Average Error: 21.0 → 0.0
Time: 12.2s
Precision: 64
\[0.0 \lt x \lt 1 \land y \lt 1\]
\[\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}\]
↓
\[\sqrt[3]{{\left(\log \left(e^{\frac{x - y}{\mathsf{hypot}\left(x, y\right)} \cdot \frac{x + y}{\mathsf{hypot}\left(x, y\right)}}\right)\right)}^{3}}\]
\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}
↓
\sqrt[3]{{\left(\log \left(e^{\frac{x - y}{\mathsf{hypot}\left(x, y\right)} \cdot \frac{x + y}{\mathsf{hypot}\left(x, y\right)}}\right)\right)}^{3}}
double f(double x, double y) {
        double r91894 = x;
        double r91895 = y;
        double r91896 = r91894 - r91895;
        double r91897 = r91894 + r91895;
        double r91898 = r91896 * r91897;
        double r91899 = r91894 * r91894;
        double r91900 = r91895 * r91895;
        double r91901 = r91899 + r91900;
        double r91902 = r91898 / r91901;
        return r91902;
}
↓
double f(double x, double y) {
        double r91903 = x;
        double r91904 = y;
        double r91905 = r91903 - r91904;
        double r91906 = hypot(r91903, r91904);
        double r91907 = r91905 / r91906;
        double r91908 = r91903 + r91904;
        double r91909 = r91908 / r91906;
        double r91910 = r91907 * r91909;
        double r91911 = exp(r91910);
        double r91912 = log(r91911);
        double r91913 = 3.0;
        double r91914 = pow(r91912, r91913);
        double r91915 = cbrt(r91914);
        return r91915;
}

Error

Bits error versus x

Bits error versus y

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original21.0
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 21.0

    \[\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}\]
  2. Using strategy rm

  3. Applied add-sqr-sqrt21.0

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

  4. Applied times-frac21.0

    \[\leadsto \color{blue}{\frac{x - y}{\sqrt{x \cdot x + y \cdot y}} \cdot \frac{x + y}{\sqrt{x \cdot x + y \cdot y}}}\]

  5. Simplified21.0

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

  6. Simplified0.0

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

  8. Applied add-cbrt-cube32.8

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

  9. Applied add-cbrt-cube32.6

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

  10. Applied cbrt-undiv32.6

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

  11. Applied add-cbrt-cube33.3

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

  12. Applied add-cbrt-cube32.6

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

  13. Applied cbrt-undiv32.6

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

  14. Applied cbrt-unprod32.6

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

  15. Simplified0.0

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

  17. Applied add-log-exp0.0

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

  18. Final simplification0.0

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

Reproduce

herbie shell --seed 2020047 +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))))