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

Average Error: 20.9 → 18.6

Time: 17.3s

Precision: 64

• Falling back on racket egraph because egg-herbie package not installed

Math

\[\left(2 \cdot \sqrt{x}\right) \cdot \cos \left(y - \frac{z \cdot t}{3}\right) - \frac{a}{b \cdot 3}\]

↓

\[\begin{array}{l} \mathbf{if}\;z \cdot t = -\infty \lor \neg \left(z \cdot t \le 5.07070334138322496 \cdot 10^{205}\right):\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \left(1 - \frac{1}{2} \cdot {y}^{2}\right) - \frac{a}{b \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(2 \cdot \sqrt{x}\right) \cdot \left(\cos y \cdot \cos \left(\frac{z \cdot t}{3}\right)\right) + \left(\sqrt[3]{\left(2 \cdot \sqrt{x}\right) \cdot \left(\sin y \cdot \sin \left(\frac{z \cdot t}{3}\right)\right)} \cdot \sqrt[3]{\left(2 \cdot \sqrt{x}\right) \cdot \left(\sin y \cdot \sin \left(\frac{z \cdot t}{3}\right)\right)}\right) \cdot \sqrt[3]{\left(2 \cdot \sqrt{x}\right) \cdot \left(\sin y \cdot \sin \left(\frac{z \cdot t}{3}\right)\right)}\right) - \frac{a}{b \cdot 3}\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a, double b) {
        double r848853 = 2.0;
        double r848854 = x;
        double r848855 = sqrt(r848854);
        double r848856 = r848853 * r848855;
        double r848857 = y;
        double r848858 = z;
        double r848859 = t;
        double r848860 = r848858 * r848859;
        double r848861 = 3.0;
        double r848862 = r848860 / r848861;
        double r848863 = r848857 - r848862;
        double r848864 = cos(r848863);
        double r848865 = r848856 * r848864;
        double r848866 = a;
        double r848867 = b;
        double r848868 = r848867 * r848861;
        double r848869 = r848866 / r848868;
        double r848870 = r848865 - r848869;
        return r848870;
}

↓

double f(double x, double y, double z, double t, double a, double b) {
        double r848871 = z;
        double r848872 = t;
        double r848873 = r848871 * r848872;
        double r848874 = -inf.0;
        bool r848875 = r848873 <= r848874;
        double r848876 = 5.070703341383225e+205;
        bool r848877 = r848873 <= r848876;
        double r848878 = !r848877;
        bool r848879 = r848875 || r848878;
        double r848880 = 2.0;
        double r848881 = x;
        double r848882 = sqrt(r848881);
        double r848883 = r848880 * r848882;
        double r848884 = 1.0;
        double r848885 = 0.5;
        double r848886 = y;
        double r848887 = 2.0;
        double r848888 = pow(r848886, r848887);
        double r848889 = r848885 * r848888;
        double r848890 = r848884 - r848889;
        double r848891 = r848883 * r848890;
        double r848892 = a;
        double r848893 = b;
        double r848894 = 3.0;
        double r848895 = r848893 * r848894;
        double r848896 = r848892 / r848895;
        double r848897 = r848891 - r848896;
        double r848898 = cos(r848886);
        double r848899 = r848873 / r848894;
        double r848900 = cos(r848899);
        double r848901 = r848898 * r848900;
        double r848902 = r848883 * r848901;
        double r848903 = sin(r848886);
        double r848904 = sin(r848899);
        double r848905 = r848903 * r848904;
        double r848906 = r848883 * r848905;
        double r848907 = cbrt(r848906);
        double r848908 = r848907 * r848907;
        double r848909 = r848908 * r848907;
        double r848910 = r848902 + r848909;
        double r848911 = r848910 - r848896;
        double r848912 = r848879 ? r848897 : r848911;
        return r848912;
}

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	20.9
Target	18.9
Herbie	18.6

\[\begin{array}{l} \mathbf{if}\;z \lt -1.379333748723514 \cdot 10^{129}:\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \cos \left(\frac{1}{y} - \frac{\frac{0.333333333333333315}{z}}{t}\right) - \frac{\frac{a}{3}}{b}\\ \mathbf{elif}\;z \lt 3.51629061355598715 \cdot 10^{106}:\\ \;\;\;\;\left(\sqrt{x} \cdot 2\right) \cdot \cos \left(y - \frac{t}{3} \cdot z\right) - \frac{\frac{a}{3}}{b}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(y - \frac{\frac{0.333333333333333315}{z}}{t}\right) \cdot \left(2 \cdot \sqrt{x}\right) - \frac{\frac{a}{b}}{3}\\ \end{array}\]

Derivation

1. Split input into 2 regimes

2. if (* z t) < -inf.0 or 5.070703341383225e+205 < (* z t)

   1. Initial program 56.0

      \[\left(2 \cdot \sqrt{x}\right) \cdot \cos \left(y - \frac{z \cdot t}{3}\right) - \frac{a}{b \cdot 3}\]

   2. Taylor expanded around 0 45.7

      \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \color{blue}{\left(1 - \frac{1}{2} \cdot {y}^{2}\right)} - \frac{a}{b \cdot 3}\]

   if -inf.0 < (* z t) < 5.070703341383225e+205

   1. Initial program 13.4

      \[\left(2 \cdot \sqrt{x}\right) \cdot \cos \left(y - \frac{z \cdot t}{3}\right) - \frac{a}{b \cdot 3}\]
   2. Using strategy rm

   3. Applied cos-diff 12.9

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

   4. Applied distribute-lft-in 12.9

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

   6. Applied add-cube-cbrt 12.9

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

4. Final simplification 18.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot t = -\infty \lor \neg \left(z \cdot t \le 5.07070334138322496 \cdot 10^{205}\right):\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \left(1 - \frac{1}{2} \cdot {y}^{2}\right) - \frac{a}{b \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(2 \cdot \sqrt{x}\right) \cdot \left(\cos y \cdot \cos \left(\frac{z \cdot t}{3}\right)\right) + \left(\sqrt[3]{\left(2 \cdot \sqrt{x}\right) \cdot \left(\sin y \cdot \sin \left(\frac{z \cdot t}{3}\right)\right)} \cdot \sqrt[3]{\left(2 \cdot \sqrt{x}\right) \cdot \left(\sin y \cdot \sin \left(\frac{z \cdot t}{3}\right)\right)}\right) \cdot \sqrt[3]{\left(2 \cdot \sqrt{x}\right) \cdot \left(\sin y \cdot \sin \left(\frac{z \cdot t}{3}\right)\right)}\right) - \frac{a}{b \cdot 3}\\ \end{array}\]

Reproduce

herbie shell --seed 2020043 +o rules:numerics
(FPCore (x y z t a b)
  :name "Diagrams.Solve.Polynomial:cubForm  from diagrams-solve-0.1, K"
  :precision binary64

  :herbie-target
  (if (< z -1.379333748723514e+129) (- (* (* 2 (sqrt x)) (cos (- (/ 1 y) (/ (/ 0.3333333333333333 z) t)))) (/ (/ a 3) b)) (if (< z 3.516290613555987e+106) (- (* (* (sqrt x) 2) (cos (- y (* (/ t 3) z)))) (/ (/ a 3) b)) (- (* (cos (- y (/ (/ 0.3333333333333333 z) t))) (* 2 (sqrt x))) (/ (/ a b) 3))))

  (- (* (* 2 (sqrt x)) (cos (- y (/ (* z t) 3)))) (/ a (* b 3))))