Average Error: 34.0 → 10.2
Time: 22.9s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -5.836476522143192884813309842834615858794 \cdot 10^{134}:\\ \;\;\;\;\frac{\mathsf{fma}\left(2, \frac{c}{b}, \frac{b \cdot -2}{a}\right)}{2}\\ \mathbf{elif}\;b \le 8.272706925888016273629504343982488312855 \cdot 10^{-43}:\\ \;\;\;\;\frac{\frac{1}{a} \cdot \left(\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{c}{b} \cdot -2}{2}\\ \end{array}\]
\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 -5.836476522143192884813309842834615858794 \cdot 10^{134}:\\
\;\;\;\;\frac{\mathsf{fma}\left(2, \frac{c}{b}, \frac{b \cdot -2}{a}\right)}{2}\\

\mathbf{elif}\;b \le 8.272706925888016273629504343982488312855 \cdot 10^{-43}:\\
\;\;\;\;\frac{\frac{1}{a} \cdot \left(\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b\right)}{2}\\

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

\end{array}
double f(double a, double b, double c) {
        double r3227334 = b;
        double r3227335 = -r3227334;
        double r3227336 = r3227334 * r3227334;
        double r3227337 = 4.0;
        double r3227338 = a;
        double r3227339 = c;
        double r3227340 = r3227338 * r3227339;
        double r3227341 = r3227337 * r3227340;
        double r3227342 = r3227336 - r3227341;
        double r3227343 = sqrt(r3227342);
        double r3227344 = r3227335 + r3227343;
        double r3227345 = 2.0;
        double r3227346 = r3227345 * r3227338;
        double r3227347 = r3227344 / r3227346;
        return r3227347;
}
↓
double f(double a, double b, double c) {
        double r3227348 = b;
        double r3227349 = -5.836476522143193e+134;
        bool r3227350 = r3227348 <= r3227349;
        double r3227351 = 2.0;
        double r3227352 = c;
        double r3227353 = r3227352 / r3227348;
        double r3227354 = -2.0;
        double r3227355 = r3227348 * r3227354;
        double r3227356 = a;
        double r3227357 = r3227355 / r3227356;
        double r3227358 = fma(r3227351, r3227353, r3227357);
        double r3227359 = r3227358 / r3227351;
        double r3227360 = 8.272706925888016e-43;
        bool r3227361 = r3227348 <= r3227360;
        double r3227362 = 1.0;
        double r3227363 = r3227362 / r3227356;
        double r3227364 = r3227348 * r3227348;
        double r3227365 = 4.0;
        double r3227366 = r3227365 * r3227356;
        double r3227367 = r3227366 * r3227352;
        double r3227368 = r3227364 - r3227367;
        double r3227369 = sqrt(r3227368);
        double r3227370 = r3227369 - r3227348;
        double r3227371 = r3227363 * r3227370;
        double r3227372 = r3227371 / r3227351;
        double r3227373 = -2.0;
        double r3227374 = r3227353 * r3227373;
        double r3227375 = r3227374 / r3227351;
        double r3227376 = r3227361 ? r3227372 : r3227375;
        double r3227377 = r3227350 ? r3227359 : r3227376;
        return r3227377;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Target

Original34.0
Target21.1
Herbie10.2
\[\begin{array}{l} \mathbf{if}\;b \lt 0.0:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if b < -5.836476522143193e+134

    1. Initial program 56.2

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

    2. Simplified56.2

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

    4. Applied div-sub56.2

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

    5. Taylor expanded around -inf 2.4

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

    6. Simplified2.5

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

    if -5.836476522143193e+134 < b < 8.272706925888016e-43

    1. Initial program 14.0

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

    2. Simplified14.0

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

    4. Applied div-inv14.1

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

    if 8.272706925888016e-43 < b

    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

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

    3. Taylor expanded around inf 7.5

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

  4. Final simplification10.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -5.836476522143192884813309842834615858794 \cdot 10^{134}:\\ \;\;\;\;\frac{\mathsf{fma}\left(2, \frac{c}{b}, \frac{b \cdot -2}{a}\right)}{2}\\ \mathbf{elif}\;b \le 8.272706925888016273629504343982488312855 \cdot 10^{-43}:\\ \;\;\;\;\frac{\frac{1}{a} \cdot \left(\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{c}{b} \cdot -2}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019169 +o rules:numerics
(FPCore (a b c)
  :name "quadp (p42, positive)"

  :herbie-target
  (if (< b 0.0) (/ (+ (- b) (sqrt (- (* b b) (* 4.0 (* a c))))) (* 2.0 a)) (/ c (* a (/ (- (- b) (sqrt (- (* b b) (* 4.0 (* a c))))) (* 2.0 a)))))

  (/ (+ (- b) (sqrt (- (* b b) (* 4.0 (* a c))))) (* 2.0 a)))