Average Error: 20.3 → 5.8
Time: 29.2s
Precision: 64
Internal Precision: 128
\[\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le -1.2032223167563086 \cdot 10^{-42}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \le -2.0118099058943753 \cdot 10^{-159}:\\ \;\;\;\;\frac{x - y}{\frac{(x \cdot x + \left(y \cdot y\right))_*}{y + x}}\\ \mathbf{elif}\;y \le -4.1330897435654156 \cdot 10^{-178}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \le 1.5107001695998868 \cdot 10^{-176}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{x - y}{\frac{(x \cdot x + \left(y \cdot y\right))_*}{y + x}}\\ \end{array}\]

Error

Bits error versus x

Bits error versus y

Target

Original20.3
Target0.1
Herbie5.8
\[\begin{array}{l} \mathbf{if}\;0.5 \lt \left|\frac{x}{y}\right| \lt 2:\\ \;\;\;\;\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{2}{1 + \frac{x}{y} \cdot \frac{x}{y}}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if y < -1.2032223167563086e-42 or -2.0118099058943753e-159 < y < -4.1330897435654156e-178

    1. Initial program 27.7

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

    2. Simplified27.7

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

    3. Taylor expanded around 0 2.3

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

    if -1.2032223167563086e-42 < y < -2.0118099058943753e-159 or 1.5107001695998868e-176 < y

    1. Initial program 1.7

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

    2. Simplified1.7

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

    4. Applied associate-/l*2.1

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

    if -4.1330897435654156e-178 < y < 1.5107001695998868e-176

    1. Initial program 30.1

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

    2. Simplified30.1

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

    4. Applied associate-/l*30.5

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

    5. Taylor expanded around inf 14.8

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

  4. Final simplification5.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -1.2032223167563086 \cdot 10^{-42}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \le -2.0118099058943753 \cdot 10^{-159}:\\ \;\;\;\;\frac{x - y}{\frac{(x \cdot x + \left(y \cdot y\right))_*}{y + x}}\\ \mathbf{elif}\;y \le -4.1330897435654156 \cdot 10^{-178}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \le 1.5107001695998868 \cdot 10^{-176}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{x - y}{\frac{(x \cdot x + \left(y \cdot y\right))_*}{y + x}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019089 +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))))