Diagrams.TwoD.Arc:arcBetween from diagrams-lib-1.3.0.3

Average Error: 31.5 → 13.5

Time: 1.3s

Precision: binary64

Math

\[\frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y}\]

↓

\[\begin{array}{l} \mathbf{if}\;x \leq -9.176639587062072 \cdot 10^{+126}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq -1.1078942387214006 \cdot 10^{-156}:\\ \;\;\;\;\frac{x \cdot x - y \cdot \left(y \cdot 4\right)}{x \cdot x + y \cdot \left(y \cdot 4\right)}\\ \mathbf{elif}\;x \leq 2.8925572567990346 \cdot 10^{+34}:\\ \;\;\;\;-1\\ \mathbf{elif}\;x \leq 5.06138741049213 \cdot 10^{+85}:\\ \;\;\;\;\frac{x \cdot x - y \cdot \left(y \cdot 4\right)}{x \cdot x + y \cdot \left(y \cdot 4\right)}\\ \mathbf{elif}\;x \leq 5.713892186588795 \cdot 10^{+103}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array}\]

FPCore

(FPCore (x y)
 :precision binary64
 (/ (- (* x x) (* (* y 4.0) y)) (+ (* x x) (* (* y 4.0) y))))

↓

(FPCore (x y)
 :precision binary64
 (if (<= x -9.176639587062072e+126)
   1.0
   (if (<= x -1.1078942387214006e-156)
     (/ (- (* x x) (* y (* y 4.0))) (+ (* x x) (* y (* y 4.0))))
     (if (<= x 2.8925572567990346e+34)
       -1.0
       (if (<= x 5.06138741049213e+85)
         (/ (- (* x x) (* y (* y 4.0))) (+ (* x x) (* y (* y 4.0))))
         (if (<= x 5.713892186588795e+103) -1.0 1.0))))))

C

double code(double x, double y) {
	return ((x * x) - ((y * 4.0) * y)) / ((x * x) + ((y * 4.0) * y));
}

↓

double code(double x, double y) {
	double tmp;
	if (x <= -9.176639587062072e+126) {
		tmp = 1.0;
	} else if (x <= -1.1078942387214006e-156) {
		tmp = ((x * x) - (y * (y * 4.0))) / ((x * x) + (y * (y * 4.0)));
	} else if (x <= 2.8925572567990346e+34) {
		tmp = -1.0;
	} else if (x <= 5.06138741049213e+85) {
		tmp = ((x * x) - (y * (y * 4.0))) / ((x * x) + (y * (y * 4.0)));
	} else if (x <= 5.713892186588795e+103) {
		tmp = -1.0;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Error

Bits error versus x

Bits error versus y

Target

Original	31.5
Target	31.2
Herbie	13.5

\[\begin{array}{l} \mathbf{if}\;\frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y} < 0.9743233849626781:\\ \;\;\;\;\frac{x \cdot x}{x \cdot x + \left(y \cdot y\right) \cdot 4} - \frac{\left(y \cdot y\right) \cdot 4}{x \cdot x + \left(y \cdot y\right) \cdot 4}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{x}{\sqrt{x \cdot x + \left(y \cdot y\right) \cdot 4}}\right)}^{2} - \frac{\left(y \cdot y\right) \cdot 4}{x \cdot x + \left(y \cdot y\right) \cdot 4}\\ \end{array}\]

Derivation

1. Split input into 3 regimes

2. if x < -9.17663958706207189e126 or 5.71389218658879528e103 < x

   1. Initial program 54.5

      \[\frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y}\]

   2. Taylor expanded around inf 10.4

      \[\leadsto \color{blue}{1}\]

   if -9.17663958706207189e126 < x < -1.10789423872140058e-156 or 2.89255725679903464e34 < x < 5.06138741049213032e85

   1. Initial program 14.4

      \[\frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y}\]

   if -1.10789423872140058e-156 < x < 2.89255725679903464e34 or 5.06138741049213032e85 < x < 5.71389218658879528e103

   1. Initial program 24.8

      \[\frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y}\]

   2. Taylor expanded around 0 15.2

      \[\leadsto \color{blue}{-1}\]
3. Recombined 3 regimes into one program.

4. Final simplification 13.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -9.176639587062072 \cdot 10^{+126}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq -1.1078942387214006 \cdot 10^{-156}:\\ \;\;\;\;\frac{x \cdot x - y \cdot \left(y \cdot 4\right)}{x \cdot x + y \cdot \left(y \cdot 4\right)}\\ \mathbf{elif}\;x \leq 2.8925572567990346 \cdot 10^{+34}:\\ \;\;\;\;-1\\ \mathbf{elif}\;x \leq 5.06138741049213 \cdot 10^{+85}:\\ \;\;\;\;\frac{x \cdot x - y \cdot \left(y \cdot 4\right)}{x \cdot x + y \cdot \left(y \cdot 4\right)}\\ \mathbf{elif}\;x \leq 5.713892186588795 \cdot 10^{+103}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array}\]

Reproduce

herbie shell --seed 2020231 
(FPCore (x y)
  :name "Diagrams.TwoD.Arc:arcBetween from diagrams-lib-1.3.0.3"
  :precision binary64

  :herbie-target
  (if (< (/ (- (* x x) (* (* y 4.0) y)) (+ (* x x) (* (* y 4.0) y))) 0.9743233849626781) (- (/ (* x x) (+ (* x x) (* (* y y) 4.0))) (/ (* (* y y) 4.0) (+ (* x x) (* (* y y) 4.0)))) (- (pow (/ x (sqrt (+ (* x x) (* (* y y) 4.0)))) 2.0) (/ (* (* y y) 4.0) (+ (* x x) (* (* y y) 4.0)))))

  (/ (- (* x x) (* (* y 4.0) y)) (+ (* x x) (* (* y 4.0) y))))