Average Error: 33.7 → 10.2
Time: 19.5s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -5.3248915655872564 \cdot 10^{+79}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \le 4.2796532586596585 \cdot 10^{-91}:\\ \;\;\;\;\left(\left(-b\right) + \sqrt{b \cdot b - c \cdot \left(4 \cdot a\right)}\right) \cdot \frac{\frac{1}{2}}{a}\\ \mathbf{else}:\\ \;\;\;\;-\frac{c}{b}\\ \end{array}\]
\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 -5.3248915655872564 \cdot 10^{+79}:\\
\;\;\;\;\frac{c}{b} - \frac{b}{a}\\

\mathbf{elif}\;b \le 4.2796532586596585 \cdot 10^{-91}:\\
\;\;\;\;\left(\left(-b\right) + \sqrt{b \cdot b - c \cdot \left(4 \cdot a\right)}\right) \cdot \frac{\frac{1}{2}}{a}\\

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

\end{array}
double f(double a, double b, double c) {
        double r1103539 = b;
        double r1103540 = -r1103539;
        double r1103541 = r1103539 * r1103539;
        double r1103542 = 4.0;
        double r1103543 = a;
        double r1103544 = r1103542 * r1103543;
        double r1103545 = c;
        double r1103546 = r1103544 * r1103545;
        double r1103547 = r1103541 - r1103546;
        double r1103548 = sqrt(r1103547);
        double r1103549 = r1103540 + r1103548;
        double r1103550 = 2.0;
        double r1103551 = r1103550 * r1103543;
        double r1103552 = r1103549 / r1103551;
        return r1103552;
}
↓
double f(double a, double b, double c) {
        double r1103553 = b;
        double r1103554 = -5.3248915655872564e+79;
        bool r1103555 = r1103553 <= r1103554;
        double r1103556 = c;
        double r1103557 = r1103556 / r1103553;
        double r1103558 = a;
        double r1103559 = r1103553 / r1103558;
        double r1103560 = r1103557 - r1103559;
        double r1103561 = 4.2796532586596585e-91;
        bool r1103562 = r1103553 <= r1103561;
        double r1103563 = -r1103553;
        double r1103564 = r1103553 * r1103553;
        double r1103565 = 4.0;
        double r1103566 = r1103565 * r1103558;
        double r1103567 = r1103556 * r1103566;
        double r1103568 = r1103564 - r1103567;
        double r1103569 = sqrt(r1103568);
        double r1103570 = r1103563 + r1103569;
        double r1103571 = 0.5;
        double r1103572 = r1103571 / r1103558;
        double r1103573 = r1103570 * r1103572;
        double r1103574 = -r1103557;
        double r1103575 = r1103562 ? r1103573 : r1103574;
        double r1103576 = r1103555 ? r1103560 : r1103575;
        return r1103576;
}

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 < -5.3248915655872564e+79

    1. Initial program 41.1

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

    3. Applied div-inv41.2

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

    4. Simplified41.2

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

    5. Taylor expanded around -inf 4.6

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

    if -5.3248915655872564e+79 < b < 4.2796532586596585e-91

    1. Initial program 13.0

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

    3. Applied div-inv13.1

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

    4. Simplified13.1

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

    if 4.2796532586596585e-91 < b

    1. Initial program 52.0

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

    2. Taylor expanded around inf 9.6

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

    3. Simplified9.6

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

  4. Final simplification10.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -5.3248915655872564 \cdot 10^{+79}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \le 4.2796532586596585 \cdot 10^{-91}:\\ \;\;\;\;\left(\left(-b\right) + \sqrt{b \cdot b - c \cdot \left(4 \cdot a\right)}\right) \cdot \frac{\frac{1}{2}}{a}\\ \mathbf{else}:\\ \;\;\;\;-\frac{c}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2019135 +o rules:numerics
(FPCore (a b c)
  :name "Quadratic roots, full range"
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4 a) c)))) (* 2 a)))