cos2 (problem 3.4.1)

Average Error: 31.6 → 0.3

Time: 11.3s

Precision: binary64

Cost: 7240

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \leq -0.0042:\\ \;\;\;\;\frac{\frac{1 - \cos x}{x}}{x}\\ \mathbf{elif}\;x \leq 0.004:\\ \;\;\;\;0.5 + -0.041666666666666664 \cdot \left(x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\cos x + -1\right) \cdot \frac{-1}{x}}{x}\\ \end{array} \]

FPCore

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

↓

(FPCore (x)
 :precision binary64
 (if (<= x -0.0042)
   (/ (/ (- 1.0 (cos x)) x) x)
   (if (<= x 0.004)
     (+ 0.5 (* -0.041666666666666664 (* x x)))
     (/ (* (+ (cos x) -1.0) (/ -1.0 x)) x))))

C

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

↓

double code(double x) {
	double tmp;
	if (x <= -0.0042) {
		tmp = ((1.0 - cos(x)) / x) / x;
	} else if (x <= 0.004) {
		tmp = 0.5 + (-0.041666666666666664 * (x * x));
	} else {
		tmp = ((cos(x) + -1.0) * (-1.0 / 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) :: tmp
    if (x <= (-0.0042d0)) then
        tmp = ((1.0d0 - cos(x)) / x) / x
    else if (x <= 0.004d0) then
        tmp = 0.5d0 + ((-0.041666666666666664d0) * (x * x))
    else
        tmp = ((cos(x) + (-1.0d0)) * ((-1.0d0) / 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 tmp;
	if (x <= -0.0042) {
		tmp = ((1.0 - Math.cos(x)) / x) / x;
	} else if (x <= 0.004) {
		tmp = 0.5 + (-0.041666666666666664 * (x * x));
	} else {
		tmp = ((Math.cos(x) + -1.0) * (-1.0 / x)) / x;
	}
	return tmp;
}

Python

def code(x):
	return (1.0 - math.cos(x)) / (x * x)

↓

def code(x):
	tmp = 0
	if x <= -0.0042:
		tmp = ((1.0 - math.cos(x)) / x) / x
	elif x <= 0.004:
		tmp = 0.5 + (-0.041666666666666664 * (x * x))
	else:
		tmp = ((math.cos(x) + -1.0) * (-1.0 / x)) / x
	return tmp

Julia

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

↓

function code(x)
	tmp = 0.0
	if (x <= -0.0042)
		tmp = Float64(Float64(Float64(1.0 - cos(x)) / x) / x);
	elseif (x <= 0.004)
		tmp = Float64(0.5 + Float64(-0.041666666666666664 * Float64(x * x)));
	else
		tmp = Float64(Float64(Float64(cos(x) + -1.0) * Float64(-1.0 / x)) / x);
	end
	return tmp
end

MATLAB

function tmp = code(x)
	tmp = (1.0 - cos(x)) / (x * x);
end

↓

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -0.0042)
		tmp = ((1.0 - cos(x)) / x) / x;
	elseif (x <= 0.004)
		tmp = 0.5 + (-0.041666666666666664 * (x * x));
	else
		tmp = ((cos(x) + -1.0) * (-1.0 / 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_] := If[LessEqual[x, -0.0042], N[(N[(N[(1.0 - N[Cos[x], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[x, 0.004], N[(0.5 + N[(-0.041666666666666664 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[Cos[x], $MachinePrecision] + -1.0), $MachinePrecision] * N[(-1.0 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.00419999999999999974

   1. Initial program 1.1

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

   2. Applied egg-rr 1.1

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

   3. Applied egg-rr 0.5

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

   if -0.00419999999999999974 < x < 0.0040000000000000001

   1. Initial program 62.3

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

   2. Taylor expanded in x around 0 0.0

      \[\leadsto \color{blue}{0.5 + -0.041666666666666664 \cdot {x}^{2}} \]

   3. Simplified 0.0

      \[\leadsto \color{blue}{0.5 + -0.041666666666666664 \cdot \left(x \cdot x\right)} \]
      Proof

      [Start] 0.0	\[ 0.5 + -0.041666666666666664 \cdot {x}^{2} \]
      unpow2 [=>] 0.0	\[ 0.5 + -0.041666666666666664 \cdot \color{blue}{\left(x \cdot x\right)} \]

   if 0.0040000000000000001 < x

   1. Initial program 1.2

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

   2. Applied egg-rr 1.2

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

   3. Applied egg-rr 0.5

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

4. Final simplification 0.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.0042:\\ \;\;\;\;\frac{\frac{1 - \cos x}{x}}{x}\\ \mathbf{elif}\;x \leq 0.004:\\ \;\;\;\;0.5 + -0.041666666666666664 \cdot \left(x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\cos x + -1\right) \cdot \frac{-1}{x}}{x}\\ \end{array} \]

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

\[\begin{array}{l} \mathbf{if}\;x \leq -0.0042 \lor \neg \left(x \leq 0.004\right):\\ \;\;\;\;\frac{1 - \cos x}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;0.5 + -0.041666666666666664 \cdot \left(x \cdot x\right)\\ \end{array} \]

Alternative 3
Error	0.3
Cost	7113

\[\begin{array}{l} \mathbf{if}\;x \leq -0.0042 \lor \neg \left(x \leq 0.004\right):\\ \;\;\;\;\frac{\frac{1 - \cos x}{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;0.5 + -0.041666666666666664 \cdot \left(x \cdot x\right)\\ \end{array} \]

Alternative 4
Error	13.6
Cost	969

\[\begin{array}{l} \mathbf{if}\;x \leq -3.5 \lor \neg \left(x \leq 3.4\right):\\ \;\;\;\;\frac{6 + \frac{-72}{x \cdot x}}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;0.5 + -0.041666666666666664 \cdot \left(x \cdot x\right)\\ \end{array} \]

Alternative 5
Error	13.6
Cost	713

\[\begin{array}{l} \mathbf{if}\;x \leq -3.3 \lor \neg \left(x \leq 3.3\right):\\ \;\;\;\;\frac{6}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;0.5 + -0.041666666666666664 \cdot \left(x \cdot x\right)\\ \end{array} \]

Alternative 6
Error	13.7
Cost	704

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

Alternative 7
Error	13.8
Cost	585

\[\begin{array}{l} \mathbf{if}\;x \leq -3.5 \lor \neg \left(x \leq 3.4\right):\\ \;\;\;\;\frac{6}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \]

Alternative 8
Error	15.3
Cost	328

\[\begin{array}{l} \mathbf{if}\;x \leq -1.1 \cdot 10^{+77}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 8 \cdot 10^{+76}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 9
Error	46.3
Cost	64

\[0 \]

Reproduce

herbie shell --seed 2023045 
(FPCore (x)
  :name "cos2 (problem 3.4.1)"
  :precision binary64
  (/ (- 1.0 (cos x)) (* x x)))