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

Average Error: 6.3 → 1.6

Time: 3.7s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;y \cdot \left(z - t\right) \le 1.0060223783074188 \cdot 10^{-122} \lor \neg \left(y \cdot \left(z - t\right) \le 1.3455688495101448 \cdot 10^{96}\right):\\ \;\;\;\;x + \left(-\frac{y}{a}\right) \cdot \left(z - t\right)\\ \mathbf{else}:\\ \;\;\;\;x - \left(y \cdot \left(z - t\right)\right) \cdot \frac{1}{a}\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a) {
        double r307746 = x;
        double r307747 = y;
        double r307748 = z;
        double r307749 = t;
        double r307750 = r307748 - r307749;
        double r307751 = r307747 * r307750;
        double r307752 = a;
        double r307753 = r307751 / r307752;
        double r307754 = r307746 - r307753;
        return r307754;
}

↓

double f(double x, double y, double z, double t, double a) {
        double r307755 = y;
        double r307756 = z;
        double r307757 = t;
        double r307758 = r307756 - r307757;
        double r307759 = r307755 * r307758;
        double r307760 = 1.0060223783074188e-122;
        bool r307761 = r307759 <= r307760;
        double r307762 = 1.3455688495101448e+96;
        bool r307763 = r307759 <= r307762;
        double r307764 = !r307763;
        bool r307765 = r307761 || r307764;
        double r307766 = x;
        double r307767 = a;
        double r307768 = r307755 / r307767;
        double r307769 = -r307768;
        double r307770 = r307769 * r307758;
        double r307771 = r307766 + r307770;
        double r307772 = 1.0;
        double r307773 = r307772 / r307767;
        double r307774 = r307759 * r307773;
        double r307775 = r307766 - r307774;
        double r307776 = r307765 ? r307771 : r307775;
        return r307776;
}

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	0.7
Herbie	1.6

\[\begin{array}{l} \mathbf{if}\;y \lt -1.07612662163899753 \cdot 10^{-10}:\\ \;\;\;\;x - \frac{1}{\frac{\frac{a}{z - t}}{y}}\\ \mathbf{elif}\;y \lt 2.8944268627920891 \cdot 10^{-49}:\\ \;\;\;\;x - \frac{y \cdot \left(z - t\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y}{\frac{a}{z - t}}\\ \end{array}\]

Derivation

1. Split input into 2 regimes

2. if (* y (- z t)) < 1.0060223783074188e-122 or 1.3455688495101448e+96 < (* y (- z t))

   1. Initial program 7.9

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

   3. Applied clear-num 8.0

      \[\leadsto x - \color{blue}{\frac{1}{\frac{a}{y \cdot \left(z - t\right)}}}\]
   4. Using strategy rm

   5. Applied sub-neg 8.0

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

   6. Simplified 2.0

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

   if 1.0060223783074188e-122 < (* y (- z t)) < 1.3455688495101448e+96

   1. Initial program 0.1

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

   3. Applied div-inv 0.2

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

4. Final simplification 1.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(z - t\right) \le 1.0060223783074188 \cdot 10^{-122} \lor \neg \left(y \cdot \left(z - t\right) \le 1.3455688495101448 \cdot 10^{96}\right):\\ \;\;\;\;x + \left(-\frac{y}{a}\right) \cdot \left(z - t\right)\\ \mathbf{else}:\\ \;\;\;\;x - \left(y \cdot \left(z - t\right)\right) \cdot \frac{1}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2020039 
(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)))