cos2 (problem 3.4.1)

Percentage Accurate: 51.1% → 99.8%

Time: 8.3s

Alternatives: 5

Speedup: 17.8×

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: 51.1% 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

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

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

Derivation

1. Initial program 50.5%

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

   1. flip-- 50.3%

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

   2. clear-num 50.2%

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

   3. metadata-eval 50.2%

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

   4. pow2 50.2%

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

4. Applied egg-rr 50.2%

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

   1. unpow2 50.2%

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

   2. 1-sub-cos 78.0%

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

6. Applied egg-rr 78.0%

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

7. Taylor expanded in x around inf 79.2%

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

   1. associate-/r* 79.2%

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

   2. unpow2 79.2%

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

   3. associate-/r* 79.9%

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

   4. *-rgt-identity 79.9%

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

   5. associate-*r/ 79.8%

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

   6. associate-/l* 78.3%

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

   7. *-rgt-identity 78.3%

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

   8. associate-*r/ 78.1%

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

   9. unpow-1 78.1%

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

   10. unpow-1 78.1%

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

   11. pow-sqr 78.4%

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

   12. metadata-eval 78.4%

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

   13. *-lft-identity 78.4%

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

   14. associate-*l/ 78.4%

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

   15. associate-*r* 78.4%

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

9. Simplified 79.2%

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

    1. associate-*r* 78.6%

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

    2. metadata-eval 78.6%

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

    3. pow-flip 78.2%

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

    4. pow2 78.2%

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

    5. div-inv 79.1%

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

    6. times-frac 99.8%

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

    7. associate-*r/ 99.8%

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

    8. div-inv 99.8%

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

    9. metadata-eval 99.8%

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

11. Applied egg-rr 99.8%

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

Alternative 2: 75.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.0037:\\ \;\;\;\;0.5 + \left(x \cdot x\right) \cdot -0.041666666666666664\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 - \cos x}{x}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 0.0037)
   (+ 0.5 (* (* x x) -0.041666666666666664))
   (/ (/ (- 1.0 (cos x)) x) x)))

C

double code(double x) {
	double tmp;
	if (x <= 0.0037) {
		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 <= 0.0037d0) 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 <= 0.0037) {
		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 <= 0.0037:
		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 <= 0.0037)
		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 <= 0.0037)
		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, 0.0037], 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 < 0.0037000000000000002

   1. Initial program 34.8%

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

      1. flip-- 34.6%

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

      2. clear-num 34.6%

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

      3. metadata-eval 34.6%

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

      4. pow2 34.6%

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

   4. Applied egg-rr 34.6%

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

      1. unpow2 34.6%

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

      2. 1-sub-cos 71.2%

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

   6. Applied egg-rr 71.2%

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

   7. Taylor expanded in x around 0 66.8%

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

      1. *-commutative 66.8%

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

   9. Simplified 66.8%

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

       1. pow2 66.8%

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

   11. Applied egg-rr 66.8%

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

   if 0.0037000000000000002 < x

   1. Initial program 99.5%

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

      1. associate-/r* 99.7%

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

      2. clear-num 99.6%

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

      3. inv-pow 99.6%

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

   4. Applied egg-rr 99.6%

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

      1. unpow-1 99.6%

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

      2. clear-num 99.7%

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

   6. Applied egg-rr 99.7%

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

Alternative 3: 75.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.0037:\\ \;\;\;\;0.5 + \left(x \cdot x\right) \cdot -0.041666666666666664\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \cos x}{x \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 0.0037)
   (+ 0.5 (* (* x x) -0.041666666666666664))
   (/ (- 1.0 (cos x)) (* x x))))

C

double code(double x) {
	double tmp;
	if (x <= 0.0037) {
		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 <= 0.0037d0) 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 <= 0.0037) {
		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 <= 0.0037:
		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 <= 0.0037)
		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 <= 0.0037)
		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, 0.0037], 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 < 0.0037000000000000002

   1. Initial program 34.8%

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

      1. flip-- 34.6%

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

      2. clear-num 34.6%

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

      3. metadata-eval 34.6%

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

      4. pow2 34.6%

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

   4. Applied egg-rr 34.6%

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

      1. unpow2 34.6%

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

      2. 1-sub-cos 71.2%

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

   6. Applied egg-rr 71.2%

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

   7. Taylor expanded in x around 0 66.8%

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

      1. *-commutative 66.8%

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

   9. Simplified 66.8%

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

       1. pow2 66.8%

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

   11. Applied egg-rr 66.8%

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

   if 0.0037000000000000002 < x

   1. Initial program 99.5%

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

Alternative 4: 63.8% accurate, 17.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 10^{+77}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (if (<= x 1e+77) 0.5 0.0))

C

double code(double x) {
	double tmp;
	if (x <= 1e+77) {
		tmp = 0.5;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 1d+77) then
        tmp = 0.5d0
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 1e+77) {
		tmp = 0.5;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 1e+77:
		tmp = 0.5
	else:
		tmp = 0.0
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 1e+77)
		tmp = 0.5;
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 1e+77)
		tmp = 0.5;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 1e+77], 0.5, 0.0]

Derivation

1. Split input into 2 regimes

2. if x < 9.99999999999999983e76

   1. Initial program 40.2%

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

   3. Taylor expanded in x around 0 62.3%

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

   if 9.99999999999999983e76 < x

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0 72.3%

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

   4. Taylor expanded in x around 0 72.3%

      \[\leadsto \color{blue}{0} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 28.0% accurate, 107.0× speedup

Math

\[\begin{array}{l} \\ 0 \end{array} \]

FPCore

(FPCore (x) :precision binary64 0.0)

C

double code(double x) {
	return 0.0;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 0.0d0
end function

Java

public static double code(double x) {
	return 0.0;
}

Python

def code(x):
	return 0.0

Julia

function code(x)
	return 0.0
end

MATLAB

function tmp = code(x)
	tmp = 0.0;
end

Wolfram

code[x_] := 0.0

Derivation

1. Initial program 50.5%

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

3. Taylor expanded in x around 0 26.7%

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

4. Taylor expanded in x around 0 27.3%

   \[\leadsto \color{blue}{0} \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2024165 
(FPCore (x)
  :name "cos2 (problem 3.4.1)"
  :precision binary64
  (/ (- 1.0 (cos x)) (* x x)))