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

Average Error: 5.6 → 1.9

Time: 24.6s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;y \le -4.8824004811172388 \cdot 10^{97} \lor \neg \left(y \le 7.21743865698620678 \cdot 10^{-28}\right):\\ \;\;\;\;\left(\left(\left(18 \cdot \left(\left(t \cdot \left(x \cdot z\right)\right) \cdot y\right) - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - j \cdot \left(27 \cdot k\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(\left(18 \cdot t\right) \cdot \left(x \cdot \left(z \cdot y\right)\right) - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a, double b, double c, double i, double j, double k) {
        double r1429940 = x;
        double r1429941 = 18.0;
        double r1429942 = r1429940 * r1429941;
        double r1429943 = y;
        double r1429944 = r1429942 * r1429943;
        double r1429945 = z;
        double r1429946 = r1429944 * r1429945;
        double r1429947 = t;
        double r1429948 = r1429946 * r1429947;
        double r1429949 = a;
        double r1429950 = 4.0;
        double r1429951 = r1429949 * r1429950;
        double r1429952 = r1429951 * r1429947;
        double r1429953 = r1429948 - r1429952;
        double r1429954 = b;
        double r1429955 = c;
        double r1429956 = r1429954 * r1429955;
        double r1429957 = r1429953 + r1429956;
        double r1429958 = r1429940 * r1429950;
        double r1429959 = i;
        double r1429960 = r1429958 * r1429959;
        double r1429961 = r1429957 - r1429960;
        double r1429962 = j;
        double r1429963 = 27.0;
        double r1429964 = r1429962 * r1429963;
        double r1429965 = k;
        double r1429966 = r1429964 * r1429965;
        double r1429967 = r1429961 - r1429966;
        return r1429967;
}

↓

double f(double x, double y, double z, double t, double a, double b, double c, double i, double j, double k) {
        double r1429968 = y;
        double r1429969 = -4.882400481117239e+97;
        bool r1429970 = r1429968 <= r1429969;
        double r1429971 = 7.217438656986207e-28;
        bool r1429972 = r1429968 <= r1429971;
        double r1429973 = !r1429972;
        bool r1429974 = r1429970 || r1429973;
        double r1429975 = 18.0;
        double r1429976 = t;
        double r1429977 = x;
        double r1429978 = z;
        double r1429979 = r1429977 * r1429978;
        double r1429980 = r1429976 * r1429979;
        double r1429981 = r1429980 * r1429968;
        double r1429982 = r1429975 * r1429981;
        double r1429983 = a;
        double r1429984 = 4.0;
        double r1429985 = r1429983 * r1429984;
        double r1429986 = r1429985 * r1429976;
        double r1429987 = r1429982 - r1429986;
        double r1429988 = b;
        double r1429989 = c;
        double r1429990 = r1429988 * r1429989;
        double r1429991 = r1429987 + r1429990;
        double r1429992 = r1429977 * r1429984;
        double r1429993 = i;
        double r1429994 = r1429992 * r1429993;
        double r1429995 = r1429991 - r1429994;
        double r1429996 = j;
        double r1429997 = 27.0;
        double r1429998 = k;
        double r1429999 = r1429997 * r1429998;
        double r1430000 = r1429996 * r1429999;
        double r1430001 = r1429995 - r1430000;
        double r1430002 = r1429975 * r1429976;
        double r1430003 = r1429978 * r1429968;
        double r1430004 = r1429977 * r1430003;
        double r1430005 = r1430002 * r1430004;
        double r1430006 = r1430005 - r1429986;
        double r1430007 = r1430006 + r1429990;
        double r1430008 = r1430007 - r1429994;
        double r1430009 = r1429996 * r1429997;
        double r1430010 = r1430009 * r1429998;
        double r1430011 = r1430008 - r1430010;
        double r1430012 = r1429974 ? r1430001 : r1430011;
        return r1430012;
}

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	5.6
Target	1.5
Herbie	1.9

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

Derivation

1. Split input into 2 regimes

2. if y < -4.882400481117239e+97 or 7.217438656986207e-28 < y

   1. Initial program 11.5

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

   2. Taylor expanded around inf 12.9

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

   4. Applied associate-*r* 7.2

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

   6. Applied associate-*r* 2.0

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

   8. Applied associate-*l* 1.9

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

   if -4.882400481117239e+97 < y < 7.217438656986207e-28

   1. Initial program 1.9

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

   2. Taylor expanded around inf 1.8

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

   4. Applied associate-*r* 1.8

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

4. Final simplification 1.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -4.8824004811172388 \cdot 10^{97} \lor \neg \left(y \le 7.21743865698620678 \cdot 10^{-28}\right):\\ \;\;\;\;\left(\left(\left(18 \cdot \left(\left(t \cdot \left(x \cdot z\right)\right) \cdot y\right) - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - j \cdot \left(27 \cdot k\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(\left(18 \cdot t\right) \cdot \left(x \cdot \left(z \cdot y\right)\right) - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k\\ \end{array}\]

Reproduce

herbie shell --seed 2019198 
(FPCore (x y z t a b c i j k)
  :name "Diagrams.Solve.Polynomial:cubForm  from diagrams-solve-0.1, E"

  :herbie-target
  (if (< t -1.6210815397541398e-69) (- (- (* (* 18.0 t) (* (* x y) z)) (* (+ (* a t) (* i x)) 4.0)) (- (* (* k j) 27.0) (* c b))) (if (< t 165.68027943805222) (+ (- (* (* 18.0 y) (* x (* z t))) (* (+ (* a t) (* i x)) 4.0)) (- (* c b) (* 27.0 (* k j)))) (- (- (* (* 18.0 t) (* (* x y) z)) (* (+ (* a t) (* i x)) 4.0)) (- (* (* k j) 27.0) (* c b)))))

  (- (- (+ (- (* (* (* (* x 18.0) y) z) t) (* (* a 4.0) t)) (* b c)) (* (* x 4.0) i)) (* (* j 27.0) k)))