Average Error: 0.5 → 0.5
Time: 11.0s
Precision: 64
\[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)}\]
↓
\[\frac{2 + \left(\frac{\sin y - \frac{\sin x}{16}}{2} \cdot \left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) + \log \left(\sqrt{e^{\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)}}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)}\]
\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 \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(\frac{\sin y - \frac{\sin x}{16}}{2} \cdot \left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) + \log \left(\sqrt{e^{\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)}}\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)}
double f(double x, double y) {
        double r210775 = 2.0;
        double r210776 = sqrt(r210775);
        double r210777 = x;
        double r210778 = sin(r210777);
        double r210779 = y;
        double r210780 = sin(r210779);
        double r210781 = 16.0;
        double r210782 = r210780 / r210781;
        double r210783 = r210778 - r210782;
        double r210784 = r210776 * r210783;
        double r210785 = r210778 / r210781;
        double r210786 = r210780 - r210785;
        double r210787 = r210784 * r210786;
        double r210788 = cos(r210777);
        double r210789 = cos(r210779);
        double r210790 = r210788 - r210789;
        double r210791 = r210787 * r210790;
        double r210792 = r210775 + r210791;
        double r210793 = 3.0;
        double r210794 = 1.0;
        double r210795 = 5.0;
        double r210796 = sqrt(r210795);
        double r210797 = r210796 - r210794;
        double r210798 = r210797 / r210775;
        double r210799 = r210798 * r210788;
        double r210800 = r210794 + r210799;
        double r210801 = r210793 - r210796;
        double r210802 = r210801 / r210775;
        double r210803 = r210802 * r210789;
        double r210804 = r210800 + r210803;
        double r210805 = r210793 * r210804;
        double r210806 = r210792 / r210805;
        return r210806;
}
↓
double f(double x, double y) {
        double r210807 = 2.0;
        double r210808 = y;
        double r210809 = sin(r210808);
        double r210810 = x;
        double r210811 = sin(r210810);
        double r210812 = 16.0;
        double r210813 = r210811 / r210812;
        double r210814 = r210809 - r210813;
        double r210815 = 2.0;
        double r210816 = r210814 / r210815;
        double r210817 = sqrt(r210807);
        double r210818 = r210809 / r210812;
        double r210819 = r210811 - r210818;
        double r210820 = r210817 * r210819;
        double r210821 = r210816 * r210820;
        double r210822 = r210820 * r210814;
        double r210823 = exp(r210822);
        double r210824 = sqrt(r210823);
        double r210825 = log(r210824);
        double r210826 = r210821 + r210825;
        double r210827 = cos(r210810);
        double r210828 = cos(r210808);
        double r210829 = r210827 - r210828;
        double r210830 = r210826 * r210829;
        double r210831 = r210807 + r210830;
        double r210832 = 3.0;
        double r210833 = 1.0;
        double r210834 = 5.0;
        double r210835 = sqrt(r210834);
        double r210836 = r210835 - r210833;
        double r210837 = r210836 / r210807;
        double r210838 = r210837 * r210827;
        double r210839 = r210833 + r210838;
        double r210840 = r210832 - r210835;
        double r210841 = r210840 / r210807;
        double r210842 = r210841 * r210828;
        double r210843 = r210839 + r210842;
        double r210844 = r210832 * r210843;
        double r210845 = r210831 / r210844;
        return r210845;
}

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 add-log-exp0.5

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

  5. Applied add-sqr-sqrt0.5

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

  6. Applied log-prod0.5

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

  8. Applied add-log-exp0.5

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

  9. Applied exp-to-pow0.5

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

  10. Applied sqrt-pow10.5

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

  11. Applied log-pow0.5

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

  12. Simplified0.5

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

  13. Final simplification0.5

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

Reproduce

herbie shell --seed 2020027 
(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))))))