Average Error: 44.0 → 11.1
Time: 6.2s
Precision: 64
\[1.11022 \cdot 10^{-16} \lt a \lt 9.0072 \cdot 10^{15} \land 1.11022 \cdot 10^{-16} \lt b \lt 9.0072 \cdot 10^{15} \land 1.11022 \cdot 10^{-16} \lt c \lt 9.0072 \cdot 10^{15}\]
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le 0.00169426965961779418:\\ \;\;\;\;\frac{\frac{{b}^{2} - \left({b}^{2} - \left(3 \cdot a\right) \cdot c\right)}{\left(-b\right) - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array}\]
\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 0.00169426965961779418:\\
\;\;\;\;\frac{\frac{{b}^{2} - \left({b}^{2} - \left(3 \cdot a\right) \cdot c\right)}{\left(-b\right) - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}}{3 \cdot a}\\

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

\end{array}
double f(double a, double b, double c) {
        double r159491 = b;
        double r159492 = -r159491;
        double r159493 = r159491 * r159491;
        double r159494 = 3.0;
        double r159495 = a;
        double r159496 = r159494 * r159495;
        double r159497 = c;
        double r159498 = r159496 * r159497;
        double r159499 = r159493 - r159498;
        double r159500 = sqrt(r159499);
        double r159501 = r159492 + r159500;
        double r159502 = r159501 / r159496;
        return r159502;
}
↓
double f(double a, double b, double c) {
        double r159503 = b;
        double r159504 = 0.0016942696596177942;
        bool r159505 = r159503 <= r159504;
        double r159506 = 2.0;
        double r159507 = pow(r159503, r159506);
        double r159508 = 3.0;
        double r159509 = a;
        double r159510 = r159508 * r159509;
        double r159511 = c;
        double r159512 = r159510 * r159511;
        double r159513 = r159507 - r159512;
        double r159514 = r159507 - r159513;
        double r159515 = -r159503;
        double r159516 = r159503 * r159503;
        double r159517 = r159516 - r159512;
        double r159518 = sqrt(r159517);
        double r159519 = r159515 - r159518;
        double r159520 = r159514 / r159519;
        double r159521 = r159520 / r159510;
        double r159522 = -0.5;
        double r159523 = r159511 / r159503;
        double r159524 = r159522 * r159523;
        double r159525 = r159505 ? r159521 : r159524;
        return r159525;
}

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 < 0.0016942696596177942

    1. Initial program 21.0

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

    3. Applied flip-+21.0

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

    4. Simplified20.0

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

    if 0.0016942696596177942 < b

    1. Initial program 46.2

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

    2. Taylor expanded around inf 10.2

      \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b}}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification11.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le 0.00169426965961779418:\\ \;\;\;\;\frac{\frac{{b}^{2} - \left({b}^{2} - \left(3 \cdot a\right) \cdot c\right)}{\left(-b\right) - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2020045 
(FPCore (a b c)
  :name "Cubic critical, medium range"
  :precision binary64
  :pre (and (< 1.11022e-16 a 9.0072e+15) (< 1.11022e-16 b 9.0072e+15) (< 1.11022e-16 c 9.0072e+15))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3 a) c)))) (* 3 a)))