Average Error: 0.5 → 0.5
Time: 19.3s
Precision: 64
\[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)}\]
↓
\[\frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(1, \cos x, -{\left(\sqrt[3]{\cos y}\right)}^{3}\right) + \left(\left(-{\left(\sqrt[3]{\cos y}\right)}^{3}\right) + {\left(\sqrt[3]{\cos y}\right)}^{3}\right), \sqrt[3]{{\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right)}^{3}} \cdot \left(\sin y - \frac{\sin x}{16}\right), 2\right)}{3}}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right)}\]
\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(\mathsf{fma}\left(1, \cos x, -{\left(\sqrt[3]{\cos y}\right)}^{3}\right) + \left(\left(-{\left(\sqrt[3]{\cos y}\right)}^{3}\right) + {\left(\sqrt[3]{\cos y}\right)}^{3}\right), \sqrt[3]{{\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right)}^{3}} \cdot \left(\sin y - \frac{\sin x}{16}\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 r214098 = 2.0;
        double r214099 = sqrt(r214098);
        double r214100 = x;
        double r214101 = sin(r214100);
        double r214102 = y;
        double r214103 = sin(r214102);
        double r214104 = 16.0;
        double r214105 = r214103 / r214104;
        double r214106 = r214101 - r214105;
        double r214107 = r214099 * r214106;
        double r214108 = r214101 / r214104;
        double r214109 = r214103 - r214108;
        double r214110 = r214107 * r214109;
        double r214111 = cos(r214100);
        double r214112 = cos(r214102);
        double r214113 = r214111 - r214112;
        double r214114 = r214110 * r214113;
        double r214115 = r214098 + r214114;
        double r214116 = 3.0;
        double r214117 = 1.0;
        double r214118 = 5.0;
        double r214119 = sqrt(r214118);
        double r214120 = r214119 - r214117;
        double r214121 = r214120 / r214098;
        double r214122 = r214121 * r214111;
        double r214123 = r214117 + r214122;
        double r214124 = r214116 - r214119;
        double r214125 = r214124 / r214098;
        double r214126 = r214125 * r214112;
        double r214127 = r214123 + r214126;
        double r214128 = r214116 * r214127;
        double r214129 = r214115 / r214128;
        return r214129;
}
↓
double f(double x, double y) {
        double r214130 = 1.0;
        double r214131 = x;
        double r214132 = cos(r214131);
        double r214133 = y;
        double r214134 = cos(r214133);
        double r214135 = cbrt(r214134);
        double r214136 = 3.0;
        double r214137 = pow(r214135, r214136);
        double r214138 = -r214137;
        double r214139 = fma(r214130, r214132, r214138);
        double r214140 = r214138 + r214137;
        double r214141 = r214139 + r214140;
        double r214142 = 2.0;
        double r214143 = sqrt(r214142);
        double r214144 = sin(r214131);
        double r214145 = sin(r214133);
        double r214146 = 16.0;
        double r214147 = r214145 / r214146;
        double r214148 = r214144 - r214147;
        double r214149 = r214143 * r214148;
        double r214150 = pow(r214149, r214136);
        double r214151 = cbrt(r214150);
        double r214152 = r214144 / r214146;
        double r214153 = r214145 - r214152;
        double r214154 = r214151 * r214153;
        double r214155 = fma(r214141, r214154, r214142);
        double r214156 = 3.0;
        double r214157 = r214155 / r214156;
        double r214158 = 5.0;
        double r214159 = sqrt(r214158);
        double r214160 = r214156 - r214159;
        double r214161 = r214160 / r214142;
        double r214162 = 1.0;
        double r214163 = r214159 - r214162;
        double r214164 = r214163 / r214142;
        double r214165 = fma(r214132, r214164, r214162);
        double r214166 = fma(r214134, r214161, r214165);
        double r214167 = r214157 / r214166;
        return r214167;
}

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

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

  4. Applied add-cube-cbrt0.5

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

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

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

  6. Applied prod-diff0.5

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

  7. Simplified0.5

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

  8. Simplified0.5

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

  10. Applied add-cbrt-cube0.5

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

  11. Applied add-cbrt-cube0.5

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

  12. Applied cbrt-unprod0.5

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

  13. Simplified0.5

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

  14. Final simplification0.5

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

Reproduce

herbie shell --seed 2020043 +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))))))