Average Error: 34.3 → 9.8
Time: 10.3s
Precision: 64
\[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b_2 \le -2.271875817960049862049445045905922921165 \cdot 10^{-81}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le 3.583649041028097672662453906912522703022 \cdot 10^{84}:\\ \;\;\;\;\frac{\left(-b_2\right) - \sqrt{\mathsf{fma}\left(-c, a, b_2 \cdot b_2\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b_2\right) - b_2}{a}\\ \end{array}\]
\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}
↓
\begin{array}{l}
\mathbf{if}\;b_2 \le -2.271875817960049862049445045905922921165 \cdot 10^{-81}:\\
\;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\

\mathbf{elif}\;b_2 \le 3.583649041028097672662453906912522703022 \cdot 10^{84}:\\
\;\;\;\;\frac{\left(-b_2\right) - \sqrt{\mathsf{fma}\left(-c, a, b_2 \cdot b_2\right)}}{a}\\

\mathbf{else}:\\
\;\;\;\;\frac{\left(-b_2\right) - b_2}{a}\\

\end{array}
double f(double a, double b_2, double c) {
        double r57178 = b_2;
        double r57179 = -r57178;
        double r57180 = r57178 * r57178;
        double r57181 = a;
        double r57182 = c;
        double r57183 = r57181 * r57182;
        double r57184 = r57180 - r57183;
        double r57185 = sqrt(r57184);
        double r57186 = r57179 - r57185;
        double r57187 = r57186 / r57181;
        return r57187;
}
↓
double f(double a, double b_2, double c) {
        double r57188 = b_2;
        double r57189 = -2.27187581796005e-81;
        bool r57190 = r57188 <= r57189;
        double r57191 = -0.5;
        double r57192 = c;
        double r57193 = r57192 / r57188;
        double r57194 = r57191 * r57193;
        double r57195 = 3.5836490410280977e+84;
        bool r57196 = r57188 <= r57195;
        double r57197 = -r57188;
        double r57198 = -r57192;
        double r57199 = a;
        double r57200 = r57188 * r57188;
        double r57201 = fma(r57198, r57199, r57200);
        double r57202 = sqrt(r57201);
        double r57203 = r57197 - r57202;
        double r57204 = r57203 / r57199;
        double r57205 = r57197 - r57188;
        double r57206 = r57205 / r57199;
        double r57207 = r57196 ? r57204 : r57206;
        double r57208 = r57190 ? r57194 : r57207;
        return r57208;
}

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 < -2.27187581796005e-81

    1. Initial program 53.4

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

    2. Simplified53.4

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

    3. Taylor expanded around -inf 8.8

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

    if -2.27187581796005e-81 < b_2 < 3.5836490410280977e+84

    1. Initial program 13.1

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

    2. Simplified13.1

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

    3. Taylor expanded around 0 13.1

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

    4. Simplified13.1

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

    if 3.5836490410280977e+84 < b_2

    1. Initial program 43.9

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

    2. Simplified43.9

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

    3. Taylor expanded around 0 3.7

      \[\leadsto \frac{\left(-b_2\right) - \color{blue}{b_2}}{a}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification9.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -2.271875817960049862049445045905922921165 \cdot 10^{-81}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le 3.583649041028097672662453906912522703022 \cdot 10^{84}:\\ \;\;\;\;\frac{\left(-b_2\right) - \sqrt{\mathsf{fma}\left(-c, a, b_2 \cdot b_2\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b_2\right) - b_2}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2019194 +o rules:numerics
(FPCore (a b_2 c)
  :name "NMSE problem 3.2.1"
  (/ (- (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))