Graphics.Rendering.Plot.Render.Plot.Axis:renderAxisTick from plot-0.2.3.4, B

Average Error: 16.3 → 11.3

Time: 13.7s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;a \le -4.62236662418381568 \cdot 10^{27}:\\ \;\;\;\;\left(x + y\right) - \frac{z - t}{\frac{a - t}{y}}\\ \mathbf{elif}\;a \le -1.30764370089987406 \cdot 10^{-192}:\\ \;\;\;\;x + \left(y - \frac{\left(z - t\right) \cdot y}{a - t}\right)\\ \mathbf{elif}\;a \le 3.13833703943278678 \cdot 10^{-175}:\\ \;\;\;\;\frac{z \cdot y}{t} + x\\ \mathbf{elif}\;a \le 2.0626164423737537 \cdot 10^{-64}:\\ \;\;\;\;\left(x + y\right) - \frac{z - t}{\frac{a - t}{y}}\\ \mathbf{elif}\;a \le 4.2051587456830085 \cdot 10^{57}:\\ \;\;\;\;\frac{z \cdot y}{t} + x\\ \mathbf{else}:\\ \;\;\;\;\left(x + y\right) - \frac{z - t}{\sqrt{a - t}} \cdot \frac{y}{\sqrt{a - t}}\\ \end{array}\]

C

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

↓

double code(double x, double y, double z, double t, double a) {
	double VAR;
	if ((a <= -4.6223666241838157e+27)) {
		VAR = ((double) (((double) (x + y)) - ((double) (((double) (z - t)) / ((double) (((double) (a - t)) / y))))));
	} else {
		double VAR_1;
		if ((a <= -1.307643700899874e-192)) {
			VAR_1 = ((double) (x + ((double) (y - ((double) (((double) (((double) (z - t)) * y)) / ((double) (a - t))))))));
		} else {
			double VAR_2;
			if ((a <= 3.138337039432787e-175)) {
				VAR_2 = ((double) (((double) (((double) (z * y)) / t)) + x));
			} else {
				double VAR_3;
				if ((a <= 2.0626164423737537e-64)) {
					VAR_3 = ((double) (((double) (x + y)) - ((double) (((double) (z - t)) / ((double) (((double) (a - t)) / y))))));
				} else {
					double VAR_4;
					if ((a <= 4.2051587456830085e+57)) {
						VAR_4 = ((double) (((double) (((double) (z * y)) / t)) + x));
					} else {
						VAR_4 = ((double) (((double) (x + y)) - ((double) (((double) (((double) (z - t)) / ((double) sqrt(((double) (a - t)))))) * ((double) (y / ((double) sqrt(((double) (a - t))))))))));
					}
					VAR_3 = VAR_4;
				}
				VAR_2 = VAR_3;
			}
			VAR_1 = VAR_2;
		}
		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	16.3
Target	8.0
Herbie	11.3

\[\begin{array}{l} \mathbf{if}\;\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t} \lt -1.3664970889390727 \cdot 10^{-7}:\\ \;\;\;\;\left(y + x\right) - \left(\left(z - t\right) \cdot \frac{1}{a - t}\right) \cdot y\\ \mathbf{elif}\;\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t} \lt 1.47542934445772333 \cdot 10^{-239}:\\ \;\;\;\;\frac{y \cdot \left(a - z\right) - x \cdot t}{a - t}\\ \mathbf{else}:\\ \;\;\;\;\left(y + x\right) - \left(\left(z - t\right) \cdot \frac{1}{a - t}\right) \cdot y\\ \end{array}\]

Derivation

1. Split input into 4 regimes

2. if a < -4.62236662418381568e27 or 3.13833703943278678e-175 < a < 2.0626164423737537e-64

   1. Initial program 14.8

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

   3. Applied associate-/l* 8.8

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

   if -4.62236662418381568e27 < a < -1.30764370089987406e-192

   1. Initial program 17.0

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

   3. Applied associate--l+ 13.8

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

   if -1.30764370089987406e-192 < a < 3.13833703943278678e-175 or 2.0626164423737537e-64 < a < 4.2051587456830085e57

   1. Initial program 19.3

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

   2. Taylor expanded around inf 14.8

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

   if 4.2051587456830085e57 < a

   1. Initial program 14.1

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

   3. Applied add-sqr-sqrt 17.0

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

   4. Applied times-frac 8.5

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

4. Final simplification 11.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \le -4.62236662418381568 \cdot 10^{27}:\\ \;\;\;\;\left(x + y\right) - \frac{z - t}{\frac{a - t}{y}}\\ \mathbf{elif}\;a \le -1.30764370089987406 \cdot 10^{-192}:\\ \;\;\;\;x + \left(y - \frac{\left(z - t\right) \cdot y}{a - t}\right)\\ \mathbf{elif}\;a \le 3.13833703943278678 \cdot 10^{-175}:\\ \;\;\;\;\frac{z \cdot y}{t} + x\\ \mathbf{elif}\;a \le 2.0626164423737537 \cdot 10^{-64}:\\ \;\;\;\;\left(x + y\right) - \frac{z - t}{\frac{a - t}{y}}\\ \mathbf{elif}\;a \le 4.2051587456830085 \cdot 10^{57}:\\ \;\;\;\;\frac{z \cdot y}{t} + x\\ \mathbf{else}:\\ \;\;\;\;\left(x + y\right) - \frac{z - t}{\sqrt{a - t}} \cdot \frac{y}{\sqrt{a - t}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020175 
(FPCore (x y z t a)
  :name "Graphics.Rendering.Plot.Render.Plot.Axis:renderAxisTick from plot-0.2.3.4, B"
  :precision binary64

  :herbie-target
  (if (< (- (+ x y) (/ (* (- z t) y) (- a t))) -1.3664970889390727e-07) (- (+ y x) (* (* (- z t) (/ 1.0 (- a t))) y)) (if (< (- (+ x y) (/ (* (- z t) y) (- a t))) 1.4754293444577233e-239) (/ (- (* y (- a z)) (* x t)) (- a t)) (- (+ y x) (* (* (- z t) (/ 1.0 (- a t))) y))))

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