Average Error: 20.4 → 5.6
Time: 17.0s
Precision: 64
\[0 \lt x \lt 1 \land y \lt 1\]
\[\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le -1.359699432060768 \cdot 10^{+154}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \le -1.2474742340765115 \cdot 10^{-174}:\\ \;\;\;\;\frac{\left(y + x\right) \cdot \left(x - y\right)}{\mathsf{fma}\left(y, y, x \cdot x\right)}\\ \mathbf{elif}\;y \le -5.766846463446123 \cdot 10^{-210}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \le 1.1093912770840933 \cdot 10^{-203}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \le 1.1960469199561086 \cdot 10^{-189}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(y + x\right) \cdot \left(x - y\right)}{\mathsf{fma}\left(y, y, x \cdot x\right)}\\ \end{array}\]
\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}
↓
\begin{array}{l}
\mathbf{if}\;y \le -1.359699432060768 \cdot 10^{+154}:\\
\;\;\;\;-1\\

\mathbf{elif}\;y \le -1.2474742340765115 \cdot 10^{-174}:\\
\;\;\;\;\frac{\left(y + x\right) \cdot \left(x - y\right)}{\mathsf{fma}\left(y, y, x \cdot x\right)}\\

\mathbf{elif}\;y \le -5.766846463446123 \cdot 10^{-210}:\\
\;\;\;\;-1\\

\mathbf{elif}\;y \le 1.1093912770840933 \cdot 10^{-203}:\\
\;\;\;\;1\\

\mathbf{elif}\;y \le 1.1960469199561086 \cdot 10^{-189}:\\
\;\;\;\;-1\\

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

\end{array}
double f(double x, double y) {
        double r3506251 = x;
        double r3506252 = y;
        double r3506253 = r3506251 - r3506252;
        double r3506254 = r3506251 + r3506252;
        double r3506255 = r3506253 * r3506254;
        double r3506256 = r3506251 * r3506251;
        double r3506257 = r3506252 * r3506252;
        double r3506258 = r3506256 + r3506257;
        double r3506259 = r3506255 / r3506258;
        return r3506259;
}
↓
double f(double x, double y) {
        double r3506260 = y;
        double r3506261 = -1.359699432060768e+154;
        bool r3506262 = r3506260 <= r3506261;
        double r3506263 = -1.0;
        double r3506264 = -1.2474742340765115e-174;
        bool r3506265 = r3506260 <= r3506264;
        double r3506266 = x;
        double r3506267 = r3506260 + r3506266;
        double r3506268 = r3506266 - r3506260;
        double r3506269 = r3506267 * r3506268;
        double r3506270 = r3506266 * r3506266;
        double r3506271 = fma(r3506260, r3506260, r3506270);
        double r3506272 = r3506269 / r3506271;
        double r3506273 = -5.766846463446123e-210;
        bool r3506274 = r3506260 <= r3506273;
        double r3506275 = 1.1093912770840933e-203;
        bool r3506276 = r3506260 <= r3506275;
        double r3506277 = 1.0;
        double r3506278 = 1.1960469199561086e-189;
        bool r3506279 = r3506260 <= r3506278;
        double r3506280 = r3506279 ? r3506263 : r3506272;
        double r3506281 = r3506276 ? r3506277 : r3506280;
        double r3506282 = r3506274 ? r3506263 : r3506281;
        double r3506283 = r3506265 ? r3506272 : r3506282;
        double r3506284 = r3506262 ? r3506263 : r3506283;
        return r3506284;
}

Error

Bits error versus x

Bits error versus y

Target

Original20.4
Target0.0
Herbie5.6
\[\begin{array}{l} \mathbf{if}\;0.5 \lt \left|\frac{x}{y}\right| \lt 2:\\ \;\;\;\;\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{2}{1 + \frac{x}{y} \cdot \frac{x}{y}}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if y < -1.359699432060768e+154 or -1.2474742340765115e-174 < y < -5.766846463446123e-210 or 1.1093912770840933e-203 < y < 1.1960469199561086e-189

    1. Initial program 55.7

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

    2. Simplified55.7

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

    3. Taylor expanded around 0 9.6

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

    if -1.359699432060768e+154 < y < -1.2474742340765115e-174 or 1.1960469199561086e-189 < y

    1. Initial program 2.0

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

    2. Simplified2.0

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

    if -5.766846463446123e-210 < y < 1.1093912770840933e-203

    1. Initial program 28.7

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

    2. Simplified28.7

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

    3. Taylor expanded around inf 10.2

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

  4. Final simplification5.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -1.359699432060768 \cdot 10^{+154}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \le -1.2474742340765115 \cdot 10^{-174}:\\ \;\;\;\;\frac{\left(y + x\right) \cdot \left(x - y\right)}{\mathsf{fma}\left(y, y, x \cdot x\right)}\\ \mathbf{elif}\;y \le -5.766846463446123 \cdot 10^{-210}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \le 1.1093912770840933 \cdot 10^{-203}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \le 1.1960469199561086 \cdot 10^{-189}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(y + x\right) \cdot \left(x - y\right)}{\mathsf{fma}\left(y, y, x \cdot x\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019146 +o rules:numerics
(FPCore (x y)
  :name "Kahan p9 Example"
  :pre (and (< 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))))