Average Error: 20.2 → 0.0
Time: 21.1s
Precision: 64
\[0.0 \lt x \lt 1 \land y \lt 1\]
\[\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}\]
↓
\[\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{x - y}{\mathsf{hypot}\left(x, y\right)} \cdot \frac{x + y}{\mathsf{hypot}\left(x, y\right)}\right)\right)\]
\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}
↓
\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{x - y}{\mathsf{hypot}\left(x, y\right)} \cdot \frac{x + y}{\mathsf{hypot}\left(x, y\right)}\right)\right)
double f(double x, double y) {
        double r2577642 = x;
        double r2577643 = y;
        double r2577644 = r2577642 - r2577643;
        double r2577645 = r2577642 + r2577643;
        double r2577646 = r2577644 * r2577645;
        double r2577647 = r2577642 * r2577642;
        double r2577648 = r2577643 * r2577643;
        double r2577649 = r2577647 + r2577648;
        double r2577650 = r2577646 / r2577649;
        return r2577650;
}
↓
double f(double x, double y) {
        double r2577651 = x;
        double r2577652 = y;
        double r2577653 = r2577651 - r2577652;
        double r2577654 = hypot(r2577651, r2577652);
        double r2577655 = r2577653 / r2577654;
        double r2577656 = r2577651 + r2577652;
        double r2577657 = r2577656 / r2577654;
        double r2577658 = r2577655 * r2577657;
        double r2577659 = expm1(r2577658);
        double r2577660 = log1p(r2577659);
        return r2577660;
}

Error

Bits error versus x

Bits error versus y

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original20.2
Target0.1
Herbie0.0
\[\begin{array}{l} \mathbf{if}\;0.5 \lt \left|\frac{x}{y}\right| \lt 2:\\ \;\;\;\;\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{2}{1 + \frac{x}{y} \cdot \frac{x}{y}}\\ \end{array}\]

Derivation

  1. Initial program 20.2

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

  2. Simplified20.2

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

  4. Applied add-sqr-sqrt20.2

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

  5. Applied times-frac20.3

    \[\leadsto \color{blue}{\frac{x - y}{\sqrt{\mathsf{fma}\left(x, x, y \cdot y\right)}} \cdot \frac{y + x}{\sqrt{\mathsf{fma}\left(x, x, y \cdot y\right)}}}\]
  6. Using strategy rm

  7. Applied log1p-expm1-u20.3

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

  8. Simplified0.0

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

  9. Final simplification0.0

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

Reproduce

herbie shell --seed 2019171 +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))))