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

Average Error: 6.1 → 0.5

Time: 14.6s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;y \cdot \left(z - t\right) \le -1.247620159404664309930925189564717520917 \cdot 10^{220} \lor \neg \left(y \cdot \left(z - t\right) \le 3.684420206893129511608175584226867275509 \cdot 10^{217}\right):\\ \;\;\;\;x + \frac{y}{\frac{a}{z - t}}\\ \mathbf{else}:\\ \;\;\;\;x + \left(y \cdot \left(z - t\right)\right) \cdot \frac{1}{a}\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a) {
        double r344387 = x;
        double r344388 = y;
        double r344389 = z;
        double r344390 = t;
        double r344391 = r344389 - r344390;
        double r344392 = r344388 * r344391;
        double r344393 = a;
        double r344394 = r344392 / r344393;
        double r344395 = r344387 + r344394;
        return r344395;
}

↓

double f(double x, double y, double z, double t, double a) {
        double r344396 = y;
        double r344397 = z;
        double r344398 = t;
        double r344399 = r344397 - r344398;
        double r344400 = r344396 * r344399;
        double r344401 = -1.2476201594046643e+220;
        bool r344402 = r344400 <= r344401;
        double r344403 = 3.6844202068931295e+217;
        bool r344404 = r344400 <= r344403;
        double r344405 = !r344404;
        bool r344406 = r344402 || r344405;
        double r344407 = x;
        double r344408 = a;
        double r344409 = r344408 / r344399;
        double r344410 = r344396 / r344409;
        double r344411 = r344407 + r344410;
        double r344412 = 1.0;
        double r344413 = r344412 / r344408;
        double r344414 = r344400 * r344413;
        double r344415 = r344407 + r344414;
        double r344416 = r344406 ? r344411 : r344415;
        return r344416;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original	6.1
Target	0.7
Herbie	0.5

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

Derivation

1. Split input into 2 regimes

2. if (* y (- z t)) < -1.2476201594046643e+220 or 3.6844202068931295e+217 < (* y (- z t))

   1. Initial program 31.5

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

   3. Applied associate-/l* 0.9

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

   if -1.2476201594046643e+220 < (* y (- z t)) < 3.6844202068931295e+217

   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 0.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(z - t\right) \le -1.247620159404664309930925189564717520917 \cdot 10^{220} \lor \neg \left(y \cdot \left(z - t\right) \le 3.684420206893129511608175584226867275509 \cdot 10^{217}\right):\\ \;\;\;\;x + \frac{y}{\frac{a}{z - t}}\\ \mathbf{else}:\\ \;\;\;\;x + \left(y \cdot \left(z - t\right)\right) \cdot \frac{1}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2019303 
(FPCore (x y z t a)
  :name "Optimisation.CirclePacking:place from circle-packing-0.1.0.4, E"
  :precision binary64

  :herbie-target
  (if (< y -1.07612662163899753e-10) (+ x (/ 1 (/ (/ a (- z t)) y))) (if (< y 2.8944268627920891e-49) (+ x (/ (* y (- z t)) a)) (+ x (/ y (/ a (- z t))))))

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