cos2 (problem 3.4.1)

Percentage Accurate: 50.9% → 99.8%

Time: 11.8s

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: 50.9% 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

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 53.0%

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

   1. flip-- 52.8%

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

   2. div-inv 52.8%

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

   3. metadata-eval 52.8%

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

   4. pow2 52.8%

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

4. Applied egg-rr 52.8%

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

   1. associate-*r/ 52.8%

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

   2. *-rgt-identity 52.8%

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

6. Simplified 52.8%

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

   1. unpow2 52.8%

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

   2. 1-sub-cos 76.2%

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

8. Applied egg-rr 76.2%

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

   1. associate-/l* 76.2%

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

   2. times-frac 99.6%

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

   3. hang-0p-tan 99.8%

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

10. Applied egg-rr 99.8%

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

    1. *-commutative 99.8%

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

    2. clear-num 99.8%

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

    3. un-div-inv 99.8%

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

    4. div-inv 99.8%

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

    5. metadata-eval 99.8%

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

12. Applied egg-rr 99.8%

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

13. Final simplification 99.8%

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

Alternative 2: 99.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \frac{\sin x}{x} \cdot \frac{\tan \left(\frac{x}{2}\right)}{x} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 53.0%

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

   1. flip-- 52.8%

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

   2. div-inv 52.8%

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

   3. metadata-eval 52.8%

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

   4. pow2 52.8%

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

4. Applied egg-rr 52.8%

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

   1. associate-*r/ 52.8%

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

   2. *-rgt-identity 52.8%

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

6. Simplified 52.8%

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

   1. unpow2 52.8%

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

   2. 1-sub-cos 76.2%

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

8. Applied egg-rr 76.2%

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

   1. associate-/l* 76.2%

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

   2. times-frac 99.6%

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

   3. hang-0p-tan 99.8%

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

10. Applied egg-rr 99.8%

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

11. Final simplification 99.8%

    \[\leadsto \frac{\sin x}{x} \cdot \frac{\tan \left(\frac{x}{2}\right)}{x} \]
12. Add Preprocessing

Alternative 3: 74.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.0058:\\ \;\;\;\;0.5 + -0.041666666666666664 \cdot {x}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\cos x + -1}{x}}{-x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 0.0058)
   (+ 0.5 (* -0.041666666666666664 (pow x 2.0)))
   (/ (/ (+ (cos x) -1.0) x) (- x))))

C

double code(double x) {
	double tmp;
	if (x <= 0.0058) {
		tmp = 0.5 + (-0.041666666666666664 * pow(x, 2.0));
	} else {
		tmp = ((cos(x) + -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 ** 2.0d0))
    else
        tmp = ((cos(x) + (-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 * Math.pow(x, 2.0));
	} else {
		tmp = ((Math.cos(x) + -1.0) / x) / -x;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 0.0058:
		tmp = 0.5 + (-0.041666666666666664 * math.pow(x, 2.0))
	else:
		tmp = ((math.cos(x) + -1.0) / x) / -x
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 0.0058)
		tmp = Float64(0.5 + Float64(-0.041666666666666664 * (x ^ 2.0)));
	else
		tmp = Float64(Float64(Float64(cos(x) + -1.0) / x) / Float64(-x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 0.0058)
		tmp = 0.5 + (-0.041666666666666664 * (x ^ 2.0));
	else
		tmp = ((cos(x) + -1.0) / x) / -x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 0.0058], N[(0.5 + N[(-0.041666666666666664 * N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[Cos[x], $MachinePrecision] + -1.0), $MachinePrecision] / x), $MachinePrecision] / (-x)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.0058

   1. Initial program 35.3%

      \[\frac{1 - \cos x}{x \cdot x} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 66.5%

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

   if 0.0058 < x

   1. Initial program 99.0%

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

      1. associate-/r* 99.1%

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

      2. div-inv 99.1%

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

   4. Applied egg-rr 99.1%

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

      1. div-sub 98.8%

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

      2. sub-neg 98.8%

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

   6. Applied egg-rr 98.8%

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

      1. sub-neg 98.8%

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

   8. Simplified 98.8%

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

      1. un-div-inv 98.8%

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

      2. frac-2neg 98.8%

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

   10. Applied egg-rr 99.1%

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

4. Final simplification 75.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.0058:\\ \;\;\;\;0.5 + -0.041666666666666664 \cdot {x}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\cos x + -1}{x}}{-x}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 74.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.0058:\\ \;\;\;\;0.5 + -0.041666666666666664 \cdot {x}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \cos x}{x \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 0.0058)
   (+ 0.5 (* -0.041666666666666664 (pow x 2.0)))
   (/ (- 1.0 (cos x)) (* x x))))

C

double code(double x) {
	double tmp;
	if (x <= 0.0058) {
		tmp = 0.5 + (-0.041666666666666664 * pow(x, 2.0));
	} 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 ** 2.0d0))
    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 * Math.pow(x, 2.0));
	} 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 * math.pow(x, 2.0))
	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 * (x ^ 2.0)));
	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 ^ 2.0));
	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[Power[x, 2.0], $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 35.3%

      \[\frac{1 - \cos x}{x \cdot x} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 66.5%

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

   if 0.0058 < x

   1. Initial program 99.0%

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

4. Final simplification 75.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.0058:\\ \;\;\;\;0.5 + -0.041666666666666664 \cdot {x}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \cos x}{x \cdot x}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 63.8% accurate, 6.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 9.2 \cdot 10^{+76}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x} \cdot \left(\frac{1}{x} - \frac{1}{x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 9.2e+76) 0.5 (* (/ 1.0 x) (- (/ 1.0 x) (/ 1.0 x)))))

C

double code(double x) {
	double tmp;
	if (x <= 9.2e+76) {
		tmp = 0.5;
	} else {
		tmp = (1.0 / x) * ((1.0 / x) - (1.0 / x));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 9.2d+76) then
        tmp = 0.5d0
    else
        tmp = (1.0d0 / x) * ((1.0d0 / x) - (1.0d0 / x))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 9.2e+76) {
		tmp = 0.5;
	} else {
		tmp = (1.0 / x) * ((1.0 / x) - (1.0 / x));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 9.2e+76:
		tmp = 0.5
	else:
		tmp = (1.0 / x) * ((1.0 / x) - (1.0 / x))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 9.2e+76)
		tmp = 0.5;
	else
		tmp = Float64(Float64(1.0 / x) * Float64(Float64(1.0 / x) - Float64(1.0 / x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 9.2e+76)
		tmp = 0.5;
	else
		tmp = (1.0 / x) * ((1.0 / x) - (1.0 / x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 9.2e+76], 0.5, N[(N[(1.0 / x), $MachinePrecision] * N[(N[(1.0 / x), $MachinePrecision] - N[(1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 9.20000000000000005e76

   1. Initial program 40.8%

      \[\frac{1 - \cos x}{x \cdot x} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 61.6%

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

   if 9.20000000000000005e76 < x

   1. Initial program 99.5%

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

      1. associate-/r* 99.6%

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

      2. div-inv 99.6%

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

   4. Applied egg-rr 99.6%

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

      1. div-sub 99.6%

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

      2. sub-neg 99.6%

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

   6. Applied egg-rr 99.6%

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

      1. sub-neg 99.6%

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

   8. Simplified 99.6%

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

   9. Taylor expanded in x around 0 73.5%

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

4. Final simplification 64.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 9.2 \cdot 10^{+76}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x} \cdot \left(\frac{1}{x} - \frac{1}{x}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 51.5% 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 53.0%

   \[\frac{1 - \cos x}{x \cdot x} \]
2. Add Preprocessing

3. Taylor expanded in x around 0 49.5%

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

4. Final simplification 49.5%

   \[\leadsto 0.5 \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2024044 
(FPCore (x)
  :name "cos2 (problem 3.4.1)"
  :precision binary64
  (/ (- 1.0 (cos x)) (* x x)))