Data.Octree.Internal:octantDistance from Octree-0.5.4.2

Average Error: 31.6 → 18.7

Time: 2.2s

Precision: binary64

Math

\[\sqrt{x \cdot x + y \cdot y}\]

↓

\[\begin{array}{l} \mathbf{if}\;y \leq -1.8899720103254926 \cdot 10^{+146}:\\ \;\;\;\;\left|x\right|\\ \mathbf{elif}\;y \leq -2.454304521609729 \cdot 10^{-75}:\\ \;\;\;\;\sqrt{x \cdot x + y \cdot y}\\ \mathbf{elif}\;y \leq 8.447014685477577 \cdot 10^{-160}:\\ \;\;\;\;\left|x\right|\\ \mathbf{elif}\;y \leq 1.6336242095797044 \cdot 10^{-143}:\\ \;\;\;\;y\\ \mathbf{elif}\;y \leq 1.3066676791772896 \cdot 10^{-35}:\\ \;\;\;\;\left|x\right|\\ \mathbf{elif}\;y \leq 4.248125126133587 \cdot 10^{+95}:\\ \;\;\;\;\sqrt{x \cdot x + y \cdot y}\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array}\]

FPCore

(FPCore (x y) :precision binary64 (sqrt (+ (* x x) (* y y))))

↓

(FPCore (x y)
 :precision binary64
 (if (<= y -1.8899720103254926e+146)
   (fabs x)
   (if (<= y -2.454304521609729e-75)
     (sqrt (+ (* x x) (* y y)))
     (if (<= y 8.447014685477577e-160)
       (fabs x)
       (if (<= y 1.6336242095797044e-143)
         y
         (if (<= y 1.3066676791772896e-35)
           (fabs x)
           (if (<= y 4.248125126133587e+95) (sqrt (+ (* x x) (* y y))) y)))))))

C

double code(double x, double y) {
	return sqrt((x * x) + (y * y));
}

↓

double code(double x, double y) {
	double tmp;
	if (y <= -1.8899720103254926e+146) {
		tmp = fabs(x);
	} else if (y <= -2.454304521609729e-75) {
		tmp = sqrt((x * x) + (y * y));
	} else if (y <= 8.447014685477577e-160) {
		tmp = fabs(x);
	} else if (y <= 1.6336242095797044e-143) {
		tmp = y;
	} else if (y <= 1.3066676791772896e-35) {
		tmp = fabs(x);
	} else if (y <= 4.248125126133587e+95) {
		tmp = sqrt((x * x) + (y * y));
	} else {
		tmp = y;
	}
	return tmp;
}

Error

Bits error versus x

Bits error versus y

Target

Original	31.6
Target	17.1
Herbie	18.7

\[\begin{array}{l} \mathbf{if}\;x < -1.1236950826599826 \cdot 10^{+145}:\\ \;\;\;\;-x\\ \mathbf{elif}\;x < 1.116557621183362 \cdot 10^{+93}:\\ \;\;\;\;\sqrt{x \cdot x + y \cdot y}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array}\]

Derivation

1. Split input into 3 regimes

2. if y < -1.88997201032549258e146 or -2.4543045216097289e-75 < y < 8.44701468547757748e-160 or 1.6336242095797044e-143 < y < 1.3066676791772896e-35

   1. Initial program 34.3

      \[\sqrt{x \cdot x + y \cdot y}\]

   2. Taylor expanded around 0 40.6

      \[\leadsto \sqrt{\color{blue}{{x}^{2}}}\]

   3. Simplified 40.6

      \[\leadsto \sqrt{\color{blue}{x \cdot x}}\]
   4. Using strategy rm

   5. Applied rem-sqrt-square_binary64_17823 22.1

      \[\leadsto \color{blue}{\left|x\right|}\]

   if -1.88997201032549258e146 < y < -2.4543045216097289e-75 or 1.3066676791772896e-35 < y < 4.24812512613358684e95

   1. Initial program 15.8

      \[\sqrt{x \cdot x + y \cdot y}\]

   if 8.44701468547757748e-160 < y < 1.6336242095797044e-143 or 4.24812512613358684e95 < y

   1. Initial program 48.1

      \[\sqrt{x \cdot x + y \cdot y}\]

   2. Taylor expanded around 0 13.6

      \[\leadsto \color{blue}{y}\]
3. Recombined 3 regimes into one program.

4. Final simplification 18.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.8899720103254926 \cdot 10^{+146}:\\ \;\;\;\;\left|x\right|\\ \mathbf{elif}\;y \leq -2.454304521609729 \cdot 10^{-75}:\\ \;\;\;\;\sqrt{x \cdot x + y \cdot y}\\ \mathbf{elif}\;y \leq 8.447014685477577 \cdot 10^{-160}:\\ \;\;\;\;\left|x\right|\\ \mathbf{elif}\;y \leq 1.6336242095797044 \cdot 10^{-143}:\\ \;\;\;\;y\\ \mathbf{elif}\;y \leq 1.3066676791772896 \cdot 10^{-35}:\\ \;\;\;\;\left|x\right|\\ \mathbf{elif}\;y \leq 4.248125126133587 \cdot 10^{+95}:\\ \;\;\;\;\sqrt{x \cdot x + y \cdot y}\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array}\]

Reproduce

herbie shell --seed 2020344 
(FPCore (x y)
  :name "Data.Octree.Internal:octantDistance  from Octree-0.5.4.2"
  :precision binary64

  :herbie-target
  (if (< x -1.1236950826599826e+145) (- x) (if (< x 1.116557621183362e+93) (sqrt (+ (* x x) (* y y))) x))

  (sqrt (+ (* x x) (* y y))))