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

Average Error: 6.1 → 0.5

Time: 15.5s

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 r243500 = x;
        double r243501 = y;
        double r243502 = z;
        double r243503 = t;
        double r243504 = r243502 - r243503;
        double r243505 = r243501 * r243504;
        double r243506 = a;
        double r243507 = r243505 / r243506;
        double r243508 = r243500 + r243507;
        return r243508;
}

↓

double f(double x, double y, double z, double t, double a) {
        double r243509 = y;
        double r243510 = z;
        double r243511 = t;
        double r243512 = r243510 - r243511;
        double r243513 = r243509 * r243512;
        double r243514 = -1.2476201594046643e+220;
        bool r243515 = r243513 <= r243514;
        double r243516 = 3.6844202068931295e+217;
        bool r243517 = r243513 <= r243516;
        double r243518 = !r243517;
        bool r243519 = r243515 || r243518;
        double r243520 = x;
        double r243521 = a;
        double r243522 = r243521 / r243512;
        double r243523 = r243509 / r243522;
        double r243524 = r243520 + r243523;
        double r243525 = 1.0;
        double r243526 = r243525 / r243521;
        double r243527 = r243513 * r243526;
        double r243528 = r243520 + r243527;
        double r243529 = r243519 ? r243524 : r243528;
        return r243529;
}

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)))