Average Error: 20.6 → 0.0
Time: 4.3s
Precision: 64
\[0.0 \lt x \lt 1 \land y \lt 1\]
\[\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}\]
↓
\[\sqrt[3]{{\left(\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{x - y}{\mathsf{hypot}\left(x, y\right) \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(\frac{\mathsf{hypot}\left(x, y\right)}{x + y}\right)\right)}\right)\right)\right)}^{3}}\]
\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}
↓
\sqrt[3]{{\left(\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{x - y}{\mathsf{hypot}\left(x, y\right) \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(\frac{\mathsf{hypot}\left(x, y\right)}{x + y}\right)\right)}\right)\right)\right)}^{3}}
double f(double x, double y) {
        double r76883 = x;
        double r76884 = y;
        double r76885 = r76883 - r76884;
        double r76886 = r76883 + r76884;
        double r76887 = r76885 * r76886;
        double r76888 = r76883 * r76883;
        double r76889 = r76884 * r76884;
        double r76890 = r76888 + r76889;
        double r76891 = r76887 / r76890;
        return r76891;
}
↓
double f(double x, double y) {
        double r76892 = x;
        double r76893 = y;
        double r76894 = r76892 - r76893;
        double r76895 = hypot(r76892, r76893);
        double r76896 = r76892 + r76893;
        double r76897 = r76895 / r76896;
        double r76898 = expm1(r76897);
        double r76899 = log1p(r76898);
        double r76900 = r76895 * r76899;
        double r76901 = r76894 / r76900;
        double r76902 = expm1(r76901);
        double r76903 = log1p(r76902);
        double r76904 = 3.0;
        double r76905 = pow(r76903, r76904);
        double r76906 = cbrt(r76905);
        return r76906;
}

Error

Bits error versus x

Bits error versus y

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original20.6
Target0.0
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 20.6

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

  2. Simplified20.8

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

  4. Applied *-un-lft-identity20.8

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

  5. Applied add-sqr-sqrt20.8

    \[\leadsto \frac{x - y}{\frac{\color{blue}{\sqrt{\mathsf{fma}\left(x, x, y \cdot y\right)} \cdot \sqrt{\mathsf{fma}\left(x, x, y \cdot y\right)}}}{1 \cdot \left(x + y\right)}}\]

  6. Applied times-frac20.7

    \[\leadsto \frac{x - y}{\color{blue}{\frac{\sqrt{\mathsf{fma}\left(x, x, y \cdot y\right)}}{1} \cdot \frac{\sqrt{\mathsf{fma}\left(x, x, y \cdot y\right)}}{x + y}}}\]

  7. Simplified20.7

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

  8. Simplified0.0

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

  10. Applied add-cbrt-cube32.8

    \[\leadsto \frac{x - y}{\mathsf{hypot}\left(x, y\right) \cdot \frac{\mathsf{hypot}\left(x, y\right)}{\color{blue}{\sqrt[3]{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(x + y\right)}}}}\]

  11. Applied add-cbrt-cube32.7

    \[\leadsto \frac{x - y}{\mathsf{hypot}\left(x, y\right) \cdot \frac{\color{blue}{\sqrt[3]{\left(\mathsf{hypot}\left(x, y\right) \cdot \mathsf{hypot}\left(x, y\right)\right) \cdot \mathsf{hypot}\left(x, y\right)}}}{\sqrt[3]{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(x + y\right)}}}\]

  12. Applied cbrt-undiv32.7

    \[\leadsto \frac{x - y}{\mathsf{hypot}\left(x, y\right) \cdot \color{blue}{\sqrt[3]{\frac{\left(\mathsf{hypot}\left(x, y\right) \cdot \mathsf{hypot}\left(x, y\right)\right) \cdot \mathsf{hypot}\left(x, y\right)}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(x + y\right)}}}}\]

  13. Applied add-cbrt-cube33.3

    \[\leadsto \frac{x - y}{\color{blue}{\sqrt[3]{\left(\mathsf{hypot}\left(x, y\right) \cdot \mathsf{hypot}\left(x, y\right)\right) \cdot \mathsf{hypot}\left(x, y\right)}} \cdot \sqrt[3]{\frac{\left(\mathsf{hypot}\left(x, y\right) \cdot \mathsf{hypot}\left(x, y\right)\right) \cdot \mathsf{hypot}\left(x, y\right)}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(x + y\right)}}}\]

  14. Applied cbrt-unprod33.3

    \[\leadsto \frac{x - y}{\color{blue}{\sqrt[3]{\left(\left(\mathsf{hypot}\left(x, y\right) \cdot \mathsf{hypot}\left(x, y\right)\right) \cdot \mathsf{hypot}\left(x, y\right)\right) \cdot \frac{\left(\mathsf{hypot}\left(x, y\right) \cdot \mathsf{hypot}\left(x, y\right)\right) \cdot \mathsf{hypot}\left(x, y\right)}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(x + y\right)}}}}\]

  15. Applied add-cbrt-cube32.7

    \[\leadsto \frac{\color{blue}{\sqrt[3]{\left(\left(x - y\right) \cdot \left(x - y\right)\right) \cdot \left(x - y\right)}}}{\sqrt[3]{\left(\left(\mathsf{hypot}\left(x, y\right) \cdot \mathsf{hypot}\left(x, y\right)\right) \cdot \mathsf{hypot}\left(x, y\right)\right) \cdot \frac{\left(\mathsf{hypot}\left(x, y\right) \cdot \mathsf{hypot}\left(x, y\right)\right) \cdot \mathsf{hypot}\left(x, y\right)}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(x + y\right)}}}\]

  16. Applied cbrt-undiv32.6

    \[\leadsto \color{blue}{\sqrt[3]{\frac{\left(\left(x - y\right) \cdot \left(x - y\right)\right) \cdot \left(x - y\right)}{\left(\left(\mathsf{hypot}\left(x, y\right) \cdot \mathsf{hypot}\left(x, y\right)\right) \cdot \mathsf{hypot}\left(x, y\right)\right) \cdot \frac{\left(\mathsf{hypot}\left(x, y\right) \cdot \mathsf{hypot}\left(x, y\right)\right) \cdot \mathsf{hypot}\left(x, y\right)}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(x + y\right)}}}}\]

  17. Simplified0.0

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

  19. Applied log1p-expm1-u0.0

    \[\leadsto \sqrt[3]{{\color{blue}{\left(\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{x - y}{\mathsf{hypot}\left(x, y\right) \cdot \frac{\mathsf{hypot}\left(x, y\right)}{x + y}}\right)\right)\right)}}^{3}}\]
  20. Using strategy rm

  21. Applied log1p-expm1-u0.0

    \[\leadsto \sqrt[3]{{\left(\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{x - y}{\mathsf{hypot}\left(x, y\right) \cdot \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{\mathsf{hypot}\left(x, y\right)}{x + y}\right)\right)}}\right)\right)\right)}^{3}}\]

  22. Final simplification0.0

    \[\leadsto \sqrt[3]{{\left(\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{x - y}{\mathsf{hypot}\left(x, y\right) \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(\frac{\mathsf{hypot}\left(x, y\right)}{x + y}\right)\right)}\right)\right)\right)}^{3}}\]

Reproduce

herbie shell --seed 2020025 +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))))