Average Error: 34.4 → 10.6
Time: 6.2s
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 r191328 = b;
        double r191329 = -r191328;
        double r191330 = r191328 * r191328;
        double r191331 = 4.0;
        double r191332 = a;
        double r191333 = r191331 * r191332;
        double r191334 = c;
        double r191335 = r191333 * r191334;
        double r191336 = r191330 - r191335;
        double r191337 = sqrt(r191336);
        double r191338 = r191329 + r191337;
        double r191339 = 2.0;
        double r191340 = r191339 * r191332;
        double r191341 = r191338 / r191340;
        return r191341;
}
↓
double f(double a, double b, double c) {
        double r191342 = b;
        double r191343 = -2.9644058459680186e+71;
        bool r191344 = r191342 <= r191343;
        double r191345 = 1.0;
        double r191346 = c;
        double r191347 = r191346 / r191342;
        double r191348 = a;
        double r191349 = r191342 / r191348;
        double r191350 = r191347 - r191349;
        double r191351 = r191345 * r191350;
        double r191352 = 1.0502924240289742e-108;
        bool r191353 = r191342 <= r191352;
        double r191354 = -r191342;
        double r191355 = r191342 * r191342;
        double r191356 = 4.0;
        double r191357 = r191356 * r191348;
        double r191358 = r191357 * r191346;
        double r191359 = r191355 - r191358;
        double r191360 = sqrt(r191359);
        double r191361 = r191354 + r191360;
        double r191362 = 1.0;
        double r191363 = 2.0;
        double r191364 = r191363 * r191348;
        double r191365 = r191362 / r191364;
        double r191366 = r191361 * r191365;
        double r191367 = -1.0;
        double r191368 = r191367 * r191347;
        double r191369 = r191353 ? r191366 : r191368;
        double r191370 = r191344 ? r191351 : r191369;
        return r191370;
}

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

Target

Original34.4
Target21.4
Herbie10.6
\[\begin{array}{l} \mathbf{if}\;b \lt 0.0:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}}\\ \end{array}\]

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 "The quadratic formula (r1)"
  :precision binary64

  :herbie-target
  (if (< b 0.0) (/ (+ (- b) (sqrt (- (* b b) (* (* 4 a) c)))) (* 2 a)) (/ c (* a (/ (- (- b) (sqrt (- (* b b) (* (* 4 a) c)))) (* 2 a)))))

  (/ (+ (- b) (sqrt (- (* b b) (* (* 4 a) c)))) (* 2 a)))