Average Error: 0.5 → 0.5
Time: 31.8s
Precision: 64
\[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)}\]
↓
\[\frac{2 + \frac{\left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \frac{\left(-\cos \left(y + y\right)\right) + \cos \left(2 \cdot x\right)}{2}}{\sqrt[3]{{\left(\cos y + \cos x\right)}^{3}}}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\frac{3 \cdot 3 - 5}{3 + \sqrt{5}}}{2} \cdot \cos y\right)}\]
\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{2 + \frac{\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 \frac{\left(-\cos \left(y + y\right)\right) + \cos \left(2 \cdot x\right)}{2}}{\sqrt[3]{{\left(\cos y + \cos x\right)}^{3}}}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\frac{3 \cdot 3 - 5}{3 + \sqrt{5}}}{2} \cdot \cos y\right)}
double f(double x, double y) {
        double r161897 = 2.0;
        double r161898 = sqrt(r161897);
        double r161899 = x;
        double r161900 = sin(r161899);
        double r161901 = y;
        double r161902 = sin(r161901);
        double r161903 = 16.0;
        double r161904 = r161902 / r161903;
        double r161905 = r161900 - r161904;
        double r161906 = r161898 * r161905;
        double r161907 = r161900 / r161903;
        double r161908 = r161902 - r161907;
        double r161909 = r161906 * r161908;
        double r161910 = cos(r161899);
        double r161911 = cos(r161901);
        double r161912 = r161910 - r161911;
        double r161913 = r161909 * r161912;
        double r161914 = r161897 + r161913;
        double r161915 = 3.0;
        double r161916 = 1.0;
        double r161917 = 5.0;
        double r161918 = sqrt(r161917);
        double r161919 = r161918 - r161916;
        double r161920 = r161919 / r161897;
        double r161921 = r161920 * r161910;
        double r161922 = r161916 + r161921;
        double r161923 = r161915 - r161918;
        double r161924 = r161923 / r161897;
        double r161925 = r161924 * r161911;
        double r161926 = r161922 + r161925;
        double r161927 = r161915 * r161926;
        double r161928 = r161914 / r161927;
        return r161928;
}
↓
double f(double x, double y) {
        double r161929 = 2.0;
        double r161930 = sqrt(r161929);
        double r161931 = x;
        double r161932 = sin(r161931);
        double r161933 = y;
        double r161934 = sin(r161933);
        double r161935 = 16.0;
        double r161936 = r161934 / r161935;
        double r161937 = r161932 - r161936;
        double r161938 = r161930 * r161937;
        double r161939 = r161932 / r161935;
        double r161940 = r161934 - r161939;
        double r161941 = r161938 * r161940;
        double r161942 = r161933 + r161933;
        double r161943 = cos(r161942);
        double r161944 = -r161943;
        double r161945 = 2.0;
        double r161946 = r161945 * r161931;
        double r161947 = cos(r161946);
        double r161948 = r161944 + r161947;
        double r161949 = r161948 / r161945;
        double r161950 = r161941 * r161949;
        double r161951 = cos(r161933);
        double r161952 = cos(r161931);
        double r161953 = r161951 + r161952;
        double r161954 = 3.0;
        double r161955 = pow(r161953, r161954);
        double r161956 = cbrt(r161955);
        double r161957 = r161950 / r161956;
        double r161958 = r161929 + r161957;
        double r161959 = 3.0;
        double r161960 = 1.0;
        double r161961 = 5.0;
        double r161962 = sqrt(r161961);
        double r161963 = r161962 - r161960;
        double r161964 = r161963 / r161929;
        double r161965 = r161964 * r161952;
        double r161966 = r161960 + r161965;
        double r161967 = r161959 * r161959;
        double r161968 = r161967 - r161961;
        double r161969 = r161959 + r161962;
        double r161970 = r161968 / r161969;
        double r161971 = r161970 / r161929;
        double r161972 = r161971 * r161951;
        double r161973 = r161966 + r161972;
        double r161974 = r161959 * r161973;
        double r161975 = r161958 / r161974;
        return r161975;
}

Error

Bits error versus x

Bits error versus y

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

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. Using strategy rm

  3. Applied flip--0.5

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

  4. Simplified0.5

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

  6. Applied flip--0.5

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

  7. Applied associate-*r/0.5

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

  9. Applied add-cbrt-cube0.5

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

  10. Simplified0.5

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

  12. Applied cos-mult0.5

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

  13. Applied cos-mult0.5

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

  14. Applied sub-div0.5

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

  15. Simplified0.5

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

  16. Final simplification0.5

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

Reproduce

herbie shell --seed 2019212 
(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))))))