Average Error: 5.9 → 1.1
Time: 1.1m
Precision: 64
Internal Precision: 384
\[x + \frac{y \cdot \left(z - t\right)}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;a \le -5.273243369695188 \cdot 10^{-34}:\\ \;\;\;\;x + \frac{y}{\frac{a}{z - t}}\\ \mathbf{if}\;a \le 1.4912760646799497 \cdot 10^{-24}:\\ \;\;\;\;x + \frac{y \cdot \left(z - t\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{z}{\frac{a}{y}} - \frac{t}{\frac{a}{y}}\right) + x\\ \end{array}\]

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original5.9
Target0.7
Herbie1.1
\[\begin{array}{l} \mathbf{if}\;y \lt -1.0761266216389975 \cdot 10^{-10}:\\ \;\;\;\;x + \frac{1}{\frac{\frac{a}{z - t}}{y}}\\ \mathbf{if}\;y \lt 2.894426862792089 \cdot 10^{-49}:\\ \;\;\;\;x + \frac{y \cdot \left(z - t\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{\frac{a}{z - t}}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if a < -5.273243369695188e-34

    1. Initial program 8.6

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

    3. Applied associate-/l*0.6

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

    if -5.273243369695188e-34 < a < 1.4912760646799497e-24

    1. Initial program 0.8

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

    if 1.4912760646799497e-24 < a

    1. Initial program 8.0

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

    2. Taylor expanded around 0 8.1

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

    3. Applied simplify1.8

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

    5. Applied div-sub1.8

      \[\leadsto \color{blue}{\left(\frac{z}{\frac{a}{y}} - \frac{t}{\frac{a}{y}}\right)} + x\]
  3. Recombined 3 regimes into one program.
  4. Removed slow pow expressions.

Runtime

Time bar (total: 1.1m)Debug log

herbie shell --seed '#(1567391828 2030694642 2833800258 828025724 3004380912 3532991858)' +o setup:early-exit +o reduce:binary-search
(FPCore (x y z t a)
  :name "Optimisation.CirclePacking:place from circle-packing-0.1.0.4, E"

  :herbie-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)))