?

Average Error: 0.5 → 0.4
Time: 30.5s
Precision: binary64
Cost: 72640

?

\[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
↓
\[\frac{2 + \left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right)}{3 \cdot \left(1 + \left(\frac{\cos y}{\sqrt{1.25} + 1.5} + \frac{\cos x}{\sqrt{1.25} + 0.5}\right)\right)} \]
(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
 (/
  (+
   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
    (+ (/ (cos y) (+ (sqrt 1.25) 1.5)) (/ (cos x) (+ (sqrt 1.25) 0.5)))))))
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 (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 + ((cos(y) / (sqrt(1.25) + 1.5)) + (cos(x) / (sqrt(1.25) + 0.5)))));
}

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = (2.0d0 + (((sqrt(2.0d0) * (sin(x) - (sin(y) / 16.0d0))) * (sin(y) - (sin(x) / 16.0d0))) * (cos(x) - cos(y)))) / (3.0d0 * ((1.0d0 + (((sqrt(5.0d0) - 1.0d0) / 2.0d0) * cos(x))) + (((3.0d0 - sqrt(5.0d0)) / 2.0d0) * cos(y))))
end function
↓
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = (2.0d0 + ((sqrt(2.0d0) * (sin(x) - (sin(y) / 16.0d0))) * ((sin(y) - (sin(x) / 16.0d0)) * (cos(x) - cos(y))))) / (3.0d0 * (1.0d0 + ((cos(y) / (sqrt(1.25d0) + 1.5d0)) + (cos(x) / (sqrt(1.25d0) + 0.5d0)))))
end function

public static double code(double x, double y) {
	return (2.0 + (((Math.sqrt(2.0) * (Math.sin(x) - (Math.sin(y) / 16.0))) * (Math.sin(y) - (Math.sin(x) / 16.0))) * (Math.cos(x) - Math.cos(y)))) / (3.0 * ((1.0 + (((Math.sqrt(5.0) - 1.0) / 2.0) * Math.cos(x))) + (((3.0 - Math.sqrt(5.0)) / 2.0) * Math.cos(y))));
}
↓
public static double code(double x, double y) {
	return (2.0 + ((Math.sqrt(2.0) * (Math.sin(x) - (Math.sin(y) / 16.0))) * ((Math.sin(y) - (Math.sin(x) / 16.0)) * (Math.cos(x) - Math.cos(y))))) / (3.0 * (1.0 + ((Math.cos(y) / (Math.sqrt(1.25) + 1.5)) + (Math.cos(x) / (Math.sqrt(1.25) + 0.5)))));
}

def code(x, y):
	return (2.0 + (((math.sqrt(2.0) * (math.sin(x) - (math.sin(y) / 16.0))) * (math.sin(y) - (math.sin(x) / 16.0))) * (math.cos(x) - math.cos(y)))) / (3.0 * ((1.0 + (((math.sqrt(5.0) - 1.0) / 2.0) * math.cos(x))) + (((3.0 - math.sqrt(5.0)) / 2.0) * math.cos(y))))
↓
def code(x, y):
	return (2.0 + ((math.sqrt(2.0) * (math.sin(x) - (math.sin(y) / 16.0))) * ((math.sin(y) - (math.sin(x) / 16.0)) * (math.cos(x) - math.cos(y))))) / (3.0 * (1.0 + ((math.cos(y) / (math.sqrt(1.25) + 1.5)) + (math.cos(x) / (math.sqrt(1.25) + 0.5)))))

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(2.0 + Float64(Float64(sqrt(2.0) * Float64(sin(x) - Float64(sin(y) / 16.0))) * Float64(Float64(sin(y) - Float64(sin(x) / 16.0)) * Float64(cos(x) - cos(y))))) / Float64(3.0 * Float64(1.0 + Float64(Float64(cos(y) / Float64(sqrt(1.25) + 1.5)) + Float64(cos(x) / Float64(sqrt(1.25) + 0.5))))))
end

function tmp = code(x, y)
	tmp = (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))));
end
↓
function tmp = code(x, y)
	tmp = (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 + ((cos(y) / (sqrt(1.25) + 1.5)) + (cos(x) / (sqrt(1.25) + 0.5)))));
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[(2.0 + N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Sin[x], $MachinePrecision] - N[(N[Sin[y], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Sin[y], $MachinePrecision] - N[(N[Sin[x], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(1.0 + N[(N[(N[Cos[y], $MachinePrecision] / N[(N[Sqrt[1.25], $MachinePrecision] + 1.5), $MachinePrecision]), $MachinePrecision] + N[(N[Cos[x], $MachinePrecision] / N[(N[Sqrt[1.25], $MachinePrecision] + 0.5), $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{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 - \cos y\right)\right)}{3 \cdot \left(1 + \left(\frac{\cos y}{\sqrt{1.25} + 1.5} + \frac{\cos x}{\sqrt{1.25} + 0.5}\right)\right)}

Error?

Try it out?

Your Program's Arguments

Results

Enter valid numbers for all inputs

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

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

    [Start]0.5

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

    associate-*l* [=>]0.5

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

  3. Applied egg-rr0.5

    \[\leadsto \frac{2 + \left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right)}{3 \cdot \left(1 + \left(\cos x \cdot \left(\frac{\sqrt{5}}{2} - 0.5\right) + \cos y \cdot \color{blue}{\frac{1}{1.5 + \sqrt{1.25}}}\right)\right)} \]

  4. Applied egg-rr0.4

    \[\leadsto \frac{2 + \left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right)}{3 \cdot \left(1 + \left(\color{blue}{\frac{\cos x}{\sqrt{1.25} + 0.5}} + \cos y \cdot \frac{1}{1.5 + \sqrt{1.25}}\right)\right)} \]

  5. Taylor expanded in x around inf 0.4

    \[\leadsto \frac{2 + \left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right)}{3 \cdot \left(1 + \color{blue}{\left(\frac{\cos y}{\sqrt{1.25} + 1.5} + \frac{\cos x}{\sqrt{1.25} + 0.5}\right)}\right)} \]

  6. Final simplification0.4

    \[\leadsto \frac{2 + \left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right)}{3 \cdot \left(1 + \left(\frac{\cos y}{\sqrt{1.25} + 1.5} + \frac{\cos x}{\sqrt{1.25} + 0.5}\right)\right)} \]

Alternatives

Alternative 1
Error12.0
Cost67016
\[\begin{array}{l} t_0 := \cos y \cdot \frac{1}{\sqrt{1.25} + 1.5}\\ t_1 := 3 \cdot \left(1 + \left(\frac{\cos x}{\sqrt{1.25} + 0.5} + t_0\right)\right)\\ t_2 := \sin y - \frac{\sin x}{16}\\ t_3 := 2 + \left(t_2 \cdot \left(\cos x - \cos y\right)\right) \cdot \left(\sqrt{2} \cdot \sin x\right)\\ \mathbf{if}\;x \leq -0.061:\\ \;\;\;\;\frac{t_3}{t_1}\\ \mathbf{elif}\;x \leq 0.152:\\ \;\;\;\;\frac{2 + \left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(t_2 \cdot \left(1 + \left(-0.5 \cdot \left(x \cdot x\right) - \cos y\right)\right)\right)}{t_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_3}{3 \cdot \left(1 + \left(t_0 + \cos x \cdot \left(-0.5 + \frac{\sqrt{5}}{2}\right)\right)\right)}\\ \end{array} \]
Alternative 2
Error12.1
Cost66632
\[\begin{array}{l} t_0 := \cos y \cdot \frac{1}{\sqrt{1.25} + 1.5}\\ t_1 := 3 \cdot \left(1 + \left(\frac{\cos x}{\sqrt{1.25} + 0.5} + t_0\right)\right)\\ t_2 := \sin y - \frac{\sin x}{16}\\ t_3 := 2 + \left(t_2 \cdot \left(\cos x - \cos y\right)\right) \cdot \left(\sqrt{2} \cdot \sin x\right)\\ \mathbf{if}\;x \leq -0.0056:\\ \;\;\;\;\frac{t_3}{t_1}\\ \mathbf{elif}\;x \leq 0.064:\\ \;\;\;\;\frac{2 + \left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(t_2 \cdot \left(1 - \cos y\right)\right)}{t_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_3}{3 \cdot \left(1 + \left(t_0 + \cos x \cdot \left(-0.5 + \frac{\sqrt{5}}{2}\right)\right)\right)}\\ \end{array} \]
Alternative 3
Error12.5
Cost66504
\[\begin{array}{l} t_0 := \cos y \cdot \frac{1}{\sqrt{1.25} + 1.5}\\ t_1 := 2 + \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right) \cdot \left(\sqrt{2} \cdot \sin x\right)\\ \mathbf{if}\;x \leq -2.1 \cdot 10^{-6}:\\ \;\;\;\;\frac{t_1}{3 \cdot \left(1 + \left(\frac{\cos x}{\sqrt{1.25} + 0.5} + t_0\right)\right)}\\ \mathbf{elif}\;x \leq 2.3 \cdot 10^{-22}:\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left(\sqrt{2} \cdot \left(\left(1 - \cos y\right) \cdot {\sin y}^{2}\right)\right)}{1.5 \cdot \left(\sqrt{5} + -1\right) + \left(3 + 1.5 \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_1}{3 \cdot \left(1 + \left(t_0 + \cos x \cdot \left(-0.5 + \frac{\sqrt{5}}{2}\right)\right)\right)}\\ \end{array} \]
Alternative 4
Error12.1
Cost66504
\[\begin{array}{l} t_0 := \sqrt{1.25} + 1.5\\ t_1 := \cos y \cdot \frac{1}{t_0}\\ t_2 := \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\\ t_3 := 2 + t_2 \cdot \left(\sqrt{2} \cdot \sin x\right)\\ t_4 := \sqrt{1.25} + 0.5\\ \mathbf{if}\;x \leq -3.6 \cdot 10^{-5}:\\ \;\;\;\;\frac{t_3}{3 \cdot \left(1 + \left(\frac{\cos x}{t_4} + t_1\right)\right)}\\ \mathbf{elif}\;x \leq 0.064:\\ \;\;\;\;\frac{2 + \left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot t_2}{3 \cdot \left(1 + \left(\frac{\cos y}{t_0} + \frac{1}{t_4}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_3}{3 \cdot \left(1 + \left(t_1 + \cos x \cdot \left(-0.5 + \frac{\sqrt{5}}{2}\right)\right)\right)}\\ \end{array} \]
Alternative 5
Error12.5
Cost66377
\[\begin{array}{l} \mathbf{if}\;x \leq -1.15 \cdot 10^{-5} \lor \neg \left(x \leq 2.3 \cdot 10^{-22}\right):\\ \;\;\;\;\frac{2 + \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right) \cdot \left(\sqrt{2} \cdot \sin x\right)}{3 \cdot \left(1 + \left(\frac{\cos x}{\sqrt{1.25} + 0.5} + \cos y \cdot \frac{1}{\sqrt{1.25} + 1.5}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left(\sqrt{2} \cdot \left(\left(1 - \cos y\right) \cdot {\sin y}^{2}\right)\right)}{1.5 \cdot \left(\sqrt{5} + -1\right) + \left(3 + 1.5 \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)\right)}\\ \end{array} \]
Alternative 6
Error13.6
Cost53385
\[\begin{array}{l} \mathbf{if}\;x \leq -5.1 \cdot 10^{-6} \lor \neg \left(x \leq 2.3 \cdot 10^{-22}\right):\\ \;\;\;\;0.3333333333333333 \cdot \frac{2 - \sqrt{2} \cdot \left(-0.0625 \cdot \left({\sin x}^{2} \cdot \left(1 - \cos x\right)\right)\right)}{1 + \left(\cos y \cdot \left(1.5 + -0.5 \cdot \sqrt{5}\right) + \cos x \cdot \left(\sqrt{1.25} + -0.5\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left(\sqrt{2} \cdot \left(\left(1 - \cos y\right) \cdot {\sin y}^{2}\right)\right)}{1.5 \cdot \left(\sqrt{5} + -1\right) + \left(3 + 1.5 \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)\right)}\\ \end{array} \]
Alternative 7
Error13.6
Cost53384
\[\begin{array}{l} t_0 := 2 - \sqrt{2} \cdot \left(-0.0625 \cdot \left({\sin x}^{2} \cdot \left(1 - \cos x\right)\right)\right)\\ t_1 := \cos y \cdot \left(1.5 + -0.5 \cdot \sqrt{5}\right)\\ \mathbf{if}\;x \leq -1.86 \cdot 10^{-6}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{t_0}{1 + \left(\frac{\cos x}{\sqrt{1.25} + 0.5} + t_1\right)}\\ \mathbf{elif}\;x \leq 2.3 \cdot 10^{-22}:\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left(\sqrt{2} \cdot \left(\left(1 - \cos y\right) \cdot {\sin y}^{2}\right)\right)}{1.5 \cdot \left(\sqrt{5} + -1\right) + \left(3 + 1.5 \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{t_0}{1 + \left(t_1 + \cos x \cdot \left(\sqrt{1.25} + -0.5\right)\right)}\\ \end{array} \]
Alternative 8
Error13.8
Cost52996
\[\begin{array}{l} t_0 := \sqrt{5} + -1\\ t_1 := 3 - \sqrt{5}\\ t_2 := {\sin x}^{2} \cdot \left(1 - \cos x\right)\\ \mathbf{if}\;x \leq -6 \cdot 10^{-6}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{2 + \left(\sqrt{2} \cdot t_2\right) \cdot 0.0625}{2.5 + \left(\cos x \cdot \mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right) + -0.5 \cdot \sqrt{5}\right)}\\ \mathbf{elif}\;x \leq 0.064:\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left(\sqrt{2} \cdot \left(\left(1 - \cos y\right) \cdot {\sin y}^{2}\right)\right)}{1.5 \cdot t_0 + \left(3 + 1.5 \cdot \left(\cos y \cdot t_1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(\sqrt{2} \cdot 0.0625\right) \cdot t_2}{3 + 1.5 \cdot \left(t_1 + \frac{\cos x}{\frac{1}{t_0}}\right)}\\ \end{array} \]
Alternative 9
Error13.9
Cost46856
\[\begin{array}{l} t_0 := 2 + \left(\sqrt{2} \cdot 0.0625\right) \cdot \left({\sin x}^{2} \cdot \left(1 - \cos x\right)\right)\\ t_1 := \cos x \cdot \left(\sqrt{5} + -1\right)\\ \mathbf{if}\;x \leq -2.4 \cdot 10^{-6}:\\ \;\;\;\;\frac{t_0}{3 + 1.5 \cdot \left(3 + \left(t_1 - \sqrt{5}\right)\right)}\\ \mathbf{elif}\;x \leq 0.064:\\ \;\;\;\;0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left(\sqrt{2} \cdot \left(\left(1 - \cos y\right) \cdot {\sin y}^{2}\right)\right)}{0.5 + \left(0.5 \cdot \sqrt{5} + \cos y \cdot \left(1.5 + -0.5 \cdot \sqrt{5}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0}{3 + 1.5 \cdot \left(\left(3 - \sqrt{5}\right) + t_1\right)}\\ \end{array} \]
Alternative 10
Error13.8
Cost46856
\[\begin{array}{l} t_0 := 3 - \sqrt{5}\\ t_1 := 2 + \left(\sqrt{2} \cdot 0.0625\right) \cdot \left({\sin x}^{2} \cdot \left(1 - \cos x\right)\right)\\ t_2 := \sqrt{5} + -1\\ t_3 := \cos x \cdot t_2\\ \mathbf{if}\;x \leq -7.2 \cdot 10^{-5}:\\ \;\;\;\;\frac{t_1}{3 + 1.5 \cdot \left(3 + \left(t_3 - \sqrt{5}\right)\right)}\\ \mathbf{elif}\;x \leq 0.064:\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left(\sqrt{2} \cdot \left(\left(1 - \cos y\right) \cdot {\sin y}^{2}\right)\right)}{1.5 \cdot t_2 + \left(3 + 1.5 \cdot \left(\cos y \cdot t_0\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_1}{3 + 1.5 \cdot \left(t_0 + t_3\right)}\\ \end{array} \]
Alternative 11
Error13.8
Cost46856
\[\begin{array}{l} t_0 := 3 - \sqrt{5}\\ t_1 := 2 + \left(\sqrt{2} \cdot 0.0625\right) \cdot \left({\sin x}^{2} \cdot \left(1 - \cos x\right)\right)\\ t_2 := \sqrt{5} + -1\\ \mathbf{if}\;x \leq -2.6 \cdot 10^{-6}:\\ \;\;\;\;\frac{t_1}{3 + 1.5 \cdot \left(3 + \left(\cos x \cdot t_2 - \sqrt{5}\right)\right)}\\ \mathbf{elif}\;x \leq 0.064:\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left(\sqrt{2} \cdot \left(\left(1 - \cos y\right) \cdot {\sin y}^{2}\right)\right)}{1.5 \cdot t_2 + \left(3 + 1.5 \cdot \left(\cos y \cdot t_0\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_1}{3 + 1.5 \cdot \left(t_0 + \frac{\cos x}{\frac{1}{t_2}}\right)}\\ \end{array} \]
Alternative 12
Error13.8
Cost46856
\[\begin{array}{l} t_0 := 2 + \left(\sqrt{2} \cdot 0.0625\right) \cdot \left({\sin x}^{2} \cdot \left(1 - \cos x\right)\right)\\ t_1 := 3 - \sqrt{5}\\ t_2 := \sqrt{5} + -1\\ \mathbf{if}\;x \leq -2.15 \cdot 10^{-5}:\\ \;\;\;\;\frac{t_0}{3 + 1.5 \cdot \left(t_1 + \frac{\cos x \cdot 4}{1 + \sqrt{5}}\right)}\\ \mathbf{elif}\;x \leq 0.064:\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left(\sqrt{2} \cdot \left(\left(1 - \cos y\right) \cdot {\sin y}^{2}\right)\right)}{1.5 \cdot t_2 + \left(3 + 1.5 \cdot \left(\cos y \cdot t_1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0}{3 + 1.5 \cdot \left(t_1 + \frac{\cos x}{\frac{1}{t_2}}\right)}\\ \end{array} \]
Alternative 13
Error24.8
Cost46729
\[\begin{array}{l} \mathbf{if}\;x \leq -2.25 \cdot 10^{-7} \lor \neg \left(x \leq 0.064\right):\\ \;\;\;\;\frac{2 + \left(\sqrt{2} \cdot 0.0625\right) \cdot \left({\sin x}^{2} \cdot \left(1 - \cos x\right)\right)}{3 + 1.5 \cdot \left(\left(3 - \sqrt{5}\right) + \cos x \cdot \left(\sqrt{5} + -1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.6666666666666666}{\mathsf{fma}\left(\mathsf{fma}\left(-0.5, \sqrt{5}, 1.5\right), \cos y, \mathsf{fma}\left(0.5, \sqrt{5}, 0.5\right)\right)}\\ \end{array} \]
Alternative 14
Error24.8
Cost46728
\[\begin{array}{l} t_0 := 2 + \left(\sqrt{2} \cdot 0.0625\right) \cdot \left({\sin x}^{2} \cdot \left(1 - \cos x\right)\right)\\ t_1 := \cos x \cdot \left(\sqrt{5} + -1\right)\\ \mathbf{if}\;x \leq -1.65 \cdot 10^{-7}:\\ \;\;\;\;\frac{t_0}{3 + 1.5 \cdot \left(3 + \left(t_1 - \sqrt{5}\right)\right)}\\ \mathbf{elif}\;x \leq 0.064:\\ \;\;\;\;\frac{0.6666666666666666}{\mathsf{fma}\left(\mathsf{fma}\left(-0.5, \sqrt{5}, 1.5\right), \cos y, \mathsf{fma}\left(0.5, \sqrt{5}, 0.5\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0}{3 + 1.5 \cdot \left(\left(3 - \sqrt{5}\right) + t_1\right)}\\ \end{array} \]
Alternative 15
Error36.9
Cost38976
\[\frac{0.6666666666666666}{\mathsf{fma}\left(\mathsf{fma}\left(-0.5, \sqrt{5}, 1.5\right), \cos y, \mathsf{fma}\left(0.5, \sqrt{5}, 0.5\right)\right)} \]
Alternative 16
Error36.9
Cost20160
\[\frac{0.6666666666666666}{0.5 + \left(0.5 \cdot \sqrt{5} + \cos y \cdot \left(1.5 + -0.5 \cdot \sqrt{5}\right)\right)} \]
Alternative 17
Error38.2
Cost64
\[0.3333333333333333 \]

Error

Reproduce?

herbie shell --seed 2023018 
(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))))))