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

Average Error: 6.0 → 0.6

Time: 10.2s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;y \cdot \left(z - t\right) \le -9.2076128670123546 \cdot 10^{171}:\\ \;\;\;\;x - y \cdot \left(\frac{z}{a} - \frac{t}{a}\right)\\ \mathbf{elif}\;y \cdot \left(z - t\right) \le 2.8461840642724514 \cdot 10^{164}:\\ \;\;\;\;x - \frac{y \cdot \left(z - t\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y}{\frac{a}{z - t}}\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a) {
        double r1445 = x;
        double r1446 = y;
        double r1447 = z;
        double r1448 = t;
        double r1449 = r1447 - r1448;
        double r1450 = r1446 * r1449;
        double r1451 = a;
        double r1452 = r1450 / r1451;
        double r1453 = r1445 - r1452;
        return r1453;
}

↓

double f(double x, double y, double z, double t, double a) {
        double r1454 = y;
        double r1455 = z;
        double r1456 = t;
        double r1457 = r1455 - r1456;
        double r1458 = r1454 * r1457;
        double r1459 = -9.207612867012355e+171;
        bool r1460 = r1458 <= r1459;
        double r1461 = x;
        double r1462 = a;
        double r1463 = r1455 / r1462;
        double r1464 = r1456 / r1462;
        double r1465 = r1463 - r1464;
        double r1466 = r1454 * r1465;
        double r1467 = r1461 - r1466;
        double r1468 = 2.8461840642724514e+164;
        bool r1469 = r1458 <= r1468;
        double r1470 = r1458 / r1462;
        double r1471 = r1461 - r1470;
        double r1472 = r1462 / r1457;
        double r1473 = r1454 / r1472;
        double r1474 = r1461 - r1473;
        double r1475 = r1469 ? r1471 : r1474;
        double r1476 = r1460 ? r1467 : r1475;
        return r1476;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original	6.0
Target	0.7
Herbie	0.6

\[\begin{array}{l} \mathbf{if}\;y \lt -1.07612662163899753 \cdot 10^{-10}:\\ \;\;\;\;x - \frac{1}{\frac{\frac{a}{z - t}}{y}}\\ \mathbf{elif}\;y \lt 2.8944268627920891 \cdot 10^{-49}:\\ \;\;\;\;x - \frac{y \cdot \left(z - t\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y}{\frac{a}{z - t}}\\ \end{array}\]

Derivation

1. Split input into 3 regimes

2. if (* y (- z t)) < -9.207612867012355e+171

   1. Initial program 22.7

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

   3. Applied associate-/l* 1.2

      \[\leadsto x - \color{blue}{\frac{y}{\frac{a}{z - t}}}\]
   4. Using strategy rm

   5. Applied div-inv 1.3

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

   6. Applied associate-/r* 1.5

      \[\leadsto x - \color{blue}{\frac{\frac{y}{a}}{\frac{1}{z - t}}}\]
   7. Using strategy rm

   8. Applied add-cube-cbrt 2.2

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

   9. Applied add-sqr-sqrt 2.2

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

   10. Applied times-frac 2.2

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

   11. Applied *-un-lft-identity 2.2

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

   12. Applied *-un-lft-identity 2.2

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

   13. Applied times-frac 2.2

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

   14. Applied times-frac 2.2

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

   15. Simplified 2.2

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

   16. Simplified 2.2

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

   17. Taylor expanded around 0 22.7

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

   18. Simplified 1.4

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

   if -9.207612867012355e+171 < (* y (- z t)) < 2.8461840642724514e+164

   1. Initial program 0.4

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

   if 2.8461840642724514e+164 < (* y (- z t))

   1. Initial program 23.2

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

   3. Applied associate-/l* 1.3

      \[\leadsto x - \color{blue}{\frac{y}{\frac{a}{z - t}}}\]
3. Recombined 3 regimes into one program.

4. Final simplification 0.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(z - t\right) \le -9.2076128670123546 \cdot 10^{171}:\\ \;\;\;\;x - y \cdot \left(\frac{z}{a} - \frac{t}{a}\right)\\ \mathbf{elif}\;y \cdot \left(z - t\right) \le 2.8461840642724514 \cdot 10^{164}:\\ \;\;\;\;x - \frac{y \cdot \left(z - t\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y}{\frac{a}{z - t}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020025 
(FPCore (x y z t a)
  :name "Optimisation.CirclePacking:place from circle-packing-0.1.0.4, F"
  :precision binary64

  :herbie-target
  (if (< y -1.0761266216389975e-10) (- x (/ 1 (/ (/ a (- z t)) y))) (if (< y 2.894426862792089e-49) (- x (/ (* y (- z t)) a)) (- x (/ y (/ a (- z t))))))

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