Numeric.Signal:interpolate from hsignal-0.2.7.1

Average Error: 14.6 → 7.9

Time: 6.2s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x + \left(y - z\right) \cdot \frac{t - x}{a - z} \le -3.0937449944064416 \cdot 10^{-295} \lor \neg \left(x + \left(y - z\right) \cdot \frac{t - x}{a - z} \le 3.7357647875255324 \cdot 10^{-237}\right):\\ \;\;\;\;x + \frac{y - z}{\sqrt[3]{a - z} \cdot \sqrt[3]{a - z}} \cdot \frac{t - x}{\sqrt[3]{a - z}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{x}{z} - \frac{t}{z}, t\right)\\ \end{array}\]

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 VAR;
	if ((((x + ((y - z) * ((t - x) / (a - z)))) <= -3.0937449944064416e-295) || !((x + ((y - z) * ((t - x) / (a - z)))) <= 3.7357647875255324e-237))) {
		VAR = (x + (((y - z) / (cbrt((a - z)) * cbrt((a - z)))) * ((t - x) / cbrt((a - z)))));
	} else {
		VAR = fma(y, ((x / z) - (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)))) < -3.0937449944064416e-295 or 3.7357647875255324e-237 < (+ x (* (- y z) (/ (- t x) (- a z))))

   1. Initial program 7.1

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

   3. Applied add-cube-cbrt 7.8

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

   4. Applied *-un-lft-identity 7.8

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

   5. Applied times-frac 7.8

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

   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

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

   if -3.0937449944064416e-295 < (+ x (* (- y z) (/ (- t x) (- a z)))) < 3.7357647875255324e-237

   1. Initial program 57.3

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

   2. Simplified 56.9

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

   3. Taylor expanded around inf 27.2

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

   4. Simplified 23.2

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

4. Final simplification 7.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;x + \left(y - z\right) \cdot \frac{t - x}{a - z} \le -3.0937449944064416 \cdot 10^{-295} \lor \neg \left(x + \left(y - z\right) \cdot \frac{t - x}{a - z} \le 3.7357647875255324 \cdot 10^{-237}\right):\\ \;\;\;\;x + \frac{y - z}{\sqrt[3]{a - z} \cdot \sqrt[3]{a - z}} \cdot \frac{t - x}{\sqrt[3]{a - z}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{x}{z} - \frac{t}{z}, t\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020103 +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)))))