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

Average Error: 5.8 → 1.0

Time: 40.0s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i \le -1.591976256934889034187455180953636735584 \cdot 10^{302} \lor \neg \left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i \le 2.618961756211518564383950228017988163092 \cdot 10^{294}\right):\\ \;\;\;\;\mathsf{fma}\left(\left(\left(z \cdot t\right) \cdot y\right) \cdot x, 18, \mathsf{fma}\left(c, b, -\mathsf{fma}\left(4, \mathsf{fma}\left(t, a, x \cdot i\right), \left(j \cdot 27\right) \cdot k\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a, double b, double c, double i, double j, double k) {
        double r515901 = x;
        double r515902 = 18.0;
        double r515903 = r515901 * r515902;
        double r515904 = y;
        double r515905 = r515903 * r515904;
        double r515906 = z;
        double r515907 = r515905 * r515906;
        double r515908 = t;
        double r515909 = r515907 * r515908;
        double r515910 = a;
        double r515911 = 4.0;
        double r515912 = r515910 * r515911;
        double r515913 = r515912 * r515908;
        double r515914 = r515909 - r515913;
        double r515915 = b;
        double r515916 = c;
        double r515917 = r515915 * r515916;
        double r515918 = r515914 + r515917;
        double r515919 = r515901 * r515911;
        double r515920 = i;
        double r515921 = r515919 * r515920;
        double r515922 = r515918 - r515921;
        double r515923 = j;
        double r515924 = 27.0;
        double r515925 = r515923 * r515924;
        double r515926 = k;
        double r515927 = r515925 * r515926;
        double r515928 = r515922 - r515927;
        return r515928;
}

↓

double f(double x, double y, double z, double t, double a, double b, double c, double i, double j, double k) {
        double r515929 = x;
        double r515930 = 18.0;
        double r515931 = r515929 * r515930;
        double r515932 = y;
        double r515933 = r515931 * r515932;
        double r515934 = z;
        double r515935 = r515933 * r515934;
        double r515936 = t;
        double r515937 = r515935 * r515936;
        double r515938 = a;
        double r515939 = 4.0;
        double r515940 = r515938 * r515939;
        double r515941 = r515940 * r515936;
        double r515942 = r515937 - r515941;
        double r515943 = b;
        double r515944 = c;
        double r515945 = r515943 * r515944;
        double r515946 = r515942 + r515945;
        double r515947 = r515929 * r515939;
        double r515948 = i;
        double r515949 = r515947 * r515948;
        double r515950 = r515946 - r515949;
        double r515951 = -1.591976256934889e+302;
        bool r515952 = r515950 <= r515951;
        double r515953 = 2.6189617562115186e+294;
        bool r515954 = r515950 <= r515953;
        double r515955 = !r515954;
        bool r515956 = r515952 || r515955;
        double r515957 = r515934 * r515936;
        double r515958 = r515957 * r515932;
        double r515959 = r515958 * r515929;
        double r515960 = r515929 * r515948;
        double r515961 = fma(r515936, r515938, r515960);
        double r515962 = j;
        double r515963 = 27.0;
        double r515964 = r515962 * r515963;
        double r515965 = k;
        double r515966 = r515964 * r515965;
        double r515967 = fma(r515939, r515961, r515966);
        double r515968 = -r515967;
        double r515969 = fma(r515944, r515943, r515968);
        double r515970 = fma(r515959, r515930, r515969);
        double r515971 = r515950 - r515966;
        double r515972 = r515956 ? r515970 : r515971;
        return r515972;
}

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	5.8
Target	1.6
Herbie	1.0

\[\begin{array}{l} \mathbf{if}\;t \lt -1.62108153975413982700795070153457058168 \cdot 10^{-69}:\\ \;\;\;\;\left(\left(18 \cdot t\right) \cdot \left(\left(x \cdot y\right) \cdot z\right) - \left(a \cdot t + i \cdot x\right) \cdot 4\right) - \left(\left(k \cdot j\right) \cdot 27 - c \cdot b\right)\\ \mathbf{elif}\;t \lt 165.6802794380522243500308832153677940369:\\ \;\;\;\;\left(\left(18 \cdot y\right) \cdot \left(x \cdot \left(z \cdot t\right)\right) - \left(a \cdot t + i \cdot x\right) \cdot 4\right) + \left(c \cdot b - 27 \cdot \left(k \cdot j\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(18 \cdot t\right) \cdot \left(\left(x \cdot y\right) \cdot z\right) - \left(a \cdot t + i \cdot x\right) \cdot 4\right) - \left(\left(k \cdot j\right) \cdot 27 - c \cdot b\right)\\ \end{array}\]

Derivation

1. Split input into 2 regimes

2. if (- (+ (- (* (* (* (* x 18.0) y) z) t) (* (* a 4.0) t)) (* b c)) (* (* x 4.0) i)) < -1.591976256934889e+302 or 2.6189617562115186e+294 < (- (+ (- (* (* (* (* x 18.0) y) z) t) (* (* a 4.0) t)) (* b c)) (* (* x 4.0) i))

   1. Initial program 49.0

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

   2. Simplified 13.5

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

   4. Applied associate-*r* 7.4

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

   6. Applied associate-*l* 16.5

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

   7. Simplified 16.5

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

   9. Applied associate-*r* 6.0

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

   10. Simplified 6.0

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

   if -1.591976256934889e+302 < (- (+ (- (* (* (* (* x 18.0) y) z) t) (* (* a 4.0) t)) (* b c)) (* (* x 4.0) i)) < 2.6189617562115186e+294

   1. Initial program 0.3

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

4. Final simplification 1.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i \le -1.591976256934889034187455180953636735584 \cdot 10^{302} \lor \neg \left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i \le 2.618961756211518564383950228017988163092 \cdot 10^{294}\right):\\ \;\;\;\;\mathsf{fma}\left(\left(\left(z \cdot t\right) \cdot y\right) \cdot x, 18, \mathsf{fma}\left(c, b, -\mathsf{fma}\left(4, \mathsf{fma}\left(t, a, x \cdot i\right), \left(j \cdot 27\right) \cdot k\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k\\ \end{array}\]

Reproduce

herbie shell --seed 2019325 +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)))