Average Error: 0.5 → 0.4
Time: 13.2s
Precision: binary64
\[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
↓
\[\begin{array}{l} t_0 := 1.5 \cdot \left(\sqrt{5} - 1\right)\\ \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 + \log \left(e^{\cos x \cdot \sqrt[3]{t_0 \cdot \left(t_0 \cdot t_0\right)} + \cos y \cdot \frac{6}{3 + \sqrt{5}}}\right)} \end{array} \]
\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)}
↓
\begin{array}{l}
t_0 := 1.5 \cdot \left(\sqrt{5} - 1\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 + \log \left(e^{\cos x \cdot \sqrt[3]{t_0 \cdot \left(t_0 \cdot t_0\right)} + \cos y \cdot \frac{6}{3 + \sqrt{5}}}\right)}
\end{array}

(FPCore (x y)
 :precision binary64
 (/
  (+
   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))))))
↓
(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* 1.5 (- (sqrt 5.0) 1.0))))
   (/
    (+
     2.0
     (*
      (*
       (* (sqrt 2.0) (- (sin x) (/ (sin y) 16.0)))
       (- (sin y) (/ (sin x) 16.0)))
      (- (cos x) (cos y))))
    (+
     3.0
     (log
      (exp
       (+
        (* (cos x) (cbrt (* t_0 (* t_0 t_0))))
        (* (cos y) (/ 6.0 (+ 3.0 (sqrt 5.0)))))))))))
double code(double x, double y) {
	return (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))));
}
↓
double code(double x, double y) {
	double t_0 = 1.5 * (sqrt(5.0) - 1.0);
	return (2.0 + (((sqrt(2.0) * (sin(x) - (sin(y) / 16.0))) * (sin(y) - (sin(x) / 16.0))) * (cos(x) - cos(y)))) / (3.0 + log(exp((cos(x) * cbrt(t_0 * (t_0 * t_0))) + (cos(y) * (6.0 / (3.0 + sqrt(5.0)))))));
}

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. Simplified0.4

    \[\leadsto \color{blue}{\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 + \left(\cos y \cdot \left(1.5 \cdot \left(3 - \sqrt{5}\right)\right) + 1.5 \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right)\right)}} \]
  3. Using strategy rm

  4. Applied add-sqr-sqrt_binary640.4

    \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 + \left(\cos y \cdot \left(1.5 \cdot \left(3 - \sqrt{5}\right)\right) + \color{blue}{\left(\sqrt{1.5} \cdot \sqrt{1.5}\right)} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right)\right)} \]

  5. Applied associate-*l*_binary640.4

    \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 + \left(\cos y \cdot \left(1.5 \cdot \left(3 - \sqrt{5}\right)\right) + \color{blue}{\sqrt{1.5} \cdot \left(\sqrt{1.5} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right)\right)}\right)} \]

  6. Simplified0.4

    \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 + \left(\cos y \cdot \left(1.5 \cdot \left(3 - \sqrt{5}\right)\right) + \sqrt{1.5} \cdot \color{blue}{\left(\left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) \cdot \sqrt{1.5}\right)}\right)} \]
  7. Using strategy rm

  8. Applied flip--_binary640.8

    \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 + \left(\cos y \cdot \left(1.5 \cdot \color{blue}{\frac{3 \cdot 3 - \sqrt{5} \cdot \sqrt{5}}{3 + \sqrt{5}}}\right) + \sqrt{1.5} \cdot \left(\left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) \cdot \sqrt{1.5}\right)\right)} \]

  9. Simplified0.4

    \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 + \left(\cos y \cdot \left(1.5 \cdot \frac{\color{blue}{4}}{3 + \sqrt{5}}\right) + \sqrt{1.5} \cdot \left(\left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) \cdot \sqrt{1.5}\right)\right)} \]

  10. Simplified0.4

    \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 + \left(\cos y \cdot \left(1.5 \cdot \frac{4}{\color{blue}{\sqrt{5} + 3}}\right) + \sqrt{1.5} \cdot \left(\left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) \cdot \sqrt{1.5}\right)\right)} \]
  11. Using strategy rm

  12. Applied add-log-exp_binary640.4

    \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 + \left(\cos y \cdot \left(1.5 \cdot \frac{4}{\sqrt{5} + 3}\right) + \color{blue}{\log \left(e^{\sqrt{1.5} \cdot \left(\left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) \cdot \sqrt{1.5}\right)}\right)}\right)} \]

  13. Applied add-log-exp_binary640.4

    \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 + \left(\color{blue}{\log \left(e^{\cos y \cdot \left(1.5 \cdot \frac{4}{\sqrt{5} + 3}\right)}\right)} + \log \left(e^{\sqrt{1.5} \cdot \left(\left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) \cdot \sqrt{1.5}\right)}\right)\right)} \]

  14. Applied sum-log_binary640.4

    \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 + \color{blue}{\log \left(e^{\cos y \cdot \left(1.5 \cdot \frac{4}{\sqrt{5} + 3}\right)} \cdot e^{\sqrt{1.5} \cdot \left(\left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) \cdot \sqrt{1.5}\right)}\right)}} \]

  15. Simplified0.4

    \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 + \log \color{blue}{\left(e^{\cos x \cdot \left(1.5 \cdot \left(\sqrt{5} - 1\right)\right) + \cos y \cdot \frac{6}{\sqrt{5} + 3}}\right)}} \]
  16. Using strategy rm

  17. Applied add-cbrt-cube_binary640.4

    \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 + \log \left(e^{\cos x \cdot \color{blue}{\sqrt[3]{\left(\left(1.5 \cdot \left(\sqrt{5} - 1\right)\right) \cdot \left(1.5 \cdot \left(\sqrt{5} - 1\right)\right)\right) \cdot \left(1.5 \cdot \left(\sqrt{5} - 1\right)\right)}} + \cos y \cdot \frac{6}{\sqrt{5} + 3}}\right)} \]

  18. Final simplification0.4

    \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 + \log \left(e^{\cos x \cdot \sqrt[3]{\left(1.5 \cdot \left(\sqrt{5} - 1\right)\right) \cdot \left(\left(1.5 \cdot \left(\sqrt{5} - 1\right)\right) \cdot \left(1.5 \cdot \left(\sqrt{5} - 1\right)\right)\right)} + \cos y \cdot \frac{6}{3 + \sqrt{5}}}\right)} \]

Reproduce

herbie shell --seed 2021190 
(FPCore (x y)
  :name "Diagrams.TwoD.Path.Metafont.Internal:hobbyF from diagrams-contrib-1.3.0.5"
  :precision binary64
  (/ (+ 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))))))