Average Error: 0.5 → 0.5
Time: 17.1s
Precision: 64
\[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)}\]
↓
\[\frac{2 + \left(\left(\mathsf{fma}\left(\sqrt[3]{\sin y} \cdot \sqrt[3]{\sin y}, \sqrt[3]{\sin y}, \frac{\sqrt[3]{\sin x}}{16} \cdot \left(-\sqrt[3]{\sin x} \cdot \sqrt[3]{\sin x}\right)\right) \cdot \sqrt{2}\right) \cdot \left(\sin x - \frac{\sin y}{16}\right) + \left(\left(\left(\sqrt[3]{\sin x} \cdot \sqrt[3]{\sin x}\right) \cdot \left(\left(-\frac{\sqrt[3]{\sin x}}{16}\right) + \frac{\sqrt[3]{\sin x}}{16}\right)\right) \cdot \sqrt{2}\right) \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)}\]
\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 \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(\mathsf{fma}\left(\sqrt[3]{\sin y} \cdot \sqrt[3]{\sin y}, \sqrt[3]{\sin y}, \frac{\sqrt[3]{\sin x}}{16} \cdot \left(-\sqrt[3]{\sin x} \cdot \sqrt[3]{\sin x}\right)\right) \cdot \sqrt{2}\right) \cdot \left(\sin x - \frac{\sin y}{16}\right) + \left(\left(\left(\sqrt[3]{\sin x} \cdot \sqrt[3]{\sin x}\right) \cdot \left(\left(-\frac{\sqrt[3]{\sin x}}{16}\right) + \frac{\sqrt[3]{\sin x}}{16}\right)\right) \cdot \sqrt{2}\right) \cdot \left(\sin x - \frac{\sin y}{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)}
double f(double x, double y) {
        double r220977 = 2.0;
        double r220978 = sqrt(r220977);
        double r220979 = x;
        double r220980 = sin(r220979);
        double r220981 = y;
        double r220982 = sin(r220981);
        double r220983 = 16.0;
        double r220984 = r220982 / r220983;
        double r220985 = r220980 - r220984;
        double r220986 = r220978 * r220985;
        double r220987 = r220980 / r220983;
        double r220988 = r220982 - r220987;
        double r220989 = r220986 * r220988;
        double r220990 = cos(r220979);
        double r220991 = cos(r220981);
        double r220992 = r220990 - r220991;
        double r220993 = r220989 * r220992;
        double r220994 = r220977 + r220993;
        double r220995 = 3.0;
        double r220996 = 1.0;
        double r220997 = 5.0;
        double r220998 = sqrt(r220997);
        double r220999 = r220998 - r220996;
        double r221000 = r220999 / r220977;
        double r221001 = r221000 * r220990;
        double r221002 = r220996 + r221001;
        double r221003 = r220995 - r220998;
        double r221004 = r221003 / r220977;
        double r221005 = r221004 * r220991;
        double r221006 = r221002 + r221005;
        double r221007 = r220995 * r221006;
        double r221008 = r220994 / r221007;
        return r221008;
}
↓
double f(double x, double y) {
        double r221009 = 2.0;
        double r221010 = y;
        double r221011 = sin(r221010);
        double r221012 = cbrt(r221011);
        double r221013 = r221012 * r221012;
        double r221014 = x;
        double r221015 = sin(r221014);
        double r221016 = cbrt(r221015);
        double r221017 = 16.0;
        double r221018 = r221016 / r221017;
        double r221019 = r221016 * r221016;
        double r221020 = -r221019;
        double r221021 = r221018 * r221020;
        double r221022 = fma(r221013, r221012, r221021);
        double r221023 = sqrt(r221009);
        double r221024 = r221022 * r221023;
        double r221025 = r221011 / r221017;
        double r221026 = r221015 - r221025;
        double r221027 = r221024 * r221026;
        double r221028 = -r221018;
        double r221029 = r221028 + r221018;
        double r221030 = r221019 * r221029;
        double r221031 = r221030 * r221023;
        double r221032 = r221031 * r221026;
        double r221033 = r221027 + r221032;
        double r221034 = cos(r221014);
        double r221035 = cos(r221010);
        double r221036 = r221034 - r221035;
        double r221037 = r221033 * r221036;
        double r221038 = r221009 + r221037;
        double r221039 = 3.0;
        double r221040 = 1.0;
        double r221041 = 5.0;
        double r221042 = sqrt(r221041);
        double r221043 = r221042 - r221040;
        double r221044 = r221043 / r221009;
        double r221045 = r221044 * r221034;
        double r221046 = r221040 + r221045;
        double r221047 = r221039 - r221042;
        double r221048 = r221047 / r221009;
        double r221049 = r221048 * r221035;
        double r221050 = r221046 + r221049;
        double r221051 = r221039 * r221050;
        double r221052 = r221038 / r221051;
        return r221052;
}

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. Using strategy rm

  3. Applied *-un-lft-identity0.5

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

  4. Applied add-cube-cbrt0.5

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

  5. Applied times-frac0.5

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

  6. Applied add-cube-cbrt0.5

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

  7. Applied prod-diff0.5

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

  8. Applied distribute-lft-in0.5

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

  9. Simplified0.5

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

  10. Simplified0.5

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

  11. Final simplification0.5

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

Reproduce

herbie shell --seed 2020035 +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))))))