cos2 (problem 3.4.1)

Average Accuracy: 50.3% → 99.8%

Time: 14.9s

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 50.3%

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

2. Applied egg-rr 75.2%

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

   [Start] 50.3	\[ \frac{1 - \cos x}{x \cdot x} \]
   flip-- [=>] 50.1	\[ \frac{\color{blue}{\frac{1 \cdot 1 - \cos x \cdot \cos x}{1 + \cos x}}}{x \cdot x} \]
   div-inv [=>] 50.1	\[ \frac{\color{blue}{\left(1 \cdot 1 - \cos x \cdot \cos x\right) \cdot \frac{1}{1 + \cos x}}}{x \cdot x} \]
   metadata-eval [=>] 50.1	\[ \frac{\left(\color{blue}{1} - \cos x \cdot \cos x\right) \cdot \frac{1}{1 + \cos x}}{x \cdot x} \]
   1-sub-cos [=>] 75.2	\[ \frac{\color{blue}{\left(\sin x \cdot \sin x\right)} \cdot \frac{1}{1 + \cos x}}{x \cdot x} \]

3. Simplified 75.5%

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

   [Start] 75.2	\[ \frac{\left(\sin x \cdot \sin x\right) \cdot \frac{1}{1 + \cos x}}{x \cdot x} \]
   associate-*l* [=>] 75.2	\[ \frac{\color{blue}{\sin x \cdot \left(\sin x \cdot \frac{1}{1 + \cos x}\right)}}{x \cdot x} \]
   associate-*r/ [=>] 75.2	\[ \frac{\sin x \cdot \color{blue}{\frac{\sin x \cdot 1}{1 + \cos x}}}{x \cdot x} \]
   *-rgt-identity [=>] 75.2	\[ \frac{\sin x \cdot \frac{\color{blue}{\sin x}}{1 + \cos x}}{x \cdot x} \]
   hang-0p-tan [=>] 75.5	\[ \frac{\sin x \cdot \color{blue}{\tan \left(\frac{x}{2}\right)}}{x \cdot x} \]

4. Applied egg-rr 99.8%

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

   [Start] 75.5	\[ \frac{\sin x \cdot \tan \left(\frac{x}{2}\right)}{x \cdot x} \]
   times-frac [=>] 99.8	\[ \color{blue}{\frac{\sin x}{x} \cdot \frac{\tan \left(\frac{x}{2}\right)}{x}} \]
   *-commutative [=>] 99.8	\[ \color{blue}{\frac{\tan \left(\frac{x}{2}\right)}{x} \cdot \frac{\sin x}{x}} \]
   div-inv [=>] 99.8	\[ \frac{\tan \color{blue}{\left(x \cdot \frac{1}{2}\right)}}{x} \cdot \frac{\sin x}{x} \]
   metadata-eval [=>] 99.8	\[ \frac{\tan \left(x \cdot \color{blue}{0.5}\right)}{x} \cdot \frac{\sin x}{x} \]

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

6. Final simplification 99.8%

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

Alternatives

Alternative 1
Accuracy	99.0%
Cost	20680

\[\begin{array}{l} t_0 := \frac{1 - \cos x}{x \cdot x}\\ t_1 := x \cdot 0.16666666666666666 + \frac{1}{x} \cdot 2\\ \mathbf{if}\;t_0 \leq 0:\\ \;\;\;\;\frac{-1}{\frac{t_1}{\frac{-1}{x}}}\\ \mathbf{elif}\;t_0 \leq 0.495:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x}}{t_1}\\ \end{array} \]

Alternative 2
Accuracy	99.4%
Cost	13641

\[\begin{array}{l} \mathbf{if}\;x \leq -0.0003 \lor \neg \left(x \leq 5 \cdot 10^{-10}\right):\\ \;\;\;\;\tan \left(x \cdot 0.5\right) \cdot \frac{\sin x}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x}}{x \cdot 0.16666666666666666 + \frac{1}{x} \cdot 2}\\ \end{array} \]

Alternative 3
Accuracy	99.8%
Cost	13376

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

Alternative 4
Accuracy	99.6%
Cost	7240

\[\begin{array}{l} t_0 := \frac{1 - \cos x}{x}\\ \mathbf{if}\;x \leq -0.005:\\ \;\;\;\;\frac{t_0}{x}\\ \mathbf{elif}\;x \leq 0.0046:\\ \;\;\;\;\frac{\frac{1}{x}}{x \cdot 0.16666666666666666 + \frac{1}{x} \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x} \cdot t_0\\ \end{array} \]

Alternative 5
Accuracy	99.5%
Cost	7240

\[\begin{array}{l} \mathbf{if}\;x \leq -0.005:\\ \;\;\;\;\frac{\frac{1}{x} \cdot \left(-1 + \cos x\right)}{-x}\\ \mathbf{elif}\;x \leq 0.0046:\\ \;\;\;\;\frac{\frac{1}{x}}{x \cdot 0.16666666666666666 + \frac{1}{x} \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x} \cdot \frac{1 - \cos x}{x}\\ \end{array} \]

Alternative 6
Accuracy	99.4%
Cost	7112

\[\begin{array}{l} t_0 := 1 - \cos x\\ \mathbf{if}\;x \leq -0.005:\\ \;\;\;\;\frac{\frac{t_0}{x}}{x}\\ \mathbf{elif}\;x \leq 0.0046:\\ \;\;\;\;\frac{\frac{1}{x}}{x \cdot 0.16666666666666666 + \frac{1}{x} \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0}{x \cdot x}\\ \end{array} \]

Alternative 7
Accuracy	77.8%
Cost	969

\[\begin{array}{l} \mathbf{if}\;x \leq -3.5 \lor \neg \left(x \leq 3.4\right):\\ \;\;\;\;\frac{6 + \frac{-72}{x \cdot x}}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;0.5 + \left(x \cdot x\right) \cdot -0.041666666666666664\\ \end{array} \]

Alternative 8
Accuracy	77.8%
Cost	960

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

Alternative 9
Accuracy	77.8%
Cost	832

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

Alternative 10
Accuracy	77.8%
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 11
Accuracy	77.6%
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 12
Accuracy	52.1%
Cost	64

\[0.5 \]

Reproduce

herbie shell --seed 2023131 
(FPCore (x)
  :name "cos2 (problem 3.4.1)"
  :precision binary64
  (/ (- 1.0 (cos x)) (* x x)))