cos2 (problem 3.4.1)

Average Error: 31.6 → 0.1

Time: 12.5s

Precision: binary64

Cost: 13376

Math

\[\frac{1 - \cos x}{x \cdot x} \]

↓

\[\frac{\frac{\sin x}{x}}{\frac{x}{\tan \left(x \cdot 0.5\right)}} \]

FPCore

(FPCore (x) :precision binary64 (/ (- 1.0 (cos x)) (* x x)))

↓

(FPCore (x) :precision binary64 (/ (/ (sin x) x) (/ x (tan (* x 0.5)))))

C

double code(double x) {
	return (1.0 - cos(x)) / (x * x);
}

↓

double code(double x) {
	return (sin(x) / x) / (x / tan((x * 0.5)));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (1.0d0 - cos(x)) / (x * x)
end function

↓

real(8) function code(x)
    real(8), intent (in) :: x
    code = (sin(x) / x) / (x / tan((x * 0.5d0)))
end function

Java

public static double code(double x) {
	return (1.0 - Math.cos(x)) / (x * x);
}

↓

public static double code(double x) {
	return (Math.sin(x) / x) / (x / Math.tan((x * 0.5)));
}

Python

def code(x):
	return (1.0 - math.cos(x)) / (x * x)

↓

def code(x):
	return (math.sin(x) / x) / (x / math.tan((x * 0.5)))

Julia

function code(x)
	return Float64(Float64(1.0 - cos(x)) / Float64(x * x))
end

↓

function code(x)
	return Float64(Float64(sin(x) / x) / Float64(x / tan(Float64(x * 0.5))))
end

MATLAB

function tmp = code(x)
	tmp = (1.0 - cos(x)) / (x * x);
end

↓

function tmp = code(x)
	tmp = (sin(x) / x) / (x / tan((x * 0.5)));
end

Wolfram

code[x_] := N[(N[(1.0 - N[Cos[x], $MachinePrecision]), $MachinePrecision] / N[(x * x), $MachinePrecision]), $MachinePrecision]

↓

code[x_] := N[(N[(N[Sin[x], $MachinePrecision] / x), $MachinePrecision] / N[(x / N[Tan[N[(x * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 31.6

   \[\frac{1 - \cos x}{x \cdot x} \]

2. Applied egg-rr 16.1

   \[\leadsto \frac{\color{blue}{\frac{\sin x}{\frac{1 + \cos x}{\sin x}}}}{x \cdot x} \]

3. Applied egg-rr 0.1

   \[\leadsto \color{blue}{\frac{\tan \left(\frac{x}{2}\right)}{x} \cdot \frac{\sin x}{x}} \]

4. Applied egg-rr 0.1

   \[\leadsto \color{blue}{\frac{\frac{\sin x}{x}}{\frac{x}{\tan \left(x \cdot 0.5\right)}}} \]

5. Final simplification 0.1

   \[\leadsto \frac{\frac{\sin x}{x}}{\frac{x}{\tan \left(x \cdot 0.5\right)}} \]

Alternatives

Alternative 1
Error	0.4
Cost	13508

\[\begin{array}{l} \mathbf{if}\;x \leq -0.0004:\\ \;\;\;\;\tan \left(\frac{x}{2}\right) \cdot \frac{\sin x}{x \cdot x}\\ \mathbf{elif}\;x \leq 0.0052:\\ \;\;\;\;0.5 + \left(x \cdot x\right) \cdot -0.041666666666666664\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x} + \cos x \cdot \frac{-1}{x}}{x}\\ \end{array} \]

Alternative 2
Error	0.1
Cost	13376

\[\frac{\sin x}{x} \cdot \frac{\tan \left(\frac{x}{2}\right)}{x} \]

Alternative 3
Error	0.4
Cost	7368

\[\begin{array}{l} \mathbf{if}\;x \leq -0.0055:\\ \;\;\;\;\frac{-1}{\frac{x \cdot x}{\cos x + -1}}\\ \mathbf{elif}\;x \leq 0.0052:\\ \;\;\;\;0.5 + \left(x \cdot x\right) \cdot -0.041666666666666664\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x} + \cos x \cdot \frac{-1}{x}}{x}\\ \end{array} \]

Alternative 4
Error	0.4
Cost	7240

\[\begin{array}{l} t_0 := 1 - \cos x\\ \mathbf{if}\;x \leq -0.0055:\\ \;\;\;\;\frac{t_0}{x \cdot x}\\ \mathbf{elif}\;x \leq 0.0055:\\ \;\;\;\;0.5 + \left(x \cdot x\right) \cdot -0.041666666666666664\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x} \cdot \frac{t_0}{x}\\ \end{array} \]

Alternative 5
Error	0.4
Cost	7240

\[\begin{array}{l} \mathbf{if}\;x \leq -0.0055:\\ \;\;\;\;\frac{-1}{\frac{x \cdot x}{\cos x + -1}}\\ \mathbf{elif}\;x \leq 0.0055:\\ \;\;\;\;0.5 + \left(x \cdot x\right) \cdot -0.041666666666666664\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x} \cdot \frac{1 - \cos x}{x}\\ \end{array} \]

Alternative 6
Error	0.5
Cost	7113

\[\begin{array}{l} \mathbf{if}\;x \leq -0.0055 \lor \neg \left(x \leq 0.0055\right):\\ \;\;\;\;\frac{1 - \cos x}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;0.5 + \left(x \cdot x\right) \cdot -0.041666666666666664\\ \end{array} \]

Alternative 7
Error	0.4
Cost	7112

\[\begin{array}{l} t_0 := 1 - \cos x\\ \mathbf{if}\;x \leq -0.0055:\\ \;\;\;\;\frac{t_0}{x \cdot x}\\ \mathbf{elif}\;x \leq 0.0055:\\ \;\;\;\;0.5 + \left(x \cdot x\right) \cdot -0.041666666666666664\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{t_0}{x}}{x}\\ \end{array} \]

Alternative 8
Error	13.3
Cost	1096

\[\begin{array}{l} \mathbf{if}\;x \leq -3.2:\\ \;\;\;\;\frac{6}{x \cdot x}\\ \mathbf{elif}\;x \leq 2.1:\\ \;\;\;\;0.5 + \left(x \cdot x\right) \cdot -0.041666666666666664\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{0.16666666666666666 + \frac{2}{x \cdot x}}}{x \cdot x}\\ \end{array} \]

Alternative 9
Error	13.3
Cost	713

\[\begin{array}{l} \mathbf{if}\;x \leq -3.2 \lor \neg \left(x \leq 3.25\right):\\ \;\;\;\;\frac{6}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;0.5 + \left(x \cdot x\right) \cdot -0.041666666666666664\\ \end{array} \]

Alternative 10
Error	13.5
Cost	585

\[\begin{array}{l} \mathbf{if}\;x \leq -3.5 \lor \neg \left(x \leq 3.5\right):\\ \;\;\;\;\frac{6}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \]

Alternative 11
Error	14.9
Cost	328

\[\begin{array}{l} \mathbf{if}\;x \leq -1.02 \cdot 10^{+77}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 1.2 \cdot 10^{+77}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 12
Error	46.0
Cost	64

\[0 \]

Reproduce

herbie shell --seed 2023032 
(FPCore (x)
  :name "cos2 (problem 3.4.1)"
  :precision binary64
  (/ (- 1.0 (cos x)) (* x x)))