Average Error: 34.0 → 10.3
Time: 27.3s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -3.177289780863109 \cdot 10^{+94}:\\ \;\;\;\;\frac{\left(\frac{c}{b} - \frac{b}{a}\right) \cdot 2}{2}\\ \mathbf{elif}\;b \le 7.296044290893796 \cdot 10^{-114}:\\ \;\;\;\;\frac{\frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}{a}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2 \cdot \frac{c}{b}}{2}\\ \end{array}\]
\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 -3.177289780863109 \cdot 10^{+94}:\\
\;\;\;\;\frac{\left(\frac{c}{b} - \frac{b}{a}\right) \cdot 2}{2}\\

\mathbf{elif}\;b \le 7.296044290893796 \cdot 10^{-114}:\\
\;\;\;\;\frac{\frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}{a}}{2}\\

\mathbf{else}:\\
\;\;\;\;\frac{-2 \cdot \frac{c}{b}}{2}\\

\end{array}
double f(double a, double b, double c) {
        double r2321517 = b;
        double r2321518 = -r2321517;
        double r2321519 = r2321517 * r2321517;
        double r2321520 = 4.0;
        double r2321521 = a;
        double r2321522 = r2321520 * r2321521;
        double r2321523 = c;
        double r2321524 = r2321522 * r2321523;
        double r2321525 = r2321519 - r2321524;
        double r2321526 = sqrt(r2321525);
        double r2321527 = r2321518 + r2321526;
        double r2321528 = 2.0;
        double r2321529 = r2321528 * r2321521;
        double r2321530 = r2321527 / r2321529;
        return r2321530;
}
↓
double f(double a, double b, double c) {
        double r2321531 = b;
        double r2321532 = -3.177289780863109e+94;
        bool r2321533 = r2321531 <= r2321532;
        double r2321534 = c;
        double r2321535 = r2321534 / r2321531;
        double r2321536 = a;
        double r2321537 = r2321531 / r2321536;
        double r2321538 = r2321535 - r2321537;
        double r2321539 = 2.0;
        double r2321540 = r2321538 * r2321539;
        double r2321541 = r2321540 / r2321539;
        double r2321542 = 7.296044290893796e-114;
        bool r2321543 = r2321531 <= r2321542;
        double r2321544 = r2321531 * r2321531;
        double r2321545 = 4.0;
        double r2321546 = r2321536 * r2321534;
        double r2321547 = r2321545 * r2321546;
        double r2321548 = r2321544 - r2321547;
        double r2321549 = sqrt(r2321548);
        double r2321550 = r2321549 - r2321531;
        double r2321551 = r2321550 / r2321536;
        double r2321552 = r2321551 / r2321539;
        double r2321553 = -2.0;
        double r2321554 = r2321553 * r2321535;
        double r2321555 = r2321554 / r2321539;
        double r2321556 = r2321543 ? r2321552 : r2321555;
        double r2321557 = r2321533 ? r2321541 : r2321556;
        return r2321557;
}

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

Original34.0
Target20.9
Herbie10.3
\[\begin{array}{l} \mathbf{if}\;b \lt 0:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if b < -3.177289780863109e+94

    1. Initial program 43.6

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

    2. Simplified43.5

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

    3. Taylor expanded around -inf 3.8

      \[\leadsto \frac{\color{blue}{2 \cdot \frac{c}{b} - 2 \cdot \frac{b}{a}}}{2}\]

    4. Simplified3.8

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

    if -3.177289780863109e+94 < b < 7.296044290893796e-114

    1. Initial program 12.4

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

    2. Simplified12.3

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

    if 7.296044290893796e-114 < b

    1. Initial program 51.6

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

    2. Simplified51.6

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

    3. Taylor expanded around inf 10.8

      \[\leadsto \frac{\color{blue}{-2 \cdot \frac{c}{b}}}{2}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification10.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -3.177289780863109 \cdot 10^{+94}:\\ \;\;\;\;\frac{\left(\frac{c}{b} - \frac{b}{a}\right) \cdot 2}{2}\\ \mathbf{elif}\;b \le 7.296044290893796 \cdot 10^{-114}:\\ \;\;\;\;\frac{\frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}{a}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2 \cdot \frac{c}{b}}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019141 
(FPCore (a b c)
  :name "The quadratic formula (r1)"

  :herbie-target
  (if (< b 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)))