Average Error: 0.5 → 0.4
Time: 15.7s
Precision: binary64
\[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
↓
\[\frac{\mathsf{fma}\left(\sqrt{2}, \left(\cos x - \cos y\right) \cdot \left(\mathsf{fma}\left(-0.0625, \sin y, \sin x\right) \cdot \mathsf{fma}\left(-0.0625, \sin x, \sin y\right)\right), 2\right)}{\mathsf{fma}\left(\cos y, \mathsf{fma}\left(\sqrt{5}, -1.5, 4.5\right), \mathsf{fma}\left(\cos x, \mathsf{fma}\left(1.5, \sqrt{5}, -1.5\right), 3\right)\right)} \]
(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)
   (*
    (- (cos x) (cos y))
    (* (fma -0.0625 (sin y) (sin x)) (fma -0.0625 (sin x) (sin y))))
   2.0)
  (fma
   (cos y)
   (fma (sqrt 5.0) -1.5 4.5)
   (fma (cos x) (fma 1.5 (sqrt 5.0) -1.5) 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), ((cos(x) - cos(y)) * (fma(-0.0625, sin(y), sin(x)) * fma(-0.0625, sin(x), sin(y)))), 2.0) / fma(cos(y), fma(sqrt(5.0), -1.5, 4.5), fma(cos(x), fma(1.5, sqrt(5.0), -1.5), 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(cos(x) - cos(y)) * Float64(fma(-0.0625, sin(y), sin(x)) * fma(-0.0625, sin(x), sin(y)))), 2.0) / fma(cos(y), fma(sqrt(5.0), -1.5, 4.5), fma(cos(x), fma(1.5, sqrt(5.0), -1.5), 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[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[(N[(-0.0625 * N[Sin[y], $MachinePrecision] + N[Sin[x], $MachinePrecision]), $MachinePrecision] * N[(-0.0625 * N[Sin[x], $MachinePrecision] + N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[Cos[y], $MachinePrecision] * N[(N[Sqrt[5.0], $MachinePrecision] * -1.5 + 4.5), $MachinePrecision] + N[(N[Cos[x], $MachinePrecision] * N[(1.5 * N[Sqrt[5.0], $MachinePrecision] + -1.5), $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(\cos x - \cos y\right) \cdot \left(\mathsf{fma}\left(-0.0625, \sin y, \sin x\right) \cdot \mathsf{fma}\left(-0.0625, \sin x, \sin y\right)\right), 2\right)}{\mathsf{fma}\left(\cos y, \mathsf{fma}\left(\sqrt{5}, -1.5, 4.5\right), \mathsf{fma}\left(\cos x, \mathsf{fma}\left(1.5, \sqrt{5}, -1.5\right), 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. Taylor expanded in x around inf 0.4

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

  4. Simplified0.4

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

  5. Final simplification0.4

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

Reproduce

herbie shell --seed 2022209 
(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))))))