Average Error: 21.9 → 0.1
Time: 1.7m
Precision: 64
Internal Precision: 1408
\[1.0 - \frac{\left(1.0 - x\right) \cdot y}{y + 1.0}\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le -105446982.68410039:\\ \;\;\;\;x - \frac{1.0 - x}{\frac{y}{\frac{1.0}{y} - 1.0}}\\ \mathbf{if}\;y \le 232001311.40530783:\\ \;\;\;\;1.0 - \frac{y}{y + 1.0} \cdot \left(1.0 - x\right)\\ \mathbf{else}:\\ \;\;\;\;x - \frac{1.0 - x}{\frac{y}{\frac{1.0}{y} - 1.0}}\\ \end{array}\]

Error

Bits error versus x

Bits error versus y

Target

Original21.9
Target0.2
Herbie0.1
\[\begin{array}{l} \mathbf{if}\;y \lt -3693.8482788297247:\\ \;\;\;\;\frac{1.0}{y} - \left(\frac{x}{y} - x\right)\\ \mathbf{if}\;y \lt 6799310503.41891:\\ \;\;\;\;1.0 - \frac{\left(1.0 - x\right) \cdot y}{y + 1.0}\\ \mathbf{else}:\\ \;\;\;\;\frac{1.0}{y} - \left(\frac{x}{y} - x\right)\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if y < -105446982.68410039 or 232001311.40530783 < y

    1. Initial program 44.7

      \[1.0 - \frac{\left(1.0 - x\right) \cdot y}{y + 1.0}\]

    2. Taylor expanded around 0 44.7

      \[\leadsto 1.0 - \frac{\color{blue}{1.0 \cdot y - y \cdot x}}{y + 1.0}\]

    3. Applied simplify29.4

      \[\leadsto \color{blue}{1.0 - \frac{y}{y + 1.0} \cdot \left(1.0 - x\right)}\]

    4. Taylor expanded around inf 29.4

      \[\leadsto 1.0 - \color{blue}{\left(\left(1 + 1.0 \cdot \frac{1}{{y}^{2}}\right) - 1.0 \cdot \frac{1}{y}\right)} \cdot \left(1.0 - x\right)\]

    5. Applied simplify15.3

      \[\leadsto \color{blue}{\left(1.0 - \left(1.0 - x\right)\right) - \frac{1.0 - x}{\frac{y}{\frac{1.0}{y} - 1.0}}}\]

    6. Applied simplify0.0

      \[\leadsto \color{blue}{x} - \frac{1.0 - x}{\frac{y}{\frac{1.0}{y} - 1.0}}\]

    if -105446982.68410039 < y < 232001311.40530783

    1. Initial program 0.1

      \[1.0 - \frac{\left(1.0 - x\right) \cdot y}{y + 1.0}\]

    2. Taylor expanded around 0 0.1

      \[\leadsto 1.0 - \frac{\color{blue}{1.0 \cdot y - y \cdot x}}{y + 1.0}\]

    3. Applied simplify0.1

      \[\leadsto \color{blue}{1.0 - \frac{y}{y + 1.0} \cdot \left(1.0 - x\right)}\]
  3. Recombined 2 regimes into one program.
  4. Removed slow pow expressions.

Runtime

Time bar (total: 1.7m)Debug log

herbie shell --seed '#(1567391828 2030694642 2833800258 828025724 3004380912 3532991858)' +o setup:early-exit +o reduce:binary-search
(FPCore (x y)
  :name "Diagrams.Trail:splitAtParam  from diagrams-lib-1.3.0.3, D"

  :herbie-target
  (if (< y -3693.8482788297247) (- (/ 1.0 y) (- (/ x y) x)) (if (< y 6799310503.41891) (- 1.0 (/ (* (- 1.0 x) y) (+ y 1.0))) (- (/ 1.0 y) (- (/ x y) x))))

  (- 1.0 (/ (* (- 1.0 x) y) (+ y 1.0))))