Average Error: 34.7 → 10.8
Time: 5.6s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -6.371698442415157100029538982618411822116 \cdot 10^{150}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \mathbf{elif}\;b \le 2.306544477380116301747543706493703838768 \cdot 10^{-129}:\\ \;\;\;\;\left(\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right) \cdot \frac{1}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array}\]
\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 -6.371698442415157100029538982618411822116 \cdot 10^{150}:\\
\;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\

\mathbf{elif}\;b \le 2.306544477380116301747543706493703838768 \cdot 10^{-129}:\\
\;\;\;\;\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 r194298 = b;
        double r194299 = -r194298;
        double r194300 = r194298 * r194298;
        double r194301 = 4.0;
        double r194302 = a;
        double r194303 = r194301 * r194302;
        double r194304 = c;
        double r194305 = r194303 * r194304;
        double r194306 = r194300 - r194305;
        double r194307 = sqrt(r194306);
        double r194308 = r194299 + r194307;
        double r194309 = 2.0;
        double r194310 = r194309 * r194302;
        double r194311 = r194308 / r194310;
        return r194311;
}
↓
double f(double a, double b, double c) {
        double r194312 = b;
        double r194313 = -6.371698442415157e+150;
        bool r194314 = r194312 <= r194313;
        double r194315 = 1.0;
        double r194316 = c;
        double r194317 = r194316 / r194312;
        double r194318 = a;
        double r194319 = r194312 / r194318;
        double r194320 = r194317 - r194319;
        double r194321 = r194315 * r194320;
        double r194322 = 2.3065444773801163e-129;
        bool r194323 = r194312 <= r194322;
        double r194324 = -r194312;
        double r194325 = r194312 * r194312;
        double r194326 = 4.0;
        double r194327 = r194326 * r194318;
        double r194328 = r194327 * r194316;
        double r194329 = r194325 - r194328;
        double r194330 = sqrt(r194329);
        double r194331 = r194324 + r194330;
        double r194332 = 1.0;
        double r194333 = 2.0;
        double r194334 = r194333 * r194318;
        double r194335 = r194332 / r194334;
        double r194336 = r194331 * r194335;
        double r194337 = -1.0;
        double r194338 = r194337 * r194317;
        double r194339 = r194323 ? r194336 : r194338;
        double r194340 = r194314 ? r194321 : r194339;
        return r194340;
}

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

Original34.7
Target21.5
Herbie10.8
\[\begin{array}{l} \mathbf{if}\;b \lt 0.0:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if b < -6.371698442415157e+150

    1. Initial program 63.0

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

    2. Taylor expanded around -inf 2.5

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

    3. Simplified2.5

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

    if -6.371698442415157e+150 < b < 2.3065444773801163e-129

    1. Initial program 11.3

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

    3. Applied div-inv11.5

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

    if 2.3065444773801163e-129 < b

    1. Initial program 51.5

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

    2. Taylor expanded around inf 12.3

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

  4. Final simplification10.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -6.371698442415157100029538982618411822116 \cdot 10^{150}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \mathbf{elif}\;b \le 2.306544477380116301747543706493703838768 \cdot 10^{-129}:\\ \;\;\;\;\left(\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right) \cdot \frac{1}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2019344 
(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)))