Average Error: 33.9 → 9.2
Time: 18.2s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -6.95366867167606799833293235113613604366 \cdot 10^{101}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \mathbf{elif}\;b \le 6.712838330010700391722571804071964139806 \cdot 10^{-280}:\\ \;\;\;\;\frac{\sqrt{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}}{2} \cdot \frac{\sqrt{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}}{a}\\ \mathbf{elif}\;b \le 3.424603050151593475509461695672638978245 \cdot 10^{-29}:\\ \;\;\;\;\frac{\frac{1}{\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{4 \cdot \left(a \cdot c\right)}}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array}\]
\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 -6.95366867167606799833293235113613604366 \cdot 10^{101}:\\
\;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\

\mathbf{elif}\;b \le 6.712838330010700391722571804071964139806 \cdot 10^{-280}:\\
\;\;\;\;\frac{\sqrt{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}}{2} \cdot \frac{\sqrt{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}}{a}\\

\mathbf{elif}\;b \le 3.424603050151593475509461695672638978245 \cdot 10^{-29}:\\
\;\;\;\;\frac{\frac{1}{\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{4 \cdot \left(a \cdot c\right)}}}{2 \cdot a}\\

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

\end{array}
double f(double a, double b, double c) {
        double r80244 = b;
        double r80245 = -r80244;
        double r80246 = r80244 * r80244;
        double r80247 = 4.0;
        double r80248 = a;
        double r80249 = c;
        double r80250 = r80248 * r80249;
        double r80251 = r80247 * r80250;
        double r80252 = r80246 - r80251;
        double r80253 = sqrt(r80252);
        double r80254 = r80245 + r80253;
        double r80255 = 2.0;
        double r80256 = r80255 * r80248;
        double r80257 = r80254 / r80256;
        return r80257;
}
↓
double f(double a, double b, double c) {
        double r80258 = b;
        double r80259 = -6.953668671676068e+101;
        bool r80260 = r80258 <= r80259;
        double r80261 = 1.0;
        double r80262 = c;
        double r80263 = r80262 / r80258;
        double r80264 = a;
        double r80265 = r80258 / r80264;
        double r80266 = r80263 - r80265;
        double r80267 = r80261 * r80266;
        double r80268 = 6.7128383300107e-280;
        bool r80269 = r80258 <= r80268;
        double r80270 = r80258 * r80258;
        double r80271 = 4.0;
        double r80272 = r80264 * r80262;
        double r80273 = r80271 * r80272;
        double r80274 = r80270 - r80273;
        double r80275 = sqrt(r80274);
        double r80276 = r80275 - r80258;
        double r80277 = sqrt(r80276);
        double r80278 = 2.0;
        double r80279 = r80277 / r80278;
        double r80280 = r80277 / r80264;
        double r80281 = r80279 * r80280;
        double r80282 = 3.4246030501515935e-29;
        bool r80283 = r80258 <= r80282;
        double r80284 = 1.0;
        double r80285 = -r80258;
        double r80286 = r80285 - r80275;
        double r80287 = r80286 / r80273;
        double r80288 = r80284 / r80287;
        double r80289 = r80278 * r80264;
        double r80290 = r80288 / r80289;
        double r80291 = -1.0;
        double r80292 = r80291 * r80263;
        double r80293 = r80283 ? r80290 : r80292;
        double r80294 = r80269 ? r80281 : r80293;
        double r80295 = r80260 ? r80267 : r80294;
        return r80295;
}

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

Original33.9
Target21.0
Herbie9.2
\[\begin{array}{l} \mathbf{if}\;b \lt 0.0:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \end{array}\]

Derivation

  1. Split input into 4 regimes

  2. if b < -6.953668671676068e+101

    1. Initial program 47.8

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

    2. Taylor expanded around -inf 3.6

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

    3. Simplified3.6

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

    if -6.953668671676068e+101 < b < 6.7128383300107e-280

    1. Initial program 8.8

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

    3. Applied add-sqr-sqrt9.1

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

    4. Applied times-frac9.1

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

    5. Simplified9.1

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

    6. Simplified9.1

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

    if 6.7128383300107e-280 < b < 3.4246030501515935e-29

    1. Initial program 24.4

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

    3. Applied flip-+24.5

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

    4. Simplified18.6

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

    6. Applied clear-num18.6

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

    7. Simplified18.6

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

    if 3.4246030501515935e-29 < b

    1. Initial program 54.8

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

    2. Taylor expanded around inf 7.2

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

  4. Final simplification9.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -6.95366867167606799833293235113613604366 \cdot 10^{101}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \mathbf{elif}\;b \le 6.712838330010700391722571804071964139806 \cdot 10^{-280}:\\ \;\;\;\;\frac{\sqrt{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}}{2} \cdot \frac{\sqrt{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}}{a}\\ \mathbf{elif}\;b \le 3.424603050151593475509461695672638978245 \cdot 10^{-29}:\\ \;\;\;\;\frac{\frac{1}{\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{4 \cdot \left(a \cdot c\right)}}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2019305 
(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)))