Average Error: 20.6 → 5.2
Time: 9.7s
Precision: 64
\[0.0 \lt x \lt 1 \land y \lt 1\]
\[\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le -1.8197648299093187 \cdot 10^{151}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \le -5.47427907991377714 \cdot 10^{-158} \lor \neg \left(y \le 6.0631474065678139 \cdot 10^{-172}\right):\\ \;\;\;\;\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array}\]
\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}
↓
\begin{array}{l}
\mathbf{if}\;y \le -1.8197648299093187 \cdot 10^{151}:\\
\;\;\;\;-1\\

\mathbf{elif}\;y \le -5.47427907991377714 \cdot 10^{-158} \lor \neg \left(y \le 6.0631474065678139 \cdot 10^{-172}\right):\\
\;\;\;\;\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}\\

\mathbf{else}:\\
\;\;\;\;1\\

\end{array}
double f(double x, double y) {
        double r82190 = x;
        double r82191 = y;
        double r82192 = r82190 - r82191;
        double r82193 = r82190 + r82191;
        double r82194 = r82192 * r82193;
        double r82195 = r82190 * r82190;
        double r82196 = r82191 * r82191;
        double r82197 = r82195 + r82196;
        double r82198 = r82194 / r82197;
        return r82198;
}
↓
double f(double x, double y) {
        double r82199 = y;
        double r82200 = -1.8197648299093187e+151;
        bool r82201 = r82199 <= r82200;
        double r82202 = -1.0;
        double r82203 = -5.474279079913777e-158;
        bool r82204 = r82199 <= r82203;
        double r82205 = 6.063147406567814e-172;
        bool r82206 = r82199 <= r82205;
        double r82207 = !r82206;
        bool r82208 = r82204 || r82207;
        double r82209 = x;
        double r82210 = r82209 - r82199;
        double r82211 = r82209 + r82199;
        double r82212 = r82210 * r82211;
        double r82213 = r82209 * r82209;
        double r82214 = r82199 * r82199;
        double r82215 = r82213 + r82214;
        double r82216 = r82212 / r82215;
        double r82217 = 1.0;
        double r82218 = r82208 ? r82216 : r82217;
        double r82219 = r82201 ? r82202 : r82218;
        return r82219;
}

Error

Bits error versus x

Bits error versus y

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original20.6
Target0.1
Herbie5.2
\[\begin{array}{l} \mathbf{if}\;0.5 \lt \left|\frac{x}{y}\right| \lt 2:\\ \;\;\;\;\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{2}{1 + \frac{x}{y} \cdot \frac{x}{y}}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if y < -1.8197648299093187e+151

    1. Initial program 62.9

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

    2. Taylor expanded around 0 0

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

    if -1.8197648299093187e+151 < y < -5.474279079913777e-158 or 6.063147406567814e-172 < y

    1. Initial program 0.6

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

    if -5.474279079913777e-158 < y < 6.063147406567814e-172

    1. Initial program 30.8

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

    2. Taylor expanded around inf 15.6

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

  4. Final simplification5.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -1.8197648299093187 \cdot 10^{151}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \le -5.47427907991377714 \cdot 10^{-158} \lor \neg \left(y \le 6.0631474065678139 \cdot 10^{-172}\right):\\ \;\;\;\;\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array}\]

Reproduce

herbie shell --seed 2019198 
(FPCore (x y)
  :name "Kahan p9 Example"
  :pre (and (< 0.0 x 1.0) (< y 1.0))

  :herbie-target
  (if (< 0.5 (fabs (/ x y)) 2.0) (/ (* (- x y) (+ x y)) (+ (* x x) (* y y))) (- 1.0 (/ 2.0 (+ 1.0 (* (/ x y) (/ x y))))))

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