Average Error: 0.5 → 0.5
Time: 28.1s
Precision: 64
\[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)}\]
↓
\[\frac{\frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right), \log \left({\left(e^{\sin y - \frac{\sin x}{16}}\right)}^{\left(\mathsf{fma}\left(\sqrt[3]{\cos x} \cdot \sqrt[3]{\cos x}, \sqrt[3]{\cos x}, -\cos y \cdot 1\right) + \mathsf{fma}\left(-\cos y, 1, \cos y\right)\right)}\right), 2\right)}{\mathsf{fma}\left(\frac{3 - \sqrt{5}}{2}, \cos y, \mathsf{fma}\left(\frac{\sqrt{5} - 1}{2}, \cos x, 1\right)\right)}}{3}\]
\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(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right), \log \left({\left(e^{\sin y - \frac{\sin x}{16}}\right)}^{\left(\mathsf{fma}\left(\sqrt[3]{\cos x} \cdot \sqrt[3]{\cos x}, \sqrt[3]{\cos x}, -\cos y \cdot 1\right) + \mathsf{fma}\left(-\cos y, 1, \cos y\right)\right)}\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 r911 = 2.0;
        double r912 = sqrt(r911);
        double r913 = x;
        double r914 = sin(r913);
        double r915 = y;
        double r916 = sin(r915);
        double r917 = 16.0;
        double r918 = r916 / r917;
        double r919 = r914 - r918;
        double r920 = r912 * r919;
        double r921 = r914 / r917;
        double r922 = r916 - r921;
        double r923 = r920 * r922;
        double r924 = cos(r913);
        double r925 = cos(r915);
        double r926 = r924 - r925;
        double r927 = r923 * r926;
        double r928 = r911 + r927;
        double r929 = 3.0;
        double r930 = 1.0;
        double r931 = 5.0;
        double r932 = sqrt(r931);
        double r933 = r932 - r930;
        double r934 = r933 / r911;
        double r935 = r934 * r924;
        double r936 = r930 + r935;
        double r937 = r929 - r932;
        double r938 = r937 / r911;
        double r939 = r938 * r925;
        double r940 = r936 + r939;
        double r941 = r929 * r940;
        double r942 = r928 / r941;
        return r942;
}
↓
double f(double x, double y) {
        double r943 = 2.0;
        double r944 = sqrt(r943);
        double r945 = x;
        double r946 = sin(r945);
        double r947 = y;
        double r948 = sin(r947);
        double r949 = 16.0;
        double r950 = r948 / r949;
        double r951 = r946 - r950;
        double r952 = r944 * r951;
        double r953 = r946 / r949;
        double r954 = r948 - r953;
        double r955 = exp(r954);
        double r956 = cos(r945);
        double r957 = cbrt(r956);
        double r958 = r957 * r957;
        double r959 = cos(r947);
        double r960 = 1.0;
        double r961 = r959 * r960;
        double r962 = -r961;
        double r963 = fma(r958, r957, r962);
        double r964 = -r959;
        double r965 = fma(r964, r960, r959);
        double r966 = r963 + r965;
        double r967 = pow(r955, r966);
        double r968 = log(r967);
        double r969 = fma(r952, r968, r943);
        double r970 = 3.0;
        double r971 = 5.0;
        double r972 = sqrt(r971);
        double r973 = r970 - r972;
        double r974 = r973 / r943;
        double r975 = 1.0;
        double r976 = r972 - r975;
        double r977 = r976 / r943;
        double r978 = fma(r977, r956, r975);
        double r979 = fma(r974, r959, r978);
        double r980 = r969 / r979;
        double r981 = r980 / r970;
        return r981;
}

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-cbrt-cube0.5

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

  5. Applied add-cbrt-cube0.5

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

  6. Applied cbrt-unprod0.5

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

  7. Simplified0.5

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

  9. Applied add-log-exp0.5

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

  10. Simplified0.5

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

  12. Applied add-cube-cbrt0.5

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

  13. Applied add-cube-cbrt0.5

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

  14. Applied prod-diff0.5

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

  15. Simplified0.5

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

  16. Simplified0.5

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

  17. Final simplification0.5

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

Reproduce

herbie shell --seed 2020025 +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))))))