Average Error: 34.0 → 7.1
Time: 5.7s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -3.71711085460076329 \cdot 10^{118}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \mathbf{elif}\;b \le -2.9300475349170912 \cdot 10^{-278}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \mathbf{elif}\;b \le 3461964491124549:\\ \;\;\;\;1 \cdot \frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array}\]
\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 -3.71711085460076329 \cdot 10^{118}:\\
\;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\

\mathbf{elif}\;b \le -2.9300475349170912 \cdot 10^{-278}:\\
\;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\

\mathbf{elif}\;b \le 3461964491124549:\\
\;\;\;\;1 \cdot \frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}\\

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

\end{array}
double f(double a, double b, double c) {
        double r79347 = b;
        double r79348 = -r79347;
        double r79349 = r79347 * r79347;
        double r79350 = 4.0;
        double r79351 = a;
        double r79352 = c;
        double r79353 = r79351 * r79352;
        double r79354 = r79350 * r79353;
        double r79355 = r79349 - r79354;
        double r79356 = sqrt(r79355);
        double r79357 = r79348 + r79356;
        double r79358 = 2.0;
        double r79359 = r79358 * r79351;
        double r79360 = r79357 / r79359;
        return r79360;
}
↓
double f(double a, double b, double c) {
        double r79361 = b;
        double r79362 = -3.7171108546007633e+118;
        bool r79363 = r79361 <= r79362;
        double r79364 = 1.0;
        double r79365 = c;
        double r79366 = r79365 / r79361;
        double r79367 = a;
        double r79368 = r79361 / r79367;
        double r79369 = r79366 - r79368;
        double r79370 = r79364 * r79369;
        double r79371 = -2.930047534917091e-278;
        bool r79372 = r79361 <= r79371;
        double r79373 = -r79361;
        double r79374 = r79361 * r79361;
        double r79375 = 4.0;
        double r79376 = r79367 * r79365;
        double r79377 = r79375 * r79376;
        double r79378 = r79374 - r79377;
        double r79379 = sqrt(r79378);
        double r79380 = r79373 + r79379;
        double r79381 = 2.0;
        double r79382 = r79381 * r79367;
        double r79383 = r79380 / r79382;
        double r79384 = 3461964491124549.0;
        bool r79385 = r79361 <= r79384;
        double r79386 = 1.0;
        double r79387 = r79381 * r79365;
        double r79388 = r79373 - r79379;
        double r79389 = r79387 / r79388;
        double r79390 = r79386 * r79389;
        double r79391 = -1.0;
        double r79392 = r79391 * r79366;
        double r79393 = r79385 ? r79390 : r79392;
        double r79394 = r79372 ? r79383 : r79393;
        double r79395 = r79363 ? r79370 : r79394;
        return r79395;
}

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

Original34.0
Target20.8
Herbie7.1
\[\begin{array}{l} \mathbf{if}\;b \lt 0.0:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \end{array}\]

Derivation

  1. Split input into 4 regimes

  2. if b < -3.7171108546007633e+118

    1. Initial program 52.0

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

    2. Taylor expanded around -inf 2.9

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

    3. Simplified2.9

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

    if -3.7171108546007633e+118 < b < -2.930047534917091e-278

    1. Initial program 8.4

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

    if -2.930047534917091e-278 < b < 3461964491124549.0

    1. Initial program 26.5

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

    3. Applied flip-+26.6

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

    4. Simplified16.3

      \[\leadsto \frac{\frac{\color{blue}{0 + 4 \cdot \left(a \cdot c\right)}}{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}}{2 \cdot a}\]
    5. Using strategy rm

    6. Applied div-inv16.4

      \[\leadsto \color{blue}{\frac{0 + 4 \cdot \left(a \cdot c\right)}{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}} \cdot \frac{1}{2 \cdot a}}\]
    7. Using strategy rm

    8. Applied *-un-lft-identity16.4

      \[\leadsto \frac{0 + 4 \cdot \left(a \cdot c\right)}{\color{blue}{1 \cdot \left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)}} \cdot \frac{1}{2 \cdot a}\]

    9. Applied *-un-lft-identity16.4

      \[\leadsto \frac{\color{blue}{1 \cdot \left(0 + 4 \cdot \left(a \cdot c\right)\right)}}{1 \cdot \left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)} \cdot \frac{1}{2 \cdot a}\]

    10. Applied times-frac16.4

      \[\leadsto \color{blue}{\left(\frac{1}{1} \cdot \frac{0 + 4 \cdot \left(a \cdot c\right)}{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}\right)} \cdot \frac{1}{2 \cdot a}\]

    11. Applied associate-*l*16.4

      \[\leadsto \color{blue}{\frac{1}{1} \cdot \left(\frac{0 + 4 \cdot \left(a \cdot c\right)}{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}} \cdot \frac{1}{2 \cdot a}\right)}\]

    12. Simplified16.2

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

    13. Taylor expanded around 0 10.0

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

    if 3461964491124549.0 < b

    1. Initial program 55.9

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

    2. Taylor expanded around inf 5.5

      \[\leadsto \color{blue}{-1 \cdot \frac{c}{b}}\]
  3. Recombined 4 regimes into one program.

  4. Final simplification7.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -3.71711085460076329 \cdot 10^{118}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \mathbf{elif}\;b \le -2.9300475349170912 \cdot 10^{-278}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \mathbf{elif}\;b \le 3461964491124549:\\ \;\;\;\;1 \cdot \frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2020027 +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)))