Average Error: 20.1 → 0.0
Time: 14.9s
Precision: 64
\[0.0 \lt x \lt 1 \land y \lt 1\]
\[\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}\]
↓
\[\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{\frac{x - y}{\mathsf{hypot}\left(x, y\right)}}{\frac{\mathsf{hypot}\left(x, y\right)}{y + x}}\right)\right)\]
\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}
↓
\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{\frac{x - y}{\mathsf{hypot}\left(x, y\right)}}{\frac{\mathsf{hypot}\left(x, y\right)}{y + x}}\right)\right)
double f(double x, double y) {
        double r53867 = x;
        double r53868 = y;
        double r53869 = r53867 - r53868;
        double r53870 = r53867 + r53868;
        double r53871 = r53869 * r53870;
        double r53872 = r53867 * r53867;
        double r53873 = r53868 * r53868;
        double r53874 = r53872 + r53873;
        double r53875 = r53871 / r53874;
        return r53875;
}
↓
double f(double x, double y) {
        double r53876 = x;
        double r53877 = y;
        double r53878 = r53876 - r53877;
        double r53879 = hypot(r53876, r53877);
        double r53880 = r53878 / r53879;
        double r53881 = r53877 + r53876;
        double r53882 = r53879 / r53881;
        double r53883 = r53880 / r53882;
        double r53884 = expm1(r53883);
        double r53885 = log1p(r53884);
        return r53885;
}

Error

Bits error versus x

Bits error versus y

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original20.1
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 20.1

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

  2. Simplified20.1

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

  4. Applied add-sqr-sqrt20.1

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

  5. Applied times-frac20.1

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

  6. Simplified20.1

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

  7. Simplified0.0

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

  9. Applied log1p-expm1-u0.0

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

  10. Simplified0.0

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

  11. Final simplification0.0

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

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))))