cos2 (problem 3.4.1)

Percentage Accurate: 51.7% → 99.8%

Time: 10.0s

Alternatives: 9

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: 51.7% 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

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 49.8%

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

   1. flip-- 49.8%

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

   2. div-inv 49.8%

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

   3. metadata-eval 49.8%

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

   4. 1-sub-cos 75.0%

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

   5. pow2 75.0%

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

3. Applied egg-rr 75.0%

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

   1. associate-*r/ 75.0%

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

   2. *-rgt-identity 75.0%

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

5. Simplified 75.0%

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

6. Taylor expanded in x around -inf 75.0%

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

   1. +-commutative 75.0%

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

   2. unpow2 75.0%

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

   3. associate-*r/ 75.0%

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

   4. hang-0p-tan 75.1%

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

8. Simplified 75.1%

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

   1. times-frac 99.8%

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

   2. div-inv 99.8%

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

   3. metadata-eval 99.8%

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

10. Applied egg-rr 99.8%

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

11. Final simplification 99.8%

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

Alternative 2: 75.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.0275:\\ \;\;\;\;0.5 + \left({x}^{4} \cdot 0.001388888888888889 + x \cdot \left(x \cdot -0.041666666666666664\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{x}^{-2} \cdot \left(1 - \cos x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 0.0275)
   (+
    0.5
    (+ (* (pow x 4.0) 0.001388888888888889) (* x (* x -0.041666666666666664))))
   (* (pow x -2.0) (- 1.0 (cos x)))))

C

double code(double x) {
	double tmp;
	if (x <= 0.0275) {
		tmp = 0.5 + ((pow(x, 4.0) * 0.001388888888888889) + (x * (x * -0.041666666666666664)));
	} else {
		tmp = pow(x, -2.0) * (1.0 - cos(x));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 0.0275d0) then
        tmp = 0.5d0 + (((x ** 4.0d0) * 0.001388888888888889d0) + (x * (x * (-0.041666666666666664d0))))
    else
        tmp = (x ** (-2.0d0)) * (1.0d0 - cos(x))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 0.0275) {
		tmp = 0.5 + ((Math.pow(x, 4.0) * 0.001388888888888889) + (x * (x * -0.041666666666666664)));
	} else {
		tmp = Math.pow(x, -2.0) * (1.0 - Math.cos(x));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 0.0275:
		tmp = 0.5 + ((math.pow(x, 4.0) * 0.001388888888888889) + (x * (x * -0.041666666666666664)))
	else:
		tmp = math.pow(x, -2.0) * (1.0 - math.cos(x))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 0.0275)
		tmp = Float64(0.5 + Float64(Float64((x ^ 4.0) * 0.001388888888888889) + Float64(x * Float64(x * -0.041666666666666664))));
	else
		tmp = Float64((x ^ -2.0) * Float64(1.0 - cos(x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 0.0275)
		tmp = 0.5 + (((x ^ 4.0) * 0.001388888888888889) + (x * (x * -0.041666666666666664)));
	else
		tmp = (x ^ -2.0) * (1.0 - cos(x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 0.0275], N[(0.5 + N[(N[(N[Power[x, 4.0], $MachinePrecision] * 0.001388888888888889), $MachinePrecision] + N[(x * N[(x * -0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $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 < 0.0275000000000000001

   1. Initial program 35.2%

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

   2. Taylor expanded in x around 0 67.5%

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

      1. +-commutative 67.5%

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

      2. *-commutative 67.5%

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

      3. fma-def 67.5%

         \[\leadsto 0.5 + \color{blue}{\mathsf{fma}\left({x}^{4}, 0.001388888888888889, -0.041666666666666664 \cdot {x}^{2}\right)} \]

      4. *-commutative 67.5%

         \[\leadsto 0.5 + \mathsf{fma}\left({x}^{4}, 0.001388888888888889, \color{blue}{{x}^{2} \cdot -0.041666666666666664}\right) \]

      5. unpow2 67.5%

         \[\leadsto 0.5 + \mathsf{fma}\left({x}^{4}, 0.001388888888888889, \color{blue}{\left(x \cdot x\right)} \cdot -0.041666666666666664\right) \]

   4. Simplified 67.5%

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

      1. fma-udef 67.5%

         \[\leadsto 0.5 + \color{blue}{\left({x}^{4} \cdot 0.001388888888888889 + \left(x \cdot x\right) \cdot -0.041666666666666664\right)} \]

      2. associate-*l* 67.5%

         \[\leadsto 0.5 + \left({x}^{4} \cdot 0.001388888888888889 + \color{blue}{x \cdot \left(x \cdot -0.041666666666666664\right)}\right) \]

   6. Applied egg-rr 67.5%

      \[\leadsto 0.5 + \color{blue}{\left({x}^{4} \cdot 0.001388888888888889 + x \cdot \left(x \cdot -0.041666666666666664\right)\right)} \]

   if 0.0275000000000000001 < x

   1. Initial program 98.6%

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

      1. clear-num 98.6%

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

      2. associate-/r/ 98.5%

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

      3. pow2 98.5%

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

      4. pow-flip 99.5%

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

      5. metadata-eval 99.5%

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

   3. Applied egg-rr 99.5%

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

4. Final simplification 74.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.0275:\\ \;\;\;\;0.5 + \left({x}^{4} \cdot 0.001388888888888889 + x \cdot \left(x \cdot -0.041666666666666664\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{x}^{-2} \cdot \left(1 - \cos x\right)\\ \end{array} \]

Alternative 3: 75.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.0275:\\ \;\;\;\;0.5 + \left({x}^{4} \cdot 0.001388888888888889 + x \cdot \left(x \cdot -0.041666666666666664\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 - \cos x}{x}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 0.0275)
   (+
    0.5
    (+ (* (pow x 4.0) 0.001388888888888889) (* x (* x -0.041666666666666664))))
   (/ (/ (- 1.0 (cos x)) x) x)))

C

double code(double x) {
	double tmp;
	if (x <= 0.0275) {
		tmp = 0.5 + ((pow(x, 4.0) * 0.001388888888888889) + (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 <= 0.0275d0) then
        tmp = 0.5d0 + (((x ** 4.0d0) * 0.001388888888888889d0) + (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 <= 0.0275) {
		tmp = 0.5 + ((Math.pow(x, 4.0) * 0.001388888888888889) + (x * (x * -0.041666666666666664)));
	} else {
		tmp = ((1.0 - Math.cos(x)) / x) / x;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 0.0275:
		tmp = 0.5 + ((math.pow(x, 4.0) * 0.001388888888888889) + (x * (x * -0.041666666666666664)))
	else:
		tmp = ((1.0 - math.cos(x)) / x) / x
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 0.0275)
		tmp = Float64(0.5 + Float64(Float64((x ^ 4.0) * 0.001388888888888889) + Float64(x * Float64(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 <= 0.0275)
		tmp = 0.5 + (((x ^ 4.0) * 0.001388888888888889) + (x * (x * -0.041666666666666664)));
	else
		tmp = ((1.0 - cos(x)) / x) / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 0.0275], N[(0.5 + N[(N[(N[Power[x, 4.0], $MachinePrecision] * 0.001388888888888889), $MachinePrecision] + N[(x * N[(x * -0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $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 < 0.0275000000000000001

   1. Initial program 35.2%

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

   2. Taylor expanded in x around 0 67.5%

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

      1. +-commutative 67.5%

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

      2. *-commutative 67.5%

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

      3. fma-def 67.5%

         \[\leadsto 0.5 + \color{blue}{\mathsf{fma}\left({x}^{4}, 0.001388888888888889, -0.041666666666666664 \cdot {x}^{2}\right)} \]

      4. *-commutative 67.5%

         \[\leadsto 0.5 + \mathsf{fma}\left({x}^{4}, 0.001388888888888889, \color{blue}{{x}^{2} \cdot -0.041666666666666664}\right) \]

      5. unpow2 67.5%

         \[\leadsto 0.5 + \mathsf{fma}\left({x}^{4}, 0.001388888888888889, \color{blue}{\left(x \cdot x\right)} \cdot -0.041666666666666664\right) \]

   4. Simplified 67.5%

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

      1. fma-udef 67.5%

         \[\leadsto 0.5 + \color{blue}{\left({x}^{4} \cdot 0.001388888888888889 + \left(x \cdot x\right) \cdot -0.041666666666666664\right)} \]

      2. associate-*l* 67.5%

         \[\leadsto 0.5 + \left({x}^{4} \cdot 0.001388888888888889 + \color{blue}{x \cdot \left(x \cdot -0.041666666666666664\right)}\right) \]

   6. Applied egg-rr 67.5%

      \[\leadsto 0.5 + \color{blue}{\left({x}^{4} \cdot 0.001388888888888889 + x \cdot \left(x \cdot -0.041666666666666664\right)\right)} \]

   if 0.0275000000000000001 < x

   1. Initial program 98.6%

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

      1. associate-/r* 99.5%

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

      2. div-inv 99.5%

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

   3. Applied egg-rr 99.5%

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

      1. un-div-inv 99.5%

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

   5. Applied egg-rr 99.5%

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

4. Final simplification 74.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.0275:\\ \;\;\;\;0.5 + \left({x}^{4} \cdot 0.001388888888888889 + x \cdot \left(x \cdot -0.041666666666666664\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 - \cos x}{x}}{x}\\ \end{array} \]

Alternative 4: 74.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.0058:\\ \;\;\;\;0.5 + -0.041666666666666664 \cdot \left(x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \cos x}{x \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 0.0058)
   (+ 0.5 (* -0.041666666666666664 (* x x)))
   (/ (- 1.0 (cos x)) (* x x))))

C

double code(double x) {
	double tmp;
	if (x <= 0.0058) {
		tmp = 0.5 + (-0.041666666666666664 * (x * x));
	} 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 <= 0.0058d0) then
        tmp = 0.5d0 + ((-0.041666666666666664d0) * (x * x))
    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 <= 0.0058) {
		tmp = 0.5 + (-0.041666666666666664 * (x * x));
	} else {
		tmp = (1.0 - Math.cos(x)) / (x * x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 0.0058:
		tmp = 0.5 + (-0.041666666666666664 * (x * x))
	else:
		tmp = (1.0 - math.cos(x)) / (x * x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 0.0058)
		tmp = Float64(0.5 + Float64(-0.041666666666666664 * Float64(x * x)));
	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 <= 0.0058)
		tmp = 0.5 + (-0.041666666666666664 * (x * x));
	else
		tmp = (1.0 - cos(x)) / (x * x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 0.0058], N[(0.5 + N[(-0.041666666666666664 * N[(x * x), $MachinePrecision]), $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 < 0.0058

   1. Initial program 34.7%

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

   2. Taylor expanded in x around 0 67.2%

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

      1. *-commutative 67.2%

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

      2. unpow2 67.2%

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

   4. Simplified 67.2%

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

   if 0.0058 < x

   1. Initial program 98.1%

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

4. Final simplification 74.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.0058:\\ \;\;\;\;0.5 + -0.041666666666666664 \cdot \left(x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \cos x}{x \cdot x}\\ \end{array} \]

Alternative 5: 75.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.0058:\\ \;\;\;\;0.5 + -0.041666666666666664 \cdot \left(x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 - \cos x}{x}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 0.0058)
   (+ 0.5 (* -0.041666666666666664 (* x x)))
   (/ (/ (- 1.0 (cos x)) x) x)))

C

double code(double x) {
	double tmp;
	if (x <= 0.0058) {
		tmp = 0.5 + (-0.041666666666666664 * (x * x));
	} 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 <= 0.0058d0) then
        tmp = 0.5d0 + ((-0.041666666666666664d0) * (x * x))
    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 <= 0.0058) {
		tmp = 0.5 + (-0.041666666666666664 * (x * x));
	} else {
		tmp = ((1.0 - Math.cos(x)) / x) / x;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 0.0058:
		tmp = 0.5 + (-0.041666666666666664 * (x * x))
	else:
		tmp = ((1.0 - math.cos(x)) / x) / x
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 0.0058)
		tmp = Float64(0.5 + Float64(-0.041666666666666664 * Float64(x * x)));
	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 <= 0.0058)
		tmp = 0.5 + (-0.041666666666666664 * (x * x));
	else
		tmp = ((1.0 - cos(x)) / x) / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 0.0058], N[(0.5 + N[(-0.041666666666666664 * N[(x * x), $MachinePrecision]), $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 < 0.0058

   1. Initial program 34.7%

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

   2. Taylor expanded in x around 0 67.2%

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

      1. *-commutative 67.2%

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

      2. unpow2 67.2%

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

   4. Simplified 67.2%

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

   if 0.0058 < x

   1. Initial program 98.1%

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

      1. associate-/r* 98.9%

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

      2. div-inv 98.8%

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

   3. Applied egg-rr 98.8%

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

      1. un-div-inv 98.9%

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

   5. Applied egg-rr 98.9%

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

4. Final simplification 74.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.0058:\\ \;\;\;\;0.5 + -0.041666666666666664 \cdot \left(x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 - \cos x}{x}}{x}\\ \end{array} \]

Alternative 6: 64.4% accurate, 7.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2:\\ \;\;\;\;0.5 + -0.041666666666666664 \cdot \left(x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-1}{-0.16666666666666666 - \frac{2}{x \cdot x}}}{x \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 2.0)
   (+ 0.5 (* -0.041666666666666664 (* x x)))
   (/ (/ -1.0 (- -0.16666666666666666 (/ 2.0 (* x x)))) (* x x))))

C

double code(double x) {
	double tmp;
	if (x <= 2.0) {
		tmp = 0.5 + (-0.041666666666666664 * (x * x));
	} else {
		tmp = (-1.0 / (-0.16666666666666666 - (2.0 / (x * x)))) / (x * x);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 2.0d0) then
        tmp = 0.5d0 + ((-0.041666666666666664d0) * (x * x))
    else
        tmp = ((-1.0d0) / ((-0.16666666666666666d0) - (2.0d0 / (x * x)))) / (x * x)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 2.0) {
		tmp = 0.5 + (-0.041666666666666664 * (x * x));
	} else {
		tmp = (-1.0 / (-0.16666666666666666 - (2.0 / (x * x)))) / (x * x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 2.0:
		tmp = 0.5 + (-0.041666666666666664 * (x * x))
	else:
		tmp = (-1.0 / (-0.16666666666666666 - (2.0 / (x * x)))) / (x * x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 2.0)
		tmp = Float64(0.5 + Float64(-0.041666666666666664 * Float64(x * x)));
	else
		tmp = Float64(Float64(-1.0 / Float64(-0.16666666666666666 - Float64(2.0 / Float64(x * x)))) / Float64(x * x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 2.0)
		tmp = 0.5 + (-0.041666666666666664 * (x * x));
	else
		tmp = (-1.0 / (-0.16666666666666666 - (2.0 / (x * x)))) / (x * x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 2.0], N[(0.5 + N[(-0.041666666666666664 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(-1.0 / N[(-0.16666666666666666 - N[(2.0 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x * x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2

   1. Initial program 35.2%

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

   2. Taylor expanded in x around 0 67.2%

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

      1. *-commutative 67.2%

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

      2. unpow2 67.2%

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

   4. Simplified 67.2%

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

   if 2 < x

   1. Initial program 98.6%

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

      1. add-log-exp 98.6%

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

   3. Applied egg-rr 98.6%

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

      1. add-log-exp 98.6%

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

      2. add-sqr-sqrt 98.5%

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

      3. sqrt-unprod 98.6%

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

      4. sqr-neg 98.6%

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

      5. sqrt-unprod 0.0%

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

      6. add-sqr-sqrt 50.2%

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

      7. flip-- 50.2%

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

      8. metadata-eval 50.2%

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

      9. 1-sub-cos 50.2%

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

      10. unpow2 50.2%

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

      11. clear-num 50.2%

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

      12. distribute-neg-frac 50.2%

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

      13. metadata-eval 50.2%

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

      14. clear-num 50.2%

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

      15. unpow2 50.2%

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

      16. 1-sub-cos 50.2%

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

      17. metadata-eval 50.2%

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

      18. flip-- 50.2%

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

      19. add-sqr-sqrt 50.2%

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

      20. sqrt-unprod 50.2%

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

   5. Applied egg-rr 98.6%

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

   6. Taylor expanded in x around 0 56.7%

      \[\leadsto \frac{\frac{-1}{\color{blue}{-\left(0.16666666666666666 + 2 \cdot \frac{1}{{x}^{2}}\right)}}}{x \cdot x} \]
   7. Step-by-step derivation

      1. distribute-neg-in 56.7%

         \[\leadsto \frac{\frac{-1}{\color{blue}{\left(-0.16666666666666666\right) + \left(-2 \cdot \frac{1}{{x}^{2}}\right)}}}{x \cdot x} \]

      2. metadata-eval 56.7%

         \[\leadsto \frac{\frac{-1}{\color{blue}{-0.16666666666666666} + \left(-2 \cdot \frac{1}{{x}^{2}}\right)}}{x \cdot x} \]

      3. associate-*r/ 56.7%

         \[\leadsto \frac{\frac{-1}{-0.16666666666666666 + \left(-\color{blue}{\frac{2 \cdot 1}{{x}^{2}}}\right)}}{x \cdot x} \]

      4. metadata-eval 56.7%

         \[\leadsto \frac{\frac{-1}{-0.16666666666666666 + \left(-\frac{\color{blue}{2}}{{x}^{2}}\right)}}{x \cdot x} \]

      5. unpow2 56.7%

         \[\leadsto \frac{\frac{-1}{-0.16666666666666666 + \left(-\frac{2}{\color{blue}{x \cdot x}}\right)}}{x \cdot x} \]

   8. Simplified 56.7%

      \[\leadsto \frac{\frac{-1}{\color{blue}{-0.16666666666666666 + \left(-\frac{2}{x \cdot x}\right)}}}{x \cdot x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 64.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2:\\ \;\;\;\;0.5 + -0.041666666666666664 \cdot \left(x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-1}{-0.16666666666666666 - \frac{2}{x \cdot x}}}{x \cdot x}\\ \end{array} \]

Alternative 7: 63.1% accurate, 11.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 3.4:\\ \;\;\;\;0.5 + -0.041666666666666664 \cdot \left(x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 3.4) (+ 0.5 (* -0.041666666666666664 (* x x))) 0.0))

C

double code(double x) {
	double tmp;
	if (x <= 3.4) {
		tmp = 0.5 + (-0.041666666666666664 * (x * x));
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 3.4d0) then
        tmp = 0.5d0 + ((-0.041666666666666664d0) * (x * x))
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 3.4) {
		tmp = 0.5 + (-0.041666666666666664 * (x * x));
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 3.4:
		tmp = 0.5 + (-0.041666666666666664 * (x * x))
	else:
		tmp = 0.0
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 3.4)
		tmp = Float64(0.5 + Float64(-0.041666666666666664 * Float64(x * x)));
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 3.4)
		tmp = 0.5 + (-0.041666666666666664 * (x * x));
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 3.4], N[(0.5 + N[(-0.041666666666666664 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0]

Derivation

1. Split input into 2 regimes

2. if x < 3.39999999999999991

   1. Initial program 35.2%

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

   2. Taylor expanded in x around 0 67.2%

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

      1. *-commutative 67.2%

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

      2. unpow2 67.2%

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

   4. Simplified 67.2%

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

   if 3.39999999999999991 < x

   1. Initial program 98.6%

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

   2. Taylor expanded in x around 0 51.0%

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

   3. Taylor expanded in x around 0 51.0%

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

4. Final simplification 63.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 3.4:\\ \;\;\;\;0.5 + -0.041666666666666664 \cdot \left(x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 8: 63.6% accurate, 34.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 3 \cdot 10^{+76}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (if (<= x 3e+76) 0.5 0.0))

C

double code(double x) {
	double tmp;
	if (x <= 3e+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 <= 3d+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 <= 3e+76) {
		tmp = 0.5;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 3e+76:
		tmp = 0.5
	else:
		tmp = 0.0
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 3e+76)
		tmp = 0.5;
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 3e+76)
		tmp = 0.5;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 3e+76], 0.5, 0.0]

Derivation

1. Split input into 2 regimes

2. if x < 2.9999999999999998e76

   1. Initial program 39.7%

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

   2. Taylor expanded in x around 0 63.2%

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

   if 2.9999999999999998e76 < x

   1. Initial program 98.7%

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

   2. Taylor expanded in x around 0 67.1%

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

   3. Taylor expanded in x around 0 67.1%

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

4. Final simplification 63.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 3 \cdot 10^{+76}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 9: 28.7% 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 49.8%

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

2. Taylor expanded in x around 0 24.7%

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

3. Taylor expanded in x around 0 25.5%

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

4. Final simplification 25.5%

   \[\leadsto 0 \]

Reproduce

herbie shell --seed 2023242 
(FPCore (x)
  :name "cos2 (problem 3.4.1)"
  :precision binary64
  (/ (- 1.0 (cos x)) (* x x)))