Average Error: 0.5 → 0.4
Time: 32.6s
Precision: 64
\[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)}\]
↓
\[\frac{\frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\left(\sqrt[3]{\sin x} \cdot \sqrt[3]{\sin x}\right) \cdot \sqrt[3]{\sin x} - {\left(\frac{\sqrt[3]{\sin y}}{\sqrt[3]{16}}\right)}^{3}\right) + \left({\left(\frac{\sqrt[3]{\sin y}}{\sqrt[3]{16}}\right)}^{3} + \left(-{\left(\frac{\sqrt[3]{\sin y}}{\sqrt[3]{16}}\right)}^{3}\right)\right) \cdot \sqrt{2}, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right), 2\right)}{3}}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right)}\]
\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 \left(\left(\sqrt[3]{\sin x} \cdot \sqrt[3]{\sin x}\right) \cdot \sqrt[3]{\sin x} - {\left(\frac{\sqrt[3]{\sin y}}{\sqrt[3]{16}}\right)}^{3}\right) + \left({\left(\frac{\sqrt[3]{\sin y}}{\sqrt[3]{16}}\right)}^{3} + \left(-{\left(\frac{\sqrt[3]{\sin y}}{\sqrt[3]{16}}\right)}^{3}\right)\right) \cdot \sqrt{2}, \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 r161164 = 2.0;
        double r161165 = sqrt(r161164);
        double r161166 = x;
        double r161167 = sin(r161166);
        double r161168 = y;
        double r161169 = sin(r161168);
        double r161170 = 16.0;
        double r161171 = r161169 / r161170;
        double r161172 = r161167 - r161171;
        double r161173 = r161165 * r161172;
        double r161174 = r161167 / r161170;
        double r161175 = r161169 - r161174;
        double r161176 = r161173 * r161175;
        double r161177 = cos(r161166);
        double r161178 = cos(r161168);
        double r161179 = r161177 - r161178;
        double r161180 = r161176 * r161179;
        double r161181 = r161164 + r161180;
        double r161182 = 3.0;
        double r161183 = 1.0;
        double r161184 = 5.0;
        double r161185 = sqrt(r161184);
        double r161186 = r161185 - r161183;
        double r161187 = r161186 / r161164;
        double r161188 = r161187 * r161177;
        double r161189 = r161183 + r161188;
        double r161190 = r161182 - r161185;
        double r161191 = r161190 / r161164;
        double r161192 = r161191 * r161178;
        double r161193 = r161189 + r161192;
        double r161194 = r161182 * r161193;
        double r161195 = r161181 / r161194;
        return r161195;
}
↓
double f(double x, double y) {
        double r161196 = 2.0;
        double r161197 = sqrt(r161196);
        double r161198 = x;
        double r161199 = sin(r161198);
        double r161200 = cbrt(r161199);
        double r161201 = r161200 * r161200;
        double r161202 = r161201 * r161200;
        double r161203 = y;
        double r161204 = sin(r161203);
        double r161205 = cbrt(r161204);
        double r161206 = 16.0;
        double r161207 = cbrt(r161206);
        double r161208 = r161205 / r161207;
        double r161209 = 3.0;
        double r161210 = pow(r161208, r161209);
        double r161211 = r161202 - r161210;
        double r161212 = r161197 * r161211;
        double r161213 = -r161210;
        double r161214 = r161210 + r161213;
        double r161215 = r161214 * r161197;
        double r161216 = r161212 + r161215;
        double r161217 = r161199 / r161206;
        double r161218 = r161204 - r161217;
        double r161219 = cos(r161198);
        double r161220 = cos(r161203);
        double r161221 = r161219 - r161220;
        double r161222 = r161218 * r161221;
        double r161223 = fma(r161216, r161222, r161196);
        double r161224 = 3.0;
        double r161225 = r161223 / r161224;
        double r161226 = 5.0;
        double r161227 = sqrt(r161226);
        double r161228 = r161224 - r161227;
        double r161229 = r161228 / r161196;
        double r161230 = 1.0;
        double r161231 = r161227 - r161230;
        double r161232 = r161231 / r161196;
        double r161233 = fma(r161219, r161232, r161230);
        double r161234 = fma(r161220, r161229, r161233);
        double r161235 = r161225 / r161234;
        return r161235;
}

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 add-cube-cbrt0.4

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

  5. Applied add-cube-cbrt0.4

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

  6. Applied times-frac0.4

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

  7. Applied add-cube-cbrt0.5

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

  8. Applied prod-diff0.5

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

  9. Applied distribute-lft-in0.5

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

  10. Simplified0.4

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

  11. Simplified0.4

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

  12. Final simplification0.4

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

Reproduce

herbie shell --seed 2019347 +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))))))