Average Error: 5.5 → 1.7
Time: 21.5s
Precision: 64
\[\left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -1.15848203748868583 \cdot 10^{-15} \lor \neg \left(x \le 5.554549893679635 \cdot 10^{-42}\right):\\ \;\;\;\;\left(\left(\left(\left(x \cdot 18\right) \cdot \left(y \cdot \left(z \cdot t\right)\right) - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(\sqrt[3]{\left(j \cdot 27\right) \cdot k} \cdot \sqrt[3]{\left(j \cdot 27\right) \cdot k}\right) \cdot \sqrt[3]{\left(j \cdot 27\right) \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t, \left(\left(x \cdot 18\right) \cdot y\right) \cdot z - a \cdot 4, b \cdot c - \mathsf{fma}\left(x, 4 \cdot i, \left(j \cdot 27\right) \cdot k\right)\right)\\ \end{array}\]
\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}\;x \le -1.15848203748868583 \cdot 10^{-15} \lor \neg \left(x \le 5.554549893679635 \cdot 10^{-42}\right):\\
\;\;\;\;\left(\left(\left(\left(x \cdot 18\right) \cdot \left(y \cdot \left(z \cdot t\right)\right) - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(\sqrt[3]{\left(j \cdot 27\right) \cdot k} \cdot \sqrt[3]{\left(j \cdot 27\right) \cdot k}\right) \cdot \sqrt[3]{\left(j \cdot 27\right) \cdot k}\\

\mathbf{else}:\\
\;\;\;\;\mathsf{fma}\left(t, \left(\left(x \cdot 18\right) \cdot y\right) \cdot z - a \cdot 4, b \cdot c - \mathsf{fma}\left(x, 4 \cdot i, \left(j \cdot 27\right) \cdot k\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 r806903 = x;
        double r806904 = 18.0;
        double r806905 = r806903 * r806904;
        double r806906 = y;
        double r806907 = r806905 * r806906;
        double r806908 = z;
        double r806909 = r806907 * r806908;
        double r806910 = t;
        double r806911 = r806909 * r806910;
        double r806912 = a;
        double r806913 = 4.0;
        double r806914 = r806912 * r806913;
        double r806915 = r806914 * r806910;
        double r806916 = r806911 - r806915;
        double r806917 = b;
        double r806918 = c;
        double r806919 = r806917 * r806918;
        double r806920 = r806916 + r806919;
        double r806921 = r806903 * r806913;
        double r806922 = i;
        double r806923 = r806921 * r806922;
        double r806924 = r806920 - r806923;
        double r806925 = j;
        double r806926 = 27.0;
        double r806927 = r806925 * r806926;
        double r806928 = k;
        double r806929 = r806927 * r806928;
        double r806930 = r806924 - r806929;
        return r806930;
}
↓
double f(double x, double y, double z, double t, double a, double b, double c, double i, double j, double k) {
        double r806931 = x;
        double r806932 = -1.1584820374886858e-15;
        bool r806933 = r806931 <= r806932;
        double r806934 = 5.554549893679635e-42;
        bool r806935 = r806931 <= r806934;
        double r806936 = !r806935;
        bool r806937 = r806933 || r806936;
        double r806938 = 18.0;
        double r806939 = r806931 * r806938;
        double r806940 = y;
        double r806941 = z;
        double r806942 = t;
        double r806943 = r806941 * r806942;
        double r806944 = r806940 * r806943;
        double r806945 = r806939 * r806944;
        double r806946 = a;
        double r806947 = 4.0;
        double r806948 = r806946 * r806947;
        double r806949 = r806948 * r806942;
        double r806950 = r806945 - r806949;
        double r806951 = b;
        double r806952 = c;
        double r806953 = r806951 * r806952;
        double r806954 = r806950 + r806953;
        double r806955 = r806931 * r806947;
        double r806956 = i;
        double r806957 = r806955 * r806956;
        double r806958 = r806954 - r806957;
        double r806959 = j;
        double r806960 = 27.0;
        double r806961 = r806959 * r806960;
        double r806962 = k;
        double r806963 = r806961 * r806962;
        double r806964 = cbrt(r806963);
        double r806965 = r806964 * r806964;
        double r806966 = r806965 * r806964;
        double r806967 = r806958 - r806966;
        double r806968 = r806939 * r806940;
        double r806969 = r806968 * r806941;
        double r806970 = r806969 - r806948;
        double r806971 = r806947 * r806956;
        double r806972 = fma(r806931, r806971, r806963);
        double r806973 = r806953 - r806972;
        double r806974 = fma(r806942, r806970, r806973);
        double r806975 = r806937 ? r806967 : r806974;
        return r806975;
}

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

Target

Original5.5
Target1.7
Herbie1.7
\[\begin{array}{l} \mathbf{if}\;t \lt -1.6210815397541398 \cdot 10^{-69}:\\ \;\;\;\;\left(\left(18 \cdot t\right) \cdot \left(\left(x \cdot y\right) \cdot z\right) - \left(a \cdot t + i \cdot x\right) \cdot 4\right) - \left(\left(k \cdot j\right) \cdot 27 - c \cdot b\right)\\ \mathbf{elif}\;t \lt 165.680279438052224:\\ \;\;\;\;\left(\left(18 \cdot y\right) \cdot \left(x \cdot \left(z \cdot t\right)\right) - \left(a \cdot t + i \cdot x\right) \cdot 4\right) + \left(c \cdot b - 27 \cdot \left(k \cdot j\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(18 \cdot t\right) \cdot \left(\left(x \cdot y\right) \cdot z\right) - \left(a \cdot t + i \cdot x\right) \cdot 4\right) - \left(\left(k \cdot j\right) \cdot 27 - c \cdot b\right)\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if x < -1.1584820374886858e-15 or 5.554549893679635e-42 < x

    1. Initial program 10.9

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

    3. Applied associate-*l*7.9

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

    5. Applied associate-*l*1.8

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

    7. Applied add-cube-cbrt1.9

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

    if -1.1584820374886858e-15 < x < 5.554549893679635e-42

    1. Initial program 1.5

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

    2. Simplified1.6

      \[\leadsto \color{blue}{\mathsf{fma}\left(t, \left(\left(x \cdot 18\right) \cdot y\right) \cdot z - a \cdot 4, b \cdot c - \mathsf{fma}\left(x, 4 \cdot i, \left(j \cdot 27\right) \cdot k\right)\right)}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification1.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.15848203748868583 \cdot 10^{-15} \lor \neg \left(x \le 5.554549893679635 \cdot 10^{-42}\right):\\ \;\;\;\;\left(\left(\left(\left(x \cdot 18\right) \cdot \left(y \cdot \left(z \cdot t\right)\right) - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(\sqrt[3]{\left(j \cdot 27\right) \cdot k} \cdot \sqrt[3]{\left(j \cdot 27\right) \cdot k}\right) \cdot \sqrt[3]{\left(j \cdot 27\right) \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t, \left(\left(x \cdot 18\right) \cdot y\right) \cdot z - a \cdot 4, b \cdot c - \mathsf{fma}\left(x, 4 \cdot i, \left(j \cdot 27\right) \cdot k\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020083 +o rules:numerics
(FPCore (x y z t a b c i j k)
  :name "Diagrams.Solve.Polynomial:cubForm  from diagrams-solve-0.1, E"
  :precision binary64

  :herbie-target
  (if (< t -1.6210815397541398e-69) (- (- (* (* 18 t) (* (* x y) z)) (* (+ (* a t) (* i x)) 4)) (- (* (* k j) 27) (* c b))) (if (< t 165.68027943805222) (+ (- (* (* 18 y) (* x (* z t))) (* (+ (* a t) (* i x)) 4)) (- (* c b) (* 27 (* k j)))) (- (- (* (* 18 t) (* (* x y) z)) (* (+ (* a t) (* i x)) 4)) (- (* (* k j) 27) (* c b)))))

  (- (- (+ (- (* (* (* (* x 18) y) z) t) (* (* a 4) t)) (* b c)) (* (* x 4) i)) (* (* j 27) k)))