Diagrams.Solve.Polynomial:cubForm from diagrams-solve-0.1

Average Error: 5.8 → 1.6

Time: 47.6s

Precision: 64

Math

\[\left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k\]

↓

\[\begin{array}{l} \mathbf{if}\;\left(\left(\left(t \cdot \left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) - \left(a \cdot 4\right) \cdot t\right) + c \cdot b\right) - \left(x \cdot 4\right) \cdot i\right) - \left(27 \cdot j\right) \cdot k \le -5.925285036171236791807809089141646750982 \cdot 10^{302}:\\ \;\;\;\;\mathsf{fma}\left(b, c, x \cdot \left(\left(\left(y \cdot t\right) \cdot 18\right) \cdot z - 4 \cdot i\right) - \left(k \cdot j\right) \cdot 27\right)\\ \mathbf{elif}\;\left(\left(\left(t \cdot \left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) - \left(a \cdot 4\right) \cdot t\right) + c \cdot b\right) - \left(x \cdot 4\right) \cdot i\right) - \left(27 \cdot j\right) \cdot k \le 1.027222039166035078175021985364091063442 \cdot 10^{298}:\\ \;\;\;\;\left(\left(\left(t \cdot \left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) - \left(a \cdot 4\right) \cdot t\right) + c \cdot b\right) - \left(x \cdot 4\right) \cdot i\right) - \left(27 \cdot j\right) \cdot k\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(b, c, \mathsf{fma}\left(-\sqrt[3]{\mathsf{fma}\left(4, \mathsf{fma}\left(t, a, x \cdot i\right), \left(k \cdot j\right) \cdot 27\right)}, \sqrt[3]{\mathsf{fma}\left(4, \mathsf{fma}\left(t, a, x \cdot i\right), \left(k \cdot j\right) \cdot 27\right)} \cdot \sqrt[3]{\mathsf{fma}\left(4, \mathsf{fma}\left(t, a, x \cdot i\right), \left(k \cdot j\right) \cdot 27\right)}, \sqrt[3]{\mathsf{fma}\left(4, \mathsf{fma}\left(t, a, x \cdot i\right), \left(k \cdot j\right) \cdot 27\right)} \cdot \left(\sqrt[3]{\mathsf{fma}\left(4, \mathsf{fma}\left(t, a, x \cdot i\right), \left(k \cdot j\right) \cdot 27\right)} \cdot \sqrt[3]{\mathsf{fma}\left(4, \mathsf{fma}\left(t, a, x \cdot i\right), \left(k \cdot j\right) \cdot 27\right)}\right)\right) + \mathsf{fma}\left(z, \left(t \cdot x\right) \cdot \left(y \cdot 18\right), \sqrt[3]{\mathsf{fma}\left(4, \mathsf{fma}\left(t, a, x \cdot i\right), \left(k \cdot j\right) \cdot 27\right)} \cdot \left(\left(-\sqrt[3]{\mathsf{fma}\left(4, \mathsf{fma}\left(t, a, x \cdot i\right), \left(k \cdot j\right) \cdot 27\right)}\right) \cdot \sqrt[3]{\mathsf{fma}\left(4, \mathsf{fma}\left(t, a, x \cdot i\right), \left(k \cdot j\right) \cdot 27\right)}\right)\right)\right)\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a, double b, double c, double i, double j, double k) {
        double r5401954 = x;
        double r5401955 = 18.0;
        double r5401956 = r5401954 * r5401955;
        double r5401957 = y;
        double r5401958 = r5401956 * r5401957;
        double r5401959 = z;
        double r5401960 = r5401958 * r5401959;
        double r5401961 = t;
        double r5401962 = r5401960 * r5401961;
        double r5401963 = a;
        double r5401964 = 4.0;
        double r5401965 = r5401963 * r5401964;
        double r5401966 = r5401965 * r5401961;
        double r5401967 = r5401962 - r5401966;
        double r5401968 = b;
        double r5401969 = c;
        double r5401970 = r5401968 * r5401969;
        double r5401971 = r5401967 + r5401970;
        double r5401972 = r5401954 * r5401964;
        double r5401973 = i;
        double r5401974 = r5401972 * r5401973;
        double r5401975 = r5401971 - r5401974;
        double r5401976 = j;
        double r5401977 = 27.0;
        double r5401978 = r5401976 * r5401977;
        double r5401979 = k;
        double r5401980 = r5401978 * r5401979;
        double r5401981 = r5401975 - r5401980;
        return r5401981;
}

↓

double f(double x, double y, double z, double t, double a, double b, double c, double i, double j, double k) {
        double r5401982 = t;
        double r5401983 = x;
        double r5401984 = 18.0;
        double r5401985 = r5401983 * r5401984;
        double r5401986 = y;
        double r5401987 = r5401985 * r5401986;
        double r5401988 = z;
        double r5401989 = r5401987 * r5401988;
        double r5401990 = r5401982 * r5401989;
        double r5401991 = a;
        double r5401992 = 4.0;
        double r5401993 = r5401991 * r5401992;
        double r5401994 = r5401993 * r5401982;
        double r5401995 = r5401990 - r5401994;
        double r5401996 = c;
        double r5401997 = b;
        double r5401998 = r5401996 * r5401997;
        double r5401999 = r5401995 + r5401998;
        double r5402000 = r5401983 * r5401992;
        double r5402001 = i;
        double r5402002 = r5402000 * r5402001;
        double r5402003 = r5401999 - r5402002;
        double r5402004 = 27.0;
        double r5402005 = j;
        double r5402006 = r5402004 * r5402005;
        double r5402007 = k;
        double r5402008 = r5402006 * r5402007;
        double r5402009 = r5402003 - r5402008;
        double r5402010 = -5.925285036171237e+302;
        bool r5402011 = r5402009 <= r5402010;
        double r5402012 = r5401986 * r5401982;
        double r5402013 = r5402012 * r5401984;
        double r5402014 = r5402013 * r5401988;
        double r5402015 = r5401992 * r5402001;
        double r5402016 = r5402014 - r5402015;
        double r5402017 = r5401983 * r5402016;
        double r5402018 = r5402007 * r5402005;
        double r5402019 = r5402018 * r5402004;
        double r5402020 = r5402017 - r5402019;
        double r5402021 = fma(r5401997, r5401996, r5402020);
        double r5402022 = 1.0272220391660351e+298;
        bool r5402023 = r5402009 <= r5402022;
        double r5402024 = r5401983 * r5402001;
        double r5402025 = fma(r5401982, r5401991, r5402024);
        double r5402026 = fma(r5401992, r5402025, r5402019);
        double r5402027 = cbrt(r5402026);
        double r5402028 = -r5402027;
        double r5402029 = r5402027 * r5402027;
        double r5402030 = r5402027 * r5402029;
        double r5402031 = fma(r5402028, r5402029, r5402030);
        double r5402032 = r5401982 * r5401983;
        double r5402033 = r5401986 * r5401984;
        double r5402034 = r5402032 * r5402033;
        double r5402035 = r5402028 * r5402027;
        double r5402036 = r5402027 * r5402035;
        double r5402037 = fma(r5401988, r5402034, r5402036);
        double r5402038 = r5402031 + r5402037;
        double r5402039 = fma(r5401997, r5401996, r5402038);
        double r5402040 = r5402023 ? r5402009 : r5402039;
        double r5402041 = r5402011 ? r5402021 : r5402040;
        return r5402041;
}

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 3 regimes

2. if (- (- (+ (- (* (* (* (* x 18.0) y) z) t) (* (* a 4.0) t)) (* b c)) (* (* x 4.0) i)) (* (* j 27.0) k)) < -5.925285036171237e+302

   1. Initial program 54.0

      \[\left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k\]

   2. Simplified 10.1

      \[\leadsto \color{blue}{\mathsf{fma}\left(b, c, z \cdot \left(\left(t \cdot x\right) \cdot \left(y \cdot 18\right)\right) - \mathsf{fma}\left(4, \mathsf{fma}\left(t, a, x \cdot i\right), 27 \cdot \left(k \cdot j\right)\right)\right)}\]

   3. Taylor expanded around inf 36.7

      \[\leadsto \mathsf{fma}\left(b, c, \color{blue}{18 \cdot \left(t \cdot \left(x \cdot \left(z \cdot y\right)\right)\right) - \left(27 \cdot \left(j \cdot k\right) + 4 \cdot \left(i \cdot x\right)\right)}\right)\]

   4. Simplified 10.6

      \[\leadsto \mathsf{fma}\left(b, c, \color{blue}{x \cdot \left(\left(\left(t \cdot y\right) \cdot 18\right) \cdot z - i \cdot 4\right) - \left(k \cdot j\right) \cdot 27}\right)\]

   if -5.925285036171237e+302 < (- (- (+ (- (* (* (* (* x 18.0) y) z) t) (* (* a 4.0) t)) (* b c)) (* (* x 4.0) i)) (* (* j 27.0) k)) < 1.0272220391660351e+298

   1. Initial program 0.2

      \[\left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k\]

   if 1.0272220391660351e+298 < (- (- (+ (- (* (* (* (* x 18.0) y) z) t) (* (* a 4.0) t)) (* b c)) (* (* x 4.0) i)) (* (* j 27.0) k))

   1. Initial program 44.2

      \[\left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k\]

   2. Simplified 12.6

      \[\leadsto \color{blue}{\mathsf{fma}\left(b, c, z \cdot \left(\left(t \cdot x\right) \cdot \left(y \cdot 18\right)\right) - \mathsf{fma}\left(4, \mathsf{fma}\left(t, a, x \cdot i\right), 27 \cdot \left(k \cdot j\right)\right)\right)}\]
   3. Using strategy rm

   4. Applied add-cube-cbrt 13.1

      \[\leadsto \mathsf{fma}\left(b, c, z \cdot \left(\left(t \cdot x\right) \cdot \left(y \cdot 18\right)\right) - \color{blue}{\left(\sqrt[3]{\mathsf{fma}\left(4, \mathsf{fma}\left(t, a, x \cdot i\right), 27 \cdot \left(k \cdot j\right)\right)} \cdot \sqrt[3]{\mathsf{fma}\left(4, \mathsf{fma}\left(t, a, x \cdot i\right), 27 \cdot \left(k \cdot j\right)\right)}\right) \cdot \sqrt[3]{\mathsf{fma}\left(4, \mathsf{fma}\left(t, a, x \cdot i\right), 27 \cdot \left(k \cdot j\right)\right)}}\right)\]

   5. Applied prod-diff 13.1

      \[\leadsto \mathsf{fma}\left(b, c, \color{blue}{\mathsf{fma}\left(z, \left(t \cdot x\right) \cdot \left(y \cdot 18\right), -\sqrt[3]{\mathsf{fma}\left(4, \mathsf{fma}\left(t, a, x \cdot i\right), 27 \cdot \left(k \cdot j\right)\right)} \cdot \left(\sqrt[3]{\mathsf{fma}\left(4, \mathsf{fma}\left(t, a, x \cdot i\right), 27 \cdot \left(k \cdot j\right)\right)} \cdot \sqrt[3]{\mathsf{fma}\left(4, \mathsf{fma}\left(t, a, x \cdot i\right), 27 \cdot \left(k \cdot j\right)\right)}\right)\right) + \mathsf{fma}\left(-\sqrt[3]{\mathsf{fma}\left(4, \mathsf{fma}\left(t, a, x \cdot i\right), 27 \cdot \left(k \cdot j\right)\right)}, \sqrt[3]{\mathsf{fma}\left(4, \mathsf{fma}\left(t, a, x \cdot i\right), 27 \cdot \left(k \cdot j\right)\right)} \cdot \sqrt[3]{\mathsf{fma}\left(4, \mathsf{fma}\left(t, a, x \cdot i\right), 27 \cdot \left(k \cdot j\right)\right)}, \sqrt[3]{\mathsf{fma}\left(4, \mathsf{fma}\left(t, a, x \cdot i\right), 27 \cdot \left(k \cdot j\right)\right)} \cdot \left(\sqrt[3]{\mathsf{fma}\left(4, \mathsf{fma}\left(t, a, x \cdot i\right), 27 \cdot \left(k \cdot j\right)\right)} \cdot \sqrt[3]{\mathsf{fma}\left(4, \mathsf{fma}\left(t, a, x \cdot i\right), 27 \cdot \left(k \cdot j\right)\right)}\right)\right)}\right)\]
3. Recombined 3 regimes into one program.

4. Final simplification 1.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(\left(t \cdot \left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) - \left(a \cdot 4\right) \cdot t\right) + c \cdot b\right) - \left(x \cdot 4\right) \cdot i\right) - \left(27 \cdot j\right) \cdot k \le -5.925285036171236791807809089141646750982 \cdot 10^{302}:\\ \;\;\;\;\mathsf{fma}\left(b, c, x \cdot \left(\left(\left(y \cdot t\right) \cdot 18\right) \cdot z - 4 \cdot i\right) - \left(k \cdot j\right) \cdot 27\right)\\ \mathbf{elif}\;\left(\left(\left(t \cdot \left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) - \left(a \cdot 4\right) \cdot t\right) + c \cdot b\right) - \left(x \cdot 4\right) \cdot i\right) - \left(27 \cdot j\right) \cdot k \le 1.027222039166035078175021985364091063442 \cdot 10^{298}:\\ \;\;\;\;\left(\left(\left(t \cdot \left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) - \left(a \cdot 4\right) \cdot t\right) + c \cdot b\right) - \left(x \cdot 4\right) \cdot i\right) - \left(27 \cdot j\right) \cdot k\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(b, c, \mathsf{fma}\left(-\sqrt[3]{\mathsf{fma}\left(4, \mathsf{fma}\left(t, a, x \cdot i\right), \left(k \cdot j\right) \cdot 27\right)}, \sqrt[3]{\mathsf{fma}\left(4, \mathsf{fma}\left(t, a, x \cdot i\right), \left(k \cdot j\right) \cdot 27\right)} \cdot \sqrt[3]{\mathsf{fma}\left(4, \mathsf{fma}\left(t, a, x \cdot i\right), \left(k \cdot j\right) \cdot 27\right)}, \sqrt[3]{\mathsf{fma}\left(4, \mathsf{fma}\left(t, a, x \cdot i\right), \left(k \cdot j\right) \cdot 27\right)} \cdot \left(\sqrt[3]{\mathsf{fma}\left(4, \mathsf{fma}\left(t, a, x \cdot i\right), \left(k \cdot j\right) \cdot 27\right)} \cdot \sqrt[3]{\mathsf{fma}\left(4, \mathsf{fma}\left(t, a, x \cdot i\right), \left(k \cdot j\right) \cdot 27\right)}\right)\right) + \mathsf{fma}\left(z, \left(t \cdot x\right) \cdot \left(y \cdot 18\right), \sqrt[3]{\mathsf{fma}\left(4, \mathsf{fma}\left(t, a, x \cdot i\right), \left(k \cdot j\right) \cdot 27\right)} \cdot \left(\left(-\sqrt[3]{\mathsf{fma}\left(4, \mathsf{fma}\left(t, a, x \cdot i\right), \left(k \cdot j\right) \cdot 27\right)}\right) \cdot \sqrt[3]{\mathsf{fma}\left(4, \mathsf{fma}\left(t, a, x \cdot i\right), \left(k \cdot j\right) \cdot 27\right)}\right)\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019171 +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)))