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

Average Error: 20.2 → 18.1

Time: 36.0s

Precision: 64

Math

\[\left(2.0 \cdot \sqrt{x}\right) \cdot \cos \left(y - \frac{z \cdot t}{3.0}\right) - \frac{a}{b \cdot 3.0}\]

↓

\[\begin{array}{l} \mathbf{if}\;z \cdot t = -\infty:\\ \;\;\;\;\left(\left(y \cdot \frac{-1}{2}\right) \cdot y + 1\right) \cdot \left(\sqrt{x} \cdot 2.0\right) - \frac{a}{3.0 \cdot b}\\ \mathbf{elif}\;z \cdot t \le 3.947993520431114 \cdot 10^{+303}:\\ \;\;\;\;\left(\sqrt{x} \cdot 2.0\right) \cdot \cos \left(y - \left(\sqrt[3]{\frac{z \cdot t}{3.0}} \cdot \sqrt[3]{\frac{z \cdot t}{3.0}}\right) \cdot \sqrt[3]{\sqrt[3]{\left(\sqrt[3]{\sqrt[3]{\frac{z \cdot t}{3.0}} \cdot \sqrt[3]{\frac{z \cdot t}{3.0}}} \cdot \sqrt[3]{\sqrt[3]{\frac{z \cdot t}{3.0}}}\right) \cdot \left(\sqrt[3]{\frac{z \cdot t}{3.0}} \cdot \sqrt[3]{\frac{z \cdot t}{3.0}}\right)} \cdot \left(\sqrt[3]{\left(\sqrt[3]{\sqrt[3]{\frac{z \cdot t}{3.0}} \cdot \sqrt[3]{\frac{z \cdot t}{3.0}}} \cdot \sqrt[3]{\sqrt[3]{\frac{z \cdot t}{3.0}}}\right) \cdot \left(\sqrt[3]{\frac{z \cdot t}{3.0}} \cdot \sqrt[3]{\frac{z \cdot t}{3.0}}\right)} \cdot \sqrt[3]{\left(\sqrt[3]{\sqrt[3]{\frac{z \cdot t}{3.0}} \cdot \sqrt[3]{\frac{z \cdot t}{3.0}}} \cdot \sqrt[3]{\sqrt[3]{\frac{z \cdot t}{3.0}}}\right) \cdot \left(\sqrt[3]{\frac{z \cdot t}{3.0}} \cdot \sqrt[3]{\frac{z \cdot t}{3.0}}\right)}\right)}\right) - \frac{a}{3.0 \cdot b}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(y \cdot \frac{-1}{2}\right) \cdot y + 1\right) \cdot \left(\sqrt{x} \cdot 2.0\right) - \frac{a}{3.0 \cdot b}\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a, double b) {
        double r37464088 = 2.0;
        double r37464089 = x;
        double r37464090 = sqrt(r37464089);
        double r37464091 = r37464088 * r37464090;
        double r37464092 = y;
        double r37464093 = z;
        double r37464094 = t;
        double r37464095 = r37464093 * r37464094;
        double r37464096 = 3.0;
        double r37464097 = r37464095 / r37464096;
        double r37464098 = r37464092 - r37464097;
        double r37464099 = cos(r37464098);
        double r37464100 = r37464091 * r37464099;
        double r37464101 = a;
        double r37464102 = b;
        double r37464103 = r37464102 * r37464096;
        double r37464104 = r37464101 / r37464103;
        double r37464105 = r37464100 - r37464104;
        return r37464105;
}

↓

double f(double x, double y, double z, double t, double a, double b) {
        double r37464106 = z;
        double r37464107 = t;
        double r37464108 = r37464106 * r37464107;
        double r37464109 = -inf.0;
        bool r37464110 = r37464108 <= r37464109;
        double r37464111 = y;
        double r37464112 = -0.5;
        double r37464113 = r37464111 * r37464112;
        double r37464114 = r37464113 * r37464111;
        double r37464115 = 1.0;
        double r37464116 = r37464114 + r37464115;
        double r37464117 = x;
        double r37464118 = sqrt(r37464117);
        double r37464119 = 2.0;
        double r37464120 = r37464118 * r37464119;
        double r37464121 = r37464116 * r37464120;
        double r37464122 = a;
        double r37464123 = 3.0;
        double r37464124 = b;
        double r37464125 = r37464123 * r37464124;
        double r37464126 = r37464122 / r37464125;
        double r37464127 = r37464121 - r37464126;
        double r37464128 = 3.947993520431114e+303;
        bool r37464129 = r37464108 <= r37464128;
        double r37464130 = r37464108 / r37464123;
        double r37464131 = cbrt(r37464130);
        double r37464132 = r37464131 * r37464131;
        double r37464133 = cbrt(r37464132);
        double r37464134 = cbrt(r37464131);
        double r37464135 = r37464133 * r37464134;
        double r37464136 = r37464135 * r37464132;
        double r37464137 = cbrt(r37464136);
        double r37464138 = r37464137 * r37464137;
        double r37464139 = r37464137 * r37464138;
        double r37464140 = cbrt(r37464139);
        double r37464141 = r37464132 * r37464140;
        double r37464142 = r37464111 - r37464141;
        double r37464143 = cos(r37464142);
        double r37464144 = r37464120 * r37464143;
        double r37464145 = r37464144 - r37464126;
        double r37464146 = r37464129 ? r37464145 : r37464127;
        double r37464147 = r37464110 ? r37464127 : r37464146;
        return r37464147;
}

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	20.2
Target	18.5
Herbie	18.1

\[\begin{array}{l} \mathbf{if}\;z \lt -1.3793337487235141 \cdot 10^{+129}:\\ \;\;\;\;\left(2.0 \cdot \sqrt{x}\right) \cdot \cos \left(\frac{1}{y} - \frac{\frac{0.3333333333333333}{z}}{t}\right) - \frac{\frac{a}{3.0}}{b}\\ \mathbf{elif}\;z \lt 3.516290613555987 \cdot 10^{+106}:\\ \;\;\;\;\left(\sqrt{x} \cdot 2.0\right) \cdot \cos \left(y - \frac{t}{3.0} \cdot z\right) - \frac{\frac{a}{3.0}}{b}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(y - \frac{\frac{0.3333333333333333}{z}}{t}\right) \cdot \left(2.0 \cdot \sqrt{x}\right) - \frac{\frac{a}{b}}{3.0}\\ \end{array}\]

Derivation

1. Split input into 2 regimes

2. if (* z t) < -inf.0 or 3.947993520431114e+303 < (* z t)

   1. Initial program 62.2

      \[\left(2.0 \cdot \sqrt{x}\right) \cdot \cos \left(y - \frac{z \cdot t}{3.0}\right) - \frac{a}{b \cdot 3.0}\]
   2. Using strategy rm

   3. Applied add-cube-cbrt 62.2

      \[\leadsto \left(2.0 \cdot \sqrt{x}\right) \cdot \cos \left(y - \color{blue}{\left(\sqrt[3]{\frac{z \cdot t}{3.0}} \cdot \sqrt[3]{\frac{z \cdot t}{3.0}}\right) \cdot \sqrt[3]{\frac{z \cdot t}{3.0}}}\right) - \frac{a}{b \cdot 3.0}\]
   4. Using strategy rm

   5. Applied add-cbrt-cube 62.2

      \[\leadsto \left(2.0 \cdot \sqrt{x}\right) \cdot \cos \left(y - \left(\sqrt[3]{\frac{z \cdot t}{3.0}} \cdot \sqrt[3]{\frac{z \cdot t}{3.0}}\right) \cdot \color{blue}{\sqrt[3]{\left(\sqrt[3]{\frac{z \cdot t}{3.0}} \cdot \sqrt[3]{\frac{z \cdot t}{3.0}}\right) \cdot \sqrt[3]{\frac{z \cdot t}{3.0}}}}\right) - \frac{a}{b \cdot 3.0}\]
   6. Using strategy rm

   7. Applied add-cube-cbrt 62.2

      \[\leadsto \left(2.0 \cdot \sqrt{x}\right) \cdot \cos \left(y - \left(\sqrt[3]{\frac{z \cdot t}{3.0}} \cdot \sqrt[3]{\frac{z \cdot t}{3.0}}\right) \cdot \sqrt[3]{\left(\sqrt[3]{\frac{z \cdot t}{3.0}} \cdot \sqrt[3]{\frac{z \cdot t}{3.0}}\right) \cdot \sqrt[3]{\color{blue}{\left(\sqrt[3]{\frac{z \cdot t}{3.0}} \cdot \sqrt[3]{\frac{z \cdot t}{3.0}}\right) \cdot \sqrt[3]{\frac{z \cdot t}{3.0}}}}}\right) - \frac{a}{b \cdot 3.0}\]

   8. Applied cbrt-prod 62.2

      \[\leadsto \left(2.0 \cdot \sqrt{x}\right) \cdot \cos \left(y - \left(\sqrt[3]{\frac{z \cdot t}{3.0}} \cdot \sqrt[3]{\frac{z \cdot t}{3.0}}\right) \cdot \sqrt[3]{\left(\sqrt[3]{\frac{z \cdot t}{3.0}} \cdot \sqrt[3]{\frac{z \cdot t}{3.0}}\right) \cdot \color{blue}{\left(\sqrt[3]{\sqrt[3]{\frac{z \cdot t}{3.0}} \cdot \sqrt[3]{\frac{z \cdot t}{3.0}}} \cdot \sqrt[3]{\sqrt[3]{\frac{z \cdot t}{3.0}}}\right)}}\right) - \frac{a}{b \cdot 3.0}\]

   9. Taylor expanded around 0 45.5

      \[\leadsto \left(2.0 \cdot \sqrt{x}\right) \cdot \color{blue}{\left(1 - \frac{1}{2} \cdot {y}^{2}\right)} - \frac{a}{b \cdot 3.0}\]

   10. Simplified 45.5

       \[\leadsto \left(2.0 \cdot \sqrt{x}\right) \cdot \color{blue}{\left(1 + y \cdot \left(\frac{-1}{2} \cdot y\right)\right)} - \frac{a}{b \cdot 3.0}\]

   if -inf.0 < (* z t) < 3.947993520431114e+303

   1. Initial program 14.3

      \[\left(2.0 \cdot \sqrt{x}\right) \cdot \cos \left(y - \frac{z \cdot t}{3.0}\right) - \frac{a}{b \cdot 3.0}\]
   2. Using strategy rm

   3. Applied add-cube-cbrt 14.2

      \[\leadsto \left(2.0 \cdot \sqrt{x}\right) \cdot \cos \left(y - \color{blue}{\left(\sqrt[3]{\frac{z \cdot t}{3.0}} \cdot \sqrt[3]{\frac{z \cdot t}{3.0}}\right) \cdot \sqrt[3]{\frac{z \cdot t}{3.0}}}\right) - \frac{a}{b \cdot 3.0}\]
   4. Using strategy rm

   5. Applied add-cbrt-cube 14.2

      \[\leadsto \left(2.0 \cdot \sqrt{x}\right) \cdot \cos \left(y - \left(\sqrt[3]{\frac{z \cdot t}{3.0}} \cdot \sqrt[3]{\frac{z \cdot t}{3.0}}\right) \cdot \color{blue}{\sqrt[3]{\left(\sqrt[3]{\frac{z \cdot t}{3.0}} \cdot \sqrt[3]{\frac{z \cdot t}{3.0}}\right) \cdot \sqrt[3]{\frac{z \cdot t}{3.0}}}}\right) - \frac{a}{b \cdot 3.0}\]
   6. Using strategy rm

   7. Applied add-cube-cbrt 14.3

      \[\leadsto \left(2.0 \cdot \sqrt{x}\right) \cdot \cos \left(y - \left(\sqrt[3]{\frac{z \cdot t}{3.0}} \cdot \sqrt[3]{\frac{z \cdot t}{3.0}}\right) \cdot \sqrt[3]{\left(\sqrt[3]{\frac{z \cdot t}{3.0}} \cdot \sqrt[3]{\frac{z \cdot t}{3.0}}\right) \cdot \sqrt[3]{\color{blue}{\left(\sqrt[3]{\frac{z \cdot t}{3.0}} \cdot \sqrt[3]{\frac{z \cdot t}{3.0}}\right) \cdot \sqrt[3]{\frac{z \cdot t}{3.0}}}}}\right) - \frac{a}{b \cdot 3.0}\]

   8. Applied cbrt-prod 14.2

      \[\leadsto \left(2.0 \cdot \sqrt{x}\right) \cdot \cos \left(y - \left(\sqrt[3]{\frac{z \cdot t}{3.0}} \cdot \sqrt[3]{\frac{z \cdot t}{3.0}}\right) \cdot \sqrt[3]{\left(\sqrt[3]{\frac{z \cdot t}{3.0}} \cdot \sqrt[3]{\frac{z \cdot t}{3.0}}\right) \cdot \color{blue}{\left(\sqrt[3]{\sqrt[3]{\frac{z \cdot t}{3.0}} \cdot \sqrt[3]{\frac{z \cdot t}{3.0}}} \cdot \sqrt[3]{\sqrt[3]{\frac{z \cdot t}{3.0}}}\right)}}\right) - \frac{a}{b \cdot 3.0}\]
   9. Using strategy rm

   10. Applied add-cbrt-cube 14.2

       \[\leadsto \left(2.0 \cdot \sqrt{x}\right) \cdot \cos \left(y - \left(\sqrt[3]{\frac{z \cdot t}{3.0}} \cdot \sqrt[3]{\frac{z \cdot t}{3.0}}\right) \cdot \color{blue}{\sqrt[3]{\left(\sqrt[3]{\left(\sqrt[3]{\frac{z \cdot t}{3.0}} \cdot \sqrt[3]{\frac{z \cdot t}{3.0}}\right) \cdot \left(\sqrt[3]{\sqrt[3]{\frac{z \cdot t}{3.0}} \cdot \sqrt[3]{\frac{z \cdot t}{3.0}}} \cdot \sqrt[3]{\sqrt[3]{\frac{z \cdot t}{3.0}}}\right)} \cdot \sqrt[3]{\left(\sqrt[3]{\frac{z \cdot t}{3.0}} \cdot \sqrt[3]{\frac{z \cdot t}{3.0}}\right) \cdot \left(\sqrt[3]{\sqrt[3]{\frac{z \cdot t}{3.0}} \cdot \sqrt[3]{\frac{z \cdot t}{3.0}}} \cdot \sqrt[3]{\sqrt[3]{\frac{z \cdot t}{3.0}}}\right)}\right) \cdot \sqrt[3]{\left(\sqrt[3]{\frac{z \cdot t}{3.0}} \cdot \sqrt[3]{\frac{z \cdot t}{3.0}}\right) \cdot \left(\sqrt[3]{\sqrt[3]{\frac{z \cdot t}{3.0}} \cdot \sqrt[3]{\frac{z \cdot t}{3.0}}} \cdot \sqrt[3]{\sqrt[3]{\frac{z \cdot t}{3.0}}}\right)}}}\right) - \frac{a}{b \cdot 3.0}\]
3. Recombined 2 regimes into one program.

4. Final simplification 18.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot t = -\infty:\\ \;\;\;\;\left(\left(y \cdot \frac{-1}{2}\right) \cdot y + 1\right) \cdot \left(\sqrt{x} \cdot 2.0\right) - \frac{a}{3.0 \cdot b}\\ \mathbf{elif}\;z \cdot t \le 3.947993520431114 \cdot 10^{+303}:\\ \;\;\;\;\left(\sqrt{x} \cdot 2.0\right) \cdot \cos \left(y - \left(\sqrt[3]{\frac{z \cdot t}{3.0}} \cdot \sqrt[3]{\frac{z \cdot t}{3.0}}\right) \cdot \sqrt[3]{\sqrt[3]{\left(\sqrt[3]{\sqrt[3]{\frac{z \cdot t}{3.0}} \cdot \sqrt[3]{\frac{z \cdot t}{3.0}}} \cdot \sqrt[3]{\sqrt[3]{\frac{z \cdot t}{3.0}}}\right) \cdot \left(\sqrt[3]{\frac{z \cdot t}{3.0}} \cdot \sqrt[3]{\frac{z \cdot t}{3.0}}\right)} \cdot \left(\sqrt[3]{\left(\sqrt[3]{\sqrt[3]{\frac{z \cdot t}{3.0}} \cdot \sqrt[3]{\frac{z \cdot t}{3.0}}} \cdot \sqrt[3]{\sqrt[3]{\frac{z \cdot t}{3.0}}}\right) \cdot \left(\sqrt[3]{\frac{z \cdot t}{3.0}} \cdot \sqrt[3]{\frac{z \cdot t}{3.0}}\right)} \cdot \sqrt[3]{\left(\sqrt[3]{\sqrt[3]{\frac{z \cdot t}{3.0}} \cdot \sqrt[3]{\frac{z \cdot t}{3.0}}} \cdot \sqrt[3]{\sqrt[3]{\frac{z \cdot t}{3.0}}}\right) \cdot \left(\sqrt[3]{\frac{z \cdot t}{3.0}} \cdot \sqrt[3]{\frac{z \cdot t}{3.0}}\right)}\right)}\right) - \frac{a}{3.0 \cdot b}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(y \cdot \frac{-1}{2}\right) \cdot y + 1\right) \cdot \left(\sqrt{x} \cdot 2.0\right) - \frac{a}{3.0 \cdot b}\\ \end{array}\]

Reproduce

herbie shell --seed 2019162 
(FPCore (x y z t a b)
  :name "Diagrams.Solve.Polynomial:cubForm  from diagrams-solve-0.1, K"

  :herbie-target
  (if (< z -1.3793337487235141e+129) (- (* (* 2.0 (sqrt x)) (cos (- (/ 1 y) (/ (/ 0.3333333333333333 z) t)))) (/ (/ a 3.0) b)) (if (< z 3.516290613555987e+106) (- (* (* (sqrt x) 2.0) (cos (- y (* (/ t 3.0) z)))) (/ (/ a 3.0) b)) (- (* (cos (- y (/ (/ 0.3333333333333333 z) t))) (* 2.0 (sqrt x))) (/ (/ a b) 3.0))))

  (- (* (* 2.0 (sqrt x)) (cos (- y (/ (* z t) 3.0)))) (/ a (* b 3.0))))