Optimisation.CirclePacking:place from circle-packing-0.1.0.4, E

Average Error: 6.4 → 1.9

Time: 4.1s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;y \leq 4.892833980010369 \cdot 10^{-60}:\\ \;\;\;\;x + \frac{y}{a} \cdot \left(z - t\right)\\ \mathbf{else}:\\ \;\;\;\;x + \left(y \cdot \frac{z}{a} - y \cdot \frac{t}{a}\right)\\ \end{array}\]

FPCore

(FPCore (x y z t a) :precision binary64 (+ x (/ (* y (- z t)) a)))

↓

(FPCore (x y z t a)
 :precision binary64
 (if (<= y 4.892833980010369e-60)
   (+ x (* (/ y a) (- z t)))
   (+ x (- (* y (/ z a)) (* y (/ t a))))))

C

double code(double x, double y, double z, double t, double a) {
	return ((double) (x + (((double) (y * ((double) (z - t)))) / a)));
}

↓

double code(double x, double y, double z, double t, double a) {
	double VAR;
	if ((y <= 4.892833980010369e-60)) {
		VAR = ((double) (x + ((double) ((y / a) * ((double) (z - t))))));
	} else {
		VAR = ((double) (x + ((double) (((double) (y * (z / a))) - ((double) (y * (t / a)))))));
	}
	return VAR;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original	6.4
Target	0.8
Herbie	1.9

\[\begin{array}{l} \mathbf{if}\;y < -1.0761266216389975 \cdot 10^{-10}:\\ \;\;\;\;x + \frac{1}{\frac{\frac{a}{z - t}}{y}}\\ \mathbf{elif}\;y < 2.894426862792089 \cdot 10^{-49}:\\ \;\;\;\;x + \frac{y \cdot \left(z - t\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{\frac{a}{z - t}}\\ \end{array}\]

Derivation

1. Split input into 2 regimes

2. if y < 4.89283398001036916e-60

   1. Initial program 4.5

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

   3. Applied clear-num 4.5

      \[\leadsto x + \color{blue}{\frac{1}{\frac{a}{y \cdot \left(z - t\right)}}}\]
   4. Using strategy rm

   5. Applied associate-/r* 2.2

      \[\leadsto x + \frac{1}{\color{blue}{\frac{\frac{a}{y}}{z - t}}}\]
   6. Using strategy rm

   7. Applied div-inv 2.3

      \[\leadsto x + \frac{1}{\color{blue}{\frac{a}{y} \cdot \frac{1}{z - t}}}\]

   8. Applied add-sqr-sqrt 2.3

      \[\leadsto x + \frac{\color{blue}{\sqrt{1} \cdot \sqrt{1}}}{\frac{a}{y} \cdot \frac{1}{z - t}}\]

   9. Applied times-frac 2.3

      \[\leadsto x + \color{blue}{\frac{\sqrt{1}}{\frac{a}{y}} \cdot \frac{\sqrt{1}}{\frac{1}{z - t}}}\]

   10. Simplified 2.2

       \[\leadsto x + \color{blue}{\frac{y}{a}} \cdot \frac{\sqrt{1}}{\frac{1}{z - t}}\]

   11. Simplified 2.2

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

   if 4.89283398001036916e-60 < y

   1. Initial program 12.0

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

   3. Applied clear-num 12.0

      \[\leadsto x + \color{blue}{\frac{1}{\frac{a}{y \cdot \left(z - t\right)}}}\]
   4. Using strategy rm

   5. Applied associate-/r* 2.5

      \[\leadsto x + \frac{1}{\color{blue}{\frac{\frac{a}{y}}{z - t}}}\]
   6. Using strategy rm

   7. Applied div-inv 2.5

      \[\leadsto x + \frac{1}{\color{blue}{\frac{a}{y} \cdot \frac{1}{z - t}}}\]

   8. Applied add-sqr-sqrt 2.5

      \[\leadsto x + \frac{\color{blue}{\sqrt{1} \cdot \sqrt{1}}}{\frac{a}{y} \cdot \frac{1}{z - t}}\]

   9. Applied times-frac 2.8

      \[\leadsto x + \color{blue}{\frac{\sqrt{1}}{\frac{a}{y}} \cdot \frac{\sqrt{1}}{\frac{1}{z - t}}}\]

   10. Simplified 2.7

       \[\leadsto x + \color{blue}{\frac{y}{a}} \cdot \frac{\sqrt{1}}{\frac{1}{z - t}}\]

   11. Simplified 2.7

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

   13. Applied sub-neg 2.7

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

   14. Applied distribute-lft-in 2.7

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

   15. Simplified 3.0

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

   16. Simplified 1.0

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

4. Final simplification 1.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 4.892833980010369 \cdot 10^{-60}:\\ \;\;\;\;x + \frac{y}{a} \cdot \left(z - t\right)\\ \mathbf{else}:\\ \;\;\;\;x + \left(y \cdot \frac{z}{a} - y \cdot \frac{t}{a}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020198 
(FPCore (x y z t a)
  :name "Optimisation.CirclePacking:place from circle-packing-0.1.0.4, E"
  :precision binary64

  :herbie-target
  (if (< y -1.0761266216389975e-10) (+ x (/ 1.0 (/ (/ a (- z t)) y))) (if (< y 2.894426862792089e-49) (+ x (/ (* y (- z t)) a)) (+ x (/ y (/ a (- z t))))))

  (+ x (/ (* y (- z t)) a)))