Average Error: 33.6 → 9.8
Time: 20.7s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -4.694684309811035 \cdot 10^{+121}:\\ \;\;\;\;\frac{\left(\frac{c}{b} - \frac{b}{a}\right) \cdot 2}{2}\\ \mathbf{elif}\;b \le 4.6659701943749105 \cdot 10^{-84}:\\ \;\;\;\;\frac{\frac{1}{a} \cdot \left(\sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)} - b\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2 \cdot \frac{c}{b}}{2}\\ \end{array}\]
\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}
↓
\begin{array}{l}
\mathbf{if}\;b \le -4.694684309811035 \cdot 10^{+121}:\\
\;\;\;\;\frac{\left(\frac{c}{b} - \frac{b}{a}\right) \cdot 2}{2}\\

\mathbf{elif}\;b \le 4.6659701943749105 \cdot 10^{-84}:\\
\;\;\;\;\frac{\frac{1}{a} \cdot \left(\sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)} - b\right)}{2}\\

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

\end{array}
double f(double a, double b, double c) {
        double r3265291 = b;
        double r3265292 = -r3265291;
        double r3265293 = r3265291 * r3265291;
        double r3265294 = 4.0;
        double r3265295 = a;
        double r3265296 = r3265294 * r3265295;
        double r3265297 = c;
        double r3265298 = r3265296 * r3265297;
        double r3265299 = r3265293 - r3265298;
        double r3265300 = sqrt(r3265299);
        double r3265301 = r3265292 + r3265300;
        double r3265302 = 2.0;
        double r3265303 = r3265302 * r3265295;
        double r3265304 = r3265301 / r3265303;
        return r3265304;
}
↓
double f(double a, double b, double c) {
        double r3265305 = b;
        double r3265306 = -4.694684309811035e+121;
        bool r3265307 = r3265305 <= r3265306;
        double r3265308 = c;
        double r3265309 = r3265308 / r3265305;
        double r3265310 = a;
        double r3265311 = r3265305 / r3265310;
        double r3265312 = r3265309 - r3265311;
        double r3265313 = 2.0;
        double r3265314 = r3265312 * r3265313;
        double r3265315 = r3265314 / r3265313;
        double r3265316 = 4.6659701943749105e-84;
        bool r3265317 = r3265305 <= r3265316;
        double r3265318 = 1.0;
        double r3265319 = r3265318 / r3265310;
        double r3265320 = -4.0;
        double r3265321 = r3265310 * r3265320;
        double r3265322 = r3265305 * r3265305;
        double r3265323 = fma(r3265308, r3265321, r3265322);
        double r3265324 = sqrt(r3265323);
        double r3265325 = r3265324 - r3265305;
        double r3265326 = r3265319 * r3265325;
        double r3265327 = r3265326 / r3265313;
        double r3265328 = -2.0;
        double r3265329 = r3265328 * r3265309;
        double r3265330 = r3265329 / r3265313;
        double r3265331 = r3265317 ? r3265327 : r3265330;
        double r3265332 = r3265307 ? r3265315 : r3265331;
        return r3265332;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Target

Original33.6
Target20.4
Herbie9.8
\[\begin{array}{l} \mathbf{if}\;b \lt 0:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if b < -4.694684309811035e+121

    1. Initial program 49.8

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

    2. Simplified49.8

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

    3. Taylor expanded around -inf 2.6

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

    4. Simplified2.6

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

    if -4.694684309811035e+121 < b < 4.6659701943749105e-84

    1. Initial program 12.2

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

    2. Simplified12.2

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

    3. Taylor expanded around -inf 12.2

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

    4. Simplified12.2

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

    6. Applied div-inv12.3

      \[\leadsto \frac{\color{blue}{\left(\sqrt{\mathsf{fma}\left(c, -4 \cdot a, b \cdot b\right)} - b\right) \cdot \frac{1}{a}}}{2}\]

    if 4.6659701943749105e-84 < b

    1. Initial program 52.2

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

    2. Simplified52.2

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

    3. Taylor expanded around inf 9.3

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

  4. Final simplification9.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -4.694684309811035 \cdot 10^{+121}:\\ \;\;\;\;\frac{\left(\frac{c}{b} - \frac{b}{a}\right) \cdot 2}{2}\\ \mathbf{elif}\;b \le 4.6659701943749105 \cdot 10^{-84}:\\ \;\;\;\;\frac{\frac{1}{a} \cdot \left(\sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)} - b\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2 \cdot \frac{c}{b}}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019144 +o rules:numerics
(FPCore (a b c)
  :name "The quadratic formula (r1)"

  :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)))