cos2 (problem 3.4.1)

Average Error: 31.6 → 0.5

Time: 14.4s

Precision: binary64

Cost: 47432

Math

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

↓

\[\begin{array}{l} t_0 := 1 - \cos x\\ t_1 := t_0 \cdot \left(\frac{1}{t_0} \cdot t_0\right)\\ \mathbf{if}\;x \leq -0.029:\\ \;\;\;\;\frac{t_0}{x \cdot x}\\ \mathbf{elif}\;x \leq 0.033:\\ \;\;\;\;0.5 + \left(-0.041666666666666664 \cdot {x}^{2} + 0.001388888888888889 \cdot {x}^{4}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{t_1} \cdot \left(t_0 \cdot t_1\right)}{x \cdot x}\\ \end{array} \]

FPCore

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

↓

(FPCore (x)
 :precision binary64
 (let* ((t_0 (- 1.0 (cos x))) (t_1 (* t_0 (* (/ 1.0 t_0) t_0))))
   (if (<= x -0.029)
     (/ t_0 (* x x))
     (if (<= x 0.033)
       (+
        0.5
        (+
         (* -0.041666666666666664 (pow x 2.0))
         (* 0.001388888888888889 (pow x 4.0))))
       (/ (* (/ 1.0 t_1) (* t_0 t_1)) (* x x))))))

C

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

↓

double code(double x) {
	double t_0 = 1.0 - cos(x);
	double t_1 = t_0 * ((1.0 / t_0) * t_0);
	double tmp;
	if (x <= -0.029) {
		tmp = t_0 / (x * x);
	} else if (x <= 0.033) {
		tmp = 0.5 + ((-0.041666666666666664 * pow(x, 2.0)) + (0.001388888888888889 * pow(x, 4.0)));
	} else {
		tmp = ((1.0 / t_1) * (t_0 * t_1)) / (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) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = 1.0d0 - cos(x)
    t_1 = t_0 * ((1.0d0 / t_0) * t_0)
    if (x <= (-0.029d0)) then
        tmp = t_0 / (x * x)
    else if (x <= 0.033d0) then
        tmp = 0.5d0 + (((-0.041666666666666664d0) * (x ** 2.0d0)) + (0.001388888888888889d0 * (x ** 4.0d0)))
    else
        tmp = ((1.0d0 / t_1) * (t_0 * t_1)) / (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 t_0 = 1.0 - Math.cos(x);
	double t_1 = t_0 * ((1.0 / t_0) * t_0);
	double tmp;
	if (x <= -0.029) {
		tmp = t_0 / (x * x);
	} else if (x <= 0.033) {
		tmp = 0.5 + ((-0.041666666666666664 * Math.pow(x, 2.0)) + (0.001388888888888889 * Math.pow(x, 4.0)));
	} else {
		tmp = ((1.0 / t_1) * (t_0 * t_1)) / (x * x);
	}
	return tmp;
}

Python

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

↓

def code(x):
	t_0 = 1.0 - math.cos(x)
	t_1 = t_0 * ((1.0 / t_0) * t_0)
	tmp = 0
	if x <= -0.029:
		tmp = t_0 / (x * x)
	elif x <= 0.033:
		tmp = 0.5 + ((-0.041666666666666664 * math.pow(x, 2.0)) + (0.001388888888888889 * math.pow(x, 4.0)))
	else:
		tmp = ((1.0 / t_1) * (t_0 * t_1)) / (x * x)
	return tmp

Julia

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

↓

function code(x)
	t_0 = Float64(1.0 - cos(x))
	t_1 = Float64(t_0 * Float64(Float64(1.0 / t_0) * t_0))
	tmp = 0.0
	if (x <= -0.029)
		tmp = Float64(t_0 / Float64(x * x));
	elseif (x <= 0.033)
		tmp = Float64(0.5 + Float64(Float64(-0.041666666666666664 * (x ^ 2.0)) + Float64(0.001388888888888889 * (x ^ 4.0))));
	else
		tmp = Float64(Float64(Float64(1.0 / t_1) * Float64(t_0 * t_1)) / 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)
	t_0 = 1.0 - cos(x);
	t_1 = t_0 * ((1.0 / t_0) * t_0);
	tmp = 0.0;
	if (x <= -0.029)
		tmp = t_0 / (x * x);
	elseif (x <= 0.033)
		tmp = 0.5 + ((-0.041666666666666664 * (x ^ 2.0)) + (0.001388888888888889 * (x ^ 4.0)));
	else
		tmp = ((1.0 / t_1) * (t_0 * t_1)) / (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_] := Block[{t$95$0 = N[(1.0 - N[Cos[x], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 * N[(N[(1.0 / t$95$0), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -0.029], N[(t$95$0 / N[(x * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.033], N[(0.5 + N[(N[(-0.041666666666666664 * N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision] + N[(0.001388888888888889 * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 / t$95$1), $MachinePrecision] * N[(t$95$0 * t$95$1), $MachinePrecision]), $MachinePrecision] / N[(x * x), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.0290000000000000015

   1. Initial program 1.0

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

   if -0.0290000000000000015 < x < 0.033000000000000002

   1. Initial program 62.2

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

   2. Taylor expanded in x around 0 0.0

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

   if 0.033000000000000002 < x

   1. Initial program 1.0

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

   2. Applied egg-rr 1.1

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

   3. Applied egg-rr 1.1

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

   4. Applied egg-rr 1.1

      \[\leadsto \frac{\frac{1}{\left(1 - \cos x\right) \cdot \left(\frac{1}{1 - \cos x} \cdot \left(1 - \cos x\right)\right)} \cdot \left(\left(1 - \cos x\right) \cdot \color{blue}{\left(\left(1 - \cos x\right) \cdot \left(\frac{1}{1 - \cos x} \cdot \left(1 - \cos x\right)\right)\right)}\right)}{x \cdot x} \]
3. Recombined 3 regimes into one program.

4. Final simplification 0.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.029:\\ \;\;\;\;\frac{1 - \cos x}{x \cdot x}\\ \mathbf{elif}\;x \leq 0.033:\\ \;\;\;\;0.5 + \left(-0.041666666666666664 \cdot {x}^{2} + 0.001388888888888889 \cdot {x}^{4}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\left(1 - \cos x\right) \cdot \left(\frac{1}{1 - \cos x} \cdot \left(1 - \cos x\right)\right)} \cdot \left(\left(1 - \cos x\right) \cdot \left(\left(1 - \cos x\right) \cdot \left(\frac{1}{1 - \cos x} \cdot \left(1 - \cos x\right)\right)\right)\right)}{x \cdot x}\\ \end{array} \]

Alternatives

Alternative 1
Error	0.5
Cost	33800

\[\begin{array}{l} t_0 := 1 - \cos x\\ \mathbf{if}\;x \leq -0.029:\\ \;\;\;\;\frac{t_0}{x \cdot x}\\ \mathbf{elif}\;x \leq 0.033:\\ \;\;\;\;0.5 + \left(-0.041666666666666664 \cdot {x}^{2} + 0.001388888888888889 \cdot {x}^{4}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{t_0 \cdot \left(\frac{1}{t_0} \cdot t_0\right)} \cdot {t_0}^{2}}{x \cdot x}\\ \end{array} \]

Alternative 2
Error	0.5
Cost	20360

\[\begin{array}{l} t_0 := 1 - \cos x\\ \mathbf{if}\;x \leq -0.029:\\ \;\;\;\;\frac{t_0}{x \cdot x}\\ \mathbf{elif}\;x \leq 0.033:\\ \;\;\;\;0.5 + \left(-0.041666666666666664 \cdot {x}^{2} + 0.001388888888888889 \cdot {x}^{4}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{t_0} \cdot {t_0}^{2}}{x \cdot x}\\ \end{array} \]

Alternative 3
Error	0.5
Cost	13768

\[\begin{array}{l} t_0 := \frac{1 - \cos x}{x \cdot x}\\ \mathbf{if}\;x \leq -0.029:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 0.033:\\ \;\;\;\;0.5 + \left(-0.041666666666666664 \cdot {x}^{2} + 0.001388888888888889 \cdot {x}^{4}\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 4
Error	0.5
Cost	7112

\[\begin{array}{l} t_0 := \frac{1 - \cos x}{x \cdot x}\\ \mathbf{if}\;x \leq -0.0055:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 0.0053:\\ \;\;\;\;0.5 + -0.041666666666666664 \cdot {x}^{2}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 5
Error	30.8
Cost	64

\[0.5 \]

Reproduce

herbie shell --seed 2023092 
(FPCore (x)
  :name "cos2 (problem 3.4.1)"
  :precision binary64
  (/ (- 1.0 (cos x)) (* x x)))