Average Error: 34.0 → 9.3
Time: 6.7s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -8.5069461462218695 \cdot 10^{125}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \mathbf{elif}\;b \le 2.2742398392973687 \cdot 10^{-86}:\\ \;\;\;\;\left(\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right) \cdot \frac{1}{2 \cdot a}\\ \mathbf{elif}\;b \le 9.16799708835065593 \cdot 10^{-7}:\\ \;\;\;\;\frac{\frac{0 + 4 \cdot \left(a \cdot c\right)}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array}\]
\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 -8.5069461462218695 \cdot 10^{125}:\\
\;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\

\mathbf{elif}\;b \le 2.2742398392973687 \cdot 10^{-86}:\\
\;\;\;\;\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 9.16799708835065593 \cdot 10^{-7}:\\
\;\;\;\;\frac{\frac{0 + 4 \cdot \left(a \cdot c\right)}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}{2 \cdot a}\\

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

\end{array}
double f(double a, double b, double c) {
        double r58300 = b;
        double r58301 = -r58300;
        double r58302 = r58300 * r58300;
        double r58303 = 4.0;
        double r58304 = a;
        double r58305 = r58303 * r58304;
        double r58306 = c;
        double r58307 = r58305 * r58306;
        double r58308 = r58302 - r58307;
        double r58309 = sqrt(r58308);
        double r58310 = r58301 + r58309;
        double r58311 = 2.0;
        double r58312 = r58311 * r58304;
        double r58313 = r58310 / r58312;
        return r58313;
}
↓
double f(double a, double b, double c) {
        double r58314 = b;
        double r58315 = -8.50694614622187e+125;
        bool r58316 = r58314 <= r58315;
        double r58317 = 1.0;
        double r58318 = c;
        double r58319 = r58318 / r58314;
        double r58320 = a;
        double r58321 = r58314 / r58320;
        double r58322 = r58319 - r58321;
        double r58323 = r58317 * r58322;
        double r58324 = 2.2742398392973687e-86;
        bool r58325 = r58314 <= r58324;
        double r58326 = -r58314;
        double r58327 = r58314 * r58314;
        double r58328 = 4.0;
        double r58329 = r58328 * r58320;
        double r58330 = r58329 * r58318;
        double r58331 = r58327 - r58330;
        double r58332 = sqrt(r58331);
        double r58333 = r58326 + r58332;
        double r58334 = 1.0;
        double r58335 = 2.0;
        double r58336 = r58335 * r58320;
        double r58337 = r58334 / r58336;
        double r58338 = r58333 * r58337;
        double r58339 = 9.167997088350656e-07;
        bool r58340 = r58314 <= r58339;
        double r58341 = 0.0;
        double r58342 = r58320 * r58318;
        double r58343 = r58328 * r58342;
        double r58344 = r58341 + r58343;
        double r58345 = r58326 - r58332;
        double r58346 = r58344 / r58345;
        double r58347 = r58346 / r58336;
        double r58348 = -1.0;
        double r58349 = r58348 * r58319;
        double r58350 = r58340 ? r58347 : r58349;
        double r58351 = r58325 ? r58338 : r58350;
        double r58352 = r58316 ? r58323 : r58351;
        return r58352;
}

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

Derivation

  1. Split input into 4 regimes

  2. if b < -8.50694614622187e+125

    1. Initial program 53.3

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

    2. Taylor expanded around -inf 2.7

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

    3. Simplified2.7

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

    if -8.50694614622187e+125 < b < 2.2742398392973687e-86

    1. Initial program 12.3

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

    3. Applied div-inv12.5

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

    if 2.2742398392973687e-86 < b < 9.167997088350656e-07

    1. Initial program 38.3

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

    3. Applied flip-+38.3

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

    4. Simplified18.8

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

    if 9.167997088350656e-07 < b

    1. Initial program 55.8

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

    2. Taylor expanded around inf 5.7

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

  4. Final simplification9.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -8.5069461462218695 \cdot 10^{125}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \mathbf{elif}\;b \le 2.2742398392973687 \cdot 10^{-86}:\\ \;\;\;\;\left(\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right) \cdot \frac{1}{2 \cdot a}\\ \mathbf{elif}\;b \le 9.16799708835065593 \cdot 10^{-7}:\\ \;\;\;\;\frac{\frac{0 + 4 \cdot \left(a \cdot c\right)}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2020003 
(FPCore (a b c)
  :name "Quadratic roots, full range"
  :precision binary64
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4 a) c)))) (* 2 a)))