Average Error: 43.7 → 11.1
Time: 27.4s
Precision: 64
\[1.1102230246251565 \cdot 10^{-16} \lt a \lt 9007199254740992.0 \land 1.1102230246251565 \cdot 10^{-16} \lt b \lt 9007199254740992.0 \land 1.1102230246251565 \cdot 10^{-16} \lt c \lt 9007199254740992.0\]
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le 0.00010986090000445213:\\ \;\;\;\;\frac{\frac{\left(\left(a \cdot c\right) \cdot -3 + b \cdot b\right) \cdot \sqrt{\left(a \cdot c\right) \cdot -3 + b \cdot b} - b \cdot \left(b \cdot b\right)}{\left(\left(a \cdot c\right) \cdot -3 + b \cdot b\right) + \left(b \cdot b + b \cdot \sqrt{\left(a \cdot c\right) \cdot -3 + b \cdot b}\right)}}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b}\\ \end{array}\]
\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}
↓
\begin{array}{l}
\mathbf{if}\;b \le 0.00010986090000445213:\\
\;\;\;\;\frac{\frac{\left(\left(a \cdot c\right) \cdot -3 + b \cdot b\right) \cdot \sqrt{\left(a \cdot c\right) \cdot -3 + b \cdot b} - b \cdot \left(b \cdot b\right)}{\left(\left(a \cdot c\right) \cdot -3 + b \cdot b\right) + \left(b \cdot b + b \cdot \sqrt{\left(a \cdot c\right) \cdot -3 + b \cdot b}\right)}}{a \cdot 3}\\

\mathbf{else}:\\
\;\;\;\;\frac{-1}{2} \cdot \frac{c}{b}\\

\end{array}
double f(double a, double b, double c) {
        double r4161271 = b;
        double r4161272 = -r4161271;
        double r4161273 = r4161271 * r4161271;
        double r4161274 = 3.0;
        double r4161275 = a;
        double r4161276 = r4161274 * r4161275;
        double r4161277 = c;
        double r4161278 = r4161276 * r4161277;
        double r4161279 = r4161273 - r4161278;
        double r4161280 = sqrt(r4161279);
        double r4161281 = r4161272 + r4161280;
        double r4161282 = r4161281 / r4161276;
        return r4161282;
}
↓
double f(double a, double b, double c) {
        double r4161283 = b;
        double r4161284 = 0.00010986090000445213;
        bool r4161285 = r4161283 <= r4161284;
        double r4161286 = a;
        double r4161287 = c;
        double r4161288 = r4161286 * r4161287;
        double r4161289 = -3.0;
        double r4161290 = r4161288 * r4161289;
        double r4161291 = r4161283 * r4161283;
        double r4161292 = r4161290 + r4161291;
        double r4161293 = sqrt(r4161292);
        double r4161294 = r4161292 * r4161293;
        double r4161295 = r4161283 * r4161291;
        double r4161296 = r4161294 - r4161295;
        double r4161297 = r4161283 * r4161293;
        double r4161298 = r4161291 + r4161297;
        double r4161299 = r4161292 + r4161298;
        double r4161300 = r4161296 / r4161299;
        double r4161301 = 3.0;
        double r4161302 = r4161286 * r4161301;
        double r4161303 = r4161300 / r4161302;
        double r4161304 = -0.5;
        double r4161305 = r4161287 / r4161283;
        double r4161306 = r4161304 * r4161305;
        double r4161307 = r4161285 ? r4161303 : r4161306;
        return r4161307;
}

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 < 0.00010986090000445213

    1. Initial program 18.3

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]

    2. Simplified18.3

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

    4. Applied flip3--18.4

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

    5. Simplified17.7

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

    6. Simplified17.7

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

    if 0.00010986090000445213 < b

    1. Initial program 45.6

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]

    2. Simplified45.6

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

    3. Taylor expanded around inf 10.7

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

  4. Final simplification11.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le 0.00010986090000445213:\\ \;\;\;\;\frac{\frac{\left(\left(a \cdot c\right) \cdot -3 + b \cdot b\right) \cdot \sqrt{\left(a \cdot c\right) \cdot -3 + b \cdot b} - b \cdot \left(b \cdot b\right)}{\left(\left(a \cdot c\right) \cdot -3 + b \cdot b\right) + \left(b \cdot b + b \cdot \sqrt{\left(a \cdot c\right) \cdot -3 + b \cdot b}\right)}}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2019141 
(FPCore (a b c)
  :name "Cubic critical, 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) (* (* 3 a) c)))) (* 3 a)))