cos2 (problem 3.4.1)

Average Error: 32.2 → 0.4

Time: 12.8s

Precision: binary64

Cost: 19780

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \leq -753.3516174146289:\\ \;\;\;\;{x}^{-2} - {x}^{-2} \cdot \cos x\\ \mathbf{elif}\;x \leq 1.2754132945467591 \cdot 10^{-5}:\\ \;\;\;\;0.5 + \left(-0.041666666666666664 \cdot \left(x \cdot x\right) + 0.001388888888888889 \cdot {x}^{4}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\tan \left(\frac{x}{2}\right) \cdot \sin x}{x \cdot x}\\ \end{array} \]

FPCore

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

↓

(FPCore (x)
 :precision binary64
 (if (<= x -753.3516174146289)
   (- (pow x -2.0) (* (pow x -2.0) (cos x)))
   (if (<= x 1.2754132945467591e-5)
     (+
      0.5
      (+
       (* -0.041666666666666664 (* x x))
       (* 0.001388888888888889 (pow x 4.0))))
     (/ (* (tan (/ x 2.0)) (sin x)) (* x x)))))

C

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

↓

double code(double x) {
	double tmp;
	if (x <= -753.3516174146289) {
		tmp = pow(x, -2.0) - (pow(x, -2.0) * cos(x));
	} else if (x <= 1.2754132945467591e-5) {
		tmp = 0.5 + ((-0.041666666666666664 * (x * x)) + (0.001388888888888889 * pow(x, 4.0)));
	} else {
		tmp = (tan((x / 2.0)) * sin(x)) / (x * x);
	}
	return tmp;
}

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
    real(8) :: tmp
    if (x <= (-753.3516174146289d0)) then
        tmp = (x ** (-2.0d0)) - ((x ** (-2.0d0)) * cos(x))
    else if (x <= 1.2754132945467591d-5) then
        tmp = 0.5d0 + (((-0.041666666666666664d0) * (x * x)) + (0.001388888888888889d0 * (x ** 4.0d0)))
    else
        tmp = (tan((x / 2.0d0)) * sin(x)) / (x * x)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	return (1.0 - Math.cos(x)) / (x * x);
}

↓

public static double code(double x) {
	double tmp;
	if (x <= -753.3516174146289) {
		tmp = Math.pow(x, -2.0) - (Math.pow(x, -2.0) * Math.cos(x));
	} else if (x <= 1.2754132945467591e-5) {
		tmp = 0.5 + ((-0.041666666666666664 * (x * x)) + (0.001388888888888889 * Math.pow(x, 4.0)));
	} else {
		tmp = (Math.tan((x / 2.0)) * Math.sin(x)) / (x * x);
	}
	return tmp;
}

Python

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

↓

def code(x):
	tmp = 0
	if x <= -753.3516174146289:
		tmp = math.pow(x, -2.0) - (math.pow(x, -2.0) * math.cos(x))
	elif x <= 1.2754132945467591e-5:
		tmp = 0.5 + ((-0.041666666666666664 * (x * x)) + (0.001388888888888889 * math.pow(x, 4.0)))
	else:
		tmp = (math.tan((x / 2.0)) * math.sin(x)) / (x * x)
	return tmp

Julia

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

↓

function code(x)
	tmp = 0.0
	if (x <= -753.3516174146289)
		tmp = Float64((x ^ -2.0) - Float64((x ^ -2.0) * cos(x)));
	elseif (x <= 1.2754132945467591e-5)
		tmp = Float64(0.5 + Float64(Float64(-0.041666666666666664 * Float64(x * x)) + Float64(0.001388888888888889 * (x ^ 4.0))));
	else
		tmp = Float64(Float64(tan(Float64(x / 2.0)) * sin(x)) / Float64(x * x));
	end
	return tmp
end

MATLAB

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

↓

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -753.3516174146289)
		tmp = (x ^ -2.0) - ((x ^ -2.0) * cos(x));
	elseif (x <= 1.2754132945467591e-5)
		tmp = 0.5 + ((-0.041666666666666664 * (x * x)) + (0.001388888888888889 * (x ^ 4.0)));
	else
		tmp = (tan((x / 2.0)) * sin(x)) / (x * x);
	end
	tmp_2 = tmp;
end

Wolfram

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

↓

code[x_] := If[LessEqual[x, -753.3516174146289], N[(N[Power[x, -2.0], $MachinePrecision] - N[(N[Power[x, -2.0], $MachinePrecision] * N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.2754132945467591e-5], N[(0.5 + N[(N[(-0.041666666666666664 * N[(x * x), $MachinePrecision]), $MachinePrecision] + N[(0.001388888888888889 * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Tan[N[(x / 2.0), $MachinePrecision]], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision] / N[(x * x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -753.35161741462889

   1. Initial program 1.2

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

   2. Applied egg-rr 0.5

      \[\leadsto \color{blue}{{x}^{-2} - \cos x \cdot {x}^{-2}} \]

   if -753.35161741462889 < x < 1.2754132945467591e-5

   1. Initial program 62.1

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

   2. Taylor expanded in x around 0 0.3

      \[\leadsto \color{blue}{0.5 + \left(-0.041666666666666664 \cdot {x}^{2} + 0.001388888888888889 \cdot {x}^{4}\right)} \]

   3. Applied egg-rr 0.3

      \[\leadsto 0.5 + \left(\color{blue}{\left(0 + -0.041666666666666664 \cdot \left(x \cdot x\right)\right)} + 0.001388888888888889 \cdot {x}^{4}\right) \]

   if 1.2754132945467591e-5 < x

   1. Initial program 1.0

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

   2. Applied egg-rr 0.9

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

   3. Applied egg-rr 0.6

      \[\leadsto \frac{\color{blue}{\tan \left(\frac{x}{2}\right) \cdot \sin x}}{x \cdot x} \]
3. Recombined 3 regimes into one program.

4. Final simplification 0.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -753.3516174146289:\\ \;\;\;\;{x}^{-2} - {x}^{-2} \cdot \cos x\\ \mathbf{elif}\;x \leq 1.2754132945467591 \cdot 10^{-5}:\\ \;\;\;\;0.5 + \left(-0.041666666666666664 \cdot \left(x \cdot x\right) + 0.001388888888888889 \cdot {x}^{4}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\tan \left(\frac{x}{2}\right) \cdot \sin x}{x \cdot x}\\ \end{array} \]

Alternatives

Alternative 1
Error	0.4
Cost	13640

\[\begin{array}{l} \mathbf{if}\;x \leq -753.3516174146289:\\ \;\;\;\;\frac{\frac{1 - \cos x}{x}}{x}\\ \mathbf{elif}\;x \leq 0.0038429952274706726:\\ \;\;\;\;0.5 + \left(-0.041666666666666664 \cdot \left(x \cdot x\right) + 0.001388888888888889 \cdot {x}^{4}\right)\\ \mathbf{else}:\\ \;\;\;\;\tan \left(\frac{x}{2}\right) \cdot \frac{\sin x}{x \cdot x}\\ \end{array} \]

Alternative 2
Error	0.4
Cost	13640

\[\begin{array}{l} \mathbf{if}\;x \leq -753.3516174146289:\\ \;\;\;\;\frac{\frac{1 - \cos x}{x}}{x}\\ \mathbf{elif}\;x \leq 1.2754132945467591 \cdot 10^{-5}:\\ \;\;\;\;0.5 + \left(-0.041666666666666664 \cdot \left(x \cdot x\right) + 0.001388888888888889 \cdot {x}^{4}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\tan \left(\frac{x}{2}\right) \cdot \sin x}{x \cdot x}\\ \end{array} \]

Alternative 3
Error	0.4
Cost	7432

\[\begin{array}{l} t_0 := 1 - \cos x\\ \mathbf{if}\;x \leq -753.3516174146289:\\ \;\;\;\;\frac{\frac{t_0}{x}}{x}\\ \mathbf{elif}\;x \leq 0.0038429952274706726:\\ \;\;\;\;0.5 + \left(-0.041666666666666664 \cdot \left(x \cdot x\right) + 0.001388888888888889 \cdot {x}^{4}\right)\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot \frac{\frac{1}{x}}{x}\\ \end{array} \]

Alternative 4
Error	0.5
Cost	7240

\[\begin{array}{l} t_0 := 1 - \cos x\\ \mathbf{if}\;x \leq -753.3516174146289:\\ \;\;\;\;\frac{\frac{t_0}{x}}{x}\\ \mathbf{elif}\;x \leq 0.0038429952274706726:\\ \;\;\;\;0.5 + -0.041666666666666664 \cdot \left(x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot \frac{\frac{1}{x}}{x}\\ \end{array} \]

Alternative 5
Error	0.4
Cost	7112

\[\begin{array}{l} t_0 := \frac{\frac{1 - \cos x}{x}}{x}\\ \mathbf{if}\;x \leq -753.3516174146289:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 0.0038429952274706726:\\ \;\;\;\;0.5 + -0.041666666666666664 \cdot \left(x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 6
Error	13.3
Cost	832

\[\frac{1}{x \cdot \left(x \cdot 0.16666666666666666 + 2 \cdot \frac{1}{x}\right)} \]

Alternative 7
Error	14.9
Cost	712

\[\begin{array}{l} \mathbf{if}\;x \leq -753.3516174146289:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 0.0038429952274706726:\\ \;\;\;\;0.5 + -0.041666666666666664 \cdot \left(x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 8
Error	14.8
Cost	328

\[\begin{array}{l} \mathbf{if}\;x \leq -8.362066599189694 \cdot 10^{+88}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 5.596090494016065 \cdot 10^{+71}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 9
Error	30.2
Cost	64

\[0.5 \]

Reproduce

herbie shell --seed 2022318 
(FPCore (x)
  :name "cos2 (problem 3.4.1)"
  :precision binary64
  (/ (- 1.0 (cos x)) (* x x)))