Average Error: 33.9 → 10.2
Time: 1.0m
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -3.157094219357017 \cdot 10^{+135}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \le 7.191626946559579 \cdot 10^{-55}:\\ \;\;\;\;\frac{\frac{\sqrt{\mathsf{fma}\left(\left(c \cdot a\right), -4, \left(b \cdot b\right)\right)} - b}{2}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array}\]
\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}
↓
\begin{array}{l}
\mathbf{if}\;b \le -3.157094219357017 \cdot 10^{+135}:\\
\;\;\;\;\frac{c}{b} - \frac{b}{a}\\

\mathbf{elif}\;b \le 7.191626946559579 \cdot 10^{-55}:\\
\;\;\;\;\frac{\frac{\sqrt{\mathsf{fma}\left(\left(c \cdot a\right), -4, \left(b \cdot b\right)\right)} - b}{2}}{a}\\

\mathbf{else}:\\
\;\;\;\;\frac{-c}{b}\\

\end{array}
double f(double a, double b, double c) {
        double r3628281 = b;
        double r3628282 = -r3628281;
        double r3628283 = r3628281 * r3628281;
        double r3628284 = 4.0;
        double r3628285 = a;
        double r3628286 = c;
        double r3628287 = r3628285 * r3628286;
        double r3628288 = r3628284 * r3628287;
        double r3628289 = r3628283 - r3628288;
        double r3628290 = sqrt(r3628289);
        double r3628291 = r3628282 + r3628290;
        double r3628292 = 2.0;
        double r3628293 = r3628292 * r3628285;
        double r3628294 = r3628291 / r3628293;
        return r3628294;
}
↓
double f(double a, double b, double c) {
        double r3628295 = b;
        double r3628296 = -3.157094219357017e+135;
        bool r3628297 = r3628295 <= r3628296;
        double r3628298 = c;
        double r3628299 = r3628298 / r3628295;
        double r3628300 = a;
        double r3628301 = r3628295 / r3628300;
        double r3628302 = r3628299 - r3628301;
        double r3628303 = 7.191626946559579e-55;
        bool r3628304 = r3628295 <= r3628303;
        double r3628305 = r3628298 * r3628300;
        double r3628306 = -4.0;
        double r3628307 = r3628295 * r3628295;
        double r3628308 = fma(r3628305, r3628306, r3628307);
        double r3628309 = sqrt(r3628308);
        double r3628310 = r3628309 - r3628295;
        double r3628311 = 2.0;
        double r3628312 = r3628310 / r3628311;
        double r3628313 = r3628312 / r3628300;
        double r3628314 = -r3628298;
        double r3628315 = r3628314 / r3628295;
        double r3628316 = r3628304 ? r3628313 : r3628315;
        double r3628317 = r3628297 ? r3628302 : r3628316;
        return r3628317;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Target

Original33.9
Target20.9
Herbie10.2
\[\begin{array}{l} \mathbf{if}\;b \lt 0:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if b < -3.157094219357017e+135

    1. Initial program 54.4

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]

    2. Simplified54.4

      \[\leadsto \color{blue}{\frac{\frac{\sqrt{\mathsf{fma}\left(b, b, \left(\left(a \cdot c\right) \cdot -4\right)\right)} - b}{2}}{a}}\]

    3. Taylor expanded around -inf 2.8

      \[\leadsto \color{blue}{\frac{c}{b} - \frac{b}{a}}\]

    if -3.157094219357017e+135 < b < 7.191626946559579e-55

    1. Initial program 13.6

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]

    2. Simplified13.6

      \[\leadsto \color{blue}{\frac{\frac{\sqrt{\mathsf{fma}\left(b, b, \left(\left(a \cdot c\right) \cdot -4\right)\right)} - b}{2}}{a}}\]

    3. Taylor expanded around inf 13.6

      \[\leadsto \frac{\frac{\sqrt{\color{blue}{{b}^{2} - 4 \cdot \left(a \cdot c\right)}} - b}{2}}{a}\]

    4. Simplified13.6

      \[\leadsto \frac{\frac{\sqrt{\color{blue}{\mathsf{fma}\left(\left(a \cdot c\right), -4, \left(b \cdot b\right)\right)}} - b}{2}}{a}\]

    if 7.191626946559579e-55 < b

    1. Initial program 53.4

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]

    2. Simplified53.4

      \[\leadsto \color{blue}{\frac{\frac{\sqrt{\mathsf{fma}\left(b, b, \left(\left(a \cdot c\right) \cdot -4\right)\right)} - b}{2}}{a}}\]

    3. Taylor expanded around inf 8.1

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

    4. Simplified8.1

      \[\leadsto \color{blue}{-\frac{c}{b}}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification10.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -3.157094219357017 \cdot 10^{+135}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \le 7.191626946559579 \cdot 10^{-55}:\\ \;\;\;\;\frac{\frac{\sqrt{\mathsf{fma}\left(\left(c \cdot a\right), -4, \left(b \cdot b\right)\right)} - b}{2}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2019120 +o rules:numerics
(FPCore (a b c)
  :name "quadp (p42, positive)"

  :herbie-target
  (if (< b 0) (/ (+ (- b) (sqrt (- (* b b) (* 4 (* a c))))) (* 2 a)) (/ c (* a (/ (- (- b) (sqrt (- (* b b) (* 4 (* a c))))) (* 2 a)))))

  (/ (+ (- b) (sqrt (- (* b b) (* 4 (* a c))))) (* 2 a)))