Average Error: 32.9 → 10.5
Time: 16.8s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -1.7512236628315378 \cdot 10^{+131}:\\ \;\;\;\;\frac{\frac{\left(\frac{a}{\frac{b}{c}} - b\right) \cdot 2}{a}}{2}\\ \mathbf{elif}\;b \le 1.489031291672483 \cdot 10^{-98}:\\ \;\;\;\;\frac{\frac{\sqrt{\mathsf{fma}\left(a, -4 \cdot c, b \cdot b\right)} - b}{a}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\frac{b}{c} \cdot \frac{-1}{2}}}{2}\\ \end{array}\]
\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 -1.7512236628315378 \cdot 10^{+131}:\\
\;\;\;\;\frac{\frac{\left(\frac{a}{\frac{b}{c}} - b\right) \cdot 2}{a}}{2}\\

\mathbf{elif}\;b \le 1.489031291672483 \cdot 10^{-98}:\\
\;\;\;\;\frac{\frac{\sqrt{\mathsf{fma}\left(a, -4 \cdot c, b \cdot b\right)} - b}{a}}{2}\\

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

\end{array}
double f(double a, double b, double c) {
        double r829394 = b;
        double r829395 = -r829394;
        double r829396 = r829394 * r829394;
        double r829397 = 4.0;
        double r829398 = a;
        double r829399 = r829397 * r829398;
        double r829400 = c;
        double r829401 = r829399 * r829400;
        double r829402 = r829396 - r829401;
        double r829403 = sqrt(r829402);
        double r829404 = r829395 + r829403;
        double r829405 = 2.0;
        double r829406 = r829405 * r829398;
        double r829407 = r829404 / r829406;
        return r829407;
}
↓
double f(double a, double b, double c) {
        double r829408 = b;
        double r829409 = -1.7512236628315378e+131;
        bool r829410 = r829408 <= r829409;
        double r829411 = a;
        double r829412 = c;
        double r829413 = r829408 / r829412;
        double r829414 = r829411 / r829413;
        double r829415 = r829414 - r829408;
        double r829416 = 2.0;
        double r829417 = r829415 * r829416;
        double r829418 = r829417 / r829411;
        double r829419 = r829418 / r829416;
        double r829420 = 1.489031291672483e-98;
        bool r829421 = r829408 <= r829420;
        double r829422 = -4.0;
        double r829423 = r829422 * r829412;
        double r829424 = r829408 * r829408;
        double r829425 = fma(r829411, r829423, r829424);
        double r829426 = sqrt(r829425);
        double r829427 = r829426 - r829408;
        double r829428 = r829427 / r829411;
        double r829429 = r829428 / r829416;
        double r829430 = 1.0;
        double r829431 = -0.5;
        double r829432 = r829413 * r829431;
        double r829433 = r829430 / r829432;
        double r829434 = r829433 / r829416;
        double r829435 = r829421 ? r829429 : r829434;
        double r829436 = r829410 ? r829419 : r829435;
        return r829436;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

  1. Split input into 3 regimes

  2. if b < -1.7512236628315378e+131

    1. Initial program 51.5

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

    2. Simplified51.5

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

    3. Taylor expanded around -inf 9.8

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

    4. Simplified3.0

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

    if -1.7512236628315378e+131 < b < 1.489031291672483e-98

    1. Initial program 11.5

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

    2. Simplified11.6

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

    3. Taylor expanded around 0 11.6

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

    4. Simplified11.6

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

    6. Applied *-un-lft-identity11.6

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

    if 1.489031291672483e-98 < b

    1. Initial program 51.5

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

    2. Simplified51.5

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

    3. Taylor expanded around 0 51.5

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

    4. Simplified51.5

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

    6. Applied clear-num51.5

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

    7. Taylor expanded around 0 11.4

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

  4. Final simplification10.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -1.7512236628315378 \cdot 10^{+131}:\\ \;\;\;\;\frac{\frac{\left(\frac{a}{\frac{b}{c}} - b\right) \cdot 2}{a}}{2}\\ \mathbf{elif}\;b \le 1.489031291672483 \cdot 10^{-98}:\\ \;\;\;\;\frac{\frac{\sqrt{\mathsf{fma}\left(a, -4 \cdot c, b \cdot b\right)} - b}{a}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\frac{b}{c} \cdot \frac{-1}{2}}}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019154 +o rules:numerics
(FPCore (a b c)
  :name "Quadratic roots, full range"
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4 a) c)))) (* 2 a)))