cos2 (problem 3.4.1)

Percentage Accurate: 51.7% → 99.8%

Time: 12.5s

Alternatives: 9

Speedup: 15.2×

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: 51.7% 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

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 47.6%

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

   1. flip-- 47.4%

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

   2. div-inv 47.4%

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

   3. metadata-eval 47.4%

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

   4. 1-sub-cos 78.3%

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

   5. pow2 78.3%

      \[\leadsto \frac{\color{blue}{{\sin x}^{2}} \cdot \frac{1}{1 + \cos x}}{x \cdot x} \]

3. Applied egg-rr 78.3%

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

   1. unpow2 78.3%

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

   2. associate-*l* 78.3%

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

   3. associate-*r/ 78.3%

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

   4. *-rgt-identity 78.3%

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

   5. hang-0p-tan 78.6%

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

5. Simplified 78.6%

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

   1. *-commutative 78.6%

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

   2. times-frac 99.8%

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

   3. div-inv 99.8%

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

   4. metadata-eval 99.8%

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

7. Applied egg-rr 99.8%

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

8. Final simplification 99.8%

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

Alternative 2: 74.6% accurate, 0.5× speedup

Math

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

FPCore

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

C

double code(double x) {
	double tmp;
	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
    real(8) :: tmp
    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) {
	double tmp;
	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):
	tmp = 0
	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

Julia

function code(x)
	tmp = 0.0
	if (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_2 = code(x)
	tmp = 0.0;
	if (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_] := 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 2 regimes

2. if x < 0.0054999999999999997

   1. Initial program 34.3%

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

   2. Taylor expanded in x around 0 67.8%

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

      1. *-commutative 67.8%

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

      2. unpow2 67.8%

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

   4. Simplified 67.8%

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

   if 0.0054999999999999997 < x

   1. Initial program 98.3%

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

      1. clear-num 98.2%

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

      2. associate-/r/ 98.3%

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

      3. pow2 98.3%

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

      4. pow-flip 99.3%

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

      5. metadata-eval 99.3%

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

   3. Applied egg-rr 99.3%

      \[\leadsto \color{blue}{{x}^{-2} \cdot \left(1 - \cos x\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 74.3%

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

Alternative 3: 74.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double x) {
	double tmp;
	if (x <= 0.0055) {
		tmp = 0.5 + ((x * x) * -0.041666666666666664);
	} else {
		tmp = ((1.0 - cos(x)) / x) * (1.0 / x);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 0.0055d0) then
        tmp = 0.5d0 + ((x * x) * (-0.041666666666666664d0))
    else
        tmp = ((1.0d0 - cos(x)) / x) * (1.0d0 / x)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 0.0055) {
		tmp = 0.5 + ((x * x) * -0.041666666666666664);
	} else {
		tmp = ((1.0 - Math.cos(x)) / x) * (1.0 / x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 0.0055:
		tmp = 0.5 + ((x * x) * -0.041666666666666664)
	else:
		tmp = ((1.0 - math.cos(x)) / x) * (1.0 / x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 0.0055)
		tmp = Float64(0.5 + Float64(Float64(x * x) * -0.041666666666666664));
	else
		tmp = Float64(Float64(Float64(1.0 - cos(x)) / x) * Float64(1.0 / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 0.0055)
		tmp = 0.5 + ((x * x) * -0.041666666666666664);
	else
		tmp = ((1.0 - cos(x)) / x) * (1.0 / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 0.0055], N[(0.5 + N[(N[(x * x), $MachinePrecision] * -0.041666666666666664), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 - N[Cos[x], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] * N[(1.0 / x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.0054999999999999997

   1. Initial program 34.3%

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

   2. Taylor expanded in x around 0 67.8%

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

      1. *-commutative 67.8%

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

      2. unpow2 67.8%

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

   4. Simplified 67.8%

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

   if 0.0054999999999999997 < x

   1. Initial program 98.3%

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

      1. associate-/r* 99.3%

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

      2. div-inv 99.3%

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

   3. Applied egg-rr 99.3%

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

4. Final simplification 74.3%

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

Alternative 4: 74.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double x) {
	double tmp;
	if (x <= 0.0055) {
		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 <= 0.0055d0) 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 <= 0.0055) {
		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 <= 0.0055:
		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 <= 0.0055)
		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 <= 0.0055)
		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, 0.0055], 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 < 0.0054999999999999997

   1. Initial program 34.3%

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

   2. Taylor expanded in x around 0 67.8%

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

      1. *-commutative 67.8%

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

      2. unpow2 67.8%

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

   4. Simplified 67.8%

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

   if 0.0054999999999999997 < x

   1. Initial program 98.3%

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

4. Final simplification 74.1%

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

Alternative 5: 74.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double x) {
	double tmp;
	if (x <= 0.0055) {
		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 <= 0.0055d0) 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 <= 0.0055) {
		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 <= 0.0055:
		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 <= 0.0055)
		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 <= 0.0055)
		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, 0.0055], 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 < 0.0054999999999999997

   1. Initial program 34.3%

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

   2. Taylor expanded in x around 0 67.8%

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

      1. *-commutative 67.8%

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

      2. unpow2 67.8%

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

   4. Simplified 67.8%

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

   if 0.0054999999999999997 < x

   1. Initial program 98.3%

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

      1. associate-/r* 99.3%

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

      2. div-inv 99.3%

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

   3. Applied egg-rr 99.3%

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

      1. un-div-inv 99.3%

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

   5. Applied egg-rr 99.3%

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

4. Final simplification 74.3%

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

Alternative 6: 77.5% accurate, 9.7× speedup

Math

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

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(Float64(-1.0 / 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[(N[(-1.0 / x), $MachinePrecision] / N[(N[(x * -0.16666666666666666), $MachinePrecision] - N[(2.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 47.6%

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

   1. frac-2neg 47.6%

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

   2. div-inv 47.5%

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

   3. distribute-rgt-neg-in 47.5%

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

3. Applied egg-rr 47.5%

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

   1. associate-*r/ 47.6%

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

   2. *-commutative 47.6%

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

   3. times-frac 48.6%

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

   4. frac-2neg 48.6%

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

   5. div-inv 48.6%

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

   6. clear-num 48.1%

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

   7. frac-2neg 48.1%

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

   8. metadata-eval 48.1%

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

   9. frac-2neg 48.1%

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

   10. add-sqr-sqrt 26.5%

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

   11. sqrt-unprod 37.4%

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

   12. sqr-neg 37.4%

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

   13. sqrt-prod 11.5%

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

   14. add-sqr-sqrt 22.8%

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

   15. distribute-frac-neg 22.8%

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

   16. frac-2neg 22.8%

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

   17. div-inv 22.8%

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

   18. clear-num 22.8%

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

   19. add-sqr-sqrt 22.8%

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

   20. sqrt-unprod 22.8%

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

   21. sqr-neg 22.8%

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

5. Applied egg-rr 48.1%

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

6. Taylor expanded in x around 0 77.6%

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

   1. expm1-log1p-u 77.6%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{-1}{x \cdot \left(-0.16666666666666666 \cdot x - 2 \cdot \frac{1}{x}\right)}\right)\right)} \]

   2. expm1-udef 74.9%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{-1}{x \cdot \left(-0.16666666666666666 \cdot x - 2 \cdot \frac{1}{x}\right)}\right)} - 1} \]

   3. *-commutative 74.9%

      \[\leadsto e^{\mathsf{log1p}\left(\frac{-1}{x \cdot \left(\color{blue}{x \cdot -0.16666666666666666} - 2 \cdot \frac{1}{x}\right)}\right)} - 1 \]

   4. un-div-inv 74.9%

      \[\leadsto e^{\mathsf{log1p}\left(\frac{-1}{x \cdot \left(x \cdot -0.16666666666666666 - \color{blue}{\frac{2}{x}}\right)}\right)} - 1 \]

8. Applied egg-rr 74.9%

   \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{-1}{x \cdot \left(x \cdot -0.16666666666666666 - \frac{2}{x}\right)}\right)} - 1} \]
9. Step-by-step derivation

   1. expm1-def 77.6%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{-1}{x \cdot \left(x \cdot -0.16666666666666666 - \frac{2}{x}\right)}\right)\right)} \]

   2. expm1-log1p 77.6%

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

   3. associate-/r* 77.7%

      \[\leadsto \color{blue}{\frac{\frac{-1}{x}}{x \cdot -0.16666666666666666 - \frac{2}{x}}} \]

10. Simplified 77.7%

    \[\leadsto \color{blue}{\frac{\frac{-1}{x}}{x \cdot -0.16666666666666666 - \frac{2}{x}}} \]

11. Final simplification 77.7%

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

Alternative 7: 63.4% accurate, 11.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 3.25:\\ \;\;\;\;0.5 + \left(x \cdot x\right) \cdot -0.041666666666666664\\ \mathbf{else}:\\ \;\;\;\;\frac{6}{x \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 3.25) (+ 0.5 (* (* x x) -0.041666666666666664)) (/ 6.0 (* x x))))

C

double code(double x) {
	double tmp;
	if (x <= 3.25) {
		tmp = 0.5 + ((x * x) * -0.041666666666666664);
	} else {
		tmp = 6.0 / (x * x);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 3.25d0) then
        tmp = 0.5d0 + ((x * x) * (-0.041666666666666664d0))
    else
        tmp = 6.0d0 / (x * x)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 3.25) {
		tmp = 0.5 + ((x * x) * -0.041666666666666664);
	} else {
		tmp = 6.0 / (x * x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 3.25:
		tmp = 0.5 + ((x * x) * -0.041666666666666664)
	else:
		tmp = 6.0 / (x * x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 3.25)
		tmp = Float64(0.5 + Float64(Float64(x * x) * -0.041666666666666664));
	else
		tmp = Float64(6.0 / Float64(x * x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 3.25)
		tmp = 0.5 + ((x * x) * -0.041666666666666664);
	else
		tmp = 6.0 / (x * x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 3.25], N[(0.5 + N[(N[(x * x), $MachinePrecision] * -0.041666666666666664), $MachinePrecision]), $MachinePrecision], N[(6.0 / N[(x * x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 3.25

   1. Initial program 34.3%

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

   2. Taylor expanded in x around 0 67.8%

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

      1. *-commutative 67.8%

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

      2. unpow2 67.8%

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

   4. Simplified 67.8%

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

   if 3.25 < x

   1. Initial program 98.3%

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

      1. frac-2neg 98.3%

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

      2. div-inv 98.3%

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

      3. distribute-rgt-neg-in 98.3%

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

   3. Applied egg-rr 98.3%

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

      1. associate-*r/ 98.3%

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

      2. *-commutative 98.3%

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

      3. times-frac 99.3%

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

      4. frac-2neg 99.3%

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

      5. div-inv 99.3%

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

      6. clear-num 98.2%

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

      7. frac-2neg 98.2%

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

      8. metadata-eval 98.2%

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

      9. frac-2neg 98.2%

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

      10. add-sqr-sqrt 0.0%

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

      11. sqrt-unprod 51.5%

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

      12. sqr-neg 51.5%

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

      13. sqrt-prod 51.5%

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

      14. add-sqr-sqrt 51.5%

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

      15. distribute-frac-neg 51.5%

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

      16. frac-2neg 51.5%

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

      17. div-inv 51.5%

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

      18. clear-num 51.5%

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

      19. add-sqr-sqrt 51.5%

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

      20. sqrt-unprod 51.5%

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

      21. sqr-neg 51.5%

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

   5. Applied egg-rr 98.2%

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

   6. Taylor expanded in x around 0 57.8%

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

   7. Taylor expanded in x around inf 57.8%

      \[\leadsto \color{blue}{\frac{6}{{x}^{2}}} \]
   8. Step-by-step derivation

      1. unpow2 57.8%

         \[\leadsto \frac{6}{\color{blue}{x \cdot x}} \]

   9. Simplified 57.8%

      \[\leadsto \color{blue}{\frac{6}{x \cdot x}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 65.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 3.25:\\ \;\;\;\;0.5 + \left(x \cdot x\right) \cdot -0.041666666666666664\\ \mathbf{else}:\\ \;\;\;\;\frac{6}{x \cdot x}\\ \end{array} \]

Alternative 8: 63.8% accurate, 15.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 3.4:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{6}{x \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (if (<= x 3.4) 0.5 (/ 6.0 (* x x))))

C

double code(double x) {
	double tmp;
	if (x <= 3.4) {
		tmp = 0.5;
	} else {
		tmp = 6.0 / (x * x);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 3.4d0) then
        tmp = 0.5d0
    else
        tmp = 6.0d0 / (x * x)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 3.4) {
		tmp = 0.5;
	} else {
		tmp = 6.0 / (x * x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 3.4:
		tmp = 0.5
	else:
		tmp = 6.0 / (x * x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 3.4)
		tmp = 0.5;
	else
		tmp = Float64(6.0 / Float64(x * x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 3.4)
		tmp = 0.5;
	else
		tmp = 6.0 / (x * x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 3.4], 0.5, N[(6.0 / N[(x * x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 3.39999999999999991

   1. Initial program 34.3%

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

   2. Taylor expanded in x around 0 68.3%

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

   if 3.39999999999999991 < x

   1. Initial program 98.3%

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

      1. frac-2neg 98.3%

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

      2. div-inv 98.3%

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

      3. distribute-rgt-neg-in 98.3%

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

   3. Applied egg-rr 98.3%

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

      1. associate-*r/ 98.3%

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

      2. *-commutative 98.3%

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

      3. times-frac 99.3%

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

      4. frac-2neg 99.3%

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

      5. div-inv 99.3%

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

      6. clear-num 98.2%

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

      7. frac-2neg 98.2%

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

      8. metadata-eval 98.2%

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

      9. frac-2neg 98.2%

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

      10. add-sqr-sqrt 0.0%

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

      11. sqrt-unprod 51.5%

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

      12. sqr-neg 51.5%

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

      13. sqrt-prod 51.5%

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

      14. add-sqr-sqrt 51.5%

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

      15. distribute-frac-neg 51.5%

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

      16. frac-2neg 51.5%

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

      17. div-inv 51.5%

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

      18. clear-num 51.5%

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

      19. add-sqr-sqrt 51.5%

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

      20. sqrt-unprod 51.5%

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

      21. sqr-neg 51.5%

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

   5. Applied egg-rr 98.2%

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

   6. Taylor expanded in x around 0 57.8%

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

   7. Taylor expanded in x around inf 57.8%

      \[\leadsto \color{blue}{\frac{6}{{x}^{2}}} \]
   8. Step-by-step derivation

      1. unpow2 57.8%

         \[\leadsto \frac{6}{\color{blue}{x \cdot x}} \]

   9. Simplified 57.8%

      \[\leadsto \color{blue}{\frac{6}{x \cdot x}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 66.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 3.4:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{6}{x \cdot x}\\ \end{array} \]

Alternative 9: 50.8% 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 47.6%

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

2. Taylor expanded in x around 0 55.1%

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

3. Final simplification 55.1%

   \[\leadsto 0.5 \]

Reproduce

herbie shell --seed 2023275 
(FPCore (x)
  :name "cos2 (problem 3.4.1)"
  :precision binary64
  (/ (- 1.0 (cos x)) (* x x)))