cos2 (problem 3.4.1)

Percentage Accurate: 50.8% → 99.7%

Time: 12.4s

Alternatives: 10

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: 50.8% 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: 99.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{\sin x}{x}}{x} \cdot \tan \left(\frac{x}{2}\right) \end{array} \]

FPCore

(FPCore (x) :precision binary64 (* (/ (/ (sin x) x) x) (tan (/ x 2.0))))

C

double code(double x) {
	return ((sin(x) / x) / x) * tan((x / 2.0));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = ((sin(x) / x) / x) * tan((x / 2.0d0))
end function

Java

public static double code(double x) {
	return ((Math.sin(x) / x) / x) * Math.tan((x / 2.0));
}

Python

def code(x):
	return ((math.sin(x) / x) / x) * math.tan((x / 2.0))

Julia

function code(x)
	return Float64(Float64(Float64(sin(x) / x) / x) * tan(Float64(x / 2.0)))
end

MATLAB

function tmp = code(x)
	tmp = ((sin(x) / x) / x) * tan((x / 2.0));
end

Wolfram

code[x_] := N[(N[(N[(N[Sin[x], $MachinePrecision] / x), $MachinePrecision] / x), $MachinePrecision] * N[Tan[N[(x / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 45.9%

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

   1. clear-num 45.9%

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

   2. associate-/r/ 45.9%

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

   3. pow2 45.9%

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

   4. pow-flip 46.3%

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

   5. metadata-eval 46.3%

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

3. Applied egg-rr 46.3%

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

   1. metadata-eval 46.3%

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

   2. pow-flip 45.9%

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

   3. pow2 45.9%

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

   4. associate-/r/ 45.9%

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

   5. clear-num 45.9%

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

   6. flip-- 45.7%

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

   7. associate-/l/ 45.7%

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

   8. metadata-eval 45.7%

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

   9. 1-sub-cos 72.2%

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

   10. pow2 72.2%

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

5. Applied egg-rr 72.2%

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

   1. unpow2 72.2%

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

   2. times-frac 72.1%

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

   3. hang-0p-tan 72.5%

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

7. Simplified 72.5%

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

   1. *-un-lft-identity 72.5%

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

   2. times-frac 99.6%

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

9. Applied egg-rr 99.6%

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

    1. associate-*l/ 99.7%

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

    2. *-lft-identity 99.7%

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

11. Simplified 99.7%

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

12. Final simplification 99.7%

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

Alternative 2: 98.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{1 - \cos x}{x \cdot x} \leq 0.04:\\ \;\;\;\;\tan \left(\frac{x}{2}\right) \cdot \frac{\sin x}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;0.5 + \left(x \cdot x\right) \cdot -0.041666666666666664\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (/ (- 1.0 (cos x)) (* x x)) 0.04)
   (* (tan (/ x 2.0)) (/ (sin x) (* x x)))
   (+ 0.5 (* (* x x) -0.041666666666666664))))

C

double code(double x) {
	double tmp;
	if (((1.0 - cos(x)) / (x * x)) <= 0.04) {
		tmp = tan((x / 2.0)) * (sin(x) / (x * x));
	} else {
		tmp = 0.5 + ((x * x) * -0.041666666666666664);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (((1.0d0 - cos(x)) / (x * x)) <= 0.04d0) then
        tmp = tan((x / 2.0d0)) * (sin(x) / (x * x))
    else
        tmp = 0.5d0 + ((x * x) * (-0.041666666666666664d0))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (((1.0 - Math.cos(x)) / (x * x)) <= 0.04) {
		tmp = Math.tan((x / 2.0)) * (Math.sin(x) / (x * x));
	} else {
		tmp = 0.5 + ((x * x) * -0.041666666666666664);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if ((1.0 - math.cos(x)) / (x * x)) <= 0.04:
		tmp = math.tan((x / 2.0)) * (math.sin(x) / (x * x))
	else:
		tmp = 0.5 + ((x * x) * -0.041666666666666664)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (Float64(Float64(1.0 - cos(x)) / Float64(x * x)) <= 0.04)
		tmp = Float64(tan(Float64(x / 2.0)) * Float64(sin(x) / Float64(x * x)));
	else
		tmp = Float64(0.5 + Float64(Float64(x * x) * -0.041666666666666664));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (((1.0 - cos(x)) / (x * x)) <= 0.04)
		tmp = tan((x / 2.0)) * (sin(x) / (x * x));
	else
		tmp = 0.5 + ((x * x) * -0.041666666666666664);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[N[(N[(1.0 - N[Cos[x], $MachinePrecision]), $MachinePrecision] / N[(x * x), $MachinePrecision]), $MachinePrecision], 0.04], N[(N[Tan[N[(x / 2.0), $MachinePrecision]], $MachinePrecision] * N[(N[Sin[x], $MachinePrecision] / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 + N[(N[(x * x), $MachinePrecision] * -0.041666666666666664), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 1 (cos.f64 x)) (*.f64 x x)) < 0.0400000000000000008

   1. Initial program 63.3%

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

      1. clear-num 63.2%

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

      2. associate-/r/ 63.2%

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

      3. pow2 63.2%

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

      4. pow-flip 63.7%

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

      5. metadata-eval 63.7%

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

   3. Applied egg-rr 63.7%

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

      1. metadata-eval 63.7%

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

      2. pow-flip 63.2%

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

      3. pow2 63.2%

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

      4. associate-/r/ 63.2%

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

      5. clear-num 63.3%

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

      6. flip-- 63.0%

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

      7. associate-/l/ 62.9%

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

      8. metadata-eval 62.9%

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

      9. 1-sub-cos 98.8%

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

      10. pow2 98.8%

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

   5. Applied egg-rr 98.8%

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

      1. unpow2 98.8%

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

      2. times-frac 97.6%

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

      3. hang-0p-tan 98.0%

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

   7. Simplified 98.0%

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

   if 0.0400000000000000008 < (/.f64 (-.f64 1 (cos.f64 x)) (*.f64 x x))

   1. Initial program 0.8%

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

   2. Taylor expanded in x around 0 100.0%

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

      1. unpow2 100.0%

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

   4. Simplified 100.0%

      \[\leadsto \color{blue}{0.5 + -0.041666666666666664 \cdot \left(x \cdot x\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1 - \cos x}{x \cdot x} \leq 0.04:\\ \;\;\;\;\tan \left(\frac{x}{2}\right) \cdot \frac{\sin x}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;0.5 + \left(x \cdot x\right) \cdot -0.041666666666666664\\ \end{array} \]

Alternative 3: 99.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 - \cos x\\ \mathbf{if}\;x \leq -0.0048:\\ \;\;\;\;t_0 \cdot {x}^{-2}\\ \mathbf{elif}\;x \leq 0.0053:\\ \;\;\;\;0.5 + \left(x \cdot x\right) \cdot -0.041666666666666664\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0}{x \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (- 1.0 (cos x))))
   (if (<= x -0.0048)
     (* t_0 (pow x -2.0))
     (if (<= x 0.0053)
       (+ 0.5 (* (* x x) -0.041666666666666664))
       (/ t_0 (* x x))))))

C

double code(double x) {
	double t_0 = 1.0 - cos(x);
	double tmp;
	if (x <= -0.0048) {
		tmp = t_0 * pow(x, -2.0);
	} else if (x <= 0.0053) {
		tmp = 0.5 + ((x * x) * -0.041666666666666664);
	} else {
		tmp = t_0 / (x * x);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 1.0d0 - cos(x)
    if (x <= (-0.0048d0)) then
        tmp = t_0 * (x ** (-2.0d0))
    else if (x <= 0.0053d0) then
        tmp = 0.5d0 + ((x * x) * (-0.041666666666666664d0))
    else
        tmp = t_0 / (x * x)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double t_0 = 1.0 - Math.cos(x);
	double tmp;
	if (x <= -0.0048) {
		tmp = t_0 * Math.pow(x, -2.0);
	} else if (x <= 0.0053) {
		tmp = 0.5 + ((x * x) * -0.041666666666666664);
	} else {
		tmp = t_0 / (x * x);
	}
	return tmp;
}

Python

def code(x):
	t_0 = 1.0 - math.cos(x)
	tmp = 0
	if x <= -0.0048:
		tmp = t_0 * math.pow(x, -2.0)
	elif x <= 0.0053:
		tmp = 0.5 + ((x * x) * -0.041666666666666664)
	else:
		tmp = t_0 / (x * x)
	return tmp

Julia

function code(x)
	t_0 = Float64(1.0 - cos(x))
	tmp = 0.0
	if (x <= -0.0048)
		tmp = Float64(t_0 * (x ^ -2.0));
	elseif (x <= 0.0053)
		tmp = Float64(0.5 + Float64(Float64(x * x) * -0.041666666666666664));
	else
		tmp = Float64(t_0 / Float64(x * x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = 1.0 - cos(x);
	tmp = 0.0;
	if (x <= -0.0048)
		tmp = t_0 * (x ^ -2.0);
	elseif (x <= 0.0053)
		tmp = 0.5 + ((x * x) * -0.041666666666666664);
	else
		tmp = t_0 / (x * x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[(1.0 - N[Cos[x], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -0.0048], N[(t$95$0 * N[Power[x, -2.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.0053], N[(0.5 + N[(N[(x * x), $MachinePrecision] * -0.041666666666666664), $MachinePrecision]), $MachinePrecision], N[(t$95$0 / N[(x * x), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.00479999999999999958

   1. Initial program 97.3%

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

      1. clear-num 97.3%

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

      2. associate-/r/ 97.3%

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

      3. pow2 97.3%

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

      4. pow-flip 98.9%

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

      5. metadata-eval 98.9%

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

   3. Applied egg-rr 98.9%

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

   if -0.00479999999999999958 < x < 0.00530000000000000002

   1. Initial program 1.9%

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

   2. Taylor expanded in x around 0 100.0%

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

      1. unpow2 100.0%

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

   4. Simplified 100.0%

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

   if 0.00530000000000000002 < x

   1. Initial program 99.3%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.0048:\\ \;\;\;\;\left(1 - \cos x\right) \cdot {x}^{-2}\\ \mathbf{elif}\;x \leq 0.0053:\\ \;\;\;\;0.5 + \left(x \cdot x\right) \cdot -0.041666666666666664\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \cos x}{x \cdot x}\\ \end{array} \]

Alternative 4: 99.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 - \cos x\\ \mathbf{if}\;x \leq -0.0048:\\ \;\;\;\;t_0 \cdot \frac{\frac{-1}{x}}{-x}\\ \mathbf{elif}\;x \leq 0.0053:\\ \;\;\;\;0.5 + \left(x \cdot x\right) \cdot -0.041666666666666664\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0}{x \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (- 1.0 (cos x))))
   (if (<= x -0.0048)
     (* t_0 (/ (/ -1.0 x) (- x)))
     (if (<= x 0.0053)
       (+ 0.5 (* (* x x) -0.041666666666666664))
       (/ t_0 (* x x))))))

C

double code(double x) {
	double t_0 = 1.0 - cos(x);
	double tmp;
	if (x <= -0.0048) {
		tmp = t_0 * ((-1.0 / x) / -x);
	} else if (x <= 0.0053) {
		tmp = 0.5 + ((x * x) * -0.041666666666666664);
	} else {
		tmp = t_0 / (x * x);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 1.0d0 - cos(x)
    if (x <= (-0.0048d0)) then
        tmp = t_0 * (((-1.0d0) / x) / -x)
    else if (x <= 0.0053d0) then
        tmp = 0.5d0 + ((x * x) * (-0.041666666666666664d0))
    else
        tmp = t_0 / (x * x)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double t_0 = 1.0 - Math.cos(x);
	double tmp;
	if (x <= -0.0048) {
		tmp = t_0 * ((-1.0 / x) / -x);
	} else if (x <= 0.0053) {
		tmp = 0.5 + ((x * x) * -0.041666666666666664);
	} else {
		tmp = t_0 / (x * x);
	}
	return tmp;
}

Python

def code(x):
	t_0 = 1.0 - math.cos(x)
	tmp = 0
	if x <= -0.0048:
		tmp = t_0 * ((-1.0 / x) / -x)
	elif x <= 0.0053:
		tmp = 0.5 + ((x * x) * -0.041666666666666664)
	else:
		tmp = t_0 / (x * x)
	return tmp

Julia

function code(x)
	t_0 = Float64(1.0 - cos(x))
	tmp = 0.0
	if (x <= -0.0048)
		tmp = Float64(t_0 * Float64(Float64(-1.0 / x) / Float64(-x)));
	elseif (x <= 0.0053)
		tmp = Float64(0.5 + Float64(Float64(x * x) * -0.041666666666666664));
	else
		tmp = Float64(t_0 / Float64(x * x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = 1.0 - cos(x);
	tmp = 0.0;
	if (x <= -0.0048)
		tmp = t_0 * ((-1.0 / x) / -x);
	elseif (x <= 0.0053)
		tmp = 0.5 + ((x * x) * -0.041666666666666664);
	else
		tmp = t_0 / (x * x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[(1.0 - N[Cos[x], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -0.0048], N[(t$95$0 * N[(N[(-1.0 / x), $MachinePrecision] / (-x)), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.0053], N[(0.5 + N[(N[(x * x), $MachinePrecision] * -0.041666666666666664), $MachinePrecision]), $MachinePrecision], N[(t$95$0 / N[(x * x), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.00479999999999999958

   1. Initial program 97.3%

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

      1. frac-2neg 97.3%

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

      2. div-inv 97.3%

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

      3. distribute-rgt-neg-in 97.3%

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

   3. Applied egg-rr 97.3%

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

      1. distribute-lft-neg-out 97.3%

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

      2. associate-/r* 98.9%

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

   5. Simplified 98.9%

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

   if -0.00479999999999999958 < x < 0.00530000000000000002

   1. Initial program 1.9%

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

   2. Taylor expanded in x around 0 100.0%

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

      1. unpow2 100.0%

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

   4. Simplified 100.0%

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

   if 0.00530000000000000002 < x

   1. Initial program 99.3%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.0048:\\ \;\;\;\;\left(1 - \cos x\right) \cdot \frac{\frac{-1}{x}}{-x}\\ \mathbf{elif}\;x \leq 0.0053:\\ \;\;\;\;0.5 + \left(x \cdot x\right) \cdot -0.041666666666666664\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \cos x}{x \cdot x}\\ \end{array} \]

Alternative 5: 99.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.0048 \lor \neg \left(x \leq 0.0053\right):\\ \;\;\;\;\frac{1 - \cos x}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;0.5 + \left(x \cdot x\right) \cdot -0.041666666666666664\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (or (<= x -0.0048) (not (<= x 0.0053)))
   (/ (- 1.0 (cos x)) (* x x))
   (+ 0.5 (* (* x x) -0.041666666666666664))))

C

double code(double x) {
	double tmp;
	if ((x <= -0.0048) || !(x <= 0.0053)) {
		tmp = (1.0 - cos(x)) / (x * x);
	} else {
		tmp = 0.5 + ((x * x) * -0.041666666666666664);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((x <= (-0.0048d0)) .or. (.not. (x <= 0.0053d0))) then
        tmp = (1.0d0 - cos(x)) / (x * x)
    else
        tmp = 0.5d0 + ((x * x) * (-0.041666666666666664d0))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if ((x <= -0.0048) || !(x <= 0.0053)) {
		tmp = (1.0 - Math.cos(x)) / (x * x);
	} else {
		tmp = 0.5 + ((x * x) * -0.041666666666666664);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if (x <= -0.0048) or not (x <= 0.0053):
		tmp = (1.0 - math.cos(x)) / (x * x)
	else:
		tmp = 0.5 + ((x * x) * -0.041666666666666664)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if ((x <= -0.0048) || !(x <= 0.0053))
		tmp = Float64(Float64(1.0 - cos(x)) / Float64(x * x));
	else
		tmp = Float64(0.5 + Float64(Float64(x * x) * -0.041666666666666664));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if ((x <= -0.0048) || ~((x <= 0.0053)))
		tmp = (1.0 - cos(x)) / (x * x);
	else
		tmp = 0.5 + ((x * x) * -0.041666666666666664);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[Or[LessEqual[x, -0.0048], N[Not[LessEqual[x, 0.0053]], $MachinePrecision]], N[(N[(1.0 - N[Cos[x], $MachinePrecision]), $MachinePrecision] / N[(x * x), $MachinePrecision]), $MachinePrecision], N[(0.5 + N[(N[(x * x), $MachinePrecision] * -0.041666666666666664), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -0.00479999999999999958 or 0.00530000000000000002 < x

   1. Initial program 98.2%

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

   if -0.00479999999999999958 < x < 0.00530000000000000002

   1. Initial program 1.9%

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

   2. Taylor expanded in x around 0 100.0%

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

      1. unpow2 100.0%

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

   4. Simplified 100.0%

      \[\leadsto \color{blue}{0.5 + -0.041666666666666664 \cdot \left(x \cdot x\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.0048 \lor \neg \left(x \leq 0.0053\right):\\ \;\;\;\;\frac{1 - \cos x}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;0.5 + \left(x \cdot x\right) \cdot -0.041666666666666664\\ \end{array} \]

Alternative 6: 99.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 - \cos x\\ \mathbf{if}\;x \leq -0.0048:\\ \;\;\;\;\frac{\frac{t_0}{x}}{x}\\ \mathbf{elif}\;x \leq 0.0053:\\ \;\;\;\;0.5 + \left(x \cdot x\right) \cdot -0.041666666666666664\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0}{x \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (- 1.0 (cos x))))
   (if (<= x -0.0048)
     (/ (/ t_0 x) x)
     (if (<= x 0.0053)
       (+ 0.5 (* (* x x) -0.041666666666666664))
       (/ t_0 (* x x))))))

C

double code(double x) {
	double t_0 = 1.0 - cos(x);
	double tmp;
	if (x <= -0.0048) {
		tmp = (t_0 / x) / x;
	} else if (x <= 0.0053) {
		tmp = 0.5 + ((x * x) * -0.041666666666666664);
	} else {
		tmp = t_0 / (x * x);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 1.0d0 - cos(x)
    if (x <= (-0.0048d0)) then
        tmp = (t_0 / x) / x
    else if (x <= 0.0053d0) then
        tmp = 0.5d0 + ((x * x) * (-0.041666666666666664d0))
    else
        tmp = t_0 / (x * x)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double t_0 = 1.0 - Math.cos(x);
	double tmp;
	if (x <= -0.0048) {
		tmp = (t_0 / x) / x;
	} else if (x <= 0.0053) {
		tmp = 0.5 + ((x * x) * -0.041666666666666664);
	} else {
		tmp = t_0 / (x * x);
	}
	return tmp;
}

Python

def code(x):
	t_0 = 1.0 - math.cos(x)
	tmp = 0
	if x <= -0.0048:
		tmp = (t_0 / x) / x
	elif x <= 0.0053:
		tmp = 0.5 + ((x * x) * -0.041666666666666664)
	else:
		tmp = t_0 / (x * x)
	return tmp

Julia

function code(x)
	t_0 = Float64(1.0 - cos(x))
	tmp = 0.0
	if (x <= -0.0048)
		tmp = Float64(Float64(t_0 / x) / x);
	elseif (x <= 0.0053)
		tmp = Float64(0.5 + Float64(Float64(x * x) * -0.041666666666666664));
	else
		tmp = Float64(t_0 / Float64(x * x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = 1.0 - cos(x);
	tmp = 0.0;
	if (x <= -0.0048)
		tmp = (t_0 / x) / x;
	elseif (x <= 0.0053)
		tmp = 0.5 + ((x * x) * -0.041666666666666664);
	else
		tmp = t_0 / (x * x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[(1.0 - N[Cos[x], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -0.0048], N[(N[(t$95$0 / x), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[x, 0.0053], N[(0.5 + N[(N[(x * x), $MachinePrecision] * -0.041666666666666664), $MachinePrecision]), $MachinePrecision], N[(t$95$0 / N[(x * x), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.00479999999999999958

   1. Initial program 97.3%

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

      1. clear-num 97.3%

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

      2. associate-/r/ 97.3%

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

      3. pow2 97.3%

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

      4. pow-flip 98.9%

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

      5. metadata-eval 98.9%

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

   3. Applied egg-rr 98.9%

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

      1. metadata-eval 98.9%

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

      2. pow-flip 97.3%

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

      3. pow2 97.3%

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

      4. associate-/r/ 97.3%

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

      5. clear-num 97.3%

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

      6. associate-/r* 98.8%

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

   5. Applied egg-rr 98.8%

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

   if -0.00479999999999999958 < x < 0.00530000000000000002

   1. Initial program 1.9%

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

   2. Taylor expanded in x around 0 100.0%

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

      1. unpow2 100.0%

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

   4. Simplified 100.0%

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

   if 0.00530000000000000002 < x

   1. Initial program 99.3%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.0048:\\ \;\;\;\;\frac{\frac{1 - \cos x}{x}}{x}\\ \mathbf{elif}\;x \leq 0.0053:\\ \;\;\;\;0.5 + \left(x \cdot x\right) \cdot -0.041666666666666664\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \cos x}{x \cdot x}\\ \end{array} \]

Alternative 7: 99.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 - \cos x\\ \mathbf{if}\;x \leq -0.0048:\\ \;\;\;\;\frac{\frac{1}{x}}{\frac{x}{t_0}}\\ \mathbf{elif}\;x \leq 0.0053:\\ \;\;\;\;0.5 + \left(x \cdot x\right) \cdot -0.041666666666666664\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0}{x \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (- 1.0 (cos x))))
   (if (<= x -0.0048)
     (/ (/ 1.0 x) (/ x t_0))
     (if (<= x 0.0053)
       (+ 0.5 (* (* x x) -0.041666666666666664))
       (/ t_0 (* x x))))))

C

double code(double x) {
	double t_0 = 1.0 - cos(x);
	double tmp;
	if (x <= -0.0048) {
		tmp = (1.0 / x) / (x / t_0);
	} else if (x <= 0.0053) {
		tmp = 0.5 + ((x * x) * -0.041666666666666664);
	} else {
		tmp = t_0 / (x * x);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 1.0d0 - cos(x)
    if (x <= (-0.0048d0)) then
        tmp = (1.0d0 / x) / (x / t_0)
    else if (x <= 0.0053d0) then
        tmp = 0.5d0 + ((x * x) * (-0.041666666666666664d0))
    else
        tmp = t_0 / (x * x)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double t_0 = 1.0 - Math.cos(x);
	double tmp;
	if (x <= -0.0048) {
		tmp = (1.0 / x) / (x / t_0);
	} else if (x <= 0.0053) {
		tmp = 0.5 + ((x * x) * -0.041666666666666664);
	} else {
		tmp = t_0 / (x * x);
	}
	return tmp;
}

Python

def code(x):
	t_0 = 1.0 - math.cos(x)
	tmp = 0
	if x <= -0.0048:
		tmp = (1.0 / x) / (x / t_0)
	elif x <= 0.0053:
		tmp = 0.5 + ((x * x) * -0.041666666666666664)
	else:
		tmp = t_0 / (x * x)
	return tmp

Julia

function code(x)
	t_0 = Float64(1.0 - cos(x))
	tmp = 0.0
	if (x <= -0.0048)
		tmp = Float64(Float64(1.0 / x) / Float64(x / t_0));
	elseif (x <= 0.0053)
		tmp = Float64(0.5 + Float64(Float64(x * x) * -0.041666666666666664));
	else
		tmp = Float64(t_0 / Float64(x * x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = 1.0 - cos(x);
	tmp = 0.0;
	if (x <= -0.0048)
		tmp = (1.0 / x) / (x / t_0);
	elseif (x <= 0.0053)
		tmp = 0.5 + ((x * x) * -0.041666666666666664);
	else
		tmp = t_0 / (x * x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[(1.0 - N[Cos[x], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -0.0048], N[(N[(1.0 / x), $MachinePrecision] / N[(x / t$95$0), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.0053], N[(0.5 + N[(N[(x * x), $MachinePrecision] * -0.041666666666666664), $MachinePrecision]), $MachinePrecision], N[(t$95$0 / N[(x * x), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.00479999999999999958

   1. Initial program 97.3%

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

      1. associate-/r* 98.8%

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

      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. clear-num 98.8%

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

      2. associate-*l/ 98.8%

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

      3. *-un-lft-identity 98.8%

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

   5. Applied egg-rr 98.8%

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

   if -0.00479999999999999958 < x < 0.00530000000000000002

   1. Initial program 1.9%

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

   2. Taylor expanded in x around 0 100.0%

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

      1. unpow2 100.0%

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

   4. Simplified 100.0%

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

   if 0.00530000000000000002 < x

   1. Initial program 99.3%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.0048:\\ \;\;\;\;\frac{\frac{1}{x}}{\frac{x}{1 - \cos x}}\\ \mathbf{elif}\;x \leq 0.0053:\\ \;\;\;\;0.5 + \left(x \cdot x\right) \cdot -0.041666666666666664\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \cos x}{x \cdot x}\\ \end{array} \]

Alternative 8: 75.9% accurate, 8.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -9.5 \cdot 10^{+76} \lor \neg \left(x \leq 5.4 \cdot 10^{+76}\right):\\ \;\;\;\;\frac{\frac{1}{x} + \frac{-1}{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (or (<= x -9.5e+76) (not (<= x 5.4e+76)))
   (/ (+ (/ 1.0 x) (/ -1.0 x)) x)
   0.5))

C

double code(double x) {
	double tmp;
	if ((x <= -9.5e+76) || !(x <= 5.4e+76)) {
		tmp = ((1.0 / x) + (-1.0 / x)) / x;
	} else {
		tmp = 0.5;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((x <= (-9.5d+76)) .or. (.not. (x <= 5.4d+76))) then
        tmp = ((1.0d0 / x) + ((-1.0d0) / x)) / x
    else
        tmp = 0.5d0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if ((x <= -9.5e+76) || !(x <= 5.4e+76)) {
		tmp = ((1.0 / x) + (-1.0 / x)) / x;
	} else {
		tmp = 0.5;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if (x <= -9.5e+76) or not (x <= 5.4e+76):
		tmp = ((1.0 / x) + (-1.0 / x)) / x
	else:
		tmp = 0.5
	return tmp

Julia

function code(x)
	tmp = 0.0
	if ((x <= -9.5e+76) || !(x <= 5.4e+76))
		tmp = Float64(Float64(Float64(1.0 / x) + Float64(-1.0 / x)) / x);
	else
		tmp = 0.5;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if ((x <= -9.5e+76) || ~((x <= 5.4e+76)))
		tmp = ((1.0 / x) + (-1.0 / x)) / x;
	else
		tmp = 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[Or[LessEqual[x, -9.5e+76], N[Not[LessEqual[x, 5.4e+76]], $MachinePrecision]], N[(N[(N[(1.0 / x), $MachinePrecision] + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], 0.5]

Derivation

1. Split input into 2 regimes

2. if x < -9.5000000000000003e76 or 5.3999999999999998e76 < x

   1. Initial program 98.3%

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

      1. clear-num 98.2%

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

      2. associate-/r/ 98.2%

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

      3. pow2 98.2%

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

      4. pow-flip 99.4%

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

      5. metadata-eval 99.4%

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

   3. Applied egg-rr 99.4%

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

      1. metadata-eval 99.4%

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

      2. pow-flip 98.2%

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

      3. pow2 98.2%

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

      4. associate-/r/ 98.2%

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

      5. clear-num 98.3%

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

      6. associate-/r* 99.3%

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

   5. Applied egg-rr 99.3%

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

      1. *-un-lft-identity 99.3%

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

      2. associate-*l/ 99.3%

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

      3. sub-neg 99.3%

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

      4. distribute-lft-in 99.3%

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

      5. associate-/r/ 99.3%

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

      6. clear-num 99.3%

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

   7. Applied egg-rr 99.3%

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

   8. Taylor expanded in x around 0 66.4%

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

   if -9.5000000000000003e76 < x < 5.3999999999999998e76

   1. Initial program 18.0%

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

   2. Taylor expanded in x around 0 84.3%

      \[\leadsto \color{blue}{0.5} \]
3. Recombined 2 regimes into one program.

4. Final simplification 78.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -9.5 \cdot 10^{+76} \lor \neg \left(x \leq 5.4 \cdot 10^{+76}\right):\\ \;\;\;\;\frac{\frac{1}{x} + \frac{-1}{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \]

Alternative 9: 78.4% accurate, 8.2× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{1}{x}}{x \cdot 0.16666666666666666 + 2 \cdot \frac{1}{x}} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/ (/ 1.0 x) (+ (* x 0.16666666666666666) (* 2.0 (/ 1.0 x)))))

C

double code(double x) {
	return (1.0 / x) / ((x * 0.16666666666666666) + (2.0 * (1.0 / x)));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (1.0d0 / x) / ((x * 0.16666666666666666d0) + (2.0d0 * (1.0d0 / x)))
end function

Java

public static double code(double x) {
	return (1.0 / x) / ((x * 0.16666666666666666) + (2.0 * (1.0 / x)));
}

Python

def code(x):
	return (1.0 / x) / ((x * 0.16666666666666666) + (2.0 * (1.0 / x)))

Julia

function code(x)
	return Float64(Float64(1.0 / x) / Float64(Float64(x * 0.16666666666666666) + Float64(2.0 * Float64(1.0 / x))))
end

MATLAB

function tmp = code(x)
	tmp = (1.0 / x) / ((x * 0.16666666666666666) + (2.0 * (1.0 / x)));
end

Wolfram

code[x_] := N[(N[(1.0 / x), $MachinePrecision] / N[(N[(x * 0.16666666666666666), $MachinePrecision] + N[(2.0 * N[(1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 45.9%

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

   1. associate-/r* 47.1%

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

   2. div-inv 47.1%

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

3. Applied egg-rr 47.1%

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

   1. clear-num 47.1%

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

   2. associate-*l/ 47.2%

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

   3. *-un-lft-identity 47.2%

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

5. Applied egg-rr 47.2%

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

6. Taylor expanded in x around 0 80.2%

   \[\leadsto \frac{\frac{1}{x}}{\color{blue}{0.16666666666666666 \cdot x + 2 \cdot \frac{1}{x}}} \]

7. Final simplification 80.2%

   \[\leadsto \frac{\frac{1}{x}}{x \cdot 0.16666666666666666 + 2 \cdot \frac{1}{x}} \]

Alternative 10: 51.7% 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 45.9%

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

2. Taylor expanded in x around 0 56.2%

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

3. Final simplification 56.2%

   \[\leadsto 0.5 \]

Reproduce

herbie shell --seed 2023195 
(FPCore (x)
  :name "cos2 (problem 3.4.1)"
  :precision binary64
  (/ (- 1.0 (cos x)) (* x x)))