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

Average Error: 20.9 → 18.4

Time: 13.8s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\cos \left(y - \frac{z \cdot t}{3}\right) \le 0.9999929107336669:\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \left(\cos y \cdot \cos \left(\frac{\frac{z \cdot t}{\sqrt[3]{3} \cdot \sqrt[3]{3}}}{\sqrt[3]{3}}\right) + \sin y \cdot \sin \left(\frac{z}{\frac{3}{t}}\right)\right) - \frac{\frac{a}{b}}{3}\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \left(1 - \frac{1}{2} \cdot {y}^{2}\right) - \frac{a}{b \cdot 3}\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a, double b) {
        double r764775 = 2.0;
        double r764776 = x;
        double r764777 = sqrt(r764776);
        double r764778 = r764775 * r764777;
        double r764779 = y;
        double r764780 = z;
        double r764781 = t;
        double r764782 = r764780 * r764781;
        double r764783 = 3.0;
        double r764784 = r764782 / r764783;
        double r764785 = r764779 - r764784;
        double r764786 = cos(r764785);
        double r764787 = r764778 * r764786;
        double r764788 = a;
        double r764789 = b;
        double r764790 = r764789 * r764783;
        double r764791 = r764788 / r764790;
        double r764792 = r764787 - r764791;
        return r764792;
}

↓

double f(double x, double y, double z, double t, double a, double b) {
        double r764793 = y;
        double r764794 = z;
        double r764795 = t;
        double r764796 = r764794 * r764795;
        double r764797 = 3.0;
        double r764798 = r764796 / r764797;
        double r764799 = r764793 - r764798;
        double r764800 = cos(r764799);
        double r764801 = 0.9999929107336669;
        bool r764802 = r764800 <= r764801;
        double r764803 = 2.0;
        double r764804 = x;
        double r764805 = sqrt(r764804);
        double r764806 = r764803 * r764805;
        double r764807 = cos(r764793);
        double r764808 = cbrt(r764797);
        double r764809 = r764808 * r764808;
        double r764810 = r764796 / r764809;
        double r764811 = r764810 / r764808;
        double r764812 = cos(r764811);
        double r764813 = r764807 * r764812;
        double r764814 = sin(r764793);
        double r764815 = r764797 / r764795;
        double r764816 = r764794 / r764815;
        double r764817 = sin(r764816);
        double r764818 = r764814 * r764817;
        double r764819 = r764813 + r764818;
        double r764820 = r764806 * r764819;
        double r764821 = a;
        double r764822 = b;
        double r764823 = r764821 / r764822;
        double r764824 = r764823 / r764797;
        double r764825 = r764820 - r764824;
        double r764826 = 1.0;
        double r764827 = 0.5;
        double r764828 = 2.0;
        double r764829 = pow(r764793, r764828);
        double r764830 = r764827 * r764829;
        double r764831 = r764826 - r764830;
        double r764832 = r764806 * r764831;
        double r764833 = r764822 * r764797;
        double r764834 = r764821 / r764833;
        double r764835 = r764832 - r764834;
        double r764836 = r764802 ? r764825 : r764835;
        return r764836;
}

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	19.1
Herbie	18.4

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

Derivation

1. Split input into 2 regimes

2. if (cos (- y (/ (* z t) 3.0))) < 0.9999929107336669

   1. Initial program 20.3

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

   3. Applied cos-diff 19.7

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

   5. Applied associate-/l* 19.7

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

   7. Applied add-cube-cbrt 19.7

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

   8. Applied associate-/r* 19.7

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

   10. Applied associate-/r* 19.7

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

   if 0.9999929107336669 < (cos (- y (/ (* z t) 3.0)))

   1. Initial program 21.9

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

   2. Taylor expanded around 0 16.2

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

4. Final simplification 18.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(y - \frac{z \cdot t}{3}\right) \le 0.9999929107336669:\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \left(\cos y \cdot \cos \left(\frac{\frac{z \cdot t}{\sqrt[3]{3} \cdot \sqrt[3]{3}}}{\sqrt[3]{3}}\right) + \sin y \cdot \sin \left(\frac{z}{\frac{3}{t}}\right)\right) - \frac{\frac{a}{b}}{3}\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \left(1 - \frac{1}{2} \cdot {y}^{2}\right) - \frac{a}{b \cdot 3}\\ \end{array}\]

Reproduce

herbie shell --seed 2020036 +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))))