Average Error: 20.0 → 7.7
Time: 40.0s
Precision: 64
\[\begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -6.918676892540579659424874054568244375692 \cdot 10^{70}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{\left(-b\right) + \left(\frac{\left(c \cdot 2\right) \cdot a}{b} - b\right)}\\ \end{array}\\ \mathbf{elif}\;b \le 1.94479993562256152078378503839473835686 \cdot 10^{143}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{\left(-b\right) + \sqrt{\sqrt{\sqrt[3]{b \cdot b - \left(a \cdot 4\right) \cdot c}} \cdot \sqrt{\sqrt[3]{b \cdot b - \left(a \cdot 4\right) \cdot c} \cdot \sqrt[3]{b \cdot b - \left(a \cdot 4\right) \cdot c}}} \cdot \sqrt{\sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c}}}\\ \end{array}\\ \mathbf{elif}\;b \ge 0.0:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{a}{b} \cdot c, 2, -2 \cdot b\right)}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{\sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c} + \left(-b\right)}\\ \end{array}\]
\begin{array}{l}
\mathbf{if}\;b \ge 0.0:\\
\;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\

\mathbf{else}:\\
\;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\

\end{array}
↓
\begin{array}{l}
\mathbf{if}\;b \le -6.918676892540579659424874054568244375692 \cdot 10^{70}:\\
\;\;\;\;\begin{array}{l}
\mathbf{if}\;b \ge 0.0:\\
\;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c}}{2 \cdot a}\\

\mathbf{else}:\\
\;\;\;\;\frac{c \cdot 2}{\left(-b\right) + \left(\frac{\left(c \cdot 2\right) \cdot a}{b} - b\right)}\\

\end{array}\\

\mathbf{elif}\;b \le 1.94479993562256152078378503839473835686 \cdot 10^{143}:\\
\;\;\;\;\begin{array}{l}
\mathbf{if}\;b \ge 0.0:\\
\;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c}}{2 \cdot a}\\

\mathbf{else}:\\
\;\;\;\;\frac{c \cdot 2}{\left(-b\right) + \sqrt{\sqrt{\sqrt[3]{b \cdot b - \left(a \cdot 4\right) \cdot c}} \cdot \sqrt{\sqrt[3]{b \cdot b - \left(a \cdot 4\right) \cdot c} \cdot \sqrt[3]{b \cdot b - \left(a \cdot 4\right) \cdot c}}} \cdot \sqrt{\sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c}}}\\

\end{array}\\

\mathbf{elif}\;b \ge 0.0:\\
\;\;\;\;\frac{\mathsf{fma}\left(\frac{a}{b} \cdot c, 2, -2 \cdot b\right)}{2 \cdot a}\\

\mathbf{else}:\\
\;\;\;\;\frac{c \cdot 2}{\sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c} + \left(-b\right)}\\

\end{array}
double f(double a, double b, double c) {
        double r1259298 = b;
        double r1259299 = 0.0;
        bool r1259300 = r1259298 >= r1259299;
        double r1259301 = -r1259298;
        double r1259302 = r1259298 * r1259298;
        double r1259303 = 4.0;
        double r1259304 = a;
        double r1259305 = r1259303 * r1259304;
        double r1259306 = c;
        double r1259307 = r1259305 * r1259306;
        double r1259308 = r1259302 - r1259307;
        double r1259309 = sqrt(r1259308);
        double r1259310 = r1259301 - r1259309;
        double r1259311 = 2.0;
        double r1259312 = r1259311 * r1259304;
        double r1259313 = r1259310 / r1259312;
        double r1259314 = r1259311 * r1259306;
        double r1259315 = r1259301 + r1259309;
        double r1259316 = r1259314 / r1259315;
        double r1259317 = r1259300 ? r1259313 : r1259316;
        return r1259317;
}
↓
double f(double a, double b, double c) {
        double r1259318 = b;
        double r1259319 = -6.91867689254058e+70;
        bool r1259320 = r1259318 <= r1259319;
        double r1259321 = 0.0;
        bool r1259322 = r1259318 >= r1259321;
        double r1259323 = -r1259318;
        double r1259324 = r1259318 * r1259318;
        double r1259325 = a;
        double r1259326 = 4.0;
        double r1259327 = r1259325 * r1259326;
        double r1259328 = c;
        double r1259329 = r1259327 * r1259328;
        double r1259330 = r1259324 - r1259329;
        double r1259331 = sqrt(r1259330);
        double r1259332 = r1259323 - r1259331;
        double r1259333 = 2.0;
        double r1259334 = r1259333 * r1259325;
        double r1259335 = r1259332 / r1259334;
        double r1259336 = r1259328 * r1259333;
        double r1259337 = r1259336 * r1259325;
        double r1259338 = r1259337 / r1259318;
        double r1259339 = r1259338 - r1259318;
        double r1259340 = r1259323 + r1259339;
        double r1259341 = r1259336 / r1259340;
        double r1259342 = r1259322 ? r1259335 : r1259341;
        double r1259343 = 1.9447999356225615e+143;
        bool r1259344 = r1259318 <= r1259343;
        double r1259345 = cbrt(r1259330);
        double r1259346 = sqrt(r1259345);
        double r1259347 = r1259345 * r1259345;
        double r1259348 = sqrt(r1259347);
        double r1259349 = r1259346 * r1259348;
        double r1259350 = sqrt(r1259349);
        double r1259351 = sqrt(r1259331);
        double r1259352 = r1259350 * r1259351;
        double r1259353 = r1259323 + r1259352;
        double r1259354 = r1259336 / r1259353;
        double r1259355 = r1259322 ? r1259335 : r1259354;
        double r1259356 = r1259325 / r1259318;
        double r1259357 = r1259356 * r1259328;
        double r1259358 = -2.0;
        double r1259359 = r1259358 * r1259318;
        double r1259360 = fma(r1259357, r1259333, r1259359);
        double r1259361 = r1259360 / r1259334;
        double r1259362 = r1259331 + r1259323;
        double r1259363 = r1259336 / r1259362;
        double r1259364 = r1259322 ? r1259361 : r1259363;
        double r1259365 = r1259344 ? r1259355 : r1259364;
        double r1259366 = r1259320 ? r1259342 : r1259365;
        return r1259366;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

  1. Split input into 3 regimes

  2. if b < -6.91867689254058e+70

    1. Initial program 27.4

      \[\begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array}\]

    2. Taylor expanded around -inf 7.0

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\color{blue}{\left(-b\right) + \left(2 \cdot \frac{a \cdot c}{b} - b\right)}}\\ \end{array}\]

    3. Simplified7.1

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\color{blue}{\left(-b\right) + \left(\frac{a \cdot \left(2 \cdot c\right)}{b} - b\right)}}\\ \end{array}\]

    if -6.91867689254058e+70 < b < 1.9447999356225615e+143

    1. Initial program 8.9

      \[\begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array}\]
    2. Using strategy rm

    3. Applied add-sqr-sqrt8.9

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\color{blue}{\left(-b\right)} + \sqrt{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} \cdot \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}\\ \end{array}\]

    4. Applied sqrt-prod9.0

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\color{blue}{\left(-b\right) + \sqrt{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}} \cdot \sqrt{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}}\\ \end{array}\]
    5. Using strategy rm

    6. Applied +-commutative9.0

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\color{blue}{\frac{2 \cdot c}{\sqrt{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}} \cdot \sqrt{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}} + \left(-b\right)}}\\ \end{array}\]
    7. Using strategy rm

    8. Applied add-cube-cbrt9.0

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\sqrt{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}} \cdot \sqrt{\sqrt{\left(\sqrt[3]{b \cdot b - \left(4 \cdot a\right) \cdot c} \cdot \sqrt[3]{b \cdot b - \left(4 \cdot a\right) \cdot c}\right) \cdot \sqrt[3]{b \cdot b - \left(4 \cdot a\right) \cdot c}}} + \left(-b\right)}\\ \end{array}\]

    9. Applied sqrt-prod9.0

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\sqrt{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}} \cdot \sqrt{\sqrt{\sqrt[3]{b \cdot b - \left(4 \cdot a\right) \cdot c} \cdot \sqrt[3]{b \cdot b - \left(4 \cdot a\right) \cdot c}} \cdot \sqrt{\sqrt[3]{b \cdot b - \left(4 \cdot a\right) \cdot c}}} + \left(-b\right)}\\ \end{array}\]

    if 1.9447999356225615e+143 < b

    1. Initial program 59.9

      \[\begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array}\]

    2. Taylor expanded around inf 9.8

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{\color{blue}{2 \cdot \frac{a \cdot c}{b} - 2 \cdot b}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array}\]

    3. Simplified2.1

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{\color{blue}{\mathsf{fma}\left(\frac{a}{b} \cdot c, 2, -2 \cdot b\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification7.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -6.918676892540579659424874054568244375692 \cdot 10^{70}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{\left(-b\right) + \left(\frac{\left(c \cdot 2\right) \cdot a}{b} - b\right)}\\ \end{array}\\ \mathbf{elif}\;b \le 1.94479993562256152078378503839473835686 \cdot 10^{143}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{\left(-b\right) + \sqrt{\sqrt{\sqrt[3]{b \cdot b - \left(a \cdot 4\right) \cdot c}} \cdot \sqrt{\sqrt[3]{b \cdot b - \left(a \cdot 4\right) \cdot c} \cdot \sqrt[3]{b \cdot b - \left(a \cdot 4\right) \cdot c}}} \cdot \sqrt{\sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c}}}\\ \end{array}\\ \mathbf{elif}\;b \ge 0.0:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{a}{b} \cdot c, 2, -2 \cdot b\right)}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{\sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c} + \left(-b\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019200 +o rules:numerics
(FPCore (a b c)
  :name "jeff quadratic root 1"
  (if (>= b 0.0) (/ (- (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)) (/ (* 2.0 c) (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))))))