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

Average Error: 6.3 → 1.6

Time: 5.3s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{y \cdot \left(z - t\right)}{a} \leq -2.45259261574801 \cdot 10^{+236}:\\ \;\;\;\;x + \left(z - t\right) \cdot \frac{y}{a}\\ \mathbf{elif}\;\frac{y \cdot \left(z - t\right)}{a} \leq -5.459359479482239 \cdot 10^{-269}:\\ \;\;\;\;\frac{y \cdot \left(z - t\right)}{a} + x\\ \mathbf{else}:\\ \;\;\;\;x + \frac{\frac{\sqrt[3]{y} \cdot \sqrt[3]{y}}{\sqrt[3]{a}} \cdot \left(\left(z - t\right) \cdot \frac{\sqrt[3]{y}}{\sqrt[3]{a}}\right)}{\sqrt[3]{a}}\\ \end{array}\]

FPCore

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

↓

(FPCore (x y z t a)
 :precision binary64
 (if (<= (/ (* y (- z t)) a) -2.45259261574801e+236)
   (+ x (* (- z t) (/ y a)))
   (if (<= (/ (* y (- z t)) a) -5.459359479482239e-269)
     (+ (/ (* y (- z t)) a) x)
     (+
      x
      (/
       (* (/ (* (cbrt y) (cbrt y)) (cbrt a)) (* (- z t) (/ (cbrt y) (cbrt a))))
       (cbrt 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 (((y * (z - t)) / a) <= -2.45259261574801e+236) {
		tmp = x + ((z - t) * (y / a));
	} else if (((y * (z - t)) / a) <= -5.459359479482239e-269) {
		tmp = ((y * (z - t)) / a) + x;
	} else {
		tmp = x + ((((cbrt(y) * cbrt(y)) / cbrt(a)) * ((z - t) * (cbrt(y) / cbrt(a)))) / cbrt(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	6.3
Target	0.9
Herbie	1.6

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

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 y (-.f64 z t)) a) < -2.45259261574801e236

   1. Initial program 33.4

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

   3. Applied add-cube-cbrt_binary64_11366 34.0

      \[\leadsto x + \frac{y \cdot \left(z - t\right)}{\color{blue}{\left(\sqrt[3]{a} \cdot \sqrt[3]{a}\right) \cdot \sqrt[3]{a}}}\]

   4. Applied associate-/r*_binary64_11275 34.0

      \[\leadsto x + \color{blue}{\frac{\frac{y \cdot \left(z - t\right)}{\sqrt[3]{a} \cdot \sqrt[3]{a}}}{\sqrt[3]{a}}}\]

   5. Simplified 10.8

      \[\leadsto x + \frac{\color{blue}{\frac{y}{\sqrt[3]{a} \cdot \sqrt[3]{a}} \cdot \left(z - t\right)}}{\sqrt[3]{a}}\]

   6. Taylor expanded around -inf 33.4

      \[\leadsto x + \color{blue}{\left(\frac{t \cdot y}{{\left(\sqrt[3]{-1}\right)}^{3} \cdot a} - \frac{z \cdot y}{a \cdot {\left(\sqrt[3]{-1}\right)}^{3}}\right)}\]

   7. Simplified 4.8

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

   8. Taylor expanded around 0 33.4

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

   9. Simplified 4.8

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

   if -2.45259261574801e236 < (/.f64 (*.f64 y (-.f64 z t)) a) < -5.459359479482239e-269

   1. Initial program 0.2

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

   if -5.459359479482239e-269 < (/.f64 (*.f64 y (-.f64 z t)) a)

   1. Initial program 5.9

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

   3. Applied add-cube-cbrt_binary64_11366 6.3

      \[\leadsto x + \frac{y \cdot \left(z - t\right)}{\color{blue}{\left(\sqrt[3]{a} \cdot \sqrt[3]{a}\right) \cdot \sqrt[3]{a}}}\]

   4. Applied associate-/r*_binary64_11275 6.3

      \[\leadsto x + \color{blue}{\frac{\frac{y \cdot \left(z - t\right)}{\sqrt[3]{a} \cdot \sqrt[3]{a}}}{\sqrt[3]{a}}}\]

   5. Simplified 3.0

      \[\leadsto x + \frac{\color{blue}{\frac{y}{\sqrt[3]{a} \cdot \sqrt[3]{a}} \cdot \left(z - t\right)}}{\sqrt[3]{a}}\]
   6. Using strategy rm

   7. Applied add-cube-cbrt_binary64_11366 3.1

      \[\leadsto x + \frac{\frac{\color{blue}{\left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) \cdot \sqrt[3]{y}}}{\sqrt[3]{a} \cdot \sqrt[3]{a}} \cdot \left(z - t\right)}{\sqrt[3]{a}}\]

   8. Applied times-frac_binary64_11337 3.1

      \[\leadsto x + \frac{\color{blue}{\left(\frac{\sqrt[3]{y} \cdot \sqrt[3]{y}}{\sqrt[3]{a}} \cdot \frac{\sqrt[3]{y}}{\sqrt[3]{a}}\right)} \cdot \left(z - t\right)}{\sqrt[3]{a}}\]

   9. Applied associate-*l*_binary64_11272 1.9

      \[\leadsto x + \frac{\color{blue}{\frac{\sqrt[3]{y} \cdot \sqrt[3]{y}}{\sqrt[3]{a}} \cdot \left(\frac{\sqrt[3]{y}}{\sqrt[3]{a}} \cdot \left(z - t\right)\right)}}{\sqrt[3]{a}}\]

   10. Simplified 1.9

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

4. Final simplification 1.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y \cdot \left(z - t\right)}{a} \leq -2.45259261574801 \cdot 10^{+236}:\\ \;\;\;\;x + \left(z - t\right) \cdot \frac{y}{a}\\ \mathbf{elif}\;\frac{y \cdot \left(z - t\right)}{a} \leq -5.459359479482239 \cdot 10^{-269}:\\ \;\;\;\;\frac{y \cdot \left(z - t\right)}{a} + x\\ \mathbf{else}:\\ \;\;\;\;x + \frac{\frac{\sqrt[3]{y} \cdot \sqrt[3]{y}}{\sqrt[3]{a}} \cdot \left(\left(z - t\right) \cdot \frac{\sqrt[3]{y}}{\sqrt[3]{a}}\right)}{\sqrt[3]{a}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020298 
(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)))