Average Error: 0.5 → 0.4
Time: 24.1s
Precision: binary64
Cost: 98048
\[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
↓
\[\frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + \sin x \cdot -0.0625\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\mathsf{fma}\left(\cos x, {\left(0.16666666666666666 \cdot \left(\sqrt{5} + 1\right)\right)}^{-1}, \mathsf{fma}\left(\cos y, 4.5 - \frac{\sqrt{5}}{0.6666666666666666}, 3\right)\right)} \]
(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
 (/
  (fma
   (sqrt 2.0)
   (*
    (+ (sin x) (* -0.0625 (sin y)))
    (* (+ (sin y) (* (sin x) -0.0625)) (- (cos x) (cos y))))
   2.0)
  (fma
   (cos x)
   (pow (* 0.16666666666666666 (+ (sqrt 5.0) 1.0)) -1.0)
   (fma (cos y) (- 4.5 (/ (sqrt 5.0) 0.6666666666666666)) 3.0))))
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) {
	return fma(sqrt(2.0), ((sin(x) + (-0.0625 * sin(y))) * ((sin(y) + (sin(x) * -0.0625)) * (cos(x) - cos(y)))), 2.0) / fma(cos(x), pow((0.16666666666666666 * (sqrt(5.0) + 1.0)), -1.0), fma(cos(y), (4.5 - (sqrt(5.0) / 0.6666666666666666)), 3.0));
}

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)
	return Float64(fma(sqrt(2.0), Float64(Float64(sin(x) + Float64(-0.0625 * sin(y))) * Float64(Float64(sin(y) + Float64(sin(x) * -0.0625)) * Float64(cos(x) - cos(y)))), 2.0) / fma(cos(x), (Float64(0.16666666666666666 * Float64(sqrt(5.0) + 1.0)) ^ -1.0), fma(cos(y), Float64(4.5 - Float64(sqrt(5.0) / 0.6666666666666666)), 3.0)))
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_] := N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[(N[Sin[x], $MachinePrecision] + N[(-0.0625 * N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Sin[y], $MachinePrecision] + N[(N[Sin[x], $MachinePrecision] * -0.0625), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[Cos[x], $MachinePrecision] * N[Power[N[(0.16666666666666666 * N[(N[Sqrt[5.0], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision] + N[(N[Cos[y], $MachinePrecision] * N[(4.5 - N[(N[Sqrt[5.0], $MachinePrecision] / 0.6666666666666666), $MachinePrecision]), $MachinePrecision] + 3.0), $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)}
↓
\frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + \sin x \cdot -0.0625\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\mathsf{fma}\left(\cos x, {\left(0.16666666666666666 \cdot \left(\sqrt{5} + 1\right)\right)}^{-1}, \mathsf{fma}\left(\cos y, 4.5 - \frac{\sqrt{5}}{0.6666666666666666}, 3\right)\right)}

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

    \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\mathsf{fma}\left(\cos x, \color{blue}{{\left(0.16666666666666666 \cdot \left(\sqrt{5} + 1\right)\right)}^{-1}}, \mathsf{fma}\left(\cos y, 4.5 - \frac{\sqrt{5}}{0.6666666666666666}, 3\right)\right)} \]

  4. Final simplification0.4

    \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + \sin x \cdot -0.0625\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\mathsf{fma}\left(\cos x, {\left(0.16666666666666666 \cdot \left(\sqrt{5} + 1\right)\right)}^{-1}, \mathsf{fma}\left(\cos y, 4.5 - \frac{\sqrt{5}}{0.6666666666666666}, 3\right)\right)} \]

Alternatives

Alternative 1
Error0.4
Cost85824
\[\begin{array}{l} t_0 := \cos x - \cos y\\ \frac{2 + \sqrt{2} \cdot \left(\left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(-0.0625 \cdot \left(\sin x \cdot t_0\right) + \sin y \cdot t_0\right)\right)}{3 + \left(9 \cdot \frac{\cos y}{4.5 + \sqrt{11.25}} + 1.5 \cdot \left(\cos x \cdot \left(\sqrt{5} + -1\right)\right)\right)} \end{array} \]
Alternative 2
Error0.5
Cost72896
\[\frac{2 + \sqrt{2} \cdot \left(\left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + \sin x \cdot -0.0625\right) \cdot \left(\cos x - \cos y\right)\right)\right)}{3 \cdot \left(\left(1 + \frac{\cos x}{0.5 \cdot \left(\sqrt{5} + 1\right)}\right) + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)} \]
Alternative 3
Error12.6
Cost66504
\[\begin{array}{l} t_0 := \sqrt{5} + -1\\ t_1 := \frac{2 + \left(\cos x - \cos y\right) \cdot \left(\left(\sqrt{2} \cdot \sin x\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right)}{3 \cdot \left(\cos y \cdot \frac{3 - \sqrt{5}}{2} + \left(1 + \cos x \cdot \frac{t_0}{2}\right)\right)}\\ \mathbf{if}\;x \leq -3470.6548157365837:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 1.1429636891040436 \cdot 10^{-16}:\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left(\sqrt{2} \cdot \left(\left(1 - \cos y\right) \cdot {\sin y}^{2}\right)\right)}{1.5 \cdot t_0 + \left(3 + 9 \cdot \frac{\cos y}{4.5 + \sqrt{11.25}}\right)}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]
Alternative 4
Error12.6
Cost66504
\[\begin{array}{l} t_0 := 2 + \left(\cos x - \cos y\right) \cdot \left(\left(\sqrt{2} \cdot \sin x\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right)\\ t_1 := \sqrt{5} + -1\\ t_2 := 1 + \cos x \cdot \frac{t_1}{2}\\ \mathbf{if}\;x \leq -3470.6548157365837:\\ \;\;\;\;\frac{t_0}{3 \cdot \left(t_2 + \cos y \cdot \frac{\frac{4}{\sqrt{5} + 3}}{2}\right)}\\ \mathbf{elif}\;x \leq 1.1429636891040436 \cdot 10^{-16}:\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left(\sqrt{2} \cdot \left(\left(1 - \cos y\right) \cdot {\sin y}^{2}\right)\right)}{1.5 \cdot t_1 + \left(3 + 9 \cdot \frac{\cos y}{4.5 + \sqrt{11.25}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0}{3 \cdot \left(\cos y \cdot \frac{3 - \sqrt{5}}{2} + t_2\right)}\\ \end{array} \]
Alternative 5
Error13.9
Cost53384
\[\begin{array}{l} t_0 := 4.5 + \sqrt{11.25}\\ t_1 := 1.5 \cdot \left(\cos x \cdot \left(\sqrt{5} + -1\right)\right)\\ t_2 := \frac{2 + \sqrt{2} \cdot \left({\sin y}^{2} \cdot \left(-0.0625 + \cos y \cdot 0.0625\right)\right)}{3 + \left(9 \cdot \frac{\cos y}{t_0} + t_1\right)}\\ \mathbf{if}\;y \leq -1748625.2095035857:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y \leq 2.3277668791324443 \cdot 10^{-25}:\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left(\sqrt{2} \cdot \left({\sin x}^{2} \cdot \left(\cos x + -1\right)\right)\right)}{3 + \left(t_1 + 9 \cdot \frac{1}{t_0}\right)}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]
Alternative 6
Error13.7
Cost47560
\[\begin{array}{l} t_0 := \sqrt{5} + -1\\ t_1 := \frac{2 + \left(\cos x + -1\right) \cdot \left(\left(0.5 + \cos \left(x + x\right) \cdot -0.5\right) \cdot \left(\sqrt{2} \cdot -0.0625\right)\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{t_0}{2}\right) + \cos y \cdot \frac{\frac{4}{\sqrt{5} + 3}}{2}\right)}\\ \mathbf{if}\;x \leq -3470.6548157365837:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 1.1429636891040436 \cdot 10^{-16}:\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left(\sqrt{2} \cdot \left(\left(1 - \cos y\right) \cdot {\sin y}^{2}\right)\right)}{1.5 \cdot t_0 + \left(3 + 9 \cdot \frac{\cos y}{4.5 + \sqrt{11.25}}\right)}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]
Alternative 7
Error13.9
Cost47368
\[\begin{array}{l} t_0 := \sqrt{5} + -1\\ t_1 := \frac{2 + -0.0625 \cdot \left(\sqrt{2} \cdot \left({\sin x}^{2} \cdot \left(\cos x + -1\right)\right)\right)}{3 + \left(1.5 \cdot \left(\cos x \cdot t_0\right) + 9 \cdot \frac{1}{4.5 + \sqrt{11.25}}\right)}\\ \mathbf{if}\;x \leq -3470.6548157365837:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 1.4085383778962206:\\ \;\;\;\;\frac{2 + \left(1 - \cos y\right) \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot -0.0625\right)\right)}{3 \cdot \left(\cos y \cdot \frac{3 - \sqrt{5}}{2} + \left(1 + t_0 \cdot \left(0.5 + -0.25 \cdot \left(x \cdot x\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]
Alternative 8
Error14.1
Cost46984
\[\begin{array}{l} t_0 := 4.5 + \sqrt{11.25}\\ t_1 := \sqrt{5} + -1\\ t_2 := \frac{2 + -0.0625 \cdot \left(\sqrt{2} \cdot \left({\sin x}^{2} \cdot \left(\cos x + -1\right)\right)\right)}{3 + \left(1.5 \cdot \left(\cos x \cdot t_1\right) + 9 \cdot \frac{1}{t_0}\right)}\\ \mathbf{if}\;x \leq -3470.6548157365837:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq 1.1429636891040436 \cdot 10^{-16}:\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left(\sqrt{2} \cdot \left(\left(1 - \cos y\right) \cdot {\sin y}^{2}\right)\right)}{1.5 \cdot t_1 + \left(3 + 9 \cdot \frac{\cos y}{t_0}\right)}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]
Alternative 9
Error25.0
Cost46856
\[\begin{array}{l} t_0 := 1 - \cos y\\ t_1 := \frac{2 + -0.0625 \cdot \left(\sqrt{2} \cdot \left(t_0 \cdot {\sin y}^{2}\right)\right)}{1.5 \cdot \left(\sqrt{5} + -1\right) + \left(3 + 9 \cdot \frac{\cos y}{4.5 + \sqrt{11.25}}\right)}\\ \mathbf{if}\;y \leq -1748625.2095035857:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 5.9525952521098926 \cdot 10^{-5}:\\ \;\;\;\;\frac{2 + t_0 \cdot \left(y \cdot \left(-0.0625 \cdot \left(\sqrt{2} \cdot y\right)\right)\right)}{3 \cdot \left(\left(1 + \frac{\cos x}{0.5 \cdot \left(\sqrt{5} + 1\right)}\right) + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]
Alternative 10
Error36.6
Cost32832
\[\frac{2}{\frac{9}{4.5 + \sqrt{11.25}} + \mathsf{fma}\left(\cos x, \mathsf{fma}\left(1.5, \sqrt{5}, -1.5\right), 3\right)} \]
Alternative 11
Error36.6
Cost20416
\[\frac{2}{3 + \left(1.5 \cdot \left(\cos x \cdot \left(\sqrt{5} + -1\right)\right) + 9 \cdot \frac{1}{4.5 + \sqrt{11.25}}\right)} \]
Alternative 12
Error44.9
Cost13760
\[\frac{2 + \left(1 - \cos y\right) \cdot \left(y \cdot \left(-0.0625 \cdot \left(\sqrt{2} \cdot y\right)\right)\right)}{6} \]

Error

Reproduce

herbie shell --seed 2022217 
(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))))))