Average Error: 34.3 → 6.6
Time: 15.6s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -1.569310777886352095486911207889814773134 \cdot 10^{111}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \mathbf{elif}\;b \le -3.353411485092052314829631610929360473517 \cdot 10^{-283}:\\ \;\;\;\;\left(\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right) \cdot \frac{1}{2 \cdot a}\\ \mathbf{elif}\;b \le 1.608284405260305560018146283533738265178 \cdot 10^{78}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array}\]
\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 -1.569310777886352095486911207889814773134 \cdot 10^{111}:\\
\;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\

\mathbf{elif}\;b \le -3.353411485092052314829631610929360473517 \cdot 10^{-283}:\\
\;\;\;\;\left(\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right) \cdot \frac{1}{2 \cdot a}\\

\mathbf{elif}\;b \le 1.608284405260305560018146283533738265178 \cdot 10^{78}:\\
\;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\

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

\end{array}
double f(double a, double b, double c) {
        double r56222 = b;
        double r56223 = -r56222;
        double r56224 = r56222 * r56222;
        double r56225 = 4.0;
        double r56226 = a;
        double r56227 = r56225 * r56226;
        double r56228 = c;
        double r56229 = r56227 * r56228;
        double r56230 = r56224 - r56229;
        double r56231 = sqrt(r56230);
        double r56232 = r56223 + r56231;
        double r56233 = 2.0;
        double r56234 = r56233 * r56226;
        double r56235 = r56232 / r56234;
        return r56235;
}
↓
double f(double a, double b, double c) {
        double r56236 = b;
        double r56237 = -1.569310777886352e+111;
        bool r56238 = r56236 <= r56237;
        double r56239 = 1.0;
        double r56240 = c;
        double r56241 = r56240 / r56236;
        double r56242 = a;
        double r56243 = r56236 / r56242;
        double r56244 = r56241 - r56243;
        double r56245 = r56239 * r56244;
        double r56246 = -3.3534114850920523e-283;
        bool r56247 = r56236 <= r56246;
        double r56248 = -r56236;
        double r56249 = r56236 * r56236;
        double r56250 = 4.0;
        double r56251 = r56250 * r56242;
        double r56252 = r56251 * r56240;
        double r56253 = r56249 - r56252;
        double r56254 = sqrt(r56253);
        double r56255 = r56248 + r56254;
        double r56256 = 1.0;
        double r56257 = 2.0;
        double r56258 = r56257 * r56242;
        double r56259 = r56256 / r56258;
        double r56260 = r56255 * r56259;
        double r56261 = 1.6082844052603056e+78;
        bool r56262 = r56236 <= r56261;
        double r56263 = r56257 * r56240;
        double r56264 = r56248 - r56254;
        double r56265 = r56263 / r56264;
        double r56266 = -1.0;
        double r56267 = r56266 * r56241;
        double r56268 = r56262 ? r56265 : r56267;
        double r56269 = r56247 ? r56260 : r56268;
        double r56270 = r56238 ? r56245 : r56269;
        return r56270;
}

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
Herbie6.6
\[\begin{array}{l} \mathbf{if}\;b \lt 0.0:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}}\\ \end{array}\]

Derivation

  1. Split input into 4 regimes

  2. if b < -1.569310777886352e+111

    1. Initial program 50.4

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

    2. Taylor expanded around -inf 3.9

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

    3. Simplified3.9

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

    if -1.569310777886352e+111 < b < -3.3534114850920523e-283

    1. Initial program 8.4

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

    3. Applied div-inv8.6

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

    if -3.3534114850920523e-283 < b < 1.6082844052603056e+78

    1. Initial program 30.3

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

    3. Applied div-inv30.3

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

    5. Applied flip-+30.4

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

    6. Applied associate-*l/30.5

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

    7. Simplified15.7

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

    8. Taylor expanded around 0 9.1

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

    if 1.6082844052603056e+78 < b

    1. Initial program 58.7

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

    2. Taylor expanded around inf 3.2

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

  4. Final simplification6.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -1.569310777886352095486911207889814773134 \cdot 10^{111}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \mathbf{elif}\;b \le -3.353411485092052314829631610929360473517 \cdot 10^{-283}:\\ \;\;\;\;\left(\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right) \cdot \frac{1}{2 \cdot a}\\ \mathbf{elif}\;b \le 1.608284405260305560018146283533738265178 \cdot 10^{78}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2019303 
(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)))