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

Average Error: 5.6 → 2.0

Time: 46.5s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \le -7.626211434592408442050935797592645449118 \cdot 10^{-36}:\\ \;\;\;\;\left(\left(\left(\left(x \cdot 18\right) \cdot \left(\left(y \cdot z\right) \cdot t\right) - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - 27 \cdot \left(k \cdot j\right)\\ \mathbf{elif}\;x \le 21885021756705142739042304:\\ \;\;\;\;\left(\left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot \left(\sqrt[3]{z} \cdot \sqrt[3]{z}\right)\right) \cdot \sqrt[3]{z}\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - j \cdot \left(27 \cdot k\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(\left(x \cdot 18\right) \cdot \left(y \cdot \left(z \cdot t\right)\right) - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - j \cdot \left(27 \cdot k\right)\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a, double b, double c, double i, double j, double k) {
        double r77056 = x;
        double r77057 = 18.0;
        double r77058 = r77056 * r77057;
        double r77059 = y;
        double r77060 = r77058 * r77059;
        double r77061 = z;
        double r77062 = r77060 * r77061;
        double r77063 = t;
        double r77064 = r77062 * r77063;
        double r77065 = a;
        double r77066 = 4.0;
        double r77067 = r77065 * r77066;
        double r77068 = r77067 * r77063;
        double r77069 = r77064 - r77068;
        double r77070 = b;
        double r77071 = c;
        double r77072 = r77070 * r77071;
        double r77073 = r77069 + r77072;
        double r77074 = r77056 * r77066;
        double r77075 = i;
        double r77076 = r77074 * r77075;
        double r77077 = r77073 - r77076;
        double r77078 = j;
        double r77079 = 27.0;
        double r77080 = r77078 * r77079;
        double r77081 = k;
        double r77082 = r77080 * r77081;
        double r77083 = r77077 - r77082;
        return r77083;
}

↓

double f(double x, double y, double z, double t, double a, double b, double c, double i, double j, double k) {
        double r77084 = x;
        double r77085 = -7.626211434592408e-36;
        bool r77086 = r77084 <= r77085;
        double r77087 = 18.0;
        double r77088 = r77084 * r77087;
        double r77089 = y;
        double r77090 = z;
        double r77091 = r77089 * r77090;
        double r77092 = t;
        double r77093 = r77091 * r77092;
        double r77094 = r77088 * r77093;
        double r77095 = a;
        double r77096 = 4.0;
        double r77097 = r77095 * r77096;
        double r77098 = r77097 * r77092;
        double r77099 = r77094 - r77098;
        double r77100 = b;
        double r77101 = c;
        double r77102 = r77100 * r77101;
        double r77103 = r77099 + r77102;
        double r77104 = r77084 * r77096;
        double r77105 = i;
        double r77106 = r77104 * r77105;
        double r77107 = r77103 - r77106;
        double r77108 = 27.0;
        double r77109 = k;
        double r77110 = j;
        double r77111 = r77109 * r77110;
        double r77112 = r77108 * r77111;
        double r77113 = r77107 - r77112;
        double r77114 = 2.1885021756705143e+25;
        bool r77115 = r77084 <= r77114;
        double r77116 = r77088 * r77089;
        double r77117 = cbrt(r77090);
        double r77118 = r77117 * r77117;
        double r77119 = r77116 * r77118;
        double r77120 = r77119 * r77117;
        double r77121 = r77120 * r77092;
        double r77122 = r77121 - r77098;
        double r77123 = r77122 + r77102;
        double r77124 = r77123 - r77106;
        double r77125 = r77108 * r77109;
        double r77126 = r77110 * r77125;
        double r77127 = r77124 - r77126;
        double r77128 = r77090 * r77092;
        double r77129 = r77089 * r77128;
        double r77130 = r77088 * r77129;
        double r77131 = r77130 - r77098;
        double r77132 = r77131 + r77102;
        double r77133 = r77132 - r77106;
        double r77134 = r77133 - r77126;
        double r77135 = r77115 ? r77127 : r77134;
        double r77136 = r77086 ? r77113 : r77135;
        return r77136;
}

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

Derivation

1. Split input into 3 regimes

2. if x < -7.626211434592408e-36

   1. Initial program 11.3

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

   3. Applied associate-*l* 11.3

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

   5. Applied associate-*l* 7.3

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

   7. Applied associate-*l* 2.6

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

   9. Applied pow1 2.6

      \[\leadsto \left(\left(\left(\left(x \cdot 18\right) \cdot \left(\left(y \cdot z\right) \cdot t\right) - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - j \cdot \left(27 \cdot \color{blue}{{k}^{1}}\right)\]

   10. Applied pow1 2.6

       \[\leadsto \left(\left(\left(\left(x \cdot 18\right) \cdot \left(\left(y \cdot z\right) \cdot t\right) - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - j \cdot \left(\color{blue}{{27}^{1}} \cdot {k}^{1}\right)\]

   11. Applied pow-prod-down 2.6

       \[\leadsto \left(\left(\left(\left(x \cdot 18\right) \cdot \left(\left(y \cdot z\right) \cdot t\right) - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - j \cdot \color{blue}{{\left(27 \cdot k\right)}^{1}}\]

   12. Applied pow1 2.6

       \[\leadsto \left(\left(\left(\left(x \cdot 18\right) \cdot \left(\left(y \cdot z\right) \cdot t\right) - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \color{blue}{{j}^{1}} \cdot {\left(27 \cdot k\right)}^{1}\]

   13. Applied pow-prod-down 2.6

       \[\leadsto \left(\left(\left(\left(x \cdot 18\right) \cdot \left(\left(y \cdot z\right) \cdot t\right) - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \color{blue}{{\left(j \cdot \left(27 \cdot k\right)\right)}^{1}}\]

   14. Simplified 2.5

       \[\leadsto \left(\left(\left(\left(x \cdot 18\right) \cdot \left(\left(y \cdot z\right) \cdot t\right) - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - {\color{blue}{\left(27 \cdot \left(k \cdot j\right)\right)}}^{1}\]

   if -7.626211434592408e-36 < x < 2.1885021756705143e+25

   1. Initial program 1.6

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

   3. Applied associate-*l* 1.7

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

   5. Applied add-cube-cbrt 1.8

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

   6. Applied associate-*r* 1.8

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

   if 2.1885021756705143e+25 < x

   1. Initial program 13.6

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

   3. Applied associate-*l* 13.8

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

   5. Applied associate-*l* 7.7

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

   7. Applied associate-*l* 1.9

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

   9. Applied associate-*l* 1.9

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

4. Final simplification 2.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -7.626211434592408442050935797592645449118 \cdot 10^{-36}:\\ \;\;\;\;\left(\left(\left(\left(x \cdot 18\right) \cdot \left(\left(y \cdot z\right) \cdot t\right) - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - 27 \cdot \left(k \cdot j\right)\\ \mathbf{elif}\;x \le 21885021756705142739042304:\\ \;\;\;\;\left(\left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot \left(\sqrt[3]{z} \cdot \sqrt[3]{z}\right)\right) \cdot \sqrt[3]{z}\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - j \cdot \left(27 \cdot k\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(\left(x \cdot 18\right) \cdot \left(y \cdot \left(z \cdot t\right)\right) - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - j \cdot \left(27 \cdot k\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019326 +o rules:numerics
(FPCore (x y z t a b c i j k)
  :name "Diagrams.Solve.Polynomial:cubForm  from diagrams-solve-0.1"
  :precision binary64
  (- (- (+ (- (* (* (* (* x 18) y) z) t) (* (* a 4) t)) (* b c)) (* (* x 4) i)) (* (* j 27) k)))