2cos (problem 3.3.5)

Average Error: 39.8 → 14.6

Time: 16.4s

Precision: binary64

Cost: 46664

Math

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

↓

\[\begin{array}{l} t_0 := \cos \varepsilon - \cos x\\ \mathbf{if}\;\varepsilon \leq -0.23:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\varepsilon \leq 0.195:\\ \;\;\;\;-0.5 \cdot \left(\cos x \cdot {\varepsilon}^{2}\right) + \left(0.041666666666666664 \cdot \left(\cos x \cdot {\varepsilon}^{4}\right) + \left(\left(-\varepsilon \cdot \sin x\right) + 0.16666666666666666 \cdot \left(\sin x \cdot {\varepsilon}^{3}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

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 -0.23)
     t_0
     (if (<= eps 0.195)
       (+
        (* -0.5 (* (cos x) (pow eps 2.0)))
        (+
         (* 0.041666666666666664 (* (cos x) (pow eps 4.0)))
         (+
          (- (* eps (sin x)))
          (* 0.16666666666666666 (* (sin x) (pow eps 3.0))))))
       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 <= -0.23) {
		tmp = t_0;
	} else if (eps <= 0.195) {
		tmp = (-0.5 * (cos(x) * pow(eps, 2.0))) + ((0.041666666666666664 * (cos(x) * pow(eps, 4.0))) + (-(eps * sin(x)) + (0.16666666666666666 * (sin(x) * pow(eps, 3.0)))));
	} 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 <= (-0.23d0)) then
        tmp = t_0
    else if (eps <= 0.195d0) then
        tmp = ((-0.5d0) * (cos(x) * (eps ** 2.0d0))) + ((0.041666666666666664d0 * (cos(x) * (eps ** 4.0d0))) + (-(eps * sin(x)) + (0.16666666666666666d0 * (sin(x) * (eps ** 3.0d0)))))
    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 <= -0.23) {
		tmp = t_0;
	} else if (eps <= 0.195) {
		tmp = (-0.5 * (Math.cos(x) * Math.pow(eps, 2.0))) + ((0.041666666666666664 * (Math.cos(x) * Math.pow(eps, 4.0))) + (-(eps * Math.sin(x)) + (0.16666666666666666 * (Math.sin(x) * Math.pow(eps, 3.0)))));
	} 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 <= -0.23:
		tmp = t_0
	elif eps <= 0.195:
		tmp = (-0.5 * (math.cos(x) * math.pow(eps, 2.0))) + ((0.041666666666666664 * (math.cos(x) * math.pow(eps, 4.0))) + (-(eps * math.sin(x)) + (0.16666666666666666 * (math.sin(x) * math.pow(eps, 3.0)))))
	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 <= -0.23)
		tmp = t_0;
	elseif (eps <= 0.195)
		tmp = Float64(Float64(-0.5 * Float64(cos(x) * (eps ^ 2.0))) + Float64(Float64(0.041666666666666664 * Float64(cos(x) * (eps ^ 4.0))) + Float64(Float64(-Float64(eps * sin(x))) + Float64(0.16666666666666666 * Float64(sin(x) * (eps ^ 3.0))))));
	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 <= -0.23)
		tmp = t_0;
	elseif (eps <= 0.195)
		tmp = (-0.5 * (cos(x) * (eps ^ 2.0))) + ((0.041666666666666664 * (cos(x) * (eps ^ 4.0))) + (-(eps * sin(x)) + (0.16666666666666666 * (sin(x) * (eps ^ 3.0)))));
	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, -0.23], t$95$0, If[LessEqual[eps, 0.195], N[(N[(-0.5 * N[(N[Cos[x], $MachinePrecision] * N[Power[eps, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(0.041666666666666664 * N[(N[Cos[x], $MachinePrecision] * N[Power[eps, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[((-N[(eps * N[Sin[x], $MachinePrecision]), $MachinePrecision]) + N[(0.16666666666666666 * N[(N[Sin[x], $MachinePrecision] * N[Power[eps, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if eps < -0.23000000000000001 or 0.19500000000000001 < eps

   1. Initial program 30.5

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

   2. Taylor expanded in x around 0 29.1

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

   if -0.23000000000000001 < eps < 0.19500000000000001

   1. Initial program 48.9

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

   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

      \[\leadsto \color{blue}{-0.5 \cdot \left(\cos x \cdot {\varepsilon}^{2}\right) + \left(0.041666666666666664 \cdot \left(\cos x \cdot {\varepsilon}^{4}\right) + \left(\left(-\varepsilon \cdot \sin x\right) + 0.16666666666666666 \cdot \left(\sin x \cdot {\varepsilon}^{3}\right)\right)\right)} \]
      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_45_simplify-80 [=>] 0.3	\[ 0.041666666666666664 \cdot \left({\varepsilon}^{4} \cdot \cos x\right) + \color{blue}{\left(-0.5 \cdot \left({\varepsilon}^{2} \cdot \cos x\right) + \left(0.16666666666666666 \cdot \left({\varepsilon}^{3} \cdot \sin x\right) + -1 \cdot \left(\varepsilon \cdot \sin x\right)\right)\right)} \]
      rational_best_45_simplify-80 [=>] 0.3	\[ \color{blue}{-0.5 \cdot \left({\varepsilon}^{2} \cdot \cos x\right) + \left(0.041666666666666664 \cdot \left({\varepsilon}^{4} \cdot \cos x\right) + \left(0.16666666666666666 \cdot \left({\varepsilon}^{3} \cdot \sin x\right) + -1 \cdot \left(\varepsilon \cdot \sin x\right)\right)\right)} \]
      rational_best_45_simplify-91 [=>] 0.3	\[ -0.5 \cdot \color{blue}{\left(\cos x \cdot {\varepsilon}^{2}\right)} + \left(0.041666666666666664 \cdot \left({\varepsilon}^{4} \cdot \cos x\right) + \left(0.16666666666666666 \cdot \left({\varepsilon}^{3} \cdot \sin x\right) + -1 \cdot \left(\varepsilon \cdot \sin x\right)\right)\right) \]
      rational_best_45_simplify-91 [=>] 0.3	\[ -0.5 \cdot \left(\cos x \cdot {\varepsilon}^{2}\right) + \left(0.041666666666666664 \cdot \color{blue}{\left(\cos x \cdot {\varepsilon}^{4}\right)} + \left(0.16666666666666666 \cdot \left({\varepsilon}^{3} \cdot \sin x\right) + -1 \cdot \left(\varepsilon \cdot \sin x\right)\right)\right) \]
      rational_best_45_simplify-73 [=>] 0.3	\[ -0.5 \cdot \left(\cos x \cdot {\varepsilon}^{2}\right) + \left(0.041666666666666664 \cdot \left(\cos x \cdot {\varepsilon}^{4}\right) + \color{blue}{\left(-1 \cdot \left(\varepsilon \cdot \sin x\right) + 0.16666666666666666 \cdot \left({\varepsilon}^{3} \cdot \sin x\right)\right)}\right) \]
      rational_best_45_simplify-91 [=>] 0.3	\[ -0.5 \cdot \left(\cos x \cdot {\varepsilon}^{2}\right) + \left(0.041666666666666664 \cdot \left(\cos x \cdot {\varepsilon}^{4}\right) + \left(\color{blue}{\left(\varepsilon \cdot \sin x\right) \cdot -1} + 0.16666666666666666 \cdot \left({\varepsilon}^{3} \cdot \sin x\right)\right)\right) \]
      rational_best_45_simplify-16 [=>] 0.3	\[ -0.5 \cdot \left(\cos x \cdot {\varepsilon}^{2}\right) + \left(0.041666666666666664 \cdot \left(\cos x \cdot {\varepsilon}^{4}\right) + \left(\color{blue}{\left(-\varepsilon \cdot \sin x\right)} + 0.16666666666666666 \cdot \left({\varepsilon}^{3} \cdot \sin x\right)\right)\right) \]
      rational_best_45_simplify-91 [=>] 0.3	\[ -0.5 \cdot \left(\cos x \cdot {\varepsilon}^{2}\right) + \left(0.041666666666666664 \cdot \left(\cos x \cdot {\varepsilon}^{4}\right) + \left(\left(-\varepsilon \cdot \sin x\right) + 0.16666666666666666 \cdot \color{blue}{\left(\sin x \cdot {\varepsilon}^{3}\right)}\right)\right) \]

3. Recombined 2 regimes into one program.

4. Final simplification 14.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.23:\\ \;\;\;\;\cos \varepsilon - \cos x\\ \mathbf{elif}\;\varepsilon \leq 0.195:\\ \;\;\;\;-0.5 \cdot \left(\cos x \cdot {\varepsilon}^{2}\right) + \left(0.041666666666666664 \cdot \left(\cos x \cdot {\varepsilon}^{4}\right) + \left(\left(-\varepsilon \cdot \sin x\right) + 0.16666666666666666 \cdot \left(\sin x \cdot {\varepsilon}^{3}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\cos \varepsilon - \cos x\\ \end{array} \]

Alternatives

Alternative 1
Error	14.7
Cost	20168

\[\begin{array}{l} t_0 := \cos \varepsilon - \cos x\\ \mathbf{if}\;\varepsilon \leq -0.0132:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\varepsilon \leq 0.00078:\\ \;\;\;\;\left(-\varepsilon \cdot \sin x\right) + -0.5 \cdot \left(\cos x \cdot {\varepsilon}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 2
Error	14.9
Cost	13640

\[\begin{array}{l} t_0 := \cos \varepsilon - \cos x\\ \mathbf{if}\;\varepsilon \leq -0.0125:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\varepsilon \leq 0.00078:\\ \;\;\;\;\left(-\varepsilon \cdot \sin x\right) + {\varepsilon}^{2} \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 3
Error	20.8
Cost	13256

\[\begin{array}{l} t_0 := \cos \varepsilon - \cos x\\ \mathbf{if}\;\varepsilon \leq -3.8 \cdot 10^{-13}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\varepsilon \leq 2.95 \cdot 10^{-6}:\\ \;\;\;\;-\varepsilon \cdot \sin x\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 4
Error	21.3
Cost	6984

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -3.8 \cdot 10^{-13}:\\ \;\;\;\;\cos \varepsilon - 1\\ \mathbf{elif}\;\varepsilon \leq 2.25 \cdot 10^{-5}:\\ \;\;\;\;-\varepsilon \cdot \sin x\\ \mathbf{else}:\\ \;\;\;\;\cos \left(x + \varepsilon\right) - 1\\ \end{array} \]

Alternative 5
Error	21.3
Cost	6920

\[\begin{array}{l} t_0 := \cos \varepsilon - 1\\ \mathbf{if}\;\varepsilon \leq -3.8 \cdot 10^{-13}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\varepsilon \leq 6 \cdot 10^{-6}:\\ \;\;\;\;-\varepsilon \cdot \sin x\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 6
Error	36.4
Cost	6856

\[\begin{array}{l} t_0 := \cos \varepsilon - 1\\ \mathbf{if}\;\varepsilon \leq -1.35 \cdot 10^{-41}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\varepsilon \leq 2.7 \cdot 10^{-36}:\\ \;\;\;\;\varepsilon \cdot \left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 7
Error	52.5
Cost	256

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

Alternative 8
Error	55.6
Cost	64

\[0 \]

Reproduce

herbie shell --seed 2023098 
(FPCore (x eps)
  :name "2cos (problem 3.3.5)"
  :precision binary64
  (- (cos (+ x eps)) (cos x)))