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

Average Error: 6.0 → 1.4

Time: 5.3s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;y \cdot \left(z - t\right) \le -3.7639044471113401 \cdot 10^{265}:\\ \;\;\;\;x + \frac{y}{a} \cdot \left(z - t\right)\\ \mathbf{elif}\;y \cdot \left(z - t\right) \le -5.3931014774220371 \cdot 10^{-184}:\\ \;\;\;\;x + \left(y \cdot \left(z - t\right)\right) \cdot \frac{1}{a}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{\frac{y}{a}}{\frac{1}{z - t}}\\ \end{array}\]

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)))) <= -3.76390444711134e+265)) {
		VAR = ((double) (x + ((double) (((double) (y / a)) * ((double) (z - t))))));
	} else {
		double VAR_1;
		if ((((double) (y * ((double) (z - t)))) <= -5.393101477422037e-184)) {
			VAR_1 = ((double) (x + ((double) (((double) (y * ((double) (z - t)))) * ((double) (1.0 / a))))));
		} else {
			VAR_1 = ((double) (x + ((double) (((double) (y / a)) / ((double) (1.0 / ((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	1.4

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

Derivation

1. Split input into 3 regimes

2. if (* y (- z t)) < -3.7639044471113401e265

   1. Initial program 44.1

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

   3. Applied associate-/l* 0.2

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

   5. Applied associate-/r/ 0.2

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

   if -3.7639044471113401e265 < (* y (- z t)) < -5.3931014774220371e-184

   1. Initial program 0.1

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

   3. Applied div-inv 0.2

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

   if -5.3931014774220371e-184 < (* y (- z t))

   1. Initial program 5.5

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

   3. Applied associate-/l* 4.8

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

   5. Applied div-inv 4.8

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

   6. Applied associate-/r* 2.2

      \[\leadsto x + \color{blue}{\frac{\frac{y}{a}}{\frac{1}{z - t}}}\]
3. Recombined 3 regimes into one program.

4. Final simplification 1.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(z - t\right) \le -3.7639044471113401 \cdot 10^{265}:\\ \;\;\;\;x + \frac{y}{a} \cdot \left(z - t\right)\\ \mathbf{elif}\;y \cdot \left(z - t\right) \le -5.3931014774220371 \cdot 10^{-184}:\\ \;\;\;\;x + \left(y \cdot \left(z - t\right)\right) \cdot \frac{1}{a}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{\frac{y}{a}}{\frac{1}{z - t}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020173 
(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)))