Average Error: 34.3 → 10.3
Time: 21.8s
Precision: 64
\[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b_2 \le -1.006239684017339546564770304051967174461 \cdot 10^{118}:\\ \;\;\;\;\mathsf{fma}\left(\frac{c}{b_2}, \frac{1}{2}, -2 \cdot \frac{b_2}{a}\right)\\ \mathbf{elif}\;b_2 \le 1.892098135471955771557857083920836890719 \cdot 10^{-53}:\\ \;\;\;\;\frac{\left(-b_2\right) + \sqrt{\mathsf{fma}\left(b_2, b_2, \left(-a\right) \cdot c\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \end{array}\]
\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}
↓
\begin{array}{l}
\mathbf{if}\;b_2 \le -1.006239684017339546564770304051967174461 \cdot 10^{118}:\\
\;\;\;\;\mathsf{fma}\left(\frac{c}{b_2}, \frac{1}{2}, -2 \cdot \frac{b_2}{a}\right)\\

\mathbf{elif}\;b_2 \le 1.892098135471955771557857083920836890719 \cdot 10^{-53}:\\
\;\;\;\;\frac{\left(-b_2\right) + \sqrt{\mathsf{fma}\left(b_2, b_2, \left(-a\right) \cdot c\right)}}{a}\\

\mathbf{else}:\\
\;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\

\end{array}
double f(double a, double b_2, double c) {
        double r25239 = b_2;
        double r25240 = -r25239;
        double r25241 = r25239 * r25239;
        double r25242 = a;
        double r25243 = c;
        double r25244 = r25242 * r25243;
        double r25245 = r25241 - r25244;
        double r25246 = sqrt(r25245);
        double r25247 = r25240 + r25246;
        double r25248 = r25247 / r25242;
        return r25248;
}
↓
double f(double a, double b_2, double c) {
        double r25249 = b_2;
        double r25250 = -1.0062396840173395e+118;
        bool r25251 = r25249 <= r25250;
        double r25252 = c;
        double r25253 = r25252 / r25249;
        double r25254 = 0.5;
        double r25255 = -2.0;
        double r25256 = a;
        double r25257 = r25249 / r25256;
        double r25258 = r25255 * r25257;
        double r25259 = fma(r25253, r25254, r25258);
        double r25260 = 1.8920981354719558e-53;
        bool r25261 = r25249 <= r25260;
        double r25262 = -r25249;
        double r25263 = -r25256;
        double r25264 = r25263 * r25252;
        double r25265 = fma(r25249, r25249, r25264);
        double r25266 = sqrt(r25265);
        double r25267 = r25262 + r25266;
        double r25268 = r25267 / r25256;
        double r25269 = -0.5;
        double r25270 = r25269 * r25253;
        double r25271 = r25261 ? r25268 : r25270;
        double r25272 = r25251 ? r25259 : r25271;
        return r25272;
}

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 < -1.0062396840173395e+118

    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 2.8

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

    3. Simplified2.8

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{c}{b_2}, \frac{1}{2}, -2 \cdot \frac{b_2}{a}\right)}\]

    if -1.0062396840173395e+118 < b_2 < 1.8920981354719558e-53

    1. Initial program 13.9

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

    3. Applied fma-neg13.9

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

    4. Simplified13.9

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

    if 1.8920981354719558e-53 < b_2

    1. Initial program 54.0

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

    2. Taylor expanded around inf 8.4

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

  4. Final simplification10.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -1.006239684017339546564770304051967174461 \cdot 10^{118}:\\ \;\;\;\;\mathsf{fma}\left(\frac{c}{b_2}, \frac{1}{2}, -2 \cdot \frac{b_2}{a}\right)\\ \mathbf{elif}\;b_2 \le 1.892098135471955771557857083920836890719 \cdot 10^{-53}:\\ \;\;\;\;\frac{\left(-b_2\right) + \sqrt{\mathsf{fma}\left(b_2, b_2, \left(-a\right) \cdot c\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019306 +o rules:numerics
(FPCore (a b_2 c)
  :name "quad2p (problem 3.2.1, positive)"
  :precision binary64
  (/ (+ (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))