Average Error: 0.5 → 0.5
Time: 36.9s
Precision: 64
\[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)}\]
↓
\[\frac{\frac{\mathsf{fma}\left(\left(\sqrt[3]{\left(\sin y - \frac{\sin x}{16}\right) \cdot \sqrt{2}} \cdot \left(\sqrt[3]{\left(\sin y - \frac{\sin x}{16}\right) \cdot \sqrt{2}} \cdot \left(\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sin y - \frac{\sin x}{16}}\right)\right)\right) \cdot \mathsf{fma}\left(\sqrt[3]{\cos x} \cdot \sqrt[3]{\cos x}, \sqrt[3]{\cos x}, -\cos y\right), \sin x - \frac{\sin y}{16}, 2\right)}{3}}{\mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, \mathsf{fma}\left(\frac{\cos y}{2}, 3 - \sqrt{5}, 1\right)\right)}\]
\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{\frac{\mathsf{fma}\left(\left(\sqrt[3]{\left(\sin y - \frac{\sin x}{16}\right) \cdot \sqrt{2}} \cdot \left(\sqrt[3]{\left(\sin y - \frac{\sin x}{16}\right) \cdot \sqrt{2}} \cdot \left(\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sin y - \frac{\sin x}{16}}\right)\right)\right) \cdot \mathsf{fma}\left(\sqrt[3]{\cos x} \cdot \sqrt[3]{\cos x}, \sqrt[3]{\cos x}, -\cos y\right), \sin x - \frac{\sin y}{16}, 2\right)}{3}}{\mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, \mathsf{fma}\left(\frac{\cos y}{2}, 3 - \sqrt{5}, 1\right)\right)}
double f(double x, double y) {
        double r6676788 = 2.0;
        double r6676789 = sqrt(r6676788);
        double r6676790 = x;
        double r6676791 = sin(r6676790);
        double r6676792 = y;
        double r6676793 = sin(r6676792);
        double r6676794 = 16.0;
        double r6676795 = r6676793 / r6676794;
        double r6676796 = r6676791 - r6676795;
        double r6676797 = r6676789 * r6676796;
        double r6676798 = r6676791 / r6676794;
        double r6676799 = r6676793 - r6676798;
        double r6676800 = r6676797 * r6676799;
        double r6676801 = cos(r6676790);
        double r6676802 = cos(r6676792);
        double r6676803 = r6676801 - r6676802;
        double r6676804 = r6676800 * r6676803;
        double r6676805 = r6676788 + r6676804;
        double r6676806 = 3.0;
        double r6676807 = 1.0;
        double r6676808 = 5.0;
        double r6676809 = sqrt(r6676808);
        double r6676810 = r6676809 - r6676807;
        double r6676811 = r6676810 / r6676788;
        double r6676812 = r6676811 * r6676801;
        double r6676813 = r6676807 + r6676812;
        double r6676814 = r6676806 - r6676809;
        double r6676815 = r6676814 / r6676788;
        double r6676816 = r6676815 * r6676802;
        double r6676817 = r6676813 + r6676816;
        double r6676818 = r6676806 * r6676817;
        double r6676819 = r6676805 / r6676818;
        return r6676819;
}
↓
double f(double x, double y) {
        double r6676820 = y;
        double r6676821 = sin(r6676820);
        double r6676822 = x;
        double r6676823 = sin(r6676822);
        double r6676824 = 16.0;
        double r6676825 = r6676823 / r6676824;
        double r6676826 = r6676821 - r6676825;
        double r6676827 = 2.0;
        double r6676828 = sqrt(r6676827);
        double r6676829 = r6676826 * r6676828;
        double r6676830 = cbrt(r6676829);
        double r6676831 = cbrt(r6676828);
        double r6676832 = cbrt(r6676826);
        double r6676833 = r6676831 * r6676832;
        double r6676834 = r6676830 * r6676833;
        double r6676835 = r6676830 * r6676834;
        double r6676836 = cos(r6676822);
        double r6676837 = cbrt(r6676836);
        double r6676838 = r6676837 * r6676837;
        double r6676839 = cos(r6676820);
        double r6676840 = -r6676839;
        double r6676841 = fma(r6676838, r6676837, r6676840);
        double r6676842 = r6676835 * r6676841;
        double r6676843 = r6676821 / r6676824;
        double r6676844 = r6676823 - r6676843;
        double r6676845 = fma(r6676842, r6676844, r6676827);
        double r6676846 = 3.0;
        double r6676847 = r6676845 / r6676846;
        double r6676848 = 5.0;
        double r6676849 = sqrt(r6676848);
        double r6676850 = 1.0;
        double r6676851 = r6676849 - r6676850;
        double r6676852 = r6676851 / r6676827;
        double r6676853 = r6676839 / r6676827;
        double r6676854 = r6676846 - r6676849;
        double r6676855 = fma(r6676853, r6676854, r6676850);
        double r6676856 = fma(r6676836, r6676852, r6676855);
        double r6676857 = r6676847 / r6676856;
        return r6676857;
}

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

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

  4. Applied add-cube-cbrt0.5

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

  6. Applied add-cube-cbrt0.5

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

  7. Applied fma-neg0.5

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

  9. Applied cbrt-prod0.5

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

  10. Final simplification0.5

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

Reproduce

herbie shell --seed 2019171 +o rules:numerics
(FPCore (x y)
  :name "Diagrams.TwoD.Path.Metafont.Internal:hobbyF from diagrams-contrib-1.3.0.5"
  (/ (+ 2.0 (* (* (* (sqrt 2.0) (- (sin x) (/ (sin y) 16.0))) (- (sin y) (/ (sin x) 16.0))) (- (cos x) (cos y)))) (* 3.0 (+ (+ 1.0 (* (/ (- (sqrt 5.0) 1.0) 2.0) (cos x))) (* (/ (- 3.0 (sqrt 5.0)) 2.0) (cos y))))))