cos2 (problem 3.4.1)

Average Error: 49.19% → 0.2%

Time: 11.7s

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 49.19

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

2. Applied egg-rr 25.4

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

3. Taylor expanded in x around inf 25.08

   \[\leadsto \color{blue}{\frac{{\sin x}^{2}}{\left(1 + \cos x\right) \cdot {x}^{2}}} \]

4. Simplified 25.45

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

   [Start] 25.08	\[ \frac{{\sin x}^{2}}{\left(1 + \cos x\right) \cdot {x}^{2}} \]
   associate-/r* [=>] 25.39	\[ \color{blue}{\frac{\frac{{\sin x}^{2}}{1 + \cos x}}{{x}^{2}}} \]
   unpow2 [=>] 25.4	\[ \frac{\frac{\color{blue}{\sin x \cdot \sin x}}{1 + \cos x}}{{x}^{2}} \]
   associate-*r/ [<=] 25.4	\[ \frac{\color{blue}{\sin x \cdot \frac{\sin x}{1 + \cos x}}}{{x}^{2}} \]
   unpow2 [=>] 25.4	\[ \frac{\sin x \cdot \frac{\sin x}{1 + \cos x}}{\color{blue}{x \cdot x}} \]
   associate-*l/ [<=] 25.69	\[ \color{blue}{\frac{\sin x}{x \cdot x} \cdot \frac{\sin x}{1 + \cos x}} \]
   hang-0p-tan [=>] 25.45	\[ \frac{\sin x}{x \cdot x} \cdot \color{blue}{\tan \left(\frac{x}{2}\right)} \]

5. Applied egg-rr 0.2

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

6. Final simplification 0.2

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

Alternatives

Alternative 1
Error	0.7%
Cost	13376

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

Alternative 2
Error	0.39%
Cost	7240

\[\begin{array}{l} t_0 := -1 + \cos x\\ \mathbf{if}\;x \leq -0.032:\\ \;\;\;\;\frac{t_0 \cdot \frac{-1}{x}}{x}\\ \mathbf{elif}\;x \leq 0.033:\\ \;\;\;\;0.5 + \left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot 0.001388888888888889 + -0.041666666666666664\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-1}{\frac{x}{t_0}}}{x}\\ \end{array} \]

Alternative 3
Error	0.86%
Cost	7113

\[\begin{array}{l} \mathbf{if}\;x \leq -0.032 \lor \neg \left(x \leq 0.033\right):\\ \;\;\;\;\frac{1 - \cos x}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;0.5 + \left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot 0.001388888888888889 + -0.041666666666666664\right)\\ \end{array} \]

Alternative 4
Error	0.37%
Cost	7113

\[\begin{array}{l} \mathbf{if}\;x \leq -0.032 \lor \neg \left(x \leq 0.033\right):\\ \;\;\;\;\frac{\frac{1 - \cos x}{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;0.5 + \left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot 0.001388888888888889 + -0.041666666666666664\right)\\ \end{array} \]

Alternative 5
Error	0.39%
Cost	7112

\[\begin{array}{l} t_0 := \frac{1 - \cos x}{x}\\ \mathbf{if}\;x \leq -0.032:\\ \;\;\;\;t_0 \cdot \frac{1}{x}\\ \mathbf{elif}\;x \leq 0.033:\\ \;\;\;\;0.5 + \left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot 0.001388888888888889 + -0.041666666666666664\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0}{x}\\ \end{array} \]

Alternative 6
Error	0.38%
Cost	7112

\[\begin{array}{l} \mathbf{if}\;x \leq -0.032:\\ \;\;\;\;\frac{\left(-1 + \cos x\right) \cdot \frac{-1}{x}}{x}\\ \mathbf{elif}\;x \leq 0.033:\\ \;\;\;\;0.5 + \left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot 0.001388888888888889 + -0.041666666666666664\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 - \cos x}{x}}{x}\\ \end{array} \]

Alternative 7
Error	21.66%
Cost	1096

\[\begin{array}{l} \mathbf{if}\;x \leq -4.8:\\ \;\;\;\;\frac{6}{x \cdot x}\\ \mathbf{elif}\;x \leq 3.9:\\ \;\;\;\;0.5 + \left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot 0.001388888888888889 + -0.041666666666666664\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{6 - \frac{72}{x \cdot x}}{x \cdot x}\\ \end{array} \]

Alternative 8
Error	21.76%
Cost	968

\[\begin{array}{l} \mathbf{if}\;x \leq -3.3:\\ \;\;\;\;\frac{6}{x \cdot x}\\ \mathbf{elif}\;x \leq 3.4:\\ \;\;\;\;0.5 + \left(x \cdot x\right) \cdot -0.041666666666666664\\ \mathbf{else}:\\ \;\;\;\;\frac{6 - \frac{72}{x \cdot x}}{x \cdot x}\\ \end{array} \]

Alternative 9
Error	21.76%
Cost	713

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

Alternative 10
Error	22.05%
Cost	585

\[\begin{array}{l} \mathbf{if}\;x \leq -3.5 \lor \neg \left(x \leq 3.4\right):\\ \;\;\;\;\frac{6}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \]

Alternative 11
Error	48.46%
Cost	64

\[0.5 \]

Reproduce

herbie shell --seed 2023088 
(FPCore (x)
  :name "cos2 (problem 3.4.1)"
  :precision binary64
  (/ (- 1.0 (cos x)) (* x x)))