cos2 (problem 3.4.1)

Average Accuracy: 50.9% → 99.4%

Time: 14.9s

Precision: binary64

Cost: 13508

Math

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

↓

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

FPCore

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

↓

(FPCore (x)
 :precision binary64
 (if (<= x -0.5)
   (* (/ (tan (* x 0.5)) (* x x)) (sin x))
   (if (<= x 0.0055)
     (+ 0.5 (* (* x x) -0.041666666666666664))
     (* (pow x -2.0) (- 1.0 (cos x))))))

C

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

↓

double code(double x) {
	double tmp;
	if (x <= -0.5) {
		tmp = (tan((x * 0.5)) / (x * x)) * sin(x);
	} else if (x <= 0.0055) {
		tmp = 0.5 + ((x * x) * -0.041666666666666664);
	} else {
		tmp = pow(x, -2.0) * (1.0 - cos(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 <= (-0.5d0)) then
        tmp = (tan((x * 0.5d0)) / (x * x)) * sin(x)
    else if (x <= 0.0055d0) then
        tmp = 0.5d0 + ((x * x) * (-0.041666666666666664d0))
    else
        tmp = (x ** (-2.0d0)) * (1.0d0 - cos(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 <= -0.5) {
		tmp = (Math.tan((x * 0.5)) / (x * x)) * Math.sin(x);
	} else if (x <= 0.0055) {
		tmp = 0.5 + ((x * x) * -0.041666666666666664);
	} else {
		tmp = Math.pow(x, -2.0) * (1.0 - Math.cos(x));
	}
	return tmp;
}

Python

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

↓

def code(x):
	tmp = 0
	if x <= -0.5:
		tmp = (math.tan((x * 0.5)) / (x * x)) * math.sin(x)
	elif x <= 0.0055:
		tmp = 0.5 + ((x * x) * -0.041666666666666664)
	else:
		tmp = math.pow(x, -2.0) * (1.0 - math.cos(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 <= -0.5)
		tmp = Float64(Float64(tan(Float64(x * 0.5)) / Float64(x * x)) * sin(x));
	elseif (x <= 0.0055)
		tmp = Float64(0.5 + Float64(Float64(x * x) * -0.041666666666666664));
	else
		tmp = Float64((x ^ -2.0) * Float64(1.0 - cos(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 <= -0.5)
		tmp = (tan((x * 0.5)) / (x * x)) * sin(x);
	elseif (x <= 0.0055)
		tmp = 0.5 + ((x * x) * -0.041666666666666664);
	else
		tmp = (x ^ -2.0) * (1.0 - cos(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, -0.5], N[(N[(N[Tan[N[(x * 0.5), $MachinePrecision]], $MachinePrecision] / N[(x * x), $MachinePrecision]), $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.0055], N[(0.5 + N[(N[(x * x), $MachinePrecision] * -0.041666666666666664), $MachinePrecision]), $MachinePrecision], N[(N[Power[x, -2.0], $MachinePrecision] * N[(1.0 - N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.5

   1. Initial program 98.4%

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

   2. Applied egg-rr 98.3%

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

      [Start] 98.4	\[ \frac{1 - \cos x}{x \cdot x} \]
      flip-- [=>] 98.1	\[ \frac{\color{blue}{\frac{1 \cdot 1 - \cos x \cdot \cos x}{1 + \cos x}}}{x \cdot x} \]
      div-inv [=>] 98.0	\[ \frac{\color{blue}{\left(1 \cdot 1 - \cos x \cdot \cos x\right) \cdot \frac{1}{1 + \cos x}}}{x \cdot x} \]
      metadata-eval [=>] 98.0	\[ \frac{\left(\color{blue}{1} - \cos x \cdot \cos x\right) \cdot \frac{1}{1 + \cos x}}{x \cdot x} \]
      1-sub-cos [=>] 98.3	\[ \frac{\color{blue}{\left(\sin x \cdot \sin x\right)} \cdot \frac{1}{1 + \cos x}}{x \cdot x} \]

   3. Simplified 98.8%

      \[\leadsto \frac{\color{blue}{\sin x \cdot \tan \left(\frac{x}{2}\right)}}{x \cdot x} \]
      Proof

      [Start] 98.3	\[ \frac{\left(\sin x \cdot \sin x\right) \cdot \frac{1}{1 + \cos x}}{x \cdot x} \]
      associate-*l* [=>] 98.3	\[ \frac{\color{blue}{\sin x \cdot \left(\sin x \cdot \frac{1}{1 + \cos x}\right)}}{x \cdot x} \]
      associate-*r/ [=>] 98.3	\[ \frac{\sin x \cdot \color{blue}{\frac{\sin x \cdot 1}{1 + \cos x}}}{x \cdot x} \]
      *-rgt-identity [=>] 98.3	\[ \frac{\sin x \cdot \frac{\color{blue}{\sin x}}{1 + \cos x}}{x \cdot x} \]
      hang-0p-tan [=>] 98.8	\[ \frac{\sin x \cdot \color{blue}{\tan \left(\frac{x}{2}\right)}}{x \cdot x} \]

   4. Applied egg-rr 98.8%

      \[\leadsto \color{blue}{\frac{\tan \left(x \cdot 0.5\right)}{x \cdot x} \cdot \sin x} \]
      Proof

      [Start] 98.8	\[ \frac{\sin x \cdot \tan \left(\frac{x}{2}\right)}{x \cdot x} \]
      *-commutative [=>] 98.8	\[ \frac{\color{blue}{\tan \left(\frac{x}{2}\right) \cdot \sin x}}{x \cdot x} \]
      associate-/l* [=>] 98.8	\[ \color{blue}{\frac{\tan \left(\frac{x}{2}\right)}{\frac{x \cdot x}{\sin x}}} \]
      associate-/r/ [=>] 98.8	\[ \color{blue}{\frac{\tan \left(\frac{x}{2}\right)}{x \cdot x} \cdot \sin x} \]
      div-inv [=>] 98.8	\[ \frac{\tan \color{blue}{\left(x \cdot \frac{1}{2}\right)}}{x \cdot x} \cdot \sin x \]
      metadata-eval [=>] 98.8	\[ \frac{\tan \left(x \cdot \color{blue}{0.5}\right)}{x \cdot x} \cdot \sin x \]

   if -0.5 < x < 0.0054999999999999997

   1. Initial program 2.6%

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

   2. Taylor expanded in x around 0 99.9%

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

   3. Simplified 99.9%

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

      [Start] 99.9	\[ 0.5 + -0.041666666666666664 \cdot {x}^{2} \]
      unpow2 [=>] 99.9	\[ 0.5 + -0.041666666666666664 \cdot \color{blue}{\left(x \cdot x\right)} \]

   if 0.0054999999999999997 < x

   1. Initial program 98.3%

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

   2. Applied egg-rr 99.2%

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

      [Start] 98.3	\[ \frac{1 - \cos x}{x \cdot x} \]
      div-inv [=>] 98.3	\[ \color{blue}{\left(1 - \cos x\right) \cdot \frac{1}{x \cdot x}} \]
      *-commutative [=>] 98.3	\[ \color{blue}{\frac{1}{x \cdot x} \cdot \left(1 - \cos x\right)} \]
      pow2 [=>] 98.3	\[ \frac{1}{\color{blue}{{x}^{2}}} \cdot \left(1 - \cos x\right) \]
      pow-flip [=>] 99.2	\[ \color{blue}{{x}^{\left(-2\right)}} \cdot \left(1 - \cos x\right) \]
      metadata-eval [=>] 99.2	\[ {x}^{\color{blue}{-2}} \cdot \left(1 - \cos x\right) \]

3. Recombined 3 regimes into one program.

4. Final simplification 99.4%

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

Alternatives

Alternative 1
Accuracy	99.6%
Cost	13448

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

Alternative 2
Accuracy	99.2%
Cost	7113

\[\begin{array}{l} \mathbf{if}\;x \leq -0.0052 \lor \neg \left(x \leq 0.0055\right):\\ \;\;\;\;\frac{1 - \cos x}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;0.5 + \left(x \cdot x\right) \cdot -0.041666666666666664\\ \end{array} \]

Alternative 3
Accuracy	99.6%
Cost	7113

\[\begin{array}{l} \mathbf{if}\;x \leq -0.0052 \lor \neg \left(x \leq 0.0055\right):\\ \;\;\;\;\frac{\frac{1 - \cos x}{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;0.5 + \left(x \cdot x\right) \cdot -0.041666666666666664\\ \end{array} \]

Alternative 4
Accuracy	76.0%
Cost	328

\[\begin{array}{l} \mathbf{if}\;x \leq -3.9 \cdot 10^{+76}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 1.25 \cdot 10^{+77}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 5
Accuracy	27.8%
Cost	64

\[0 \]

Reproduce

herbie shell --seed 2023135 
(FPCore (x)
  :name "cos2 (problem 3.4.1)"
  :precision binary64
  (/ (- 1.0 (cos x)) (* x x)))