cos2 (problem 3.4.1)

Percentage Accurate: 50.6% → 99.8%

Time: 9.0s

Alternatives: 8

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: 50.6% 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

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 53.4%

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

   1. flip-- 53.2%

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

   2. div-inv 53.2%

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

   3. metadata-eval 53.2%

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

   4. 1-sub-cos 77.5%

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

   5. pow2 77.5%

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

3. Applied egg-rr 77.5%

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

   1. unpow2 77.5%

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

   2. associate-*l* 77.4%

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

   3. associate-*r/ 77.5%

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

   4. *-rgt-identity 77.5%

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

   5. hang-0p-tan 77.7%

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

5. Simplified 77.7%

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

   1. associate-/r* 78.4%

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

   2. div-inv 78.3%

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

   3. associate-/l* 99.7%

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

   4. div-inv 99.7%

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

   5. metadata-eval 99.7%

      \[\leadsto \frac{\sin x}{\frac{x}{\tan \left(x \cdot \color{blue}{0.5}\right)}} \cdot \frac{1}{x} \]

7. Applied egg-rr 99.7%

   \[\leadsto \color{blue}{\frac{\sin x}{\frac{x}{\tan \left(x \cdot 0.5\right)}} \cdot \frac{1}{x}} \]
8. Step-by-step derivation

   1. associate-*l/ 99.7%

      \[\leadsto \color{blue}{\frac{\sin x \cdot \frac{1}{x}}{\frac{x}{\tan \left(x \cdot 0.5\right)}}} \]

   2. div-inv 99.8%

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

9. Applied egg-rr 99.8%

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

10. Final simplification 99.8%

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

Alternative 2: 75.5% accurate, 0.5× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 5e-5)
   (+ 0.5 (* (* x x) -0.041666666666666664))
   (* (tan (* x 0.5)) (/ (sin x) (* x x)))))

C

double code(double x) {
	double tmp;
	if (x <= 5e-5) {
		tmp = 0.5 + ((x * x) * -0.041666666666666664);
	} else {
		tmp = tan((x * 0.5)) * (sin(x) / (x * x));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 5d-5) then
        tmp = 0.5d0 + ((x * x) * (-0.041666666666666664d0))
    else
        tmp = tan((x * 0.5d0)) * (sin(x) / (x * x))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 5e-5) {
		tmp = 0.5 + ((x * x) * -0.041666666666666664);
	} else {
		tmp = Math.tan((x * 0.5)) * (Math.sin(x) / (x * x));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 5e-5:
		tmp = 0.5 + ((x * x) * -0.041666666666666664)
	else:
		tmp = math.tan((x * 0.5)) * (math.sin(x) / (x * x))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 5e-5)
		tmp = Float64(0.5 + Float64(Float64(x * x) * -0.041666666666666664));
	else
		tmp = Float64(tan(Float64(x * 0.5)) * Float64(sin(x) / Float64(x * x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 5e-5)
		tmp = 0.5 + ((x * x) * -0.041666666666666664);
	else
		tmp = tan((x * 0.5)) * (sin(x) / (x * x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 5e-5], N[(0.5 + N[(N[(x * x), $MachinePrecision] * -0.041666666666666664), $MachinePrecision]), $MachinePrecision], N[(N[Tan[N[(x * 0.5), $MachinePrecision]], $MachinePrecision] * N[(N[Sin[x], $MachinePrecision] / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 5.00000000000000024e-5

   1. Initial program 37.9%

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

   2. Taylor expanded in x around 0 64.1%

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

      1. *-commutative 64.1%

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

      2. unpow2 64.1%

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

   4. Simplified 64.1%

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

   if 5.00000000000000024e-5 < x

   1. Initial program 98.8%

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

      1. flip-- 98.3%

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

      2. div-inv 98.3%

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

      3. metadata-eval 98.3%

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

      4. 1-sub-cos 99.2%

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

      5. pow2 99.2%

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

   3. Applied egg-rr 99.2%

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

      1. unpow2 99.2%

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

      2. associate-*l* 99.1%

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

      3. associate-*r/ 99.1%

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

      4. *-rgt-identity 99.1%

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

      5. hang-0p-tan 99.7%

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

   5. Simplified 99.7%

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

      1. associate-/l* 99.6%

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

      2. associate-/r/ 99.7%

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

      3. div-inv 99.7%

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

      4. metadata-eval 99.7%

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

   7. Applied egg-rr 99.7%

      \[\leadsto \color{blue}{\frac{\sin x}{x \cdot x} \cdot \tan \left(x \cdot 0.5\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 73.2%

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

Alternative 3: 75.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.0044:\\ \;\;\;\;0.5 + \left(x \cdot x\right) \cdot -0.041666666666666664\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x} \cdot \frac{1 - \cos x}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 0.0044)
   (+ 0.5 (* (* x x) -0.041666666666666664))
   (* (/ 1.0 x) (/ (- 1.0 (cos x)) x))))

C

double code(double x) {
	double tmp;
	if (x <= 0.0044) {
		tmp = 0.5 + ((x * x) * -0.041666666666666664);
	} else {
		tmp = (1.0 / x) * ((1.0 - cos(x)) / x);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 0.0044d0) then
        tmp = 0.5d0 + ((x * x) * (-0.041666666666666664d0))
    else
        tmp = (1.0d0 / x) * ((1.0d0 - cos(x)) / x)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 0.0044) {
		tmp = 0.5 + ((x * x) * -0.041666666666666664);
	} else {
		tmp = (1.0 / x) * ((1.0 - Math.cos(x)) / x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 0.0044:
		tmp = 0.5 + ((x * x) * -0.041666666666666664)
	else:
		tmp = (1.0 / x) * ((1.0 - math.cos(x)) / x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 0.0044)
		tmp = Float64(0.5 + Float64(Float64(x * x) * -0.041666666666666664));
	else
		tmp = Float64(Float64(1.0 / x) * Float64(Float64(1.0 - cos(x)) / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 0.0044)
		tmp = 0.5 + ((x * x) * -0.041666666666666664);
	else
		tmp = (1.0 / x) * ((1.0 - cos(x)) / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 0.0044], N[(0.5 + N[(N[(x * x), $MachinePrecision] * -0.041666666666666664), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / x), $MachinePrecision] * N[(N[(1.0 - N[Cos[x], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.00440000000000000027

   1. Initial program 37.9%

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

   2. Taylor expanded in x around 0 64.1%

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

      1. *-commutative 64.1%

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

      2. unpow2 64.1%

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

   4. Simplified 64.1%

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

   if 0.00440000000000000027 < x

   1. Initial program 98.8%

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

      1. associate-/r* 98.7%

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

      2. div-inv 98.8%

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

   3. Applied egg-rr 98.8%

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

4. Final simplification 72.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.0044:\\ \;\;\;\;0.5 + \left(x \cdot x\right) \cdot -0.041666666666666664\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x} \cdot \frac{1 - \cos x}{x}\\ \end{array} \]

Alternative 4: 75.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.0044:\\ \;\;\;\;0.5 + \left(x \cdot x\right) \cdot -0.041666666666666664\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x}}{\frac{x}{1 - \cos x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 0.0044)
   (+ 0.5 (* (* x x) -0.041666666666666664))
   (/ (/ 1.0 x) (/ x (- 1.0 (cos x))))))

C

double code(double x) {
	double tmp;
	if (x <= 0.0044) {
		tmp = 0.5 + ((x * x) * -0.041666666666666664);
	} else {
		tmp = (1.0 / x) / (x / (1.0 - cos(x)));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 0.0044d0) then
        tmp = 0.5d0 + ((x * x) * (-0.041666666666666664d0))
    else
        tmp = (1.0d0 / x) / (x / (1.0d0 - cos(x)))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 0.0044) {
		tmp = 0.5 + ((x * x) * -0.041666666666666664);
	} else {
		tmp = (1.0 / x) / (x / (1.0 - Math.cos(x)));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 0.0044:
		tmp = 0.5 + ((x * x) * -0.041666666666666664)
	else:
		tmp = (1.0 / x) / (x / (1.0 - math.cos(x)))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 0.0044)
		tmp = Float64(0.5 + Float64(Float64(x * x) * -0.041666666666666664));
	else
		tmp = Float64(Float64(1.0 / x) / Float64(x / Float64(1.0 - cos(x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 0.0044)
		tmp = 0.5 + ((x * x) * -0.041666666666666664);
	else
		tmp = (1.0 / x) / (x / (1.0 - cos(x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 0.0044], N[(0.5 + N[(N[(x * x), $MachinePrecision] * -0.041666666666666664), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / x), $MachinePrecision] / N[(x / N[(1.0 - N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.00440000000000000027

   1. Initial program 37.9%

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

   2. Taylor expanded in x around 0 64.1%

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

      1. *-commutative 64.1%

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

      2. unpow2 64.1%

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

   4. Simplified 64.1%

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

   if 0.00440000000000000027 < x

   1. Initial program 98.8%

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

      1. associate-/r* 98.7%

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

      2. div-inv 98.8%

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

   3. Applied egg-rr 98.8%

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

      1. *-commutative 98.8%

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

      2. clear-num 98.8%

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

      3. un-div-inv 98.7%

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

   5. Applied egg-rr 98.7%

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

4. Final simplification 72.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.0044:\\ \;\;\;\;0.5 + \left(x \cdot x\right) \cdot -0.041666666666666664\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x}}{\frac{x}{1 - \cos x}}\\ \end{array} \]

Alternative 5: 75.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.0044:\\ \;\;\;\;0.5 + \left(x \cdot x\right) \cdot -0.041666666666666664\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \cos x}{x \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 0.0044)
   (+ 0.5 (* (* x x) -0.041666666666666664))
   (/ (- 1.0 (cos x)) (* x x))))

C

double code(double x) {
	double tmp;
	if (x <= 0.0044) {
		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 <= 0.0044d0) 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 <= 0.0044) {
		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 <= 0.0044:
		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 <= 0.0044)
		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 <= 0.0044)
		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, 0.0044], 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 < 0.00440000000000000027

   1. Initial program 37.9%

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

   2. Taylor expanded in x around 0 64.1%

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

      1. *-commutative 64.1%

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

      2. unpow2 64.1%

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

   4. Simplified 64.1%

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

   if 0.00440000000000000027 < x

   1. Initial program 98.8%

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

4. Final simplification 72.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.0044:\\ \;\;\;\;0.5 + \left(x \cdot x\right) \cdot -0.041666666666666664\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \cos x}{x \cdot x}\\ \end{array} \]

Alternative 6: 75.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.0044:\\ \;\;\;\;0.5 + \left(x \cdot x\right) \cdot -0.041666666666666664\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 - \cos x}{x}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 0.0044)
   (+ 0.5 (* (* x x) -0.041666666666666664))
   (/ (/ (- 1.0 (cos x)) x) x)))

C

double code(double x) {
	double tmp;
	if (x <= 0.0044) {
		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 <= 0.0044d0) 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 <= 0.0044) {
		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 <= 0.0044:
		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 <= 0.0044)
		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 <= 0.0044)
		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, 0.0044], 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 < 0.00440000000000000027

   1. Initial program 37.9%

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

   2. Taylor expanded in x around 0 64.1%

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

      1. *-commutative 64.1%

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

      2. unpow2 64.1%

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

   4. Simplified 64.1%

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

   if 0.00440000000000000027 < x

   1. Initial program 98.8%

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

      1. associate-/r* 98.7%

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

      2. div-inv 98.8%

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

   3. Applied egg-rr 98.8%

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

      1. un-div-inv 98.7%

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

   5. Applied egg-rr 98.7%

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

4. Final simplification 72.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.0044:\\ \;\;\;\;0.5 + \left(x \cdot x\right) \cdot -0.041666666666666664\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 - \cos x}{x}}{x}\\ \end{array} \]

Alternative 7: 52.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{\sin x}{x}}{2} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ (/ (sin x) x) 2.0))

C

double code(double x) {
	return (sin(x) / x) / 2.0;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (sin(x) / x) / 2.0d0
end function

Java

public static double code(double x) {
	return (Math.sin(x) / x) / 2.0;
}

Python

def code(x):
	return (math.sin(x) / x) / 2.0

Julia

function code(x)
	return Float64(Float64(sin(x) / x) / 2.0)
end

MATLAB

function tmp = code(x)
	tmp = (sin(x) / x) / 2.0;
end

Wolfram

code[x_] := N[(N[(N[Sin[x], $MachinePrecision] / x), $MachinePrecision] / 2.0), $MachinePrecision]

Derivation

1. Initial program 53.4%

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

   1. flip-- 53.2%

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

   2. div-inv 53.2%

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

   3. metadata-eval 53.2%

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

   4. 1-sub-cos 77.5%

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

   5. pow2 77.5%

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

3. Applied egg-rr 77.5%

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

   1. unpow2 77.5%

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

   2. associate-*l* 77.4%

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

   3. associate-*r/ 77.5%

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

   4. *-rgt-identity 77.5%

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

   5. hang-0p-tan 77.7%

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

5. Simplified 77.7%

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

   1. associate-/r* 78.4%

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

   2. div-inv 78.3%

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

   3. associate-/l* 99.7%

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

   4. div-inv 99.7%

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

   5. metadata-eval 99.7%

      \[\leadsto \frac{\sin x}{\frac{x}{\tan \left(x \cdot \color{blue}{0.5}\right)}} \cdot \frac{1}{x} \]

7. Applied egg-rr 99.7%

   \[\leadsto \color{blue}{\frac{\sin x}{\frac{x}{\tan \left(x \cdot 0.5\right)}} \cdot \frac{1}{x}} \]
8. Step-by-step derivation

   1. associate-*l/ 99.7%

      \[\leadsto \color{blue}{\frac{\sin x \cdot \frac{1}{x}}{\frac{x}{\tan \left(x \cdot 0.5\right)}}} \]

   2. div-inv 99.8%

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

9. Applied egg-rr 99.8%

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

10. Taylor expanded in x around 0 50.1%

    \[\leadsto \frac{\frac{\sin x}{x}}{\color{blue}{2}} \]

11. Final simplification 50.1%

    \[\leadsto \frac{\frac{\sin x}{x}}{2} \]

Alternative 8: 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 53.4%

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

2. Taylor expanded in x around 0 49.2%

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

3. Final simplification 49.2%

   \[\leadsto 0.5 \]

Reproduce

herbie shell --seed 2023223 
(FPCore (x)
  :name "cos2 (problem 3.4.1)"
  :precision binary64
  (/ (- 1.0 (cos x)) (* x x)))