Average Error: 34.3 → 9.3
Time: 6.4s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -2.14194017547317126 \cdot 10^{130}:\\ \;\;\;\;0.5 \cdot \frac{c}{b} - 0.66666666666666663 \cdot \frac{b}{a}\\ \mathbf{elif}\;b \le 7.481934651249181 \cdot 10^{-117}:\\ \;\;\;\;\frac{1}{\frac{3 \cdot a}{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}}\\ \mathbf{elif}\;b \le 7.24024992671430264 \cdot 10^{121}:\\ \;\;\;\;\frac{\frac{3 \cdot \left(a \cdot c\right)}{3 \cdot \left(\left(-b\right) - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array}\]
\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 -2.14194017547317126 \cdot 10^{130}:\\
\;\;\;\;0.5 \cdot \frac{c}{b} - 0.66666666666666663 \cdot \frac{b}{a}\\

\mathbf{elif}\;b \le 7.481934651249181 \cdot 10^{-117}:\\
\;\;\;\;\frac{1}{\frac{3 \cdot a}{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}}\\

\mathbf{elif}\;b \le 7.24024992671430264 \cdot 10^{121}:\\
\;\;\;\;\frac{\frac{3 \cdot \left(a \cdot c\right)}{3 \cdot \left(\left(-b\right) - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}\right)}}{a}\\

\mathbf{else}:\\
\;\;\;\;-0.5 \cdot \frac{c}{b}\\

\end{array}
double f(double a, double b, double c) {
        double r127419 = b;
        double r127420 = -r127419;
        double r127421 = r127419 * r127419;
        double r127422 = 3.0;
        double r127423 = a;
        double r127424 = r127422 * r127423;
        double r127425 = c;
        double r127426 = r127424 * r127425;
        double r127427 = r127421 - r127426;
        double r127428 = sqrt(r127427);
        double r127429 = r127420 + r127428;
        double r127430 = r127429 / r127424;
        return r127430;
}
↓
double f(double a, double b, double c) {
        double r127431 = b;
        double r127432 = -2.1419401754731713e+130;
        bool r127433 = r127431 <= r127432;
        double r127434 = 0.5;
        double r127435 = c;
        double r127436 = r127435 / r127431;
        double r127437 = r127434 * r127436;
        double r127438 = 0.6666666666666666;
        double r127439 = a;
        double r127440 = r127431 / r127439;
        double r127441 = r127438 * r127440;
        double r127442 = r127437 - r127441;
        double r127443 = 7.481934651249181e-117;
        bool r127444 = r127431 <= r127443;
        double r127445 = 1.0;
        double r127446 = 3.0;
        double r127447 = r127446 * r127439;
        double r127448 = -r127431;
        double r127449 = r127431 * r127431;
        double r127450 = r127447 * r127435;
        double r127451 = r127449 - r127450;
        double r127452 = sqrt(r127451);
        double r127453 = r127448 + r127452;
        double r127454 = r127447 / r127453;
        double r127455 = r127445 / r127454;
        double r127456 = 7.240249926714303e+121;
        bool r127457 = r127431 <= r127456;
        double r127458 = r127439 * r127435;
        double r127459 = r127446 * r127458;
        double r127460 = r127448 - r127452;
        double r127461 = r127446 * r127460;
        double r127462 = r127459 / r127461;
        double r127463 = r127462 / r127439;
        double r127464 = -0.5;
        double r127465 = r127464 * r127436;
        double r127466 = r127457 ? r127463 : r127465;
        double r127467 = r127444 ? r127455 : r127466;
        double r127468 = r127433 ? r127442 : r127467;
        return r127468;
}

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 4 regimes

  2. if b < -2.1419401754731713e+130

    1. Initial program 57.3

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

    2. Taylor expanded around -inf 3.6

      \[\leadsto \color{blue}{0.5 \cdot \frac{c}{b} - 0.66666666666666663 \cdot \frac{b}{a}}\]

    if -2.1419401754731713e+130 < b < 7.481934651249181e-117

    1. Initial program 11.0

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

    3. Applied clear-num11.0

      \[\leadsto \color{blue}{\frac{1}{\frac{3 \cdot a}{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}}}\]

    if 7.481934651249181e-117 < b < 7.240249926714303e+121

    1. Initial program 42.1

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

    3. Applied flip-+42.1

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

    4. Simplified16.6

      \[\leadsto \frac{\frac{\color{blue}{0 + 3 \cdot \left(a \cdot c\right)}}{\left(-b\right) - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}}{3 \cdot a}\]
    5. Using strategy rm

    6. Applied associate-/r*16.6

      \[\leadsto \color{blue}{\frac{\frac{\frac{0 + 3 \cdot \left(a \cdot c\right)}{\left(-b\right) - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}}{3}}{a}}\]

    7. Simplified16.7

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

    if 7.240249926714303e+121 < b

    1. Initial program 61.0

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

    2. Taylor expanded around inf 2.1

      \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b}}\]
  3. Recombined 4 regimes into one program.

  4. Final simplification9.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -2.14194017547317126 \cdot 10^{130}:\\ \;\;\;\;0.5 \cdot \frac{c}{b} - 0.66666666666666663 \cdot \frac{b}{a}\\ \mathbf{elif}\;b \le 7.481934651249181 \cdot 10^{-117}:\\ \;\;\;\;\frac{1}{\frac{3 \cdot a}{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}}\\ \mathbf{elif}\;b \le 7.24024992671430264 \cdot 10^{121}:\\ \;\;\;\;\frac{\frac{3 \cdot \left(a \cdot c\right)}{3 \cdot \left(\left(-b\right) - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2020036 +o rules:numerics
(FPCore (a b c)
  :name "Cubic critical"
  :precision binary64
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3 a) c)))) (* 3 a)))