Average Error: 34.3 → 8.6
Time: 17.2s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -1.569310777886352095486911207889814773134 \cdot 10^{111}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \mathbf{elif}\;b \le -2.075943821136515538074933331988827259408 \cdot 10^{-290}:\\ \;\;\;\;\left(\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} + \left(-b\right)\right) \cdot \frac{1}{2 \cdot a}\\ \mathbf{elif}\;b \le 1.447939350868406385811948663168665665979 \cdot 10^{78}:\\ \;\;\;\;\frac{\frac{\frac{\left(a \cdot c\right) \cdot 4}{2}}{a}}{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array}\]
\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(\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} + \left(-b\right)\right) \cdot \frac{1}{2 \cdot a}\\

\mathbf{elif}\;b \le 1.447939350868406385811948663168665665979 \cdot 10^{78}:\\
\;\;\;\;\frac{\frac{\frac{\left(a \cdot c\right) \cdot 4}{2}}{a}}{\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 r62205 = b;
        double r62206 = -r62205;
        double r62207 = r62205 * r62205;
        double r62208 = 4.0;
        double r62209 = a;
        double r62210 = c;
        double r62211 = r62209 * r62210;
        double r62212 = r62208 * r62211;
        double r62213 = r62207 - r62212;
        double r62214 = sqrt(r62213);
        double r62215 = r62206 + r62214;
        double r62216 = 2.0;
        double r62217 = r62216 * r62209;
        double r62218 = r62215 / r62217;
        return r62218;
}
↓
double f(double a, double b, double c) {
        double r62219 = b;
        double r62220 = -1.569310777886352e+111;
        bool r62221 = r62219 <= r62220;
        double r62222 = 1.0;
        double r62223 = c;
        double r62224 = r62223 / r62219;
        double r62225 = a;
        double r62226 = r62219 / r62225;
        double r62227 = r62224 - r62226;
        double r62228 = r62222 * r62227;
        double r62229 = -2.0759438211365155e-290;
        bool r62230 = r62219 <= r62229;
        double r62231 = r62219 * r62219;
        double r62232 = 4.0;
        double r62233 = r62225 * r62223;
        double r62234 = r62232 * r62233;
        double r62235 = r62231 - r62234;
        double r62236 = sqrt(r62235);
        double r62237 = -r62219;
        double r62238 = r62236 + r62237;
        double r62239 = 1.0;
        double r62240 = 2.0;
        double r62241 = r62240 * r62225;
        double r62242 = r62239 / r62241;
        double r62243 = r62238 * r62242;
        double r62244 = 1.4479393508684064e+78;
        bool r62245 = r62219 <= r62244;
        double r62246 = r62233 * r62232;
        double r62247 = r62246 / r62240;
        double r62248 = r62247 / r62225;
        double r62249 = r62237 - r62236;
        double r62250 = r62248 / r62249;
        double r62251 = -1.0;
        double r62252 = r62251 * r62224;
        double r62253 = r62245 ? r62250 : r62252;
        double r62254 = r62230 ? r62243 : r62253;
        double r62255 = r62221 ? r62228 : r62254;
        return r62255;
}

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. 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 < -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 *-un-lft-identity8.5

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

    5. Applied div-inv8.6

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

    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 *-un-lft-identity30.6

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

    5. Applied div-inv30.6

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

    7. Applied flip-+30.7

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

    8. Applied associate-*r/30.7

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

    9. Applied associate-*l/30.7

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

    10. Simplified15.8

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

    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. Taylor expanded around inf 3.2

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

  4. Final simplification8.6

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

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)))