2cos (problem 3.3.5)

Average Error: 39.4 → 14.8

Time: 20.7s

Precision: binary64

Cost: 33608

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.035:\\ \;\;\;\;\cos \varepsilon - \cos x\\ \mathbf{elif}\;\varepsilon \leq 0.062:\\ \;\;\;\;\sin x \cdot \left(0.16666666666666666 \cdot {\varepsilon}^{3} + \left(-\varepsilon\right)\right) + \cos x \cdot \left(-0.5 \cdot {\varepsilon}^{2} + 0.041666666666666664 \cdot {\varepsilon}^{4}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\cos \varepsilon + \left(1 - \cos x\right)\right) + -1\\ \end{array} \]

FPCore

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

↓

(FPCore (x eps)
 :precision binary64
 (if (<= eps -0.035)
   (- (cos eps) (cos x))
   (if (<= eps 0.062)
     (+
      (* (sin x) (+ (* 0.16666666666666666 (pow eps 3.0)) (- eps)))
      (*
       (cos x)
       (+ (* -0.5 (pow eps 2.0)) (* 0.041666666666666664 (pow eps 4.0)))))
     (+ (+ (cos eps) (- 1.0 (cos x))) -1.0))))

C

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

↓

double code(double x, double eps) {
	double tmp;
	if (eps <= -0.035) {
		tmp = cos(eps) - cos(x);
	} else if (eps <= 0.062) {
		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 = (cos(eps) + (1.0 - cos(x))) + -1.0;
	}
	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) :: tmp
    if (eps <= (-0.035d0)) then
        tmp = cos(eps) - cos(x)
    else if (eps <= 0.062d0) then
        tmp = (sin(x) * ((0.16666666666666666d0 * (eps ** 3.0d0)) + -eps)) + (cos(x) * (((-0.5d0) * (eps ** 2.0d0)) + (0.041666666666666664d0 * (eps ** 4.0d0))))
    else
        tmp = (cos(eps) + (1.0d0 - cos(x))) + (-1.0d0)
    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 tmp;
	if (eps <= -0.035) {
		tmp = Math.cos(eps) - Math.cos(x);
	} else if (eps <= 0.062) {
		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 = (Math.cos(eps) + (1.0 - Math.cos(x))) + -1.0;
	}
	return tmp;
}

Python

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

↓

def code(x, eps):
	tmp = 0
	if eps <= -0.035:
		tmp = math.cos(eps) - math.cos(x)
	elif eps <= 0.062:
		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 = (math.cos(eps) + (1.0 - math.cos(x))) + -1.0
	return tmp

Julia

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

↓

function code(x, eps)
	tmp = 0.0
	if (eps <= -0.035)
		tmp = Float64(cos(eps) - cos(x));
	elseif (eps <= 0.062)
		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 = Float64(Float64(cos(eps) + Float64(1.0 - cos(x))) + -1.0);
	end
	return tmp
end

MATLAB

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

↓

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (eps <= -0.035)
		tmp = cos(eps) - cos(x);
	elseif (eps <= 0.062)
		tmp = (sin(x) * ((0.16666666666666666 * (eps ^ 3.0)) + -eps)) + (cos(x) * ((-0.5 * (eps ^ 2.0)) + (0.041666666666666664 * (eps ^ 4.0))));
	else
		tmp = (cos(eps) + (1.0 - cos(x))) + -1.0;
	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_] := If[LessEqual[eps, -0.035], N[(N[Cos[eps], $MachinePrecision] - N[Cos[x], $MachinePrecision]), $MachinePrecision], If[LessEqual[eps, 0.062], 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], N[(N[(N[Cos[eps], $MachinePrecision] + N[(1.0 - N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if eps < -0.035000000000000003

   1. Initial program 29.5

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

   2. Taylor expanded in x around 0 28.1

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

   if -0.035000000000000003 < eps < 0.062

   1. Initial program 48.9

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

   2. Taylor expanded in eps around 0 0.2

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

   3. Simplified 0.2

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

      [Start] 0.2	\[ 0.041666666666666664 \cdot \left({\varepsilon}^{4} \cdot \cos x\right) + \left(0.16666666666666666 \cdot \left({\varepsilon}^{3} \cdot \sin x\right) + \left(-0.5 \cdot \left({\varepsilon}^{2} \cdot \cos x\right) + -1 \cdot \left(\varepsilon \cdot \sin x\right)\right)\right) \]
      rational.json-simplify-41 [=>] 0.2	\[ 0.041666666666666664 \cdot \left({\varepsilon}^{4} \cdot \cos x\right) + \color{blue}{\left(-0.5 \cdot \left({\varepsilon}^{2} \cdot \cos x\right) + \left(-1 \cdot \left(\varepsilon \cdot \sin x\right) + 0.16666666666666666 \cdot \left({\varepsilon}^{3} \cdot \sin x\right)\right)\right)} \]
      rational.json-simplify-1 [=>] 0.2	\[ 0.041666666666666664 \cdot \left({\varepsilon}^{4} \cdot \cos x\right) + \color{blue}{\left(\left(-1 \cdot \left(\varepsilon \cdot \sin x\right) + 0.16666666666666666 \cdot \left({\varepsilon}^{3} \cdot \sin x\right)\right) + -0.5 \cdot \left({\varepsilon}^{2} \cdot \cos x\right)\right)} \]
      rational.json-simplify-41 [=>] 0.2	\[ \color{blue}{\left(-1 \cdot \left(\varepsilon \cdot \sin x\right) + 0.16666666666666666 \cdot \left({\varepsilon}^{3} \cdot \sin x\right)\right) + \left(-0.5 \cdot \left({\varepsilon}^{2} \cdot \cos x\right) + 0.041666666666666664 \cdot \left({\varepsilon}^{4} \cdot \cos x\right)\right)} \]
      rational.json-simplify-43 [<=] 0.2	\[ \left(\color{blue}{\sin x \cdot \left(-1 \cdot \varepsilon\right)} + 0.16666666666666666 \cdot \left({\varepsilon}^{3} \cdot \sin x\right)\right) + \left(-0.5 \cdot \left({\varepsilon}^{2} \cdot \cos x\right) + 0.041666666666666664 \cdot \left({\varepsilon}^{4} \cdot \cos x\right)\right) \]
      rational.json-simplify-2 [=>] 0.2	\[ \left(\sin x \cdot \color{blue}{\left(\varepsilon \cdot -1\right)} + 0.16666666666666666 \cdot \left({\varepsilon}^{3} \cdot \sin x\right)\right) + \left(-0.5 \cdot \left({\varepsilon}^{2} \cdot \cos x\right) + 0.041666666666666664 \cdot \left({\varepsilon}^{4} \cdot \cos x\right)\right) \]
      rational.json-simplify-8 [<=] 0.2	\[ \left(\sin x \cdot \color{blue}{\left(-\varepsilon\right)} + 0.16666666666666666 \cdot \left({\varepsilon}^{3} \cdot \sin x\right)\right) + \left(-0.5 \cdot \left({\varepsilon}^{2} \cdot \cos x\right) + 0.041666666666666664 \cdot \left({\varepsilon}^{4} \cdot \cos x\right)\right) \]
      rational.json-simplify-2 [=>] 0.2	\[ \left(\sin x \cdot \left(-\varepsilon\right) + 0.16666666666666666 \cdot \color{blue}{\left(\sin x \cdot {\varepsilon}^{3}\right)}\right) + \left(-0.5 \cdot \left({\varepsilon}^{2} \cdot \cos x\right) + 0.041666666666666664 \cdot \left({\varepsilon}^{4} \cdot \cos x\right)\right) \]
      rational.json-simplify-43 [=>] 0.2	\[ \left(\sin x \cdot \left(-\varepsilon\right) + \color{blue}{\sin x \cdot \left({\varepsilon}^{3} \cdot 0.16666666666666666\right)}\right) + \left(-0.5 \cdot \left({\varepsilon}^{2} \cdot \cos x\right) + 0.041666666666666664 \cdot \left({\varepsilon}^{4} \cdot \cos x\right)\right) \]
      rational.json-simplify-2 [=>] 0.2	\[ \left(\sin x \cdot \left(-\varepsilon\right) + \color{blue}{\left({\varepsilon}^{3} \cdot 0.16666666666666666\right) \cdot \sin x}\right) + \left(-0.5 \cdot \left({\varepsilon}^{2} \cdot \cos x\right) + 0.041666666666666664 \cdot \left({\varepsilon}^{4} \cdot \cos x\right)\right) \]
      rational.json-simplify-51 [=>] 0.2	\[ \color{blue}{\sin x \cdot \left({\varepsilon}^{3} \cdot 0.16666666666666666 + \left(-\varepsilon\right)\right)} + \left(-0.5 \cdot \left({\varepsilon}^{2} \cdot \cos x\right) + 0.041666666666666664 \cdot \left({\varepsilon}^{4} \cdot \cos x\right)\right) \]
      rational.json-simplify-2 [=>] 0.2	\[ \sin x \cdot \left(\color{blue}{0.16666666666666666 \cdot {\varepsilon}^{3}} + \left(-\varepsilon\right)\right) + \left(-0.5 \cdot \left({\varepsilon}^{2} \cdot \cos x\right) + 0.041666666666666664 \cdot \left({\varepsilon}^{4} \cdot \cos x\right)\right) \]

   if 0.062 < eps

   1. Initial program 31.0

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

   2. Applied egg-rr 31.0

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

   3. Taylor expanded in x around 0 29.7

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

4. Final simplification 14.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.035:\\ \;\;\;\;\cos \varepsilon - \cos x\\ \mathbf{elif}\;\varepsilon \leq 0.062:\\ \;\;\;\;\sin x \cdot \left(0.16666666666666666 \cdot {\varepsilon}^{3} + \left(-\varepsilon\right)\right) + \cos x \cdot \left(-0.5 \cdot {\varepsilon}^{2} + 0.041666666666666664 \cdot {\varepsilon}^{4}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\cos \varepsilon + \left(1 - \cos x\right)\right) + -1\\ \end{array} \]

Alternatives

Alternative 1
Error	15.0
Cost	20168

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.0075:\\ \;\;\;\;\cos \varepsilon - \cos x\\ \mathbf{elif}\;\varepsilon \leq 0.31:\\ \;\;\;\;\sin x \cdot \left(-\varepsilon\right) + \cos x \cdot \left({\varepsilon}^{2} \cdot -0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\cos \varepsilon + \left(1 - \cos x\right)\right) + -1\\ \end{array} \]

Alternative 2
Error	20.9
Cost	13644

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -6.5 \cdot 10^{-5}:\\ \;\;\;\;\cos \varepsilon - \cos x\\ \mathbf{elif}\;\varepsilon \leq -1.85 \cdot 10^{-46}:\\ \;\;\;\;\varepsilon \cdot \left(-x\right) + -0.5 \cdot {\varepsilon}^{2}\\ \mathbf{elif}\;\varepsilon \leq 3.4 \cdot 10^{-6}:\\ \;\;\;\;\sin x \cdot \left(-\varepsilon\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\cos \varepsilon + \left(1 - \cos x\right)\right) + -1\\ \end{array} \]

Alternative 3
Error	20.9
Cost	13388

\[\begin{array}{l} t_0 := \cos \varepsilon - \cos x\\ \mathbf{if}\;\varepsilon \leq -0.000145:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\varepsilon \leq -1.65 \cdot 10^{-46}:\\ \;\;\;\;\varepsilon \cdot \left(-x\right) + -0.5 \cdot {\varepsilon}^{2}\\ \mathbf{elif}\;\varepsilon \leq 4 \cdot 10^{-6}:\\ \;\;\;\;\sin x \cdot \left(-\varepsilon\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 4
Error	21.4
Cost	7240

\[\begin{array}{l} t_0 := \cos \varepsilon - 1\\ \mathbf{if}\;\varepsilon \leq -0.00015:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\varepsilon \leq -8.6 \cdot 10^{-47}:\\ \;\;\;\;\varepsilon \cdot \left(-x\right) + -0.5 \cdot {\varepsilon}^{2}\\ \mathbf{elif}\;\varepsilon \leq 1.4 \cdot 10^{-6}:\\ \;\;\;\;\sin x \cdot \left(-\varepsilon\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 5
Error	21.5
Cost	7052

\[\begin{array}{l} t_0 := \cos \varepsilon - 1\\ \mathbf{if}\;\varepsilon \leq -0.00015:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\varepsilon \leq -7 \cdot 10^{-47}:\\ \;\;\;\;-0.5 \cdot {\varepsilon}^{2}\\ \mathbf{elif}\;\varepsilon \leq 8 \cdot 10^{-7}:\\ \;\;\;\;\sin x \cdot \left(-\varepsilon\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 6
Error	33.8
Cost	6920

\[\begin{array}{l} t_0 := \cos \varepsilon - 1\\ \mathbf{if}\;\varepsilon \leq -0.00015:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\varepsilon \leq 0.00019:\\ \;\;\;\;-0.5 \cdot {\varepsilon}^{2}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 7
Error	54.6
Cost	6592

\[1 - \cos x \]

Alternative 8
Error	39.2
Cost	6592

\[\cos \varepsilon - 1 \]

Alternative 9
Error	55.7
Cost	64

\[0 \]

Reproduce

herbie shell --seed 2023074 
(FPCore (x eps)
  :name "2cos (problem 3.3.5)"
  :precision binary64
  (- (cos (+ x eps)) (cos x)))