cos2 (problem 3.4.1)

Percentage Accurate: 50.7% → 99.8%

Time: 11.9s

Alternatives: 6

Speedup: 107.0×

Specification

Math

\[\begin{array}{l} \\ \frac{1 - \cos x}{x \cdot x} \end{array} \]

FPCore

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

C

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

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (1.0d0 - cos(x)) / (x * x)
end function

Java

public static double code(double x) {
	return (1.0 - Math.cos(x)) / (x * x);
}

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := N[(N[(1.0 - N[Cos[x], $MachinePrecision]), $MachinePrecision] / N[(x * x), $MachinePrecision]), $MachinePrecision]

Initial Program: 50.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{1 - \cos x}{x \cdot x} \end{array} \]

FPCore

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

C

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

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (1.0d0 - cos(x)) / (x * x)
end function

Java

public static double code(double x) {
	return (1.0 - Math.cos(x)) / (x * x);
}

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := N[(N[(1.0 - N[Cos[x], $MachinePrecision]), $MachinePrecision] / N[(x * x), $MachinePrecision]), $MachinePrecision]

Alternative 1: 99.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{\sin x}{\frac{x}{\tan \left(x \cdot 0.5\right)}}}{x} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ (/ (sin x) (/ x (tan (* x 0.5)))) x))

C

double code(double x) {
	return (sin(x) / (x / tan((x * 0.5)))) / x;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (sin(x) / (x / tan((x * 0.5d0)))) / x
end function

Java

public static double code(double x) {
	return (Math.sin(x) / (x / Math.tan((x * 0.5)))) / x;
}

Python

def code(x):
	return (math.sin(x) / (x / math.tan((x * 0.5)))) / x

Julia

function code(x)
	return Float64(Float64(sin(x) / Float64(x / tan(Float64(x * 0.5)))) / x)
end

MATLAB

function tmp = code(x)
	tmp = (sin(x) / (x / tan((x * 0.5)))) / x;
end

Wolfram

code[x_] := N[(N[(N[Sin[x], $MachinePrecision] / N[(x / N[Tan[N[(x * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]

Derivation

1. Initial program 51.8%

   \[\frac{1 - \cos x}{x \cdot x} \]
2. Step-by-step derivation

   1. clear-num 51.7%

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

   2. inv-pow 51.7%

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

   3. associate-/l* 52.5%

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

3. Applied egg-rr 52.5%

   \[\leadsto \color{blue}{{\left(\frac{x}{\frac{1 - \cos x}{x}}\right)}^{-1}} \]
4. Step-by-step derivation

   1. flip-- 52.4%

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

   2. div-inv 52.4%

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

   3. metadata-eval 52.4%

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

   4. 1-sub-cos 74.1%

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

   5. pow2 74.1%

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

5. Applied egg-rr 74.1%

   \[\leadsto {\left(\frac{x}{\frac{\color{blue}{{\sin x}^{2} \cdot \frac{1}{1 + \cos x}}}{x}}\right)}^{-1} \]
6. Step-by-step derivation

   1. unpow2 74.1%

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

   2. associate-*l* 74.1%

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

   3. associate-*r/ 74.1%

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

   4. *-rgt-identity 74.1%

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

   5. hang-0p-tan 74.3%

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

7. Simplified 74.3%

   \[\leadsto {\left(\frac{x}{\frac{\color{blue}{\sin x \cdot \tan \left(\frac{x}{2}\right)}}{x}}\right)}^{-1} \]
8. Step-by-step derivation

   1. unpow-1 74.3%

      \[\leadsto \color{blue}{\frac{1}{\frac{x}{\frac{\sin x \cdot \tan \left(\frac{x}{2}\right)}{x}}}} \]

   2. clear-num 74.6%

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

   3. associate-/l* 99.8%

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

   4. div-inv 99.8%

      \[\leadsto \frac{\frac{\sin x}{\frac{x}{\tan \color{blue}{\left(x \cdot \frac{1}{2}\right)}}}}{x} \]

   5. metadata-eval 99.8%

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

9. Applied egg-rr 99.8%

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

10. Final simplification 99.8%

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

Alternative 2: 75.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 5 \cdot 10^{-156}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin x \cdot \tan \left(\frac{x}{2}\right)}{x \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 5e-156) 0.5 (/ (* (sin x) (tan (/ x 2.0))) (* x x))))

C

double code(double x) {
	double tmp;
	if (x <= 5e-156) {
		tmp = 0.5;
	} else {
		tmp = (sin(x) * tan((x / 2.0))) / (x * x);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 5d-156) then
        tmp = 0.5d0
    else
        tmp = (sin(x) * tan((x / 2.0d0))) / (x * x)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 5e-156) {
		tmp = 0.5;
	} else {
		tmp = (Math.sin(x) * Math.tan((x / 2.0))) / (x * x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 5e-156:
		tmp = 0.5
	else:
		tmp = (math.sin(x) * math.tan((x / 2.0))) / (x * x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 5e-156)
		tmp = 0.5;
	else
		tmp = Float64(Float64(sin(x) * tan(Float64(x / 2.0))) / Float64(x * x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 5e-156)
		tmp = 0.5;
	else
		tmp = (sin(x) * tan((x / 2.0))) / (x * x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 5e-156], 0.5, N[(N[(N[Sin[x], $MachinePrecision] * N[Tan[N[(x / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(x * x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 5.00000000000000007e-156

   1. Initial program 38.8%

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

   2. Taylor expanded in x around 0 63.4%

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

   if 5.00000000000000007e-156 < x

   1. Initial program 72.7%

      \[\frac{1 - \cos x}{x \cdot x} \]
   2. Step-by-step derivation

      1. flip-- 72.4%

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

      2. div-inv 72.4%

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

      3. metadata-eval 72.4%

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

      4. 1-sub-cos 99.0%

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

      5. pow2 99.0%

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

   3. Applied egg-rr 99.0%

      \[\leadsto \frac{\color{blue}{{\sin x}^{2} \cdot \frac{1}{1 + \cos x}}}{x \cdot x} \]
   4. Step-by-step derivation

      1. unpow2 99.0%

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

      2. associate-*l* 99.0%

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

      3. associate-*r/ 99.0%

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

      4. *-rgt-identity 99.0%

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

      5. hang-0p-tan 99.3%

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

   5. Simplified 99.4%

      \[\leadsto \frac{\color{blue}{\sin x \cdot \tan \left(\frac{x}{2}\right)}}{x \cdot x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 77.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 5 \cdot 10^{-156}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin x \cdot \tan \left(\frac{x}{2}\right)}{x \cdot x}\\ \end{array} \]

Alternative 3: 74.9% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 0.0052)
   (+ 0.5 (* (* x x) -0.041666666666666664))
   (/ (- 1.0 (cos x)) (* x x))))

C

double code(double x) {
	double tmp;
	if (x <= 0.0052) {
		tmp = 0.5 + ((x * x) * -0.041666666666666664);
	} else {
		tmp = (1.0 - cos(x)) / (x * x);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 0.0052d0) then
        tmp = 0.5d0 + ((x * x) * (-0.041666666666666664d0))
    else
        tmp = (1.0d0 - cos(x)) / (x * x)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 0.0052) {
		tmp = 0.5 + ((x * x) * -0.041666666666666664);
	} else {
		tmp = (1.0 - Math.cos(x)) / (x * x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 0.0052:
		tmp = 0.5 + ((x * x) * -0.041666666666666664)
	else:
		tmp = (1.0 - math.cos(x)) / (x * x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 0.0052)
		tmp = Float64(0.5 + Float64(Float64(x * x) * -0.041666666666666664));
	else
		tmp = Float64(Float64(1.0 - cos(x)) / Float64(x * x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 0.0052)
		tmp = 0.5 + ((x * x) * -0.041666666666666664);
	else
		tmp = (1.0 - cos(x)) / (x * x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 0.0052], N[(0.5 + N[(N[(x * x), $MachinePrecision] * -0.041666666666666664), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - N[Cos[x], $MachinePrecision]), $MachinePrecision] / N[(x * x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.0051999999999999998

   1. Initial program 34.1%

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

   2. Taylor expanded in x around 0 67.9%

      \[\leadsto \color{blue}{0.5 + -0.041666666666666664 \cdot {x}^{2}} \]
   3. Step-by-step derivation

      1. unpow2 67.9%

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

   4. Simplified 67.9%

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

   if 0.0051999999999999998 < x

   1. Initial program 98.7%

      \[\frac{1 - \cos x}{x \cdot x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 76.3%

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

Alternative 4: 75.2% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 0.0052)
   (+ 0.5 (* (* x x) -0.041666666666666664))
   (/ (/ (- 1.0 (cos x)) x) x)))

C

double code(double x) {
	double tmp;
	if (x <= 0.0052) {
		tmp = 0.5 + ((x * x) * -0.041666666666666664);
	} else {
		tmp = ((1.0 - cos(x)) / x) / x;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 0.0052d0) then
        tmp = 0.5d0 + ((x * x) * (-0.041666666666666664d0))
    else
        tmp = ((1.0d0 - cos(x)) / x) / x
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 0.0052) {
		tmp = 0.5 + ((x * x) * -0.041666666666666664);
	} else {
		tmp = ((1.0 - Math.cos(x)) / x) / x;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 0.0052:
		tmp = 0.5 + ((x * x) * -0.041666666666666664)
	else:
		tmp = ((1.0 - math.cos(x)) / x) / x
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 0.0052)
		tmp = Float64(0.5 + Float64(Float64(x * x) * -0.041666666666666664));
	else
		tmp = Float64(Float64(Float64(1.0 - cos(x)) / x) / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 0.0052)
		tmp = 0.5 + ((x * x) * -0.041666666666666664);
	else
		tmp = ((1.0 - cos(x)) / x) / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 0.0052], N[(0.5 + N[(N[(x * x), $MachinePrecision] * -0.041666666666666664), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 - N[Cos[x], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.0051999999999999998

   1. Initial program 34.1%

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

   2. Taylor expanded in x around 0 67.9%

      \[\leadsto \color{blue}{0.5 + -0.041666666666666664 \cdot {x}^{2}} \]
   3. Step-by-step derivation

      1. unpow2 67.9%

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

   4. Simplified 67.9%

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

   if 0.0051999999999999998 < x

   1. Initial program 98.7%

      \[\frac{1 - \cos x}{x \cdot x} \]
   2. Step-by-step derivation

      1. frac-2neg 98.7%

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

      2. div-inv 98.7%

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

      3. distribute-rgt-neg-in 98.7%

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

   3. Applied egg-rr 98.7%

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

   4. Taylor expanded in x around 0 98.7%

      \[\leadsto \left(-\left(1 - \cos x\right)\right) \cdot \color{blue}{\frac{-1}{{x}^{2}}} \]
   5. Step-by-step derivation

      1. unpow2 98.7%

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

      2. associate-/r* 99.0%

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

   6. Simplified 99.0%

      \[\leadsto \left(-\left(1 - \cos x\right)\right) \cdot \color{blue}{\frac{\frac{-1}{x}}{x}} \]
   7. Step-by-step derivation

      1. div-inv 98.8%

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

      2. frac-2neg 98.8%

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

      3. metadata-eval 98.8%

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

      4. inv-pow 98.8%

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

      5. inv-pow 98.8%

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

      6. unpow-prod-down 98.7%

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

      7. *-commutative 98.7%

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

      8. unpow-prod-down 98.8%

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

      9. inv-pow 98.8%

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

      10. inv-pow 98.8%

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

      11. metadata-eval 98.8%

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

      12. frac-2neg 98.8%

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

   8. Applied egg-rr 98.8%

      \[\leadsto \left(-\left(1 - \cos x\right)\right) \cdot \color{blue}{\left(\frac{1}{x} \cdot \frac{-1}{x}\right)} \]
   9. Step-by-step derivation

      1. associate-*r* 98.9%

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

      2. clear-num 98.9%

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

      3. un-div-inv 99.1%

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

      4. un-div-inv 99.1%

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

      5. add-sqr-sqrt 0.0%

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

      6. sqrt-unprod 43.5%

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

      7. sqr-neg 43.5%

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

      8. sqrt-unprod 43.5%

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

      9. add-sqr-sqrt 43.5%

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

      10. clear-num 43.5%

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

      11. add-sqr-sqrt 0.0%

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

      12. sqrt-unprod 99.1%

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

      13. frac-times 98.7%

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

      14. metadata-eval 98.7%

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

      15. metadata-eval 98.7%

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

      16. frac-times 99.1%

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

      17. sqrt-unprod 99.1%

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

      18. add-sqr-sqrt 99.1%

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

      19. clear-num 99.1%

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

      20. /-rgt-identity 99.1%

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

   10. Applied egg-rr 99.1%

       \[\leadsto \color{blue}{\frac{\frac{1 - \cos x}{x}}{x}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 76.4%

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

Alternative 5: 78.6% accurate, 9.7× speedup

Math

\[\begin{array}{l} \\ \frac{-1}{x \cdot \left(x \cdot -0.16666666666666666 - \frac{2}{x}\right)} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/ -1.0 (* x (- (* x -0.16666666666666666) (/ 2.0 x)))))

C

double code(double x) {
	return -1.0 / (x * ((x * -0.16666666666666666) - (2.0 / x)));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (-1.0d0) / (x * ((x * (-0.16666666666666666d0)) - (2.0d0 / x)))
end function

Java

public static double code(double x) {
	return -1.0 / (x * ((x * -0.16666666666666666) - (2.0 / x)));
}

Python

def code(x):
	return -1.0 / (x * ((x * -0.16666666666666666) - (2.0 / x)))

Julia

function code(x)
	return Float64(-1.0 / Float64(x * Float64(Float64(x * -0.16666666666666666) - Float64(2.0 / x))))
end

MATLAB

function tmp = code(x)
	tmp = -1.0 / (x * ((x * -0.16666666666666666) - (2.0 / x)));
end

Wolfram

code[x_] := N[(-1.0 / N[(x * N[(N[(x * -0.16666666666666666), $MachinePrecision] - N[(2.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 51.8%

   \[\frac{1 - \cos x}{x \cdot x} \]
2. Step-by-step derivation

   1. frac-2neg 51.8%

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

   2. div-inv 51.8%

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

   3. distribute-rgt-neg-in 51.8%

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

3. Applied egg-rr 51.8%

   \[\leadsto \color{blue}{\left(-\left(1 - \cos x\right)\right) \cdot \frac{1}{x \cdot \left(-x\right)}} \]
4. Step-by-step derivation

   1. associate-/r* 51.9%

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

   2. associate-*r/ 52.8%

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

   3. distribute-lft-neg-in 52.8%

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

   4. div-inv 52.8%

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

   5. frac-2neg 52.8%

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

   6. clear-num 52.5%

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

   7. frac-2neg 52.5%

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

   8. metadata-eval 52.5%

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

   9. frac-2neg 52.5%

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

   10. add-sqr-sqrt 24.2%

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

   11. sqrt-unprod 36.2%

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

   12. sqr-neg 36.2%

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

   13. sqrt-prod 12.7%

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

   14. add-sqr-sqrt 23.9%

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

   15. distribute-frac-neg 23.9%

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

   16. frac-2neg 23.9%

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

5. Applied egg-rr 52.5%

   \[\leadsto \color{blue}{\frac{-1}{\frac{x}{\frac{-1 + \cos x}{x}}}} \]
6. Step-by-step derivation

   1. associate-/r/ 52.5%

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

7. Applied egg-rr 52.5%

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

8. Taylor expanded in x around 0 74.9%

   \[\leadsto \frac{-1}{\color{blue}{\left(-0.16666666666666666 \cdot x - 2 \cdot \frac{1}{x}\right)} \cdot x} \]
9. Step-by-step derivation

   1. *-commutative 74.9%

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

   2. associate-*r/ 74.9%

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

   3. metadata-eval 74.9%

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

10. Simplified 74.9%

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

11. Final simplification 74.9%

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

Alternative 6: 51.8% accurate, 107.0× speedup

Math

\[\begin{array}{l} \\ 0.5 \end{array} \]

FPCore

(FPCore (x) :precision binary64 0.5)

C

double code(double x) {
	return 0.5;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 0.5d0
end function

Java

public static double code(double x) {
	return 0.5;
}

Python

def code(x):
	return 0.5

Julia

function code(x)
	return 0.5
end

MATLAB

function tmp = code(x)
	tmp = 0.5;
end

Wolfram

code[x_] := 0.5

Derivation

1. Initial program 51.8%

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

2. Taylor expanded in x around 0 50.8%

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

3. Final simplification 50.8%

   \[\leadsto 0.5 \]

Reproduce

herbie shell --seed 2023280 
(FPCore (x)
  :name "cos2 (problem 3.4.1)"
  :precision binary64
  (/ (- 1.0 (cos x)) (* x x)))