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

Average Error: 6.2 → 0.6

Time: 9.5s

Precision: 64

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;y \cdot \left(z - t\right) \le -1.1054956616853247 \cdot 10^{118} \lor \neg \left(y \cdot \left(z - t\right) \le 2.6579274057747119 \cdot 10^{209}\right):\\ \;\;\;\;x + \frac{t - z}{\frac{a}{y}}\\ \mathbf{else}:\\ \;\;\;\;x - \left(y \cdot \left(z - t\right)\right) \cdot \frac{1}{a}\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a) {
        double r354283 = x;
        double r354284 = y;
        double r354285 = z;
        double r354286 = t;
        double r354287 = r354285 - r354286;
        double r354288 = r354284 * r354287;
        double r354289 = a;
        double r354290 = r354288 / r354289;
        double r354291 = r354283 - r354290;
        return r354291;
}

↓

double f(double x, double y, double z, double t, double a) {
        double r354292 = y;
        double r354293 = z;
        double r354294 = t;
        double r354295 = r354293 - r354294;
        double r354296 = r354292 * r354295;
        double r354297 = -1.1054956616853247e+118;
        bool r354298 = r354296 <= r354297;
        double r354299 = 2.657927405774712e+209;
        bool r354300 = r354296 <= r354299;
        double r354301 = !r354300;
        bool r354302 = r354298 || r354301;
        double r354303 = x;
        double r354304 = r354294 - r354293;
        double r354305 = a;
        double r354306 = r354305 / r354292;
        double r354307 = r354304 / r354306;
        double r354308 = r354303 + r354307;
        double r354309 = 1.0;
        double r354310 = r354309 / r354305;
        double r354311 = r354296 * r354310;
        double r354312 = r354303 - r354311;
        double r354313 = r354302 ? r354308 : r354312;
        return r354313;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original	6.2
Target	0.6
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 2 regimes

2. if (* y (- z t)) < -1.1054956616853247e+118 or 2.657927405774712e+209 < (* y (- z t))

   1. Initial program 22.5

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

   3. Applied sub-neg 22.5

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

   4. Simplified 1.0

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

   if -1.1054956616853247e+118 < (* y (- z t)) < 2.657927405774712e+209

   1. Initial program 0.4

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

   3. Applied div-inv 0.4

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

4. Final simplification 0.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(z - t\right) \le -1.1054956616853247 \cdot 10^{118} \lor \neg \left(y \cdot \left(z - t\right) \le 2.6579274057747119 \cdot 10^{209}\right):\\ \;\;\;\;x + \frac{t - z}{\frac{a}{y}}\\ \mathbf{else}:\\ \;\;\;\;x - \left(y \cdot \left(z - t\right)\right) \cdot \frac{1}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2020043 
(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)))