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

Average Error: 6.7 → 0.7

Time: 3.3s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x + \frac{y \cdot \left(z - x\right)}{t} = -\infty:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{t}, z - x, x\right)\\ \mathbf{elif}\;x + \frac{y \cdot \left(z - x\right)}{t} \le 1.1104617369215329 \cdot 10^{308}:\\ \;\;\;\;x + \frac{y \cdot \left(z - x\right)}{t}\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \frac{z - x}{t}\\ \end{array}\]

C

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

↓

double code(double x, double y, double z, double t) {
	double VAR;
	if ((((double) (x + ((double) (((double) (y * ((double) (z - x)))) / t)))) <= -inf.0)) {
		VAR = ((double) fma(((double) (y / t)), ((double) (z - x)), x));
	} else {
		double VAR_1;
		if ((((double) (x + ((double) (((double) (y * ((double) (z - x)))) / t)))) <= 1.110461736921533e+308)) {
			VAR_1 = ((double) (x + ((double) (((double) (y * ((double) (z - x)))) / t))));
		} else {
			VAR_1 = ((double) (x + ((double) (y * ((double) (((double) (z - x)) / t))))));
		}
		VAR = VAR_1;
	}
	return VAR;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original	6.7
Target	2.1
Herbie	0.7

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

Derivation

1. Split input into 3 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. Simplified 0.2

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

   if -inf.0 < (+ x (/ (* y (- z x)) t)) < 1.110461736921533e+308

   1. Initial program 0.8

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

   if 1.110461736921533e+308 < (+ x (/ (* y (- z x)) t))

   1. Initial program 63.8

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

   3. Applied *-un-lft-identity 63.8

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

   4. Applied times-frac 0.2

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

   5. Simplified 0.2

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

4. Final simplification 0.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;x + \frac{y \cdot \left(z - x\right)}{t} = -\infty:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{t}, z - x, x\right)\\ \mathbf{elif}\;x + \frac{y \cdot \left(z - x\right)}{t} \le 1.1104617369215329 \cdot 10^{308}:\\ \;\;\;\;x + \frac{y \cdot \left(z - x\right)}{t}\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \frac{z - x}{t}\\ \end{array}\]

Reproduce

herbie shell --seed 2020120 +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)))