Average Error: 19.4 → 13.1
Time: 1.4m
Precision: 64
\[\begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le 5.199715807044694 \cdot 10^{+77}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{\mathsf{fma}\left(\left(a \cdot -4\right), c, \left(b \cdot b\right)\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{\sqrt{\mathsf{fma}\left(\left(a \cdot -4\right), c, \left(b \cdot b\right)\right)} - b} \cdot c\right) \cdot 2\\ \end{array}\\ \mathbf{elif}\;b \ge 0:\\ \;\;\;\;\frac{2 \cdot \left(\frac{c}{b} \cdot a - b\right)}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{\sqrt{\mathsf{fma}\left(\left(a \cdot -4\right), c, \left(b \cdot b\right)\right)} - b} \cdot 2\\ \end{array}\]
\begin{array}{l}
\mathbf{if}\;b \ge 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 5.199715807044694 \cdot 10^{+77}:\\
\;\;\;\;\begin{array}{l}
\mathbf{if}\;b \ge 0:\\
\;\;\;\;\frac{\left(-b\right) - \sqrt{\mathsf{fma}\left(\left(a \cdot -4\right), c, \left(b \cdot b\right)\right)}}{2 \cdot a}\\

\mathbf{else}:\\
\;\;\;\;\left(\frac{1}{\sqrt{\mathsf{fma}\left(\left(a \cdot -4\right), c, \left(b \cdot b\right)\right)} - b} \cdot c\right) \cdot 2\\

\end{array}\\

\mathbf{elif}\;b \ge 0:\\
\;\;\;\;\frac{2 \cdot \left(\frac{c}{b} \cdot a - b\right)}{2 \cdot a}\\

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

\end{array}
double f(double a, double b, double c) {
        double r2077298 = b;
        double r2077299 = 0.0;
        bool r2077300 = r2077298 >= r2077299;
        double r2077301 = -r2077298;
        double r2077302 = r2077298 * r2077298;
        double r2077303 = 4.0;
        double r2077304 = a;
        double r2077305 = r2077303 * r2077304;
        double r2077306 = c;
        double r2077307 = r2077305 * r2077306;
        double r2077308 = r2077302 - r2077307;
        double r2077309 = sqrt(r2077308);
        double r2077310 = r2077301 - r2077309;
        double r2077311 = 2.0;
        double r2077312 = r2077311 * r2077304;
        double r2077313 = r2077310 / r2077312;
        double r2077314 = r2077311 * r2077306;
        double r2077315 = r2077301 + r2077309;
        double r2077316 = r2077314 / r2077315;
        double r2077317 = r2077300 ? r2077313 : r2077316;
        return r2077317;
}
↓
double f(double a, double b, double c) {
        double r2077318 = b;
        double r2077319 = 5.199715807044694e+77;
        bool r2077320 = r2077318 <= r2077319;
        double r2077321 = 0.0;
        bool r2077322 = r2077318 >= r2077321;
        double r2077323 = -r2077318;
        double r2077324 = a;
        double r2077325 = -4.0;
        double r2077326 = r2077324 * r2077325;
        double r2077327 = c;
        double r2077328 = r2077318 * r2077318;
        double r2077329 = fma(r2077326, r2077327, r2077328);
        double r2077330 = sqrt(r2077329);
        double r2077331 = r2077323 - r2077330;
        double r2077332 = 2.0;
        double r2077333 = r2077332 * r2077324;
        double r2077334 = r2077331 / r2077333;
        double r2077335 = 1.0;
        double r2077336 = r2077330 - r2077318;
        double r2077337 = r2077335 / r2077336;
        double r2077338 = r2077337 * r2077327;
        double r2077339 = r2077338 * r2077332;
        double r2077340 = r2077322 ? r2077334 : r2077339;
        double r2077341 = r2077327 / r2077318;
        double r2077342 = r2077341 * r2077324;
        double r2077343 = r2077342 - r2077318;
        double r2077344 = r2077332 * r2077343;
        double r2077345 = r2077344 / r2077333;
        double r2077346 = r2077327 / r2077336;
        double r2077347 = r2077346 * r2077332;
        double r2077348 = r2077322 ? r2077345 : r2077347;
        double r2077349 = r2077320 ? r2077340 : r2077348;
        return r2077349;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

  1. Split input into 2 regimes

  2. if b < 5.199715807044694e+77

    1. Initial program 14.9

      \[\begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array}\]

    2. Simplified14.9

      \[\leadsto \color{blue}{\begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{\mathsf{fma}\left(\left(-4 \cdot a\right), c, \left(b \cdot b\right)\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{c}{\sqrt{\mathsf{fma}\left(\left(-4 \cdot a\right), c, \left(b \cdot b\right)\right)} - b}\\ \end{array}}\]
    3. Using strategy rm

    4. Applied div-inv14.9

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{\mathsf{fma}\left(\left(-4 \cdot a\right), c, \left(b \cdot b\right)\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\color{blue}{2 \cdot \left(c \cdot \frac{1}{\sqrt{\mathsf{fma}\left(\left(-4 \cdot a\right), c, \left(b \cdot b\right)\right)} - b}\right)}\\ \end{array}\]

    if 5.199715807044694e+77 < b

    1. Initial program 40.5

      \[\begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array}\]

    2. Simplified40.5

      \[\leadsto \color{blue}{\begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{\mathsf{fma}\left(\left(-4 \cdot a\right), c, \left(b \cdot b\right)\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{c}{\sqrt{\mathsf{fma}\left(\left(-4 \cdot a\right), c, \left(b \cdot b\right)\right)} - b}\\ \end{array}}\]

    3. Taylor expanded around inf 10.3

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{\color{blue}{2 \cdot \frac{a \cdot c}{b} - 2 \cdot b}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{c}{\sqrt{\mathsf{fma}\left(\left(-4 \cdot a\right), c, \left(b \cdot b\right)\right)} - b}\\ \end{array}\]

    4. Simplified4.9

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{\color{blue}{\left(\frac{c}{b} \cdot a - b\right) \cdot 2}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{c}{\sqrt{\mathsf{fma}\left(\left(-4 \cdot a\right), c, \left(b \cdot b\right)\right)} - b}\\ \end{array}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification13.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le 5.199715807044694 \cdot 10^{+77}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{\mathsf{fma}\left(\left(a \cdot -4\right), c, \left(b \cdot b\right)\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{\sqrt{\mathsf{fma}\left(\left(a \cdot -4\right), c, \left(b \cdot b\right)\right)} - b} \cdot c\right) \cdot 2\\ \end{array}\\ \mathbf{elif}\;b \ge 0:\\ \;\;\;\;\frac{2 \cdot \left(\frac{c}{b} \cdot a - b\right)}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{\sqrt{\mathsf{fma}\left(\left(a \cdot -4\right), c, \left(b \cdot b\right)\right)} - b} \cdot 2\\ \end{array}\]

Reproduce

herbie shell --seed 2019124 +o rules:numerics
(FPCore (a b c)
  :name "jeff quadratic root 1"
  (if (>= b 0) (/ (- (- b) (sqrt (- (* b b) (* (* 4 a) c)))) (* 2 a)) (/ (* 2 c) (+ (- b) (sqrt (- (* b b) (* (* 4 a) c)))))))