Average Error: 20.1 → 5.7
Time: 13.2s
Precision: 64
\[0.0 \lt x \lt 1 \land y \lt 1\]
\[\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le -6208632077.0414142608642578125:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \le -4.057021676631311060519344937448265135952 \cdot 10^{-160}:\\ \;\;\;\;\frac{x - y}{\frac{x \cdot x + y \cdot y}{x + y}}\\ \mathbf{elif}\;y \le 2.835103494848979070409532780741214855767 \cdot 10^{-200}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \le 3.401055982211937821476509685336375382237 \cdot 10^{-168}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;\frac{x - y}{\frac{x \cdot x + y \cdot y}{x + y}}\\ \end{array}\]
\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}
↓
\begin{array}{l}
\mathbf{if}\;y \le -6208632077.0414142608642578125:\\
\;\;\;\;-1\\

\mathbf{elif}\;y \le -4.057021676631311060519344937448265135952 \cdot 10^{-160}:\\
\;\;\;\;\frac{x - y}{\frac{x \cdot x + y \cdot y}{x + y}}\\

\mathbf{elif}\;y \le 2.835103494848979070409532780741214855767 \cdot 10^{-200}:\\
\;\;\;\;1\\

\mathbf{elif}\;y \le 3.401055982211937821476509685336375382237 \cdot 10^{-168}:\\
\;\;\;\;-1\\

\mathbf{else}:\\
\;\;\;\;\frac{x - y}{\frac{x \cdot x + y \cdot y}{x + y}}\\

\end{array}
double f(double x, double y) {
        double r52496 = x;
        double r52497 = y;
        double r52498 = r52496 - r52497;
        double r52499 = r52496 + r52497;
        double r52500 = r52498 * r52499;
        double r52501 = r52496 * r52496;
        double r52502 = r52497 * r52497;
        double r52503 = r52501 + r52502;
        double r52504 = r52500 / r52503;
        return r52504;
}
↓
double f(double x, double y) {
        double r52505 = y;
        double r52506 = -6208632077.041414;
        bool r52507 = r52505 <= r52506;
        double r52508 = -1.0;
        double r52509 = -4.057021676631311e-160;
        bool r52510 = r52505 <= r52509;
        double r52511 = x;
        double r52512 = r52511 - r52505;
        double r52513 = r52511 * r52511;
        double r52514 = r52505 * r52505;
        double r52515 = r52513 + r52514;
        double r52516 = r52511 + r52505;
        double r52517 = r52515 / r52516;
        double r52518 = r52512 / r52517;
        double r52519 = 2.835103494848979e-200;
        bool r52520 = r52505 <= r52519;
        double r52521 = 1.0;
        double r52522 = 3.401055982211938e-168;
        bool r52523 = r52505 <= r52522;
        double r52524 = r52523 ? r52508 : r52518;
        double r52525 = r52520 ? r52521 : r52524;
        double r52526 = r52510 ? r52518 : r52525;
        double r52527 = r52507 ? r52508 : r52526;
        return r52527;
}

Error

Bits error versus x

Bits error versus y

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original20.1
Target0.1
Herbie5.7
\[\begin{array}{l} \mathbf{if}\;0.5 \lt \left|\frac{x}{y}\right| \lt 2:\\ \;\;\;\;\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{2}{1 + \frac{x}{y} \cdot \frac{x}{y}}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if y < -6208632077.041414 or 2.835103494848979e-200 < y < 3.401055982211938e-168

    1. Initial program 32.5

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

    2. Taylor expanded around 0 3.6

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

    if -6208632077.041414 < y < -4.057021676631311e-160 or 3.401055982211938e-168 < y

    1. Initial program 0.6

      \[\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}\]
    2. Using strategy rm

    3. Applied associate-/l*1.1

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

    if -4.057021676631311e-160 < y < 2.835103494848979e-200

    1. Initial program 29.2

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

    2. Taylor expanded around inf 14.0

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

  4. Final simplification5.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -6208632077.0414142608642578125:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \le -4.057021676631311060519344937448265135952 \cdot 10^{-160}:\\ \;\;\;\;\frac{x - y}{\frac{x \cdot x + y \cdot y}{x + y}}\\ \mathbf{elif}\;y \le 2.835103494848979070409532780741214855767 \cdot 10^{-200}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \le 3.401055982211937821476509685336375382237 \cdot 10^{-168}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;\frac{x - y}{\frac{x \cdot x + y \cdot y}{x + y}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019326 
(FPCore (x y)
  :name "Kahan p9 Example"
  :precision binary64
  :pre (and (< 0.0 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))))