Average Error: 43.7 → 11.5
Time: 11.7s
Precision: 64
\[1.1102230246251565404236316680908203125 \cdot 10^{-16} \lt a \lt 9007199254740992 \land 1.1102230246251565404236316680908203125 \cdot 10^{-16} \lt b \lt 9007199254740992 \land 1.1102230246251565404236316680908203125 \cdot 10^{-16} \lt c \lt 9007199254740992\]
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le 0.1273142573446926772984255649134865961969:\\ \;\;\;\;\frac{\frac{\left(b \cdot b - \left(4 \cdot a\right) \cdot c\right) - b \cdot b}{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} + b}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array}\]
\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 0.1273142573446926772984255649134865961969:\\
\;\;\;\;\frac{\frac{\left(b \cdot b - \left(4 \cdot a\right) \cdot c\right) - b \cdot b}{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} + b}}{2 \cdot a}\\

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

\end{array}
double f(double a, double b, double c) {
        double r30642 = b;
        double r30643 = -r30642;
        double r30644 = r30642 * r30642;
        double r30645 = 4.0;
        double r30646 = a;
        double r30647 = r30645 * r30646;
        double r30648 = c;
        double r30649 = r30647 * r30648;
        double r30650 = r30644 - r30649;
        double r30651 = sqrt(r30650);
        double r30652 = r30643 + r30651;
        double r30653 = 2.0;
        double r30654 = r30653 * r30646;
        double r30655 = r30652 / r30654;
        return r30655;
}
↓
double f(double a, double b, double c) {
        double r30656 = b;
        double r30657 = 0.12731425734469268;
        bool r30658 = r30656 <= r30657;
        double r30659 = r30656 * r30656;
        double r30660 = 4.0;
        double r30661 = a;
        double r30662 = r30660 * r30661;
        double r30663 = c;
        double r30664 = r30662 * r30663;
        double r30665 = r30659 - r30664;
        double r30666 = r30665 - r30659;
        double r30667 = sqrt(r30665);
        double r30668 = r30667 + r30656;
        double r30669 = r30666 / r30668;
        double r30670 = 2.0;
        double r30671 = r30670 * r30661;
        double r30672 = r30669 / r30671;
        double r30673 = -1.0;
        double r30674 = r30663 / r30656;
        double r30675 = r30673 * r30674;
        double r30676 = r30658 ? r30672 : r30675;
        return r30676;
}

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 2 regimes

  2. if b < 0.12731425734469268

    1. Initial program 23.3

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

    2. Simplified23.3

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

    4. Applied flip--23.3

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

    5. Simplified22.3

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

    if 0.12731425734469268 < b

    1. Initial program 46.7

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

    2. Simplified46.7

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

    3. Taylor expanded around inf 9.9

      \[\leadsto \color{blue}{-1 \cdot \frac{c}{b}}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification11.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le 0.1273142573446926772984255649134865961969:\\ \;\;\;\;\frac{\frac{\left(b \cdot b - \left(4 \cdot a\right) \cdot c\right) - b \cdot b}{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} + b}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2019209 
(FPCore (a b c)
  :name "Quadratic roots, medium range"
  :precision binary64
  :pre (and (< 1.11022e-16 a 9.0072e15) (< 1.11022e-16 b 9.0072e15) (< 1.11022e-16 c 9.0072e15))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4 a) c)))) (* 2 a)))