Average Error: 0.5 → 0.4
Time: 14.0s
Precision: binary64
\[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
↓
\[\frac{\mathsf{fma}\left(\sqrt{2}, \log \left(e^{\left(\left(\sin y + \sin x \cdot -0.0625\right) \cdot \left(\sin x + \sin y \cdot -0.0625\right)\right) \cdot \left(\cos x - \cos y\right)}\right), 2\right)}{\cos x \cdot \mathsf{fma}\left(1.5, \sqrt{5}, -1.5\right) + \mathsf{fma}\left(\cos y, 4.5 - \frac{\sqrt{5}}{0.6666666666666666}, 3\right)} \]
(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)
   (log
    (exp
     (*
      (* (+ (sin y) (* (sin x) -0.0625)) (+ (sin x) (* (sin y) -0.0625)))
      (- (cos x) (cos y)))))
   2.0)
  (+
   (* (cos x) (fma 1.5 (sqrt 5.0) -1.5))
   (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), log(exp((((sin(y) + (sin(x) * -0.0625)) * (sin(x) + (sin(y) * -0.0625))) * (cos(x) - cos(y))))), 2.0) / ((cos(x) * fma(1.5, sqrt(5.0), -1.5)) + 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), log(exp(Float64(Float64(Float64(sin(y) + Float64(sin(x) * -0.0625)) * Float64(sin(x) + Float64(sin(y) * -0.0625))) * Float64(cos(x) - cos(y))))), 2.0) / Float64(Float64(cos(x) * fma(1.5, sqrt(5.0), -1.5)) + 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[Log[N[Exp[N[(N[(N[(N[Sin[y], $MachinePrecision] + N[(N[Sin[x], $MachinePrecision] * -0.0625), $MachinePrecision]), $MachinePrecision] * N[(N[Sin[x], $MachinePrecision] + N[(N[Sin[y], $MachinePrecision] * -0.0625), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(N[Cos[x], $MachinePrecision] * N[(1.5 * N[Sqrt[5.0], $MachinePrecision] + -1.5), $MachinePrecision]), $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}, \log \left(e^{\left(\left(\sin y + \sin x \cdot -0.0625\right) \cdot \left(\sin x + \sin y \cdot -0.0625\right)\right) \cdot \left(\cos x - \cos y\right)}\right), 2\right)}{\cos x \cdot \mathsf{fma}\left(1.5, \sqrt{5}, -1.5\right) + \mathsf{fma}\left(\cos y, 4.5 - \frac{\sqrt{5}}{0.6666666666666666}, 3\right)}

Error

Bits error versus x

Bits error versus y

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. Simplified0.5

    \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\frac{\cos x \cdot \left(\sqrt{5} + -1\right)}{0.6666666666666666} + \mathsf{fma}\left(\cos y, 4.5 - \frac{\sqrt{5}}{0.6666666666666666}, 3\right)}} \]

  3. Applied egg-rr0.5

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

  4. Taylor expanded in x around inf 0.5

    \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \log \left(e^{\left(\left(\sin y - \sin x \cdot 0.0625\right) \cdot \left(\sin x - \sin y \cdot 0.0625\right)\right) \cdot \left(\cos x - \cos y\right)}\right), 2\right)}{\color{blue}{1.5 \cdot \left(\left(\sqrt{5} - 1\right) \cdot \cos x\right)} + \mathsf{fma}\left(\cos y, 4.5 - \frac{\sqrt{5}}{0.6666666666666666}, 3\right)} \]

  5. Simplified0.4

    \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \log \left(e^{\left(\left(\sin y - \sin x \cdot 0.0625\right) \cdot \left(\sin x - \sin y \cdot 0.0625\right)\right) \cdot \left(\cos x - \cos y\right)}\right), 2\right)}{\color{blue}{\cos x \cdot \mathsf{fma}\left(1.5, \sqrt{5}, -1.5\right)} + \mathsf{fma}\left(\cos y, 4.5 - \frac{\sqrt{5}}{0.6666666666666666}, 3\right)} \]

  6. Final simplification0.4

    \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \log \left(e^{\left(\left(\sin y + \sin x \cdot -0.0625\right) \cdot \left(\sin x + \sin y \cdot -0.0625\right)\right) \cdot \left(\cos x - \cos y\right)}\right), 2\right)}{\cos x \cdot \mathsf{fma}\left(1.5, \sqrt{5}, -1.5\right) + \mathsf{fma}\left(\cos y, 4.5 - \frac{\sqrt{5}}{0.6666666666666666}, 3\right)} \]

Reproduce

herbie shell --seed 2022162 
(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))))))