cos2 (problem 3.4.1)

Percentage Accurate: 50.3% → 99.6%

Time: 11.3s

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: 50.3% 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: 99.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \sin x \cdot \left(\frac{1}{x} \cdot \frac{\tan \left(x \cdot 0.5\right)}{x}\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (* (sin x) (* (/ 1.0 x) (/ (tan (* x 0.5)) x))))

C

double code(double x) {
	return sin(x) * ((1.0 / x) * (tan((x * 0.5)) / x));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = sin(x) * ((1.0d0 / x) * (tan((x * 0.5d0)) / x))
end function

Java

public static double code(double x) {
	return Math.sin(x) * ((1.0 / x) * (Math.tan((x * 0.5)) / x));
}

Python

def code(x):
	return math.sin(x) * ((1.0 / x) * (math.tan((x * 0.5)) / x))

Julia

function code(x)
	return Float64(sin(x) * Float64(Float64(1.0 / x) * Float64(tan(Float64(x * 0.5)) / x)))
end

MATLAB

function tmp = code(x)
	tmp = sin(x) * ((1.0 / x) * (tan((x * 0.5)) / x));
end

Wolfram

code[x_] := N[(N[Sin[x], $MachinePrecision] * N[(N[(1.0 / x), $MachinePrecision] * N[(N[Tan[N[(x * 0.5), $MachinePrecision]], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 49.1%

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

   1. flip-- 49.0%

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

   2. div-inv 49.0%

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

   3. metadata-eval 49.0%

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

   4. pow2 49.0%

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

4. Applied egg-rr 49.0%

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

   1. associate-*r/ 49.0%

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

   2. *-rgt-identity 49.0%

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

6. Simplified 49.0%

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

7. Taylor expanded in x around inf 49.0%

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

   1. rem-square-sqrt 49.0%

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

   2. unpow2 49.0%

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

   3. unpow2 49.0%

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

   4. 1-sub-cos 76.7%

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

   5. rem-sqrt-square 76.7%

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

9. Simplified 76.7%

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

    1. div-inv 74.7%

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

    2. unpow2 74.7%

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

    3. associate-*l* 75.3%

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

    4. add-sqr-sqrt 34.4%

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

    5. fabs-sqr 34.4%

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

    6. add-sqr-sqrt 47.5%

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

    7. add-sqr-sqrt 34.4%

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

    8. fabs-sqr 34.4%

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

    9. add-sqr-sqrt 75.3%

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

    10. pow2 75.3%

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

    11. *-commutative 75.3%

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

    12. pow2 75.3%

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

11. Applied egg-rr 75.3%

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

    1. associate-*r/ 76.6%

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

    2. *-rgt-identity 76.6%

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

    3. associate-/r* 76.7%

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

    4. hang-0p-tan 76.8%

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

13. Simplified 76.8%

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

    1. *-un-lft-identity 76.8%

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

    2. unpow2 76.8%

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

    3. times-frac 99.6%

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

    4. div-inv 99.6%

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

    5. metadata-eval 99.6%

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

15. Applied egg-rr 99.6%

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

16. Final simplification 99.6%

    \[\leadsto \sin x \cdot \left(\frac{1}{x} \cdot \frac{\tan \left(x \cdot 0.5\right)}{x}\right) \]
17. Add Preprocessing

Alternative 2: 74.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.0045:\\ \;\;\;\;0.5 + {x}^{2} \cdot -0.041666666666666664\\ \mathbf{else}:\\ \;\;\;\;{x}^{-2} \cdot \left(1 - \cos x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 0.0045)
   (+ 0.5 (* (pow x 2.0) -0.041666666666666664))
   (* (pow x -2.0) (- 1.0 (cos x)))))

C

double code(double x) {
	double tmp;
	if (x <= 0.0045) {
		tmp = 0.5 + (pow(x, 2.0) * -0.041666666666666664);
	} else {
		tmp = pow(x, -2.0) * (1.0 - cos(x));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 0.0045d0) then
        tmp = 0.5d0 + ((x ** 2.0d0) * (-0.041666666666666664d0))
    else
        tmp = (x ** (-2.0d0)) * (1.0d0 - cos(x))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 0.0045) {
		tmp = 0.5 + (Math.pow(x, 2.0) * -0.041666666666666664);
	} else {
		tmp = Math.pow(x, -2.0) * (1.0 - Math.cos(x));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 0.0045:
		tmp = 0.5 + (math.pow(x, 2.0) * -0.041666666666666664)
	else:
		tmp = math.pow(x, -2.0) * (1.0 - math.cos(x))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 0.0045)
		tmp = Float64(0.5 + Float64((x ^ 2.0) * -0.041666666666666664));
	else
		tmp = Float64((x ^ -2.0) * Float64(1.0 - cos(x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 0.0045)
		tmp = 0.5 + ((x ^ 2.0) * -0.041666666666666664);
	else
		tmp = (x ^ -2.0) * (1.0 - cos(x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 0.0045], N[(0.5 + N[(N[Power[x, 2.0], $MachinePrecision] * -0.041666666666666664), $MachinePrecision]), $MachinePrecision], N[(N[Power[x, -2.0], $MachinePrecision] * N[(1.0 - N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.00449999999999999966

   1. Initial program 30.5%

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

   3. Taylor expanded in x around 0 70.9%

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

      1. *-commutative 70.9%

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

   5. Simplified 70.9%

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

   if 0.00449999999999999966 < x

   1. Initial program 98.5%

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

      1. clear-num 98.5%

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

      2. associate-/r/ 98.5%

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

      3. pow2 98.5%

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

      4. pow-flip 99.0%

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

      5. metadata-eval 99.0%

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

   4. Applied egg-rr 99.0%

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

4. Final simplification 78.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.0045:\\ \;\;\;\;0.5 + {x}^{2} \cdot -0.041666666666666664\\ \mathbf{else}:\\ \;\;\;\;{x}^{-2} \cdot \left(1 - \cos x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 74.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.0045:\\ \;\;\;\;0.5 + {x}^{2} \cdot -0.041666666666666664\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \cos x}{x \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 0.0045)
   (+ 0.5 (* (pow x 2.0) -0.041666666666666664))
   (/ (- 1.0 (cos x)) (* x x))))

C

double code(double x) {
	double tmp;
	if (x <= 0.0045) {
		tmp = 0.5 + (pow(x, 2.0) * -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 <= 0.0045d0) then
        tmp = 0.5d0 + ((x ** 2.0d0) * (-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 <= 0.0045) {
		tmp = 0.5 + (Math.pow(x, 2.0) * -0.041666666666666664);
	} else {
		tmp = (1.0 - Math.cos(x)) / (x * x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 0.0045:
		tmp = 0.5 + (math.pow(x, 2.0) * -0.041666666666666664)
	else:
		tmp = (1.0 - math.cos(x)) / (x * x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 0.0045)
		tmp = Float64(0.5 + Float64((x ^ 2.0) * -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 <= 0.0045)
		tmp = 0.5 + ((x ^ 2.0) * -0.041666666666666664);
	else
		tmp = (1.0 - cos(x)) / (x * x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 0.0045], N[(0.5 + N[(N[Power[x, 2.0], $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 < 0.00449999999999999966

   1. Initial program 30.5%

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

   3. Taylor expanded in x around 0 70.9%

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

      1. *-commutative 70.9%

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

   5. Simplified 70.9%

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

   if 0.00449999999999999966 < x

   1. Initial program 98.5%

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

4. Final simplification 78.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.0045:\\ \;\;\;\;0.5 + {x}^{2} \cdot -0.041666666666666664\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \cos x}{x \cdot x}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 74.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.0045:\\ \;\;\;\;0.5 + {x}^{2} \cdot -0.041666666666666664\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 - \cos x}{x}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 0.0045)
   (+ 0.5 (* (pow x 2.0) -0.041666666666666664))
   (/ (/ (- 1.0 (cos x)) x) x)))

C

double code(double x) {
	double tmp;
	if (x <= 0.0045) {
		tmp = 0.5 + (pow(x, 2.0) * -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 <= 0.0045d0) then
        tmp = 0.5d0 + ((x ** 2.0d0) * (-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 <= 0.0045) {
		tmp = 0.5 + (Math.pow(x, 2.0) * -0.041666666666666664);
	} else {
		tmp = ((1.0 - Math.cos(x)) / x) / x;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 0.0045:
		tmp = 0.5 + (math.pow(x, 2.0) * -0.041666666666666664)
	else:
		tmp = ((1.0 - math.cos(x)) / x) / x
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 0.0045)
		tmp = Float64(0.5 + Float64((x ^ 2.0) * -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 <= 0.0045)
		tmp = 0.5 + ((x ^ 2.0) * -0.041666666666666664);
	else
		tmp = ((1.0 - cos(x)) / x) / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 0.0045], N[(0.5 + N[(N[Power[x, 2.0], $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 < 0.00449999999999999966

   1. Initial program 30.5%

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

   3. Taylor expanded in x around 0 70.9%

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

      1. *-commutative 70.9%

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

   5. Simplified 70.9%

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

   if 0.00449999999999999966 < x

   1. Initial program 98.5%

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

      1. associate-/r* 99.0%

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

      2. div-inv 99.0%

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

   4. Applied egg-rr 99.0%

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

      1. un-div-inv 99.0%

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

   6. Applied egg-rr 99.0%

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

4. Final simplification 78.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.0045:\\ \;\;\;\;0.5 + {x}^{2} \cdot -0.041666666666666664\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 - \cos x}{x}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 52.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \sin x \cdot \frac{0.5}{x} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 49.1%

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

   1. flip-- 49.0%

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

   2. div-inv 49.0%

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

   3. metadata-eval 49.0%

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

   4. pow2 49.0%

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

4. Applied egg-rr 49.0%

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

   1. associate-*r/ 49.0%

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

   2. *-rgt-identity 49.0%

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

6. Simplified 49.0%

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

7. Taylor expanded in x around inf 49.0%

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

   1. rem-square-sqrt 49.0%

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

   2. unpow2 49.0%

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

   3. unpow2 49.0%

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

   4. 1-sub-cos 76.7%

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

   5. rem-sqrt-square 76.7%

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

9. Simplified 76.7%

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

    1. div-inv 74.7%

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

    2. unpow2 74.7%

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

    3. associate-*l* 75.3%

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

    4. add-sqr-sqrt 34.4%

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

    5. fabs-sqr 34.4%

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

    6. add-sqr-sqrt 47.5%

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

    7. add-sqr-sqrt 34.4%

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

    8. fabs-sqr 34.4%

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

    9. add-sqr-sqrt 75.3%

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

    10. pow2 75.3%

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

    11. *-commutative 75.3%

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

    12. pow2 75.3%

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

11. Applied egg-rr 75.3%

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

    1. associate-*r/ 76.6%

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

    2. *-rgt-identity 76.6%

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

    3. associate-/r* 76.7%

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

    4. hang-0p-tan 76.8%

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

13. Simplified 76.8%

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

14. Taylor expanded in x around 0 54.0%

    \[\leadsto \sin x \cdot \color{blue}{\frac{0.5}{x}} \]

15. Final simplification 54.0%

    \[\leadsto \sin x \cdot \frac{0.5}{x} \]
16. Add Preprocessing

Alternative 6: 64.0% accurate, 10.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 8.2 \cdot 10^{+76}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{0}{x \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (if (<= x 8.2e+76) 0.5 (/ 0.0 (* x x))))

C

double code(double x) {
	double tmp;
	if (x <= 8.2e+76) {
		tmp = 0.5;
	} else {
		tmp = 0.0 / (x * x);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 8.2d+76) then
        tmp = 0.5d0
    else
        tmp = 0.0d0 / (x * x)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 8.2e+76) {
		tmp = 0.5;
	} else {
		tmp = 0.0 / (x * x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 8.2e+76:
		tmp = 0.5
	else:
		tmp = 0.0 / (x * x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 8.2e+76)
		tmp = 0.5;
	else
		tmp = Float64(0.0 / Float64(x * x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 8.2e+76)
		tmp = 0.5;
	else
		tmp = 0.0 / (x * x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 8.2e+76], 0.5, N[(0.0 / N[(x * x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 8.1999999999999997e76

   1. Initial program 36.1%

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

   3. Taylor expanded in x around 0 66.1%

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

   if 8.1999999999999997e76 < x

   1. Initial program 98.9%

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

      1. add-log-exp 98.8%

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

   4. Applied egg-rr 98.8%

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

   5. Taylor expanded in x around 0 78.3%

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

4. Final simplification 68.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 8.2 \cdot 10^{+76}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{0}{x \cdot x}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 52.2% 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 49.1%

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

3. Taylor expanded in x around 0 53.1%

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

4. Final simplification 53.1%

   \[\leadsto 0.5 \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2024100 
(FPCore (x)
  :name "cos2 (problem 3.4.1)"
  :precision binary64
  (/ (- 1.0 (cos x)) (* x x)))