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

Average Error: 0.5 → 0.5

Time: 2.1m

Precision: 64

Math

\[\frac{2.0 + \left(\left(\sqrt{2.0} \cdot \left(\sin x - \frac{\sin y}{16.0}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16.0}\right)\right) \cdot \left(\cos x - \cos y\right)}{3.0 \cdot \left(\left(1.0 + \frac{\sqrt{5.0} - 1.0}{2.0} \cdot \cos x\right) + \frac{3.0 - \sqrt{5.0}}{2.0} \cdot \cos y\right)}\]

↓

\[\left(\left(\left(\cos y \cdot \frac{\frac{3.0 \cdot 3.0 - 5.0}{\sqrt{5.0} + 3.0}}{2.0}\right) \cdot \left(\cos y \cdot \frac{\frac{3.0 \cdot 3.0 - 5.0}{\sqrt{5.0} + 3.0}}{2.0}\right) - \left(1.0 + \frac{\sqrt{5.0} - 1.0}{2.0} \cdot \cos x\right) \cdot \left(\cos y \cdot \frac{\frac{3.0 \cdot 3.0 - 5.0}{\sqrt{5.0} + 3.0}}{2.0}\right)\right) + \left(1.0 + \frac{\sqrt{5.0} - 1.0}{2.0} \cdot \cos x\right) \cdot \left(1.0 + \frac{\sqrt{5.0} - 1.0}{2.0} \cdot \cos x\right)\right) \cdot \frac{\frac{\sqrt[3]{\left(\left(\sin x - \frac{\sin y}{16.0}\right) \cdot \sqrt{2.0}\right) \cdot \left(\left(\left(\sin x - \frac{\sin y}{16.0}\right) \cdot \sqrt{2.0}\right) \cdot \left(\left(\sin x - \frac{\sin y}{16.0}\right) \cdot \sqrt{2.0}\right)\right)} \cdot \left(\left(\cos x - \cos y\right) \cdot \left(\sin y - \frac{\sin x}{16.0}\right)\right) + 2.0}{3.0}}{\left(\left(1.0 + \frac{\sqrt{5.0} - 1.0}{2.0} \cdot \cos x\right) \cdot \left(1.0 + \frac{\sqrt{5.0} - 1.0}{2.0} \cdot \cos x\right)\right) \cdot \left(1.0 + \frac{\sqrt{5.0} - 1.0}{2.0} \cdot \cos x\right) + \left(\cos y \cdot \frac{3.0 \cdot 3.0 - 5.0}{\left(\sqrt{5.0} + 3.0\right) \cdot 2.0}\right) \cdot \left(\left(\cos y \cdot \frac{3.0 \cdot 3.0 - 5.0}{\left(\sqrt{5.0} + 3.0\right) \cdot 2.0}\right) \cdot \left(\cos y \cdot \frac{3.0 \cdot 3.0 - 5.0}{\left(\sqrt{5.0} + 3.0\right) \cdot 2.0}\right)\right)}\]

C

double f(double x, double y) {
        double r13939034 = 2.0;
        double r13939035 = sqrt(r13939034);
        double r13939036 = x;
        double r13939037 = sin(r13939036);
        double r13939038 = y;
        double r13939039 = sin(r13939038);
        double r13939040 = 16.0;
        double r13939041 = r13939039 / r13939040;
        double r13939042 = r13939037 - r13939041;
        double r13939043 = r13939035 * r13939042;
        double r13939044 = r13939037 / r13939040;
        double r13939045 = r13939039 - r13939044;
        double r13939046 = r13939043 * r13939045;
        double r13939047 = cos(r13939036);
        double r13939048 = cos(r13939038);
        double r13939049 = r13939047 - r13939048;
        double r13939050 = r13939046 * r13939049;
        double r13939051 = r13939034 + r13939050;
        double r13939052 = 3.0;
        double r13939053 = 1.0;
        double r13939054 = 5.0;
        double r13939055 = sqrt(r13939054);
        double r13939056 = r13939055 - r13939053;
        double r13939057 = r13939056 / r13939034;
        double r13939058 = r13939057 * r13939047;
        double r13939059 = r13939053 + r13939058;
        double r13939060 = r13939052 - r13939055;
        double r13939061 = r13939060 / r13939034;
        double r13939062 = r13939061 * r13939048;
        double r13939063 = r13939059 + r13939062;
        double r13939064 = r13939052 * r13939063;
        double r13939065 = r13939051 / r13939064;
        return r13939065;
}

↓

double f(double x, double y) {
        double r13939066 = y;
        double r13939067 = cos(r13939066);
        double r13939068 = 3.0;
        double r13939069 = r13939068 * r13939068;
        double r13939070 = 5.0;
        double r13939071 = r13939069 - r13939070;
        double r13939072 = sqrt(r13939070);
        double r13939073 = r13939072 + r13939068;
        double r13939074 = r13939071 / r13939073;
        double r13939075 = 2.0;
        double r13939076 = r13939074 / r13939075;
        double r13939077 = r13939067 * r13939076;
        double r13939078 = r13939077 * r13939077;
        double r13939079 = 1.0;
        double r13939080 = r13939072 - r13939079;
        double r13939081 = r13939080 / r13939075;
        double r13939082 = x;
        double r13939083 = cos(r13939082);
        double r13939084 = r13939081 * r13939083;
        double r13939085 = r13939079 + r13939084;
        double r13939086 = r13939085 * r13939077;
        double r13939087 = r13939078 - r13939086;
        double r13939088 = r13939085 * r13939085;
        double r13939089 = r13939087 + r13939088;
        double r13939090 = sin(r13939082);
        double r13939091 = sin(r13939066);
        double r13939092 = 16.0;
        double r13939093 = r13939091 / r13939092;
        double r13939094 = r13939090 - r13939093;
        double r13939095 = sqrt(r13939075);
        double r13939096 = r13939094 * r13939095;
        double r13939097 = r13939096 * r13939096;
        double r13939098 = r13939096 * r13939097;
        double r13939099 = cbrt(r13939098);
        double r13939100 = r13939083 - r13939067;
        double r13939101 = r13939090 / r13939092;
        double r13939102 = r13939091 - r13939101;
        double r13939103 = r13939100 * r13939102;
        double r13939104 = r13939099 * r13939103;
        double r13939105 = r13939104 + r13939075;
        double r13939106 = r13939105 / r13939068;
        double r13939107 = r13939088 * r13939085;
        double r13939108 = r13939073 * r13939075;
        double r13939109 = r13939071 / r13939108;
        double r13939110 = r13939067 * r13939109;
        double r13939111 = r13939110 * r13939110;
        double r13939112 = r13939110 * r13939111;
        double r13939113 = r13939107 + r13939112;
        double r13939114 = r13939106 / r13939113;
        double r13939115 = r13939089 * r13939114;
        return r13939115;
}

Error

Bits error versus x

Bits error versus y

Derivation

1. Initial program 0.5

   \[\frac{2.0 + \left(\left(\sqrt{2.0} \cdot \left(\sin x - \frac{\sin y}{16.0}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16.0}\right)\right) \cdot \left(\cos x - \cos y\right)}{3.0 \cdot \left(\left(1.0 + \frac{\sqrt{5.0} - 1.0}{2.0} \cdot \cos x\right) + \frac{3.0 - \sqrt{5.0}}{2.0} \cdot \cos y\right)}\]
2. Using strategy rm

3. Applied flip-- 0.5

   \[\leadsto \frac{2.0 + \left(\left(\sqrt{2.0} \cdot \left(\sin x - \frac{\sin y}{16.0}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16.0}\right)\right) \cdot \left(\cos x - \cos y\right)}{3.0 \cdot \left(\left(1.0 + \frac{\sqrt{5.0} - 1.0}{2.0} \cdot \cos x\right) + \frac{\color{blue}{\frac{3.0 \cdot 3.0 - \sqrt{5.0} \cdot \sqrt{5.0}}{3.0 + \sqrt{5.0}}}}{2.0} \cdot \cos y\right)}\]

4. Simplified 0.5

   \[\leadsto \frac{2.0 + \left(\left(\sqrt{2.0} \cdot \left(\sin x - \frac{\sin y}{16.0}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16.0}\right)\right) \cdot \left(\cos x - \cos y\right)}{3.0 \cdot \left(\left(1.0 + \frac{\sqrt{5.0} - 1.0}{2.0} \cdot \cos x\right) + \frac{\frac{\color{blue}{3.0 \cdot 3.0 - 5.0}}{3.0 + \sqrt{5.0}}}{2.0} \cdot \cos y\right)}\]
5. Using strategy rm

6. Applied flip3-+ 0.6

   \[\leadsto \frac{2.0 + \left(\left(\sqrt{2.0} \cdot \left(\sin x - \frac{\sin y}{16.0}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16.0}\right)\right) \cdot \left(\cos x - \cos y\right)}{3.0 \cdot \color{blue}{\frac{{\left(1.0 + \frac{\sqrt{5.0} - 1.0}{2.0} \cdot \cos x\right)}^{3} + {\left(\frac{\frac{3.0 \cdot 3.0 - 5.0}{3.0 + \sqrt{5.0}}}{2.0} \cdot \cos y\right)}^{3}}{\left(1.0 + \frac{\sqrt{5.0} - 1.0}{2.0} \cdot \cos x\right) \cdot \left(1.0 + \frac{\sqrt{5.0} - 1.0}{2.0} \cdot \cos x\right) + \left(\left(\frac{\frac{3.0 \cdot 3.0 - 5.0}{3.0 + \sqrt{5.0}}}{2.0} \cdot \cos y\right) \cdot \left(\frac{\frac{3.0 \cdot 3.0 - 5.0}{3.0 + \sqrt{5.0}}}{2.0} \cdot \cos y\right) - \left(1.0 + \frac{\sqrt{5.0} - 1.0}{2.0} \cdot \cos x\right) \cdot \left(\frac{\frac{3.0 \cdot 3.0 - 5.0}{3.0 + \sqrt{5.0}}}{2.0} \cdot \cos y\right)\right)}}}\]

7. Applied associate-*r/ 0.6

   \[\leadsto \frac{2.0 + \left(\left(\sqrt{2.0} \cdot \left(\sin x - \frac{\sin y}{16.0}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16.0}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\frac{3.0 \cdot \left({\left(1.0 + \frac{\sqrt{5.0} - 1.0}{2.0} \cdot \cos x\right)}^{3} + {\left(\frac{\frac{3.0 \cdot 3.0 - 5.0}{3.0 + \sqrt{5.0}}}{2.0} \cdot \cos y\right)}^{3}\right)}{\left(1.0 + \frac{\sqrt{5.0} - 1.0}{2.0} \cdot \cos x\right) \cdot \left(1.0 + \frac{\sqrt{5.0} - 1.0}{2.0} \cdot \cos x\right) + \left(\left(\frac{\frac{3.0 \cdot 3.0 - 5.0}{3.0 + \sqrt{5.0}}}{2.0} \cdot \cos y\right) \cdot \left(\frac{\frac{3.0 \cdot 3.0 - 5.0}{3.0 + \sqrt{5.0}}}{2.0} \cdot \cos y\right) - \left(1.0 + \frac{\sqrt{5.0} - 1.0}{2.0} \cdot \cos x\right) \cdot \left(\frac{\frac{3.0 \cdot 3.0 - 5.0}{3.0 + \sqrt{5.0}}}{2.0} \cdot \cos y\right)\right)}}}\]

8. Applied associate-/r/ 0.5

   \[\leadsto \color{blue}{\frac{2.0 + \left(\left(\sqrt{2.0} \cdot \left(\sin x - \frac{\sin y}{16.0}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16.0}\right)\right) \cdot \left(\cos x - \cos y\right)}{3.0 \cdot \left({\left(1.0 + \frac{\sqrt{5.0} - 1.0}{2.0} \cdot \cos x\right)}^{3} + {\left(\frac{\frac{3.0 \cdot 3.0 - 5.0}{3.0 + \sqrt{5.0}}}{2.0} \cdot \cos y\right)}^{3}\right)} \cdot \left(\left(1.0 + \frac{\sqrt{5.0} - 1.0}{2.0} \cdot \cos x\right) \cdot \left(1.0 + \frac{\sqrt{5.0} - 1.0}{2.0} \cdot \cos x\right) + \left(\left(\frac{\frac{3.0 \cdot 3.0 - 5.0}{3.0 + \sqrt{5.0}}}{2.0} \cdot \cos y\right) \cdot \left(\frac{\frac{3.0 \cdot 3.0 - 5.0}{3.0 + \sqrt{5.0}}}{2.0} \cdot \cos y\right) - \left(1.0 + \frac{\sqrt{5.0} - 1.0}{2.0} \cdot \cos x\right) \cdot \left(\frac{\frac{3.0 \cdot 3.0 - 5.0}{3.0 + \sqrt{5.0}}}{2.0} \cdot \cos y\right)\right)\right)}\]

9. Simplified 0.5

   \[\leadsto \color{blue}{\frac{\frac{\left(\left(\cos x - \cos y\right) \cdot \left(\sin y - \frac{\sin x}{16.0}\right)\right) \cdot \left(\left(\sin x - \frac{\sin y}{16.0}\right) \cdot \sqrt{2.0}\right) + 2.0}{3.0}}{\left(\cos y \cdot \frac{3.0 \cdot 3.0 - 5.0}{2.0 \cdot \left(3.0 + \sqrt{5.0}\right)}\right) \cdot \left(\left(\cos y \cdot \frac{3.0 \cdot 3.0 - 5.0}{2.0 \cdot \left(3.0 + \sqrt{5.0}\right)}\right) \cdot \left(\cos y \cdot \frac{3.0 \cdot 3.0 - 5.0}{2.0 \cdot \left(3.0 + \sqrt{5.0}\right)}\right)\right) + \left(\cos x \cdot \frac{\sqrt{5.0} - 1.0}{2.0} + 1.0\right) \cdot \left(\left(\cos x \cdot \frac{\sqrt{5.0} - 1.0}{2.0} + 1.0\right) \cdot \left(\cos x \cdot \frac{\sqrt{5.0} - 1.0}{2.0} + 1.0\right)\right)}} \cdot \left(\left(1.0 + \frac{\sqrt{5.0} - 1.0}{2.0} \cdot \cos x\right) \cdot \left(1.0 + \frac{\sqrt{5.0} - 1.0}{2.0} \cdot \cos x\right) + \left(\left(\frac{\frac{3.0 \cdot 3.0 - 5.0}{3.0 + \sqrt{5.0}}}{2.0} \cdot \cos y\right) \cdot \left(\frac{\frac{3.0 \cdot 3.0 - 5.0}{3.0 + \sqrt{5.0}}}{2.0} \cdot \cos y\right) - \left(1.0 + \frac{\sqrt{5.0} - 1.0}{2.0} \cdot \cos x\right) \cdot \left(\frac{\frac{3.0 \cdot 3.0 - 5.0}{3.0 + \sqrt{5.0}}}{2.0} \cdot \cos y\right)\right)\right)\]
10. Using strategy rm

11. Applied add-cbrt-cube 0.5

    \[\leadsto \frac{\frac{\left(\left(\cos x - \cos y\right) \cdot \left(\sin y - \frac{\sin x}{16.0}\right)\right) \cdot \color{blue}{\sqrt[3]{\left(\left(\left(\sin x - \frac{\sin y}{16.0}\right) \cdot \sqrt{2.0}\right) \cdot \left(\left(\sin x - \frac{\sin y}{16.0}\right) \cdot \sqrt{2.0}\right)\right) \cdot \left(\left(\sin x - \frac{\sin y}{16.0}\right) \cdot \sqrt{2.0}\right)}} + 2.0}{3.0}}{\left(\cos y \cdot \frac{3.0 \cdot 3.0 - 5.0}{2.0 \cdot \left(3.0 + \sqrt{5.0}\right)}\right) \cdot \left(\left(\cos y \cdot \frac{3.0 \cdot 3.0 - 5.0}{2.0 \cdot \left(3.0 + \sqrt{5.0}\right)}\right) \cdot \left(\cos y \cdot \frac{3.0 \cdot 3.0 - 5.0}{2.0 \cdot \left(3.0 + \sqrt{5.0}\right)}\right)\right) + \left(\cos x \cdot \frac{\sqrt{5.0} - 1.0}{2.0} + 1.0\right) \cdot \left(\left(\cos x \cdot \frac{\sqrt{5.0} - 1.0}{2.0} + 1.0\right) \cdot \left(\cos x \cdot \frac{\sqrt{5.0} - 1.0}{2.0} + 1.0\right)\right)} \cdot \left(\left(1.0 + \frac{\sqrt{5.0} - 1.0}{2.0} \cdot \cos x\right) \cdot \left(1.0 + \frac{\sqrt{5.0} - 1.0}{2.0} \cdot \cos x\right) + \left(\left(\frac{\frac{3.0 \cdot 3.0 - 5.0}{3.0 + \sqrt{5.0}}}{2.0} \cdot \cos y\right) \cdot \left(\frac{\frac{3.0 \cdot 3.0 - 5.0}{3.0 + \sqrt{5.0}}}{2.0} \cdot \cos y\right) - \left(1.0 + \frac{\sqrt{5.0} - 1.0}{2.0} \cdot \cos x\right) \cdot \left(\frac{\frac{3.0 \cdot 3.0 - 5.0}{3.0 + \sqrt{5.0}}}{2.0} \cdot \cos y\right)\right)\right)\]

12. Final simplification 0.5

    \[\leadsto \left(\left(\left(\cos y \cdot \frac{\frac{3.0 \cdot 3.0 - 5.0}{\sqrt{5.0} + 3.0}}{2.0}\right) \cdot \left(\cos y \cdot \frac{\frac{3.0 \cdot 3.0 - 5.0}{\sqrt{5.0} + 3.0}}{2.0}\right) - \left(1.0 + \frac{\sqrt{5.0} - 1.0}{2.0} \cdot \cos x\right) \cdot \left(\cos y \cdot \frac{\frac{3.0 \cdot 3.0 - 5.0}{\sqrt{5.0} + 3.0}}{2.0}\right)\right) + \left(1.0 + \frac{\sqrt{5.0} - 1.0}{2.0} \cdot \cos x\right) \cdot \left(1.0 + \frac{\sqrt{5.0} - 1.0}{2.0} \cdot \cos x\right)\right) \cdot \frac{\frac{\sqrt[3]{\left(\left(\sin x - \frac{\sin y}{16.0}\right) \cdot \sqrt{2.0}\right) \cdot \left(\left(\left(\sin x - \frac{\sin y}{16.0}\right) \cdot \sqrt{2.0}\right) \cdot \left(\left(\sin x - \frac{\sin y}{16.0}\right) \cdot \sqrt{2.0}\right)\right)} \cdot \left(\left(\cos x - \cos y\right) \cdot \left(\sin y - \frac{\sin x}{16.0}\right)\right) + 2.0}{3.0}}{\left(\left(1.0 + \frac{\sqrt{5.0} - 1.0}{2.0} \cdot \cos x\right) \cdot \left(1.0 + \frac{\sqrt{5.0} - 1.0}{2.0} \cdot \cos x\right)\right) \cdot \left(1.0 + \frac{\sqrt{5.0} - 1.0}{2.0} \cdot \cos x\right) + \left(\cos y \cdot \frac{3.0 \cdot 3.0 - 5.0}{\left(\sqrt{5.0} + 3.0\right) \cdot 2.0}\right) \cdot \left(\left(\cos y \cdot \frac{3.0 \cdot 3.0 - 5.0}{\left(\sqrt{5.0} + 3.0\right) \cdot 2.0}\right) \cdot \left(\cos y \cdot \frac{3.0 \cdot 3.0 - 5.0}{\left(\sqrt{5.0} + 3.0\right) \cdot 2.0}\right)\right)}\]

Reproduce

herbie shell --seed 2019165 
(FPCore (x y)
  :name "Diagrams.TwoD.Path.Metafont.Internal:hobbyF from diagrams-contrib-1.3.0.5"
  (/ (+ 2.0 (* (* (* (sqrt 2.0) (- (sin x) (/ (sin y) 16.0))) (- (sin y) (/ (sin x) 16.0))) (- (cos x) (cos y)))) (* 3.0 (+ (+ 1.0 (* (/ (- (sqrt 5.0) 1.0) 2.0) (cos x))) (* (/ (- 3.0 (sqrt 5.0)) 2.0) (cos y))))))