b_2

    1. Initial program 38.9

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

    3. Applied add-sqr-sqrt38.9

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

    4. Applied sqrt-prod39.0

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

    5. Taylor expanded around inf 10.1

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

    6. Simplified4.3

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(-2, b_2, \left(\frac{1}{2} \cdot \frac{a}{\frac{b_2}{c}}\right)\right)}}{a}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification10.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -3.2092322739463293 \cdot 10^{-86}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le 2.891777552454845 \cdot 10^{+74}:\\ \;\;\;\;-\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} + b_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-2, b_2, \left(\frac{a}{\frac{b_2}{c}} \cdot \frac{1}{2}\right)\right)}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2019130 +o rules:numerics
(FPCore (a b_2 c)
  :name "NMSE problem 3.2.1"
  (/ (- (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))