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

Average Error: 20.7 → 17.8

Time: 31.6s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\cos \left(y - \frac{t \cdot z}{3}\right) \le 0.999999999991291077527932884549954906106:\\ \;\;\;\;\left(\sqrt{x} \cdot 2\right) \cdot \left(\cos y \cdot \cos \left(0.3333333333333333148296162562473909929395 \cdot \left(t \cdot z\right)\right) + \sin y \cdot \left(\sqrt[3]{\sin \left(0.3333333333333333148296162562473909929395 \cdot \left(t \cdot z\right)\right)} \cdot \left(\sqrt[3]{\sin \left(0.3333333333333333148296162562473909929395 \cdot \left(t \cdot z\right)\right)} \cdot \sqrt[3]{\sin \left(0.3333333333333333148296162562473909929395 \cdot \left(t \cdot z\right)\right)}\right)\right)\right) - \frac{a}{b \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\left(1 - \frac{1}{2} \cdot \left(y \cdot y\right)\right) \cdot \left(\sqrt{x} \cdot 2\right) - \frac{a}{b \cdot 3}\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a, double b) {
        double r34912816 = 2.0;
        double r34912817 = x;
        double r34912818 = sqrt(r34912817);
        double r34912819 = r34912816 * r34912818;
        double r34912820 = y;
        double r34912821 = z;
        double r34912822 = t;
        double r34912823 = r34912821 * r34912822;
        double r34912824 = 3.0;
        double r34912825 = r34912823 / r34912824;
        double r34912826 = r34912820 - r34912825;
        double r34912827 = cos(r34912826);
        double r34912828 = r34912819 * r34912827;
        double r34912829 = a;
        double r34912830 = b;
        double r34912831 = r34912830 * r34912824;
        double r34912832 = r34912829 / r34912831;
        double r34912833 = r34912828 - r34912832;
        return r34912833;
}

↓

double f(double x, double y, double z, double t, double a, double b) {
        double r34912834 = y;
        double r34912835 = t;
        double r34912836 = z;
        double r34912837 = r34912835 * r34912836;
        double r34912838 = 3.0;
        double r34912839 = r34912837 / r34912838;
        double r34912840 = r34912834 - r34912839;
        double r34912841 = cos(r34912840);
        double r34912842 = 0.9999999999912911;
        bool r34912843 = r34912841 <= r34912842;
        double r34912844 = x;
        double r34912845 = sqrt(r34912844);
        double r34912846 = 2.0;
        double r34912847 = r34912845 * r34912846;
        double r34912848 = cos(r34912834);
        double r34912849 = 0.3333333333333333;
        double r34912850 = r34912849 * r34912837;
        double r34912851 = cos(r34912850);
        double r34912852 = r34912848 * r34912851;
        double r34912853 = sin(r34912834);
        double r34912854 = sin(r34912850);
        double r34912855 = cbrt(r34912854);
        double r34912856 = r34912855 * r34912855;
        double r34912857 = r34912855 * r34912856;
        double r34912858 = r34912853 * r34912857;
        double r34912859 = r34912852 + r34912858;
        double r34912860 = r34912847 * r34912859;
        double r34912861 = a;
        double r34912862 = b;
        double r34912863 = r34912862 * r34912838;
        double r34912864 = r34912861 / r34912863;
        double r34912865 = r34912860 - r34912864;
        double r34912866 = 1.0;
        double r34912867 = 0.5;
        double r34912868 = r34912834 * r34912834;
        double r34912869 = r34912867 * r34912868;
        double r34912870 = r34912866 - r34912869;
        double r34912871 = r34912870 * r34912847;
        double r34912872 = r34912871 - r34912864;
        double r34912873 = r34912843 ? r34912865 : r34912872;
        return r34912873;
}

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	20.7
Target	18.7
Herbie	17.8

\[\begin{array}{l} \mathbf{if}\;z \lt -1.379333748723514136852843173740882251575 \cdot 10^{129}:\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \cos \left(\frac{1}{y} - \frac{\frac{0.3333333333333333148296162562473909929395}{z}}{t}\right) - \frac{\frac{a}{3}}{b}\\ \mathbf{elif}\;z \lt 3.516290613555987147199887107423758623887 \cdot 10^{106}:\\ \;\;\;\;\left(\sqrt{x} \cdot 2\right) \cdot \cos \left(y - \frac{t}{3} \cdot z\right) - \frac{\frac{a}{3}}{b}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(y - \frac{\frac{0.3333333333333333148296162562473909929395}{z}}{t}\right) \cdot \left(2 \cdot \sqrt{x}\right) - \frac{\frac{a}{b}}{3}\\ \end{array}\]

Derivation

1. Split input into 2 regimes

2. if (cos (- y (/ (* z t) 3.0))) < 0.9999999999912911

   1. Initial program 19.7

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

   3. Applied cos-diff 19.0

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

   4. Taylor expanded around inf 19.0

      \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \left(\cos y \cdot \color{blue}{\cos \left(0.3333333333333333148296162562473909929395 \cdot \left(t \cdot z\right)\right)} + \sin y \cdot \sin \left(\frac{z \cdot t}{3}\right)\right) - \frac{a}{b \cdot 3}\]

   5. Simplified 19.0

      \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \left(\cos y \cdot \color{blue}{\cos \left(\left(z \cdot t\right) \cdot 0.3333333333333333148296162562473909929395\right)} + \sin y \cdot \sin \left(\frac{z \cdot t}{3}\right)\right) - \frac{a}{b \cdot 3}\]

   6. Taylor expanded around inf 19.0

      \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \left(\cos y \cdot \cos \left(\left(z \cdot t\right) \cdot 0.3333333333333333148296162562473909929395\right) + \sin y \cdot \color{blue}{\sin \left(0.3333333333333333148296162562473909929395 \cdot \left(t \cdot z\right)\right)}\right) - \frac{a}{b \cdot 3}\]

   7. Simplified 19.0

      \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \left(\cos y \cdot \cos \left(\left(z \cdot t\right) \cdot 0.3333333333333333148296162562473909929395\right) + \sin y \cdot \color{blue}{\sin \left(\left(z \cdot t\right) \cdot 0.3333333333333333148296162562473909929395\right)}\right) - \frac{a}{b \cdot 3}\]
   8. Using strategy rm

   9. Applied add-cube-cbrt 19.0

      \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \left(\cos y \cdot \cos \left(\left(z \cdot t\right) \cdot 0.3333333333333333148296162562473909929395\right) + \sin y \cdot \color{blue}{\left(\left(\sqrt[3]{\sin \left(\left(z \cdot t\right) \cdot 0.3333333333333333148296162562473909929395\right)} \cdot \sqrt[3]{\sin \left(\left(z \cdot t\right) \cdot 0.3333333333333333148296162562473909929395\right)}\right) \cdot \sqrt[3]{\sin \left(\left(z \cdot t\right) \cdot 0.3333333333333333148296162562473909929395\right)}\right)}\right) - \frac{a}{b \cdot 3}\]

   if 0.9999999999912911 < (cos (- y (/ (* z t) 3.0)))

   1. Initial program 22.3

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

   2. Taylor expanded around 0 15.6

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

   3. Simplified 15.6

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

4. Final simplification 17.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(y - \frac{t \cdot z}{3}\right) \le 0.999999999991291077527932884549954906106:\\ \;\;\;\;\left(\sqrt{x} \cdot 2\right) \cdot \left(\cos y \cdot \cos \left(0.3333333333333333148296162562473909929395 \cdot \left(t \cdot z\right)\right) + \sin y \cdot \left(\sqrt[3]{\sin \left(0.3333333333333333148296162562473909929395 \cdot \left(t \cdot z\right)\right)} \cdot \left(\sqrt[3]{\sin \left(0.3333333333333333148296162562473909929395 \cdot \left(t \cdot z\right)\right)} \cdot \sqrt[3]{\sin \left(0.3333333333333333148296162562473909929395 \cdot \left(t \cdot z\right)\right)}\right)\right)\right) - \frac{a}{b \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\left(1 - \frac{1}{2} \cdot \left(y \cdot y\right)\right) \cdot \left(\sqrt{x} \cdot 2\right) - \frac{a}{b \cdot 3}\\ \end{array}\]

Reproduce

herbie shell --seed 2019192 
(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 (- (/ 1.0 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))))