Average Error: 43.8 → 11.5
Time: 17.9s
Precision: 64
\[1.1102230246251565 \cdot 10^{-16} \lt a \lt 9007199254740992.0 \land 1.1102230246251565 \cdot 10^{-16} \lt b \lt 9007199254740992.0 \land 1.1102230246251565 \cdot 10^{-16} \lt c \lt 9007199254740992.0\]
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le 0.2404309694818497:\\ \;\;\;\;\frac{\frac{\frac{\left(b \cdot b - \left(a \cdot 4\right) \cdot c\right) \cdot \sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c} - \left(b \cdot b\right) \cdot b}{\left(b \cdot b - \left(a \cdot 4\right) \cdot c\right) + b \cdot \left(b + \sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c}\right)}}{2}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array}\]
\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 0.2404309694818497:\\
\;\;\;\;\frac{\frac{\frac{\left(b \cdot b - \left(a \cdot 4\right) \cdot c\right) \cdot \sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c} - \left(b \cdot b\right) \cdot b}{\left(b \cdot b - \left(a \cdot 4\right) \cdot c\right) + b \cdot \left(b + \sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c}\right)}}{2}}{a}\\

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

\end{array}
double f(double a, double b, double c) {
        double r2040432 = b;
        double r2040433 = -r2040432;
        double r2040434 = r2040432 * r2040432;
        double r2040435 = 4.0;
        double r2040436 = a;
        double r2040437 = r2040435 * r2040436;
        double r2040438 = c;
        double r2040439 = r2040437 * r2040438;
        double r2040440 = r2040434 - r2040439;
        double r2040441 = sqrt(r2040440);
        double r2040442 = r2040433 + r2040441;
        double r2040443 = 2.0;
        double r2040444 = r2040443 * r2040436;
        double r2040445 = r2040442 / r2040444;
        return r2040445;
}
↓
double f(double a, double b, double c) {
        double r2040446 = b;
        double r2040447 = 0.2404309694818497;
        bool r2040448 = r2040446 <= r2040447;
        double r2040449 = r2040446 * r2040446;
        double r2040450 = a;
        double r2040451 = 4.0;
        double r2040452 = r2040450 * r2040451;
        double r2040453 = c;
        double r2040454 = r2040452 * r2040453;
        double r2040455 = r2040449 - r2040454;
        double r2040456 = sqrt(r2040455);
        double r2040457 = r2040455 * r2040456;
        double r2040458 = r2040449 * r2040446;
        double r2040459 = r2040457 - r2040458;
        double r2040460 = r2040446 + r2040456;
        double r2040461 = r2040446 * r2040460;
        double r2040462 = r2040455 + r2040461;
        double r2040463 = r2040459 / r2040462;
        double r2040464 = 2.0;
        double r2040465 = r2040463 / r2040464;
        double r2040466 = r2040465 / r2040450;
        double r2040467 = -r2040453;
        double r2040468 = r2040467 / r2040446;
        double r2040469 = r2040448 ? r2040466 : r2040468;
        return r2040469;
}

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 < 0.2404309694818497

    1. Initial program 24.0

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

    2. Simplified24.0

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

    4. Applied flip3--24.0

      \[\leadsto \frac{\frac{\color{blue}{\frac{{\left(\sqrt{b \cdot b - \left(c \cdot a\right) \cdot 4}\right)}^{3} - {b}^{3}}{\sqrt{b \cdot b - \left(c \cdot a\right) \cdot 4} \cdot \sqrt{b \cdot b - \left(c \cdot a\right) \cdot 4} + \left(b \cdot b + \sqrt{b \cdot b - \left(c \cdot a\right) \cdot 4} \cdot b\right)}}}{2}}{a}\]

    5. Simplified23.3

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

    6. Simplified23.3

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

    if 0.2404309694818497 < b

    1. Initial program 47.1

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

    2. Simplified47.1

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

    3. Taylor expanded around inf 9.6

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

    4. Simplified9.6

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

  4. Final simplification11.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le 0.2404309694818497:\\ \;\;\;\;\frac{\frac{\frac{\left(b \cdot b - \left(a \cdot 4\right) \cdot c\right) \cdot \sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c} - \left(b \cdot b\right) \cdot b}{\left(b \cdot b - \left(a \cdot 4\right) \cdot c\right) + b \cdot \left(b + \sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c}\right)}}{2}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2019165 
(FPCore (a b c)
  :name "Quadratic roots, medium range"
  :pre (and (< 1.1102230246251565e-16 a 9007199254740992.0) (< 1.1102230246251565e-16 b 9007199254740992.0) (< 1.1102230246251565e-16 c 9007199254740992.0))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4 a) c)))) (* 2 a)))