cos2 (problem 3.4.1)

Average Error: 32.0 → 0.4

Time: 12.0s

Precision: binary64

Cost: 13316

Math

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

↓

\[\begin{array}{l} t_0 := 1 - \cos x\\ \mathbf{if}\;x \leq -0.00013:\\ \;\;\;\;{x}^{-2} \cdot t_0\\ \mathbf{elif}\;x \leq 0.00018:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{t_0}{x}}{x}\\ \end{array} \]

FPCore

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

↓

(FPCore (x)
 :precision binary64
 (let* ((t_0 (- 1.0 (cos x))))
   (if (<= x -0.00013)
     (* (pow x -2.0) t_0)
     (if (<= x 0.00018) 0.5 (/ (/ t_0 x) x)))))

C

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

↓

double code(double x) {
	double t_0 = 1.0 - cos(x);
	double tmp;
	if (x <= -0.00013) {
		tmp = pow(x, -2.0) * t_0;
	} else if (x <= 0.00018) {
		tmp = 0.5;
	} else {
		tmp = (t_0 / x) / 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)
    if (x <= (-0.00013d0)) then
        tmp = (x ** (-2.0d0)) * t_0
    else if (x <= 0.00018d0) then
        tmp = 0.5d0
    else
        tmp = (t_0 / x) / 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);
	double tmp;
	if (x <= -0.00013) {
		tmp = Math.pow(x, -2.0) * t_0;
	} else if (x <= 0.00018) {
		tmp = 0.5;
	} else {
		tmp = (t_0 / x) / 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)
	tmp = 0
	if x <= -0.00013:
		tmp = math.pow(x, -2.0) * t_0
	elif x <= 0.00018:
		tmp = 0.5
	else:
		tmp = (t_0 / x) / x
	return tmp

Julia

function code(x)
	return Float64(Float64(1.0 - cos(x)) / Float64(x * x))
end

↓

function code(x)
	t_0 = Float64(1.0 - cos(x))
	tmp = 0.0
	if (x <= -0.00013)
		tmp = Float64((x ^ -2.0) * t_0);
	elseif (x <= 0.00018)
		tmp = 0.5;
	else
		tmp = Float64(Float64(t_0 / x) / 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);
	tmp = 0.0;
	if (x <= -0.00013)
		tmp = (x ^ -2.0) * t_0;
	elseif (x <= 0.00018)
		tmp = 0.5;
	else
		tmp = (t_0 / x) / 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[(1.0 - N[Cos[x], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -0.00013], N[(N[Power[x, -2.0], $MachinePrecision] * t$95$0), $MachinePrecision], If[LessEqual[x, 0.00018], 0.5, N[(N[(t$95$0 / x), $MachinePrecision] / x), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.29999999999999989e-4

   1. Initial program 1.2

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

   2. Applied egg-rr 0.7

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

   if -1.29999999999999989e-4 < x < 1.80000000000000011e-4

   1. Initial program 62.6

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

   2. Taylor expanded in x around 0 0.1

      \[\leadsto \color{blue}{0.5} \]

   if 1.80000000000000011e-4 < x

   1. Initial program 1.1

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

   2. Applied egg-rr 0.6

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

   3. Applied egg-rr 0.6

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

4. Final simplification 0.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.00013:\\ \;\;\;\;{x}^{-2} \cdot \left(1 - \cos x\right)\\ \mathbf{elif}\;x \leq 0.00018:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 - \cos x}{x}}{x}\\ \end{array} \]

Alternatives

Alternative 1
Error	0.4
Cost	13376

\[\frac{\tan \left(x \cdot 0.5\right)}{x \cdot \frac{x}{\sin x}} \]

Alternative 2
Error	0.6
Cost	7113

\[\begin{array}{l} \mathbf{if}\;x \leq -0.00013 \lor \neg \left(x \leq 0.00018\right):\\ \;\;\;\;\frac{1 - \cos x}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \]

Alternative 3
Error	0.4
Cost	7113

\[\begin{array}{l} \mathbf{if}\;x \leq -0.00013 \lor \neg \left(x \leq 0.00018\right):\\ \;\;\;\;\frac{\frac{1 - \cos x}{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \]

Alternative 4
Error	13.8
Cost	832

\[\frac{-1}{x \cdot \left(x \cdot -0.16666666666666666 + 2 \cdot \frac{-1}{x}\right)} \]

Alternative 5
Error	13.8
Cost	768

\[\frac{\frac{1}{x}}{x \cdot 0.16666666666666666 + \frac{-2}{-x}} \]

Alternative 6
Error	13.9
Cost	585

\[\begin{array}{l} \mathbf{if}\;x \leq -3.45 \lor \neg \left(x \leq 3.5\right):\\ \;\;\;\;\frac{6}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \]

Alternative 7
Error	15.4
Cost	328

\[\begin{array}{l} \mathbf{if}\;x \leq -1.5 \cdot 10^{+77}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 5.5 \cdot 10^{+76}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 8
Error	46.8
Cost	64

\[0 \]

Reproduce

herbie shell --seed 2023066 
(FPCore (x)
  :name "cos2 (problem 3.4.1)"
  :precision binary64
  (/ (- 1.0 (cos x)) (* x x)))