Numeric.Signal:interpolate from hsignal-0.2.7.1

Average Error: 14.6 → 7.4

Time: 7.1s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x + \left(y - z\right) \cdot \frac{t - x}{a - z} \le -2.4098482626461414 \cdot 10^{-305}:\\ \;\;\;\;x + \frac{\frac{y - z}{\sqrt[3]{a - z} \cdot \sqrt[3]{a - z}}}{\sqrt[3]{\sqrt[3]{a - z} \cdot \sqrt[3]{a - z}}} \cdot \frac{t - x}{\sqrt[3]{\sqrt[3]{a - z}}}\\ \mathbf{elif}\;x + \left(y - z\right) \cdot \frac{t - x}{a - z} \le 0.0:\\ \;\;\;\;\left(\frac{x \cdot y}{z} + t\right) - \frac{t \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{\frac{\sqrt[3]{y - z} \cdot \sqrt[3]{y - z}}{\sqrt[3]{a - z}}}{\sqrt[3]{\sqrt[3]{a - z}}} \cdot \left(\frac{\frac{\sqrt[3]{y - z}}{\sqrt[3]{a - z}}}{\sqrt[3]{\sqrt[3]{a - z}}} \cdot \frac{t - x}{\sqrt[3]{\sqrt[3]{a - z}}}\right)\\ \end{array}\]

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)))))))) <= -2.4098482626461414e-305)) {
		VAR = ((double) (x + ((double) (((double) (((double) (((double) (y - z)) / ((double) (((double) cbrt(((double) (a - z)))) * ((double) cbrt(((double) (a - z)))))))) / ((double) cbrt(((double) (((double) cbrt(((double) (a - z)))) * ((double) cbrt(((double) (a - z)))))))))) * ((double) (((double) (t - x)) / ((double) cbrt(((double) cbrt(((double) (a - z))))))))))));
	} else {
		double VAR_1;
		if ((((double) (x + ((double) (((double) (y - z)) * ((double) (((double) (t - x)) / ((double) (a - z)))))))) <= 0.0)) {
			VAR_1 = ((double) (((double) (((double) (((double) (x * y)) / z)) + t)) - ((double) (((double) (t * y)) / z))));
		} else {
			VAR_1 = ((double) (x + ((double) (((double) (((double) (((double) (((double) cbrt(((double) (y - z)))) * ((double) cbrt(((double) (y - z)))))) / ((double) cbrt(((double) (a - z)))))) / ((double) cbrt(((double) cbrt(((double) (a - z)))))))) * ((double) (((double) (((double) (((double) cbrt(((double) (y - z)))) / ((double) cbrt(((double) (a - z)))))) / ((double) cbrt(((double) cbrt(((double) (a - z)))))))) * ((double) (((double) (t - x)) / ((double) cbrt(((double) cbrt(((double) (a - z))))))))))))));
		}
		VAR = VAR_1;
	}
	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 3 regimes

2. if (+ x (* (- y z) (/ (- t x) (- a z)))) < -2.4098482626461414e-305

   1. Initial program 7.6

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

   3. Applied add-cube-cbrt 8.2

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

   4. Applied *-un-lft-identity 8.2

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

   5. Applied times-frac 8.2

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

   6. Applied associate-*r* 5.2

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

   7. Simplified 5.2

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

   9. Applied add-cube-cbrt 5.2

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

   10. Applied cbrt-prod 5.3

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

   11. Applied *-un-lft-identity 5.3

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

   12. Applied times-frac 5.3

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

   13. Applied associate-*r* 4.9

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

   14. Simplified 4.9

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

   if -2.4098482626461414e-305 < (+ x (* (- y z) (/ (- t x) (- a z)))) < 0.0

   1. Initial program 61.6

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

   2. Taylor expanded around inf 25.7

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

   if 0.0 < (+ x (* (- y z) (/ (- t x) (- a z))))

   1. Initial program 7.4

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

   3. Applied add-cube-cbrt 8.0

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

   4. Applied *-un-lft-identity 8.0

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

   5. Applied times-frac 8.1

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

   6. Applied associate-*r* 4.9

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

   7. Simplified 4.9

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

   9. Applied add-cube-cbrt 5.0

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

   10. Applied cbrt-prod 5.0

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

   11. Applied *-un-lft-identity 5.0

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

   12. Applied times-frac 5.0

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

   13. Applied associate-*r* 4.6

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

   14. Simplified 4.6

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

   16. Applied cbrt-prod 4.7

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

   17. Applied add-cube-cbrt 4.7

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

   18. Applied times-frac 4.7

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

   19. Applied times-frac 4.7

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

   20. Applied associate-*l* 4.3

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

4. Final simplification 7.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;x + \left(y - z\right) \cdot \frac{t - x}{a - z} \le -2.4098482626461414 \cdot 10^{-305}:\\ \;\;\;\;x + \frac{\frac{y - z}{\sqrt[3]{a - z} \cdot \sqrt[3]{a - z}}}{\sqrt[3]{\sqrt[3]{a - z} \cdot \sqrt[3]{a - z}}} \cdot \frac{t - x}{\sqrt[3]{\sqrt[3]{a - z}}}\\ \mathbf{elif}\;x + \left(y - z\right) \cdot \frac{t - x}{a - z} \le 0.0:\\ \;\;\;\;\left(\frac{x \cdot y}{z} + t\right) - \frac{t \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{\frac{\sqrt[3]{y - z} \cdot \sqrt[3]{y - z}}{\sqrt[3]{a - z}}}{\sqrt[3]{\sqrt[3]{a - z}}} \cdot \left(\frac{\frac{\sqrt[3]{y - z}}{\sqrt[3]{a - z}}}{\sqrt[3]{\sqrt[3]{a - z}}} \cdot \frac{t - x}{\sqrt[3]{\sqrt[3]{a - z}}}\right)\\ \end{array}\]

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)))))