Numeric.Signal:interpolate from hsignal-0.2.7.1

Average Error: 15.4 → 8.3

Time: 8.2s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x + \left(y - z\right) \cdot \frac{t - x}{a - z} \le -4.9891520750712892 \cdot 10^{-292} \lor \neg \left(x + \left(y - z\right) \cdot \frac{t - x}{a - z} \le 1.0233613271197179 \cdot 10^{-256}\right):\\ \;\;\;\;x + \left(\frac{y - z}{\sqrt[3]{a - z} \cdot \sqrt[3]{a - z}} \cdot \left(\sqrt[3]{\frac{t - x}{\sqrt[3]{a - z}}} \cdot \sqrt[3]{\frac{t - x}{\sqrt[3]{a - z}}}\right)\right) \cdot \left(\sqrt[3]{\frac{\sqrt[3]{t - x} \cdot \sqrt[3]{t - x}}{\sqrt[3]{\sqrt[3]{a - z}} \cdot \sqrt[3]{\sqrt[3]{a - z}}}} \cdot \sqrt[3]{\frac{\sqrt[3]{t - x}}{\sqrt[3]{\sqrt[3]{a - z}}}}\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(\frac{x}{z} - \frac{t}{z}\right) + t\\ \end{array}\]

C

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

↓

double code(double x, double y, double z, double t, double a) {
	double VAR;
	if (((((double) (x + ((double) (((double) (y - z)) * ((double) (((double) (t - x)) / ((double) (a - z)))))))) <= -4.989152075071289e-292) || !(((double) (x + ((double) (((double) (y - z)) * ((double) (((double) (t - x)) / ((double) (a - z)))))))) <= 1.0233613271197179e-256))) {
		VAR = ((double) (x + ((double) (((double) (((double) (((double) (y - z)) / ((double) (((double) cbrt(((double) (a - z)))) * ((double) cbrt(((double) (a - z)))))))) * ((double) (((double) cbrt(((double) (((double) (t - x)) / ((double) cbrt(((double) (a - z)))))))) * ((double) cbrt(((double) (((double) (t - x)) / ((double) cbrt(((double) (a - z)))))))))))) * ((double) (((double) cbrt(((double) (((double) (((double) cbrt(((double) (t - x)))) * ((double) cbrt(((double) (t - x)))))) / ((double) (((double) cbrt(((double) cbrt(((double) (a - z)))))) * ((double) cbrt(((double) cbrt(((double) (a - z)))))))))))) * ((double) cbrt(((double) (((double) cbrt(((double) (t - x)))) / ((double) cbrt(((double) cbrt(((double) (a - z))))))))))))))));
	} else {
		VAR = ((double) (((double) (y * ((double) (((double) (x / z)) - ((double) (t / z)))))) + t));
	}
	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 (+ x (* (- y z) (/ (- t x) (- a z)))) < -4.9891520750712892e-292 or 1.0233613271197179e-256 < (+ x (* (- y z) (/ (- t x) (- a z))))

   1. Initial program 7.7

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

   3. Applied add-cube-cbrt 8.4

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

   4. Applied *-un-lft-identity 8.4

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

   5. Applied times-frac 8.4

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

   6. Applied associate-*r* 5.4

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

   7. Simplified 5.4

      \[\leadsto x + \color{blue}{\frac{y - z}{\sqrt[3]{a - z} \cdot \sqrt[3]{a - z}}} \cdot \frac{t - x}{\sqrt[3]{a - z}}\]
   8. Using strategy rm

   9. Applied add-cube-cbrt 5.6

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

   10. Applied associate-*r* 5.6

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

   12. Applied add-cube-cbrt 5.6

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

   13. Applied add-cube-cbrt 5.6

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

   14. Applied times-frac 5.6

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

   15. Applied cbrt-prod 5.6

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

   if -4.9891520750712892e-292 < (+ x (* (- y z) (/ (- t x) (- a z)))) < 1.0233613271197179e-256

   1. Initial program 59.3

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

   3. Applied add-cube-cbrt 59.0

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

   4. Applied *-un-lft-identity 59.0

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

   5. Applied times-frac 59.0

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

   6. Applied associate-*r* 56.5

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

   7. Simplified 56.5

      \[\leadsto x + \color{blue}{\frac{y - z}{\sqrt[3]{a - z} \cdot \sqrt[3]{a - z}}} \cdot \frac{t - x}{\sqrt[3]{a - z}}\]
   8. Using strategy rm

   9. Applied add-cube-cbrt 56.6

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

   10. Applied associate-*r* 56.6

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

   11. Taylor expanded around inf 27.7

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

   12. Simplified 23.3

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

4. Final simplification 8.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;x + \left(y - z\right) \cdot \frac{t - x}{a - z} \le -4.9891520750712892 \cdot 10^{-292} \lor \neg \left(x + \left(y - z\right) \cdot \frac{t - x}{a - z} \le 1.0233613271197179 \cdot 10^{-256}\right):\\ \;\;\;\;x + \left(\frac{y - z}{\sqrt[3]{a - z} \cdot \sqrt[3]{a - z}} \cdot \left(\sqrt[3]{\frac{t - x}{\sqrt[3]{a - z}}} \cdot \sqrt[3]{\frac{t - x}{\sqrt[3]{a - z}}}\right)\right) \cdot \left(\sqrt[3]{\frac{\sqrt[3]{t - x} \cdot \sqrt[3]{t - x}}{\sqrt[3]{\sqrt[3]{a - z}} \cdot \sqrt[3]{\sqrt[3]{a - z}}}} \cdot \sqrt[3]{\frac{\sqrt[3]{t - x}}{\sqrt[3]{\sqrt[3]{a - z}}}}\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(\frac{x}{z} - \frac{t}{z}\right) + t\\ \end{array}\]

Reproduce

herbie shell --seed 2020162 
(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)))))