Diagrams.Trail:splitAtParam from diagrams-lib-1.3.0.3, A

Average Error: 7.6 → 2.6

Time: 4.2s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \le -1.9015920711265839 \cdot 10^{227} \lor \neg \left(\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \le 1.281878617467965 \cdot 10^{268}\right):\\ \;\;\;\;\frac{x + \frac{y}{t}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + \frac{1}{\frac{t \cdot z - x}{y \cdot z - x}}}{x + 1}\\ \end{array}\]

C

double code(double x, double y, double z, double t) {
	return ((double) (((double) (x + ((double) (((double) (((double) (y * z)) - x)) / ((double) (((double) (t * z)) - x)))))) / ((double) (x + 1.0))));
}

↓

double code(double x, double y, double z, double t) {
	double VAR;
	if (((((double) (((double) (x + ((double) (((double) (((double) (y * z)) - x)) / ((double) (((double) (t * z)) - x)))))) / ((double) (x + 1.0)))) <= -1.901592071126584e+227) || !(((double) (((double) (x + ((double) (((double) (((double) (y * z)) - x)) / ((double) (((double) (t * z)) - x)))))) / ((double) (x + 1.0)))) <= 1.2818786174679648e+268))) {
		VAR = ((double) (((double) (x + ((double) (y / t)))) / ((double) (x + 1.0))));
	} else {
		VAR = ((double) (((double) (x + ((double) (1.0 / ((double) (((double) (((double) (t * z)) - x)) / ((double) (((double) (y * z)) - x)))))))) / ((double) (x + 1.0))));
	}
	return VAR;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original	7.6
Target	0.3
Herbie	2.6

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

Derivation

1. Split input into 2 regimes

2. if (/ (+ x (/ (- (* y z) x) (- (* t z) x))) (+ x 1.0)) < -1.901592071126584e+227 or 1.2818786174679648e+268 < (/ (+ x (/ (- (* y z) x) (- (* t z) x))) (+ x 1.0))

   1. Initial program 55.5

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

   2. Taylor expanded around inf 15.7

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

   if -1.901592071126584e+227 < (/ (+ x (/ (- (* y z) x) (- (* t z) x))) (+ x 1.0)) < 1.2818786174679648e+268

   1. Initial program 0.6

      \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1}\]
   2. Using strategy rm

   3. Applied clear-num 0.7

      \[\leadsto \frac{x + \color{blue}{\frac{1}{\frac{t \cdot z - x}{y \cdot z - x}}}}{x + 1}\]
3. Recombined 2 regimes into one program.

4. Final simplification 2.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \le -1.9015920711265839 \cdot 10^{227} \lor \neg \left(\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \le 1.281878617467965 \cdot 10^{268}\right):\\ \;\;\;\;\frac{x + \frac{y}{t}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + \frac{1}{\frac{t \cdot z - x}{y \cdot z - x}}}{x + 1}\\ \end{array}\]

Reproduce

herbie shell --seed 2020121 
(FPCore (x y z t)
  :name "Diagrams.Trail:splitAtParam  from diagrams-lib-1.3.0.3, A"
  :precision binary64

  :herbie-target
  (/ (+ x (- (/ y (- t (/ x z))) (/ x (- (* t z) x)))) (+ x 1))

  (/ (+ x (/ (- (* y z) x) (- (* t z) x))) (+ x 1)))