Average Error: 20.0 → 5.1
Time: 1.4s
Precision: 64
\[0.0 \lt x \lt 1 \land y \lt 1\]
\[\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le -1.66952962063310841 \cdot 10^{152}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \le -4.28774228849345812 \cdot 10^{-158}:\\ \;\;\;\;\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}\\ \mathbf{elif}\;y \le 2.9977670369096852 \cdot 10^{-162}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}\\ \end{array}\]
\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}
↓
\begin{array}{l}
\mathbf{if}\;y \le -1.66952962063310841 \cdot 10^{152}:\\
\;\;\;\;-1\\

\mathbf{elif}\;y \le -4.28774228849345812 \cdot 10^{-158}:\\
\;\;\;\;\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}\\

\mathbf{elif}\;y \le 2.9977670369096852 \cdot 10^{-162}:\\
\;\;\;\;1\\

\mathbf{else}:\\
\;\;\;\;\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}\\

\end{array}
double f(double x, double y) {
        double r77484 = x;
        double r77485 = y;
        double r77486 = r77484 - r77485;
        double r77487 = r77484 + r77485;
        double r77488 = r77486 * r77487;
        double r77489 = r77484 * r77484;
        double r77490 = r77485 * r77485;
        double r77491 = r77489 + r77490;
        double r77492 = r77488 / r77491;
        return r77492;
}
↓
double f(double x, double y) {
        double r77493 = y;
        double r77494 = -1.6695296206331084e+152;
        bool r77495 = r77493 <= r77494;
        double r77496 = -1.0;
        double r77497 = -4.287742288493458e-158;
        bool r77498 = r77493 <= r77497;
        double r77499 = x;
        double r77500 = r77499 - r77493;
        double r77501 = r77499 + r77493;
        double r77502 = r77500 * r77501;
        double r77503 = r77499 * r77499;
        double r77504 = r77493 * r77493;
        double r77505 = r77503 + r77504;
        double r77506 = r77502 / r77505;
        double r77507 = 2.997767036909685e-162;
        bool r77508 = r77493 <= r77507;
        double r77509 = 1.0;
        double r77510 = r77508 ? r77509 : r77506;
        double r77511 = r77498 ? r77506 : r77510;
        double r77512 = r77495 ? r77496 : r77511;
        return r77512;
}

Error

Bits error versus x

Bits error versus y

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original20.0
Target0.1
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 < -1.6695296206331084e+152

    1. Initial program 63.3

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

    2. Taylor expanded around 0 0

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

    if -1.6695296206331084e+152 < y < -4.287742288493458e-158 or 2.997767036909685e-162 < y

    1. Initial program 0.0

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

    if -4.287742288493458e-158 < y < 2.997767036909685e-162

    1. Initial program 30.2

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

    2. Taylor expanded around inf 16.1

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

  4. Final simplification5.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -1.66952962063310841 \cdot 10^{152}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \le -4.28774228849345812 \cdot 10^{-158}:\\ \;\;\;\;\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}\\ \mathbf{elif}\;y \le 2.9977670369096852 \cdot 10^{-162}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}\\ \end{array}\]

Reproduce

herbie shell --seed 2020018 
(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))))