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

Average Error: 6.0 → 0.4

Time: 3.3s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;y \cdot \left(z - t\right) \le -1.37784695377484635 \cdot 10^{240}:\\ \;\;\;\;x + y \cdot \frac{z - t}{a}\\ \mathbf{elif}\;y \cdot \left(z - t\right) \le 9.00014865074114473 \cdot 10^{239}:\\ \;\;\;\;x + \left(y \cdot \left(z - t\right)\right) \cdot \frac{1}{a}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{a} \cdot \left(z - t\right)\\ \end{array}\]

C

double code(double x, double y, double z, double t, double a) {
	return ((double) (x + ((double) (((double) (y * ((double) (z - t)))) / a))));
}

↓

double code(double x, double y, double z, double t, double a) {
	double VAR;
	if ((((double) (y * ((double) (z - t)))) <= -1.3778469537748463e+240)) {
		VAR = ((double) (x + ((double) (y * ((double) (((double) (z - t)) / a))))));
	} else {
		double VAR_1;
		if ((((double) (y * ((double) (z - t)))) <= 9.000148650741145e+239)) {
			VAR_1 = ((double) (x + ((double) (((double) (y * ((double) (z - t)))) * ((double) (1.0 / a))))));
		} else {
			VAR_1 = ((double) (x + ((double) (((double) (y / a)) * ((double) (z - t))))));
		}
		VAR = VAR_1;
	}
	return VAR;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original	6.0
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 3 regimes

2. if (* y (- z t)) < -1.3778469537748463e+240

   1. Initial program 38.2

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

   3. Applied *-un-lft-identity 38.2

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

   4. Applied times-frac 0.3

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

   5. Simplified 0.3

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

   if -1.3778469537748463e+240 < (* y (- z t)) < 9.000148650741145e+239

   1. Initial program 0.4

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

   3. Applied div-inv 0.4

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

   if 9.000148650741145e+239 < (* y (- z t))

   1. Initial program 37.0

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

   3. Applied associate-/l* 0.5

      \[\leadsto x + \color{blue}{\frac{y}{\frac{a}{z - t}}}\]
   4. Using strategy rm

   5. Applied associate-/r/ 0.5

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

4. Final simplification 0.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(z - t\right) \le -1.37784695377484635 \cdot 10^{240}:\\ \;\;\;\;x + y \cdot \frac{z - t}{a}\\ \mathbf{elif}\;y \cdot \left(z - t\right) \le 9.00014865074114473 \cdot 10^{239}:\\ \;\;\;\;x + \left(y \cdot \left(z - t\right)\right) \cdot \frac{1}{a}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{a} \cdot \left(z - t\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020128 
(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 (/ 1.0 (/ (/ a (- z t)) y))) (if (< y 2.894426862792089e-49) (+ x (/ (* y (- z t)) a)) (+ x (/ y (/ a (- z t))))))

  (+ x (/ (* y (- z t)) a)))