cos2 (problem 3.4.1)

Average Error: 31.4 → 0.1

Time: 10.6s

Precision: binary64

Cost: 13376

Math

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

↓

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

FPCore

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

↓

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

C

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

↓

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

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 = (tan((x * 0.5d0)) / x) / (x / sin(x))
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.tan((x * 0.5)) / x) / (x / Math.sin(x));
}

Python

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

↓

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

Julia

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

↓

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

MATLAB

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

↓

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

Wolfram

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

↓

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

Derivation

1. Initial program 31.4

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

2. Applied egg-rr 15.9

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

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

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

5. Final simplification 0.1

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

Alternatives

Alternative 1
Error	0.4
Cost	13448

\[\begin{array}{l} t_0 := 1 - \cos x\\ \mathbf{if}\;x \leq -0.0046:\\ \;\;\;\;\frac{\frac{1}{\frac{1}{t_0}}}{x \cdot x}\\ \mathbf{elif}\;x \leq 0.0037:\\ \;\;\;\;0.5 + \left(x \cdot x\right) \cdot -0.041666666666666664\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot {x}^{-2}\\ \end{array} \]

Alternative 2
Error	0.1
Cost	13376

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

Alternative 3
Error	0.4
Cost	7236

\[\begin{array}{l} t_0 := 1 - \cos x\\ \mathbf{if}\;x \leq -0.0046:\\ \;\;\;\;\frac{\frac{1}{\frac{1}{t_0}}}{x \cdot x}\\ \mathbf{elif}\;x \leq 0.0037:\\ \;\;\;\;0.5 + \left(x \cdot x\right) \cdot -0.041666666666666664\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{t_0}{x}}{x}\\ \end{array} \]

Alternative 4
Error	0.6
Cost	7113

\[\begin{array}{l} \mathbf{if}\;x \leq -0.0046 \lor \neg \left(x \leq 0.0037\right):\\ \;\;\;\;\frac{1 - \cos x}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;0.5 + \left(x \cdot x\right) \cdot -0.041666666666666664\\ \end{array} \]

Alternative 5
Error	0.4
Cost	7112

\[\begin{array}{l} t_0 := 1 - \cos x\\ \mathbf{if}\;x \leq -0.0046:\\ \;\;\;\;\frac{t_0}{x \cdot x}\\ \mathbf{elif}\;x \leq 0.0037:\\ \;\;\;\;0.5 + \left(x \cdot x\right) \cdot -0.041666666666666664\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{t_0}{x}}{x}\\ \end{array} \]

Alternative 6
Error	14.0
Cost	1096

\[\begin{array}{l} \mathbf{if}\;x \leq -3.25:\\ \;\;\;\;\frac{6}{x \cdot x}\\ \mathbf{elif}\;x \leq 10^{-6}:\\ \;\;\;\;0.5 + \left(x \cdot x\right) \cdot -0.041666666666666664\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{0.16666666666666666 + \frac{2}{x \cdot x}}}{x \cdot x}\\ \end{array} \]

Alternative 7
Error	14.0
Cost	969

\[\begin{array}{l} \mathbf{if}\;x \leq -3.25 \lor \neg \left(x \leq 3.3\right):\\ \;\;\;\;\frac{6}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;0.5 + \left(\left(1 + \left(x \cdot x\right) \cdot -0.041666666666666664\right) + -1\right)\\ \end{array} \]

Alternative 8
Error	14.0
Cost	968

\[\begin{array}{l} \mathbf{if}\;x \leq -3.25:\\ \;\;\;\;\frac{6}{x \cdot x}\\ \mathbf{elif}\;x \leq 3.45:\\ \;\;\;\;0.5 + \left(\left(1 + \left(x \cdot x\right) \cdot -0.041666666666666664\right) + -1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{6 + \frac{-72}{x \cdot x}}{x \cdot x}\\ \end{array} \]

Alternative 9
Error	14.0
Cost	713

\[\begin{array}{l} \mathbf{if}\;x \leq -3.25 \lor \neg \left(x \leq 3.3\right):\\ \;\;\;\;\frac{6}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;0.5 + \left(x \cdot x\right) \cdot -0.041666666666666664\\ \end{array} \]

Alternative 10
Error	14.2
Cost	585

\[\begin{array}{l} \mathbf{if}\;x \leq -3.5 \lor \neg \left(x \leq 3.45\right):\\ \;\;\;\;\frac{6}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \]

Alternative 11
Error	31.1
Cost	64

\[0.5 \]

Reproduce

herbie shell --seed 2023054 
(FPCore (x)
  :name "cos2 (problem 3.4.1)"
  :precision binary64
  (/ (- 1.0 (cos x)) (* x x)))