Linear.Projection:perspective from linear-1.19.1.3, B

Average Error: 14.9 → 0.9

Time: 1.7s

Precision: binary64

Math

\[\frac{\left(x \cdot 2\right) \cdot y}{x - y}\]

↓

\[\begin{array}{l} \mathbf{if}\;\frac{\left(x \cdot 2\right) \cdot y}{x - y} \le -4.6783977535483921 \cdot 10^{-66}:\\ \;\;\;\;x \cdot \left(2 \cdot \frac{y}{x - y}\right)\\ \mathbf{elif}\;\frac{\left(x \cdot 2\right) \cdot y}{x - y} \le -2.44862790981823397 \cdot 10^{-302}:\\ \;\;\;\;\frac{\left(x \cdot 2\right) \cdot y}{x - y}\\ \mathbf{elif}\;\frac{\left(x \cdot 2\right) \cdot y}{x - y} \le 0.0:\\ \;\;\;\;\frac{x \cdot 2}{\frac{x}{y} - 1}\\ \mathbf{elif}\;\frac{\left(x \cdot 2\right) \cdot y}{x - y} \le 4.98076392673537 \cdot 10^{-81}:\\ \;\;\;\;\frac{\left(x \cdot 2\right) \cdot y}{x - y}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(2 \cdot \frac{y}{x - y}\right)\\ \end{array}\]

C

double code(double x, double y) {
	return ((double) (((double) (((double) (x * 2.0)) * y)) / ((double) (x - y))));
}

↓

double code(double x, double y) {
	double VAR;
	if ((((double) (((double) (((double) (x * 2.0)) * y)) / ((double) (x - y)))) <= -4.678397753548392e-66)) {
		VAR = ((double) (x * ((double) (2.0 * ((double) (y / ((double) (x - y))))))));
	} else {
		double VAR_1;
		if ((((double) (((double) (((double) (x * 2.0)) * y)) / ((double) (x - y)))) <= -2.448627909818234e-302)) {
			VAR_1 = ((double) (((double) (((double) (x * 2.0)) * y)) / ((double) (x - y))));
		} else {
			double VAR_2;
			if ((((double) (((double) (((double) (x * 2.0)) * y)) / ((double) (x - y)))) <= 0.0)) {
				VAR_2 = ((double) (((double) (x * 2.0)) / ((double) (((double) (x / y)) - 1.0))));
			} else {
				double VAR_3;
				if ((((double) (((double) (((double) (x * 2.0)) * y)) / ((double) (x - y)))) <= 4.98076392673537e-81)) {
					VAR_3 = ((double) (((double) (((double) (x * 2.0)) * y)) / ((double) (x - y))));
				} else {
					VAR_3 = ((double) (x * ((double) (2.0 * ((double) (y / ((double) (x - y))))))));
				}
				VAR_2 = VAR_3;
			}
			VAR_1 = VAR_2;
		}
		VAR = VAR_1;
	}
	return VAR;
}

Error

Bits error versus x

Bits error versus y

Target

Original	14.9
Target	0.3
Herbie	0.9

\[\begin{array}{l} \mathbf{if}\;x \lt -1.7210442634149447 \cdot 10^{81}:\\ \;\;\;\;\frac{2 \cdot x}{x - y} \cdot y\\ \mathbf{elif}\;x \lt 83645045635564432:\\ \;\;\;\;\frac{x \cdot 2}{\frac{x - y}{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot x}{x - y} \cdot y\\ \end{array}\]

Derivation

1. Split input into 3 regimes

2. if (/ (* (* x 2.0) y) (- x y)) < -4.6783977535483921e-66 or 4.98076392673537e-81 < (/ (* (* x 2.0) y) (- x y))

   1. Initial program 21.3

      \[\frac{\left(x \cdot 2\right) \cdot y}{x - y}\]

   2. Simplified 1.1

      \[\leadsto \color{blue}{x \cdot \left(2 \cdot \frac{y}{x - y}\right)}\]

   if -4.6783977535483921e-66 < (/ (* (* x 2.0) y) (- x y)) < -2.44862790981823397e-302 or 0.0 < (/ (* (* x 2.0) y) (- x y)) < 4.98076392673537e-81

   1. Initial program 0.8

      \[\frac{\left(x \cdot 2\right) \cdot y}{x - y}\]

   if -2.44862790981823397e-302 < (/ (* (* x 2.0) y) (- x y)) < 0.0

   1. Initial program 57.0

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

   3. Applied associate-/l* 0.8

      \[\leadsto \color{blue}{\frac{x \cdot 2}{\frac{x - y}{y}}}\]

   4. Simplified 0.8

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

4. Final simplification 0.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(x \cdot 2\right) \cdot y}{x - y} \le -4.6783977535483921 \cdot 10^{-66}:\\ \;\;\;\;x \cdot \left(2 \cdot \frac{y}{x - y}\right)\\ \mathbf{elif}\;\frac{\left(x \cdot 2\right) \cdot y}{x - y} \le -2.44862790981823397 \cdot 10^{-302}:\\ \;\;\;\;\frac{\left(x \cdot 2\right) \cdot y}{x - y}\\ \mathbf{elif}\;\frac{\left(x \cdot 2\right) \cdot y}{x - y} \le 0.0:\\ \;\;\;\;\frac{x \cdot 2}{\frac{x}{y} - 1}\\ \mathbf{elif}\;\frac{\left(x \cdot 2\right) \cdot y}{x - y} \le 4.98076392673537 \cdot 10^{-81}:\\ \;\;\;\;\frac{\left(x \cdot 2\right) \cdot y}{x - y}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(2 \cdot \frac{y}{x - y}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020179 
(FPCore (x y)
  :name "Linear.Projection:perspective from linear-1.19.1.3, B"
  :precision binary64

  :herbie-target
  (if (< x -1.7210442634149447e+81) (* (/ (* 2.0 x) (- x y)) y) (if (< x 8.364504563556443e+16) (/ (* x 2.0) (/ (- x y) y)) (* (/ (* 2.0 x) (- x y)) y)))

  (/ (* (* x 2.0) y) (- x y)))