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

Average Error: 6.4 → 0.4

Time: 8.0s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;y \cdot \left(z - t\right) \le -2.96367916998019457 \cdot 10^{273}:\\ \;\;\;\;x + y \cdot \frac{z - t}{a}\\ \mathbf{elif}\;y \cdot \left(z - t\right) \le 7.0733190662448602 \cdot 10^{232}:\\ \;\;\;\;x + \frac{y \cdot z + y \cdot \left(-t\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{\frac{a}{z - t}}\\ \end{array}\]

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	0.7
Herbie	0.4

\[\begin{array}{l} \mathbf{if}\;y \lt -1.07612662163899753 \cdot 10^{-10}:\\ \;\;\;\;x + \frac{1}{\frac{\frac{a}{z - t}}{y}}\\ \mathbf{elif}\;y \lt 2.8944268627920891 \cdot 10^{-49}:\\ \;\;\;\;x + \frac{y \cdot \left(z - t\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{\frac{a}{z - t}}\\ \end{array}\]

Derivation

1. Split input into 3 regimes

2. if (* y (- z t)) < -2.96367916998019457e273

   1. Initial program 48.9

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

   3. Applied *-un-lft-identity 48.9

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

   4. Applied times-frac 0.2

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

   5. Simplified 0.2

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

   if -2.96367916998019457e273 < (* y (- z t)) < 7.0733190662448602e232

   1. Initial program 0.4

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

   3. Applied sub-neg 0.4

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

   4. Applied distribute-lft-in 0.4

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

   if 7.0733190662448602e232 < (* y (- z t))

   1. Initial program 37.3

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

   3. Applied associate-/l* 0.4

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

4. Final simplification 0.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(z - t\right) \le -2.96367916998019457 \cdot 10^{273}:\\ \;\;\;\;x + y \cdot \frac{z - t}{a}\\ \mathbf{elif}\;y \cdot \left(z - t\right) \le 7.0733190662448602 \cdot 10^{232}:\\ \;\;\;\;x + \frac{y \cdot z + y \cdot \left(-t\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{\frac{a}{z - t}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020168 
(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)))