Average Error: 0.5 → 0.5
Time: 32.5s
Precision: 64
\[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)}\]
↓
\[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \sqrt[3]{{\left(\cos x - \cos y\right)}^{3}}}{3 \cdot \left(\left(\left(\frac{\sqrt{\sqrt{5} - 1}}{2} \cdot \cos x\right) \cdot \sqrt{\sqrt{5} - 1} + 1\right) + \frac{\frac{3 \cdot 3 - 5}{3 + \sqrt{5}}}{2} \cdot \cos y\right)}\]
\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 + \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 \sqrt[3]{{\left(\cos x - \cos y\right)}^{3}}}{3 \cdot \left(\left(\left(\frac{\sqrt{\sqrt{5} - 1}}{2} \cdot \cos x\right) \cdot \sqrt{\sqrt{5} - 1} + 1\right) + \frac{\frac{3 \cdot 3 - 5}{3 + \sqrt{5}}}{2} \cdot \cos y\right)}
double f(double x, double y) {
        double r171763 = 2.0;
        double r171764 = sqrt(r171763);
        double r171765 = x;
        double r171766 = sin(r171765);
        double r171767 = y;
        double r171768 = sin(r171767);
        double r171769 = 16.0;
        double r171770 = r171768 / r171769;
        double r171771 = r171766 - r171770;
        double r171772 = r171764 * r171771;
        double r171773 = r171766 / r171769;
        double r171774 = r171768 - r171773;
        double r171775 = r171772 * r171774;
        double r171776 = cos(r171765);
        double r171777 = cos(r171767);
        double r171778 = r171776 - r171777;
        double r171779 = r171775 * r171778;
        double r171780 = r171763 + r171779;
        double r171781 = 3.0;
        double r171782 = 1.0;
        double r171783 = 5.0;
        double r171784 = sqrt(r171783);
        double r171785 = r171784 - r171782;
        double r171786 = r171785 / r171763;
        double r171787 = r171786 * r171776;
        double r171788 = r171782 + r171787;
        double r171789 = r171781 - r171784;
        double r171790 = r171789 / r171763;
        double r171791 = r171790 * r171777;
        double r171792 = r171788 + r171791;
        double r171793 = r171781 * r171792;
        double r171794 = r171780 / r171793;
        return r171794;
}
↓
double f(double x, double y) {
        double r171795 = 2.0;
        double r171796 = sqrt(r171795);
        double r171797 = x;
        double r171798 = sin(r171797);
        double r171799 = y;
        double r171800 = sin(r171799);
        double r171801 = 16.0;
        double r171802 = r171800 / r171801;
        double r171803 = r171798 - r171802;
        double r171804 = r171796 * r171803;
        double r171805 = r171798 / r171801;
        double r171806 = r171800 - r171805;
        double r171807 = r171804 * r171806;
        double r171808 = cos(r171797);
        double r171809 = cos(r171799);
        double r171810 = r171808 - r171809;
        double r171811 = 3.0;
        double r171812 = pow(r171810, r171811);
        double r171813 = cbrt(r171812);
        double r171814 = r171807 * r171813;
        double r171815 = r171795 + r171814;
        double r171816 = 3.0;
        double r171817 = 5.0;
        double r171818 = sqrt(r171817);
        double r171819 = 1.0;
        double r171820 = r171818 - r171819;
        double r171821 = sqrt(r171820);
        double r171822 = r171821 / r171795;
        double r171823 = r171822 * r171808;
        double r171824 = r171823 * r171821;
        double r171825 = r171824 + r171819;
        double r171826 = r171816 * r171816;
        double r171827 = r171826 - r171817;
        double r171828 = r171816 + r171818;
        double r171829 = r171827 / r171828;
        double r171830 = r171829 / r171795;
        double r171831 = r171830 * r171809;
        double r171832 = r171825 + r171831;
        double r171833 = r171816 * r171832;
        double r171834 = r171815 / r171833;
        return r171834;
}

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 *-un-lft-identity0.5

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

  4. Applied add-sqr-sqrt0.5

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

  5. Applied times-frac0.5

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

  6. Applied associate-*l*0.5

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

  8. Applied add-cbrt-cube0.5

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

  9. Simplified0.5

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

  11. Applied flip--0.5

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

  12. Simplified0.5

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

  13. Final simplification0.5

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

Reproduce

herbie shell --seed 2019235 
(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))))))