cos2 (problem 3.4.1)

Average Error: 31.2 → 0.3

Time: 12.3s

Precision: binary64

Cost: 13768

Math

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

↓

\[\begin{array}{l} t_0 := 1 - \cos x\\ \mathbf{if}\;x \leq -0.032:\\ \;\;\;\;\frac{\frac{1}{x}}{x} \cdot t_0\\ \mathbf{elif}\;x \leq 0.032:\\ \;\;\;\;0.5 + \left(-0.041666666666666664 \cdot {x}^{2} + 0.001388888888888889 \cdot {x}^{4}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{t_0}{x}}{x}\\ \end{array} \]

FPCore

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

↓

(FPCore (x)
 :precision binary64
 (let* ((t_0 (- 1.0 (cos x))))
   (if (<= x -0.032)
     (* (/ (/ 1.0 x) x) t_0)
     (if (<= x 0.032)
       (+
        0.5
        (+
         (* -0.041666666666666664 (pow x 2.0))
         (* 0.001388888888888889 (pow x 4.0))))
       (/ (/ 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 <= -0.032) {
		tmp = ((1.0 / x) / x) * t_0;
	} else if (x <= 0.032) {
		tmp = 0.5 + ((-0.041666666666666664 * pow(x, 2.0)) + (0.001388888888888889 * pow(x, 4.0)));
	} 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 <= (-0.032d0)) then
        tmp = ((1.0d0 / x) / x) * t_0
    else if (x <= 0.032d0) then
        tmp = 0.5d0 + (((-0.041666666666666664d0) * (x ** 2.0d0)) + (0.001388888888888889d0 * (x ** 4.0d0)))
    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 <= -0.032) {
		tmp = ((1.0 / x) / x) * t_0;
	} else if (x <= 0.032) {
		tmp = 0.5 + ((-0.041666666666666664 * Math.pow(x, 2.0)) + (0.001388888888888889 * Math.pow(x, 4.0)));
	} 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 <= -0.032:
		tmp = ((1.0 / x) / x) * t_0
	elif x <= 0.032:
		tmp = 0.5 + ((-0.041666666666666664 * math.pow(x, 2.0)) + (0.001388888888888889 * math.pow(x, 4.0)))
	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 <= -0.032)
		tmp = Float64(Float64(Float64(1.0 / x) / x) * t_0);
	elseif (x <= 0.032)
		tmp = Float64(0.5 + Float64(Float64(-0.041666666666666664 * (x ^ 2.0)) + Float64(0.001388888888888889 * (x ^ 4.0))));
	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 <= -0.032)
		tmp = ((1.0 / x) / x) * t_0;
	elseif (x <= 0.032)
		tmp = 0.5 + ((-0.041666666666666664 * (x ^ 2.0)) + (0.001388888888888889 * (x ^ 4.0)));
	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, -0.032], N[(N[(N[(1.0 / x), $MachinePrecision] / x), $MachinePrecision] * t$95$0), $MachinePrecision], If[LessEqual[x, 0.032], N[(0.5 + N[(N[(-0.041666666666666664 * N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision] + N[(0.001388888888888889 * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$0 / x), $MachinePrecision] / x), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.032000000000000001

   1. Initial program 1.0

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

   2. Applied egg-rr 1.1

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

   3. Applied egg-rr 1.1

      \[\leadsto \color{blue}{\left(\frac{1}{x \cdot x} + 0\right)} \cdot \left(1 - \cos x\right) \]

   4. Simplified 0.5

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

      [Start] 1.1	\[ \left(\frac{1}{x \cdot x} + 0\right) \cdot \left(1 - \cos x\right) \]
      rational.json-simplify-4 [=>] 1.1	\[ \color{blue}{\frac{1}{x \cdot x}} \cdot \left(1 - \cos x\right) \]
      rational.json-simplify-46 [=>] 0.5	\[ \color{blue}{\frac{\frac{1}{x}}{x}} \cdot \left(1 - \cos x\right) \]

   if -0.032000000000000001 < x < 0.032000000000000001

   1. Initial program 62.2

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

   2. Taylor expanded in x around 0 0.0

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

   if 0.032000000000000001 < x

   1. Initial program 1.0

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

   2. Applied egg-rr 1.0

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

   3. Applied egg-rr 0.5

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

4. Final simplification 0.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.032:\\ \;\;\;\;\frac{\frac{1}{x}}{x} \cdot \left(1 - \cos x\right)\\ \mathbf{elif}\;x \leq 0.032:\\ \;\;\;\;0.5 + \left(-0.041666666666666664 \cdot {x}^{2} + 0.001388888888888889 \cdot {x}^{4}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 - \cos x}{x}}{x}\\ \end{array} \]

Alternatives

Alternative 1
Error	0.6
Cost	7112

\[\begin{array}{l} t_0 := \frac{1 - \cos x}{x \cdot x}\\ \mathbf{if}\;x \leq -0.0057:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 0.0056:\\ \;\;\;\;0.5 + -0.041666666666666664 \cdot {x}^{2}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 2
Error	0.3
Cost	7112

\[\begin{array}{l} t_0 := \frac{\frac{1 - \cos x}{x}}{x}\\ \mathbf{if}\;x \leq -0.0057:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 0.0056:\\ \;\;\;\;0.5 + -0.041666666666666664 \cdot {x}^{2}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 3
Error	0.3
Cost	7112

\[\begin{array}{l} t_0 := \frac{1 - \cos x}{x}\\ \mathbf{if}\;x \leq -0.0057:\\ \;\;\;\;t_0 \cdot \frac{1}{x}\\ \mathbf{elif}\;x \leq 0.0056:\\ \;\;\;\;0.5 + -0.041666666666666664 \cdot {x}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0}{x}\\ \end{array} \]

Alternative 4
Error	0.3
Cost	7112

\[\begin{array}{l} t_0 := 1 - \cos x\\ \mathbf{if}\;x \leq -0.0057:\\ \;\;\;\;\frac{\frac{1}{x}}{x} \cdot t_0\\ \mathbf{elif}\;x \leq 0.0056:\\ \;\;\;\;0.5 + -0.041666666666666664 \cdot {x}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{t_0}{x}}{x}\\ \end{array} \]

Alternative 5
Error	15.6
Cost	840

\[\begin{array}{l} t_0 := \frac{x \cdot 0.5}{\frac{x \cdot x}{x}}\\ \mathbf{if}\;x \leq -3.25:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 3.2:\\ \;\;\;\;x \cdot \left(\frac{0.5}{x} + -0.041666666666666664 \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 6
Error	13.8
Cost	832

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

Alternative 7
Error	13.9
Cost	704

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

Alternative 8
Error	13.8
Cost	704

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

Alternative 9
Error	31.2
Cost	64

\[0.5 \]

Reproduce

herbie shell --seed 2023064 
(FPCore (x)
  :name "cos2 (problem 3.4.1)"
  :precision binary64
  (/ (- 1.0 (cos x)) (* x x)))