Average Error: 21.0 → 5.3
Time: 1.8s
Precision: 64
\[0.0 \lt x \lt 1 \land y \lt 1\]
\[\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le -2.17401776624398403 \cdot 10^{153}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \le -2.03359079106827272 \cdot 10^{-162}:\\ \;\;\;\;\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}\\ \mathbf{elif}\;y \le 1.5900983176733966 \cdot 10^{-155}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}\\ \end{array}\]
\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}
↓
\begin{array}{l}
\mathbf{if}\;y \le -2.17401776624398403 \cdot 10^{153}:\\
\;\;\;\;-1\\

\mathbf{elif}\;y \le -2.03359079106827272 \cdot 10^{-162}:\\
\;\;\;\;\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}\\

\mathbf{elif}\;y \le 1.5900983176733966 \cdot 10^{-155}:\\
\;\;\;\;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 r95555 = x;
        double r95556 = y;
        double r95557 = r95555 - r95556;
        double r95558 = r95555 + r95556;
        double r95559 = r95557 * r95558;
        double r95560 = r95555 * r95555;
        double r95561 = r95556 * r95556;
        double r95562 = r95560 + r95561;
        double r95563 = r95559 / r95562;
        return r95563;
}
↓
double f(double x, double y) {
        double r95564 = y;
        double r95565 = -2.174017766243984e+153;
        bool r95566 = r95564 <= r95565;
        double r95567 = -1.0;
        double r95568 = -2.0335907910682727e-162;
        bool r95569 = r95564 <= r95568;
        double r95570 = x;
        double r95571 = r95570 - r95564;
        double r95572 = r95570 + r95564;
        double r95573 = r95571 * r95572;
        double r95574 = r95570 * r95570;
        double r95575 = r95564 * r95564;
        double r95576 = r95574 + r95575;
        double r95577 = r95573 / r95576;
        double r95578 = 1.5900983176733966e-155;
        bool r95579 = r95564 <= r95578;
        double r95580 = 1.0;
        double r95581 = r95579 ? r95580 : r95577;
        double r95582 = r95569 ? r95577 : r95581;
        double r95583 = r95566 ? r95567 : r95582;
        return r95583;
}

Error

Bits error versus x

Bits error versus y

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original21.0
Target0.1
Herbie5.3
\[\begin{array}{l} \mathbf{if}\;0.5 \lt \left|\frac{x}{y}\right| \lt 2:\\ \;\;\;\;\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{2}{1 + \frac{x}{y} \cdot \frac{x}{y}}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if y < -2.174017766243984e+153

    1. Initial program 63.6

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

    2. Taylor expanded around 0 0

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

    if -2.174017766243984e+153 < y < -2.0335907910682727e-162 or 1.5900983176733966e-155 < y

    1. Initial program 0.0

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

    if -2.0335907910682727e-162 < y < 1.5900983176733966e-155

    1. Initial program 30.7

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

    2. Taylor expanded around inf 16.6

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

  4. Final simplification5.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -2.17401776624398403 \cdot 10^{153}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \le -2.03359079106827272 \cdot 10^{-162}:\\ \;\;\;\;\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}\\ \mathbf{elif}\;y \le 1.5900983176733966 \cdot 10^{-155}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}\\ \end{array}\]

Reproduce

herbie shell --seed 2020047 
(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))))