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

Average Error: 6.2 → 1.3

Time: 5.3s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x + \frac{y \cdot \left(z - t\right)}{a} \leq -3.133221453962233 \cdot 10^{+111}:\\ \;\;\;\;x + \left(z - t\right) \cdot \frac{y}{a}\\ \mathbf{elif}\;x + \frac{y \cdot \left(z - t\right)}{a} \leq 8.572792260787035 \cdot 10^{+296}:\\ \;\;\;\;x + \left(y \cdot \left(z - t\right)\right) \cdot \frac{1}{a}\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \frac{z - t}{a}\\ \end{array}\]

FPCore

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

↓

(FPCore (x y z t a)
 :precision binary64
 (if (<= (+ x (/ (* y (- z t)) a)) -3.133221453962233e+111)
   (+ x (* (- z t) (/ y a)))
   (if (<= (+ x (/ (* y (- z t)) a)) 8.572792260787035e+296)
     (+ x (* (* y (- z t)) (/ 1.0 a)))
     (+ x (* y (/ (- z t) a))))))

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 ((x + ((y * (z - t)) / a)) <= -3.133221453962233e+111) {
		tmp = x + ((z - t) * (y / a));
	} else if ((x + ((y * (z - t)) / a)) <= 8.572792260787035e+296) {
		tmp = x + ((y * (z - t)) * (1.0 / a));
	} else {
		tmp = x + (y * ((z - t) / a));
	}
	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	6.2
Target	0.7
Herbie	1.3

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

Derivation

1. Split input into 3 regimes

2. if (+.f64 x (/.f64 (*.f64 y (-.f64 z t)) a)) < -3.13322145396223321e111

   1. Initial program 10.3

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

   3. Applied associate-/l*_binary64_10253 8.6

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

   5. Applied associate-/r/_binary64_10254 2.5

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

   if -3.13322145396223321e111 < (+.f64 x (/.f64 (*.f64 y (-.f64 z t)) a)) < 8.5727922607870355e296

   1. Initial program 0.5

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

   3. Applied div-inv_binary64_10305 0.6

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

   if 8.5727922607870355e296 < (+.f64 x (/.f64 (*.f64 y (-.f64 z t)) a))

   1. Initial program 50.0

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

   3. Applied *-un-lft-identity_binary64_10308 50.0

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

   4. Applied times-frac_binary64_10314 3.7

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

   5. Simplified 3.7

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

4. Final simplification 1.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;x + \frac{y \cdot \left(z - t\right)}{a} \leq -3.133221453962233 \cdot 10^{+111}:\\ \;\;\;\;x + \left(z - t\right) \cdot \frac{y}{a}\\ \mathbf{elif}\;x + \frac{y \cdot \left(z - t\right)}{a} \leq 8.572792260787035 \cdot 10^{+296}:\\ \;\;\;\;x + \left(y \cdot \left(z - t\right)\right) \cdot \frac{1}{a}\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \frac{z - t}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2021014 
(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)))