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

Average Error: 3.8 → 0.7

Time: 17.4s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;y \cdot 9 \le -3.338893372507656821369864426932132802203 \cdot 10^{-32}:\\ \;\;\;\;\left(x \cdot 2 - y \cdot \left(9 \cdot \left(t \cdot z\right)\right)\right) + 27 \cdot \left(a \cdot b\right)\\ \mathbf{elif}\;y \cdot 9 \le 7.055620513394500257077151321056658993069 \cdot 10^{57}:\\ \;\;\;\;\left(x \cdot 2 - \left(y \cdot \left(z \cdot 9\right)\right) \cdot t\right) + \left(a \cdot 27\right) \cdot b\\ \mathbf{else}:\\ \;\;\;\;\sqrt{27} \cdot \left(\sqrt{27} \cdot \left(a \cdot b\right)\right) + \left(x \cdot 2 - \left(y \cdot 9\right) \cdot \left(t \cdot z\right)\right)\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a, double b) {
        double r578864 = x;
        double r578865 = 2.0;
        double r578866 = r578864 * r578865;
        double r578867 = y;
        double r578868 = 9.0;
        double r578869 = r578867 * r578868;
        double r578870 = z;
        double r578871 = r578869 * r578870;
        double r578872 = t;
        double r578873 = r578871 * r578872;
        double r578874 = r578866 - r578873;
        double r578875 = a;
        double r578876 = 27.0;
        double r578877 = r578875 * r578876;
        double r578878 = b;
        double r578879 = r578877 * r578878;
        double r578880 = r578874 + r578879;
        return r578880;
}

↓

double f(double x, double y, double z, double t, double a, double b) {
        double r578881 = y;
        double r578882 = 9.0;
        double r578883 = r578881 * r578882;
        double r578884 = -3.338893372507657e-32;
        bool r578885 = r578883 <= r578884;
        double r578886 = x;
        double r578887 = 2.0;
        double r578888 = r578886 * r578887;
        double r578889 = t;
        double r578890 = z;
        double r578891 = r578889 * r578890;
        double r578892 = r578882 * r578891;
        double r578893 = r578881 * r578892;
        double r578894 = r578888 - r578893;
        double r578895 = 27.0;
        double r578896 = a;
        double r578897 = b;
        double r578898 = r578896 * r578897;
        double r578899 = r578895 * r578898;
        double r578900 = r578894 + r578899;
        double r578901 = 7.0556205133945e+57;
        bool r578902 = r578883 <= r578901;
        double r578903 = r578890 * r578882;
        double r578904 = r578881 * r578903;
        double r578905 = r578904 * r578889;
        double r578906 = r578888 - r578905;
        double r578907 = r578896 * r578895;
        double r578908 = r578907 * r578897;
        double r578909 = r578906 + r578908;
        double r578910 = sqrt(r578895);
        double r578911 = r578910 * r578898;
        double r578912 = r578910 * r578911;
        double r578913 = r578883 * r578891;
        double r578914 = r578888 - r578913;
        double r578915 = r578912 + r578914;
        double r578916 = r578902 ? r578909 : r578915;
        double r578917 = r578885 ? r578900 : r578916;
        return r578917;
}

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	2.6
Herbie	0.7

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

Derivation

1. Split input into 3 regimes

2. if (* y 9.0) < -3.338893372507657e-32

   1. Initial program 7.1

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

   3. Applied associate-*l* 0.8

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

   4. Simplified 0.8

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

   6. Applied pow1 0.8

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

   7. Applied pow1 0.8

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

   8. Applied pow1 0.8

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

   9. Applied pow-prod-down 0.8

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

   10. Applied pow-prod-down 0.8

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

   11. Simplified 0.8

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

   13. Applied associate-*l* 0.7

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

   if -3.338893372507657e-32 < (* y 9.0) < 7.0556205133945e+57

   1. Initial program 0.7

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

   3. Applied associate-*l* 0.7

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

   4. Simplified 0.7

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

   if 7.0556205133945e+57 < (* y 9.0)

   1. Initial program 10.4

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

   3. Applied associate-*l* 0.9

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

   4. Simplified 0.9

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

   6. Applied pow1 0.9

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

   7. Applied pow1 0.9

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

   8. Applied pow1 0.9

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

   9. Applied pow-prod-down 0.9

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

   10. Applied pow-prod-down 0.9

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

   11. Simplified 0.8

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

   13. Applied add-sqr-sqrt 0.8

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

   14. Applied associate-*l* 0.9

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

4. Final simplification 0.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot 9 \le -3.338893372507656821369864426932132802203 \cdot 10^{-32}:\\ \;\;\;\;\left(x \cdot 2 - y \cdot \left(9 \cdot \left(t \cdot z\right)\right)\right) + 27 \cdot \left(a \cdot b\right)\\ \mathbf{elif}\;y \cdot 9 \le 7.055620513394500257077151321056658993069 \cdot 10^{57}:\\ \;\;\;\;\left(x \cdot 2 - \left(y \cdot \left(z \cdot 9\right)\right) \cdot t\right) + \left(a \cdot 27\right) \cdot b\\ \mathbf{else}:\\ \;\;\;\;\sqrt{27} \cdot \left(\sqrt{27} \cdot \left(a \cdot b\right)\right) + \left(x \cdot 2 - \left(y \cdot 9\right) \cdot \left(t \cdot z\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019323 +o rules:numerics
(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)))