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

Average Error: 3.8 → 0.5

Time: 4.3s

Precision: 64

Math

\[\left(x \cdot 2 - \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right) + \left(a \cdot 27\right) \cdot b\]

↓

\[\begin{array}{l} \mathbf{if}\;\left(y \cdot 9\right) \cdot z \le -1.238690280796900705766198006768324442173 \cdot 10^{177}:\\ \;\;\;\;\left(2 \cdot x - {\left(\left(9 \cdot \left(t \cdot y\right)\right) \cdot z\right)}^{1}\right) + 27 \cdot \left(a \cdot b\right)\\ \mathbf{elif}\;\left(y \cdot 9\right) \cdot z \le 9.72585044449842875475358544069428986135 \cdot 10^{252}:\\ \;\;\;\;\left(x \cdot 2 - \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right) + a \cdot \left(27 \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot x - 9 \cdot \left(\left(t \cdot z\right) \cdot y\right)\right) + 27 \cdot \left(a \cdot b\right)\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a, double b) {
        double r1020961 = x;
        double r1020962 = 2.0;
        double r1020963 = r1020961 * r1020962;
        double r1020964 = y;
        double r1020965 = 9.0;
        double r1020966 = r1020964 * r1020965;
        double r1020967 = z;
        double r1020968 = r1020966 * r1020967;
        double r1020969 = t;
        double r1020970 = r1020968 * r1020969;
        double r1020971 = r1020963 - r1020970;
        double r1020972 = a;
        double r1020973 = 27.0;
        double r1020974 = r1020972 * r1020973;
        double r1020975 = b;
        double r1020976 = r1020974 * r1020975;
        double r1020977 = r1020971 + r1020976;
        return r1020977;
}

↓

double f(double x, double y, double z, double t, double a, double b) {
        double r1020978 = y;
        double r1020979 = 9.0;
        double r1020980 = r1020978 * r1020979;
        double r1020981 = z;
        double r1020982 = r1020980 * r1020981;
        double r1020983 = -1.2386902807969007e+177;
        bool r1020984 = r1020982 <= r1020983;
        double r1020985 = 2.0;
        double r1020986 = x;
        double r1020987 = r1020985 * r1020986;
        double r1020988 = t;
        double r1020989 = r1020988 * r1020978;
        double r1020990 = r1020979 * r1020989;
        double r1020991 = r1020990 * r1020981;
        double r1020992 = 1.0;
        double r1020993 = pow(r1020991, r1020992);
        double r1020994 = r1020987 - r1020993;
        double r1020995 = 27.0;
        double r1020996 = a;
        double r1020997 = b;
        double r1020998 = r1020996 * r1020997;
        double r1020999 = r1020995 * r1020998;
        double r1021000 = r1020994 + r1020999;
        double r1021001 = 9.725850444498429e+252;
        bool r1021002 = r1020982 <= r1021001;
        double r1021003 = r1020986 * r1020985;
        double r1021004 = r1020982 * r1020988;
        double r1021005 = r1021003 - r1021004;
        double r1021006 = r1020995 * r1020997;
        double r1021007 = r1020996 * r1021006;
        double r1021008 = r1021005 + r1021007;
        double r1021009 = r1020988 * r1020981;
        double r1021010 = r1021009 * r1020978;
        double r1021011 = r1020979 * r1021010;
        double r1021012 = r1020987 - r1021011;
        double r1021013 = r1021012 + r1020999;
        double r1021014 = r1021002 ? r1021008 : r1021013;
        double r1021015 = r1020984 ? r1021000 : r1021014;
        return r1021015;
}

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

Target

Original	3.8
Target	2.8
Herbie	0.5

\[\begin{array}{l} \mathbf{if}\;y \lt 7.590524218811188954625810696587370427881 \cdot 10^{-161}:\\ \;\;\;\;\left(x \cdot 2 - \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right) + a \cdot \left(27 \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot 2 - 9 \cdot \left(y \cdot \left(t \cdot z\right)\right)\right) + \left(a \cdot 27\right) \cdot b\\ \end{array}\]

Derivation

1. Split input into 3 regimes

2. if (* (* y 9.0) z) < -1.2386902807969007e+177

   1. Initial program 22.4

      \[\left(x \cdot 2 - \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right) + \left(a \cdot 27\right) \cdot b\]

   2. Taylor expanded around inf 21.5

      \[\leadsto \color{blue}{\left(2 \cdot x - 9 \cdot \left(t \cdot \left(z \cdot y\right)\right)\right)} + \left(a \cdot 27\right) \cdot b\]

   3. Taylor expanded around 0 21.4

      \[\leadsto \left(2 \cdot x - 9 \cdot \left(t \cdot \left(z \cdot y\right)\right)\right) + \color{blue}{27 \cdot \left(a \cdot b\right)}\]
   4. Using strategy rm

   5. Applied add-cube-cbrt 21.4

      \[\leadsto \left(2 \cdot x - \color{blue}{\left(\left(\sqrt[3]{9} \cdot \sqrt[3]{9}\right) \cdot \sqrt[3]{9}\right)} \cdot \left(t \cdot \left(z \cdot y\right)\right)\right) + 27 \cdot \left(a \cdot b\right)\]

   6. Applied associate-*l* 21.4

      \[\leadsto \left(2 \cdot x - \color{blue}{\left(\sqrt[3]{9} \cdot \sqrt[3]{9}\right) \cdot \left(\sqrt[3]{9} \cdot \left(t \cdot \left(z \cdot y\right)\right)\right)}\right) + 27 \cdot \left(a \cdot b\right)\]
   7. Using strategy rm

   8. Applied pow1 21.4

      \[\leadsto \left(2 \cdot x - \left(\sqrt[3]{9} \cdot \sqrt[3]{9}\right) \cdot \left(\sqrt[3]{9} \cdot \left(t \cdot \left(z \cdot \color{blue}{{y}^{1}}\right)\right)\right)\right) + 27 \cdot \left(a \cdot b\right)\]

   9. Applied pow1 21.4

      \[\leadsto \left(2 \cdot x - \left(\sqrt[3]{9} \cdot \sqrt[3]{9}\right) \cdot \left(\sqrt[3]{9} \cdot \left(t \cdot \left(\color{blue}{{z}^{1}} \cdot {y}^{1}\right)\right)\right)\right) + 27 \cdot \left(a \cdot b\right)\]

   10. Applied pow-prod-down 21.4

       \[\leadsto \left(2 \cdot x - \left(\sqrt[3]{9} \cdot \sqrt[3]{9}\right) \cdot \left(\sqrt[3]{9} \cdot \left(t \cdot \color{blue}{{\left(z \cdot y\right)}^{1}}\right)\right)\right) + 27 \cdot \left(a \cdot b\right)\]

   11. Applied pow1 21.4

       \[\leadsto \left(2 \cdot x - \left(\sqrt[3]{9} \cdot \sqrt[3]{9}\right) \cdot \left(\sqrt[3]{9} \cdot \left(\color{blue}{{t}^{1}} \cdot {\left(z \cdot y\right)}^{1}\right)\right)\right) + 27 \cdot \left(a \cdot b\right)\]

   12. Applied pow-prod-down 21.4

       \[\leadsto \left(2 \cdot x - \left(\sqrt[3]{9} \cdot \sqrt[3]{9}\right) \cdot \left(\sqrt[3]{9} \cdot \color{blue}{{\left(t \cdot \left(z \cdot y\right)\right)}^{1}}\right)\right) + 27 \cdot \left(a \cdot b\right)\]

   13. Applied pow1 21.4

       \[\leadsto \left(2 \cdot x - \left(\sqrt[3]{9} \cdot \sqrt[3]{9}\right) \cdot \left(\color{blue}{{\left(\sqrt[3]{9}\right)}^{1}} \cdot {\left(t \cdot \left(z \cdot y\right)\right)}^{1}\right)\right) + 27 \cdot \left(a \cdot b\right)\]

   14. Applied pow-prod-down 21.4

       \[\leadsto \left(2 \cdot x - \left(\sqrt[3]{9} \cdot \sqrt[3]{9}\right) \cdot \color{blue}{{\left(\sqrt[3]{9} \cdot \left(t \cdot \left(z \cdot y\right)\right)\right)}^{1}}\right) + 27 \cdot \left(a \cdot b\right)\]

   15. Applied pow1 21.4

       \[\leadsto \left(2 \cdot x - \left(\sqrt[3]{9} \cdot \color{blue}{{\left(\sqrt[3]{9}\right)}^{1}}\right) \cdot {\left(\sqrt[3]{9} \cdot \left(t \cdot \left(z \cdot y\right)\right)\right)}^{1}\right) + 27 \cdot \left(a \cdot b\right)\]

   16. Applied pow1 21.4

       \[\leadsto \left(2 \cdot x - \left(\color{blue}{{\left(\sqrt[3]{9}\right)}^{1}} \cdot {\left(\sqrt[3]{9}\right)}^{1}\right) \cdot {\left(\sqrt[3]{9} \cdot \left(t \cdot \left(z \cdot y\right)\right)\right)}^{1}\right) + 27 \cdot \left(a \cdot b\right)\]

   17. Applied pow-prod-down 21.4

       \[\leadsto \left(2 \cdot x - \color{blue}{{\left(\sqrt[3]{9} \cdot \sqrt[3]{9}\right)}^{1}} \cdot {\left(\sqrt[3]{9} \cdot \left(t \cdot \left(z \cdot y\right)\right)\right)}^{1}\right) + 27 \cdot \left(a \cdot b\right)\]

   18. Applied pow-prod-down 21.4

       \[\leadsto \left(2 \cdot x - \color{blue}{{\left(\left(\sqrt[3]{9} \cdot \sqrt[3]{9}\right) \cdot \left(\sqrt[3]{9} \cdot \left(t \cdot \left(z \cdot y\right)\right)\right)\right)}^{1}}\right) + 27 \cdot \left(a \cdot b\right)\]

   19. Simplified 1.3

       \[\leadsto \left(2 \cdot x - {\color{blue}{\left(\left(9 \cdot \left(t \cdot y\right)\right) \cdot z\right)}}^{1}\right) + 27 \cdot \left(a \cdot b\right)\]

   if -1.2386902807969007e+177 < (* (* y 9.0) z) < 9.725850444498429e+252

   1. Initial program 0.4

      \[\left(x \cdot 2 - \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right) + \left(a \cdot 27\right) \cdot b\]
   2. Using strategy rm

   3. Applied associate-*l* 0.4

      \[\leadsto \left(x \cdot 2 - \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right) + \color{blue}{a \cdot \left(27 \cdot b\right)}\]

   if 9.725850444498429e+252 < (* (* y 9.0) z)

   1. Initial program 39.8

      \[\left(x \cdot 2 - \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right) + \left(a \cdot 27\right) \cdot b\]

   2. Taylor expanded around inf 38.3

      \[\leadsto \color{blue}{\left(2 \cdot x - 9 \cdot \left(t \cdot \left(z \cdot y\right)\right)\right)} + \left(a \cdot 27\right) \cdot b\]

   3. Taylor expanded around 0 38.3

      \[\leadsto \left(2 \cdot x - 9 \cdot \left(t \cdot \left(z \cdot y\right)\right)\right) + \color{blue}{27 \cdot \left(a \cdot b\right)}\]
   4. Using strategy rm

   5. Applied associate-*r* 0.8

      \[\leadsto \left(2 \cdot x - 9 \cdot \color{blue}{\left(\left(t \cdot z\right) \cdot y\right)}\right) + 27 \cdot \left(a \cdot b\right)\]
3. Recombined 3 regimes into one program.

4. Final simplification 0.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(y \cdot 9\right) \cdot z \le -1.238690280796900705766198006768324442173 \cdot 10^{177}:\\ \;\;\;\;\left(2 \cdot x - {\left(\left(9 \cdot \left(t \cdot y\right)\right) \cdot z\right)}^{1}\right) + 27 \cdot \left(a \cdot b\right)\\ \mathbf{elif}\;\left(y \cdot 9\right) \cdot z \le 9.72585044449842875475358544069428986135 \cdot 10^{252}:\\ \;\;\;\;\left(x \cdot 2 - \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right) + a \cdot \left(27 \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot x - 9 \cdot \left(\left(t \cdot z\right) \cdot y\right)\right) + 27 \cdot \left(a \cdot b\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019353 
(FPCore (x y z t a b)
  :name "Diagrams.Solve.Polynomial:cubForm  from diagrams-solve-0.1, A"
  :precision binary64

  :herbie-target
  (if (< y 7.590524218811189e-161) (+ (- (* x 2) (* (* (* y 9) z) t)) (* a (* 27 b))) (+ (- (* x 2) (* 9 (* y (* t z)))) (* (* a 27) b)))

  (+ (- (* x 2) (* (* (* y 9) z) t)) (* (* a 27) b)))