?

Average Accuracy: 99.3% → 99.3%
Time: 59.3s
Precision: binary64
Cost: 104384

?

\[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
↓
\[\frac{\mathsf{fma}\left(\sqrt{2}, \log \left({\left({\left(e^{\sin x - \sin y \cdot 0.0625}\right)}^{\left(\cos x - \cos y\right)}\right)}^{\left(\sin y - \sin x \cdot 0.0625\right)}\right), 2\right)}{3 + \left(\cos y \cdot \left(4.5 - \sqrt{11.25}\right) + \frac{\cos x \cdot 6}{\sqrt{5} + 1}\right)} \]
(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
    (pow
     (pow (exp (- (sin x) (* (sin y) 0.0625))) (- (cos x) (cos y)))
     (- (sin y) (* (sin x) 0.0625))))
   2.0)
  (+
   3.0
   (+
    (* (cos y) (- 4.5 (sqrt 11.25)))
    (/ (* (cos x) 6.0) (+ (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) {
	return fma(sqrt(2.0), log(pow(pow(exp((sin(x) - (sin(y) * 0.0625))), (cos(x) - cos(y))), (sin(y) - (sin(x) * 0.0625)))), 2.0) / (3.0 + ((cos(y) * (4.5 - sqrt(11.25))) + ((cos(x) * 6.0) / (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)
	return Float64(fma(sqrt(2.0), log(((exp(Float64(sin(x) - Float64(sin(y) * 0.0625))) ^ Float64(cos(x) - cos(y))) ^ Float64(sin(y) - Float64(sin(x) * 0.0625)))), 2.0) / Float64(3.0 + Float64(Float64(cos(y) * Float64(4.5 - sqrt(11.25))) + Float64(Float64(cos(x) * 6.0) / 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_] := N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[Log[N[Power[N[Power[N[Exp[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], N[(N[Sin[y], $MachinePrecision] - N[(N[Sin[x], $MachinePrecision] * 0.0625), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] + 2.0), $MachinePrecision] / N[(3.0 + N[(N[(N[Cos[y], $MachinePrecision] * N[(4.5 - N[Sqrt[11.25], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[Cos[x], $MachinePrecision] * 6.0), $MachinePrecision] / N[(N[Sqrt[5.0], $MachinePrecision] + 1.0), $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)}
↓
\frac{\mathsf{fma}\left(\sqrt{2}, \log \left({\left({\left(e^{\sin x - \sin y \cdot 0.0625}\right)}^{\left(\cos x - \cos y\right)}\right)}^{\left(\sin y - \sin x \cdot 0.0625\right)}\right), 2\right)}{3 + \left(\cos y \cdot \left(4.5 - \sqrt{11.25}\right) + \frac{\cos x \cdot 6}{\sqrt{5} + 1}\right)}

Error?

Bogosity?

Bogosity

Derivation?

  1. Initial program 99.3%

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

  2. Simplified99.2%

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

    [Start]99.3

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

  3. Applied egg-rr99.3%

    \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \color{blue}{\left(\left(4.5 - \sqrt{11.25}\right) \cdot \cos y + \left(\sqrt{5} + -1\right) \cdot \left(\cos x \cdot 1.5\right)\right)}} \]
    Proof

    [Start]99.2

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

    fma-udef [=>]99.2

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

    *-commutative [=>]99.2

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

    add-sqr-sqrt [=>]99.2

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

    sqrt-unprod [=>]99.2

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

    frac-times [=>]99.2

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

    add-sqr-sqrt [<=]99.2

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

    metadata-eval [=>]99.2

    \[ \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \left(\left(4.5 - \sqrt{\frac{5}{\color{blue}{0.4444444444444444}}}\right) \cdot \cos y + \frac{\cos x \cdot \left(\sqrt{5} + -1\right)}{0.6666666666666666}\right)} \]

    metadata-eval [=>]99.2

    \[ \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \left(\left(4.5 - \sqrt{\color{blue}{11.25}}\right) \cdot \cos y + \frac{\cos x \cdot \left(\sqrt{5} + -1\right)}{0.6666666666666666}\right)} \]

    div-inv [=>]99.3

    \[ \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \left(\left(4.5 - \sqrt{11.25}\right) \cdot \cos y + \color{blue}{\left(\cos x \cdot \left(\sqrt{5} + -1\right)\right) \cdot \frac{1}{0.6666666666666666}}\right)} \]

    *-commutative [=>]99.3

    \[ \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \left(\left(4.5 - \sqrt{11.25}\right) \cdot \cos y + \color{blue}{\left(\left(\sqrt{5} + -1\right) \cdot \cos x\right)} \cdot \frac{1}{0.6666666666666666}\right)} \]

    associate-*l* [=>]99.3

    \[ \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \left(\left(4.5 - \sqrt{11.25}\right) \cdot \cos y + \color{blue}{\left(\sqrt{5} + -1\right) \cdot \left(\cos x \cdot \frac{1}{0.6666666666666666}\right)}\right)} \]

    metadata-eval [=>]99.3

    \[ \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \left(\left(4.5 - \sqrt{11.25}\right) \cdot \cos y + \left(\sqrt{5} + -1\right) \cdot \left(\cos x \cdot \color{blue}{1.5}\right)\right)} \]

  4. Applied egg-rr99.3%

    \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \left(\left(4.5 - \sqrt{11.25}\right) \cdot \cos y + \color{blue}{\frac{\left(\cos x \cdot 1.5\right) \cdot 4}{\sqrt{5} + 1}}\right)} \]
    Proof

    [Start]99.3

    \[ \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \left(\left(4.5 - \sqrt{11.25}\right) \cdot \cos y + \left(\sqrt{5} + -1\right) \cdot \left(\cos x \cdot 1.5\right)\right)} \]

    *-commutative [=>]99.3

    \[ \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \left(\left(4.5 - \sqrt{11.25}\right) \cdot \cos y + \color{blue}{\left(\cos x \cdot 1.5\right) \cdot \left(\sqrt{5} + -1\right)}\right)} \]

    flip-+ [=>]99.0

    \[ \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \left(\left(4.5 - \sqrt{11.25}\right) \cdot \cos y + \left(\cos x \cdot 1.5\right) \cdot \color{blue}{\frac{\sqrt{5} \cdot \sqrt{5} - -1 \cdot -1}{\sqrt{5} - -1}}\right)} \]

    associate-*r/ [=>]99.2

    \[ \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \left(\left(4.5 - \sqrt{11.25}\right) \cdot \cos y + \color{blue}{\frac{\left(\cos x \cdot 1.5\right) \cdot \left(\sqrt{5} \cdot \sqrt{5} - -1 \cdot -1\right)}{\sqrt{5} - -1}}\right)} \]

    add-sqr-sqrt [<=]99.3

    \[ \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \left(\left(4.5 - \sqrt{11.25}\right) \cdot \cos y + \frac{\left(\cos x \cdot 1.5\right) \cdot \left(\color{blue}{5} - -1 \cdot -1\right)}{\sqrt{5} - -1}\right)} \]

    metadata-eval [=>]99.3

    \[ \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \left(\left(4.5 - \sqrt{11.25}\right) \cdot \cos y + \frac{\left(\cos x \cdot 1.5\right) \cdot \left(5 - \color{blue}{1}\right)}{\sqrt{5} - -1}\right)} \]

    metadata-eval [=>]99.3

    \[ \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \left(\left(4.5 - \sqrt{11.25}\right) \cdot \cos y + \frac{\left(\cos x \cdot 1.5\right) \cdot \color{blue}{4}}{\sqrt{5} - -1}\right)} \]

    sub-neg [=>]99.3

    \[ \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \left(\left(4.5 - \sqrt{11.25}\right) \cdot \cos y + \frac{\left(\cos x \cdot 1.5\right) \cdot 4}{\color{blue}{\sqrt{5} + \left(--1\right)}}\right)} \]

    metadata-eval [=>]99.3

    \[ \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \left(\left(4.5 - \sqrt{11.25}\right) \cdot \cos y + \frac{\left(\cos x \cdot 1.5\right) \cdot 4}{\sqrt{5} + \color{blue}{1}}\right)} \]

  5. Simplified99.3%

    \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \left(\left(4.5 - \sqrt{11.25}\right) \cdot \cos y + \color{blue}{\frac{\cos x \cdot 6}{\sqrt{5} + 1}}\right)} \]
    Proof

    [Start]99.3

    \[ \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \left(\left(4.5 - \sqrt{11.25}\right) \cdot \cos y + \frac{\left(\cos x \cdot 1.5\right) \cdot 4}{\sqrt{5} + 1}\right)} \]

    associate-*l* [=>]99.3

    \[ \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \left(\left(4.5 - \sqrt{11.25}\right) \cdot \cos y + \frac{\color{blue}{\cos x \cdot \left(1.5 \cdot 4\right)}}{\sqrt{5} + 1}\right)} \]

    metadata-eval [=>]99.3

    \[ \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \left(\left(4.5 - \sqrt{11.25}\right) \cdot \cos y + \frac{\cos x \cdot \color{blue}{6}}{\sqrt{5} + 1}\right)} \]

  6. Applied egg-rr99.4%

    \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \color{blue}{\log \left({\left({\left(e^{\sin x - \sin y \cdot 0.0625}\right)}^{\left(\cos x - \cos y\right)}\right)}^{\left(\sin y - \sin x \cdot 0.0625\right)}\right)}, 2\right)}{3 + \left(\left(4.5 - \sqrt{11.25}\right) \cdot \cos y + \frac{\cos x \cdot 6}{\sqrt{5} + 1}\right)} \]
    Proof

    [Start]99.3

    \[ \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \left(\left(4.5 - \sqrt{11.25}\right) \cdot \cos y + \frac{\cos x \cdot 6}{\sqrt{5} + 1}\right)} \]

    add-log-exp [=>]99.4

    \[ \frac{\mathsf{fma}\left(\sqrt{2}, \color{blue}{\log \left(e^{\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\cos x - \cos y\right)\right)}\right)}, 2\right)}{3 + \left(\left(4.5 - \sqrt{11.25}\right) \cdot \cos y + \frac{\cos x \cdot 6}{\sqrt{5} + 1}\right)} \]

    *-commutative [=>]99.4

    \[ \frac{\mathsf{fma}\left(\sqrt{2}, \log \left(e^{\color{blue}{\left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\cos x - \cos y\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)}}\right), 2\right)}{3 + \left(\left(4.5 - \sqrt{11.25}\right) \cdot \cos y + \frac{\cos x \cdot 6}{\sqrt{5} + 1}\right)} \]

    exp-prod [=>]99.4

    \[ \frac{\mathsf{fma}\left(\sqrt{2}, \log \color{blue}{\left({\left(e^{\left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\cos x - \cos y\right)}\right)}^{\left(\sin y - \frac{\sin x}{16}\right)}\right)}, 2\right)}{3 + \left(\left(4.5 - \sqrt{11.25}\right) \cdot \cos y + \frac{\cos x \cdot 6}{\sqrt{5} + 1}\right)} \]

    exp-prod [=>]99.4

    \[ \frac{\mathsf{fma}\left(\sqrt{2}, \log \left({\color{blue}{\left({\left(e^{\sin x - \frac{\sin y}{16}}\right)}^{\left(\cos x - \cos y\right)}\right)}}^{\left(\sin y - \frac{\sin x}{16}\right)}\right), 2\right)}{3 + \left(\left(4.5 - \sqrt{11.25}\right) \cdot \cos y + \frac{\cos x \cdot 6}{\sqrt{5} + 1}\right)} \]

    div-inv [=>]99.4

    \[ \frac{\mathsf{fma}\left(\sqrt{2}, \log \left({\left({\left(e^{\sin x - \color{blue}{\sin y \cdot \frac{1}{16}}}\right)}^{\left(\cos x - \cos y\right)}\right)}^{\left(\sin y - \frac{\sin x}{16}\right)}\right), 2\right)}{3 + \left(\left(4.5 - \sqrt{11.25}\right) \cdot \cos y + \frac{\cos x \cdot 6}{\sqrt{5} + 1}\right)} \]

    metadata-eval [=>]99.4

    \[ \frac{\mathsf{fma}\left(\sqrt{2}, \log \left({\left({\left(e^{\sin x - \sin y \cdot \color{blue}{0.0625}}\right)}^{\left(\cos x - \cos y\right)}\right)}^{\left(\sin y - \frac{\sin x}{16}\right)}\right), 2\right)}{3 + \left(\left(4.5 - \sqrt{11.25}\right) \cdot \cos y + \frac{\cos x \cdot 6}{\sqrt{5} + 1}\right)} \]

    div-inv [=>]99.4

    \[ \frac{\mathsf{fma}\left(\sqrt{2}, \log \left({\left({\left(e^{\sin x - \sin y \cdot 0.0625}\right)}^{\left(\cos x - \cos y\right)}\right)}^{\left(\sin y - \color{blue}{\sin x \cdot \frac{1}{16}}\right)}\right), 2\right)}{3 + \left(\left(4.5 - \sqrt{11.25}\right) \cdot \cos y + \frac{\cos x \cdot 6}{\sqrt{5} + 1}\right)} \]

    metadata-eval [=>]99.4

    \[ \frac{\mathsf{fma}\left(\sqrt{2}, \log \left({\left({\left(e^{\sin x - \sin y \cdot 0.0625}\right)}^{\left(\cos x - \cos y\right)}\right)}^{\left(\sin y - \sin x \cdot \color{blue}{0.0625}\right)}\right), 2\right)}{3 + \left(\left(4.5 - \sqrt{11.25}\right) \cdot \cos y + \frac{\cos x \cdot 6}{\sqrt{5} + 1}\right)} \]

  7. Final simplification99.4%

    \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \log \left({\left({\left(e^{\sin x - \sin y \cdot 0.0625}\right)}^{\left(\cos x - \cos y\right)}\right)}^{\left(\sin y - \sin x \cdot 0.0625\right)}\right), 2\right)}{3 + \left(\cos y \cdot \left(4.5 - \sqrt{11.25}\right) + \frac{\cos x \cdot 6}{\sqrt{5} + 1}\right)} \]

Alternatives

Alternative 1
Accuracy99.3%
Cost91712
\[\frac{\mathsf{fma}\left(\sqrt{2}, \left(\cos x - \cos y\right) \cdot \left(\left(\sin y + \sin x \cdot -0.0625\right) \cdot \left(\sin x + \sin y \cdot -0.0625\right)\right), 2\right)}{3 \cdot \mathsf{fma}\left(\cos x, -0.5 + \sqrt{5} \cdot 0.5, \mathsf{fma}\left(\cos y, \frac{2}{3 + \sqrt{5}}, 1\right)\right)} \]
Alternative 2
Accuracy99.4%
Cost79040
\[\frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\cos x - \cos y\right) \cdot \left(\sin x - \frac{\sin y}{16}\right)\right), 2\right)}{3 + \left(\frac{\cos x \cdot 6}{\sqrt{5} + 1} + \cos y \cdot \frac{9}{4.5 + \sqrt{11.25}}\right)} \]
Alternative 3
Accuracy99.4%
Cost78912
\[\frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x - \sin y \cdot 0.0625\right) \cdot \left(\left(\cos x - \cos y\right) \cdot \left(\sin y - \sin x \cdot 0.0625\right)\right), 2\right)}{3 + \left(\cos y \cdot \left(4.5 - \sqrt{11.25}\right) + \frac{\cos x \cdot 6}{\sqrt{5} + 1}\right)} \]
Alternative 4
Accuracy99.3%
Cost72896
\[\frac{2 + \sqrt{2} \cdot \left(\left(\sin x - \sin y \cdot 0.0625\right) \cdot \left(\left(\cos x - \cos y\right) \cdot \left(\sin y - \sin x \cdot 0.0625\right)\right)\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)} \]
Alternative 5
Accuracy99.3%
Cost72896
\[\frac{2 + \left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\left(\cos x - \cos y\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right)}{3 \cdot \left(1 + \left(\cos x \cdot \left(\frac{\sqrt{5}}{2} - 0.5\right) + \cos y \cdot \frac{1}{1.5 + \sqrt{1.25}}\right)\right)} \]
Alternative 6
Accuracy99.3%
Cost72768
\[\frac{2 + \left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\left(\cos x - \cos y\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right)}{3 \cdot \left(1 + \left(\cos x \cdot \left(\frac{\sqrt{5}}{2} - 0.5\right) + \cos y \cdot \left(1.5 - \sqrt{1.25}\right)\right)\right)} \]
Alternative 7
Accuracy81.1%
Cost67272
\[\begin{array}{l} t_0 := \sqrt{2} \cdot \sin x\\ t_1 := 1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\\ t_2 := \cos x - \cos y\\ t_3 := \frac{\sqrt{5}}{2}\\ t_4 := \sin y - \frac{\sin x}{16}\\ \mathbf{if}\;x \leq -0.06:\\ \;\;\;\;\frac{2 + \left(t_2 \cdot t_4\right) \cdot t_0}{3 \cdot \left(1 + \left(\cos x \cdot \left(t_3 - 0.5\right) + \cos y \cdot \left(1.5 - t_3\right)\right)\right)}\\ \mathbf{elif}\;x \leq 3.6 \cdot 10^{-26}:\\ \;\;\;\;\frac{2 + \left(t_4 \cdot \left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right)\right) \cdot \left(1 + \left(-0.5 \cdot \left(x \cdot x\right) - \cos y\right)\right)}{3 \cdot \left(t_1 + \cos y \cdot \frac{\frac{4}{3 + \sqrt{5}}}{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + t_2 \cdot \left(t_4 \cdot t_0\right)}{3 \cdot \left(t_1 + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)}\\ \end{array} \]
Alternative 8
Accuracy81.1%
Cost67144
\[\begin{array}{l} t_0 := 3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)\\ t_1 := \sqrt{2} \cdot \sin x\\ t_2 := \cos x - \cos y\\ t_3 := \frac{\sqrt{5}}{2}\\ t_4 := \sin y - \frac{\sin x}{16}\\ \mathbf{if}\;x \leq -0.032:\\ \;\;\;\;\frac{2 + \left(t_2 \cdot t_4\right) \cdot t_1}{3 \cdot \left(1 + \left(\cos x \cdot \left(t_3 - 0.5\right) + \cos y \cdot \left(1.5 - t_3\right)\right)\right)}\\ \mathbf{elif}\;x \leq 3.6 \cdot 10^{-26}:\\ \;\;\;\;\frac{2 + \left(t_4 \cdot \left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right)\right) \cdot \left(1 + \left(-0.5 \cdot \left(x \cdot x\right) - \cos y\right)\right)}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + t_2 \cdot \left(t_4 \cdot t_1\right)}{t_0}\\ \end{array} \]
Alternative 9
Accuracy81.0%
Cost67016
\[\begin{array}{l} t_0 := 3 - \sqrt{5}\\ t_1 := \sqrt{2} \cdot \sin x\\ t_2 := \cos x - \cos y\\ t_3 := \sin y - \frac{\sin x}{16}\\ t_4 := \frac{\sqrt{5}}{2}\\ t_5 := \sqrt{5} + -1\\ \mathbf{if}\;x \leq -0.052:\\ \;\;\;\;\frac{2 + \left(t_2 \cdot t_3\right) \cdot t_1}{3 \cdot \left(1 + \left(\cos x \cdot \left(t_4 - 0.5\right) + \cos y \cdot \left(1.5 - t_4\right)\right)\right)}\\ \mathbf{elif}\;x \leq 3.6 \cdot 10^{-26}:\\ \;\;\;\;\frac{0.3333333333333333 \cdot \left(2 + \sqrt{2} \cdot \left(\left(1 + \left(-0.5 \cdot \left(x \cdot x\right) - \cos y\right)\right) \cdot \left(\left(\sin y - \sin x \cdot 0.0625\right) \cdot \left(\sin x + \sin y \cdot -0.0625\right)\right)\right)\right)}{1 + 0.5 \cdot \left(\cos x \cdot t_5 + \cos y \cdot t_0\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + t_2 \cdot \left(t_3 \cdot t_1\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{t_5}{2}\right) + \cos y \cdot \frac{t_0}{2}\right)}\\ \end{array} \]
Alternative 10
Accuracy81.0%
Cost66505
\[\begin{array}{l} t_0 := \frac{\sqrt{5}}{2}\\ t_1 := \sin y - \frac{\sin x}{16}\\ \mathbf{if}\;x \leq -0.019 \lor \neg \left(x \leq 3.6 \cdot 10^{-26}\right):\\ \;\;\;\;\frac{2 + \left(\left(\cos x - \cos y\right) \cdot t_1\right) \cdot \left(\sqrt{2} \cdot \sin x\right)}{3 \cdot \left(1 + \left(\cos x \cdot \left(t_0 - 0.5\right) + \cos y \cdot \left(1.5 - t_0\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(1 + \left(-0.5 \cdot \left(x \cdot x\right) - \cos y\right)\right) \cdot \left(t_1 \cdot \left(\sqrt{2} \cdot \left(x + \sin y \cdot -0.0625\right)\right)\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)}\\ \end{array} \]
Alternative 11
Accuracy81.0%
Cost66504
\[\begin{array}{l} t_0 := \cos x - \cos y\\ t_1 := \frac{\sqrt{5}}{2}\\ t_2 := \sin y - \frac{\sin x}{16}\\ t_3 := \sqrt{2} \cdot \sin x\\ t_4 := 3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)\\ \mathbf{if}\;x \leq -0.027:\\ \;\;\;\;\frac{2 + \left(t_0 \cdot t_2\right) \cdot t_3}{3 \cdot \left(1 + \left(\cos x \cdot \left(t_1 - 0.5\right) + \cos y \cdot \left(1.5 - t_1\right)\right)\right)}\\ \mathbf{elif}\;x \leq 3.6 \cdot 10^{-26}:\\ \;\;\;\;\frac{2 + \left(1 + \left(-0.5 \cdot \left(x \cdot x\right) - \cos y\right)\right) \cdot \left(t_2 \cdot \left(\sqrt{2} \cdot \left(x + \sin y \cdot -0.0625\right)\right)\right)}{t_4}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + t_0 \cdot \left(t_2 \cdot t_3\right)}{t_4}\\ \end{array} \]
Alternative 12
Accuracy79.9%
Cost60744
\[\begin{array}{l} t_0 := 3 - \sqrt{5}\\ t_1 := \cos x + -1\\ t_2 := \sqrt{5} + -1\\ t_3 := {\sin x}^{2}\\ \mathbf{if}\;x \leq -0.038:\\ \;\;\;\;\frac{0.3333333333333333 \cdot \mathsf{fma}\left(-0.0625, \left(\sqrt{2} \cdot t_3\right) \cdot t_1, 2\right)}{1 + 0.5 \cdot \left(\cos x \cdot t_2 + \cos y \cdot t_0\right)}\\ \mathbf{elif}\;x \leq 0.028:\\ \;\;\;\;\frac{2 + \left(1 + \left(-0.5 \cdot \left(x \cdot x\right) - \cos y\right)\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\sqrt{2} \cdot \left(x + \sin y \cdot -0.0625\right)\right)\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{t_2}{2}\right) + \cos y \cdot \frac{t_0}{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{2}, -0.0625 \cdot \left(t_3 \cdot t_1\right), 2\right)}{3 + \left(\cos y \cdot \left(4.5 - \sqrt{11.25}\right) + \frac{\cos x \cdot 6}{\sqrt{5} + 1}\right)}\\ \end{array} \]
Alternative 13
Accuracy79.6%
Cost60488
\[\begin{array}{l} t_0 := \sqrt{5} \cdot 0.5\\ t_1 := \sin y - \frac{\sin x}{16}\\ t_2 := 1 - \cos y\\ \mathbf{if}\;y \leq -0.00385:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{2}, -0.0625 \cdot \left(t_2 \cdot {\sin y}^{2}\right), 2\right)}{3 + \left(\cos y \cdot \left(4.5 - \sqrt{11.25}\right) + \frac{\cos x \cdot 6}{\sqrt{5} + 1}\right)}\\ \mathbf{elif}\;y \leq 0.0029:\\ \;\;\;\;\frac{2 + \left(t_1 \cdot \left(\sqrt{2} \cdot \sin x\right)\right) \cdot \left(\cos x + \left(-1 + 0.5 \cdot \left(y \cdot y\right)\right)\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(t_1 \cdot t_2\right)}{3 \cdot \left(1 + \left(\left(t_0 + \cos y \cdot \left(1.5 - t_0\right)\right) - 0.5\right)\right)}\\ \end{array} \]
Alternative 14
Accuracy79.9%
Cost60296
\[\begin{array}{l} t_0 := 3 - \sqrt{5}\\ t_1 := \cos x + -1\\ t_2 := \sqrt{5} + -1\\ t_3 := {\sin x}^{2}\\ \mathbf{if}\;x \leq -0.0039:\\ \;\;\;\;\frac{0.3333333333333333 \cdot \mathsf{fma}\left(-0.0625, \left(\sqrt{2} \cdot t_3\right) \cdot t_1, 2\right)}{1 + 0.5 \cdot \left(\cos x \cdot t_2 + \cos y \cdot t_0\right)}\\ \mathbf{elif}\;x \leq 0.00355:\\ \;\;\;\;\frac{2 + \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right) \cdot \left(-0.0625 \cdot {\sin y}^{2} + x \cdot \left(\sin y \cdot 1.00390625\right)\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{t_2}{2}\right) + \cos y \cdot \frac{t_0}{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{2}, -0.0625 \cdot \left(t_3 \cdot t_1\right), 2\right)}{3 + \left(\cos y \cdot \left(4.5 - \sqrt{11.25}\right) + \frac{\cos x \cdot 6}{\sqrt{5} + 1}\right)}\\ \end{array} \]
Alternative 15
Accuracy79.2%
Cost59528
\[\begin{array}{l} t_0 := \sqrt{5} \cdot 0.5\\ t_1 := {\sin x}^{2} \cdot \left(\cos x + -1\right)\\ \mathbf{if}\;x \leq -0.00048:\\ \;\;\;\;0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left(\sqrt{2} \cdot t_1\right)}{1 + \left(0.5 \cdot \left(\cos x \cdot \left(\sqrt{5} + -1\right)\right) + 0.5 \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)\right)}\\ \mathbf{elif}\;x \leq 3.6 \cdot 10^{-26}:\\ \;\;\;\;\frac{2 + \left(\sqrt{2} \cdot \left(x + \sin y \cdot -0.0625\right)\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(1 + \left(-0.5 \cdot \left(x \cdot x\right) - \cos y\right)\right)\right)}{3 \cdot \left(1 + \left(\left(t_0 + \cos y \cdot \left(1.5 - t_0\right)\right) - 0.5\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{2}, -0.0625 \cdot t_1, 2\right)}{3 + \left(\cos y \cdot \left(4.5 - \sqrt{11.25}\right) + \frac{\cos x \cdot 6}{\sqrt{5} + 1}\right)}\\ \end{array} \]
Alternative 16
Accuracy79.2%
Cost59528
\[\begin{array}{l} t_0 := \cos x + -1\\ t_1 := \sqrt{5} \cdot 0.5\\ t_2 := {\sin x}^{2}\\ \mathbf{if}\;x \leq -0.00015:\\ \;\;\;\;\frac{0.3333333333333333 \cdot \mathsf{fma}\left(-0.0625, \left(\sqrt{2} \cdot t_2\right) \cdot t_0, 2\right)}{1 + 0.5 \cdot \left(\cos x \cdot \left(\sqrt{5} + -1\right) + \cos y \cdot \left(3 - \sqrt{5}\right)\right)}\\ \mathbf{elif}\;x \leq 3.6 \cdot 10^{-26}:\\ \;\;\;\;\frac{2 + \left(\sqrt{2} \cdot \left(x + \sin y \cdot -0.0625\right)\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(1 + \left(-0.5 \cdot \left(x \cdot x\right) - \cos y\right)\right)\right)}{3 \cdot \left(1 + \left(\left(t_1 + \cos y \cdot \left(1.5 - t_1\right)\right) - 0.5\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{2}, -0.0625 \cdot \left(t_2 \cdot t_0\right), 2\right)}{3 + \left(\cos y \cdot \left(4.5 - \sqrt{11.25}\right) + \frac{\cos x \cdot 6}{\sqrt{5} + 1}\right)}\\ \end{array} \]
Alternative 17
Accuracy79.2%
Cost54216
\[\begin{array}{l} t_0 := 2 + -0.0625 \cdot \left(\sqrt{2} \cdot \left({\sin x}^{2} \cdot \left(\cos x + -1\right)\right)\right)\\ t_1 := \sqrt{5} + -1\\ t_2 := \sqrt{5} \cdot 0.5\\ \mathbf{if}\;x \leq -0.000145:\\ \;\;\;\;0.3333333333333333 \cdot \frac{t_0}{1 + \left(0.5 \cdot \left(\cos x \cdot t_1\right) + 0.5 \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)\right)}\\ \mathbf{elif}\;x \leq 3.6 \cdot 10^{-26}:\\ \;\;\;\;\frac{2 + \left(\sqrt{2} \cdot \left(x + \sin y \cdot -0.0625\right)\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(1 + \left(-0.5 \cdot \left(x \cdot x\right) - \cos y\right)\right)\right)}{3 \cdot \left(1 + \left(\left(t_2 + \cos y \cdot \left(1.5 - t_2\right)\right) - 0.5\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0}{3 \cdot \left(\left(1 + \cos x \cdot \frac{t_1}{2}\right) + \cos y \cdot \frac{\frac{4}{3 + \sqrt{5}}}{2}\right)}\\ \end{array} \]
Alternative 18
Accuracy79.2%
Cost53832
\[\begin{array}{l} t_0 := \sqrt{5} \cdot 0.5\\ t_1 := 2 + -0.0625 \cdot \left(\sqrt{2} \cdot \left({\sin x}^{2} \cdot \left(\cos x + -1\right)\right)\right)\\ t_2 := \sqrt{5} + -1\\ \mathbf{if}\;x \leq -0.000225:\\ \;\;\;\;0.3333333333333333 \cdot \frac{t_1}{1 + \left(0.5 \cdot \left(\cos x \cdot t_2\right) + 0.5 \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)\right)}\\ \mathbf{elif}\;x \leq 3.6 \cdot 10^{-26}:\\ \;\;\;\;\frac{2 + \left(\sqrt{2} \cdot \left(x + \sin y \cdot -0.0625\right)\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(1 - \cos y\right)\right)}{3 \cdot \left(1 + \left(\left(t_0 + \cos y \cdot \left(1.5 - t_0\right)\right) - 0.5\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_1}{3 \cdot \left(\left(1 + \cos x \cdot \frac{t_2}{2}\right) + \cos y \cdot \frac{\frac{4}{3 + \sqrt{5}}}{2}\right)}\\ \end{array} \]
Alternative 19
Accuracy79.1%
Cost53513
\[\begin{array}{l} t_0 := \sqrt{5} \cdot 0.5\\ \mathbf{if}\;x \leq -0.000145 \lor \neg \left(x \leq 3.6 \cdot 10^{-26}\right):\\ \;\;\;\;0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left(\sqrt{2} \cdot \left({\sin x}^{2} \cdot \left(\cos x + -1\right)\right)\right)}{1 + \left(0.5 \cdot \left(\cos x \cdot \left(\sqrt{5} + -1\right)\right) + 0.5 \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left(\left(1 - \cos y\right) \cdot \left(\sqrt{2} \cdot {\sin y}^{2}\right)\right)}{3 \cdot \left(1 + \left(\left(t_0 + \cos y \cdot \left(1.5 - t_0\right)\right) - 0.5\right)\right)}\\ \end{array} \]
Alternative 20
Accuracy79.6%
Cost53513
\[\begin{array}{l} t_0 := 3 - \sqrt{5}\\ t_1 := \sqrt{5} + -1\\ \mathbf{if}\;y \leq -0.000105 \lor \neg \left(y \leq 1.35 \cdot 10^{-6}\right):\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left(\left(1 - \cos y\right) \cdot \left(\sqrt{2} \cdot {\sin y}^{2}\right)\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{t_1}{2}\right) + \cos y \cdot \frac{t_0}{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.3333333333333333 \cdot \mathsf{fma}\left(-0.0625, \left(\sqrt{2} \cdot {\sin x}^{2}\right) \cdot \left(\cos x + -1\right), 2\right)}{1 + 0.5 \cdot \left(t_0 + \cos x \cdot t_1\right)}\\ \end{array} \]
Alternative 21
Accuracy78.5%
Cost52996
\[\begin{array}{l} t_0 := \sqrt{5} \cdot 0.5\\ t_1 := \cos x + -1\\ t_2 := \cos x \cdot \left(\sqrt{5} + -1\right)\\ t_3 := {\sin x}^{2}\\ \mathbf{if}\;x \leq -0.000145:\\ \;\;\;\;\frac{0.3333333333333333 \cdot \mathsf{fma}\left(-0.0625, \left(\sqrt{2} \cdot t_3\right) \cdot t_1, 2\right)}{1 + 0.5 \cdot \left(\left(3 - \sqrt{5}\right) + t_2\right)}\\ \mathbf{elif}\;x \leq 3.6 \cdot 10^{-26}:\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left(\left(1 - \cos y\right) \cdot \left(\sqrt{2} \cdot {\sin y}^{2}\right)\right)}{3 \cdot \left(1 + \left(\left(t_0 + \cos y \cdot \left(1.5 - t_0\right)\right) - 0.5\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left(\sqrt{2} \cdot \left(t_3 \cdot t_1\right)\right)}{1 + \left(0.5 \cdot t_2 + 2 \cdot \frac{1}{3 + \sqrt{5}}\right)}\\ \end{array} \]
Alternative 22
Accuracy78.5%
Cost47112
\[\begin{array}{l} t_0 := \sqrt{5} \cdot 0.5\\ t_1 := 0.5 \cdot \left(\cos x \cdot \left(\sqrt{5} + -1\right)\right)\\ t_2 := 2 + -0.0625 \cdot \left(\sqrt{2} \cdot \left({\sin x}^{2} \cdot \left(\cos x + -1\right)\right)\right)\\ \mathbf{if}\;x \leq -0.000145:\\ \;\;\;\;0.3333333333333333 \cdot \frac{t_2}{1 + \left(t_1 + 0.5 \cdot \left(3 - \sqrt{5}\right)\right)}\\ \mathbf{elif}\;x \leq 3.6 \cdot 10^{-26}:\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left(\left(1 - \cos y\right) \cdot \left(\sqrt{2} \cdot {\sin y}^{2}\right)\right)}{3 \cdot \left(1 + \left(\left(t_0 + \cos y \cdot \left(1.5 - t_0\right)\right) - 0.5\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{t_2}{1 + \left(t_1 + 2 \cdot \frac{1}{3 + \sqrt{5}}\right)}\\ \end{array} \]
Alternative 23
Accuracy78.4%
Cost46985
\[\begin{array}{l} \mathbf{if}\;x \leq -0.000145 \lor \neg \left(x \leq 3.6 \cdot 10^{-26}\right):\\ \;\;\;\;0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left(\sqrt{2} \cdot \left({\sin x}^{2} \cdot \left(\cos x + -1\right)\right)\right)}{1 + \left(0.5 \cdot \left(\cos x \cdot \left(\sqrt{5} + -1\right)\right) + 0.5 \cdot \left(3 - \sqrt{5}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left(\sqrt{2} \cdot \left(\left(1 - \cos y\right) \cdot {\sin y}^{2}\right)\right)}{0.5 + \left(\sqrt{5} \cdot 0.5 + 2 \cdot \frac{\cos y}{3 + \sqrt{5}}\right)}\\ \end{array} \]
Alternative 24
Accuracy78.5%
Cost46985
\[\begin{array}{l} t_0 := \sqrt{5} \cdot 0.5\\ \mathbf{if}\;x \leq -0.000145 \lor \neg \left(x \leq 3.6 \cdot 10^{-26}\right):\\ \;\;\;\;0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left(\sqrt{2} \cdot \left({\sin x}^{2} \cdot \left(\cos x + -1\right)\right)\right)}{1 + \left(0.5 \cdot \left(\cos x \cdot \left(\sqrt{5} + -1\right)\right) + 0.5 \cdot \left(3 - \sqrt{5}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left(\left(1 - \cos y\right) \cdot \left(\sqrt{2} \cdot {\sin y}^{2}\right)\right)}{3 \cdot \left(1 + \left(\left(t_0 + \cos y \cdot \left(1.5 - t_0\right)\right) - 0.5\right)\right)}\\ \end{array} \]
Alternative 25
Accuracy59.4%
Cost46592
\[0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left(\sqrt{2} \cdot \left(\left(1 - \cos y\right) \cdot {\sin y}^{2}\right)\right)}{0.5 + \left(\sqrt{5} \cdot 0.5 + 2 \cdot \frac{\cos y}{3 + \sqrt{5}}\right)} \]
Alternative 26
Accuracy42.3%
Cost45504
\[\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{0.6666666666666666}{\mathsf{fma}\left(-1 + \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5}\right), 0.5, 1\right)}\right)\right) \]
Alternative 27
Accuracy42.3%
Cost33088
\[\frac{0.6666666666666666}{1 + 0.5 \cdot \left(\left(\sqrt{5} + -1\right) + \log \left(1 + \mathsf{expm1}\left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)\right)\right)} \]
Alternative 28
Accuracy42.3%
Cost32832
\[0.6666666666666666 \cdot \frac{1}{\mathsf{fma}\left(-1 + \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5}\right), 0.5, 1\right)} \]
Alternative 29
Accuracy42.3%
Cost32704
\[\frac{0.6666666666666666}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} + -1\right), 1\right)} \]
Alternative 30
Accuracy42.3%
Cost20032
\[\frac{0.6666666666666666}{0.5 + 0.5 \cdot \left(\sqrt{5} + \cos y \cdot \left(3 - \sqrt{5}\right)\right)} \]
Alternative 31
Accuracy40.4%
Cost64
\[0.3333333333333333 \]

Error

Reproduce?

herbie shell --seed 2023160 
(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))))))