Numeric.Signal:interpolate from hsignal-0.2.7.1

Average Error: 14.9 → 10.7

Time: 14.6min

Precision: binary64

• Falling back on racket egraph because egg-herbie package not installed

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;z \le -2.13818177103117769 \cdot 10^{136}:\\ \;\;\;\;\frac{y}{z} \cdot \left(x - t\right) + t\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y - z}{a - z} \cdot \left(t - x\right)\\ \end{array}\]

C

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

↓

double code(double x, double y, double z, double t, double a) {
	double VAR;
	if ((z <= -2.1381817710311777e+136)) {
		VAR = ((double) (((double) ((y / z) * ((double) (x - t)))) + t));
	} else {
		VAR = ((double) (x + ((double) ((((double) (y - z)) / ((double) (a - z))) * ((double) (t - x))))));
	}
	return VAR;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Derivation

1. Split input into 2 regimes

2. if z < -2.13818177103117769e136

   1. Initial program 27.8

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

   2. Taylor expanded around inf 24.2

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

   3. Simplified 15.6

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

   if -2.13818177103117769e136 < z

   1. Initial program 12.5

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

   3. Applied add-cube-cbrt 13.1

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

   4. Applied associate-*l* 13.1

      \[\leadsto x + \color{blue}{\left(\sqrt[3]{y - z} \cdot \sqrt[3]{y - z}\right) \cdot \left(\sqrt[3]{y - z} \cdot \frac{t - x}{a - z}\right)}\]
   5. Using strategy rm

   6. Applied pow1 13.1

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

   7. Applied pow1 13.1

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

   8. Applied pow-prod-down 13.1

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

   9. Applied pow1 13.1

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

   10. Applied pow1 13.1

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

   11. Applied pow-prod-down 13.1

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

   12. Applied pow-prod-down 13.1

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

   13. Simplified 9.8

       \[\leadsto x + {\color{blue}{\left(\frac{y - z}{a - z} \cdot \left(t - x\right)\right)}}^{1}\]
3. Recombined 2 regimes into one program.

4. Final simplification 10.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -2.13818177103117769 \cdot 10^{136}:\\ \;\;\;\;\frac{y}{z} \cdot \left(x - t\right) + t\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y - z}{a - z} \cdot \left(t - x\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020181 
(FPCore (x y z t a)
  :name "Numeric.Signal:interpolate   from hsignal-0.2.7.1"
  :precision binary64
  (+ x (* (- y z) (/ (- t x) (- a z)))))