Average Error: 34.4 → 10.6
Time: 6.0s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -2.9644058459680186 \cdot 10^{71}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \mathbf{elif}\;b \le 1.05029242402897421 \cdot 10^{-108}:\\ \;\;\;\;\left(\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right) \cdot \frac{1}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array}\]
\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 -2.9644058459680186 \cdot 10^{71}:\\
\;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\

\mathbf{elif}\;b \le 1.05029242402897421 \cdot 10^{-108}:\\
\;\;\;\;\left(\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right) \cdot \frac{1}{2 \cdot a}\\

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

\end{array}
double f(double a, double b, double c) {
        double r52308 = b;
        double r52309 = -r52308;
        double r52310 = r52308 * r52308;
        double r52311 = 4.0;
        double r52312 = a;
        double r52313 = r52311 * r52312;
        double r52314 = c;
        double r52315 = r52313 * r52314;
        double r52316 = r52310 - r52315;
        double r52317 = sqrt(r52316);
        double r52318 = r52309 + r52317;
        double r52319 = 2.0;
        double r52320 = r52319 * r52312;
        double r52321 = r52318 / r52320;
        return r52321;
}
↓
double f(double a, double b, double c) {
        double r52322 = b;
        double r52323 = -2.9644058459680186e+71;
        bool r52324 = r52322 <= r52323;
        double r52325 = 1.0;
        double r52326 = c;
        double r52327 = r52326 / r52322;
        double r52328 = a;
        double r52329 = r52322 / r52328;
        double r52330 = r52327 - r52329;
        double r52331 = r52325 * r52330;
        double r52332 = 1.0502924240289742e-108;
        bool r52333 = r52322 <= r52332;
        double r52334 = -r52322;
        double r52335 = r52322 * r52322;
        double r52336 = 4.0;
        double r52337 = r52336 * r52328;
        double r52338 = r52337 * r52326;
        double r52339 = r52335 - r52338;
        double r52340 = sqrt(r52339);
        double r52341 = r52334 + r52340;
        double r52342 = 1.0;
        double r52343 = 2.0;
        double r52344 = r52343 * r52328;
        double r52345 = r52342 / r52344;
        double r52346 = r52341 * r52345;
        double r52347 = -1.0;
        double r52348 = r52347 * r52327;
        double r52349 = r52333 ? r52346 : r52348;
        double r52350 = r52324 ? r52331 : r52349;
        return r52350;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 3 regimes

  2. if b < -2.9644058459680186e+71

    1. Initial program 42.3

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

    2. Taylor expanded around -inf 4.7

      \[\leadsto \color{blue}{1 \cdot \frac{c}{b} - 1 \cdot \frac{b}{a}}\]

    3. Simplified4.7

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

    if -2.9644058459680186e+71 < b < 1.0502924240289742e-108

    1. Initial program 13.1

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

    3. Applied div-inv13.2

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

    if 1.0502924240289742e-108 < b

    1. Initial program 51.6

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

    2. Taylor expanded around inf 10.7

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

  4. Final simplification10.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -2.9644058459680186 \cdot 10^{71}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \mathbf{elif}\;b \le 1.05029242402897421 \cdot 10^{-108}:\\ \;\;\;\;\left(\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right) \cdot \frac{1}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2020089 +o rules:numerics
(FPCore (a b c)
  :name "Quadratic roots, full range"
  :precision binary64
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4 a) c)))) (* 2 a)))