cos2 (problem 3.4.1)

Average Error: 31.5 → 0.1

Time: 8.2s

Precision: binary64

Cost: 13376

Math

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

↓

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

FPCore

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

↓

(FPCore (x) :precision binary64 (* (/ (sin x) x) (/ (tan (/ x 2.0)) x)))

C

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

↓

double code(double x) {
	return (sin(x) / x) * (tan((x / 2.0)) / 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 = (sin(x) / x) * (tan((x / 2.0d0)) / 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.sin(x) / x) * (Math.tan((x / 2.0)) / x);
}

Python

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

↓

def code(x):
	return (math.sin(x) / x) * (math.tan((x / 2.0)) / x)

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(tan(Float64(x / 2.0)) / x))
end

MATLAB

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

↓

function tmp = code(x)
	tmp = (sin(x) / x) * (tan((x / 2.0)) / x);
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[(N[Tan[N[(x / 2.0), $MachinePrecision]], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 31.5

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

2. Applied egg-rr 15.9

   \[\leadsto \frac{\color{blue}{\left(\sin x \cdot \sin x\right) \cdot \frac{1}{1 + \cos x}}}{x \cdot x} \]

3. Applied egg-rr 0.1

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

4. Final simplification 0.1

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

Alternatives

Alternative 1
Error	0.6
Cost	7816

\[\begin{array}{l} t_0 := 1 - \cos x\\ \mathbf{if}\;x \leq -916.5785249190432:\\ \;\;\;\;\frac{\frac{1}{x}}{x \cdot \frac{1}{t_0}}\\ \mathbf{elif}\;x \leq 1.3195296957559512 \cdot 10^{-8}:\\ \;\;\;\;\frac{\frac{1}{x}}{0.008333333333333333 \cdot {x}^{3} + \left(x \cdot 0.16666666666666666 + 2 \cdot \frac{1}{x}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{t_0}{x}}{x}\\ \end{array} \]

Alternative 2
Error	0.7
Cost	7236

\[\begin{array}{l} t_0 := 1 - \cos x\\ \mathbf{if}\;x \leq -916.5785249190432:\\ \;\;\;\;\frac{\frac{1}{x}}{x \cdot \frac{1}{t_0}}\\ \mathbf{elif}\;x \leq 1.3195296957559512 \cdot 10^{-8}:\\ \;\;\;\;0.5 + x \cdot \left(x \cdot -0.041666666666666664\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{t_0}{x}}{x}\\ \end{array} \]

Alternative 3
Error	0.9
Cost	7112

\[\begin{array}{l} t_0 := \frac{1 - \cos x}{x \cdot x}\\ \mathbf{if}\;x \leq -916.5785249190432:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 1.3195296957559512 \cdot 10^{-8}:\\ \;\;\;\;0.5 + x \cdot \left(x \cdot -0.041666666666666664\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 4
Error	0.7
Cost	7112

\[\begin{array}{l} t_0 := \frac{\frac{1 - \cos x}{x}}{x}\\ \mathbf{if}\;x \leq -916.5785249190432:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 1.3195296957559512 \cdot 10^{-8}:\\ \;\;\;\;0.5 + x \cdot \left(x \cdot -0.041666666666666664\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 5
Error	0.7
Cost	7112

\[\begin{array}{l} t_0 := 1 - \cos x\\ \mathbf{if}\;x \leq -916.5785249190432:\\ \;\;\;\;\frac{\frac{1}{x}}{\frac{x}{t_0}}\\ \mathbf{elif}\;x \leq 1.3195296957559512 \cdot 10^{-8}:\\ \;\;\;\;0.5 + x \cdot \left(x \cdot -0.041666666666666664\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{t_0}{x}}{x}\\ \end{array} \]

Alternative 6
Error	14.0
Cost	832

\[\frac{\frac{1}{x}}{x \cdot 0.16666666666666666 + 2 \cdot \frac{1}{x}} \]

Alternative 7
Error	30.9
Cost	64

\[0.5 \]

Reproduce

herbie shell --seed 2022228 
(FPCore (x)
  :name "cos2 (problem 3.4.1)"
  :precision binary64
  (/ (- 1.0 (cos x)) (* x x)))