Average Error: 28.7 → 15.2
Time: 4.6s
Precision: 64
\[1.05367121277235087 \cdot 10^{-8} \lt a \lt 94906265.6242515594 \land 1.05367121277235087 \cdot 10^{-8} \lt b \lt 94906265.6242515594 \land 1.05367121277235087 \cdot 10^{-8} \lt c \lt 94906265.6242515594\]
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \le -1.0303071264349629 \cdot 10^{-4}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(b, b, -\left(b \cdot b - \left(3 \cdot a\right) \cdot c\right)\right)}{\left(-b\right) - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{a \cdot c}{b}}{\frac{3}{-1.5}}}{a}\\ \end{array}\]
\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}
↓
\begin{array}{l}
\mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \le -1.0303071264349629 \cdot 10^{-4}:\\
\;\;\;\;\frac{\frac{\mathsf{fma}\left(b, b, -\left(b \cdot b - \left(3 \cdot a\right) \cdot c\right)\right)}{\left(-b\right) - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}}{3 \cdot a}\\

\mathbf{else}:\\
\;\;\;\;\frac{\frac{\frac{a \cdot c}{b}}{\frac{3}{-1.5}}}{a}\\

\end{array}
double f(double a, double b, double c) {
        double r83739 = b;
        double r83740 = -r83739;
        double r83741 = r83739 * r83739;
        double r83742 = 3.0;
        double r83743 = a;
        double r83744 = r83742 * r83743;
        double r83745 = c;
        double r83746 = r83744 * r83745;
        double r83747 = r83741 - r83746;
        double r83748 = sqrt(r83747);
        double r83749 = r83740 + r83748;
        double r83750 = r83749 / r83744;
        return r83750;
}
↓
double f(double a, double b, double c) {
        double r83751 = b;
        double r83752 = -r83751;
        double r83753 = r83751 * r83751;
        double r83754 = 3.0;
        double r83755 = a;
        double r83756 = r83754 * r83755;
        double r83757 = c;
        double r83758 = r83756 * r83757;
        double r83759 = r83753 - r83758;
        double r83760 = sqrt(r83759);
        double r83761 = r83752 + r83760;
        double r83762 = r83761 / r83756;
        double r83763 = -0.00010303071264349629;
        bool r83764 = r83762 <= r83763;
        double r83765 = -r83759;
        double r83766 = fma(r83751, r83751, r83765);
        double r83767 = r83752 - r83760;
        double r83768 = r83766 / r83767;
        double r83769 = r83768 / r83756;
        double r83770 = r83755 * r83757;
        double r83771 = r83770 / r83751;
        double r83772 = -1.5;
        double r83773 = r83754 / r83772;
        double r83774 = r83771 / r83773;
        double r83775 = r83774 / r83755;
        double r83776 = r83764 ? r83769 : r83775;
        return r83776;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

  1. Split input into 2 regimes

  2. if (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)) < -0.00010303071264349629

    1. Initial program 16.0

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

    3. Applied flip-+16.0

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

    4. Simplified15.3

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

    if -0.00010303071264349629 < (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a))

    1. Initial program 37.8

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

    2. Taylor expanded around inf 15.1

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

    4. Applied clear-num15.1

      \[\leadsto \frac{-1.5 \cdot \color{blue}{\frac{1}{\frac{b}{a \cdot c}}}}{3 \cdot a}\]
    5. Using strategy rm

    6. Applied associate-/r*15.1

      \[\leadsto \color{blue}{\frac{\frac{-1.5 \cdot \frac{1}{\frac{b}{a \cdot c}}}{3}}{a}}\]

    7. Simplified15.1

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

  4. Final simplification15.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \le -1.0303071264349629 \cdot 10^{-4}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(b, b, -\left(b \cdot b - \left(3 \cdot a\right) \cdot c\right)\right)}{\left(-b\right) - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{a \cdot c}{b}}{\frac{3}{-1.5}}}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2020046 +o rules:numerics
(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)))