cos2 (problem 3.4.1)

Average Error: 31.7 → 0.3

Time: 12.8s

Precision: binary64

Cost: 13640

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \leq -0.033:\\ \;\;\;\;{x}^{-2} \cdot \left(1 - \cos x\right)\\ \mathbf{elif}\;x \leq 2 \cdot 10^{-5}:\\ \;\;\;\;0.5 + \left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot 0.001388888888888889 + -0.041666666666666664\right)\\ \mathbf{else}:\\ \;\;\;\;\tan \left(x \cdot 0.5\right) \cdot \frac{\sin x}{x \cdot x}\\ \end{array} \]

FPCore

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

↓

(FPCore (x)
 :precision binary64
 (if (<= x -0.033)
   (* (pow x -2.0) (- 1.0 (cos x)))
   (if (<= x 2e-5)
     (+
      0.5
      (* (* x x) (+ (* (* x x) 0.001388888888888889) -0.041666666666666664)))
     (* (tan (* x 0.5)) (/ (sin x) (* x x))))))

C

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

↓

double code(double x) {
	double tmp;
	if (x <= -0.033) {
		tmp = pow(x, -2.0) * (1.0 - cos(x));
	} else if (x <= 2e-5) {
		tmp = 0.5 + ((x * x) * (((x * x) * 0.001388888888888889) + -0.041666666666666664));
	} else {
		tmp = tan((x * 0.5)) * (sin(x) / (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 <= (-0.033d0)) then
        tmp = (x ** (-2.0d0)) * (1.0d0 - cos(x))
    else if (x <= 2d-5) then
        tmp = 0.5d0 + ((x * x) * (((x * x) * 0.001388888888888889d0) + (-0.041666666666666664d0)))
    else
        tmp = tan((x * 0.5d0)) * (sin(x) / (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 <= -0.033) {
		tmp = Math.pow(x, -2.0) * (1.0 - Math.cos(x));
	} else if (x <= 2e-5) {
		tmp = 0.5 + ((x * x) * (((x * x) * 0.001388888888888889) + -0.041666666666666664));
	} else {
		tmp = Math.tan((x * 0.5)) * (Math.sin(x) / (x * x));
	}
	return tmp;
}

Python

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

↓

def code(x):
	tmp = 0
	if x <= -0.033:
		tmp = math.pow(x, -2.0) * (1.0 - math.cos(x))
	elif x <= 2e-5:
		tmp = 0.5 + ((x * x) * (((x * x) * 0.001388888888888889) + -0.041666666666666664))
	else:
		tmp = math.tan((x * 0.5)) * (math.sin(x) / (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 <= -0.033)
		tmp = Float64((x ^ -2.0) * Float64(1.0 - cos(x)));
	elseif (x <= 2e-5)
		tmp = Float64(0.5 + Float64(Float64(x * x) * Float64(Float64(Float64(x * x) * 0.001388888888888889) + -0.041666666666666664)));
	else
		tmp = Float64(tan(Float64(x * 0.5)) * Float64(sin(x) / Float64(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 <= -0.033)
		tmp = (x ^ -2.0) * (1.0 - cos(x));
	elseif (x <= 2e-5)
		tmp = 0.5 + ((x * x) * (((x * x) * 0.001388888888888889) + -0.041666666666666664));
	else
		tmp = tan((x * 0.5)) * (sin(x) / (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, -0.033], N[(N[Power[x, -2.0], $MachinePrecision] * N[(1.0 - N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 2e-5], N[(0.5 + N[(N[(x * x), $MachinePrecision] * N[(N[(N[(x * x), $MachinePrecision] * 0.001388888888888889), $MachinePrecision] + -0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Tan[N[(x * 0.5), $MachinePrecision]], $MachinePrecision] * N[(N[Sin[x], $MachinePrecision] / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.033000000000000002

   1. Initial program 1.1

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

   2. Applied egg-rr 0.4

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

   if -0.033000000000000002 < x < 2.00000000000000016e-5

   1. Initial program 62.5

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

   2. Taylor expanded in x around 0 0.0

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

   3. Simplified 0.0

      \[\leadsto \color{blue}{0.5 + \mathsf{fma}\left(0.001388888888888889, {x}^{4}, x \cdot \left(x \cdot -0.041666666666666664\right)\right)} \]
      Proof

      [Start] 0.0	\[ 0.5 + \left(-0.041666666666666664 \cdot {x}^{2} + 0.001388888888888889 \cdot {x}^{4}\right) \]
      +-commutative [=>] 0.0	\[ 0.5 + \color{blue}{\left(0.001388888888888889 \cdot {x}^{4} + -0.041666666666666664 \cdot {x}^{2}\right)} \]
      fma-def [=>] 0.0	\[ 0.5 + \color{blue}{\mathsf{fma}\left(0.001388888888888889, {x}^{4}, -0.041666666666666664 \cdot {x}^{2}\right)} \]
      *-commutative [=>] 0.0	\[ 0.5 + \mathsf{fma}\left(0.001388888888888889, {x}^{4}, \color{blue}{{x}^{2} \cdot -0.041666666666666664}\right) \]
      unpow2 [=>] 0.0	\[ 0.5 + \mathsf{fma}\left(0.001388888888888889, {x}^{4}, \color{blue}{\left(x \cdot x\right)} \cdot -0.041666666666666664\right) \]
      associate-*l* [=>] 0.0	\[ 0.5 + \mathsf{fma}\left(0.001388888888888889, {x}^{4}, \color{blue}{x \cdot \left(x \cdot -0.041666666666666664\right)}\right) \]

   4. Applied egg-rr 0.0

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

   5. Applied egg-rr 0.0

      \[\leadsto 0.5 + \color{blue}{\left(x \cdot x\right) \cdot \left(0.001388888888888889 \cdot \left(x \cdot x\right) + -0.041666666666666664\right)} \]

   if 2.00000000000000016e-5 < x

   1. Initial program 1.1

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

   2. Applied egg-rr 1.1

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

   3. Simplified 0.7

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

      [Start] 1.1	\[ \frac{\left(\sin x \cdot \sin x\right) \cdot \frac{1}{1 + \cos x}}{x \cdot x} \]
      associate-*l* [=>] 1.1	\[ \frac{\color{blue}{\sin x \cdot \left(\sin x \cdot \frac{1}{1 + \cos x}\right)}}{x \cdot x} \]
      associate-*r/ [=>] 1.0	\[ \frac{\sin x \cdot \color{blue}{\frac{\sin x \cdot 1}{1 + \cos x}}}{x \cdot x} \]
      *-rgt-identity [=>] 1.0	\[ \frac{\sin x \cdot \frac{\color{blue}{\sin x}}{1 + \cos x}}{x \cdot x} \]
      hang-0p-tan [=>] 0.7	\[ \frac{\sin x \cdot \color{blue}{\tan \left(\frac{x}{2}\right)}}{x \cdot x} \]

   4. Applied egg-rr 0.7

      \[\leadsto \color{blue}{\frac{\sin x}{x \cdot x} \cdot \tan \left(x \cdot 0.5\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 0.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.033:\\ \;\;\;\;{x}^{-2} \cdot \left(1 - \cos x\right)\\ \mathbf{elif}\;x \leq 2 \cdot 10^{-5}:\\ \;\;\;\;0.5 + \left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot 0.001388888888888889 + -0.041666666666666664\right)\\ \mathbf{else}:\\ \;\;\;\;\tan \left(x \cdot 0.5\right) \cdot \frac{\sin x}{x \cdot x}\\ \end{array} \]

Alternatives

Alternative 1
Error	0.4
Cost	13576

\[\begin{array}{l} \mathbf{if}\;x \leq -0.033:\\ \;\;\;\;{x}^{-2} \cdot \left(1 - \cos x\right)\\ \mathbf{elif}\;x \leq 0.0285:\\ \;\;\;\;0.5 + \left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot 0.001388888888888889 + -0.041666666666666664\right)\\ \mathbf{else}:\\ \;\;\;\;{x}^{-2} - \frac{\cos x}{x \cdot x}\\ \end{array} \]

Alternative 2
Error	0.4
Cost	13376

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

Alternative 3
Error	0.4
Cost	13316

\[\begin{array}{l} \mathbf{if}\;x \leq -0.033:\\ \;\;\;\;{x}^{-2} \cdot \left(1 - \cos x\right)\\ \mathbf{elif}\;x \leq 0.0285:\\ \;\;\;\;0.5 + \left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot 0.001388888888888889 + -0.041666666666666664\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x \cdot x} + \cos x \cdot \frac{-1}{x \cdot x}\\ \end{array} \]

Alternative 4
Error	0.4
Cost	7496

\[\begin{array}{l} \mathbf{if}\;x \leq -0.033:\\ \;\;\;\;\left(\cos x + -1\right) \cdot \frac{\frac{-1}{x}}{x}\\ \mathbf{elif}\;x \leq 0.0285:\\ \;\;\;\;0.5 + \left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot 0.001388888888888889 + -0.041666666666666664\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x \cdot x} + \cos x \cdot \frac{-1}{x \cdot x}\\ \end{array} \]

Alternative 5
Error	0.3
Cost	7241

\[\begin{array}{l} \mathbf{if}\;x \leq -0.033 \lor \neg \left(x \leq 0.035\right):\\ \;\;\;\;\left(\cos x + -1\right) \cdot \frac{\frac{-1}{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;0.5 + \left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot 0.001388888888888889 + -0.041666666666666664\right)\\ \end{array} \]

Alternative 6
Error	0.5
Cost	7113

\[\begin{array}{l} \mathbf{if}\;x \leq -0.033 \lor \neg \left(x \leq 0.035\right):\\ \;\;\;\;\frac{1 - \cos x}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;0.5 + \left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot 0.001388888888888889 + -0.041666666666666664\right)\\ \end{array} \]

Alternative 7
Error	0.4
Cost	7112

\[\begin{array}{l} t_0 := 1 - \cos x\\ \mathbf{if}\;x \leq -0.033:\\ \;\;\;\;\frac{\frac{t_0}{x}}{x}\\ \mathbf{elif}\;x \leq 0.035:\\ \;\;\;\;0.5 + \left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot 0.001388888888888889 + -0.041666666666666664\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0}{x \cdot x}\\ \end{array} \]

Alternative 8
Error	15.1
Cost	1097

\[\begin{array}{l} \mathbf{if}\;x \leq -7.5 \cdot 10^{+38} \lor \neg \left(x \leq 6.7 \cdot 10^{+38}\right):\\ \;\;\;\;\frac{0}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;0.5 + \left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot 0.001388888888888889 + -0.041666666666666664\right)\\ \end{array} \]

Alternative 9
Error	13.7
Cost	896

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

Alternative 10
Error	15.3
Cost	713

\[\begin{array}{l} \mathbf{if}\;x \leq -3.5 \lor \neg \left(x \leq 3.5\right):\\ \;\;\;\;\frac{0}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;0.5 + \left(x \cdot x\right) \cdot -0.041666666666666664\\ \end{array} \]

Alternative 11
Error	30.8
Cost	64

\[0.5 \]

Reproduce

herbie shell --seed 2023060 
(FPCore (x)
  :name "cos2 (problem 3.4.1)"
  :precision binary64
  (/ (- 1.0 (cos x)) (* x x)))