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

Average Error: 2.1 → 1.8

Time: 4.3s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \le -1.5507018103804069 \cdot 10^{54}:\\ \;\;\;\;\frac{x}{\sqrt[3]{y} \cdot \sqrt[3]{y}} \cdot \frac{z - t}{\sqrt[3]{y}} + t\\ \mathbf{elif}\;x \le 8.515468159734346 \cdot 10^{-92}:\\ \;\;\;\;\frac{1}{\sqrt[3]{y} \cdot \sqrt[3]{y}} \cdot \left(\frac{x}{\sqrt[3]{y}} \cdot \left(z - t\right)\right) + t\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{z - t}{y} + t\\ \end{array}\]

C

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

↓

double code(double x, double y, double z, double t) {
	double VAR;
	if ((x <= -1.550701810380407e+54)) {
		VAR = (((x / (cbrt(y) * cbrt(y))) * ((z - t) / cbrt(y))) + t);
	} else {
		double VAR_1;
		if ((x <= 8.515468159734346e-92)) {
			VAR_1 = (((1.0 / (cbrt(y) * cbrt(y))) * ((x / cbrt(y)) * (z - t))) + t);
		} else {
			VAR_1 = ((x * ((z - t) / y)) + t);
		}
		VAR = VAR_1;
	}
	return VAR;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original	2.1
Target	2.4
Herbie	1.8

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

Derivation

1. Split input into 3 regimes

2. if x < -1.550701810380407e+54

   1. Initial program 4.7

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

   3. Applied div-inv 4.8

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

   4. Applied associate-*l* 2.2

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

   5. Simplified 2.1

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

   7. Applied add-cube-cbrt 2.9

      \[\leadsto x \cdot \frac{z - t}{\color{blue}{\left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) \cdot \sqrt[3]{y}}} + t\]

   8. Applied *-un-lft-identity 2.9

      \[\leadsto x \cdot \frac{\color{blue}{1 \cdot \left(z - t\right)}}{\left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) \cdot \sqrt[3]{y}} + t\]

   9. Applied times-frac 2.9

      \[\leadsto x \cdot \color{blue}{\left(\frac{1}{\sqrt[3]{y} \cdot \sqrt[3]{y}} \cdot \frac{z - t}{\sqrt[3]{y}}\right)} + t\]

   10. Applied associate-*r* 3.6

       \[\leadsto \color{blue}{\left(x \cdot \frac{1}{\sqrt[3]{y} \cdot \sqrt[3]{y}}\right) \cdot \frac{z - t}{\sqrt[3]{y}}} + t\]

   11. Simplified 3.6

       \[\leadsto \color{blue}{\frac{x}{\sqrt[3]{y} \cdot \sqrt[3]{y}}} \cdot \frac{z - t}{\sqrt[3]{y}} + t\]

   if -1.550701810380407e+54 < x < 8.515468159734346e-92

   1. Initial program 1.3

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

   3. Applied add-cube-cbrt 1.7

      \[\leadsto \frac{x}{\color{blue}{\left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) \cdot \sqrt[3]{y}}} \cdot \left(z - t\right) + t\]

   4. Applied *-un-lft-identity 1.7

      \[\leadsto \frac{\color{blue}{1 \cdot x}}{\left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) \cdot \sqrt[3]{y}} \cdot \left(z - t\right) + t\]

   5. Applied times-frac 1.7

      \[\leadsto \color{blue}{\left(\frac{1}{\sqrt[3]{y} \cdot \sqrt[3]{y}} \cdot \frac{x}{\sqrt[3]{y}}\right)} \cdot \left(z - t\right) + t\]

   6. Applied associate-*l* 1.0

      \[\leadsto \color{blue}{\frac{1}{\sqrt[3]{y} \cdot \sqrt[3]{y}} \cdot \left(\frac{x}{\sqrt[3]{y}} \cdot \left(z - t\right)\right)} + t\]

   if 8.515468159734346e-92 < x

   1. Initial program 2.5

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

   3. Applied div-inv 2.6

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

   4. Applied associate-*l* 2.3

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

   5. Simplified 2.3

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

4. Final simplification 1.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.5507018103804069 \cdot 10^{54}:\\ \;\;\;\;\frac{x}{\sqrt[3]{y} \cdot \sqrt[3]{y}} \cdot \frac{z - t}{\sqrt[3]{y}} + t\\ \mathbf{elif}\;x \le 8.515468159734346 \cdot 10^{-92}:\\ \;\;\;\;\frac{1}{\sqrt[3]{y} \cdot \sqrt[3]{y}} \cdot \left(\frac{x}{\sqrt[3]{y}} \cdot \left(z - t\right)\right) + t\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{z - t}{y} + t\\ \end{array}\]

Reproduce

herbie shell --seed 2020106 
(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))