cos2 (problem 3.4.1)

Percentage Accurate: 50.7% → 99.8%

Time: 10.1s

Alternatives: 8

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: 50.7% 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 47.2%

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

   1. clear-num 47.2%

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

   2. inv-pow 47.2%

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

   3. flip-- 47.1%

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

   4. associate-/r/ 47.1%

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

   5. unpow-prod-down 47.1%

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

   6. pow2 47.1%

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

   7. metadata-eval 47.1%

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

   8. pow2 47.1%

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

   9. inv-pow 47.1%

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

4. Applied egg-rr 47.1%

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

   1. associate-*r/ 47.1%

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

   2. *-rgt-identity 47.1%

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

   3. unpow-1 47.1%

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

   4. associate-/r/ 47.0%

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

6. Simplified 47.0%

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

   1. unpow2 47.0%

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

   2. 1-sub-cos 73.7%

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

8. Applied egg-rr 73.7%

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

9. Taylor expanded in x around inf 74.9%

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

    1. associate-/l/ 74.1%

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

    2. unpow2 74.1%

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

    3. associate-*r/ 74.1%

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

    4. associate-/l* 74.9%

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

    5. hang-0p-tan 75.1%

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

11. Simplified 75.1%

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

    1. associate-*r/ 74.3%

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

    2. pow2 74.3%

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

    3. times-frac 99.8%

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

    4. clear-num 99.8%

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

    5. associate-*l/ 99.8%

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

    6. *-un-lft-identity 99.8%

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

    7. div-inv 99.8%

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

    8. metadata-eval 99.8%

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

13. Applied egg-rr 99.8%

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

Alternative 2: 75.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.000125:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;{x}^{-2} \cdot \left(1 - \cos x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 0.000125) 0.5 (* (pow x -2.0) (- 1.0 (cos x)))))

C

double code(double x) {
	double tmp;
	if (x <= 0.000125) {
		tmp = 0.5;
	} 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 <= 0.000125d0) then
        tmp = 0.5d0
    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 <= 0.000125) {
		tmp = 0.5;
	} else {
		tmp = Math.pow(x, -2.0) * (1.0 - Math.cos(x));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 0.000125:
		tmp = 0.5
	else:
		tmp = math.pow(x, -2.0) * (1.0 - math.cos(x))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 0.000125)
		tmp = 0.5;
	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 <= 0.000125)
		tmp = 0.5;
	else
		tmp = (x ^ -2.0) * (1.0 - cos(x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 0.000125], 0.5, 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 < 1.25e-4

   1. Initial program 28.0%

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

   3. Taylor expanded in x around 0 73.5%

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

   if 1.25e-4 < x

   1. Initial program 98.4%

      \[\frac{1 - \cos x}{x \cdot x} \]
   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/ 98.4%

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

      3. pow2 98.4%

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

      4. pow-flip 99.3%

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

      5. metadata-eval 99.3%

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

   4. Applied egg-rr 99.3%

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

Alternative 3: 99.8% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 47.2%

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

   1. add-cube-cbrt 46.9%

      \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{1 - \cos x} \cdot \sqrt[3]{1 - \cos x}\right) \cdot \sqrt[3]{1 - \cos x}}}{x \cdot x} \]

   2. pow3 46.9%

      \[\leadsto \frac{\color{blue}{{\left(\sqrt[3]{1 - \cos x}\right)}^{3}}}{x \cdot x} \]

4. Applied egg-rr 46.9%

   \[\leadsto \frac{\color{blue}{{\left(\sqrt[3]{1 - \cos x}\right)}^{3}}}{x \cdot x} \]
5. Step-by-step derivation

   1. rem-cube-cbrt 47.2%

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

   2. flip-- 47.1%

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

   3. metadata-eval 47.1%

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

   4. div-sub 47.1%

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

   5. pow2 47.1%

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

6. Applied egg-rr 47.1%

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

   1. remove-double-neg 47.1%

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

   2. sub-neg 47.1%

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

   3. remove-double-neg 47.1%

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

   4. sub-neg 47.1%

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

   5. div-sub 47.1%

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

   6. unpow2 47.1%

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

   7. sqr-neg 47.1%

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

   8. sqr-neg 47.1%

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

   9. 1-sub-cos 74.1%

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

   10. sub-neg 74.1%

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

   11. remove-double-neg 74.1%

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

   12. associate-*r/ 74.1%

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

   13. hang-0p-tan 74.3%

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

8. Simplified 74.3%

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

   1. *-commutative 74.3%

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

   2. times-frac 99.8%

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

   3. div-inv 99.8%

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

   4. metadata-eval 99.8%

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

10. Applied egg-rr 99.8%

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

Alternative 4: 75.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.000125:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\left(1 - \cos x\right) \cdot \frac{\frac{1}{x}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 0.000125) 0.5 (* (- 1.0 (cos x)) (/ (/ 1.0 x) x))))

C

double code(double x) {
	double tmp;
	if (x <= 0.000125) {
		tmp = 0.5;
	} else {
		tmp = (1.0 - cos(x)) * ((1.0 / x) / x);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 0.000125d0) then
        tmp = 0.5d0
    else
        tmp = (1.0d0 - cos(x)) * ((1.0d0 / x) / x)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 0.000125) {
		tmp = 0.5;
	} else {
		tmp = (1.0 - Math.cos(x)) * ((1.0 / x) / x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 0.000125:
		tmp = 0.5
	else:
		tmp = (1.0 - math.cos(x)) * ((1.0 / x) / x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 0.000125)
		tmp = 0.5;
	else
		tmp = Float64(Float64(1.0 - cos(x)) * Float64(Float64(1.0 / x) / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 0.000125)
		tmp = 0.5;
	else
		tmp = (1.0 - cos(x)) * ((1.0 / x) / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 0.000125], 0.5, N[(N[(1.0 - N[Cos[x], $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 / x), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.25e-4

   1. Initial program 28.0%

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

   3. Taylor expanded in x around 0 73.5%

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

   if 1.25e-4 < x

   1. Initial program 98.4%

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

      1. add-cube-cbrt 97.8%

         \[\leadsto \color{blue}{\left(\sqrt[3]{\frac{1 - \cos x}{x \cdot x}} \cdot \sqrt[3]{\frac{1 - \cos x}{x \cdot x}}\right) \cdot \sqrt[3]{\frac{1 - \cos x}{x \cdot x}}} \]

      2. pow3 97.8%

         \[\leadsto \color{blue}{{\left(\sqrt[3]{\frac{1 - \cos x}{x \cdot x}}\right)}^{3}} \]

      3. div-inv 97.7%

         \[\leadsto {\left(\sqrt[3]{\color{blue}{\left(1 - \cos x\right) \cdot \frac{1}{x \cdot x}}}\right)}^{3} \]

      4. cbrt-prod 97.7%

         \[\leadsto {\color{blue}{\left(\sqrt[3]{1 - \cos x} \cdot \sqrt[3]{\frac{1}{x \cdot x}}\right)}}^{3} \]

      5. unpow-prod-down 97.7%

         \[\leadsto \color{blue}{{\left(\sqrt[3]{1 - \cos x}\right)}^{3} \cdot {\left(\sqrt[3]{\frac{1}{x \cdot x}}\right)}^{3}} \]

      6. pow3 97.8%

         \[\leadsto \color{blue}{\left(\left(\sqrt[3]{1 - \cos x} \cdot \sqrt[3]{1 - \cos x}\right) \cdot \sqrt[3]{1 - \cos x}\right)} \cdot {\left(\sqrt[3]{\frac{1}{x \cdot x}}\right)}^{3} \]

      7. add-cube-cbrt 98.1%

         \[\leadsto \color{blue}{\left(1 - \cos x\right)} \cdot {\left(\sqrt[3]{\frac{1}{x \cdot x}}\right)}^{3} \]

      8. pow2 98.1%

         \[\leadsto \left(1 - \cos x\right) \cdot {\left(\sqrt[3]{\frac{1}{\color{blue}{{x}^{2}}}}\right)}^{3} \]

      9. pow-flip 98.8%

         \[\leadsto \left(1 - \cos x\right) \cdot {\left(\sqrt[3]{\color{blue}{{x}^{\left(-2\right)}}}\right)}^{3} \]

      10. metadata-eval 98.8%

          \[\leadsto \left(1 - \cos x\right) \cdot {\left(\sqrt[3]{{x}^{\color{blue}{-2}}}\right)}^{3} \]

   4. Applied egg-rr 98.8%

      \[\leadsto \color{blue}{\left(1 - \cos x\right) \cdot {\left(\sqrt[3]{{x}^{-2}}\right)}^{3}} \]
   5. Step-by-step derivation

      1. rem-cube-cbrt 99.3%

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

      2. metadata-eval 99.3%

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

      3. pow-flip 98.4%

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

      4. pow2 98.4%

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

      5. associate-/r* 99.3%

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

   6. Applied egg-rr 99.3%

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

Alternative 5: 75.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.000125:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 - \cos x}{x}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 0.000125) 0.5 (/ (/ (- 1.0 (cos x)) x) x)))

C

double code(double x) {
	double tmp;
	if (x <= 0.000125) {
		tmp = 0.5;
	} 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 <= 0.000125d0) then
        tmp = 0.5d0
    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 <= 0.000125) {
		tmp = 0.5;
	} else {
		tmp = ((1.0 - Math.cos(x)) / x) / x;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 0.000125:
		tmp = 0.5
	else:
		tmp = ((1.0 - math.cos(x)) / x) / x
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 0.000125)
		tmp = 0.5;
	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 <= 0.000125)
		tmp = 0.5;
	else
		tmp = ((1.0 - cos(x)) / x) / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 0.000125], 0.5, N[(N[(N[(1.0 - N[Cos[x], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.25e-4

   1. Initial program 28.0%

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

   3. Taylor expanded in x around 0 73.5%

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

   if 1.25e-4 < x

   1. Initial program 98.4%

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

      1. add-cube-cbrt 97.8%

         \[\leadsto \color{blue}{\left(\sqrt[3]{\frac{1 - \cos x}{x \cdot x}} \cdot \sqrt[3]{\frac{1 - \cos x}{x \cdot x}}\right) \cdot \sqrt[3]{\frac{1 - \cos x}{x \cdot x}}} \]

      2. pow3 97.8%

         \[\leadsto \color{blue}{{\left(\sqrt[3]{\frac{1 - \cos x}{x \cdot x}}\right)}^{3}} \]

      3. div-inv 97.7%

         \[\leadsto {\left(\sqrt[3]{\color{blue}{\left(1 - \cos x\right) \cdot \frac{1}{x \cdot x}}}\right)}^{3} \]

      4. cbrt-prod 97.7%

         \[\leadsto {\color{blue}{\left(\sqrt[3]{1 - \cos x} \cdot \sqrt[3]{\frac{1}{x \cdot x}}\right)}}^{3} \]

      5. unpow-prod-down 97.7%

         \[\leadsto \color{blue}{{\left(\sqrt[3]{1 - \cos x}\right)}^{3} \cdot {\left(\sqrt[3]{\frac{1}{x \cdot x}}\right)}^{3}} \]

      6. pow3 97.8%

         \[\leadsto \color{blue}{\left(\left(\sqrt[3]{1 - \cos x} \cdot \sqrt[3]{1 - \cos x}\right) \cdot \sqrt[3]{1 - \cos x}\right)} \cdot {\left(\sqrt[3]{\frac{1}{x \cdot x}}\right)}^{3} \]

      7. add-cube-cbrt 98.1%

         \[\leadsto \color{blue}{\left(1 - \cos x\right)} \cdot {\left(\sqrt[3]{\frac{1}{x \cdot x}}\right)}^{3} \]

      8. pow2 98.1%

         \[\leadsto \left(1 - \cos x\right) \cdot {\left(\sqrt[3]{\frac{1}{\color{blue}{{x}^{2}}}}\right)}^{3} \]

      9. pow-flip 98.8%

         \[\leadsto \left(1 - \cos x\right) \cdot {\left(\sqrt[3]{\color{blue}{{x}^{\left(-2\right)}}}\right)}^{3} \]

      10. metadata-eval 98.8%

          \[\leadsto \left(1 - \cos x\right) \cdot {\left(\sqrt[3]{{x}^{\color{blue}{-2}}}\right)}^{3} \]

   4. Applied egg-rr 98.8%

      \[\leadsto \color{blue}{\left(1 - \cos x\right) \cdot {\left(\sqrt[3]{{x}^{-2}}\right)}^{3}} \]
   5. Step-by-step derivation

      1. rem-cube-cbrt 99.3%

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

      2. metadata-eval 99.3%

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

      3. pow-flip 98.4%

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

      4. div-inv 98.4%

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

      5. pow2 98.4%

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

      6. associate-/r* 99.3%

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

   6. Applied egg-rr 99.3%

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

Alternative 6: 75.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.000125:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \cos x}{x \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 0.000125) 0.5 (/ (- 1.0 (cos x)) (* x x))))

C

double code(double x) {
	double tmp;
	if (x <= 0.000125) {
		tmp = 0.5;
	} 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 <= 0.000125d0) then
        tmp = 0.5d0
    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 <= 0.000125) {
		tmp = 0.5;
	} else {
		tmp = (1.0 - Math.cos(x)) / (x * x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 0.000125:
		tmp = 0.5
	else:
		tmp = (1.0 - math.cos(x)) / (x * x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 0.000125)
		tmp = 0.5;
	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 <= 0.000125)
		tmp = 0.5;
	else
		tmp = (1.0 - cos(x)) / (x * x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 0.000125], 0.5, N[(N[(1.0 - N[Cos[x], $MachinePrecision]), $MachinePrecision] / N[(x * x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.25e-4

   1. Initial program 28.0%

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

   3. Taylor expanded in x around 0 73.5%

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

   if 1.25e-4 < x

   1. Initial program 98.4%

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

Alternative 7: 63.6% accurate, 17.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.25 \cdot 10^{+77}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (if (<= x 1.25e+77) 0.5 0.0))

C

double code(double x) {
	double tmp;
	if (x <= 1.25e+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 <= 1.25d+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 <= 1.25e+77) {
		tmp = 0.5;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 1.25e+77:
		tmp = 0.5
	else:
		tmp = 0.0
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.25e+77)
		tmp = 0.5;
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 1.25e+77)
		tmp = 0.5;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 1.25e+77], 0.5, 0.0]

Derivation

1. Split input into 2 regimes

2. if x < 1.25000000000000001e77

   1. Initial program 33.6%

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

   3. Taylor expanded in x around 0 68.2%

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

   if 1.25000000000000001e77 < x

   1. Initial program 98.4%

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

   3. Taylor expanded in x around 0 66.2%

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

   4. Taylor expanded in x around 0 66.2%

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

Alternative 8: 27.2% 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 47.2%

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

3. Taylor expanded in x around 0 24.4%

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

4. Taylor expanded in x around 0 25.1%

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

Reproduce

herbie shell --seed 2024117 
(FPCore (x)
  :name "cos2 (problem 3.4.1)"
  :precision binary64
  (/ (- 1.0 (cos x)) (* x x)))