Average Error: 19.6 → 5.1
Time: 7.2s
Precision: 64
Internal Precision: 128
\[\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le -1.3337852543374606 \cdot 10^{+154}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \le -5.003071307679352 \cdot 10^{-158}:\\ \;\;\;\;\frac{\left(x - y\right) \cdot \left(y + x\right)}{(x \cdot x + \left(y \cdot y\right))_*}\\ \mathbf{elif}\;y \le 3.682302512490061 \cdot 10^{-197}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \le 2.1250373042840873 \cdot 10^{-173}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(x - y\right) \cdot \left(y + x\right)}{(x \cdot x + \left(y \cdot y\right))_*}\\ \end{array}\]

Error

Bits error versus x

Bits error versus y

Target

Original19.6
Target0.0
Herbie5.1
\[\begin{array}{l} \mathbf{if}\;0.5 \lt \left|\frac{x}{y}\right| \lt 2:\\ \;\;\;\;\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{2}{1 + \frac{x}{y} \cdot \frac{x}{y}}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if y < -1.3337852543374606e+154 or 3.682302512490061e-197 < y < 2.1250373042840873e-173

    1. Initial program 59.6

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

    2. Simplified59.6

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

    3. Taylor expanded around 0 4.4

      \[\leadsto \color{blue}{-1}\]

    if -1.3337852543374606e+154 < y < -5.003071307679352e-158 or 2.1250373042840873e-173 < y

    1. Initial program 0.6

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

    2. Simplified0.6

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

    if -5.003071307679352e-158 < y < 3.682302512490061e-197

    1. Initial program 29.5

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

    2. Simplified29.5

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

    3. Taylor expanded around -inf 14.5

      \[\leadsto \color{blue}{1}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification5.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -1.3337852543374606 \cdot 10^{+154}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \le -5.003071307679352 \cdot 10^{-158}:\\ \;\;\;\;\frac{\left(x - y\right) \cdot \left(y + x\right)}{(x \cdot x + \left(y \cdot y\right))_*}\\ \mathbf{elif}\;y \le 3.682302512490061 \cdot 10^{-197}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \le 2.1250373042840873 \cdot 10^{-173}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(x - y\right) \cdot \left(y + x\right)}{(x \cdot x + \left(y \cdot y\right))_*}\\ \end{array}\]

Reproduce

herbie shell --seed 2019090 +o rules:numerics
(FPCore (x y)
  :name "Kahan p9 Example"
  :pre (and (< 0 x 1) (< y 1))

  :herbie-target
  (if (< 0.5 (fabs (/ x y)) 2) (/ (* (- x y) (+ x y)) (+ (* x x) (* y y))) (- 1 (/ 2 (+ 1 (* (/ x y) (/ x y))))))

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