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

Average Error: 6.1 → 0.4

Time: 13.2s

Precision: binary64

Cost: 1730

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;y \cdot \left(z - t\right) \leq -\infty:\\ \;\;\;\;x + \left(z - t\right) \cdot \frac{y}{a}\\ \mathbf{elif}\;y \cdot \left(z - t\right) \leq 3.9254137765804485 \cdot 10^{+209}:\\ \;\;\;\;x + \frac{y \cdot \left(z - t\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{\frac{a}{z - t}}\\ \end{array}\]

FPCore

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

↓

(FPCore (x y z t a)
 :precision binary64
 (if (<= (* y (- z t)) (- INFINITY))
   (+ x (* (- z t) (/ y a)))
   (if (<= (* y (- z t)) 3.9254137765804485e+209)
     (+ x (/ (* y (- z t)) a))
     (+ x (/ y (/ a (- z t)))))))

C

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

↓

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((y * (z - t)) <= -((double) INFINITY)) {
		tmp = x + ((z - t) * (y / a));
	} else if ((y * (z - t)) <= 3.9254137765804485e+209) {
		tmp = x + ((y * (z - t)) / a);
	} else {
		tmp = x + (y / (a / (z - t)));
	}
	return tmp;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original	6.1
Target	0.8
Herbie	0.4

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

Alternatives

Alternative 1
Error	0.4
Cost	1730

\[\begin{array}{l} \mathbf{if}\;y \cdot \left(z - t\right) \leq -\infty:\\ \;\;\;\;x + y \cdot \frac{z - t}{a}\\ \mathbf{elif}\;y \cdot \left(z - t\right) \leq 3.9254137765804485 \cdot 10^{+209}:\\ \;\;\;\;x + \frac{y \cdot \left(z - t\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{\frac{a}{z - t}}\\ \end{array}\]

Alternative 2
Error	0.4
Cost	1416

\[\begin{array}{l} \mathbf{if}\;y \cdot \left(z - t\right) \leq -\infty \lor \neg \left(y \cdot \left(z - t\right) \leq 1.4623540897987464 \cdot 10^{+225}\right):\\ \;\;\;\;x + y \cdot \frac{z - t}{a}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y \cdot \left(z - t\right)}{a}\\ \end{array}\]

Alternative 3
Error	4.0
Cost	2705

\[\begin{array}{l} \mathbf{if}\;\frac{y \cdot \left(z - t\right)}{a} \leq -\infty \lor \neg \left(\frac{y \cdot \left(z - t\right)}{a} \leq -2.5690609144704082 \cdot 10^{+138} \lor \neg \left(\frac{y \cdot \left(z - t\right)}{a} \leq 2.0839191885777027 \cdot 10^{+196}\right) \land \frac{y \cdot \left(z - t\right)}{a} \leq 6.950418515138569 \cdot 10^{+301}\right):\\ \;\;\;\;x + y \cdot \frac{z - t}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot \left(z - t\right)}{a}\\ \end{array}\]

Alternative 4
Error	12.8
Cost	6088

\[\begin{array}{l} \mathbf{if}\;\frac{y \cdot \left(z - t\right)}{a} \leq -1.5869757133031571 \cdot 10^{+137}:\\ \;\;\;\;\left(z - t\right) \cdot \frac{y}{a}\\ \mathbf{elif}\;\frac{y \cdot \left(z - t\right)}{a} \leq -6.978901212274857 \cdot 10^{+68}:\\ \;\;\;\;x - \frac{y \cdot t}{a}\\ \mathbf{elif}\;\frac{y \cdot \left(z - t\right)}{a} \leq -2.5680070502093675 \cdot 10^{+20}:\\ \;\;\;\;\left(z - t\right) \cdot \frac{y}{a}\\ \mathbf{elif}\;\frac{y \cdot \left(z - t\right)}{a} \leq 1.186588078276849 \cdot 10^{+100}:\\ \;\;\;\;x - \frac{y \cdot t}{a}\\ \mathbf{elif}\;\frac{y \cdot \left(z - t\right)}{a} \leq 2.6629160219080746 \cdot 10^{+142}:\\ \;\;\;\;\frac{y \cdot \left(z - t\right)}{a}\\ \mathbf{elif}\;\frac{y \cdot \left(z - t\right)}{a} \leq 5.0994858525912255 \cdot 10^{+157}:\\ \;\;\;\;x - \frac{t}{\frac{a}{y}}\\ \mathbf{elif}\;\frac{y \cdot \left(z - t\right)}{a} \leq 1.1531213554742953 \cdot 10^{+209}:\\ \;\;\;\;x + \frac{1}{\frac{a}{y \cdot z}}\\ \mathbf{elif}\;\frac{y \cdot \left(z - t\right)}{a} \leq 6.950418515138569 \cdot 10^{+301}:\\ \;\;\;\;\frac{y \cdot \left(z - t\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{z}{\frac{a}{y}}\\ \end{array}\]

Alternative 5
Error	12.8
Cost	6088

\[\begin{array}{l} \mathbf{if}\;\frac{y \cdot \left(z - t\right)}{a} \leq -1.5869757133031571 \cdot 10^{+137}:\\ \;\;\;\;\left(z - t\right) \cdot \frac{y}{a}\\ \mathbf{elif}\;\frac{y \cdot \left(z - t\right)}{a} \leq -6.978901212274857 \cdot 10^{+68}:\\ \;\;\;\;x - \frac{y \cdot t}{a}\\ \mathbf{elif}\;\frac{y \cdot \left(z - t\right)}{a} \leq -2.5680070502093675 \cdot 10^{+20}:\\ \;\;\;\;\left(z - t\right) \cdot \frac{y}{a}\\ \mathbf{elif}\;\frac{y \cdot \left(z - t\right)}{a} \leq 1.186588078276849 \cdot 10^{+100}:\\ \;\;\;\;x - \frac{y \cdot t}{a}\\ \mathbf{elif}\;\frac{y \cdot \left(z - t\right)}{a} \leq 2.6629160219080746 \cdot 10^{+142}:\\ \;\;\;\;\frac{y \cdot \left(z - t\right)}{a}\\ \mathbf{elif}\;\frac{y \cdot \left(z - t\right)}{a} \leq 5.0994858525912255 \cdot 10^{+157}:\\ \;\;\;\;x - \frac{t}{\frac{a}{y}}\\ \mathbf{elif}\;\frac{y \cdot \left(z - t\right)}{a} \leq 1.1531213554742953 \cdot 10^{+209}:\\ \;\;\;\;x + \frac{y \cdot z}{a}\\ \mathbf{elif}\;\frac{y \cdot \left(z - t\right)}{a} \leq 6.950418515138569 \cdot 10^{+301}:\\ \;\;\;\;\frac{y \cdot \left(z - t\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{z}{\frac{a}{y}}\\ \end{array}\]

Alternative 6
Error	12.2
Cost	6793

\[\begin{array}{l} \mathbf{if}\;\frac{y \cdot \left(z - t\right)}{a} \leq -\infty:\\ \;\;\;\;\frac{y}{\frac{a}{z - t}}\\ \mathbf{elif}\;\frac{y \cdot \left(z - t\right)}{a} \leq -1.5869757133031571 \cdot 10^{+137}:\\ \;\;\;\;\frac{y \cdot \left(z - t\right)}{a}\\ \mathbf{elif}\;\frac{y \cdot \left(z - t\right)}{a} \leq -6.978901212274857 \cdot 10^{+68}:\\ \;\;\;\;x - \frac{y \cdot t}{a}\\ \mathbf{elif}\;\frac{y \cdot \left(z - t\right)}{a} \leq -1.0456727484995792 \cdot 10^{+28}:\\ \;\;\;\;\frac{y \cdot \left(z - t\right)}{a}\\ \mathbf{elif}\;\frac{y \cdot \left(z - t\right)}{a} \leq 1.186588078276849 \cdot 10^{+100}:\\ \;\;\;\;x - \frac{y \cdot t}{a}\\ \mathbf{elif}\;\frac{y \cdot \left(z - t\right)}{a} \leq 2.6629160219080746 \cdot 10^{+142}:\\ \;\;\;\;\frac{y \cdot \left(z - t\right)}{a}\\ \mathbf{elif}\;\frac{y \cdot \left(z - t\right)}{a} \leq 5.0994858525912255 \cdot 10^{+157}:\\ \;\;\;\;x - \frac{t}{\frac{a}{y}}\\ \mathbf{elif}\;\frac{y \cdot \left(z - t\right)}{a} \leq 1.1531213554742953 \cdot 10^{+209}:\\ \;\;\;\;x + \frac{y \cdot z}{a}\\ \mathbf{elif}\;\frac{y \cdot \left(z - t\right)}{a} \leq 6.950418515138569 \cdot 10^{+301}:\\ \;\;\;\;\frac{y \cdot \left(z - t\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{z}{\frac{a}{y}}\\ \end{array}\]

Alternative 7
Error	12.3
Cost	6479

\[\begin{array}{l} \mathbf{if}\;\frac{y \cdot \left(z - t\right)}{a} \leq -\infty:\\ \;\;\;\;\frac{y}{\frac{a}{z - t}}\\ \mathbf{elif}\;\frac{y \cdot \left(z - t\right)}{a} \leq -1.5869757133031571 \cdot 10^{+137}:\\ \;\;\;\;\frac{y \cdot \left(z - t\right)}{a}\\ \mathbf{elif}\;\frac{y \cdot \left(z - t\right)}{a} \leq -6.978901212274857 \cdot 10^{+68}:\\ \;\;\;\;x - \frac{y \cdot t}{a}\\ \mathbf{elif}\;\frac{y \cdot \left(z - t\right)}{a} \leq -1.0456727484995792 \cdot 10^{+28}:\\ \;\;\;\;\frac{y \cdot \left(z - t\right)}{a}\\ \mathbf{elif}\;\frac{y \cdot \left(z - t\right)}{a} \leq 1.186588078276849 \cdot 10^{+100}:\\ \;\;\;\;x - \frac{y \cdot t}{a}\\ \mathbf{elif}\;\frac{y \cdot \left(z - t\right)}{a} \leq 2.6629160219080746 \cdot 10^{+142}:\\ \;\;\;\;\frac{y \cdot \left(z - t\right)}{a}\\ \mathbf{elif}\;\frac{y \cdot \left(z - t\right)}{a} \leq 5.0994858525912255 \cdot 10^{+157}:\\ \;\;\;\;x - \frac{t}{\frac{a}{y}}\\ \mathbf{elif}\;\frac{y \cdot \left(z - t\right)}{a} \leq 1.2602329876702486 \cdot 10^{+211} \lor \neg \left(\frac{y \cdot \left(z - t\right)}{a} \leq 6.950418515138569 \cdot 10^{+301}\right):\\ \;\;\;\;x + \frac{z}{\frac{a}{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot \left(z - t\right)}{a}\\ \end{array}\]

Alternative 8
Error	11.0
Cost	4678

\[\begin{array}{l} \mathbf{if}\;\frac{y \cdot \left(z - t\right)}{a} \leq -\infty:\\ \;\;\;\;\frac{y}{\frac{a}{z - t}}\\ \mathbf{elif}\;\frac{y \cdot \left(z - t\right)}{a} \leq -1.5869757133031571 \cdot 10^{+137}:\\ \;\;\;\;\frac{y \cdot \left(z - t\right)}{a}\\ \mathbf{elif}\;\frac{y \cdot \left(z - t\right)}{a} \leq -6.978901212274857 \cdot 10^{+68}:\\ \;\;\;\;x - \frac{y \cdot t}{a}\\ \mathbf{elif}\;\frac{y \cdot \left(z - t\right)}{a} \leq -1.0456727484995792 \cdot 10^{+28}:\\ \;\;\;\;\frac{y \cdot \left(z - t\right)}{a}\\ \mathbf{elif}\;\frac{y \cdot \left(z - t\right)}{a} \leq 1.186588078276849 \cdot 10^{+100}:\\ \;\;\;\;x - \frac{y \cdot t}{a}\\ \mathbf{elif}\;\frac{y \cdot \left(z - t\right)}{a} \leq 6.950418515138569 \cdot 10^{+301}:\\ \;\;\;\;\frac{y \cdot \left(z - t\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\frac{a}{z - t}}\\ \end{array}\]

Alternative 9
Error	12.5
Cost	4678

\[\begin{array}{l} \mathbf{if}\;\frac{y \cdot \left(z - t\right)}{a} \leq -\infty:\\ \;\;\;\;x - \frac{t}{\frac{a}{y}}\\ \mathbf{elif}\;\frac{y \cdot \left(z - t\right)}{a} \leq -1.5869757133031571 \cdot 10^{+137}:\\ \;\;\;\;\frac{y \cdot \left(z - t\right)}{a}\\ \mathbf{elif}\;\frac{y \cdot \left(z - t\right)}{a} \leq -6.978901212274857 \cdot 10^{+68}:\\ \;\;\;\;x - \frac{y \cdot t}{a}\\ \mathbf{elif}\;\frac{y \cdot \left(z - t\right)}{a} \leq -1.0456727484995792 \cdot 10^{+28}:\\ \;\;\;\;\frac{y \cdot \left(z - t\right)}{a}\\ \mathbf{elif}\;\frac{y \cdot \left(z - t\right)}{a} \leq 1.186588078276849 \cdot 10^{+100}:\\ \;\;\;\;x - \frac{y \cdot t}{a}\\ \mathbf{elif}\;\frac{y \cdot \left(z - t\right)}{a} \leq 3.6794851738591936 \cdot 10^{+291}:\\ \;\;\;\;\frac{y \cdot \left(z - t\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{t}{\frac{a}{y}}\\ \end{array}\]

Alternative 10
Error	16.4
Cost	769

\[\begin{array}{l} \mathbf{if}\;z \leq 2.5645447844495938 \cdot 10^{+184}:\\ \;\;\;\;x - \frac{t}{\frac{a}{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{\frac{a}{y}}\\ \end{array}\]

Alternative 11
Error	28.1
Cost	1026

\[\begin{array}{l} \mathbf{if}\;x \leq -2.4501930820528447 \cdot 10^{-116}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 7.363952084645234 \cdot 10^{-40}:\\ \;\;\;\;\frac{-t}{\frac{a}{y}}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array}\]

Alternative 12
Error	28.4
Cost	962

\[\begin{array}{l} \mathbf{if}\;x \leq -1.1186627941643973 \cdot 10^{-179}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 3.45452832059587 \cdot 10^{-94}:\\ \;\;\;\;\frac{y \cdot z}{a}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array}\]

Alternative 13
Error	31.3
Cost	64

\[x\]

Alternative 14
Error	61.9
Cost	64

\[-1\]

Alternative 15
Error	61.8
Cost	64

\[1\]

Derivation

1. Split input into 3 regimes

2. if (*.f64 y (-.f64 z t)) < -inf.0

   1. Initial program 64.0

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

   3. Applied clear-num_binary64_10989 64.0

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

   5. Applied associate-/r*_binary64_10934 0.2

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

   7. Applied div-inv_binary64_10987 0.3

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

   8. Applied add-sqr-sqrt_binary64_11012 0.3

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

   9. Applied times-frac_binary64_10996 0.4

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

   10. Simplified 0.3

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

   11. Simplified 0.2

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

   12. Simplified 0.2

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

   if -inf.0 < (*.f64 y (-.f64 z t)) < 3.92541377658044846e209

   1. Initial program 0.3

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

   2. Simplified 0.3

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

   if 3.92541377658044846e209 < (*.f64 y (-.f64 z t))

   1. Initial program 30.0

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

   3. Applied associate-/l*_binary64_10935 1.0

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

   4. Simplified 1.0

      \[\leadsto \color{blue}{x + \frac{y}{\frac{a}{z - t}}}\]
3. Recombined 3 regimes into one program.

4. Final simplification 0.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(z - t\right) \leq -\infty:\\ \;\;\;\;x + \left(z - t\right) \cdot \frac{y}{a}\\ \mathbf{elif}\;y \cdot \left(z - t\right) \leq 3.9254137765804485 \cdot 10^{+209}:\\ \;\;\;\;x + \frac{y \cdot \left(z - t\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{\frac{a}{z - t}}\\ \end{array}\]

Reproduce

herbie shell --seed 2021044 
(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)))