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

Average Error: 6.3 → 1.0

Time: 7.7s

Precision: 64

• Falling back on racket egraph because egg-herbie package not installed

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;y \cdot \left(z - t\right) \le -1.8638720380733864 \cdot 10^{137} \lor \neg \left(y \cdot \left(z - t\right) \le 1.10810879622775848 \cdot 10^{56}\right):\\ \;\;\;\;x + \frac{z - t}{\frac{a}{y}}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y \cdot \left(z - t\right)}{a}\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a) {
        double r216017 = x;
        double r216018 = y;
        double r216019 = z;
        double r216020 = t;
        double r216021 = r216019 - r216020;
        double r216022 = r216018 * r216021;
        double r216023 = a;
        double r216024 = r216022 / r216023;
        double r216025 = r216017 + r216024;
        return r216025;
}

↓

double f(double x, double y, double z, double t, double a) {
        double r216026 = y;
        double r216027 = z;
        double r216028 = t;
        double r216029 = r216027 - r216028;
        double r216030 = r216026 * r216029;
        double r216031 = -1.8638720380733864e+137;
        bool r216032 = r216030 <= r216031;
        double r216033 = 1.1081087962277585e+56;
        bool r216034 = r216030 <= r216033;
        double r216035 = !r216034;
        bool r216036 = r216032 || r216035;
        double r216037 = x;
        double r216038 = a;
        double r216039 = r216038 / r216026;
        double r216040 = r216029 / r216039;
        double r216041 = r216037 + r216040;
        double r216042 = r216030 / r216038;
        double r216043 = r216037 + r216042;
        double r216044 = r216036 ? r216041 : r216043;
        return r216044;
}

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	0.7
Herbie	1.0

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

Derivation

1. Split input into 2 regimes

2. if (* y (- z t)) < -1.8638720380733864e+137 or 1.1081087962277585e+56 < (* y (- z t))

   1. Initial program 16.1

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

   3. Applied associate-/l* 3.2

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

   5. Applied associate-/r/ 1.9

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

   6. Taylor expanded around 0 16.1

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

   7. Simplified 1.9

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

   if -1.8638720380733864e+137 < (* y (- z t)) < 1.1081087962277585e+56

   1. Initial program 0.5

      \[x + \frac{y \cdot \left(z - t\right)}{a}\]
3. Recombined 2 regimes into one program.

4. Final simplification 1.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(z - t\right) \le -1.8638720380733864 \cdot 10^{137} \lor \neg \left(y \cdot \left(z - t\right) \le 1.10810879622775848 \cdot 10^{56}\right):\\ \;\;\;\;x + \frac{z - t}{\frac{a}{y}}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y \cdot \left(z - t\right)}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2020046 
(FPCore (x y z t a)
  :name "Optimisation.CirclePacking:place from circle-packing-0.1.0.4, E"
  :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)))