2cos (problem 3.3.5)

Average Error: 39.1 → 14.2

Time: 19.9s

Precision: binary64

Cost: 40136

Math

\[\cos \left(x + \varepsilon\right) - \cos x \]

↓

\[\begin{array}{l} t_0 := \cos \varepsilon - \cos x\\ \mathbf{if}\;\varepsilon \leq -0.06:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\varepsilon \leq 0.092:\\ \;\;\;\;\left(\cos x \cdot \left(-0.5 \cdot {\varepsilon}^{2}\right) + \sin x \cdot \left(0.16666666666666666 \cdot {\varepsilon}^{3} + \left(-\varepsilon\right)\right)\right) + \cos x \cdot \left(0.041666666666666664 \cdot {\varepsilon}^{4}\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

FPCore

(FPCore (x eps) :precision binary64 (- (cos (+ x eps)) (cos x)))

↓

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- (cos eps) (cos x))))
   (if (<= eps -0.06)
     t_0
     (if (<= eps 0.092)
       (+
        (+
         (* (cos x) (* -0.5 (pow eps 2.0)))
         (* (sin x) (+ (* 0.16666666666666666 (pow eps 3.0)) (- eps))))
        (* (cos x) (* 0.041666666666666664 (pow eps 4.0))))
       t_0))))

C

double code(double x, double eps) {
	return cos((x + eps)) - cos(x);
}

↓

double code(double x, double eps) {
	double t_0 = cos(eps) - cos(x);
	double tmp;
	if (eps <= -0.06) {
		tmp = t_0;
	} else if (eps <= 0.092) {
		tmp = ((cos(x) * (-0.5 * pow(eps, 2.0))) + (sin(x) * ((0.16666666666666666 * pow(eps, 3.0)) + -eps))) + (cos(x) * (0.041666666666666664 * pow(eps, 4.0)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = cos((x + eps)) - cos(x)
end function

↓

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: t_0
    real(8) :: tmp
    t_0 = cos(eps) - cos(x)
    if (eps <= (-0.06d0)) then
        tmp = t_0
    else if (eps <= 0.092d0) then
        tmp = ((cos(x) * ((-0.5d0) * (eps ** 2.0d0))) + (sin(x) * ((0.16666666666666666d0 * (eps ** 3.0d0)) + -eps))) + (cos(x) * (0.041666666666666664d0 * (eps ** 4.0d0)))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	return Math.cos((x + eps)) - Math.cos(x);
}

↓

public static double code(double x, double eps) {
	double t_0 = Math.cos(eps) - Math.cos(x);
	double tmp;
	if (eps <= -0.06) {
		tmp = t_0;
	} else if (eps <= 0.092) {
		tmp = ((Math.cos(x) * (-0.5 * Math.pow(eps, 2.0))) + (Math.sin(x) * ((0.16666666666666666 * Math.pow(eps, 3.0)) + -eps))) + (Math.cos(x) * (0.041666666666666664 * Math.pow(eps, 4.0)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, eps):
	return math.cos((x + eps)) - math.cos(x)

↓

def code(x, eps):
	t_0 = math.cos(eps) - math.cos(x)
	tmp = 0
	if eps <= -0.06:
		tmp = t_0
	elif eps <= 0.092:
		tmp = ((math.cos(x) * (-0.5 * math.pow(eps, 2.0))) + (math.sin(x) * ((0.16666666666666666 * math.pow(eps, 3.0)) + -eps))) + (math.cos(x) * (0.041666666666666664 * math.pow(eps, 4.0)))
	else:
		tmp = t_0
	return tmp

Julia

function code(x, eps)
	return Float64(cos(Float64(x + eps)) - cos(x))
end

↓

function code(x, eps)
	t_0 = Float64(cos(eps) - cos(x))
	tmp = 0.0
	if (eps <= -0.06)
		tmp = t_0;
	elseif (eps <= 0.092)
		tmp = Float64(Float64(Float64(cos(x) * Float64(-0.5 * (eps ^ 2.0))) + Float64(sin(x) * Float64(Float64(0.16666666666666666 * (eps ^ 3.0)) + Float64(-eps)))) + Float64(cos(x) * Float64(0.041666666666666664 * (eps ^ 4.0))));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp = code(x, eps)
	tmp = cos((x + eps)) - cos(x);
end

↓

function tmp_2 = code(x, eps)
	t_0 = cos(eps) - cos(x);
	tmp = 0.0;
	if (eps <= -0.06)
		tmp = t_0;
	elseif (eps <= 0.092)
		tmp = ((cos(x) * (-0.5 * (eps ^ 2.0))) + (sin(x) * ((0.16666666666666666 * (eps ^ 3.0)) + -eps))) + (cos(x) * (0.041666666666666664 * (eps ^ 4.0)));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := N[(N[Cos[N[(x + eps), $MachinePrecision]], $MachinePrecision] - N[Cos[x], $MachinePrecision]), $MachinePrecision]

↓

code[x_, eps_] := Block[{t$95$0 = N[(N[Cos[eps], $MachinePrecision] - N[Cos[x], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[eps, -0.06], t$95$0, If[LessEqual[eps, 0.092], N[(N[(N[(N[Cos[x], $MachinePrecision] * N[(-0.5 * N[Power[eps, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Sin[x], $MachinePrecision] * N[(N[(0.16666666666666666 * N[Power[eps, 3.0], $MachinePrecision]), $MachinePrecision] + (-eps)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Cos[x], $MachinePrecision] * N[(0.041666666666666664 * N[Power[eps, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if eps < -0.059999999999999998 or 0.091999999999999998 < eps

   1. Initial program 29.5

      \[\cos \left(x + \varepsilon\right) - \cos x \]

   2. Taylor expanded in x around 0 28.2

      \[\leadsto \color{blue}{\cos \varepsilon} - \cos x \]

   if -0.059999999999999998 < eps < 0.091999999999999998

   1. Initial program 48.8

      \[\cos \left(x + \varepsilon\right) - \cos x \]

   2. Taylor expanded in eps around 0 0.3

      \[\leadsto \color{blue}{0.041666666666666664 \cdot \left({\varepsilon}^{4} \cdot \cos x\right) + \left(0.16666666666666666 \cdot \left({\varepsilon}^{3} \cdot \sin x\right) + \left(-0.5 \cdot \left({\varepsilon}^{2} \cdot \cos x\right) + -1 \cdot \left(\varepsilon \cdot \sin x\right)\right)\right)} \]

   3. Simplified 0.3

      \[\leadsto \color{blue}{\left(\cos x \cdot \left(-0.5 \cdot {\varepsilon}^{2}\right) + \sin x \cdot \left(0.16666666666666666 \cdot {\varepsilon}^{3} + \left(-\varepsilon\right)\right)\right) + \cos x \cdot \left(0.041666666666666664 \cdot {\varepsilon}^{4}\right)} \]
      Proof

      [Start] 0.3	\[ 0.041666666666666664 \cdot \left({\varepsilon}^{4} \cdot \cos x\right) + \left(0.16666666666666666 \cdot \left({\varepsilon}^{3} \cdot \sin x\right) + \left(-0.5 \cdot \left({\varepsilon}^{2} \cdot \cos x\right) + -1 \cdot \left(\varepsilon \cdot \sin x\right)\right)\right) \]
      rational_best-simplify-1 [=>] 0.3	\[ \color{blue}{\left(0.16666666666666666 \cdot \left({\varepsilon}^{3} \cdot \sin x\right) + \left(-0.5 \cdot \left({\varepsilon}^{2} \cdot \cos x\right) + -1 \cdot \left(\varepsilon \cdot \sin x\right)\right)\right) + 0.041666666666666664 \cdot \left({\varepsilon}^{4} \cdot \cos x\right)} \]

3. Recombined 2 regimes into one program.

4. Final simplification 14.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.06:\\ \;\;\;\;\cos \varepsilon - \cos x\\ \mathbf{elif}\;\varepsilon \leq 0.092:\\ \;\;\;\;\left(\cos x \cdot \left(-0.5 \cdot {\varepsilon}^{2}\right) + \sin x \cdot \left(0.16666666666666666 \cdot {\varepsilon}^{3} + \left(-\varepsilon\right)\right)\right) + \cos x \cdot \left(0.041666666666666664 \cdot {\varepsilon}^{4}\right)\\ \mathbf{else}:\\ \;\;\;\;\cos \varepsilon - \cos x\\ \end{array} \]

Alternatives

Alternative 1
Error	14.4
Cost	26824

\[\begin{array}{l} t_0 := \cos \varepsilon - \cos x\\ \mathbf{if}\;\varepsilon \leq -0.012:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\varepsilon \leq 0.024:\\ \;\;\;\;{\varepsilon}^{2} \cdot \left(-0.5 \cdot \cos x\right) + \left(0.16666666666666666 \cdot {\varepsilon}^{3} - \varepsilon\right) \cdot \sin x\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 2
Error	14.4
Cost	20168

\[\begin{array}{l} t_0 := \cos \varepsilon - \cos x\\ \mathbf{if}\;\varepsilon \leq -0.0078:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\varepsilon \leq 0.0102:\\ \;\;\;\;\sin x \cdot \left(-\varepsilon\right) + \cos x \cdot \left(-0.5 \cdot {\varepsilon}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 3
Error	19.9
Cost	13772

\[\begin{array}{l} t_0 := \cos \varepsilon - \cos x\\ \mathbf{if}\;\varepsilon \leq -0.0042:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\varepsilon \leq 1.7 \cdot 10^{-45}:\\ \;\;\;\;\sin x \cdot \left(-\varepsilon\right)\\ \mathbf{elif}\;\varepsilon \leq 0.011:\\ \;\;\;\;0.041666666666666664 \cdot {\varepsilon}^{4} + -0.5 \cdot {\varepsilon}^{2}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 4
Error	19.9
Cost	13388

\[\begin{array}{l} t_0 := \cos \varepsilon - \cos x\\ \mathbf{if}\;\varepsilon \leq -0.0042:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\varepsilon \leq 8 \cdot 10^{-49}:\\ \;\;\;\;\sin x \cdot \left(-\varepsilon\right)\\ \mathbf{elif}\;\varepsilon \leq 0.000165:\\ \;\;\;\;-0.5 \cdot {\varepsilon}^{2}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 5
Error	32.1
Cost	7184

\[\begin{array}{l} t_0 := \cos \varepsilon - 1\\ t_1 := -0.5 \cdot {\varepsilon}^{2}\\ \mathbf{if}\;\varepsilon \leq -0.00018:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\varepsilon \leq -2.7 \cdot 10^{-142}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\varepsilon \leq 2.15 \cdot 10^{-157}:\\ \;\;\;\;\varepsilon \cdot \left(-x\right)\\ \mathbf{elif}\;\varepsilon \leq 0.000118:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 6
Error	20.4
Cost	7052

\[\begin{array}{l} t_0 := \cos \varepsilon - 1\\ \mathbf{if}\;\varepsilon \leq -0.0042:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\varepsilon \leq 1.45 \cdot 10^{-49}:\\ \;\;\;\;\sin x \cdot \left(-\varepsilon\right)\\ \mathbf{elif}\;\varepsilon \leq 0.000118:\\ \;\;\;\;-0.5 \cdot {\varepsilon}^{2}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 7
Error	35.3
Cost	6856

\[\begin{array}{l} t_0 := \cos \varepsilon - 1\\ \mathbf{if}\;\varepsilon \leq -2.05 \cdot 10^{-34}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\varepsilon \leq 9.5 \cdot 10^{-22}:\\ \;\;\;\;\varepsilon \cdot \left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 8
Error	52.0
Cost	256

\[\varepsilon \cdot \left(-x\right) \]

Reproduce

herbie shell --seed 2023092 
(FPCore (x eps)
  :name "2cos (problem 3.3.5)"
  :precision binary64
  (- (cos (+ x eps)) (cos x)))