cos2 (problem 3.4.1)

Average Error: 31.2 → 0.5

Time: 21.1s

Precision: binary64

Cost: 13768

Math

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

↓

\[\begin{array}{l} t_0 := \frac{1 - \cos x}{x \cdot x}\\ \mathbf{if}\;x \leq -0.03:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 0.035:\\ \;\;\;\;-0.041666666666666664 \cdot {x}^{2} + \left(0.5 + 0.001388888888888889 \cdot {x}^{4}\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

FPCore

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

↓

(FPCore (x)
 :precision binary64
 (let* ((t_0 (/ (- 1.0 (cos x)) (* x x))))
   (if (<= x -0.03)
     t_0
     (if (<= x 0.035)
       (+
        (* -0.041666666666666664 (pow x 2.0))
        (+ 0.5 (* 0.001388888888888889 (pow x 4.0))))
       t_0))))

C

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

↓

double code(double x) {
	double t_0 = (1.0 - cos(x)) / (x * x);
	double tmp;
	if (x <= -0.03) {
		tmp = t_0;
	} else if (x <= 0.035) {
		tmp = (-0.041666666666666664 * pow(x, 2.0)) + (0.5 + (0.001388888888888889 * pow(x, 4.0)));
	} else {
		tmp = t_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)) / (x * x)
    if (x <= (-0.03d0)) then
        tmp = t_0
    else if (x <= 0.035d0) then
        tmp = ((-0.041666666666666664d0) * (x ** 2.0d0)) + (0.5d0 + (0.001388888888888889d0 * (x ** 4.0d0)))
    else
        tmp = t_0
    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)) / (x * x);
	double tmp;
	if (x <= -0.03) {
		tmp = t_0;
	} else if (x <= 0.035) {
		tmp = (-0.041666666666666664 * Math.pow(x, 2.0)) + (0.5 + (0.001388888888888889 * Math.pow(x, 4.0)));
	} else {
		tmp = t_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)) / (x * x)
	tmp = 0
	if x <= -0.03:
		tmp = t_0
	elif x <= 0.035:
		tmp = (-0.041666666666666664 * math.pow(x, 2.0)) + (0.5 + (0.001388888888888889 * math.pow(x, 4.0)))
	else:
		tmp = t_0
	return tmp

Julia

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

↓

function code(x)
	t_0 = Float64(Float64(1.0 - cos(x)) / Float64(x * x))
	tmp = 0.0
	if (x <= -0.03)
		tmp = t_0;
	elseif (x <= 0.035)
		tmp = Float64(Float64(-0.041666666666666664 * (x ^ 2.0)) + Float64(0.5 + Float64(0.001388888888888889 * (x ^ 4.0))));
	else
		tmp = t_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)) / (x * x);
	tmp = 0.0;
	if (x <= -0.03)
		tmp = t_0;
	elseif (x <= 0.035)
		tmp = (-0.041666666666666664 * (x ^ 2.0)) + (0.5 + (0.001388888888888889 * (x ^ 4.0)));
	else
		tmp = t_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[(N[(1.0 - N[Cos[x], $MachinePrecision]), $MachinePrecision] / N[(x * x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -0.03], t$95$0, If[LessEqual[x, 0.035], N[(N[(-0.041666666666666664 * N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision] + N[(0.5 + N[(0.001388888888888889 * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -0.029999999999999999 or 0.035000000000000003 < x

   1. Initial program 0.9

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

   if -0.029999999999999999 < x < 0.035000000000000003

   1. Initial program 62.3

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

   2. Taylor expanded in x around 0 0.0

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

   3. Simplified 0.0

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

      [Start] 0.0	\[ 0.5 + \left(-0.041666666666666664 \cdot {x}^{2} + 0.001388888888888889 \cdot {x}^{4}\right) \]
      rational.json-simplify-41 [=>] 0.0	\[ \color{blue}{-0.041666666666666664 \cdot {x}^{2} + \left(0.001388888888888889 \cdot {x}^{4} + 0.5\right)} \]
      rational.json-simplify-1 [=>] 0.0	\[ -0.041666666666666664 \cdot {x}^{2} + \color{blue}{\left(0.5 + 0.001388888888888889 \cdot {x}^{4}\right)} \]

3. Recombined 2 regimes into one program.

4. Final simplification 0.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.03:\\ \;\;\;\;\frac{1 - \cos x}{x \cdot x}\\ \mathbf{elif}\;x \leq 0.035:\\ \;\;\;\;-0.041666666666666664 \cdot {x}^{2} + \left(0.5 + 0.001388888888888889 \cdot {x}^{4}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \cos x}{x \cdot x}\\ \end{array} \]

Alternatives

Alternative 1
Error	0.5
Cost	7112

\[\begin{array}{l} t_0 := \frac{1 - \cos x}{x \cdot x}\\ \mathbf{if}\;x \leq -0.0048:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 0.0046:\\ \;\;\;\;0.5 + -0.041666666666666664 \cdot {x}^{2}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 2
Error	16.1
Cost	328

\[\begin{array}{l} \mathbf{if}\;x \leq -1.02 \cdot 10^{+77}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 1.06 \cdot 10^{+77}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 3
Error	46.8
Cost	64

\[0 \]

Reproduce

herbie shell --seed 2023077 
(FPCore (x)
  :name "cos2 (problem 3.4.1)"
  :precision binary64
  (/ (- 1.0 (cos x)) (* x x)))