cos2 (problem 3.4.1)

Average Error: 31.7 → 0.1

Time: 10.5s

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 31.7

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

2. Applied egg-rr 16.3

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

3. Applied egg-rr 16.2

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

4. Applied egg-rr 0.1

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

5. Applied egg-rr 0.1

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

6. Final simplification 0.1

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

Alternatives

Alternative 1
Error	0.5
Cost	13508

\[\begin{array}{l} \mathbf{if}\;x \leq -71.91497133033104:\\ \;\;\;\;\frac{\tan \left(x \cdot 0.5\right)}{\frac{x \cdot x}{\sin x}}\\ \mathbf{elif}\;x \leq 0.0002529585668099084:\\ \;\;\;\;0.5 + \left(x \cdot x\right) \cdot -0.041666666666666664\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 - \cos x}{x}}{x}\\ \end{array} \]

Alternative 2
Error	0.4
Cost	13316

\[\begin{array}{l} t_0 := 1 - \cos x\\ \mathbf{if}\;x \leq -71.91497133033104:\\ \;\;\;\;t_0 \cdot {x}^{-2}\\ \mathbf{elif}\;x \leq 0.0002529585668099084:\\ \;\;\;\;0.5 + \left(x \cdot x\right) \cdot -0.041666666666666664\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{t_0}{x}}{x}\\ \end{array} \]

Alternative 3
Error	0.4
Cost	7112

\[\begin{array}{l} t_0 := \frac{\frac{1 - \cos x}{x}}{x}\\ \mathbf{if}\;x \leq -71.91497133033104:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 0.0002529585668099084:\\ \;\;\;\;0.5 + \left(x \cdot x\right) \cdot -0.041666666666666664\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 4
Error	0.6
Cost	7112

\[\begin{array}{l} t_0 := 1 - \cos x\\ \mathbf{if}\;x \leq -71.91497133033104:\\ \;\;\;\;\frac{1}{x \cdot \frac{x}{t_0}}\\ \mathbf{elif}\;x \leq 0.0002529585668099084:\\ \;\;\;\;0.5 + \left(x \cdot x\right) \cdot -0.041666666666666664\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{t_0}{x}}{x}\\ \end{array} \]

Alternative 5
Error	15.6
Cost	968

\[\begin{array}{l} t_0 := \frac{1}{\frac{\frac{x \cdot x}{x \cdot 0.5}}{x}}\\ \mathbf{if}\;x \leq -71.91497133033104:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 0.0002529585668099084:\\ \;\;\;\;0.5 + \left(x \cdot x\right) \cdot -0.041666666666666664\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 6
Error	13.9
Cost	832

\[\frac{1}{x \cdot \left(x \cdot 0.16666666666666666 + 2 \cdot \frac{1}{x}\right)} \]

Alternative 7
Error	30.8
Cost	64

\[0.5 \]

Reproduce

herbie shell --seed 2022296 
(FPCore (x)
  :name "cos2 (problem 3.4.1)"
  :precision binary64
  (/ (- 1.0 (cos x)) (* x x)))