Average Error: 33.3 → 9.8
Time: 16.6s
Precision: 64
\[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b_2 \le -5.7874989996849275 \cdot 10^{-40}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le 1.7665622931893247 \cdot 10^{+83}:\\ \;\;\;\;\frac{1}{\frac{a}{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - c \cdot a}}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{b_2}{a}, -2, \frac{\frac{1}{2}}{\frac{b_2}{c}}\right)\\ \end{array}\]
\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}
↓
\begin{array}{l}
\mathbf{if}\;b_2 \le -5.7874989996849275 \cdot 10^{-40}:\\
\;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\

\mathbf{elif}\;b_2 \le 1.7665622931893247 \cdot 10^{+83}:\\
\;\;\;\;\frac{1}{\frac{a}{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - c \cdot a}}}\\

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

\end{array}
double f(double a, double b_2, double c) {
        double r1200256 = b_2;
        double r1200257 = -r1200256;
        double r1200258 = r1200256 * r1200256;
        double r1200259 = a;
        double r1200260 = c;
        double r1200261 = r1200259 * r1200260;
        double r1200262 = r1200258 - r1200261;
        double r1200263 = sqrt(r1200262);
        double r1200264 = r1200257 - r1200263;
        double r1200265 = r1200264 / r1200259;
        return r1200265;
}
↓
double f(double a, double b_2, double c) {
        double r1200266 = b_2;
        double r1200267 = -5.7874989996849275e-40;
        bool r1200268 = r1200266 <= r1200267;
        double r1200269 = -0.5;
        double r1200270 = c;
        double r1200271 = r1200270 / r1200266;
        double r1200272 = r1200269 * r1200271;
        double r1200273 = 1.7665622931893247e+83;
        bool r1200274 = r1200266 <= r1200273;
        double r1200275 = 1.0;
        double r1200276 = a;
        double r1200277 = -r1200266;
        double r1200278 = r1200266 * r1200266;
        double r1200279 = r1200270 * r1200276;
        double r1200280 = r1200278 - r1200279;
        double r1200281 = sqrt(r1200280);
        double r1200282 = r1200277 - r1200281;
        double r1200283 = r1200276 / r1200282;
        double r1200284 = r1200275 / r1200283;
        double r1200285 = r1200266 / r1200276;
        double r1200286 = -2.0;
        double r1200287 = 0.5;
        double r1200288 = r1200266 / r1200270;
        double r1200289 = r1200287 / r1200288;
        double r1200290 = fma(r1200285, r1200286, r1200289);
        double r1200291 = r1200274 ? r1200284 : r1200290;
        double r1200292 = r1200268 ? r1200272 : r1200291;
        return r1200292;
}

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 < -5.7874989996849275e-40

    1. Initial program 53.7

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

    2. Taylor expanded around -inf 7.3

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

    if -5.7874989996849275e-40 < b_2 < 1.7665622931893247e+83

    1. Initial program 13.8

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

    3. Applied clear-num13.9

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

    if 1.7665622931893247e+83 < b_2

    1. Initial program 42.5

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

    2. Taylor expanded around inf 3.8

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

    3. Simplified3.8

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

  4. Final simplification9.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -5.7874989996849275 \cdot 10^{-40}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le 1.7665622931893247 \cdot 10^{+83}:\\ \;\;\;\;\frac{1}{\frac{a}{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - c \cdot a}}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{b_2}{a}, -2, \frac{\frac{1}{2}}{\frac{b_2}{c}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019156 +o rules:numerics
(FPCore (a b_2 c)
  :name "NMSE problem 3.2.1"
  (/ (- (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))