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

Average Error: 20.7 → 18.4

Time: 28.4s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\left(2 \cdot \sqrt{x}\right) \cdot \cos \left(y - \frac{z \cdot t}{3}\right) \le 1.0306985206888286 \cdot 10^{151}:\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \cos \left(y - z \cdot \frac{t}{3}\right) - \frac{1}{\frac{3}{a} \cdot b}\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \left(1 - \frac{1}{2} \cdot {y}^{2}\right) - \frac{\frac{a}{b}}{3}\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a, double b) {
        double r613692 = 2.0;
        double r613693 = x;
        double r613694 = sqrt(r613693);
        double r613695 = r613692 * r613694;
        double r613696 = y;
        double r613697 = z;
        double r613698 = t;
        double r613699 = r613697 * r613698;
        double r613700 = 3.0;
        double r613701 = r613699 / r613700;
        double r613702 = r613696 - r613701;
        double r613703 = cos(r613702);
        double r613704 = r613695 * r613703;
        double r613705 = a;
        double r613706 = b;
        double r613707 = r613706 * r613700;
        double r613708 = r613705 / r613707;
        double r613709 = r613704 - r613708;
        return r613709;
}

↓

double f(double x, double y, double z, double t, double a, double b) {
        double r613710 = 2.0;
        double r613711 = x;
        double r613712 = sqrt(r613711);
        double r613713 = r613710 * r613712;
        double r613714 = y;
        double r613715 = z;
        double r613716 = t;
        double r613717 = r613715 * r613716;
        double r613718 = 3.0;
        double r613719 = r613717 / r613718;
        double r613720 = r613714 - r613719;
        double r613721 = cos(r613720);
        double r613722 = r613713 * r613721;
        double r613723 = 1.0306985206888286e+151;
        bool r613724 = r613722 <= r613723;
        double r613725 = r613716 / r613718;
        double r613726 = r613715 * r613725;
        double r613727 = r613714 - r613726;
        double r613728 = cos(r613727);
        double r613729 = r613713 * r613728;
        double r613730 = 1.0;
        double r613731 = a;
        double r613732 = r613718 / r613731;
        double r613733 = b;
        double r613734 = r613732 * r613733;
        double r613735 = r613730 / r613734;
        double r613736 = r613729 - r613735;
        double r613737 = 0.5;
        double r613738 = 2.0;
        double r613739 = pow(r613714, r613738);
        double r613740 = r613737 * r613739;
        double r613741 = r613730 - r613740;
        double r613742 = r613713 * r613741;
        double r613743 = r613731 / r613733;
        double r613744 = r613743 / r613718;
        double r613745 = r613742 - r613744;
        double r613746 = r613724 ? r613736 : r613745;
        return r613746;
}

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	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 (* (* 2.0 (sqrt x)) (cos (- y (/ (* z t) 3.0)))) < 1.0306985206888286e+151

   1. Initial program 14.2

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

   3. Applied *-un-lft-identity 14.2

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

   4. Applied times-frac 14.2

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

   5. Simplified 14.2

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

   7. Applied associate-/r* 14.2

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

   9. Applied *-un-lft-identity 14.2

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

   10. Applied *-un-lft-identity 14.2

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

   11. Applied times-frac 14.2

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

   12. Applied associate-/l* 14.3

       \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \cos \left(y - z \cdot \frac{t}{3}\right) - \color{blue}{\frac{\frac{1}{1}}{\frac{3}{\frac{a}{b}}}}\]
   13. Using strategy rm

   14. Applied associate-/r/ 14.2

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

   if 1.0306985206888286e+151 < (* (* 2.0 (sqrt x)) (cos (- y (/ (* z t) 3.0))))

   1. Initial program 61.6

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

   3. Applied *-un-lft-identity 61.6

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

   4. Applied times-frac 61.5

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

   5. Simplified 61.5

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

   7. Applied associate-/r* 61.5

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

   8. Taylor expanded around 0 44.8

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

4. Final simplification 18.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(2 \cdot \sqrt{x}\right) \cdot \cos \left(y - \frac{z \cdot t}{3}\right) \le 1.0306985206888286 \cdot 10^{151}:\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \cos \left(y - z \cdot \frac{t}{3}\right) - \frac{1}{\frac{3}{a} \cdot b}\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \left(1 - \frac{1}{2} \cdot {y}^{2}\right) - \frac{\frac{a}{b}}{3}\\ \end{array}\]

Reproduce

herbie shell --seed 2019198 
(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))))