cos2 (problem 3.4.1)

Percentage Accurate: 50.3% → 99.8%

Time: 9.5s

Alternatives: 7

Speedup: 17.8×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 50.3% 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

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 54.2%

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

   1. associate-/r* 55.2%

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

   2. div-inv 55.2%

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

4. Applied egg-rr 55.2%

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

   1. clear-num 55.2%

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

   2. associate-/r/ 55.1%

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

6. Applied egg-rr 55.1%

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

   1. flip-- 54.8%

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

   2. metadata-eval 54.8%

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

   3. unpow2 54.8%

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

   4. unpow2 54.8%

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

   5. 1-sub-cos 76.3%

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

   6. frac-times 76.4%

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

   7. *-un-lft-identity 76.4%

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

   8. pow2 76.4%

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

8. Applied egg-rr 76.4%

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

   1. associate-/l/ 76.4%

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

   2. unpow2 76.4%

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

   3. associate-*r/ 76.4%

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

   4. hang-0p-tan 76.8%

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

10. Simplified 76.8%

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

    1. un-div-inv 76.9%

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

    2. associate-/l* 99.8%

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

    3. div-inv 99.8%

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

    4. metadata-eval 99.8%

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

12. Applied egg-rr 99.8%

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

Alternative 2: 76.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.0295:\\ \;\;\;\;0.5 + {x}^{2} \cdot \left(0.001388888888888889 \cdot \left(x \cdot x\right) - 0.041666666666666664\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x}}{\frac{x}{1 - \cos x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 0.0295)
   (+
    0.5
    (* (pow x 2.0) (- (* 0.001388888888888889 (* x x)) 0.041666666666666664)))
   (/ (/ 1.0 x) (/ x (- 1.0 (cos x))))))

C

double code(double x) {
	double tmp;
	if (x <= 0.0295) {
		tmp = 0.5 + (pow(x, 2.0) * ((0.001388888888888889 * (x * x)) - 0.041666666666666664));
	} else {
		tmp = (1.0 / x) / (x / (1.0 - cos(x)));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 0.0295d0) then
        tmp = 0.5d0 + ((x ** 2.0d0) * ((0.001388888888888889d0 * (x * x)) - 0.041666666666666664d0))
    else
        tmp = (1.0d0 / x) / (x / (1.0d0 - cos(x)))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 0.0295) {
		tmp = 0.5 + (Math.pow(x, 2.0) * ((0.001388888888888889 * (x * x)) - 0.041666666666666664));
	} else {
		tmp = (1.0 / x) / (x / (1.0 - Math.cos(x)));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 0.0295:
		tmp = 0.5 + (math.pow(x, 2.0) * ((0.001388888888888889 * (x * x)) - 0.041666666666666664))
	else:
		tmp = (1.0 / x) / (x / (1.0 - math.cos(x)))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 0.0295)
		tmp = Float64(0.5 + Float64((x ^ 2.0) * Float64(Float64(0.001388888888888889 * Float64(x * x)) - 0.041666666666666664)));
	else
		tmp = Float64(Float64(1.0 / x) / Float64(x / Float64(1.0 - cos(x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 0.0295)
		tmp = 0.5 + ((x ^ 2.0) * ((0.001388888888888889 * (x * x)) - 0.041666666666666664));
	else
		tmp = (1.0 / x) / (x / (1.0 - cos(x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 0.0295], N[(0.5 + N[(N[Power[x, 2.0], $MachinePrecision] * N[(N[(0.001388888888888889 * N[(x * x), $MachinePrecision]), $MachinePrecision] - 0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / x), $MachinePrecision] / N[(x / N[(1.0 - N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.029499999999999998

   1. Initial program 37.6%

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

   3. Taylor expanded in x around 0 65.0%

      \[\leadsto \color{blue}{0.5 + {x}^{2} \cdot \left(0.001388888888888889 \cdot {x}^{2} - 0.041666666666666664\right)} \]
   4. Step-by-step derivation

      1. unpow2 65.0%

         \[\leadsto 0.5 + {x}^{2} \cdot \left(0.001388888888888889 \cdot \color{blue}{\left(x \cdot x\right)} - 0.041666666666666664\right) \]

   5. Applied egg-rr 65.0%

      \[\leadsto 0.5 + {x}^{2} \cdot \left(0.001388888888888889 \cdot \color{blue}{\left(x \cdot x\right)} - 0.041666666666666664\right) \]

   if 0.029499999999999998 < x

   1. Initial program 98.3%

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

      1. clear-num 98.2%

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

      2. inv-pow 98.2%

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

      3. flip-- 97.7%

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

      4. associate-/r/ 97.7%

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

      5. unpow-prod-down 97.6%

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

      6. pow2 97.6%

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

      7. metadata-eval 97.6%

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

      8. pow2 97.6%

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

      9. inv-pow 97.6%

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

   4. Applied egg-rr 97.6%

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

      1. unpow-1 97.6%

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

   6. Simplified 97.6%

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

      1. frac-times 97.7%

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

      2. associate-/r/ 97.7%

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

      3. metadata-eval 97.7%

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

      4. unpow2 97.7%

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

      5. flip-- 98.2%

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

      6. unpow2 98.2%

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

      7. associate-/l* 98.2%

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

      8. frac-times 99.2%

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

      9. un-div-inv 99.4%

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

   8. Applied egg-rr 99.4%

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

Alternative 3: 76.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.0054:\\ \;\;\;\;0.5 + {x}^{2} \cdot -0.041666666666666664\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x}}{\frac{x}{1 - \cos x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 0.0054)
   (+ 0.5 (* (pow x 2.0) -0.041666666666666664))
   (/ (/ 1.0 x) (/ x (- 1.0 (cos x))))))

C

double code(double x) {
	double tmp;
	if (x <= 0.0054) {
		tmp = 0.5 + (pow(x, 2.0) * -0.041666666666666664);
	} else {
		tmp = (1.0 / x) / (x / (1.0 - cos(x)));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 0.0054d0) then
        tmp = 0.5d0 + ((x ** 2.0d0) * (-0.041666666666666664d0))
    else
        tmp = (1.0d0 / x) / (x / (1.0d0 - cos(x)))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 0.0054) {
		tmp = 0.5 + (Math.pow(x, 2.0) * -0.041666666666666664);
	} else {
		tmp = (1.0 / x) / (x / (1.0 - Math.cos(x)));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 0.0054:
		tmp = 0.5 + (math.pow(x, 2.0) * -0.041666666666666664)
	else:
		tmp = (1.0 / x) / (x / (1.0 - math.cos(x)))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 0.0054)
		tmp = Float64(0.5 + Float64((x ^ 2.0) * -0.041666666666666664));
	else
		tmp = Float64(Float64(1.0 / x) / Float64(x / Float64(1.0 - cos(x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 0.0054)
		tmp = 0.5 + ((x ^ 2.0) * -0.041666666666666664);
	else
		tmp = (1.0 / x) / (x / (1.0 - cos(x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 0.0054], N[(0.5 + N[(N[Power[x, 2.0], $MachinePrecision] * -0.041666666666666664), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / x), $MachinePrecision] / N[(x / N[(1.0 - N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.0054000000000000003

   1. Initial program 37.6%

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

   3. Taylor expanded in x around 0 64.4%

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

   if 0.0054000000000000003 < x

   1. Initial program 98.3%

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

      1. clear-num 98.2%

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

      2. inv-pow 98.2%

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

      3. flip-- 97.7%

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

      4. associate-/r/ 97.7%

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

      5. unpow-prod-down 97.6%

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

      6. pow2 97.6%

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

      7. metadata-eval 97.6%

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

      8. pow2 97.6%

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

      9. inv-pow 97.6%

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

   4. Applied egg-rr 97.6%

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

      1. unpow-1 97.6%

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

   6. Simplified 97.6%

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

      1. frac-times 97.7%

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

      2. associate-/r/ 97.7%

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

      3. metadata-eval 97.7%

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

      4. unpow2 97.7%

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

      5. flip-- 98.2%

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

      6. unpow2 98.2%

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

      7. associate-/l* 98.2%

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

      8. frac-times 99.2%

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

      9. un-div-inv 99.4%

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

   8. Applied egg-rr 99.4%

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

4. Final simplification 74.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.0054:\\ \;\;\;\;0.5 + {x}^{2} \cdot -0.041666666666666664\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x}}{\frac{x}{1 - \cos x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 76.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.0054:\\ \;\;\;\;0.5 + {x}^{2} \cdot -0.041666666666666664\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 - \cos x}{x}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 0.0054)
   (+ 0.5 (* (pow x 2.0) -0.041666666666666664))
   (/ (/ (- 1.0 (cos x)) x) x)))

C

double code(double x) {
	double tmp;
	if (x <= 0.0054) {
		tmp = 0.5 + (pow(x, 2.0) * -0.041666666666666664);
	} else {
		tmp = ((1.0 - cos(x)) / x) / x;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 0.0054d0) then
        tmp = 0.5d0 + ((x ** 2.0d0) * (-0.041666666666666664d0))
    else
        tmp = ((1.0d0 - cos(x)) / x) / x
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 0.0054) {
		tmp = 0.5 + (Math.pow(x, 2.0) * -0.041666666666666664);
	} else {
		tmp = ((1.0 - Math.cos(x)) / x) / x;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 0.0054:
		tmp = 0.5 + (math.pow(x, 2.0) * -0.041666666666666664)
	else:
		tmp = ((1.0 - math.cos(x)) / x) / x
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 0.0054)
		tmp = Float64(0.5 + Float64((x ^ 2.0) * -0.041666666666666664));
	else
		tmp = Float64(Float64(Float64(1.0 - cos(x)) / x) / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 0.0054)
		tmp = 0.5 + ((x ^ 2.0) * -0.041666666666666664);
	else
		tmp = ((1.0 - cos(x)) / x) / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 0.0054], N[(0.5 + N[(N[Power[x, 2.0], $MachinePrecision] * -0.041666666666666664), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 - N[Cos[x], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.0054000000000000003

   1. Initial program 37.6%

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

   3. Taylor expanded in x around 0 64.4%

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

   if 0.0054000000000000003 < x

   1. Initial program 98.3%

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

      1. clear-num 98.2%

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

      2. inv-pow 98.2%

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

      3. flip-- 97.7%

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

      4. associate-/r/ 97.7%

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

      5. unpow-prod-down 97.6%

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

      6. pow2 97.6%

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

      7. metadata-eval 97.6%

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

      8. pow2 97.6%

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

      9. inv-pow 97.6%

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

   4. Applied egg-rr 97.6%

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

      1. unpow-1 97.6%

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

   6. Simplified 97.6%

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

      1. frac-times 97.7%

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

      2. associate-/r/ 97.7%

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

      3. metadata-eval 97.7%

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

      4. unpow2 97.7%

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

      5. flip-- 98.2%

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

      6. metadata-eval 98.2%

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

      7. clear-num 98.3%

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

      8. unpow2 98.3%

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

      9. associate-/r* 99.3%

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

   8. Applied egg-rr 99.3%

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

4. Final simplification 74.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.0054:\\ \;\;\;\;0.5 + {x}^{2} \cdot -0.041666666666666664\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 - \cos x}{x}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 75.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.0054:\\ \;\;\;\;0.5 + {x}^{2} \cdot -0.041666666666666664\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \cos x}{x \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 0.0054)
   (+ 0.5 (* (pow x 2.0) -0.041666666666666664))
   (/ (- 1.0 (cos x)) (* x x))))

C

double code(double x) {
	double tmp;
	if (x <= 0.0054) {
		tmp = 0.5 + (pow(x, 2.0) * -0.041666666666666664);
	} else {
		tmp = (1.0 - cos(x)) / (x * x);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 0.0054d0) then
        tmp = 0.5d0 + ((x ** 2.0d0) * (-0.041666666666666664d0))
    else
        tmp = (1.0d0 - cos(x)) / (x * x)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 0.0054) {
		tmp = 0.5 + (Math.pow(x, 2.0) * -0.041666666666666664);
	} else {
		tmp = (1.0 - Math.cos(x)) / (x * x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 0.0054:
		tmp = 0.5 + (math.pow(x, 2.0) * -0.041666666666666664)
	else:
		tmp = (1.0 - math.cos(x)) / (x * x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 0.0054)
		tmp = Float64(0.5 + Float64((x ^ 2.0) * -0.041666666666666664));
	else
		tmp = Float64(Float64(1.0 - cos(x)) / Float64(x * x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 0.0054)
		tmp = 0.5 + ((x ^ 2.0) * -0.041666666666666664);
	else
		tmp = (1.0 - cos(x)) / (x * x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 0.0054], N[(0.5 + N[(N[Power[x, 2.0], $MachinePrecision] * -0.041666666666666664), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - N[Cos[x], $MachinePrecision]), $MachinePrecision] / N[(x * x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.0054000000000000003

   1. Initial program 37.6%

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

   3. Taylor expanded in x around 0 64.4%

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

   if 0.0054000000000000003 < x

   1. Initial program 98.3%

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

4. Final simplification 73.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.0054:\\ \;\;\;\;0.5 + {x}^{2} \cdot -0.041666666666666664\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \cos x}{x \cdot x}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 64.1% accurate, 17.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 9.8 \cdot 10^{+76}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (if (<= x 9.8e+76) 0.5 0.0))

C

double code(double x) {
	double tmp;
	if (x <= 9.8e+76) {
		tmp = 0.5;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 9.8d+76) then
        tmp = 0.5d0
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 9.8e+76) {
		tmp = 0.5;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 9.8e+76:
		tmp = 0.5
	else:
		tmp = 0.0
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 9.8e+76)
		tmp = 0.5;
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 9.8e+76)
		tmp = 0.5;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 9.8e+76], 0.5, 0.0]

Derivation

1. Split input into 2 regimes

2. if x < 9.80000000000000053e76

   1. Initial program 43.8%

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

   3. Taylor expanded in x around 0 59.2%

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

   if 9.80000000000000053e76 < x

   1. Initial program 98.1%

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

   3. Taylor expanded in x around 0 66.0%

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

   4. Taylor expanded in x around 0 66.0%

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

Alternative 7: 27.1% accurate, 107.0× speedup

Math

\[\begin{array}{l} \\ 0 \end{array} \]

FPCore

(FPCore (x) :precision binary64 0.0)

C

double code(double x) {
	return 0.0;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 0.0d0
end function

Java

public static double code(double x) {
	return 0.0;
}

Python

def code(x):
	return 0.0

Julia

function code(x)
	return 0.0
end

MATLAB

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

Wolfram

code[x_] := 0.0

Derivation

1. Initial program 54.2%

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

3. Taylor expanded in x around 0 24.7%

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

4. Taylor expanded in x around 0 25.5%

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

Reproduce

herbie shell --seed 2024180 
(FPCore (x)
  :name "cos2 (problem 3.4.1)"
  :precision binary64
  (/ (- 1.0 (cos x)) (* x x)))