cos2 (problem 3.4.1)

Percentage Accurate: 50.9% → 99.8%

Time: 9.7s

Alternatives: 7

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

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 49.6%

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

   1. associate-/r* 50.5%

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

   2. div-inv 50.5%

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

3. Applied egg-rr 50.5%

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

   1. div-sub 50.5%

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

   2. sub-neg 50.5%

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

5. Applied egg-rr 50.5%

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

   1. sub-neg 50.5%

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

7. Simplified 50.5%

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

   1. sub-div 50.5%

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

   2. flip-- 50.3%

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

   3. metadata-eval 50.3%

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

   4. 1-sub-cos 74.0%

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

   5. div-inv 74.0%

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

   6. *-un-lft-identity 74.0%

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

   7. times-frac 73.9%

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

   8. pow2 73.9%

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

9. Applied egg-rr 73.9%

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

    1. /-rgt-identity 73.9%

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

    2. associate-*r/ 74.0%

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

    3. associate-*r/ 74.0%

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

    4. *-rgt-identity 74.0%

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

    5. unpow2 74.0%

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

    6. associate-*r/ 74.0%

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

    7. hang-0p-tan 74.4%

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

11. Simplified 74.4%

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

    1. un-div-inv 74.5%

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

    2. associate-/l* 99.8%

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

    3. div-inv 99.8%

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

    4. metadata-eval 99.8%

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

13. Applied egg-rr 99.8%

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

14. Final simplification 99.8%

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

Alternative 2: 99.5% accurate, 0.5× speedup

Math

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

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(sin(x) / Float64(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[Sin[x], $MachinePrecision] / N[(x * N[(x / N[Tan[N[(x * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 49.6%

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

   1. associate-/r* 50.5%

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

   2. div-inv 50.5%

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

3. Applied egg-rr 50.5%

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

   1. div-sub 50.5%

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

   2. sub-neg 50.5%

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

5. Applied egg-rr 50.5%

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

   1. sub-neg 50.5%

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

7. Simplified 50.5%

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

   1. sub-div 50.5%

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

   2. flip-- 50.3%

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

   3. metadata-eval 50.3%

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

   4. 1-sub-cos 74.0%

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

   5. div-inv 74.0%

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

   6. *-un-lft-identity 74.0%

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

   7. times-frac 73.9%

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

   8. pow2 73.9%

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

9. Applied egg-rr 73.9%

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

    1. /-rgt-identity 73.9%

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

    2. associate-*r/ 74.0%

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

    3. associate-*r/ 74.0%

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

    4. *-rgt-identity 74.0%

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

    5. unpow2 74.0%

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

    6. associate-*r/ 74.0%

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

    7. hang-0p-tan 74.4%

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

11. Simplified 74.4%

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

    1. *-commutative 74.4%

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

    2. associate-/l* 99.7%

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

    3. frac-times 99.7%

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

    4. *-un-lft-identity 99.7%

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

    5. div-inv 99.7%

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

    6. metadata-eval 99.7%

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

13. Applied egg-rr 99.7%

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

14. Final simplification 99.7%

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

Alternative 3: 74.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.005:\\ \;\;\;\;0.5 + -0.041666666666666664 \cdot {x}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x} \cdot \left(\cos x + -1\right)}{-x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 0.005)
   (+ 0.5 (* -0.041666666666666664 (pow x 2.0)))
   (/ (* (/ 1.0 x) (+ (cos x) -1.0)) (- x))))

C

double code(double x) {
	double tmp;
	if (x <= 0.005) {
		tmp = 0.5 + (-0.041666666666666664 * pow(x, 2.0));
	} else {
		tmp = ((1.0 / x) * (cos(x) + -1.0)) / -x;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 0.005d0) then
        tmp = 0.5d0 + ((-0.041666666666666664d0) * (x ** 2.0d0))
    else
        tmp = ((1.0d0 / x) * (cos(x) + (-1.0d0))) / -x
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 0.005) {
		tmp = 0.5 + (-0.041666666666666664 * Math.pow(x, 2.0));
	} else {
		tmp = ((1.0 / x) * (Math.cos(x) + -1.0)) / -x;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 0.005:
		tmp = 0.5 + (-0.041666666666666664 * math.pow(x, 2.0))
	else:
		tmp = ((1.0 / x) * (math.cos(x) + -1.0)) / -x
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 0.005)
		tmp = Float64(0.5 + Float64(-0.041666666666666664 * (x ^ 2.0)));
	else
		tmp = Float64(Float64(Float64(1.0 / x) * Float64(cos(x) + -1.0)) / Float64(-x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 0.005)
		tmp = 0.5 + (-0.041666666666666664 * (x ^ 2.0));
	else
		tmp = ((1.0 / x) * (cos(x) + -1.0)) / -x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 0.005], N[(0.5 + N[(-0.041666666666666664 * N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 / x), $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision] / (-x)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.0050000000000000001

   1. Initial program 31.2%

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

   2. Taylor expanded in x around 0 70.4%

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

   if 0.0050000000000000001 < x

   1. Initial program 98.6%

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

      1. associate-/r* 99.1%

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

      2. div-inv 99.1%

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

   3. Applied egg-rr 99.1%

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

      1. *-commutative 99.1%

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

      2. frac-2neg 99.1%

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

      3. associate-*r/ 99.1%

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

   5. Applied egg-rr 99.1%

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

4. Final simplification 78.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.005:\\ \;\;\;\;0.5 + -0.041666666666666664 \cdot {x}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x} \cdot \left(\cos x + -1\right)}{-x}\\ \end{array} \]

Alternative 4: 74.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.005:\\ \;\;\;\;0.5 + -0.041666666666666664 \cdot {x}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \cos x}{x \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 0.005)
   (+ 0.5 (* -0.041666666666666664 (pow x 2.0)))
   (/ (- 1.0 (cos x)) (* x x))))

C

double code(double x) {
	double tmp;
	if (x <= 0.005) {
		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 <= 0.005d0) 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 <= 0.005) {
		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 <= 0.005:
		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 <= 0.005)
		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 <= 0.005)
		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, 0.005], 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 < 0.0050000000000000001

   1. Initial program 31.2%

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

   2. Taylor expanded in x around 0 70.4%

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

   if 0.0050000000000000001 < x

   1. Initial program 98.6%

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

4. Final simplification 78.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.005:\\ \;\;\;\;0.5 + -0.041666666666666664 \cdot {x}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \cos x}{x \cdot x}\\ \end{array} \]

Alternative 5: 74.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.005:\\ \;\;\;\;0.5 + -0.041666666666666664 \cdot {x}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 - \cos x}{x}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 0.005)
   (+ 0.5 (* -0.041666666666666664 (pow x 2.0)))
   (/ (/ (- 1.0 (cos x)) x) x)))

C

double code(double x) {
	double tmp;
	if (x <= 0.005) {
		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 <= 0.005d0) 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 <= 0.005) {
		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 <= 0.005:
		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 <= 0.005)
		tmp = Float64(0.5 + Float64(-0.041666666666666664 * (x ^ 2.0)));
	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 <= 0.005)
		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, 0.005], N[(0.5 + N[(-0.041666666666666664 * N[Power[x, 2.0], $MachinePrecision]), $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 < 0.0050000000000000001

   1. Initial program 31.2%

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

   2. Taylor expanded in x around 0 70.4%

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

   if 0.0050000000000000001 < x

   1. Initial program 98.6%

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

      1. associate-/r* 99.1%

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

      2. div-inv 99.1%

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

   3. Applied egg-rr 99.1%

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

      1. div-sub 99.1%

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

      2. sub-neg 99.1%

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

   5. Applied egg-rr 99.1%

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

      1. sub-neg 99.1%

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

   7. Simplified 99.1%

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

      1. un-div-inv 99.1%

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

      2. sub-div 99.1%

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

   9. Applied egg-rr 99.1%

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

4. Final simplification 78.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.005:\\ \;\;\;\;0.5 + -0.041666666666666664 \cdot {x}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 - \cos x}{x}}{x}\\ \end{array} \]

Alternative 6: 63.8% accurate, 8.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.3 \cdot 10^{+77}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x} \cdot \left(\frac{1}{x} + \frac{-1}{x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.3e+77) 0.5 (* (/ 1.0 x) (+ (/ 1.0 x) (/ -1.0 x)))))

C

double code(double x) {
	double tmp;
	if (x <= 1.3e+77) {
		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 <= 1.3d+77) 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 <= 1.3e+77) {
		tmp = 0.5;
	} else {
		tmp = (1.0 / x) * ((1.0 / x) + (-1.0 / x));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 1.3e+77:
		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 <= 1.3e+77)
		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 <= 1.3e+77)
		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, 1.3e+77], 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 < 1.3000000000000001e77

   1. Initial program 33.8%

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

   2. Taylor expanded in x around 0 68.5%

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

   if 1.3000000000000001e77 < x

   1. Initial program 99.0%

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

      1. associate-/r* 99.7%

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

      2. div-inv 99.6%

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

   3. Applied egg-rr 99.6%

      \[\leadsto \color{blue}{\frac{1 - \cos x}{x} \cdot \frac{1}{x}} \]
   4. 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%

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

   5. Applied egg-rr 99.6%

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

      1. sub-neg 99.6%

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

   7. Simplified 99.6%

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

   8. Taylor expanded in x around 0 69.1%

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

4. Final simplification 68.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.3 \cdot 10^{+77}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x} \cdot \left(\frac{1}{x} + \frac{-1}{x}\right)\\ \end{array} \]

Alternative 7: 51.6% 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 49.6%

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

2. Taylor expanded in x around 0 52.7%

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

3. Final simplification 52.7%

   \[\leadsto 0.5 \]

Reproduce

herbie shell --seed 2023333 
(FPCore (x)
  :name "cos2 (problem 3.4.1)"
  :precision binary64
  (/ (- 1.0 (cos x)) (* x x)))