cos2 (problem 3.4.1)

Average Error: 49.86% → 0.19%

Time: 12.6s

Precision: binary64

Cost: 13376

Math

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

↓

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

FPCore

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

↓

(FPCore (x) :precision binary64 (/ (* (tan (* x 0.5)) (/ (sin x) x)) x))

C

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

↓

double code(double x) {
	return (tan((x * 0.5)) * (sin(x) / x)) / 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 = (tan((x * 0.5d0)) * (sin(x) / x)) / 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.tan((x * 0.5)) * (Math.sin(x) / x)) / x;
}

Python

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

↓

def code(x):
	return (math.tan((x * 0.5)) * (math.sin(x) / x)) / x

Julia

function code(x)
	return Float64(Float64(1.0 - cos(x)) / Float64(x * x))
end

↓

function code(x)
	return Float64(Float64(tan(Float64(x * 0.5)) * Float64(sin(x) / x)) / x)
end

MATLAB

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

↓

function tmp = code(x)
	tmp = (tan((x * 0.5)) * (sin(x) / x)) / x;
end

Wolfram

code[x_] := N[(N[(1.0 - N[Cos[x], $MachinePrecision]), $MachinePrecision] / N[(x * x), $MachinePrecision]), $MachinePrecision]

↓

code[x_] := N[(N[(N[Tan[N[(x * 0.5), $MachinePrecision]], $MachinePrecision] * N[(N[Sin[x], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]

Derivation

1. Initial program 49.86

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

2. Applied egg-rr 25.5

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

3. Simplified 25.25

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

   [Start] 25.5	\[ \frac{\left(\sin x \cdot \sin x\right) \cdot \frac{1}{1 + \cos x}}{x \cdot x} \]
   associate-*l* [=>] 25.5	\[ \frac{\color{blue}{\sin x \cdot \left(\sin x \cdot \frac{1}{1 + \cos x}\right)}}{x \cdot x} \]
   associate-*r/ [=>] 25.49	\[ \frac{\sin x \cdot \color{blue}{\frac{\sin x \cdot 1}{1 + \cos x}}}{x \cdot x} \]
   *-rgt-identity [=>] 25.49	\[ \frac{\sin x \cdot \frac{\color{blue}{\sin x}}{1 + \cos x}}{x \cdot x} \]
   hang-0p-tan [=>] 25.25	\[ \frac{\sin x \cdot \color{blue}{\tan \left(\frac{x}{2}\right)}}{x \cdot x} \]

4. Applied egg-rr 25.59

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

5. Applied egg-rr 0.19

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

6. Final simplification 0.19

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

Alternatives

Alternative 1
Error	0.57%
Cost	13640

\[\begin{array}{l} \mathbf{if}\;x \leq -0.032:\\ \;\;\;\;\frac{\frac{1}{\frac{1}{1 - \cos x}}}{x} \cdot \frac{1}{x}\\ \mathbf{elif}\;x \leq 2 \cdot 10^{-23}:\\ \;\;\;\;0.5 + \left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot 0.001388888888888889 + -0.041666666666666664\right)\\ \mathbf{else}:\\ \;\;\;\;\tan \left(x \cdot 0.5\right) \cdot \frac{\sin x}{x \cdot x}\\ \end{array} \]

Alternative 2
Error	0.38%
Cost	13448

\[\begin{array}{l} t_0 := 1 - \cos x\\ \mathbf{if}\;x \leq -0.032:\\ \;\;\;\;\frac{\frac{1}{\frac{1}{t_0}}}{x} \cdot \frac{1}{x}\\ \mathbf{elif}\;x \leq 0.03:\\ \;\;\;\;0.5 + \left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot 0.001388888888888889 + -0.041666666666666664\right)\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot {x}^{-2}\\ \end{array} \]

Alternative 3
Error	0.19%
Cost	13376

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

Alternative 4
Error	0.38%
Cost	7364

\[\begin{array}{l} t_0 := 1 - \cos x\\ \mathbf{if}\;x \leq -0.032:\\ \;\;\;\;\frac{\frac{1}{\frac{1}{t_0}}}{x} \cdot \frac{1}{x}\\ \mathbf{elif}\;x \leq 0.03:\\ \;\;\;\;0.5 + \left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot 0.001388888888888889 + -0.041666666666666664\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{t_0}{x}}{x}\\ \end{array} \]

Alternative 5
Error	0.85%
Cost	7113

\[\begin{array}{l} \mathbf{if}\;x \leq -0.032 \lor \neg \left(x \leq 0.03\right):\\ \;\;\;\;\frac{1 - \cos x}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;0.5 + \left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot 0.001388888888888889 + -0.041666666666666664\right)\\ \end{array} \]

Alternative 6
Error	0.36%
Cost	7113

\[\begin{array}{l} \mathbf{if}\;x \leq -0.032 \lor \neg \left(x \leq 0.03\right):\\ \;\;\;\;\frac{\frac{1 - \cos x}{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;0.5 + \left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot 0.001388888888888889 + -0.041666666666666664\right)\\ \end{array} \]

Alternative 7
Error	0.37%
Cost	7112

\[\begin{array}{l} t_0 := \frac{1 - \cos x}{x}\\ \mathbf{if}\;x \leq -0.032:\\ \;\;\;\;\frac{1}{x} \cdot t_0\\ \mathbf{elif}\;x \leq 0.03:\\ \;\;\;\;0.5 + \left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot 0.001388888888888889 + -0.041666666666666664\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0}{x}\\ \end{array} \]

Alternative 8
Error	20.83%
Cost	1097

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

Alternative 9
Error	20.92%
Cost	969

\[\begin{array}{l} \mathbf{if}\;x \leq -3.2 \lor \neg \left(x \leq 3.3\right):\\ \;\;\;\;\frac{6}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;0.5 + \left(\left(1 + \left(x \cdot x\right) \cdot -0.041666666666666664\right) + -1\right)\\ \end{array} \]

Alternative 10
Error	20.92%
Cost	969

\[\begin{array}{l} \mathbf{if}\;x \leq -3.5 \lor \neg \left(x \leq 3.4\right):\\ \;\;\;\;\frac{6 - \frac{72}{x \cdot x}}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;0.5 + \left(\left(1 + \left(x \cdot x\right) \cdot -0.041666666666666664\right) + -1\right)\\ \end{array} \]

Alternative 11
Error	20.92%
Cost	713

\[\begin{array}{l} \mathbf{if}\;x \leq -3.2 \lor \neg \left(x \leq 3.3\right):\\ \;\;\;\;\frac{6}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;0.5 + \left(x \cdot x\right) \cdot -0.041666666666666664\\ \end{array} \]

Alternative 12
Error	21.14%
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 13
Error	47.81%
Cost	64

\[0.5 \]

Reproduce

herbie shell --seed 2023090 
(FPCore (x)
  :name "cos2 (problem 3.4.1)"
  :precision binary64
  (/ (- 1.0 (cos x)) (* x x)))