Numeric.Signal.Multichannel:$cget from hsignal-0.2.7.1

Average Error: 2.2 → 1.1

Time: 3.8s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;y \le -1020.47037932589012:\\ \;\;\;\;x \cdot \frac{z - t}{y} + t\\ \mathbf{elif}\;y \le 2.2155056236690332 \cdot 10^{-6}:\\ \;\;\;\;\frac{x \cdot \left(z - t\right)}{y} + t\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\frac{1}{y} \cdot z\right) + \left(t - t \cdot \frac{x}{y}\right)\\ \end{array}\]

C

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

↓

double code(double x, double y, double z, double t) {
	double VAR;
	if ((y <= -1020.4703793258901)) {
		VAR = ((double) (((double) (x * ((double) (((double) (z - t)) / y)))) + t));
	} else {
		double VAR_1;
		if ((y <= 2.215505623669033e-06)) {
			VAR_1 = ((double) (((double) (((double) (x * ((double) (z - t)))) / y)) + t));
		} else {
			VAR_1 = ((double) (((double) (x * ((double) (((double) (1.0 / y)) * z)))) + ((double) (t - ((double) (t * ((double) (x / y))))))));
		}
		VAR = VAR_1;
	}
	return VAR;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original	2.2
Target	2.5
Herbie	1.1

\[\begin{array}{l} \mathbf{if}\;z \lt 2.7594565545626922 \cdot 10^{-282}:\\ \;\;\;\;\frac{x}{y} \cdot \left(z - t\right) + t\\ \mathbf{elif}\;z \lt 2.326994450874436 \cdot 10^{-110}:\\ \;\;\;\;x \cdot \frac{z - t}{y} + t\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} \cdot \left(z - t\right) + t\\ \end{array}\]

Derivation

1. Split input into 3 regimes

2. if y < -1020.4703793258901

   1. Initial program 1.1

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

   3. Applied div-inv 1.2

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

   4. Applied associate-*l* 1.0

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

   5. Simplified 1.0

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

   if -1020.4703793258901 < y < 2.215505623669033e-06

   1. Initial program 3.9

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

   3. Applied associate-*l/ 1.6

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

   if 2.215505623669033e-06 < y

   1. Initial program 1.2

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

   3. Applied sub-neg 1.2

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

   4. Applied distribute-lft-in 1.2

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

   5. Applied associate-+l+ 1.2

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

   6. Simplified 1.2

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

   8. Applied div-inv 1.2

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

   9. Applied associate-*l* 0.7

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

4. Final simplification 1.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -1020.47037932589012:\\ \;\;\;\;x \cdot \frac{z - t}{y} + t\\ \mathbf{elif}\;y \le 2.2155056236690332 \cdot 10^{-6}:\\ \;\;\;\;\frac{x \cdot \left(z - t\right)}{y} + t\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\frac{1}{y} \cdot z\right) + \left(t - t \cdot \frac{x}{y}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020121 
(FPCore (x y z t)
  :name "Numeric.Signal.Multichannel:$cget from hsignal-0.2.7.1"
  :precision binary64

  :herbie-target
  (if (< z 2.759456554562692e-282) (+ (* (/ x y) (- z t)) t) (if (< z 2.326994450874436e-110) (+ (* x (/ (- z t) y)) t) (+ (* (/ x y) (- z t)) t)))

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