Numeric.Signal:interpolate from hsignal-0.2.7.1

Average Error: 14.9 → 8.6

Time: 5.8s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x + \left(y - z\right) \cdot \frac{t - x}{a - z} \le -2.72946199750767504 \cdot 10^{-281}:\\ \;\;\;\;\mathsf{fma}\left(\sqrt[3]{x} \cdot \sqrt[3]{x}, \sqrt[3]{x}, \left(\sqrt[3]{\frac{y - z}{\sqrt[3]{a - z} \cdot \sqrt[3]{a - z}} \cdot \frac{t - x}{\sqrt[3]{a - z}}} \cdot \sqrt[3]{\frac{y - z}{\sqrt[3]{a - z} \cdot \sqrt[3]{a - z}} \cdot \frac{t - x}{\sqrt[3]{a - z}}}\right) \cdot \sqrt[3]{\frac{y - z}{\sqrt[3]{a - z} \cdot \sqrt[3]{a - z}} \cdot \frac{t - x}{\sqrt[3]{a - z}}}\right)\\ \mathbf{elif}\;x + \left(y - z\right) \cdot \frac{t - x}{a - z} \le 2.76338921123335085 \cdot 10^{-287}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{x}{z} - \frac{t}{z}, t\right)\\ \mathbf{else}:\\ \;\;\;\;x + \left(y - z\right) \cdot \frac{t - x}{a - z}\\ \end{array}\]

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 VAR;
	if (((x + ((y - z) * ((t - x) / (a - z)))) <= -2.729461997507675e-281)) {
		VAR = fma((cbrt(x) * cbrt(x)), cbrt(x), ((cbrt((((y - z) / (cbrt((a - z)) * cbrt((a - z)))) * ((t - x) / cbrt((a - z))))) * cbrt((((y - z) / (cbrt((a - z)) * cbrt((a - z)))) * ((t - x) / cbrt((a - z)))))) * cbrt((((y - z) / (cbrt((a - z)) * cbrt((a - z)))) * ((t - x) / cbrt((a - z)))))));
	} else {
		double VAR_1;
		if (((x + ((y - z) * ((t - x) / (a - z)))) <= 2.763389211233351e-287)) {
			VAR_1 = fma(y, ((x / z) - (t / z)), t);
		} else {
			VAR_1 = (x + ((y - z) * ((t - x) / (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)))) < -2.729461997507675e-281

   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 8.4

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

   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* 5.1

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

   7. Simplified 5.1

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

   9. Applied add-cube-cbrt 5.6

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

   10. Applied fma-def 5.6

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

   12. Applied add-cube-cbrt 5.7

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

   if -2.729461997507675e-281 < (+ x (* (- y z) (/ (- t x) (- a z)))) < 2.763389211233351e-287

   1. Initial program 59.7

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

   2. Simplified 59.3

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

   3. Taylor expanded around inf 26.6

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

   4. Simplified 21.2

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

   if 2.763389211233351e-287 < (+ x (* (- y z) (/ (- t x) (- a z))))

   1. Initial program 7.1

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

4. Final simplification 8.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;x + \left(y - z\right) \cdot \frac{t - x}{a - z} \le -2.72946199750767504 \cdot 10^{-281}:\\ \;\;\;\;\mathsf{fma}\left(\sqrt[3]{x} \cdot \sqrt[3]{x}, \sqrt[3]{x}, \left(\sqrt[3]{\frac{y - z}{\sqrt[3]{a - z} \cdot \sqrt[3]{a - z}} \cdot \frac{t - x}{\sqrt[3]{a - z}}} \cdot \sqrt[3]{\frac{y - z}{\sqrt[3]{a - z} \cdot \sqrt[3]{a - z}} \cdot \frac{t - x}{\sqrt[3]{a - z}}}\right) \cdot \sqrt[3]{\frac{y - z}{\sqrt[3]{a - z} \cdot \sqrt[3]{a - z}} \cdot \frac{t - x}{\sqrt[3]{a - z}}}\right)\\ \mathbf{elif}\;x + \left(y - z\right) \cdot \frac{t - x}{a - z} \le 2.76338921123335085 \cdot 10^{-287}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{x}{z} - \frac{t}{z}, t\right)\\ \mathbf{else}:\\ \;\;\;\;x + \left(y - z\right) \cdot \frac{t - x}{a - z}\\ \end{array}\]

Reproduce

herbie shell --seed 2020079 +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)))))