Numeric.Signal:interpolate from hsignal-0.2.7.1

Average Error: 14.9 → 12.3

Time: 4.0m

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;z \le -2.6105916431147138 \cdot 10^{171}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{z}, y, t - \frac{t \cdot y}{z}\right)\\ \mathbf{elif}\;z \le -1.061022533733094 \cdot 10^{-165}:\\ \;\;\;\;x + \frac{y - z}{\frac{a - z}{t - x}}\\ \mathbf{elif}\;z \le 9.7047982793877566 \cdot 10^{127}:\\ \;\;\;\;\mathsf{fma}\left(\sqrt[3]{x} \cdot \sqrt[3]{x}, \sqrt[3]{x}, \left(\left(\left(y - z\right) \cdot \frac{\sqrt[3]{t - x} \cdot \left(\sqrt[3]{\sqrt[3]{t - x}} \cdot \sqrt[3]{\sqrt[3]{t - x}}\right)}{\sqrt[3]{a - z} \cdot \sqrt[3]{1}}\right) \cdot \frac{\sqrt[3]{\sqrt[3]{t - x}}}{\sqrt[3]{a - z}}\right) \cdot \frac{\sqrt[3]{t - x}}{\sqrt[3]{a - z}}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{z}, y, t - \frac{t \cdot y}{z}\right)\\ \end{array}\]

C

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

↓

double code(double x, double y, double z, double t, double a) {
	double temp;
	if ((z <= -2.610591643114714e+171)) {
		temp = fma((x / z), y, (t - ((t * y) / z)));
	} else {
		double temp_1;
		if ((z <= -1.061022533733094e-165)) {
			temp_1 = (x + ((y - z) / ((a - z) / (t - x))));
		} else {
			double temp_2;
			if ((z <= 9.704798279387757e+127)) {
				temp_2 = fma((cbrt(x) * cbrt(x)), cbrt(x), ((((y - z) * ((cbrt((t - x)) * (cbrt(cbrt((t - x))) * cbrt(cbrt((t - x))))) / (cbrt((a - z)) * cbrt(1.0)))) * (cbrt(cbrt((t - x))) / cbrt((a - z)))) * (cbrt((t - x)) / cbrt((a - z)))));
			} else {
				temp_2 = fma((x / z), y, (t - ((t * y) / z)));
			}
			temp_1 = temp_2;
		}
		temp = temp_1;
	}
	return temp;
}

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 z < -2.610591643114714e+171 or 9.704798279387757e+127 < z

   1. Initial program 27.8

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

   2. Taylor expanded around inf 25.3

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

   3. Simplified 22.2

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

   if -2.610591643114714e+171 < z < -1.061022533733094e-165

   1. Initial program 11.0

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

   3. Applied clear-num 11.2

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

   4. Applied un-div-inv 11.0

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

   if -1.061022533733094e-165 < z < 9.704798279387757e+127

   1. Initial program 8.9

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

   3. Applied add-cube-cbrt 9.4

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

   4. Applied add-cube-cbrt 9.5

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

   5. Applied times-frac 9.5

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

   6. Applied associate-*r* 5.7

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

   8. Applied *-un-lft-identity 5.7

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

   9. Applied cbrt-prod 5.7

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

   10. Applied associate-*r* 5.7

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

   11. Applied add-cube-cbrt 5.8

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

   12. Applied associate-*r* 5.8

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

   13. Applied times-frac 5.8

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

   14. Applied associate-*r* 6.0

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

   16. Applied add-cube-cbrt 6.6

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

   17. Applied fma-def 6.5

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

4. Final simplification 12.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -2.6105916431147138 \cdot 10^{171}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{z}, y, t - \frac{t \cdot y}{z}\right)\\ \mathbf{elif}\;z \le -1.061022533733094 \cdot 10^{-165}:\\ \;\;\;\;x + \frac{y - z}{\frac{a - z}{t - x}}\\ \mathbf{elif}\;z \le 9.7047982793877566 \cdot 10^{127}:\\ \;\;\;\;\mathsf{fma}\left(\sqrt[3]{x} \cdot \sqrt[3]{x}, \sqrt[3]{x}, \left(\left(\left(y - z\right) \cdot \frac{\sqrt[3]{t - x} \cdot \left(\sqrt[3]{\sqrt[3]{t - x}} \cdot \sqrt[3]{\sqrt[3]{t - x}}\right)}{\sqrt[3]{a - z} \cdot \sqrt[3]{1}}\right) \cdot \frac{\sqrt[3]{\sqrt[3]{t - x}}}{\sqrt[3]{a - z}}\right) \cdot \frac{\sqrt[3]{t - x}}{\sqrt[3]{a - z}}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{z}, y, t - \frac{t \cdot y}{z}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020066 +o rules:numerics
(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)))))