Diagrams.TwoD.Path.Metafont.Internal:hobbyF from diagrams-contrib-1.3.0.5

Average Error: 0.5 → 0.4

Time: 20.5s

Precision: 64

• Falling back on racket egraph because egg-herbie package not installed

Math

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

↓

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

C

double f(double x, double y) {
        double r252186 = 2.0;
        double r252187 = sqrt(r252186);
        double r252188 = x;
        double r252189 = sin(r252188);
        double r252190 = y;
        double r252191 = sin(r252190);
        double r252192 = 16.0;
        double r252193 = r252191 / r252192;
        double r252194 = r252189 - r252193;
        double r252195 = r252187 * r252194;
        double r252196 = r252189 / r252192;
        double r252197 = r252191 - r252196;
        double r252198 = r252195 * r252197;
        double r252199 = cos(r252188);
        double r252200 = cos(r252190);
        double r252201 = r252199 - r252200;
        double r252202 = r252198 * r252201;
        double r252203 = r252186 + r252202;
        double r252204 = 3.0;
        double r252205 = 1.0;
        double r252206 = 5.0;
        double r252207 = sqrt(r252206);
        double r252208 = r252207 - r252205;
        double r252209 = r252208 / r252186;
        double r252210 = r252209 * r252199;
        double r252211 = r252205 + r252210;
        double r252212 = r252204 - r252207;
        double r252213 = r252212 / r252186;
        double r252214 = r252213 * r252200;
        double r252215 = r252211 + r252214;
        double r252216 = r252204 * r252215;
        double r252217 = r252203 / r252216;
        return r252217;
}

↓

double f(double x, double y) {
        double r252218 = x;
        double r252219 = sin(r252218);
        double r252220 = y;
        double r252221 = sin(r252220);
        double r252222 = 16.0;
        double r252223 = r252221 / r252222;
        double r252224 = r252219 - r252223;
        double r252225 = 2.0;
        double r252226 = sqrt(r252225);
        double r252227 = r252224 * r252226;
        double r252228 = r252219 / r252222;
        double r252229 = r252221 - r252228;
        double r252230 = cos(r252218);
        double r252231 = cos(r252220);
        double r252232 = r252230 - r252231;
        double r252233 = r252229 * r252232;
        double r252234 = fma(r252227, r252233, r252225);
        double r252235 = 3.0;
        double r252236 = 5.0;
        double r252237 = sqrt(r252236);
        double r252238 = r252235 - r252237;
        double r252239 = r252238 / r252225;
        double r252240 = 1.0;
        double r252241 = r252237 - r252240;
        double r252242 = r252241 / r252225;
        double r252243 = fma(r252242, r252230, r252240);
        double r252244 = fma(r252239, r252231, r252243);
        double r252245 = r252234 / r252244;
        double r252246 = r252245 / r252235;
        return r252246;
}

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 add-log-exp 0.5

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

5. Applied add-cube-cbrt 0.5

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

6. Applied add-cube-cbrt 0.5

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

7. Applied times-frac 0.5

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

8. Applied add-sqr-sqrt 31.9

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

9. Applied prod-diff 31.9

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

10. Applied distribute-lft-in 31.9

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

11. Simplified 0.5

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

12. Simplified 0.5

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

13. Simplified 0.4

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

15. Applied associate-/r* 0.4

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

16. Final simplification 0.4

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

Reproduce

herbie shell --seed 2019350 +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))))))