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

Average Error: 6.3 → 0.6

Time: 4.4s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;a \le -5.386011167286209797538090785742809716412 \cdot 10^{-28} \lor \neg \left(a \le 4635323820368954973487104\right):\\ \;\;\;\;x + y \cdot \frac{z - t}{a}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y \cdot z + y \cdot \left(-t\right)}{a}\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a) {
        double r285304 = x;
        double r285305 = y;
        double r285306 = z;
        double r285307 = t;
        double r285308 = r285306 - r285307;
        double r285309 = r285305 * r285308;
        double r285310 = a;
        double r285311 = r285309 / r285310;
        double r285312 = r285304 + r285311;
        return r285312;
}

↓

double f(double x, double y, double z, double t, double a) {
        double r285313 = a;
        double r285314 = -5.38601116728621e-28;
        bool r285315 = r285313 <= r285314;
        double r285316 = 4.635323820368955e+24;
        bool r285317 = r285313 <= r285316;
        double r285318 = !r285317;
        bool r285319 = r285315 || r285318;
        double r285320 = x;
        double r285321 = y;
        double r285322 = z;
        double r285323 = t;
        double r285324 = r285322 - r285323;
        double r285325 = r285324 / r285313;
        double r285326 = r285321 * r285325;
        double r285327 = r285320 + r285326;
        double r285328 = r285321 * r285322;
        double r285329 = -r285323;
        double r285330 = r285321 * r285329;
        double r285331 = r285328 + r285330;
        double r285332 = r285331 / r285313;
        double r285333 = r285320 + r285332;
        double r285334 = r285319 ? r285327 : r285333;
        return r285334;
}

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	0.6
Herbie	0.6

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

Derivation

1. Split input into 2 regimes

2. if a < -5.38601116728621e-28 or 4.635323820368955e+24 < a

   1. Initial program 9.8

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

   3. Applied *-un-lft-identity 9.8

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

   4. Applied times-frac 0.6

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

   5. Simplified 0.6

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

   if -5.38601116728621e-28 < a < 4.635323820368955e+24

   1. Initial program 0.6

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

   3. Applied sub-neg 0.6

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

   4. Applied distribute-lft-in 0.6

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

4. Final simplification 0.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \le -5.386011167286209797538090785742809716412 \cdot 10^{-28} \lor \neg \left(a \le 4635323820368954973487104\right):\\ \;\;\;\;x + y \cdot \frac{z - t}{a}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y \cdot z + y \cdot \left(-t\right)}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2019322 
(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)))