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

Average Error: 20.6 → 17.8

Time: 9.8s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\cos \left(y - \frac{z \cdot t}{3}\right) \le 0.9999999429473585:\\ \;\;\;\;\left(\left(2 \cdot \sqrt{x}\right) \cdot \left(\cos y \cdot \cos \left(\frac{z \cdot t}{3}\right)\right) + \left(2 \cdot \sqrt{x}\right) \cdot \left(\sin y \cdot \left(\left(\sqrt[3]{\sin \left(0.333333333333333315 \cdot \left(t \cdot z\right)\right)} \cdot \sqrt[3]{\sin \left(0.333333333333333315 \cdot \left(t \cdot z\right)\right)}\right) \cdot \sqrt[3]{\sin \left(0.333333333333333315 \cdot \left(t \cdot z\right)\right)}\right)\right)\right) - \frac{a}{b \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \left(1 - \frac{1}{2} \cdot {y}^{2}\right) - \frac{a}{b \cdot 3}\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a, double b) {
        double r842655 = 2.0;
        double r842656 = x;
        double r842657 = sqrt(r842656);
        double r842658 = r842655 * r842657;
        double r842659 = y;
        double r842660 = z;
        double r842661 = t;
        double r842662 = r842660 * r842661;
        double r842663 = 3.0;
        double r842664 = r842662 / r842663;
        double r842665 = r842659 - r842664;
        double r842666 = cos(r842665);
        double r842667 = r842658 * r842666;
        double r842668 = a;
        double r842669 = b;
        double r842670 = r842669 * r842663;
        double r842671 = r842668 / r842670;
        double r842672 = r842667 - r842671;
        return r842672;
}

↓

double f(double x, double y, double z, double t, double a, double b) {
        double r842673 = y;
        double r842674 = z;
        double r842675 = t;
        double r842676 = r842674 * r842675;
        double r842677 = 3.0;
        double r842678 = r842676 / r842677;
        double r842679 = r842673 - r842678;
        double r842680 = cos(r842679);
        double r842681 = 0.9999999429473585;
        bool r842682 = r842680 <= r842681;
        double r842683 = 2.0;
        double r842684 = x;
        double r842685 = sqrt(r842684);
        double r842686 = r842683 * r842685;
        double r842687 = cos(r842673);
        double r842688 = cos(r842678);
        double r842689 = r842687 * r842688;
        double r842690 = r842686 * r842689;
        double r842691 = sin(r842673);
        double r842692 = 0.3333333333333333;
        double r842693 = r842675 * r842674;
        double r842694 = r842692 * r842693;
        double r842695 = sin(r842694);
        double r842696 = cbrt(r842695);
        double r842697 = r842696 * r842696;
        double r842698 = r842697 * r842696;
        double r842699 = r842691 * r842698;
        double r842700 = r842686 * r842699;
        double r842701 = r842690 + r842700;
        double r842702 = a;
        double r842703 = b;
        double r842704 = r842703 * r842677;
        double r842705 = r842702 / r842704;
        double r842706 = r842701 - r842705;
        double r842707 = 1.0;
        double r842708 = 0.5;
        double r842709 = 2.0;
        double r842710 = pow(r842673, r842709);
        double r842711 = r842708 * r842710;
        double r842712 = r842707 - r842711;
        double r842713 = r842686 * r842712;
        double r842714 = r842713 - r842705;
        double r842715 = r842682 ? r842706 : r842714;
        return r842715;
}

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	20.6
Target	18.7
Herbie	17.8

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

Derivation

1. Split input into 2 regimes

2. if (cos (- y (/ (* z t) 3.0))) < 0.9999999429473585

   1. Initial program 19.9

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

   3. Applied cos-diff 19.2

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

   4. Applied distribute-lft-in 19.2

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

   5. Taylor expanded around inf 19.2

      \[\leadsto \left(\left(2 \cdot \sqrt{x}\right) \cdot \left(\cos y \cdot \cos \left(\frac{z \cdot t}{3}\right)\right) + \left(2 \cdot \sqrt{x}\right) \cdot \left(\sin y \cdot \color{blue}{\sin \left(0.333333333333333315 \cdot \left(t \cdot z\right)\right)}\right)\right) - \frac{a}{b \cdot 3}\]
   6. Using strategy rm

   7. Applied add-cube-cbrt 19.2

      \[\leadsto \left(\left(2 \cdot \sqrt{x}\right) \cdot \left(\cos y \cdot \cos \left(\frac{z \cdot t}{3}\right)\right) + \left(2 \cdot \sqrt{x}\right) \cdot \left(\sin y \cdot \color{blue}{\left(\left(\sqrt[3]{\sin \left(0.333333333333333315 \cdot \left(t \cdot z\right)\right)} \cdot \sqrt[3]{\sin \left(0.333333333333333315 \cdot \left(t \cdot z\right)\right)}\right) \cdot \sqrt[3]{\sin \left(0.333333333333333315 \cdot \left(t \cdot z\right)\right)}\right)}\right)\right) - \frac{a}{b \cdot 3}\]

   if 0.9999999429473585 < (cos (- y (/ (* z t) 3.0)))

   1. Initial program 21.8

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

   2. Taylor expanded around 0 15.5

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

4. Final simplification 17.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(y - \frac{z \cdot t}{3}\right) \le 0.9999999429473585:\\ \;\;\;\;\left(\left(2 \cdot \sqrt{x}\right) \cdot \left(\cos y \cdot \cos \left(\frac{z \cdot t}{3}\right)\right) + \left(2 \cdot \sqrt{x}\right) \cdot \left(\sin y \cdot \left(\left(\sqrt[3]{\sin \left(0.333333333333333315 \cdot \left(t \cdot z\right)\right)} \cdot \sqrt[3]{\sin \left(0.333333333333333315 \cdot \left(t \cdot z\right)\right)}\right) \cdot \sqrt[3]{\sin \left(0.333333333333333315 \cdot \left(t \cdot z\right)\right)}\right)\right)\right) - \frac{a}{b \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \left(1 - \frac{1}{2} \cdot {y}^{2}\right) - \frac{a}{b \cdot 3}\\ \end{array}\]

Reproduce

herbie shell --seed 2020060 
(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))))