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

Average Error: 5.5 → 0.9

Time: 1.4m

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\left(\left(t \cdot \left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) - \left(a \cdot 4\right) \cdot t\right) + c \cdot b\right) - \left(x \cdot 4\right) \cdot i = -\infty:\\ \;\;\;\;\left(\left(c \cdot b + \left(x \cdot \left(18 \cdot \left(\left(t \cdot z\right) \cdot y\right)\right) - \left(a \cdot 4\right) \cdot t\right)\right) - \left(x \cdot 4\right) \cdot i\right) - k \cdot \left(j \cdot 27\right)\\ \mathbf{elif}\;\left(\left(t \cdot \left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) - \left(a \cdot 4\right) \cdot t\right) + c \cdot b\right) - \left(x \cdot 4\right) \cdot i \le 2.091817559392553810929566838550160622791 \cdot 10^{304}:\\ \;\;\;\;\left(\left(\left(t \cdot \left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) - \left(a \cdot 4\right) \cdot t\right) + c \cdot b\right) - \left(x \cdot 4\right) \cdot i\right) - \left(k \cdot 27\right) \cdot j\\ \mathbf{else}:\\ \;\;\;\;\left(\left(c \cdot b + \left(x \cdot \left(\left(y \cdot 18\right) \cdot \left(t \cdot z\right)\right) - \left(a \cdot 4\right) \cdot t\right)\right) - \left(x \cdot 4\right) \cdot i\right) - k \cdot \left(j \cdot 27\right)\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a, double b, double c, double i, double j, double k) {
        double r36649834 = x;
        double r36649835 = 18.0;
        double r36649836 = r36649834 * r36649835;
        double r36649837 = y;
        double r36649838 = r36649836 * r36649837;
        double r36649839 = z;
        double r36649840 = r36649838 * r36649839;
        double r36649841 = t;
        double r36649842 = r36649840 * r36649841;
        double r36649843 = a;
        double r36649844 = 4.0;
        double r36649845 = r36649843 * r36649844;
        double r36649846 = r36649845 * r36649841;
        double r36649847 = r36649842 - r36649846;
        double r36649848 = b;
        double r36649849 = c;
        double r36649850 = r36649848 * r36649849;
        double r36649851 = r36649847 + r36649850;
        double r36649852 = r36649834 * r36649844;
        double r36649853 = i;
        double r36649854 = r36649852 * r36649853;
        double r36649855 = r36649851 - r36649854;
        double r36649856 = j;
        double r36649857 = 27.0;
        double r36649858 = r36649856 * r36649857;
        double r36649859 = k;
        double r36649860 = r36649858 * r36649859;
        double r36649861 = r36649855 - r36649860;
        return r36649861;
}

↓

double f(double x, double y, double z, double t, double a, double b, double c, double i, double j, double k) {
        double r36649862 = t;
        double r36649863 = x;
        double r36649864 = 18.0;
        double r36649865 = r36649863 * r36649864;
        double r36649866 = y;
        double r36649867 = r36649865 * r36649866;
        double r36649868 = z;
        double r36649869 = r36649867 * r36649868;
        double r36649870 = r36649862 * r36649869;
        double r36649871 = a;
        double r36649872 = 4.0;
        double r36649873 = r36649871 * r36649872;
        double r36649874 = r36649873 * r36649862;
        double r36649875 = r36649870 - r36649874;
        double r36649876 = c;
        double r36649877 = b;
        double r36649878 = r36649876 * r36649877;
        double r36649879 = r36649875 + r36649878;
        double r36649880 = r36649863 * r36649872;
        double r36649881 = i;
        double r36649882 = r36649880 * r36649881;
        double r36649883 = r36649879 - r36649882;
        double r36649884 = -inf.0;
        bool r36649885 = r36649883 <= r36649884;
        double r36649886 = r36649862 * r36649868;
        double r36649887 = r36649886 * r36649866;
        double r36649888 = r36649864 * r36649887;
        double r36649889 = r36649863 * r36649888;
        double r36649890 = r36649889 - r36649874;
        double r36649891 = r36649878 + r36649890;
        double r36649892 = r36649891 - r36649882;
        double r36649893 = k;
        double r36649894 = j;
        double r36649895 = 27.0;
        double r36649896 = r36649894 * r36649895;
        double r36649897 = r36649893 * r36649896;
        double r36649898 = r36649892 - r36649897;
        double r36649899 = 2.0918175593925538e+304;
        bool r36649900 = r36649883 <= r36649899;
        double r36649901 = r36649893 * r36649895;
        double r36649902 = r36649901 * r36649894;
        double r36649903 = r36649883 - r36649902;
        double r36649904 = r36649866 * r36649864;
        double r36649905 = r36649904 * r36649886;
        double r36649906 = r36649863 * r36649905;
        double r36649907 = r36649906 - r36649874;
        double r36649908 = r36649878 + r36649907;
        double r36649909 = r36649908 - r36649882;
        double r36649910 = r36649909 - r36649897;
        double r36649911 = r36649900 ? r36649903 : r36649910;
        double r36649912 = r36649885 ? r36649898 : r36649911;
        return r36649912;
}

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	5.5
Target	1.7
Herbie	0.9

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

Derivation

1. Split input into 3 regimes

2. if (- (+ (- (* (* (* (* x 18.0) y) z) t) (* (* a 4.0) t)) (* b c)) (* (* x 4.0) i)) < -inf.0

   1. Initial program 64.0

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

   3. Applied associate-*l* 37.0

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

   5. Applied associate-*l* 5.1

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

   7. Applied associate-*l* 5.1

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

   if -inf.0 < (- (+ (- (* (* (* (* x 18.0) y) z) t) (* (* a 4.0) t)) (* b c)) (* (* x 4.0) i)) < 2.0918175593925538e+304

   1. Initial program 0.3

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

   3. Applied associate-*l* 0.3

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

   if 2.0918175593925538e+304 < (- (+ (- (* (* (* (* x 18.0) y) z) t) (* (* a 4.0) t)) (* b c)) (* (* x 4.0) i))

   1. Initial program 56.2

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

   3. Applied associate-*l* 31.4

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

   5. Applied associate-*l* 9.1

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

   7. Applied associate-*l* 8.4

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

   9. Applied associate-*r* 8.8

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

4. Final simplification 0.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(t \cdot \left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) - \left(a \cdot 4\right) \cdot t\right) + c \cdot b\right) - \left(x \cdot 4\right) \cdot i = -\infty:\\ \;\;\;\;\left(\left(c \cdot b + \left(x \cdot \left(18 \cdot \left(\left(t \cdot z\right) \cdot y\right)\right) - \left(a \cdot 4\right) \cdot t\right)\right) - \left(x \cdot 4\right) \cdot i\right) - k \cdot \left(j \cdot 27\right)\\ \mathbf{elif}\;\left(\left(t \cdot \left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) - \left(a \cdot 4\right) \cdot t\right) + c \cdot b\right) - \left(x \cdot 4\right) \cdot i \le 2.091817559392553810929566838550160622791 \cdot 10^{304}:\\ \;\;\;\;\left(\left(\left(t \cdot \left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) - \left(a \cdot 4\right) \cdot t\right) + c \cdot b\right) - \left(x \cdot 4\right) \cdot i\right) - \left(k \cdot 27\right) \cdot j\\ \mathbf{else}:\\ \;\;\;\;\left(\left(c \cdot b + \left(x \cdot \left(\left(y \cdot 18\right) \cdot \left(t \cdot z\right)\right) - \left(a \cdot 4\right) \cdot t\right)\right) - \left(x \cdot 4\right) \cdot i\right) - k \cdot \left(j \cdot 27\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019168 
(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)))