Average Error: 20.1 → 5.7
Time: 9.6s
Precision: 64
\[0.0 \lt x \lt 1 \land y \lt 1\]
\[\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le -1.42434775145189198 \cdot 10^{153}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \le -2.34847619210042794 \cdot 10^{-141} \lor \neg \left(y \le 4.1321470702090296 \cdot 10^{-169}\right):\\ \;\;\;\;\frac{\mathsf{fma}\left(y, -y, x \cdot x\right)}{\mathsf{fma}\left(y, y, x \cdot x\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array}\]
\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}
↓
\begin{array}{l}
\mathbf{if}\;y \le -1.42434775145189198 \cdot 10^{153}:\\
\;\;\;\;-1\\

\mathbf{elif}\;y \le -2.34847619210042794 \cdot 10^{-141} \lor \neg \left(y \le 4.1321470702090296 \cdot 10^{-169}\right):\\
\;\;\;\;\frac{\mathsf{fma}\left(y, -y, x \cdot x\right)}{\mathsf{fma}\left(y, y, x \cdot x\right)}\\

\mathbf{else}:\\
\;\;\;\;1\\

\end{array}
double f(double x, double y) {
        double r44598 = x;
        double r44599 = y;
        double r44600 = r44598 - r44599;
        double r44601 = r44598 + r44599;
        double r44602 = r44600 * r44601;
        double r44603 = r44598 * r44598;
        double r44604 = r44599 * r44599;
        double r44605 = r44603 + r44604;
        double r44606 = r44602 / r44605;
        return r44606;
}
↓
double f(double x, double y) {
        double r44607 = y;
        double r44608 = -1.424347751451892e+153;
        bool r44609 = r44607 <= r44608;
        double r44610 = -1.0;
        double r44611 = -2.348476192100428e-141;
        bool r44612 = r44607 <= r44611;
        double r44613 = 4.1321470702090296e-169;
        bool r44614 = r44607 <= r44613;
        double r44615 = !r44614;
        bool r44616 = r44612 || r44615;
        double r44617 = -r44607;
        double r44618 = x;
        double r44619 = r44618 * r44618;
        double r44620 = fma(r44607, r44617, r44619);
        double r44621 = fma(r44607, r44607, r44619);
        double r44622 = r44620 / r44621;
        double r44623 = 1.0;
        double r44624 = r44616 ? r44622 : r44623;
        double r44625 = r44609 ? r44610 : r44624;
        return r44625;
}

Error

Bits error versus x

Bits error versus y

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 < -1.424347751451892e+153

    1. Initial program 63.8

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

    2. Simplified63.8

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

    3. Taylor expanded around inf 0

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

    if -1.424347751451892e+153 < y < -2.348476192100428e-141 or 4.1321470702090296e-169 < y

    1. Initial program 0.5

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

    2. Simplified0.5

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

    3. Taylor expanded around 0 0.5

      \[\leadsto \frac{\mathsf{fma}\left(y, -y, x \cdot x\right)}{\color{blue}{{y}^{2} + {x}^{2}}}\]

    4. Simplified0.5

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

    if -2.348476192100428e-141 < y < 4.1321470702090296e-169

    1. Initial program 27.8

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

    2. Simplified27.8

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

    3. Taylor expanded around 0 16.3

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

  4. Final simplification5.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -1.42434775145189198 \cdot 10^{153}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \le -2.34847619210042794 \cdot 10^{-141} \lor \neg \left(y \le 4.1321470702090296 \cdot 10^{-169}\right):\\ \;\;\;\;\frac{\mathsf{fma}\left(y, -y, x \cdot x\right)}{\mathsf{fma}\left(y, y, x \cdot x\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array}\]

Reproduce

herbie shell --seed 2019195 +o rules:numerics
(FPCore (x y)
  :name "Kahan p9 Example"
  :pre (and (< 0.0 x 1.0) (< y 1.0))

  :herbie-target
  (if (< 0.5 (fabs (/ x y)) 2.0) (/ (* (- x y) (+ x y)) (+ (* x x) (* y y))) (- 1.0 (/ 2.0 (+ 1.0 (* (/ x y) (/ x y))))))

  (/ (* (- x y) (+ x y)) (+ (* x x) (* y y))))