Average Error: 0.5 → 0.5
Time: 11.1s
Precision: 64
\[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)}\]
↓
\[\frac{{2}^{3} + {\left(\left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)\right)}^{3}}{\left(\left(\left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \sqrt[3]{{\left(\cos x - \cos y\right)}^{3}}\right) \cdot \left(\left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right) - 2\right) + 2 \cdot 2\right) \cdot \left(3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)\right)}\]
\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{{2}^{3} + {\left(\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)\right)}^{3}}{\left(\left(\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 \sqrt[3]{{\left(\cos x - \cos y\right)}^{3}}\right) \cdot \left(\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) - 2\right) + 2 \cdot 2\right) \cdot \left(3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)\right)}
double f(double x, double y) {
        double r251092 = 2.0;
        double r251093 = sqrt(r251092);
        double r251094 = x;
        double r251095 = sin(r251094);
        double r251096 = y;
        double r251097 = sin(r251096);
        double r251098 = 16.0;
        double r251099 = r251097 / r251098;
        double r251100 = r251095 - r251099;
        double r251101 = r251093 * r251100;
        double r251102 = r251095 / r251098;
        double r251103 = r251097 - r251102;
        double r251104 = r251101 * r251103;
        double r251105 = cos(r251094);
        double r251106 = cos(r251096);
        double r251107 = r251105 - r251106;
        double r251108 = r251104 * r251107;
        double r251109 = r251092 + r251108;
        double r251110 = 3.0;
        double r251111 = 1.0;
        double r251112 = 5.0;
        double r251113 = sqrt(r251112);
        double r251114 = r251113 - r251111;
        double r251115 = r251114 / r251092;
        double r251116 = r251115 * r251105;
        double r251117 = r251111 + r251116;
        double r251118 = r251110 - r251113;
        double r251119 = r251118 / r251092;
        double r251120 = r251119 * r251106;
        double r251121 = r251117 + r251120;
        double r251122 = r251110 * r251121;
        double r251123 = r251109 / r251122;
        return r251123;
}
↓
double f(double x, double y) {
        double r251124 = 2.0;
        double r251125 = 3.0;
        double r251126 = pow(r251124, r251125);
        double r251127 = sqrt(r251124);
        double r251128 = x;
        double r251129 = sin(r251128);
        double r251130 = y;
        double r251131 = sin(r251130);
        double r251132 = 16.0;
        double r251133 = r251131 / r251132;
        double r251134 = r251129 - r251133;
        double r251135 = r251127 * r251134;
        double r251136 = r251129 / r251132;
        double r251137 = r251131 - r251136;
        double r251138 = r251135 * r251137;
        double r251139 = cos(r251128);
        double r251140 = cos(r251130);
        double r251141 = r251139 - r251140;
        double r251142 = r251138 * r251141;
        double r251143 = pow(r251142, r251125);
        double r251144 = r251126 + r251143;
        double r251145 = pow(r251141, r251125);
        double r251146 = cbrt(r251145);
        double r251147 = r251138 * r251146;
        double r251148 = r251142 - r251124;
        double r251149 = r251147 * r251148;
        double r251150 = r251124 * r251124;
        double r251151 = r251149 + r251150;
        double r251152 = 3.0;
        double r251153 = 1.0;
        double r251154 = 5.0;
        double r251155 = sqrt(r251154);
        double r251156 = r251155 - r251153;
        double r251157 = r251156 / r251124;
        double r251158 = r251157 * r251139;
        double r251159 = r251153 + r251158;
        double r251160 = r251152 - r251155;
        double r251161 = r251160 / r251124;
        double r251162 = r251161 * r251140;
        double r251163 = r251159 + r251162;
        double r251164 = r251152 * r251163;
        double r251165 = r251151 * r251164;
        double r251166 = r251144 / r251165;
        return r251166;
}

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 flip3-+0.5

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

  4. Applied associate-/l/0.5

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

  5. Simplified0.5

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

  7. Applied add-cbrt-cube0.5

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

  8. Simplified0.5

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

  9. Final simplification0.5

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

Reproduce

herbie shell --seed 2020065 
(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))))))