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

Average Error: 6.0 → 0.4

Time: 4.3s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{y \cdot \left(z - t\right)}{a} = -inf.0:\\ \;\;\;\;x - \frac{y}{\frac{a}{z - t}}\\ \mathbf{elif}\;\frac{y \cdot \left(z - t\right)}{a} \le 3.85712386761330808 \cdot 10^{295}:\\ \;\;\;\;x - \frac{y \cdot \left(z - t\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;x - \left(z - t\right) \cdot \frac{y}{a}\\ \end{array}\]

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) (((double) (y * ((double) (z - t)))) / a)) <= -inf.0)) {
		VAR = ((double) (x - ((double) (y / ((double) (a / ((double) (z - t))))))));
	} else {
		double VAR_1;
		if ((((double) (((double) (y * ((double) (z - t)))) / a)) <= 3.857123867613308e+295)) {
			VAR_1 = ((double) (x - ((double) (((double) (y * ((double) (z - t)))) / a))));
		} else {
			VAR_1 = ((double) (x - ((double) (((double) (z - t)) * ((double) (y / a))))));
		}
		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	0.6
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)) a) < -inf.0

   1. Initial program 64.0

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

   3. Applied associate-/l* 0.2

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

   if -inf.0 < (/ (* y (- z t)) a) < 3.85712386761330808e295

   1. Initial program 0.3

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

   if 3.85712386761330808e295 < (/ (* y (- z t)) a)

   1. Initial program 55.6

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

   3. Applied add-cube-cbrt 55.7

      \[\leadsto x - \frac{y \cdot \left(z - t\right)}{\color{blue}{\left(\sqrt[3]{a} \cdot \sqrt[3]{a}\right) \cdot \sqrt[3]{a}}}\]

   4. Applied associate-/r* 55.7

      \[\leadsto x - \color{blue}{\frac{\frac{y \cdot \left(z - t\right)}{\sqrt[3]{a} \cdot \sqrt[3]{a}}}{\sqrt[3]{a}}}\]

   5. Simplified 16.4

      \[\leadsto x - \frac{\color{blue}{\frac{y}{\sqrt[3]{a} \cdot \sqrt[3]{a}} \cdot \left(z - t\right)}}{\sqrt[3]{a}}\]

   6. Taylor expanded around 0 55.6

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

   7. Simplified 2.3

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

4. Final simplification 0.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y \cdot \left(z - t\right)}{a} = -inf.0:\\ \;\;\;\;x - \frac{y}{\frac{a}{z - t}}\\ \mathbf{elif}\;\frac{y \cdot \left(z - t\right)}{a} \le 3.85712386761330808 \cdot 10^{295}:\\ \;\;\;\;x - \frac{y \cdot \left(z - t\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;x - \left(z - t\right) \cdot \frac{y}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2020179 
(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 (/ 1.0 (/ (/ a (- z t)) y))) (if (< y 2.894426862792089e-49) (- x (/ (* y (- z t)) a)) (- x (/ y (/ a (- z t))))))

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