cos2 (problem 3.4.1)

Average Error: 31.4 → 0.3

Time: 11.6s

Precision: binary64

Cost: 20488

Math

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

↓

\[\begin{array}{l} t_0 := 1 - \cos x\\ \mathbf{if}\;x \leq -2.6408359392783645:\\ \;\;\;\;\frac{\frac{t_0}{x}}{x}\\ \mathbf{elif}\;x \leq 0.0007295408284379838:\\ \;\;\;\;0.5 + \left(-0.041666666666666664 \cdot {x}^{2} + \left(-2.48015873015873 \cdot 10^{-5} \cdot {x}^{6} + 0.001388888888888889 \cdot {x}^{4}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot {x}^{-2}\\ \end{array} \]

FPCore

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

↓

(FPCore (x)
 :precision binary64
 (let* ((t_0 (- 1.0 (cos x))))
   (if (<= x -2.6408359392783645)
     (/ (/ t_0 x) x)
     (if (<= x 0.0007295408284379838)
       (+
        0.5
        (+
         (* -0.041666666666666664 (pow x 2.0))
         (+
          (* -2.48015873015873e-5 (pow x 6.0))
          (* 0.001388888888888889 (pow x 4.0)))))
       (* t_0 (pow x -2.0))))))

C

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

↓

double code(double x) {
	double t_0 = 1.0 - cos(x);
	double tmp;
	if (x <= -2.6408359392783645) {
		tmp = (t_0 / x) / x;
	} else if (x <= 0.0007295408284379838) {
		tmp = 0.5 + ((-0.041666666666666664 * pow(x, 2.0)) + ((-2.48015873015873e-5 * pow(x, 6.0)) + (0.001388888888888889 * pow(x, 4.0))));
	} else {
		tmp = t_0 * pow(x, -2.0);
	}
	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) :: tmp
    t_0 = 1.0d0 - cos(x)
    if (x <= (-2.6408359392783645d0)) then
        tmp = (t_0 / x) / x
    else if (x <= 0.0007295408284379838d0) then
        tmp = 0.5d0 + (((-0.041666666666666664d0) * (x ** 2.0d0)) + (((-2.48015873015873d-5) * (x ** 6.0d0)) + (0.001388888888888889d0 * (x ** 4.0d0))))
    else
        tmp = t_0 * (x ** (-2.0d0))
    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 tmp;
	if (x <= -2.6408359392783645) {
		tmp = (t_0 / x) / x;
	} else if (x <= 0.0007295408284379838) {
		tmp = 0.5 + ((-0.041666666666666664 * Math.pow(x, 2.0)) + ((-2.48015873015873e-5 * Math.pow(x, 6.0)) + (0.001388888888888889 * Math.pow(x, 4.0))));
	} else {
		tmp = t_0 * Math.pow(x, -2.0);
	}
	return tmp;
}

Python

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

↓

def code(x):
	t_0 = 1.0 - math.cos(x)
	tmp = 0
	if x <= -2.6408359392783645:
		tmp = (t_0 / x) / x
	elif x <= 0.0007295408284379838:
		tmp = 0.5 + ((-0.041666666666666664 * math.pow(x, 2.0)) + ((-2.48015873015873e-5 * math.pow(x, 6.0)) + (0.001388888888888889 * math.pow(x, 4.0))))
	else:
		tmp = t_0 * math.pow(x, -2.0)
	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))
	tmp = 0.0
	if (x <= -2.6408359392783645)
		tmp = Float64(Float64(t_0 / x) / x);
	elseif (x <= 0.0007295408284379838)
		tmp = Float64(0.5 + Float64(Float64(-0.041666666666666664 * (x ^ 2.0)) + Float64(Float64(-2.48015873015873e-5 * (x ^ 6.0)) + Float64(0.001388888888888889 * (x ^ 4.0)))));
	else
		tmp = Float64(t_0 * (x ^ -2.0));
	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);
	tmp = 0.0;
	if (x <= -2.6408359392783645)
		tmp = (t_0 / x) / x;
	elseif (x <= 0.0007295408284379838)
		tmp = 0.5 + ((-0.041666666666666664 * (x ^ 2.0)) + ((-2.48015873015873e-5 * (x ^ 6.0)) + (0.001388888888888889 * (x ^ 4.0))));
	else
		tmp = t_0 * (x ^ -2.0);
	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]}, If[LessEqual[x, -2.6408359392783645], N[(N[(t$95$0 / x), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[x, 0.0007295408284379838], N[(0.5 + N[(N[(-0.041666666666666664 * N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision] + N[(N[(-2.48015873015873e-5 * N[Power[x, 6.0], $MachinePrecision]), $MachinePrecision] + N[(0.001388888888888889 * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[Power[x, -2.0], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.6408359392783645

   1. Initial program 1.0

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

   2. Applied egg-rr 29.3

      \[\leadsto \color{blue}{\left(1 + \left(1 - \cos x\right) \cdot {x}^{-2}\right) - 1} \]

   3. Applied egg-rr 0.5

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

   4. Applied egg-rr 28.5

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

   5. Applied egg-rr 0.4

      \[\leadsto \color{blue}{\frac{\frac{1 - \cos x}{x}}{x}} \]

   if -2.6408359392783645 < x < 7.29540828437983779e-4

   1. Initial program 62.1

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

   2. Taylor expanded in x around 0 0.1

      \[\leadsto \color{blue}{0.5 + \left(-0.041666666666666664 \cdot {x}^{2} + \left(-2.48015873015873 \cdot 10^{-5} \cdot {x}^{6} + 0.001388888888888889 \cdot {x}^{4}\right)\right)} \]

   if 7.29540828437983779e-4 < x

   1. Initial program 1.2

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

   2. Applied egg-rr 1.2

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

   3. Applied egg-rr 0.6

      \[\leadsto \color{blue}{{x}^{-2} \cdot \left(1 - \cos x\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 0.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.6408359392783645:\\ \;\;\;\;\frac{\frac{1 - \cos x}{x}}{x}\\ \mathbf{elif}\;x \leq 0.0007295408284379838:\\ \;\;\;\;0.5 + \left(-0.041666666666666664 \cdot {x}^{2} + \left(-2.48015873015873 \cdot 10^{-5} \cdot {x}^{6} + 0.001388888888888889 \cdot {x}^{4}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 - \cos x\right) \cdot {x}^{-2}\\ \end{array} \]

Alternatives

Alternative 1
Error	0.3
Cost	13768

\[\begin{array}{l} t_0 := 1 - \cos x\\ \mathbf{if}\;x \leq -0.049718773505307026:\\ \;\;\;\;\frac{\frac{t_0}{x}}{x}\\ \mathbf{elif}\;x \leq 0.0007295408284379838:\\ \;\;\;\;0.5 + \left(-0.041666666666666664 \cdot {x}^{2} + 0.001388888888888889 \cdot {x}^{4}\right)\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot {x}^{-2}\\ \end{array} \]

Alternative 2
Error	0.3
Cost	13448

\[\begin{array}{l} t_0 := 1 - \cos x\\ \mathbf{if}\;x \leq -0.049718773505307026:\\ \;\;\;\;\frac{\frac{t_0}{x}}{x}\\ \mathbf{elif}\;x \leq 0.0007295408284379838:\\ \;\;\;\;\frac{\frac{1}{x}}{0.008333333333333333 \cdot {x}^{3} + \left(x \cdot 0.16666666666666666 + 2 \cdot \frac{1}{x}\right)}\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot {x}^{-2}\\ \end{array} \]

Alternative 3
Error	0.3
Cost	7816

\[\begin{array}{l} t_0 := \frac{1 - \cos x}{x}\\ \mathbf{if}\;x \leq -0.049718773505307026:\\ \;\;\;\;\frac{t_0}{x}\\ \mathbf{elif}\;x \leq 0.0007295408284379838:\\ \;\;\;\;\frac{\frac{1}{x}}{0.008333333333333333 \cdot {x}^{3} + \left(x \cdot 0.16666666666666666 + 2 \cdot \frac{1}{x}\right)}\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot \frac{1}{x}\\ \end{array} \]

Alternative 4
Error	0.3
Cost	7240

\[\begin{array}{l} t_0 := \frac{1 - \cos x}{x}\\ \mathbf{if}\;x \leq -0.049718773505307026:\\ \;\;\;\;\frac{t_0}{x}\\ \mathbf{elif}\;x \leq 0.0007295408284379838:\\ \;\;\;\;0.5 + -0.041666666666666664 \cdot \left(x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot \frac{1}{x}\\ \end{array} \]

Alternative 5
Error	0.3
Cost	7112

\[\begin{array}{l} t_0 := \frac{\frac{1 - \cos x}{x}}{x}\\ \mathbf{if}\;x \leq -0.049718773505307026:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 0.0007295408284379838:\\ \;\;\;\;0.5 + -0.041666666666666664 \cdot \left(x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 6
Error	13.9
Cost	712

\[\begin{array}{l} t_0 := \frac{6}{x \cdot x}\\ \mathbf{if}\;x \leq -11469.135466714282:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 0.0007295408284379838:\\ \;\;\;\;0.5 + -0.041666666666666664 \cdot \left(x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 7
Error	14.0
Cost	584

\[\begin{array}{l} t_0 := \frac{6}{x \cdot x}\\ \mathbf{if}\;x \leq -11469.135466714282:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 0.0007295408284379838:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 8
Error	13.8
Cost	576

\[\frac{-0.25}{-0.041666666666666664 \cdot \left(x \cdot x\right) + -0.5} \]

Alternative 9
Error	15.4
Cost	328

\[\begin{array}{l} \mathbf{if}\;x \leq -3.2835276102229484 \cdot 10^{+78}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 1.5121850658158162 \cdot 10^{+59}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 10
Error	31.0
Cost	64

\[0.5 \]

Reproduce

herbie shell --seed 2022294 
(FPCore (x)
  :name "cos2 (problem 3.4.1)"
  :precision binary64
  (/ (- 1.0 (cos x)) (* x x)))