cos2 (problem 3.4.1)

Percentage Accurate: 51.0% → 99.8%

Time: 16.4s

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: 51.0% 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, 0.3× speedup

Math

\[\begin{array}{l} x = |x|\\ \\ \begin{array}{l} \mathbf{if}\;x \leq 0.0005:\\ \;\;\;\;0.5 + \left(x \cdot x\right) \cdot -0.041666666666666664\\ \mathbf{else}:\\ \;\;\;\;\left(\sin x \cdot \tan \left(\frac{x}{2}\right)\right) \cdot {x}^{-2}\\ \end{array} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
(FPCore (x)
 :precision binary64
 (if (<= x 0.0005)
   (+ 0.5 (* (* x x) -0.041666666666666664))
   (* (* (sin x) (tan (/ x 2.0))) (pow x -2.0))))

C

x = abs(x);
double code(double x) {
	double tmp;
	if (x <= 0.0005) {
		tmp = 0.5 + ((x * x) * -0.041666666666666664);
	} else {
		tmp = (sin(x) * tan((x / 2.0))) * pow(x, -2.0);
	}
	return tmp;
}

Fortran

NOTE: x should be positive before calling this function
real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 0.0005d0) then
        tmp = 0.5d0 + ((x * x) * (-0.041666666666666664d0))
    else
        tmp = (sin(x) * tan((x / 2.0d0))) * (x ** (-2.0d0))
    end if
    code = tmp
end function

Java

x = Math.abs(x);
public static double code(double x) {
	double tmp;
	if (x <= 0.0005) {
		tmp = 0.5 + ((x * x) * -0.041666666666666664);
	} else {
		tmp = (Math.sin(x) * Math.tan((x / 2.0))) * Math.pow(x, -2.0);
	}
	return tmp;
}

Python

x = abs(x)
def code(x):
	tmp = 0
	if x <= 0.0005:
		tmp = 0.5 + ((x * x) * -0.041666666666666664)
	else:
		tmp = (math.sin(x) * math.tan((x / 2.0))) * math.pow(x, -2.0)
	return tmp

Julia

x = abs(x)
function code(x)
	tmp = 0.0
	if (x <= 0.0005)
		tmp = Float64(0.5 + Float64(Float64(x * x) * -0.041666666666666664));
	else
		tmp = Float64(Float64(sin(x) * tan(Float64(x / 2.0))) * (x ^ -2.0));
	end
	return tmp
end

MATLAB

x = abs(x)
function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 0.0005)
		tmp = 0.5 + ((x * x) * -0.041666666666666664);
	else
		tmp = (sin(x) * tan((x / 2.0))) * (x ^ -2.0);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x should be positive before calling this function
code[x_] := If[LessEqual[x, 0.0005], N[(0.5 + N[(N[(x * x), $MachinePrecision] * -0.041666666666666664), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sin[x], $MachinePrecision] * N[Tan[N[(x / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Power[x, -2.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 5.0000000000000001e-4

   1. Initial program 31.6%

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

   2. Taylor expanded in x around 0 69.6%

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

      1. *-commutative 69.6%

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

   4. Simplified 69.6%

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

      1. pow2 69.6%

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

   6. Applied egg-rr 69.6%

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

   if 5.0000000000000001e-4 < x

   1. Initial program 98.8%

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

      1. flip-- 98.3%

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

      2. clear-num 98.2%

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

      3. metadata-eval 98.2%

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

      4. pow2 98.2%

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

   3. Applied egg-rr 98.2%

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

      1. unpow2 98.2%

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

      2. 1-sub-cos 98.5%

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

   5. Applied egg-rr 98.5%

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

      1. div-inv 98.5%

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

      2. associate-/r/ 98.5%

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

      3. associate-*l* 98.4%

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

      4. pow2 98.4%

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

      5. pow2 98.4%

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

      6. pow-flip 99.0%

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

      7. metadata-eval 99.0%

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

   7. Applied egg-rr 99.0%

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

      1. associate-*l/ 99.1%

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

      2. *-lft-identity 99.1%

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

      3. associate-*l/ 99.0%

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

      4. unpow2 99.0%

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

      5. associate-*r/ 99.0%

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

      6. hang-0p-tan 99.6%

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

   9. Simplified 99.6%

      \[\leadsto \color{blue}{\left(\sin x \cdot \tan \left(\frac{x}{2}\right)\right) \cdot {x}^{-2}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 77.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.0005:\\ \;\;\;\;0.5 + \left(x \cdot x\right) \cdot -0.041666666666666664\\ \mathbf{else}:\\ \;\;\;\;\left(\sin x \cdot \tan \left(\frac{x}{2}\right)\right) \cdot {x}^{-2}\\ \end{array} \]

Alternative 2: 99.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} x = |x|\\ \\ \begin{array}{l} \mathbf{if}\;x \leq 0.0004:\\ \;\;\;\;0.5 + \left(x \cdot x\right) \cdot -0.041666666666666664\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin x \cdot \tan \left(\frac{x}{2}\right)}{x} \cdot \frac{1}{x}\\ \end{array} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
(FPCore (x)
 :precision binary64
 (if (<= x 0.0004)
   (+ 0.5 (* (* x x) -0.041666666666666664))
   (* (/ (* (sin x) (tan (/ x 2.0))) x) (/ 1.0 x))))

C

x = abs(x);
double code(double x) {
	double tmp;
	if (x <= 0.0004) {
		tmp = 0.5 + ((x * x) * -0.041666666666666664);
	} else {
		tmp = ((sin(x) * tan((x / 2.0))) / x) * (1.0 / x);
	}
	return tmp;
}

Fortran

NOTE: x should be positive before calling this function
real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 0.0004d0) then
        tmp = 0.5d0 + ((x * x) * (-0.041666666666666664d0))
    else
        tmp = ((sin(x) * tan((x / 2.0d0))) / x) * (1.0d0 / x)
    end if
    code = tmp
end function

Java

x = Math.abs(x);
public static double code(double x) {
	double tmp;
	if (x <= 0.0004) {
		tmp = 0.5 + ((x * x) * -0.041666666666666664);
	} else {
		tmp = ((Math.sin(x) * Math.tan((x / 2.0))) / x) * (1.0 / x);
	}
	return tmp;
}

Python

x = abs(x)
def code(x):
	tmp = 0
	if x <= 0.0004:
		tmp = 0.5 + ((x * x) * -0.041666666666666664)
	else:
		tmp = ((math.sin(x) * math.tan((x / 2.0))) / x) * (1.0 / x)
	return tmp

Julia

x = abs(x)
function code(x)
	tmp = 0.0
	if (x <= 0.0004)
		tmp = Float64(0.5 + Float64(Float64(x * x) * -0.041666666666666664));
	else
		tmp = Float64(Float64(Float64(sin(x) * tan(Float64(x / 2.0))) / x) * Float64(1.0 / x));
	end
	return tmp
end

MATLAB

x = abs(x)
function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 0.0004)
		tmp = 0.5 + ((x * x) * -0.041666666666666664);
	else
		tmp = ((sin(x) * tan((x / 2.0))) / x) * (1.0 / x);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x should be positive before calling this function
code[x_] := If[LessEqual[x, 0.0004], N[(0.5 + N[(N[(x * x), $MachinePrecision] * -0.041666666666666664), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[Sin[x], $MachinePrecision] * N[Tan[N[(x / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] * N[(1.0 / x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 4.00000000000000019e-4

   1. Initial program 31.6%

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

   2. Taylor expanded in x around 0 69.6%

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

      1. *-commutative 69.6%

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

   4. Simplified 69.6%

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

      1. pow2 69.6%

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

   6. Applied egg-rr 69.6%

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

   if 4.00000000000000019e-4 < x

   1. Initial program 98.8%

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

      1. associate-/r* 99.4%

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

      2. div-inv 99.3%

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

   3. Applied egg-rr 99.3%

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

      1. clear-num 99.2%

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

      2. associate-/r/ 99.2%

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

   5. Applied egg-rr 99.2%

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

      1. flip-- 98.6%

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

      2. metadata-eval 98.6%

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

      3. 1-sub-cos 98.9%

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

      4. frac-times 99.0%

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

      5. *-un-lft-identity 99.0%

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

      6. pow2 99.0%

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

   7. Applied egg-rr 99.0%

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

      1. *-commutative 99.0%

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

      2. associate-/r* 98.9%

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

      3. unpow2 98.9%

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

      4. associate-*r/ 99.0%

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

      5. hang-0p-tan 99.6%

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

   9. Simplified 99.6%

      \[\leadsto \color{blue}{\frac{\sin x \cdot \tan \left(\frac{x}{2}\right)}{x}} \cdot \frac{1}{x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 77.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.0004:\\ \;\;\;\;0.5 + \left(x \cdot x\right) \cdot -0.041666666666666664\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin x \cdot \tan \left(\frac{x}{2}\right)}{x} \cdot \frac{1}{x}\\ \end{array} \]

Alternative 3: 99.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} x = |x|\\ \\ \begin{array}{l} \mathbf{if}\;x \leq 0.005:\\ \;\;\;\;0.5 + \left(x \cdot x\right) \cdot -0.041666666666666664\\ \mathbf{else}:\\ \;\;\;\;{x}^{-2} \cdot \left(1 - \cos x\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
(FPCore (x)
 :precision binary64
 (if (<= x 0.005)
   (+ 0.5 (* (* x x) -0.041666666666666664))
   (* (pow x -2.0) (- 1.0 (cos x)))))

C

x = abs(x);
double code(double x) {
	double tmp;
	if (x <= 0.005) {
		tmp = 0.5 + ((x * x) * -0.041666666666666664);
	} else {
		tmp = pow(x, -2.0) * (1.0 - cos(x));
	}
	return tmp;
}

Fortran

NOTE: x should be positive before calling this function
real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 0.005d0) then
        tmp = 0.5d0 + ((x * x) * (-0.041666666666666664d0))
    else
        tmp = (x ** (-2.0d0)) * (1.0d0 - cos(x))
    end if
    code = tmp
end function

Java

x = Math.abs(x);
public static double code(double x) {
	double tmp;
	if (x <= 0.005) {
		tmp = 0.5 + ((x * x) * -0.041666666666666664);
	} else {
		tmp = Math.pow(x, -2.0) * (1.0 - Math.cos(x));
	}
	return tmp;
}

Python

x = abs(x)
def code(x):
	tmp = 0
	if x <= 0.005:
		tmp = 0.5 + ((x * x) * -0.041666666666666664)
	else:
		tmp = math.pow(x, -2.0) * (1.0 - math.cos(x))
	return tmp

Julia

x = abs(x)
function code(x)
	tmp = 0.0
	if (x <= 0.005)
		tmp = Float64(0.5 + Float64(Float64(x * x) * -0.041666666666666664));
	else
		tmp = Float64((x ^ -2.0) * Float64(1.0 - cos(x)));
	end
	return tmp
end

MATLAB

x = abs(x)
function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 0.005)
		tmp = 0.5 + ((x * x) * -0.041666666666666664);
	else
		tmp = (x ^ -2.0) * (1.0 - cos(x));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x should be positive before calling this function
code[x_] := If[LessEqual[x, 0.005], N[(0.5 + N[(N[(x * x), $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 < 0.0050000000000000001

   1. Initial program 31.6%

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

   2. Taylor expanded in x around 0 69.6%

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

      1. *-commutative 69.6%

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

   4. Simplified 69.6%

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

      1. pow2 69.6%

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

   6. Applied egg-rr 69.6%

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

   if 0.0050000000000000001 < x

   1. Initial program 98.8%

      \[\frac{1 - \cos x}{x \cdot x} \]
   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 99.3%

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

      5. metadata-eval 99.3%

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

   3. Applied egg-rr 99.3%

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

4. Final simplification 77.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.005:\\ \;\;\;\;0.5 + \left(x \cdot x\right) \cdot -0.041666666666666664\\ \mathbf{else}:\\ \;\;\;\;{x}^{-2} \cdot \left(1 - \cos x\right)\\ \end{array} \]

Alternative 4: 99.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} x = |x|\\ \\ \begin{array}{l} \mathbf{if}\;x \leq 0.005:\\ \;\;\;\;0.5 + \left(x \cdot x\right) \cdot -0.041666666666666664\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \cos x}{x \cdot x}\\ \end{array} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
(FPCore (x)
 :precision binary64
 (if (<= x 0.005)
   (+ 0.5 (* (* x x) -0.041666666666666664))
   (/ (- 1.0 (cos x)) (* x x))))

C

x = abs(x);
double code(double x) {
	double tmp;
	if (x <= 0.005) {
		tmp = 0.5 + ((x * x) * -0.041666666666666664);
	} else {
		tmp = (1.0 - cos(x)) / (x * x);
	}
	return tmp;
}

Fortran

NOTE: x should be positive before calling this function
real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 0.005d0) then
        tmp = 0.5d0 + ((x * x) * (-0.041666666666666664d0))
    else
        tmp = (1.0d0 - cos(x)) / (x * x)
    end if
    code = tmp
end function

Java

x = Math.abs(x);
public static double code(double x) {
	double tmp;
	if (x <= 0.005) {
		tmp = 0.5 + ((x * x) * -0.041666666666666664);
	} else {
		tmp = (1.0 - Math.cos(x)) / (x * x);
	}
	return tmp;
}

Python

x = abs(x)
def code(x):
	tmp = 0
	if x <= 0.005:
		tmp = 0.5 + ((x * x) * -0.041666666666666664)
	else:
		tmp = (1.0 - math.cos(x)) / (x * x)
	return tmp

Julia

x = abs(x)
function code(x)
	tmp = 0.0
	if (x <= 0.005)
		tmp = Float64(0.5 + Float64(Float64(x * x) * -0.041666666666666664));
	else
		tmp = Float64(Float64(1.0 - cos(x)) / Float64(x * x));
	end
	return tmp
end

MATLAB

x = abs(x)
function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 0.005)
		tmp = 0.5 + ((x * x) * -0.041666666666666664);
	else
		tmp = (1.0 - cos(x)) / (x * x);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x should be positive before calling this function
code[x_] := If[LessEqual[x, 0.005], N[(0.5 + N[(N[(x * x), $MachinePrecision] * -0.041666666666666664), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - N[Cos[x], $MachinePrecision]), $MachinePrecision] / N[(x * x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.0050000000000000001

   1. Initial program 31.6%

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

   2. Taylor expanded in x around 0 69.6%

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

      1. *-commutative 69.6%

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

   4. Simplified 69.6%

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

      1. pow2 69.6%

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

   6. Applied egg-rr 69.6%

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

   if 0.0050000000000000001 < x

   1. Initial program 98.8%

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

4. Final simplification 77.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.005:\\ \;\;\;\;0.5 + \left(x \cdot x\right) \cdot -0.041666666666666664\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \cos x}{x \cdot x}\\ \end{array} \]

Alternative 5: 99.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} x = |x|\\ \\ \begin{array}{l} \mathbf{if}\;x \leq 0.005:\\ \;\;\;\;0.5 + \left(x \cdot x\right) \cdot -0.041666666666666664\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 - \cos x}{x}}{x}\\ \end{array} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
(FPCore (x)
 :precision binary64
 (if (<= x 0.005)
   (+ 0.5 (* (* x x) -0.041666666666666664))
   (/ (/ (- 1.0 (cos x)) x) x)))

C

x = abs(x);
double code(double x) {
	double tmp;
	if (x <= 0.005) {
		tmp = 0.5 + ((x * x) * -0.041666666666666664);
	} else {
		tmp = ((1.0 - cos(x)) / x) / x;
	}
	return tmp;
}

Fortran

NOTE: x should be positive before calling this function
real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 0.005d0) then
        tmp = 0.5d0 + ((x * x) * (-0.041666666666666664d0))
    else
        tmp = ((1.0d0 - cos(x)) / x) / x
    end if
    code = tmp
end function

Java

x = Math.abs(x);
public static double code(double x) {
	double tmp;
	if (x <= 0.005) {
		tmp = 0.5 + ((x * x) * -0.041666666666666664);
	} else {
		tmp = ((1.0 - Math.cos(x)) / x) / x;
	}
	return tmp;
}

Python

x = abs(x)
def code(x):
	tmp = 0
	if x <= 0.005:
		tmp = 0.5 + ((x * x) * -0.041666666666666664)
	else:
		tmp = ((1.0 - math.cos(x)) / x) / x
	return tmp

Julia

x = abs(x)
function code(x)
	tmp = 0.0
	if (x <= 0.005)
		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

x = abs(x)
function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 0.005)
		tmp = 0.5 + ((x * x) * -0.041666666666666664);
	else
		tmp = ((1.0 - cos(x)) / x) / x;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x should be positive before calling this function
code[x_] := If[LessEqual[x, 0.005], N[(0.5 + N[(N[(x * x), $MachinePrecision] * -0.041666666666666664), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 - N[Cos[x], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.0050000000000000001

   1. Initial program 31.6%

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

   2. Taylor expanded in x around 0 69.6%

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

      1. *-commutative 69.6%

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

   4. Simplified 69.6%

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

      1. pow2 69.6%

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

   6. Applied egg-rr 69.6%

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

   if 0.0050000000000000001 < x

   1. Initial program 98.8%

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

      1. associate-/r* 99.4%

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

      2. div-inv 99.3%

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

   3. Applied egg-rr 99.3%

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

      1. un-div-inv 99.4%

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

   5. Applied egg-rr 99.4%

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

4. Final simplification 77.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.005:\\ \;\;\;\;0.5 + \left(x \cdot x\right) \cdot -0.041666666666666664\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 - \cos x}{x}}{x}\\ \end{array} \]

Alternative 6: 76.0% accurate, 34.9× speedup

Math

\[\begin{array}{l} x = |x|\\ \\ \begin{array}{l} \mathbf{if}\;x \leq 8.8 \cdot 10^{+76}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
(FPCore (x) :precision binary64 (if (<= x 8.8e+76) 0.5 0.0))

C

x = abs(x);
double code(double x) {
	double tmp;
	if (x <= 8.8e+76) {
		tmp = 0.5;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

NOTE: x should be positive before calling this function
real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 8.8d+76) then
        tmp = 0.5d0
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

x = Math.abs(x);
public static double code(double x) {
	double tmp;
	if (x <= 8.8e+76) {
		tmp = 0.5;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

x = abs(x)
def code(x):
	tmp = 0
	if x <= 8.8e+76:
		tmp = 0.5
	else:
		tmp = 0.0
	return tmp

Julia

x = abs(x)
function code(x)
	tmp = 0.0
	if (x <= 8.8e+76)
		tmp = 0.5;
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

x = abs(x)
function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 8.8e+76)
		tmp = 0.5;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x should be positive before calling this function
code[x_] := If[LessEqual[x, 8.8e+76], 0.5, 0.0]

Derivation

1. Split input into 2 regimes

2. if x < 8.8000000000000002e76

   1. Initial program 36.8%

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

   2. Taylor expanded in x around 0 65.2%

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

   if 8.8000000000000002e76 < x

   1. Initial program 98.8%

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

      1. add-log-exp 98.8%

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

   3. Applied egg-rr 98.8%

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

   4. Taylor expanded in x around 0 69.0%

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

   5. Taylor expanded in x around 0 69.0%

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

4. Final simplification 65.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 8.8 \cdot 10^{+76}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 7: 27.9% accurate, 107.0× speedup

Math

\[\begin{array}{l} x = |x|\\ \\ 0 \end{array} \]

FPCore

NOTE: x should be positive before calling this function
(FPCore (x) :precision binary64 0.0)

C

x = abs(x);
double code(double x) {
	return 0.0;
}

Fortran

NOTE: x should be positive before calling this function
real(8) function code(x)
    real(8), intent (in) :: x
    code = 0.0d0
end function

Java

x = Math.abs(x);
public static double code(double x) {
	return 0.0;
}

Python

x = abs(x)
def code(x):
	return 0.0

Julia

x = abs(x)
function code(x)
	return 0.0
end

MATLAB

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

Wolfram

NOTE: x should be positive before calling this function
code[x_] := 0.0

Derivation

1. Initial program 48.6%

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

   1. add-log-exp 48.6%

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

3. Applied egg-rr 48.6%

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

4. Taylor expanded in x around 0 27.9%

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

5. Taylor expanded in x around 0 28.6%

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

6. Final simplification 28.6%

   \[\leadsto 0 \]

Reproduce

herbie shell --seed 2023311 
(FPCore (x)
  :name "cos2 (problem 3.4.1)"
  :precision binary64
  (/ (- 1.0 (cos x)) (* x x)))