Average Error: 0.5 → 0.5
Time: 38.2s
Precision: 64
\[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)}\]
↓
\[\frac{\frac{\mathsf{fma}\left(\sqrt{2} \cdot \mathsf{fma}\left(\sqrt[3]{\sin x} \cdot \sqrt[3]{\sin x}, \sqrt[3]{\sin x}, -\frac{{\left(\sqrt[3]{\sin y}\right)}^{3}}{16}\right) + \sqrt{2} \cdot \left(\left(-\frac{{\left(\sqrt[3]{\sin y}\right)}^{3}}{16}\right) + \frac{{\left(\sqrt[3]{\sin y}\right)}^{3}}{16}\right), \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right), 2\right)}{3}}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right)}\]
\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(\sqrt{2} \cdot \mathsf{fma}\left(\sqrt[3]{\sin x} \cdot \sqrt[3]{\sin x}, \sqrt[3]{\sin x}, -\frac{{\left(\sqrt[3]{\sin y}\right)}^{3}}{16}\right) + \sqrt{2} \cdot \left(\left(-\frac{{\left(\sqrt[3]{\sin y}\right)}^{3}}{16}\right) + \frac{{\left(\sqrt[3]{\sin y}\right)}^{3}}{16}\right), \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right), 2\right)}{3}}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right)}
double f(double x, double y) {
        double r159179 = 2.0;
        double r159180 = sqrt(r159179);
        double r159181 = x;
        double r159182 = sin(r159181);
        double r159183 = y;
        double r159184 = sin(r159183);
        double r159185 = 16.0;
        double r159186 = r159184 / r159185;
        double r159187 = r159182 - r159186;
        double r159188 = r159180 * r159187;
        double r159189 = r159182 / r159185;
        double r159190 = r159184 - r159189;
        double r159191 = r159188 * r159190;
        double r159192 = cos(r159181);
        double r159193 = cos(r159183);
        double r159194 = r159192 - r159193;
        double r159195 = r159191 * r159194;
        double r159196 = r159179 + r159195;
        double r159197 = 3.0;
        double r159198 = 1.0;
        double r159199 = 5.0;
        double r159200 = sqrt(r159199);
        double r159201 = r159200 - r159198;
        double r159202 = r159201 / r159179;
        double r159203 = r159202 * r159192;
        double r159204 = r159198 + r159203;
        double r159205 = r159197 - r159200;
        double r159206 = r159205 / r159179;
        double r159207 = r159206 * r159193;
        double r159208 = r159204 + r159207;
        double r159209 = r159197 * r159208;
        double r159210 = r159196 / r159209;
        return r159210;
}
↓
double f(double x, double y) {
        double r159211 = 2.0;
        double r159212 = sqrt(r159211);
        double r159213 = x;
        double r159214 = sin(r159213);
        double r159215 = cbrt(r159214);
        double r159216 = r159215 * r159215;
        double r159217 = y;
        double r159218 = sin(r159217);
        double r159219 = cbrt(r159218);
        double r159220 = 3.0;
        double r159221 = pow(r159219, r159220);
        double r159222 = 16.0;
        double r159223 = r159221 / r159222;
        double r159224 = -r159223;
        double r159225 = fma(r159216, r159215, r159224);
        double r159226 = r159212 * r159225;
        double r159227 = r159224 + r159223;
        double r159228 = r159212 * r159227;
        double r159229 = r159226 + r159228;
        double r159230 = r159214 / r159222;
        double r159231 = r159218 - r159230;
        double r159232 = cos(r159213);
        double r159233 = cos(r159217);
        double r159234 = r159232 - r159233;
        double r159235 = r159231 * r159234;
        double r159236 = fma(r159229, r159235, r159211);
        double r159237 = 3.0;
        double r159238 = r159236 / r159237;
        double r159239 = 5.0;
        double r159240 = sqrt(r159239);
        double r159241 = r159237 - r159240;
        double r159242 = r159241 / r159211;
        double r159243 = 1.0;
        double r159244 = r159240 - r159243;
        double r159245 = r159244 / r159211;
        double r159246 = fma(r159232, r159245, r159243);
        double r159247 = fma(r159233, r159242, r159246);
        double r159248 = r159238 / r159247;
        return r159248;
}

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. Simplified0.4

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

  4. Applied *-un-lft-identity0.4

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

  5. Applied add-cube-cbrt0.4

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

  6. Applied times-frac0.4

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

  7. Applied add-cube-cbrt0.5

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

  8. Applied prod-diff0.5

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

  9. Applied distribute-lft-in0.5

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

  10. Simplified0.5

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

  11. Simplified0.5

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

  12. Final simplification0.5

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

Reproduce

herbie shell --seed 2019323 +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))))))