Average Error: 5.1 → 1.0
Time: 26.6s
Precision: 64
\[\left(\left(\left(\left(\left(\left(x \cdot 18.0\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4.0\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4.0\right) \cdot i\right) - \left(j \cdot 27.0\right) \cdot k\]
↓
\[\begin{array}{l} \mathbf{if}\;\left(\left(t \cdot \left(\left(\left(x \cdot 18.0\right) \cdot y\right) \cdot z\right) - \left(a \cdot 4.0\right) \cdot t\right) + c \cdot b\right) - \left(x \cdot 4.0\right) \cdot i = -\infty:\\ \;\;\;\;\mathsf{fma}\left(b, c, \left(\left(t \cdot x\right) \cdot z\right) \cdot \left(y \cdot 18.0\right) - \mathsf{fma}\left(4.0, \mathsf{fma}\left(t, a, x \cdot i\right), \left(\sqrt{27.0} \cdot \left(k \cdot j\right)\right) \cdot \sqrt{27.0}\right)\right)\\ \mathbf{elif}\;\left(\left(t \cdot \left(\left(\left(x \cdot 18.0\right) \cdot y\right) \cdot z\right) - \left(a \cdot 4.0\right) \cdot t\right) + c \cdot b\right) - \left(x \cdot 4.0\right) \cdot i \le 5.9022068027212885 \cdot 10^{+286}:\\ \;\;\;\;\left(\left(\left(t \cdot \left(\left(\left(x \cdot 18.0\right) \cdot y\right) \cdot z\right) - \left(a \cdot 4.0\right) \cdot t\right) + c \cdot b\right) - \left(x \cdot 4.0\right) \cdot i\right) - \left(27.0 \cdot j\right) \cdot k\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(b, c, \left(\left(t \cdot x\right) \cdot z\right) \cdot \left(y \cdot 18.0\right) - \mathsf{fma}\left(4.0, \mathsf{fma}\left(t, a, x \cdot i\right), \left(\sqrt{27.0} \cdot \left(k \cdot j\right)\right) \cdot \sqrt{27.0}\right)\right)\\ \end{array}\]
\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}\;\left(\left(t \cdot \left(\left(\left(x \cdot 18.0\right) \cdot y\right) \cdot z\right) - \left(a \cdot 4.0\right) \cdot t\right) + c \cdot b\right) - \left(x \cdot 4.0\right) \cdot i = -\infty:\\
\;\;\;\;\mathsf{fma}\left(b, c, \left(\left(t \cdot x\right) \cdot z\right) \cdot \left(y \cdot 18.0\right) - \mathsf{fma}\left(4.0, \mathsf{fma}\left(t, a, x \cdot i\right), \left(\sqrt{27.0} \cdot \left(k \cdot j\right)\right) \cdot \sqrt{27.0}\right)\right)\\

\mathbf{elif}\;\left(\left(t \cdot \left(\left(\left(x \cdot 18.0\right) \cdot y\right) \cdot z\right) - \left(a \cdot 4.0\right) \cdot t\right) + c \cdot b\right) - \left(x \cdot 4.0\right) \cdot i \le 5.9022068027212885 \cdot 10^{+286}:\\
\;\;\;\;\left(\left(\left(t \cdot \left(\left(\left(x \cdot 18.0\right) \cdot y\right) \cdot z\right) - \left(a \cdot 4.0\right) \cdot t\right) + c \cdot b\right) - \left(x \cdot 4.0\right) \cdot i\right) - \left(27.0 \cdot j\right) \cdot k\\

\mathbf{else}:\\
\;\;\;\;\mathsf{fma}\left(b, c, \left(\left(t \cdot x\right) \cdot z\right) \cdot \left(y \cdot 18.0\right) - \mathsf{fma}\left(4.0, \mathsf{fma}\left(t, a, x \cdot i\right), \left(\sqrt{27.0} \cdot \left(k \cdot j\right)\right) \cdot \sqrt{27.0}\right)\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 r4672934 = x;
        double r4672935 = 18.0;
        double r4672936 = r4672934 * r4672935;
        double r4672937 = y;
        double r4672938 = r4672936 * r4672937;
        double r4672939 = z;
        double r4672940 = r4672938 * r4672939;
        double r4672941 = t;
        double r4672942 = r4672940 * r4672941;
        double r4672943 = a;
        double r4672944 = 4.0;
        double r4672945 = r4672943 * r4672944;
        double r4672946 = r4672945 * r4672941;
        double r4672947 = r4672942 - r4672946;
        double r4672948 = b;
        double r4672949 = c;
        double r4672950 = r4672948 * r4672949;
        double r4672951 = r4672947 + r4672950;
        double r4672952 = r4672934 * r4672944;
        double r4672953 = i;
        double r4672954 = r4672952 * r4672953;
        double r4672955 = r4672951 - r4672954;
        double r4672956 = j;
        double r4672957 = 27.0;
        double r4672958 = r4672956 * r4672957;
        double r4672959 = k;
        double r4672960 = r4672958 * r4672959;
        double r4672961 = r4672955 - r4672960;
        return r4672961;
}
↓
double f(double x, double y, double z, double t, double a, double b, double c, double i, double j, double k) {
        double r4672962 = t;
        double r4672963 = x;
        double r4672964 = 18.0;
        double r4672965 = r4672963 * r4672964;
        double r4672966 = y;
        double r4672967 = r4672965 * r4672966;
        double r4672968 = z;
        double r4672969 = r4672967 * r4672968;
        double r4672970 = r4672962 * r4672969;
        double r4672971 = a;
        double r4672972 = 4.0;
        double r4672973 = r4672971 * r4672972;
        double r4672974 = r4672973 * r4672962;
        double r4672975 = r4672970 - r4672974;
        double r4672976 = c;
        double r4672977 = b;
        double r4672978 = r4672976 * r4672977;
        double r4672979 = r4672975 + r4672978;
        double r4672980 = r4672963 * r4672972;
        double r4672981 = i;
        double r4672982 = r4672980 * r4672981;
        double r4672983 = r4672979 - r4672982;
        double r4672984 = -inf.0;
        bool r4672985 = r4672983 <= r4672984;
        double r4672986 = r4672962 * r4672963;
        double r4672987 = r4672986 * r4672968;
        double r4672988 = r4672966 * r4672964;
        double r4672989 = r4672987 * r4672988;
        double r4672990 = r4672963 * r4672981;
        double r4672991 = fma(r4672962, r4672971, r4672990);
        double r4672992 = 27.0;
        double r4672993 = sqrt(r4672992);
        double r4672994 = k;
        double r4672995 = j;
        double r4672996 = r4672994 * r4672995;
        double r4672997 = r4672993 * r4672996;
        double r4672998 = r4672997 * r4672993;
        double r4672999 = fma(r4672972, r4672991, r4672998);
        double r4673000 = r4672989 - r4672999;
        double r4673001 = fma(r4672977, r4672976, r4673000);
        double r4673002 = 5.9022068027212885e+286;
        bool r4673003 = r4672983 <= r4673002;
        double r4673004 = r4672992 * r4672995;
        double r4673005 = r4673004 * r4672994;
        double r4673006 = r4672983 - r4673005;
        double r4673007 = r4673003 ? r4673006 : r4673001;
        double r4673008 = r4672985 ? r4673001 : r4673007;
        return r4673008;
}

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 18.0) y) z) t) (* (* a 4.0) t)) (* b c)) (* (* x 4.0) i)) < -inf.0 or 5.9022068027212885e+286 < (- (+ (- (* (* (* (* x 18.0) y) z) t) (* (* a 4.0) t)) (* b c)) (* (* x 4.0) i))

    1. Initial program 44.6

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

    2. Simplified10.6

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

    4. Applied associate-*r*6.7

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

    6. Applied add-sqr-sqrt6.7

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

    7. Applied associate-*l*6.7

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

    if -inf.0 < (- (+ (- (* (* (* (* x 18.0) y) z) t) (* (* a 4.0) t)) (* b c)) (* (* x 4.0) i)) < 5.9022068027212885e+286

    1. Initial program 0.4

      \[\left(\left(\left(\left(\left(\left(x \cdot 18.0\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4.0\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4.0\right) \cdot i\right) - \left(j \cdot 27.0\right) \cdot k\]
  3. Recombined 2 regimes into one program.

  4. Final simplification1.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(t \cdot \left(\left(\left(x \cdot 18.0\right) \cdot y\right) \cdot z\right) - \left(a \cdot 4.0\right) \cdot t\right) + c \cdot b\right) - \left(x \cdot 4.0\right) \cdot i = -\infty:\\ \;\;\;\;\mathsf{fma}\left(b, c, \left(\left(t \cdot x\right) \cdot z\right) \cdot \left(y \cdot 18.0\right) - \mathsf{fma}\left(4.0, \mathsf{fma}\left(t, a, x \cdot i\right), \left(\sqrt{27.0} \cdot \left(k \cdot j\right)\right) \cdot \sqrt{27.0}\right)\right)\\ \mathbf{elif}\;\left(\left(t \cdot \left(\left(\left(x \cdot 18.0\right) \cdot y\right) \cdot z\right) - \left(a \cdot 4.0\right) \cdot t\right) + c \cdot b\right) - \left(x \cdot 4.0\right) \cdot i \le 5.9022068027212885 \cdot 10^{+286}:\\ \;\;\;\;\left(\left(\left(t \cdot \left(\left(\left(x \cdot 18.0\right) \cdot y\right) \cdot z\right) - \left(a \cdot 4.0\right) \cdot t\right) + c \cdot b\right) - \left(x \cdot 4.0\right) \cdot i\right) - \left(27.0 \cdot j\right) \cdot k\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(b, c, \left(\left(t \cdot x\right) \cdot z\right) \cdot \left(y \cdot 18.0\right) - \mathsf{fma}\left(4.0, \mathsf{fma}\left(t, a, x \cdot i\right), \left(\sqrt{27.0} \cdot \left(k \cdot j\right)\right) \cdot \sqrt{27.0}\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019164 +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)))