Average Error: 0.5 → 0.5
Time: 12.2s
Precision: 64
\[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)}\]
↓
\[\frac{\frac{\mathsf{fma}\left(\left(\sqrt{2} \cdot \left(\sqrt[3]{\sin x - \frac{\sin y}{16}} \cdot \frac{\sqrt[3]{\sqrt[3]{\sin x \cdot \sin x - \frac{\sin y}{16} \cdot \frac{\sin y}{16}} \cdot \sqrt[3]{\sin x \cdot \sin x - \frac{\sin y}{16} \cdot \frac{\sin y}{16}}} \cdot \sqrt[3]{\sqrt[3]{\sin x \cdot \sin x - \frac{\sin y}{16} \cdot \frac{\sin y}{16}}}}{\sqrt[3]{\sin x + \frac{\sin y}{16}}}\right)\right) \cdot \sqrt[3]{\sin x - \frac{\sin y}{16}}, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right), 2\right)}{\mathsf{fma}\left(\frac{3 - \sqrt{5}}{2}, \cos y, \mathsf{fma}\left(\frac{\sqrt{5} - 1}{2}, \cos x, 1\right)\right)}}{3}\]
\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)}
↓
\frac{\frac{\mathsf{fma}\left(\left(\sqrt{2} \cdot \left(\sqrt[3]{\sin x - \frac{\sin y}{16}} \cdot \frac{\sqrt[3]{\sqrt[3]{\sin x \cdot \sin x - \frac{\sin y}{16} \cdot \frac{\sin y}{16}} \cdot \sqrt[3]{\sin x \cdot \sin x - \frac{\sin y}{16} \cdot \frac{\sin y}{16}}} \cdot \sqrt[3]{\sqrt[3]{\sin x \cdot \sin x - \frac{\sin y}{16} \cdot \frac{\sin y}{16}}}}{\sqrt[3]{\sin x + \frac{\sin y}{16}}}\right)\right) \cdot \sqrt[3]{\sin x - \frac{\sin y}{16}}, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right), 2\right)}{\mathsf{fma}\left(\frac{3 - \sqrt{5}}{2}, \cos y, \mathsf{fma}\left(\frac{\sqrt{5} - 1}{2}, \cos x, 1\right)\right)}}{3}
double f(double x, double y) {
        double r170878 = 2.0;
        double r170879 = sqrt(r170878);
        double r170880 = x;
        double r170881 = sin(r170880);
        double r170882 = y;
        double r170883 = sin(r170882);
        double r170884 = 16.0;
        double r170885 = r170883 / r170884;
        double r170886 = r170881 - r170885;
        double r170887 = r170879 * r170886;
        double r170888 = r170881 / r170884;
        double r170889 = r170883 - r170888;
        double r170890 = r170887 * r170889;
        double r170891 = cos(r170880);
        double r170892 = cos(r170882);
        double r170893 = r170891 - r170892;
        double r170894 = r170890 * r170893;
        double r170895 = r170878 + r170894;
        double r170896 = 3.0;
        double r170897 = 1.0;
        double r170898 = 5.0;
        double r170899 = sqrt(r170898);
        double r170900 = r170899 - r170897;
        double r170901 = r170900 / r170878;
        double r170902 = r170901 * r170891;
        double r170903 = r170897 + r170902;
        double r170904 = r170896 - r170899;
        double r170905 = r170904 / r170878;
        double r170906 = r170905 * r170892;
        double r170907 = r170903 + r170906;
        double r170908 = r170896 * r170907;
        double r170909 = r170895 / r170908;
        return r170909;
}
↓
double f(double x, double y) {
        double r170910 = 2.0;
        double r170911 = sqrt(r170910);
        double r170912 = x;
        double r170913 = sin(r170912);
        double r170914 = y;
        double r170915 = sin(r170914);
        double r170916 = 16.0;
        double r170917 = r170915 / r170916;
        double r170918 = r170913 - r170917;
        double r170919 = cbrt(r170918);
        double r170920 = r170913 * r170913;
        double r170921 = r170917 * r170917;
        double r170922 = r170920 - r170921;
        double r170923 = cbrt(r170922);
        double r170924 = r170923 * r170923;
        double r170925 = cbrt(r170924);
        double r170926 = cbrt(r170923);
        double r170927 = r170925 * r170926;
        double r170928 = r170913 + r170917;
        double r170929 = cbrt(r170928);
        double r170930 = r170927 / r170929;
        double r170931 = r170919 * r170930;
        double r170932 = r170911 * r170931;
        double r170933 = r170932 * r170919;
        double r170934 = r170913 / r170916;
        double r170935 = r170915 - r170934;
        double r170936 = cos(r170912);
        double r170937 = cos(r170914);
        double r170938 = r170936 - r170937;
        double r170939 = r170935 * r170938;
        double r170940 = fma(r170933, r170939, r170910);
        double r170941 = 3.0;
        double r170942 = 5.0;
        double r170943 = sqrt(r170942);
        double r170944 = r170941 - r170943;
        double r170945 = r170944 / r170910;
        double r170946 = 1.0;
        double r170947 = r170943 - r170946;
        double r170948 = r170947 / r170910;
        double r170949 = fma(r170948, r170936, r170946);
        double r170950 = fma(r170945, r170937, r170949);
        double r170951 = r170940 / r170950;
        double r170952 = r170951 / r170941;
        return r170952;
}

Error

Bits error versus x

Bits error versus y

Derivation

  1. Initial program 0.5

    \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)}\]

  2. Simplified0.5

    \[\leadsto \color{blue}{\frac{\frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right), \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right), 2\right)}{\mathsf{fma}\left(\frac{3 - \sqrt{5}}{2}, \cos y, \mathsf{fma}\left(\frac{\sqrt{5} - 1}{2}, \cos x, 1\right)\right)}}{3}}\]
  3. Using strategy rm

  4. Applied add-cube-cbrt0.5

    \[\leadsto \frac{\frac{\mathsf{fma}\left(\sqrt{2} \cdot \color{blue}{\left(\left(\sqrt[3]{\sin x - \frac{\sin y}{16}} \cdot \sqrt[3]{\sin x - \frac{\sin y}{16}}\right) \cdot \sqrt[3]{\sin x - \frac{\sin y}{16}}\right)}, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right), 2\right)}{\mathsf{fma}\left(\frac{3 - \sqrt{5}}{2}, \cos y, \mathsf{fma}\left(\frac{\sqrt{5} - 1}{2}, \cos x, 1\right)\right)}}{3}\]

  5. Applied associate-*r*0.5

    \[\leadsto \frac{\frac{\mathsf{fma}\left(\color{blue}{\left(\sqrt{2} \cdot \left(\sqrt[3]{\sin x - \frac{\sin y}{16}} \cdot \sqrt[3]{\sin x - \frac{\sin y}{16}}\right)\right) \cdot \sqrt[3]{\sin x - \frac{\sin y}{16}}}, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right), 2\right)}{\mathsf{fma}\left(\frac{3 - \sqrt{5}}{2}, \cos y, \mathsf{fma}\left(\frac{\sqrt{5} - 1}{2}, \cos x, 1\right)\right)}}{3}\]
  6. Using strategy rm

  7. Applied flip--0.5

    \[\leadsto \frac{\frac{\mathsf{fma}\left(\left(\sqrt{2} \cdot \left(\sqrt[3]{\sin x - \frac{\sin y}{16}} \cdot \sqrt[3]{\color{blue}{\frac{\sin x \cdot \sin x - \frac{\sin y}{16} \cdot \frac{\sin y}{16}}{\sin x + \frac{\sin y}{16}}}}\right)\right) \cdot \sqrt[3]{\sin x - \frac{\sin y}{16}}, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right), 2\right)}{\mathsf{fma}\left(\frac{3 - \sqrt{5}}{2}, \cos y, \mathsf{fma}\left(\frac{\sqrt{5} - 1}{2}, \cos x, 1\right)\right)}}{3}\]

  8. Applied cbrt-div0.5

    \[\leadsto \frac{\frac{\mathsf{fma}\left(\left(\sqrt{2} \cdot \left(\sqrt[3]{\sin x - \frac{\sin y}{16}} \cdot \color{blue}{\frac{\sqrt[3]{\sin x \cdot \sin x - \frac{\sin y}{16} \cdot \frac{\sin y}{16}}}{\sqrt[3]{\sin x + \frac{\sin y}{16}}}}\right)\right) \cdot \sqrt[3]{\sin x - \frac{\sin y}{16}}, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right), 2\right)}{\mathsf{fma}\left(\frac{3 - \sqrt{5}}{2}, \cos y, \mathsf{fma}\left(\frac{\sqrt{5} - 1}{2}, \cos x, 1\right)\right)}}{3}\]
  9. Using strategy rm

  10. Applied add-cube-cbrt0.5

    \[\leadsto \frac{\frac{\mathsf{fma}\left(\left(\sqrt{2} \cdot \left(\sqrt[3]{\sin x - \frac{\sin y}{16}} \cdot \frac{\sqrt[3]{\color{blue}{\left(\sqrt[3]{\sin x \cdot \sin x - \frac{\sin y}{16} \cdot \frac{\sin y}{16}} \cdot \sqrt[3]{\sin x \cdot \sin x - \frac{\sin y}{16} \cdot \frac{\sin y}{16}}\right) \cdot \sqrt[3]{\sin x \cdot \sin x - \frac{\sin y}{16} \cdot \frac{\sin y}{16}}}}}{\sqrt[3]{\sin x + \frac{\sin y}{16}}}\right)\right) \cdot \sqrt[3]{\sin x - \frac{\sin y}{16}}, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right), 2\right)}{\mathsf{fma}\left(\frac{3 - \sqrt{5}}{2}, \cos y, \mathsf{fma}\left(\frac{\sqrt{5} - 1}{2}, \cos x, 1\right)\right)}}{3}\]

  11. Applied cbrt-prod0.5

    \[\leadsto \frac{\frac{\mathsf{fma}\left(\left(\sqrt{2} \cdot \left(\sqrt[3]{\sin x - \frac{\sin y}{16}} \cdot \frac{\color{blue}{\sqrt[3]{\sqrt[3]{\sin x \cdot \sin x - \frac{\sin y}{16} \cdot \frac{\sin y}{16}} \cdot \sqrt[3]{\sin x \cdot \sin x - \frac{\sin y}{16} \cdot \frac{\sin y}{16}}} \cdot \sqrt[3]{\sqrt[3]{\sin x \cdot \sin x - \frac{\sin y}{16} \cdot \frac{\sin y}{16}}}}}{\sqrt[3]{\sin x + \frac{\sin y}{16}}}\right)\right) \cdot \sqrt[3]{\sin x - \frac{\sin y}{16}}, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right), 2\right)}{\mathsf{fma}\left(\frac{3 - \sqrt{5}}{2}, \cos y, \mathsf{fma}\left(\frac{\sqrt{5} - 1}{2}, \cos x, 1\right)\right)}}{3}\]

  12. Final simplification0.5

    \[\leadsto \frac{\frac{\mathsf{fma}\left(\left(\sqrt{2} \cdot \left(\sqrt[3]{\sin x - \frac{\sin y}{16}} \cdot \frac{\sqrt[3]{\sqrt[3]{\sin x \cdot \sin x - \frac{\sin y}{16} \cdot \frac{\sin y}{16}} \cdot \sqrt[3]{\sin x \cdot \sin x - \frac{\sin y}{16} \cdot \frac{\sin y}{16}}} \cdot \sqrt[3]{\sqrt[3]{\sin x \cdot \sin x - \frac{\sin y}{16} \cdot \frac{\sin y}{16}}}}{\sqrt[3]{\sin x + \frac{\sin y}{16}}}\right)\right) \cdot \sqrt[3]{\sin x - \frac{\sin y}{16}}, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right), 2\right)}{\mathsf{fma}\left(\frac{3 - \sqrt{5}}{2}, \cos y, \mathsf{fma}\left(\frac{\sqrt{5} - 1}{2}, \cos x, 1\right)\right)}}{3}\]

Reproduce

herbie shell --seed 2020065 +o rules:numerics
(FPCore (x y)
  :name "Diagrams.TwoD.Path.Metafont.Internal:hobbyF from diagrams-contrib-1.3.0.5"
  :precision binary64
  (/ (+ 2 (* (* (* (sqrt 2) (- (sin x) (/ (sin y) 16))) (- (sin y) (/ (sin x) 16))) (- (cos x) (cos y)))) (* 3 (+ (+ 1 (* (/ (- (sqrt 5) 1) 2) (cos x))) (* (/ (- 3 (sqrt 5)) 2) (cos y))))))