Average Error: 33.0 → 10.9
Time: 21.2s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -9.348931433494438 \cdot 10^{+39}:\\ \;\;\;\;\frac{\left(\frac{c}{b} - \frac{b}{a}\right) \cdot 2}{2}\\ \mathbf{elif}\;b \le 1.3353078790738604 \cdot 10^{-121}:\\ \;\;\;\;\frac{\frac{1}{\frac{a}{\sqrt{\mathsf{fma}\left(b, b, \left(-4 \cdot a\right) \cdot c\right)} - b}}}{2}\\ \mathbf{elif}\;b \le 1.6168702840263923 \cdot 10^{-79}:\\ \;\;\;\;\frac{\frac{c}{b} \cdot -2}{2}\\ \mathbf{elif}\;b \le 1.546013236023957 \cdot 10^{-67}:\\ \;\;\;\;\frac{\frac{1}{a} \cdot \left(\sqrt{\mathsf{fma}\left(b, b, \left(-4 \cdot a\right) \cdot c\right)} - b\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{c}{b} \cdot -2}{2}\\ \end{array}\]
\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 -9.348931433494438 \cdot 10^{+39}:\\
\;\;\;\;\frac{\left(\frac{c}{b} - \frac{b}{a}\right) \cdot 2}{2}\\

\mathbf{elif}\;b \le 1.3353078790738604 \cdot 10^{-121}:\\
\;\;\;\;\frac{\frac{1}{\frac{a}{\sqrt{\mathsf{fma}\left(b, b, \left(-4 \cdot a\right) \cdot c\right)} - b}}}{2}\\

\mathbf{elif}\;b \le 1.6168702840263923 \cdot 10^{-79}:\\
\;\;\;\;\frac{\frac{c}{b} \cdot -2}{2}\\

\mathbf{elif}\;b \le 1.546013236023957 \cdot 10^{-67}:\\
\;\;\;\;\frac{\frac{1}{a} \cdot \left(\sqrt{\mathsf{fma}\left(b, b, \left(-4 \cdot a\right) \cdot c\right)} - b\right)}{2}\\

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

\end{array}
double f(double a, double b, double c) {
        double r1616370 = b;
        double r1616371 = -r1616370;
        double r1616372 = r1616370 * r1616370;
        double r1616373 = 4.0;
        double r1616374 = a;
        double r1616375 = r1616373 * r1616374;
        double r1616376 = c;
        double r1616377 = r1616375 * r1616376;
        double r1616378 = r1616372 - r1616377;
        double r1616379 = sqrt(r1616378);
        double r1616380 = r1616371 + r1616379;
        double r1616381 = 2.0;
        double r1616382 = r1616381 * r1616374;
        double r1616383 = r1616380 / r1616382;
        return r1616383;
}
↓
double f(double a, double b, double c) {
        double r1616384 = b;
        double r1616385 = -9.348931433494438e+39;
        bool r1616386 = r1616384 <= r1616385;
        double r1616387 = c;
        double r1616388 = r1616387 / r1616384;
        double r1616389 = a;
        double r1616390 = r1616384 / r1616389;
        double r1616391 = r1616388 - r1616390;
        double r1616392 = 2.0;
        double r1616393 = r1616391 * r1616392;
        double r1616394 = r1616393 / r1616392;
        double r1616395 = 1.3353078790738604e-121;
        bool r1616396 = r1616384 <= r1616395;
        double r1616397 = 1.0;
        double r1616398 = -4.0;
        double r1616399 = r1616398 * r1616389;
        double r1616400 = r1616399 * r1616387;
        double r1616401 = fma(r1616384, r1616384, r1616400);
        double r1616402 = sqrt(r1616401);
        double r1616403 = r1616402 - r1616384;
        double r1616404 = r1616389 / r1616403;
        double r1616405 = r1616397 / r1616404;
        double r1616406 = r1616405 / r1616392;
        double r1616407 = 1.6168702840263923e-79;
        bool r1616408 = r1616384 <= r1616407;
        double r1616409 = -2.0;
        double r1616410 = r1616388 * r1616409;
        double r1616411 = r1616410 / r1616392;
        double r1616412 = 1.546013236023957e-67;
        bool r1616413 = r1616384 <= r1616412;
        double r1616414 = r1616397 / r1616389;
        double r1616415 = r1616414 * r1616403;
        double r1616416 = r1616415 / r1616392;
        double r1616417 = r1616413 ? r1616416 : r1616411;
        double r1616418 = r1616408 ? r1616411 : r1616417;
        double r1616419 = r1616396 ? r1616406 : r1616418;
        double r1616420 = r1616386 ? r1616394 : r1616419;
        return r1616420;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

  1. Split input into 4 regimes

  2. if b < -9.348931433494438e+39

    1. Initial program 34.0

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

    2. Simplified34.0

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

    3. Taylor expanded around -inf 6.2

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

    4. Simplified6.2

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

    if -9.348931433494438e+39 < b < 1.3353078790738604e-121

    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. Using strategy rm

    4. Applied clear-num12.3

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

    if 1.3353078790738604e-121 < b < 1.6168702840263923e-79 or 1.546013236023957e-67 < b

    1. Initial program 50.8

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

    2. Simplified50.8

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

    3. Taylor expanded around inf 11.2

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

    if 1.6168702840263923e-79 < b < 1.546013236023957e-67

    1. Initial program 35.8

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

    2. Simplified35.8

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

    4. Applied div-inv35.9

      \[\leadsto \frac{\color{blue}{\left(\sqrt{\mathsf{fma}\left(b, b, \left(a \cdot -4\right) \cdot c\right)} - b\right) \cdot \frac{1}{a}}}{2}\]
  3. Recombined 4 regimes into one program.

  4. Final simplification10.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -9.348931433494438 \cdot 10^{+39}:\\ \;\;\;\;\frac{\left(\frac{c}{b} - \frac{b}{a}\right) \cdot 2}{2}\\ \mathbf{elif}\;b \le 1.3353078790738604 \cdot 10^{-121}:\\ \;\;\;\;\frac{\frac{1}{\frac{a}{\sqrt{\mathsf{fma}\left(b, b, \left(-4 \cdot a\right) \cdot c\right)} - b}}}{2}\\ \mathbf{elif}\;b \le 1.6168702840263923 \cdot 10^{-79}:\\ \;\;\;\;\frac{\frac{c}{b} \cdot -2}{2}\\ \mathbf{elif}\;b \le 1.546013236023957 \cdot 10^{-67}:\\ \;\;\;\;\frac{\frac{1}{a} \cdot \left(\sqrt{\mathsf{fma}\left(b, b, \left(-4 \cdot a\right) \cdot c\right)} - b\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{c}{b} \cdot -2}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019158 +o rules:numerics
(FPCore (a b c)
  :name "Quadratic roots, full range"
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4 a) c)))) (* 2 a)))