Average Error: 24.9 → 0.9
Time: 19.3s
Precision: 64
Internal Precision: 576
\[\frac{1 - \left(x - y\right) \cdot \left(x - y\right)}{\left(4 \cdot x\right) \cdot y}\]
\[\begin{array}{l} \mathbf{if}\;y \le -86.726910540641 \lor \neg \left(y \le 1.1813564328514398 \cdot 10^{-06}\right):\\ \;\;\;\;\frac{1}{2} - \frac{1}{4} \cdot \left(\frac{y}{x} + \frac{x}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2} - \frac{\frac{1}{4}}{y} \cdot \left(x - \frac{1}{x}\right)\\ \end{array}\]

Error

Bits error versus x

Bits error versus y

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 2 regimes
  2. if y < -86.726910540641 or 1.1813564328514398e-06 < y

    1. Initial program 35.9

      \[\frac{1 - \left(x - y\right) \cdot \left(x - y\right)}{\left(4 \cdot x\right) \cdot y}\]
    2. Initial simplification35.9

      \[\leadsto \frac{1 - \left(x - y\right) \cdot \left(x - y\right)}{x \cdot \left(y \cdot 4\right)}\]
    3. Taylor expanded around -inf 0.9

      \[\leadsto \color{blue}{\frac{1}{2} - \left(\frac{1}{4} \cdot \frac{y}{x} + \frac{1}{4} \cdot \frac{x}{y}\right)}\]
    4. Simplified0.9

      \[\leadsto \color{blue}{\frac{1}{2} - \left(\frac{y}{x} + \frac{x}{y}\right) \cdot \frac{1}{4}}\]

    if -86.726910540641 < y < 1.1813564328514398e-06

    1. Initial program 8.0

      \[\frac{1 - \left(x - y\right) \cdot \left(x - y\right)}{\left(4 \cdot x\right) \cdot y}\]
    2. Initial simplification8.0

      \[\leadsto \frac{1 - \left(x - y\right) \cdot \left(x - y\right)}{x \cdot \left(y \cdot 4\right)}\]
    3. Taylor expanded around 0 0.9

      \[\leadsto \color{blue}{\left(\frac{1}{4} \cdot \frac{1}{x \cdot y} + \frac{1}{2}\right) - \frac{1}{4} \cdot \frac{x}{y}}\]
    4. Simplified0.9

      \[\leadsto \color{blue}{\frac{1}{2} - \left(x - \frac{1}{x}\right) \cdot \frac{\frac{1}{4}}{y}}\]
  3. Recombined 2 regimes into one program.
  4. Final simplification0.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -86.726910540641 \lor \neg \left(y \le 1.1813564328514398 \cdot 10^{-06}\right):\\ \;\;\;\;\frac{1}{2} - \frac{1}{4} \cdot \left(\frac{y}{x} + \frac{x}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2} - \frac{\frac{1}{4}}{y} \cdot \left(x - \frac{1}{x}\right)\\ \end{array}\]

Runtime

Time bar (total: 19.3s)Debug log

herbie shell --seed '#(2775764126 3555076145 3898259844 1891440260 2599947619 1948460636)' 
(FPCore (x y)
  :name "(1-(x-y)*(x-y))/(4*x*y)"
  (/ (- 1 (* (- x y) (- x y))) (* (* 4 x) y)))