Average Error: 0.5 → 0.5
Time: 15.4s
Precision: 64
\[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)}\]
↓
\[\frac{2 + \left(\frac{\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)}{\sqrt[3]{\sin y + \frac{\sin x}{16}} \cdot \sqrt[3]{\sin y + \frac{\sin x}{16}}} \cdot \left(\frac{1}{\sqrt[3]{\sqrt[3]{\sin y + \frac{\sin x}{16}}} \cdot \sqrt[3]{\sqrt[3]{\sin y + \frac{\sin x}{16}}}} \cdot \frac{\sin y \cdot \sin y - \frac{\sin x}{16} \cdot \frac{\sin x}{16}}{\sqrt[3]{\sqrt[3]{\sin y + \frac{\sin x}{16}}}}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\frac{3 \cdot 3 + \left(-5\right)}{3 + \sqrt{5}}}{2} \cdot \cos y\right)}\]
\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{\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)}{\sqrt[3]{\sin y + \frac{\sin x}{16}} \cdot \sqrt[3]{\sin y + \frac{\sin x}{16}}} \cdot \left(\frac{1}{\sqrt[3]{\sqrt[3]{\sin y + \frac{\sin x}{16}}} \cdot \sqrt[3]{\sqrt[3]{\sin y + \frac{\sin x}{16}}}} \cdot \frac{\sin y \cdot \sin y - \frac{\sin x}{16} \cdot \frac{\sin x}{16}}{\sqrt[3]{\sqrt[3]{\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{\frac{3 \cdot 3 + \left(-5\right)}{3 + \sqrt{5}}}{2} \cdot \cos y\right)}
double f(double x, double y) {
        double r240203 = 2.0;
        double r240204 = sqrt(r240203);
        double r240205 = x;
        double r240206 = sin(r240205);
        double r240207 = y;
        double r240208 = sin(r240207);
        double r240209 = 16.0;
        double r240210 = r240208 / r240209;
        double r240211 = r240206 - r240210;
        double r240212 = r240204 * r240211;
        double r240213 = r240206 / r240209;
        double r240214 = r240208 - r240213;
        double r240215 = r240212 * r240214;
        double r240216 = cos(r240205);
        double r240217 = cos(r240207);
        double r240218 = r240216 - r240217;
        double r240219 = r240215 * r240218;
        double r240220 = r240203 + r240219;
        double r240221 = 3.0;
        double r240222 = 1.0;
        double r240223 = 5.0;
        double r240224 = sqrt(r240223);
        double r240225 = r240224 - r240222;
        double r240226 = r240225 / r240203;
        double r240227 = r240226 * r240216;
        double r240228 = r240222 + r240227;
        double r240229 = r240221 - r240224;
        double r240230 = r240229 / r240203;
        double r240231 = r240230 * r240217;
        double r240232 = r240228 + r240231;
        double r240233 = r240221 * r240232;
        double r240234 = r240220 / r240233;
        return r240234;
}
↓
double f(double x, double y) {
        double r240235 = 2.0;
        double r240236 = sqrt(r240235);
        double r240237 = x;
        double r240238 = sin(r240237);
        double r240239 = y;
        double r240240 = sin(r240239);
        double r240241 = 16.0;
        double r240242 = r240240 / r240241;
        double r240243 = r240238 - r240242;
        double r240244 = r240236 * r240243;
        double r240245 = r240238 / r240241;
        double r240246 = r240240 + r240245;
        double r240247 = cbrt(r240246);
        double r240248 = r240247 * r240247;
        double r240249 = r240244 / r240248;
        double r240250 = 1.0;
        double r240251 = cbrt(r240247);
        double r240252 = r240251 * r240251;
        double r240253 = r240250 / r240252;
        double r240254 = r240240 * r240240;
        double r240255 = r240245 * r240245;
        double r240256 = r240254 - r240255;
        double r240257 = r240256 / r240251;
        double r240258 = r240253 * r240257;
        double r240259 = r240249 * r240258;
        double r240260 = cos(r240237);
        double r240261 = cos(r240239);
        double r240262 = r240260 - r240261;
        double r240263 = r240259 * r240262;
        double r240264 = r240235 + r240263;
        double r240265 = 3.0;
        double r240266 = 1.0;
        double r240267 = 5.0;
        double r240268 = sqrt(r240267);
        double r240269 = r240268 - r240266;
        double r240270 = r240269 / r240235;
        double r240271 = r240270 * r240260;
        double r240272 = r240266 + r240271;
        double r240273 = r240265 * r240265;
        double r240274 = -r240267;
        double r240275 = r240273 + r240274;
        double r240276 = r240265 + r240268;
        double r240277 = r240275 / r240276;
        double r240278 = r240277 / r240235;
        double r240279 = r240278 * r240261;
        double r240280 = r240272 + r240279;
        double r240281 = r240265 * r240280;
        double r240282 = r240264 / r240281;
        return r240282;
}

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 flip--0.5

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

  4. Applied associate-*r/0.5

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

  6. Applied add-cube-cbrt0.5

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

  7. Applied times-frac0.5

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

  9. Applied add-cube-cbrt0.5

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

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

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

  11. Applied times-frac0.5

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

  13. Applied flip--0.6

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

  14. Simplified0.5

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

  15. Final simplification0.5

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

Reproduce

herbie shell --seed 2020035 
(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))))))