Average Error: 5.6 → 2.2
Time: 1.9m
Precision: 64
\[\left(\left(\left(\left(\left(\left(x \cdot 18.0\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4.0\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4.0\right) \cdot i\right) - \left(j \cdot 27.0\right) \cdot k\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -1.3352120350312125 \cdot 10^{+54}:\\ \;\;\;\;\left(\left(b \cdot c + \left(\left(\left(t \cdot z\right) \cdot \left(18.0 \cdot y\right)\right) \cdot x - t \cdot \left(a \cdot 4.0\right)\right)\right) - \left(4.0 \cdot x\right) \cdot i\right) - k \cdot \left(j \cdot 27.0\right)\\ \mathbf{elif}\;x \le 4.211241197926193 \cdot 10^{-142}:\\ \;\;\;\;\left(\mathsf{fma}\left(b, c, \left(t \cdot \left(18.0 \cdot \left(\left(y \cdot x\right) \cdot z\right) - a \cdot 4.0\right)\right)\right) - \left(4.0 \cdot i\right) \cdot x\right) - k \cdot \left(j \cdot 27.0\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(b \cdot c + \left(\left(\left(t \cdot z\right) \cdot \left(18.0 \cdot y\right)\right) \cdot x - t \cdot \left(a \cdot 4.0\right)\right)\right) - \left(4.0 \cdot x\right) \cdot i\right) - k \cdot \left(j \cdot 27.0\right)\\ \end{array}\]
\left(\left(\left(\left(\left(\left(x \cdot 18.0\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4.0\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4.0\right) \cdot i\right) - \left(j \cdot 27.0\right) \cdot k
↓
\begin{array}{l}
\mathbf{if}\;x \le -1.3352120350312125 \cdot 10^{+54}:\\
\;\;\;\;\left(\left(b \cdot c + \left(\left(\left(t \cdot z\right) \cdot \left(18.0 \cdot y\right)\right) \cdot x - t \cdot \left(a \cdot 4.0\right)\right)\right) - \left(4.0 \cdot x\right) \cdot i\right) - k \cdot \left(j \cdot 27.0\right)\\

\mathbf{elif}\;x \le 4.211241197926193 \cdot 10^{-142}:\\
\;\;\;\;\left(\mathsf{fma}\left(b, c, \left(t \cdot \left(18.0 \cdot \left(\left(y \cdot x\right) \cdot z\right) - a \cdot 4.0\right)\right)\right) - \left(4.0 \cdot i\right) \cdot x\right) - k \cdot \left(j \cdot 27.0\right)\\

\mathbf{else}:\\
\;\;\;\;\left(\left(b \cdot c + \left(\left(\left(t \cdot z\right) \cdot \left(18.0 \cdot y\right)\right) \cdot x - t \cdot \left(a \cdot 4.0\right)\right)\right) - \left(4.0 \cdot x\right) \cdot i\right) - k \cdot \left(j \cdot 27.0\right)\\

\end{array}
double f(double x, double y, double z, double t, double a, double b, double c, double i, double j, double k) {
        double r32386887 = x;
        double r32386888 = 18.0;
        double r32386889 = r32386887 * r32386888;
        double r32386890 = y;
        double r32386891 = r32386889 * r32386890;
        double r32386892 = z;
        double r32386893 = r32386891 * r32386892;
        double r32386894 = t;
        double r32386895 = r32386893 * r32386894;
        double r32386896 = a;
        double r32386897 = 4.0;
        double r32386898 = r32386896 * r32386897;
        double r32386899 = r32386898 * r32386894;
        double r32386900 = r32386895 - r32386899;
        double r32386901 = b;
        double r32386902 = c;
        double r32386903 = r32386901 * r32386902;
        double r32386904 = r32386900 + r32386903;
        double r32386905 = r32386887 * r32386897;
        double r32386906 = i;
        double r32386907 = r32386905 * r32386906;
        double r32386908 = r32386904 - r32386907;
        double r32386909 = j;
        double r32386910 = 27.0;
        double r32386911 = r32386909 * r32386910;
        double r32386912 = k;
        double r32386913 = r32386911 * r32386912;
        double r32386914 = r32386908 - r32386913;
        return r32386914;
}
↓
double f(double x, double y, double z, double t, double a, double b, double c, double i, double j, double k) {
        double r32386915 = x;
        double r32386916 = -1.3352120350312125e+54;
        bool r32386917 = r32386915 <= r32386916;
        double r32386918 = b;
        double r32386919 = c;
        double r32386920 = r32386918 * r32386919;
        double r32386921 = t;
        double r32386922 = z;
        double r32386923 = r32386921 * r32386922;
        double r32386924 = 18.0;
        double r32386925 = y;
        double r32386926 = r32386924 * r32386925;
        double r32386927 = r32386923 * r32386926;
        double r32386928 = r32386927 * r32386915;
        double r32386929 = a;
        double r32386930 = 4.0;
        double r32386931 = r32386929 * r32386930;
        double r32386932 = r32386921 * r32386931;
        double r32386933 = r32386928 - r32386932;
        double r32386934 = r32386920 + r32386933;
        double r32386935 = r32386930 * r32386915;
        double r32386936 = i;
        double r32386937 = r32386935 * r32386936;
        double r32386938 = r32386934 - r32386937;
        double r32386939 = k;
        double r32386940 = j;
        double r32386941 = 27.0;
        double r32386942 = r32386940 * r32386941;
        double r32386943 = r32386939 * r32386942;
        double r32386944 = r32386938 - r32386943;
        double r32386945 = 4.211241197926193e-142;
        bool r32386946 = r32386915 <= r32386945;
        double r32386947 = r32386925 * r32386915;
        double r32386948 = r32386947 * r32386922;
        double r32386949 = r32386924 * r32386948;
        double r32386950 = r32386949 - r32386931;
        double r32386951 = r32386921 * r32386950;
        double r32386952 = fma(r32386918, r32386919, r32386951);
        double r32386953 = r32386930 * r32386936;
        double r32386954 = r32386953 * r32386915;
        double r32386955 = r32386952 - r32386954;
        double r32386956 = r32386955 - r32386943;
        double r32386957 = r32386946 ? r32386956 : r32386944;
        double r32386958 = r32386917 ? r32386944 : r32386957;
        return r32386958;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Bits error versus b

Bits error versus c

Bits error versus i

Bits error versus j

Bits error versus k

Derivation

  1. Split input into 2 regimes

  2. if x < -1.3352120350312125e+54 or 4.211241197926193e-142 < x

    1. Initial program 10.3

      \[\left(\left(\left(\left(\left(\left(x \cdot 18.0\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4.0\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4.0\right) \cdot i\right) - \left(j \cdot 27.0\right) \cdot k\]
    2. Using strategy rm

    3. Applied associate-*l*7.8

      \[\leadsto \left(\left(\left(\color{blue}{\left(\left(x \cdot 18.0\right) \cdot y\right) \cdot \left(z \cdot t\right)} - \left(a \cdot 4.0\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4.0\right) \cdot i\right) - \left(j \cdot 27.0\right) \cdot k\]
    4. Using strategy rm

    5. Applied associate-*l*2.7

      \[\leadsto \left(\left(\left(\color{blue}{\left(x \cdot 18.0\right) \cdot \left(y \cdot \left(z \cdot t\right)\right)} - \left(a \cdot 4.0\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4.0\right) \cdot i\right) - \left(j \cdot 27.0\right) \cdot k\]
    6. Using strategy rm

    7. Applied associate-*l*2.6

      \[\leadsto \left(\left(\left(\color{blue}{x \cdot \left(18.0 \cdot \left(y \cdot \left(z \cdot t\right)\right)\right)} - \left(a \cdot 4.0\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4.0\right) \cdot i\right) - \left(j \cdot 27.0\right) \cdot k\]
    8. Using strategy rm

    9. Applied associate-*r*2.7

      \[\leadsto \left(\left(\left(x \cdot \color{blue}{\left(\left(18.0 \cdot y\right) \cdot \left(z \cdot t\right)\right)} - \left(a \cdot 4.0\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4.0\right) \cdot i\right) - \left(j \cdot 27.0\right) \cdot k\]

    if -1.3352120350312125e+54 < x < 4.211241197926193e-142

    1. Initial program 1.7

      \[\left(\left(\left(\left(\left(\left(x \cdot 18.0\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4.0\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4.0\right) \cdot i\right) - \left(j \cdot 27.0\right) \cdot k\]
    2. Using strategy rm

    3. Applied associate-*l*3.5

      \[\leadsto \left(\left(\left(\color{blue}{\left(\left(x \cdot 18.0\right) \cdot y\right) \cdot \left(z \cdot t\right)} - \left(a \cdot 4.0\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4.0\right) \cdot i\right) - \left(j \cdot 27.0\right) \cdot k\]
    4. Using strategy rm

    5. Applied associate-*l*6.6

      \[\leadsto \left(\left(\left(\color{blue}{\left(x \cdot 18.0\right) \cdot \left(y \cdot \left(z \cdot t\right)\right)} - \left(a \cdot 4.0\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4.0\right) \cdot i\right) - \left(j \cdot 27.0\right) \cdot k\]
    6. Using strategy rm

    7. Applied add-sqr-sqrt34.5

      \[\leadsto \left(\color{blue}{\sqrt{\left(\left(x \cdot 18.0\right) \cdot \left(y \cdot \left(z \cdot t\right)\right) - \left(a \cdot 4.0\right) \cdot t\right) + b \cdot c} \cdot \sqrt{\left(\left(x \cdot 18.0\right) \cdot \left(y \cdot \left(z \cdot t\right)\right) - \left(a \cdot 4.0\right) \cdot t\right) + b \cdot c}} - \left(x \cdot 4.0\right) \cdot i\right) - \left(j \cdot 27.0\right) \cdot k\]

    8. Applied prod-diff34.5

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(\left(\sqrt{\left(\left(x \cdot 18.0\right) \cdot \left(y \cdot \left(z \cdot t\right)\right) - \left(a \cdot 4.0\right) \cdot t\right) + b \cdot c}\right), \left(\sqrt{\left(\left(x \cdot 18.0\right) \cdot \left(y \cdot \left(z \cdot t\right)\right) - \left(a \cdot 4.0\right) \cdot t\right) + b \cdot c}\right), \left(-i \cdot \left(x \cdot 4.0\right)\right)\right) + \mathsf{fma}\left(\left(-i\right), \left(x \cdot 4.0\right), \left(i \cdot \left(x \cdot 4.0\right)\right)\right)\right)} - \left(j \cdot 27.0\right) \cdot k\]

    9. Simplified1.8

      \[\leadsto \left(\color{blue}{\left(\mathsf{fma}\left(b, c, \left(t \cdot \left(18.0 \cdot \left(\left(y \cdot x\right) \cdot z\right) - a \cdot 4.0\right)\right)\right) - \left(4.0 \cdot i\right) \cdot x\right)} + \mathsf{fma}\left(\left(-i\right), \left(x \cdot 4.0\right), \left(i \cdot \left(x \cdot 4.0\right)\right)\right)\right) - \left(j \cdot 27.0\right) \cdot k\]

    10. Simplified1.8

      \[\leadsto \left(\left(\mathsf{fma}\left(b, c, \left(t \cdot \left(18.0 \cdot \left(\left(y \cdot x\right) \cdot z\right) - a \cdot 4.0\right)\right)\right) - \left(4.0 \cdot i\right) \cdot x\right) + \color{blue}{0}\right) - \left(j \cdot 27.0\right) \cdot k\]
  3. Recombined 2 regimes into one program.

  4. Final simplification2.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.3352120350312125 \cdot 10^{+54}:\\ \;\;\;\;\left(\left(b \cdot c + \left(\left(\left(t \cdot z\right) \cdot \left(18.0 \cdot y\right)\right) \cdot x - t \cdot \left(a \cdot 4.0\right)\right)\right) - \left(4.0 \cdot x\right) \cdot i\right) - k \cdot \left(j \cdot 27.0\right)\\ \mathbf{elif}\;x \le 4.211241197926193 \cdot 10^{-142}:\\ \;\;\;\;\left(\mathsf{fma}\left(b, c, \left(t \cdot \left(18.0 \cdot \left(\left(y \cdot x\right) \cdot z\right) - a \cdot 4.0\right)\right)\right) - \left(4.0 \cdot i\right) \cdot x\right) - k \cdot \left(j \cdot 27.0\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(b \cdot c + \left(\left(\left(t \cdot z\right) \cdot \left(18.0 \cdot y\right)\right) \cdot x - t \cdot \left(a \cdot 4.0\right)\right)\right) - \left(4.0 \cdot x\right) \cdot i\right) - k \cdot \left(j \cdot 27.0\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019125 +o rules:numerics
(FPCore (x y z t a b c i j k)
  :name "Diagrams.Solve.Polynomial:cubForm  from diagrams-solve-0.1"
  (- (- (+ (- (* (* (* (* x 18.0) y) z) t) (* (* a 4.0) t)) (* b c)) (* (* x 4.0) i)) (* (* j 27.0) k)))