Optimisation.CirclePacking:place from circle-packing-0.1.0.4, E

Average Error: 6.3 → 1.3

Time: 21.9s

Precision: 64

Math

\[x + \frac{y \cdot \left(z - t\right)}{a}\]

↓

\[\frac{y}{\frac{\sqrt[3]{a}}{\sqrt[3]{z - t}} \cdot \frac{\sqrt[3]{a}}{\sqrt[3]{z - t}}} \cdot \frac{\sqrt[3]{z - t}}{\sqrt[3]{a}} + x\]

C

double f(double x, double y, double z, double t, double a) {
        double r17805598 = x;
        double r17805599 = y;
        double r17805600 = z;
        double r17805601 = t;
        double r17805602 = r17805600 - r17805601;
        double r17805603 = r17805599 * r17805602;
        double r17805604 = a;
        double r17805605 = r17805603 / r17805604;
        double r17805606 = r17805598 + r17805605;
        return r17805606;
}

↓

double f(double x, double y, double z, double t, double a) {
        double r17805607 = y;
        double r17805608 = a;
        double r17805609 = cbrt(r17805608);
        double r17805610 = z;
        double r17805611 = t;
        double r17805612 = r17805610 - r17805611;
        double r17805613 = cbrt(r17805612);
        double r17805614 = r17805609 / r17805613;
        double r17805615 = r17805614 * r17805614;
        double r17805616 = r17805607 / r17805615;
        double r17805617 = r17805613 / r17805609;
        double r17805618 = r17805616 * r17805617;
        double r17805619 = x;
        double r17805620 = r17805618 + r17805619;
        return r17805620;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original	6.3
Target	0.8
Herbie	1.3

\[\begin{array}{l} \mathbf{if}\;y \lt -1.07612662163899753216593153715602325729 \cdot 10^{-10}:\\ \;\;\;\;x + \frac{1}{\frac{\frac{a}{z - t}}{y}}\\ \mathbf{elif}\;y \lt 2.894426862792089097262541964056085749132 \cdot 10^{-49}:\\ \;\;\;\;x + \frac{y \cdot \left(z - t\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{\frac{a}{z - t}}\\ \end{array}\]

Derivation

1. Initial program 6.3

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

3. Applied add-cube-cbrt 6.8

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

4. Applied times-frac 3.2

   \[\leadsto x + \color{blue}{\frac{y}{\sqrt[3]{a} \cdot \sqrt[3]{a}} \cdot \frac{z - t}{\sqrt[3]{a}}}\]
5. Using strategy rm

6. Applied *-un-lft-identity 3.2

   \[\leadsto x + \frac{y}{\sqrt[3]{a} \cdot \sqrt[3]{a}} \cdot \frac{z - t}{\sqrt[3]{\color{blue}{1 \cdot a}}}\]

7. Applied cbrt-prod 3.2

   \[\leadsto x + \frac{y}{\sqrt[3]{a} \cdot \sqrt[3]{a}} \cdot \frac{z - t}{\color{blue}{\sqrt[3]{1} \cdot \sqrt[3]{a}}}\]

8. Applied add-cube-cbrt 3.3

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

9. Applied times-frac 3.3

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

10. Applied associate-*r* 1.9

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

11. Simplified 1.3

    \[\leadsto x + \color{blue}{\frac{y}{\frac{\sqrt[3]{a}}{\sqrt[3]{z - t}} \cdot \frac{\sqrt[3]{a}}{\sqrt[3]{z - t}}}} \cdot \frac{\sqrt[3]{z - t}}{\sqrt[3]{a}}\]

12. Final simplification 1.3

    \[\leadsto \frac{y}{\frac{\sqrt[3]{a}}{\sqrt[3]{z - t}} \cdot \frac{\sqrt[3]{a}}{\sqrt[3]{z - t}}} \cdot \frac{\sqrt[3]{z - t}}{\sqrt[3]{a}} + x\]

Reproduce

herbie shell --seed 2019170 
(FPCore (x y z t a)
  :name "Optimisation.CirclePacking:place from circle-packing-0.1.0.4, E"

  :herbie-target
  (if (< y -1.0761266216389975e-10) (+ x (/ 1.0 (/ (/ a (- z t)) y))) (if (< y 2.894426862792089e-49) (+ x (/ (* y (- z t)) a)) (+ x (/ y (/ a (- z t))))))

  (+ x (/ (* y (- z t)) a)))