Average Error: 43.8 → 11.1
Time: 6.0s
Precision: 64
\[1.11022 \cdot 10^{-16} \lt a \lt 9.0072 \cdot 10^{15} \land 1.11022 \cdot 10^{-16} \lt b \lt 9.0072 \cdot 10^{15} \land 1.11022 \cdot 10^{-16} \lt c \lt 9.0072 \cdot 10^{15}\]
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le 1.7936624356974993 \cdot 10^{-4}:\\ \;\;\;\;\frac{\frac{\left({b}^{2} - 3 \cdot \left(a \cdot c\right)\right) - {b}^{2}}{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} + b}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array}\]
\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 1.7936624356974993 \cdot 10^{-4}:\\
\;\;\;\;\frac{\frac{\left({b}^{2} - 3 \cdot \left(a \cdot c\right)\right) - {b}^{2}}{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} + b}}{3 \cdot a}\\

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

\end{array}
double f(double a, double b, double c) {
        double r71544 = b;
        double r71545 = -r71544;
        double r71546 = r71544 * r71544;
        double r71547 = 3.0;
        double r71548 = a;
        double r71549 = r71547 * r71548;
        double r71550 = c;
        double r71551 = r71549 * r71550;
        double r71552 = r71546 - r71551;
        double r71553 = sqrt(r71552);
        double r71554 = r71545 + r71553;
        double r71555 = r71554 / r71549;
        return r71555;
}
↓
double f(double a, double b, double c) {
        double r71556 = b;
        double r71557 = 0.00017936624356974993;
        bool r71558 = r71556 <= r71557;
        double r71559 = 2.0;
        double r71560 = pow(r71556, r71559);
        double r71561 = 3.0;
        double r71562 = a;
        double r71563 = c;
        double r71564 = r71562 * r71563;
        double r71565 = r71561 * r71564;
        double r71566 = r71560 - r71565;
        double r71567 = r71566 - r71560;
        double r71568 = r71556 * r71556;
        double r71569 = r71561 * r71562;
        double r71570 = r71569 * r71563;
        double r71571 = r71568 - r71570;
        double r71572 = sqrt(r71571);
        double r71573 = r71572 + r71556;
        double r71574 = r71567 / r71573;
        double r71575 = r71574 / r71569;
        double r71576 = -0.5;
        double r71577 = r71563 / r71556;
        double r71578 = r71576 * r71577;
        double r71579 = r71558 ? r71575 : r71578;
        return r71579;
}

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 < 0.00017936624356974993

    1. Initial program 18.4

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

    2. Simplified18.4

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

    4. Applied flip--18.4

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

    5. Simplified17.4

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

    if 0.00017936624356974993 < b

    1. Initial program 45.7

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

    2. Simplified45.7

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

    3. Taylor expanded around inf 10.6

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

  4. Final simplification11.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le 1.7936624356974993 \cdot 10^{-4}:\\ \;\;\;\;\frac{\frac{\left({b}^{2} - 3 \cdot \left(a \cdot c\right)\right) - {b}^{2}}{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} + b}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2020047 
(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)))