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

Average Error: 31.8 → 12.9

Time: 1.8s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;y \cdot \left(y \cdot 4\right) \leq 4.347777683403 \cdot 10^{-322}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \cdot \left(y \cdot 4\right) \leq 3.8018787503186285 \cdot 10^{-283}:\\ \;\;\;\;\frac{x \cdot x - y \cdot \left(y \cdot 4\right)}{y \cdot \left(y \cdot 4\right) + x \cdot x}\\ \mathbf{elif}\;y \cdot \left(y \cdot 4\right) \leq 4.0242122759004074 \cdot 10^{-85}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \cdot \left(y \cdot 4\right) \leq 1.33662301455546 \cdot 10^{+282}:\\ \;\;\;\;\log \left(e^{\frac{x \cdot x - y \cdot \left(y \cdot 4\right)}{y \cdot \left(y \cdot 4\right) + x \cdot x}}\right)\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array}\]

FPCore

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

↓

(FPCore (x y)
 :precision binary64
 (if (<= (* y (* y 4.0)) 4.347777683403e-322)
   1.0
   (if (<= (* y (* y 4.0)) 3.8018787503186285e-283)
     (/ (- (* x x) (* y (* y 4.0))) (+ (* y (* y 4.0)) (* x x)))
     (if (<= (* y (* y 4.0)) 4.0242122759004074e-85)
       1.0
       (if (<= (* y (* y 4.0)) 1.33662301455546e+282)
         (log
          (exp (/ (- (* x x) (* y (* y 4.0))) (+ (* y (* y 4.0)) (* x x)))))
         -1.0)))))

C

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

↓

double code(double x, double y) {
	double tmp;
	if ((((double) (y * ((double) (y * 4.0)))) <= 4.347777683403e-322)) {
		tmp = 1.0;
	} else {
		double tmp_1;
		if ((((double) (y * ((double) (y * 4.0)))) <= 3.8018787503186285e-283)) {
			tmp_1 = (((double) (((double) (x * x)) - ((double) (y * ((double) (y * 4.0)))))) / ((double) (((double) (y * ((double) (y * 4.0)))) + ((double) (x * x)))));
		} else {
			double tmp_2;
			if ((((double) (y * ((double) (y * 4.0)))) <= 4.0242122759004074e-85)) {
				tmp_2 = 1.0;
			} else {
				double tmp_3;
				if ((((double) (y * ((double) (y * 4.0)))) <= 1.33662301455546e+282)) {
					tmp_3 = ((double) log(((double) exp((((double) (((double) (x * x)) - ((double) (y * ((double) (y * 4.0)))))) / ((double) (((double) (y * ((double) (y * 4.0)))) + ((double) (x * x)))))))));
				} else {
					tmp_3 = -1.0;
				}
				tmp_2 = tmp_3;
			}
			tmp_1 = tmp_2;
		}
		tmp = tmp_1;
	}
	return tmp;
}

Error

Bits error versus x

Bits error versus y

Target

Original	31.8
Target	31.5
Herbie	12.9

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

Derivation

1. Split input into 4 regimes

2. if (*.f64 (*.f64 y 4.0) y) < 4.34778e-322 or 3.8018787503186285e-283 < (*.f64 (*.f64 y 4.0) y) < 4.0242122759004074e-85

   1. Initial program 25.5

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

   2. Taylor expanded around inf 13.0

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

   if 4.34778e-322 < (*.f64 (*.f64 y 4.0) y) < 3.8018787503186285e-283

   1. Initial program 13.9

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

   if 4.0242122759004074e-85 < (*.f64 (*.f64 y 4.0) y) < 1.3366230145554599e282

   1. Initial program 16.6

      \[\frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y}\]
   2. Using strategy rm

   3. Applied add-log-exp_binary64 16.6

      \[\leadsto \color{blue}{\log \left(e^{\frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y}}\right)}\]

   4. Simplified 16.6

      \[\leadsto \log \color{blue}{\left(e^{\frac{x \cdot x - y \cdot \left(y \cdot 4\right)}{x \cdot x + y \cdot \left(y \cdot 4\right)}}\right)}\]

   if 1.3366230145554599e282 < (*.f64 (*.f64 y 4.0) y)

   1. Initial program 60.3

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

   2. Taylor expanded around 0 8.5

      \[\leadsto \color{blue}{-1}\]
3. Recombined 4 regimes into one program.

4. Final simplification 12.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(y \cdot 4\right) \leq 4.347777683403 \cdot 10^{-322}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \cdot \left(y \cdot 4\right) \leq 3.8018787503186285 \cdot 10^{-283}:\\ \;\;\;\;\frac{x \cdot x - y \cdot \left(y \cdot 4\right)}{y \cdot \left(y \cdot 4\right) + x \cdot x}\\ \mathbf{elif}\;y \cdot \left(y \cdot 4\right) \leq 4.0242122759004074 \cdot 10^{-85}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \cdot \left(y \cdot 4\right) \leq 1.33662301455546 \cdot 10^{+282}:\\ \;\;\;\;\log \left(e^{\frac{x \cdot x - y \cdot \left(y \cdot 4\right)}{y \cdot \left(y \cdot 4\right) + x \cdot x}}\right)\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array}\]

Reproduce

herbie shell --seed 2020210 
(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))))