cos2 (problem 3.4.1)

Percentage Accurate: 50.5% → 99.8%

Time: 9.9s

Alternatives: 7

Speedup: 107.0×

Specification

Math

\[\begin{array}{l} \\ \frac{1 - \cos x}{x \cdot x} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ (- 1.0 (cos x)) (* x x)))

C

double code(double x) {
	return (1.0 - cos(x)) / (x * x);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (1.0d0 - cos(x)) / (x * x)
end function

Java

public static double code(double x) {
	return (1.0 - Math.cos(x)) / (x * x);
}

Python

def code(x):
	return (1.0 - math.cos(x)) / (x * x)

Julia

function code(x)
	return Float64(Float64(1.0 - cos(x)) / Float64(x * x))
end

MATLAB

function tmp = code(x)
	tmp = (1.0 - cos(x)) / (x * x);
end

Wolfram

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

Initial Program: 50.5% 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

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

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 51.0%

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

   1. flip-- 50.7%

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

   2. div-inv 50.6%

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

   3. metadata-eval 50.6%

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

   4. 1-sub-cos 75.6%

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

   5. pow2 75.6%

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

3. Applied egg-rr 75.6%

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

   1. unpow2 75.6%

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

   2. associate-*l* 75.6%

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

   3. associate-*r/ 75.6%

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

   4. *-rgt-identity 75.6%

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

   5. hang-0p-tan 76.0%

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

5. Simplified 76.0%

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

   1. *-commutative 76.0%

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

   2. times-frac 99.7%

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

   3. div-inv 99.7%

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

   4. metadata-eval 99.7%

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

7. Applied egg-rr 99.7%

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

8. Final simplification 99.7%

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

Alternative 2: 99.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.0054:\\ \;\;\;\;\frac{\frac{1}{x}}{-x} \cdot \left(\cos x + -1\right)\\ \mathbf{elif}\;x \leq 0.0053:\\ \;\;\;\;0.5 + \left(x \cdot x\right) \cdot -0.041666666666666664\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 - \cos x}{x}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -0.0054)
   (* (/ (/ 1.0 x) (- x)) (+ (cos x) -1.0))
   (if (<= x 0.0053)
     (+ 0.5 (* (* x x) -0.041666666666666664))
     (/ (/ (- 1.0 (cos x)) x) x))))

C

double code(double x) {
	double tmp;
	if (x <= -0.0054) {
		tmp = ((1.0 / x) / -x) * (cos(x) + -1.0);
	} else if (x <= 0.0053) {
		tmp = 0.5 + ((x * x) * -0.041666666666666664);
	} else {
		tmp = ((1.0 - cos(x)) / x) / x;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-0.0054d0)) then
        tmp = ((1.0d0 / x) / -x) * (cos(x) + (-1.0d0))
    else if (x <= 0.0053d0) then
        tmp = 0.5d0 + ((x * x) * (-0.041666666666666664d0))
    else
        tmp = ((1.0d0 - cos(x)) / x) / x
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= -0.0054) {
		tmp = ((1.0 / x) / -x) * (Math.cos(x) + -1.0);
	} else if (x <= 0.0053) {
		tmp = 0.5 + ((x * x) * -0.041666666666666664);
	} else {
		tmp = ((1.0 - Math.cos(x)) / x) / x;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -0.0054:
		tmp = ((1.0 / x) / -x) * (math.cos(x) + -1.0)
	elif x <= 0.0053:
		tmp = 0.5 + ((x * x) * -0.041666666666666664)
	else:
		tmp = ((1.0 - math.cos(x)) / x) / x
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -0.0054)
		tmp = Float64(Float64(Float64(1.0 / x) / Float64(-x)) * Float64(cos(x) + -1.0));
	elseif (x <= 0.0053)
		tmp = Float64(0.5 + Float64(Float64(x * x) * -0.041666666666666664));
	else
		tmp = Float64(Float64(Float64(1.0 - cos(x)) / x) / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -0.0054)
		tmp = ((1.0 / x) / -x) * (cos(x) + -1.0);
	elseif (x <= 0.0053)
		tmp = 0.5 + ((x * x) * -0.041666666666666664);
	else
		tmp = ((1.0 - cos(x)) / x) / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -0.0054], N[(N[(N[(1.0 / x), $MachinePrecision] / (-x)), $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.0053], N[(0.5 + N[(N[(x * x), $MachinePrecision] * -0.041666666666666664), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 - N[Cos[x], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] / x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.0054000000000000003

   1. Initial program 98.8%

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

      1. frac-2neg 98.8%

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

      2. div-inv 98.8%

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

      3. distribute-rgt-neg-in 98.8%

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

   3. Applied egg-rr 98.8%

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

      1. distribute-lft-neg-out 98.8%

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

      2. associate-/r* 98.8%

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

   5. Simplified 98.8%

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

   if -0.0054000000000000003 < x < 0.00530000000000000002

   1. Initial program 2.6%

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

   2. Taylor expanded in x around 0 99.9%

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

      1. *-commutative 99.9%

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

      2. unpow2 99.9%

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

   4. Simplified 99.9%

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

   if 0.00530000000000000002 < x

   1. Initial program 97.0%

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

      1. frac-2neg 97.0%

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

      2. div-inv 96.8%

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

      3. distribute-rgt-neg-in 96.8%

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

   3. Applied egg-rr 96.8%

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

      1. flip-- 96.2%

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

      2. metadata-eval 96.2%

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

      3. 1-sub-cos 96.2%

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

      4. unpow2 96.2%

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

      5. un-div-inv 96.3%

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

      6. div-inv 96.3%

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

      7. distribute-rgt-neg-out 96.3%

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

      8. frac-2neg 96.3%

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

      9. associate-/r* 98.7%

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

      10. un-div-inv 98.6%

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

      11. unpow2 98.6%

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

      12. 1-sub-cos 98.6%

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

      13. metadata-eval 98.6%

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

      14. flip-- 99.2%

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

   5. Applied egg-rr 99.2%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.0054:\\ \;\;\;\;\frac{\frac{1}{x}}{-x} \cdot \left(\cos x + -1\right)\\ \mathbf{elif}\;x \leq 0.0053:\\ \;\;\;\;0.5 + \left(x \cdot x\right) \cdot -0.041666666666666664\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 - \cos x}{x}}{x}\\ \end{array} \]

Alternative 3: 99.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.0054 \lor \neg \left(x \leq 0.0053\right):\\ \;\;\;\;\frac{1 - \cos x}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;0.5 + \left(x \cdot x\right) \cdot -0.041666666666666664\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (or (<= x -0.0054) (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 <= -0.0054) || !(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 <= (-0.0054d0)) .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 <= -0.0054) || !(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 <= -0.0054) 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 <= -0.0054) || !(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 <= -0.0054) || ~((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, -0.0054], 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 < -0.0054000000000000003 or 0.00530000000000000002 < x

   1. Initial program 97.9%

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

   if -0.0054000000000000003 < x < 0.00530000000000000002

   1. Initial program 2.6%

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

   2. Taylor expanded in x around 0 99.9%

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

      1. *-commutative 99.9%

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

      2. unpow2 99.9%

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

   4. Simplified 99.9%

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

4. Final simplification 98.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.0054 \lor \neg \left(x \leq 0.0053\right):\\ \;\;\;\;\frac{1 - \cos x}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;0.5 + \left(x \cdot x\right) \cdot -0.041666666666666664\\ \end{array} \]

Alternative 4: 99.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 - \cos x\\ \mathbf{if}\;x \leq -0.0054:\\ \;\;\;\;\frac{t_0}{x \cdot x}\\ \mathbf{elif}\;x \leq 0.0053:\\ \;\;\;\;0.5 + \left(x \cdot x\right) \cdot -0.041666666666666664\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{t_0}{x}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (- 1.0 (cos x))))
   (if (<= x -0.0054)
     (/ 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 <= -0.0054) {
		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 <= (-0.0054d0)) 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 <= -0.0054) {
		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 <= -0.0054:
		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 <= -0.0054)
		tmp = Float64(t_0 / Float64(x * x));
	elseif (x <= 0.0053)
		tmp = Float64(0.5 + Float64(Float64(x * x) * -0.041666666666666664));
	else
		tmp = Float64(Float64(t_0 / x) / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = 1.0 - cos(x);
	tmp = 0.0;
	if (x <= -0.0054)
		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, -0.0054], N[(t$95$0 / N[(x * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.0053], N[(0.5 + N[(N[(x * x), $MachinePrecision] * -0.041666666666666664), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$0 / x), $MachinePrecision] / x), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.0054000000000000003

   1. Initial program 98.8%

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

   if -0.0054000000000000003 < x < 0.00530000000000000002

   1. Initial program 2.6%

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

   2. Taylor expanded in x around 0 99.9%

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

      1. *-commutative 99.9%

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

      2. unpow2 99.9%

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

   4. Simplified 99.9%

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

   if 0.00530000000000000002 < x

   1. Initial program 97.0%

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

      1. frac-2neg 97.0%

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

      2. div-inv 96.8%

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

      3. distribute-rgt-neg-in 96.8%

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

   3. Applied egg-rr 96.8%

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

      1. flip-- 96.2%

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

      2. metadata-eval 96.2%

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

      3. 1-sub-cos 96.2%

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

      4. unpow2 96.2%

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

      5. un-div-inv 96.3%

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

      6. div-inv 96.3%

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

      7. distribute-rgt-neg-out 96.3%

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

      8. frac-2neg 96.3%

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

      9. associate-/r* 98.7%

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

      10. un-div-inv 98.6%

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

      11. unpow2 98.6%

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

      12. 1-sub-cos 98.6%

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

      13. metadata-eval 98.6%

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

      14. flip-- 99.2%

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

   5. Applied egg-rr 99.2%

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

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.0054:\\ \;\;\;\;\frac{1 - \cos x}{x \cdot x}\\ \mathbf{elif}\;x \leq 0.0053:\\ \;\;\;\;0.5 + \left(x \cdot x\right) \cdot -0.041666666666666664\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 - \cos x}{x}}{x}\\ \end{array} \]

Alternative 5: 99.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.0054:\\ \;\;\;\;\left(\cos x + -1\right) \cdot \frac{-1}{x \cdot x}\\ \mathbf{elif}\;x \leq 0.0053:\\ \;\;\;\;0.5 + \left(x \cdot x\right) \cdot -0.041666666666666664\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 - \cos x}{x}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -0.0054)
   (* (+ (cos x) -1.0) (/ -1.0 (* x x)))
   (if (<= x 0.0053)
     (+ 0.5 (* (* x x) -0.041666666666666664))
     (/ (/ (- 1.0 (cos x)) x) x))))

C

double code(double x) {
	double tmp;
	if (x <= -0.0054) {
		tmp = (cos(x) + -1.0) * (-1.0 / (x * x));
	} else if (x <= 0.0053) {
		tmp = 0.5 + ((x * x) * -0.041666666666666664);
	} else {
		tmp = ((1.0 - cos(x)) / x) / x;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-0.0054d0)) then
        tmp = (cos(x) + (-1.0d0)) * ((-1.0d0) / (x * x))
    else if (x <= 0.0053d0) then
        tmp = 0.5d0 + ((x * x) * (-0.041666666666666664d0))
    else
        tmp = ((1.0d0 - cos(x)) / x) / x
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= -0.0054) {
		tmp = (Math.cos(x) + -1.0) * (-1.0 / (x * x));
	} else if (x <= 0.0053) {
		tmp = 0.5 + ((x * x) * -0.041666666666666664);
	} else {
		tmp = ((1.0 - Math.cos(x)) / x) / x;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -0.0054:
		tmp = (math.cos(x) + -1.0) * (-1.0 / (x * x))
	elif x <= 0.0053:
		tmp = 0.5 + ((x * x) * -0.041666666666666664)
	else:
		tmp = ((1.0 - math.cos(x)) / x) / x
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -0.0054)
		tmp = Float64(Float64(cos(x) + -1.0) * Float64(-1.0 / Float64(x * x)));
	elseif (x <= 0.0053)
		tmp = Float64(0.5 + Float64(Float64(x * x) * -0.041666666666666664));
	else
		tmp = Float64(Float64(Float64(1.0 - cos(x)) / x) / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -0.0054)
		tmp = (cos(x) + -1.0) * (-1.0 / (x * x));
	elseif (x <= 0.0053)
		tmp = 0.5 + ((x * x) * -0.041666666666666664);
	else
		tmp = ((1.0 - cos(x)) / x) / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -0.0054], N[(N[(N[Cos[x], $MachinePrecision] + -1.0), $MachinePrecision] * N[(-1.0 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.0053], N[(0.5 + N[(N[(x * x), $MachinePrecision] * -0.041666666666666664), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 - N[Cos[x], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] / x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.0054000000000000003

   1. Initial program 98.8%

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

      1. frac-2neg 98.8%

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

      2. div-inv 98.8%

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

      3. distribute-rgt-neg-in 98.8%

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

   3. Applied egg-rr 98.8%

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

   4. Taylor expanded in x around 0 98.8%

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

      1. unpow2 98.8%

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

   6. Simplified 98.8%

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

   if -0.0054000000000000003 < x < 0.00530000000000000002

   1. Initial program 2.6%

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

   2. Taylor expanded in x around 0 99.9%

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

      1. *-commutative 99.9%

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

      2. unpow2 99.9%

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

   4. Simplified 99.9%

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

   if 0.00530000000000000002 < x

   1. Initial program 97.0%

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

      1. frac-2neg 97.0%

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

      2. div-inv 96.8%

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

      3. distribute-rgt-neg-in 96.8%

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

   3. Applied egg-rr 96.8%

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

      1. flip-- 96.2%

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

      2. metadata-eval 96.2%

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

      3. 1-sub-cos 96.2%

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

      4. unpow2 96.2%

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

      5. un-div-inv 96.3%

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

      6. div-inv 96.3%

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

      7. distribute-rgt-neg-out 96.3%

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

      8. frac-2neg 96.3%

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

      9. associate-/r* 98.7%

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

      10. un-div-inv 98.6%

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

      11. unpow2 98.6%

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

      12. 1-sub-cos 98.6%

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

      13. metadata-eval 98.6%

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

      14. flip-- 99.2%

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

   5. Applied egg-rr 99.2%

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

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.0054:\\ \;\;\;\;\left(\cos x + -1\right) \cdot \frac{-1}{x \cdot x}\\ \mathbf{elif}\;x \leq 0.0053:\\ \;\;\;\;0.5 + \left(x \cdot x\right) \cdot -0.041666666666666664\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 - \cos x}{x}}{x}\\ \end{array} \]

Alternative 6: 52.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 51.0%

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

   1. flip-- 50.7%

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

   2. div-inv 50.6%

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

   3. metadata-eval 50.6%

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

   4. 1-sub-cos 75.6%

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

   5. pow2 75.6%

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

3. Applied egg-rr 75.6%

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

   1. unpow2 75.6%

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

   2. associate-*l* 75.6%

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

   3. associate-*r/ 75.6%

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

   4. *-rgt-identity 75.6%

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

   5. hang-0p-tan 76.0%

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

5. Simplified 76.0%

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

   1. *-commutative 76.0%

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

   2. times-frac 99.7%

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

   3. div-inv 99.7%

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

   4. metadata-eval 99.7%

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

7. Applied egg-rr 99.7%

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

8. Taylor expanded in x around 0 51.8%

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

9. Final simplification 51.8%

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

Alternative 7: 51.8% 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 51.0%

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

2. Taylor expanded in x around 0 51.1%

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

3. Final simplification 51.1%

   \[\leadsto 0.5 \]

Reproduce

herbie shell --seed 2023187 
(FPCore (x)
  :name "cos2 (problem 3.4.1)"
  :precision binary64
  (/ (- 1.0 (cos x)) (* x x)))