cos2 (problem 3.4.1)

Percentage Accurate: 50.7% → 99.8%

Time: 9.1s

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

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

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 52.6%

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

   1. flip-- 52.5%

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

   2. div-inv 52.4%

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

   3. metadata-eval 52.4%

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

   4. 1-sub-cos 74.8%

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

   5. pow2 74.8%

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

3. Applied egg-rr 74.8%

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

   1. unpow2 74.8%

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

   2. associate-*l* 74.9%

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

   3. associate-*r/ 74.9%

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

   4. *-rgt-identity 74.9%

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

   5. hang-0p-tan 75.1%

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

5. Simplified 75.1%

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

   1. *-commutative 75.1%

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

   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, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.0058:\\ \;\;\;\;0.5 + -0.041666666666666664 \cdot \left(x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \cos x}{x} \cdot \frac{1}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 0.0058)
   (+ 0.5 (* -0.041666666666666664 (* x x)))
   (* (/ (- 1.0 (cos x)) x) (/ 1.0 x))))

C

double code(double x) {
	double tmp;
	if (x <= 0.0058) {
		tmp = 0.5 + (-0.041666666666666664 * (x * x));
	} 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 <= 0.0058d0) then
        tmp = 0.5d0 + ((-0.041666666666666664d0) * (x * x))
    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 <= 0.0058) {
		tmp = 0.5 + (-0.041666666666666664 * (x * x));
	} else {
		tmp = ((1.0 - Math.cos(x)) / x) * (1.0 / x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 0.0058:
		tmp = 0.5 + (-0.041666666666666664 * (x * x))
	else:
		tmp = ((1.0 - math.cos(x)) / x) * (1.0 / x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 0.0058)
		tmp = Float64(0.5 + Float64(-0.041666666666666664 * Float64(x * x)));
	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 <= 0.0058)
		tmp = 0.5 + (-0.041666666666666664 * (x * x));
	else
		tmp = ((1.0 - cos(x)) / x) * (1.0 / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 0.0058], N[(0.5 + N[(-0.041666666666666664 * N[(x * x), $MachinePrecision]), $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 < 0.0058

   1. Initial program 36.6%

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

   2. Taylor expanded in x around 0 64.8%

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

      1. unpow2 64.8%

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

   4. Simplified 64.8%

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

   if 0.0058 < x

   1. Initial program 98.5%

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

      1. associate-/r* 99.5%

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

      2. div-inv 99.4%

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

   3. Applied egg-rr 99.4%

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

4. Final simplification 73.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.0058:\\ \;\;\;\;0.5 + -0.041666666666666664 \cdot \left(x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \cos x}{x} \cdot \frac{1}{x}\\ \end{array} \]

Alternative 3: 74.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.0058:\\ \;\;\;\;0.5 + -0.041666666666666664 \cdot \left(x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\cos x + -1\right) \cdot \frac{\frac{-1}{x}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 0.0058)
   (+ 0.5 (* -0.041666666666666664 (* x x)))
   (* (+ (cos x) -1.0) (/ (/ -1.0 x) x))))

C

double code(double x) {
	double tmp;
	if (x <= 0.0058) {
		tmp = 0.5 + (-0.041666666666666664 * (x * x));
	} else {
		tmp = (cos(x) + -1.0) * ((-1.0 / x) / x);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 0.0058d0) then
        tmp = 0.5d0 + ((-0.041666666666666664d0) * (x * x))
    else
        tmp = (cos(x) + (-1.0d0)) * (((-1.0d0) / x) / x)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 0.0058) {
		tmp = 0.5 + (-0.041666666666666664 * (x * x));
	} else {
		tmp = (Math.cos(x) + -1.0) * ((-1.0 / x) / x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 0.0058:
		tmp = 0.5 + (-0.041666666666666664 * (x * x))
	else:
		tmp = (math.cos(x) + -1.0) * ((-1.0 / x) / x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 0.0058)
		tmp = Float64(0.5 + Float64(-0.041666666666666664 * Float64(x * x)));
	else
		tmp = Float64(Float64(cos(x) + -1.0) * Float64(Float64(-1.0 / x) / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 0.0058)
		tmp = 0.5 + (-0.041666666666666664 * (x * x));
	else
		tmp = (cos(x) + -1.0) * ((-1.0 / x) / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 0.0058], N[(0.5 + N[(-0.041666666666666664 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Cos[x], $MachinePrecision] + -1.0), $MachinePrecision] * N[(N[(-1.0 / x), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.0058

   1. Initial program 36.6%

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

   2. Taylor expanded in x around 0 64.8%

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

      1. unpow2 64.8%

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

   4. Simplified 64.8%

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

   if 0.0058 < x

   1. Initial program 98.5%

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

      1. frac-2neg 98.5%

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

      2. div-inv 98.4%

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

      3. distribute-rgt-neg-in 98.4%

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

   3. Applied egg-rr 98.4%

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

   4. Taylor expanded in x around 0 98.4%

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

      1. unpow2 98.4%

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

      2. associate-/r* 99.5%

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

   6. Simplified 99.5%

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

4. Final simplification 73.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.0058:\\ \;\;\;\;0.5 + -0.041666666666666664 \cdot \left(x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\cos x + -1\right) \cdot \frac{\frac{-1}{x}}{x}\\ \end{array} \]

Alternative 4: 74.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.0058:\\ \;\;\;\;0.5 + -0.041666666666666664 \cdot \left(x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \cos x}{x \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 0.0058)
   (+ 0.5 (* -0.041666666666666664 (* x x)))
   (/ (- 1.0 (cos x)) (* x x))))

C

double code(double x) {
	double tmp;
	if (x <= 0.0058) {
		tmp = 0.5 + (-0.041666666666666664 * (x * x));
	} 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 <= 0.0058d0) then
        tmp = 0.5d0 + ((-0.041666666666666664d0) * (x * x))
    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 <= 0.0058) {
		tmp = 0.5 + (-0.041666666666666664 * (x * x));
	} else {
		tmp = (1.0 - Math.cos(x)) / (x * x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 0.0058:
		tmp = 0.5 + (-0.041666666666666664 * (x * x))
	else:
		tmp = (1.0 - math.cos(x)) / (x * x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 0.0058)
		tmp = Float64(0.5 + Float64(-0.041666666666666664 * Float64(x * x)));
	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 <= 0.0058)
		tmp = 0.5 + (-0.041666666666666664 * (x * x));
	else
		tmp = (1.0 - cos(x)) / (x * x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 0.0058], N[(0.5 + N[(-0.041666666666666664 * N[(x * x), $MachinePrecision]), $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 < 0.0058

   1. Initial program 36.6%

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

   2. Taylor expanded in x around 0 64.8%

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

      1. unpow2 64.8%

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

   4. Simplified 64.8%

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

   if 0.0058 < x

   1. Initial program 98.5%

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

4. Final simplification 73.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.0058:\\ \;\;\;\;0.5 + -0.041666666666666664 \cdot \left(x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \cos x}{x \cdot x}\\ \end{array} \]

Alternative 5: 78.0% 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 52.6%

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

   1. frac-2neg 52.6%

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

   2. div-inv 52.5%

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

   3. distribute-rgt-neg-in 52.5%

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

3. Applied egg-rr 52.5%

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

   1. associate-/r* 53.3%

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

   2. associate-*r/ 54.0%

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

   3. distribute-lft-neg-in 54.0%

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

   4. div-inv 54.0%

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

   5. frac-2neg 54.0%

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

   6. clear-num 53.3%

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

   7. frac-2neg 53.3%

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

   8. metadata-eval 53.3%

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

   9. frac-2neg 53.3%

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

   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 40.9%

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

   12. sqr-neg 40.9%

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

   13. sqrt-prod 15.1%

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

   14. add-sqr-sqrt 29.1%

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

   15. distribute-frac-neg 29.1%

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

   16. frac-2neg 29.1%

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

5. Applied egg-rr 53.3%

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

   1. associate-/r/ 53.3%

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

7. Applied egg-rr 53.3%

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

8. Taylor expanded in x around 0 78.2%

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

   1. *-commutative 78.2%

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

   2. associate-*r/ 78.2%

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

   3. metadata-eval 78.2%

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

10. Simplified 78.2%

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

11. Final simplification 78.2%

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

Alternative 6: 51.7% 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 52.6%

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

2. Taylor expanded in x around 0 49.6%

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

3. Final simplification 49.6%

   \[\leadsto 0.5 \]

Reproduce

herbie shell --seed 2023297 
(FPCore (x)
  :name "cos2 (problem 3.4.1)"
  :precision binary64
  (/ (- 1.0 (cos x)) (* x x)))