cos2 (problem 3.4.1)

Average Error: 31.2 → 0.2

Time: 55.4s

Precision: binary64

Cost: 20488

Math

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

↓

\[\begin{array}{l} t_0 := \frac{1 - \cos x}{x}\\ \mathbf{if}\;x \leq -0.1:\\ \;\;\;\;t_0 \cdot \frac{1}{x}\\ \mathbf{elif}\;x \leq 0.102:\\ \;\;\;\;0.5 + \left(0.001388888888888889 \cdot {x}^{4} + \left(-2.48015873015873 \cdot 10^{-5} \cdot {x}^{6} + -0.041666666666666664 \cdot {x}^{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0}{x}\\ \end{array} \]

FPCore

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

↓

(FPCore (x)
 :precision binary64
 (let* ((t_0 (/ (- 1.0 (cos x)) x)))
   (if (<= x -0.1)
     (* t_0 (/ 1.0 x))
     (if (<= x 0.102)
       (+
        0.5
        (+
         (* 0.001388888888888889 (pow x 4.0))
         (+
          (* -2.48015873015873e-5 (pow x 6.0))
          (* -0.041666666666666664 (pow x 2.0)))))
       (/ t_0 x)))))

C

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

↓

double code(double x) {
	double t_0 = (1.0 - cos(x)) / x;
	double tmp;
	if (x <= -0.1) {
		tmp = t_0 * (1.0 / x);
	} else if (x <= 0.102) {
		tmp = 0.5 + ((0.001388888888888889 * pow(x, 4.0)) + ((-2.48015873015873e-5 * pow(x, 6.0)) + (-0.041666666666666664 * pow(x, 2.0))));
	} else {
		tmp = t_0 / x;
	}
	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
    if (x <= (-0.1d0)) then
        tmp = t_0 * (1.0d0 / x)
    else if (x <= 0.102d0) then
        tmp = 0.5d0 + ((0.001388888888888889d0 * (x ** 4.0d0)) + (((-2.48015873015873d-5) * (x ** 6.0d0)) + ((-0.041666666666666664d0) * (x ** 2.0d0))))
    else
        tmp = t_0 / x
    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;
	double tmp;
	if (x <= -0.1) {
		tmp = t_0 * (1.0 / x);
	} else if (x <= 0.102) {
		tmp = 0.5 + ((0.001388888888888889 * Math.pow(x, 4.0)) + ((-2.48015873015873e-5 * Math.pow(x, 6.0)) + (-0.041666666666666664 * Math.pow(x, 2.0))));
	} else {
		tmp = t_0 / x;
	}
	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
	tmp = 0
	if x <= -0.1:
		tmp = t_0 * (1.0 / x)
	elif x <= 0.102:
		tmp = 0.5 + ((0.001388888888888889 * math.pow(x, 4.0)) + ((-2.48015873015873e-5 * math.pow(x, 6.0)) + (-0.041666666666666664 * math.pow(x, 2.0))))
	else:
		tmp = t_0 / x
	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)) / x)
	tmp = 0.0
	if (x <= -0.1)
		tmp = Float64(t_0 * Float64(1.0 / x));
	elseif (x <= 0.102)
		tmp = Float64(0.5 + Float64(Float64(0.001388888888888889 * (x ^ 4.0)) + Float64(Float64(-2.48015873015873e-5 * (x ^ 6.0)) + Float64(-0.041666666666666664 * (x ^ 2.0)))));
	else
		tmp = Float64(t_0 / x);
	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;
	tmp = 0.0;
	if (x <= -0.1)
		tmp = t_0 * (1.0 / x);
	elseif (x <= 0.102)
		tmp = 0.5 + ((0.001388888888888889 * (x ^ 4.0)) + ((-2.48015873015873e-5 * (x ^ 6.0)) + (-0.041666666666666664 * (x ^ 2.0))));
	else
		tmp = t_0 / x;
	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] / x), $MachinePrecision]}, If[LessEqual[x, -0.1], N[(t$95$0 * N[(1.0 / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.102], N[(0.5 + N[(N[(0.001388888888888889 * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision] + N[(N[(-2.48015873015873e-5 * N[Power[x, 6.0], $MachinePrecision]), $MachinePrecision] + N[(-0.041666666666666664 * N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 / x), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.10000000000000001

   1. Initial program 1.1

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

   2. Applied egg-rr 0.5

      \[\leadsto \color{blue}{\frac{1 - \cos x}{x} \cdot \frac{1}{x}} \]

   if -0.10000000000000001 < x < 0.101999999999999993

   1. Initial program 62.3

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

   2. Taylor expanded in x around 0 0.0

      \[\leadsto \color{blue}{0.5 + \left(-0.041666666666666664 \cdot {x}^{2} + \left(-2.48015873015873 \cdot 10^{-5} \cdot {x}^{6} + 0.001388888888888889 \cdot {x}^{4}\right)\right)} \]

   3. Simplified 0.0

      \[\leadsto \color{blue}{0.5 + \left(0.001388888888888889 \cdot {x}^{4} + \left(-2.48015873015873 \cdot 10^{-5} \cdot {x}^{6} + -0.041666666666666664 \cdot {x}^{2}\right)\right)} \]
      Proof

      [Start] 0.0	\[ 0.5 + \left(-0.041666666666666664 \cdot {x}^{2} + \left(-2.48015873015873 \cdot 10^{-5} \cdot {x}^{6} + 0.001388888888888889 \cdot {x}^{4}\right)\right) \]
      rational_best-simplify-47 [=>] 0.0	\[ 0.5 + \color{blue}{\left(0.001388888888888889 \cdot {x}^{4} + \left(-2.48015873015873 \cdot 10^{-5} \cdot {x}^{6} + -0.041666666666666664 \cdot {x}^{2}\right)\right)} \]

   if 0.101999999999999993 < x

   1. Initial program 1.0

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

   2. Applied egg-rr 10.8

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

   3. Applied egg-rr 0.4

      \[\leadsto \color{blue}{\frac{\frac{1 - \cos x}{x}}{x}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 0.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.1:\\ \;\;\;\;\frac{1 - \cos x}{x} \cdot \frac{1}{x}\\ \mathbf{elif}\;x \leq 0.102:\\ \;\;\;\;0.5 + \left(0.001388888888888889 \cdot {x}^{4} + \left(-2.48015873015873 \cdot 10^{-5} \cdot {x}^{6} + -0.041666666666666664 \cdot {x}^{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 - \cos x}{x}}{x}\\ \end{array} \]

Alternatives

Alternative 1
Error	0.3
Cost	14408

\[\begin{array}{l} t_0 := \frac{1 - \cos x}{x}\\ \mathbf{if}\;x \leq -0.1:\\ \;\;\;\;t_0 \cdot \frac{1}{x}\\ \mathbf{elif}\;x \leq 0.102:\\ \;\;\;\;\frac{\frac{1}{x} + \left(x \cdot -0.08333333333333333 + \left(0.002777777777777778 \cdot {x}^{3} + -4.96031746031746 \cdot 10^{-5} \cdot {x}^{5}\right)\right)}{\frac{2}{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0}{x}\\ \end{array} \]

Alternative 2
Error	0.6
Cost	7112

\[\begin{array}{l} t_0 := \frac{1 - \cos x}{x \cdot x}\\ \mathbf{if}\;x \leq -0.004:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 0.0045:\\ \;\;\;\;\frac{x \cdot -0.041666666666666664}{\frac{1}{x}} - -0.5\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 3
Error	0.2
Cost	7112

\[\begin{array}{l} t_0 := \frac{\frac{1 - \cos x}{x}}{x}\\ \mathbf{if}\;x \leq -0.004:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 0.0045:\\ \;\;\;\;\frac{x \cdot -0.041666666666666664}{\frac{1}{x}} - -0.5\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 4
Error	0.3
Cost	7112

\[\begin{array}{l} t_0 := \frac{1 - \cos x}{x}\\ \mathbf{if}\;x \leq -0.004:\\ \;\;\;\;t_0 \cdot \frac{1}{x}\\ \mathbf{elif}\;x \leq 0.0045:\\ \;\;\;\;\frac{x \cdot -0.041666666666666664}{\frac{1}{x}} - -0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0}{x}\\ \end{array} \]

Alternative 5
Error	16.2
Cost	840

\[\begin{array}{l} t_0 := \frac{\frac{x - x}{x}}{x \cdot x}\\ \mathbf{if}\;x \leq -3.45:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 3.5:\\ \;\;\;\;\frac{x \cdot -0.041666666666666664}{\frac{1}{x}} - -0.5\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 6
Error	31.2
Cost	64

\[0.5 \]

Reproduce

herbie shell --seed 2023099 
(FPCore (x)
  :name "cos2 (problem 3.4.1)"
  :precision binary64
  (/ (- 1.0 (cos x)) (* x x)))