Average Error: 28.6 → 17.2
Time: 6.8s
Precision: 64
\[1.05367121277235087 \cdot 10^{-8} \lt a \lt 94906265.6242515594 \land 1.05367121277235087 \cdot 10^{-8} \lt b \lt 94906265.6242515594 \land 1.05367121277235087 \cdot 10^{-8} \lt c \lt 94906265.6242515594\]
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le 17.714642954298647:\\ \;\;\;\;\frac{\frac{\frac{\left({b}^{2} - 4 \cdot \left(a \cdot c\right)\right) - {b}^{2}}{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} + b}}{2}}{a}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array}\]
\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 17.714642954298647:\\
\;\;\;\;\frac{\frac{\frac{\left({b}^{2} - 4 \cdot \left(a \cdot c\right)\right) - {b}^{2}}{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} + b}}{2}}{a}\\

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

\end{array}
double f(double a, double b, double c) {
        double r24391 = b;
        double r24392 = -r24391;
        double r24393 = r24391 * r24391;
        double r24394 = 4.0;
        double r24395 = a;
        double r24396 = r24394 * r24395;
        double r24397 = c;
        double r24398 = r24396 * r24397;
        double r24399 = r24393 - r24398;
        double r24400 = sqrt(r24399);
        double r24401 = r24392 + r24400;
        double r24402 = 2.0;
        double r24403 = r24402 * r24395;
        double r24404 = r24401 / r24403;
        return r24404;
}
↓
double f(double a, double b, double c) {
        double r24405 = b;
        double r24406 = 17.714642954298647;
        bool r24407 = r24405 <= r24406;
        double r24408 = 2.0;
        double r24409 = pow(r24405, r24408);
        double r24410 = 4.0;
        double r24411 = a;
        double r24412 = c;
        double r24413 = r24411 * r24412;
        double r24414 = r24410 * r24413;
        double r24415 = r24409 - r24414;
        double r24416 = r24415 - r24409;
        double r24417 = r24405 * r24405;
        double r24418 = r24410 * r24411;
        double r24419 = r24418 * r24412;
        double r24420 = r24417 - r24419;
        double r24421 = sqrt(r24420);
        double r24422 = r24421 + r24405;
        double r24423 = r24416 / r24422;
        double r24424 = 2.0;
        double r24425 = r24423 / r24424;
        double r24426 = r24425 / r24411;
        double r24427 = -1.0;
        double r24428 = r24412 / r24405;
        double r24429 = r24427 * r24428;
        double r24430 = r24407 ? r24426 : r24429;
        return r24430;
}

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 < 17.714642954298647

    1. Initial program 13.9

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

    2. Simplified13.9

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

    4. Applied flip--13.9

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

    5. Simplified12.9

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

    if 17.714642954298647 < b

    1. Initial program 33.3

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

    2. Simplified33.3

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

    3. Taylor expanded around inf 18.6

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

  4. Final simplification17.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le 17.714642954298647:\\ \;\;\;\;\frac{\frac{\frac{\left({b}^{2} - 4 \cdot \left(a \cdot c\right)\right) - {b}^{2}}{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} + b}}{2}}{a}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2020046 
(FPCore (a b c)
  :name "Quadratic roots, narrow range"
  :precision binary64
  :pre (and (< 1.0536712127723509e-08 a 94906265.62425156) (< 1.0536712127723509e-08 b 94906265.62425156) (< 1.0536712127723509e-08 c 94906265.62425156))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4 a) c)))) (* 2 a)))