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

Average Error: 6.2 → 0.4

Time: 10.0s

Precision: 64

• Falling back on racket egraph because egg-herbie package not installed

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;y \cdot \left(z - t\right) \le -1.4829459359694524 \cdot 10^{214} \lor \neg \left(y \cdot \left(z - t\right) \le 1.7601478476406493 \cdot 10^{275}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, z - t, x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot \left(z - t\right)}{a} + x\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a) {
        double r437716 = x;
        double r437717 = y;
        double r437718 = z;
        double r437719 = t;
        double r437720 = r437718 - r437719;
        double r437721 = r437717 * r437720;
        double r437722 = a;
        double r437723 = r437721 / r437722;
        double r437724 = r437716 + r437723;
        return r437724;
}

↓

double f(double x, double y, double z, double t, double a) {
        double r437725 = y;
        double r437726 = z;
        double r437727 = t;
        double r437728 = r437726 - r437727;
        double r437729 = r437725 * r437728;
        double r437730 = -1.4829459359694524e+214;
        bool r437731 = r437729 <= r437730;
        double r437732 = 1.7601478476406493e+275;
        bool r437733 = r437729 <= r437732;
        double r437734 = !r437733;
        bool r437735 = r437731 || r437734;
        double r437736 = a;
        double r437737 = r437725 / r437736;
        double r437738 = x;
        double r437739 = fma(r437737, r437728, r437738);
        double r437740 = r437729 / r437736;
        double r437741 = r437740 + r437738;
        double r437742 = r437735 ? r437739 : r437741;
        return r437742;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original	6.2
Target	0.7
Herbie	0.4

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

Derivation

1. Split input into 2 regimes

2. if (* y (- z t)) < -1.4829459359694524e+214 or 1.7601478476406493e+275 < (* y (- z t))

   1. Initial program 38.3

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

   2. Simplified 0.5

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y}{a}, z - t, x\right)}\]

   if -1.4829459359694524e+214 < (* y (- z t)) < 1.7601478476406493e+275

   1. Initial program 0.4

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

   2. Simplified 2.9

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y}{a}, z - t, x\right)}\]
   3. Using strategy rm

   4. Applied fma-udef 2.9

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

   5. Simplified 0.4

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

4. Final simplification 0.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(z - t\right) \le -1.4829459359694524 \cdot 10^{214} \lor \neg \left(y \cdot \left(z - t\right) \le 1.7601478476406493 \cdot 10^{275}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, z - t, x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot \left(z - t\right)}{a} + x\\ \end{array}\]

Reproduce

herbie shell --seed 2020047 +o rules:numerics
(FPCore (x y z t a)
  :name "Optimisation.CirclePacking:place from circle-packing-0.1.0.4, E"
  :precision binary64

  :herbie-target
  (if (< y -1.0761266216389975e-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)))