Average Error: 19.6 → 5.5
Time: 2.7s
Precision: 64
\[0.0 \lt x \lt 1 \land y \lt 1\]
\[\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le -3.847434879443193393633321330852372755378 \cdot 10^{151}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \le -1.694063246808970668811003130641552295003 \cdot 10^{-162}:\\ \;\;\;\;e^{\mathsf{log1p}\left(\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}\right)} - 1\\ \mathbf{elif}\;y \le 6.451959238219760281097758056287369172736 \cdot 10^{-208}:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(1\right)\right)\\ \mathbf{elif}\;y \le 1.336866892806927934140174920733392856435 \cdot 10^{-166}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;e^{\mathsf{log1p}\left(\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}\right)} - 1\\ \end{array}\]
\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}
↓
\begin{array}{l}
\mathbf{if}\;y \le -3.847434879443193393633321330852372755378 \cdot 10^{151}:\\
\;\;\;\;-1\\

\mathbf{elif}\;y \le -1.694063246808970668811003130641552295003 \cdot 10^{-162}:\\
\;\;\;\;e^{\mathsf{log1p}\left(\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}\right)} - 1\\

\mathbf{elif}\;y \le 6.451959238219760281097758056287369172736 \cdot 10^{-208}:\\
\;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(1\right)\right)\\

\mathbf{elif}\;y \le 1.336866892806927934140174920733392856435 \cdot 10^{-166}:\\
\;\;\;\;-1\\

\mathbf{else}:\\
\;\;\;\;e^{\mathsf{log1p}\left(\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}\right)} - 1\\

\end{array}
double f(double x, double y) {
        double r97489 = x;
        double r97490 = y;
        double r97491 = r97489 - r97490;
        double r97492 = r97489 + r97490;
        double r97493 = r97491 * r97492;
        double r97494 = r97489 * r97489;
        double r97495 = r97490 * r97490;
        double r97496 = r97494 + r97495;
        double r97497 = r97493 / r97496;
        return r97497;
}
↓
double f(double x, double y) {
        double r97498 = y;
        double r97499 = -3.8474348794431934e+151;
        bool r97500 = r97498 <= r97499;
        double r97501 = -1.0;
        double r97502 = -1.6940632468089707e-162;
        bool r97503 = r97498 <= r97502;
        double r97504 = x;
        double r97505 = r97504 - r97498;
        double r97506 = r97504 + r97498;
        double r97507 = r97505 * r97506;
        double r97508 = r97504 * r97504;
        double r97509 = r97498 * r97498;
        double r97510 = r97508 + r97509;
        double r97511 = r97507 / r97510;
        double r97512 = log1p(r97511);
        double r97513 = exp(r97512);
        double r97514 = 1.0;
        double r97515 = r97513 - r97514;
        double r97516 = 6.45195923821976e-208;
        bool r97517 = r97498 <= r97516;
        double r97518 = expm1(r97514);
        double r97519 = log1p(r97518);
        double r97520 = 1.336866892806928e-166;
        bool r97521 = r97498 <= r97520;
        double r97522 = r97521 ? r97501 : r97515;
        double r97523 = r97517 ? r97519 : r97522;
        double r97524 = r97503 ? r97515 : r97523;
        double r97525 = r97500 ? r97501 : r97524;
        return r97525;
}

Error

Bits error versus x

Bits error versus y

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original19.6
Target0.0
Herbie5.5
\[\begin{array}{l} \mathbf{if}\;0.5 \lt \left|\frac{x}{y}\right| \lt 2:\\ \;\;\;\;\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{2}{1 + \frac{x}{y} \cdot \frac{x}{y}}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if y < -3.8474348794431934e+151 or 6.45195923821976e-208 < y < 1.336866892806928e-166

    1. Initial program 55.7

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

    2. Taylor expanded around 0 8.5

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

    if -3.8474348794431934e+151 < y < -1.6940632468089707e-162 or 1.336866892806928e-166 < y

    1. Initial program 0.3

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

    3. Applied log1p-expm1-u0.3

      \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}\right)\right)}\]
    4. Using strategy rm

    5. Applied expm1-log1p-u0.3

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}\right)\right)\right)\right)}\]

    6. Simplified0.3

      \[\leadsto \mathsf{expm1}\left(\color{blue}{\mathsf{log1p}\left(\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}\right)}\right)\]
    7. Using strategy rm

    8. Applied expm1-udef0.3

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}\right)} - 1}\]

    if -1.6940632468089707e-162 < y < 6.45195923821976e-208

    1. Initial program 28.7

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

    3. Applied log1p-expm1-u28.7

      \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}\right)\right)}\]

    4. Taylor expanded around inf 13.4

      \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\color{blue}{1}\right)\right)\]
  3. Recombined 3 regimes into one program.

  4. Final simplification5.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -3.847434879443193393633321330852372755378 \cdot 10^{151}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \le -1.694063246808970668811003130641552295003 \cdot 10^{-162}:\\ \;\;\;\;e^{\mathsf{log1p}\left(\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}\right)} - 1\\ \mathbf{elif}\;y \le 6.451959238219760281097758056287369172736 \cdot 10^{-208}:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(1\right)\right)\\ \mathbf{elif}\;y \le 1.336866892806927934140174920733392856435 \cdot 10^{-166}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;e^{\mathsf{log1p}\left(\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}\right)} - 1\\ \end{array}\]

Reproduce

herbie shell --seed 2019353 +o rules:numerics
(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))))