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

Average Error: 5.6 → 0.7

Time: 8.8s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;y \le -2.0994806737492123 \cdot 10^{-51}:\\ \;\;\;\;x + y \cdot \frac{z - t}{a}\\ \mathbf{elif}\;y \le 1.3603908647454576 \cdot 10^{-28}:\\ \;\;\;\;x + \frac{\left(-t \cdot y\right) + z \cdot y}{a}\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \frac{z - t}{a}\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a) {
        double r5828510 = x;
        double r5828511 = y;
        double r5828512 = z;
        double r5828513 = t;
        double r5828514 = r5828512 - r5828513;
        double r5828515 = r5828511 * r5828514;
        double r5828516 = a;
        double r5828517 = r5828515 / r5828516;
        double r5828518 = r5828510 + r5828517;
        return r5828518;
}

↓

double f(double x, double y, double z, double t, double a) {
        double r5828519 = y;
        double r5828520 = -2.0994806737492123e-51;
        bool r5828521 = r5828519 <= r5828520;
        double r5828522 = x;
        double r5828523 = z;
        double r5828524 = t;
        double r5828525 = r5828523 - r5828524;
        double r5828526 = a;
        double r5828527 = r5828525 / r5828526;
        double r5828528 = r5828519 * r5828527;
        double r5828529 = r5828522 + r5828528;
        double r5828530 = 1.3603908647454576e-28;
        bool r5828531 = r5828519 <= r5828530;
        double r5828532 = r5828524 * r5828519;
        double r5828533 = -r5828532;
        double r5828534 = r5828523 * r5828519;
        double r5828535 = r5828533 + r5828534;
        double r5828536 = r5828535 / r5828526;
        double r5828537 = r5828522 + r5828536;
        double r5828538 = r5828531 ? r5828537 : r5828529;
        double r5828539 = r5828521 ? r5828529 : r5828538;
        return r5828539;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original	5.6
Target	0.6
Herbie	0.7

\[\begin{array}{l} \mathbf{if}\;y \lt -1.0761266216389975 \cdot 10^{-10}:\\ \;\;\;\;x + \frac{1}{\frac{\frac{a}{z - t}}{y}}\\ \mathbf{elif}\;y \lt 2.894426862792089 \cdot 10^{-49}:\\ \;\;\;\;x + \frac{y \cdot \left(z - t\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{\frac{a}{z - t}}\\ \end{array}\]

Derivation

1. Split input into 2 regimes

2. if y < -2.0994806737492123e-51 or 1.3603908647454576e-28 < y

   1. Initial program 11.8

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

   3. Applied *-un-lft-identity 11.8

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

   4. Applied times-frac 1.0

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

   5. Simplified 1.0

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

   if -2.0994806737492123e-51 < y < 1.3603908647454576e-28

   1. Initial program 0.4

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

   3. Applied sub-neg 0.4

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

   4. Applied distribute-lft-in 0.4

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

4. Final simplification 0.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -2.0994806737492123 \cdot 10^{-51}:\\ \;\;\;\;x + y \cdot \frac{z - t}{a}\\ \mathbf{elif}\;y \le 1.3603908647454576 \cdot 10^{-28}:\\ \;\;\;\;x + \frac{\left(-t \cdot y\right) + z \cdot y}{a}\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \frac{z - t}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2019156 +o rules:numerics
(FPCore (x y z t a)
  :name "Optimisation.CirclePacking:place from circle-packing-0.1.0.4, E"

  :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)))