Average Error: 0.5 → 0.4
Time: 21.0s
Precision: binary64
Cost: 104896
\[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
↓
\[\begin{array}{l} t_0 := \sqrt[3]{\cos x}\\ \frac{2 + \sqrt{2} \cdot \left(\mathsf{fma}\left({t_0}^{2}, t_0, -\cos y\right) \cdot \left(\left(-0.0625 \cdot \sin y + \sin x\right) \cdot \left(\sin y + -0.0625 \cdot \sin x\right)\right)\right)}{3 + \left(9 \cdot \frac{\cos y}{4.5 + \sqrt{11.25}} + 1.5 \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right)\right)} \end{array} \]
(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 (cbrt (cos x))))
   (/
    (+
     2.0
     (*
      (sqrt 2.0)
      (*
       (fma (pow t_0 2.0) t_0 (- (cos y)))
       (* (+ (* -0.0625 (sin y)) (sin x)) (+ (sin y) (* -0.0625 (sin x)))))))
    (+
     3.0
     (+
      (* 9.0 (/ (cos y) (+ 4.5 (sqrt 11.25))))
      (* 1.5 (* (cos x) (- (sqrt 5.0) 1.0))))))))
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 = cbrt(cos(x));
	return (2.0 + (sqrt(2.0) * (fma(pow(t_0, 2.0), t_0, -cos(y)) * (((-0.0625 * sin(y)) + sin(x)) * (sin(y) + (-0.0625 * sin(x))))))) / (3.0 + ((9.0 * (cos(y) / (4.5 + sqrt(11.25)))) + (1.5 * (cos(x) * (sqrt(5.0) - 1.0)))));
}

function code(x, y)
	return Float64(Float64(2.0 + Float64(Float64(Float64(sqrt(2.0) * Float64(sin(x) - Float64(sin(y) / 16.0))) * Float64(sin(y) - Float64(sin(x) / 16.0))) * Float64(cos(x) - cos(y)))) / Float64(3.0 * Float64(Float64(1.0 + Float64(Float64(Float64(sqrt(5.0) - 1.0) / 2.0) * cos(x))) + Float64(Float64(Float64(3.0 - sqrt(5.0)) / 2.0) * cos(y)))))
end
↓
function code(x, y)
	t_0 = cbrt(cos(x))
	return Float64(Float64(2.0 + Float64(sqrt(2.0) * Float64(fma((t_0 ^ 2.0), t_0, Float64(-cos(y))) * Float64(Float64(Float64(-0.0625 * sin(y)) + sin(x)) * Float64(sin(y) + Float64(-0.0625 * sin(x))))))) / Float64(3.0 + Float64(Float64(9.0 * Float64(cos(y) / Float64(4.5 + sqrt(11.25)))) + Float64(1.5 * Float64(cos(x) * Float64(sqrt(5.0) - 1.0))))))
end

code[x_, y_] := N[(N[(2.0 + N[(N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Sin[x], $MachinePrecision] - N[(N[Sin[y], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] - N[(N[Sin[x], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(N[(1.0 + N[(N[(N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision] / 2.0), $MachinePrecision] * N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision] * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]
↓
code[x_, y_] := Block[{t$95$0 = N[Power[N[Cos[x], $MachinePrecision], 1/3], $MachinePrecision]}, N[(N[(2.0 + N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[(N[Power[t$95$0, 2.0], $MachinePrecision] * t$95$0 + (-N[Cos[y], $MachinePrecision])), $MachinePrecision] * N[(N[(N[(-0.0625 * N[Sin[y], $MachinePrecision]), $MachinePrecision] + N[Sin[x], $MachinePrecision]), $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] + N[(-0.0625 * N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 + N[(N[(9.0 * N[(N[Cos[y], $MachinePrecision] / N[(4.5 + N[Sqrt[11.25], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.5 * N[(N[Cos[x], $MachinePrecision] * N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

\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 := \sqrt[3]{\cos x}\\
\frac{2 + \sqrt{2} \cdot \left(\mathsf{fma}\left({t_0}^{2}, t_0, -\cos y\right) \cdot \left(\left(-0.0625 \cdot \sin y + \sin x\right) \cdot \left(\sin y + -0.0625 \cdot \sin x\right)\right)\right)}{3 + \left(9 \cdot \frac{\cos y}{4.5 + \sqrt{11.25}} + 1.5 \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right)\right)}
\end{array}

Error

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

    \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\mathsf{fma}\left(\cos x, \frac{\sqrt{5} + -1}{0.6666666666666666}, \mathsf{fma}\left(\cos y, 4.5 - \frac{\sqrt{5}}{0.6666666666666666}, 3\right)\right)}} \]

  3. Applied egg-rr0.4

    \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\mathsf{fma}\left(\cos x, \frac{\sqrt{5} + -1}{0.6666666666666666}, \mathsf{fma}\left(\cos y, \color{blue}{\frac{9}{4.5 + \sqrt{11.25}}}, 3\right)\right)} \]

  4. Taylor expanded in x around inf 0.4

    \[\leadsto \color{blue}{\frac{\sqrt{2} \cdot \left(\left(\cos x - \cos y\right) \cdot \left(\left(-0.0625 \cdot \sin y + \sin x\right) \cdot \left(-0.0625 \cdot \sin x + \sin y\right)\right)\right) + 2}{3 + \left(9 \cdot \frac{\cos y}{4.5 + \sqrt{11.25}} + 1.5 \cdot \left(\left(\sqrt{5} - 1\right) \cdot \cos x\right)\right)}} \]

  5. Applied egg-rr0.4

    \[\leadsto \frac{\sqrt{2} \cdot \left(\color{blue}{\mathsf{fma}\left({\left(\sqrt[3]{\cos x}\right)}^{2}, \sqrt[3]{\cos x}, -\cos y\right)} \cdot \left(\left(-0.0625 \cdot \sin y + \sin x\right) \cdot \left(-0.0625 \cdot \sin x + \sin y\right)\right)\right) + 2}{3 + \left(9 \cdot \frac{\cos y}{4.5 + \sqrt{11.25}} + 1.5 \cdot \left(\left(\sqrt{5} - 1\right) \cdot \cos x\right)\right)} \]

  6. Final simplification0.4

    \[\leadsto \frac{2 + \sqrt{2} \cdot \left(\mathsf{fma}\left({\left(\sqrt[3]{\cos x}\right)}^{2}, \sqrt[3]{\cos x}, -\cos y\right) \cdot \left(\left(-0.0625 \cdot \sin y + \sin x\right) \cdot \left(\sin y + -0.0625 \cdot \sin x\right)\right)\right)}{3 + \left(9 \cdot \frac{\cos y}{4.5 + \sqrt{11.25}} + 1.5 \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right)\right)} \]

Alternatives

Alternative 1
Error0.4
Cost72768
\[\frac{2 + \sqrt{2} \cdot \left(\left(\left(-0.0625 \cdot \sin y + \sin x\right) \cdot \left(\sin y + -0.0625 \cdot \sin x\right)\right) \cdot \left(\cos x - \cos y\right)\right)}{3 + \left(9 \cdot \frac{\cos y}{4.5 + \sqrt{11.25}} + 1.5 \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right)\right)} \]
Alternative 2
Error12.1
Cost66504
\[\begin{array}{l} t_0 := 3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} - 1}{2}\right) + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)\\ t_1 := \frac{2 + \left(\cos x - \cos y\right) \cdot \left(\left(\sqrt{2} \cdot \sin x\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right)}{t_0}\\ \mathbf{if}\;x \leq -439.8571803332426:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 2.308301847371468 \cdot 10^{-12}:\\ \;\;\;\;\frac{2 + \left(\sqrt{2} \cdot \left(\sin y \cdot \left(-0.0625 \cdot \sin y + x \cdot 1.00390625\right)\right)\right) \cdot \left(1 - \cos y\right)}{t_0}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]
Alternative 3
Error12.1
Cost66504
\[\begin{array}{l} t_0 := \cos x - \cos y\\ t_1 := \sqrt{5} - 1\\ t_2 := \frac{2 + t_0 \cdot \left(\left(\sqrt{2} \cdot \sin x\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{t_1}{2}\right) + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)}\\ \mathbf{if}\;x \leq -439.8571803332426:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq 2.308301847371468 \cdot 10^{-12}:\\ \;\;\;\;\frac{2 + \sqrt{2} \cdot \left(\left(\left(-0.0625 \cdot \sin y + \sin x\right) \cdot \left(\sin y + -0.0625 \cdot \sin x\right)\right) \cdot t_0\right)}{3 + \left(9 \cdot \frac{\cos y}{4.5 + \sqrt{11.25}} + 1.5 \cdot t_1\right)}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]
Alternative 4
Error13.1
Cost60104
\[\begin{array}{l} t_0 := 3 + \left(9 \cdot \frac{\cos y}{4.5 + \sqrt{11.25}} + 1.5 \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right)\right)\\ t_1 := \cos x - \cos y\\ t_2 := \frac{2 + \sqrt{2} \cdot \left(t_1 \cdot \left(-0.0625 \cdot {\sin y}^{2}\right)\right)}{t_0}\\ \mathbf{if}\;y \leq -2.337175874611913:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y \leq 6.643285079246632 \cdot 10^{-13}:\\ \;\;\;\;\frac{2 + \sqrt{2} \cdot \left(t_1 \cdot \left(\sin x \cdot \left(-0.0625 \cdot \sin x + y \cdot 1.00390625\right)\right)\right)}{t_0}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]
Alternative 5
Error13.1
Cost59784
\[\begin{array}{l} t_0 := \sqrt{5} - 1\\ t_1 := 1 + \cos x \cdot \frac{t_0}{2}\\ t_2 := -0.0625 \cdot {\sin x}^{2}\\ \mathbf{if}\;x \leq -439.8571803332426:\\ \;\;\;\;\frac{2 + \left(\sqrt{2} \cdot t_2\right) \cdot \left(\cos x + -1\right)}{3 \cdot \left(t_1 + \cos y \cdot \frac{\frac{4}{3 + \sqrt{5}}}{2}\right)}\\ \mathbf{elif}\;x \leq 2.308301847371468 \cdot 10^{-12}:\\ \;\;\;\;\frac{2 + \left(\sqrt{2} \cdot \left(\sin y \cdot \left(-0.0625 \cdot \sin y + x \cdot 1.00390625\right)\right)\right) \cdot \left(1 - \cos y\right)}{3 \cdot \left(t_1 + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \sqrt{2} \cdot \left(\left(\cos x - \cos y\right) \cdot t_2\right)}{3 + \left(9 \cdot \frac{\cos y}{4.5 + \sqrt{11.25}} + 1.5 \cdot \left(\cos x \cdot t_0\right)\right)}\\ \end{array} \]
Alternative 6
Error13.2
Cost53832
\[\begin{array}{l} t_0 := \cos y \cdot \frac{3 - \sqrt{5}}{2}\\ t_1 := 1 + \cos x \cdot \frac{\sqrt{5} - 1}{2}\\ t_2 := 2 + \left(\sqrt{2} \cdot \left(-0.0625 \cdot {\sin x}^{2}\right)\right) \cdot \left(\cos x + -1\right)\\ \mathbf{if}\;x \leq -439.8571803332426:\\ \;\;\;\;\frac{t_2}{3 \cdot \left(t_1 + \cos y \cdot \frac{\frac{4}{3 + \sqrt{5}}}{2}\right)}\\ \mathbf{elif}\;x \leq 2.308301847371468 \cdot 10^{-12}:\\ \;\;\;\;\frac{2 + \left(\sqrt{2} \cdot \left(\sin y \cdot \left(-0.0625 \cdot \sin y + x \cdot 1.00390625\right)\right)\right) \cdot \left(1 - \cos y\right)}{3 \cdot \left(t_1 + t_0\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_2}{3 \cdot \left(t_0 + \left(1 + \frac{\cos x}{0.5 \cdot \left(\sqrt{5} + 1\right)}\right)\right)}\\ \end{array} \]
Alternative 7
Error13.4
Cost53512
\[\begin{array}{l} t_0 := \sqrt{5} + 1\\ t_1 := \frac{2 + \left(\sqrt{2} \cdot \left(-0.0625 \cdot {\sin x}^{2}\right)\right) \cdot \left(\cos x + -1\right)}{3 \cdot \left(\cos y \cdot \frac{3 - \sqrt{5}}{2} + \left(1 + \frac{\cos x}{0.5 \cdot t_0}\right)\right)}\\ \mathbf{if}\;x \leq -439.8571803332426:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 292550.4281132488:\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left(\sqrt{2} \cdot \left(\left(1 - \cos y\right) \cdot {\sin y}^{2}\right)\right)}{\frac{6}{t_0} + \left(3 + 9 \cdot \frac{\cos y}{4.5 + \sqrt{11.25}}\right)}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]
Alternative 8
Error13.2
Cost53512
\[\begin{array}{l} t_0 := \sqrt{5} - 1\\ t_1 := \frac{2 + \left(1 - \cos y\right) \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot -0.0625\right)\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{t_0}{2}\right) + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)}\\ \mathbf{if}\;y \leq -1.2486277545555302 \cdot 10^{-6}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 6.643285079246632 \cdot 10^{-13}:\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left(\sqrt{2} \cdot \left({\sin x}^{2} \cdot \left(\cos x + -1\right)\right)\right)}{3 + \left(1.5 \cdot \left(\cos x \cdot t_0\right) + 9 \cdot \frac{1}{4.5 + \sqrt{11.25}}\right)}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]
Alternative 9
Error13.6
Cost46984
\[\begin{array}{l} t_0 := 4.5 + \sqrt{11.25}\\ t_1 := \frac{2 + -0.0625 \cdot \left(\sqrt{2} \cdot \left({\sin x}^{2} \cdot \left(\cos x + -1\right)\right)\right)}{3 + \left(1.5 \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + 9 \cdot \frac{1}{t_0}\right)}\\ \mathbf{if}\;x \leq -439.8571803332426:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 2.308301847371468 \cdot 10^{-12}:\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left(\sqrt{2} \cdot \left(\left(1 - \cos y\right) \cdot {\sin y}^{2}\right)\right)}{\frac{6}{\sqrt{5} + 1} + \left(3 + 9 \cdot \frac{\cos y}{t_0}\right)}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]
Alternative 10
Error25.7
Cost46592
\[\frac{2 + -0.0625 \cdot \left(\sqrt{2} \cdot \left(\left(1 - \cos y\right) \cdot {\sin y}^{2}\right)\right)}{\frac{6}{\sqrt{5} + 1} + \left(3 + 9 \cdot \frac{\cos y}{4.5 + \sqrt{11.25}}\right)} \]
Alternative 11
Error25.7
Cost40512
\[\frac{2 + -0.0625 \cdot \left(\sqrt{2} \cdot \left(\left(1 - \cos y\right) \cdot \left(0.5 + \cos \left(y + y\right) \cdot -0.5\right)\right)\right)}{1.5 \cdot \left(\sqrt{5} - 1\right) + \left(3 + 9 \cdot \frac{\cos y}{4.5 + \sqrt{11.25}}\right)} \]
Alternative 12
Error44.7
Cost33664
\[\frac{2 + \sqrt{0.00390625 \cdot \left(2 \cdot \left({y}^{8} \cdot 0.25\right)\right)}}{1.5 \cdot \left(\sqrt{5} - 1\right) + \left(3 + 9 \cdot \frac{\cos y}{4.5 + \sqrt{11.25}}\right)} \]
Alternative 13
Error44.8
Cost33536
\[\frac{2 + \sqrt{2} \cdot \left(-0.03125 \cdot {y}^{4}\right)}{1.5 \cdot \left(\sqrt{5} - 1\right) + \left(3 + 9 \cdot \frac{\cos y}{4.5 + \sqrt{11.25}}\right)} \]
Alternative 14
Error45.0
Cost27264
\[\frac{2 + -0.0625 \cdot \left(\sqrt{2} \cdot \left(0.5 \cdot {y}^{4}\right)\right)}{1.5 \cdot \left(\sqrt{5} - 1\right) + \left(3 + 9 \cdot \frac{1}{4.5 + \sqrt{11.25}}\right)} \]

Error

Reproduce

herbie shell --seed 2022325 
(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))))))