Average Error: 34.3 → 8.6
Time: 17.3s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -1.569310777886352095486911207889814773134 \cdot 10^{111}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \mathbf{elif}\;b \le -2.075943821136515538074933331988827259408 \cdot 10^{-290}:\\ \;\;\;\;\left(\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right) \cdot \frac{1}{2 \cdot a}\\ \mathbf{elif}\;b \le 1.447939350868406385811948663168665665979 \cdot 10^{78}:\\ \;\;\;\;\frac{\frac{\frac{c \cdot \left(4 \cdot a\right)}{a}}{2}}{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array}\]
\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 -1.569310777886352095486911207889814773134 \cdot 10^{111}:\\
\;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\

\mathbf{elif}\;b \le -2.075943821136515538074933331988827259408 \cdot 10^{-290}:\\
\;\;\;\;\left(\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right) \cdot \frac{1}{2 \cdot a}\\

\mathbf{elif}\;b \le 1.447939350868406385811948663168665665979 \cdot 10^{78}:\\
\;\;\;\;\frac{\frac{\frac{c \cdot \left(4 \cdot a\right)}{a}}{2}}{\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 r69339 = b;
        double r69340 = -r69339;
        double r69341 = r69339 * r69339;
        double r69342 = 4.0;
        double r69343 = a;
        double r69344 = c;
        double r69345 = r69343 * r69344;
        double r69346 = r69342 * r69345;
        double r69347 = r69341 - r69346;
        double r69348 = sqrt(r69347);
        double r69349 = r69340 + r69348;
        double r69350 = 2.0;
        double r69351 = r69350 * r69343;
        double r69352 = r69349 / r69351;
        return r69352;
}
↓
double f(double a, double b, double c) {
        double r69353 = b;
        double r69354 = -1.569310777886352e+111;
        bool r69355 = r69353 <= r69354;
        double r69356 = 1.0;
        double r69357 = c;
        double r69358 = r69357 / r69353;
        double r69359 = a;
        double r69360 = r69353 / r69359;
        double r69361 = r69358 - r69360;
        double r69362 = r69356 * r69361;
        double r69363 = -2.0759438211365155e-290;
        bool r69364 = r69353 <= r69363;
        double r69365 = -r69353;
        double r69366 = r69353 * r69353;
        double r69367 = 4.0;
        double r69368 = r69359 * r69357;
        double r69369 = r69367 * r69368;
        double r69370 = r69366 - r69369;
        double r69371 = sqrt(r69370);
        double r69372 = r69365 + r69371;
        double r69373 = 1.0;
        double r69374 = 2.0;
        double r69375 = r69374 * r69359;
        double r69376 = r69373 / r69375;
        double r69377 = r69372 * r69376;
        double r69378 = 1.4479393508684064e+78;
        bool r69379 = r69353 <= r69378;
        double r69380 = r69367 * r69359;
        double r69381 = r69357 * r69380;
        double r69382 = r69381 / r69359;
        double r69383 = r69382 / r69374;
        double r69384 = r69365 - r69371;
        double r69385 = r69383 / r69384;
        double r69386 = -1.0;
        double r69387 = r69386 * r69358;
        double r69388 = r69379 ? r69385 : r69387;
        double r69389 = r69364 ? r69377 : r69388;
        double r69390 = r69355 ? r69362 : r69389;
        return r69390;
}

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

Original34.3
Target21.1
Herbie8.6
\[\begin{array}{l} \mathbf{if}\;b \lt 0.0:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \end{array}\]

Derivation

  1. Split input into 4 regimes

  2. if b < -1.569310777886352e+111

    1. Initial program 50.4

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

    3. Applied pow150.4

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

    4. Taylor expanded around -inf 3.9

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

    5. Simplified3.9

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

    if -1.569310777886352e+111 < b < -2.0759438211365155e-290

    1. Initial program 8.5

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

    3. Applied pow18.5

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

    5. Applied div-inv8.6

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

    if -2.0759438211365155e-290 < b < 1.4479393508684064e+78

    1. Initial program 30.6

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

    3. Applied pow130.6

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

    5. Applied div-inv30.6

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

    7. Applied flip-+30.7

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

    8. Applied associate-*l/30.7

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

    9. Simplified15.8

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

    if 1.4479393508684064e+78 < b

    1. Initial program 58.7

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

    3. Applied pow158.7

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

    4. Taylor expanded around inf 3.2

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

  4. Final simplification8.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -1.569310777886352095486911207889814773134 \cdot 10^{111}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \mathbf{elif}\;b \le -2.075943821136515538074933331988827259408 \cdot 10^{-290}:\\ \;\;\;\;\left(\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right) \cdot \frac{1}{2 \cdot a}\\ \mathbf{elif}\;b \le 1.447939350868406385811948663168665665979 \cdot 10^{78}:\\ \;\;\;\;\frac{\frac{\frac{c \cdot \left(4 \cdot a\right)}{a}}{2}}{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2019303 +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)))