Average Error: 28.5 → 16.2
Time: 6.7s
Precision: 64
\[1.05367121277235087 \cdot 10^{-8} \lt a \lt 94906265.6242515594 \land 1.05367121277235087 \cdot 10^{-8} \lt b \lt 94906265.6242515594 \land 1.05367121277235087 \cdot 10^{-8} \lt c \lt 94906265.6242515594\]
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le 904.955392750566375:\\ \;\;\;\;\frac{\frac{\left({b}^{2} - 3 \cdot \left(a \cdot c\right)\right) - {b}^{2}}{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} + b}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1.5}{3} \cdot \frac{\frac{a \cdot c}{b}}{a}\\ \end{array}\]
\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 904.955392750566375:\\
\;\;\;\;\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}:\\
\;\;\;\;\frac{-1.5}{3} \cdot \frac{\frac{a \cdot c}{b}}{a}\\

\end{array}
double f(double a, double b, double c) {
        double r78617 = b;
        double r78618 = -r78617;
        double r78619 = r78617 * r78617;
        double r78620 = 3.0;
        double r78621 = a;
        double r78622 = r78620 * r78621;
        double r78623 = c;
        double r78624 = r78622 * r78623;
        double r78625 = r78619 - r78624;
        double r78626 = sqrt(r78625);
        double r78627 = r78618 + r78626;
        double r78628 = r78627 / r78622;
        return r78628;
}
↓
double f(double a, double b, double c) {
        double r78629 = b;
        double r78630 = 904.9553927505664;
        bool r78631 = r78629 <= r78630;
        double r78632 = 2.0;
        double r78633 = pow(r78629, r78632);
        double r78634 = 3.0;
        double r78635 = a;
        double r78636 = c;
        double r78637 = r78635 * r78636;
        double r78638 = r78634 * r78637;
        double r78639 = r78633 - r78638;
        double r78640 = r78639 - r78633;
        double r78641 = r78629 * r78629;
        double r78642 = r78634 * r78635;
        double r78643 = r78642 * r78636;
        double r78644 = r78641 - r78643;
        double r78645 = sqrt(r78644);
        double r78646 = r78645 + r78629;
        double r78647 = r78640 / r78646;
        double r78648 = r78647 / r78642;
        double r78649 = -1.5;
        double r78650 = r78649 / r78634;
        double r78651 = r78637 / r78629;
        double r78652 = r78651 / r78635;
        double r78653 = r78650 * r78652;
        double r78654 = r78631 ? r78648 : r78653;
        return r78654;
}

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 < 904.9553927505664

    1. Initial program 17.2

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

    2. Simplified17.2

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

    4. Applied flip--17.2

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

    5. Simplified16.1

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

    if 904.9553927505664 < b

    1. Initial program 36.2

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

    2. Simplified36.2

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

    3. Taylor expanded around inf 16.4

      \[\leadsto \frac{\color{blue}{-1.5 \cdot \frac{a \cdot c}{b}}}{3 \cdot a}\]
    4. Using strategy rm

    5. Applied times-frac16.3

      \[\leadsto \color{blue}{\frac{-1.5}{3} \cdot \frac{\frac{a \cdot c}{b}}{a}}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification16.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le 904.955392750566375:\\ \;\;\;\;\frac{\frac{\left({b}^{2} - 3 \cdot \left(a \cdot c\right)\right) - {b}^{2}}{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} + b}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1.5}{3} \cdot \frac{\frac{a \cdot c}{b}}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2020042 
(FPCore (a b c)
  :name "Cubic critical, narrow range"
  :precision binary64
  :pre (and (< 1.0536712127723509e-08 a 94906265.62425156) (< 1.0536712127723509e-08 b 94906265.62425156) (< 1.0536712127723509e-08 c 94906265.62425156))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3 a) c)))) (* 3 a)))