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

Average Error: 6.8 → 0.8

Time: 2.4s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x + \frac{y \cdot \left(z - x\right)}{t} = -\infty:\\ \;\;\;\;x + \frac{y}{\frac{t}{z - x}}\\ \mathbf{elif}\;x + \frac{y \cdot \left(z - x\right)}{t} \le -9.69488790304777811 \cdot 10^{55}:\\ \;\;\;\;x + \frac{1}{\frac{t}{y \cdot \left(z - x\right)}}\\ \mathbf{elif}\;x + \frac{y \cdot \left(z - x\right)}{t} \le 8.0502450144782888 \cdot 10^{-243}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{t}, z - x, x\right)\\ \mathbf{elif}\;x + \frac{y \cdot \left(z - x\right)}{t} \le 5.6700868646061108 \cdot 10^{271}:\\ \;\;\;\;x + \frac{y \cdot \left(z - x\right)}{t}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{t}, z - x, x\right)\\ \end{array}\]

C

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

↓

double code(double x, double y, double z, double t) {
	double temp;
	if (((x + ((y * (z - x)) / t)) <= -inf.0)) {
		temp = (x + (y / (t / (z - x))));
	} else {
		double temp_1;
		if (((x + ((y * (z - x)) / t)) <= -9.694887903047778e+55)) {
			temp_1 = (x + (1.0 / (t / (y * (z - x)))));
		} else {
			double temp_2;
			if (((x + ((y * (z - x)) / t)) <= 8.050245014478289e-243)) {
				temp_2 = fma((y / t), (z - x), x);
			} else {
				double temp_3;
				if (((x + ((y * (z - x)) / t)) <= 5.670086864606111e+271)) {
					temp_3 = (x + ((y * (z - x)) / t));
				} else {
					temp_3 = fma((y / t), (z - x), x);
				}
				temp_2 = temp_3;
			}
			temp_1 = temp_2;
		}
		temp = temp_1;
	}
	return temp;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original	6.8
Target	2.1
Herbie	0.8

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

Derivation

1. Split input into 4 regimes

2. if (+ x (/ (* y (- z x)) t)) < -inf.0

   1. Initial program 64.0

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

   3. Applied associate-/l* 0.2

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

   if -inf.0 < (+ x (/ (* y (- z x)) t)) < -9.694887903047778e+55

   1. Initial program 0.1

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

   3. Applied clear-num 0.2

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

   if -9.694887903047778e+55 < (+ x (/ (* y (- z x)) t)) < 8.050245014478289e-243 or 5.670086864606111e+271 < (+ x (/ (* y (- z x)) t))

   1. Initial program 11.8

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

   2. Simplified 1.7

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

   if 8.050245014478289e-243 < (+ x (/ (* y (- z x)) t)) < 5.670086864606111e+271

   1. Initial program 0.6

      \[x + \frac{y \cdot \left(z - x\right)}{t}\]
3. Recombined 4 regimes into one program.

4. Final simplification 0.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;x + \frac{y \cdot \left(z - x\right)}{t} = -\infty:\\ \;\;\;\;x + \frac{y}{\frac{t}{z - x}}\\ \mathbf{elif}\;x + \frac{y \cdot \left(z - x\right)}{t} \le -9.69488790304777811 \cdot 10^{55}:\\ \;\;\;\;x + \frac{1}{\frac{t}{y \cdot \left(z - x\right)}}\\ \mathbf{elif}\;x + \frac{y \cdot \left(z - x\right)}{t} \le 8.0502450144782888 \cdot 10^{-243}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{t}, z - x, x\right)\\ \mathbf{elif}\;x + \frac{y \cdot \left(z - x\right)}{t} \le 5.6700868646061108 \cdot 10^{271}:\\ \;\;\;\;x + \frac{y \cdot \left(z - x\right)}{t}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{t}, z - x, x\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020060 +o rules:numerics
(FPCore (x y z t)
  :name "Optimisation.CirclePacking:place from circle-packing-0.1.0.4, D"
  :precision binary64

  :herbie-target
  (- x (+ (* x (/ y t)) (* (- z) (/ y t))))

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