?

Average Accuracy: 99.3% → 99.3%
Time: 40.7s
Precision: binary64
Cost: 79040

?

\[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
↓
\[\frac{0.3333333333333333 \cdot \mathsf{fma}\left(\sqrt{2} \cdot \left(\sin x + -0.0625 \cdot \sin y\right), \left(\cos x - \cos y\right) \cdot \left(\sin y + \sin x \cdot -0.0625\right), 2\right)}{1 + \left(\cos y \cdot \left(1.5 + -0.5 \cdot \sqrt{5}\right) + \frac{\cos x}{0.5 + \sqrt{1.25}}\right)} \]
(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
 (/
  (*
   0.3333333333333333
   (fma
    (* (sqrt 2.0) (+ (sin x) (* -0.0625 (sin y))))
    (* (- (cos x) (cos y)) (+ (sin y) (* (sin x) -0.0625)))
    2.0))
  (+
   1.0
   (+
    (* (cos y) (+ 1.5 (* -0.5 (sqrt 5.0))))
    (/ (cos x) (+ 0.5 (sqrt 1.25)))))))
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 (0.3333333333333333 * fma((sqrt(2.0) * (sin(x) + (-0.0625 * sin(y)))), ((cos(x) - cos(y)) * (sin(y) + (sin(x) * -0.0625))), 2.0)) / (1.0 + ((cos(y) * (1.5 + (-0.5 * sqrt(5.0)))) + (cos(x) / (0.5 + sqrt(1.25)))));
}

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(Float64(0.3333333333333333 * fma(Float64(sqrt(2.0) * Float64(sin(x) + Float64(-0.0625 * sin(y)))), Float64(Float64(cos(x) - cos(y)) * Float64(sin(y) + Float64(sin(x) * -0.0625))), 2.0)) / Float64(1.0 + Float64(Float64(cos(y) * Float64(1.5 + Float64(-0.5 * sqrt(5.0)))) + Float64(cos(x) / Float64(0.5 + sqrt(1.25))))))
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[(0.3333333333333333 * N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Sin[x], $MachinePrecision] + N[(-0.0625 * N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] + N[(N[Sin[x], $MachinePrecision] * -0.0625), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(N[(N[Cos[y], $MachinePrecision] * N[(1.5 + N[(-0.5 * N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Cos[x], $MachinePrecision] / N[(0.5 + N[Sqrt[1.25], $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)}
↓
\frac{0.3333333333333333 \cdot \mathsf{fma}\left(\sqrt{2} \cdot \left(\sin x + -0.0625 \cdot \sin y\right), \left(\cos x - \cos y\right) \cdot \left(\sin y + \sin x \cdot -0.0625\right), 2\right)}{1 + \left(\cos y \cdot \left(1.5 + -0.5 \cdot \sqrt{5}\right) + \frac{\cos x}{0.5 + \sqrt{1.25}}\right)}

Error?

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. Simplified99.3%

    \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right), \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right), 2\right)}{3 \cdot \mathsf{fma}\left(\cos y, 1.5 - \frac{\sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5}}{2} + -0.5, 1\right)\right)}} \]
    Proof

    [Start]99.3

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

    +-commutative [=>]99.3

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

    associate-*l* [=>]99.3

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

    fma-def [=>]99.3

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

    +-commutative [=>]99.3

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

    *-commutative [=>]99.3

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

    fma-def [=>]99.3

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

  3. Taylor expanded in x around inf 99.2%

    \[\leadsto \color{blue}{0.3333333333333333 \cdot \frac{2 + \sqrt{2} \cdot \left(\left(\cos x - \cos y\right) \cdot \left(\left(\sin y - 0.0625 \cdot \sin x\right) \cdot \left(\sin x - 0.0625 \cdot \sin y\right)\right)\right)}{1 + \left(\left(1.5 - 0.5 \cdot \sqrt{5}\right) \cdot \cos y + \cos x \cdot \left(0.5 \cdot \sqrt{5} - 0.5\right)\right)}} \]

  4. Simplified99.2%

    \[\leadsto \color{blue}{\frac{0.3333333333333333 \cdot \mathsf{fma}\left(\sqrt{2} \cdot \left(\sin x + -0.0625 \cdot \sin y\right), \left(\cos x - \cos y\right) \cdot \left(\sin y + -0.0625 \cdot \sin x\right), 2\right)}{1 + \mathsf{fma}\left(1.5 + \sqrt{5} \cdot -0.5, \cos y, \cos x \cdot \mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right)\right)}} \]
    Proof

    [Start]99.2

    \[ 0.3333333333333333 \cdot \frac{2 + \sqrt{2} \cdot \left(\left(\cos x - \cos y\right) \cdot \left(\left(\sin y - 0.0625 \cdot \sin x\right) \cdot \left(\sin x - 0.0625 \cdot \sin y\right)\right)\right)}{1 + \left(\left(1.5 - 0.5 \cdot \sqrt{5}\right) \cdot \cos y + \cos x \cdot \left(0.5 \cdot \sqrt{5} - 0.5\right)\right)} \]

    associate-*r/ [=>]99.2

    \[ \color{blue}{\frac{0.3333333333333333 \cdot \left(2 + \sqrt{2} \cdot \left(\left(\cos x - \cos y\right) \cdot \left(\left(\sin y - 0.0625 \cdot \sin x\right) \cdot \left(\sin x - 0.0625 \cdot \sin y\right)\right)\right)\right)}{1 + \left(\left(1.5 - 0.5 \cdot \sqrt{5}\right) \cdot \cos y + \cos x \cdot \left(0.5 \cdot \sqrt{5} - 0.5\right)\right)}} \]

  5. Taylor expanded in y around inf 99.2%

    \[\leadsto \frac{0.3333333333333333 \cdot \mathsf{fma}\left(\sqrt{2} \cdot \left(\sin x + -0.0625 \cdot \sin y\right), \left(\cos x - \cos y\right) \cdot \left(\sin y + -0.0625 \cdot \sin x\right), 2\right)}{1 + \color{blue}{\left(\cos y \cdot \left(-0.5 \cdot \sqrt{5} + 1.5\right) + \cos x \cdot \left(0.5 \cdot \sqrt{5} - 0.5\right)\right)}} \]

  6. Applied egg-rr99.3%

    \[\leadsto \frac{0.3333333333333333 \cdot \mathsf{fma}\left(\sqrt{2} \cdot \left(\sin x + -0.0625 \cdot \sin y\right), \left(\cos x - \cos y\right) \cdot \left(\sin y + -0.0625 \cdot \sin x\right), 2\right)}{1 + \color{blue}{\mathsf{fma}\left(-0.5, \sqrt{5} \cdot \cos y, \mathsf{fma}\left(1.5, \cos y, \frac{\cos x}{\sqrt{1.25} + 0.5}\right)\right)}} \]
    Proof

    [Start]99.2

    \[ \frac{0.3333333333333333 \cdot \mathsf{fma}\left(\sqrt{2} \cdot \left(\sin x + -0.0625 \cdot \sin y\right), \left(\cos x - \cos y\right) \cdot \left(\sin y + -0.0625 \cdot \sin x\right), 2\right)}{1 + \left(\cos y \cdot \left(-0.5 \cdot \sqrt{5} + 1.5\right) + \cos x \cdot \left(0.5 \cdot \sqrt{5} - 0.5\right)\right)} \]

    distribute-lft-in [=>]99.2

    \[ \frac{0.3333333333333333 \cdot \mathsf{fma}\left(\sqrt{2} \cdot \left(\sin x + -0.0625 \cdot \sin y\right), \left(\cos x - \cos y\right) \cdot \left(\sin y + -0.0625 \cdot \sin x\right), 2\right)}{1 + \left(\color{blue}{\left(\cos y \cdot \left(-0.5 \cdot \sqrt{5}\right) + \cos y \cdot 1.5\right)} + \cos x \cdot \left(0.5 \cdot \sqrt{5} - 0.5\right)\right)} \]

    associate-+l+ [=>]99.1

    \[ \frac{0.3333333333333333 \cdot \mathsf{fma}\left(\sqrt{2} \cdot \left(\sin x + -0.0625 \cdot \sin y\right), \left(\cos x - \cos y\right) \cdot \left(\sin y + -0.0625 \cdot \sin x\right), 2\right)}{1 + \color{blue}{\left(\cos y \cdot \left(-0.5 \cdot \sqrt{5}\right) + \left(\cos y \cdot 1.5 + \cos x \cdot \left(0.5 \cdot \sqrt{5} - 0.5\right)\right)\right)}} \]

    *-commutative [=>]99.1

    \[ \frac{0.3333333333333333 \cdot \mathsf{fma}\left(\sqrt{2} \cdot \left(\sin x + -0.0625 \cdot \sin y\right), \left(\cos x - \cos y\right) \cdot \left(\sin y + -0.0625 \cdot \sin x\right), 2\right)}{1 + \left(\color{blue}{\left(-0.5 \cdot \sqrt{5}\right) \cdot \cos y} + \left(\cos y \cdot 1.5 + \cos x \cdot \left(0.5 \cdot \sqrt{5} - 0.5\right)\right)\right)} \]

    associate-*l* [=>]99.1

    \[ \frac{0.3333333333333333 \cdot \mathsf{fma}\left(\sqrt{2} \cdot \left(\sin x + -0.0625 \cdot \sin y\right), \left(\cos x - \cos y\right) \cdot \left(\sin y + -0.0625 \cdot \sin x\right), 2\right)}{1 + \left(\color{blue}{-0.5 \cdot \left(\sqrt{5} \cdot \cos y\right)} + \left(\cos y \cdot 1.5 + \cos x \cdot \left(0.5 \cdot \sqrt{5} - 0.5\right)\right)\right)} \]

    fma-def [=>]99.1

    \[ \frac{0.3333333333333333 \cdot \mathsf{fma}\left(\sqrt{2} \cdot \left(\sin x + -0.0625 \cdot \sin y\right), \left(\cos x - \cos y\right) \cdot \left(\sin y + -0.0625 \cdot \sin x\right), 2\right)}{1 + \color{blue}{\mathsf{fma}\left(-0.5, \sqrt{5} \cdot \cos y, \cos y \cdot 1.5 + \cos x \cdot \left(0.5 \cdot \sqrt{5} - 0.5\right)\right)}} \]

    *-commutative [=>]99.1

    \[ \frac{0.3333333333333333 \cdot \mathsf{fma}\left(\sqrt{2} \cdot \left(\sin x + -0.0625 \cdot \sin y\right), \left(\cos x - \cos y\right) \cdot \left(\sin y + -0.0625 \cdot \sin x\right), 2\right)}{1 + \mathsf{fma}\left(-0.5, \sqrt{5} \cdot \cos y, \color{blue}{1.5 \cdot \cos y} + \cos x \cdot \left(0.5 \cdot \sqrt{5} - 0.5\right)\right)} \]

    fma-def [=>]99.2

    \[ \frac{0.3333333333333333 \cdot \mathsf{fma}\left(\sqrt{2} \cdot \left(\sin x + -0.0625 \cdot \sin y\right), \left(\cos x - \cos y\right) \cdot \left(\sin y + -0.0625 \cdot \sin x\right), 2\right)}{1 + \mathsf{fma}\left(-0.5, \sqrt{5} \cdot \cos y, \color{blue}{\mathsf{fma}\left(1.5, \cos y, \cos x \cdot \left(0.5 \cdot \sqrt{5} - 0.5\right)\right)}\right)} \]

    flip-- [=>]98.9

    \[ \frac{0.3333333333333333 \cdot \mathsf{fma}\left(\sqrt{2} \cdot \left(\sin x + -0.0625 \cdot \sin y\right), \left(\cos x - \cos y\right) \cdot \left(\sin y + -0.0625 \cdot \sin x\right), 2\right)}{1 + \mathsf{fma}\left(-0.5, \sqrt{5} \cdot \cos y, \mathsf{fma}\left(1.5, \cos y, \cos x \cdot \color{blue}{\frac{\left(0.5 \cdot \sqrt{5}\right) \cdot \left(0.5 \cdot \sqrt{5}\right) - 0.5 \cdot 0.5}{0.5 \cdot \sqrt{5} + 0.5}}\right)\right)} \]

    associate-*r/ [=>]99.0

    \[ \frac{0.3333333333333333 \cdot \mathsf{fma}\left(\sqrt{2} \cdot \left(\sin x + -0.0625 \cdot \sin y\right), \left(\cos x - \cos y\right) \cdot \left(\sin y + -0.0625 \cdot \sin x\right), 2\right)}{1 + \mathsf{fma}\left(-0.5, \sqrt{5} \cdot \cos y, \mathsf{fma}\left(1.5, \cos y, \color{blue}{\frac{\cos x \cdot \left(\left(0.5 \cdot \sqrt{5}\right) \cdot \left(0.5 \cdot \sqrt{5}\right) - 0.5 \cdot 0.5\right)}{0.5 \cdot \sqrt{5} + 0.5}}\right)\right)} \]

  7. Simplified99.3%

    \[\leadsto \frac{0.3333333333333333 \cdot \mathsf{fma}\left(\sqrt{2} \cdot \left(\sin x + -0.0625 \cdot \sin y\right), \left(\cos x - \cos y\right) \cdot \left(\sin y + -0.0625 \cdot \sin x\right), 2\right)}{1 + \color{blue}{\left(\mathsf{fma}\left(-0.5, \sqrt{5}, 1.5\right) \cdot \cos y + \frac{\cos x}{0.5 + \sqrt{1.25}}\right)}} \]
    Proof

    [Start]99.3

    \[ \frac{0.3333333333333333 \cdot \mathsf{fma}\left(\sqrt{2} \cdot \left(\sin x + -0.0625 \cdot \sin y\right), \left(\cos x - \cos y\right) \cdot \left(\sin y + -0.0625 \cdot \sin x\right), 2\right)}{1 + \mathsf{fma}\left(-0.5, \sqrt{5} \cdot \cos y, \mathsf{fma}\left(1.5, \cos y, \frac{\cos x}{\sqrt{1.25} + 0.5}\right)\right)} \]

    fma-udef [=>]99.3

    \[ \frac{0.3333333333333333 \cdot \mathsf{fma}\left(\sqrt{2} \cdot \left(\sin x + -0.0625 \cdot \sin y\right), \left(\cos x - \cos y\right) \cdot \left(\sin y + -0.0625 \cdot \sin x\right), 2\right)}{1 + \color{blue}{\left(-0.5 \cdot \left(\sqrt{5} \cdot \cos y\right) + \mathsf{fma}\left(1.5, \cos y, \frac{\cos x}{\sqrt{1.25} + 0.5}\right)\right)}} \]

    fma-udef [=>]99.2

    \[ \frac{0.3333333333333333 \cdot \mathsf{fma}\left(\sqrt{2} \cdot \left(\sin x + -0.0625 \cdot \sin y\right), \left(\cos x - \cos y\right) \cdot \left(\sin y + -0.0625 \cdot \sin x\right), 2\right)}{1 + \left(-0.5 \cdot \left(\sqrt{5} \cdot \cos y\right) + \color{blue}{\left(1.5 \cdot \cos y + \frac{\cos x}{\sqrt{1.25} + 0.5}\right)}\right)} \]

    associate-+r+ [=>]99.3

    \[ \frac{0.3333333333333333 \cdot \mathsf{fma}\left(\sqrt{2} \cdot \left(\sin x + -0.0625 \cdot \sin y\right), \left(\cos x - \cos y\right) \cdot \left(\sin y + -0.0625 \cdot \sin x\right), 2\right)}{1 + \color{blue}{\left(\left(-0.5 \cdot \left(\sqrt{5} \cdot \cos y\right) + 1.5 \cdot \cos y\right) + \frac{\cos x}{\sqrt{1.25} + 0.5}\right)}} \]

    associate-*r* [=>]99.3

    \[ \frac{0.3333333333333333 \cdot \mathsf{fma}\left(\sqrt{2} \cdot \left(\sin x + -0.0625 \cdot \sin y\right), \left(\cos x - \cos y\right) \cdot \left(\sin y + -0.0625 \cdot \sin x\right), 2\right)}{1 + \left(\left(\color{blue}{\left(-0.5 \cdot \sqrt{5}\right) \cdot \cos y} + 1.5 \cdot \cos y\right) + \frac{\cos x}{\sqrt{1.25} + 0.5}\right)} \]

    distribute-rgt-in [<=]99.3

    \[ \frac{0.3333333333333333 \cdot \mathsf{fma}\left(\sqrt{2} \cdot \left(\sin x + -0.0625 \cdot \sin y\right), \left(\cos x - \cos y\right) \cdot \left(\sin y + -0.0625 \cdot \sin x\right), 2\right)}{1 + \left(\color{blue}{\cos y \cdot \left(-0.5 \cdot \sqrt{5} + 1.5\right)} + \frac{\cos x}{\sqrt{1.25} + 0.5}\right)} \]

    +-commutative [<=]99.3

    \[ \frac{0.3333333333333333 \cdot \mathsf{fma}\left(\sqrt{2} \cdot \left(\sin x + -0.0625 \cdot \sin y\right), \left(\cos x - \cos y\right) \cdot \left(\sin y + -0.0625 \cdot \sin x\right), 2\right)}{1 + \left(\cos y \cdot \color{blue}{\left(1.5 + -0.5 \cdot \sqrt{5}\right)} + \frac{\cos x}{\sqrt{1.25} + 0.5}\right)} \]

    *-commutative [<=]99.3

    \[ \frac{0.3333333333333333 \cdot \mathsf{fma}\left(\sqrt{2} \cdot \left(\sin x + -0.0625 \cdot \sin y\right), \left(\cos x - \cos y\right) \cdot \left(\sin y + -0.0625 \cdot \sin x\right), 2\right)}{1 + \left(\color{blue}{\left(1.5 + -0.5 \cdot \sqrt{5}\right) \cdot \cos y} + \frac{\cos x}{\sqrt{1.25} + 0.5}\right)} \]

    +-commutative [=>]99.3

    \[ \frac{0.3333333333333333 \cdot \mathsf{fma}\left(\sqrt{2} \cdot \left(\sin x + -0.0625 \cdot \sin y\right), \left(\cos x - \cos y\right) \cdot \left(\sin y + -0.0625 \cdot \sin x\right), 2\right)}{1 + \left(\color{blue}{\left(-0.5 \cdot \sqrt{5} + 1.5\right)} \cdot \cos y + \frac{\cos x}{\sqrt{1.25} + 0.5}\right)} \]

    fma-udef [<=]99.3

    \[ \frac{0.3333333333333333 \cdot \mathsf{fma}\left(\sqrt{2} \cdot \left(\sin x + -0.0625 \cdot \sin y\right), \left(\cos x - \cos y\right) \cdot \left(\sin y + -0.0625 \cdot \sin x\right), 2\right)}{1 + \left(\color{blue}{\mathsf{fma}\left(-0.5, \sqrt{5}, 1.5\right)} \cdot \cos y + \frac{\cos x}{\sqrt{1.25} + 0.5}\right)} \]

    +-commutative [=>]99.3

    \[ \frac{0.3333333333333333 \cdot \mathsf{fma}\left(\sqrt{2} \cdot \left(\sin x + -0.0625 \cdot \sin y\right), \left(\cos x - \cos y\right) \cdot \left(\sin y + -0.0625 \cdot \sin x\right), 2\right)}{1 + \left(\mathsf{fma}\left(-0.5, \sqrt{5}, 1.5\right) \cdot \cos y + \frac{\cos x}{\color{blue}{0.5 + \sqrt{1.25}}}\right)} \]

  8. Applied egg-rr99.3%

    \[\leadsto \frac{0.3333333333333333 \cdot \mathsf{fma}\left(\sqrt{2} \cdot \left(\sin x + -0.0625 \cdot \sin y\right), \left(\cos x - \cos y\right) \cdot \left(\sin y + -0.0625 \cdot \sin x\right), 2\right)}{1 + \left(\color{blue}{\left(-0.5 \cdot \sqrt{5} + 1.5\right)} \cdot \cos y + \frac{\cos x}{0.5 + \sqrt{1.25}}\right)} \]
    Proof

    [Start]99.3

    \[ \frac{0.3333333333333333 \cdot \mathsf{fma}\left(\sqrt{2} \cdot \left(\sin x + -0.0625 \cdot \sin y\right), \left(\cos x - \cos y\right) \cdot \left(\sin y + -0.0625 \cdot \sin x\right), 2\right)}{1 + \left(\mathsf{fma}\left(-0.5, \sqrt{5}, 1.5\right) \cdot \cos y + \frac{\cos x}{0.5 + \sqrt{1.25}}\right)} \]

    fma-udef [=>]99.3

    \[ \frac{0.3333333333333333 \cdot \mathsf{fma}\left(\sqrt{2} \cdot \left(\sin x + -0.0625 \cdot \sin y\right), \left(\cos x - \cos y\right) \cdot \left(\sin y + -0.0625 \cdot \sin x\right), 2\right)}{1 + \left(\color{blue}{\left(-0.5 \cdot \sqrt{5} + 1.5\right)} \cdot \cos y + \frac{\cos x}{0.5 + \sqrt{1.25}}\right)} \]

  9. Final simplification99.3%

    \[\leadsto \frac{0.3333333333333333 \cdot \mathsf{fma}\left(\sqrt{2} \cdot \left(\sin x + -0.0625 \cdot \sin y\right), \left(\cos x - \cos y\right) \cdot \left(\sin y + \sin x \cdot -0.0625\right), 2\right)}{1 + \left(\cos y \cdot \left(1.5 + -0.5 \cdot \sqrt{5}\right) + \frac{\cos x}{0.5 + \sqrt{1.25}}\right)} \]

Alternatives

Alternative 1
Accuracy99.3%
Cost73024
\[\frac{2 + \left(\cos x - \cos y\right) \cdot \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{\frac{4}{\sqrt{5} + 3}}{2}\right)} \]
Alternative 2
Accuracy99.2%
Cost72896
\[\begin{array}{l} t_0 := \sqrt{5} \cdot 0.5\\ 0.3333333333333333 \cdot \frac{2 + \sqrt{2} \cdot \left(\left(\sin y - \sin x \cdot 0.0625\right) \cdot \left(\left(\cos x - \cos y\right) \cdot \left(\sin x - \sin y \cdot 0.0625\right)\right)\right)}{1 + \left(\cos x \cdot \left(t_0 - 0.5\right) + \cos y \cdot \left(1.5 - t_0\right)\right)} \end{array} \]
Alternative 3
Accuracy99.2%
Cost72896
\[\begin{array}{l} t_0 := \sqrt{5} \cdot 0.5\\ 0.3333333333333333 \cdot \frac{2 + \sqrt{2} \cdot \left(\left(\sin y - \sin x \cdot 0.0625\right) \cdot \left(\left(\cos x - \cos y\right) \cdot \left(\sin x - \sin y \cdot 0.0625\right)\right)\right)}{1 + \left(\frac{\cos y}{1.5 + t_0} + \cos x \cdot \left(t_0 - 0.5\right)\right)} \end{array} \]
Alternative 4
Accuracy99.3%
Cost72896
\[\frac{2 + \left(\cos x - \cos y\right) \cdot \left(\sqrt{2} \cdot \left(\left(\sin y - \sin x \cdot 0.0625\right) \cdot \left(\sin x - \sin y \cdot 0.0625\right)\right)\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)} \]
Alternative 5
Accuracy99.3%
Cost72896
\[\begin{array}{l} t_0 := \frac{\sqrt{5}}{2}\\ \frac{2 + \left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\left(\cos x - \cos y\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right)}{3 \cdot \left(1 + \left(\cos x \cdot \left(t_0 - 0.5\right) + \cos y \cdot \left(1.5 - t_0\right)\right)\right)} \end{array} \]
Alternative 6
Accuracy99.3%
Cost72896
\[\frac{2 + \left(\cos x - \cos y\right) \cdot \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)} \]
Alternative 7
Accuracy81.2%
Cost67145
\[\begin{array}{l} t_0 := \sin y - \frac{\sin x}{16}\\ t_1 := 1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\\ \mathbf{if}\;x \leq -0.07 \lor \neg \left(x \leq 0.0005\right):\\ \;\;\;\;\frac{2 + \left(\cos x - \cos y\right) \cdot \left(t_0 \cdot \left(\sqrt{2} \cdot \sin x\right)\right)}{3 \cdot \left(t_1 + \cos y \cdot \frac{\frac{4}{\sqrt{5} + 3}}{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot t_0\right) \cdot \left(1 + \left(-0.5 \cdot \left(x \cdot x\right) - \cos y\right)\right)}{3 \cdot \left(t_1 + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)}\\ \end{array} \]
Alternative 8
Accuracy81.1%
Cost66761
\[\begin{array}{l} t_0 := \sin y - \frac{\sin x}{16}\\ t_1 := 1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\\ \mathbf{if}\;x \leq -0.0024 \lor \neg \left(x \leq 0.0005\right):\\ \;\;\;\;\frac{2 + \left(\cos x - \cos y\right) \cdot \left(t_0 \cdot \left(\sqrt{2} \cdot \sin x\right)\right)}{3 \cdot \left(t_1 + \cos y \cdot \frac{\frac{4}{\sqrt{5} + 3}}{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot t_0\right) \cdot \left(1 - \cos y\right)}{3 \cdot \left(t_1 + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)}\\ \end{array} \]
Alternative 9
Accuracy81.1%
Cost66697
\[\begin{array}{l} t_0 := 1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\\ t_1 := \cos x - \cos y\\ \mathbf{if}\;x \leq -0.0039 \lor \neg \left(x \leq 0.0005\right):\\ \;\;\;\;\frac{2 + t_1 \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\sqrt{2} \cdot \sin x\right)\right)}{3 \cdot \left(t_0 + \cos y \cdot \frac{\frac{4}{\sqrt{5} + 3}}{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + t_1 \cdot \left(\sqrt{2} \cdot \left(-0.0625 \cdot {\sin y}^{2} + \sin y \cdot \left(x \cdot 1.00390625\right)\right)\right)}{3 \cdot \left(t_0 + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)}\\ \end{array} \]
Alternative 10
Accuracy80.9%
Cost66633
\[\begin{array}{l} t_0 := \sin y - \frac{\sin x}{16}\\ t_1 := \sqrt{5} \cdot 0.5\\ \mathbf{if}\;x \leq -1.45 \cdot 10^{-7} \lor \neg \left(x \leq 0.0001\right):\\ \;\;\;\;\frac{2 + \left(\cos x - \cos y\right) \cdot \left(t_0 \cdot \left(\sqrt{2} \cdot \sin x\right)\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{\frac{4}{\sqrt{5} + 3}}{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(t_0 \cdot \left(1 - \cos y\right)\right)}{3 \cdot \left(1 + \left(\left(t_1 + \cos y \cdot \left(1.5 - t_1\right)\right) - 0.5\right)\right)}\\ \end{array} \]
Alternative 11
Accuracy80.9%
Cost66505
\[\begin{array}{l} t_0 := \frac{\sqrt{5}}{2}\\ t_1 := \sin y - \frac{\sin x}{16}\\ t_2 := \sqrt{5} \cdot 0.5\\ \mathbf{if}\;x \leq -1.45 \cdot 10^{-7} \lor \neg \left(x \leq 1.05 \cdot 10^{-5}\right):\\ \;\;\;\;\frac{2 + \left(\left(\cos x - \cos y\right) \cdot t_1\right) \cdot \left(\sqrt{2} \cdot \sin x\right)}{3 \cdot \left(1 + \left(\cos x \cdot \left(t_0 - 0.5\right) + \cos y \cdot \left(1.5 - t_0\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(t_1 \cdot \left(1 - \cos y\right)\right)}{3 \cdot \left(1 + \left(\left(t_2 + \cos y \cdot \left(1.5 - t_2\right)\right) - 0.5\right)\right)}\\ \end{array} \]
Alternative 12
Accuracy79.3%
Cost66377
\[\begin{array}{l} t_0 := \cos x - \cos y\\ \mathbf{if}\;y \leq -7.5 \cdot 10^{-7} \lor \neg \left(y \leq 0.00043\right):\\ \;\;\;\;\frac{2 + t_0 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot -0.0625\right)\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{\frac{4}{\sqrt{5} + 3}}{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(t_0 \cdot \left(\sin y - \frac{\sin x}{16}\right)\right)}{3 \cdot \left(1 + \left(1.5 + \left(\frac{\cos x}{0.5 + \sqrt{1.25}} - \sqrt{1.25}\right)\right)\right)}\\ \end{array} \]
Alternative 13
Accuracy79.2%
Cost60233
\[\begin{array}{l} t_0 := \sqrt{5} \cdot 0.5\\ \mathbf{if}\;x \leq -1.45 \cdot 10^{-7} \lor \neg \left(x \leq 1.9 \cdot 10^{-5}\right):\\ \;\;\;\;\frac{2 + \left(\cos x - \cos y\right) \cdot \left(\left(\sqrt{2} \cdot -0.0625\right) \cdot {\sin x}^{2}\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{\frac{4}{\sqrt{5} + 3}}{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(1 - \cos y\right)\right)}{3 \cdot \left(1 + \left(\left(t_0 + \cos y \cdot \left(1.5 - t_0\right)\right) - 0.5\right)\right)}\\ \end{array} \]
Alternative 14
Accuracy79.3%
Cost60233
\[\begin{array}{l} t_0 := \sqrt{5} \cdot 0.5\\ \mathbf{if}\;y \leq -7.5 \cdot 10^{-7} \lor \neg \left(y \leq 0.00058\right):\\ \;\;\;\;\frac{2 + \left(\cos x - \cos y\right) \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot -0.0625\right)\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{\frac{4}{\sqrt{5} + 3}}{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x + -1\right)\right)}{3 \cdot \left(1 + \left(\left(1.5 + \cos x \cdot \left(t_0 - 0.5\right)\right) - t_0\right)\right)}\\ \end{array} \]
Alternative 15
Accuracy79.1%
Cost60041
\[\begin{array}{l} t_0 := \sqrt{5} \cdot 0.5\\ t_1 := \cos x - \cos y\\ \mathbf{if}\;y \leq -7.5 \cdot 10^{-7} \lor \neg \left(y \leq 0.00064\right):\\ \;\;\;\;\frac{2 + t_1 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot -0.0625\right)\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{\frac{4}{\sqrt{5} + 3}}{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(t_1 \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\sqrt{2} \cdot \sin x\right)}{3 \cdot \left(1 + \left(\left(1.5 + \cos x \cdot \left(t_0 - 0.5\right)\right) - t_0\right)\right)}\\ \end{array} \]
Alternative 16
Accuracy79.1%
Cost59977
\[\begin{array}{l} t_0 := \sqrt{5} \cdot 0.5\\ t_1 := \cos x - \cos y\\ \mathbf{if}\;y \leq -7.5 \cdot 10^{-7} \lor \neg \left(y \leq 0.00031\right):\\ \;\;\;\;\frac{2 + t_1 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot -0.0625\right)\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(t_1 \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\sqrt{2} \cdot \sin x\right)}{3 \cdot \left(1 + \left(\left(1.5 + \cos x \cdot \left(t_0 - 0.5\right)\right) - t_0\right)\right)}\\ \end{array} \]
Alternative 17
Accuracy79.0%
Cost59913
\[\begin{array}{l} \mathbf{if}\;x \leq -1.45 \cdot 10^{-7} \lor \neg \left(x \leq 1.5 \cdot 10^{-6}\right):\\ \;\;\;\;\frac{2 + \left(\cos x - \cos y\right) \cdot \left(\left(\sqrt{2} \cdot -0.0625\right) \cdot {\sin x}^{2}\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.3333333333333333 \cdot \mathsf{fma}\left(-0.0625, \sqrt{2} \cdot \left(\left(1 - \cos y\right) \cdot {\sin y}^{2}\right), 2\right)}{0.5 + \mathsf{fma}\left(0.5, \sqrt{5}, \cos y \cdot \left(1.5 + -0.5 \cdot \sqrt{5}\right)\right)}\\ \end{array} \]
Alternative 18
Accuracy79.1%
Cost59913
\[\begin{array}{l} t_0 := \sqrt{5} \cdot 0.5\\ \mathbf{if}\;y \leq -7.5 \cdot 10^{-7} \lor \neg \left(y \leq 0.00031\right):\\ \;\;\;\;\frac{2 + \left(\cos x - \cos y\right) \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot -0.0625\right)\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left(\sqrt{2} \cdot \left(\left(\cos x + -1\right) \cdot {\sin x}^{2}\right)\right)}{1 + \left(\cos x \cdot \left(t_0 - 0.5\right) + \frac{1}{1.5 + t_0}\right)}\\ \end{array} \]
Alternative 19
Accuracy78.3%
Cost59400
\[\begin{array}{l} t_0 := 2 + -0.0625 \cdot \left(\sqrt{2} \cdot \left(\left(\cos x + -1\right) \cdot {\sin x}^{2}\right)\right)\\ t_1 := \sqrt{5} \cdot 0.5\\ \mathbf{if}\;x \leq -1.45 \cdot 10^{-7}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{t_0}{1 + \left(\cos x \cdot \left(t_1 - 0.5\right) + \frac{1}{1.5 + t_1}\right)}\\ \mathbf{elif}\;x \leq 4.2 \cdot 10^{-5}:\\ \;\;\;\;\frac{0.3333333333333333 \cdot \mathsf{fma}\left(-0.0625, \sqrt{2} \cdot \left(\left(1 - \cos y\right) \cdot {\sin y}^{2}\right), 2\right)}{0.5 + \mathsf{fma}\left(0.5, \sqrt{5}, \cos y \cdot \left(1.5 + -0.5 \cdot \sqrt{5}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{t_0}{\left(2.5 + 0.3333333333333333 \cdot \left(3 \cdot \left(\cos x \cdot \left(-0.5 + \sqrt{1.25}\right)\right)\right)\right) - t_1}\\ \end{array} \]
Alternative 20
Accuracy78.3%
Cost46984
\[\begin{array}{l} t_0 := {\sin x}^{2}\\ t_1 := \sqrt{5} \cdot 0.5\\ t_2 := \cos x + -1\\ \mathbf{if}\;x \leq -1.45 \cdot 10^{-7}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left(t_2 \cdot \left(\sqrt{2} \cdot t_0\right)\right)}{2.5 + \left(\cos x \cdot \left(t_1 - 0.5\right) - t_1\right)}\\ \mathbf{elif}\;x \leq 10^{-5}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left(\sqrt{2} \cdot \left(\left(1 - \cos y\right) \cdot {\sin y}^{2}\right)\right)}{0.5 + \left(t_1 + \frac{\cos y}{1.5 + t_1}\right)}\\ \mathbf{else}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left(\sqrt{2} \cdot \left(t_2 \cdot t_0\right)\right)}{\left(2.5 + 0.3333333333333333 \cdot \left(3 \cdot \left(\cos x \cdot \left(-0.5 + \sqrt{1.25}\right)\right)\right)\right) - t_1}\\ \end{array} \]
Alternative 21
Accuracy78.3%
Cost46984
\[\begin{array}{l} t_0 := 2 + -0.0625 \cdot \left(\sqrt{2} \cdot \left(\left(\cos x + -1\right) \cdot {\sin x}^{2}\right)\right)\\ t_1 := \sqrt{5} \cdot 0.5\\ t_2 := 1.5 + t_1\\ \mathbf{if}\;x \leq -1.45 \cdot 10^{-7}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{t_0}{1 + \left(\cos x \cdot \left(t_1 - 0.5\right) + \frac{1}{t_2}\right)}\\ \mathbf{elif}\;x \leq 4 \cdot 10^{-7}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left(\sqrt{2} \cdot \left(\left(1 - \cos y\right) \cdot {\sin y}^{2}\right)\right)}{0.5 + \left(t_1 + \frac{\cos y}{t_2}\right)}\\ \mathbf{else}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{t_0}{\left(2.5 + 0.3333333333333333 \cdot \left(3 \cdot \left(\cos x \cdot \left(-0.5 + \sqrt{1.25}\right)\right)\right)\right) - t_1}\\ \end{array} \]
Alternative 22
Accuracy78.3%
Cost46856
\[\begin{array}{l} t_0 := {\sin x}^{2}\\ t_1 := \sqrt{5} \cdot 0.5\\ t_2 := \cos x + -1\\ \mathbf{if}\;x \leq -1.45 \cdot 10^{-7}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left(t_2 \cdot \left(\sqrt{2} \cdot t_0\right)\right)}{2.5 + \left(\cos x \cdot \left(t_1 - 0.5\right) - t_1\right)}\\ \mathbf{elif}\;x \leq 4.5 \cdot 10^{-5}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left(\sqrt{2} \cdot \left(\left(1 - \cos y\right) \cdot {\sin y}^{2}\right)\right)}{0.5 + \left(t_1 + \cos y \cdot \left(1.5 - t_1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left(\sqrt{2} \cdot \left(t_2 \cdot t_0\right)\right)}{\left(\frac{\cos x}{0.5 + \sqrt{1.25}} + 2.5\right) - t_1}\\ \end{array} \]
Alternative 23
Accuracy78.3%
Cost46856
\[\begin{array}{l} t_0 := {\sin x}^{2}\\ t_1 := \sqrt{5} \cdot 0.5\\ t_2 := \cos x + -1\\ \mathbf{if}\;x \leq -1.45 \cdot 10^{-7}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left(t_2 \cdot \left(\sqrt{2} \cdot t_0\right)\right)}{2.5 + \left(\cos x \cdot \left(t_1 - 0.5\right) - t_1\right)}\\ \mathbf{elif}\;x \leq 5.4 \cdot 10^{-6}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left(\sqrt{2} \cdot \left(\left(1 - \cos y\right) \cdot {\sin y}^{2}\right)\right)}{0.5 + \left(t_1 + \frac{\cos y}{1.5 + t_1}\right)}\\ \mathbf{else}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left(\sqrt{2} \cdot \left(t_2 \cdot t_0\right)\right)}{\left(\frac{\cos x}{0.5 + \sqrt{1.25}} + 2.5\right) - t_1}\\ \end{array} \]
Alternative 24
Accuracy60.1%
Cost46592
\[\begin{array}{l} t_0 := \sqrt{5} \cdot 0.5\\ 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left(\left(\cos x + -1\right) \cdot \left(\sqrt{2} \cdot {\sin x}^{2}\right)\right)}{2.5 + \left(\cos x \cdot \left(t_0 - 0.5\right) - t_0\right)} \end{array} \]
Alternative 25
Accuracy60.0%
Cost46464
\[0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left(\sqrt{2} \cdot \left(\left(\cos x + -1\right) \cdot {\sin x}^{2}\right)\right)}{\left(\frac{\cos x}{0.5 + \sqrt{1.25}} + 2.5\right) - \sqrt{5} \cdot 0.5} \]
Alternative 26
Accuracy60.0%
Cost40512
\[\begin{array}{l} t_0 := \sqrt{5} \cdot 0.5\\ 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left(\sqrt{2} \cdot \left(\left(\cos x + -1\right) \cdot \left(0.5 - \frac{\cos \left(x + x\right)}{2}\right)\right)\right)}{\left(\cos x \cdot \left(t_0 - 0.5\right) + 2.5\right) - t_0} \end{array} \]
Alternative 27
Accuracy41.8%
Cost32704
\[\frac{0.6666666666666666}{0.5 + \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, -0.5, 1.5\right), \cos y, \sqrt{5} \cdot 0.5\right)} \]
Alternative 28
Accuracy41.8%
Cost20160
\[\begin{array}{l} t_0 := \sqrt{5} \cdot 0.5\\ \frac{0.6666666666666666}{0.5 + \left(t_0 + \cos y \cdot \left(1.5 - t_0\right)\right)} \end{array} \]
Alternative 29
Accuracy39.9%
Cost64
\[0.3333333333333333 \]

Error

Reproduce?

herbie shell --seed 2023151 
(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))))))