cos2 (problem 3.4.1)

Percentage Accurate: 50.8% → 99.8%

Time: 15.8s

Alternatives: 7

Speedup: 34.9×

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: 50.8% 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

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

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

Derivation

1. Initial program 52.3%

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

   1. associate-/r* 53.0%

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

   2. div-inv 53.0%

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

3. Applied egg-rr 53.0%

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

   1. clear-num 53.0%

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

   2. associate-/r/ 52.9%

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

5. Applied egg-rr 52.9%

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

   1. flip-- 52.9%

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

   2. metadata-eval 52.9%

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

   3. 1-sub-cos 79.5%

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

   4. frac-times 79.8%

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

   5. *-un-lft-identity 79.8%

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

   6. pow2 79.8%

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

7. Applied egg-rr 79.8%

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

   1. associate-/l/ 79.7%

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

   2. *-rgt-identity 79.7%

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

   3. associate-*r/ 79.6%

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

   4. unpow2 79.6%

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

   5. associate-*r* 79.6%

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

   6. associate-*r/ 79.7%

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

   7. *-rgt-identity 79.7%

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

   8. hang-0p-tan 79.8%

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

9. Simplified 79.8%

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

    1. associate-/l* 99.7%

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

    2. associate-*l/ 99.7%

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

    3. div-inv 99.8%

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

    4. div-inv 99.8%

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

    5. metadata-eval 99.8%

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

11. Applied egg-rr 99.8%

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

12. Final simplification 99.8%

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

Alternative 2: 99.8% accurate, 0.5× speedup

Math

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

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

Derivation

1. Initial program 52.3%

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

   1. flip-- 52.1%

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

   2. div-inv 52.1%

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

   3. metadata-eval 52.1%

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

   4. pow2 52.1%

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

3. Applied egg-rr 52.1%

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

   1. associate-*r/ 52.1%

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

   2. *-rgt-identity 52.1%

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

5. Simplified 52.1%

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

   1. unpow2 52.1%

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

   2. 1-sub-cos 79.0%

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

7. Applied egg-rr 79.0%

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

8. Taylor expanded in x around inf 79.0%

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

   1. unpow2 79.0%

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

   2. associate-*r/ 79.0%

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

   3. hang-0p-tan 79.2%

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

10. Simplified 79.2%

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

    1. *-commutative 79.2%

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

    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%

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

12. Applied egg-rr 99.8%

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

13. Final simplification 99.8%

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

Alternative 3: 75.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.0041:\\ \;\;\;\;0.5 + {x}^{2} \cdot -0.041666666666666664\\ \mathbf{else}:\\ \;\;\;\;{x}^{-2} \cdot \left(1 - \cos x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 0.0041)
   (+ 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 <= 0.0041) {
		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 <= 0.0041d0) 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 <= 0.0041) {
		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 <= 0.0041:
		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 <= 0.0041)
		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 <= 0.0041)
		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, 0.0041], 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 < 0.00410000000000000035

   1. Initial program 35.5%

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

   2. Taylor expanded in x around 0 67.0%

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

      1. *-commutative 67.0%

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

   4. Simplified 67.0%

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

   if 0.00410000000000000035 < x

   1. Initial program 98.6%

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

      1. clear-num 98.6%

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

      2. associate-/r/ 98.7%

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

      3. pow2 98.7%

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

      4. pow-flip 99.0%

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

      5. metadata-eval 99.0%

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

   3. 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 75.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.0041:\\ \;\;\;\;0.5 + {x}^{2} \cdot -0.041666666666666664\\ \mathbf{else}:\\ \;\;\;\;{x}^{-2} \cdot \left(1 - \cos x\right)\\ \end{array} \]

Alternative 4: 74.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.0041:\\ \;\;\;\;0.5 + {x}^{2} \cdot -0.041666666666666664\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \cos x}{x \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 0.0041)
   (+ 0.5 (* (pow x 2.0) -0.041666666666666664))
   (/ (- 1.0 (cos x)) (* x x))))

C

double code(double x) {
	double tmp;
	if (x <= 0.0041) {
		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 <= 0.0041d0) 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 <= 0.0041) {
		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 <= 0.0041:
		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 <= 0.0041)
		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 <= 0.0041)
		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, 0.0041], 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 < 0.00410000000000000035

   1. Initial program 35.5%

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

   2. Taylor expanded in x around 0 67.0%

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

      1. *-commutative 67.0%

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

   4. Simplified 67.0%

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

   if 0.00410000000000000035 < x

   1. Initial program 98.6%

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

4. Final simplification 75.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.0041:\\ \;\;\;\;0.5 + {x}^{2} \cdot -0.041666666666666664\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \cos x}{x \cdot x}\\ \end{array} \]

Alternative 5: 75.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.0041:\\ \;\;\;\;0.5 + {x}^{2} \cdot -0.041666666666666664\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 - \cos x}{x}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 0.0041)
   (+ 0.5 (* (pow x 2.0) -0.041666666666666664))
   (/ (/ (- 1.0 (cos x)) x) x)))

C

double code(double x) {
	double tmp;
	if (x <= 0.0041) {
		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 <= 0.0041d0) 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 <= 0.0041) {
		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 <= 0.0041:
		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 <= 0.0041)
		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 <= 0.0041)
		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, 0.0041], 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 < 0.00410000000000000035

   1. Initial program 35.5%

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

   2. Taylor expanded in x around 0 67.0%

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

      1. *-commutative 67.0%

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

   4. Simplified 67.0%

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

   if 0.00410000000000000035 < x

   1. Initial program 98.6%

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

      1. associate-/r* 98.9%

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

      2. div-inv 98.8%

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

   3. Applied egg-rr 98.8%

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

      1. un-div-inv 98.9%

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

   5. Applied egg-rr 98.9%

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

4. Final simplification 75.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.0041:\\ \;\;\;\;0.5 + {x}^{2} \cdot -0.041666666666666664\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 - \cos x}{x}}{x}\\ \end{array} \]

Alternative 6: 63.8% accurate, 34.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 7.5 \cdot 10^{+76}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (if (<= x 7.5e+76) 0.5 0.0))

C

double code(double x) {
	double tmp;
	if (x <= 7.5e+76) {
		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 <= 7.5d+76) 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 <= 7.5e+76) {
		tmp = 0.5;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 7.5e+76:
		tmp = 0.5
	else:
		tmp = 0.0
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 7.5e+76)
		tmp = 0.5;
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 7.5e+76)
		tmp = 0.5;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 7.5e+76], 0.5, 0.0]

Derivation

1. Split input into 2 regimes

2. if x < 7.4999999999999995e76

   1. Initial program 41.2%

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

   2. Taylor expanded in x around 0 62.0%

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

   if 7.4999999999999995e76 < x

   1. Initial program 99.0%

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

   2. Taylor expanded in x around 0 69.0%

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

   3. Taylor expanded in x around 0 69.0%

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

4. Final simplification 63.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 7.5 \cdot 10^{+76}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 7: 27.5% 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 52.3%

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

2. Taylor expanded in x around 0 28.5%

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

3. Taylor expanded in x around 0 29.1%

   \[\leadsto \color{blue}{0} \]

4. Final simplification 29.1%

   \[\leadsto 0 \]

Reproduce

herbie shell --seed 2023318 
(FPCore (x)
  :name "cos2 (problem 3.4.1)"
  :precision binary64
  (/ (- 1.0 (cos x)) (* x x)))