Average Error: 34.6 → 10.7
Time: 27.7s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -2.766818940874854722177248139872145176232 \cdot 10^{100}:\\ \;\;\;\;\left(\frac{c}{b} - \frac{b}{a}\right) \cdot 1\\ \mathbf{elif}\;b \le 7.923524897992036987166355557663274472861 \cdot 10^{-153}:\\ \;\;\;\;\frac{\frac{1}{a \cdot 2}}{\frac{1}{\sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c} - b}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -1\\ \end{array}\]
\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 -2.766818940874854722177248139872145176232 \cdot 10^{100}:\\
\;\;\;\;\left(\frac{c}{b} - \frac{b}{a}\right) \cdot 1\\

\mathbf{elif}\;b \le 7.923524897992036987166355557663274472861 \cdot 10^{-153}:\\
\;\;\;\;\frac{\frac{1}{a \cdot 2}}{\frac{1}{\sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c} - b}}\\

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

\end{array}
double f(double a, double b, double c) {
        double r2018397 = b;
        double r2018398 = -r2018397;
        double r2018399 = r2018397 * r2018397;
        double r2018400 = 4.0;
        double r2018401 = a;
        double r2018402 = r2018400 * r2018401;
        double r2018403 = c;
        double r2018404 = r2018402 * r2018403;
        double r2018405 = r2018399 - r2018404;
        double r2018406 = sqrt(r2018405);
        double r2018407 = r2018398 + r2018406;
        double r2018408 = 2.0;
        double r2018409 = r2018408 * r2018401;
        double r2018410 = r2018407 / r2018409;
        return r2018410;
}
↓
double f(double a, double b, double c) {
        double r2018411 = b;
        double r2018412 = -2.7668189408748547e+100;
        bool r2018413 = r2018411 <= r2018412;
        double r2018414 = c;
        double r2018415 = r2018414 / r2018411;
        double r2018416 = a;
        double r2018417 = r2018411 / r2018416;
        double r2018418 = r2018415 - r2018417;
        double r2018419 = 1.0;
        double r2018420 = r2018418 * r2018419;
        double r2018421 = 7.923524897992037e-153;
        bool r2018422 = r2018411 <= r2018421;
        double r2018423 = 1.0;
        double r2018424 = 2.0;
        double r2018425 = r2018416 * r2018424;
        double r2018426 = r2018423 / r2018425;
        double r2018427 = r2018411 * r2018411;
        double r2018428 = 4.0;
        double r2018429 = r2018416 * r2018428;
        double r2018430 = r2018429 * r2018414;
        double r2018431 = r2018427 - r2018430;
        double r2018432 = sqrt(r2018431);
        double r2018433 = r2018432 - r2018411;
        double r2018434 = r2018423 / r2018433;
        double r2018435 = r2018426 / r2018434;
        double r2018436 = -1.0;
        double r2018437 = r2018415 * r2018436;
        double r2018438 = r2018422 ? r2018435 : r2018437;
        double r2018439 = r2018413 ? r2018420 : r2018438;
        return r2018439;
}

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 3 regimes

  2. if b < -2.7668189408748547e+100

    1. Initial program 47.2

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

    2. Simplified47.2

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

    3. Taylor expanded around 0 47.2

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

    4. Simplified47.2

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

    5. Taylor expanded around 0 47.2

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

    6. Simplified47.2

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

    7. Taylor expanded around -inf 4.0

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

    8. Simplified4.0

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

    if -2.7668189408748547e+100 < b < 7.923524897992037e-153

    1. Initial program 10.8

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

    2. Simplified10.8

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

    3. Taylor expanded around 0 10.9

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

    4. Simplified10.8

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

    5. Taylor expanded around 0 10.9

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

    6. Simplified10.8

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

    8. Applied clear-num11.0

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

    10. Applied div-inv11.0

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

    11. Applied associate-/r*11.0

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

    if 7.923524897992037e-153 < b

    1. Initial program 50.5

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

    2. Simplified50.5

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

    3. Taylor expanded around inf 12.7

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

  4. Final simplification10.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -2.766818940874854722177248139872145176232 \cdot 10^{100}:\\ \;\;\;\;\left(\frac{c}{b} - \frac{b}{a}\right) \cdot 1\\ \mathbf{elif}\;b \le 7.923524897992036987166355557663274472861 \cdot 10^{-153}:\\ \;\;\;\;\frac{\frac{1}{a \cdot 2}}{\frac{1}{\sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c} - b}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -1\\ \end{array}\]

Reproduce

herbie shell --seed 2019200 +o rules:numerics
(FPCore (a b c)
  :name "Quadratic roots, full range"
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)))