Average Error: 33.8 → 10.4
Time: 16.8s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -3.412776568687283300932456834981587297891 \cdot 10^{126}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \mathbf{elif}\;b \le 4.603517726908400645968266248286182254745 \cdot 10^{-74}:\\ \;\;\;\;\frac{1}{\frac{2 \cdot a}{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array}\]
\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 -3.412776568687283300932456834981587297891 \cdot 10^{126}:\\
\;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\

\mathbf{elif}\;b \le 4.603517726908400645968266248286182254745 \cdot 10^{-74}:\\
\;\;\;\;\frac{1}{\frac{2 \cdot a}{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}}\\

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

\end{array}
double f(double a, double b, double c) {
        double r81287 = b;
        double r81288 = -r81287;
        double r81289 = r81287 * r81287;
        double r81290 = 4.0;
        double r81291 = a;
        double r81292 = c;
        double r81293 = r81291 * r81292;
        double r81294 = r81290 * r81293;
        double r81295 = r81289 - r81294;
        double r81296 = sqrt(r81295);
        double r81297 = r81288 + r81296;
        double r81298 = 2.0;
        double r81299 = r81298 * r81291;
        double r81300 = r81297 / r81299;
        return r81300;
}
↓
double f(double a, double b, double c) {
        double r81301 = b;
        double r81302 = -3.4127765686872833e+126;
        bool r81303 = r81301 <= r81302;
        double r81304 = 1.0;
        double r81305 = c;
        double r81306 = r81305 / r81301;
        double r81307 = a;
        double r81308 = r81301 / r81307;
        double r81309 = r81306 - r81308;
        double r81310 = r81304 * r81309;
        double r81311 = 4.603517726908401e-74;
        bool r81312 = r81301 <= r81311;
        double r81313 = 1.0;
        double r81314 = 2.0;
        double r81315 = r81314 * r81307;
        double r81316 = r81301 * r81301;
        double r81317 = 4.0;
        double r81318 = r81307 * r81305;
        double r81319 = r81317 * r81318;
        double r81320 = r81316 - r81319;
        double r81321 = sqrt(r81320);
        double r81322 = r81321 - r81301;
        double r81323 = r81315 / r81322;
        double r81324 = r81313 / r81323;
        double r81325 = -1.0;
        double r81326 = r81325 * r81306;
        double r81327 = r81312 ? r81324 : r81326;
        double r81328 = r81303 ? r81310 : r81327;
        return r81328;
}

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

Original33.8
Target20.4
Herbie10.4
\[\begin{array}{l} \mathbf{if}\;b \lt 0.0:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if b < -3.4127765686872833e+126

    1. Initial program 53.6

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

    2. Taylor expanded around -inf 3.2

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

    3. Simplified3.2

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

    if -3.4127765686872833e+126 < b < 4.603517726908401e-74

    1. Initial program 13.2

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

    3. Applied clear-num13.3

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

    4. Simplified13.3

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

    if 4.603517726908401e-74 < b

    1. Initial program 53.2

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

    2. Taylor expanded around inf 9.1

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

  4. Final simplification10.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -3.412776568687283300932456834981587297891 \cdot 10^{126}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \mathbf{elif}\;b \le 4.603517726908400645968266248286182254745 \cdot 10^{-74}:\\ \;\;\;\;\frac{1}{\frac{2 \cdot a}{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2019208 +o rules:numerics
(FPCore (a b c)
  :name "quadp (p42, positive)"
  :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)))