Average Error: 5.9 → 1.5

Time: 31.6s

Precision: 64

Internal precision: 128

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

⬇

\[\begin{array}{l} \mathbf{if}\;y \le -2.930244047323321 \cdot 10^{+18}:\\ \;\;\;\;x - y \cdot \frac{z - t}{a}\\ \mathbf{if}\;y \le 8.166728288538418 \cdot 10^{-294}:\\ \;\;\;\;x - \frac{1}{\frac{a}{y \cdot \left(z - t\right)}}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y}{a} \cdot \left(z - t\right)\\ \end{array}\]

Error

The X axis uses an exponential scale — Bits error versus `x`

Target

Original	5.9
Comparison	0.7
Herbie	1.5

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

Derivation

Split input into 3 regimes.
if y < -2.930244047323321e+18
1. Initial program 16.2
  \[x - \frac{y \cdot \left(z - t\right)}{a}\]
2. Applied taylor 16.3
  \[\leadsto x - \left(\frac{y \cdot z}{a} - \frac{t \cdot y}{a}\right)\]
3. Taylor expanded around 0 16.3
  
  \[\leadsto x - \color{blue}{\left(\frac{y \cdot z}{a} - \frac{t \cdot y}{a}\right)}\]
4. Applied simplify 3.6
  \[\leadsto \color{blue}{x - \frac{y}{a} \cdot \left(z - t\right)}\]
5. Using strategy rm
6. Applied div-inv 3.7
  \[\leadsto x - \color{blue}{\left(y \cdot \frac{1}{a}\right)} \cdot \left(z - t\right)\]
7. Applied associate-*l* 0.8
  \[\leadsto x - \color{blue}{y \cdot \left(\frac{1}{a} \cdot \left(z - t\right)\right)}\]
8. Applied simplify 0.7
  \[\leadsto x - y \cdot \color{blue}{\frac{z - t}{a}}\]
if -2.930244047323321e+18 < y < 8.166728288538418e-294
1. Initial program 0.5
  \[x - \frac{y \cdot \left(z - t\right)}{a}\]
2. Using strategy rm
3. Applied clear-num 0.5
  \[\leadsto x - \color{blue}{\frac{1}{\frac{a}{y \cdot \left(z - t\right)}}}\]
if 8.166728288538418e-294 < y
1. Initial program 5.9
  \[x - \frac{y \cdot \left(z - t\right)}{a}\]
2. Applied taylor 5.9
  \[\leadsto x - \left(\frac{y \cdot z}{a} - \frac{t \cdot y}{a}\right)\]
3. Taylor expanded around 0 5.9
  
  \[\leadsto x - \color{blue}{\left(\frac{y \cdot z}{a} - \frac{t \cdot y}{a}\right)}\]
4. Applied simplify 2.5
  \[\leadsto \color{blue}{x - \frac{y}{a} \cdot \left(z - t\right)}\]
Recombined 3 regimes into one program.
Removed slow pow expressions

Runtime

Please include this information when filing a bug report:

herbie --seed '#(3663857942 159484730 3367875525 4143958929 2381550403 3748760641)'
(FPCore (x y z t a)
  :name "Optimisation.CirclePacking:place from circle-packing-0.1.0.4, F"

  :target
  (if (< y -1.0761266216389975e-10) (- x (/ 1 (/ (/ a (- z t)) y))) (if (< y 2.894426862792089e-49) (- x (/ (* y (- z t)) a)) (- x (/ y (/ a (- z t))))))

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

Error

Target

Derivation

`if y < -2.930244047323321e+18`

`if -2.930244047323321e+18 < y < 8.166728288538418e-294`

`if 8.166728288538418e-294 < y`

Runtime