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

Average Error: 2.4 → 1.5

Time: 3.8s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{x - y}{z - y} \le -1.8737237687632686 \cdot 10^{-150}:\\ \;\;\;\;\left(\frac{x}{z - y} - \frac{y}{z - y}\right) \cdot t\\ \mathbf{elif}\;\frac{x - y}{z - y} \le -0.0:\\ \;\;\;\;{\left(\frac{\frac{t}{z - y}}{\frac{1}{x - y}}\right)}^{1}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{t}{\frac{z - y}{x - y}}\right)}^{1}\\ \end{array}\]

C

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

↓

double code(double x, double y, double z, double t) {
	double VAR;
	if ((((x - y) / (z - y)) <= -1.8737237687632686e-150)) {
		VAR = (((x / (z - y)) - (y / (z - y))) * t);
	} else {
		double VAR_1;
		if ((((x - y) / (z - y)) <= -0.0)) {
			VAR_1 = pow(((t / (z - y)) / (1.0 / (x - y))), 1.0);
		} else {
			VAR_1 = pow((t / ((z - y) / (x - y))), 1.0);
		}
		VAR = VAR_1;
	}
	return VAR;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original	2.4
Target	2.4
Herbie	1.5

\[\frac{t}{\frac{z - y}{x - y}}\]

Derivation

1. Split input into 3 regimes

2. if (/ (- x y) (- z y)) < -1.8737237687632686e-150

   1. Initial program 2.6

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

   3. Applied div-sub 2.6

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

   if -1.8737237687632686e-150 < (/ (- x y) (- z y)) < -0.0

   1. Initial program 10.3

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

   3. Applied clear-num 10.9

      \[\leadsto \color{blue}{\frac{1}{\frac{z - y}{x - y}}} \cdot t\]
   4. Using strategy rm

   5. Applied pow1 10.9

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

   6. Applied pow1 10.9

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

   7. Applied pow-prod-down 10.9

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

   8. Simplified 10.8

      \[\leadsto {\color{blue}{\left(\frac{t}{\frac{z - y}{x - y}}\right)}}^{1}\]
   9. Using strategy rm

   10. Applied div-inv 10.9

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

   11. Applied associate-/r* 1.4

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

   if -0.0 < (/ (- x y) (- z y))

   1. Initial program 1.1

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

   3. Applied clear-num 1.2

      \[\leadsto \color{blue}{\frac{1}{\frac{z - y}{x - y}}} \cdot t\]
   4. Using strategy rm

   5. Applied pow1 1.2

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

   6. Applied pow1 1.2

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

   7. Applied pow-prod-down 1.2

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

   8. Simplified 1.1

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

4. Final simplification 1.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - y}{z - y} \le -1.8737237687632686 \cdot 10^{-150}:\\ \;\;\;\;\left(\frac{x}{z - y} - \frac{y}{z - y}\right) \cdot t\\ \mathbf{elif}\;\frac{x - y}{z - y} \le -0.0:\\ \;\;\;\;{\left(\frac{\frac{t}{z - y}}{\frac{1}{x - y}}\right)}^{1}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{t}{\frac{z - y}{x - y}}\right)}^{1}\\ \end{array}\]

Reproduce

herbie shell --seed 2020079 +o rules:numerics
(FPCore (x y z t)
  :name "Numeric.Signal.Multichannel:$cput from hsignal-0.2.7.1"
  :precision binary64

  :herbie-target
  (/ t (/ (- z y) (- x y)))

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