2cos (problem 3.3.5)

Average Error: 39.1 → 13.9

Time: 25.9s

Precision: binary64

Cost: 33608

Math

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

↓

\[\begin{array}{l} t_0 := \cos \varepsilon - \cos x\\ \mathbf{if}\;\varepsilon \leq -0.075:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\varepsilon \leq 0.13:\\ \;\;\;\;\sin x \cdot \left(0.16666666666666666 \cdot {\varepsilon}^{3} + \left(-\varepsilon\right)\right) + \cos x \cdot \left(-0.5 \cdot {\varepsilon}^{2} + 0.041666666666666664 \cdot {\varepsilon}^{4}\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

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 -0.075)
     t_0
     (if (<= eps 0.13)
       (+
        (* (sin x) (+ (* 0.16666666666666666 (pow eps 3.0)) (- eps)))
        (*
         (cos x)
         (+ (* -0.5 (pow eps 2.0)) (* 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 <= -0.075) {
		tmp = t_0;
	} else if (eps <= 0.13) {
		tmp = (sin(x) * ((0.16666666666666666 * pow(eps, 3.0)) + -eps)) + (cos(x) * ((-0.5 * pow(eps, 2.0)) + (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 <= (-0.075d0)) then
        tmp = t_0
    else if (eps <= 0.13d0) then
        tmp = (sin(x) * ((0.16666666666666666d0 * (eps ** 3.0d0)) + -eps)) + (cos(x) * (((-0.5d0) * (eps ** 2.0d0)) + (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 <= -0.075) {
		tmp = t_0;
	} else if (eps <= 0.13) {
		tmp = (Math.sin(x) * ((0.16666666666666666 * Math.pow(eps, 3.0)) + -eps)) + (Math.cos(x) * ((-0.5 * Math.pow(eps, 2.0)) + (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 <= -0.075:
		tmp = t_0
	elif eps <= 0.13:
		tmp = (math.sin(x) * ((0.16666666666666666 * math.pow(eps, 3.0)) + -eps)) + (math.cos(x) * ((-0.5 * math.pow(eps, 2.0)) + (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 <= -0.075)
		tmp = t_0;
	elseif (eps <= 0.13)
		tmp = Float64(Float64(sin(x) * Float64(Float64(0.16666666666666666 * (eps ^ 3.0)) + Float64(-eps))) + Float64(cos(x) * Float64(Float64(-0.5 * (eps ^ 2.0)) + 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 <= -0.075)
		tmp = t_0;
	elseif (eps <= 0.13)
		tmp = (sin(x) * ((0.16666666666666666 * (eps ^ 3.0)) + -eps)) + (cos(x) * ((-0.5 * (eps ^ 2.0)) + (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, -0.075], t$95$0, If[LessEqual[eps, 0.13], N[(N[(N[Sin[x], $MachinePrecision] * N[(N[(0.16666666666666666 * N[Power[eps, 3.0], $MachinePrecision]), $MachinePrecision] + (-eps)), $MachinePrecision]), $MachinePrecision] + N[(N[Cos[x], $MachinePrecision] * N[(N[(-0.5 * N[Power[eps, 2.0], $MachinePrecision]), $MachinePrecision] + N[(0.041666666666666664 * N[Power[eps, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if eps < -0.0749999999999999972 or 0.13 < eps

   1. Initial program 29.3

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

   2. Taylor expanded in x around 0 28.0

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

   if -0.0749999999999999972 < eps < 0.13

   1. Initial program 48.6

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

   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

      \[\leadsto \color{blue}{\sin x \cdot \left(0.16666666666666666 \cdot {\varepsilon}^{3} + \left(-\varepsilon\right)\right) + \cos x \cdot \left(-0.5 \cdot {\varepsilon}^{2} + 0.041666666666666664 \cdot {\varepsilon}^{4}\right)} \]
      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.json-simplify-41 [=>] 0.3	\[ 0.041666666666666664 \cdot \left({\varepsilon}^{4} \cdot \cos x\right) + \color{blue}{\left(-0.5 \cdot \left({\varepsilon}^{2} \cdot \cos x\right) + \left(-1 \cdot \left(\varepsilon \cdot \sin x\right) + 0.16666666666666666 \cdot \left({\varepsilon}^{3} \cdot \sin x\right)\right)\right)} \]
      rational.json-simplify-41 [<=] 0.3	\[ \color{blue}{\left(-1 \cdot \left(\varepsilon \cdot \sin x\right) + 0.16666666666666666 \cdot \left({\varepsilon}^{3} \cdot \sin x\right)\right) + \left(0.041666666666666664 \cdot \left({\varepsilon}^{4} \cdot \cos x\right) + -0.5 \cdot \left({\varepsilon}^{2} \cdot \cos x\right)\right)} \]
      rational.json-simplify-1 [=>] 0.3	\[ \left(-1 \cdot \left(\varepsilon \cdot \sin x\right) + 0.16666666666666666 \cdot \left({\varepsilon}^{3} \cdot \sin x\right)\right) + \color{blue}{\left(-0.5 \cdot \left({\varepsilon}^{2} \cdot \cos x\right) + 0.041666666666666664 \cdot \left({\varepsilon}^{4} \cdot \cos x\right)\right)} \]

3. Recombined 2 regimes into one program.

4. Final simplification 13.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.075:\\ \;\;\;\;\cos \varepsilon - \cos x\\ \mathbf{elif}\;\varepsilon \leq 0.13:\\ \;\;\;\;\sin x \cdot \left(0.16666666666666666 \cdot {\varepsilon}^{3} + \left(-\varepsilon\right)\right) + \cos x \cdot \left(-0.5 \cdot {\varepsilon}^{2} + 0.041666666666666664 \cdot {\varepsilon}^{4}\right)\\ \mathbf{else}:\\ \;\;\;\;\cos \varepsilon - \cos x\\ \end{array} \]

Alternatives

Alternative 1
Error	14.0
Cost	26824

\[\begin{array}{l} t_0 := \cos \varepsilon - \cos x\\ \mathbf{if}\;\varepsilon \leq -0.016:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\varepsilon \leq 0.0105:\\ \;\;\;\;{\varepsilon}^{2} \cdot \left(-0.5 \cdot \cos x\right) + \left(0.16666666666666666 \cdot {\varepsilon}^{3} - \varepsilon\right) \cdot \sin x\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 2
Error	14.1
Cost	20168

\[\begin{array}{l} t_0 := \cos \varepsilon - \cos x\\ \mathbf{if}\;\varepsilon \leq -0.045:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\varepsilon \leq 0.014:\\ \;\;\;\;\varepsilon \cdot \left(-\sin x\right) + {\varepsilon}^{2} \cdot \left(-0.5 \cdot \cos x\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 3
Error	17.9
Cost	13960

\[\begin{array}{l} t_0 := \cos \varepsilon - \cos x\\ \mathbf{if}\;\varepsilon \leq -0.00172:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\varepsilon \leq 0.0033:\\ \;\;\;\;\varepsilon \cdot \left(-\left(-1 + \left(1 - \left(-\sin x\right)\right)\right)\right) + -0.5 \cdot {\varepsilon}^{2}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 4
Error	20.9
Cost	13520

\[\begin{array}{l} t_0 := \cos \varepsilon - \cos x\\ t_1 := \varepsilon \cdot \left(-\sin x\right)\\ \mathbf{if}\;\varepsilon \leq -1.35 \cdot 10^{-6}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\varepsilon \leq 2.8 \cdot 10^{-156}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\varepsilon \leq 10^{-19}:\\ \;\;\;\;x \cdot \left(-\varepsilon\right) + -0.5 \cdot {\varepsilon}^{2}\\ \mathbf{elif}\;\varepsilon \leq 1.9 \cdot 10^{-7}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 5
Error	21.3
Cost	7372

\[\begin{array}{l} t_0 := \cos \varepsilon - 1\\ t_1 := \varepsilon \cdot \left(-\sin x\right)\\ \mathbf{if}\;\varepsilon \leq -1.85 \cdot 10^{-5}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\varepsilon \leq 2.05 \cdot 10^{-156}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\varepsilon \leq 2.3 \cdot 10^{-21}:\\ \;\;\;\;x \cdot \left(-\varepsilon\right) + -0.5 \cdot {\varepsilon}^{2}\\ \mathbf{elif}\;\varepsilon \leq 2 \cdot 10^{-6}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 6
Error	20.6
Cost	7184

\[\begin{array}{l} t_0 := \cos \varepsilon - 1\\ t_1 := \varepsilon \cdot \left(-\sin x\right)\\ \mathbf{if}\;\varepsilon \leq -5.5 \cdot 10^{-6}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\varepsilon \leq 3.8 \cdot 10^{-79}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\varepsilon \leq 2.4 \cdot 10^{-49}:\\ \;\;\;\;-0.5 \cdot {\varepsilon}^{2}\\ \mathbf{elif}\;\varepsilon \leq 7.5 \cdot 10^{-7}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 7
Error	33.4
Cost	6920

\[\begin{array}{l} t_0 := \cos \varepsilon - 1\\ \mathbf{if}\;\varepsilon \leq -0.00015:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\varepsilon \leq 0.000155:\\ \;\;\;\;-0.5 \cdot {\varepsilon}^{2}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 8
Error	35.6
Cost	6856

\[\begin{array}{l} t_0 := \cos \varepsilon - 1\\ \mathbf{if}\;\varepsilon \leq -5 \cdot 10^{-58}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\varepsilon \leq 1.9 \cdot 10^{-21}:\\ \;\;\;\;x \cdot \left(-\varepsilon\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 9
Error	52.2
Cost	256

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

Reproduce

herbie shell --seed 2023064 
(FPCore (x eps)
  :name "2cos (problem 3.3.5)"
  :precision binary64
  (- (cos (+ x eps)) (cos x)))