Average Error: 28.5 → 16.3
Time: 12.3s
Precision: 64
\[1.0536712127723509 \cdot 10^{-08} \lt a \lt 94906265.62425156 \land 1.0536712127723509 \cdot 10^{-08} \lt b \lt 94906265.62425156 \land 1.0536712127723509 \cdot 10^{-08} \lt c \lt 94906265.62425156\]
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le 1935.0528928047481:\\ \;\;\;\;\frac{\frac{\frac{\left(b \cdot b - \left(a \cdot c\right) \cdot 4\right) \cdot \sqrt{b \cdot b - \left(a \cdot c\right) \cdot 4} - \left(b \cdot b\right) \cdot b}{\left(b \cdot b - \left(a \cdot c\right) \cdot 4\right) + b \cdot \left(b + \sqrt{b \cdot b - \left(a \cdot c\right) \cdot 4}\right)}}{a}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2 \cdot \frac{c}{b}}{2}\\ \end{array}\]
\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 1935.0528928047481:\\
\;\;\;\;\frac{\frac{\frac{\left(b \cdot b - \left(a \cdot c\right) \cdot 4\right) \cdot \sqrt{b \cdot b - \left(a \cdot c\right) \cdot 4} - \left(b \cdot b\right) \cdot b}{\left(b \cdot b - \left(a \cdot c\right) \cdot 4\right) + b \cdot \left(b + \sqrt{b \cdot b - \left(a \cdot c\right) \cdot 4}\right)}}{a}}{2}\\

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

\end{array}
double f(double a, double b, double c) {
        double r795383 = b;
        double r795384 = -r795383;
        double r795385 = r795383 * r795383;
        double r795386 = 4.0;
        double r795387 = a;
        double r795388 = r795386 * r795387;
        double r795389 = c;
        double r795390 = r795388 * r795389;
        double r795391 = r795385 - r795390;
        double r795392 = sqrt(r795391);
        double r795393 = r795384 + r795392;
        double r795394 = 2.0;
        double r795395 = r795394 * r795387;
        double r795396 = r795393 / r795395;
        return r795396;
}
↓
double f(double a, double b, double c) {
        double r795397 = b;
        double r795398 = 1935.0528928047481;
        bool r795399 = r795397 <= r795398;
        double r795400 = r795397 * r795397;
        double r795401 = a;
        double r795402 = c;
        double r795403 = r795401 * r795402;
        double r795404 = 4.0;
        double r795405 = r795403 * r795404;
        double r795406 = r795400 - r795405;
        double r795407 = sqrt(r795406);
        double r795408 = r795406 * r795407;
        double r795409 = r795400 * r795397;
        double r795410 = r795408 - r795409;
        double r795411 = r795397 + r795407;
        double r795412 = r795397 * r795411;
        double r795413 = r795406 + r795412;
        double r795414 = r795410 / r795413;
        double r795415 = r795414 / r795401;
        double r795416 = 2.0;
        double r795417 = r795415 / r795416;
        double r795418 = -2.0;
        double r795419 = r795402 / r795397;
        double r795420 = r795418 * r795419;
        double r795421 = r795420 / r795416;
        double r795422 = r795399 ? r795417 : r795421;
        return r795422;
}

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 < 1935.0528928047481

    1. Initial program 17.3

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

    2. Simplified17.3

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

    4. Applied flip3--17.4

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

    5. Simplified16.8

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

    6. Simplified16.7

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

    if 1935.0528928047481 < b

    1. Initial program 36.6

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

    2. Simplified36.6

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

    3. Taylor expanded around inf 16.0

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

  4. Final simplification16.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le 1935.0528928047481:\\ \;\;\;\;\frac{\frac{\frac{\left(b \cdot b - \left(a \cdot c\right) \cdot 4\right) \cdot \sqrt{b \cdot b - \left(a \cdot c\right) \cdot 4} - \left(b \cdot b\right) \cdot b}{\left(b \cdot b - \left(a \cdot c\right) \cdot 4\right) + b \cdot \left(b + \sqrt{b \cdot b - \left(a \cdot c\right) \cdot 4}\right)}}{a}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2 \cdot \frac{c}{b}}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019156 
(FPCore (a b c)
  :name "Quadratic roots, narrow range"
  :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)))