cos2 (problem 3.4.1)

Average Error: 31.2 → 0.6

Time: 11.2s

Precision: binary64

Cost: 20488

Math

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

↓

\[\begin{array}{l} t_0 := \frac{1 - \cos x}{x \cdot x}\\ \mathbf{if}\;x \leq -0.1:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 0.096:\\ \;\;\;\;\left(0.5 + -0.041666666666666664 \cdot {x}^{2}\right) + \left(0.001388888888888889 \cdot {x}^{4} + -2.48015873015873 \cdot 10^{-5} \cdot {x}^{6}\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

FPCore

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

↓

(FPCore (x)
 :precision binary64
 (let* ((t_0 (/ (- 1.0 (cos x)) (* x x))))
   (if (<= x -0.1)
     t_0
     (if (<= x 0.096)
       (+
        (+ 0.5 (* -0.041666666666666664 (pow x 2.0)))
        (+
         (* 0.001388888888888889 (pow x 4.0))
         (* -2.48015873015873e-5 (pow x 6.0))))
       t_0))))

C

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

↓

double code(double x) {
	double t_0 = (1.0 - cos(x)) / (x * x);
	double tmp;
	if (x <= -0.1) {
		tmp = t_0;
	} else if (x <= 0.096) {
		tmp = (0.5 + (-0.041666666666666664 * pow(x, 2.0))) + ((0.001388888888888889 * pow(x, 4.0)) + (-2.48015873015873e-5 * pow(x, 6.0)));
	} else {
		tmp = t_0;
	}
	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 * x)
    if (x <= (-0.1d0)) then
        tmp = t_0
    else if (x <= 0.096d0) then
        tmp = (0.5d0 + ((-0.041666666666666664d0) * (x ** 2.0d0))) + ((0.001388888888888889d0 * (x ** 4.0d0)) + ((-2.48015873015873d-5) * (x ** 6.0d0)))
    else
        tmp = t_0
    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 * x);
	double tmp;
	if (x <= -0.1) {
		tmp = t_0;
	} else if (x <= 0.096) {
		tmp = (0.5 + (-0.041666666666666664 * Math.pow(x, 2.0))) + ((0.001388888888888889 * Math.pow(x, 4.0)) + (-2.48015873015873e-5 * Math.pow(x, 6.0)));
	} else {
		tmp = t_0;
	}
	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 * x)
	tmp = 0
	if x <= -0.1:
		tmp = t_0
	elif x <= 0.096:
		tmp = (0.5 + (-0.041666666666666664 * math.pow(x, 2.0))) + ((0.001388888888888889 * math.pow(x, 4.0)) + (-2.48015873015873e-5 * math.pow(x, 6.0)))
	else:
		tmp = t_0
	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)) / Float64(x * x))
	tmp = 0.0
	if (x <= -0.1)
		tmp = t_0;
	elseif (x <= 0.096)
		tmp = Float64(Float64(0.5 + Float64(-0.041666666666666664 * (x ^ 2.0))) + Float64(Float64(0.001388888888888889 * (x ^ 4.0)) + Float64(-2.48015873015873e-5 * (x ^ 6.0))));
	else
		tmp = t_0;
	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 * x);
	tmp = 0.0;
	if (x <= -0.1)
		tmp = t_0;
	elseif (x <= 0.096)
		tmp = (0.5 + (-0.041666666666666664 * (x ^ 2.0))) + ((0.001388888888888889 * (x ^ 4.0)) + (-2.48015873015873e-5 * (x ^ 6.0)));
	else
		tmp = t_0;
	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] / N[(x * x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -0.1], t$95$0, If[LessEqual[x, 0.096], N[(N[(0.5 + N[(-0.041666666666666664 * N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(0.001388888888888889 * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision] + N[(-2.48015873015873e-5 * N[Power[x, 6.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -0.10000000000000001 or 0.096000000000000002 < x

   1. Initial program 1.2

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

   if -0.10000000000000001 < x < 0.096000000000000002

   1. Initial program 62.2

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

   2. Applied egg-rr 62.1

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

   3. Simplified 62.1

      \[\leadsto \color{blue}{\frac{1}{x \cdot x} + \left(\frac{\cos x + -1}{x \cdot x} + \frac{-1}{x \cdot x} \cdot \left(-1 + 2 \cdot \cos x\right)\right)} \]
      Proof

      [Start] 62.1	\[ \frac{-1}{x \cdot x} \cdot \left(\cos x + \left(\cos x + -1\right)\right) + \left(\frac{1}{x \cdot x} + \frac{\cos x + -1}{x \cdot x}\right) \]
      rational_best-simplify-1 [=>] 62.1	\[ \frac{-1}{x \cdot x} \cdot \left(\cos x + \left(\cos x + -1\right)\right) + \color{blue}{\left(\frac{\cos x + -1}{x \cdot x} + \frac{1}{x \cdot x}\right)} \]
      rational_best-simplify-43 [=>] 62.1	\[ \color{blue}{\frac{1}{x \cdot x} + \left(\frac{\cos x + -1}{x \cdot x} + \frac{-1}{x \cdot x} \cdot \left(\cos x + \left(\cos x + -1\right)\right)\right)} \]
      rational_best-simplify-43 [=>] 62.1	\[ \frac{1}{x \cdot x} + \left(\frac{\cos x + -1}{x \cdot x} + \frac{-1}{x \cdot x} \cdot \color{blue}{\left(-1 + \left(\cos x + \cos x\right)\right)}\right) \]
      rational_best-simplify-5 [<=] 62.1	\[ \frac{1}{x \cdot x} + \left(\frac{\cos x + -1}{x \cdot x} + \frac{-1}{x \cdot x} \cdot \left(-1 + \left(\color{blue}{1 \cdot \cos x} + \cos x\right)\right)\right) \]
      rational_best-simplify-5 [<=] 62.1	\[ \frac{1}{x \cdot x} + \left(\frac{\cos x + -1}{x \cdot x} + \frac{-1}{x \cdot x} \cdot \left(-1 + \left(1 \cdot \cos x + \color{blue}{1 \cdot \cos x}\right)\right)\right) \]
      rational_best-simplify-2 [=>] 62.1	\[ \frac{1}{x \cdot x} + \left(\frac{\cos x + -1}{x \cdot x} + \frac{-1}{x \cdot x} \cdot \left(-1 + \left(\color{blue}{\cos x \cdot 1} + 1 \cdot \cos x\right)\right)\right) \]
      rational_best-simplify-2 [=>] 62.1	\[ \frac{1}{x \cdot x} + \left(\frac{\cos x + -1}{x \cdot x} + \frac{-1}{x \cdot x} \cdot \left(-1 + \left(\cos x \cdot 1 + \color{blue}{\cos x \cdot 1}\right)\right)\right) \]
      rational_best-simplify-53 [=>] 62.1	\[ \frac{1}{x \cdot x} + \left(\frac{\cos x + -1}{x \cdot x} + \frac{-1}{x \cdot x} \cdot \left(-1 + \color{blue}{\cos x \cdot \left(1 + 1\right)}\right)\right) \]
      metadata-eval [=>] 62.1	\[ \frac{1}{x \cdot x} + \left(\frac{\cos x + -1}{x \cdot x} + \frac{-1}{x \cdot x} \cdot \left(-1 + \cos x \cdot \color{blue}{2}\right)\right) \]
      rational_best-simplify-2 [=>] 62.1	\[ \frac{1}{x \cdot x} + \left(\frac{\cos x + -1}{x \cdot x} + \frac{-1}{x \cdot x} \cdot \left(-1 + \color{blue}{2 \cdot \cos x}\right)\right) \]

   4. Taylor expanded in x around 0 0.0

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

   5. Simplified 0.0

      \[\leadsto \color{blue}{\left(0.5 + -0.041666666666666664 \cdot {x}^{2}\right) + \left(0.001388888888888889 \cdot {x}^{4} + -2.48015873015873 \cdot 10^{-5} \cdot {x}^{6}\right)} \]
      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-43 [=>] 0.0	\[ \color{blue}{\left(-2.48015873015873 \cdot 10^{-5} \cdot {x}^{6} + 0.001388888888888889 \cdot {x}^{4}\right) + \left(-0.041666666666666664 \cdot {x}^{2} + 0.5\right)} \]
      rational_best-simplify-1 [<=] 0.0	\[ \left(-2.48015873015873 \cdot 10^{-5} \cdot {x}^{6} + 0.001388888888888889 \cdot {x}^{4}\right) + \color{blue}{\left(0.5 + -0.041666666666666664 \cdot {x}^{2}\right)} \]
      rational_best-simplify-1 [=>] 0.0	\[ \color{blue}{\left(0.5 + -0.041666666666666664 \cdot {x}^{2}\right) + \left(-2.48015873015873 \cdot 10^{-5} \cdot {x}^{6} + 0.001388888888888889 \cdot {x}^{4}\right)} \]
      rational_best-simplify-1 [=>] 0.0	\[ \left(0.5 + -0.041666666666666664 \cdot {x}^{2}\right) + \color{blue}{\left(0.001388888888888889 \cdot {x}^{4} + -2.48015873015873 \cdot 10^{-5} \cdot {x}^{6}\right)} \]

3. Recombined 2 regimes into one program.

4. Final simplification 0.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.1:\\ \;\;\;\;\frac{1 - \cos x}{x \cdot x}\\ \mathbf{elif}\;x \leq 0.096:\\ \;\;\;\;\left(0.5 + -0.041666666666666664 \cdot {x}^{2}\right) + \left(0.001388888888888889 \cdot {x}^{4} + -2.48015873015873 \cdot 10^{-5} \cdot {x}^{6}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \cos x}{x \cdot x}\\ \end{array} \]

Alternatives

Alternative 1
Error	0.6
Cost	20488

\[\begin{array}{l} t_0 := \frac{1 - \cos x}{x \cdot x}\\ \mathbf{if}\;x \leq -0.1:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 0.096:\\ \;\;\;\;0.5 + \left(0.001388888888888889 \cdot {x}^{4} + \left(-0.041666666666666664 \cdot {x}^{2} + -2.48015873015873 \cdot 10^{-5} \cdot {x}^{6}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 2
Error	0.6
Cost	13768

\[\begin{array}{l} t_0 := \frac{1 - \cos x}{x \cdot x}\\ \mathbf{if}\;x \leq -0.034:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 0.028:\\ \;\;\;\;-0.041666666666666664 \cdot {x}^{2} + \left(0.5 + 0.001388888888888889 \cdot {x}^{4}\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 3
Error	0.6
Cost	7236

\[\begin{array}{l} t_0 := 1 - \cos x\\ \mathbf{if}\;x \leq -0.0046:\\ \;\;\;\;\frac{t_0}{-1 + \left(x \cdot x + 1\right)}\\ \mathbf{elif}\;x \leq 0.0055:\\ \;\;\;\;0.5 + -0.041666666666666664 \cdot {x}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0}{x \cdot x}\\ \end{array} \]

Alternative 4
Error	0.6
Cost	7112

\[\begin{array}{l} t_0 := \frac{1 - \cos x}{x \cdot x}\\ \mathbf{if}\;x \leq -0.0052:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 0.0055:\\ \;\;\;\;0.5 + -0.041666666666666664 \cdot {x}^{2}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 5
Error	16.0
Cost	7048

\[\begin{array}{l} \mathbf{if}\;x \leq -3.5:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 3.5:\\ \;\;\;\;0.5 + -0.041666666666666664 \cdot {x}^{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 6
Error	15.9
Cost	328

\[\begin{array}{l} \mathbf{if}\;x \leq -1.2 \cdot 10^{+77}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 1.06 \cdot 10^{+77}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 7
Error	46.5
Cost	64

\[0 \]

Reproduce

herbie shell --seed 2023097 
(FPCore (x)
  :name "cos2 (problem 3.4.1)"
  :precision binary64
  (/ (- 1.0 (cos x)) (* x x)))