cos2 (problem 3.4.1)

Average Accuracy: 51.4% → 99.8%

Time: 13.5s

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 51.4%

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

2. Applied egg-rr 75.7%

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

   [Start] 51.4	\[ \frac{1 - \cos x}{x \cdot x} \]
   flip-- [=>] 51.2	\[ \frac{\color{blue}{\frac{1 \cdot 1 - \cos x \cdot \cos x}{1 + \cos x}}}{x \cdot x} \]
   metadata-eval [=>] 51.2	\[ \frac{\frac{\color{blue}{1} - \cos x \cdot \cos x}{1 + \cos x}}{x \cdot x} \]
   1-sub-cos [=>] 75.8	\[ \frac{\frac{\color{blue}{\sin x \cdot \sin x}}{1 + \cos x}}{x \cdot x} \]
   associate-/l* [=>] 75.7	\[ \frac{\color{blue}{\frac{\sin x}{\frac{1 + \cos x}{\sin x}}}}{x \cdot x} \]

3. Applied egg-rr 99.8%

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

   [Start] 75.7	\[ \frac{\frac{\sin x}{\frac{1 + \cos x}{\sin x}}}{x \cdot x} \]
   associate-/r/ [=>] 75.8	\[ \frac{\color{blue}{\frac{\sin x}{1 + \cos x} \cdot \sin x}}{x \cdot x} \]
   times-frac [=>] 99.5	\[ \color{blue}{\frac{\frac{\sin x}{1 + \cos x}}{x} \cdot \frac{\sin x}{x}} \]
   hang-0p-tan [=>] 99.8	\[ \frac{\color{blue}{\tan \left(\frac{x}{2}\right)}}{x} \cdot \frac{\sin x}{x} \]

4. Applied egg-rr 99.8%

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

   [Start] 99.8	\[ \frac{\tan \left(\frac{x}{2}\right)}{x} \cdot \frac{\sin x}{x} \]
   clear-num [=>] 99.8	\[ \color{blue}{\frac{1}{\frac{x}{\tan \left(\frac{x}{2}\right)}}} \cdot \frac{\sin x}{x} \]
   associate-*l/ [=>] 99.8	\[ \color{blue}{\frac{1 \cdot \frac{\sin x}{x}}{\frac{x}{\tan \left(\frac{x}{2}\right)}}} \]
   *-un-lft-identity [<=] 99.8	\[ \frac{\color{blue}{\frac{\sin x}{x}}}{\frac{x}{\tan \left(\frac{x}{2}\right)}} \]
   div-inv [=>] 99.8	\[ \frac{\frac{\sin x}{x}}{\frac{x}{\tan \color{blue}{\left(x \cdot \frac{1}{2}\right)}}} \]
   metadata-eval [=>] 99.8	\[ \frac{\frac{\sin x}{x}}{\frac{x}{\tan \left(x \cdot \color{blue}{0.5}\right)}} \]

5. Final simplification 99.8%

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

Alternatives

Alternative 1
Accuracy	99.8%
Cost	13376

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

Alternative 2
Accuracy	99.1%
Cost	7113

\[\begin{array}{l} \mathbf{if}\;x \leq -0.0285 \lor \neg \left(x \leq 0.0295\right):\\ \;\;\;\;\frac{1 - \cos x}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;0.5 + \left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot 0.001388888888888889 + -0.041666666666666664\right)\\ \end{array} \]

Alternative 3
Accuracy	99.6%
Cost	7113

\[\begin{array}{l} \mathbf{if}\;x \leq -0.0285 \lor \neg \left(x \leq 0.0295\right):\\ \;\;\;\;\frac{\frac{1 - \cos x}{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;0.5 + \left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot 0.001388888888888889 + -0.041666666666666664\right)\\ \end{array} \]

Alternative 4
Accuracy	75.5%
Cost	905

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

Alternative 5
Accuracy	78.2%
Cost	704

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

Alternative 6
Accuracy	51.1%
Cost	64

\[0.5 \]

Reproduce

herbie shell --seed 2023133 
(FPCore (x)
  :name "cos2 (problem 3.4.1)"
  :precision binary64
  (/ (- 1.0 (cos x)) (* x x)))