Average Error: 33.6 → 10.4
Time: 24.2s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -1.3282248930815427 \cdot 10^{+154}:\\ \;\;\;\;\frac{1}{\frac{a}{c \cdot \left(a \cdot -3\right)} \cdot \left(\left(\frac{3}{2} \cdot \frac{c \cdot a}{b}\right) \cdot 3\right)}\\ \mathbf{elif}\;b \le 1.1685637944370405 \cdot 10^{-201}:\\ \;\;\;\;\frac{\frac{\sqrt{\mathsf{fma}\left(a \cdot -3, c, b \cdot b\right)} - b}{3}}{a}\\ \mathbf{elif}\;b \le 1.5879900645620077 \cdot 10^{+150}:\\ \;\;\;\;\frac{1}{\left(3 \cdot \left(\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)} + b\right)\right) \cdot \frac{\frac{-1}{3}}{c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{a}{c \cdot \left(a \cdot -3\right)} \cdot \left(3 \cdot \left(b + \mathsf{fma}\left(\frac{-3}{2}, \frac{a}{\frac{b}{c}}, b\right)\right)\right)}\\ \end{array}\]
\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 -1.3282248930815427 \cdot 10^{+154}:\\
\;\;\;\;\frac{1}{\frac{a}{c \cdot \left(a \cdot -3\right)} \cdot \left(\left(\frac{3}{2} \cdot \frac{c \cdot a}{b}\right) \cdot 3\right)}\\

\mathbf{elif}\;b \le 1.1685637944370405 \cdot 10^{-201}:\\
\;\;\;\;\frac{\frac{\sqrt{\mathsf{fma}\left(a \cdot -3, c, b \cdot b\right)} - b}{3}}{a}\\

\mathbf{elif}\;b \le 1.5879900645620077 \cdot 10^{+150}:\\
\;\;\;\;\frac{1}{\left(3 \cdot \left(\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)} + b\right)\right) \cdot \frac{\frac{-1}{3}}{c}}\\

\mathbf{else}:\\
\;\;\;\;\frac{1}{\frac{a}{c \cdot \left(a \cdot -3\right)} \cdot \left(3 \cdot \left(b + \mathsf{fma}\left(\frac{-3}{2}, \frac{a}{\frac{b}{c}}, b\right)\right)\right)}\\

\end{array}
double f(double a, double b, double c) {
        double r6582759 = b;
        double r6582760 = -r6582759;
        double r6582761 = r6582759 * r6582759;
        double r6582762 = 3.0;
        double r6582763 = a;
        double r6582764 = r6582762 * r6582763;
        double r6582765 = c;
        double r6582766 = r6582764 * r6582765;
        double r6582767 = r6582761 - r6582766;
        double r6582768 = sqrt(r6582767);
        double r6582769 = r6582760 + r6582768;
        double r6582770 = r6582769 / r6582764;
        return r6582770;
}
↓
double f(double a, double b, double c) {
        double r6582771 = b;
        double r6582772 = -1.3282248930815427e+154;
        bool r6582773 = r6582771 <= r6582772;
        double r6582774 = 1.0;
        double r6582775 = a;
        double r6582776 = c;
        double r6582777 = -3.0;
        double r6582778 = r6582775 * r6582777;
        double r6582779 = r6582776 * r6582778;
        double r6582780 = r6582775 / r6582779;
        double r6582781 = 1.5;
        double r6582782 = r6582776 * r6582775;
        double r6582783 = r6582782 / r6582771;
        double r6582784 = r6582781 * r6582783;
        double r6582785 = 3.0;
        double r6582786 = r6582784 * r6582785;
        double r6582787 = r6582780 * r6582786;
        double r6582788 = r6582774 / r6582787;
        double r6582789 = 1.1685637944370405e-201;
        bool r6582790 = r6582771 <= r6582789;
        double r6582791 = r6582771 * r6582771;
        double r6582792 = fma(r6582778, r6582776, r6582791);
        double r6582793 = sqrt(r6582792);
        double r6582794 = r6582793 - r6582771;
        double r6582795 = r6582794 / r6582785;
        double r6582796 = r6582795 / r6582775;
        double r6582797 = 1.5879900645620077e+150;
        bool r6582798 = r6582771 <= r6582797;
        double r6582799 = fma(r6582771, r6582771, r6582779);
        double r6582800 = sqrt(r6582799);
        double r6582801 = r6582800 + r6582771;
        double r6582802 = r6582785 * r6582801;
        double r6582803 = -0.3333333333333333;
        double r6582804 = r6582803 / r6582776;
        double r6582805 = r6582802 * r6582804;
        double r6582806 = r6582774 / r6582805;
        double r6582807 = -1.5;
        double r6582808 = r6582771 / r6582776;
        double r6582809 = r6582775 / r6582808;
        double r6582810 = fma(r6582807, r6582809, r6582771);
        double r6582811 = r6582771 + r6582810;
        double r6582812 = r6582785 * r6582811;
        double r6582813 = r6582780 * r6582812;
        double r6582814 = r6582774 / r6582813;
        double r6582815 = r6582798 ? r6582806 : r6582814;
        double r6582816 = r6582790 ? r6582796 : r6582815;
        double r6582817 = r6582773 ? r6582788 : r6582816;
        return r6582817;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

  1. Split input into 4 regimes

  2. if b < -1.3282248930815427e+154

    1. Initial program 60.9

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

    2. Simplified60.9

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

    4. Applied associate-/r*60.9

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

    6. Applied flip--62.3

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

    7. Simplified62.5

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

    9. Applied clear-num62.5

      \[\leadsto \color{blue}{\frac{1}{\frac{a}{\frac{\frac{\mathsf{fma}\left(c, -3 \cdot a, 0\right)}{\sqrt{\mathsf{fma}\left(-3 \cdot a, c, b \cdot b\right)} + b}}{3}}}}\]

    10. Simplified62.5

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

    11. Taylor expanded around -inf 21.3

      \[\leadsto \frac{1}{\frac{a}{\left(-3 \cdot a\right) \cdot c} \cdot \left(3 \cdot \color{blue}{\left(\frac{3}{2} \cdot \frac{a \cdot c}{b}\right)}\right)}\]

    if -1.3282248930815427e+154 < b < 1.1685637944370405e-201

    1. Initial program 10.6

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

    2. Simplified10.6

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

    4. Applied associate-/r*10.6

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

    if 1.1685637944370405e-201 < b < 1.5879900645620077e+150

    1. Initial program 37.2

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

    2. Simplified37.2

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

    4. Applied associate-/r*37.2

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

    6. Applied flip--37.3

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

    7. Simplified15.6

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

    9. Applied clear-num15.9

      \[\leadsto \color{blue}{\frac{1}{\frac{a}{\frac{\frac{\mathsf{fma}\left(c, -3 \cdot a, 0\right)}{\sqrt{\mathsf{fma}\left(-3 \cdot a, c, b \cdot b\right)} + b}}{3}}}}\]

    10. Simplified14.3

      \[\leadsto \frac{1}{\color{blue}{\frac{a}{\left(-3 \cdot a\right) \cdot c} \cdot \left(3 \cdot \left(\sqrt{\mathsf{fma}\left(b, b, \left(-3 \cdot a\right) \cdot c\right)} + b\right)\right)}}\]
    11. Using strategy rm

    12. Applied associate-/r*7.6

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

    13. Simplified7.5

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

    if 1.5879900645620077e+150 < b

    1. Initial program 62.4

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

    2. Simplified62.4

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

    4. Applied associate-/r*62.4

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

    6. Applied flip--62.5

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

    7. Simplified37.8

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

    9. Applied clear-num37.8

      \[\leadsto \color{blue}{\frac{1}{\frac{a}{\frac{\frac{\mathsf{fma}\left(c, -3 \cdot a, 0\right)}{\sqrt{\mathsf{fma}\left(-3 \cdot a, c, b \cdot b\right)} + b}}{3}}}}\]

    10. Simplified37.8

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

    11. Taylor expanded around inf 8.2

      \[\leadsto \frac{1}{\frac{a}{\left(-3 \cdot a\right) \cdot c} \cdot \left(3 \cdot \left(\color{blue}{\left(b - \frac{3}{2} \cdot \frac{a \cdot c}{b}\right)} + b\right)\right)}\]

    12. Simplified8.2

      \[\leadsto \frac{1}{\frac{a}{\left(-3 \cdot a\right) \cdot c} \cdot \left(3 \cdot \left(\color{blue}{\mathsf{fma}\left(\frac{-3}{2}, \frac{a}{\frac{b}{c}}, b\right)} + b\right)\right)}\]
  3. Recombined 4 regimes into one program.

  4. Final simplification10.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -1.3282248930815427 \cdot 10^{+154}:\\ \;\;\;\;\frac{1}{\frac{a}{c \cdot \left(a \cdot -3\right)} \cdot \left(\left(\frac{3}{2} \cdot \frac{c \cdot a}{b}\right) \cdot 3\right)}\\ \mathbf{elif}\;b \le 1.1685637944370405 \cdot 10^{-201}:\\ \;\;\;\;\frac{\frac{\sqrt{\mathsf{fma}\left(a \cdot -3, c, b \cdot b\right)} - b}{3}}{a}\\ \mathbf{elif}\;b \le 1.5879900645620077 \cdot 10^{+150}:\\ \;\;\;\;\frac{1}{\left(3 \cdot \left(\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)} + b\right)\right) \cdot \frac{\frac{-1}{3}}{c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{a}{c \cdot \left(a \cdot -3\right)} \cdot \left(3 \cdot \left(b + \mathsf{fma}\left(\frac{-3}{2}, \frac{a}{\frac{b}{c}}, b\right)\right)\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019163 +o rules:numerics
(FPCore (a b c)
  :name "Cubic critical"
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3 a) c)))) (* 3 a)))