Average Error: 20.8 → 12.8
Time: 11.3s
Precision: 64
\[0.0 \lt x \lt 1 \land y \lt 1\]
\[\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le -3.306610229065552697766015508018533353152 \cdot 10^{-6}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \le -1.040149902759510101461916731014611092466 \cdot 10^{-136}:\\ \;\;\;\;\frac{\left(x - y\right) \cdot \left(x + y\right)}{\frac{{x}^{4} - {y}^{4}}{\left(x + y\right) \cdot \left(x - y\right)}}\\ \mathbf{elif}\;y \le -4.002205660206990449557576569903637397741 \cdot 10^{-171}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \le 2.245790742829554397735960283721886784496 \cdot 10^{-222}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \le 5.875513051791536100179706901856990854373 \cdot 10^{-203}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \le 8.954732963342977511317338914679377322524 \cdot 10^{-75}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(x - y\right) \cdot \left(x + y\right)}{\frac{{x}^{4} - {y}^{4}}{\left(x + y\right) \cdot \left(x - y\right)}}\\ \end{array}\]
\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}
↓
\begin{array}{l}
\mathbf{if}\;y \le -3.306610229065552697766015508018533353152 \cdot 10^{-6}:\\
\;\;\;\;-1\\

\mathbf{elif}\;y \le -1.040149902759510101461916731014611092466 \cdot 10^{-136}:\\
\;\;\;\;\frac{\left(x - y\right) \cdot \left(x + y\right)}{\frac{{x}^{4} - {y}^{4}}{\left(x + y\right) \cdot \left(x - y\right)}}\\

\mathbf{elif}\;y \le -4.002205660206990449557576569903637397741 \cdot 10^{-171}:\\
\;\;\;\;-1\\

\mathbf{elif}\;y \le 2.245790742829554397735960283721886784496 \cdot 10^{-222}:\\
\;\;\;\;1\\

\mathbf{elif}\;y \le 5.875513051791536100179706901856990854373 \cdot 10^{-203}:\\
\;\;\;\;-1\\

\mathbf{elif}\;y \le 8.954732963342977511317338914679377322524 \cdot 10^{-75}:\\
\;\;\;\;1\\

\mathbf{else}:\\
\;\;\;\;\frac{\left(x - y\right) \cdot \left(x + y\right)}{\frac{{x}^{4} - {y}^{4}}{\left(x + y\right) \cdot \left(x - y\right)}}\\

\end{array}
double f(double x, double y) {
        double r58384 = x;
        double r58385 = y;
        double r58386 = r58384 - r58385;
        double r58387 = r58384 + r58385;
        double r58388 = r58386 * r58387;
        double r58389 = r58384 * r58384;
        double r58390 = r58385 * r58385;
        double r58391 = r58389 + r58390;
        double r58392 = r58388 / r58391;
        return r58392;
}
↓
double f(double x, double y) {
        double r58393 = y;
        double r58394 = -3.3066102290655527e-06;
        bool r58395 = r58393 <= r58394;
        double r58396 = -1.0;
        double r58397 = -1.0401499027595101e-136;
        bool r58398 = r58393 <= r58397;
        double r58399 = x;
        double r58400 = r58399 - r58393;
        double r58401 = r58399 + r58393;
        double r58402 = r58400 * r58401;
        double r58403 = 4.0;
        double r58404 = pow(r58399, r58403);
        double r58405 = pow(r58393, r58403);
        double r58406 = r58404 - r58405;
        double r58407 = r58401 * r58400;
        double r58408 = r58406 / r58407;
        double r58409 = r58402 / r58408;
        double r58410 = -4.0022056602069904e-171;
        bool r58411 = r58393 <= r58410;
        double r58412 = 2.2457907428295544e-222;
        bool r58413 = r58393 <= r58412;
        double r58414 = 1.0;
        double r58415 = 5.875513051791536e-203;
        bool r58416 = r58393 <= r58415;
        double r58417 = 8.954732963342978e-75;
        bool r58418 = r58393 <= r58417;
        double r58419 = r58418 ? r58414 : r58409;
        double r58420 = r58416 ? r58396 : r58419;
        double r58421 = r58413 ? r58414 : r58420;
        double r58422 = r58411 ? r58396 : r58421;
        double r58423 = r58398 ? r58409 : r58422;
        double r58424 = r58395 ? r58396 : r58423;
        return r58424;
}

Error

Bits error versus x

Bits error versus y

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original20.8
Target0.0
Herbie12.8
\[\begin{array}{l} \mathbf{if}\;0.5 \lt \left|\frac{x}{y}\right| \lt 2:\\ \;\;\;\;\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{2}{1 + \frac{x}{y} \cdot \frac{x}{y}}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if y < -3.3066102290655527e-06 or -1.0401499027595101e-136 < y < -4.0022056602069904e-171 or 2.2457907428295544e-222 < y < 5.875513051791536e-203

    1. Initial program 30.0

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

    2. Taylor expanded around 0 4.6

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

    if -3.3066102290655527e-06 < y < -1.0401499027595101e-136 or 8.954732963342978e-75 < y

    1. Initial program 0.0

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

    3. Applied flip-+13.4

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

    4. Simplified13.7

      \[\leadsto \frac{\left(x - y\right) \cdot \left(x + y\right)}{\frac{\color{blue}{{x}^{4} - {y}^{4}}}{x \cdot x - y \cdot y}}\]

    5. Simplified13.8

      \[\leadsto \frac{\left(x - y\right) \cdot \left(x + y\right)}{\frac{{x}^{4} - {y}^{4}}{\color{blue}{\left(x + y\right) \cdot \left(x - y\right)}}}\]

    if -4.0022056602069904e-171 < y < 2.2457907428295544e-222 or 5.875513051791536e-203 < y < 8.954732963342978e-75

    1. Initial program 22.6

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

    2. Taylor expanded around inf 20.6

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

  4. Final simplification12.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -3.306610229065552697766015508018533353152 \cdot 10^{-6}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \le -1.040149902759510101461916731014611092466 \cdot 10^{-136}:\\ \;\;\;\;\frac{\left(x - y\right) \cdot \left(x + y\right)}{\frac{{x}^{4} - {y}^{4}}{\left(x + y\right) \cdot \left(x - y\right)}}\\ \mathbf{elif}\;y \le -4.002205660206990449557576569903637397741 \cdot 10^{-171}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \le 2.245790742829554397735960283721886784496 \cdot 10^{-222}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \le 5.875513051791536100179706901856990854373 \cdot 10^{-203}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \le 8.954732963342977511317338914679377322524 \cdot 10^{-75}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(x - y\right) \cdot \left(x + y\right)}{\frac{{x}^{4} - {y}^{4}}{\left(x + y\right) \cdot \left(x - y\right)}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019325 
(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))))