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

Average Error: 5.7 → 2.3

Time: 7.4s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \le -2.5984752602216353 \cdot 10^{86} \lor \neg \left(x \le 1.33716421209114374 \cdot 10^{-48}\right):\\ \;\;\;\;\left(\left(\left(\left(x \cdot 18\right) \cdot \left(\left(y \cdot z\right) \cdot t\right) - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - j \cdot \left(27 \cdot k\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - j \cdot \left(27 \cdot k\right)\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a, double b, double c, double i, double j, double k) {
        double r704883 = x;
        double r704884 = 18.0;
        double r704885 = r704883 * r704884;
        double r704886 = y;
        double r704887 = r704885 * r704886;
        double r704888 = z;
        double r704889 = r704887 * r704888;
        double r704890 = t;
        double r704891 = r704889 * r704890;
        double r704892 = a;
        double r704893 = 4.0;
        double r704894 = r704892 * r704893;
        double r704895 = r704894 * r704890;
        double r704896 = r704891 - r704895;
        double r704897 = b;
        double r704898 = c;
        double r704899 = r704897 * r704898;
        double r704900 = r704896 + r704899;
        double r704901 = r704883 * r704893;
        double r704902 = i;
        double r704903 = r704901 * r704902;
        double r704904 = r704900 - r704903;
        double r704905 = j;
        double r704906 = 27.0;
        double r704907 = r704905 * r704906;
        double r704908 = k;
        double r704909 = r704907 * r704908;
        double r704910 = r704904 - r704909;
        return r704910;
}

↓

double f(double x, double y, double z, double t, double a, double b, double c, double i, double j, double k) {
        double r704911 = x;
        double r704912 = -2.5984752602216353e+86;
        bool r704913 = r704911 <= r704912;
        double r704914 = 1.3371642120911437e-48;
        bool r704915 = r704911 <= r704914;
        double r704916 = !r704915;
        bool r704917 = r704913 || r704916;
        double r704918 = 18.0;
        double r704919 = r704911 * r704918;
        double r704920 = y;
        double r704921 = z;
        double r704922 = r704920 * r704921;
        double r704923 = t;
        double r704924 = r704922 * r704923;
        double r704925 = r704919 * r704924;
        double r704926 = a;
        double r704927 = 4.0;
        double r704928 = r704926 * r704927;
        double r704929 = r704928 * r704923;
        double r704930 = r704925 - r704929;
        double r704931 = b;
        double r704932 = c;
        double r704933 = r704931 * r704932;
        double r704934 = r704930 + r704933;
        double r704935 = r704911 * r704927;
        double r704936 = i;
        double r704937 = r704935 * r704936;
        double r704938 = r704934 - r704937;
        double r704939 = j;
        double r704940 = 27.0;
        double r704941 = k;
        double r704942 = r704940 * r704941;
        double r704943 = r704939 * r704942;
        double r704944 = r704938 - r704943;
        double r704945 = r704919 * r704920;
        double r704946 = r704945 * r704921;
        double r704947 = r704946 * r704923;
        double r704948 = r704947 - r704929;
        double r704949 = r704948 + r704933;
        double r704950 = r704949 - r704937;
        double r704951 = r704950 - r704943;
        double r704952 = r704917 ? r704944 : r704951;
        return r704952;
}

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

Original	5.7
Target	1.6
Herbie	2.3

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

Derivation

1. Split input into 2 regimes

2. if x < -2.5984752602216353e+86 or 1.3371642120911437e-48 < x

   1. Initial program 12.4

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

   3. Applied associate-*l* 12.4

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

   5. Applied associate-*l* 7.3

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

   7. Applied associate-*l* 2.4

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

   if -2.5984752602216353e+86 < x < 1.3371642120911437e-48

   1. Initial program 2.2

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

   3. Applied associate-*l* 2.2

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

4. Final simplification 2.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -2.5984752602216353 \cdot 10^{86} \lor \neg \left(x \le 1.33716421209114374 \cdot 10^{-48}\right):\\ \;\;\;\;\left(\left(\left(\left(x \cdot 18\right) \cdot \left(\left(y \cdot z\right) \cdot t\right) - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - j \cdot \left(27 \cdot k\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - j \cdot \left(27 \cdot k\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020081 
(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)))